arXiv:0807.3270v3 [hep-th] 25 Nov 2008
LYCEN 2008-11
Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative φ4 Theory Daniel N. Blaschke∗, Fran¸cois Gieres†, Erwin Kronberger∗, Thomas Reis∗, Manfred Schweda∗ and Ren´e I.P. Sedmik∗ November 20, 2008 ∗
Institute for Theoretical Physics, Vienna University of Technology Wiedner Hauptstrasse 8-10, A-1040 Vienna (Austria) †
Universit´e de Lyon, Universit´e Lyon 1 and CNRS/IN2P3, Institut de Physique Nucl´eaire, Bat. P. Dirac, 4 rue Enrico Fermi, F - 69622 - Villeurbanne (France)
E-mail:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected]
Abstract In this paper we elaborate on the translation-invariant renormalizable φ4 theory in 4-dimensional non-commutative space which was recently introduced by the Orsay group. By explicitly performing Feynman graph calculations at one loop and higher orders we illustrate the mechanism which overcomes the UV/IR mixing problem and ultimately leads to a renormalizable model. The obtained results show that the IR divergences are also suppressed in the massless case, which is of importance for the gauge field theoretic generalization of the scalar field model.
Contents 1 Introduction
2
2 Propagator and vertex
4
3 One-loop corrections for propagator and vertex 3.1 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5 7
4 One-loop renormalization
8
5 Two and higher loops
10
6 Concluding remarks
12
A 1-loop integrals A.1 Correction to the propagator . . . . . . . . . . . . . . . . . . . . . . . . A.2 Vertex correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 13 14
B 2-loop integrals
14
C n-loop integrals
15
1
Introduction
The simplest generalization of the φ4 theory from ordinary space to non-commutative space is given by the action [1]-[3] Z λ 1 µ 2 4 ∂ φ ⋆ ∂µ φ + m φ ⋆ φ + φ ⋆ φ ⋆ φ ⋆ φ . (1) Snaive [φ] ≡ d x 2 4! R4
Here, φ denotes a relativistic scalar field in 4-dimensional Euclidean space, and the non-commutativity is implemented by the Weyl-Moyal star product [4], [xµ ⋆, xν ] ≡ xµ ⋆ xν − xν ⋆ xµ = iθµν ,
(2)
where the parameters θµν = −θνµ are real constants. In the following, we assume that the deformation matrix (θµν ) has the simple block-diagonal form 0 1 0 0 −1 0 0 0 with θ ∈ R . (3) (θµν ) = θ 0 0 0 1 , 0 0 −1 0 2
We note that the parameter θ is necessarily quite small on physical grounds, e.g. see reference [5] for a recent discussion of experimental aspects. By now it is well established that the non-commutative model described by the action (1) is not renormalizable due to the fact that it suffers from the infamous UV/IR mixing problem that plagues Moyal deformed field theories [1, 3]. For this reason, we refer to the action (1) as the na¨ıve model [6]. A procedure to obtain a renormalizable theory consists of adding a properly chosen term to the action (1) in order to overcome the UV/IR mixing problem. In doing so, one clearly changes the initial model, but one obtains a consistent quantum field theory. Recently, three such proposals have been made and each of them has been proved to provide a renormalizable theory (see reference [7] for a short review). The first proposal, put forward by Grosse and Wulkenhaar [8], consists of adding a harmonic oscillator-like potential of the form x˜2 φ2 (where x˜µ ≡ θµν xν ). A notable shortcoming of this model is that it breaks translation invariance. We also remark that the corresponding quantum corrections are hard to evaluate due to the occurrence of the Mehler kernel which is quite involved. The generalization of this model to noncommutative gauge theories is not obvious, though some work in this direction