Bilan Synthese de PNR Badis Ydri Departement de Physique, Facult´e des Sciences, Universit´e d’Annaba, Annaba, Algerie. January 13, 2014

Contents 1 IDENTIFICATION DU PROJET

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2 EQUIPE DE RECHERCHE

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3 CV: BADIS YDRI

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4 DESCRIPTION DU PROJET 4.1 MOTS-CLES . . . . . . . . . 4.2 RESUME . . . . . . . . . . . 4.3 RAPPEL DES OBJECTIFS . 4.4 IMPACTS . . . . . . . . . . .

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5 BILAN SYNTHESE 5.1 TRAVAUX PUBLIES . . . . . . . . . . . . 5.2 TRAVAIL SOUMIS POUR PUBLICATION 5.3 COMMUNICATIONS . . . . . . . . . . . . 5.4 AUTRES REMARQUES . . . . . . . . . . .

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6 INFORMATION FINANCIERE

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7 PUBLICATIONS: RESUMES ET BIBLIOGRAHIES

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IDENTIFICATION DU PROJET • PNR 8: Sciences Fondamentales. • Domaine: Physique. • Organisme Pilote: ANDRU. • Intitul´e du Projet: Th´eories des champs non-commutatives a´ partir des mod´eles des matrices et physique ´emergente. • Dur´ee: 05/11-05/13. • Chef du Projet: Badis Ydri. • Etablissement de Domiciliation: Universit´e Badji Mokhtar Annaba.

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EQUIPE DE RECHERCHE • Porteur du Projet: Badis Ydri, Ph.D, MCA, Tel:0669465033, Universit´e de Annaba. • Partenaire Socio-Economique: Djamel Dou, Ph.D, Professor, Tel:0662106726, Centre Universitaire d’Eloued. • Equipe de Recherche: A.Bouchareb, M.Moumni, R.Ahmim, C.Soudani.

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CV: BADIS YDRI • Ph.D: Syracuse University, New York, USA, 2001. • Post-Doctoral: – Hamilton Post-Doctoral Fellow, School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland, 2001-2005. – Marie Curie Post-Doctoral Fellow, Institut f¨ ur Physik, Humboldt-Universit¨at zu Berlin, Berlin, Germany, 2006-2008. • Habilitation: Universit´e Badji Mokhtar Annaba, Algeria, 2011. • Faculty Position: D´epartment de Physique, Universit´e Badji Mokhtar Annaba, Algeria, January 2009. • Research Interests: Quantum Field Theory and Quantum Gravity.

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• Publications: Search the inSPIRE website: http://inspirehep.net • Emails, Website and Phone: [email protected], [email protected] http://www.stp.dias.ie/~ ydri/ 213(0)669465033.

4 4.1

DESCRIPTION DU PROJET MOTS-CLES

Noncommutative geometry,Noncommutative field theory,Yang-Mills models, Supersymmetry, Fuzzy spaces, Matrix models, Emergent geometry, The matrix and fuzzy sphere phases,The nonuniform ordered phase,Renormalization group methods, Monte Carlo methods, Cohomolgical methods.

4.2

RESUME

The four fondamental forces of nature are the electromagnetic force, the strong force, the weak force and the gravitational force. The non-gravitational forces are described by quantum Yang-Mills field theories on a flat Minkowski four dimensional spacetime and they are unified in the standard model of particle physics. Gravity can be unified with the other forces only within the context of string theory which is not a field theory. However at low energies string theory in a magnetic field becomes a field theory on a noncommutative spacetime, i.e a spacetime in which the coordinates are no longer numbers but operators. Noncommutative spacetimes arise also from combining the principles of general relativity which describe classical gravity and quantum mechanics at the very short scales. Thus the connection between noncommutativity and quantum gravity seems to be a very plausible hypothesis. Noncommutativity is also the only known extension preserving supersymmetry which is a symmetry central to string theory. String theory can not be studied in computer simulations. The basic problem is that supersymmetry can not be preserved in a naive spacetime discretization. The only exception is the Yang-Mills matrix model due to Ishibashi, Kawai, Kitazawa and Tsuchiya which is postulated to give a nonperturbative regularization of type IIB string theory. The IKKT Yang-Mills matrix models admit noncommutative spaces as solutions and as such they can be used to regularize nonperturbatively noncommutative gauge theories. In this project we will study non-trivial phenomena which emerge dynamically in the IKKT Yang-Mills matrix models such as emergent geometry, spontaneous supersymmetry breaking and emergent gravity. This is expected to shed important light on the nonperturbative physics of noncommutative gauge theories. We 3

will also use the matrix formalism to study quantum aspects of noncommutative scalar field theories.

4.3

RAPPEL DES OBJECTIFS

• 1)-Phase diagram of non-commutative scalar phi-four using Monte Carlo methods. • 2)-Renormalization group equations for non-commutative field theory a la Wilson and a la Polchinski . • 3)-Emergent geometry and non-commutative gauge theories from Yang-Mills matrix models .

4.4

IMPACTS

The expected scientific impacts may include the following: 1) The conjecture that large N IKKT Yang-Mills matrix models are of central importance to fundamental physics can be verified explicitly in several interrelated physical contexts. The relevance of these large N IKKT Yang-Mills matrix models to fundamental physics can be summarized as follows: • a) They may help us understand nonperturbative physics of noncommutative gauge theories by means of cohomological and random matrix models. • b) They may provide a nonperturbative definition of supersymmetry. Indeed supersymmetry in this language may prove to be tractable in Monte Carlo simulation. • c) They provide concrete models for emergent geometry and possibly emergent gravity. Indeed geometry in transition seems to be possible in all these matrix models. These matrix models seems also very suited to test the gauge/gravity duality. 2) A more coherent understanding of the quantum dynamics of noncommutative scalar field theories by means of matrix models techniques and renormalization group and Monte Carlo methods. The expected educatif impacts may include the following: 1) Supervision of doctoral theses which are based on different parts of this project. 2) Initiation to different methods and techniques of theoretical physics. 3) Introduction to new area of research in theoretical physics such as supersymmetric field theory and gauge/gravity duality. 4

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BILAN SYNTHESE

5.1

TRAVAUX PUBLIES

• We have published during the period 2011- 2013 in total six articles in internationally renowned journals of theoretical particle physics and cosmology such as Physical Review D, Journal of Mathematical Physics, International Journal of Modern Physics A and Modern Physics Letters A. The six published articles between 2011 and 2013 can be found in the bibliography at the end of this report and the first page of each article is included in section 5. • Three articles (articles 1, 2 and 3) deals directly with the original objectives of our PNR project whereas article 4 is indirectly related to our project. Only articles 5 and 6 are on different matters. • Article 1: Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models . This is the longest paper which deals with the problem of emergent geometry and emergent gauge theory (the third objective of our PNR project) from Yang-Mills matrix models using the Monte Carlo method and matrix models techniques. This axis of research is still ongoing with two doctoral (LMD) students. • Article 2: Matrix Model Fixed Point of Noncommutative phi-four Theory. This is the second longest paper which deals with the problem of the construction of viable and solvable renormalization group equations for matrix models and noncommutative field theories (the second objective of our PNR project). The collaborating researcher RACHID AHMIM is a doctoral student who is now published and hence can submit and defend his thesis at any time he chooses. • Article 3: The fate of the Wilson-Fisher fixed point in non-commutative φ4 This article deals also with the problem of the renormalization group equations for noncommutative field theories in the limit of small noncommutativity. • Article 4, 5, 6: . As mentioned earlier Articles 4, 5, 6 are indirectly related to the original objectives of our PNR project since they deal with some phenomenological consequences of noncommutative field theory in atomic physics and various issues of quantum gravity, general relativity and cosmology.

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– Article 4: Effects of Non-Commutativity on Light-Hydrogen-Like Atoms and Proton Radius – Article 5: On the Problem of Vacuum Energy in FLRW Universes and Dark Energy. – Article 6: Bertotti-Robinson solutions of D=5 Einstein-Maxwell-Chern-Simons-Lambda theory.

5.2

TRAVAIL SOUMIS POUR PUBLICATION

• Article 7: New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere. Article 7 is a completed preprint submitted for publication in the Journal of High Eneregy Physics. This preprint contains a summary of ongoing work on the first objective of our PNR project, i.e. our work on the phase diagram of noncommutative phi-four on the fuzzy sphere using the Monte Carlo approach. This, as it turns out, is a very hard problem which required us to construct a completely new algorithm in order to handle properly the complex phase structure of this theory. There are two doctoral students involved in this work among them figure our collaborator C.SOUDANI.

5.3

COMMUNICATIONS

• Our contributions to the The 9th International Conference in Subatomic Physics and Applications held in Constantine from September 30th to October 2nd 2013 are: – M.Moumeni, Lyman α Spectroscopy in Non-Commutative Space-Time. – A.Bouchereb, On the problem of vacuum energy in FLRW universes and dark energy. – B.Ydri, Aspects of noncommutative φ4 on the fuzzy sphere.

5.4

AUTRES REMARQUES

• Re:Ad´ equation des Actions Men´ ees et Niveau des Objectifs Atteints: We hope it is clear from the above brief outline of our results during the past two years that our work and action were largely adequate vis-a-vis the stated original goals of our PNR project. In fact we estimate that our work and action were very relevant to the stated goals. Also we estimate that we have in fact achieved at least 75 per cent of our original plan. • Re:Le Projet a-t-il Donn´ e lieu ´ a un Produit et Valorisations: 6

If by ”product” it is meant scientific publications in well established refereed journals then the answer is certainly YES. As of the valorization aspect of our PNR project it consists solely in scientific contribution and getting recognition for it and we are continuously working hard towards these two goals within this PNR and outside it. With regard to this point we should always be mindful that theoretical physics is a fundamental science which addresses largely basic problems of nature and as such the only aim we strive to reach is to continuously improve our understanding of the world we live in at both the large and small scales using mostly the language of mathematics but also in recent years numerical simulations. In theoretical physics scientific contribution is the goal and the prize at the same time. • Re:Le Projet a-t-il Contribu´ e´ a la Formation de Chercheurs: There are two research members (AHMIM and SOUDANI) who are preparing their doctoral dissertations within the axes of this PNR project. As stated earlier AHMIM has already published an article (article 2) and have started writing his thesis whereas the work of SOUDANI on the phase diagram of noncommutative phi-four on the fuzzy sphere is still ongoing. There are further four LMD doctoral students working on different aspects of matrix field theory and matrix geometry initiated within this PNR project.

References [1] B. Ydri, “Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II,” Int. J. Mod. Phys. A, Vol. 27, No. 17 (2012) 1250088 (35 pages), DOI: 10.1142/S0217751X12500881, arXiv:1206.6375 [hep-th]. [2] B. Ydri and R. Ahmim, “Matrix Model Fixed Point of Noncommutative Phi-Four,” Phys. Rev. D 88, 106001 (2013) (30 pages), DOI: 10.1103/PhysRevD.88.106001, arXiv:1304.7303 [hep-th]. [3] B. Ydri and A. Bouchareb, “The fate of the Wilson-Fisher fixed point in non-commutative φ4 ,” J. Math. Phys. 53, 102301 (2012) (14 pages), DOI: 10.1063/1.4754816, arXiv:1206.5653 [hep-th]. 7

[4] M. Moumni and A. BenSlama, “Effects of Non-Commutativity on Light-Hydrogen-Like Atoms and Proton Radius,” Int. J. Mod. Phys. A 28, 1350139 (2013) (15 pages), DOI: 10.1142/S0217751X1350139X, arXiv:1305.3508 [hep-ph]. [5] B. Ydri and A. Bouchareb, “On the Problem of Vacuum Energy in FLRW Universes and Dark Energy,” Mod. Phys. Lett. A, Vol. 28, No. 36 (2013) 1350166 (9 pages), DOI: 10.1142/S0217732313501666, arXiv:1307.2749 [hep-th]. [6] A. Bouchareb, C. -M. Chen, G´er. Cl´ement and D. V. Gal’tsov, “Bertotti-Robinson solutions of D=5 Einstein-Maxwell-Chern-Simons-Lambda theory,” Phys. Rev. D 88, 084048 (2013) (17 pages), DOI: 10.1103/PhysRevD.88.084048, arXiv:1308.6461 [gr-qc]. [7] B. Ydri, “New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere,” arXiv:1401.1529 [hep-th].

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INFORMATION FINANCIERE Consult the appendix at the end of this report.

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PUBLICATIONS: RESUMES ET BIBLIOGRAHIES

We include in the following the abstracts, introductions, conclusions and bibliographies of our 7 finished articles.

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International Journal of Modern Physics A Vol. 27, No. 17 (2012) 1250088 (35 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217751X12500881

IMPACT OF SUPERSYMMETRY ON EMERGENT GEOMETRY IN YANG MILLS MATRIX MODELS II

BADIS YDRI Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria [email protected] [email protected] Received 1 May 2012 Revised 7 May 2012 Accepted 8 May 2012 Published 18 June 2012 We present a study of D = 4 supersymmetric Yang–Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N . We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law. Keywords: Noncommutative geometry; gauge theory; Yang–Mills matrix models; emergent geometry; the fuzzy sphere; supersymmetry.

1. Introduction Reduced Yang–Mills theories play a central role in the nonperturbative definitions of M -theory and superstrings. The Banks–Fischler–Shenker–Susskind (BFSS) conjecture1 relates discrete light-cone quantization (DLCQ) of M -theory to the theory of N coincident D0 branes which at low energy (small velocities and/or string coupling) is the reduction to 0 + 1 dimension of the ten-dimensional U (N ) supersymmetric Yang–Mills gauge theory.2 The BFSS model is therefore a Yang–Mills quantum mechanics which is supposed to be the UV completion of 11-dimensional 1250088-1

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supergravity. As it turns out, BFSS action is nothing else but the regularization of the supermembrane action in the light cone gauge.3 The Berenstein–Maldacena–Nastase (BMN) model4 is a generalization of BFSS model to curved backgrounds. It is obtained by adding to BFSS action a oneparameter mass deformation corresponding to the maximally supersymmetric ppwave background of 11-dimensional supergravity.5–7 We note, in passing, that all maximally supersymmetric pp-wave geometries can arise as Penrose limits of AdSp × S q spaces.8 The Ishibashi–Kawai–Kitazawa–Tsuchiya (IKKT) model9 is, on the other hand, a Yang–Mills matrix model obtained by dimensionally reducing ten-dimensional U (N ) supersymmetric Yang–Mills gauge theory to 0 + 0 dimensions. The IKKT model is postulated to provide a constructive definition of type IIB superstring theory and for this reason it is also called type IIB matrix model. Supersymmetric analogue of IKKT model also exists in dimensions d = 3, 4 and 6 while the partition functions converge only in dimensions d = 4, 6.48,49 The IKKT Yang–Mills matrix models can be thought of as continuum Eguchi– Kawai reduced models as opposed to the usual lattice Eguchi–Kawai reduced model formulated in Ref. 11. We point out here the similarity between the conjecture that the lattice Eguchi–Kawai reduced model allows us to recover the full gauge theory in the large N theory and the conjecture that IKKT matrix model allows us to recover type IIB superstring. The relation between BFSS Yang–Mills quantum mechanics and IKKT Yang– Mills matrix model is discussed at length in the seminal paper10 where it is also shown that toroidal compactification of the D-instanton action (the bosonic part of IKKT action) yields, in a very natural way, a noncommutative Yang–Mills theory on a dual noncommutative torus.22 From the other hand, we can easily check that the ground state of the D-instanton action is given by commuting matrices which can be diagonalized simultaneously with the eigenvalues giving the coordinates of the D-branes. Thus at tree-level an ordinary space–time emerges from the bosonic truncation of IKKT action while higher-order quantum corrections will define a noncommutative space–time. In summary, Yang–Mills matrix models which provide a constructive definition of string theories will naturally lead to emergent geometry18 and noncommutative gauge theory.19,20 Furthermore, noncommutative geometry21,23 and their noncommutative field theories24,25 play an essential role in the nonperturbative dynamics of superstrings and M -theory. Thus the connections between noncommutative field theories, emergent geometry and matrix models from one side and string theory from the other side run deep. It seems therefore natural that Yang–Mills matrix models provide a nonperturbative framework for emergent space–time geometry and noncommutative gauge theories. Since noncommutativity is the only extension which preserves maximal supersymmetry, we also hope that Yang–Mills matrix models will provide a regularization which preserves supersymmetry.26 1250088-2

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Impact of Supersymmetry on Emergent Geometry

In this paper we will explore in particular the possibility of using IKKT Yang– Mills matrix models in dimensions 4 and 3 to provide a nonperturbative definition of emergent space–time geometry, noncommutative gauge theory and supersymmetry in two dimensions. From our perspective in this paper, the phase of commuting matrices has no geometry in the sense of Connes and thus we need to modify the models so that a geometry with a well defined spectral triple can also emerge alongside the phase of commuting matrices. There are two solutions to this problem. The first solution is given by adding mass deformations which preserve supersymmetry to the flat IKKT Yang–Mills matrix models12 or alternatively by an Eguchi–Kawai reduction of the mass deformed BFSS Yang–Mills quantum mechanics constructed in.13,60–62 The second solution, which we have also considered in this paper, is given by deforming the flat Yang–Mills matrix model in D = 4 using the powerful formalism of cohomological Yang–Mills theory.30–32,63 These mass deformed or cohomologically deformed IKKT Yang–Mills matrix models are the analogue of BMN model and they typically include a Myers term14 and thus they will sustain the geometry of the fuzzy sphere,27,28 as a ground state which at large N will approach the geometry of the ordinary sphere, the ordinary plane or the noncommutative plane depending on the scaling limit. Thus a nonperturbative formulation of noncommutative gauge theory in two dimensions can be captured rigorously within these models15–17 (see also Refs. 64 and 65). This can in principle be generalized to other fuzzy spaces29 and higher dimensional noncommutative gauge theories by considering appropriate mass deformations of the flat IKKT Yang–Mills matrix models. The problem or virtue of this construction, depending on the perspective, is that in these Yang–Mills matrix models the geometry of the fuzzy sphere collapses under quantum fluctuations into the phase of commuting matrices. Equivalently, it is seen that the geometry of the fuzzy sphere emerges from the dynamics of a random matrix theory.52,54 Supersymmetry is naturally expected to stabilize the space–time geometry, and in fact the nonstability of the nonsupersymmetric vacuum should have come as no surprise to us.33 We should mention here the approach of Ref. 34 in which a noncommutative Yang–Mills gauge theory on the fuzzy sphere emerges also from the dynamics of a random matrix theory. The fuzzy sphere is stable in the sense that the transition to commuting matrices is pushed towards infinite gauge coupling at large N .53 This was achieved by considering a very special nonsupersymmetric mass deformation which is quartic in the bosonic matrices. This construction was extended to a noncommutative gauge theory on the fuzzy sphere based on co-adjoint orbits.35 Let us also note here that the instability and the phase transition discussed here were also observed on the noncommutative torus,39–41,69,70 where the twisted Eguchi–Kawai model was employed as a nonperturbative regularization of noncommutative Yang–Mills gauge theory.36–38 1250088-3

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In this paper, we will then study using the Monte Carlo method the mass deformed Yang–Mills matrix model in D = 4 as well as a particular truncation to D = 3. We will also derive and study a one-parameter cohomological deformation of the Yang–Mills matrix model which coincides with the mass deformed model in D = 4 when the parameter is tuned appropriately. We will show that the first-/ second-order phase transition from the fuzzy sphere to the phase of commuting matrices observed in the bosonic models is converted in the supersymmetric models into a very slow crossover transition with an arbitrary small transition point in the large N limit. We will determine the eigenvalues distributions for both D = 4 and D = 3 throughout the phase diagram. This paper is organized as follows. In Sec. 2 we will derive the mass deformed Yang–Mills quantum mechanics from the requirement of supersymmetry and then reduce it further to obtain Yang–Mills matrix model in D = 4 dimensions. In Sec. 3 we will derive a one-parameter family of cohomologically deformed models and then show that the mass deformed model constructed in Sec. 2 can be obtained for a particular value of the parameter. In Sec. 4 we report our first Monte Carlo results for the model D = 4 including the eigenvalues distributions and also comment on the D = 3 model obtained by simply setting the fourth matrix to 0. We conclude in Sec. 5 with a comprehensive summary of the results and discuss future directions. 2. Mass Deformation of D = 4 Super-Yang Mills Matrix Model 2.1. Deformed Yang Mills quantum mechanics in 4D The N = 1 supersymmetric Yang–Mills theory reduced to one dimension is given by the supersymmetric Yang–Mills quantum mechanics (with D0 = ∂0 − i[X0 , · ]) 1 1 1 1¯ 0 i ¯ i 1 L0 = 2 Tr (D0 Xi )2 + [Xi , Xj ]2 − ψγ D0 ψ + ψγ [Xi , ψ] + F 2 . (2.1) g 2 4 2 2 2 The corresponding supersymmetric transformations are δ0 X0 = ¯ǫ γ0 ψ , δ0 Xi = ǫ¯γi ψ , 1 i δ0 ψ = − [γ 0 , γ i ]D0 Xi + [γ i , γ j ][Xi , Xj ] + iγ5 F ǫ , 2 4

(2.2)

δ0 F = −i¯ ǫ γ5 γ0 D0 ψ + ǫ¯γ5 γi [Xi , ψ] . Let µ be a constant mass parameter. A mass deformation of the Lagrangian density L0 takes the form Lµ = L0 +

µ µ2 L + L2 + · · · . 1 g2 g2

(2.3)

The Lagrangian density L0 has mass dimension 4. The corrections L1 and L2 must have mass dimension 3 and 2, respectively. We recall that the Bosonic matrices X0 and Xa have mass dimension 1 whereas the Fermionic matrices ψi have mass dimension 23 . A typical term in the Lagrangian densities L1 and L2 will contain nf 1250088-4

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Impact of Supersymmetry on Emergent Geometry ∼

∼

EVs of the N=10 bosonic models with β =2/9 and α =0.5

Eigenvalues Distributions

0.8

D=3 uniform in D=3 uniform in D=4 D=4 uniform in D=3 uniform in D=4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3

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-1 0 1 Eigenvalues of X3

2

3

∼

EVs of the N=10 bosonic models with β =2/9

Eigenvalues Distributions

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∼ D=3, α =1.0 ∼ α =0.1 ∼ D=4, ∼α =1.0 α =0.1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3

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-1 0 1 Eigenvalues of X3

2

3

Fig. 9. The eigenvalues distributions of X3 of the = 3, 4 bosonic models with β˜ = 2/9 in the matrix phase.

5. Summary and Future Directions In this paper we employed the Monte Carlo method to study nonperturbatively Yang–Mills matrix models in D = 4 with mass terms. We can summarize the main results, findings and conjectures of this work as follows: • By imposing the requirement of supersymmetry and SO(3) covariance we have shown that there exists a single mass deformed Yang–Mills quantum mechanics in D = 4 which preserves all four real supersymmetries of the original theory although in a deformed form. This is the four-dimensional analogue of the tendimensional BMN model. Full reduction yields a unique mass deformed D = 4 Yang–Mills matrix model. This latter four-dimensional model is the analogue of the ten-dimensional IKKT model. 1250088-29

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• By using cohomological deformation of supersymmetry we constructed a oneparameter (ζ0 ) family of cohomologically deformed D = 4 Yang–Mills matrix models which preserve two supercharges. The mass deformed model is one limit (ζ0 → 0) of this one-parameter family of cohomologically deformed Yang–Mills models. • We studied the models with the values β˜ = 0 and β˜ = 2/9 where β˜ is the mass parameter of the bosonic matrices Xa . The second model is special in the sense that classically the configurations Xa ∼ La , X4 = 0 is degenerate with the configuration Xa = 0 and X4 = 0. • The Monte Carlo simulation of the bosonic D = 4 Yang–Mills matrix model with mass terms shows the existence of an exotic first-/second-order transition from a phase with a well defined background geometry given by the famous fuzzy sphere to a phase with commuting matrices with no geometry in the sense of Connes. The transition looks first-order due to the jump in the action whereas it looks second-order due to the divergent peak in the specific heat. • The fuzzy sphere is less stable as we increase the mass term of the bosonic ˜ For β˜ = 2/9 we find the critical value α matrices Xa , i.e. as we increase β. ˜ ∗ = 4.9 whereas for β˜ = 0 we find the critical value β˜ = 2.55. • The measured critical line in the plane α ˜ − β˜ agrees well with the theoretical prediction coming from the effective potential calculation. • The order parameter of the transition is given by the inverse radius of the sphere defined by 1/r = Tr Xa2 /(˜ α2 c2 ). The radius is equal to 1/φ2 (where φ is the classical configuration) in the fuzzy sphere phase. At the transition point the sphere expands abruptly to infinite size. Then as we decrease the inverse temperature (the inverse gauge coupling constant) α ˜ , the size of the sphere shrinks fast to 0, i.e. the sphere evaporates. • The fermion determinant is positive definite for all gauge configurations in D = 4. We have conjectured that the path integral is convergent as long as the scalar curvature (the mass of the fermionic matrices) is zero. • We have simulated the two models β˜ = 0 and β˜ = 2/9 with dynamical fermions. The model with β˜ = 0 has two supercharges while the model β˜ = 2/9 has a softly broken supersymmetry since in this case we needed to set by hand the scalar curvature to zero in order to regularize the path integral. Thus β˜ = 0 is amongst the very few models (which we known of) with exact supersymmetry which can be probed and accessed with the Monte Carlo method. • The fuzzy sphere is stable for the supersymmetric D = 4 Yang–Mills matrix model with mass terms in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point α ˜ is found to scale to zero with N . There is no jump in the action nor a peak in the specific heat. • The fuzzy sphere is stable also in the sense that the radius is equals 1/φ2 over a much larger region then it starts to decrease slowly as we decrease the inverse temperature α ˜ until it reaches 0 at α ˜ = 0. We claim that the value where the radius starts decreasing becomes smaller as we increase N . 1250088-30

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Impact of Supersymmetry on Emergent Geometry

The model at α ˜ = 0 can never sustain the geometry of the fuzzy sphere since it is the nondeformed model so in some sense the transition to commuting matrices always occurs and in the limit N → ∞ it will occur at α ˜ ∗ → 0. • We have spent a lot of time in trying to determine the eigenvalues distributions of the matrices Xµ in both the bosonic and supersymmetric theories. A universal behavior seems to emerge with many subtleties. These can be summarized as follows: – In the fuzzy sphere the matrices Xa are given by the SU (2) irreducible representations La . For example diagonalizing the matrix X3 gives N eigenvalues between (N − 1)/2 and −(N − 1)/2 with a step equal 1, viz m = (N − 1)/2, (N − 3)/2, . . . , −(N − 3)/2, −(N − 1)/2. – In the matrix phase the matrices Xµ become commuting. More explicitly the eigenvalues distribution of any of the matrices Xµ in the matrix phase is given by the nonpolynomial law ρ4 (x) =

–

– –

– –

–

–

8 3 (R2 − x2 ) 2 . 3πR4

(5.1)

This can be obtained from the conjecture that the joint probability distribution of the four matrices Xµ is uniform inside a solid ball with radius R. In the matrix phase the eigenvalues distribution of any of the Xa , say X3 , is given by the above nonpolynomial law with a radius R independent of α ˜ and N . This is also confirmed by computing the radius in this distribution and comparing to the Monte Carlo data. A very precise measurement of the transition point can be made by observing the point at which the eigenvalues distribution of X3 undergoes the transition from the N -cut distribution to the above nonpolynomial law. The eigenvalues distribution of X4 is always given by the above nonpolynomial law, i.e. for all values of α ˜ , with a radius R which depends on α ˜ and N . Another signal that the matrix phase is fully reached is when the eigenvalues distribution of X4 coincides with that of X3 . From this point downward the eigenvalues distribution of X4 ceases to depend on α ˜ and N . Monte Carlo measurements seems to indicate that R = 1.8 for bosonic models and R = 2.8 for supersymmetric models. The distribution becomes wider in the supersymmetric case. We have also observed that the eigenvalues of the normal scalar field Xa2 − c2 in the fuzzy sphere are also distributed according to the above nonpolynomial law. This led us to the conjecture that the eigenvalues of the gauge field on the background geometry are also distributed according to the above nonpolynomial law. Recall that the normal scalar field is the normal component of the gauge field to the background geometry which is the sphere here. 1250088-31

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• In the D = 3 Yang–Mills matrix model with mass terms the eigenvalues distribution becomes polynomial (parabolic) given by 3 (R2 − x2 ) . (5.2) 4R3 It was difficult for us in this paper to differentiate with certainty between the two distributions ρ4 and ρ3 in the three-dimensional setting. • Finally, we conjecture that the transition from a background geometry to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. Indeed mass deformed supersymmetry preserves the fuzzy sphere configuration but not diagonal matrices. ρ3 (x) =

Among the future directions that can be considered we will simply mention the following four points: • Higher precision Monte Carlo simulations of the models studied in this paper is the first obvious direction for future investigation. The most urgent question (in our view) is the precise determination of the behavior of the eigenvalues distributions in D = 4 and D = 3. An analytical derivation of ρ3 and especially ρ4 is an outstanding problem. • Finding matrix models with emergent four-dimensional background geometry is also an outstanding problem. • Models for emergent time, and to a lesser extent emergent gravity, and as a consequence emergent cosmology are very rare. • Monte Carlo simulation of supersymmetry based on matrix models seems to be a very promising goal. • A complete analytical understanding of the emergent geometry transition observed in Yang–Mills matrix models with mass terms is also an outstanding problem. In Ref. 68, we have attempted to compute the above eigenvalues distributions analytically. Using localization techniques we were able to find a special set of parameters for which the D = 4 Yang–Mills matrix model with mass terms can be reduced to the three-dimensional Chern–Simons (CS) matrix model. The saddlepoint method leads then immediately to the eigenvalues distributions ρ3 . We believe that our theoretical prediction for the value of R is reasonable compared to the Monte Carlo value. We have also made a preliminary comparison between the dependence of R on α in the Hermitian and anti-Hermitian CS matrix models. The Hermitian case seems more appropriate for the description of the eigenvalues of X3 whereas the anti-Hermitian case may be relevant to the description of the eigenvalues of X4 . Acknowledgments I would like to thank Denjoe O’Connor for extensive discussions at various stages of this project. I would also like to thank R. Delgadillo-Blando and Adel Bouchareb 1250088-32

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Matrix model fixed point of noncommutative 4 theory Badis Ydri1,* and Rachid Ahmim2 1

Institute of Physics, BM Annaba University, BP 12, 23000 Annaba, Algeria Department of Physics, El-Oued University, BP 789, 39000 El-Oued, Algeria (Received 9 May 2013; published 5 November 2013)

2

In this article, we exhibit explicitly the matrix model ( ¼ 1) fixed point of 4 theory on noncommutative spacetime with only two noncommuting directions, using the Wilson renormalization group recursion formula and the 1=N expansion of the zero-dimensional reduction, and then calculate the mass critical exponent and the anomalous dimension in various dimensions. DOI: 10.1103/PhysRevD.88.106001

PACS numbers: 11.10.Nx, 11.15.Pg

I. INTRODUCTION AND SUMMARY OF RESULTS A. Introduction The Wilson recursion formula is the oldest, simplest, and most intuitive renormalization group approach, which although approximate, agrees very well with high-temperature expansions [1]. The goal in this article is to apply this method to the self-dual noncommutative 4 in the matrix basis [2], which after appropriate nonperturbative definition becomes an N N matrix model, where N is a regulator in the noncommutative directions. More precisely, we propose to employ, following Refs. [3–5], a combination of i) the Wilson approximate renormalization group recursion formula and ii) the solution to the corresponding zerodimensional large-N counting problem given in our case by the Hermitian Penner matrix model, which can be turned into a multitrace Hermitian quartic matrix model for large values of . As discussed neatly in Ref. [3], the virtue and power of combining these two methods lies in the crucial fact that all leading Feynman diagrams in 1=N will be counted correctly in this scheme, including the so-called ‘‘setting sun’’ diagrams. As it turns out, the recursion formula can also be integrated explicitly in the large-N limit, which in itself is a very desirable property. In a previous work [6], a nonperturbative study of the Ising universality class fixed point in a noncommutative OðNÞ model was carried out using precisely a combination of the above two methods. It was found that the WilsonFisher fixed point makes good sense only for sufficiently small values of up to a certain maximal noncommutativity. In the current work, we focus on the opposite limit of large , although the 1=N expansion invoked in this article is different from the 1=N expansion of the OðNÞ vector model, since N here has direct connection with noncommutativity itself. The central aim of this article, as we will see, is to exhibit as explicitly as possible the matrix-model fixed point which describes the transition from the one-cut (disordered) phase to the two-cut (nonuniform ordered, stripe) phase in the same way that the Wilson-Fisher fixed *[email protected]; [email protected]

1550-7998= 2013=88(10)=106001(30)

point describes the transition from the disordered phase to the uniform ordered phase. B. Summary of results We start by summarizing the main statements and results of this paper. We will be interested in 4 theory on a degenerate noncommutative Moyal-Weyl space with only two noncommuting coordinates Rd ¼ RD R2 , where D ¼ d 2 with commutation relations ½x^ i ; x^ j ¼ iij , ½x^ i ; x ¼ ½x ; x ¼ 0. The action takes the form Z 1 ^ 2 1 2 ~2 1 2 2 ^ þ D ^ S½ ¼ 2 d xTrH @i þ X i @ þ 2 2 2 2 ^þ^ ^þ^ : (1.1) þ 4! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ In the above equation, ¼ ðBÞ=2, 2 ¼ det ð2Þ, and X~i ¼ 2ð1 Þij Xj , where Xi ¼ ðx^ i þ x^ Ri Þ=2. This action is covariant under a duality transformation which exchanges, among other things, positions and momenta as xi $ k~i ¼ 2 B1 ij kj . The value ¼ 1 in particular gives an action which is invariant under this duality transformation. By expanding the field in an appropriate basis (for example, the Landau basis), introducing a cutoff N in the noncommuting directions and a cutoff in the commuting directions, and setting 2 ¼ 1, we obtain the action Z 1 1 D S½M ¼ d xTrN @ Mþ @ M þ 2 Mþ M 2 2 1 2 u þ þ 2 þ r EfM; M g þ ðM MÞ ; (1.2) 2 N r2

8 ¼ ; 2

N u¼ ; 4! 2

Elm

1 ¼ l lm : 2

(1.3)

We will consider in the remainder only the case of Hermitian matrices, viz. M ¼ Mþ :

(1.4)

There are three independent parameters in this theory. These are the usual mass parameter 2 and the quartic coupling constant u plus the inverse noncommutativity

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r ¼ 4=. The free propagator of this theory is simply given by 2

mn ðpÞ ¼

1 : p2 þ 2 þ r2 ðm þ n 1Þ

(1.5)

renormalization group approach consists in rescaling the field in such a way that the kinetic term is brought to its canonical form, we obtain the effective action [Eq. (4.39)]. In position space, this effective action takes the form 02 Z 1Z D ~ 0 Þ2 þ ~ 02 d xTrN ð@ M dD xTrN M 2 2 0 Z Z ~ 02 þ u ~ 04 : dD xTrN M þ r02 dD xTrN EM N (1.8)

In the limit !1, this propagator behaves as 1=ðp2 þ2 Þ. More precisely, we have in this limit the useful properties X r 1 2 mj ðp1 Þrmj ðp2 Þ... ! Nrn10 j1 ðp1 Þrn20 j2 ðp2 Þ... (1.6) 1 2

S þ S ¼

The Wilson renormalization group approach consists in general of three main steps: 1) integration, 2) rescaling, and 3) normalization. In our case, here we will supplement the first step of integration with two approximations: a) truncation and b) the Wilson recursion formula. We start by decomposing the N N matrix M into an ~ and an N N fluctuation N N background matrix M ~ ~ contains matrix m, viz. M ¼ M þ m. The background M slow modes, i.e., modes with momenta less than or equal to , while the fluctuation m contains fast modes, i.e., modes with momenta larger than , where 0 < < 1. The integration step involves performing the path integral over the fluctuation m to obtain an effective path integral ~ alone. We find over the background M Z ~ SðMÞ ~ ~ S½M Z ¼ dMe e : (1.7)

~ 0 is related to the bare field M ~ as The renormalized field M 0 and M are the Fourier transforms of M ~ and follows: If M ~ 0 , respectively, then M sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 N 2þD 0 ðpÞ ¼ 2 Zðg; 2 Þ þ Zðg; 2 ÞMð pÞ: M ðc2 þ 2 Þ2

m

~ The main goal is to compute the effective action SðMÞ, which contains corrections to the operators already ~ together with all present in the original action S½M possible effective interactions generated by the integration ~ up to the fourth process. An exact formula for SðMÞ ~ is given by the cumulant expansion power in the field M [Eq. (4.8)]. The formula in Eq. (4.8) is still very complicated. To simplify it and to get explicit equations, we employ the so-called Wilson truncation and Wilson recursion formula. This is usually thought of as part of the integration step. Wilson truncation means that we calculate quantum corrections to only those terms which appear in the starting action. Wilson recursion formula is completely equivalent to the use in perturbation theory of the Polyakov-Wilson rules given by the following two approximations: 1) We replace every internal propagator 1=ðk2 þ 2 Þ with 1=ðc2 þ 2 Þ, where c is a constant taken to be equal to the cutoff R, and 2) We replace every D D momentum loop integral with another d k=ð2Þ D constant vD ¼ v^ D , where the definition of v^ D is obvious. This is a very long and tedious calculation. The end result is given by the sum of the three equations [Eqs. (4.18), (4.29), and (4.35)]. By performing the second step of the Wilson renormalization group approach, i.e., by scaling momenta as p ! p= so that the cutoff returns to its original value , and the third and final step of the Wilson

(1.9) The renormalized mass 02 , the renormalized quartic coupling constant u0 , and the renormalized inverse noncommutativity r02 are given by 2 02 ¼ ðg; 2 Þ þ r2 NðgÞ Zðg; 2 Þ r2 N ðg; 2 ÞZðg; 2 Þ 2 ; (1.10) ðc þ 2 Þ2 Zðg; 2 Þ r02 ¼

2 r2 r2 ðN þ 1Þ ðgÞ þ e ðgÞ e c2 þ 2 Zðg; 2 Þ r2 N e ðgÞZðg; 2 Þ 2 ; ðc þ 2 Þ2 Zðg; 2 Þ

1 r2 N 4 ðgÞ þ 2 4 ðgÞ 2 2 Z ðg; Þ 4g c þ 2 2r2 N 4 ðgÞZðg; 2 Þ : 2 ðc þ 2 Þ2 Zðg; 2 Þ

(1.11)

u0 ¼ u

The effective coupling g is defined by vD u g¼ 2 : ðc þ 2 þ r2 NÞ2

(1.12)

(1.13)

The various functions Zðg; 2 Þ, ðg; 2 Þ, 2 ðgÞ, and 4 ðgÞ are known nonperturbatively, whereas we were able to determine the functions Zðg; 2 Þ, ðgÞ, 4 ðgÞ, e ðgÞ, and e ðgÞ only perturbatively. These functions are summarized in Table I. The process which led from the bare coupling constants 2 , r2 , and u to the renormalized coupling constants 02 , r02 , and u0 can be repeated an arbitrary number of times. The bare coupling constants will be denoted by 20 , r20 , and u0 , whereas the the renormalized coupling constants at the first step of the renormalization group procedure will be denoted by 21 , r21 , and u1 . At a generic

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step l þ 1 of the renormalization group process, the renormalized coupling constants 2lþ1 , r2lþ1 , and ulþ1 are related to their previous values 2l , r2l , and ul by precisely the above renormalization group equations. The effective coupling constant gl will, of course, be given in terms of 2l , r2l , and ul by the same formula that related g0 to 20 , r20 , and u0 . We are therefore interested in renormalization group flow in a three-dimensional parameter space generated by the mass 2 , the quartic coupling constant u, and the harmonic oscillator coupling constant (inverse noncommutativity) r2 . We have reached the stage where it is very hard to push any further by pure analytical means, and therefore we have to turn to numerical tools. In any case, the renormalization group approach was originally devised with numerical approximations in mind [1]; see also Ref. [7]. A renormalization group fixed point is a point in the space parameter which is invariant under the renormalization group flow. If we denote the fixed point by 2 , r2 , and

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u , then we must set ¼ 0 ¼ , r ¼ r0 ¼ r , and u ¼ u0 ¼ u , as well as g ¼ g0 ¼ g in the above renormalization group equations (1.10), (1.11), and (1.12). The matrix model fixed point corresponding to infinite noncommutativity is given by the following equations: r2 ¼ 0;

(1.14)

fðg Þ ¼ 0;

(1.15)

; 1

(1.16)

g ð1 þ ^ 2 Þ2 : v^ D

(1.17)

^ 2 ¼

u^ ¼

The functions f and are given by (with 0 ¼ =D and

¼ 4 D)

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 8 ðgÞ 2 fðgÞ ¼ 1 þ 2 ð1 2 2 0 Z2 ðgÞÞ þ 2 ð1 þ 0 ÞZ2 ðgÞð2 ðgÞ 1Þ þ 2 4 2Z2 ðgÞ 0 ; 4g 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ðgÞ 42 2 4 þ 4Z2 ðgÞ 0 5: ðgÞ ¼ 4ð1 þ 0 ÞZ2 ðgÞ 4g We can immediately see that this fixed point is fully determined by the functions 2 , Z2 , and Z4 , which are known nonperturbatively. These functions are the twopoint proper vertex, the wave function renormalization and the four-point proper vertex of the quartic matrix model, respectively. The results of this calculation are shown in Table II. In our approximation, we have checked that there is always a nontrivial fixed point for any value of in the interval 0 < < 1. We can also compute the mass critical exponent and the anomalous dimension within this scheme. From the wave function renormalization [Eq. (1.9)], we compute immediately the anomalous dimension. We find ¼

ln ð4 ðg Þ=4g Þ : 2 2 ln

(1.20)

The computation of the mass critical exponent requires linearization of the renormalization group equations (1.10), (1.11), and (1.12). The linearized renormalization group equations are of the form [with G ¼ G G , where G ¼ ðG1 ¼ 2 ; G2 ¼ u; G3 ¼ r2 Þ] G0 ¼ MðG ; ÞG: The matrix M in our case is of the form

(1.21)

0

M11

B B @ M21 0

(1.18)

(1.19)

M12 M22 0

M13

1

C M23 C A:

(1.22)

M33

We find that 3 ¼ 2 e ðG Þ=ZðG ; 2 Þ > 1, and hence r2 is a relevant coupling constant like the mass. However, the function e ðgÞ used in this formula is only known perturbatively, and hence this conclusion should be taken with care. The two remaining eigenvalues are determined from the linearized renormalization group equations in the twodimensional space generated by G1 ¼ 2 and G2 ¼ u. As it turns out, this problem depends only on functions which are fully known nonperturbatively. The eigenvalues 1 ð Þ and 2 ð Þ can be determined from the trace and determinant of M in an obvious way. It is not difficult to convince ourselves that these renormalization group eigenvalues must scale as ð Þ ¼ ð1Þ y :

(1.23)

This formula [Eq. (1.23)] was used as a crucial test for our numerical calculations. In particular, we have determined by means of this formula the range of the dilatation parameter over which the logarithm of the eigenvalues scales linearly with ln . It is natural to expect this behavior to hold only if the renormalization group steps are

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sufficiently small so not to alter drastically the infrared physics of the problem. For D ¼ 2, the renormalization group steps can be thought of as small, and the behavior in Eq. (1.23) holds in the regime ln < 1. As it happens, this is the most important case corresponding to d ¼ 4. We find explicitly the following fits: ln j1 j ¼ 1:296 ln þ 0:412; ln j2 j ¼ 0:435 ln þ 2:438:

(1.24)

We conclude immediately that the scaling field u1 corresponding to the mass is relevant, while the scaling field u2 corresponding to the quartic coupling constant is irrelevant. This is the usual conclusion in d ¼ 4. The critical exponents in D ¼ 2 (d ¼ 4) are given, respectively, by y1 ¼ 1:296;

y2 ¼ 0:435:

(1.25)

The mass critical exponent is given by the inverse of the critical exponent y1 associated with the relevant direction, viz. ¼ 1=y1 . The corresponding results for y1 and are included in Table IV. We note that the results shown in Table IV are very close to the average value of 2=d and 2=D, viz. ¼

1 1 þ : d D

(1.26)

C. Outline This article is organized as follows: In Sec. II, we write down noncommutative 4 in the matrix basis and discuss some useful approximations involving the propagator at the self-dual point which are valid for large . In Sec. III, we consider the dimensional reduction of noncommutative 4 and some of its properties. In particular, we will derive a very simple nonperturbative equation for the two-point proper vertex, which will be used to test the results obtained later using the 1=N expansion and the recursion formula. In Sec. IV, we perform the tedious task of deriving the renormalization group equations which control the flow of the coupling constants of the model (three in this case) using the Wilson renormalization group recursion formula and 1=N expansion. The most difficult piece of the calculation, as we will see, is wave function renormalization. In Sec. V, we derive the nontrivial fixed point and the associated critical exponents and by solving numerically via the Newton-Raphson algorithm the renormalization group equations and discuss some of the relevant physics. In Sec. VI, we extend the analysis to the Grosse-VignesTourneret model, which involves an extra term, the double trace operator ðTrN MÞ2 , required for the renormalizability of the theory. We conclude in Sec. VII with a summary of the obtained results and a brief outlook. We have also included three appendixes for completeness and for the convenience of interested readers.

