On the Landau–Khalatnikov–Fradkin transformation in quenched $\mathrm{QED}_3$

Abstract

We present the results of studies of the gauge covariance of the massless fermion propagator in three- dimensional quenched quantum electrodynamics in the framework of dimensional regularization in ${d=3-2\varepsilon}$. Assuming the finiteness of the perturbative expansion, i.e., the existence of the limit $\varepsilon\to 0$, we show that exactly for $d=3$ all odd perturbative coefficients starting from the third order must be equal to zero in any gauge. To test this, we calculate three- and four-loop corrections to the massless fermion propagator. Three-loop corrections are finite and gauge invariant, while four-loop corrections have singularities. The terms depending on the gauge parameter are completely determined by the lower orders in accordance with the Landau–Khalatnikov–Fradkin transformation.

Appendix: Details of calculations

To calculate the unrenormalized fermion self-energy in QED${}_{3}$ up to the four-loop order, the authors of [24] used the QCD result for the unnormalized quark propagator. The QCD results valid in an arbitrary space–time dimension $d$ and for an arbitrary gauge are given by a set of master integrals [26], and are also part of the FORCER package [27] designed to reduce the four-loop integrals of the massless propagator. The QED${}_{d}$ limit follows from the QCD result by substituting

$$ C_A=d_A^{abcd}d_A^{abcd}=d_A^{abcd}d_{\mathrm F}^{abcd} =0,\qquad C_{\mathrm F}=d_{\mathrm F}^{abcd}d_{\mathrm F}^{abcd}=T_{\mathrm F}=1.$$

(23)

After that, the quenched QED${}_{d}$ limit is obtained by setting $n_{\mathrm f}=0$, which discards all diagrams with closed fermion loops.

To calculate the required four-loop propagator integrals in $d=3-2\varepsilon$, the dimensional recurrence and analyticity (DRA) method [28] was used, giving results in the form of fast convergent sums. After performing the summation, high-precision numerical values for integrals in an arbitrary space–time dimension can be reconstructed using the PSLQ algorithm [29] if a suitable basis of transcendental constants is specified; this yields analytic results.

We note that near $d=4$, such calculations give expansions of all the necessary master integrals [30]. The results are well known and are available as the input in the SummerTime package [31] along with the package itself (see [32]).

The case $d=3-2\varepsilon$ is less known and was recently considered in [31], where the $\varepsilon$-expansion of the majority of the master integrals needed for our calculation is given. Successful reconstructions around $d=3$ were performed there using a basis of transcendental constants consisting of only multiple zeta values (MZV) and alternating MZVs. As already noted in [31], such a basis is too restrictive to represent all master integrals, and therefore some of them remained unreconstructed.

In [24], all necessary integrals were successfully reconstructed and agreement was found with the results in [30] using a basis consisting of MZVs and alternating MZVs. In addition, one more constant, unknown in [31], was determined after a careful analysis of one of the integrals, namely, $D_1(p)$,

$$ D_1(p)=\int\frac{d^dk_1\,d^dk_2}{(2\pi)^{2d}} \frac{L(k_2)L(p-k_1)}{k_1^{2}(p-k_2)^{2}(k_1-k_2)^{2}},$$

(24)

where

$$ L(p)=\int\frac{d^dk}{(2\pi)^d}\frac{1}{k^{2}(p-k)^{2}}$$

(25)

is a simple loop. Evaluating the loops, we have obtain

$$ D_1=\frac{\pi^3}{(4\pi)^{d}}\,G\biggl(p\kern1pt;1,\frac{1}{2},1,\frac{1}{2},1\biggr)+O(\varepsilon),$$

(26)

where

$$G(p\kern1pt;\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)=\int\frac{d^dk_1\,d^dk_2}{(2\pi)^{2d}} \frac{1}{k_1^{2\alpha_1}k_2^{2\alpha_2}(p-k_2)^{2\alpha_3}(p-k_1)^{2\alpha_4}(k_1-k_2)^{2\alpha_5}}.$$

The diagram $G(p\kern1pt;\alpha,1,\beta,1,1)$ was studied in [25], where several of its representations were presented as combinations of ${}_3F_2$-hypergeometric functions with argument 1.Footnote

This analysis is based on [33], where a class of more complicated diagrams with three arbitrary indices was studied and the corresponding results were also given as combinations of ${}_3F_2$-hypergeometric functions with argument 1.

Considering representations in the case $\alpha=\beta=1/2$, Andrey Pikelner noticed that only generalized polylogarithms appear with the argument given by a fourth root of unity. By extending the PSLQ framework to include the full set of polylogarithms at a fourth root of unity, he successfully reconstructed the analytic result for $G(p\kern1pt;1,1/2,1,1/2,1)$ as

$$ G\biggl(p\kern1pt;1,\frac{1}{2},1,\frac{1}{2},1\biggr)=\frac{1}{(4\pi)^d}\frac{8}{3\pi} \biggl(\mathrm C\pi^2+24\operatorname{Cl}_4\biggl(\frac{\pi}{2}\biggr)+O(\varepsilon^1)\biggr)\frac{\mu^{2\varepsilon}}{p^{2(1+2\varepsilon)}}.$$

(27)

Here, $\mathrm C=\operatorname{Cl}_2(\pi/2)$ is the Catalan constant and $\operatorname{Cl}_n(\theta)$ is the Clausen function, which for an even weight can be expressed in terms of the classical polylogarithm as $\operatorname{Cl}_{2k}(\theta)=\operatorname{Im}\operatorname{Li}_{2k}(e^{i\theta})$. As can be understood from the above result, the required extension of the basis of transcendental constants includes polylogarithms at a fourth root of unity, in this case, the Clausen function (see, e.g., [34], where polylogarithms appear with arguments given by the second, fourth, and sixth roots of unity).

This analysis is based on [33], where a class of more complicated diagrams with three arbitrary indices was studied and the corresponding results were also given as combinations of ${}_3F_2$-hypergeometric functions with argument 1.

The diagram $G(p\kern1pt;1,1/2,1,1/2,1)$ is useful in calculations in effective theories. Moreover, following the transformations discussed in [33], we see that for $d=3$,

$$\begin{aligned} \, G\biggl(p\kern1pt;1,\frac{1}{2},1,\frac{1}{2},1\biggr)\bigg|_{p=1}&= G\bigg(p\kern1pt;\frac{1}{2},1,\frac{1}{2},1,\frac{1}{2}\biggr)\bigg|_{p=1}=G\biggl(p\kern1pt;1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1\biggr)\bigg|_{p=1}= G\biggl(p\kern1pt;\frac{1}{2},1,1,1,\frac{1}{2}\biggr)\bigg|_{p=1}, \end{aligned}$$

and hence the result in (27) is already applicable to several diagrams.