Absence of π2 terms in physical anomalous dimensions in DIS: Verification and resulting predictions

J Davies,A Vogt

doi:10.1016/j.physletb.2017.11.036

Abstract

We study the higher-order corrections to structure functions in inclusive deep-inelastic scattering (DIS) in massless perturbative QCD, in the context of the conjectured absence of even-n values of the Riemann zeta-function ζn, i.e., of powers of π2, in Euclidean physical quantities. We provide substantial additional support for this conjecture by demonstrating that it holds, as far as it can be tested by the results of diagram computations, for the physical anomalous dimensions of structure functions at the fourth and fifth order in the strong coupling constant αs. The conjecture is then employed to predict hitherto unknown ζ4 and ζ6 contributions to the anomalous dimensions for parton distributions and to the coefficient functions for the longitudinal structure function FL.

Highlights

As far as they are presently known, the anomalous dimensions – i.e., the even-N or odd-N Mellin moments of the splitting functions – for the scale dependence of parton distributions can be expressed in terms of rational numbers and integer-n values ζn of the Riemann ζ -function
It is an old observation that the above ‘spacelike’ quantities do not include terms linear in ζ2 = π 2/6.1 Terms with ζ22 or ζ4 = 2/5 ζ22 = π 4/90 do occur in the three-loop coefficient functions in deepinelastic scattering (DIS) [1,2,3] and the four-loop anomalous dimensions [9,10,11], which together define the next-to-next-to-next-to-leading order
In the standard MS renormalization scheme, this rule is violated at N4LO by the scalar quark and gluon correlators that enter the hadronic decays of the Higgs boson [14]

Summary

Introduction

As far as they are presently known, the anomalous dimensions – i.e., the even-N or odd-N Mellin moments of the splitting functions – for the scale dependence (evolution) of parton distributions can be expressed in terms of rational numbers and integer-n values ζn of the Riemann ζ -function. Absence of π 2 terms in physical anomalous dimensions in DIS: Verification and resulting predictions It is an old observation that the above ‘spacelike’ quantities do not include terms linear in ζ2 = π 2/6.1 Terms with ζ22 or ζ4 = 2/5 ζ22 = π 4/90 do occur in the three-loop coefficient functions in DIS (with the exception of the longitudinal structure function F L ) [1,2,3] and the four-loop anomalous dimensions [9,10,11], which together define the next-to-next-to-next-to-leading order

Results

Conclusion