has been done [9]. Another proposal, due to Grosse and Vignes-Tourneret [10], consists of adding a R 4 2 non-local term of the form θµ4 d x φ(x) to the action (1). This approach yields a minimalist translation-invariant φ4 -theory in non-commutative space. The third proposal, made by Gurau, Magnen, Rivasseau and Tanasa [6], consists of adding a non-local counterterm φ θ21 φ for the quadratic IR divergence of the na¨ıve model. This procedure provides a solution for the UV/IR mixing problem while maintaining translation invariance. In the present work, we focus on the latter model, i.e. on the action Z a2 λ 1 µ 2 4 ∂ φ ⋆ ∂µ φ + m φ ⋆ φ − φ ⋆ φ + φ ⋆ φ ⋆ φ ⋆ φ , (4) S[φ] ≡ d x 2 4! R4
in Euclidean space. In expression (4), the parameter a is assumed to have the form a = a′ /θ where a′ represents a real dimensionless constant. One of our motivations is that the non-local term has been generalized [11] to U(1) gauge theories whose renormalizability still remains an open problem. Therefore, it should be quite useful to gain deeper insight into the quantum corrections for the model (4) since the multiscale analysis [12] which was used to establish its renormalizability cannot be applied to gauge field theories as it breaks gauge invariance. Accordingly we will study perturbative corrections to the propagators and vertices for the theory (4), and in particular the vanishing mass limit which is of interest for gauge theories. In doing so, we will explicitly exhibit the improvements for the quantum theory brought about by the non-local term. Our paper is organized as follows: After introducing the Feynman rules for the 3
model in the next section, we deal with the renormalization procedure at the oneloop level in Sections 3 and 4. Section 5 is devoted to two and more loops, and exhibits the mechanism that removes the potential IR divergences, which ultimately leads to the renormalizability of the model [6]. In this respect, the scaling behaviour of the propagator determined by the action (4) plays an essential role. The appendices describe the techniques for handling the loop calculations and should apply to generic higher loop graphs. Although our investigations concern the massive theory, we also consider the limit m → 0 in several instances and find well defined results. This issue is of importance for the gauge field theoretic generalization of the model (4) which was recently proposed [11] and which is to be further discussed elsewhere [13].
2
Propagator and vertex
Concerning the action (4) we recall [11] that the operator 1/ denotes the Green function associated to the 4-dimensional Laplacian ≡ ∂ µ ∂µ = ∂12 + . . . + ∂42 and that it is given by −1/k 2 in momentum space. Accordingly, the propagator in momentum space reads k
= G(k) =
1 k 2 + m2 +
a2 k2
.
(5)
Note that this propagator has a “damping” behaviour for vanishing momentum [6], lim G(k) = 0 ,
k→0
which allows to avoid potential IR divergences in higher loop graphs (see Section 5). 2 This property is due to the non-local term φ a φ in the action (4) and represents a crucial difference (and improvement) compared to the na¨ıve model (1). In terms of the notation k˜µ ≡ θµν k ν , the vertex in momentum space has the following form (see reference [2]): k1 k4 k2 k3
= V (k1 , k2 , k3 , k4 ) " λ 1 ˜ 1 ˜ 4 4 = (2π) δ (k1 + k2 + k3 + k4 ) cos k1 k2 cos k3 k4 3 2 2 # 1 ˜ 1 ˜ 1 ˜ 1 ˜ k1 k3 cos k2 k4 + cos k1 k4 cos k2 k3 . + cos 2 2 2 2
4
(6)
3
One-loop corrections for propagator and vertex
To start with, we determine the relevant corrections for the 1PI two-point and fourpoint functions at one-loop level.
