Landau-Khalatnikov-Fradkin transformation in three-dimensional quenched QED

We study the gauge-covariance of the massless fermion propagator in three-dimensional quenched Quantum Electrodynamics in the framework of dimensional regularization in d=3-2\ep. Assuming the finiteness of the quenched perturbative expansion, that is the existence of the limit \ep \to 0, we state that, exactly in d=3, all odd perturbative coefficients, starting with the third order one, should be zero in any gauge.


I. INTRODUCTION
Quantum electrodynamics in three space-time dimensions (QED 3 ) with N flavors of four-component massless Dirac fermions has been attracting continuous attention as a useful field-theoretical model for the last forty years. It served as a toy model to study several key quantum field theory problems such as infrared singularities in low-dimensional theories with massless particles, non-analyticity in coupling constant, dynamical symmetry breaking and fermion mass generation, phase transition and relation between chiral symmetry breaking and confinement.
In the last three decades, QED 3 found many applications in condensed matter physics, in particular, in high-T c superconductivity [1-3], planar antiferromagnets [4], and graphene [5] where quasiparticle excitations have a linear dispersion at low energies and are described by the massless Dirac equation in 2+1 dimensions (for graphene, see reviews in Ref. [6]). QED 3 is described by the action (in Euclidean formulation) where D µ = ∂ µ − ieA µ , i = 1, 2, . . . N , Euclidean gamma matrices satisfy γ † µ = γ µ , {γ µ , γ ν } = δ µν , and the gauge coupling constant has a dimension [e] = (mass) 1/2 . It is super-renormalizable with a mass scale e 2 N/8. As such, the model is plagued with severe infrared (IR) singularities in the loop expansion in e 2 of various Green's functions since, for dimensional reasons, higher-order diagrams contain higher powers of momentum in the denominator. For example, the fermion propagator is affected by IR divergences starting at two-loops. The problem of IR divergences in QED 3 has been intensively investigated by various methods in the 80s [7][8][9][10][11].
In order to better appreciate how these IR singularities may be cured, let's recall that the effective dimensionless coupling of QED 3 may be written as: α(q) = e 2 q (1 + Π(q 2 )) = e 2 /q q ≫ e 2 N/8 8/N q ≪ e 2 N/8 , where the one-loop polarization operator was used: Π(q 2 ) = N e 2 /(8q). The corresponding beta function reads: and displays two stable fixed points: an asymptotically free UV fixed point (α → 0) and an interacting IR fixed point (α → 8/N ). 1 The presence of an infrared fixed point in QED 3 is intriguing especially because of the possible existence of an analogous fixed point in four-dimensional SU (N c ) gauge theories with N massless fermions. In this case, there exists a so-called conformal window: a region of values of colors N c and flavors N , for which the beta function has a form resembling Eq. (3). Thus, the theory is asymptotically free at short distances while the long distance physics is governed by a non-trivial fixed-point [15] (for studies on the lattice and an extensive list of references see Ref. [16]).
At the fixed points of the beta-function (3), IR singularities of QED 3 are cured. In the IR limit, q ≪ e 2 N/8, this goes through an infrared softening of the photon propagator, due to vacuum-polarization insertions, together with the disappearance of the dimensionful e 2 in favour of a dimensionless coupling constant proportional to 1/N . The theory has then the same power counting as the (renormalizable) four-dimensional one. In the following, we shall refer to this limit as the large-N limit of QED 3 . In this limit, the fermion propagator is affected by (standard) UV singularities that can be renormalized [17].
Massless QED 3 also plays an important role in studying the problems of dynamical symmetry breaking and fermion mass generation in gauge theories. This comes from the fact that the properties (3) are reminiscent of those of QCD: the intrinsic dimensionful parameter e 2 N/8 plays a role similar to the QCD scale Λ QCD , and the effective coupling α(q) approaches zero at large momenta q. The main question that has been debated for a long time is then whether there exists a critical fermion flavor number, N cr , separating the chiral-symmetric and the chiral-symmetry-broken phases [17][18][19][20][21][22][23][24][25][26][27] (for recent studies in this direction see, e.g., Refs. [28][29][30][31][32]).
