Landau-Khalatnikov-Fradkin transformation of the fermion propagator in massless reduced QED

We study the gauge-covariance of the massless fermion propagator in reduced Quantum Electrodynamics (QED). Starting from its value in some gauge, we evaluate an all order expression for it in another gauge by means of the Landau-Khalatnikov-Fradkin transformation. We find that the weak coupling expansions thus derived are in perfect agreement with the exact calculations. We also prove that the fermion anomalous dimension of reduced QED is gauge invariant to all orders of perturbation theory except for the first one.


I. INTRODUCTION
The Landau-Khalatnikov-Fradkin (LKF) transformation [1] (see also [2,3]) is an elegant and powerful transformation allowing one to study the gauge covariance of Green's functions in gauge theories. In the latter, gauge freedom is implemented by a covariant gauge fixing procedure that introduces an explicit dependence of Green's functions on a gauge fixing parameter ξ. The LKF transformation then relates the Green's functions in two different ξ-gauges. Of course, physical quantities should not depend on ξ. But important information can be obtained by studying the ξ-dependence of various correlation functions.
In its original form, the LKF transformation was applied to the fermion propagator (and also to the fermionphoton vertex that will not be discussed here) of fourdimensional quantum electrodynamics (QED 4 ) which is the primary example of an Abelian gauge field theory. Since then, it has been extensively used in studies of QED in various dimensions, see, e.g., [4][5][6][7][8][9][10][11] and, more recently, in their generalization to brane worlds [12] that we shall come back to in the following and also to non-abelian SU (N ) gauge field theories [13,14].
As a well known application, let us first mention its crucial role within the study of QED Schwinger-Dyson equations see, e.g., [4][5][6], where any viable chargedparticle-photon vertex ansatz has to satisfy the LKF transformation, both for scalar [7] and spinor QED [8]. Another notable application [9,10] that will be closer to our present concerns is devoted to estimating the large order behaviour of perturbative expansions. Namely, the non-perturbative nature of the LKF transformation fixes certain coefficients appearing in the all-order expansion of the fermion propagator. Given a perturbative propagator written for some fixed gauge parameter, say η, all the coefficients depending on the difference between the gauge fixing parameters of the two propagators, i.e. ξ − η, get fixed by the weak coupling expansion of the LKF-transformed initial propagator. Recently, such a procedure allowed to prove [11] the so-called "noπ theorem" [15][16][17][18][19], e.g., cancellations involving ζ 2n (or equivalently π 2n ) values in the perturbative expansion of Euclidean fermion propagator in massless QED 4 , thereby clarifying the transcendental structure of the latter.
In the present paper, we apply the LKF transformation to the fermion propagator of massless reduced QED or RQED dγ,de , see Refs. [20][21][22] and references therein. The latter is an Abelian gauge theory where the photon and fermion fields live in different space-time dimensionalities, namely the photon is in d γ dimensions whereas the fermion fields are confined to d e dimensions, where we take d e ≤ d γ . We shall focus on the special case of RQED 4,3 which is an effective field theory for the socalled planar Dirac liquids, i.e., condensed matter physics systems whose low-energy excitations have a gapless linear, relativistic-like linear dispersion relation and where electrons are confined to a plane (d e = 2 + 1) while interacting via the exchange of photons that can travel through a d γ = (3 + 1)-dimensional bulk. A prototypical example includes graphene [23][24][25]. Nowadays, planar Dirac liquids are well observed experimentally and are under active study in, e.g., (artificial) graphene-like materials [26], surface states of topological insulators [27], and half-filled fractional quantum Hall systems [28]. Interest in RQED 4,3 also comes from its connections to QED 3 [29] which is quite often used as an effective field theory of high temperature superconductors [30][31][32].
More specifically, we will focus on the case of massless RQED 4,3 . Within the condensed matter context, a vanishing fermion mass implies long-ranged (unscreened) interactions among the electrons in the absence of doping (the so called intrinsic case). These interactions in turn enforce the flow of the Fermi velocity, e.g., v ≈ c/300 at experimentally accessible scales for graphene, to the velocity of light, c, deep in the infra-red (IR) with a corresponding flow of the fine structure constant, e.g., α g ≈ e 2 /4πǫ v ≈ 2.2 for graphene, to the usual fine structure constant, α em ≈ 1/137. Within this context, it is this IR Lorentz invariant fixed point [33] that can be described by massless RQED 4,3 [22]. A thorough understanding of this fixed point is a prerequisite to set on a firm ground the study of the physics away from the fixed point which is closer to the experimental reality. But this is more difficult to study theoretically, see e.g., [34], the interesting new work [35] and also the recent reviews in Refs. [36,37].
The gauge-covariance of the fermion propagator of massless reduced QED has already been considered in [12]. Here, we carefully reconsider this problem using the LKF transformation in the framework of dimensional regularization. We not only focus on the bare propagator but also on the renormalized one and provide a detailed comparison between the weak coupling expansion of LKF transformed quantities and earlier exact perturbative calculations [21,22]. This paper is organized as follows. In Sec. II, we start by introducing the position space LKF transformation for the general case of reduced QED theories and then derive its momentum space representation for QED 4,3 that will be the main subject of focus from there on. In Sec. III, a weak coupling expansion of this transformation is performed up to two loops in the MS-scheme and its matching with existing perturbative results are discussed. A similar task is carried out in Sec. IV for the renormalization constant and the renormalized propagator. Additionally, we present a proof of the purely one-loop gauge dependence of the fermion anomalous dimension in reduced QED. Finally, in Sec. V, we summarize our results and conclude. For completeness, various other choices of scales are presented in App. A and in App. B the LKF transformation for reduced scalar QED is derived.