3.1
Propagator k
p
k
p
Figure 1: Planar and non-planar one-loop corrections for the propagator The one-loop correction to the propagator, which is described by the Feynman diagrams in Fig. 1, corresponds to the following integral (including a symmetry factor 1/2): Z λ d4 k 2 + cos(k p˜) Π(p) = − ≡ Πplan + Πn-pl (p) . (7) 6 (2π)4 k 2 + m2 + ka22 R4
Here, Πplan and Πn-pl denote the planar and non-planar parts, respectively. We note that 1 X iηkp˜ e , (8) cos(k p˜) = 2 η=±1 and 1 2 k + m2 +
a2 k2
=
k2 +
k2 m2 2 2
m2 1 + ζ 2M 1 X 2 , = 2 m 2 2 ζ=±1 k + 2 + ζM 2 − M4
(9)
q m4 − a2 (which may be real or purely imaginary depending on the where M 2 ≡ 4 value of a). Taking into account these identities, the non-planar part can be evaluated straightforwardly by using Schwinger’s exponential parametrization (see Appendix A
5
for details): m2 1 + ζ 2M d4 k X 2 Π eiηkp˜ 2 4 m 2 2 (2π) η,ζ=±1 k + 2 + ζM s ! r m2 2 + ζM λ X 2 m2 2 1 + ζ 2M K1 , = − p˜ 2 m2 + ζM 2 2 48π 2 ζ=±1 p˜ 2 n-pl
λ (p) ≡ − 24
Z
(10)
where K1 is the modified Bessel function. The result is finite for p˜ 2 6= 0, i.e. if θ 6= 0 and p 6= 0. In the following, we will focus on the IR behaviour of the model, i.e. the limit 2 p˜ → 0. For small z, the function 1z K1 (z) admits the expansion 1 1 z2 5 1 1 1 γE − ln 2 − + ln z + γE − ln 2 − + O(z 4 ) , K1 (z) = 2 + ln z + z z 2 2 2 16 4 (11) where γE denotes the Euler-Mascheroni constant. Thus, for p˜ 2 ≪ 1, the expression (10) behaves like " q 4 −λ 2 2 n-pl m4 4 + m ln p˜ Π (p) = −M 4 6(4π)2 p˜ 2 s 2 # m 2 + M 4 m + O(1) , (12) ln m22 + M 2 + 4M 2 2 − M 2 and thereby involves a quadratic IR divergence (and a subleading logarithmic IR diver2 gence). For a → 0 (i.e. M 2 → m2 ) this result reduces to the one which was previously found [1, 2] for a = 0, i.e. for the na¨ıve model. The integral defining the planar part does not contain a phase factor eiηkp˜ and is therefore UV divergent. It can be regularized by introducing a cutoff Λ and subsequently taking the limit p˜ 2 → 0, as explained in Appendix A.1. The final result can be expanded for large values of Λ, yielding " q −λ 2 2 plan 1 m4 4 4Λ + m ln Λ2 −M Π (Λ) = 4 regul. 3(4π)2 s 2 # m 2 + M m4 + M 2 + 4M + O(1). (13) ln m22 2 − M2 2
6
3.2
Vertex
The basic one-loop correction to the vertex is given by the three connected graphs that can be constructed with four external legs [14, 2, 15]: V1-loop (p1 , p2 , p3 , p4 ) =
1 3
"
p1
k
p4
1 2
p1
p3
p2
k p4
+ p2
p3
p1
p4
p2
k p3
+
#
.
