Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(13)

Infrared renormalons and $1/Q^2$ power corrections in deep-inelastic sum rules are studied. The renormalization of operators with power divergence are discussed. The higher-twist terms in the operator product expansion are shown to account for the residual

in Wilson’s operator product expansion,and radiative corrections to jet production,Drell-Yan cross section,etc.

However,it was pointed out by Shifman,Vainshtein,and Zakharov(SVZ)that using “infrared-safe”as a criteria to decide a perturbative calculation is inadequate[24].They argued,in the case of current correlation functions in the vacuum,that the part of Feynman diagrams involving low-virtuality quarks and gluons cannot be calculated perturbatively despite such a calculation contains no infrared divergence and the contribution might be small.A legitimate calculation for the part must be done non-perturbatively,by using, in their case,various vacuum condensates.These vacuum condensates correspond to the higher-twist terms in Wilson’s OPE.Thus,the role of higher-twist contributions is to take into account properly the residual soft contributions in Feynman diagrams after the KLN type of cancellation.

Thus to devise a twist expansion,one can start by separating the hard and soft contri-butions in Feynman diagrams.This is precisely what Mueller did for vacuum correlation functions[16].In this way,he obtained coe?cient functions that are entirely perturbative and free of IR renormalons,and the higher-twist operators that are free of ultraviolet diver-gence due to an explicit cut-o?.Both contributions now depends on the twist-separation scale but the sum does not.Notice the separation scale here plays the same role as the prescriptions of regularizing the higher-twist operators in David’s paper.In this section,I follow Mueller’s approach in Ref.[16]and attempt to construct a twist expansion for the Bjorken sum rule.To motivate the approach,let me come back to the physical origin of the renormalon singularities in the perturbation series.

The appearance of n!in the coe?cient functions is due to loop integrations in small momentum regions in Feynman diagrams[16,27].In fact,consider a one-loop Feynman diagram with the gluon momentum k(chosen to be the loop momentum).Replace the gluon couplingαs(Q2)(again,µ2=Q2)with a running couplingαs(k2),

αs(k2)=

αs(Q2)

4π

.ln k2

4π)n ln n(

Q2