Analytical studies of chiral-symmetry breaking and mass generation in QED 3 are based on using truncated Schwinger-Dyson (SD) equations with some approximations (ansatzes) for the full fermion-photon vertex. For example, the simplest one is the replacement of the full vertex by the bare vertex γ µ (the ladder approximation). A more sophisticated way is to use the vertex consistent with the vector Ward-Takahashi identity and satisfying several other requirements such as the absence of kinematic poles and multiplicative renormalizability. The most known among them are the Ball-Chiu [33] and the Curtis-Pennington [34] vertices constructed for the quenched approximation of four-dimensional QED (QED 4 ) -for similar vertices in QED 3 , see [35].
Another crucial requirement for truncated SD equations, not implemented yet, is the covariance of the fermion propagator and the vertex under the Landau-Khalatnikov-Fradkin (LKF) transformations [36][37][38]. These transformations have a simple form in the coordinate space representation and allow us to evaluate Green's functions in an arbitrary covariant gauge if we know their value in any particular gauge. However, in momentum space, the LKF transformations have a rather complicated form for the vertex, and this is the reason why they were not fully involved yet for restricting the form of the vertex even in the quenched approximation (for work in this direction, see papers by Bashir and collaborators [35,39,40] and the review in Ref. [41]).
In the present paper, we study the gauge-covariance of the massless fermion propagator in quenched QED 3 in a linear covariant gauge. The LKF transformation for unquenched QED 3 including a non-local gauge will be considered in the future. At this point, let's recall that the quenched limit of QED is the approximation in which we can neglect the effects of closed fermion loops. The approximation came from investigations of the lattice representation of QED 4 (see [42,43]) which showed that a reasonable estimate of the hadron spectrum could be obtained by eliminating all internal quark loops. Moreover, the quenched approximation in QED 4 is now used to include QED effects within lattice QCD calculations (see the recent paper [44] and references and discussions therein). Just after its introduction, the quenched approximation in QED 4 has also been used in Refs. [45][46][47] within the formalism of the SD equations.
The paper is organized as follows. In Sec. II, we con-sider the LKF transformation for the fermion propagator in momentum space in exactly three dimensions. We notice that it cannot be applied to terms higher than e 4 in the perturbative expansion since they lead to infrared singularities in the LKF relation. Therefore, in Sec. III, we apply dimensional regularization (following [47][48][49]) to deal with higher order terms. The obtained general expressions, valid for arbitrary order terms, are then analyzed thus revealing the self-consistency of the LKF transformation in Secs. IV and V. The analysis leads to the conclusion that exactly in three dimensions, d = 3, all odd perturbative coefficients, starting with third order one, should be zero in any gauge. The results are summarized and discussed in Sec. VI. In three Appendices A, B, and C we present the details of the calculations.

II. LKF TRANSFORMATION
In the following, we shall consider a Euclidean space of dimension d = 3 − 2ε. The general form of the fermion propagator S F (p, ξ) in some gauge ξ reads: where the tensorial structure, e.g., the factorp containing Dirac γ-matrices, has been extracted and P (p, ξ) is a scalar function of p = p 2 . It is also convenient to introduce the x-space representation S F (x, ξ) of the fermion propagator as: The two representations, S F (x, ξ) and S F (p, ξ), are related by the Fourier transform which is defined as: In momentum space, the photon propagator can be written in the following general form: where the functions D T (q 2 ) and D L (q 2 ) encode the transverse and longitudinal parts of the photon propagator, respectively, and the ξ-dependence was made explicit. In linear covariant gauges that we shall focus on in this paper, radiative corrections affect transverse photons and not the longitudinal ones, i.e., where Π(q 2 ) is the polarization part.