II. THE NONCOMMUTATIVE 4 THEORY A. The model Let us consider a 4 theory on a generic noncommutative Moyal-Weyl space Rd . We introduce noncommutativity in momentum space by introducing a minimal coupling to a constant background magnetic field Bij as was done originally by Langmann, Szabo, and Zarembo in Ref. [2]. The most general action with a quartic potential takes in the operator basis the form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ^ ^ þ D^ 2i S ¼ det ð2ÞTrH ~ C^ 2i þ 2 ^þ^ ^þ^ 0 ^ þ ^ þ ^ ^ þ þ : 4! 4!

(2.1)

In this equation, D^ i ¼ @^ i iBij Xj and C^ i ¼ @^ i þ iBij Xj , where Xi ¼ ðx^ i þ x^ Ri Þ=2. In the original Langmann-Szabo model, we choose ¼ 1, ~ ¼ 0, and 0 ¼ 0, which, as it turns out, leads to a trivial model [8]. The famous Grosse-Wulkenhaar model corresponds to ¼ ~ and 0 ¼ 0. We choose without any loss of generality ¼ ~ ¼ 1=4. The Grosse-Wulkenhaar model corresponds to the addition of a harmonic oscillator potential to the kinetic action which modifies and thus allows us to control the IR behavior of the theory. A particular version of this theory was shown to be renormalizable by Grosse and Wulkenhaar in Ref. [9]. The action of interest in terms of the star product is given by S¼

1 1 2 dd x þ @2i þ ðBij xj Þ2 þ 2 2 2 þ þ þ : 4!

Z

(2.2)

Equivalently,

1 2 1 2 2 2 @i þ x~i þ S¼ 2 2 2 þ þ þ : 4! Z

dd x

þ

(2.3)

The harmonic oscillator coupling constant is defined by 2 ¼ B2 2 =4, whereas the coordinate x~i is defined by x~i ¼ 2ð1 Þij xj . It was shown in Ref. [8] that this action is covariant under a duality transformation which exchanges, among other things, positions and momenta 2 as xi $ k~i ¼ B1 ij kj . The value ¼ 1 in particular gives an action which is invariant under this duality transformation. The theory at 2 ¼ 1 is essentially the original Langmann-Szabo model. Let us consider now a 4 theory on a noncommutative Moyal-Weyl space with only two noncommuting coordinates, viz. Rd ¼ RD R2 , where D ¼ d 2. This is the degenerate case. The above action generalizes to

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possibility of the existence of other fixed-point solutions even within this scheme. (6) The renormalization group eigenvalue 3 was determined to be larger than 1, and hence r2 is a relevant coupling similar to the mass. This statement was, however, derived from the available perturbative knowledge of the proper vertex e , and thus should be taken with care. (7) We have found that the renormalization group eigenvalues 1 (corresponding to the mass, relevant) and 2 (corresponding to the quartic coupling, irrelevant) become complex conjugates of each other at the value of the dilatation parameter ln ’ 1. After extracting the dependence on , we obtain the mass critical exponent

double trace term is renormalizable in the limit considered. A similar conclusion is obtained in Ref. [13]. This matrix model fixed point seems to be different from the Ising universality class fixed point.5 A more comprehensive scheme which seems to be more appropriate to describing the matrix transition is to consider instead of the GrosseVignes-Tourneret model the most general quartic action containing all multitrace operators consistent with the symmetry M ! M. Thus, the extra terms ðTrN M2 Þ2 , ðTrN MÞ4 , and ðTrN MÞ2 TrN M2 should also be included. For example, the term ðTrN M2 Þ2 was found to play a crucial role in the renormalizability of noncommutative 4 theory in the large- limit in Ref. [13]. We hope to return to this point as well as other points (see towards the end of the next section) in the future.

1 1 þ : (7.2) d D In D ¼ 2 (the most important case for us) it is found that both the eigenvalues ln 1 and ln 2 scale linearly with ln over the entire real range ln < 1. We obtain the indices ’

VII. CONCLUSION AND OUTLOOK In this article, we have presented a study of 4 theory on noncommutative spaces with only two noncommuting directions at the self-dual point using a combination of the Wilson approximate renormalization group recursion formula and the solution to the corresponding zerodimensional matrix model at large N. The most important results of this work are as follows: (1) The Penner matrix model TrN ðM2 =2 þ m2 EM2 þ gM4 =NÞ can be systematically solved by the multitrace approach of Ref. [15], as illustrated by the computation of the two-point proper vertex in Sec. III. This might be an alternative approach to the one pursued in Ref. [23]. (2) The action studied in this article is a generalization of the Penner matrix model of the form S½M ¼

1 1 ð@ MÞ2 þ 2 M2 2 2 u þ r2 EM2 þ M4 : N

Z

dD xTrN

(7.1)

(3) The renormalizations of the mass term, the harmonic oscillator term, and the quartic interaction are straightforward to obtain in this scheme. (4) The most difficult calculation of all is wave function renormalization. We have conjectured in this article that wave function renormalization within the scheme of the recursion formula is fully encoded in the approximation (4.23). (5) We have found a fixed-point solution of the renormalization group equations (4.44), (4.45), and (4.46) with r2 ¼ 0, 2 < 0 and u > 0 in all dimensions D ¼ 2, 3, 4 corresponding to the dimensions d ¼ 4, 5, 6, respectively. We do not exclude here the 5

The d-dimensional physics is described in terms of a critical behavior in D ¼ d 2 dimensions.

y1 ¼ 1:296ð ¼ 0:772Þ;

y2 ¼ 0:435:

(7.3)

(8) The anomalous dimension was found to scale as ’

4D ; 2

! 0:

(7.4)

(9) An extension of the above results to the GrosseVignes-Tourneret model, which involves an extra term given by the double trace operator ðTrN MÞ2 , was also given. We conclude this article by indicating that the most natural extension of this work should be to repeat the same analysis of the matrix model fixed point by replacing the recursion formula with the exact functional renormalization group method. The functional renormalization group (as opposed to other forms of the renormalization group) is the most direct implementation of the Wilson idea which goes along the lines of the recursion formula but is exact, although explicit calculation will undoubtedly involve various truncations which by their nature are also approximate. ACKNOWLEDGMENTS This research was supported by the National Agency for the Development of University Research (ANDRU) under PNR Contract No. U23/Av58 (8/u23/2723). APPENDIX A: DEGENERATE DUALITY TRANSFORMATIONS Let us start with the quadratic action. We have (with Di ¼ @i iBij xj and Ci ¼ @i þ iBij xj )

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The propagator 0mt satisfies the equation X ðp2 þ 2 þ r2 ð2m 1ÞÞ0mt þ r2 2 0lt ¼ mt :

APPENDIX C: THE PROPAGATOR We start with the Laplacian Gmn;kl ¼ ðp2 þ 2 þ r2 ðm þ n 1ÞÞml nk þ r2 2 mn kl : The propagator is defined by X X nm;lk Gkl;ts ¼ Gmn;kl lk;st ¼ ns mt : k;l

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l

(C9)

(C1) This can be solved by the ansatz (C2)

0mt ¼

k;l

Since the Laplacian Gmn;kl is nonzero only when m þ k ¼ n þ l, the propagator mn;kl is nonzero only when m þ k ¼ n þ l. We introduce n ¼ m þ and s ¼ t þ , then X Gmmþ;lþl llþ;tþt ¼ mt : (C3)

For ¼ , we obtain an ordinary matrix inversion for every value of , viz. X Gmmþ;lþl llþ;tþt ¼ mt : (C5)

r2 2 P 1 þ r2 2 l ½p2 þ 2 þ r2 ð2l 1Þ1 1 2 : 2 p þ þ r2 ð2t 1Þ

(C11)

In this paper, we are dealing with the limit ! 1, and hence we do not really need the above exact solution of 0mt . In fact, all we need is the leading correction in the new parameter r2 2 (recall that r2 ¼ 4=). From the above solution, we obtain in a straightforward way the result

l

We introduce Gml ¼ Gmmþ;lþl and lt ¼ llþ;tþt , writing this as X Gml lt ¼ mt : (C6)

(C10)

Xt ¼

We remark that for Þ , we have immediately (C4)

2

We can find immediately

l

llþ;tþt ¼ 0:

1 ½mt þ Xt : p þ þ r2 ð2m 1Þ 2

0mt ¼

l

1 p2 þ 2 þ r2 ð2m 1Þ r2 2 4 4 Þ : þ Oðr mt 2 p þ 2 þ r2 ð2t 1Þ (C12)

We have From Eqs. (C4), (C8), and (C12), we get the propagator

Gml ¼ ðp2 þ 2 þ r2 ð2m þ 1ÞÞml þ r2 2 0 : (C7)

mn;kl ¼

Thus, for ¼ and Þ 0, we have immediately ml

1 ml : ¼ 2 2 2 p þ þ r ð2m þ 1Þ

2

r2 2 p2 þ 2 þ r2 ðk þ l 1Þ 4 4 mn kl þ Oðr Þ :

(C8)

The only value which requires special attention is ¼ 0.

[1] K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974). [2] E. Langmann, R. J. Szabo, and K. Zarembo, J. High Energy Phys. 01 (2004) 017; Phys. Lett. B 569, 95 (2003). [3] G. Ferretti, Nucl. Phys. B487, 739 (1997). [4] G. Ferretti, Nucl. Phys. B450, 713 (1995). [5] S. Nishigaki, Phys. Lett. B 376, 73 (1996). [6] B. Ydri and A. Bouchareb, J. Math. Phys. (N.Y.) 53, 102301 (2012). [7] C. Bagnuls and C. Bervillier, Phys. Rep. 348, 91 (2001). [8] E. Langmann and R. J. Szabo, Phys. Lett. B 533, 168 (2002).

1 þ r2 ðm þ n 1Þ

þ ml nk p2

(C13)

[9] H. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005); J. High Energy Phys. 12 (2003) 019; Commun. Math. Phys. 254, 91 (2005). [10] H. Grosse and F. Vignes-Tourneret, J. Noncommut. Geom. 4, 555 (2010). [11] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 2002), ed. 4. [12] J. M. Gracia-Bondia and J. C. Varilly, J. Math. Phys. (N.Y.) 29, 869 (1988). [13] C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B664, 371 (2003).

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[14] M. Disertori, R. Gurau, J. Magnen, and V. Rivasseau, Phys. Lett. B 649, 95 (2007). [15] D. O’Connor and C. Saemann, J. High Energy Phys. 08 (2007) 066. [16] E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Commun. Math. Phys. 59, 35 (1978). [17] Y. Shimamune, Phys. Lett. 108B, 407 (1982). [18] X. Martin, J. High Energy Phys. 04 (2004) 077; F. G. Flores, X. Martin, and D. O’Connor, Int. J. Mod. Phys. A 24, 3917 (2009); M. Panero, J. High Energy Phys. 05 (2007) 082. [19] S. S. Gubser and S. L. Sondhi, Nucl. Phys. B605, 395 (2001). [20] W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 05 (2004) 047.

[21] J. Ambjorn and S. Catterall, Phys. Lett. B 549, 253 (2002); W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 06 (2004) 042; J. Medina, W. Bietenholz, and D. O’Connor, J. High Energy Phys. 04 (2008) 041; C. R. Das, S. Digal, and T. R. Govindarajan, Mod. Phys. Lett. A 23, 1781 (2008); F. Lizzi and B. Spisso, Int. J. Mod. Phys. A 27, 1250137 (2012). [22] A. P. Polychronakos, Phys. Rev. D 88, 065010 (2013). [23] H. Grosse and R. Wulkenhaar, arXiv:1205.0465. [24] G. R. Golner, Phys. Rev. B 8, 339 (1973). [25] P. Kopietz, L. Bartosch, and F. Schutz, Lect. Notes Phys. 798, 1 (2010).

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The fate of the Wilson-Fisher fixed point in non-commutative φ 4 Badis Ydria) and Adel Boucharebb) Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria (Received 12 July 2012; accepted 6 September 2012; published online 4 October 2012)

In this article we study non-commutative vector sigma model with the most general φ 4 interaction on Moyal-Weyl spaces. We compute the 2- and 4-point functions to all orders in the large N limit and then apply the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. The non-commutative Wilson-Fisher fixed point interpolates between the commutative Wilson-Fisher fixed point of the Ising universality class which is found to lie at zero value of the critical coupling constant a* of the zero dimensional reduction of the theory, and a novel strongly interacting fixed point which lies at infinite value of a* corresponding to maximal non-commutativity beyond which the two-sheeted C 2012 American structure of a* as a function of the dilation parameter disappears. Institute of Physics. [http://dx.doi.org/10.1063/1.4754816]

I. INTRODUCTION

A non-commutative scalar field theory is a non-local field theory in which the ordinary local point-wise multiplication of fields is replaced by a non-local star product such as the Moyal-Weyl star product.1, 2 We suggest3 and references therein for an elementary and illuminating discussion of the Moyal-Weyl product and other star products and their relation to the Weyl map,4 coherent states,5–7 Berezin quantization,8 and deformation quantization.9 The first study of the quantum theory of a non-commutative φ 4 is found in Ref. 10, where it is shown that planar diagrams in the non-commutative theory are essentially identical to the planar diagrams in the commutative theory. More interestingly, it was found in Ref. 11 that the renormalized one-loop action of a non-commutative φ 4 suffers from an infrared divergence which is obtained when we send either the external momentum or the non-commutativity to zero. This non-analyticity at small momenta or small non-commutativity (IR) which is due to the high energy modes (UV) in virtual loops is termed the UV-IR mixing and it is intimately related to the structure of the non-planar diagrams of the theory. As it turns out this effect is expected in a non-local theory such as a non-commutative φ 4 . Renormalization of the non-commutative φ 4 was studied, for example, in Refs. 12–18. The main observation of Ref. 14 is that we can control the UV-IR mixing found in non-commutative φ 4 by modifying the large distance behavior of the free propagator through adding a harmonic oscillator potential to the kinetic term. More precisely, the UV-IR mixing of the theory is implemented precisely in terms of a certain duality symmetry of the new action which connects momenta and positions.19 The corresponding Wilson-Polchinski renormalization group equation20, 21 of the theory can then be solved in terms of ribbon graphs drawn on Riemann surfaces. The existence of a regular solution of the Wilson-Polchinski equation together with the fact that we can scale to zero the coefficient of the harmonic oscillator potential in two dimensions leads to the conclusion that the standard non-commutative φ 4 in two dimensions is renormalizable.16 In four dimensions, the harmonic oscillator term seems to be essential for the renormalizability of the theory.15 a) E-mail addresses: [email protected] and [email protected] b) E-mail: [email protected]

0022-2488/2012/53(10)/102301/14/$30.00

53, 102301-1

C 2012 American Institute of Physics

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There are many other approaches to renormalization of quantum non-commutative φ 4 . See, for example, Refs. 50–54 and 58. A very remarkable property of quantum non-commutative φ 4 is the appearance of a new order in the theory termed the striped phase which was first computed in a one-loop self-consistent HartreeFock approximation in the seminal paper.22 For alternative derivations of this order see, for example, Refs. 48 and 49. It is believed that the perturbative UV-IR mixing is only a manifestation of this more profound property. As it turns out, this order should be called more appropriately a non-uniform ordered phase in contrast with the usual uniform ordered phase of the Ising universality class and it is related to spontaneous breaking of translational invariance. It was numerically observed in d = 4 in Ref. 23 and in d = 3 in Ref. 24 where the Moyal-Weyl space was approximated by a non-commutative fuzzy torus.25 The beautiful result of Ref. 24 shows explicitly that the minimum of the model shifts to a non-zero value of the momentum indicating a non-trivial condensation and hence spontaneous breaking of translational invariance. In summary the phase diagram of quantum non-commutative φ 4 consists of three phases. The usual disordered and uniform ordered phases together with the non-uniform (striped) ordered phase and thus a triple point must exist. In Refs. 22 and 48, it is conjectured that this point is a Lifshitz point which is a multi-critical point at which a disordered, a homogeneous (uniform) ordered and a spatially modulated (non-uniform) ordered phases meet.26 We note that Ref. 48 uses Wilson renormalization group approach44 to derive the Wilson-Fisher fixed point of the theory at one-loop which is found to suffer from an instability at large non-commutativity. The non-commutative fuzzy torus used to regularized non-commutative φ 4 is a matrix model which can be mapped to a lattice. Another matrix regularization of non-commutative φ 4 can be found in Refs. 27 and 28 which emphasizes connection to fuzzy spaces.29, 30 The phase structure of non-commutative φ 4 in d = 2 and d = 3 using the fuzzy sphere31, 32 regularization was studied extensively in Refs. 33–37. Again the phase diagram consists of three phases: a disordered phase, a uniform ordered phases, and a non-uniform ordered phase which meet at a triple point. In this case it is well established that the transitions from the disordered phase to the non-uniform ordered phase and from the non-uniform ordered phase to the uniform ordered phase originate from the one-cut/two-cut transition in the quartic hermitian matrix model.40, 41 See also Refs. 42 and 43. This was also confirmed analytically by the multi-trace approach of Refs. 38 and 39 which relies on the expansion of the kinetic term in the action instead of the usual expansion in the interaction which is very reminiscent to the hopping parameter expansion on the lattice.55, 56 The non-uniform ordered phase on the fuzzy sphere (sometimes also called the matrix phase) goes to the striped phase on the Moyal-Weyl plane in appropriate flattening limit. Finally, there is a strong evidence that the non-uniform ordered phase should be present on all non-commutative spaces regardless of the dimension. In this article we will attempt to understand the phase structure of non-commutative φ 4 near the Wilson-Fisher fixed point, i.e., the transition disordered/uniform ordered and how this critical behavior changes with the non-commutativity until it merges with the transition disordered/nonuniform ordered. We will consider a large vector O(N) sigma model where all leading Feynman diagrams can be taken into consideration and employ the approximate Wilson renormalization group equation to study the renormalized action of the theory. This article is organized as follows. In Sec. II we will introduce the non-commutative vector sigma model with the most general φ 4 interaction on the Moyal-Weyl spaces Rdθ , and then we will compute in the large N limit the 2- and 4-point functions to all orders. In Sec. III we will apply the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. In particular we will derive the renormalization group equations and then calculate the fixed points of the theory. In Sec. IV we give our conclusion and outlook. II. THE NON-COMMUTATIVE O(N) SIGMA MODEL

The cumulant expansion: We will consider in this note the φ 4 action S = d d xa (−∂i2 + μ2 )a + Sint ,

(1)

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J. Math. Phys. 53, 102301 (2012)

IV. CONCLUSION AND OUTLOOK

In this article we have studied the non-commutative vector sigma model with the most general φ 4 interaction on the Moyal-Weyl spaces Rdθ in the large N limit. We computed the 2- and 4-point functions to all orders and then applied the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. More precisely, we have taken into account all leading Feynman diagrams in the large N limit, and then estimated them using the so-called Wilson contraction which consists of three rules. We have argued that the first rule of Wilson contraction is certainly valid in the limit θ¯ −→ 0. In the commutative theory we observe that the non-perturbative Wilson-Fisher fixed point is located on the second sheet of the parameter a which is the coupling constant of the zero dimensional reduction of the theory. The Wilson-Fisher fixed point can be reached in the limit in which we send the dilation parameter ρ to zero which corresponds to a single step of the renormalization group transformation. The Wilson-Fisher fixed point is also found to scale to zero in the limit ρ −→ 0 although differently then the perturbative fixed point in the limit ρ −→ 1. In the non-commutative theory the Wilson-Fisher fixed point does not scale to zero in the limit ρ −→ 0 in contrast with the perturbative fixed point which still scales to zero as ρ −→ 1. The main obstacle comes from the fact that the critical coupling constant a* which starts from 0 at ρ = 1, increases to ∞ at ρ = 1 − t/2, does not return to zero as we decrease ρ back from ρ = 1 − t/2 to 0. The non-perturbative window (sheet) shrinks as we increase the non-commutativity until it disappears at t = 2. At this point, the critical value a* diverges and the non-commutative WilsonFisher point becomes very different. It is natural to conjecture that the point at t = 2 is completely different (lies in a different universality class) then the commutative Wilson-Fisher fixed point at t = 1 which is in the Ising universality class. Therefore, the non-commutative Wilson-Fisher fixed point interpolates between these two classes. A thorough study of the renormalized action (including the coupling constant v) and the noncommutative Wilson-Fisher fixed point (including the range t > 2), together with the computation of the critical exponents, and improvement of the approximate Wilson renormalization group recursion formula will be reported elsewhere.57 For example, already we observe from our study here that we can compute the 2-point function without any resort to the first rule of Wilson contraction resulting in Eq. (40) which indicates the presence of a wave function renormalization in the theory. As already discussed in Ref. 48 this leads to an instability of the theory. Solving the renormalization group equations in the case of a non-zero wave function may/will require a numerical approach. Another approach to the Wilson renormalization group recursion formula of non-commutative φ 4 will be in terms of the matrix model associated with the theory which will allow us to access the limit θ¯ −→ ∞ instead. We also hope to return to this point in future communication. 57 ACKNOWLEDGMENTS

This research was supported by “The National Agency for the Development of University Research (ANDRU)” under PNR Contract No. U23/Av58 (8/u23/2723). 1 H.

J. Groenewold, Physica 12, 405 (1946). E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949). 3 G. Alexanian, A. Pinzul, and A. Stern, Nucl. Phys. B 600, 531 (2001); e-print arXiv:hep-th/0010187. 4 H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). 5 V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Scr. 55, 528 (1997); e-print arXiv:quant-ph/9612006. 6 A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986). 7 J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985). 8 F. A. Berezin, Commun. Math. Phys. 40, 153 (1975). 9 M. Kontsevich, Lett. Math. Phys. 66, 157 (2003); e-print arXiv:q-alg/9709040 [q-alg]. 10 T. Filk, Phys. Lett. B 376, 53 (1996). 11 S. Minwalla, M. Van Raamsdonk, and N. Seiberg, J. High Energy Phys. 0002, 020 (2000); e-print arXiv:hep-th/9912072. 12 I. Chepelev and R. Roiban, J. High Energy Phys. 0005, 037 (2000); e-print arXiv:hep-th/9911098. 13 I. Chepelev and R. Roiban, J. High Energy Phys. 0103, 001 (2001); e-print arXiv:hep-th/0008090. 2 J.

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J. Math. Phys. 53, 102301 (2012)

14 H.

Grosse and R. Wulkenhaar, Commun. Math. Phys. 254, 91 (2005); e-print arXiv:hep-th/0305066. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005); e-print arXiv:hep-th/0401128. 16 H. Grosse and R. Wulkenhaar, J. High Energy Phys. 0312, 019 (2003); e-print arXiv:hep-th/0307017. 17 V. Rivasseau, F. Vignes-Tourneret, and R. Wulkenhaar, Commun. Math. Phys. 262, 565 (2006); e-print arXiv:hep-th/0501036. 18 R. Gurau, J. Magnen, V. Rivasseau, and F. Vignes-Tourneret, Commun. Math. Phys. 267, 515 (2006); e-print arXiv:hep-th/0512271. 19 E. Langmann and R. J. Szabo, Phys. Lett. B 533, 168 (2002); e-print arXiv:hep-th/0202039. 20 J. Polchinski, Nucl. Phys. B 231, 269 (1984). 21 G. Keller, C. Kopper, and M. Salmhofer, Helv. Phys. Acta 65, 32 (1992). 22 S. S. Gubser and S. L. Sondhi, Nucl. Phys. B 605, 395 (2001); e-print arXiv:hep-th/0006119. 23 J. Ambjorn and S. Catterall, Phys. Lett. B 549, 253 (2002); e-print arXiv:hep-lat/0209106. 24 W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 0406, 042 (2004); e-print arXiv:hep-th/0404020. 25 J. Ambjorn, Y. M. Makeenko, J. Nishimura, and R. J. Szabo, J. High Energy Phys. 0005, 023 (2000); e-print arXiv:hep-th/0004147. 26 R. M. Hornreich, M. Luban, and S. Shtrikman, Phys. Rev. Lett. 35, 1678 (1975). 27 E. Langmann, R. J. Szabo, and K. Zarembo, J. High Energy Phys. 0401, 017 (2004); e-print arXiv:hep-th/0308043. 28 H. Steinacker, J. High Energy Phys. 0503, 075 (2005); e-print arXiv:hep-th/0501174. 29 A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya (Singapore World Scientific, Singapore, 2007) p. 191; e-print arXiv:hep-th/0511114. 30 D. O’Connor, Mod. Phys. Lett. A 18, 2423 (2003). 31 J. Hoppe, Ph.D dissertation, MIT, 1982. 32 J. Madore, Class. Quantum Grav. 9, 69 (1992). 33 X. Martin, J. High Energy Phys. 0404, 077 (2004); e-print arXiv:hep-th/0402230. 34 F. Garcia Flores, X. Martin, and D. O’Connor, Int. J. Mod. Phys. A 24, 3917 (2009); e-print arXiv:0903.1986 [hep-lat]. 35 M. Panero, J. High Energy Phys. 0705, 082 (2007); e-print arXiv:hep-th/0608202. 36 J. Medina, W. Bietenholz, and D. O’Connor, J. High Energy Phys. 0804, 041 (2008); e-print arXiv:0712.3366 [hep-th]. 37 C. R. Das, S. Digal, and T. R. Govindarajan, Mod. Phys. Lett. A 23, 1781 (2008); e-print arXiv:0706.0695 [hep-th]. 38 D. O’Connor and C. Saemann, J. High Energy Phys. 0708, 066 (2007); e-print arXiv:0706.2493 [hep-th]. 39 C. Saemann, SIGMA 6, 050 (2010); e-print arXiv:1003.4683 [hep-th]. 40 E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Commun. Math. Phys. 59, 35 (1978). 41 Y. Shimamune, Phys. Lett. B 108, 407 (1982). 42 P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, Phys. Rep. 254, 1 (1995); e-print arXiv:hep-th/9306153. 43 B. Eynard, Cours de Physique Theorique de Saclay (unpublished). 44 K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974). 45 S. Hikami and E. Brezin, J. Phys. A 12, 759 (1979). 46 G. Ferretti, Nucl. Phys. B 450, 713 (1995); e-print arXiv:hep-th/9504013. 47 S. Nishigaki, Phys. Lett. B 376, 73 (1996); e-print arXiv:hep-th/9601043. 48 G.-H. Chen and Y.-S. Wu, Nucl. Phys. B 622, 189 (2002); e-print arXiv:hep-th/0110134. 49 P. Castorina and D. Zappala, Phys. Rev. D 68, 065008 (2003); e-print arXiv:hep-th/0303030. 50 C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B 633, 250 (2002); e-print arXiv:hep-th/0202155. 51 C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B 664, 371 (2003); e-print arXiv:hep-th/0304159. 52 R. Gurau and O. J. Rosten, J. High Energy Phys. 0907, 064 (2009); e-print arXiv:0902.4888 [hep-th]. 53 L. Griguolo and M. Pietroni, J. High Energy Phys. 0105, 032 (2001); e-print arXiv:hep-th/0104217. 54 R. Gurau, J. Magnen, V. Rivasseau, and A. Tanasa, Commun. Math. Phys. 287, 275 (2009); e-print arXiv:0802.0791 [math-ph]. 55 I. Montvay and G. Munster, Quantum Fields on a Lattice, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1994), p. 491. 56 J. Smit, Cambridge Lect. Notes Phys. 15, 1 (2002). 57 Badis Ydri, Rachid Ahmim, and Adel Bouchareb, work in progress. 58 A. Sfondrini, T. A. Koslowski, Int. J. Mod. Phys. A 26, 4009 (2011); e-print arXiv:1006.5145. 15 H.

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International Journal of Modern Physics A Vol. 28, No. 28 (2013) 1350139 (15 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217751X1350139X

EFFECTS OF NONCOMMUTATIVITY ON LIGHT HYDROGEN-LIKE ATOMS AND PROTON RADIUS

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M. MOUMNI Department of Matter Sciences, University of Biskra, Algeria [email protected] A. BENSLAMA Department of Physics, University Constantine 1, Algeria [email protected] Received 21 August 2013 Revised 3 October 2013 Accepted 3 October 2013 Published 31 October 2013 We study the corrections induced by the theory of noncommutativity, in both space– space and space–time versions, on the spectrum of hydrogen-like atoms. For this, we use the relativistic theory of two-particle systems to take into account the effects of the reduced mass, and we use perturbation methods to study the effects of noncommutativity. We apply our study to the muon hydrogen with the aim to solve the puzzle of proton radius [R. Pohl et al., Nature 466, 213 (2010) and A. Antognini et al., Science 339, 417 (2013)]. The shifts in the spectrum are found more noticeable in muon H (µH) than in electron H (eH) because the corrections depend on the mass to the third power. This explains the discrepancy between µH and eH results. In space–space noncommutativity, the parameter required to resolve the puzzle θss ≈ (0.35 GeV)−2 , exceeds the limit obtained for this parameter from various studies on eH Lamb shift. For space– time noncommutativity, the value θst ≈ (14.3 GeV)−2 has been obtained and it is in agreement with the limit determined by Lamb shift spectroscopy in eH. We have also found that this value fills the gap between theory and experiment in the case of µD and improves the agreement between theoretical and experimental values in the case of hydrogen–deuterium isotope shift. Keywords: Noncommutativity; hydrogen-like atoms; proton radius. PACS numbers: 02.40.Gh, 67.63.Gh, 31.30.jr

1. Introduction Historically, experimental spectroscopy was the perfect test for any theory having any connection with matter and it played the leading role in calibrating the values of physical constants. But lately, it has reached such precision that it accessed the role of indicator of new theories; and last experience on the Lamb shift in muonic hydrogen1 is the perfect example. 1350139-1

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The discrepancy between these experimental results rµH = 0.84169(66) fm and those extracted from electronic hydrogen or elastic electron–proton scattering and recorded in CODATA reH = 0.8775(51) fm (Ref. 2) (the values are at 7σ variance with respect to each other) has had an impact in the whole scientific community and this raised many questions about the cause of such disagreement. The experimental methods used to obtain the two results are very elaborated. This is why many studies have investigated how to explain this difference and to reconcile the two results by trying to rectify the theory. But the difference between the two experiments remains a puzzle until now and especially after being reinforced recently with a more accurate value rµH = 0.84087(39).3 Nonperturbative numerical computations of the Dirac equation confirmed the validity of perturbation methods used to compute the radius.4,5 No significant QED correction has been found yet, which would explain the discrepancy.6,7 Using electron scattering experiments,8 found that data rules out values of the third Zemach moment large enough to explain the puzzle. Three-body physics does not solve the problem as demonstrated in Ref. 9. Constraints from low energy data disfavor new spin-0, spin-1 and spin-2 particles as an explanation.10 There are some claims that proton polarizability contribution in the Lamb shift may explain the discrepancy because it is proportional to the lepton mass to the fourth power.11,12 These effects could be probed in scattering experiment planned to run at Paul Scherrer Institute (PSI). For more information about the different approaches to the problem, see Refs. 6, 13–15. Because Pohl et al. used an indirect method to calculate rµH that involves comparing the frequency measured experimentally with that given theoretically according to the radius:1 rp2 rp3 ∆(2P1/2 → 2S1/2 ) = 206.0573(45) − 5.2262 2 + 0.0347 3 , meV fm fm

(1a)

rp2 rp3 ∆(2P3/2 → 2S1/2 ) = 209.9779(49) − 5.2262 2 + 0.0347 3 , (1b) meV fm fm we propose, in this paper, to modify the precedent theoretical expressions of the transition frequency by incorporating the corrections induced by the noncommutative structure of space–time. The idea of taking noncommutative space–time coordinates dates from the 1930s. It had as objective to avoid infinities in Coulomb potentials (gravitation and electricity) by introducing a lower bound for the measurement of length. Despite the fact that the concept was suffering from some problems with unitarity and causality, the theory evolved from the mathematical point of view, especially after the work of Connes in the eighties of last century.16 In 1999, the work of Seiberg and Witten on string theory17 has aroused new interest in the theory. They showed that the dynamics of the endpoints of an open string on a D-brane in the presence of a magnetic background field is described by a theory of Yang–Mills on a noncommutative space–time. 1350139-2

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Effects of Noncommutativity on Light Hydrogen-like Atoms and Proton Radius

Today, we find noncommutativity in various fields of physics such as solid state physics, where it was shown that is the framework in which Hall conductivity is quantized18 and that it is the proper tool replacing Bloch’s theory whenever translation invariance is broken in aperiodic solids.19 In fluid mechanics, noncommutative fluids are introduced by studying the quantum Hall effect20 or bosonization of collective fermion states.21 There is also some connection with quantum statistical physics,22 and it is also an interpretation of Ising-type models.23 One can even find a manifestation of the noncommutativity in the physiology of the brain, where noncommutative computation in the vestibulo-ocular reflex was demonstrated in a way that is unattainable by any commutative system24,25 is an excellent reference for the different manifestations and applications of noncommutative field theory. The theory is a distortion of space–time where the coordinates xµ become Hermitian operators and thus do not commute: µ ν xnc , xnc = iθµν , µ, ν = 0, 1, 2, 3 . (2) The nc indices denote noncommutative coordinates. θµν is the parameter of the deformation and it is an antisymmetric real matrix. We distinguish two types of noncommutativity; the first one is the space–space case, where the deformation is introduced between the spatial coordinates only, and the second is when the spatial coordinates commute with one another but not with time coordinate and it is noted space–time case. (For a review, one can see Ref. 26.) In the literature, there are a lot of studies on hydrogen atom in noncommutativity. For space–space noncommutativity, we cite Refs. 27–30. For the space–time case, one can see Refs. 31–33. We have found in Refs. 31 and 32, that the corrections induced by noncommutativity on the spectrum of the hydrogen atom are proportional to the lepton mass to the third power (the result is confirmed by Ref. 33), and this is exactly the shape of the corrections induced by the nuclear size as demonstrated in Refs. 34 and 35. We will apply our result to the muonic hydrogen, and we will incorporate therein the effects of the finite mass of the nucleus. We start by computing the corrections to the energies in both space–space and space–time cases of noncommutativity using perturbation methods in the Dirac theory of two particles systems. Then, we compare the difference between theoretical and experimental results obtained in µH experience. This allows us to obtain values of the noncommutative parameter that resolve the puzzle. Then we will discuss the possible effects of these corrections on the Lamb shift of muonic deuterium µD and on the difference between the radii of the proton and deuteron via the 2S–1S transition. 2. Coulomb Potential in Noncommutative Space Time We start by rewriting (2) for the two versions to consider of the noncommutativity: j 0 xst , xst = iθj0 , (3a) j k xss , xss = iθjk , (3b) 1350139-3

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We will evaluate the contribution coming from space–time noncommutativity to this shift. We use the relations from (15) to (20), to compute the noncommutative corrections to the transition in both hydrogen and deuterium: (st)

(st)

(38a)

(st)

(st)

(38b)

1S−2S ∆E2S (H) − ∆E1S (H) = hfH = 9.06690 × 1010 (θst eV2 ) eV , 1S−2S ∆E2S (D) − ∆E1S (D) = hfD = 9.07060 × 1010 (θst eV2 ) eV .

Inserting the value of the parameter θst found for µH (31a) in these expressions, we find the noncommutative correction to the hydrogen–deuterium isotope shift: (nc)

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1S−2S 1S−2S ∆fth (H − D) = fH − fD = 275.488 Hz .

(39)

This contribution does not fill the gap of 12 kHz between ∆fth and ∆fex , but it improves slightly the agreement between the two values. This confirms the fact that there are no doubts about the results of experiments on eH spectroscopy because they are so accurate and one has to look the side of muonic systems. 5. Conclusion In this paper, we studied the corrections induced by a noncommutative structure of space–time, in its two versions space–space and space–time, on the spectrum of hydrogen-like atoms. We have applied our study to the muonic hydrogen and this with the aim to solve the puzzle of proton radius, because we think that the experiments used to study this phenomenon are so developed that we cannot doubt of their results, and therefore one must look on the side of theory of atomic spectroscopy used to compute the radius. In this study, we considered the effects of the mass of the nucleus of the noncommutative corrections and thus we have improved previous works in this area. This allowed us to consider the difference caused by changing the nucleus (from proton to deuteron) in addition to that which occurs when changing the orbiting particle (from electron to muon). It should be noted that the effects of the nucleus shape on the energy levels of the atom are proportional to the third power of the mass of the orbiting particle; this is easily understood by the fact that the Bohr radius (a0 = ~2 /me2 ) is inversely proportional to the mass and thus the particle is that much nearer the nucleus, that its mass is greater; and this makes it more sensitive to these effects. We have demonstrated that the effects of noncommutativity are also proportional to the third power of the mass of the particle because it distorts the Coulomb potential and adds a term proportional to r−2 in space–time case and to r−3 in space–space case. It is for this reason also that the effects decreases with increasing values of quantum numbers as can be seen in the different relations of the spectrum corrections (because the term r−n with n > 1 is very steep for small values of r). This is why we use this theory to explain the puzzle because its effects are different depending on whether it is applied to muonic hydrogen or electronic hydrogen. The shifts in the spectrum are more noticeable in muon H than in ordinary ones, and 1350139-12

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Effects of Noncommutativity on Light Hydrogen-like Atoms and Proton Radius

this explains the fact that experiments on µH spectroscopy give results that are different from those obtained with eH. In the case of space–space noncommutativity, the parameter required to resolve the puzzle is θss ≈ (0.35 GeV)−2 . This value exceeds the limit obtained for this parameter from studies on eH Lamb shift θss ≤ (0.6 GeV)−2 .33 If we use this limit to compute the radius, we find 0.86409 fm which is outside the experimental limits for rµH .1,3 Another problem arises with in this case; it is the ± sign in the corrections (due to the presence of the azimuthal quantum number in their expressions). This sign means that the corrected value of the radius ranges from 0.84169 fm to 0.91192 fm in violation of µH experiences. In the case of the space–time noncommutativity, the value θst ≈ (14.3 GeV)−2 has been obtained and it is in agreement with the limit determined by Lamb shift spectroscopy in eH θst ≤ (6 GeV)−2 .33 It was also found in this case, that the corrections are the same for both levels 2P1/2 and 2P3/2 although we found that corrections remove the degeneracy of the Dirac energies with respect to the total angular momentum quantum number j = l +1/2 = (l +1)−1/2 (noncommutativity acts like the Lamb shift here). This is in agreement with the two relations (27a) and (27b), where the terms in r are equal for both transitions. This is not true for the space–space case because the corrections of the two levels differ from one another (29a) and (29b). We say that this is a consequence of the fact that the correction term to the Coulomb potential is proportional to r−2 in the space–time case, and so as we have previously mentioned in Ref. 31, we assimilate it to the field of a central dipole. In other words, the action of space–time noncommutativity is equivalent to consider the extended charged nature of the proton in the nucleus, which is the principal characteristic studied in µH experiments. When applying the result obtained from the study of µH in µD, we found a correction of 0.348 meV which is almost exactly equal to the difference between theory and experiment for this system. The very close values of the corrections in these two systems µH and µD are easily explained by the fact that the ratio between the reduced masses, of the two is 0.95 ≈ 1. (The disagreements between theory and experiment in both systems are almost equal.) Using the same result coming from µH for the eH–eD isotope shift, the correction found improves the agreement between theoretical and experimental results (which was already excellent). The same reasoning as above is used, and we say that the corrections in electronic systems either eH or eD are infinitely small compared to those of muonic systems; the ratio between the reduced masses of the two is ≈ 10−7 . Eyes are now turned to the results of experimental on µp scattering and µHe spectroscopy to see whether the phenomenon is spectroscopic or is it due to the nature of the particles. On the side of electronic systems, there is practically no doubt on their veracity; the radius of the proton was even calculated in a model independent way from ep scattering51,52 and the results confirm CODATA value. 1350139-13

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It was reported in our work,31 that the limit on the parameter θst ≈ (1 TeV)−2 , and so our result here is greater than the latter; however it was pointed out to us that such a limit (θst ≈ (1 TeV)−2 ) should not only be estimated according to experimental precision calculations but rather on the disagreement between experiment and theory, which requires a correction of the limit of Ref. 31 and this is what is done in this work for the muon hydrogen. We can also evoke that the proton raises other questions about its properties in addition to the one discussed in this paper and we can mention as an example the origin of its spin or what is called “spin crisis” in Ref. 53 or “proton spin puzzle” in Ref. 54.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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R. Pohl et al., Nature 466, 213 (2010). P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. 84, 1527 (2012). A. Antognini et al., Science 339, 417 (2013). J. D. Carroll, A. W. Thomas, J. Rafelski and G. A. Miller, Phys. Rev. A 84, 012506 (2011). P. Indelicato, Phys. Rev. A 87, 022501 (2013). R. Pohl, R. Gilman, G. A. Miller and K. Pachucki, Annu. Rev. Nucl. Part. Sci. 63, 175 (2013). A. Antognini et al., Ann. Phys. 331, 127 (2013). I. C. Clo¨et and G. A. Miller, Phys. Rev. C 83, 012201(R) (2011). J. P. Karr and L. Hillico, Phys. Rev. Lett. 109, 103401 (2012). V. Barger, C. W. Chiang, W. Y. Keung and D. Marfatia, Phys. Rev. Lett. 106, 153001 (2011). G. A. Miller, A. W. Thomas, J. D. Carroll and J. Rafelski, Phys. Rev. A 84, 020101(R) (2011). G. A. Miller, Phys. Lett. B 718, 1078 (2013). U. D. Jentschura, Ann. Phys. 326, 500 (2011). U. D. Jentschura, Ann. Phys. 326, 516 (2011). Gilman et al., Studying the proton “radius” puzzle with µρ elastic scattering, arXiv:1303.2160. A. Connes, Noncommutative Geometry (Academic Press, CA, 1994). N. Seiberg and E. Witten, J. High Energy Phys. 09, 032 (1999). J. Bellissard, H. Schulz-Baldes and A. Van Elst, J. Math. Phys. 35, 5373 (1994). J. Bellissard, Noncommutative geometry of aperiodic solids, in Proc. of the Summer School — Geometric and Topological Methods for Quantum Field Theory, eds. H. Ocampo, S. Paycha and A. Cardona (World Scientific, 2003), pp. 86–156. A. P. Polychronakos, Prog. Math. Phys. 53, 109 (2007), and references therein. A. Connes and M. Marcolli, From Physics to Number Theory via Noncommutative Geometry, Frontiers in Numbers Theory, Physics, and Geometry I, Part II (Springer, 2006), pp. 269–349. A. Sitarz, J. Phys. A: Math. Gen. 26, 5305 (1993). A. B´erard and H. Mohrbach, Phys. Lett. A 352, 190 (2006). D. B. Tweed, T. P. Halswanter, V. Happe and M. Fetter, Nature 399, 261 (1999). M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001). R. J. Szabo, Phys. Rep. 378, 207 (2003). 1350139-14

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27. M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001). 28. T. C. Adorno, M. C. Baldiotti, M. Chaichian, D. M. Gitman and A. Tureanu, Phys. Lett. B 682, 235 (2009). 29. W. O. Santos and A. M. C. Souza, Int. J. Theor. Phys. 51, 3882 (2011). 30. L. Khodja and S. Zaim, Int. J. Mod. Phys. A 27, 1250100 (2012). 31. M. Moumni, A. BenSlama and S. Zaim, Afr. Rev. Phys. 07, 83 (2012). 32. M. Moumni, A. BenSlama and S. Zaim, Relativistic spectrum of hydrogen atom in the space–time non-commutativity, in Proc. 8th Int. Conf. on Progress in Theoretical Physics, AIP Conf. Proc., Vol. 1444, eds. Mebarki et al. (2012), pp. 253–257. 33. A. Stern, Phys. Rev. D 78, 065006 (2008). 34. J. L. Friar, Ann. Phys. 122, 151 (1979). 35. M. I. Eides, H. Grotcha and V. A. Shelyuto, Phys. Rep. 342, 63 (2001). 36. S. Dulat and K. Li, Eur. Phys. J. C 54, 333 (2008). 37. S. Fabi, B. Harms and A. Stern, Phys. Rev. D 78, 065037 (2008). 38. M. Chaichian, A. Tureanu, M. R. Setare and G. Zet, J. High Energy Phys. 0804, 064 (2008). 39. S. K. Suslov and B. Trey, J. Math. Phys. 49, 012104 (2008). 40. J. Sapirstein, Quantum electrodynamics, in Atomic, Molecular, and Optical Physics Handbook, ed. G. W. F. Drake (AIP Press, New York, 1996), pp. 327–340. 41. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic Press, New York, 1957). 42. W. Greiner, Relativistic Quantum Mechanics Wave Equations, 3rd edn. (Springer, Berlin, 2000). 43. E. Borie, Ann. Phys. 327, 733 (2012). 44. M. Gorshteyn, Nuclear and hadronic contributions to Lamb shift in muonic deuterium, Talk given at Workshop to Explore Physics Opportunities with Intense, Polarized Electron Beams up to 300 MeV, MIT, Cambridge, Massachusetts, March 14–16, 2013. 45. K. Pachucki, Phys. Rev. Lett. 106, 193007 (2011). 46. A. A. Krutov and A. P. Martynenko, Phys. Rev. A 84, 052514 (2011). 47. R. Pohl, Muonic news, talk given at ECT Workshop on the “Proton Radius Puzzle”, Trento, Italy, Oct 29–Nov 2 2012. 48. C. G. Parthey et al., Phys. Rev. Lett. 107, 203001 (2011). 49. C. G. Parthey et al., Phys. Rev. Lett. 104, 233001 (2010). 50. U. D. Jentschura et al., Phys. Rev. A 83, 042505 (2011). 51. R. J. Hill and G. Paz, Phys. Rev. D 82, 113005 (2010). 52. I. Sick, Prog. Partic. Nucl. Phys. 67, 473 (2012). 53. A. W. Thomas, A. Casey and H. H. Matevosyan, Int. J. Mod. Phys. A 25, 4149 (2010). 54. C. A. Aidala, S. D. Bass, D. Hasch and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013).

