Riemann $\zeta(3)$- terms in perturbative QED series, conformal symmetry and the analogies with structures of multiloop effects in N=4 supersymmetric Yang-Mills theory

As was discovered recently, 5-loop perturbative quenched QED approximation to the QED $\beta$-function consist from the rational term and the term proportional to $\zeta(3)$-function. It is stressed, that this feature is also manifesting itself in the conformal invariant pqQED series for the 4-loop approximation to the anomalous mass dimension. The 4-loop pqQED expression for the singlet contribution into the Ellis-Jaffe polarized sum rule is obtained. It coincides with the similar approximation for the non-singlet coefficient function of Ellis-Jaffe sum rule and of Bjorken polarized sum rule. It is stressed that this property is valid in all orders of perturbation theory thanks to the conformal symmetry of pqQED series and to Crewther relation, which relates non-singlet and singlet coefficient functions of Ellis-Jaffe sum rule with the coefficient functions of non-singlet and singlet Adler D-functions. The basic steps of derivation of Crewther relation in the singlet channel from the AVV triangle diagram are outlined. The similarities between analytical structure of asymptotic series for the coefficient functions in pqQED and for the anomalous dimensions in N=4 conformal invariant supersymmetric Yang-Mills theory are observed. The guess is proposed that the appearance of $\zeta(3)$-terms in the pqQED expressions and the absence of $\zeta(5)$-terms at the same level is the indication of absence of"wrapping"interactions in pqQED.


Introduction
calculations of Ref. [19] clarified the origin of rationality of this order A 4 term.
In view of the 3 and 4-loop cancellations of the transcendental Riemann functions at the 3-and 4-loop levels, the explicit manifestation of the ζ(3)-function at the 5-loop level was considered as a puzzle. In this paper we will show that this is not the puzzle at all, but the regular feature, which is consistent with the structure of perturbative series in another conformal invariant model, namely N = 4 supersymmetric (SUSY) Yang-Mills theory.
2. Manifestations of ζ(3) functions in perturbative quenched QED series and its cancellation in the 4-loop expression for Ellis-Jaffe sum rule.
It is important to have a look, how in perturbative quenched QED series ζ(3)-terms are manifesting themselves. Consider first the original result of Refs. [1], [3] for the 5-loop expression of the pqQED β-function, namely β [1] QED (A) = The coefficient function is defined from the QCD expression for the non-singlet contribution to the e + e − -annihilation Adler D-function The QED and QCD perturbation-theory expansion parameters are normalized as A = α/(4π) and A s = α s /(4π) with α and α s being the renormalized QED and QCD coupling constants. It is interesting to have a look whether in perturbative quenched QED there are any other renormalization group function for the gauge-invariant operators, which contain ζ(3)function in high order corrections.
Consider first perturbative series for the anomalous mass dimension in pqQED. Its expression differs from the anomalous dimension of the operator ΨΨ by overall sign only, and therefore, for the reason of rigour it is better not to introduce mass term in the QED lagrangian, and consider massless conformal invariant limit of the QED series for the anomalous dimension function γ ΨΨ (A) = −γ m (A). Its expression can be obtained from the 4-loop QCD calculations of the mass anomalous dimension function γ m (α s ), performed in Ref. [20] and in Ref. [21] independently. It is more convenient to use the results of [20], since this work contains the explicit dependence of the 4-loop expression for γ m (α s ) from Casimir operators C F , C A , normalization factor T F and the number of quarks flavours N F . The choice C F = 1, C A = 0, T F = 1 and N F = 0 corresponds to the case of pqQED approximation. The pqQED expression for the anomalous dimension of the gauge-invariant operator ΨΨ has the following form γ pqQED The analytical structure of this series was already investigated in Ref. [18] using the Shwinger-Dyson approach. In view of the appearance of ζ(3)-term in the pqQED part for the QED β-function (see Eq.(1)), it is worth to attract more attention to the appearance of ζ(3)-term in the 4-loop correction in Eq. (4). Moreover, the 4-loop manifestation of ζ(3)term in the conformal invariant expression of Eq.(4) indicate that the similar feature may manifest itself in other pqQED series as well. The anticipating its manifestation cancellations of ζ(3)-terms at the intermediate stages of lower order calculation should also hold in the series of Eq.(4). This statement is the consequence of the experience gained in the process of evaluation of 3-loop counter-terms in QCD during the 4-loop calculations, which result in the publications of the works of Refs. [17], [22], [23].