The LKF transformation then expresses the covariance of the fermion propagator under a gauge transformation. It can be derived by standard arguments, see, e.g., [36][37][38] and its most general form can be written as: thereby relating the fermion propagator in two arbitrary covariant gauges ξ and η. For d = 3 and D L (q 2 ) given by Eq. (8), the LKF transformation takes the form [39,41]: As can be seen from (11), the transformation has a very simple form in configuration space. It is much more complicated in momentum space where the Fourier transform of Eq. (9) is given by a convolution Here G(k, ∆) is the Fourier transform of G(x, ∆) which, for d = 3, is given by: and is such that G(k, ∆ = 0) = (2π) 3 δ (3) (k). In terms of the (scalar) function P of Eq. (4), the LKF transform can be written as (still in d = 3): Integrating over the angles in Eq. (14) yields a more explicit equation relating the P -functions in two different gauges: It is well known that the perturbation theory in α in massless QED 3 suffers from infrared divergences since, in fact, for dimensional reasons, the expansion is in terms of α/p and infrared divergences are encountered as a consequence of the momenta in the denominator. Potentially infrared divergent diagrams contain insertions of the vacuum polarization diagram into the gauge propagator. For the fermion propagator, an infrared divergent contribution appears first at the order α 2 . However, there are weighty arguments that quenched perturbation theory is infrared finite [7] and they have been recently confirmed by lattice studies of the quenched approximation [50]. In the present paper, we therefore focus on the quenched case only.
Let us use as an initial η-gauge 2 the Landau gauge in Eq. (15). At two loop order in quenched perturbation theory, we have [39]: where the second term comes from two-loop self-energy diagram with crossed photon lines. Inserting the last expression in Eq. (15) and evaluating the integral we get Expanding the r.h.s. up to O(α 2 ), we reproduce the perturbative expansion for the massless fermion propagator in an arbitrary covariant gauge (18) Eqs. (17), (18) are in agreement with Ref. [39]. Beyond α 2 , the terms of the P (p, ξ) expansion are not yet calculated in perturbation theory. The LKF transformation, being non-perturbative in nature, contains, at a given order of the loop-expansion, important information about higher order terms. Note, for example, the absence of the α 2 ξ term and it has been suggested [39] that the contributions α 3 ξ m (m = 1, 2, 3) are also absent upon further expanding the expression (17).
Although Eq. (15) is an exact one, it cannot be applied to the terms of order (α/p) 3 and higher ones in the loop-expansion of the massless fermion propagator. This is because the kernel of Eq. (15) (in square brackets) behaves as ∼ k 2 as k → 0 which leads to infrared divergences for higher order terms in the expansion of P . In the following, we will show that studying the LKF transformation in dimensional regularization allows one to get around this difficulty and obtain explicit finite expressions at any order in quenched perturbation theory for d = 3. In general, this approach can also be applied to the unquenched expansion with dimensionally regularized perturbation theory that we postpone for a future study.
Following Refs. [48] and [49], we use from now on dimensional regularization for which D(0) = 0 (D(0) is a massless tadpole and, thus, it is eliminated in dimensional regularization) and Eq. (9) simplifies as: In order to proceed, it is useful to first present the Euclidean space Fourier transforms of massless propagators (see, for example, the recent review [51]), which have very simple and symmetric forms: With the help of (20a), we may evaluate D(x) which is expressed as in d = 3 − 2ε. The calculation yields [47]: which, for ε = 0, leads to the exponent in Eq. (11). Notice that Eq. (23) is finite in the limit ε → 0, i.e., QED 3 is free from UV singularities. Moreover, in the quenched case that we consider, IR singularities arising from fermion loops, e.g., in gauge-invariant contributions that are in the pre-exponential factor of (19), are suppressed. Nevertheless, as anticipated in the Introduction, the super-renormalizable nature of quenched QED 3 may still give rise to IR singularities at high orders. In dimensional regularization, these IR singularities take the form of poles in 1/ε just as UV singularities. In principle, in order to keep track of them, the full ε-dependence of Eq. (23) has to be taken into account.

B. Momentum space LKF transformation
Following the usual steps for the derivation of the LKF transformation, we then consider the fermion propagator S F (p, η) with the external momentum p in some gauge η. The latter has the form (4) with P (p, η) as where a m (η) are coefficients of the loop expansion of the propagator andμ is the scalẽ which is intermediate between the MS scale µ and the MS scale µ. In Eq. (24), the expansion has been written in terms of the dimensionless ratio α/p with an additional conventional factor of 1/(2 √ π). Its exact form is coming from the consideration of four-dimensional QED in [48] (see also App. A and discussions just after Eq. (A2)). Now we would like to find the exact formulas for the transformation a m (η) → a m (ξ). We will do it in two different ways. Following Refs. [48,49], in the following first sub-subsection we will obtain it from the x-space LKF evolution (19) transforming the ansatz (24) to xspace. Since only few details of the approach were given in [48], the full demonstration will be given in App. A in the case of an Euclidean space of arbitrary dimension d. The second sub-subsection, together with App. B for a more detailed analysis, will provide the corresponding derivation in momentum-space with the help of a direct evaluation of G(p, ∆) and then the evaluation Eq. (12).

