cal errors), the value obtained by a simple-minded continuum extrapolation is considerably smaller than the experimental value. Unquenching does not seem to ...
gA on the lattice M. G¨ockeler1 2 , R. Horsley3 , D. Pleiter4 , P.E.L. Rakow5 , A. Sch¨afer2 , and G. Schierholz4 6 1 2 3 4 5
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Institut f¨ ur Theoretische Physik, Universit¨ at Leipzig, D-04109 Leipzig, Germany Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, D-93040 Regensburg, Germany School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK John von Neumann-Institut f¨ ur Computing NIC / DESY Zeuthen, D-15738 Zeuthen, Germany Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK Deutsches Elektronensynchrotron DESY, D-22603 Hamburg, Germany
Summary. We describe the techniques used in lattice evaluations of hadronic matrix elements like the neutron decay constant gA . Recent results for gA are presented and the influence of the finite quark mass and the finite volume on the determination of gA is briefly discussed.
1 What do we want to compute? Lattice evaluations of gA make use of nucleon matrix elements of the axial vector current. For the quark flavors q = u, d we have hproton, p, s|¯ q γµ γ5 q|proton, p, si = 2∆q · sµ ,
(1)
where p denotes the momentum of the proton and s is its spin vector. In parton model language, ∆q is the fraction of the proton spin carried by the quarks of flavor q. Assuming perfect isospin symmetry we can write ¯ µ γ5 d|proton, p, si = hproton, p, s|¯ hproton, p, s|¯ uγµ γ5 u − dγ uγµ γ5 d|neutron, p, si = 2gA · sµ (2) and hence gA = ∆u − ∆d.
2 What can we compute on the lattice? The basic observables in lattice QCD are Euclidean n-point correlation functions. Since space-time has been discretised (with lattice spacing a) the path integral has become a high-dimensional integral over a discrete set of field variables. As the (Grassmann valued) quark fields appear bilinearly in the action, they can be integrated out analytically leaving behind the determinant of the lattice Dirac operator and products of quark propagators. The remaining integrals over the gluon fields can then be evaluated by Monte Carlo methods. In the quenched approximation, which will be employed throughout most of this paper, the determinant of the Dirac operator is replaced by 1. This approximation saves a lot of computer time, but it is hardly possible to estimate its accuracy. Let us briefly sketch how hadronic matrix elements can be extracted from ratios of threepoint functions over two-point functions. First, one has to choose suitable interpolating fields for the particle to be studied. For a proton with momentum p one may take X (3) Bα (t, p) = e−ip·x ²ijk uiα (x)ujβ (x)(Cγ5 )βγ dkγ (x) x;x4 =t
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¯ where i, j, . . . are color indices, α, β, . . . are Dirac indices and C and the corresponding B, is the charge conjugation matrix. As the time extent T of our lattice tends to ∞, the two-point correlation function becomes the vacuum expectation value of the corresponding Hilbert space operators with the Euclidean evolution operator e−Ht in between, i.e. we have, omitting Dirac indices and momenta for simplicity: T →∞ ¯ ¯ . hB(t)B(0)i = h0|Be−Ht B|0i (4) If in addition the time t gets large, the ground state |protoni of the proton will dominate ¯ and the two-point function will decay the sum over intermediate states between B and B, exponentially with a decay rate given by the proton energy Eprot : T →∞
t→∞
¯ ¯ ¯ + ··· hB(t)B(0)i = h0|Be−Ht B|0i = h0|B|protonie−Eprot t hproton|B|0i
(5)
Of course, if the momentum vanishes, we have Eprot = mprot , the proton mass. Similarly we have for a three-point function with the operator O whose matrix elements we want to calculate: ¯ hB(t)O(τ )B(0)i
T →∞
¯ = h0|Be−H(t−τ ) Oe−Hτ B|0i −Eprot (t−τ ) ¯ + ··· = h0|B|protonie hproton|O|protonie−Eprot τ hproton|B|0i ¯ = h0|B|protonie−Eprot t hproton|B|0ihproton|O|protoni + ··· (6)
if t > τ > 0. Hence the ratio R≡
¯ hB(t)O(τ )B(0)i = hproton|O|protoni + · · · ¯ hB(t)B(0)i
(7)
will be independent of the times τ and t, if all time differences are so large that excited states can be neglected, and then R yields the desired matrix element. The proton three-point function for a two-quark operator contains quark-line connected as well as quark-line disconnected pieces. In the quark-line connected contributions the operator is inserted in one of the quark lines of the nucleon propagator, while in the disconnected pieces the operator is attached to an additional closed quark line which communicates with the valence quarks in the proton only via gluon exchange. In the limit of exact isospin invariance considered in this paper, the disconnected contributions of the u quarks and the d quarks cancel in the case of non-singlet two-quark operators. Fortunately, the operator needed for the evaluation of gA in Eq.(2) is of this type so that we do not have to cope with the disconnected contributions, which are very hard to compute.