(14)
This expression can be evaluated by proceeding along the lines of reference [2]: by applying the Feynman rules (5) and (6), and by taking advantage of the identity (9) we find that (14) reads " ! m2 m2 4 4 4 2 X Z 1 + ζ 1 + χ X X 2M 2 2M 2 dk λ 1 + 14 eik(˜p1 +˜pi ) + 12 eikp˜i m2 2 2 27 ζ,χ=±1 (2π)4 k + 2 + ζM i=2 i=1 1 1 1 + + × 2 2 2 (p1 +p2 −k)2 + m2 +χM 2 (p1 +p3 −k)2 + m2 +χM 2 (p1 +p4 −k)2 + m2 +χM 2 # ik(˜ p1 +˜ p2 ) ik(˜ p1 +˜ p3 ) ik(˜ p1 +˜ p4 ) e e e +3 + + . (15) 4
2
(p1 +p2 −k)2 + m2 +χM 2
2
(p1 +p3 −k)2 + m2 +χM 2
2
(p1 +p4 −k)2 + m2 +χM 2
Thus, we again have an expression involving planar and non-planar parts (the latter involving a phase factor of the form eikq˜). The generic integral for the non-planar part is given by m2 m2 Z 1 + ζ 2M 2 1 + χ 2M 2 eik(˜p+˜q) X d4 k I(p, q) ≡ 2 2 (2π)4 k 2 + m2 + ζM 2 (p − k)2 + m2 + χM 2 ζ,χ=±1 =
X ζ,χ
Z1 ei(1−ξ)p˜q 2 m 1 + ζ 2M 2 1 + χ 2M dξ 2 8π 2 0 s ! 2 m (˜ p + q˜)2 ξ(1 − ξ)p2 + × K0 + (χ + ξ(ζ − χ)) M 2 . 2 m2
(16)
Here, p denotes the total incoming momentum, and q represents one of the variables pi (see Appendix A for calculational details). For small arguments the modified Bessel function K0 can be expanded according to K0 (z) = − ln z + ln 2 − γE + O z 2 , (17)
7
from which we can derive the following estimation for small external momenta p and q: q 1 (˜ p + q˜)2 m4 1 m4 4 1 − |I(p, q)| ≤ ln − M + 2γ − E 4 4M 4 (2π)2 4 2 s 2 m 2 + M m4 m2 + 1 + 4M (18) ln m22 . 4 4M 2 2 − M 2
The planar part of expression (14) can again be evaluated by introducing a cut-off Λ (as was discussed for the propagator in Appendix A.1): the final result directly follows from (16) and (18) by replacing (˜ p + q˜)2 with 1/Λ2 .
4
One-loop renormalization
According to the standard renormalization procedure, the dressed propagator at oneloop level is given by ≡ ∆′ (p) =
p
1 1 1 + Σ(Λ, p) , A A A
(19)
where a2 , p2 Σ(Λ, p) ≡ Πplan regul. (Λ) + Πn-pl (p) .
A ≡ p2 + m2 +
Since A 6= 0, we can apply the formula
1 1 1 1 1 1 1 = − B = − B + O(B 2 ) , A+B A A A+B A A A which allows us to rewrite expression (19) to order Σ (i.e. to order λ) as ∆′ (p) =
1 p2
+
m2
+
a2 p2
− Σ(Λ, p)
.
(20)
The contribution Πn-pl (p) to Σ(Λ, p) is finite except for vanishing external momentum p. The expansion for small values of p˜ 2 , as given in Eq. (12), reveals a quadratic and a logarithmic IR divergence1 at p˜ 2 = 0. The quadratically divergent term obviously has 1
In this respect, we should emphasize that these IR divergences are fundamentally different from the ones encountered in quantum field theories on commutative space since they are tied to the UV divergences and only appear in non-planar diagrams which are not present in usual QFT [4]. Thus, these divergences cannot be regularized by introducing an infrared regulator (like an additional mass parameter).