x-space analysis
Using the ansatz (24) and the Fourier transform (A4), we obtain that (see App. A for a more extended analysis in arbitrary d): Expanding the LKF exponent, we have where Factorizing all the x-dependence yields: Hence, from the correspondence between the results for the propagators P (p, η) and S F (x, η) in (24) and (26), respectively, together with the result (30) for S F (x, η), we have, for P (p, ξ): where In this way, we have derived the expression of a k (ξ) using a simple expansion of the LKF exponent in x-space. From this representation of the LKF transformation, we see that the magnitude a k (ξ) is determined by a m (η) with 0 < m < k.
We would like to note that Eq. (32) exactly reproduces our initial ansatz (24). It shows that the initial ansatz (24) is correct. Moreover, the representation (32) can be used as starting point for the evolution to another gauge (this will be discussed in Sec. V).
Very often, however (see Eqs. (17) and (18) and Ref. [39]), the subject of the study is not the magnitude a m (ξ) but the p-and ∆-dependencies of each magnitude a l (η) as it evolves from the η to the ξ gauge. The corresponding result for the p-and ∆-dependencies ofâ m (ξ, p) can be obtained interchanging the order of the sums in the r.h.s. of (32). Performing such interchange yields: where, now, the coefficients transform aŝ

p-space analysis
Here we present the basic steps of the direct derivation of Eq. (36). A more extended analysis can be found in Appendix B.
To evaluate the Fourier transform of e D(x) we use the Mellin integral representation, integrate over x and get (see App. B for the details): The two-parameter function E ν,α (z) is a special case of the generalized Wright function 1 Ψ 1 [52]. Since the LKF transformation is non-perturbative in nature, the exact expression (39) can be used when solving, for example, truncated Schwinger-Dyson equations.
To evaluate the convolution integral Eq. (12) we use the Mellin-Barnes representation of this function where the contour separates the poles of the gamma functions in the numerator. By using the expansion (24), together with Eqs. (39) and (40), we can integrate over k and get the final answer for P (p, ξ) in the form of (35), with the coefficientsâ m (ξ, p) representing the gauge-evolution of the magnitude a m (η): Using the asymptotic expansion of the Wright function at large values of its argument, we can write the LKF relation between the coefficientsâ m (ξ, p) and a m (η) in the form (36), (37).
A. Coefficientsâ k (ξ, p) at ε = 0 As discussed in Sec. I, quenched QED 3 is a priori free from both IR and UV singularities. Because of this finiteness, it is tempting to set ε = 0 in Eq. (41). We consider this possibility here as a prerequisite to the more complete analysis of the following (sub-)sections.
In the case ε = 0, the fermion propagator (35) takes the simpler form where the coefficients (41) are greatly simplified since the function 1 Ψ 1 reduces to the Gauss hypergeometric function. Hence, we get: The last expression can be rewritten in a slightly different form using analytic continuation through connection formulas for 2 F 1 hypergeometric functions (see, e.g., [53]) : 3â Eqs. (43) and (44) are suitable for describing series expansions at small and large values of p, respectively. Let's end this subsection by giving some explicit formulas for the first values of m from Eq. (43) (see, e.g., [54] for the evaluation of 2 F 1 hypergeometric functions): a 2 (ξ, p) = a 2 (η) 1 (1 + B 2 ) 2 3 We would like to note that these equations can also be obtained by considering even and odd l-values in Eq. (36) at ε = 0.