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Modern Physics Letters A Vol. 28, No. 36 (2013) 1350166 (9 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217732313501666

ON THE PROBLEM OF VACUUM ENERGY IN FLRW UNIVERSES AND DARK ENERGY

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BADIS YDRI∗ and ADEL BOUCHAREB† Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria ∗[email protected] ∗[email protected] †[email protected]

Received 3 October 2013 Accepted 9 October 2013 Published 29 October 2013 We present a (hopefully) novel calculation of the vacuum energy in expanding FLRW spacetimes based on the renormalization of quantum field theory in nonzero backgrounds. We compute the renormalized effective action up to the two-point function and then apply the formalism to the cosmological backgrounds of interest. As an example we calculate for quasi de Sitter spacetimes the leading correction to the vacuum energy given by the tadpole diagram and show that it behaves as ∼ H02 ΛP where H0 is the Hubble constant and ΛP is the Planck constant. This is of the same order of magnitude as the observed dark energy density in the universe. Keywords: Cosmological constant; vacuum energy; FLRW metrics; QFT on curved backgrounds; renormalization; dark energy.

1. Introduction In recent years it has been established, to a very reasonable level of confidence, that the universe is spatially flat and is composed of 4% ordinary matter, 23% dark matter and 73% dark energy. This state of affair is summarized in the cosmological concordance ΛCDM model.1 The dominant component, dark energy, is believed to be the same thing as the cosmological constant introduced by Einstein in 1917 which in turn is believed to originate in the energy of the vacuum.2 We note that dark energy is characterized mainly by a negative pressure and no dependence on the 2 scale factor and √ its density behaves as 3∼ H0 ΛP where H0 is the Hubble parametera and ΛP = 1/ 8πG is the Planck mass. More precisely we have (with ΩΛ ≃ 0.73) ρΛ = 3ΩΛ H02 Λ2P ≃ 39ΩΛ (10−12 GeV)4 .

a The

(1.1)

vacuum energy density is a constant which can be expressed as something which is proportional to the square of the Hubble parameter at the current epoch (Hubble constant). We are not saying that the vacuum energy density falls with the square of the Hubble parameter. 1350166-1

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The reality of the energy of the vacuum is exhibited, as is well known, in a dramatic way in the Casimir force. See, for example, Ref. 4 for a systematic presentation of this subject. In this paper we will adopt the usual line of thought outlined in Refs. 2 and 3 and identify dark energy with vacuum energy. The calculation of vacuum energy in curved spacetimes such as FLRW universes and de Sitter spacetime requires the use of quantum field theory in the presence of a non zero gravitational background.5,6 de Sitter spacetime is of particular interest since we know that both the early universe as well as its future evolution is dominated by vacuum, i.e. FLRW universes may be understood as a perturbation V of de Sitter spacetime which vanishes in the early universe as well as in the limit a → ∞. The main difficulties in doing quantum field theory on curved background is the definition of the vacuum state and renormalization. In an expanding de Sitter spacetime and also in quasi de Sitter spacetimes, a natural choice of the vacuum is given by the so-called Euclidean or Bunch–Davies state.7–9 It can be shown10,11 that the vacuum energy in de Sitter spacetime with the Bunch–Davies state behaves in the right way as ∼ H02 Λ20 where H0 is the de Sitter Hubble parameter and Λ0 is a physical cutoff introduced for example along the lines of Ref. 12. As discussed above FLRW spacetimes may be treated as quasi de Sitter spacetimes. The usual in–out formalism may then be used to extend the calculation of vacuum energy to these spacetimes.11 A more systematic and more fundamental approach to the calculation of vacuum energy in FLRW spacetimes is based on the renormalization of quantum field theory in nonzero backgrounds. This is the approach we will discuss in this paper which is inspired by the recent original work on the Casimir force found in Refs. 13–15 in which the starting point is a renormalizable quantum field theory in a nonzero background. The main ingredients of this approach are as follows: (1) We reinterpret scalar field theory coupled to FLRW metric as a scalar field theory in a flat spacetime interacting with a particular time-dependent (effective graviton) background. (2) We regularize and then renormalize the resulting scalar field theory for arbitrary backgrounds in the usual way. Renormalization, if possible, removes all divergences from all proper vertices of the effective action. (3) We compute the vacuum energy for arbitrary backgrounds. (4) We substitute the particular background found in (1). (5) There is always the possibility that the vacuum energy still diverges for particular profiles of the background configuration. This is indeed the case for the Casimir force14 as well as for the FLRW spacetimes considered here. We thus regularize with a cutoff to obtain an estimate for the vacuum energy. Although we think that this approach is novel, potential and possible overlap with other approaches is certainly expected. A systematic investigation of this point is still underway. 1350166-2

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It is obvious that with these parameters we have Λ0 ≫ 1. H0 We can then approximate the above integral byb ! " # Z Λ 2 2 p Λ Λ0 dp p2 ln 2 − 4 sin pη ≃ ηa4 H02 Λ20 4 − ln 2 cos . µ µ H 0 0

(3.10)

(3.11)

The energy density then becomes

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" # Hφ 1 Λ2 Λ0 2 2 = (1 − 6ξ)H0 Λ0 ln 2 − 4 cos . v 16π 2 µ H0

(3.12)

The mass scale µ2 is also comoving and therefore the physical mass scale must be defined by µ2 = µ20 a. The energy density is then time-independent given by " # H02 Λ20 Λ20 Λ0 Hφ = (1 − 6ξ) ln 2 − 4 cos . (3.13) 2 v 16π µ0 H0 The mass scale µ20 may be taken to be of the order of particle physics mass scale, say µ0 ≃ 102 GeV .

(3.14)

By using the parameters (3.8), (3.9) and (3.14) we obtain a numerical estimation for the vacuum energy density given by Hφ = (1 − 6ξ)(0.08 × (10−12 GeV)4 )(71.45)(0.94) v = (1 − 6ξ)(5.37 × (10−12 GeV)4 ) .

(3.15)

This is of the same order of magnitude as the experimental value (1.1). Corrections due to deviation from a perfect de Sitter spacetime are of order V while corrections due to the contribution of the two-point function are of the order of (1 − 6ξ)2 . A quantitative discussion of these effects will be done elsewhere.11 4. Conclusion In this paper we have presented a new calculation of the vacuum energy in a certain class of FLRW spacetimes which can be viewed as perturbed de Sitter spacetime. This calculation is based on the renormalization of quantum field theory in nonzero (effective graviton) backgrounds in analogy with the recent treatment of the Casimir force found in Refs. 13 and 14. It is found that the vacuum energy still diverges, after renormalization of the quantum field theory proper vertices, for the timedependent cosmological backgrounds of interest. Indeed these backgrounds do not b We remark that although Λ /H ≫ 1 the cosine remains bounded and therefore the remaining 0 0 divergence, which is due to the special cosmological shapes, is really quadratic.

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vanish sufficiently fast for large momenta. By introducing an appropriate comoving cutoff,c an estimation of the vacuum energy is obtained which is compared favorably with the experimental value. Acknowledgment This research was supported by The National Agency for the Development of University Research (ANDRU) under PNR contract number U23/Av58 (8/u23/2723).

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Particle Data Group Collab. (J. Beringer et al.), Phys. Rev. D 86, 010001 (2012). S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). S. M. Carroll, Living Rev. Relativ. 4, 1 (2001). K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, 2001). N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge Univ. Press, 1982). S. A. Fulling, London Math. Soc. Student Texts 17, 1 (1989). T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360, 117 (1978). E. Mottola, Phys. Rev. D 31, 754 (1985). B. Allen, Phys. Rev. D 32, 3136 (1985). A. Prain, Vacuum energy in expanding spacetime and superoscillation induced resonance, Master thesis (2008). Work in progress. A. Kempf, Phys. Rev. D 63, 083514 (2001). R. L. Jaffe, Phys. Rev. D 72, 021301 (2005). N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, O. Schroeder and H. Weigel, Nucl. Phys. B 677, 379 (2004). K. A. Milton, Phys. Rev. D 68, 065020 (2003). J. Zinn-Justin, Int. Ser. Monogr. Phys. 113, 1 (2002). S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley, 2004).

cA

comoving cutoff breaks Lorentz invariance but we are only here trying to obtain a rough, albeit reasonable, estimation of the vacuum energy density. 1350166-9

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Bertotti-Robinson solutions of D ¼ 5 Einstein-Maxwell-Chern-Simons-Lambda theory Adel Bouchareb* Institute of Physics, Badji Mokhtar Annaba University, B.P. 11, 23000 Annaba, Algeria

Chiang-Mei Chen† Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli 320, Taiwan

Ge´rard Cle´ment‡ LAPTh, Universite´ de Savoie, CNRS, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France

Dmitri V. Gal’tsov§ Department of Theoretical Physics, Moscow State University, 119899 Moscow, Russia (Received 11 September 2013; published 28 October 2013) We present a series of new solutions in five-dimensional Einstein-Maxwell-Chern-Simons theory with an arbitrary Chern-Simons coupling and a cosmological constant . For general and we give various generalizations of the Bertotti-Robinson solutions supported by electric and magnetic fluxes, some of which presumably describe the near-horizon regions of black strings or black rings. Among them there is a solution which could apply to the horizon of a topological anti–de Sitter black ring in gauged minimal supergravity. Others are horizonless and geodesically complete. We also construct extremal asymptotically flat multistring solutions for ¼ 0 and arbitrary . DOI: 10.1103/PhysRevD.88.084048

PACS numbers: 04.20.Jb, 04.65.+e

S5 ¼

I. INTRODUCTION Five-dimensional supergravity is an interesting proving ground for string theory. Its Lagrangian, obtainable by toroidal dimensional reduction of eleven-dimensional supergravity, contains a Chern-Simons term for the Maxwell field inherited from reduction of the corresponding four-form term. Though it does not influence the Einstein equations, it modifies the Maxwell equations and consequently the gravitational field too. Surprisingly enough, its presence is crucial for the enhancement of hidden symmetries of five-dimensional Einstein-Maxwell (EM) theory in the case of field configurations possessing two commuting spacetime Killing vectors. For such configurations the theory reduces to a three-dimensional sigma model realizing a harmonic map from the spacetime manifold to the homogeneous space G=H ¼ G2ðþ2Þ = ððSLð2; RÞ SLð2; RÞÞ [1–4]. Owing to this hidden symmetry, a generating technique had been developed [1,2], which opened the way to derive new charged rotating black rings and general black strings [5–10]. In various physical contexts one is also interested in the more general Einstein-Maxwell theory containing a Chern-Simons term with an arbitrary coupling constant and a cosmological constant : *[email protected] † [email protected] ‡ [email protected] § [email protected]

1550-7998= 2013=88(8)=084048(12)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 1 1 d5 x jgð5Þ j Rð5Þ Fð5Þ Fð5Þ 2 16G5 4 (1.1) Fð5Þ Fð5Þ Að5Þ ; pﬃﬃﬃ 12 3

where Fð5Þ ¼ dAð5Þ , ; ; ¼ 1; ; 5, and is an antisymmetric symbol whose signs will be detailed later. For ¼ 1, ¼ 0 this is the action of minimal fivedimensional supergravity, for ¼ 1, < 0 the action of minimal gauged supergravity, while for ¼ 0 it is the EM action. Restricted to field configurations possessing two commuting Killing vectors, this theory reduces to a threedimensional gravitating sigma model coupled to a potential originating from the cosmological constant term. For 1 the target space of this sigma model is not a symmetric space (the isometry group is solvable), so there are no nontrivial hidden symmetries which could be used to generate exact solutions. Moreover, the potential term is not invariant under target space isometries apart from some trivial ones. Not surprisingly, no exact charged rotating black hole solutions are known in the pure EinsteinMaxwell ( ¼ 0) theory even for ¼ 0, though their existence was demonstrated perturbatively [11–13] and numerically [14]. A similar situation holds for charged black rings: static solutions in EM theory have been found by Ida and Uchida [15], and generalized to EM-dilaton theory in [16,17]. But stationary charged black ring solutions of pure EM theory are not known in closed form, though their existence again was confirmed both

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perturbatively (for small charges [18]) and numerically [19]. For 1, 0 exact charged black hole/ring solutions are not known either, while perturbative [20] and numerical solutions have been constructed [21] and shown to have unusual properties for > 1 such as rotation in the sense opposite to the angular momentum and a negative horizon mass with positive asymptotic mass. It is therefore of interest to explore other tools to generate exact solutions of the general Einstein-MaxwellChern-Simons-Lambda (EMCS) action (1.1) which could shed some light on the nature of black objects in this theory. A particular motivation for this lies in the still unsolved question about the existence of asymptotically anti–de Sitter (AdS) and de Sitter (dS) black rings in presence of the cosmological constant. The issue of charged five-dimensional black objects with the cosmological constant was discussed previously in a number of papers. Supersymmetric AdS5 black holes were obtained by several authors [22]. General nonextremal rotating black holes in minimal five-dimensional gauged supergravity were constructed by Chong, Cvetic˘, Lu¨, and Pope [23]. Charged squashed black holes in EM theory ( ¼ 0) with the cosmological constant were constructed numerically in [24]. Black strings with the cosmological constant were studied in [25]. The issue of black rings turns out to be more subtle. General considerations do not prevent their existence both for positive and negative cosmological constants: the additional centripetal/centrifugal force acting on the S1 can be balanced by tuning the angular momentum along the S1 . And indeed, Chu and Dai [26] have found analytically asymptotically dS black rings within the N ¼ 4 de Sitter supergravity (see also [27]). As in the asymptotically flat (AF) case the asymptotically de Sitter black holes/rings may have horizon topologies S3 (or a quotient), S1 S2 and T 3 . For a negative cosmological constant other topologies can be anticipated, namely, S1 H 2 , where H 2 stands for a negative curvature hyperbolic two-surface. In the meantime, no analytical solutions are known for black rings with a negative cosmological constant. Approximate ‘‘thin’’ black rings were obtained by Caldarelli, Emparan, and Rodriguez [28] using the ‘‘blackfolds’’ approach applicable in arbitrary dimensions [29]. These approximate solutions exist for both signs of the cosmological constant and smoothly go into a straight black string in the limit of an infinite S1 radius. Moreover, the possibility of black Saturns with nonflat asymptotics was also indicated. Supersymmetric black rings with AdS5 asymptotics and compact horizons were investigated in [30] for a negative cosmological constant. Lacking exact globally defined solutions, it is tempting to explore local solutions in the vicinity of the event horizons of the presumed black objects. This is a particularly fruitful approach in the extremal case. Usually the near-horizon limits are themselves exact solutions of the same theory, as in the case of the near-horizon limit of

extremal four-dimensional EM black holes, which is the Bertotti-Robinson (BR) solution with geometry AdS2 S2 supported by a monopole electric or magnetic field. Typically, the near-horizon solutions possess a larger isometry group than the full black hole solutions, therefore they can be obtained by different solution-generating techniques. All known exactly five-dimensional supergravity solutions exhibit enhancement of isometries in the nearhorizon region to SOð2; 1Þ Uð1Þ2 , with Uð1Þ factors standing for rotational symmetries [31] (for a review see [32]). This is valid for stationary and asymptotically AdS solutions, and also persists in the presence of scalar fields with a potential [33] and with an account for highercurvature corrections. In D dimensions the enhanced symmetry includes the Uð1ÞD3 rotational symmetry. Note that in this and more general theories certain properties of extremal black holes including topology and thermodynamics can be extracted from the near-horizon solutions using Sen’s entropy function approach [34]. Keeping in mind the importance of the BR solution in the four-dimensional EM theory, we explore here similar exact solutions within the five-dimensional EMCS theory with arbitrary and . Presumably these could be near-horizon limits of black holes/rings with various topologies, including the above-mentioned case of the hyperbolic topology S1 H 2 . As was observed several decades ago, the n-dimensional Einstein-Maxwell theories can be compactified to (n 2)-dimensional spacetime, the two extra space dimensions being curved into a two-sphere through the action of a monopole magnetic field living on that two-sphere [35]. This mechanism was later generalized to the Freund-Rubin compactification of (d s) or s dimensions by an s-form field [36]. The consistency of general sphere compactifications of supergravity actions was extensively discussed in the past (see, e.g., [37]) showing that a consistent sphere truncation of gravity coupled to a form field and (possibly) to a dilaton is possible only in a limited number of cases. In a nonsupersymmetric setting, solutions of the D-dimensional Einstein-Maxwell theory with the cosmological constant that are the topological product of two manifolds of a constant curvature were studied in [38]. Our approach consists in compactifying the theory (1.1) on a constant curvature two-space 2 of a positive, zero, or negative curvature. We consider only compactification through a direct product ansatz, as in the Freund-Rubin case, thereby avoiding the consistency problems associated with non-Abelian Kaluza-Klein gauge fields. This compactification, carried out in Sec. II, results in the threedimensional EMCS gravity, with an additional constraint on the scalar curvature. In Sec. III we present a number of new nontrivial solutions of the five-dimensional EMCS theory of the BR-type which are obtained by uplifting Banados-Teitelboim-Zanelli (BTZ), self-dual, and Go¨del solutions of the constrained three-dimensional theory.

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These solutions are applicable, in particular, to gauged and ungauged D ¼ 5 supergravities, but they are also valid in the EMCS theory with more general values of parameters. In Sec. IV we perform an alternative toroidal reduction of the five-dimensional EMCS to three dimensions, deriving a gravity coupled sigma model with a potential. Some of the solutions listed in Sec. III correspond to null geodesics of the target space. In the case of a vanishing cosmological constant, the reduced three-space is flat, enabling the generalization of these solutions to new classes of nonasymptotically flat or asymptotically flat multicenter solutions of the five-dimensional theory with arbitrary . The technical proof that the solutions of Sec. III are the only solutions with constant curvature two-space sections is outlined in the Appendix. II. REDUCTION ON CONSTANT CURVATURE TWO-SPACES The main idea underlying the present paper is that the five-dimensional theory (1.1) may be reduced, by monopole compactification on constant curvature twosurfaces 2 , to three-dimensional EMCS: 1 Z 3 qﬃﬃﬃﬃﬃﬃ 1 d x jgj R F F 2 Sð3Þ ¼ 2 4 F A ; (2.1) 4 with an additional constraint, which admits several classes of nontrivial exact stationary solutions, leading to nonasymptotically flat stationary solutions of the original five-dimensional theory with the structure M3 2 . The five-dimensional Maxwell-Chern-Simons and Einstein equations following from the action (1.1) are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ @ jgð5Þ jFð5Þ ¼ pﬃﬃﬃ Fð5Þ Fð5Þ ; (2.2) 4 3 1 1 1 2 R ð5Þ 2 Rð5Þ ¼ 2 Fð5Þ Fð5Þ 8 Fð5Þ : (2.3) In Eq. (2.2) we need to fix a sign convention for the fivedimensional antisymmetric symbol. Throughout this paper we will assume that 12345 ¼ þ1, with the spacetime coordinates numbered according to their order of appearance in the relevant five-dimensional metric. Let us assume for the five-dimensional metric and the vector potential the direct product ansatze¨ ds2ð5Þ ¼ g ðx Þdx dx þ a2 dk ; Að5Þ ¼ A ðx Þdx þ efk d’;

(2.4)

where , , ¼ 1, 2, 3, and the two-metrics for k ¼ 1, 0 are

d1 ¼ d2 þ sin 2 d’2 ;

d0 ¼ d2 þ 2 d’2 ;

d1 ¼ d2 þ sinh 2 d’2 ; 1 f0 ¼ 2 ; 2

f1 ¼ cos ;

(2.5)

f1 ¼ cosh ; (2.6)

with ’ 2 ½0; 2 and 2 ½0; for k ¼ 1 and 2 ½0; 1 for k ¼ 0, 1. Here the moduli e and a2 are taken to be constant and real (though for generality we do not assume outright a2 to be positive). The corresponding five-dimensional geometric quantities are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ Rð5Þ ¼ R’ð5Þ’ ¼ ka2 ; (2.7) jgð5Þ j ¼ jgjja2 [email protected] fk ; and the Maxwell tensor decomposes as e ’ Fð5Þ ¼ 4 ; Fð5Þ’ ¼ [email protected] fk ; a @ fk

F ¼ dA: (2.8)

Inserting the ansatze¨ (2.4) in the field equations [39], we find that the equations (2.2) for ¼ 4, 5 are trivially satisfied, while for ¼ they reduce to qﬃﬃﬃﬃﬃﬃ e @ jgjF ¼ pﬃﬃﬃ 2 F : (2.9) 3ja j The Einstein equations (2.3) lead to the system 1 1 1 R R ¼ F F F2 2 2 8 4ka2 e2 þ

; 4a4 4ðe2 3ka2 Þ þ 8: F2 ¼ a4

(2.10)

It is straightforward to check that the reduced equations derive from the action (2.1) with ¼ 2G5 =ja2 j and the following identification of parameters: pﬃﬃﬃ ¼ þ ðe2 4ka2 Þ=4a4 ; ¼ g=ja2 j; ðg ¼ 2e= 3Þ; (2.11) with the additional constraint on the three-dimensional scalar curvature R ¼ ðe2 6ka2 Þ=2a4 þ 4:

(2.12)

Inverting the above relations for k 0 and 0 leads to k2 ; þ ð Þ2 pﬃﬃﬃ 3 ; e¼ 2j32 =16 þ ð Þ2 j

a2 ¼

32 =16

(2.13)

and the constraint R ¼ þ 3 32 =162 ¼ þ 3 e2 =4a4 :

(2.14)

The equations (2.13) break down for k ¼ 0, in which case the parameters , are related by

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the fact that the form of the solution (4.18) breaks down for ¼ 0 and ¼ 1=4], the AF solution (4.18) is an extreme black string for the down sign and ¼ ðn þ 2Þ=4, where n is an integer (or all real for c ¼ 0). For the upper sign, geodesics with v 0 are reflected by an infinite potential barrier, while spacelike geodesics with v ¼ 0 are either reflected or attain r ¼ 0 only asymptotically, so that the spacetime is geodesically complete. For ¼ 1=4 and the lower sign the solution (4.17) is replaced by 3c2 1 2 1 dsð5Þ ¼ dudv þ M þ ln du2 þ 2 dx~ 2 ; 4 pﬃﬃﬃ Að5Þ ¼ 3½c1=2 du A3 ; (4.21) and for ¼ 0 (Einstein-Maxwell theory) it is replaced by ds2ð5Þ ¼ 1 dudv þ ðM 3c2 ln Þdu2 þ 2 dx~ 2 ; pﬃﬃﬃ Að5Þ ¼ 3½c ln du A3 :

(4.22)

In the case of the special vacuum solution (3.8), r is one of the Cartesian coordinates of the reduced three-space and is a harmonic function on that space, so that the generalization to a multicenter solution is the vacuum solution [56,60] ds2ð5Þ ¼ 2 dudv du2 þ dx~ 2

(4.23)

(with u ¼ t, v ¼ z). C. Go¨del solutions The first obvious candidate multicenter solution in the Go¨del class is the ¼ 1=2, 2 ¼ 1 solution (3.17) with k ¼ þ1, g2 ¼ a2 . Actually, it turns out that, after a trivial coordinate transformation z / u, t / v, this is just an instance of the self-dual solution (4.17) for ¼ 1=2 and the lower sign, with M / . The Einstein-Maxwell solution (3.16) ( ¼ 0) with m2 ¼ 0, ¼ 0 leads to the multicenter solution ds2ð5Þ ¼ 2 dt2 þ dz2 þ 2 dx~ 2 ; pﬃﬃﬃ Að5Þ ¼ 2½1 dt þ A3 ;

2 r2 dt2 þ R2 ðdz N z dtÞ2 ðr þ aÞ2 R2 ðr þ aÞ2 2 þ dr þ ðr þ aÞ2 d22 ; r2 pﬃﬃﬃ pﬃﬃﬃ 2 3 ðdt $dzÞ cos d’ ; ¼a rþa 2

ds2ð5Þ ¼

Að5Þ

(4.26)

with R2 ¼ 2 ð1 $Þ2 Nz ¼

2a$ð1 $Þ a2 $2 þ ; rþa ðr þ aÞ2

r½ð1 $Þr þ a ; ðr þ aÞ2 R2

is a dyonic extremepblack string. ﬃﬃﬃ Finally, for > 3=2, ds2ð5Þ ¼ ð1 dt þ dzÞ2 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃ pﬃﬃﬃ 1 3 3 1 2 A ¼ Að5Þ ¼ 2 dt ; 3 2 82 2

(4.27)

~ is harmonic on the Minkowskian reduced where ðxÞ metric dx~2 ¼ ij dxi dxj . The signature of the metric (4.27) is ( þ þ ). A Minkowskian multicenter solution may be obtained from the ¼ 0 Go¨del solution (3.22) with k ¼ 1, g 2 2 ¼ a2 . By transforming the H 2 coordinates ð; ’Þ to coordinates ðr; zÞ such that d1 ¼

dr2 þ r2 dz2 ; r2

f1 ¼ r;

(4.28)

we obtain the multicenter solution with Newman-UntiTamburino parameters (NUTs): ds2ð5Þ ¼ ðdt 1 A3 Þ2 þ 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2ð1 þ 2 Þ 3 1 Að5Þ ¼ dz: A3 2 2

(4.29)

(4.24) V. SUMMARY

which is the trivial five-dimensional embedding of a dyonic Majumdar-Papapetrou solution. 2 ¨ del solution (3.15) for From p the ﬃﬃﬃ generic m ¼ 0 Go 0 < < 3=2 with k ¼ þ1, g2 2 ¼ a2 , we derive the multicenter solution ds2ð5Þ ¼ ð1 dt þ dzÞ2 þ 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃ pﬃﬃﬃ 3 1 3 1 A3 2 ¼ 2 ; Að5Þ ¼ 2 dt 2 8 2

solution, generalized by the local coordinate transformation t ! t $z (with $ a second parameter)

(4.25)

which may be viewed as a deformation of (4.24). For 2 > 1 ( < 1=2), the one-center asymptotically flat

In this paper we have investigated solutions of the general five-dimensional EMCS theory—containing as particular cases minimal ungauged and gauged supergravities—known exact solutions of three-dimensional Einstein-Maxwell gravity with a Chern-Simons term. Our main tool was dimensional reduction, assuming the five-dimensional spacetime to be the direct product of a constant curvature surface 2 of positive, zero, or negative curvature (k ¼ 1, 0 or 1) and a three-dimensional spacetime. The reduced theory is then the three-dimensional EMCS3 with a constraint on the scalar curvature, which can be satisfied by three classes of solutions found earlier,

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namely, the BTZ, self-dual, and Go¨del classes. Promoting these to five dimensions, we have constructed new EMCS5 solutions of the generalized Bertotti-Robinson– type which could be near-horizon limits of black string and black ring solutions. The three-parameter BTZ class of solutions is geodesically complete and exists for an arbitrary (irrelevant) Chern-Simons coupling . Generically these solutions are nonvacuum, being supported by magnetic fluxes along the constant curvature two-surfaces. Their isometry groups are SOð2; 2Þ SOð3Þ for k ¼ þ1, SOð2; 2Þ GLð2; RÞ for k ¼ 0, and SOð2; 2Þ SOð2; 1Þ for k ¼ 1. In the minimal supergravity case ¼ 0, ¼ 1 only spherical sections are possible, and the corresponding solutions are near-horizon limits of black strings. For a negative cosmological constant this class of solutions exists in all the three versions k ¼ 1, 0, or 1, with horizon topologies S1 S2 , S1 R2 , and S1 H 2 respectively. The latter two could be near-horizon limits of topological asymptotically AdS black rings, though the existence of the global solutions remains to be checked. There is a special case with zero magnetic flux, when we get a two-parameter family of vacuum solutions which can be seen as nonasymptotically flat topological black rings with the horizon S1 H 2 . These are (presumably new) locally AdS3 H 2 solutions of vacuum five-dimensional Einstein equations. For a positive cosmological constant, the solutions exists in the k ¼ 1 version only. Their interpretation is similar, but the relevant asymptotics is de Sitter. The second class of EMCS5 solutions, depending also on three parameters, generalize the extremal solutions of the BTZ class, which they asymptote. They are supported by a magnetic flux together with an independent dyonic electromagnetic field. The solutions of the third threeparameter class are generated from three-dimensional Go p ﬃﬃﬃ¨ del black holes. They pﬃﬃﬃ exist with k ¼ þ1 forpﬃﬃﬃ < 3=2, k ¼ 0 for ¼ 3=2, and k ¼ 1 for > 3=2. The isometry groups are SOð2; 1Þ SOð2Þ SOð3Þ, SOð2; 1Þ SOð2Þ GLð2; RÞ, and SOð2; 1Þ SOð2Þ SOð2; 1Þ, respectively. Subclasses include solutions analogue to previously found rotating BR solutions in dilaton-axion gravity, horizonless and geodesically complete solutions with spacetime topology H2 2 R, and NUTty solutions with spacetime topology S2 2 R. In the case of minimal D ¼ 5 supergravity, the NUTty S2 H 2 R solution (3.24) has been shown [53] to be a near-extreme, near-bolt limit of an asymptotically flat solitonic string solution of EMCS5. We have then shown that some of the BR solutions thus constructed can be promoted to NAF or AF multicenter solutions. For this purpose we have performed an alternative toroidal compactification of EMCS5 to three dimensions, leading to a sigma-model representation with a potential. For ¼ 0 this potential vanishes, in which case null geodesics of the target space give rise

to exact solutions of the five-dimensional equations with a flat reduced three-space. The corresponding BR solutions may be promoted to multicenter solutions by redefinition of the associated harmonic functions. We have thus identified three families of multistring solutions of EMCS5, the two self-dualpfamilies (4.17), ﬃﬃﬃ ¨ﬃﬃﬃ del family (4.25) (for < 3=2) or (4.29) and the p Go (for > 3=2). An unexpected by-product of our analysis is the construction of new closed-form asymptotically flat solutions (4.18) generated by a dyonic electromagnetic field along with a magnetic flux. For the lower sign these are regular extreme black strings for a discrete set of values of the Chern-Simons coupling constant (including the minimal supergravity case ¼ 1), while for the upper sign they are geodesically complete. ACKNOWLEDGMENTS Three of the authors are grateful to the LAPTh Annecy for hospitality in July (C.-M. C.) and December (A. B. and D. G.) 2012 while the paper was in progress. The work of D. G. was supported in part by the RFBR Grant No. 11-0201371-a. The work of C.-M. C. was supported by the National Science Council of the R.O.C. under Grant No. NSC 102-2112-M-008 -015 -MY3, and in part by the National Center of Theoretical Sciences (NCTS). The work of A. B. was supported by the Agence The´matique de Recherche en Sciences et Technologie (ATRST) under Contract No. U23/Av58 (PNR 8/u23/2723) and in part by the CNEPRU. APPENDIX: CONSTANT SCALAR CURVATURE SOLUTIONS TO THREE-DIMENSIONAL EMCS THEORY The ansatz [42] ds2 ¼ ab ðÞdxa dxb þ 2 ðÞR2 ðÞd2 ; A ¼ c a ðÞdxa ;

(A1)

where is the 2 2 matrix ¼

TþX

Y

Y

TX

! ;

(A2)

and R2 X2 is the Minkowski pseudonorm of the ‘‘vector’’ XðÞ ¼ ðT; X; YÞ, X2 ¼ ij X i X j ¼ T 2 þ X 2 þ Y 2 ;

(A3)

reduces the equations of three-dimensional EMCS theory to the Maxwell-Chern-Simons, Einstein and Hamiltonian constraint equations

084048-10

S0 ¼

2 X ^ S; R2

(A4)

BERTOTTI-ROBINSON SOLUTIONS OF D ¼ 5 . . .

X00 ¼

PHYSICAL REVIEW D 88, 084048 (2013)

2 2 ðS XÞX S ; R2 R2

X02 þ 2X X00 þ 4 2 ¼ 0;

(A5)

(A6)

where S is the null vector 1 S ¼ ð c 20 þ c 21 ; c 20 c 21 ; 2 c 0 c 1 Þ; 4

ðS2 ¼ 0Þ;

(A7)

_ ¼ @[email protected], and the wedge product is defined by ðX ^ YÞi ¼ ij jkl Xk Y l (with 012 ¼ þ1). To the above equations we must add the constraint (2.14) 1 R 2 X02 ðX2 Þ00 ¼ þ 3 32 =162 : (A8) 2 Combining equations (A6) and (A8), we obtain the relations ðR2 Þ00 ¼ 2b;

X X00 ¼ c;

X02 ¼ b c;

(A9)

c ¼ ð 3Þ= with b ¼ ð þ Þ=2 þ 3=162 , 2 3=162 . From (A5) it follows that c S X ¼ R2 : 2

(A10)

Noting that ðS XÞ0 ¼ S X0 from (A4), we derive from (A10) ðS XÞ00 ¼ S X00 þ S0 X0 ¼ c2

2 L S; R2

(A11)

where we have defined L ¼ X ^ X0 :

(A12)

Comparing (A11), (A10), and (A9), we obtain LS¼

cðc bÞ 2 R: 2

(A13)

Next, noting that L0 S ¼ 0 from (A4), we obtain 2 ðL SÞ ¼ 2 ðX ^ X0 Þ ðX ^ SÞ R 2 ¼ 2 ½ðS XÞðX X0 Þ R2 ðS X0 Þ R c ¼ ðR2 Þ0 : 2

Consider now the second possibility, b c ¼ 1. Squaring Eq. (A5), we find that X002 ¼ 0; (A16) 02 while the last equation (A9), which now reads X ¼ 1, implies that X0 X00 ¼ 0. The fact that the null vector X00 is orthogonal to X 0 implies that the wedge product X0 ^ X00 is collinear with X00 : (A17) X0 ^ X00 ¼ X00 : This relation, together with Eq. (A5), may be used to transform Eq. (A4) to S0 ¼ X ^ X00 ¼ X ^ ðX0 ^ X00 Þ 1 ¼ ðX X0 ÞX00 ðX X00 ÞX0 ¼ ðR2 Þ0 X00 cX0 : 2 (A18) Taking the scalar product of (A18) with the vector L and using (A5) and (A13), we obtain c L S0 ¼ ðR2 Þ0 ; (A19) 2 which upon comparison with (A14) shows that we should take the upper sign in the preceding equations. Finally, Eq. (A18) with the upper sign may be compared with the direct derivative of Eq. (A5), 1 1 S0 ¼ ðR2 ÞX000 ðR2 Þ0 X00 þ cX0 ; 2 2 leading to the conclusion that

(A20)

X000 ¼ 0: This equation is integrated by

(A21)

ð 2 ¼ 0; ^ ¼ Þ; (A22) X ¼ 2 þ þ ; the vector relations between the constant vectors , , following from (A16) and (A17). We recognize in (A22) the quadratic ansatz which was used in [42] to derive the Go¨del solutions to three-dimensional EMCS. For the last possibility, ðR2 Þ0 ¼ 0, we can adapt the above argument to show that X0 ^ X00 ¼ qX00

0

Equation (A18) is now replaced by c S0 ¼ X0 : q (A14)

Comparing with (A13), we derive the constraint cðc b þ 1ÞðR2 Þ0 ¼ 0:

ðq2 ¼ b cÞ:

(A15)

The first possibility c ¼ 0 means that S X ¼ 0, which is equivalent to the self-duality condition [46] F F ¼ 0, which leads either to the self-dual solutions (3.9) for < 0 and (3.11) for ¼ 0, or to the vacuum solutions of the BTZ class.

(A23)

(A24)

Taking the scalar product with X0 and using (A11) and (A13), we obtain S0 X0 ¼ cq ¼ cq2 :

(A25)

The first solution, c ¼ 0, leads to a subcase of the self-dual class. The second solution, q ¼ 0, means that X00 is collinear with X0 so that X X00 ¼ c ¼ 0, leading again to a subcase of the self-dual class. And the last solution, q ¼ 1, means b c ¼ 1, corresponding to a subclass of the Go¨del class.

084048-11

BOUCHAREB et al.

PHYSICAL REVIEW D 88, 084048 (2013)

[1] A. Bouchareb, C. M. Chen, G. Cle´ment, D. V. Gal’tsov, N. G. Scherbluk and T. Wolf, Phys. Rev. D 76, 104032 (2007); 78, 029901(E) (2008). [2] G. Cle´ment, J. Math. Phys. (N.Y.) 49, 042503 (2008); 49, 079901(E) (2008). [3] M. Berkooz and B. Pioline, J. High Energy Phys. 05 (2008) 045. [4] D. V. Gal’tsov and N. G. Scherbluk, Phys. Rev. D 78, 064033 (2008); 79, 064020 (2009). [5] D. V. Gal’tsov and N. G. Scherbluk, Phys. Rev. D 81, 044028 (2010). [6] S. Tomizawa, Y. Yasuii, and Y. Morisawa, Classical Quantum Gravity 26, 145006 (2009). [7] G. Compe`re, S. de Buyl, E. Jamsin, and A. Virmani, Classical Quantum Gravity 26, 125016 (2009). [8] P. Figueras, E. Jamsin, J. V. Rocha, and A. Virmani, Classical Quantum Gravity 27, 135011 (2010). [9] S. S. Kim, J. L. Ho¨rnlund, J. Palmkvist, and A. Virmani, J. High Energy Phys. 08 (2010) 072. [10] G. Compe`re, S. de Buyl, S. Stotyn, and A. Virmani, J. High Energy Phys. 11 (2010) 133. [11] A. N. Aliev and V. P. Frolov, Phys. Rev. D 69, 084022 (2004); A. N. Aliev, Phys. Rev. D 74, 024011 (2006). [12] F. Navarro-Lerida, Gen. Relativ. Gravit. 42, 2891 (2010). [13] M. Allahverdizadeh, J. Kunz, and F. Navarro-Lerida, Phys. Rev. D 82, 024030 (2010). [14] J. Kunz, F. Navarro-Lerida, and A. K. Petersen, Phys. Lett. B 614, 104 (2005). [15] D. Ida and Y. Uchida, Phys. Rev. D 68, 104014 (2003). [16] S. S. Yazadjiev, arXiv:hep-th/0507097. [17] H. K. Kunduri and J. Lucietti, Phys. Lett. B 609, 143 (2005). [18] M. Ortaggio and V. Pravda, J. High Energy Phys. 12 (2006) 054. [19] B. Kleihaus, J. Kunz, and K. Schnulle, Phys. Lett. B 699, 192 (2011). [20] A. N. Aliev and D. K. Ciftci, Phys. Rev. D 79, 044004 (2009). [21] J. Kunz and F. Navarro-Lerida, Mod. Phys. Lett. A 21, 2621 (2006). [22] D. Klemm and W. A. Sabra, Phys. Lett. B 503, 147 (2001); J. P. Gauntlett and J. B. Gutowski, Phys. Rev. D 68, 105009 (2003); 70, 089901(E) (2004); J. B. Gutowski and H. S. Reall, J. High Energy Phys. 04 (2004) 048. [23] Z. W. Chong, M. Cvetic˘, H. Lu¨, and C. N. Pope, Phys. Lett. B 644, 192 (2007). [24] K. Murata, T. Nishioka, and N. Tanahashi, Prog. Theor. Phys. 121, 941 (2009); Y. Brihaye, J. Kunz, and E. Radu, J. High Energy Phys. 08 (2009) 025; Y. Brihaye, E. Radu, and C. Stelea, Phys. Lett. B 709, 293 (2012). [25] R. B. Mann, E. Radu, and C. Stelea, J. High Energy Phys. 09 (2006) 073; Y. Brihaye, E. Radu, and C. Stelea, Classical Quantum Gravity 24, 4839 (2007); A. Bernamonti, M. M. Caldarelli, D. Klemm, R. Olea, C. Sieg, and E. Zorzan, J. High Energy Phys. 01 (2008) 061. [26] C.-S. Chu and S.-H. Dai, Phys. Rev. D 75, 064016 (2007). [27] J. Gutowski and W. A. Sabra, J. High Energy Phys. 05 (2011) 020. [28] M. M. Caldarelli, R. Emparan, and M. J. Rodriguez, J. High Energy Phys. 11 (2008) 011. [29] R. Emparan, T. Harmark, V. Niarchos, and N. A. Obers, Phys. Rev. Lett. 102, 191301 (2009); J. High Energy Phys. 03 (2010) 063.