8
2
the same structure as the term ap2 appearing in the bare propagator (5). Henceforth, we will absorb it by a finite renormalization of the parameter a2 . From the expansions (12) and (13) it follows that, to order λ, we have ∆′ (p) =
Z p2 + m2r +
a2r p2
+ f (p2 )
,
(21)
where Z ≡ 1 + λαθ2 , (α ∈ R) , q λ 2 2 2 2 m4 1 4 mr ≡ m + 4Λ + m ln Λ2 −M + regular for Λ → ∞ , 4 3(4π)2 2 2 2 2 2 + αa θ , ar ≡ a + λ 3(4πθ)2 λ 2 f (p2 ) ≡ (22) m ln θ2 p2 + O((θp)4 ) . 2 6(4π)
The quantities mr and ar represent the renormalized mass and a-parameter to oneloop order, and the function f (p2 ) is analytic for θ 6= 0 and p2 > 0. The expression Z amounts to a finite wave function renormalization2 . The logarithmic singularity of f (p2 ) for vanishing external momentum p represents a mild divergence which is unproblematic for the amplitudes (see also the multiscale analysis [6]). The constant α appearing in Z and in 2 2 ′2 ′2 2 2 ′2 (23) + αθ a ar ≡ θ ar = a + λ 3(4π)2 is determined by the numerical factor that occurs in the expansion of Σ(Λ, p) at order p˜ 2 (see equations (11) and (12)). We have 2 5 2 αθ = ln 2 + − γE θ2 m4 − a′2 , (24) 2 3(16π) 4
which is positive for θ2 m4 > a′2 . However, even in the case where α < 0, the one-loop renormalized parameter a′2 r is positive provided s 2 1 B 1 B ′2 2 4 2 4 a < +θ m + +θ m +B, 2 λA 4 λA 16 2 , B≡ > 0. (25) where A≡ 2 3(4π) ln 2 + 45 − γE 2
For the φ4 -theory on commutative space, there is no wave function renormalization at one-loop order, but this is a peculiarity of this theory [14].
9
Since θ is necessarily quite small on physical grounds, the dominating factor in the previous inequality is 1/λ. Hence, even for m = 0, the parameter a′2 r is positive for small values of the coupling constant λ (more precisely for a′2 < 103 /λ). The renormalized coupling constant λr at one-loop order is obtained by considering the planar part of (14). One finds an expression which is similar to the one in the commutative theory. The non-planar part of (14) again involves a logarithmic singularity (see Equation (18)).
5
Two and higher loops
In order to exhibit how the non-local term φ θ21 φ improves the IR behaviour of the na¨ıve model at the higher loop level, we consider a non-planar tadpole graph with non-planar insertions (see Fig. 2a for one insertion and Fig. 2b for several insertions). Since we q2 q3
q
q1
k p
p
p
p
k (a) with 1 insertion
(b) with 3 insertions
Figure 2: Non-planar 2-loop and 4-loop graphs. are only concerned about the IR divergences, we limit ourselves to the first (i.e. most singular) term in the expansion (12) of Πn-pl , hence [1] we consider the approximation Πn-pl (k) ∝ 1/k˜2 . Within this approximation, a graph with n non-planar insertions is described by the expression n npl-ins.
Π
X Z d4 k eiηkp˜ (p) ≡ λ n (2π)4 k˜2 k 2 + m2 + η=±1 2
a2 n+1 k2
.
(26)
For the na¨ıve model (where a = 0), the integral (26) involves an IR divergence for n ≥ 2, because the integrand behaves like (k 2 )−n for k 2 → 0. In contrast, for the model 10
under consideration (where a 6= 0), the integrand behaves like k˜ 2 1 n n+1 = . ′2 )n+1 a2 2 ˜ (a k k2
(27)
Thus, the propagator (5) “damps” the IR-dangerous insertions and therefore cures potential IR problems in the integral (26). This is a nice demonstration of the mechanism leading to the renormalizability of the present model. In this respect we recall that its renormalizability has been proved quite generally in reference [6] using multiscale analysis. In the following we will present a more detailed mathematical analysis of the IR behaviour of the graph with n non-planar insertions. In fact, the integral (26) can be evaluated by the same techniques as those applied in the previous section (see Appendices B and C for details). For n = 1, one finds that ( s q q s 2 2 2 −λ m+ m− Π1 npl-ins. (p) = m2 K1 K1 m2+ p˜ 2 − m2− p˜ 2 2 2 6 2 16π θ M p˜ p˜ 2 q q ) + M 2 m2+ K0 m2+ p˜ 2 + m2− K0 m2− p˜ 2 , (28) where m2± ≡