II. LKF TRANSFORMATION FOR REDUCED QED
We have the following action for reduced QED dγ ,de where ξ is the gauge fixing parameter and the flavour index σ runs from 1 to N F . In Eq. (1), the volume elements show that the fermion fields ψ σ are confined to d e dimensions whereas the gauge field mediates the interaction through d γ space-time dimensions. In explicit form, the (Euclidean space) photon propagator in reduced QEDs reads [20] [49]: whereξ = ε e + (1 − ε e )ξ is the reduced gauge fixing parameter while we may refer to the original gauge fixing parameter ξ as the bulk one. In the following, all results will be presented in Euclidean space (d γ = 4 − 2ε γ , d e = d γ −2ε e ) for QED dγ ,de by analogy with the case of QED 4 .

A. LKF transformation in position space
We assume that the fermion propagator S F (p, ξ) in some gauge ξ takes the following general form wherep = γ µ p µ , which contains Dirac γ-matrices, has been factored out and P (p, ξ) is a scalar function, i.e., its momentum dependence is only via p 2 . By analogy, the position-space representation S F (x, ξ) of the fermion propagator can be written as where S F (x, ξ) and S F (p, ξ) are related to each other with the help of the Fourier transform In position space, the LKF transformation [1, 2] connects in a very simple way the representations of fermion propagators written for different gauge parameters ξ and η. In dimensional regularization, it takes the following form where [12] D and the prefactor is given by and follows from the longitudinal part of the photon propagator in Eq. (2) above. As in the case of QED 4 , see [11], D(0) is proportional to the massless tadpole and therefore vanishes in dimensional regularization. Hence, Eq. (6) takes the simpler form and the remaining task is to compute D(x). This can be achieved using the following simple formulas for the Fourier transform of massless propagators (see, for example Ref. [38]): We would like to note that the use of the Euclidean metric simplifies the Fourier transforms thereby illuminating the appearance of additional factors such as i k , where i is the imaginary unit and the factor k is ε-independent. So, for RQED dγ ,de , we have: Making the dependence on the parameter ε explicit (here and below we shall set ε γ ≡ ε and d γ ≡ d) we finally arrive at the expression Remarkably, the parameter ε e has completely disappeared and Eq. (12) has exactly the same form as in QED 4 with a common factor ∆A, accompanied by the singularity ε −1 , contributing to D(x).
Hereafter, we shall only consider the case d e = 3, i.e., RQED dγ ,3 , which corresponds (as d γ → 4) to the ultra relativistic limit of graphene (see Ref. [20]). Note that, as it was shown in [34], an application of dimensional regularization is very convenient in the non-relativistic limit as well, i.e., where the particles interact via the (instantaneous) Coulomb interaction.