Note, that the expression for Eq.(4) follows from the calculations of the renormalization group function of "vertex operator". In the case of calculations of renormalization-group quantities, related to two-point functions, ζ(3)-term is appearing one loop later, namely at the 5-loop order (see Eq.(1)). It enters the expressions for the non-singlet coefficient functions of the 5-loop O(A 4 )-corrections to the e + e − -annihilation Adler D-function and the Bjorken polarized deep-inelastic scattering sum rule, defined in QCD as Indeed, comparing Eq.(1) with Eq. (2), one can get: The similar 5-loop expression for the coefficient function of the Bjorken sum rule, given in Ref. [8] and confirmed by diagram-by-diagram calculations in Ref. [3], reads: These quantities do not contain anomalous dimension terms. The logic of the discussions presented above leads to the conclusion that in the pqQED series for the quantities, which are related with the non-zero anomalous renormalization constant, ζ(3)-should cancel down 1-loop prior their manifestation in Eq.(6) and Eq. (7), namely at the level of O(A 3 )-corrections.
To verify this statement consider now the Ellis-Jaffe sum rule of the deep-inelastic scattering of polarized leptons on protons. In QCD it is defined as where a 3 = ∆u−∆d, a 8 = ∆u+∆d−2∆s, ∆u, ∆d and ∆s are the polarized distributions and ∆Σ depends from the scheme choice. In the MS-scheme it is defined as ∆Σ = ∆u+∆d+∆s, while in the Adler-Bardeen scheme it contains the additional additive contribution from polarized gluon distribution ∆G. The 4-loop QCD corrections to the coefficient function of the singlet part of Ellis-Jaffe sum rule were calculated in Ref. [24] using the method of the dimensional regularization. In the framework of the dimensional regularization the final expression for the singlet coefficient function can be presented as the ratio of two functions [24]: Here Z s 5 is the finite singlet renormalization constant of the operator Ψγ µ γ 5 Ψ, which should be calculated within dimensional regularization and the MS-scheme. This finite constant is similar to the finite constant Z ns 5 in the definition of the non-singlet axial operator Ψγ µ γ 5 (λ a /2)Ψ within dimensional regularization. It enters in the procedure of calculations of high order QCD corrections to the Bjorken polarized sum rule at the 3-loop [25], [26] and 4-loop [27] levels. In view of the property, that the expression for Z 5 differs from Z ns 5 by the corrections, which come from the light-by-light-type scattering graphs [28], in the pqQED limit these constants coincide. Therefore, the 4-loop corrections in Eq.(9) are determined by the ratio of the following pqQED expressions for the coefficient function and for the finite renormalization constants, namely The expressions of Eq.(10) and of Eq.(11) are extracted from the results of calculations of Ref. [24] and Ref. [27] correspondingly. Notice the appearances of ζ(3)-terms in the coefficients of the O(A 3 )-corrections to Eq.(10) and Eq. (11). However, these terms cancel each other in our new ultimate 4-loop pqQED result for the coefficient of order A 3 approximation to the singlet coefficient function: and coincide with the similar expression for the pqQED series of Eq. (7). At the possible next step of analytical calculations of Eq.(10) ζ(5) must manifest itself. Indeed, not presented yet next term in the result of Eq.(11) for the renormalization constant holds in pqQED in all orders of perturbation theory and is the consequence of the of the axial variant of Crewther relation. Let me outline the basic steps of the proof of this statement in the momentum space. These steps were first discussed in Ref. [29] together with more detailed proof of the original non-singlet Crewther relation in the momentum space [30] The proof is based on the application of the operator product expansion approach to the 3-point function with the axial singlet current : where A µ = ψγ µ γ 5 ψ. Keeping the singlet structure in the operator-product expansion of the two non-singlet vector currents, one can get where C SI,ab µνα ∼ iδ ab ǫ µναβ The second ingredient in the singlet version of the Crewther relation appears after consideration of vacuum expectation value of the product of two axial currents In this channel one can also define Adler function and its coefficient function C s D (A s ) as well. Taking now the conformal symmetry limit, it is possible to get the singlet variant of the Crewther relation [29], namely This expression should be compared with the similar expression for the non-singlet Crewther relation [30], which reads In the massless pqQED approximation the following identity takes place Comparing now Eq.(18) with Eq. (19) and taking into account Eq. (20), I find that the expression of Eq. (13) is indeed valid in all orders of perturbation theory. Note once more that pqQED is the conformal invariant version of massless perturbative QED. Therefore, in order to understand deeper the status and nature of the manifestation of odd ζ-functions, it is important to have a look to the structure of perturbative series in some other conformal invariant theory and N = 4 SUSY Yang-Mills theory in particular. 3. Analytic structure for the anomalous dimension of the Konishi operator in N=4 SUSY Yang-Mills theory.
To demonstrate that the explicit manifestation of transcendental ζ(3)-terms in high order perturbation theory corrections to renormalization group quantities does not contradict conformal symmetry let us turn to the behaviour of perturbative series for the anomalous dimensions in the massless N = 4 SUSY Yang-Mills theory. Since its renormalization group β-function is identically equal to zero, this theory possess the property of explicit conformal symmetry. The validity of this property at the 3-loop level was discovered in Ref. [31] by perturbative methods. Soon aftewards the absence of renormalization of the coupling constant in this theory was proved within the light-cone quantization approach [32].