m2 2
± M 2 . For generic n, we get Πn npl-ins. (p) = λ2
P
Jn (p) with Jn (p)
η=±1
given by the integral (45). By expanding the expressions for n = 1 and n = 2 for small external momentum p˜ 2 , one obtains s " # 2 λ m2+ 1 npl-ins. 4 2 m2 m4 Π (p) = M − 4 ln + M 2 + O(˜ p 2) , (29) 2 2 2 6 16 π θ M m− s " # 2 4 4 2 2 m m 3m − 4M λ + p 2) . ln − 6 8 + O(˜ Π2 npl-ins. (p) = 256 π 2θ4 M 10 m2− M We note that the latter result diverges in the limit a → 0. This fact again illustrates how the propagator (5) regularizes graphs which diverge in the na¨ıve model. In the limit m → 0 (i.e. for a massless field), the expressions (29) reduce to finite quantities: λ2 1 npl-ins. = Π (p) + O(˜ p 2) , 32 π |θa′ | m=0 λ2 + O(˜ p 2) . (30) = Π2 npl-ins. (p) 128 π|θa′3 | m=0 11
Henceforth, in contrast to the na¨ıve model (e.g. see reference [2]), the higher-loop graphs in Fig. 2b do not diverge for m → 0. In other words, the IR divergent insertion 1/k˜2 does not cause any harm in these higher-loop graphs, even for a massless field. This is an important feature for the gauge field theoretic generalization of the model (4) which was introduced in reference [11], and which is to be addressed elsewhere [13].
6
Concluding remarks
Concerning the quantum corrections for the model under consideration, we note the quite recent work [16, 17, 18] which is complementary to our study: the second work is devoted to the calculation of the one-loop beta function and to the parametric representation of Feynman amplitudes (according to the topology of the considered graphs). The approach of references [16, 17] is based on the assumption that 4a2 < m4 whereas our analysis does not require any restrictions on the parameters of the theory (as was mentioned in Section 3). While some of the obtained expressions allow for a smooth limit a → 0 towards the corresponding results of the na¨ıve (non-renormalizable) theory, this is — as expected — not the case for the higher loop corrections. As is apparent from the calculations outlined in the appendices, the Schwinger parametrization is quite convenient for determining the quantum corrections for the scalar field model. Since the gauge field theoretic generalization of this model extends the propagator for a (massless) scalar field, the same techniques should allow us to tackle the problem of IR divergences and discuss the issue of renormalizability for translation-invariant gauge field theories on non-commutative space.
Acknowledgments The work of D. N. Blaschke, E. Kronberger, T. Reis and R. I. P. Sedmik was supported by the “Fonds zur F¨orderung der Wissenschaftlichen Forschung” (FWF) under contracts P20507-N1 and P19513-N16. F. Gieres also acknowledges the financial support of this program for his visit to the Vienna University of Technology, during which part of the present work has been carried out. He furthermore acknowledges partial support by the “Agence Nationale de la Recherche” under contract ANR-07-CEXC-010. The authors would like to thank F. Delduc, H. Grosse and M. Wohlgenannt for enlightening discussions.
12
A
1-loop integrals
A.1
Correction to the propagator
By virtue of equations (7)-(9), thePnon-planar part of the one-loop correction to the λ I(p) with propagator reads Πn-pl (p) = − 24 η=±1
Z d4 k X eiηkp˜ m2 1 + ζ 2M 2 I(p) ≡ . 2 (2π)4 k 2 + m2 + ζM 2 ζ=±1
For m > 0 and a 6= 0, the combination can use the Schwinger parametrization: 1 k2 +
m2 2
+ ζM 2
m2 2
=
(31)
+ ζM 2 has a positive real part so that we
Z∞
dα e−α(k
2 2 + m +ζM 2 ) 2
.