[30] H. K. Kunduri, J. Lucietti, and H. S. Reall, J. High Energy Phys. 02 (2007) 026. [31] H. K. Kunduri and J. Lucietti, Classical Quantum Gravity 26, 245010 (2009); Commun. Math. Phys.303, 31 (2011); H. K. Kunduri, Classical Quantum Gravity 28, 114010 (2011); H. K. Kunduri and J. Lucietti, J. High Energy Phys. 07 (2011) 107. [32] H. K. Kunduri and J. Lucietti, Living Rev. Relativity 16, 8 (2013). [33] H. K. Kunduri, J. Lucietti, and H. S. Reall, Classical Quantum Gravity 24, 4169 (2007). [34] A. Sen, J. High Energy Phys. 03 (2006) 008. [35] Z. Horvath, L. Palla, E. Cremmer, and J. Scherk, Nucl. Phys. B127, 57 (1977). [36] P. G. O. Freund and M. A. Rubin, Phys. Lett. B 97, 233 (1980). [37] M. Cvetic˘, H. Lu¨, and C. N. Pope, Nucl. Phys. B597, 172 (2001). [38] V. Cardoso, O. J. C. Dias, and J. P. S. Lemos, Phys. Rev. D 70, 024002 (2004). [39] Dimensional reduction in the action (1.1) does not produce a correct three-dimensional action. [40] G. Cle´ment, Geometry of Constrained Dynamical Systems, edited by J. M. Charap (Cambridge University Press, Cambridge, 1995), pp. 17–22. [41] M. Ban˜ados, G. Barnich, G. Compe`re, and A. Gomberoff, Phys. Rev. D 73, 044006 (2006). [42] K. Ait Moussa, G. Cle´ment, H. Guennoune, and C. Leygnac, Phys. Rev. D 78, 064065 (2008). [43] J. Grover, J. B. Gutowski, G. Papadopoulos, and W. A. Sabra, arXiv:1303.0853. [44] G. Cle´ment, Int. J. Theor. Phys. 24, 267 (1985). [45] G. Cle´ment and I. Zouzou, Phys. Rev. D 50, 7271 (1994). [46] G. Cle´ment, Phys. Lett. B 367, 70 (1996). [47] G. Cle´ment, Classical Quantum Gravity 10, L49 (1993). [48] K. Ait Moussa, G. Cle´ment, and C. Leygnac, Classical Quantum Gravity 20, L277 (2003). [49] A. Bouchareb and G. Cle´ment, Classical Quantum Gravity 24, 5581 (2007). [50] D. Anninos, W. Li, M. Padi, W. Song, and A. Strominger, J. High Energy Phys. 03 (2009) 130. [51] These solutions were obtained in [42] under the assumption > 0, i.e., on account of the second equation in Eq. (2.13) e > 0. [52] G. Cle´ment and D. Gal’tsov, Phys. Rev. D 63, 124011 (2001); Nucl. Phys. B619, 741 (2001). [53] A. Bouchareb, C. M. Chen, G. Cle´ment, and D. V. Gal’tsov (to be published). [54] The magnetic potential of [1,2] is noted here to avoid confusion with the Chern-Simons coupling constant. [55] G. Neugebauer and D. Kramer, Ann. Phys. (Leipzig) 24, 2 (1969). [56] G. Cle´ment, Gen. Relativ. Gravit. 18, 861 (1986); Phys. Lett. A 118, 11 (1986). [57] G. Cle´ment and D. Gal’tsov, Phys. Rev. D 54, 6136 (1996). [58] We have changed a sign in Að5Þ because our coordinate transformation implies uv ¼ tz . [59] K. Matsuno, H. Ishihara, M. Kimura, and T. Tatsuoka, Phys. Rev. D 86, 104054 (2012). [60] G. W. Gibbons, Nucl. Phys. B207, 337 (1982).

084048-12

arXiv:1401.1529v1 [hep-th] 7 Jan 2014

New Algorithm and Phase Diagram of Noncommutative Φ4 on the Fuzzy Sphere Badis Ydri∗ Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria. January 9, 2014

Abstract We propose a new algorithm for simulating noncommutative phi-four theory on the fuzzy sphere based on i) coupling the scalar field to a U (1) gauge field in such a way that in the commutative limit N −→ ∞ the two modes decouple and we are left with pure scalar phi-four on the sphere and ii) diagonalizing the scalar field by means of a U (N ) unitary matrix and then integrating out the unitary group from the partition function. The number of degrees of freedom in the scalar sector reduces therefore from N 2 to the N eigenvalues of the scalar field whereas the dynamics of the U (1) gauge field is given by D = 3 Yang-Mills matrix model with a Myers term. As an application the phase diagram, including the triple point, of noncommutative phi-four theory on the fuzzy sphere is reconstructed with small values of N up to N = 10 and large numbers of statistics.

Contents 1 Introduction

2

2 Model and Algorithm 2.1 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Algorithm and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6

3 The 3.1 3.2 3.3 3.4 ∗

Phase Diagram The Ising Phase Transition . . . . . . . . . . . The Uniform-to-Non-Uniform Phase Transition The Matrix Phase Transition . . . . . . . . . . Triple Point and Phase Diagram . . . . . . . .

Email:[email protected], [email protected]

1

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8 8 9 13 15

4 The Self-Dual Noncommutative Φ4 on the Fuzzy Sphere 4.1 Self-Dual Noncommutative Φ4 . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Sphere as a Regulator . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17

5 Conclusion and Outlook

19

1

Introduction

The goal of this article is to reconstruct by means of a (hopefully) novel and efficient Monte Carlo method the phase diagram of noncommutative phi-four on the fuzzy sphere which was originally done in [1]. The basic theory is given by the following two-parameter matrix model S0 = TrH − a[La , Φ]2 + bΦ2 + cΦ4 .

(1)

In this equation La are the SU (2) generators in the irreducible representation with spin s = (N − 1)/2, TrH 1 = N , b is the mass parameter and c is the coupling constant. The parameter a can always be chosen to be equal to 1. There are three known phases in this model. The usual Ising transition between disorder and uniform order, a matrix transition between disorder and a non-uniform ordered phase and a (very hard to observe) transition between uniform order and non-uniform order. The three phases meet at a triple point [1,2]. The non-uniform phase in which rotational invariance is spontaneously broken is simply absent in the commutative theory. The non-uniform phase is the analogue of the stripe phase observed on the Moyal-Weyl spaces [14] whereas the disorder-to-non-uniformorder transition is the generalization of the one-cut-to-two-cut transition observed in the Hermitian quartic matrix model [15, 16] to the fuzzy sphere. This is a highly non-trivial problem which is due mainly to the more complicated phase structure of matrix scalar phi-four which involves transitions between vacuum states with very low probability distributions and as a consequence they are extremely difficult to sample correctly with the Metropolis algorithm. In particular the uniform-to-uniform transition is virtually unobservable in ordinary Metropolis due to the absence of tunneling between the identity matrix corresponding to the uniform phase and the other idempotent matrices corresponding to the non-uniform phase. This means simply that the Metropolis updating procedure does not sample correctly and equally, i.e. according to the Boltzmann weight the entire phase space which includes an infinite number of vacuum states. This was circumvented in [1, 2] by a complicated variant of the Metropolis algorithm in which detailed balance is broken. This problem was also studied in [3, 4, 7, 8]. The analytic derivation of the phase diagram of noncommutative phi-four on the fuzzy sphere was attempted in [17–19]. The related problem of Monte Carlo simulation of noncommutative phi-four on the fuzzy torus and the fuzzy disc was considered in [5, 6] and [9] respectively. The main strategy employed in this article towards a better resolution of this problem is to reduce the model down to its eigenvalues without actually altering it. This is achieved by

2

1) coupling the scalar field to a U (1) gauge field in such a way that in the commutative limit N −→ ∞ the two modes decouple completely and thus we return to an ordinary phi-four theory and 2) diagonalizing the scalar field by means of a U (N ) gauge transformation, viz Φ = U ΛU + and then integrating out the unitary matrices U and U + from the path integral. In this algorithm we thus trade off the Monte Carlo simulation of the unitary matrix U and U + in the original model (1) with the Monte Carlo simulation of a U (1) gauge field on the fuzzy sphere which we know is very efficient using ordinary Metropolis [10]. The primary interest of this article is therefore Monte Carlo simulation of a noncommutative phi-four theory coupled to a U (1) gauge field on the fuzzy sphere using the Metropolis algorithm with exact detailed balance. The scalar field transforms in the adjoint representation of the U (1) gauge group and as a consequence the scalar and gauge degrees of freedom decouple in the commutative limit N −→ ∞. In other words this theory becomes an ordinary phi-four theory in the commutative limit. In this theory the usual scalar kinetic action ∼ −T r[La , Φ]2 is replaced with ∼ −T r[Xa , Φ]2 where Xa is itself obtained by Monte Carlo simulation of an appropriate gauge action which will be centered around ∼ La in the so-called fuzzy sphere phase1 . The pure gauge action is given by D = 3 Yang-Mills action with a Chern-Simons (Myers) term. For b = c = 0 the full action is in fact D = 4 Yang-Mills action with a Chern-Simons (Myers) term. This article is organized as follows. In section 2 we present the detail of the U (1) gauge covariant noncommutative phi-four theory on the fuzzy sphere and also explain the Metropolis algorithm employed in our Monte Carlo simulations. In section 3 we report our first numerical results on the phase diagram of noncommutative phi-four on the fuzzy sphere using our new algorithm. We give independent measurements of the three transition lines discussed above and then derive our estimation of the triple point. These results are obtained with small values of N up to N = 10 and large numbers of statistics. In section 4 we give a construction of a one-parameter family of noncommutative phi-four models on the fuzzy sphere which define a regularization of duality covariant noncommutative phi-four on the Moyal-Weyl plane. We conclude in section 5 with a brief summary and outlook.

2 2.1

Model and Algorithm The Action

Instead of the basic model (1), which is the primary interest in this article, we consider a four matrix model given by the action

1

S = Sg + Sm .

(2)

2iα 1 ǫabc Xa Xb Xc + N T r M T r(Xa2 )2 + βXa2 . Sg = N T r − [Xa , Xb ]2 + 4 3

(3)

The behavior in the matrix phase is very different and is not treated in here.

3

Equivalently K 4πR2

=

− −

N N 1 + Ω2 X X 2(i − 1)(j − 1) ˜ ˜ )φij φji (i + j − 1 − 2πθ N −1 i=1 j=1 N N r i−2 1 − Ω2 X X j−2 ˜ ˜ (i − 1)(j − 1)(1 − )(1 − )φij φj−1i−1 2πθ N −1 N −1 i=1 j=1 N N r j−1 ˜ ˜ i−1 1 − Ω2 X X )(1 − )φij φj+1i+1 . (56) ij(1 − 2πθ N −1 N −1 i=1 j=1

We can now include a mass term and a phi-four coupling in a trivial way. The full action on the fuzzy sphere will read 1ˆ 2 µ2 ˆ2 λ ˆ4 4Ω2 2 2 2 2 2 ˆ SΩ = 4πR TrH − φ La + Ω L3 − 2 (Xa − X3 ) φ + φ + φ . (57) 2 θ 2 4! √ ˜ 2π and also introduce the parameters We scale the field as φˆ = φ/ r = µ2 R 2 , u =

λR2 . 4!π

(58)

The full action can then be rewritten as

SΩ = TrH

2 2 2 2 2 2 4 ˜ ˜ ˜ ˜ ˜ − [La , φ] − Ω [L3 , φ] + Ω {La , φ} + r φ + uφ .

(59)

This is a one-parameter family of phi-four models on the fuzzy sphere which generalizes (1). Coupling to a U (1) gauge field is straightforward, i.e. we make the replacement p La −→ N/2Xa . The analogue of (4) is obviously given by SΩ = −

5

2 2 N ˜ 2 − N Ω T r[X3 , φ] ˜ 2 + N Ω T r{Xa , φ} ˜ 2 + T rV (φ). ˜ T r[Xa , φ] 2 2 2

(60)

Conclusion and Outlook

In this article we have proposed a new algorithm for the Monte Carlo simulation of noncommutative phi-four on the fuzzy sphere and also reported our first numerical results on the corresponding phase diagram obtained with small values of N up to N = 10 and large numbers of statistics. Basically the new algorithm employs gauge invariance in order to reduce the scalar sector to the core eigenvalues problem. The phase diagram is complex consisting of three transition lines (the Ising or uniform-to-disorder, the matrix or nonuniform-to-disorder and the uniform-to-non-uniform transition lines) which intersect at a triple point. The measurement of the uniform-to-non-uniform transition line using our algorithm remains very demanding but tractable. The measurements included in this article are largely consistent with those reported originally in [1]. The first immediate extension of this work is to optimize the algorithm further and push the calculation of the phase diagram to higher values of N with reasonably large numbers

19

of statistics especially in the case of the uniform-to-non-uniform transition line. A major improvement of the algorithm may be achievable by replacing the Metropolis updating procedure for the scalar eigenvalues problem by the Hybrid Monte Carlo algorithm whereas we may keep using the very efficient Metropolis for the gauge sector. Another immediate line of investigation is the calculation of the phase diagram of the self-dual noncommutative phi-four on the fuzzy sphere constructed in the last section. The main question here is what happens to the Ising transition line as Ω goes from Ω = 0 to Ω = 1 and as a consequence what is the fate of the triple point.

Acknowledgments: This research was supported by ”The National Agency for the Development of University Research (ANDRU), MESRS, Algeria” under PNR contract number U23/Av58 (8/u23/2723). The Monte Carlo simulations reported in this article were largely performed on the machines of the School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland.

References [1] F. Garcia Flores, X. Martin and D. O’Connor, “Simulation of a scalar field on a fuzzy sphere,” Int. J. Mod. Phys. A 24, 3917 (2009) [arXiv:0903.1986 [hep-lat]]. See also [2] [2] F. Garcia Flores, D. O’Connor and X. Martin, “Simulating the scalar field on the fuzzy sphere,” PoS LAT 2005, 262 (2006) [hep-lat/0601012]. [3] X. Martin, “A matrix phase for the phi**4 scalar field on the fuzzy sphere,” JHEP 0404, 077 (2004) [hep-th/0402230]. [4] M. Panero, “Numerical simulations of a non-commutative theory: The Scalar model on the fuzzy sphere,” JHEP 0705, 082 (2007) [hep-th/0608202]. [5] J. Ambjorn and S. Catterall, “Stripes from (noncommutative) stars,” Phys. Lett. B 549, 253 (2002) [hep-lat/0209106]. [6] W. Bietenholz, F. Hofheinz and J. Nishimura, “Phase diagram and dispersion relation of the noncommutative lambda phi**4 model in d = 3,” JHEP 0406, 042 (2004) [hep-th/0404020]. [7] J. Medina, W. Bietenholz and D. O’Connor, “Probing the fuzzy sphere regularisation in simulations of the 3d lambda phi**4 model,” JHEP 0804, 041 (2008) [arXiv:0712.3366 [hep-th]]. [8] C. R. Das, S. Digal and T. R. Govindarajan, “Finite temperature phase transition of a single scalar field on a fuzzy sphere,” Mod. Phys. Lett. A 23, 1781 (2008) [arXiv:0706.0695 [hep-th]]. [9] F. Lizzi and B. Spisso, “Noncommutative Field Theory: Numerical Analysis with the Fuzzy Disc,” Int. J. Mod. Phys. A 27, 1250137 (2012) [arXiv:1207.4998 [hep-th]]. [10] B. Ydri, “Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II,” Int. J. Mod. Phys. A 27, 1250088 (2012) [arXiv:1206.6375 [hep-th]].

20

[11] M.P. Vachovski, “Numerical studies of the critical behaviour of non-commutative field theories,” Ph.D thesis, private communication by Denjoe O’Connor. [12] H. Grosse and R. Wulkenhaar, “Renormalization of phi**4 theory on noncommutative R**4 in the matrix base,” Commun. Math. Phys. 256, 305 (2005) [hepth/0401128],“Renormalization of phi**4 theory on noncommutative R**2 in the matrix base,” JHEP 0312, 019 (2003) [hep-th/0307017],“Power counting theorem for non-local matrix models and renormalization,” Commun. Math. Phys. 254, 91 (2005) [hep-th/0305066]. [13] E. Langmann and R. J. Szabo, “Duality in scalar field theory on noncommutative phase spaces,” Phys. Lett. B 533, 168 (2002) [hep-th/0202039]. [14] S. S. Gubser and S. L. Sondhi, “Phase structure of noncommutative scalar field theories,” Nucl. Phys. B 605, 395 (2001) [hep-th/0006119]. [15] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978). [16] Y. Shimamune, “On The Phase Structure Of Large N Matrix Models And Gauge Models,” Phys. Lett. B 108, 407 (1982). [17] D. O’Connor and C. Saemann, “Fuzzy Scalar Field Theory as a Multitrace Matrix Model,” JHEP 0708, 066 (2007) [arXiv:0706.2493 [hep-th]]. [18] C. Saemann, “The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP n ,” SIGMA 6, 050 (2010) [arXiv:1003.4683 [hep-th]]. [19] A. P. Polychronakos, “Effective action and phase transitions of scalar field on the fuzzy sphere,” Phys. Rev. D 88, 065010 (2013) [arXiv:1306.6645 [hep-th]].

21

• Information financière

Université Badji Mokhtar - Annaba

Intitulé du projet PNR : Théories

des Champs Non-Commutatives à partir des Modèles des Matrices et

Physique Emergente Chef du projet PNR : Badis YDRI Laboratoire de rattachement du PNR : Laboratoire

Directeur du Laboratoire : MERADJI

de Physique et Rayonnements

(LPR)

HOCINE Unité En ; DA

Ventilation du Budget relatif à un Projet PNR Arrêté interministériel du 01 Mars 2012 fixant la nomenclature des dépenses consacrées à la recherche scientifique Et au développement Chapitres 01 1

Intitulés

3 4 5

Remboursement de frais Frais de mission et de déplacement en Algérie, à l'étranger. Rencontres Scientifiques : Frais d'organisation, d’hébergement, de restauration et de transport. Honoraires des enquêteurs. Honoraires des guides. Honoraires des experts et consultants. Frais d’études, de travaux et de prestations réalisés pour le compte de l’entité. SOUS TOTAL

02

Matériels et mobiliers

1

3 4

Matériels et instruments scientifiques et audiovisuels. Renouvellement du matériel informatique, achat accessoires, logiciels et consommables informatiques. Mobilier de laboratoire. Entretiens et réparations.

03

Fournitures

1 2 3

Produits chimiques. Produits consommables. Composants électroniques, mécaniques et audio-visuels.

4

Papeterie et fournitures de bureau.

5 6

Périodiques. Documentation et ouvrages de recherche. Fournitures des besoins de laboratoires (animaux, plantes etc.)

2

6

2

SOUS TOTAL

SOUS TOTAL 04

Charges Annexes

1

Impression et édition.

2

Affranchissements postaux.

répartition des crédits /

/ /

462 150 ,00 100 795 ,50

562945,50

179885,00

705868,30

885753,3

3 4 5

Communications téléphoniques, Fax, télex, télégramme, Internet. Autre frais (impôts et taxes, droits de douane, frais financiers, assurances, frais de stockage, et autre). Banque de données (acquisitions et abonnements). SOUS TOTAL TOTAL GENERAL

1448698,80

PRESENTATION DU PROJET SITUATION ACTUELLE DU PROJET: Intitulé du PNR

Code du Projet (Réservé à l’administration) D7

Sciences Fondamentales X

Nouveau projet : Projet reformule:

(Joindre une copie de la notification de l’avis de reformulation)

1.1. Domiciliation du projet Laboratoire de Physique des Rayonnements, Département de Physique, Faculté des Sciences, Université Badji Mokhtar, Annaba

1.2. Identification du projet 1.2.1‐ Nature de la recherche

Fondamentale

X

Appliquée

Titre du projet : Acronyme du projet : Intitulé du thème : Intitulé de l’axe : Intitulé du domaine :

Mots‐clés (12 max)

Durée estimée du projet

X Développement Formation Théories des Champs Non‐Commutatives à Partir des Modèles des Matrices et Physique Emergente NCQFT Théorie quantique des champs Particules élémentaires, champs et cosmologie Physique Géométrie Non‐Commutative, Théories des Champs Non‐Commutatives, Espaces Non‐Commutatifs, Sphère ‘Fuzzy’, Modèles de Yang‐Mills, Super symétrie, Modèles des Matrices, Géométrie et Gravitation Emergente, Méthodes de Groupes de Renormalisation, Méthodes Monte‐Carlo, Méthodes Cohomologiques, Transitions des Phases exotiques. 24 mois

1.2.2 Résumé du projet (250 mots)

Les quatre forces fondamentales de l’univers sont l’électromagnétisme, l’interaction forte, l’interaction faible et la force gravitationnelle. Les forces non‐gravitationnelles sont décrites par des théories quantiques des champs de Yang‐Mills sur un espace‐temps plat à quatre dimensions et elles sont unifiées dans le cadre du modèle standard de la physique des particules. La gravitation ne peut être unifiée avec les autres forces que dans le contexte de la théorie des cordes qui n'est pas une théorie des champs. Cependant à basses énergies la théorie des cordes dans un champ magnétique devient une théorie des champs sur un espace‐temps non‐ commutatif, i.e. un espace‐temps dont les coordonnées ne sont plus des nombres mais des opérateurs. Les espaces‐temps non‐commutatifs émergent également à des échelles de longueurs très petites comparables à l’échelle de Planck de la combinaison des principes de la relativité générale qui décrit la gravitation classique et la mécanique quantique. Ainsi le lien entre la non‐commutativité et la gravitation quantique semble être une hypothèse très raisonnable. La non‐commutativité est aussi la seule extension qui préserve la super symétrie qui est une symétrie centrale de la théorie des cordes.

1

La théorie des cordes ne peut pas être étudiée par simulations numériques. La difficulté majeure vient du fait qu’on ne peut pas maintenir la super symétrie dans un espace‐temps discret. Le seul modèle qui fait exception est le modèle des matrices de Yang‐Mills proposé par Ishibashi, Kawai, Kitazawa et Tsuchiya (IKKT) qui est supposé donner une régularisation non‐perturbative de la théorie des cordes de type IIB. Les modèles des matrices IKKT admettent des espaces non‐commutatifs comme solutions et donc ils peuvent aussi être utilisés pour régulariser non‐perturbativement les théories des champs de jauge non‐commutatives. Dans ce projet nous allons étudier les phénomènes qui émergent de façon dynamique dans les modèles des matrices IKKT tels que la géométrie émergente, la brisure spontanée de la supersymétrie et la gravitation émergente. Nous allons également utiliser le formalisme des matrices pour étudier les aspects quantiques des théories des champs scalaires non‐ commutatives.

1.3. Problématique du projet Sommaire (250 mots)

We propose to investigate two nonperturbative phenomena typical of noncommutative field theories which are known to lead to the perturbative instability known as the UV‐IR mixing which is absent in conventional field theory. The first phenomenon concerns the emergence of spacetime geometries in matrix models which describe perturbative noncommutative gauge theories on fuzzy backgrounds. These transitions from geometrical backgrounds to matrix phases make the description of noncommutative gauge theories in terms of fields only valid below a critical value of the coupling constant. We propose here to investigate this effect in four dimensional mass deformed supersymmetric Yang‐Mills matrix models and study the impact of supersymmetry on the geometry in transition and vice versa. This will also shed crucial light on the nonperturbative physics of the emergent noncommutative gauge theories and may open the door for Monte Carlo treatment of supersymmetry. The second phenomenon concerns the emergence of a nonuniform ordered phase in noncommutative scalar phi four field theory and the spontaneous symmetry breaking of translational/rotational invariance. This phenomena originates in the underlying matrix degrees of freedom of the noncommutative theory. We expect that in addition to the Wilson‐Fisher fixed point at zero noncommutativity there exists a novel fixed point at infinite noncommutativity corresponding to the quartic hermitian matrix model. We propose here to consider Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries and compute their nonperturbative phase structure. A renormalization group analysis which will determine the structure of the fixed points is also planned. A related question is to establish the renormalizability or lack of it of the phi four models on fuzzy spherical geometries.

1.4. Objectifs du projet Lister les objectifs scientifiques, techniques, technologiques, socio‐économiques et/ou socioculturels. (250 mots)

The scientific goals of this project can be summarized as follows: •

Gauge Fields, Supersymmetry and Emergent Geometry

1. The calculation of the phase structure of four dimensional mass deformed supersymmetric Yang‐Mills matrix models by means of the Monte Carlo method. 2. A nonperturbative analytical analysis of the phase transition associated with the emergent geometry by means of the deformation approach of cohomological models.

2

In the process we may clarify the impact of supersymmetry on the geometry in transition and vice versa. In particular to establish whether or not supersymmetry is spontaneously broken across the transition point is of paramount importance. This also may help us explain some of the nonperturbative physics of noncommutative gauge theory and allow us understand more fully the matrix phase. •

Noncommutative Scalar Fields

1. The calculation of the phase structure of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries by means of the Monte Carlo method.

2.

To establish renormalizability or lack of it of scalar phi four theories on fuzzy spherical geometries by means of the exact Polchinski renormalization group equation formulated in the matrix basis. The ultimate objective here is to determine the structure of the fixed points of noncommutative scalar phi four models. This is essentially an analytical question and as such the renormalization group equation in the matrix basis is the central tool.

1.5. Description du projet 1.5‐1‐ Etat des connaissances sur le sujet (500 mots)

Noncommutative field theories emerge in low energy dynamics of string theories. Seiberg and Witten showed that the dynamics of open strings moving in a flat space in the presence of a Neveu‐Schwarz B‐field and with Dp‐branes is equivalent to leading order in the string tension to gauge theory on Moyal‐Weyl space. In the case of open strings moving in a curved space with a three‐sphere metric the resulting effective gauge theory lives on a noncommutative fuzzy two‐sphere (Alekseev, Recknagel and Schomerus). Quantization of noncommutative field theories is plagued with stringy effects such as the notorious UV‐IR mixing (Minwalla, Van Raamsdonk and Seiberg) which translates into an instability of the geometry of spacetime itself (Bietenholz, Nishimura, Susaki and Volkholz). Noncommutative field theory can be expressed either in terms of star products or in terms of operator algebras. In the case of spherical geometries the operator algebras are isomorphic to finite dimensional matrix algebras. The IKKT Yang‐Mills matrix model in ten dimensions, i.e. the IIB matrix model is postulated to give a constructive definition of type IIB superstring theory (Ishibashi, Kawai, Kitazawa and Tsuchiya). The IKKT model exists also in four and six dimensions. These models are obtained from the reduction of supersymmetric Yang‐ Mills actions to zero dimensions. The most important mass deformation is provided by the Myers effect (Myers). Mass deformed IKKT Yang‐Mills matrix models in various dimensions admit the fuzzy sphere (Hoppe,Madore) as a vacuum solution. Noncommutative gauge theories can be realized by expanding the IKKT models around vacuum solutions. The connection between the UV‐IR mixing and the instability of the geometry of spacetime was also observed in three dimensional IKKT matrix models by Delgadillo Blando, O’Connor and Ydri. Thus connections between noncommutative gauge theories and matrix models run deep. It seems to indicate that matrices are the fundamental objects and that gauge fields, noncommutativity and geometry are derived concepts. The central hypothesis here will be that geometry, noncommutative gauge theory and supersymmetry should be nonperturbatively regularized with finite dimensional matrix models. For scalar field theory on Moyal‐Weyl spaces the UV‐IR mixing was shown to destroy the perturbative renormalizability of the theory (Chepelev and Roiban). The physics at very large distances is altered by the

3

noncommutativity which is supposed to be relevant only at very short distances. Grosse and Wulkenhaar found a modification of the free propagator which makes the theory renormalizable. It consist in adding a harmonic oscillator potential to the kinetic term. This is the only known renormalizable noncommutative scalar field theory. It is important to note the crucial role played by the matrix formulation of noncommutative Moyal‐Weyl spaces in their proof of renormalizability. The phase diagram of scalar phi four field theory consists of three phases instead of the usual two phases found in commutative scalar field theory. It was determined using the Monte Carlo method on the noncommutative torus by Ambjorn, Catterall from one hand and Bietenholz,Hofheinz,Nishimura from the other hand and on the fuzzy sphere by Garcia Flores, O'Connor and Martin. They observed a disordered phase, a uniform ordered phase and a novel nonuniform ordered phase which meet at a triple point. In the nonuniform ordered phase we have spontaneous breakdown of translational/rotational invariance. The most important analytical calculation of the phase structure of scalar phi four theory was done by Gubser and Sondhi. 1.5‐2‐ Méthodologie détaillée (300 mots)

The first goal is to investigate the exotic phase transitions from geometrical phases to matrix phases. The geometrical phases are generically given in terms of noncommutative gauge theories on fuzzy backgrounds. We propose to investigate this effect in four dimensional mass deformed supersymmetric IKKT Yang‐Mills matrix models and study the impact of supersymmetry on the geometry in transition and vice versa. The main tool of investigation is the Monte Carlo method. A nonperturbative analytical derivation of the critical properties based on matrix models techniques is unknown. We plan to apply the powerful deformation approach of cohomological models of Moore, Nekrasov and Shatashvili which will reduce the problem to a random matrix theory with one single hermitian matrix. This will shed crucial light on the nonperturbative physics of noncommutative gauge theories near the critical point. It is also an outstanding problem to determine precisely the structure of the matrix phase. It seems that this phase is universally dominated by commuting matrices with uniform eigenvalues distribution regardless of all other detail. It is not clear what happens to supersymmetry as we go from the matrix phase to the geometrical phase and vice versa. It may be spontaneously broken in one of the phases in analogy with the spontaneously broken rotational invariance in the nonuniform ordered phase in noncommutative scalar phi four theories. This whole program may thus open the door for a nonperturbative Monte Carlo treatment of supersymmetry. The second goal is to investigate the phase structure of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries and to establish renormalizability or lack of it of scalar phi four theories on fuzzy spherical geometries. The Grosse‐Wulkenhaar models are more general than conventional noncommutative scalar models as they involve an extra parameter in the phase diagram. The Monte Carlo method will be used to probe the phase diagram of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries. In order to study renormalizability of the phi four models on fuzzy spherical geometries we will use the Polchinski renormalization group equation in the matrix basis. A renormalization group analysis a la Wilson applied in the matrix basis may allow us to determine the structure of the fixed points. The Polchinski equation can also be used to compute the relevant, irrelevant and marginal interactions and as a consequence to determine the structure of the fixed points. 1.5‐3‐ Principales références bibliographiques

1. N.Seiberg and E.Witten,``String theory and noncommutative geometry,'' JHEP 9909, 032 (1999). 2. A.Y.Alekseev, A.Recknagel and V.Schomerus,``Non‐commutative world‐volume geometries: Branes on SU(2) and fuzzy spheres,JHEP 9909, 023 (1999). 3. J.Hoppe, ``Quantum theory of a massless relativistic surface and a two‐dimensional bound state problem,'' Ph.D thesis,MIT,1982. 4. J.Madore, ``the fuzzy sphere,''Class.Quant. Grav.9, 69 (1992). 5. R.C.Myers, ``Dielectric‐branes,''JHEP 9912, 022 (1999). 6. N.Ishibashi, H.Kawai, Y.Kitazawa and A.Tsuchiya, ``A large‐N reduced model as superstring,''Nucl. Phys. B 498, 467 (1997).

4

7. H.Grosse and R.Wulkenhaar, ``Power‐counting theorem for non‐local matrix models and renormalisation,''Commun. Math. Phys.254, 91 (2005); ``Renormalisation of phi four theory on noncommutative R four in the matrix base,'' Commun. Math. Phys. 256, 305 (2005). 8. F.Garcia Flores, D.O'Connor and X.Martin, ``Simulating the scalar field on the fuzzy sphere,''PoS LAT2005, 262 (2006). 9. J.Ambjorn and S.Catterall, ``Stripes from (noncommutative) stars,''Phys. Lett. B 549, 253 (2002). 10. W.Bietenholz, F.Hofheinz and J.Nishimura,```Phase diagram and dispersion relation of the non‐ commutative lambda phi four model in d = 3,'' JHEP 0406, 042 (2004). 11. S.S.Gubser and S.L.Sondhi, ``Phase structure of non‐commutative scalar field theories,'' Nucl. Phys. B 605, 395 (2001). 12. W.Bietenholz, J.Nishimura, Y.Susaki and J.Volkholz, ``A non‐perturbative study of 4d U(1) non‐ commutative gauge theory: The fate of one‐loop instability,'' JHEP 0610, 042 (2006). 13. S.Minwalla, M.Van Raamsdonk and N.Seiberg, ``Noncommutative perturbative dynamics,''JHEP 0002, 020 (2000). 14. I.Chepelev and R.Roiban, ``Convergence theorem for non‐commutative Feynman graphs and renormalization,''JHEP 0103, 001 (2001). 15. G.W.Moore, N.Nekrasov and S.Shatashvili, ``D‐particle bound states and generalized instantons,'' Commun. Math. Phys. 209, 77 (2000). 16. R.Delgadillo‐Blando, D.O'Connor and B.Ydri, ``Geometry in transition: A model of emergent geometry,'' Phys. Rev. Lett. 100, 201601 (2008).

1.6. Impacts attendus Impacts directs et indirects (Scientifiques, socio‐économiques, socioculturels) The expected scientific impacts may include the following: 1. The conjecture that large N IKKT Yang‐Mills matrix models are of central importance to fundamental physics can be verified explicitly in several interrelated physical contexts. The relevance of these large N IKKT Yang‐Mills matrix models to fundamental physics can be summarized as follows: •

They may help us understand nonperturbative physics of noncommutative gauge theories by means of cohomological and random matrix models.

•

They may provide a nonperturbative definition of supersymmetry. Indeed supersymmetry in this language may prove to be tractable in Monte Carlo simulation.

•

They provide concrete models for emergent geometry and possibly emergent gravity. Indeed geometry in transition seems to be possible in all these matrix models. These matrix models seem also very suited to test the gauge/gravity duality.

2. A more coherent understanding of the quantum dynamics of noncommutative scalar field theories by means of matrix models techniques and renormalization group and Monte Carlo methods. The expected educative impacts include the following: 3. Supervision of two doctoral students.

1.7. Planning des taches / année Taches

semestre 1

1)Monte Carlo Study of IKKT Yang‐Mills Matrix Models in Dimension Four.

5

semestre 2

semestre 3

semestre 4

2)Cohomological Study of IKKT Yang‐Mills Matrix Models in Dimension Four. 3) Phase Structure of Grosse‐Wulkenhaar Scalar Phi Four Models by Means of Monte Carlo. 4) Renormalization of Grosse Wulkenhaar Scalar Phi Four Models on Fuzzy Spheres in Dimensions 2 and 4 by Means of the Polchinski Renormalization Group Equation.

6

1.8. Identification du porteur (chef) de projet Nom & Prénom Grade Spécialité Statut

YDRI Badis Maitre de Conférences B Physique Théorique Enseignant chercheur(1)

x Chercheur permanent(2)

Email

[email protected]

Adresse professionnelle

Département de Physique Faculté des Sciences Université Badji Mokhtar B.P.12,23000 Annaba

Contacts

Tel : 038 87 53 99

Fax : 038 87 53 99

Associé(3)

Autre(4)

GSM :

Diplômes Obtenus (Graduation, Post‐Graduation)

Année

1 (Bacc.)

Bac. Mathématiques

1989

Lycée Mubarek El‐Mili

2 (L,M,Ing)

D.E.S Physique Théorique

1993

Université de Constantine

2001 2010

Université de Syracuse, New‐York, USA Université Badji Mokhtar Annaba

3 (doct.)

Ph.D. Physique Théorique Habilitation Universitaire

Etablissement

Participation à des programmes de recherche (nationaux, Internationaux, multisectoriels) Intitulé du Programme Origine des rayons cosmiques de très haute énergie (Code:D01120090017)

2010

Ministère (CNEPRU)

de

l'Enseignement

Supérieure

Lister vos trois derniers travaux les plus importants (recherche/recherche développement) 1

R.Delgadillo‐Blando, D.O’Connor, B.Ydri, Geometry in Transition: A Model of Emergent Geometry, Phys.Rev.Lett.100:201601, 2008.

2

D.Dou, B.Ydri, Topology Change from Quantum Instability of Gauge Theory on Fuzzy CP Two, Nucl.Phys.B771:167‐189, 2007.

3

B.Ydri, Quantum Equivalence of NC and YM Gauge Theories in 2D and Matrix Theory, Phys.Rev.D75:105008, 2007.

Visa du Chef d’établissement De rattachement :

Date : Signature :

7

2. Identification du partenaire socio‐économique du projet Nom & Prénom Grade Spécialité Statut

Enseignant chercheur(1)

x

Chercheur permanent(2)

Associé(3)

Autre (4)

Email Adresse professionnelle Contacts Tel : Fax : GSM : Diplômes Obtenus (Graduation, Post‐ Année Etablissement Graduation) 1(Lic,M,Ing) 2(Doct.) Participation à des programmes de recherche (nationaux, Internat., Sectoriels) Intitulé du Programme Année Organisme

A) Lister vos deux derniers travaux d’intérêt socio‐économiques 1 2 B) Autres Projets dans lesquels le partenaire du projet est impliqué Type de Durée du Ministère Projet(*) Intitulé Année de démarrage projet concerné A B C D

(1) Concerne les chercheurs universitaires (université, centre de recherche, école, institut). (2) Concerne les chercheurs permanents (centre, unité, institut de recherche) (3) Concerne les chercheurs associés (établissement de rattachement où le chef du projet exerce les fonctions de chercheur associé). (4)Préciser la fonction des personnels administratifs (cadre supérieur, fonctionnaire supérieur, etc. (*) Cocher la case correspondante : A : Projet par voie d’avis d’appel à proposition de projets (PNR.). B : Projet de recherche universitaire relevant de la CNEPRU. C : Projet de recherche sectorielle relevant des centres et unités de recherche sous tutelle du MESRS et hors MESRS. D : Projet de coopération.

Visa du Chef d’établissement de rattachement :

Date : Signature :

8

3. Chercheurs impliqués dans le projet (une fiche par chercheur) Nom & Prénom Grade

BOUCHAREB Adel Maitre Assistant A

Spécialité

Physique Théorique Enseignant chercheur(1) X Chercheur permanent(2)

Statut Email

[email protected]

Adresse professionnelle

Département de Physique Université Badji Mokhtar B.P.12, Annaba 23000

Associé(3)

Autre (4)

GSM : Tel : 038 87 53 99 Fax : 038 87 53 99 Diplômes Obtenus (Graduation, Post‐ Année Etablissement Graduation) Bac. Techniques Mathématiques Technicum Ali Boushaba (Constantine) 1 1989 D.E.S Université de Constantine 2 1993 Magister Université de Constantine 3 1996 Participation à des programmes de recherche Intitulé du Programme Année Organisme Contacts tel :

Adaptation du programme CORSIKA de simulation Monte Carlo des grandes gerbes de l’air aux énergies de l’expérience Pierre Auger (Code : D2301/08/04)

2004‐2006

Ministère de l’Enseignement Supérieur (CNEPRU)

Etude de l’interaction du rayonnement cosmique de très haute énergie avec l’atmosphère terrestre (Code : D01120060024)

2007‐2009

Ministère de l’Enseignement Supérieur (CNEPRU)

Origine des rayons cosmiques de très haute énergie (Code : D01120090017)

2010

Ministère de l’Enseignement Supérieur (CNEPRU)

A) Lister vos deux derniers travaux les plus importants 1 2

A.Bouchareb, G.Clement, C.M.Chen, D.V.Gal'tsov, N.G.Scherbluk and T.Wolf, G(2) generating technique for minimal D=5 supergravity and black rings, Phys.Rev.D76:104032,2007, Erratum‐ibid.D78:029901,2008. A.Bouchareb and G.Clement, Black hole mass and angular momentum in topologically massive gravity, Class.Quant.Grav.24:5581-5594,2007.

C) Tâches affectées au chercheur (à mentionner clairement): 1

Calculation of the phase diagram of Grosse-Wulkenhaar scalar models in two dimensions by means of the Monte Carlo method.

2

Cohomological models transition.

approach to the phenomena of geometry in

3 Visa du Chef d’établissement de rattachement :

Date : Signature :

9

4. Composante de l’équipe de recherche (Tableau anonyme : six personnes au maximum dont 3 chercheurs confirmés. Inscrire le responsable du projet en début de liste, ne pas inscrire de nom, ni l’intitulé de l’établissement de rattachement) Grade universitaire ou scientifique

Dernier diplôme obtenu

Tâche principale affectée dans le projet

1‐Maitre de Conférences B 2‐Maitre de conférences A

Ph.D

Supervision of the research project.

Ph.D

3‐Maitre Assistant A

Magister

4‐ Maitre Assistant A

Magister

5‐Maitre Assistant A

Magister

6‐Maitre Assistant A

Magister

Renormalization of Grosse‐Wulkenhaar Scalar Models on Fuzzy Spherical Geometries in 2 and 4 Dimensions. Cohomological and Monte Carlo Approaches of 4D IKKT Models. Monte Carlo Simulation of Grosse‐ Wulkenhaar Scalar Models and Cohomological Approach of 4D IKKT Models. Monte Carlo Simulation of 4D IKKT and Grosse‐Wulkenhaar Scalar Models. Renormalization of Grosse‐Wulkenhaar Scalar Models on Fuzzy Spherical Geometries in 2 and 4 Dimensions.

Emargement

‐Ne pas inscrire dans ce tableau les noms des membres de l’équipe, ni leurs établissements de rattachement. ‐Indiquer en tête de liste les informations relatives au porteur (chef) de projet.

10

5. Equipements scientifiques disponibles 5.1‐ Matériel existant pouvant être utilisé dans l’exécution du projet Nature

Localisation

Observations

2 micros+1 imprimante laser

Laboratoire de Physique des Rayonnements U. d’Annaba

5.2 – Matériel et Mobilier de Bureau à acquérir pour l’exécution du projet Nature Montant en DA Destination Observations 3 micro‐ ordinateurs

300 000,00

2 disques durs externes de grande capacité

30 000,00

Photocopieur

100 000,00

Meubles de bureau

40 000,00

Imprimante

50 000,00

Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba

Pour simulations numériques.

Pour simulations numériques.

Détailler la liste des matériels et mobiliers dont les montants sont mentionnés dans l’annexe financière.