B. LKF transformation in momentum space
Let S F (p, η), the fermion propagator for some gauge parameter η and external momentum p, take the form (3) with P (p, η) having an expansion which is appropriate for the massless case relevant to the present study (as explained in the Introduction, it corresponds to the ultrarelativistic limit of planar Dirac liquids) [50]. In Eq. (13) the a m (η) are coefficients of the loop expansion andμ the renormalization scalẽ which lies somewhere between the MS scale µ and MS scale µ. Then, the LKF transformation shows that for another gauge parameter ξ the result has the form where now In order to derive Eq. (16), we used the fermion propagator S F (p, η) with P (p, η) given by Eq. (13), took the inverse Fourier transform to S F (x, η) and applied the LKF transformation (9) in position space. As a final step, we took the Fourier transform back to momentum space and obtained S F (p, ξ) with P (p, ξ) in (15). Let us also note that expansions similar to (13) and (16) can also be expressed in Minkowski space with the help of the replacement p 2 → −p 2 .

C. MS scheme
Now let us focus on the MS scale µ, which is equal (in the most standard definition) to where γ E is the Euler-Mascheroni constant. As is well known, the MS scale completely subtracts out the universal factors of γ E from the ε-expansions.
In the MS-scheme we can rewrite the result (16) in the following form where we have purposefully extracted the factor (1 − 2(m + 1)ε)/(1 − 2(m + l + 1)ε) fromΦ(m, l, ε) in order to have equal transcendental level, i.e., the same s values of ζ s in the ε-expansion ofΦ(m, l, ε) (see below). As will be shown below, the factorΦ(m, l, ε) reading can be written as an expansion in the ζ i (i ≥ 2) Euler constants. Note that the γ E -dependent term arises from the redefinition (17) of the scaleμ → µ.
At this point, it is convenient to re-express the Γfunctions with arguments close to half-integer ones using the standard property (Legendre duplication formula): which leads to the following relation Then, we may writẽ and Eq. (18) can be represented as with We would like to draw the attention of the reader to the redefinition of the argument in the r.h.s. of (24): µ 2 /p 2 → µ 2 /(4p 2 ). Such a redefinition amounts to subtracting factors of ln 2. As we shall see below, it agrees with the exact perturbative calculations done with the MS scale (see [22]). Note that the latter include an additional negative sign for momentum squared in the denominator because the results of [22] were given in Minkowski space. Therefore, p 2 E = −p 2 M under Wick rotations, in the mostly minus signature that was used in that paper.

D. ε-expansion
Let us recall that the Γ-function Γ(1 + βε) has the following expansion around 1: So, the factor Φ(m, l, ε) can be written as: where (but now including s ≥ 1) p s (m, l) = (2 s − 1) (m + 1) s − (m + l + 1) s (δ 1 s is the Kronecker symbol) and, indeed, i.e., the MS-scale takes out the Euler constant γ E from consideration. As can be seen from Eq. (27), the factor Φ(m, l, ε) contains ζ s -values with the same weight s in front of ε s . This is rather similar to what was found in Ref. [39]. In some cases, such a property allows to derive results without any detailed calculations (as in Ref. [40]). In other cases, it simplifies the structure of the results, which can be predicted as an ansatz in a very simple way (see Refs. [41,42]). For recent applications of this property to QCD and super Yang-Mills, see the papers [43] and references and discussions therein.
Recently, this property was also applied to the LKF transformation of QED 4 in [11] by some of the present authors. Combined with an appropriate choice of scale, it led to an all-order proof [11] that the perturbative series can be exactly expressed in terms of a hatted transcendental basis that eliminates all even ζ-values, i.e., the noπ theorem [15][16][17][18][19]. In the case of QED 4,3 , the situation is not so simple. As can be seen from Eq. (29), the fact that p 2 (m, l) = 0 means that ζ 2 -values cannot be subtracted out, unlike in the even-dimensional case [11]. As shown in App. A, other choices of scale are possible but do not further simplify the transcendental structure of this (partially) odd dimensional theory, see also Ref. [48] for an early study.
Thus, in this section, we have obtained a series representation, Eq. (24), for the LKF transformation of the fermion propagator of reduced QED 4,3 in the MS-scheme (see App. A for other choices of scales and App. B for an analogous expression in the case of reduced scalar QED). We now need to verify that the gauge dependence produced by this transformation agrees with exact perturbative results (known in the literature up to the 2-loop order). The next two sections are devoted to this task.

