Landau-Khalatnikov-Fradkin Transformation and Even zeta Functions
aa r X i v : . [ h e p - t h ] J a n Landau-Khalatnikov-Fradkin Transformation and Even ζ Functions
A. V. Kotikov and S. Teber Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,141980 Dubna, Russia. Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies,LPTHE, F-75005 Paris, France.
Abstract
An exact formula that relates standard ζ functions and so-called hatted ζ (ˆ ζ ) func-tions in all orders of perturbation theory is presented. This formula is based on theLandau-Khalatnikov-Fradkin transformation. We consider properties of multiloop massless functions of the propagator type. There isan ever growing number of indications (see, for example, [1]) that, in the calculations ofvarious quantities in the Euclidean region, striking regularities arise in terms proportionalto ζ n –that is, to even Euler ζ functions. These regularities are thought [2] to be due to thefact that ε -dependent combinations of ζ functions, such asˆ ζ ≡ ζ + 3 ε ζ − ε ζ , ˆ ζ ≡ ζ + 5 ε ζ , ˆ ζ ≡ ζ , (1)rather than the ζ functions themselves are dominant objects that eliminate ζ n in the ε expansions of four-loop functions belonging to the propagator type. A generalization ofcombinations in (1) to the cases of five, six, and seven loops can be found in [3]. Theresults in (1) and their generalization in [3] make it possible to predict π n terms in higherorders 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) trans-formation [6], which relates the fermion propagators in quantum electrodynamics (QED) intwo different gauges. It should be noted that the most important applications of the LKFtransformation are generally associated with the predictions of some terms in high ordersof perturbation theory in QED [7], its generalizations [8], and more general SU(N) gaugetheories.In the present article, we give a brief survey of the results reported in [4], placing emphasison how the LKF transformation demonstrates in a natural way the existence of ˆ ζ functionsand makes it possible to extend the results in (1) to any order in ε . We note that the results in [3] also contain multiple ζ functions (multi-zeta values), but their analysisis beyond the scope of the present article. LKF transformation
Let us consider QED in d -dimensional ( d = 4 − ε ) Euclidean space. In general, the fermionpropagator in a gauge involving the parameter ξ in the p and x representations has the form S F ( p, ξ ) = 1 i ˆ p P ( p, ξ ) , S F ( x, ξ ) = ˆ x X ( x, ξ ) , (2)where there are explicit expressions for the factors ˆ p and ˆ x , which involve the Dirac γ matrices.Within the dimensional regularization, the LKF transformation relates the fermion prop-agator in these two gauges with parameters ξ and η , respectively, as [4] S F ( x, ξ ) = S F ( x, η ) e i D ( x ) , (3)where D ( x ) = i ∆ Aε Γ(1 − ε ) ( πµ x ) ε , ∆ = ξ − η, A = α em π = e (4 π ) . (4)This means that D ( x ) makes a contribution proportional to ∆ A and the pole ε − .Suppose that, for a gauge-fixing parameter η , the fermion propagator S F ( p, η ) with anexternal momentum p has the form (2), where P ( p, η ) = ∞ X m =0 a m ( η ) A m ˜ µ p ! mε , ˜ µ = 4 πµ . (5)Here, a m ( η ) are the coefficients in the loop expansion of the propagator and ˜ µ is the renor-malization scale lying between the scales of the MS (minimal-subtraction) and MS (modified-minimal-subtraction) schemes. The LKF transformation determines the fermion propagatorfor another gauge parameter ξ as P ( p, ξ ) = ∞ X m =0 a m ( η ) A m ˜ µ p ! mε ∞ X l =0 − ( m + 1) ε − ( m + l + 1) ε Φ MV ( m, l, ε ) (∆ A ) l ( − ε ) l l ! µ p ! lε , (6)where Φ MV ( m, l, ε ) = Γ(1 − ( m + 1) ε )Γ(1 + ( m + l ) ε )Γ l (1 − ε )Γ(1 + mε )Γ(1 − ( m + l + 1) ε ) . (7)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 givenin 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 LKFtransformation (3). As a final step, we performed the inverse Fourier transformation andobtained 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 theΓ function in the formΓ(1 + βε ) = exp h − γβε + ∞ X s =2 ( − s η s β s ε s i , η s = ζ s s , (8)2here γ is the Euler constant. Substituting this expansion into expression (7), we recast thefactor Φ MV ( m, l, ε ) into the formΦ MV ( m, l, ε ) = exp h ∞ X s =2 η s p s ( m, l ) ε s i , (9)where p s ( m, l ) = ( m +1) s − ( m + l +1) s +2 l +( − s n ( m + l ) s − m s o , p ( m, l ) = 0 , p ( 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 constrainsstrongly the coefficients, thereby simplifying the ensuing analysis (the authors of the articlesquoted in [11] also used this property). ˆ ζ n − p s ( m, l ) in Eq. (10). It is convenient to partition it intocomponents featuring even and odd values of s. The following recursion relations hold: p k = p k − + Lp k − + p , p k − = p k − + Lp k − + p , L = l ( l + 1) . (11)Expressing even components, p k , in terms of odd ones as p k = k X s =2 p s − C k, s − = k − X m =1 p k − m +1 C k, k − m +1 (12)we can determine the exact structure of C k, k − m +1 in the form C k, k − m +1 = b m − (2 k )!(2 m − k − m + 1)! , b m − = (2 m − m B m , (13)where B m are well-known Bernoulli numbers.It is now convenient to represent the argument of the exponential form on the right-handside of Eq. (9) in the form ∞ X s =3 η s p s ε s = ∞ X k =2 η k p k ε k + ∞ X k =2 η k − p k − ε k − . (14)With the aid of Eq. (12), the first term on the right-hand side of (14) can be represented inthe form ∞ X k =2 η k p k ε k = ∞ X k =2 η k ε k k X s =2 p s − C k, s − = ∞ X s =2 p s − ∞ X k = s η k C k, s − ε k . Relation (14) can then be recast into the form ∞ X s =2 ˆ η s − p s − ε s − = ∞ X s =2 [ ˆ ζ s − / (2 s − p s − ε s − , (15)3here ˆ ζ s − = ζ s − + ∞ X k = s ζ k ˆ C k, s − ε k − s )+1 (16)with ˆ C k, s − = 2 s − k C k, s − = b k − s +1 (2 k − s − k − s + 1)! . (17)Relations (16), (17), and (13) lead to an expression for ˆ ζ s − in terms of standard ζ functionsthat is valid in all orders of the expansion in ε . The recursion relations in (11) between the even and odd components of the polynomialassociated with the factor Φ MV ( m, l, ε ) (7) have been deduced from the result in (6) ob-tained by means of the LKF transformation for the fermion propagator. These recursionrelations make it possible to express all results for the factor Φ MV ( m, l, ε ) in terms of ˆ ζ s − .Expressions (16) and (17) for them are valid in any order of perturbation theory.A.V. 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