5. Annexe financière : Budget et postes de dépenses prévisionnels (exprimés en DA) 1ère

Intitulés des postes de dépenses par année

Frais de séjour scientifique et de déplacement 500 000,00 à l’étranger Frais de séjour scientifique et de déplacement 100 000,00 en Algérie Frais d'organisation de rencontres scientifiques Honoraires des enquêteurs Honoraires des guides Frais de travaux et de prestations Matériels et instruments scientifiques Matériel informatique

480 000,00

Matériels d'expérience (animaux, végétaux, etc…) 40 000,00 Mobilier de bureau et de laboratoire Entretien et réparation 11

2ème 500 000,00 100 000,00

Produits chimiques Produits consommables Composants électroniques, mécaniques et audio‐ visuels 50 000,00 Accessoires et consommables informatiques 50 000,00 Papeterie et fournitures de bureau Périodiques Ouvrages et documentation scientifiques et 250 000,00 techniques Logiciels

50 000,00 50 000,00

300 000,00

Impression et Edition Affranchissements Postaux Communications téléphoniques, Fax, Internet Droits de douanes, Assurances Carburant TOTAL DES CREDITS OUVERTS :

1470 000,00

1000 000,00

Remarque : Les besoins financiers en devises doivent être exprimés en Dinars Algériens, après conversion au taux de change en cours.

12

Contents 1 IDENTIFICATION DU PROJET

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2 EQUIPE DE RECHERCHE

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3 CV: BADIS YDRI

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4 DESCRIPTION DU PROJET 4.1 MOTS-CLES . . . . . . . . . 4.2 RESUME . . . . . . . . . . . 4.3 RAPPEL DES OBJECTIFS . 4.4 IMPACTS . . . . . . . . . . .

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5 BILAN SYNTHESE 5.1 TRAVAUX PUBLIES . . . . . . . . . . . . 5.2 TRAVAIL SOUMIS POUR PUBLICATION 5.3 COMMUNICATIONS . . . . . . . . . . . . 5.4 AUTRES REMARQUES . . . . . . . . . . .

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6 INFORMATION FINANCIERE

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7 PUBLICATIONS: RESUMES ET BIBLIOGRAHIES

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1

1

IDENTIFICATION DU PROJET • PNR 8: Sciences Fondamentales. • Domaine: Physique. • Organisme Pilote: ANDRU. • Intitul´e du Projet: Th´eories des champs non-commutatives a´ partir des mod´eles des matrices et physique ´emergente. • Dur´ee: 05/11-05/13. • Chef du Projet: Badis Ydri. • Etablissement de Domiciliation: Universit´e Badji Mokhtar Annaba.

2

EQUIPE DE RECHERCHE • Porteur du Projet: Badis Ydri, Ph.D, MCA, Tel:0669465033, Universit´e de Annaba. • Partenaire Socio-Economique: Djamel Dou, Ph.D, Professor, Tel:0662106726, Centre Universitaire d’Eloued. • Equipe de Recherche: A.Bouchareb, M.Moumni, R.Ahmim, C.Soudani.

3

CV: BADIS YDRI • Ph.D: Syracuse University, New York, USA, 2001. • Post-Doctoral: – Hamilton Post-Doctoral Fellow, School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland, 2001-2005. – Marie Curie Post-Doctoral Fellow, Institut f¨ ur Physik, Humboldt-Universit¨at zu Berlin, Berlin, Germany, 2006-2008. • Habilitation: Universit´e Badji Mokhtar Annaba, Algeria, 2011. • Faculty Position: D´epartment de Physique, Universit´e Badji Mokhtar Annaba, Algeria, January 2009. • Research Interests: Quantum Field Theory and Quantum Gravity.

2

• Publications: Search the inSPIRE website: http://inspirehep.net • Emails, Website and Phone: [email protected], [email protected] http://www.stp.dias.ie/~ ydri/ 213(0)669465033.

4 4.1

DESCRIPTION DU PROJET MOTS-CLES

Noncommutative geometry,Noncommutative field theory,Yang-Mills models, Supersymmetry, Fuzzy spaces, Matrix models, Emergent geometry, The matrix and fuzzy sphere phases,The nonuniform ordered phase,Renormalization group methods, Monte Carlo methods, Cohomolgical methods.

4.2

RESUME

The four fondamental forces of nature are the electromagnetic force, the strong force, the weak force and the gravitational force. The non-gravitational forces are described by quantum Yang-Mills field theories on a flat Minkowski four dimensional spacetime and they are unified in the standard model of particle physics. Gravity can be unified with the other forces only within the context of string theory which is not a field theory. However at low energies string theory in a magnetic field becomes a field theory on a noncommutative spacetime, i.e a spacetime in which the coordinates are no longer numbers but operators. Noncommutative spacetimes arise also from combining the principles of general relativity which describe classical gravity and quantum mechanics at the very short scales. Thus the connection between noncommutativity and quantum gravity seems to be a very plausible hypothesis. Noncommutativity is also the only known extension preserving supersymmetry which is a symmetry central to string theory. String theory can not be studied in computer simulations. The basic problem is that supersymmetry can not be preserved in a naive spacetime discretization. The only exception is the Yang-Mills matrix model due to Ishibashi, Kawai, Kitazawa and Tsuchiya which is postulated to give a nonperturbative regularization of type IIB string theory. The IKKT Yang-Mills matrix models admit noncommutative spaces as solutions and as such they can be used to regularize nonperturbatively noncommutative gauge theories. In this project we will study non-trivial phenomena which emerge dynamically in the IKKT Yang-Mills matrix models such as emergent geometry, spontaneous supersymmetry breaking and emergent gravity. This is expected to shed important light on the nonperturbative physics of noncommutative gauge theories. We 3

will also use the matrix formalism to study quantum aspects of noncommutative scalar field theories.

4.3

RAPPEL DES OBJECTIFS

• 1)-Phase diagram of non-commutative scalar phi-four using Monte Carlo methods. • 2)-Renormalization group equations for non-commutative field theory a la Wilson and a la Polchinski . • 3)-Emergent geometry and non-commutative gauge theories from Yang-Mills matrix models .

4.4

IMPACTS

The expected scientific impacts may include the following: 1) The conjecture that large N IKKT Yang-Mills matrix models are of central importance to fundamental physics can be verified explicitly in several interrelated physical contexts. The relevance of these large N IKKT Yang-Mills matrix models to fundamental physics can be summarized as follows: • a) They may help us understand nonperturbative physics of noncommutative gauge theories by means of cohomological and random matrix models. • b) They may provide a nonperturbative definition of supersymmetry. Indeed supersymmetry in this language may prove to be tractable in Monte Carlo simulation. • c) They provide concrete models for emergent geometry and possibly emergent gravity. Indeed geometry in transition seems to be possible in all these matrix models. These matrix models seems also very suited to test the gauge/gravity duality. 2) A more coherent understanding of the quantum dynamics of noncommutative scalar field theories by means of matrix models techniques and renormalization group and Monte Carlo methods. The expected educatif impacts may include the following: 1) Supervision of doctoral theses which are based on different parts of this project. 2) Initiation to different methods and techniques of theoretical physics. 3) Introduction to new area of research in theoretical physics such as supersymmetric field theory and gauge/gravity duality. 4

5

BILAN SYNTHESE

5.1

TRAVAUX PUBLIES

• We have published during the period 2011- 2013 in total six articles in internationally renowned journals of theoretical particle physics and cosmology such as Physical Review D, Journal of Mathematical Physics, International Journal of Modern Physics A and Modern Physics Letters A. The six published articles between 2011 and 2013 can be found in the bibliography at the end of this report and the first page of each article is included in section 5. • Three articles (articles 1, 2 and 3) deals directly with the original objectives of our PNR project whereas article 4 is indirectly related to our project. Only articles 5 and 6 are on different matters. • Article 1: Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models . This is the longest paper which deals with the problem of emergent geometry and emergent gauge theory (the third objective of our PNR project) from Yang-Mills matrix models using the Monte Carlo method and matrix models techniques. This axis of research is still ongoing with two doctoral (LMD) students. • Article 2: Matrix Model Fixed Point of Noncommutative phi-four Theory. This is the second longest paper which deals with the problem of the construction of viable and solvable renormalization group equations for matrix models and noncommutative field theories (the second objective of our PNR project). The collaborating researcher RACHID AHMIM is a doctoral student who is now published and hence can submit and defend his thesis at any time he chooses. • Article 3: The fate of the Wilson-Fisher fixed point in non-commutative φ4 This article deals also with the problem of the renormalization group equations for noncommutative field theories in the limit of small noncommutativity. • Article 4, 5, 6: . As mentioned earlier Articles 4, 5, 6 are indirectly related to the original objectives of our PNR project since they deal with some phenomenological consequences of noncommutative field theory in atomic physics and various issues of quantum gravity, general relativity and cosmology.

5

– Article 4: Effects of Non-Commutativity on Light-Hydrogen-Like Atoms and Proton Radius – Article 5: On the Problem of Vacuum Energy in FLRW Universes and Dark Energy. – Article 6: Bertotti-Robinson solutions of D=5 Einstein-Maxwell-Chern-Simons-Lambda theory.

5.2

TRAVAIL SOUMIS POUR PUBLICATION

• Article 7: New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere. Article 7 is a completed preprint submitted for publication in the Journal of High Eneregy Physics. This preprint contains a summary of ongoing work on the first objective of our PNR project, i.e. our work on the phase diagram of noncommutative phi-four on the fuzzy sphere using the Monte Carlo approach. This, as it turns out, is a very hard problem which required us to construct a completely new algorithm in order to handle properly the complex phase structure of this theory. There are two doctoral students involved in this work among them figure our collaborator C.SOUDANI.

5.3

COMMUNICATIONS

• Our contributions to the The 9th International Conference in Subatomic Physics and Applications held in Constantine from September 30th to October 2nd 2013 are: – M.Moumeni, Lyman α Spectroscopy in Non-Commutative Space-Time. – A.Bouchereb, On the problem of vacuum energy in FLRW universes and dark energy. – B.Ydri, Aspects of noncommutative φ4 on the fuzzy sphere.

5.4

AUTRES REMARQUES

• Re:Ad´ equation des Actions Men´ ees et Niveau des Objectifs Atteints: We hope it is clear from the above brief outline of our results during the past two years that our work and action were largely adequate vis-a-vis the stated original goals of our PNR project. In fact we estimate that our work and action were very relevant to the stated goals. Also we estimate that we have in fact achieved at least 75 per cent of our original plan. • Re:Le Projet a-t-il Donn´ e lieu ´ a un Produit et Valorisations: 6

If by ”product” it is meant scientific publications in well established refereed journals then the answer is certainly YES. As of the valorization aspect of our PNR project it consists solely in scientific contribution and getting recognition for it and we are continuously working hard towards these two goals within this PNR and outside it. With regard to this point we should always be mindful that theoretical physics is a fundamental science which addresses largely basic problems of nature and as such the only aim we strive to reach is to continuously improve our understanding of the world we live in at both the large and small scales using mostly the language of mathematics but also in recent years numerical simulations. In theoretical physics scientific contribution is the goal and the prize at the same time. • Re:Le Projet a-t-il Contribu´ e´ a la Formation de Chercheurs: There are two research members (AHMIM and SOUDANI) who are preparing their doctoral dissertations within the axes of this PNR project. As stated earlier AHMIM has already published an article (article 2) and have started writing his thesis whereas the work of SOUDANI on the phase diagram of noncommutative phi-four on the fuzzy sphere is still ongoing. There are further four LMD doctoral students working on different aspects of matrix field theory and matrix geometry initiated within this PNR project.

References [1] B. Ydri, “Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II,” Int. J. Mod. Phys. A, Vol. 27, No. 17 (2012) 1250088 (35 pages), DOI: 10.1142/S0217751X12500881, arXiv:1206.6375 [hep-th]. [2] B. Ydri and R. Ahmim, “Matrix Model Fixed Point of Noncommutative Phi-Four,” Phys. Rev. D 88, 106001 (2013) (30 pages), DOI: 10.1103/PhysRevD.88.106001, arXiv:1304.7303 [hep-th]. [3] B. Ydri and A. Bouchareb, “The fate of the Wilson-Fisher fixed point in non-commutative φ4 ,” J. Math. Phys. 53, 102301 (2012) (14 pages), DOI: 10.1063/1.4754816, arXiv:1206.5653 [hep-th]. 7

[4] M. Moumni and A. BenSlama, “Effects of Non-Commutativity on Light-Hydrogen-Like Atoms and Proton Radius,” Int. J. Mod. Phys. A 28, 1350139 (2013) (15 pages), DOI: 10.1142/S0217751X1350139X, arXiv:1305.3508 [hep-ph]. [5] B. Ydri and A. Bouchareb, “On the Problem of Vacuum Energy in FLRW Universes and Dark Energy,” Mod. Phys. Lett. A, Vol. 28, No. 36 (2013) 1350166 (9 pages), DOI: 10.1142/S0217732313501666, arXiv:1307.2749 [hep-th]. [6] A. Bouchareb, C. -M. Chen, G´er. Cl´ement and D. V. Gal’tsov, “Bertotti-Robinson solutions of D=5 Einstein-Maxwell-Chern-Simons-Lambda theory,” Phys. Rev. D 88, 084048 (2013) (17 pages), DOI: 10.1103/PhysRevD.88.084048, arXiv:1308.6461 [gr-qc]. [7] B. Ydri, “New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere,” arXiv:1401.1529 [hep-th].

6

INFORMATION FINANCIERE Consult the appendix at the end of this report.

7

PUBLICATIONS: RESUMES ET BIBLIOGRAHIES

We include in the following the abstracts, introductions, conclusions and bibliographies of our 7 finished articles.

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International Journal of Modern Physics A Vol. 27, No. 17 (2012) 1250088 (35 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217751X12500881

IMPACT OF SUPERSYMMETRY ON EMERGENT GEOMETRY IN YANG MILLS MATRIX MODELS II

BADIS YDRI Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria [email protected] [email protected] Received 1 May 2012 Revised 7 May 2012 Accepted 8 May 2012 Published 18 June 2012 We present a study of D = 4 supersymmetric Yang–Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N . We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law. Keywords: Noncommutative geometry; gauge theory; Yang–Mills matrix models; emergent geometry; the fuzzy sphere; supersymmetry.

1. Introduction Reduced Yang–Mills theories play a central role in the nonperturbative definitions of M -theory and superstrings. The Banks–Fischler–Shenker–Susskind (BFSS) conjecture1 relates discrete light-cone quantization (DLCQ) of M -theory to the theory of N coincident D0 branes which at low energy (small velocities and/or string coupling) is the reduction to 0 + 1 dimension of the ten-dimensional U (N ) supersymmetric Yang–Mills gauge theory.2 The BFSS model is therefore a Yang–Mills quantum mechanics which is supposed to be the UV completion of 11-dimensional 1250088-1

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supergravity. As it turns out, BFSS action is nothing else but the regularization of the supermembrane action in the light cone gauge.3 The Berenstein–Maldacena–Nastase (BMN) model4 is a generalization of BFSS model to curved backgrounds. It is obtained by adding to BFSS action a oneparameter mass deformation corresponding to the maximally supersymmetric ppwave background of 11-dimensional supergravity.5–7 We note, in passing, that all maximally supersymmetric pp-wave geometries can arise as Penrose limits of AdSp × S q spaces.8 The Ishibashi–Kawai–Kitazawa–Tsuchiya (IKKT) model9 is, on the other hand, a Yang–Mills matrix model obtained by dimensionally reducing ten-dimensional U (N ) supersymmetric Yang–Mills gauge theory to 0 + 0 dimensions. The IKKT model is postulated to provide a constructive definition of type IIB superstring theory and for this reason it is also called type IIB matrix model. Supersymmetric analogue of IKKT model also exists in dimensions d = 3, 4 and 6 while the partition functions converge only in dimensions d = 4, 6.48,49 The IKKT Yang–Mills matrix models can be thought of as continuum Eguchi– Kawai reduced models as opposed to the usual lattice Eguchi–Kawai reduced model formulated in Ref. 11. We point out here the similarity between the conjecture that the lattice Eguchi–Kawai reduced model allows us to recover the full gauge theory in the large N theory and the conjecture that IKKT matrix model allows us to recover type IIB superstring. The relation between BFSS Yang–Mills quantum mechanics and IKKT Yang– Mills matrix model is discussed at length in the seminal paper10 where it is also shown that toroidal compactification of the D-instanton action (the bosonic part of IKKT action) yields, in a very natural way, a noncommutative Yang–Mills theory on a dual noncommutative torus.22 From the other hand, we can easily check that the ground state of the D-instanton action is given by commuting matrices which can be diagonalized simultaneously with the eigenvalues giving the coordinates of the D-branes. Thus at tree-level an ordinary space–time emerges from the bosonic truncation of IKKT action while higher-order quantum corrections will define a noncommutative space–time. In summary, Yang–Mills matrix models which provide a constructive definition of string theories will naturally lead to emergent geometry18 and noncommutative gauge theory.19,20 Furthermore, noncommutative geometry21,23 and their noncommutative field theories24,25 play an essential role in the nonperturbative dynamics of superstrings and M -theory. Thus the connections between noncommutative field theories, emergent geometry and matrix models from one side and string theory from the other side run deep. It seems therefore natural that Yang–Mills matrix models provide a nonperturbative framework for emergent space–time geometry and noncommutative gauge theories. Since noncommutativity is the only extension which preserves maximal supersymmetry, we also hope that Yang–Mills matrix models will provide a regularization which preserves supersymmetry.26 1250088-2

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Impact of Supersymmetry on Emergent Geometry

In this paper we will explore in particular the possibility of using IKKT Yang– Mills matrix models in dimensions 4 and 3 to provide a nonperturbative definition of emergent space–time geometry, noncommutative gauge theory and supersymmetry in two dimensions. From our perspective in this paper, the phase of commuting matrices has no geometry in the sense of Connes and thus we need to modify the models so that a geometry with a well defined spectral triple can also emerge alongside the phase of commuting matrices. There are two solutions to this problem. The first solution is given by adding mass deformations which preserve supersymmetry to the flat IKKT Yang–Mills matrix models12 or alternatively by an Eguchi–Kawai reduction of the mass deformed BFSS Yang–Mills quantum mechanics constructed in.13,60–62 The second solution, which we have also considered in this paper, is given by deforming the flat Yang–Mills matrix model in D = 4 using the powerful formalism of cohomological Yang–Mills theory.30–32,63 These mass deformed or cohomologically deformed IKKT Yang–Mills matrix models are the analogue of BMN model and they typically include a Myers term14 and thus they will sustain the geometry of the fuzzy sphere,27,28 as a ground state which at large N will approach the geometry of the ordinary sphere, the ordinary plane or the noncommutative plane depending on the scaling limit. Thus a nonperturbative formulation of noncommutative gauge theory in two dimensions can be captured rigorously within these models15–17 (see also Refs. 64 and 65). This can in principle be generalized to other fuzzy spaces29 and higher dimensional noncommutative gauge theories by considering appropriate mass deformations of the flat IKKT Yang–Mills matrix models. The problem or virtue of this construction, depending on the perspective, is that in these Yang–Mills matrix models the geometry of the fuzzy sphere collapses under quantum fluctuations into the phase of commuting matrices. Equivalently, it is seen that the geometry of the fuzzy sphere emerges from the dynamics of a random matrix theory.52,54 Supersymmetry is naturally expected to stabilize the space–time geometry, and in fact the nonstability of the nonsupersymmetric vacuum should have come as no surprise to us.33 We should mention here the approach of Ref. 34 in which a noncommutative Yang–Mills gauge theory on the fuzzy sphere emerges also from the dynamics of a random matrix theory. The fuzzy sphere is stable in the sense that the transition to commuting matrices is pushed towards infinite gauge coupling at large N .53 This was achieved by considering a very special nonsupersymmetric mass deformation which is quartic in the bosonic matrices. This construction was extended to a noncommutative gauge theory on the fuzzy sphere based on co-adjoint orbits.35 Let us also note here that the instability and the phase transition discussed here were also observed on the noncommutative torus,39–41,69,70 where the twisted Eguchi–Kawai model was employed as a nonperturbative regularization of noncommutative Yang–Mills gauge theory.36–38 1250088-3

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In this paper, we will then study using the Monte Carlo method the mass deformed Yang–Mills matrix model in D = 4 as well as a particular truncation to D = 3. We will also derive and study a one-parameter cohomological deformation of the Yang–Mills matrix model which coincides with the mass deformed model in D = 4 when the parameter is tuned appropriately. We will show that the first-/ second-order phase transition from the fuzzy sphere to the phase of commuting matrices observed in the bosonic models is converted in the supersymmetric models into a very slow crossover transition with an arbitrary small transition point in the large N limit. We will determine the eigenvalues distributions for both D = 4 and D = 3 throughout the phase diagram. This paper is organized as follows. In Sec. 2 we will derive the mass deformed Yang–Mills quantum mechanics from the requirement of supersymmetry and then reduce it further to obtain Yang–Mills matrix model in D = 4 dimensions. In Sec. 3 we will derive a one-parameter family of cohomologically deformed models and then show that the mass deformed model constructed in Sec. 2 can be obtained for a particular value of the parameter. In Sec. 4 we report our first Monte Carlo results for the model D = 4 including the eigenvalues distributions and also comment on the D = 3 model obtained by simply setting the fourth matrix to 0. We conclude in Sec. 5 with a comprehensive summary of the results and discuss future directions. 2. Mass Deformation of D = 4 Super-Yang Mills Matrix Model 2.1. Deformed Yang Mills quantum mechanics in 4D The N = 1 supersymmetric Yang–Mills theory reduced to one dimension is given by the supersymmetric Yang–Mills quantum mechanics (with D0 = ∂0 − i[X0 , · ]) 1 1 1 1¯ 0 i ¯ i 1 L0 = 2 Tr (D0 Xi )2 + [Xi , Xj ]2 − ψγ D0 ψ + ψγ [Xi , ψ] + F 2 . (2.1) g 2 4 2 2 2 The corresponding supersymmetric transformations are δ0 X0 = ¯ǫ γ0 ψ , δ0 Xi = ǫ¯γi ψ , 1 i δ0 ψ = − [γ 0 , γ i ]D0 Xi + [γ i , γ j ][Xi , Xj ] + iγ5 F ǫ , 2 4

(2.2)

δ0 F = −i¯ ǫ γ5 γ0 D0 ψ + ǫ¯γ5 γi [Xi , ψ] . Let µ be a constant mass parameter. A mass deformation of the Lagrangian density L0 takes the form Lµ = L0 +

µ µ2 L + L2 + · · · . 1 g2 g2

(2.3)

The Lagrangian density L0 has mass dimension 4. The corrections L1 and L2 must have mass dimension 3 and 2, respectively. We recall that the Bosonic matrices X0 and Xa have mass dimension 1 whereas the Fermionic matrices ψi have mass dimension 23 . A typical term in the Lagrangian densities L1 and L2 will contain nf 1250088-4

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Impact of Supersymmetry on Emergent Geometry ∼

∼

EVs of the N=10 bosonic models with β =2/9 and α =0.5

Eigenvalues Distributions

0.8

D=3 uniform in D=3 uniform in D=4 D=4 uniform in D=3 uniform in D=4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3

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-1 0 1 Eigenvalues of X3

2

3

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EVs of the N=10 bosonic models with β =2/9

Eigenvalues Distributions

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∼ D=3, α =1.0 ∼ α =0.1 ∼ D=4, ∼α =1.0 α =0.1

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3

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-1 0 1 Eigenvalues of X3

2

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Fig. 9. The eigenvalues distributions of X3 of the = 3, 4 bosonic models with β˜ = 2/9 in the matrix phase.

5. Summary and Future Directions In this paper we employed the Monte Carlo method to study nonperturbatively Yang–Mills matrix models in D = 4 with mass terms. We can summarize the main results, findings and conjectures of this work as follows: • By imposing the requirement of supersymmetry and SO(3) covariance we have shown that there exists a single mass deformed Yang–Mills quantum mechanics in D = 4 which preserves all four real supersymmetries of the original theory although in a deformed form. This is the four-dimensional analogue of the tendimensional BMN model. Full reduction yields a unique mass deformed D = 4 Yang–Mills matrix model. This latter four-dimensional model is the analogue of the ten-dimensional IKKT model. 1250088-29

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• By using cohomological deformation of supersymmetry we constructed a oneparameter (ζ0 ) family of cohomologically deformed D = 4 Yang–Mills matrix models which preserve two supercharges. The mass deformed model is one limit (ζ0 → 0) of this one-parameter family of cohomologically deformed Yang–Mills models. • We studied the models with the values β˜ = 0 and β˜ = 2/9 where β˜ is the mass parameter of the bosonic matrices Xa . The second model is special in the sense that classically the configurations Xa ∼ La , X4 = 0 is degenerate with the configuration Xa = 0 and X4 = 0. • The Monte Carlo simulation of the bosonic D = 4 Yang–Mills matrix model with mass terms shows the existence of an exotic first-/second-order transition from a phase with a well defined background geometry given by the famous fuzzy sphere to a phase with commuting matrices with no geometry in the sense of Connes. The transition looks first-order due to the jump in the action whereas it looks second-order due to the divergent peak in the specific heat. • The fuzzy sphere is less stable as we increase the mass term of the bosonic ˜ For β˜ = 2/9 we find the critical value α matrices Xa , i.e. as we increase β. ˜ ∗ = 4.9 whereas for β˜ = 0 we find the critical value β˜ = 2.55. • The measured critical line in the plane α ˜ − β˜ agrees well with the theoretical prediction coming from the effective potential calculation. • The order parameter of the transition is given by the inverse radius of the sphere defined by 1/r = Tr Xa2 /(˜ α2 c2 ). The radius is equal to 1/φ2 (where φ is the classical configuration) in the fuzzy sphere phase. At the transition point the sphere expands abruptly to infinite size. Then as we decrease the inverse temperature (the inverse gauge coupling constant) α ˜ , the size of the sphere shrinks fast to 0, i.e. the sphere evaporates. • The fermion determinant is positive definite for all gauge configurations in D = 4. We have conjectured that the path integral is convergent as long as the scalar curvature (the mass of the fermionic matrices) is zero. • We have simulated the two models β˜ = 0 and β˜ = 2/9 with dynamical fermions. The model with β˜ = 0 has two supercharges while the model β˜ = 2/9 has a softly broken supersymmetry since in this case we needed to set by hand the scalar curvature to zero in order to regularize the path integral. Thus β˜ = 0 is amongst the very few models (which we known of) with exact supersymmetry which can be probed and accessed with the Monte Carlo method. • The fuzzy sphere is stable for the supersymmetric D = 4 Yang–Mills matrix model with mass terms in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point α ˜ is found to scale to zero with N . There is no jump in the action nor a peak in the specific heat. • The fuzzy sphere is stable also in the sense that the radius is equals 1/φ2 over a much larger region then it starts to decrease slowly as we decrease the inverse temperature α ˜ until it reaches 0 at α ˜ = 0. We claim that the value where the radius starts decreasing becomes smaller as we increase N . 1250088-30

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The model at α ˜ = 0 can never sustain the geometry of the fuzzy sphere since it is the nondeformed model so in some sense the transition to commuting matrices always occurs and in the limit N → ∞ it will occur at α ˜ ∗ → 0. • We have spent a lot of time in trying to determine the eigenvalues distributions of the matrices Xµ in both the bosonic and supersymmetric theories. A universal behavior seems to emerge with many subtleties. These can be summarized as follows: – In the fuzzy sphere the matrices Xa are given by the SU (2) irreducible representations La . For example diagonalizing the matrix X3 gives N eigenvalues between (N − 1)/2 and −(N − 1)/2 with a step equal 1, viz m = (N − 1)/2, (N − 3)/2, . . . , −(N − 3)/2, −(N − 1)/2. – In the matrix phase the matrices Xµ become commuting. More explicitly the eigenvalues distribution of any of the matrices Xµ in the matrix phase is given by the nonpolynomial law ρ4 (x) =

–

– –

– –

–

–

8 3 (R2 − x2 ) 2 . 3πR4

(5.1)

This can be obtained from the conjecture that the joint probability distribution of the four matrices Xµ is uniform inside a solid ball with radius R. In the matrix phase the eigenvalues distribution of any of the Xa , say X3 , is given by the above nonpolynomial law with a radius R independent of α ˜ and N . This is also confirmed by computing the radius in this distribution and comparing to the Monte Carlo data. A very precise measurement of the transition point can be made by observing the point at which the eigenvalues distribution of X3 undergoes the transition from the N -cut distribution to the above nonpolynomial law. The eigenvalues distribution of X4 is always given by the above nonpolynomial law, i.e. for all values of α ˜ , with a radius R which depends on α ˜ and N . Another signal that the matrix phase is fully reached is when the eigenvalues distribution of X4 coincides with that of X3 . From this point downward the eigenvalues distribution of X4 ceases to depend on α ˜ and N . Monte Carlo measurements seems to indicate that R = 1.8 for bosonic models and R = 2.8 for supersymmetric models. The distribution becomes wider in the supersymmetric case. We have also observed that the eigenvalues of the normal scalar field Xa2 − c2 in the fuzzy sphere are also distributed according to the above nonpolynomial law. This led us to the conjecture that the eigenvalues of the gauge field on the background geometry are also distributed according to the above nonpolynomial law. Recall that the normal scalar field is the normal component of the gauge field to the background geometry which is the sphere here. 1250088-31

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• In the D = 3 Yang–Mills matrix model with mass terms the eigenvalues distribution becomes polynomial (parabolic) given by 3 (R2 − x2 ) . (5.2) 4R3 It was difficult for us in this paper to differentiate with certainty between the two distributions ρ4 and ρ3 in the three-dimensional setting. • Finally, we conjecture that the transition from a background geometry to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. Indeed mass deformed supersymmetry preserves the fuzzy sphere configuration but not diagonal matrices. ρ3 (x) =

Among the future directions that can be considered we will simply mention the following four points: • Higher precision Monte Carlo simulations of the models studied in this paper is the first obvious direction for future investigation. The most urgent question (in our view) is the precise determination of the behavior of the eigenvalues distributions in D = 4 and D = 3. An analytical derivation of ρ3 and especially ρ4 is an outstanding problem. • Finding matrix models with emergent four-dimensional background geometry is also an outstanding problem. • Models for emergent time, and to a lesser extent emergent gravity, and as a consequence emergent cosmology are very rare. • Monte Carlo simulation of supersymmetry based on matrix models seems to be a very promising goal. • A complete analytical understanding of the emergent geometry transition observed in Yang–Mills matrix models with mass terms is also an outstanding problem. In Ref. 68, we have attempted to compute the above eigenvalues distributions analytically. Using localization techniques we were able to find a special set of parameters for which the D = 4 Yang–Mills matrix model with mass terms can be reduced to the three-dimensional Chern–Simons (CS) matrix model. The saddlepoint method leads then immediately to the eigenvalues distributions ρ3 . We believe that our theoretical prediction for the value of R is reasonable compared to the Monte Carlo value. We have also made a preliminary comparison between the dependence of R on α in the Hermitian and anti-Hermitian CS matrix models. The Hermitian case seems more appropriate for the description of the eigenvalues of X3 whereas the anti-Hermitian case may be relevant to the description of the eigenvalues of X4 . Acknowledgments I would like to thank Denjoe O’Connor for extensive discussions at various stages of this project. I would also like to thank R. Delgadillo-Blando and Adel Bouchareb 1250088-32

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PHYSICAL REVIEW D 88, 106001 (2013)

Matrix model fixed point of noncommutative 4 theory Badis Ydri1,* and Rachid Ahmim2 1

Institute of Physics, BM Annaba University, BP 12, 23000 Annaba, Algeria Department of Physics, El-Oued University, BP 789, 39000 El-Oued, Algeria (Received 9 May 2013; published 5 November 2013)

2

In this article, we exhibit explicitly the matrix model ( ¼ 1) fixed point of 4 theory on noncommutative spacetime with only two noncommuting directions, using the Wilson renormalization group recursion formula and the 1=N expansion of the zero-dimensional reduction, and then calculate the mass critical exponent and the anomalous dimension in various dimensions. DOI: 10.1103/PhysRevD.88.106001

PACS numbers: 11.10.Nx, 11.15.Pg

I. INTRODUCTION AND SUMMARY OF RESULTS A. Introduction The Wilson recursion formula is the oldest, simplest, and most intuitive renormalization group approach, which although approximate, agrees very well with high-temperature expansions [1]. The goal in this article is to apply this method to the self-dual noncommutative 4 in the matrix basis [2], which after appropriate nonperturbative definition becomes an N N matrix model, where N is a regulator in the noncommutative directions. More precisely, we propose to employ, following Refs. [3–5], a combination of i) the Wilson approximate renormalization group recursion formula and ii) the solution to the corresponding zerodimensional large-N counting problem given in our case by the Hermitian Penner matrix model, which can be turned into a multitrace Hermitian quartic matrix model for large values of . As discussed neatly in Ref. [3], the virtue and power of combining these two methods lies in the crucial fact that all leading Feynman diagrams in 1=N will be counted correctly in this scheme, including the so-called ‘‘setting sun’’ diagrams. As it turns out, the recursion formula can also be integrated explicitly in the large-N limit, which in itself is a very desirable property. In a previous work [6], a nonperturbative study of the Ising universality class fixed point in a noncommutative OðNÞ model was carried out using precisely a combination of the above two methods. It was found that the WilsonFisher fixed point makes good sense only for sufficiently small values of up to a certain maximal noncommutativity. In the current work, we focus on the opposite limit of large , although the 1=N expansion invoked in this article is different from the 1=N expansion of the OðNÞ vector model, since N here has direct connection with noncommutativity itself. The central aim of this article, as we will see, is to exhibit as explicitly as possible the matrix-model fixed point which describes the transition from the one-cut (disordered) phase to the two-cut (nonuniform ordered, stripe) phase in the same way that the Wilson-Fisher fixed *[email protected]; [email protected]

1550-7998= 2013=88(10)=106001(30)

point describes the transition from the disordered phase to the uniform ordered phase. B. Summary of results We start by summarizing the main statements and results of this paper. We will be interested in 4 theory on a degenerate noncommutative Moyal-Weyl space with only two noncommuting coordinates Rd ¼ RD R2 , where D ¼ d 2 with commutation relations ½x^ i ; x^ j ¼ iij , ½x^ i ; x ¼ ½x ; x ¼ 0. The action takes the form Z 1 ^ 2 1 2 ~2 1 2 2 ^ þ D ^ S½ ¼ 2 d xTrH @i þ X i @ þ 2 2 2 2 ^þ^ ^þ^ : (1.1) þ 4! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ In the above equation, ¼ ðBÞ=2, 2 ¼ det ð2Þ, and X~i ¼ 2ð1 Þij Xj , where Xi ¼ ðx^ i þ x^ Ri Þ=2. This action is covariant under a duality transformation which exchanges, among other things, positions and momenta as xi $ k~i ¼ 2 B1 ij kj . The value ¼ 1 in particular gives an action which is invariant under this duality transformation. By expanding the field in an appropriate basis (for example, the Landau basis), introducing a cutoff N in the noncommuting directions and a cutoff in the commuting directions, and setting 2 ¼ 1, we obtain the action Z 1 1 D S½M ¼ d xTrN @ Mþ @ M þ 2 Mþ M 2 2 1 2 u þ þ 2 þ r EfM; M g þ ðM MÞ ; (1.2) 2 N r2

8 ¼ ; 2

N u¼ ; 4! 2

Elm

1 ¼ l lm : 2

(1.3)

We will consider in the remainder only the case of Hermitian matrices, viz. M ¼ Mþ :

(1.4)

There are three independent parameters in this theory. These are the usual mass parameter 2 and the quartic coupling constant u plus the inverse noncommutativity

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PHYSICAL REVIEW D 88, 106001 (2013)

r ¼ 4=. The free propagator of this theory is simply given by 2

mn ðpÞ ¼

1 : p2 þ 2 þ r2 ðm þ n 1Þ

(1.5)

renormalization group approach consists in rescaling the field in such a way that the kinetic term is brought to its canonical form, we obtain the effective action [Eq. (4.39)]. In position space, this effective action takes the form 02 Z 1Z D ~ 0 Þ2 þ ~ 02 d xTrN ð@ M dD xTrN M 2 2 0 Z Z ~ 02 þ u ~ 04 : dD xTrN M þ r02 dD xTrN EM N (1.8)

In the limit !1, this propagator behaves as 1=ðp2 þ2 Þ. More precisely, we have in this limit the useful properties X r 1 2 mj ðp1 Þrmj ðp2 Þ... ! Nrn10 j1 ðp1 Þrn20 j2 ðp2 Þ... (1.6) 1 2

S þ S ¼

The Wilson renormalization group approach consists in general of three main steps: 1) integration, 2) rescaling, and 3) normalization. In our case, here we will supplement the first step of integration with two approximations: a) truncation and b) the Wilson recursion formula. We start by decomposing the N N matrix M into an ~ and an N N fluctuation N N background matrix M ~ ~ contains matrix m, viz. M ¼ M þ m. The background M slow modes, i.e., modes with momenta less than or equal to , while the fluctuation m contains fast modes, i.e., modes with momenta larger than , where 0 < < 1. The integration step involves performing the path integral over the fluctuation m to obtain an effective path integral ~ alone. We find over the background M Z ~ SðMÞ ~ ~ S½M Z ¼ dMe e : (1.7)

~ 0 is related to the bare field M ~ as The renormalized field M 0 and M are the Fourier transforms of M ~ and follows: If M ~ 0 , respectively, then M sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 N 2þD 0 ðpÞ ¼ 2 Zðg; 2 Þ þ Zðg; 2 ÞMð pÞ: M ðc2 þ 2 Þ2

m

~ The main goal is to compute the effective action SðMÞ, which contains corrections to the operators already ~ together with all present in the original action S½M possible effective interactions generated by the integration ~ up to the fourth process. An exact formula for SðMÞ ~ is given by the cumulant expansion power in the field M [Eq. (4.8)]. The formula in Eq. (4.8) is still very complicated. To simplify it and to get explicit equations, we employ the so-called Wilson truncation and Wilson recursion formula. This is usually thought of as part of the integration step. Wilson truncation means that we calculate quantum corrections to only those terms which appear in the starting action. Wilson recursion formula is completely equivalent to the use in perturbation theory of the Polyakov-Wilson rules given by the following two approximations: 1) We replace every internal propagator 1=ðk2 þ 2 Þ with 1=ðc2 þ 2 Þ, where c is a constant taken to be equal to the cutoff R, and 2) We replace every D D momentum loop integral with another d k=ð2Þ D constant vD ¼ v^ D , where the definition of v^ D is obvious. This is a very long and tedious calculation. The end result is given by the sum of the three equations [Eqs. (4.18), (4.29), and (4.35)]. By performing the second step of the Wilson renormalization group approach, i.e., by scaling momenta as p ! p= so that the cutoff returns to its original value , and the third and final step of the Wilson

(1.9) The renormalized mass 02 , the renormalized quartic coupling constant u0 , and the renormalized inverse noncommutativity r02 are given by 2 02 ¼ ðg; 2 Þ þ r2 NðgÞ Zðg; 2 Þ r2 N ðg; 2 ÞZðg; 2 Þ 2 ; (1.10) ðc þ 2 Þ2 Zðg; 2 Þ r02 ¼

2 r2 r2 ðN þ 1Þ ðgÞ þ e ðgÞ e c2 þ 2 Zðg; 2 Þ r2 N e ðgÞZðg; 2 Þ 2 ; ðc þ 2 Þ2 Zðg; 2 Þ

1 r2 N 4 ðgÞ þ 2 4 ðgÞ 2 2 Z ðg; Þ 4g c þ 2 2r2 N 4 ðgÞZðg; 2 Þ : 2 ðc þ 2 Þ2 Zðg; 2 Þ

(1.11)

u0 ¼ u

The effective coupling g is defined by vD u g¼ 2 : ðc þ 2 þ r2 NÞ2

(1.12)

(1.13)

The various functions Zðg; 2 Þ, ðg; 2 Þ, 2 ðgÞ, and 4 ðgÞ are known nonperturbatively, whereas we were able to determine the functions Zðg; 2 Þ, ðgÞ, 4 ðgÞ, e ðgÞ, and e ðgÞ only perturbatively. These functions are summarized in Table I. The process which led from the bare coupling constants 2 , r2 , and u to the renormalized coupling constants 02 , r02 , and u0 can be repeated an arbitrary number of times. The bare coupling constants will be denoted by 20 , r20 , and u0 , whereas the the renormalized coupling constants at the first step of the renormalization group procedure will be denoted by 21 , r21 , and u1 . At a generic

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step l þ 1 of the renormalization group process, the renormalized coupling constants 2lþ1 , r2lþ1 , and ulþ1 are related to their previous values 2l , r2l , and ul by precisely the above renormalization group equations. The effective coupling constant gl will, of course, be given in terms of 2l , r2l , and ul by the same formula that related g0 to 20 , r20 , and u0 . We are therefore interested in renormalization group flow in a three-dimensional parameter space generated by the mass 2 , the quartic coupling constant u, and the harmonic oscillator coupling constant (inverse noncommutativity) r2 . We have reached the stage where it is very hard to push any further by pure analytical means, and therefore we have to turn to numerical tools. In any case, the renormalization group approach was originally devised with numerical approximations in mind [1]; see also Ref. [7]. A renormalization group fixed point is a point in the space parameter which is invariant under the renormalization group flow. If we denote the fixed point by 2 , r2 , and

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u , then we must set ¼ 0 ¼ , r ¼ r0 ¼ r , and u ¼ u0 ¼ u , as well as g ¼ g0 ¼ g in the above renormalization group equations (1.10), (1.11), and (1.12). The matrix model fixed point corresponding to infinite noncommutativity is given by the following equations: r2 ¼ 0;

(1.14)

fðg Þ ¼ 0;

(1.15)

; 1

(1.16)

g ð1 þ ^ 2 Þ2 : v^ D

(1.17)

^ 2 ¼

u^ ¼

The functions f and are given by (with 0 ¼ =D and

¼ 4 D)

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 8 ðgÞ 2 fðgÞ ¼ 1 þ 2 ð1 2 2 0 Z2 ðgÞÞ þ 2 ð1 þ 0 ÞZ2 ðgÞð2 ðgÞ 1Þ þ 2 4 2Z2 ðgÞ 0 ; 4g 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ðgÞ 42 2 4 þ 4Z2 ðgÞ 0 5: ðgÞ ¼ 4ð1 þ 0 ÞZ2 ðgÞ 4g We can immediately see that this fixed point is fully determined by the functions 2 , Z2 , and Z4 , which are known nonperturbatively. These functions are the twopoint proper vertex, the wave function renormalization and the four-point proper vertex of the quartic matrix model, respectively. The results of this calculation are shown in Table II. In our approximation, we have checked that there is always a nontrivial fixed point for any value of in the interval 0 < < 1. We can also compute the mass critical exponent and the anomalous dimension within this scheme. From the wave function renormalization [Eq. (1.9)], we compute immediately the anomalous dimension. We find ¼

ln ð4 ðg Þ=4g Þ : 2 2 ln

(1.20)

The computation of the mass critical exponent requires linearization of the renormalization group equations (1.10), (1.11), and (1.12). The linearized renormalization group equations are of the form [with G ¼ G G , where G ¼ ðG1 ¼ 2 ; G2 ¼ u; G3 ¼ r2 Þ] G0 ¼ MðG ; ÞG: The matrix M in our case is of the form

(1.21)

0

M11

B B @ M21 0

(1.18)

(1.19)

M12 M22 0

M13

1

C M23 C A:

(1.22)

M33

We find that 3 ¼ 2 e ðG Þ=ZðG ; 2 Þ > 1, and hence r2 is a relevant coupling constant like the mass. However, the function e ðgÞ used in this formula is only known perturbatively, and hence this conclusion should be taken with care. The two remaining eigenvalues are determined from the linearized renormalization group equations in the twodimensional space generated by G1 ¼ 2 and G2 ¼ u. As it turns out, this problem depends only on functions which are fully known nonperturbatively. The eigenvalues 1 ð Þ and 2 ð Þ can be determined from the trace and determinant of M in an obvious way. It is not difficult to convince ourselves that these renormalization group eigenvalues must scale as ð Þ ¼ ð1Þ y :

(1.23)

This formula [Eq. (1.23)] was used as a crucial test for our numerical calculations. In particular, we have determined by means of this formula the range of the dilatation parameter over which the logarithm of the eigenvalues scales linearly with ln . It is natural to expect this behavior to hold only if the renormalization group steps are

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sufficiently small so not to alter drastically the infrared physics of the problem. For D ¼ 2, the renormalization group steps can be thought of as small, and the behavior in Eq. (1.23) holds in the regime ln < 1. As it happens, this is the most important case corresponding to d ¼ 4. We find explicitly the following fits: ln j1 j ¼ 1:296 ln þ 0:412; ln j2 j ¼ 0:435 ln þ 2:438:

(1.24)

We conclude immediately that the scaling field u1 corresponding to the mass is relevant, while the scaling field u2 corresponding to the quartic coupling constant is irrelevant. This is the usual conclusion in d ¼ 4. The critical exponents in D ¼ 2 (d ¼ 4) are given, respectively, by y1 ¼ 1:296;

y2 ¼ 0:435:

(1.25)

The mass critical exponent is given by the inverse of the critical exponent y1 associated with the relevant direction, viz. ¼ 1=y1 . The corresponding results for y1 and are included in Table IV. We note that the results shown in Table IV are very close to the average value of 2=d and 2=D, viz. ¼

1 1 þ : d D

(1.26)

C. Outline This article is organized as follows: In Sec. II, we write down noncommutative 4 in the matrix basis and discuss some useful approximations involving the propagator at the self-dual point which are valid for large . In Sec. III, we consider the dimensional reduction of noncommutative 4 and some of its properties. In particular, we will derive a very simple nonperturbative equation for the two-point proper vertex, which will be used to test the results obtained later using the 1=N expansion and the recursion formula. In Sec. IV, we perform the tedious task of deriving the renormalization group equations which control the flow of the coupling constants of the model (three in this case) using the Wilson renormalization group recursion formula and 1=N expansion. The most difficult piece of the calculation, as we will see, is wave function renormalization. In Sec. V, we derive the nontrivial fixed point and the associated critical exponents and by solving numerically via the Newton-Raphson algorithm the renormalization group equations and discuss some of the relevant physics. In Sec. VI, we extend the analysis to the Grosse-VignesTourneret model, which involves an extra term, the double trace operator ðTrN MÞ2 , required for the renormalizability of the theory. We conclude in Sec. VII with a summary of the obtained results and a brief outlook. We have also included three appendixes for completeness and for the convenience of interested readers.

