Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(14)

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

To identify such contributions in the Bjorken sum rule,consider the one-loop contribution to the coe?cient function shown in Fig.2.The one-loop integrations are free of ultraviolet divergence after the standard QCD renormalization.The external quark momentum is set to zero after the collinear singularity in the box diagram(Fig.2d)is subtracted by the one-loop matrix element of the axial current[21].Now focus on contributions from the region where the gluon momentum is soft(k2is small).In this region,one can expand the integrand in k2/Q2and keep the leading-order contribution.Applying an upper cut-o?Λ2 on the gluon virtuality,I?nd the perturbative part of c1from the soft region,

?c soft pert.

1=

8g2

(2π)4

?i

9

αs(Q2)

Q2

,(14)

where in the second line have rotated the integration to the Euclidean space.The factor 8/3in the?rst line is the sum of2,1,and?1/3,coming from diagrams a),b),and c) in Fig.2,respectively.The cut-o?Λ2should be on the order of0.5to1GeV2,above which perturbative calculation is justi?able.Clearly,the contribution in Eq.(14)must subtracted from c1and the corresponding non-perturbative contribution should be added as a higher-twist contribution.

To?nd the correct non-perturbative contribution,I re-examine the one loop diagrams in Fig. 2.When the gluons are soft,we cannot contract the?eld operators in the per-turbative vacuum,instead we must keep them and evaluate the matrix elements in the non-perturbative zero-momentum quark state.[The same thing can be said of any quark lines that carry soft momenta.]For example,the diagrams(b)and(c)in Fig.2contribute to the quark Compton amplitude,

Tµνq= d4k(2π)4Hµνα(k,k1)Sα(k,k1),(15)

where k and k1are quark and gluon momenta,respectively,with Euclidean k21restricted to ≤Λ2.Hµναrepresents part of the diagrams above the dashed lines and is perturbative.Sαrepresents that below and can be obtained from the?rst-order calculation of

Sα(k,k1)= d4ξd4ξ1e ik·ξe ik1·ξ1 q(0)|T¯ψ(ξ)Aα(ξ1)ψ(0)|q(0) ,(16) whereψand Aµare quark and gluon?eld operators and|q(0) is a zero-momentum quark state.Thus the one-loop diagrams arise from a perturbative expansion of the zero-momentum-quark wave function,which of course is incorrect as the gluon momentum be-comes soft.However,so long as one refrains from such a perturbative expansion,one gets the correct contribution from the soft-gluon region.