(32)
0
The integral over k can be carried out after completing the square in the exponent: I(p) =
X ζ
=
2 Z d4 k Z∞ iηk p˜ m 2 2 1 + ζ 2M 2 dα exp −α k − −α + ζM α 2 (2π)4 m2
0
X1+ζ ζ
m2 2M 2
(4π)2
Z∞ 0
2 dα p˜ 2 m 2 . exp − −α + ζM α2 4α 2
(33)
Although it is not necessary here, we introduce a cutoff Λ into I(p) by making the replacement p˜ 2 → p˜ 2 + Λ12 . In fact [2], this will allow us to evaluate the planar part Πpl below. The integral (33) can be looked up [19] and the result depends upon the modified Bessel function K−1 = K1 : s ! r m2 2 2 X 1 + ζ m22 + ζM 2M 2 Iregul. (p, Λ) = p˜ 2 + Λ12 m2 + ζM 2 . (34) K1 2 2+ 1 (2π) p ˜ 2 Λ ζ Taking the limit Λ → ∞ leads to the result (10) for the non-planar part Πn-pl (p). The result (13) for the regularized planar part Πplan regul. (Λ) also follows from the expression (34) by taking the limit p˜ 2 → 0.
13
A.2
Vertex correction
The non-planar integrals in the vertex correction (15) have the following form: m2 m2 1 + ζ 2M 1 + χ eik(˜p+˜q) X Z d4 k 2 2M 2 I(p, q) ≡ 2 4 m2 2 + m2 + ζM 2 2 (p − k) + + χM (2π) k 2 2 ζ,χ =
X ζ,χ
Z d4 k Z∞ Z∞ h m2 m2 1 + χ 2M 2 dα dβ exp − (α + β) 1+ 2 (2π)4 0 0 i ik(˜ p + q˜) + 2βkp 2 − αζM 2 − β p2 + χM 2 . (35) − (α + β) k − α+β m2 ζ 2M 2
After carrying out the integration over k, and performing the change of variables (α, β) → (λ, ξ) with α = λξ and β = λ(1 − ξ) (where λ ∈ [0, ∞[ and ξ ∈ [0, 1]), one obtains an integral over λ, Z1 Z∞ dλ 1 X 2 2 m m 1 + ζ 2M I(p, q) = 1 + χ 2M dξ 2 2 (4π)2 λ ζ,χ 0 0 h n q )2 + i(1 − ξ)p˜ q − λ ξ(1 − ξ)p2 + × exp − (˜p+˜ 4λ
(36) m2 2
+ (χ + ξ(ζ − χ)) M
2
io
,
which can be expressed [19] in terms of the modified Bessel function K0 , see Eq. (16).
B
2-loop integrals
We use the decomposition (9) and the Schwinger P parametrization to evaluate the inteJ1 (p) with gral (26) for n = 1. Hence Π1 npl-ins. (p) = λ2 η=±1
1X J1 (p) ≡ 4 ζ,χ
Z
2
2
m m 1 + χ 2M 1 + ζ 2M d4 k eiηkp˜ 2 2 4 ˜2 m2 m2 2 2 2 (2π) k k + 2 + ζM k + 2 + χM 2
Z d4 k Z∞ Z∞ Z∞ 1 X 2 2 m m 1 + χ 2M = 2 dα dβ dγ 1 + ζ 2M 2 2 4θ ζ,χ (2π)4 0 0 0 h i 2 × exp − (α + β + γ)k 2 − (α + β) m2 − (αζ + βχ)M 2 + iηk p˜ .