II. THE NONCOMMUTATIVE 4 THEORY A. The model Let us consider a 4 theory on a generic noncommutative Moyal-Weyl space Rd . We introduce noncommutativity in momentum space by introducing a minimal coupling to a constant background magnetic field Bij as was done originally by Langmann, Szabo, and Zarembo in Ref. [2]. The most general action with a quartic potential takes in the operator basis the form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ^ ^ þ D^ 2i S ¼ det ð2ÞTrH ~ C^ 2i þ 2 ^þ^ ^þ^ 0 ^ þ ^ þ ^ ^ þ þ : 4! 4!

(2.1)

In this equation, D^ i ¼ @^ i iBij Xj and C^ i ¼ @^ i þ iBij Xj , where Xi ¼ ðx^ i þ x^ Ri Þ=2. In the original Langmann-Szabo model, we choose ¼ 1, ~ ¼ 0, and 0 ¼ 0, which, as it turns out, leads to a trivial model [8]. The famous Grosse-Wulkenhaar model corresponds to ¼ ~ and 0 ¼ 0. We choose without any loss of generality ¼ ~ ¼ 1=4. The Grosse-Wulkenhaar model corresponds to the addition of a harmonic oscillator potential to the kinetic action which modifies and thus allows us to control the IR behavior of the theory. A particular version of this theory was shown to be renormalizable by Grosse and Wulkenhaar in Ref. [9]. The action of interest in terms of the star product is given by S¼

1 1 2 dd x þ @2i þ ðBij xj Þ2 þ 2 2 2 þ þ þ : 4!

Z

(2.2)

Equivalently,

1 2 1 2 2 2 @i þ x~i þ S¼ 2 2 2 þ þ þ : 4! Z

dd x

þ

(2.3)

The harmonic oscillator coupling constant is defined by 2 ¼ B2 2 =4, whereas the coordinate x~i is defined by x~i ¼ 2ð1 Þij xj . It was shown in Ref. [8] that this action is covariant under a duality transformation which exchanges, among other things, positions and momenta 2 as xi $ k~i ¼ B1 ij kj . The value ¼ 1 in particular gives an action which is invariant under this duality transformation. The theory at 2 ¼ 1 is essentially the original Langmann-Szabo model. Let us consider now a 4 theory on a noncommutative Moyal-Weyl space with only two noncommuting coordinates, viz. Rd ¼ RD R2 , where D ¼ d 2. This is the degenerate case. The above action generalizes to

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possibility of the existence of other fixed-point solutions even within this scheme. (6) The renormalization group eigenvalue 3 was determined to be larger than 1, and hence r2 is a relevant coupling similar to the mass. This statement was, however, derived from the available perturbative knowledge of the proper vertex e , and thus should be taken with care. (7) We have found that the renormalization group eigenvalues 1 (corresponding to the mass, relevant) and 2 (corresponding to the quartic coupling, irrelevant) become complex conjugates of each other at the value of the dilatation parameter ln ’ 1. After extracting the dependence on , we obtain the mass critical exponent

double trace term is renormalizable in the limit considered. A similar conclusion is obtained in Ref. [13]. This matrix model fixed point seems to be different from the Ising universality class fixed point.5 A more comprehensive scheme which seems to be more appropriate to describing the matrix transition is to consider instead of the GrosseVignes-Tourneret model the most general quartic action containing all multitrace operators consistent with the symmetry M ! M. Thus, the extra terms ðTrN M2 Þ2 , ðTrN MÞ4 , and ðTrN MÞ2 TrN M2 should also be included. For example, the term ðTrN M2 Þ2 was found to play a crucial role in the renormalizability of noncommutative 4 theory in the large- limit in Ref. [13]. We hope to return to this point as well as other points (see towards the end of the next section) in the future.

1 1 þ : (7.2) d D In D ¼ 2 (the most important case for us) it is found that both the eigenvalues ln 1 and ln 2 scale linearly with ln over the entire real range ln < 1. We obtain the indices ’

VII. CONCLUSION AND OUTLOOK In this article, we have presented a study of 4 theory on noncommutative spaces with only two noncommuting directions at the self-dual point using a combination of the Wilson approximate renormalization group recursion formula and the solution to the corresponding zerodimensional matrix model at large N. The most important results of this work are as follows: (1) The Penner matrix model TrN ðM2 =2 þ m2 EM2 þ gM4 =NÞ can be systematically solved by the multitrace approach of Ref. [15], as illustrated by the computation of the two-point proper vertex in Sec. III. This might be an alternative approach to the one pursued in Ref. [23]. (2) The action studied in this article is a generalization of the Penner matrix model of the form S½M ¼

1 1 ð@ MÞ2 þ 2 M2 2 2 u þ r2 EM2 þ M4 : N

Z

dD xTrN

(7.1)

(3) The renormalizations of the mass term, the harmonic oscillator term, and the quartic interaction are straightforward to obtain in this scheme. (4) The most difficult calculation of all is wave function renormalization. We have conjectured in this article that wave function renormalization within the scheme of the recursion formula is fully encoded in the approximation (4.23). (5) We have found a fixed-point solution of the renormalization group equations (4.44), (4.45), and (4.46) with r2 ¼ 0, 2 < 0 and u > 0 in all dimensions D ¼ 2, 3, 4 corresponding to the dimensions d ¼ 4, 5, 6, respectively. We do not exclude here the 5

The d-dimensional physics is described in terms of a critical behavior in D ¼ d 2 dimensions.

y1 ¼ 1:296ð ¼ 0:772Þ;

y2 ¼ 0:435:

(7.3)

(8) The anomalous dimension was found to scale as ’

4D ; 2

! 0:

(7.4)

(9) An extension of the above results to the GrosseVignes-Tourneret model, which involves an extra term given by the double trace operator ðTrN MÞ2 , was also given. We conclude this article by indicating that the most natural extension of this work should be to repeat the same analysis of the matrix model fixed point by replacing the recursion formula with the exact functional renormalization group method. The functional renormalization group (as opposed to other forms of the renormalization group) is the most direct implementation of the Wilson idea which goes along the lines of the recursion formula but is exact, although explicit calculation will undoubtedly involve various truncations which by their nature are also approximate. ACKNOWLEDGMENTS This research was supported by the National Agency for the Development of University Research (ANDRU) under PNR Contract No. U23/Av58 (8/u23/2723). APPENDIX A: DEGENERATE DUALITY TRANSFORMATIONS Let us start with the quadratic action. We have (with Di ¼ @i iBij xj and Ci ¼ @i þ iBij xj )

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The propagator 0mt satisfies the equation X ðp2 þ 2 þ r2 ð2m 1ÞÞ0mt þ r2 2 0lt ¼ mt :

APPENDIX C: THE PROPAGATOR We start with the Laplacian Gmn;kl ¼ ðp2 þ 2 þ r2 ðm þ n 1ÞÞml nk þ r2 2 mn kl : The propagator is defined by X X nm;lk Gkl;ts ¼ Gmn;kl lk;st ¼ ns mt : k;l

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l

(C9)

(C1) This can be solved by the ansatz (C2)

0mt ¼

k;l

Since the Laplacian Gmn;kl is nonzero only when m þ k ¼ n þ l, the propagator mn;kl is nonzero only when m þ k ¼ n þ l. We introduce n ¼ m þ and s ¼ t þ , then X Gmmþ;lþl llþ;tþt ¼ mt : (C3)

For ¼ , we obtain an ordinary matrix inversion for every value of , viz. X Gmmþ;lþl llþ;tþt ¼ mt : (C5)

r2 2 P 1 þ r2 2 l ½p2 þ 2 þ r2 ð2l 1Þ1 1 2 : 2 p þ þ r2 ð2t 1Þ

(C11)

In this paper, we are dealing with the limit ! 1, and hence we do not really need the above exact solution of 0mt . In fact, all we need is the leading correction in the new parameter r2 2 (recall that r2 ¼ 4=). From the above solution, we obtain in a straightforward way the result

l

We introduce Gml ¼ Gmmþ;lþl and lt ¼ llþ;tþt , writing this as X Gml lt ¼ mt : (C6)

(C10)

Xt ¼

We remark that for Þ , we have immediately (C4)

2

We can find immediately

l

llþ;tþt ¼ 0:

1 ½mt þ Xt : p þ þ r2 ð2m 1Þ 2

0mt ¼

l

1 p2 þ 2 þ r2 ð2m 1Þ r2 2 4 4 Þ : þ Oðr mt 2 p þ 2 þ r2 ð2t 1Þ (C12)

We have From Eqs. (C4), (C8), and (C12), we get the propagator

Gml ¼ ðp2 þ 2 þ r2 ð2m þ 1ÞÞml þ r2 2 0 : (C7)

mn;kl ¼

Thus, for ¼ and Þ 0, we have immediately ml

1 ml : ¼ 2 2 2 p þ þ r ð2m þ 1Þ

2

r2 2 p2 þ 2 þ r2 ðk þ l 1Þ 4 4 mn kl þ Oðr Þ :

(C8)

The only value which requires special attention is ¼ 0.

[1] K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974). [2] E. Langmann, R. J. Szabo, and K. Zarembo, J. High Energy Phys. 01 (2004) 017; Phys. Lett. B 569, 95 (2003). [3] G. Ferretti, Nucl. Phys. B487, 739 (1997). [4] G. Ferretti, Nucl. Phys. B450, 713 (1995). [5] S. Nishigaki, Phys. Lett. B 376, 73 (1996). [6] B. Ydri and A. Bouchareb, J. Math. Phys. (N.Y.) 53, 102301 (2012). [7] C. Bagnuls and C. Bervillier, Phys. Rep. 348, 91 (2001). [8] E. Langmann and R. J. Szabo, Phys. Lett. B 533, 168 (2002).

1 þ r2 ðm þ n 1Þ

þ ml nk p2

(C13)

[9] H. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005); J. High Energy Phys. 12 (2003) 019; Commun. Math. Phys. 254, 91 (2005). [10] H. Grosse and F. Vignes-Tourneret, J. Noncommut. Geom. 4, 555 (2010). [11] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 2002), ed. 4. [12] J. M. Gracia-Bondia and J. C. Varilly, J. Math. Phys. (N.Y.) 29, 869 (1988). [13] C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B664, 371 (2003).

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[14] M. Disertori, R. Gurau, J. Magnen, and V. Rivasseau, Phys. Lett. B 649, 95 (2007). [15] D. O’Connor and C. Saemann, J. High Energy Phys. 08 (2007) 066. [16] E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Commun. Math. Phys. 59, 35 (1978). [17] Y. Shimamune, Phys. Lett. 108B, 407 (1982). [18] X. Martin, J. High Energy Phys. 04 (2004) 077; F. G. Flores, X. Martin, and D. O’Connor, Int. J. Mod. Phys. A 24, 3917 (2009); M. Panero, J. High Energy Phys. 05 (2007) 082. [19] S. S. Gubser and S. L. Sondhi, Nucl. Phys. B605, 395 (2001). [20] W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 05 (2004) 047.

[21] J. Ambjorn and S. Catterall, Phys. Lett. B 549, 253 (2002); W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 06 (2004) 042; J. Medina, W. Bietenholz, and D. O’Connor, J. High Energy Phys. 04 (2008) 041; C. R. Das, S. Digal, and T. R. Govindarajan, Mod. Phys. Lett. A 23, 1781 (2008); F. Lizzi and B. Spisso, Int. J. Mod. Phys. A 27, 1250137 (2012). [22] A. P. Polychronakos, Phys. Rev. D 88, 065010 (2013). [23] H. Grosse and R. Wulkenhaar, arXiv:1205.0465. [24] G. R. Golner, Phys. Rev. B 8, 339 (1973). [25] P. Kopietz, L. Bartosch, and F. Schutz, Lect. Notes Phys. 798, 1 (2010).

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The fate of the Wilson-Fisher fixed point in non-commutative φ 4 Badis Ydria) and Adel Boucharebb) Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria (Received 12 July 2012; accepted 6 September 2012; published online 4 October 2012)

In this article we study non-commutative vector sigma model with the most general φ 4 interaction on Moyal-Weyl spaces. We compute the 2- and 4-point functions to all orders in the large N limit and then apply the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. The non-commutative Wilson-Fisher fixed point interpolates between the commutative Wilson-Fisher fixed point of the Ising universality class which is found to lie at zero value of the critical coupling constant a* of the zero dimensional reduction of the theory, and a novel strongly interacting fixed point which lies at infinite value of a* corresponding to maximal non-commutativity beyond which the two-sheeted C 2012 American structure of a* as a function of the dilation parameter disappears. Institute of Physics. [http://dx.doi.org/10.1063/1.4754816]

I. INTRODUCTION

A non-commutative scalar field theory is a non-local field theory in which the ordinary local point-wise multiplication of fields is replaced by a non-local star product such as the Moyal-Weyl star product.1, 2 We suggest3 and references therein for an elementary and illuminating discussion of the Moyal-Weyl product and other star products and their relation to the Weyl map,4 coherent states,5–7 Berezin quantization,8 and deformation quantization.9 The first study of the quantum theory of a non-commutative φ 4 is found in Ref. 10, where it is shown that planar diagrams in the non-commutative theory are essentially identical to the planar diagrams in the commutative theory. More interestingly, it was found in Ref. 11 that the renormalized one-loop action of a non-commutative φ 4 suffers from an infrared divergence which is obtained when we send either the external momentum or the non-commutativity to zero. This non-analyticity at small momenta or small non-commutativity (IR) which is due to the high energy modes (UV) in virtual loops is termed the UV-IR mixing and it is intimately related to the structure of the non-planar diagrams of the theory. As it turns out this effect is expected in a non-local theory such as a non-commutative φ 4 . Renormalization of the non-commutative φ 4 was studied, for example, in Refs. 12–18. The main observation of Ref. 14 is that we can control the UV-IR mixing found in non-commutative φ 4 by modifying the large distance behavior of the free propagator through adding a harmonic oscillator potential to the kinetic term. More precisely, the UV-IR mixing of the theory is implemented precisely in terms of a certain duality symmetry of the new action which connects momenta and positions.19 The corresponding Wilson-Polchinski renormalization group equation20, 21 of the theory can then be solved in terms of ribbon graphs drawn on Riemann surfaces. The existence of a regular solution of the Wilson-Polchinski equation together with the fact that we can scale to zero the coefficient of the harmonic oscillator potential in two dimensions leads to the conclusion that the standard non-commutative φ 4 in two dimensions is renormalizable.16 In four dimensions, the harmonic oscillator term seems to be essential for the renormalizability of the theory.15 a) E-mail addresses: [email protected] and [email protected] b) E-mail: [email protected]

0022-2488/2012/53(10)/102301/14/$30.00

53, 102301-1

C 2012 American Institute of Physics

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There are many other approaches to renormalization of quantum non-commutative φ 4 . See, for example, Refs. 50–54 and 58. A very remarkable property of quantum non-commutative φ 4 is the appearance of a new order in the theory termed the striped phase which was first computed in a one-loop self-consistent HartreeFock approximation in the seminal paper.22 For alternative derivations of this order see, for example, Refs. 48 and 49. It is believed that the perturbative UV-IR mixing is only a manifestation of this more profound property. As it turns out, this order should be called more appropriately a non-uniform ordered phase in contrast with the usual uniform ordered phase of the Ising universality class and it is related to spontaneous breaking of translational invariance. It was numerically observed in d = 4 in Ref. 23 and in d = 3 in Ref. 24 where the Moyal-Weyl space was approximated by a non-commutative fuzzy torus.25 The beautiful result of Ref. 24 shows explicitly that the minimum of the model shifts to a non-zero value of the momentum indicating a non-trivial condensation and hence spontaneous breaking of translational invariance. In summary the phase diagram of quantum non-commutative φ 4 consists of three phases. The usual disordered and uniform ordered phases together with the non-uniform (striped) ordered phase and thus a triple point must exist. In Refs. 22 and 48, it is conjectured that this point is a Lifshitz point which is a multi-critical point at which a disordered, a homogeneous (uniform) ordered and a spatially modulated (non-uniform) ordered phases meet.26 We note that Ref. 48 uses Wilson renormalization group approach44 to derive the Wilson-Fisher fixed point of the theory at one-loop which is found to suffer from an instability at large non-commutativity. The non-commutative fuzzy torus used to regularized non-commutative φ 4 is a matrix model which can be mapped to a lattice. Another matrix regularization of non-commutative φ 4 can be found in Refs. 27 and 28 which emphasizes connection to fuzzy spaces.29, 30 The phase structure of non-commutative φ 4 in d = 2 and d = 3 using the fuzzy sphere31, 32 regularization was studied extensively in Refs. 33–37. Again the phase diagram consists of three phases: a disordered phase, a uniform ordered phases, and a non-uniform ordered phase which meet at a triple point. In this case it is well established that the transitions from the disordered phase to the non-uniform ordered phase and from the non-uniform ordered phase to the uniform ordered phase originate from the one-cut/two-cut transition in the quartic hermitian matrix model.40, 41 See also Refs. 42 and 43. This was also confirmed analytically by the multi-trace approach of Refs. 38 and 39 which relies on the expansion of the kinetic term in the action instead of the usual expansion in the interaction which is very reminiscent to the hopping parameter expansion on the lattice.55, 56 The non-uniform ordered phase on the fuzzy sphere (sometimes also called the matrix phase) goes to the striped phase on the Moyal-Weyl plane in appropriate flattening limit. Finally, there is a strong evidence that the non-uniform ordered phase should be present on all non-commutative spaces regardless of the dimension. In this article we will attempt to understand the phase structure of non-commutative φ 4 near the Wilson-Fisher fixed point, i.e., the transition disordered/uniform ordered and how this critical behavior changes with the non-commutativity until it merges with the transition disordered/nonuniform ordered. We will consider a large vector O(N) sigma model where all leading Feynman diagrams can be taken into consideration and employ the approximate Wilson renormalization group equation to study the renormalized action of the theory. This article is organized as follows. In Sec. II we will introduce the non-commutative vector sigma model with the most general φ 4 interaction on the Moyal-Weyl spaces Rdθ , and then we will compute in the large N limit the 2- and 4-point functions to all orders. In Sec. III we will apply the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. In particular we will derive the renormalization group equations and then calculate the fixed points of the theory. In Sec. IV we give our conclusion and outlook. II. THE NON-COMMUTATIVE O(N) SIGMA MODEL

The cumulant expansion: We will consider in this note the φ 4 action S = d d xa (−∂i2 + μ2 )a + Sint ,

(1)

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IV. CONCLUSION AND OUTLOOK

In this article we have studied the non-commutative vector sigma model with the most general φ 4 interaction on the Moyal-Weyl spaces Rdθ in the large N limit. We computed the 2- and 4-point functions to all orders and then applied the approximate Wilson renormalization group recursion formula to study the renormalized coupling constants of the theory. More precisely, we have taken into account all leading Feynman diagrams in the large N limit, and then estimated them using the so-called Wilson contraction which consists of three rules. We have argued that the first rule of Wilson contraction is certainly valid in the limit θ¯ −→ 0. In the commutative theory we observe that the non-perturbative Wilson-Fisher fixed point is located on the second sheet of the parameter a which is the coupling constant of the zero dimensional reduction of the theory. The Wilson-Fisher fixed point can be reached in the limit in which we send the dilation parameter ρ to zero which corresponds to a single step of the renormalization group transformation. The Wilson-Fisher fixed point is also found to scale to zero in the limit ρ −→ 0 although differently then the perturbative fixed point in the limit ρ −→ 1. In the non-commutative theory the Wilson-Fisher fixed point does not scale to zero in the limit ρ −→ 0 in contrast with the perturbative fixed point which still scales to zero as ρ −→ 1. The main obstacle comes from the fact that the critical coupling constant a* which starts from 0 at ρ = 1, increases to ∞ at ρ = 1 − t/2, does not return to zero as we decrease ρ back from ρ = 1 − t/2 to 0. The non-perturbative window (sheet) shrinks as we increase the non-commutativity until it disappears at t = 2. At this point, the critical value a* diverges and the non-commutative WilsonFisher point becomes very different. It is natural to conjecture that the point at t = 2 is completely different (lies in a different universality class) then the commutative Wilson-Fisher fixed point at t = 1 which is in the Ising universality class. Therefore, the non-commutative Wilson-Fisher fixed point interpolates between these two classes. A thorough study of the renormalized action (including the coupling constant v) and the noncommutative Wilson-Fisher fixed point (including the range t > 2), together with the computation of the critical exponents, and improvement of the approximate Wilson renormalization group recursion formula will be reported elsewhere.57 For example, already we observe from our study here that we can compute the 2-point function without any resort to the first rule of Wilson contraction resulting in Eq. (40) which indicates the presence of a wave function renormalization in the theory. As already discussed in Ref. 48 this leads to an instability of the theory. Solving the renormalization group equations in the case of a non-zero wave function may/will require a numerical approach. Another approach to the Wilson renormalization group recursion formula of non-commutative φ 4 will be in terms of the matrix model associated with the theory which will allow us to access the limit θ¯ −→ ∞ instead. We also hope to return to this point in future communication. 57 ACKNOWLEDGMENTS

This research was supported by “The National Agency for the Development of University Research (ANDRU)” under PNR Contract No. U23/Av58 (8/u23/2723). 1 H.

J. Groenewold, Physica 12, 405 (1946). E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949). 3 G. Alexanian, A. Pinzul, and A. Stern, Nucl. Phys. B 600, 531 (2001); e-print arXiv:hep-th/0010187. 4 H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). 5 V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Scr. 55, 528 (1997); e-print arXiv:quant-ph/9612006. 6 A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986). 7 J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985). 8 F. A. Berezin, Commun. Math. Phys. 40, 153 (1975). 9 M. Kontsevich, Lett. Math. Phys. 66, 157 (2003); e-print arXiv:q-alg/9709040 [q-alg]. 10 T. Filk, Phys. Lett. B 376, 53 (1996). 11 S. Minwalla, M. Van Raamsdonk, and N. Seiberg, J. High Energy Phys. 0002, 020 (2000); e-print arXiv:hep-th/9912072. 12 I. Chepelev and R. Roiban, J. High Energy Phys. 0005, 037 (2000); e-print arXiv:hep-th/9911098. 13 I. Chepelev and R. Roiban, J. High Energy Phys. 0103, 001 (2001); e-print arXiv:hep-th/0008090. 2 J.

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J. Math. Phys. 53, 102301 (2012)

14 H.

Grosse and R. Wulkenhaar, Commun. Math. Phys. 254, 91 (2005); e-print arXiv:hep-th/0305066. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005); e-print arXiv:hep-th/0401128. 16 H. Grosse and R. Wulkenhaar, J. High Energy Phys. 0312, 019 (2003); e-print arXiv:hep-th/0307017. 17 V. Rivasseau, F. Vignes-Tourneret, and R. Wulkenhaar, Commun. Math. Phys. 262, 565 (2006); e-print arXiv:hep-th/0501036. 18 R. Gurau, J. Magnen, V. Rivasseau, and F. Vignes-Tourneret, Commun. Math. Phys. 267, 515 (2006); e-print arXiv:hep-th/0512271. 19 E. Langmann and R. J. Szabo, Phys. Lett. B 533, 168 (2002); e-print arXiv:hep-th/0202039. 20 J. Polchinski, Nucl. Phys. B 231, 269 (1984). 21 G. Keller, C. Kopper, and M. Salmhofer, Helv. Phys. Acta 65, 32 (1992). 22 S. S. Gubser and S. L. Sondhi, Nucl. Phys. B 605, 395 (2001); e-print arXiv:hep-th/0006119. 23 J. Ambjorn and S. Catterall, Phys. Lett. B 549, 253 (2002); e-print arXiv:hep-lat/0209106. 24 W. Bietenholz, F. Hofheinz, and J. Nishimura, J. High Energy Phys. 0406, 042 (2004); e-print arXiv:hep-th/0404020. 25 J. Ambjorn, Y. M. Makeenko, J. Nishimura, and R. J. Szabo, J. High Energy Phys. 0005, 023 (2000); e-print arXiv:hep-th/0004147. 26 R. M. Hornreich, M. Luban, and S. Shtrikman, Phys. Rev. Lett. 35, 1678 (1975). 27 E. Langmann, R. J. Szabo, and K. Zarembo, J. High Energy Phys. 0401, 017 (2004); e-print arXiv:hep-th/0308043. 28 H. Steinacker, J. High Energy Phys. 0503, 075 (2005); e-print arXiv:hep-th/0501174. 29 A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya (Singapore World Scientific, Singapore, 2007) p. 191; e-print arXiv:hep-th/0511114. 30 D. O’Connor, Mod. Phys. Lett. A 18, 2423 (2003). 31 J. Hoppe, Ph.D dissertation, MIT, 1982. 32 J. Madore, Class. Quantum Grav. 9, 69 (1992). 33 X. Martin, J. High Energy Phys. 0404, 077 (2004); e-print arXiv:hep-th/0402230. 34 F. Garcia Flores, X. Martin, and D. O’Connor, Int. J. Mod. Phys. A 24, 3917 (2009); e-print arXiv:0903.1986 [hep-lat]. 35 M. Panero, J. High Energy Phys. 0705, 082 (2007); e-print arXiv:hep-th/0608202. 36 J. Medina, W. Bietenholz, and D. O’Connor, J. High Energy Phys. 0804, 041 (2008); e-print arXiv:0712.3366 [hep-th]. 37 C. R. Das, S. Digal, and T. R. Govindarajan, Mod. Phys. Lett. A 23, 1781 (2008); e-print arXiv:0706.0695 [hep-th]. 38 D. O’Connor and C. Saemann, J. High Energy Phys. 0708, 066 (2007); e-print arXiv:0706.2493 [hep-th]. 39 C. Saemann, SIGMA 6, 050 (2010); e-print arXiv:1003.4683 [hep-th]. 40 E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Commun. Math. Phys. 59, 35 (1978). 41 Y. Shimamune, Phys. Lett. B 108, 407 (1982). 42 P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, Phys. Rep. 254, 1 (1995); e-print arXiv:hep-th/9306153. 43 B. Eynard, Cours de Physique Theorique de Saclay (unpublished). 44 K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974). 45 S. Hikami and E. Brezin, J. Phys. A 12, 759 (1979). 46 G. Ferretti, Nucl. Phys. B 450, 713 (1995); e-print arXiv:hep-th/9504013. 47 S. Nishigaki, Phys. Lett. B 376, 73 (1996); e-print arXiv:hep-th/9601043. 48 G.-H. Chen and Y.-S. Wu, Nucl. Phys. B 622, 189 (2002); e-print arXiv:hep-th/0110134. 49 P. Castorina and D. Zappala, Phys. Rev. D 68, 065008 (2003); e-print arXiv:hep-th/0303030. 50 C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B 633, 250 (2002); e-print arXiv:hep-th/0202155. 51 C. Becchi, S. Giusto, and C. Imbimbo, Nucl. Phys. B 664, 371 (2003); e-print arXiv:hep-th/0304159. 52 R. Gurau and O. J. Rosten, J. High Energy Phys. 0907, 064 (2009); e-print arXiv:0902.4888 [hep-th]. 53 L. Griguolo and M. Pietroni, J. High Energy Phys. 0105, 032 (2001); e-print arXiv:hep-th/0104217. 54 R. Gurau, J. Magnen, V. Rivasseau, and A. Tanasa, Commun. Math. Phys. 287, 275 (2009); e-print arXiv:0802.0791 [math-ph]. 55 I. Montvay and G. Munster, Quantum Fields on a Lattice, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1994), p. 491. 56 J. Smit, Cambridge Lect. Notes Phys. 15, 1 (2002). 57 Badis Ydri, Rachid Ahmim, and Adel Bouchareb, work in progress. 58 A. Sfondrini, T. A. Koslowski, Int. J. Mod. Phys. A 26, 4009 (2011); e-print arXiv:1006.5145. 15 H.

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International Journal of Modern Physics A Vol. 28, No. 28 (2013) 1350139 (15 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217751X1350139X

EFFECTS OF NONCOMMUTATIVITY ON LIGHT HYDROGEN-LIKE ATOMS AND PROTON RADIUS

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M. MOUMNI Department of Matter Sciences, University of Biskra, Algeria [email protected] A. BENSLAMA Department of Physics, University Constantine 1, Algeria [email protected] Received 21 August 2013 Revised 3 October 2013 Accepted 3 October 2013 Published 31 October 2013 We study the corrections induced by the theory of noncommutativity, in both space– space and space–time versions, on the spectrum of hydrogen-like atoms. For this, we use the relativistic theory of two-particle systems to take into account the effects of the reduced mass, and we use perturbation methods to study the effects of noncommutativity. We apply our study to the muon hydrogen with the aim to solve the puzzle of proton radius [R. Pohl et al., Nature 466, 213 (2010) and A. Antognini et al., Science 339, 417 (2013)]. The shifts in the spectrum are found more noticeable in muon H (µH) than in electron H (eH) because the corrections depend on the mass to the third power. This explains the discrepancy between µH and eH results. In space–space noncommutativity, the parameter required to resolve the puzzle θss ≈ (0.35 GeV)−2 , exceeds the limit obtained for this parameter from various studies on eH Lamb shift. For space– time noncommutativity, the value θst ≈ (14.3 GeV)−2 has been obtained and it is in agreement with the limit determined by Lamb shift spectroscopy in eH. We have also found that this value fills the gap between theory and experiment in the case of µD and improves the agreement between theoretical and experimental values in the case of hydrogen–deuterium isotope shift. Keywords: Noncommutativity; hydrogen-like atoms; proton radius. PACS numbers: 02.40.Gh, 67.63.Gh, 31.30.jr

1. Introduction Historically, experimental spectroscopy was the perfect test for any theory having any connection with matter and it played the leading role in calibrating the values of physical constants. But lately, it has reached such precision that it accessed the role of indicator of new theories; and last experience on the Lamb shift in muonic hydrogen1 is the perfect example. 1350139-1

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The discrepancy between these experimental results rµH = 0.84169(66) fm and those extracted from electronic hydrogen or elastic electron–proton scattering and recorded in CODATA reH = 0.8775(51) fm (Ref. 2) (the values are at 7σ variance with respect to each other) has had an impact in the whole scientific community and this raised many questions about the cause of such disagreement. The experimental methods used to obtain the two results are very elaborated. This is why many studies have investigated how to explain this difference and to reconcile the two results by trying to rectify the theory. But the difference between the two experiments remains a puzzle until now and especially after being reinforced recently with a more accurate value rµH = 0.84087(39).3 Nonperturbative numerical computations of the Dirac equation confirmed the validity of perturbation methods used to compute the radius.4,5 No significant QED correction has been found yet, which would explain the discrepancy.6,7 Using electron scattering experiments,8 found that data rules out values of the third Zemach moment large enough to explain the puzzle. Three-body physics does not solve the problem as demonstrated in Ref. 9. Constraints from low energy data disfavor new spin-0, spin-1 and spin-2 particles as an explanation.10 There are some claims that proton polarizability contribution in the Lamb shift may explain the discrepancy because it is proportional to the lepton mass to the fourth power.11,12 These effects could be probed in scattering experiment planned to run at Paul Scherrer Institute (PSI). For more information about the different approaches to the problem, see Refs. 6, 13–15. Because Pohl et al. used an indirect method to calculate rµH that involves comparing the frequency measured experimentally with that given theoretically according to the radius:1 rp2 rp3 ∆(2P1/2 → 2S1/2 ) = 206.0573(45) − 5.2262 2 + 0.0347 3 , meV fm fm

(1a)

rp2 rp3 ∆(2P3/2 → 2S1/2 ) = 209.9779(49) − 5.2262 2 + 0.0347 3 , (1b) meV fm fm we propose, in this paper, to modify the precedent theoretical expressions of the transition frequency by incorporating the corrections induced by the noncommutative structure of space–time. The idea of taking noncommutative space–time coordinates dates from the 1930s. It had as objective to avoid infinities in Coulomb potentials (gravitation and electricity) by introducing a lower bound for the measurement of length. Despite the fact that the concept was suffering from some problems with unitarity and causality, the theory evolved from the mathematical point of view, especially after the work of Connes in the eighties of last century.16 In 1999, the work of Seiberg and Witten on string theory17 has aroused new interest in the theory. They showed that the dynamics of the endpoints of an open string on a D-brane in the presence of a magnetic background field is described by a theory of Yang–Mills on a noncommutative space–time. 1350139-2

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Effects of Noncommutativity on Light Hydrogen-like Atoms and Proton Radius

Today, we find noncommutativity in various fields of physics such as solid state physics, where it was shown that is the framework in which Hall conductivity is quantized18 and that it is the proper tool replacing Bloch’s theory whenever translation invariance is broken in aperiodic solids.19 In fluid mechanics, noncommutative fluids are introduced by studying the quantum Hall effect20 or bosonization of collective fermion states.21 There is also some connection with quantum statistical physics,22 and it is also an interpretation of Ising-type models.23 One can even find a manifestation of the noncommutativity in the physiology of the brain, where noncommutative computation in the vestibulo-ocular reflex was demonstrated in a way that is unattainable by any commutative system24,25 is an excellent reference for the different manifestations and applications of noncommutative field theory. The theory is a distortion of space–time where the coordinates xµ become Hermitian operators and thus do not commute: µ ν xnc , xnc = iθµν , µ, ν = 0, 1, 2, 3 . (2) The nc indices denote noncommutative coordinates. θµν is the parameter of the deformation and it is an antisymmetric real matrix. We distinguish two types of noncommutativity; the first one is the space–space case, where the deformation is introduced between the spatial coordinates only, and the second is when the spatial coordinates commute with one another but not with time coordinate and it is noted space–time case. (For a review, one can see Ref. 26.) In the literature, there are a lot of studies on hydrogen atom in noncommutativity. For space–space noncommutativity, we cite Refs. 27–30. For the space–time case, one can see Refs. 31–33. We have found in Refs. 31 and 32, that the corrections induced by noncommutativity on the spectrum of the hydrogen atom are proportional to the lepton mass to the third power (the result is confirmed by Ref. 33), and this is exactly the shape of the corrections induced by the nuclear size as demonstrated in Refs. 34 and 35. We will apply our result to the muonic hydrogen, and we will incorporate therein the effects of the finite mass of the nucleus. We start by computing the corrections to the energies in both space–space and space–time cases of noncommutativity using perturbation methods in the Dirac theory of two particles systems. Then, we compare the difference between theoretical and experimental results obtained in µH experience. This allows us to obtain values of the noncommutative parameter that resolve the puzzle. Then we will discuss the possible effects of these corrections on the Lamb shift of muonic deuterium µD and on the difference between the radii of the proton and deuteron via the 2S–1S transition. 2. Coulomb Potential in Noncommutative Space Time We start by rewriting (2) for the two versions to consider of the noncommutativity: j 0 xst , xst = iθj0 , (3a) j k xss , xss = iθjk , (3b) 1350139-3

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We will evaluate the contribution coming from space–time noncommutativity to this shift. We use the relations from (15) to (20), to compute the noncommutative corrections to the transition in both hydrogen and deuterium: (st)

(st)

(38a)

(st)

(st)

(38b)

1S−2S ∆E2S (H) − ∆E1S (H) = hfH = 9.06690 × 1010 (θst eV2 ) eV , 1S−2S ∆E2S (D) − ∆E1S (D) = hfD = 9.07060 × 1010 (θst eV2 ) eV .

Inserting the value of the parameter θst found for µH (31a) in these expressions, we find the noncommutative correction to the hydrogen–deuterium isotope shift: (nc)

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1S−2S 1S−2S ∆fth (H − D) = fH − fD = 275.488 Hz .

(39)

This contribution does not fill the gap of 12 kHz between ∆fth and ∆fex , but it improves slightly the agreement between the two values. This confirms the fact that there are no doubts about the results of experiments on eH spectroscopy because they are so accurate and one has to look the side of muonic systems. 5. Conclusion In this paper, we studied the corrections induced by a noncommutative structure of space–time, in its two versions space–space and space–time, on the spectrum of hydrogen-like atoms. We have applied our study to the muonic hydrogen and this with the aim to solve the puzzle of proton radius, because we think that the experiments used to study this phenomenon are so developed that we cannot doubt of their results, and therefore one must look on the side of theory of atomic spectroscopy used to compute the radius. In this study, we considered the effects of the mass of the nucleus of the noncommutative corrections and thus we have improved previous works in this area. This allowed us to consider the difference caused by changing the nucleus (from proton to deuteron) in addition to that which occurs when changing the orbiting particle (from electron to muon). It should be noted that the effects of the nucleus shape on the energy levels of the atom are proportional to the third power of the mass of the orbiting particle; this is easily understood by the fact that the Bohr radius (a0 = ~2 /me2 ) is inversely proportional to the mass and thus the particle is that much nearer the nucleus, that its mass is greater; and this makes it more sensitive to these effects. We have demonstrated that the effects of noncommutativity are also proportional to the third power of the mass of the particle because it distorts the Coulomb potential and adds a term proportional to r−2 in space–time case and to r−3 in space–space case. It is for this reason also that the effects decreases with increasing values of quantum numbers as can be seen in the different relations of the spectrum corrections (because the term r−n with n > 1 is very steep for small values of r). This is why we use this theory to explain the puzzle because its effects are different depending on whether it is applied to muonic hydrogen or electronic hydrogen. The shifts in the spectrum are more noticeable in muon H than in ordinary ones, and 1350139-12

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Effects of Noncommutativity on Light Hydrogen-like Atoms and Proton Radius

this explains the fact that experiments on µH spectroscopy give results that are different from those obtained with eH. In the case of space–space noncommutativity, the parameter required to resolve the puzzle is θss ≈ (0.35 GeV)−2 . This value exceeds the limit obtained for this parameter from studies on eH Lamb shift θss ≤ (0.6 GeV)−2 .33 If we use this limit to compute the radius, we find 0.86409 fm which is outside the experimental limits for rµH .1,3 Another problem arises with in this case; it is the ± sign in the corrections (due to the presence of the azimuthal quantum number in their expressions). This sign means that the corrected value of the radius ranges from 0.84169 fm to 0.91192 fm in violation of µH experiences. In the case of the space–time noncommutativity, the value θst ≈ (14.3 GeV)−2 has been obtained and it is in agreement with the limit determined by Lamb shift spectroscopy in eH θst ≤ (6 GeV)−2 .33 It was also found in this case, that the corrections are the same for both levels 2P1/2 and 2P3/2 although we found that corrections remove the degeneracy of the Dirac energies with respect to the total angular momentum quantum number j = l +1/2 = (l +1)−1/2 (noncommutativity acts like the Lamb shift here). This is in agreement with the two relations (27a) and (27b), where the terms in r are equal for both transitions. This is not true for the space–space case because the corrections of the two levels differ from one another (29a) and (29b). We say that this is a consequence of the fact that the correction term to the Coulomb potential is proportional to r−2 in the space–time case, and so as we have previously mentioned in Ref. 31, we assimilate it to the field of a central dipole. In other words, the action of space–time noncommutativity is equivalent to consider the extended charged nature of the proton in the nucleus, which is the principal characteristic studied in µH experiments. When applying the result obtained from the study of µH in µD, we found a correction of 0.348 meV which is almost exactly equal to the difference between theory and experiment for this system. The very close values of the corrections in these two systems µH and µD are easily explained by the fact that the ratio between the reduced masses, of the two is 0.95 ≈ 1. (The disagreements between theory and experiment in both systems are almost equal.) Using the same result coming from µH for the eH–eD isotope shift, the correction found improves the agreement between theoretical and experimental results (which was already excellent). The same reasoning as above is used, and we say that the corrections in electronic systems either eH or eD are infinitely small compared to those of muonic systems; the ratio between the reduced masses of the two is ≈ 10−7 . Eyes are now turned to the results of experimental on µp scattering and µHe spectroscopy to see whether the phenomenon is spectroscopic or is it due to the nature of the particles. On the side of electronic systems, there is practically no doubt on their veracity; the radius of the proton was even calculated in a model independent way from ep scattering51,52 and the results confirm CODATA value. 1350139-13

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It was reported in our work,31 that the limit on the parameter θst ≈ (1 TeV)−2 , and so our result here is greater than the latter; however it was pointed out to us that such a limit (θst ≈ (1 TeV)−2 ) should not only be estimated according to experimental precision calculations but rather on the disagreement between experiment and theory, which requires a correction of the limit of Ref. 31 and this is what is done in this work for the muon hydrogen. We can also evoke that the proton raises other questions about its properties in addition to the one discussed in this paper and we can mention as an example the origin of its spin or what is called “spin crisis” in Ref. 53 or “proton spin puzzle” in Ref. 54.