The results (45) are in exact agreement with those derived from the integral relation (15). Thus, substituting back (45) in the fermion propagator (42), the LKF transformation is found to be in perfect agreement with the perturbative result (18). Starting from m ≥ 3, both Eqs. (44) and (43) are subject to singularities. This agrees with the analysis based on (15) which shows that such singularities are of IR nature and arise from the super-renormalizability of (quenched) QED 3 . Moreover, from Eq. (43) -and equivalently from (44) -a parity effect is clearly displayed whereby even coefficients,â 2s (ξ, p), are finite for all s while odd coefficients,â 2s+1 (ξ, p), are singular for all s ≥ 1.
Such an observation calls for a more careful treatment of quenched QED 3 . In particular, the parameter ε has to be kept non-zero in order to regulate the singularities. This will be the subject of the following subsection.
We would like to complete the results of the previous subsection by computing the coefficientsâ m (ξ, p) at ε → 0. A subtlety in the related ε-expansion is that only the leading order term in ε → 0 (whether a constant, a pole 1/ε and/or a contribution ∼ ε) is needed in order to analyze the previously described parity effect (between odd and even coefficientsâ m (ξ, p)). Such an expansion strategy will be fully appreciated (and detailed) in the next subsection.
Interestingly, from Eq. (41), we see quite clearly that singularities do affect odd coefficients, m = 2s + 1 for s ≥ 1, and that these singularities are entirely located in the first gamma function in the numerator. Other terms including the Wright function, 1 Ψ 1 , are finite. Postponing the details of the derivations to App. B, in the following we shall be brief and present only the final results.
The finite even coefficients,â 2s , may be re-written for ε = 0 as (see, Eq. (44)): where δ s 0 is the Kronecker symbol. In the case of the odd coefficients, m = 2s + 1 (s ≥ 1), the leading order result takes the form of a simple pole in ε. It follows from Eq. (41) that: This can be re-written in slightly different form as: a 2s+1 (ξ, p) = − 2sB (2s + 1)πε a 2s+1 (η) In agreement with the previous subsection, the above results confirm that odd coefficientsâ 2s+1 (ξ, p) are singular for s ≥ 1, see (47), while even coefficients are finite. Assuming that quenched QED 3 is finite (see Refs . [7] and discussion therein), we find that the requirement for the limit ε = 0 to be well-defined is: i.e., the identical vanishing of all odd coefficients (magnitudes) other than a 1 (ξ). In the next section, we will provide a refined proof of (49) based on the direct analysis of the coefficients a k (ξ) (rather than their gauge evolution a m (ξ, p)).
A. Coefficients a k (ξ) at ε → 0 The analysis of the parity effect may be pushed further with the help of the first representation of the LKF transform, Eq. (32), that allows to study the momentum-independent magnitudes, a k (ξ), of Eq. (33). As for the study of their gauge-evolution,â m (ξ, p), the ε-expansion of a k (ξ) will be carried out at leading order in ε → 0. In the following, we shall justify such a procedure by showing that this accuracy is enough to prove the self-consistency of the LKF transformation.
The analysis of the coefficients a k (ξ) requires considering the cases of even and odd values of k separately. The complete evaluation is carried out in App. C. Here we present only the final results.
1. In the case of even k values, i.e., k = 2r, the final results for a 2r (ξ) can be expressed as a sum of the contributions a The latter come in-turn from the corresponding contributions of the initial amplitudes a 2s (η), a 1 (η) and a 2s+1 (η) as a (1) where δ = √ π∆.

2.
In the case of odd k values, i.e., k = 2r + 1, we should consider the cases r = 0 and r ≥ 1 separately.
In the case k = 1, we have the following result: The final result for a 2r+1 (ξ) (for r ≥ 1) can be expressed as a sum of the contributions a The latter come in-turn from the corresponding contributions of the initial amplitudes a 2s (η), a 1 (η) and a 2s+1 (η) as a 2s+1 (η) (r + 1) (s + 1) We note that, as anticipated above, the contributions (51), (52) and (54) correspond to the first terms of the ε-expansion, which is sufficient to analyze the self-consistency given in the next subsection.