III. LKF TRANSFORMATION FOR THE BARE FERMION PROPAGATOR
A. Bare fermion propagator The calculations of the photon and fermion propagators in the framework of the reduced QED have been done in Refs. [20] and [21,22], respectively (see also the recent reviews in [36,37]).

B. LKF transformation
With the help of the results of Secs. II C and III A above, we can deduce that the one-and two-loop results for the fermion propagator in two different gauges are related to each other in the following way: with Σ 0V (η) = 1. Taking η = 0, i.e., starting from the Landau gauge and the fact that the contribution Σ 2aV is gauge invariant, we have that where the results for Σ 1V (ξ = 0) and Σ 2bcV (ξ = 0) can be obtained from Eqs. (33) and (41) after setting ξ = 0. This yields: With the help of Eqs. (25), (27) and (28), we find that the expansions of Φ(m, l, ε) for the cases of interest read: Φ(1, 1, ε) = 1 + 2ε + 12 − 11 2 ζ 2 ε 2 , (45b) Then Eqs. (43), together with the expressions of Σ 1V (ξ = 0) and Σ 2bcV (ξ = 0) in (44a) and (44b) as well as the εexpansions of Eqs. (45) immediately allow to reproduce the full results for Σ 1V (ξ) and Σ 2bcV (ξ) presented in the previous sections, Eqs. (33) and (41). Thus, we have verified that the bare results for Σ 1V (ξ) and Σ 2bcV (ξ) are exactly in agreement with the LKF transformation (using dimensional regularization, our derivations proceed without any replacements involving a cut-off parameter Λ and the scale µ as in the case of Ref. [12]).

A. Renormalized fermion propagator in momentum space
Since all renormalized results are constructed from the bare ones through the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) procedure (for a definition of the procedure, see for example, Refs. [22,37]), all results including the renormalized ones must be in agreement with the LKF transformation, too.
In order to show this explicitly to the 2-loop order, let us first note that the fermion propagator given by Eq. (3) is the unrenormalized one. It can be conveniently factored as: where we have taken into account the fact that A and ξ are not renormalized in QED 4,3 , i.e., A r ≡ A and ξ r ≡ ξ. In Eq. (46), the renormalization constant Z ψ (A, ξ) and the renormalized fermion propagator P r (p, ξ) can be expanded as: The renormalization constant and renormalized propagator have been computed [22] up to two loops in reduced QED, for arbitrary ε at one-loop and to O(ε 0 ) for the propagator. The one-loop expressions read [22]: where L p = ln(4p 2 /µ 2 ). The two-loop expressions are given by [22]: Let us further note that these expressions allow one to compute the fermion anomalous dimension up to two loops with the help of the relation: yielding [22]:

B. LKF transformation in momentum space
We shall now determine the LKF transformations of Z lψ (ξ) and P lr (p, ξ) up to two-loop and compare the obtained results with those of the last subsection. In order to proceed, we first note that, at NLO, Eq. (46) can be written as: Comparing (52) with (39) and using the notations of (32) then yields: where Σ lV (ξ) has the following ε-expansion: The LKF transformations of Z lψ (ξ) and P lr (p, ξ) can then be obtained by identifying identical powers of ε on both sides of Eqs. (53). At one loop, this straightforwardly yields: where As for the two loop case, we first note that: which in component form can be written as: where we restricted to Σ (2,j) V with j ≤ 0. Then, using Eq. (43b) yields: We are now in a position to use (53b) and first focus on the renormalization constant. In component form, we obtain: where In Eq. (60b) we have used a renormalization constraint arising from the finiteness of the fermion anomalous dimension in the limit ε → 0 whereby the coefficients Z (l,−k) ψ for l > 1 and k = 2, · · · , l may be expressed in terms of coefficients of lower l and k. At two-loop, there is only one constraint: Z (2,−2) ψ (ξ) = (Z (1,−1) ψ (ξ)) 2 /2 which, when applied to (60b), insures that the renormalization constant does not depend on L p . This agrees with the fact that renormalization constants should not depend on masses and external momenta in the MS scheme [44].
We may proceed in a similar way with the 2-loop renormalized fermion propagator. To leading order in the εexpansion, it has the form: We are now in a position to compare the above derived LKF expressions with the exact results presented in Sec. IV A. At one-loop, we find a perfect agreement for the terms proportional to ξ between Eqs. (55) and (48). At two-loop, we also find a perfect agreement for the terms proportional to ξ 2 between Eqs. (60) and (49a) on the one hand and between Eqs. (62a) and (49b) on the other hand. These results are in accordance with the fact that at l-loops, the LKF transformation allows to fix exactly all terms proportional to ξ l . Moreover, by extracting the values of the coefficients Σ (1,j) V (0) from Eqs. (44a) and (44b) and substituting them in Eqs. (56), (61) and (62b), we immediately recover from (55), (60) and (62a) the full results of Eqs. (48) and (49).
Finally, we note the remarkable fact that Eq. (60b) does not depend on the gauge fixing parameter. From Eq. (50), this implies that the 2-loop fermion anomalous dimension is gauge invariant and is in agreement with (51). Actually, we may extend such a remark to 3-loops though no exact result is available yet at this order. All calculations done this yields (in the MS scheme): [52] where the first two terms are easily derived from renormalization constraints [53] while in the third term the Landau gauge coefficient Σ is not known at the time of writing. Nevertheless, Eq. (63c) is clearly gauge invariant and so is the 3-loop fermion anomalous dimension.

C. Gauge dependence of γ ψ
In the last subsection, the LKF transformation revealed that both the 2-loop and 3-loop fermion anomalous dimensions are gauge invariant in reduced QED. We will now show that this gauge invariance extends to all higher orders, see Refs. [45,46], for similar proofs in the case of QED 4 .
We proceed in x-space starting from the unrenormalized fermion propagator of Eq. (9). Similar to the p-space case, it is conveniently factored as: Taking the logarithm of Eq. (9) with D(x) given by Eq. (12) and identifying powers of 1/ε straightforwardly yields: which simply translates an exponentiation of the gaugedependence at the level of the renormalization constant. At this point, let us recall that: where β(A) is the beta function and γ(A) is the gaugefield anomalous dimension. The latter can be expressed as: where the coefficients satisfy: β l = −γ l (actually, they even vanish in the case of RQED 4,3 ). Substituting Eq. (65) in (66) and using (67), yields: showing that all the gauge dependence is contained in the one-loop contribution while all higher order corrections are indeed gauge-invariant.

V. SUMMARY AND CONCLUSION
In this paper, we have studied the gauge-covariance of the fermion propagator of reduced QED with the help of the LKF transformation in dimensional regularization. The x-space transformation has been derived in the general case of QED dγ ,de and its structure, Eq. (12), was found to be similar to QED 4 . Focusing on the odddimensional case, d e = 3 (together with d γ = 4 − 2ε), we have then derived the p-space LKF transformation in the form of a series representation for the coefficients of the loop expansion of the propagator in the MS-scheme, Eq. (24) (see also App. A for other choices of scales and App. B for an analogous expression in the case of reduced scalar QED). The series has been expressed in terms of a uniform transcendental factor Φ(m, l, ε), Eq. (23). The ζ-structure of the latter (see Eq. (28) and discussion below it) is transcendentally more complicated than in the four-dimensional case [11] as expected from an odddimensional theory [48]. We have then performed a twoloop expansion of the transformation for the bare fermion propagator, Eqs. (42), and also for the renormalization constant and renormalized propagator, Eqs. (55), (60) and (62). Starting from the Landau gauge (ξ = 0) to a general ξ-gauge, all these weak-coupling expansions were found to fully agree with previously known exact perturbative results up to the 2-loop order. In particular, we have checked that the LKF predicted coefficients of the form (Aξ) l match with the perturbative results. Additionally, we have presented a proof of the purely one-loop gauge dependence of the fermion anomalous dimension in reduced QED, Eq. (68). In conclusion, our analysis and in particular our all order series representations, Eq. (24) and equivalent ones in App. A, can of course be used beyond the present 2-loop accuracy of perturbative results. They should provide some stringent constraints on future higher order calculations in reduced QED.