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Effects of Noncommutativity on Light Hydrogen-like Atoms and Proton Radius

27. M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001). 28. T. C. Adorno, M. C. Baldiotti, M. Chaichian, D. M. Gitman and A. Tureanu, Phys. Lett. B 682, 235 (2009). 29. W. O. Santos and A. M. C. Souza, Int. J. Theor. Phys. 51, 3882 (2011). 30. L. Khodja and S. Zaim, Int. J. Mod. Phys. A 27, 1250100 (2012). 31. M. Moumni, A. BenSlama and S. Zaim, Afr. Rev. Phys. 07, 83 (2012). 32. M. Moumni, A. BenSlama and S. Zaim, Relativistic spectrum of hydrogen atom in the space–time non-commutativity, in Proc. 8th Int. Conf. on Progress in Theoretical Physics, AIP Conf. Proc., Vol. 1444, eds. Mebarki et al. (2012), pp. 253–257. 33. A. Stern, Phys. Rev. D 78, 065006 (2008). 34. J. L. Friar, Ann. Phys. 122, 151 (1979). 35. M. I. Eides, H. Grotcha and V. A. Shelyuto, Phys. Rep. 342, 63 (2001). 36. S. Dulat and K. Li, Eur. Phys. J. C 54, 333 (2008). 37. S. Fabi, B. Harms and A. Stern, Phys. Rev. D 78, 065037 (2008). 38. M. Chaichian, A. Tureanu, M. R. Setare and G. Zet, J. High Energy Phys. 0804, 064 (2008). 39. S. K. Suslov and B. Trey, J. Math. Phys. 49, 012104 (2008). 40. J. Sapirstein, Quantum electrodynamics, in Atomic, Molecular, and Optical Physics Handbook, ed. G. W. F. Drake (AIP Press, New York, 1996), pp. 327–340. 41. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic Press, New York, 1957). 42. W. Greiner, Relativistic Quantum Mechanics Wave Equations, 3rd edn. (Springer, Berlin, 2000). 43. E. Borie, Ann. Phys. 327, 733 (2012). 44. M. Gorshteyn, Nuclear and hadronic contributions to Lamb shift in muonic deuterium, Talk given at Workshop to Explore Physics Opportunities with Intense, Polarized Electron Beams up to 300 MeV, MIT, Cambridge, Massachusetts, March 14–16, 2013. 45. K. Pachucki, Phys. Rev. Lett. 106, 193007 (2011). 46. A. A. Krutov and A. P. Martynenko, Phys. Rev. A 84, 052514 (2011). 47. R. Pohl, Muonic news, talk given at ECT Workshop on the “Proton Radius Puzzle”, Trento, Italy, Oct 29–Nov 2 2012. 48. C. G. Parthey et al., Phys. Rev. Lett. 107, 203001 (2011). 49. C. G. Parthey et al., Phys. Rev. Lett. 104, 233001 (2010). 50. U. D. Jentschura et al., Phys. Rev. A 83, 042505 (2011). 51. R. J. Hill and G. Paz, Phys. Rev. D 82, 113005 (2010). 52. I. Sick, Prog. Partic. Nucl. Phys. 67, 473 (2012). 53. A. W. Thomas, A. Casey and H. H. Matevosyan, Int. J. Mod. Phys. A 25, 4149 (2010). 54. C. A. Aidala, S. D. Bass, D. Hasch and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013).

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Modern Physics Letters A Vol. 28, No. 36 (2013) 1350166 (9 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217732313501666

ON THE PROBLEM OF VACUUM ENERGY IN FLRW UNIVERSES AND DARK ENERGY

Mod. Phys. Lett. A Downloaded from www.worldscientific.com by WSPC on 11/11/13. For personal use only.

BADIS YDRI∗ and ADEL BOUCHAREB† Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria ∗[email protected] ∗[email protected] †[email protected]

Received 3 October 2013 Accepted 9 October 2013 Published 29 October 2013 We present a (hopefully) novel calculation of the vacuum energy in expanding FLRW spacetimes based on the renormalization of quantum field theory in nonzero backgrounds. We compute the renormalized effective action up to the two-point function and then apply the formalism to the cosmological backgrounds of interest. As an example we calculate for quasi de Sitter spacetimes the leading correction to the vacuum energy given by the tadpole diagram and show that it behaves as ∼ H02 ΛP where H0 is the Hubble constant and ΛP is the Planck constant. This is of the same order of magnitude as the observed dark energy density in the universe. Keywords: Cosmological constant; vacuum energy; FLRW metrics; QFT on curved backgrounds; renormalization; dark energy.

1. Introduction In recent years it has been established, to a very reasonable level of confidence, that the universe is spatially flat and is composed of 4% ordinary matter, 23% dark matter and 73% dark energy. This state of affair is summarized in the cosmological concordance ΛCDM model.1 The dominant component, dark energy, is believed to be the same thing as the cosmological constant introduced by Einstein in 1917 which in turn is believed to originate in the energy of the vacuum.2 We note that dark energy is characterized mainly by a negative pressure and no dependence on the 2 scale factor and √ its density behaves as 3∼ H0 ΛP where H0 is the Hubble parametera and ΛP = 1/ 8πG is the Planck mass. More precisely we have (with ΩΛ ≃ 0.73) ρΛ = 3ΩΛ H02 Λ2P ≃ 39ΩΛ (10−12 GeV)4 .

a The

(1.1)

vacuum energy density is a constant which can be expressed as something which is proportional to the square of the Hubble parameter at the current epoch (Hubble constant). We are not saying that the vacuum energy density falls with the square of the Hubble parameter. 1350166-1

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The reality of the energy of the vacuum is exhibited, as is well known, in a dramatic way in the Casimir force. See, for example, Ref. 4 for a systematic presentation of this subject. In this paper we will adopt the usual line of thought outlined in Refs. 2 and 3 and identify dark energy with vacuum energy. The calculation of vacuum energy in curved spacetimes such as FLRW universes and de Sitter spacetime requires the use of quantum field theory in the presence of a non zero gravitational background.5,6 de Sitter spacetime is of particular interest since we know that both the early universe as well as its future evolution is dominated by vacuum, i.e. FLRW universes may be understood as a perturbation V of de Sitter spacetime which vanishes in the early universe as well as in the limit a → ∞. The main difficulties in doing quantum field theory on curved background is the definition of the vacuum state and renormalization. In an expanding de Sitter spacetime and also in quasi de Sitter spacetimes, a natural choice of the vacuum is given by the so-called Euclidean or Bunch–Davies state.7–9 It can be shown10,11 that the vacuum energy in de Sitter spacetime with the Bunch–Davies state behaves in the right way as ∼ H02 Λ20 where H0 is the de Sitter Hubble parameter and Λ0 is a physical cutoff introduced for example along the lines of Ref. 12. As discussed above FLRW spacetimes may be treated as quasi de Sitter spacetimes. The usual in–out formalism may then be used to extend the calculation of vacuum energy to these spacetimes.11 A more systematic and more fundamental approach to the calculation of vacuum energy in FLRW spacetimes is based on the renormalization of quantum field theory in nonzero backgrounds. This is the approach we will discuss in this paper which is inspired by the recent original work on the Casimir force found in Refs. 13–15 in which the starting point is a renormalizable quantum field theory in a nonzero background. The main ingredients of this approach are as follows: (1) We reinterpret scalar field theory coupled to FLRW metric as a scalar field theory in a flat spacetime interacting with a particular time-dependent (effective graviton) background. (2) We regularize and then renormalize the resulting scalar field theory for arbitrary backgrounds in the usual way. Renormalization, if possible, removes all divergences from all proper vertices of the effective action. (3) We compute the vacuum energy for arbitrary backgrounds. (4) We substitute the particular background found in (1). (5) There is always the possibility that the vacuum energy still diverges for particular profiles of the background configuration. This is indeed the case for the Casimir force14 as well as for the FLRW spacetimes considered here. We thus regularize with a cutoff to obtain an estimate for the vacuum energy. Although we think that this approach is novel, potential and possible overlap with other approaches is certainly expected. A systematic investigation of this point is still underway. 1350166-2

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It is obvious that with these parameters we have Λ0 ≫ 1. H0 We can then approximate the above integral byb ! " # Z Λ 2 2 p Λ Λ0 dp p2 ln 2 − 4 sin pη ≃ ηa4 H02 Λ20 4 − ln 2 cos . µ µ H 0 0

(3.10)

(3.11)

The energy density then becomes

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" # Hφ 1 Λ2 Λ0 2 2 = (1 − 6ξ)H0 Λ0 ln 2 − 4 cos . v 16π 2 µ H0

(3.12)

The mass scale µ2 is also comoving and therefore the physical mass scale must be defined by µ2 = µ20 a. The energy density is then time-independent given by " # H02 Λ20 Λ20 Λ0 Hφ = (1 − 6ξ) ln 2 − 4 cos . (3.13) 2 v 16π µ0 H0 The mass scale µ20 may be taken to be of the order of particle physics mass scale, say µ0 ≃ 102 GeV .

(3.14)

By using the parameters (3.8), (3.9) and (3.14) we obtain a numerical estimation for the vacuum energy density given by Hφ = (1 − 6ξ)(0.08 × (10−12 GeV)4 )(71.45)(0.94) v = (1 − 6ξ)(5.37 × (10−12 GeV)4 ) .

(3.15)

This is of the same order of magnitude as the experimental value (1.1). Corrections due to deviation from a perfect de Sitter spacetime are of order V while corrections due to the contribution of the two-point function are of the order of (1 − 6ξ)2 . A quantitative discussion of these effects will be done elsewhere.11 4. Conclusion In this paper we have presented a new calculation of the vacuum energy in a certain class of FLRW spacetimes which can be viewed as perturbed de Sitter spacetime. This calculation is based on the renormalization of quantum field theory in nonzero (effective graviton) backgrounds in analogy with the recent treatment of the Casimir force found in Refs. 13 and 14. It is found that the vacuum energy still diverges, after renormalization of the quantum field theory proper vertices, for the timedependent cosmological backgrounds of interest. Indeed these backgrounds do not b We remark that although Λ /H ≫ 1 the cosine remains bounded and therefore the remaining 0 0 divergence, which is due to the special cosmological shapes, is really quadratic.

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vanish sufficiently fast for large momenta. By introducing an appropriate comoving cutoff,c an estimation of the vacuum energy is obtained which is compared favorably with the experimental value. Acknowledgment This research was supported by The National Agency for the Development of University Research (ANDRU) under PNR contract number U23/Av58 (8/u23/2723).

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Particle Data Group Collab. (J. Beringer et al.), Phys. Rev. D 86, 010001 (2012). S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). S. M. Carroll, Living Rev. Relativ. 4, 1 (2001). K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, 2001). N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge Univ. Press, 1982). S. A. Fulling, London Math. Soc. Student Texts 17, 1 (1989). T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360, 117 (1978). E. Mottola, Phys. Rev. D 31, 754 (1985). B. Allen, Phys. Rev. D 32, 3136 (1985). A. Prain, Vacuum energy in expanding spacetime and superoscillation induced resonance, Master thesis (2008). Work in progress. A. Kempf, Phys. Rev. D 63, 083514 (2001). R. L. Jaffe, Phys. Rev. D 72, 021301 (2005). N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, O. Schroeder and H. Weigel, Nucl. Phys. B 677, 379 (2004). K. A. Milton, Phys. Rev. D 68, 065020 (2003). J. Zinn-Justin, Int. Ser. Monogr. Phys. 113, 1 (2002). S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley, 2004).

cA

comoving cutoff breaks Lorentz invariance but we are only here trying to obtain a rough, albeit reasonable, estimation of the vacuum energy density. 1350166-9

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Bertotti-Robinson solutions of D ¼ 5 Einstein-Maxwell-Chern-Simons-Lambda theory Adel Bouchareb* Institute of Physics, Badji Mokhtar Annaba University, B.P. 11, 23000 Annaba, Algeria

Chiang-Mei Chen† Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli 320, Taiwan

Ge´rard Cle´ment‡ LAPTh, Universite´ de Savoie, CNRS, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France

Dmitri V. Gal’tsov§ Department of Theoretical Physics, Moscow State University, 119899 Moscow, Russia (Received 11 September 2013; published 28 October 2013) We present a series of new solutions in five-dimensional Einstein-Maxwell-Chern-Simons theory with an arbitrary Chern-Simons coupling and a cosmological constant . For general and we give various generalizations of the Bertotti-Robinson solutions supported by electric and magnetic fluxes, some of which presumably describe the near-horizon regions of black strings or black rings. Among them there is a solution which could apply to the horizon of a topological anti–de Sitter black ring in gauged minimal supergravity. Others are horizonless and geodesically complete. We also construct extremal asymptotically flat multistring solutions for ¼ 0 and arbitrary . DOI: 10.1103/PhysRevD.88.084048

PACS numbers: 04.20.Jb, 04.65.+e

S5 ¼

I. INTRODUCTION Five-dimensional supergravity is an interesting proving ground for string theory. Its Lagrangian, obtainable by toroidal dimensional reduction of eleven-dimensional supergravity, contains a Chern-Simons term for the Maxwell field inherited from reduction of the corresponding four-form term. Though it does not influence the Einstein equations, it modifies the Maxwell equations and consequently the gravitational field too. Surprisingly enough, its presence is crucial for the enhancement of hidden symmetries of five-dimensional Einstein-Maxwell (EM) theory in the case of field configurations possessing two commuting spacetime Killing vectors. For such configurations the theory reduces to a three-dimensional sigma model realizing a harmonic map from the spacetime manifold to the homogeneous space G=H ¼ G2ðþ2Þ = ððSLð2; RÞ SLð2; RÞÞ [1–4]. Owing to this hidden symmetry, a generating technique had been developed [1,2], which opened the way to derive new charged rotating black rings and general black strings [5–10]. In various physical contexts one is also interested in the more general Einstein-Maxwell theory containing a Chern-Simons term with an arbitrary coupling constant and a cosmological constant : *[email protected] † [email protected] ‡ [email protected] § [email protected]

1550-7998= 2013=88(8)=084048(12)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 1 1 d5 x jgð5Þ j Rð5Þ Fð5Þ Fð5Þ 2 16G5 4 (1.1) Fð5Þ Fð5Þ Að5Þ ; pﬃﬃﬃ 12 3

where Fð5Þ ¼ dAð5Þ , ; ; ¼ 1; ; 5, and is an antisymmetric symbol whose signs will be detailed later. For ¼ 1, ¼ 0 this is the action of minimal fivedimensional supergravity, for ¼ 1, < 0 the action of minimal gauged supergravity, while for ¼ 0 it is the EM action. Restricted to field configurations possessing two commuting Killing vectors, this theory reduces to a threedimensional gravitating sigma model coupled to a potential originating from the cosmological constant term. For 1 the target space of this sigma model is not a symmetric space (the isometry group is solvable), so there are no nontrivial hidden symmetries which could be used to generate exact solutions. Moreover, the potential term is not invariant under target space isometries apart from some trivial ones. Not surprisingly, no exact charged rotating black hole solutions are known in the pure EinsteinMaxwell ( ¼ 0) theory even for ¼ 0, though their existence was demonstrated perturbatively [11–13] and numerically [14]. A similar situation holds for charged black rings: static solutions in EM theory have been found by Ida and Uchida [15], and generalized to EM-dilaton theory in [16,17]. But stationary charged black ring solutions of pure EM theory are not known in closed form, though their existence again was confirmed both

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perturbatively (for small charges [18]) and numerically [19]. For 1, 0 exact charged black hole/ring solutions are not known either, while perturbative [20] and numerical solutions have been constructed [21] and shown to have unusual properties for > 1 such as rotation in the sense opposite to the angular momentum and a negative horizon mass with positive asymptotic mass. It is therefore of interest to explore other tools to generate exact solutions of the general Einstein-MaxwellChern-Simons-Lambda (EMCS) action (1.1) which could shed some light on the nature of black objects in this theory. A particular motivation for this lies in the still unsolved question about the existence of asymptotically anti–de Sitter (AdS) and de Sitter (dS) black rings in presence of the cosmological constant. The issue of charged five-dimensional black objects with the cosmological constant was discussed previously in a number of papers. Supersymmetric AdS5 black holes were obtained by several authors [22]. General nonextremal rotating black holes in minimal five-dimensional gauged supergravity were constructed by Chong, Cvetic˘, Lu¨, and Pope [23]. Charged squashed black holes in EM theory ( ¼ 0) with the cosmological constant were constructed numerically in [24]. Black strings with the cosmological constant were studied in [25]. The issue of black rings turns out to be more subtle. General considerations do not prevent their existence both for positive and negative cosmological constants: the additional centripetal/centrifugal force acting on the S1 can be balanced by tuning the angular momentum along the S1 . And indeed, Chu and Dai [26] have found analytically asymptotically dS black rings within the N ¼ 4 de Sitter supergravity (see also [27]). As in the asymptotically flat (AF) case the asymptotically de Sitter black holes/rings may have horizon topologies S3 (or a quotient), S1 S2 and T 3 . For a negative cosmological constant other topologies can be anticipated, namely, S1 H 2 , where H 2 stands for a negative curvature hyperbolic two-surface. In the meantime, no analytical solutions are known for black rings with a negative cosmological constant. Approximate ‘‘thin’’ black rings were obtained by Caldarelli, Emparan, and Rodriguez [28] using the ‘‘blackfolds’’ approach applicable in arbitrary dimensions [29]. These approximate solutions exist for both signs of the cosmological constant and smoothly go into a straight black string in the limit of an infinite S1 radius. Moreover, the possibility of black Saturns with nonflat asymptotics was also indicated. Supersymmetric black rings with AdS5 asymptotics and compact horizons were investigated in [30] for a negative cosmological constant. Lacking exact globally defined solutions, it is tempting to explore local solutions in the vicinity of the event horizons of the presumed black objects. This is a particularly fruitful approach in the extremal case. Usually the near-horizon limits are themselves exact solutions of the same theory, as in the case of the near-horizon limit of

extremal four-dimensional EM black holes, which is the Bertotti-Robinson (BR) solution with geometry AdS2 S2 supported by a monopole electric or magnetic field. Typically, the near-horizon solutions possess a larger isometry group than the full black hole solutions, therefore they can be obtained by different solution-generating techniques. All known exactly five-dimensional supergravity solutions exhibit enhancement of isometries in the nearhorizon region to SOð2; 1Þ Uð1Þ2 , with Uð1Þ factors standing for rotational symmetries [31] (for a review see [32]). This is valid for stationary and asymptotically AdS solutions, and also persists in the presence of scalar fields with a potential [33] and with an account for highercurvature corrections. In D dimensions the enhanced symmetry includes the Uð1ÞD3 rotational symmetry. Note that in this and more general theories certain properties of extremal black holes including topology and thermodynamics can be extracted from the near-horizon solutions using Sen’s entropy function approach [34]. Keeping in mind the importance of the BR solution in the four-dimensional EM theory, we explore here similar exact solutions within the five-dimensional EMCS theory with arbitrary and . Presumably these could be near-horizon limits of black holes/rings with various topologies, including the above-mentioned case of the hyperbolic topology S1 H 2 . As was observed several decades ago, the n-dimensional Einstein-Maxwell theories can be compactified to (n 2)-dimensional spacetime, the two extra space dimensions being curved into a two-sphere through the action of a monopole magnetic field living on that two-sphere [35]. This mechanism was later generalized to the Freund-Rubin compactification of (d s) or s dimensions by an s-form field [36]. The consistency of general sphere compactifications of supergravity actions was extensively discussed in the past (see, e.g., [37]) showing that a consistent sphere truncation of gravity coupled to a form field and (possibly) to a dilaton is possible only in a limited number of cases. In a nonsupersymmetric setting, solutions of the D-dimensional Einstein-Maxwell theory with the cosmological constant that are the topological product of two manifolds of a constant curvature were studied in [38]. Our approach consists in compactifying the theory (1.1) on a constant curvature two-space 2 of a positive, zero, or negative curvature. We consider only compactification through a direct product ansatz, as in the Freund-Rubin case, thereby avoiding the consistency problems associated with non-Abelian Kaluza-Klein gauge fields. This compactification, carried out in Sec. II, results in the threedimensional EMCS gravity, with an additional constraint on the scalar curvature. In Sec. III we present a number of new nontrivial solutions of the five-dimensional EMCS theory of the BR-type which are obtained by uplifting Banados-Teitelboim-Zanelli (BTZ), self-dual, and Go¨del solutions of the constrained three-dimensional theory.

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These solutions are applicable, in particular, to gauged and ungauged D ¼ 5 supergravities, but they are also valid in the EMCS theory with more general values of parameters. In Sec. IV we perform an alternative toroidal reduction of the five-dimensional EMCS to three dimensions, deriving a gravity coupled sigma model with a potential. Some of the solutions listed in Sec. III correspond to null geodesics of the target space. In the case of a vanishing cosmological constant, the reduced three-space is flat, enabling the generalization of these solutions to new classes of nonasymptotically flat or asymptotically flat multicenter solutions of the five-dimensional theory with arbitrary . The technical proof that the solutions of Sec. III are the only solutions with constant curvature two-space sections is outlined in the Appendix. II. REDUCTION ON CONSTANT CURVATURE TWO-SPACES The main idea underlying the present paper is that the five-dimensional theory (1.1) may be reduced, by monopole compactification on constant curvature twosurfaces 2 , to three-dimensional EMCS: 1 Z 3 qﬃﬃﬃﬃﬃﬃ 1 d x jgj R F F 2 Sð3Þ ¼ 2 4 F A ; (2.1) 4 with an additional constraint, which admits several classes of nontrivial exact stationary solutions, leading to nonasymptotically flat stationary solutions of the original five-dimensional theory with the structure M3 2 . The five-dimensional Maxwell-Chern-Simons and Einstein equations following from the action (1.1) are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ @ jgð5Þ jFð5Þ ¼ pﬃﬃﬃ Fð5Þ Fð5Þ ; (2.2) 4 3 1 1 1 2 R ð5Þ 2 Rð5Þ ¼ 2 Fð5Þ Fð5Þ 8 Fð5Þ : (2.3) In Eq. (2.2) we need to fix a sign convention for the fivedimensional antisymmetric symbol. Throughout this paper we will assume that 12345 ¼ þ1, with the spacetime coordinates numbered according to their order of appearance in the relevant five-dimensional metric. Let us assume for the five-dimensional metric and the vector potential the direct product ansatze¨ ds2ð5Þ ¼ g ðx Þdx dx þ a2 dk ; Að5Þ ¼ A ðx Þdx þ efk d’;

(2.4)

where , , ¼ 1, 2, 3, and the two-metrics for k ¼ 1, 0 are

d1 ¼ d2 þ sin 2 d’2 ;

d0 ¼ d2 þ 2 d’2 ;

d1 ¼ d2 þ sinh 2 d’2 ; 1 f0 ¼ 2 ; 2

f1 ¼ cos ;

(2.5)

f1 ¼ cosh ; (2.6)

with ’ 2 ½0; 2 and 2 ½0; for k ¼ 1 and 2 ½0; 1 for k ¼ 0, 1. Here the moduli e and a2 are taken to be constant and real (though for generality we do not assume outright a2 to be positive). The corresponding five-dimensional geometric quantities are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ Rð5Þ ¼ R’ð5Þ’ ¼ ka2 ; (2.7) jgð5Þ j ¼ jgjja2 [email protected] fk ; and the Maxwell tensor decomposes as e ’ Fð5Þ ¼ 4 ; Fð5Þ’ ¼ [email protected] fk ; a @ fk

F ¼ dA: (2.8)

Inserting the ansatze¨ (2.4) in the field equations [39], we find that the equations (2.2) for ¼ 4, 5 are trivially satisfied, while for ¼ they reduce to qﬃﬃﬃﬃﬃﬃ e @ jgjF ¼ pﬃﬃﬃ 2 F : (2.9) 3ja j The Einstein equations (2.3) lead to the system 1 1 1 R R ¼ F F F2 2 2 8 4ka2 e2 þ

; 4a4 4ðe2 3ka2 Þ þ 8: F2 ¼ a4

(2.10)

It is straightforward to check that the reduced equations derive from the action (2.1) with ¼ 2G5 =ja2 j and the following identification of parameters: pﬃﬃﬃ ¼ þ ðe2 4ka2 Þ=4a4 ; ¼ g=ja2 j; ðg ¼ 2e= 3Þ; (2.11) with the additional constraint on the three-dimensional scalar curvature R ¼ ðe2 6ka2 Þ=2a4 þ 4:

(2.12)

Inverting the above relations for k 0 and 0 leads to k2 ; þ ð Þ2 pﬃﬃﬃ 3 ; e¼ 2j32 =16 þ ð Þ2 j

a2 ¼

32 =16

(2.13)

and the constraint R ¼ þ 3 32 =162 ¼ þ 3 e2 =4a4 :

(2.14)

The equations (2.13) break down for k ¼ 0, in which case the parameters , are related by

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the fact that the form of the solution (4.18) breaks down for ¼ 0 and ¼ 1=4], the AF solution (4.18) is an extreme black string for the down sign and ¼ ðn þ 2Þ=4, where n is an integer (or all real for c ¼ 0). For the upper sign, geodesics with v 0 are reflected by an infinite potential barrier, while spacelike geodesics with v ¼ 0 are either reflected or attain r ¼ 0 only asymptotically, so that the spacetime is geodesically complete. For ¼ 1=4 and the lower sign the solution (4.17) is replaced by 3c2 1 2 1 dsð5Þ ¼ dudv þ M þ ln du2 þ 2 dx~ 2 ; 4 pﬃﬃﬃ Að5Þ ¼ 3½c1=2 du A3 ; (4.21) and for ¼ 0 (Einstein-Maxwell theory) it is replaced by ds2ð5Þ ¼ 1 dudv þ ðM 3c2 ln Þdu2 þ 2 dx~ 2 ; pﬃﬃﬃ Að5Þ ¼ 3½c ln du A3 :

(4.22)

In the case of the special vacuum solution (3.8), r is one of the Cartesian coordinates of the reduced three-space and is a harmonic function on that space, so that the generalization to a multicenter solution is the vacuum solution [56,60] ds2ð5Þ ¼ 2 dudv du2 þ dx~ 2

(4.23)

(with u ¼ t, v ¼ z). C. Go¨del solutions The first obvious candidate multicenter solution in the Go¨del class is the ¼ 1=2, 2 ¼ 1 solution (3.17) with k ¼ þ1, g2 ¼ a2 . Actually, it turns out that, after a trivial coordinate transformation z / u, t / v, this is just an instance of the self-dual solution (4.17) for ¼ 1=2 and the lower sign, with M / . The Einstein-Maxwell solution (3.16) ( ¼ 0) with m2 ¼ 0, ¼ 0 leads to the multicenter solution ds2ð5Þ ¼ 2 dt2 þ dz2 þ 2 dx~ 2 ; pﬃﬃﬃ Að5Þ ¼ 2½1 dt þ A3 ;

2 r2 dt2 þ R2 ðdz N z dtÞ2 ðr þ aÞ2 R2 ðr þ aÞ2 2 þ dr þ ðr þ aÞ2 d22 ; r2 pﬃﬃﬃ pﬃﬃﬃ 2 3 ðdt $dzÞ cos d’ ; ¼a rþa 2

ds2ð5Þ ¼

Að5Þ

(4.26)

with R2 ¼ 2 ð1 $Þ2 Nz ¼

2a$ð1 $Þ a2 $2 þ ; rþa ðr þ aÞ2

r½ð1 $Þr þ a ; ðr þ aÞ2 R2

is a dyonic extremepblack string. ﬃﬃﬃ Finally, for > 3=2, ds2ð5Þ ¼ ð1 dt þ dzÞ2 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃ pﬃﬃﬃ 1 3 3 1 2 A ¼ Að5Þ ¼ 2 dt ; 3 2 82 2

(4.27)

~ is harmonic on the Minkowskian reduced where ðxÞ metric dx~2 ¼ ij dxi dxj . The signature of the metric (4.27) is ( þ þ ). A Minkowskian multicenter solution may be obtained from the ¼ 0 Go¨del solution (3.22) with k ¼ 1, g 2 2 ¼ a2 . By transforming the H 2 coordinates ð; ’Þ to coordinates ðr; zÞ such that d1 ¼

dr2 þ r2 dz2 ; r2

f1 ¼ r;

(4.28)

we obtain the multicenter solution with Newman-UntiTamburino parameters (NUTs): ds2ð5Þ ¼ ðdt 1 A3 Þ2 þ 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2ð1 þ 2 Þ 3 1 Að5Þ ¼ dz: A3 2 2

(4.29)

(4.24) V. SUMMARY

which is the trivial five-dimensional embedding of a dyonic Majumdar-Papapetrou solution. 2 ¨ del solution (3.15) for From p the ﬃﬃﬃ generic m ¼ 0 Go 0 < < 3=2 with k ¼ þ1, g2 2 ¼ a2 , we derive the multicenter solution ds2ð5Þ ¼ ð1 dt þ dzÞ2 þ 2 dz2 þ 2 dx~ 2 ; pﬃﬃﬃ pﬃﬃﬃ 3 1 3 1 A3 2 ¼ 2 ; Að5Þ ¼ 2 dt 2 8 2

solution, generalized by the local coordinate transformation t ! t $z (with $ a second parameter)

(4.25)

which may be viewed as a deformation of (4.24). For 2 > 1 ( < 1=2), the one-center asymptotically flat

In this paper we have investigated solutions of the general five-dimensional EMCS theory—containing as particular cases minimal ungauged and gauged supergravities—known exact solutions of three-dimensional Einstein-Maxwell gravity with a Chern-Simons term. Our main tool was dimensional reduction, assuming the five-dimensional spacetime to be the direct product of a constant curvature surface 2 of positive, zero, or negative curvature (k ¼ 1, 0 or 1) and a three-dimensional spacetime. The reduced theory is then the three-dimensional EMCS3 with a constraint on the scalar curvature, which can be satisfied by three classes of solutions found earlier,

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PHYSICAL REVIEW D 88, 084048 (2013)

namely, the BTZ, self-dual, and Go¨del classes. Promoting these to five dimensions, we have constructed new EMCS5 solutions of the generalized Bertotti-Robinson– type which could be near-horizon limits of black string and black ring solutions. The three-parameter BTZ class of solutions is geodesically complete and exists for an arbitrary (irrelevant) Chern-Simons coupling . Generically these solutions are nonvacuum, being supported by magnetic fluxes along the constant curvature two-surfaces. Their isometry groups are SOð2; 2Þ SOð3Þ for k ¼ þ1, SOð2; 2Þ GLð2; RÞ for k ¼ 0, and SOð2; 2Þ SOð2; 1Þ for k ¼ 1. In the minimal supergravity case ¼ 0, ¼ 1 only spherical sections are possible, and the corresponding solutions are near-horizon limits of black strings. For a negative cosmological constant this class of solutions exists in all the three versions k ¼ 1, 0, or 1, with horizon topologies S1 S2 , S1 R2 , and S1 H 2 respectively. The latter two could be near-horizon limits of topological asymptotically AdS black rings, though the existence of the global solutions remains to be checked. There is a special case with zero magnetic flux, when we get a two-parameter family of vacuum solutions which can be seen as nonasymptotically flat topological black rings with the horizon S1 H 2 . These are (presumably new) locally AdS3 H 2 solutions of vacuum five-dimensional Einstein equations. For a positive cosmological constant, the solutions exists in the k ¼ 1 version only. Their interpretation is similar, but the relevant asymptotics is de Sitter. The second class of EMCS5 solutions, depending also on three parameters, generalize the extremal solutions of the BTZ class, which they asymptote. They are supported by a magnetic flux together with an independent dyonic electromagnetic field. The solutions of the third threeparameter class are generated from three-dimensional Go p ﬃﬃﬃ¨ del black holes. They pﬃﬃﬃ exist with k ¼ þ1 forpﬃﬃﬃ < 3=2, k ¼ 0 for ¼ 3=2, and k ¼ 1 for > 3=2. The isometry groups are SOð2; 1Þ SOð2Þ SOð3Þ, SOð2; 1Þ SOð2Þ GLð2; RÞ, and SOð2; 1Þ SOð2Þ SOð2; 1Þ, respectively. Subclasses include solutions analogue to previously found rotating BR solutions in dilaton-axion gravity, horizonless and geodesically complete solutions with spacetime topology H2 2 R, and NUTty solutions with spacetime topology S2 2 R. In the case of minimal D ¼ 5 supergravity, the NUTty S2 H 2 R solution (3.24) has been shown [53] to be a near-extreme, near-bolt limit of an asymptotically flat solitonic string solution of EMCS5. We have then shown that some of the BR solutions thus constructed can be promoted to NAF or AF multicenter solutions. For this purpose we have performed an alternative toroidal compactification of EMCS5 to three dimensions, leading to a sigma-model representation with a potential. For ¼ 0 this potential vanishes, in which case null geodesics of the target space give rise

to exact solutions of the five-dimensional equations with a flat reduced three-space. The corresponding BR solutions may be promoted to multicenter solutions by redefinition of the associated harmonic functions. We have thus identified three families of multistring solutions of EMCS5, the two self-dualpfamilies (4.17), ﬃﬃﬃ ¨ﬃﬃﬃ del family (4.25) (for < 3=2) or (4.29) and the p Go (for > 3=2). An unexpected by-product of our analysis is the construction of new closed-form asymptotically flat solutions (4.18) generated by a dyonic electromagnetic field along with a magnetic flux. For the lower sign these are regular extreme black strings for a discrete set of values of the Chern-Simons coupling constant (including the minimal supergravity case ¼ 1), while for the upper sign they are geodesically complete. ACKNOWLEDGMENTS Three of the authors are grateful to the LAPTh Annecy for hospitality in July (C.-M. C.) and December (A. B. and D. G.) 2012 while the paper was in progress. The work of D. G. was supported in part by the RFBR Grant No. 11-0201371-a. The work of C.-M. C. was supported by the National Science Council of the R.O.C. under Grant No. NSC 102-2112-M-008 -015 -MY3, and in part by the National Center of Theoretical Sciences (NCTS). The work of A. B. was supported by the Agence The´matique de Recherche en Sciences et Technologie (ATRST) under Contract No. U23/Av58 (PNR 8/u23/2723) and in part by the CNEPRU. APPENDIX: CONSTANT SCALAR CURVATURE SOLUTIONS TO THREE-DIMENSIONAL EMCS THEORY The ansatz [42] ds2 ¼ ab ðÞdxa dxb þ 2 ðÞR2 ðÞd2 ; A ¼ c a ðÞdxa ;

(A1)

where is the 2 2 matrix ¼

TþX

Y

Y

TX

! ;

(A2)

and R2 X2 is the Minkowski pseudonorm of the ‘‘vector’’ XðÞ ¼ ðT; X; YÞ, X2 ¼ ij X i X j ¼ T 2 þ X 2 þ Y 2 ;

(A3)

reduces the equations of three-dimensional EMCS theory to the Maxwell-Chern-Simons, Einstein and Hamiltonian constraint equations

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S0 ¼

2 X ^ S; R2

(A4)

BERTOTTI-ROBINSON SOLUTIONS OF D ¼ 5 . . .

X00 ¼

PHYSICAL REVIEW D 88, 084048 (2013)

2 2 ðS XÞX S ; R2 R2

X02 þ 2X X00 þ 4 2 ¼ 0;

(A5)

(A6)

where S is the null vector 1 S ¼ ð c 20 þ c 21 ; c 20 c 21 ; 2 c 0 c 1 Þ; 4

ðS2 ¼ 0Þ;

(A7)

_ ¼ @[email protected], and the wedge product is defined by ðX ^ YÞi ¼ ij jkl Xk Y l (with 012 ¼ þ1). To the above equations we must add the constraint (2.14) 1 R 2 X02 ðX2 Þ00 ¼ þ 3 32 =162 : (A8) 2 Combining equations (A6) and (A8), we obtain the relations ðR2 Þ00 ¼ 2b;

X X00 ¼ c;

X02 ¼ b c;

(A9)

c ¼ ð 3Þ= with b ¼ ð þ Þ=2 þ 3=162 , 2 3=162 . From (A5) it follows that c S X ¼ R2 : 2

(A10)

Noting that ðS XÞ0 ¼ S X0 from (A4), we derive from (A10) ðS XÞ00 ¼ S X00 þ S0 X0 ¼ c2

2 L S; R2

(A11)

where we have defined L ¼ X ^ X0 :

(A12)

Comparing (A11), (A10), and (A9), we obtain LS¼

cðc bÞ 2 R: 2

(A13)

Next, noting that L0 S ¼ 0 from (A4), we obtain 2 ðL SÞ ¼ 2 ðX ^ X0 Þ ðX ^ SÞ R 2 ¼ 2 ½ðS XÞðX X0 Þ R2 ðS X0 Þ R c ¼ ðR2 Þ0 : 2

Consider now the second possibility, b c ¼ 1. Squaring Eq. (A5), we find that X002 ¼ 0; (A16) 02 while the last equation (A9), which now reads X ¼ 1, implies that X0 X00 ¼ 0. The fact that the null vector X00 is orthogonal to X 0 implies that the wedge product X0 ^ X00 is collinear with X00 : (A17) X0 ^ X00 ¼ X00 : This relation, together with Eq. (A5), may be used to transform Eq. (A4) to S0 ¼ X ^ X00 ¼ X ^ ðX0 ^ X00 Þ 1 ¼ ðX X0 ÞX00 ðX X00 ÞX0 ¼ ðR2 Þ0 X00 cX0 : 2 (A18) Taking the scalar product of (A18) with the vector L and using (A5) and (A13), we obtain c L S0 ¼ ðR2 Þ0 ; (A19) 2 which upon comparison with (A14) shows that we should take the upper sign in the preceding equations. Finally, Eq. (A18) with the upper sign may be compared with the direct derivative of Eq. (A5), 1 1 S0 ¼ ðR2 ÞX000 ðR2 Þ0 X00 þ cX0 ; 2 2 leading to the conclusion that

(A20)

X000 ¼ 0: This equation is integrated by

(A21)

ð 2 ¼ 0; ^ ¼ Þ; (A22) X ¼ 2 þ þ ; the vector relations between the constant vectors , , following from (A16) and (A17). We recognize in (A22) the quadratic ansatz which was used in [42] to derive the Go¨del solutions to three-dimensional EMCS. For the last possibility, ðR2 Þ0 ¼ 0, we can adapt the above argument to show that X0 ^ X00 ¼ qX00

0

Equation (A18) is now replaced by c S0 ¼ X0 : q (A14)

Comparing with (A13), we derive the constraint cðc b þ 1ÞðR2 Þ0 ¼ 0:

ðq2 ¼ b cÞ:

(A15)

The first possibility c ¼ 0 means that S X ¼ 0, which is equivalent to the self-duality condition [46] F F ¼ 0, which leads either to the self-dual solutions (3.9) for < 0 and (3.11) for ¼ 0, or to the vacuum solutions of the BTZ class.

(A23)

(A24)

Taking the scalar product with X0 and using (A11) and (A13), we obtain S0 X0 ¼ cq ¼ cq2 :

(A25)

The first solution, c ¼ 0, leads to a subcase of the self-dual class. The second solution, q ¼ 0, means that X00 is collinear with X0 so that X X00 ¼ c ¼ 0, leading again to a subcase of the self-dual class. And the last solution, q ¼ 1, means b c ¼ 1, corresponding to a subclass of the Go¨del class.

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PHYSICAL REVIEW D 88, 084048 (2013)

[1] A. Bouchareb, C. M. Chen, G. Cle´ment, D. V. Gal’tsov, N. G. Scherbluk and T. Wolf, Phys. Rev. D 76, 104032 (2007); 78, 029901(E) (2008). [2] G. Cle´ment, J. Math. Phys. (N.Y.) 49, 042503 (2008); 49, 079901(E) (2008). [3] M. Berkooz and B. Pioline, J. High Energy Phys. 05 (2008) 045. [4] D. V. Gal’tsov and N. G. Scherbluk, Phys. Rev. D 78, 064033 (2008); 79, 064020 (2009). [5] D. V. Gal’tsov and N. G. Scherbluk, Phys. Rev. D 81, 044028 (2010). [6] S. Tomizawa, Y. Yasuii, and Y. Morisawa, Classical Quantum Gravity 26, 145006 (2009). [7] G. Compe`re, S. de Buyl, E. Jamsin, and A. Virmani, Classical Quantum Gravity 26, 125016 (2009). [8] P. Figueras, E. Jamsin, J. V. Rocha, and A. Virmani, Classical Quantum Gravity 27, 135011 (2010). [9] S. S. Kim, J. L. Ho¨rnlund, J. Palmkvist, and A. Virmani, J. High Energy Phys. 08 (2010) 072. [10] G. Compe`re, S. de Buyl, S. Stotyn, and A. Virmani, J. High Energy Phys. 11 (2010) 133. [11] A. N. Aliev and V. P. Frolov, Phys. Rev. D 69, 084022 (2004); A. N. Aliev, Phys. Rev. D 74, 024011 (2006). [12] F. Navarro-Lerida, Gen. Relativ. Gravit. 42, 2891 (2010). [13] M. Allahverdizadeh, J. Kunz, and F. Navarro-Lerida, Phys. Rev. D 82, 024030 (2010). [14] J. Kunz, F. Navarro-Lerida, and A. K. Petersen, Phys. Lett. B 614, 104 (2005). [15] D. Ida and Y. Uchida, Phys. Rev. D 68, 104014 (2003). [16] S. S. Yazadjiev, arXiv:hep-th/0507097. [17] H. K. Kunduri and J. Lucietti, Phys. Lett. B 609, 143 (2005). [18] M. Ortaggio and V. Pravda, J. High Energy Phys. 12 (2006) 054. [19] B. Kleihaus, J. Kunz, and K. Schnulle, Phys. Lett. B 699, 192 (2011). [20] A. N. Aliev and D. K. Ciftci, Phys. Rev. D 79, 044004 (2009). [21] J. Kunz and F. Navarro-Lerida, Mod. Phys. Lett. A 21, 2621 (2006). [22] D. Klemm and W. A. Sabra, Phys. Lett. B 503, 147 (2001); J. P. Gauntlett and J. B. Gutowski, Phys. Rev. D 68, 105009 (2003); 70, 089901(E) (2004); J. B. Gutowski and H. S. Reall, J. High Energy Phys. 04 (2004) 048. [23] Z. W. Chong, M. Cvetic˘, H. Lu¨, and C. N. Pope, Phys. Lett. B 644, 192 (2007). [24] K. Murata, T. Nishioka, and N. Tanahashi, Prog. Theor. Phys. 121, 941 (2009); Y. Brihaye, J. Kunz, and E. Radu, J. High Energy Phys. 08 (2009) 025; Y. Brihaye, E. Radu, and C. Stelea, Phys. Lett. B 709, 293 (2012). [25] R. B. Mann, E. Radu, and C. Stelea, J. High Energy Phys. 09 (2006) 073; Y. Brihaye, E. Radu, and C. Stelea, Classical Quantum Gravity 24, 4839 (2007); A. Bernamonti, M. M. Caldarelli, D. Klemm, R. Olea, C. Sieg, and E. Zorzan, J. High Energy Phys. 01 (2008) 061. [26] C.-S. Chu and S.-H. Dai, Phys. Rev. D 75, 064016 (2007). [27] J. Gutowski and W. A. Sabra, J. High Energy Phys. 05 (2011) 020. [28] M. M. Caldarelli, R. Emparan, and M. J. Rodriguez, J. High Energy Phys. 11 (2008) 011. [29] R. Emparan, T. Harmark, V. Niarchos, and N. A. Obers, Phys. Rev. Lett. 102, 191301 (2009); J. High Energy Phys. 03 (2010) 063.