For the coefficient a 1 (ξ 1 ), we have (hereafter δ 1 = √ π(ξ 1 − ξ), δ 1 = √ π(ξ 1 − η)): because So, we obtain the expression of a 1 (ξ 1 ) and it coincides with the one obtained directly from the η-gauge with help of Eq. (55b) (with the replacements ξ → ξ 1 and η → ξ). Similarly, the coefficient a 2 (ξ 1 ) changes as: The term in factor of a 1 (η) corresponds to: because of (58). The term in factor of a 2 (η) corresponds to: Taking all the results together, we have: Thus, we derive the expression of a 2 (ξ 1 ) and it coincides with the one obtained directly from the η-gauge with help of Eq. (55c) (with the replacements ξ → ξ 1 and η → ξ). Similar transformations can also be performed for the other coefficients a i (ξ 1 ) (i ≥ 2) in a similar way. So, we can obtain a full agreement between the transformation and the results for a i (ξ 1 ) obtained directly from the ηgauge with help of Eqs. (55) (with the replacements ξ → ξ 1 and η → ξ).
With the purpose of checking the statement a 2m+1 (ξ) = 0 for (m ≥ 1) for d = 3, we plan to perform a direct calculation of the coefficient a 3 in the Feynman and/or Landau gauge. Moreover, with the help of modern methods of calculations (see [51] for a recent review), we may calculate exactly the ε-dependence of a k (ξ) (k = 1, 2, 3) in d = 3 − 2ε (or, at least, obtain the first few coefficients in the expansion with respect to ε) and compare it with the ε-dependence coming from LKF transformation, see Eqs. (32) and (35).

VI. SUMMARY AND CONCLUSION
In this work we have studied the LKF transformation for the massless fermion propagator of three-dimensional QED in the quenched approximation to all orders in the coupling α. 4 Previous studies in the literature were limited to the order α 2 . Our investigations were performed in dimensional regularization in d = 3 − 2ε Euclidean space.
We have formulated two equivalent transformations: Eq. (32) together with (33) and (34) on the one hand, and Eq. (35) together with (36) and (37) on the other hand. Moreover, for the coefficientsâ m (ξ, p) in the transformation (35), we managed to obtain the closed expression (41) in terms of the generalized Wright function 1 Ψ 1 whose asymptotic expansion gives Eqs. (36) and (37).
The transformation (35), which is similar to the ones used at lower orders in other papers [35,[39][40][41], allowed us to study the gauge-evolution (from the η-gauge to the ξ-gauge) of each initial magnitude a m (η) and to reproduce all results of the previous studies [35,39].
There are arguments in favor of ultraviolet and infrared perturbative finiteness of massless quenched QED 3 [7,50]. Hence, assuming the existence of a finite limit as ε → 0, we find that, exactly in d = 3, all odd terms a 2t+1 (ξ) in perturbation theory, except a 1 , should be exactly zero in any gauge, i.e., even in the Landau gauge.
This statement is very strong and needs a further check. At the order α 2 , analytical expressions for the fermion self-energy diagrams are well known. However, to the best of our knowledge, such results are absent at three-loop order. We plan to study the a 3 term, i.e., three-loop diagrams, directly in the framework of perturbation theory in our future investigations.
Moreover, in our future studies, we also plan to consider the LKF transformation in unquenched QED 3 as well as in the large-N limit of QED 3 (see Refs. [28][29][30] and the recent review [31]). In particular, this latter study will be performed in a non-local gauge where we plan to apply results from previous studies [14,56,57] and [49,58] in reduced QED 4,3 which is similar (see Ref. [59]) to QED 3 in the 1/N -expansion.
In this Appendix, we shall present the LKF transformation in a Euclidean space of dimension d. In the course of the evaluation, we shall also use the representation d = 4 − 2ε, which is natural in four-dimensional space. Such a representation is useful because the derived expressions are rather compact when expressed in terms of the ε-dependence. Accordingly, we do not make any decomposition in ε and, therefore, at any stage of the calculation in this Appendix, all results have an exact d-dependence, when ε is replaced by d as ε = (4 − d)/2.
where a m (η) are coefficients of the loop expansion of the propagator andμ is the scale displayed in Eq. (25) which is intermediate between the MS scale µ and the MS scale µ.