(37)
14
After carrying out the integration over k and after the change of variables (α, β, γ) → (λ, ξ, σ) with α = λξσ , β = λ(1 − ξ)σ , γ = λ(1 − σ) , and λ ∈ [0, ∞[ , ξ ∈ [0, 1] , σ ∈ [0, 1] ,
(38)
one obtains J1 (p) =
2
m X 1 + ζ 2M 2 ζ,χ
m2 Z∞ 1 + χ 2M 2
4θ2 (4π)2
dλ
0
2
Z1
dξ
0
Z1
dσσ
0
i p˜ 2 2 m2 × exp − − λσ 2 + ξζM + (1 − ξ)χM 4λ Z∞ Z1 Z1 m2 1 2 2 dλ dξ dσσ cosh λσξM − sinh λσξM = 2 θ (4π)2 2M 2 0 0 0 − p˜ 2 −λσ m2 m2 2 2 2 . e 4λ × cosh λσ(1 − ξ)M − sinh λσ(1 − ξ)M 2M 2 (39) h
After integrating out ξ and σ, one is left with sums of integrals over λ which are again given by modified Bessel functions, see Eq. (28).
C
n-loop integrals
The calculation proceeds along the lines of the 2-loop P integral discussed in Appendix n npl-ins. 2 B. The integral (26) is given by Π (p) = λ Jn (p) with η=±1
eiηkp˜ d4 k n 2 n+1 (2π)4 k˜ 2 k 2 + m2 + ka2 Z 4 n+1 X d k eiηkp˜ Y 1 = n+1 2n n 2 θ (2π)4 (k 2 ) i=1 ζ1 ,...,ζn+1 =±1
Jn (p) ≡
Z
15
2
m 1 + ζi 2M 2
k2 +
m2 2
+ ζi M 2
!
.
(40)
We need a total of n + 2 Schwinger parameters αi to parameterize the denominators of the integrand [20]: 1 = m2 2 k + 2 + ζi M 2
Z∞
“ ” 2 −αi k 2 + m2 +ζi M 2
dαi e
,
for i ∈ {1, . . . , n + 1} ,
0
1 1 = 2 n (k ) Γ(n)
Z∞
2
for k 2 > 0 .
dαn+2 (αn+2 )n−1 e−αn+2 k ,
(41)
0
We perform the change of variables (α1 , . . . , αn+2 ) → (ξ1 , . . . , ξn+1, λ) with α1 = λ
n+1 Q
ξi ,
α2 = λ(1 − ξ1 )
i=1
n+1 Q
ξi ,
... ,
αk = λ(1 − ξk−1 )
i=2
n+1 Q
ξi ,
i=k
... ,
(42)
αn+2 = λ(1 − ξn+1) ,
where ξi ∈ [0, 1] and λ ∈ [0, ∞[. The integration measure transforms as n+2 Y
dαi = λ
n+1
i=1
and we have
n+2 P
n Y
l
(ξl+1 ) dλ
n+1 Y
dξj ,
(43)
j=1
l=1
αi = λ. The integration over k can be carried out by completing the
i=1
square in the exponent so that we arrive at X 1 Jn = 2n n+1 θ 2 (4π)2 Γ(n) ζ ,...,ζ
n+1
1
n+1 Y
m2
1 + ζi 2M 2
i=1
Z∞
dλ λ
2n−2
1 n+1 YZ
dξj
j=1
n Y
(ξl+1)l
l=1
0 0 2 2 p˜ m n−1 2 × (1 − ξn+1 ) exp − − λξn+1 − (ζ1 α1 + . . . + ζn+1αn+1 ) M , 4λ 2 (44)
where the coefficients α1 , . . . , αn+1 in the exponent are functions of the variables λ, ξ1 , . . . , ξn+1 according to Eq. (42). The sum over the ζi can be expressed in terms of hyperbolic functions: 1
∞
Z n n+1 Y YZ 1 l n−1 (ξl+1 ) dξj (1 − ξn+1 ) dλ λ2n−2 Jn = 2n θ (4π)2 Γ(n) j=1 l=1 0
p ˜2
− 4λ −λξn+1
×e
(45)
0
m2 2
n+1 Y
cosh αi M
i=1
2
m2 2 . sinh αi M − 2M 2
Integration over ξ1 , . . . , ξn+1 yields a sum of integrals over λ which are once more given by modified Bessel functions. 16
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