[30] H. K. Kunduri, J. Lucietti, and H. S. Reall, J. High Energy Phys. 02 (2007) 026. [31] H. K. Kunduri and J. Lucietti, Classical Quantum Gravity 26, 245010 (2009); Commun. Math. Phys.303, 31 (2011); H. K. Kunduri, Classical Quantum Gravity 28, 114010 (2011); H. K. Kunduri and J. Lucietti, J. High Energy Phys. 07 (2011) 107. [32] H. K. Kunduri and J. Lucietti, Living Rev. Relativity 16, 8 (2013). [33] H. K. Kunduri, J. Lucietti, and H. S. Reall, Classical Quantum Gravity 24, 4169 (2007). [34] A. Sen, J. High Energy Phys. 03 (2006) 008. [35] Z. Horvath, L. Palla, E. Cremmer, and J. Scherk, Nucl. Phys. B127, 57 (1977). [36] P. G. O. Freund and M. A. Rubin, Phys. Lett. B 97, 233 (1980). [37] M. Cvetic˘, H. Lu¨, and C. N. Pope, Nucl. Phys. B597, 172 (2001). [38] V. Cardoso, O. J. C. Dias, and J. P. S. Lemos, Phys. Rev. D 70, 024002 (2004). [39] Dimensional reduction in the action (1.1) does not produce a correct three-dimensional action. [40] G. Cle´ment, Geometry of Constrained Dynamical Systems, edited by J. M. Charap (Cambridge University Press, Cambridge, 1995), pp. 17–22. [41] M. Ban˜ados, G. Barnich, G. Compe`re, and A. Gomberoff, Phys. Rev. D 73, 044006 (2006). [42] K. Ait Moussa, G. Cle´ment, H. Guennoune, and C. Leygnac, Phys. Rev. D 78, 064065 (2008). [43] J. Grover, J. B. Gutowski, G. Papadopoulos, and W. A. Sabra, arXiv:1303.0853. [44] G. Cle´ment, Int. J. Theor. Phys. 24, 267 (1985). [45] G. Cle´ment and I. Zouzou, Phys. Rev. D 50, 7271 (1994). [46] G. Cle´ment, Phys. Lett. B 367, 70 (1996). [47] G. Cle´ment, Classical Quantum Gravity 10, L49 (1993). [48] K. Ait Moussa, G. Cle´ment, and C. Leygnac, Classical Quantum Gravity 20, L277 (2003). [49] A. Bouchareb and G. Cle´ment, Classical Quantum Gravity 24, 5581 (2007). [50] D. Anninos, W. Li, M. Padi, W. Song, and A. Strominger, J. High Energy Phys. 03 (2009) 130. [51] These solutions were obtained in [42] under the assumption > 0, i.e., on account of the second equation in Eq. (2.13) e > 0. [52] G. Cle´ment and D. Gal’tsov, Phys. Rev. D 63, 124011 (2001); Nucl. Phys. B619, 741 (2001). [53] A. Bouchareb, C. M. Chen, G. Cle´ment, and D. V. Gal’tsov (to be published). [54] The magnetic potential of [1,2] is noted here to avoid confusion with the Chern-Simons coupling constant. [55] G. Neugebauer and D. Kramer, Ann. Phys. (Leipzig) 24, 2 (1969). [56] G. Cle´ment, Gen. Relativ. Gravit. 18, 861 (1986); Phys. Lett. A 118, 11 (1986). [57] G. Cle´ment and D. Gal’tsov, Phys. Rev. D 54, 6136 (1996). [58] We have changed a sign in Að5Þ because our coordinate transformation implies uv ¼ tz . [59] K. Matsuno, H. Ishihara, M. Kimura, and T. Tatsuoka, Phys. Rev. D 86, 104054 (2012). [60] G. W. Gibbons, Nucl. Phys. B207, 337 (1982).

084048-12

arXiv:1401.1529v1 [hep-th] 7 Jan 2014

New Algorithm and Phase Diagram of Noncommutative Φ4 on the Fuzzy Sphere Badis Ydri∗ Institute of Physics, BM Annaba University, BP 12, 23000, Annaba, Algeria. January 9, 2014

Abstract We propose a new algorithm for simulating noncommutative phi-four theory on the fuzzy sphere based on i) coupling the scalar field to a U (1) gauge field in such a way that in the commutative limit N −→ ∞ the two modes decouple and we are left with pure scalar phi-four on the sphere and ii) diagonalizing the scalar field by means of a U (N ) unitary matrix and then integrating out the unitary group from the partition function. The number of degrees of freedom in the scalar sector reduces therefore from N 2 to the N eigenvalues of the scalar field whereas the dynamics of the U (1) gauge field is given by D = 3 Yang-Mills matrix model with a Myers term. As an application the phase diagram, including the triple point, of noncommutative phi-four theory on the fuzzy sphere is reconstructed with small values of N up to N = 10 and large numbers of statistics.

Contents 1 Introduction

2

2 Model and Algorithm 2.1 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Algorithm and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6

3 The 3.1 3.2 3.3 3.4 ∗

Phase Diagram The Ising Phase Transition . . . . . . . . . . . The Uniform-to-Non-Uniform Phase Transition The Matrix Phase Transition . . . . . . . . . . Triple Point and Phase Diagram . . . . . . . .

Email:[email protected], [email protected]

1

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8 8 9 13 15

4 The Self-Dual Noncommutative Φ4 on the Fuzzy Sphere 4.1 Self-Dual Noncommutative Φ4 . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Sphere as a Regulator . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17

5 Conclusion and Outlook

19

1

Introduction

The goal of this article is to reconstruct by means of a (hopefully) novel and efficient Monte Carlo method the phase diagram of noncommutative phi-four on the fuzzy sphere which was originally done in [1]. The basic theory is given by the following two-parameter matrix model S0 = TrH − a[La , Φ]2 + bΦ2 + cΦ4 .

(1)

In this equation La are the SU (2) generators in the irreducible representation with spin s = (N − 1)/2, TrH 1 = N , b is the mass parameter and c is the coupling constant. The parameter a can always be chosen to be equal to 1. There are three known phases in this model. The usual Ising transition between disorder and uniform order, a matrix transition between disorder and a non-uniform ordered phase and a (very hard to observe) transition between uniform order and non-uniform order. The three phases meet at a triple point [1,2]. The non-uniform phase in which rotational invariance is spontaneously broken is simply absent in the commutative theory. The non-uniform phase is the analogue of the stripe phase observed on the Moyal-Weyl spaces [14] whereas the disorder-to-non-uniformorder transition is the generalization of the one-cut-to-two-cut transition observed in the Hermitian quartic matrix model [15, 16] to the fuzzy sphere. This is a highly non-trivial problem which is due mainly to the more complicated phase structure of matrix scalar phi-four which involves transitions between vacuum states with very low probability distributions and as a consequence they are extremely difficult to sample correctly with the Metropolis algorithm. In particular the uniform-to-uniform transition is virtually unobservable in ordinary Metropolis due to the absence of tunneling between the identity matrix corresponding to the uniform phase and the other idempotent matrices corresponding to the non-uniform phase. This means simply that the Metropolis updating procedure does not sample correctly and equally, i.e. according to the Boltzmann weight the entire phase space which includes an infinite number of vacuum states. This was circumvented in [1, 2] by a complicated variant of the Metropolis algorithm in which detailed balance is broken. This problem was also studied in [3, 4, 7, 8]. The analytic derivation of the phase diagram of noncommutative phi-four on the fuzzy sphere was attempted in [17–19]. The related problem of Monte Carlo simulation of noncommutative phi-four on the fuzzy torus and the fuzzy disc was considered in [5, 6] and [9] respectively. The main strategy employed in this article towards a better resolution of this problem is to reduce the model down to its eigenvalues without actually altering it. This is achieved by

2

1) coupling the scalar field to a U (1) gauge field in such a way that in the commutative limit N −→ ∞ the two modes decouple completely and thus we return to an ordinary phi-four theory and 2) diagonalizing the scalar field by means of a U (N ) gauge transformation, viz Φ = U ΛU + and then integrating out the unitary matrices U and U + from the path integral. In this algorithm we thus trade off the Monte Carlo simulation of the unitary matrix U and U + in the original model (1) with the Monte Carlo simulation of a U (1) gauge field on the fuzzy sphere which we know is very efficient using ordinary Metropolis [10]. The primary interest of this article is therefore Monte Carlo simulation of a noncommutative phi-four theory coupled to a U (1) gauge field on the fuzzy sphere using the Metropolis algorithm with exact detailed balance. The scalar field transforms in the adjoint representation of the U (1) gauge group and as a consequence the scalar and gauge degrees of freedom decouple in the commutative limit N −→ ∞. In other words this theory becomes an ordinary phi-four theory in the commutative limit. In this theory the usual scalar kinetic action ∼ −T r[La , Φ]2 is replaced with ∼ −T r[Xa , Φ]2 where Xa is itself obtained by Monte Carlo simulation of an appropriate gauge action which will be centered around ∼ La in the so-called fuzzy sphere phase1 . The pure gauge action is given by D = 3 Yang-Mills action with a Chern-Simons (Myers) term. For b = c = 0 the full action is in fact D = 4 Yang-Mills action with a Chern-Simons (Myers) term. This article is organized as follows. In section 2 we present the detail of the U (1) gauge covariant noncommutative phi-four theory on the fuzzy sphere and also explain the Metropolis algorithm employed in our Monte Carlo simulations. In section 3 we report our first numerical results on the phase diagram of noncommutative phi-four on the fuzzy sphere using our new algorithm. We give independent measurements of the three transition lines discussed above and then derive our estimation of the triple point. These results are obtained with small values of N up to N = 10 and large numbers of statistics. In section 4 we give a construction of a one-parameter family of noncommutative phi-four models on the fuzzy sphere which define a regularization of duality covariant noncommutative phi-four on the Moyal-Weyl plane. We conclude in section 5 with a brief summary and outlook.

2 2.1

Model and Algorithm The Action

Instead of the basic model (1), which is the primary interest in this article, we consider a four matrix model given by the action

1

S = Sg + Sm .

(2)

2iα 1 ǫabc Xa Xb Xc + N T r M T r(Xa2 )2 + βXa2 . Sg = N T r − [Xa , Xb ]2 + 4 3

(3)

The behavior in the matrix phase is very different and is not treated in here.

3

Equivalently K 4πR2

=

− −

N N 1 + Ω2 X X 2(i − 1)(j − 1) ˜ ˜ )φij φji (i + j − 1 − 2πθ N −1 i=1 j=1 N N r i−2 1 − Ω2 X X j−2 ˜ ˜ (i − 1)(j − 1)(1 − )(1 − )φij φj−1i−1 2πθ N −1 N −1 i=1 j=1 N N r j−1 ˜ ˜ i−1 1 − Ω2 X X )(1 − )φij φj+1i+1 . (56) ij(1 − 2πθ N −1 N −1 i=1 j=1

We can now include a mass term and a phi-four coupling in a trivial way. The full action on the fuzzy sphere will read 1ˆ 2 µ2 ˆ2 λ ˆ4 4Ω2 2 2 2 2 2 ˆ SΩ = 4πR TrH − φ La + Ω L3 − 2 (Xa − X3 ) φ + φ + φ . (57) 2 θ 2 4! √ ˜ 2π and also introduce the parameters We scale the field as φˆ = φ/ r = µ2 R 2 , u =

λR2 . 4!π

(58)

The full action can then be rewritten as

SΩ = TrH

2 2 2 2 2 2 4 ˜ ˜ ˜ ˜ ˜ − [La , φ] − Ω [L3 , φ] + Ω {La , φ} + r φ + uφ .

(59)

This is a one-parameter family of phi-four models on the fuzzy sphere which generalizes (1). Coupling to a U (1) gauge field is straightforward, i.e. we make the replacement p La −→ N/2Xa . The analogue of (4) is obviously given by SΩ = −

5

2 2 N ˜ 2 − N Ω T r[X3 , φ] ˜ 2 + N Ω T r{Xa , φ} ˜ 2 + T rV (φ). ˜ T r[Xa , φ] 2 2 2

(60)

Conclusion and Outlook

In this article we have proposed a new algorithm for the Monte Carlo simulation of noncommutative phi-four on the fuzzy sphere and also reported our first numerical results on the corresponding phase diagram obtained with small values of N up to N = 10 and large numbers of statistics. Basically the new algorithm employs gauge invariance in order to reduce the scalar sector to the core eigenvalues problem. The phase diagram is complex consisting of three transition lines (the Ising or uniform-to-disorder, the matrix or nonuniform-to-disorder and the uniform-to-non-uniform transition lines) which intersect at a triple point. The measurement of the uniform-to-non-uniform transition line using our algorithm remains very demanding but tractable. The measurements included in this article are largely consistent with those reported originally in [1]. The first immediate extension of this work is to optimize the algorithm further and push the calculation of the phase diagram to higher values of N with reasonably large numbers

19

of statistics especially in the case of the uniform-to-non-uniform transition line. A major improvement of the algorithm may be achievable by replacing the Metropolis updating procedure for the scalar eigenvalues problem by the Hybrid Monte Carlo algorithm whereas we may keep using the very efficient Metropolis for the gauge sector. Another immediate line of investigation is the calculation of the phase diagram of the self-dual noncommutative phi-four on the fuzzy sphere constructed in the last section. The main question here is what happens to the Ising transition line as Ω goes from Ω = 0 to Ω = 1 and as a consequence what is the fate of the triple point.

Acknowledgments: This research was supported by ”The National Agency for the Development of University Research (ANDRU), MESRS, Algeria” under PNR contract number U23/Av58 (8/u23/2723). The Monte Carlo simulations reported in this article were largely performed on the machines of the School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland.

References [1] F. Garcia Flores, X. Martin and D. O’Connor, “Simulation of a scalar field on a fuzzy sphere,” Int. J. Mod. Phys. A 24, 3917 (2009) [arXiv:0903.1986 [hep-lat]]. See also [2] [2] F. Garcia Flores, D. O’Connor and X. Martin, “Simulating the scalar field on the fuzzy sphere,” PoS LAT 2005, 262 (2006) [hep-lat/0601012]. [3] X. Martin, “A matrix phase for the phi**4 scalar field on the fuzzy sphere,” JHEP 0404, 077 (2004) [hep-th/0402230]. [4] M. Panero, “Numerical simulations of a non-commutative theory: The Scalar model on the fuzzy sphere,” JHEP 0705, 082 (2007) [hep-th/0608202]. [5] J. Ambjorn and S. Catterall, “Stripes from (noncommutative) stars,” Phys. Lett. B 549, 253 (2002) [hep-lat/0209106]. [6] W. Bietenholz, F. Hofheinz and J. Nishimura, “Phase diagram and dispersion relation of the noncommutative lambda phi**4 model in d = 3,” JHEP 0406, 042 (2004) [hep-th/0404020]. [7] J. Medina, W. Bietenholz and D. O’Connor, “Probing the fuzzy sphere regularisation in simulations of the 3d lambda phi**4 model,” JHEP 0804, 041 (2008) [arXiv:0712.3366 [hep-th]]. [8] C. R. Das, S. Digal and T. R. Govindarajan, “Finite temperature phase transition of a single scalar field on a fuzzy sphere,” Mod. Phys. Lett. A 23, 1781 (2008) [arXiv:0706.0695 [hep-th]]. [9] F. Lizzi and B. Spisso, “Noncommutative Field Theory: Numerical Analysis with the Fuzzy Disc,” Int. J. Mod. Phys. A 27, 1250137 (2012) [arXiv:1207.4998 [hep-th]]. [10] B. Ydri, “Impact of Supersymmetry on Emergent Geometry in Yang-Mills Matrix Models II,” Int. J. Mod. Phys. A 27, 1250088 (2012) [arXiv:1206.6375 [hep-th]].

20

[11] M.P. Vachovski, “Numerical studies of the critical behaviour of non-commutative field theories,” Ph.D thesis, private communication by Denjoe O’Connor. [12] H. Grosse and R. Wulkenhaar, “Renormalization of phi**4 theory on noncommutative R**4 in the matrix base,” Commun. Math. Phys. 256, 305 (2005) [hepth/0401128],“Renormalization of phi**4 theory on noncommutative R**2 in the matrix base,” JHEP 0312, 019 (2003) [hep-th/0307017],“Power counting theorem for non-local matrix models and renormalization,” Commun. Math. Phys. 254, 91 (2005) [hep-th/0305066]. [13] E. Langmann and R. J. Szabo, “Duality in scalar field theory on noncommutative phase spaces,” Phys. Lett. B 533, 168 (2002) [hep-th/0202039]. [14] S. S. Gubser and S. L. Sondhi, “Phase structure of noncommutative scalar field theories,” Nucl. Phys. B 605, 395 (2001) [hep-th/0006119]. [15] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978). [16] Y. Shimamune, “On The Phase Structure Of Large N Matrix Models And Gauge Models,” Phys. Lett. B 108, 407 (1982). [17] D. O’Connor and C. Saemann, “Fuzzy Scalar Field Theory as a Multitrace Matrix Model,” JHEP 0708, 066 (2007) [arXiv:0706.2493 [hep-th]]. [18] C. Saemann, “The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP n ,” SIGMA 6, 050 (2010) [arXiv:1003.4683 [hep-th]]. [19] A. P. Polychronakos, “Effective action and phase transitions of scalar field on the fuzzy sphere,” Phys. Rev. D 88, 065010 (2013) [arXiv:1306.6645 [hep-th]].

21

• Information financière

Université Badji Mokhtar - Annaba

Intitulé du projet PNR : Théories

des Champs Non-Commutatives à partir des Modèles des Matrices et

Physique Emergente Chef du projet PNR : Badis YDRI Laboratoire de rattachement du PNR : Laboratoire

Directeur du Laboratoire : MERADJI

de Physique et Rayonnements

(LPR)

HOCINE Unité En ; DA

Ventilation du Budget relatif à un Projet PNR Arrêté interministériel du 01 Mars 2012 fixant la nomenclature des dépenses consacrées à la recherche scientifique Et au développement Chapitres 01 1

Intitulés

3 4 5

Remboursement de frais Frais de mission et de déplacement en Algérie, à l'étranger. Rencontres Scientifiques : Frais d'organisation, d’hébergement, de restauration et de transport. Honoraires des enquêteurs. Honoraires des guides. Honoraires des experts et consultants. Frais d’études, de travaux et de prestations réalisés pour le compte de l’entité. SOUS TOTAL

02

Matériels et mobiliers

1

3 4

Matériels et instruments scientifiques et audiovisuels. Renouvellement du matériel informatique, achat accessoires, logiciels et consommables informatiques. Mobilier de laboratoire. Entretiens et réparations.

03

Fournitures

1 2 3

Produits chimiques. Produits consommables. Composants électroniques, mécaniques et audio-visuels.

4

Papeterie et fournitures de bureau.

5 6

Périodiques. Documentation et ouvrages de recherche. Fournitures des besoins de laboratoires (animaux, plantes etc.)

2

6

2

SOUS TOTAL

SOUS TOTAL 04

Charges Annexes

1

Impression et édition.

2

Affranchissements postaux.

répartition des crédits /

/ /

462 150 ,00 100 795 ,50

562945,50

179885,00

705868,30

885753,3

3 4 5

Communications téléphoniques, Fax, télex, télégramme, Internet. Autre frais (impôts et taxes, droits de douane, frais financiers, assurances, frais de stockage, et autre). Banque de données (acquisitions et abonnements). SOUS TOTAL TOTAL GENERAL

1448698,80

PRESENTATION DU PROJET SITUATION ACTUELLE DU PROJET: Intitulé du PNR

Code du Projet (Réservé à l’administration) D7

Sciences Fondamentales X

Nouveau projet : Projet reformule:

(Joindre une copie de la notification de l’avis de reformulation)

1.1. Domiciliation du projet Laboratoire de Physique des Rayonnements, Département de Physique, Faculté des Sciences, Université Badji Mokhtar, Annaba

1.2. Identification du projet 1.2.1‐ Nature de la recherche

Fondamentale

X

Appliquée

Titre du projet : Acronyme du projet : Intitulé du thème : Intitulé de l’axe : Intitulé du domaine :

Mots‐clés (12 max)

Durée estimée du projet

X Développement Formation Théories des Champs Non‐Commutatives à Partir des Modèles des Matrices et Physique Emergente NCQFT Théorie quantique des champs Particules élémentaires, champs et cosmologie Physique Géométrie Non‐Commutative, Théories des Champs Non‐Commutatives, Espaces Non‐Commutatifs, Sphère ‘Fuzzy’, Modèles de Yang‐Mills, Super symétrie, Modèles des Matrices, Géométrie et Gravitation Emergente, Méthodes de Groupes de Renormalisation, Méthodes Monte‐Carlo, Méthodes Cohomologiques, Transitions des Phases exotiques. 24 mois

1.2.2 Résumé du projet (250 mots)

Les quatre forces fondamentales de l’univers sont l’électromagnétisme, l’interaction forte, l’interaction faible et la force gravitationnelle. Les forces non‐gravitationnelles sont décrites par des théories quantiques des champs de Yang‐Mills sur un espace‐temps plat à quatre dimensions et elles sont unifiées dans le cadre du modèle standard de la physique des particules. La gravitation ne peut être unifiée avec les autres forces que dans le contexte de la théorie des cordes qui n'est pas une théorie des champs. Cependant à basses énergies la théorie des cordes dans un champ magnétique devient une théorie des champs sur un espace‐temps non‐ commutatif, i.e. un espace‐temps dont les coordonnées ne sont plus des nombres mais des opérateurs. Les espaces‐temps non‐commutatifs émergent également à des échelles de longueurs très petites comparables à l’échelle de Planck de la combinaison des principes de la relativité générale qui décrit la gravitation classique et la mécanique quantique. Ainsi le lien entre la non‐commutativité et la gravitation quantique semble être une hypothèse très raisonnable. La non‐commutativité est aussi la seule extension qui préserve la super symétrie qui est une symétrie centrale de la théorie des cordes.

1

La théorie des cordes ne peut pas être étudiée par simulations numériques. La difficulté majeure vient du fait qu’on ne peut pas maintenir la super symétrie dans un espace‐temps discret. Le seul modèle qui fait exception est le modèle des matrices de Yang‐Mills proposé par Ishibashi, Kawai, Kitazawa et Tsuchiya (IKKT) qui est supposé donner une régularisation non‐perturbative de la théorie des cordes de type IIB. Les modèles des matrices IKKT admettent des espaces non‐commutatifs comme solutions et donc ils peuvent aussi être utilisés pour régulariser non‐perturbativement les théories des champs de jauge non‐commutatives. Dans ce projet nous allons étudier les phénomènes qui émergent de façon dynamique dans les modèles des matrices IKKT tels que la géométrie émergente, la brisure spontanée de la supersymétrie et la gravitation émergente. Nous allons également utiliser le formalisme des matrices pour étudier les aspects quantiques des théories des champs scalaires non‐ commutatives.

1.3. Problématique du projet Sommaire (250 mots)

We propose to investigate two nonperturbative phenomena typical of noncommutative field theories which are known to lead to the perturbative instability known as the UV‐IR mixing which is absent in conventional field theory. The first phenomenon concerns the emergence of spacetime geometries in matrix models which describe perturbative noncommutative gauge theories on fuzzy backgrounds. These transitions from geometrical backgrounds to matrix phases make the description of noncommutative gauge theories in terms of fields only valid below a critical value of the coupling constant. We propose here to investigate this effect in four dimensional mass deformed supersymmetric Yang‐Mills matrix models and study the impact of supersymmetry on the geometry in transition and vice versa. This will also shed crucial light on the nonperturbative physics of the emergent noncommutative gauge theories and may open the door for Monte Carlo treatment of supersymmetry. The second phenomenon concerns the emergence of a nonuniform ordered phase in noncommutative scalar phi four field theory and the spontaneous symmetry breaking of translational/rotational invariance. This phenomena originates in the underlying matrix degrees of freedom of the noncommutative theory. We expect that in addition to the Wilson‐Fisher fixed point at zero noncommutativity there exists a novel fixed point at infinite noncommutativity corresponding to the quartic hermitian matrix model. We propose here to consider Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries and compute their nonperturbative phase structure. A renormalization group analysis which will determine the structure of the fixed points is also planned. A related question is to establish the renormalizability or lack of it of the phi four models on fuzzy spherical geometries.

1.4. Objectifs du projet Lister les objectifs scientifiques, techniques, technologiques, socio‐économiques et/ou socioculturels. (250 mots)

The scientific goals of this project can be summarized as follows: •

Gauge Fields, Supersymmetry and Emergent Geometry

1. The calculation of the phase structure of four dimensional mass deformed supersymmetric Yang‐Mills matrix models by means of the Monte Carlo method. 2. A nonperturbative analytical analysis of the phase transition associated with the emergent geometry by means of the deformation approach of cohomological models.

2

In the process we may clarify the impact of supersymmetry on the geometry in transition and vice versa. In particular to establish whether or not supersymmetry is spontaneously broken across the transition point is of paramount importance. This also may help us explain some of the nonperturbative physics of noncommutative gauge theory and allow us understand more fully the matrix phase. •

Noncommutative Scalar Fields

1. The calculation of the phase structure of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries by means of the Monte Carlo method.

2.

To establish renormalizability or lack of it of scalar phi four theories on fuzzy spherical geometries by means of the exact Polchinski renormalization group equation formulated in the matrix basis. The ultimate objective here is to determine the structure of the fixed points of noncommutative scalar phi four models. This is essentially an analytical question and as such the renormalization group equation in the matrix basis is the central tool.

1.5. Description du projet 1.5‐1‐ Etat des connaissances sur le sujet (500 mots)

Noncommutative field theories emerge in low energy dynamics of string theories. Seiberg and Witten showed that the dynamics of open strings moving in a flat space in the presence of a Neveu‐Schwarz B‐field and with Dp‐branes is equivalent to leading order in the string tension to gauge theory on Moyal‐Weyl space. In the case of open strings moving in a curved space with a three‐sphere metric the resulting effective gauge theory lives on a noncommutative fuzzy two‐sphere (Alekseev, Recknagel and Schomerus). Quantization of noncommutative field theories is plagued with stringy effects such as the notorious UV‐IR mixing (Minwalla, Van Raamsdonk and Seiberg) which translates into an instability of the geometry of spacetime itself (Bietenholz, Nishimura, Susaki and Volkholz). Noncommutative field theory can be expressed either in terms of star products or in terms of operator algebras. In the case of spherical geometries the operator algebras are isomorphic to finite dimensional matrix algebras. The IKKT Yang‐Mills matrix model in ten dimensions, i.e. the IIB matrix model is postulated to give a constructive definition of type IIB superstring theory (Ishibashi, Kawai, Kitazawa and Tsuchiya). The IKKT model exists also in four and six dimensions. These models are obtained from the reduction of supersymmetric Yang‐ Mills actions to zero dimensions. The most important mass deformation is provided by the Myers effect (Myers). Mass deformed IKKT Yang‐Mills matrix models in various dimensions admit the fuzzy sphere (Hoppe,Madore) as a vacuum solution. Noncommutative gauge theories can be realized by expanding the IKKT models around vacuum solutions. The connection between the UV‐IR mixing and the instability of the geometry of spacetime was also observed in three dimensional IKKT matrix models by Delgadillo Blando, O’Connor and Ydri. Thus connections between noncommutative gauge theories and matrix models run deep. It seems to indicate that matrices are the fundamental objects and that gauge fields, noncommutativity and geometry are derived concepts. The central hypothesis here will be that geometry, noncommutative gauge theory and supersymmetry should be nonperturbatively regularized with finite dimensional matrix models. For scalar field theory on Moyal‐Weyl spaces the UV‐IR mixing was shown to destroy the perturbative renormalizability of the theory (Chepelev and Roiban). The physics at very large distances is altered by the

3

noncommutativity which is supposed to be relevant only at very short distances. Grosse and Wulkenhaar found a modification of the free propagator which makes the theory renormalizable. It consist in adding a harmonic oscillator potential to the kinetic term. This is the only known renormalizable noncommutative scalar field theory. It is important to note the crucial role played by the matrix formulation of noncommutative Moyal‐Weyl spaces in their proof of renormalizability. The phase diagram of scalar phi four field theory consists of three phases instead of the usual two phases found in commutative scalar field theory. It was determined using the Monte Carlo method on the noncommutative torus by Ambjorn, Catterall from one hand and Bietenholz,Hofheinz,Nishimura from the other hand and on the fuzzy sphere by Garcia Flores, O'Connor and Martin. They observed a disordered phase, a uniform ordered phase and a novel nonuniform ordered phase which meet at a triple point. In the nonuniform ordered phase we have spontaneous breakdown of translational/rotational invariance. The most important analytical calculation of the phase structure of scalar phi four theory was done by Gubser and Sondhi. 1.5‐2‐ Méthodologie détaillée (300 mots)

The first goal is to investigate the exotic phase transitions from geometrical phases to matrix phases. The geometrical phases are generically given in terms of noncommutative gauge theories on fuzzy backgrounds. We propose to investigate this effect in four dimensional mass deformed supersymmetric IKKT Yang‐Mills matrix models and study the impact of supersymmetry on the geometry in transition and vice versa. The main tool of investigation is the Monte Carlo method. A nonperturbative analytical derivation of the critical properties based on matrix models techniques is unknown. We plan to apply the powerful deformation approach of cohomological models of Moore, Nekrasov and Shatashvili which will reduce the problem to a random matrix theory with one single hermitian matrix. This will shed crucial light on the nonperturbative physics of noncommutative gauge theories near the critical point. It is also an outstanding problem to determine precisely the structure of the matrix phase. It seems that this phase is universally dominated by commuting matrices with uniform eigenvalues distribution regardless of all other detail. It is not clear what happens to supersymmetry as we go from the matrix phase to the geometrical phase and vice versa. It may be spontaneously broken in one of the phases in analogy with the spontaneously broken rotational invariance in the nonuniform ordered phase in noncommutative scalar phi four theories. This whole program may thus open the door for a nonperturbative Monte Carlo treatment of supersymmetry. The second goal is to investigate the phase structure of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries and to establish renormalizability or lack of it of scalar phi four theories on fuzzy spherical geometries. The Grosse‐Wulkenhaar models are more general than conventional noncommutative scalar models as they involve an extra parameter in the phase diagram. The Monte Carlo method will be used to probe the phase diagram of Grosse‐Wulkenhaar scalar phi four models on planar and spherical geometries. In order to study renormalizability of the phi four models on fuzzy spherical geometries we will use the Polchinski renormalization group equation in the matrix basis. A renormalization group analysis a la Wilson applied in the matrix basis may allow us to determine the structure of the fixed points. The Polchinski equation can also be used to compute the relevant, irrelevant and marginal interactions and as a consequence to determine the structure of the fixed points. 1.5‐3‐ Principales références bibliographiques

1. N.Seiberg and E.Witten,``String theory and noncommutative geometry,'' JHEP 9909, 032 (1999). 2. A.Y.Alekseev, A.Recknagel and V.Schomerus,``Non‐commutative world‐volume geometries: Branes on SU(2) and fuzzy spheres,JHEP 9909, 023 (1999). 3. J.Hoppe, ``Quantum theory of a massless relativistic surface and a two‐dimensional bound state problem,'' Ph.D thesis,MIT,1982. 4. J.Madore, ``the fuzzy sphere,''Class.Quant. Grav.9, 69 (1992). 5. R.C.Myers, ``Dielectric‐branes,''JHEP 9912, 022 (1999). 6. N.Ishibashi, H.Kawai, Y.Kitazawa and A.Tsuchiya, ``A large‐N reduced model as superstring,''Nucl. Phys. B 498, 467 (1997).

4

7. H.Grosse and R.Wulkenhaar, ``Power‐counting theorem for non‐local matrix models and renormalisation,''Commun. Math. Phys.254, 91 (2005); ``Renormalisation of phi four theory on noncommutative R four in the matrix base,'' Commun. Math. Phys. 256, 305 (2005). 8. F.Garcia Flores, D.O'Connor and X.Martin, ``Simulating the scalar field on the fuzzy sphere,''PoS LAT2005, 262 (2006). 9. J.Ambjorn and S.Catterall, ``Stripes from (noncommutative) stars,''Phys. Lett. B 549, 253 (2002). 10. W.Bietenholz, F.Hofheinz and J.Nishimura,```Phase diagram and dispersion relation of the non‐ commutative lambda phi four model in d = 3,'' JHEP 0406, 042 (2004). 11. S.S.Gubser and S.L.Sondhi, ``Phase structure of non‐commutative scalar field theories,'' Nucl. Phys. B 605, 395 (2001). 12. W.Bietenholz, J.Nishimura, Y.Susaki and J.Volkholz, ``A non‐perturbative study of 4d U(1) non‐ commutative gauge theory: The fate of one‐loop instability,'' JHEP 0610, 042 (2006). 13. S.Minwalla, M.Van Raamsdonk and N.Seiberg, ``Noncommutative perturbative dynamics,''JHEP 0002, 020 (2000). 14. I.Chepelev and R.Roiban, ``Convergence theorem for non‐commutative Feynman graphs and renormalization,''JHEP 0103, 001 (2001). 15. G.W.Moore, N.Nekrasov and S.Shatashvili, ``D‐particle bound states and generalized instantons,'' Commun. Math. Phys. 209, 77 (2000). 16. R.Delgadillo‐Blando, D.O'Connor and B.Ydri, ``Geometry in transition: A model of emergent geometry,'' Phys. Rev. Lett. 100, 201601 (2008).

1.6. Impacts attendus Impacts directs et indirects (Scientifiques, socio‐économiques, socioculturels) The expected scientific impacts may include the following: 1. The conjecture that large N IKKT Yang‐Mills matrix models are of central importance to fundamental physics can be verified explicitly in several interrelated physical contexts. The relevance of these large N IKKT Yang‐Mills matrix models to fundamental physics can be summarized as follows: •

They may help us understand nonperturbative physics of noncommutative gauge theories by means of cohomological and random matrix models.

•

They may provide a nonperturbative definition of supersymmetry. Indeed supersymmetry in this language may prove to be tractable in Monte Carlo simulation.

•

They provide concrete models for emergent geometry and possibly emergent gravity. Indeed geometry in transition seems to be possible in all these matrix models. These matrix models seem also very suited to test the gauge/gravity duality.

2. A more coherent understanding of the quantum dynamics of noncommutative scalar field theories by means of matrix models techniques and renormalization group and Monte Carlo methods. The expected educative impacts include the following: 3. Supervision of two doctoral students.

1.7. Planning des taches / année Taches

semestre 1

1)Monte Carlo Study of IKKT Yang‐Mills Matrix Models in Dimension Four.

5

semestre 2

semestre 3

semestre 4

2)Cohomological Study of IKKT Yang‐Mills Matrix Models in Dimension Four. 3) Phase Structure of Grosse‐Wulkenhaar Scalar Phi Four Models by Means of Monte Carlo. 4) Renormalization of Grosse Wulkenhaar Scalar Phi Four Models on Fuzzy Spheres in Dimensions 2 and 4 by Means of the Polchinski Renormalization Group Equation.

6

1.8. Identification du porteur (chef) de projet Nom & Prénom Grade Spécialité Statut

YDRI Badis Maitre de Conférences B Physique Théorique Enseignant chercheur(1)

x Chercheur permanent(2)

[email protected]

Adresse professionnelle

Département de Physique Faculté des Sciences Université Badji Mokhtar B.P.12,23000 Annaba

Contacts

Tel : 038 87 53 99

Fax : 038 87 53 99

Associé(3)

Autre(4)

GSM :

Diplômes Obtenus (Graduation, Post‐Graduation)

Année

1 (Bacc.)

Bac. Mathématiques

1989

Lycée Mubarek El‐Mili

2 (L,M,Ing)

D.E.S Physique Théorique

1993

Université de Constantine

2001 2010

Université de Syracuse, New‐York, USA Université Badji Mokhtar Annaba

3 (doct.)

Ph.D. Physique Théorique Habilitation Universitaire

Etablissement

Participation à des programmes de recherche (nationaux, Internationaux, multisectoriels) Intitulé du Programme Origine des rayons cosmiques de très haute énergie (Code:D01120090017)

2010

Ministère (CNEPRU)

de

l'Enseignement

Supérieure

Lister vos trois derniers travaux les plus importants (recherche/recherche développement) 1

R.Delgadillo‐Blando, D.O’Connor, B.Ydri, Geometry in Transition: A Model of Emergent Geometry, Phys.Rev.Lett.100:201601, 2008.

2

D.Dou, B.Ydri, Topology Change from Quantum Instability of Gauge Theory on Fuzzy CP Two, Nucl.Phys.B771:167‐189, 2007.

3

B.Ydri, Quantum Equivalence of NC and YM Gauge Theories in 2D and Matrix Theory, Phys.Rev.D75:105008, 2007.

Visa du Chef d’établissement De rattachement :

Date : Signature :

7

2. Identification du partenaire socio‐économique du projet Nom & Prénom Grade Spécialité Statut

Enseignant chercheur(1)

x

Chercheur permanent(2)

Associé(3)

Autre (4)

Email Adresse professionnelle Contacts Tel : Fax : GSM : Diplômes Obtenus (Graduation, Post‐ Année Etablissement Graduation) 1(Lic,M,Ing) 2(Doct.) Participation à des programmes de recherche (nationaux, Internat., Sectoriels) Intitulé du Programme Année Organisme

A) Lister vos deux derniers travaux d’intérêt socio‐économiques 1 2 B) Autres Projets dans lesquels le partenaire du projet est impliqué Type de Durée du Ministère Projet(*) Intitulé Année de démarrage projet concerné A B C D

(1) Concerne les chercheurs universitaires (université, centre de recherche, école, institut). (2) Concerne les chercheurs permanents (centre, unité, institut de recherche) (3) Concerne les chercheurs associés (établissement de rattachement où le chef du projet exerce les fonctions de chercheur associé). (4)Préciser la fonction des personnels administratifs (cadre supérieur, fonctionnaire supérieur, etc. (*) Cocher la case correspondante : A : Projet par voie d’avis d’appel à proposition de projets (PNR.). B : Projet de recherche universitaire relevant de la CNEPRU. C : Projet de recherche sectorielle relevant des centres et unités de recherche sous tutelle du MESRS et hors MESRS. D : Projet de coopération.

Visa du Chef d’établissement de rattachement :

Date : Signature :

8

3. Chercheurs impliqués dans le projet (une fiche par chercheur) Nom & Prénom Grade

BOUCHAREB Adel Maitre Assistant A

Spécialité

Physique Théorique Enseignant chercheur(1) X Chercheur permanent(2)

Statut Email

[email protected]

Adresse professionnelle

Département de Physique Université Badji Mokhtar B.P.12, Annaba 23000

Associé(3)

Autre (4)

GSM : Tel : 038 87 53 99 Fax : 038 87 53 99 Diplômes Obtenus (Graduation, Post‐ Année Etablissement Graduation) Bac. Techniques Mathématiques Technicum Ali Boushaba (Constantine) 1 1989 D.E.S Université de Constantine 2 1993 Magister Université de Constantine 3 1996 Participation à des programmes de recherche Intitulé du Programme Année Organisme Contacts tel :

Adaptation du programme CORSIKA de simulation Monte Carlo des grandes gerbes de l’air aux énergies de l’expérience Pierre Auger (Code : D2301/08/04)

2004‐2006

Ministère de l’Enseignement Supérieur (CNEPRU)

Etude de l’interaction du rayonnement cosmique de très haute énergie avec l’atmosphère terrestre (Code : D01120060024)

2007‐2009

Ministère de l’Enseignement Supérieur (CNEPRU)

Origine des rayons cosmiques de très haute énergie (Code : D01120090017)

2010

Ministère de l’Enseignement Supérieur (CNEPRU)

A) Lister vos deux derniers travaux les plus importants 1 2

A.Bouchareb, G.Clement, C.M.Chen, D.V.Gal'tsov, N.G.Scherbluk and T.Wolf, G(2) generating technique for minimal D=5 supergravity and black rings, Phys.Rev.D76:104032,2007, Erratum‐ibid.D78:029901,2008. A.Bouchareb and G.Clement, Black hole mass and angular momentum in topologically massive gravity, Class.Quant.Grav.24:5581-5594,2007.

C) Tâches affectées au chercheur (à mentionner clairement): 1

Calculation of the phase diagram of Grosse-Wulkenhaar scalar models in two dimensions by means of the Monte Carlo method.

2

Cohomological models transition.

approach to the phenomena of geometry in

3 Visa du Chef d’établissement de rattachement :

Date : Signature :

9

4. Composante de l’équipe de recherche (Tableau anonyme : six personnes au maximum dont 3 chercheurs confirmés. Inscrire le responsable du projet en début de liste, ne pas inscrire de nom, ni l’intitulé de l’établissement de rattachement) Grade universitaire ou scientifique

Dernier diplôme obtenu

Tâche principale affectée dans le projet

1‐Maitre de Conférences B 2‐Maitre de conférences A

Ph.D

Supervision of the research project.

Ph.D

3‐Maitre Assistant A

Magister

4‐ Maitre Assistant A

Magister

5‐Maitre Assistant A

Magister

6‐Maitre Assistant A

Magister

Renormalization of Grosse‐Wulkenhaar Scalar Models on Fuzzy Spherical Geometries in 2 and 4 Dimensions. Cohomological and Monte Carlo Approaches of 4D IKKT Models. Monte Carlo Simulation of Grosse‐ Wulkenhaar Scalar Models and Cohomological Approach of 4D IKKT Models. Monte Carlo Simulation of 4D IKKT and Grosse‐Wulkenhaar Scalar Models. Renormalization of Grosse‐Wulkenhaar Scalar Models on Fuzzy Spherical Geometries in 2 and 4 Dimensions.

Emargement

‐Ne pas inscrire dans ce tableau les noms des membres de l’équipe, ni leurs établissements de rattachement. ‐Indiquer en tête de liste les informations relatives au porteur (chef) de projet.

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5. Equipements scientifiques disponibles 5.1‐ Matériel existant pouvant être utilisé dans l’exécution du projet Nature

Localisation

Observations

2 micros+1 imprimante laser

Laboratoire de Physique des Rayonnements U. d’Annaba

5.2 – Matériel et Mobilier de Bureau à acquérir pour l’exécution du projet Nature Montant en DA Destination Observations 3 micro‐ ordinateurs

300 000,00

2 disques durs externes de grande capacité

30 000,00

Photocopieur

100 000,00

Meubles de bureau

40 000,00

Imprimante

50 000,00

Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba Laboratoire de Physique des Rayonnements U. d’Annaba

Pour simulations numériques.

Pour simulations numériques.

Détailler la liste des matériels et mobiliers dont les montants sont mentionnés dans l’annexe financière.

5. Annexe financière : Budget et postes de dépenses prévisionnels (exprimés en DA) 1ère

Intitulés des postes de dépenses par année

Frais de séjour scientifique et de déplacement 500 000,00 à l’étranger Frais de séjour scientifique et de déplacement 100 000,00 en Algérie Frais d'organisation de rencontres scientifiques Honoraires des enquêteurs Honoraires des guides Frais de travaux et de prestations Matériels et instruments scientifiques Matériel informatique

480 000,00

Matériels d'expérience (animaux, végétaux, etc…) 40 000,00 Mobilier de bureau et de laboratoire Entretien et réparation 11

2ème 500 000,00 100 000,00

Produits chimiques Produits consommables Composants électroniques, mécaniques et audio‐ visuels 50 000,00 Accessoires et consommables informatiques 50 000,00 Papeterie et fournitures de bureau Périodiques Ouvrages et documentation scientifiques et 250 000,00 techniques Logiciels

50 000,00 50 000,00

300 000,00

Impression et Edition Affranchissements Postaux Communications téléphoniques, Fax, Internet Droits de douanes, Assurances Carburant TOTAL DES CREDITS OUVERTS :

1470 000,00

1000 000,00

Remarque : Les besoins financiers en devises doivent être exprimés en Dinars Algériens, après conversion au taux de change en cours.

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