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Landau-Khalatnikov-Fradkin Transformation and Even zeta Functions

An exact formula that relates standard zeta functions and so-called hatted zeta functions in all orders of perturbation theory is presented. This formula is based on the Landau-Khalatnikov-Fradkin transformation


Introduction
We consider properties of multiloop massless functions of the propagator type. There is an ever growing number of indications (see, for example, [1]) that, in the calculations of various quantities in the Euclidean region, striking regularities arise in terms proportional to ζ 2n -that is, to even Euler ζ functions. These regularities are thought [2] to be due to the fact that ε-dependent combinations of ζ functions, such aŝ rather than the ζ functions themselves are dominant objects that eliminate ζ 2n in the ε expansions of four-loop functions belonging to the propagator type. A generalization of combinations in (1) to the cases of five, six, and seven loops can be found in [3]. 1 The results in (1) and their generalization in [3] make it possible to predict π 2n terms in higher orders of perturbation theory. In [4] (see also [5]), the present authors extended the results in (1) to any order in ε in a rather unexpected way-by means of the Landau-Khalatnikov-Fradkin (LKF) transformation [6], which relates the fermion propagators in quantum electrodynamics (QED) in two different gauges. It should be noted that the most important applications of the LKF transformation are generally associated with the predictions of some terms in high orders of perturbation theory in QED [7], its generalizations [8], and more general SU(N) gauge theories.
In the present article, we give a brief survey of the results reported in [4], placing emphasis on how the LKF transformation demonstrates in a natural way the existence ofζ functions and makes it possible to extend the results in (1) to any order in ε.

LKF transformation
Let us consider QED in d-dimensional (d = 4 − 2ε) Euclidean space. In general, the fermion propagator in a gauge involving the parameter ξ in the p and x representations has the form where there are explicit expressions for the factorsp andx, which involve the Dirac γ matrices.
Within the dimensional regularization, the LKF transformation relates the fermion propagator in these two gauges with parameters ξ and η, respectively, as [4] where This means that D(x) makes a contribution proportional to ∆A and the pole ε −1 . Suppose that, for a gauge-fixing parameter η, the fermion propagator S F (p, η) with an external momentum p has the form (2), where Here, a m (η) are the coefficients in the loop expansion of the propagator andμ is the renormalization scale lying between the scales of the MS (minimal-subtraction) and MS (modifiedminimal-subtraction) schemes. The LKF transformation determines the fermion propagator for another gauge parameter ξ as where Here, the symbol MV stands for the so-called minimal Vladimirov scale introduced in [4]. We note that, in [4], the use of the popular G scale [10] led to the same final results given in Eqs. (16) and (17) below. In order to derive expression (6), we employed the fermion propagator S F (p, η) with P (p, η) given by (5), applied the Fourier transformation to S F (x, η), and made the LKF transformation (3). As a final step, we performed the inverse Fourier transformation and obtained the fermion propagator S F (p, ξ) with P (p, ξ) given in (6).
Let us now study the factor Φ MV (m, l, ε). For this, we make use of the expansion of the Γ function in the form where γ is the Euler constant. Substituting this expansion into expression (7), we recast the factor Φ MV (m, l, ε) into the form where p s (m, l) = (m+1) s −(m+l+1) s +2l+(−1) s (m+l) s −m s , p 1 (m, l) = 0, p 2 (m, l) = 0 . (10) One can readily see from Eq. (9) that the factor Φ MV (m, l, ε) involves values of the ζ s function of given weight s (or transcendental level) in front of ε s . This property constrains strongly the coefficients, thereby simplifying the ensuing analysis (the authors of the articles quoted in [11] also used this property).
We now focus on the polynomial p s (m, l) in Eq. (10). It is convenient to partition it into components featuring even and odd values of s. The following recursion relations hold: Expressing even components, p 2k , in terms of odd ones as we can determine the exact structure of C 2k,2k−2m+1 in the form where B m are well-known Bernoulli numbers.
It is now convenient to represent the argument of the exponential form on the right-hand side of Eq. (9) in the form With the aid of Eq. (12), the first term on the right-hand side of (14) can be represented in the form Relations (16), (17), and (13) lead to an expression forζ 2s−1 in terms of standard ζ functions that is valid in all orders of the expansion in ε.