3 Chiral symmetry Chiral symmetry plays an important role in hadronic physics. Unfortunately, it is not straightforward to implement it on the lattice. “Traditional” formulations, like (improved) Wilson fermions break chiral symmetry explicitly at finite lattice spacing a. This has the consequence that the axial vector current has to be renormalized and chiral symmetry is only restored in the continuum limit a → 0. However, the last years have seen a remarkable progress in this field. We have now “chirally symmetric” lattice formulations of the Dirac operator based on solutions of the Ginsparg-Wilson relation. These enjoy a lattice version of chiral symmetry even at finite lattice spacing such that physical consequences of chiral
gA on the lattice
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symmetry (e.g. Ward identities) hold already at finite a. In particular, there is an axial vector current which is not renormalized. However, there is a price to be paid for these nice properties: Chirally symmetric lattice fermions need considerably more computer time than the “traditional” formulations. Therefore phenomenologically interesting results obtained with lattice fermions of this kind are only slowly beginning to appear. A related problem is the chiral extrapolation: In the foreseeable future it will not be possible to perform simulations at the physical values of the masses of the u and d quarks. Hence results obtained at higher masses have to be extrapolated to the physical mass values. This extrapolation is, of course, the more reliable the smaller the masses in the simulation are. Since Ginsparg-Wilson fermions allow us to work with considerably lighter quarks than most other lattice fermions, their use will improve the quality of the results also in this respect.
4 Results In Fig.1 taken from the review [1] we show our own results (QCDSF and UKQCD collaborations) obtained with quenched and unquenched O(a)-improved Wilson fermions as well as results from the LHPC and SESAM collaborations [2] who work with quenched and unquenched unimproved Wilson fermions. The gA values from the simulations have been extrapolated linearly in the quark mass to the chiral limit and are plotted versus a 2 in units of the “force scale” r0 whose phenomenological value is ≈ 0.5 fm. Although the agreement between the various simulations is rather good (within the partially quite large statistical errors), the value obtained by a simple-minded continuum extrapolation is considerably smaller than the experimental value. Unquenching does not seem to have a big effect, which may be due to the rather large quark masses in the unquenched simulations. Could the discrepancy between the simulations and experiment be caused by the chiral extrapolation? The data at finite masses which are behind the results displayed in Fig.1 1.3
gA 1.2
1.1
1.0
0.9
QCDSF quenched,O(a)-improved LHPC+SESAM quenched,unimproved QCDSF+UKQCD unquenched,O(a)-improved LHPC+SESAM unquenched,unimproved
0.00
0.01
0.02 0.03 2 (a/r 0)
0.04
0.05
Fig. 1. gA from quenched and unquenched simulations versus a2 in units of r0 ≈ 0.5 fm. The experimental value is indicated by the asterisk.
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Fig. 2. QCDSF results for gA from quenched improved Wilson fermions plotted versus the square of the pseudoscalar mass (proportional to the quark mass). The line shows a linear chiral extrapolation. The experimental value is indicated by the asterisk.
do not show any deviation from linearity when plotted versus the quark mass. This is exemplified in Fig. 2 where the quenched QCDSF data with lattice artefacts subtracted are plotted versus the square of the pseudoscalar mass in units of r0−1 . But will this linearity persist down to the physical mass? At small masses one has to worry about finite size effects, and simulations with unimproved Wilson fermions which we are performing show indeed some indications of such effects, but no significant deviation from linearity yet. Clearer evidence for finite size effects in gA has been found by the RBC collaboration [3] in quenched simulations with domain wall fermions (an approximate realization of GinspargWilson fermions). For a more theoretical discussion of the volume dependence of g A we refer to the recent papers by Jaffe and Cohen [4]. Hopefully, the influence of the finite volume will soon be better understood leading also to a clearer picture of the quark mass dependence. Recent results from chiral perturbation theory [5] shed more light on the problem of the chiral extrapolation. These developments should eventually enable us to increase the reliability of the lattice computations of gA .
References 1. M. G¨ ockeler, R. Horsley, D. Pleiter, P.E.L. Rakow, A. Sch¨ afer and G. Schierholz, hep-lat/0209160 2. D. Dolgov, R. Brower, S. Capitani, P. Dreher, J.W. Negele, A. Pochinsky, D.B. Renner, N. Eicker, T. Lippert, K. Schilling, R.G. Edwards and U.M. Heller, hep-lat/0201021 3. S. Sasaki, T. Blum, S. Ohta and K. Orginos, Nucl. Phys. B (Proc. Suppl.) 106 (2002) 302; S. Ohta (RBC collaboration), hep-lat/0210006
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4. R.L. Jaffe, Phys. Lett. B529 (2002) 105; T.D. Cohen, Phys. Lett. B529 (2002) 50 5. T.R. Hemmert, M. Procura and W. Weise, preprint TUM-T39-02-22 (2002); W. Weise, talk at PANIC02, Osaka, preprint TUM-T39-02-23 (2002)