In the following, we will derive exact formulas for the transformation a m (η) → a m (ξ) following the LKF transformation (19) which is compact in x-space. In order to do so, it is convenient to first derive an expression of S F (x, η) based on the ansatz (A3) for P (p, η). Using the Fourier transform (20a), we have where a n (β) is defined in (21). Then, using (6b), we obtain that: With the help of (A5) together with an expansion of the LKF exponent, we have Factorizing all x-dependence yields: Hence, taking the correspondence between the results for propagators P (p, η) and S F (x, η) in (A3) and (A5), respectively, together with the result (A7) for S F (x, η), we have for P (p, ξ): where In this way, we have derived the expression of a m (ξ) using a simple expansion of the LKF exponent in x-space. From this representation of the LKF transformation, we see that the magnitude a m (ξ) is determined by a l (η) with 0 ≤ l ≤ m. The corresponding result for the p-and ∆-dependencies ofâ m (ξ, p) (see definition ofâ m (ξ, p) in Sec. III B 1) can be obtained by interchanging the order in the sums in the r.h.s. of (A8). So, we have We would like to note that all of the above results may be expressed in d = 3 − 2ε with the help of the substitutions ε = 1/2 + ε and e 2 d µ = e 2 . The last replacement can also be expressed asᾱ d µ = α/(4π), with the dimensionful α = e 2 /(4π) defined in (11).
With the help of the ansatz (24) and Eq. (B3) together with the Mellin-Barnes representation (B4) for the function E ν,α (−z), this yields: The resulting momentum integral is of the simple massless-propagator type with numerator and can be computed with the help of (see, e.g., Ref. [51]): where k µ1···µn denotes the traceless symmetric tensor, and G (n,0) (α, β) = a n (α)a 0 (β) a n (α + β − d/2) , a n (α) = Performing the integral yields: Hence we obtain The last series has the form (35) and, for the coefficientsâ m (ξ, p), we obtain the expression which after the change of the variable, s → s + d/2 − 1 − mν, can be written in terms of the generalized Wright functionâ .

(B16)
Here we used the Mellin-Barnes representation for this function where the contour separates the poles of gamma functions in the numerator. Eq. (B16) is Eq. (41) from the main text.
The Wright functions p Ψ q (z) are studied rather well in the literature, see, for example, [52,60,63]. Their series expansions can be derived from Mellin-Barnes representation. For 1 Ψ 1 deforming the contour to the left we obtain the small z expansion, On the other hand, deforming the contour to the right one going from ∞ − iδ to ∞ + iδ and enclosing the poles of Γ(a − As), we can evaluate the residues at s = l+a A , l = 0, 1, . . . and obtain the asymptotic expansion at large z ≫ 1, Correspondingly, the relation between the coefficientsâ m (ξ, p) and a m (η) takes the following form at large momentâ which corresponds to Eq. (36) in the main text. For d = 3 (ǫ = 0) the function 1 Ψ 1 in Eq. (41) takes the form where we used the series representation (B18) and the duplication formula for the gamma function. Eqs. (B21) and (41) reproduce Eq. (43) in the main text. Interestingly, Eq. (B21) is finite and so the singularities appearing in (43) are entirely due to the first gamma function in the numerator of (B16). From the latter, we see that these singularities only affect odd coefficients with m = 2s + 1 and s ≥ 1. These singularities can be regularized with the help of a leading order in ε expansion. To this end, it is enough to keep ε = 0 in the singular gamma function and set ε = 0 in all other terms. This yields: For even m = 2s the coefficientsâ 2s (ξ, p) are finite when ε = 0, they are obtained from Eq. (B22) and take the formâ in accordance with Eq. (43). The odd coefficientsâ 2s+1 (ξ, p) are singular for s ≥ 1. Their leading behavior when ǫ → 0 is easily obtained from Eq. (B22) and gives Eq. (47) in the main text.
Appendix C: Evaluation of the coefficients a k (ξ) at ε → 0 As it was shown in subsection V A, the accurate analysis of the coefficients a k (ξ) requires considering the cases of even and odd values of k separately.
where δ was defined after Eq. (51c). So, at the end, we have Eq. (51a) in the main text.
With the help of (C19), we obtain the final expression for a (2) 2r+1 (ξ) and a