The absence of the coupling constant renormalization does not mean that there are no ultraviolet divergencies in the massless N = 4 Yang-Mills theory. Indeed, calculations of anomalous dimensions of various operators in this quantum field theory give non-zero results (see e.g. Refs. [33]- [45]).
Among the most interesting are the ones, related to analytical evaluation of the anomalous dimension of the Konishi operator in high levels of perturbation theory. The operator is defined as where Φ i is the complex adjoint scalar field. The expression for the anomalous dimension of this operator obey the interesting property, namely the transcendental functions ζ(3) and ζ(5) are manifesting themselves starting from the 4-loop perturbative corrections. Indeed, the direct quantum field theory perturbative calculation, performed in terms of Feynman diagrammatic technique [38], gave the following result where λ = g 2 N c /(4π) 2 and N c is the "number of colours" of SU(N c ) gauge group. Note, that in N = 4 SUSY Yang Mills gauge theory the values of Casimir operators are fixed as C F =C A =T F N F . Another interesting feature of N = 4 SUSY Yang-Mills theory is that the property of AdS/CFT correspondence [46]- [48] links N = 4 SUSY Yang-Mills with the theory of superstings in AdS 5 × S 5 . This property opens the second way for the calculations of anomalous dimensions in N = 4 SUSY Yang-Mills theory using quantum field theory of the superstring in AdS 5 × S 5 and taking into account its integrability property. This was done in Ref. [37], where the coefficients of the series in Eq. (22) were calculated prior the work of Ref. [38].
This calculation is based on the application of the Bethe Anzatz quantization. Note, that using this ansatz it is possible to separate pure weak-coupling contribution from the one, which interpolates between strong and weak coupling [49] and is responsible for the contribution of the Lücher corrections [50].
In other words, its application allowed to demonstrate that at the level of order λ 4 extra contributions, which describe "wrapping effects" of Lücher corrections [50], are manifesting themselves. These effects are detectable both at strong coupling constant regime (see e.g. Ref. [51]) and weak coupling constant regime [36].
Perturbation-theory oriented clarification of these words is encoded in the results of Ref. [37]. Indeed, the 4-loop expression for γ K can be decomposed into two terms, namely where The result of Eq.(24) was first obtained in Ref. [35]. The analytical calculations of overall order λ 4 -contribution and of its two parts are in agreement with the calculations performed with superspace diagrammatic formalism [36]. In its turn, the total expression for the order λ 4 -approximation of Eq.(23), obtained in Ref. [37] and Ref. [36], coincide with the result of Eq. (22), obtained in Ref. [38] from direct Feynman diagrams calculations. This independent calculation gave real confidence in the correctness of final analytical expression and in the fact that the asymptotic part of 4-loop result for γ K (see Eq.(24)) does not contain ζ(5)-contribution, which, together with additional pure rational and ζ(3)-contributions, enter into 4-loop "wrapping" effects (for the diagrammatic explanation of the appearance of ζ(5) in Eq.(24) see Ref. [39]) The results of Eq. (24) should be compared with the pqQED ones, given in Sec.2. Compared with each other they indicate, that Riemann ζ(3)-puzzle is not the puzzle, but the regular feature of the asymptotic series in the conformal-invariant theories. Following this conclusion, one should expect manifestation of ζ(3) and ζ(5) terms in the next-to-presented above coefficients of the corresponding asymptotic perturbative series in the conformal invariant theories. This feature is realized in the results of calculations of 5-loop corrections to the anomalous dimension of Eq. (24) in N = 4 SUSY Yang-Mills theory [41]- [42]. Note that ζ(7)-terms are appearing in the 5-loop "wrapping contributions" only (see e.g. [43], [44]). Moreover, ζ-functions counting rule, namely the appearance of extra ζ-functions in high order wrapping contributions, is supported by the results of six loop calculations (see Ref. [40] and Ref. [45] ), which demonstrate the appearance of ζ(9)-terms.

Conclusions.
In this work we introduce the way of explaining the structure of analytical expression for high order corrections in asymptotic perturbative series for the anomalous dimensions and coefficient functions of gauge-invariant operators in pqQED. The arguments, presented in this work, are useful for realizing that the appearance of ζ(3)-terms in the pqQED series is rather regular feature. This feature is supported by the property of conformal symmetry.
Indeed, the ζ-functions counting rules are also satisfied for dealing with coefficients of the asymptotic perturbative series for the anomalous dimensions of operators in superconformal N = 4 SUSY Yang-Mills gauge theory in the case when "wrapping interactions" are not taken into account. These interactions are responsible for the interpolation into the regime of large values of coupling constant.
At present I do not know whether it is possible to find the signals of the existence of these interactions in the strong-coupling phase of quenched QED. In the case if these interactions do exist, they may signal about themselves through the explicit manifestations of higher transcendentalities, and ζ(5) in particular.