Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(8)

?Operators with logarithmic divergence:All twist-two and twist-three operators have logarithmic,and only logarithmic,divergence.Here I restrict myself to just local op-erators.Let me remind the reader that the twist of an operator is de?ned by the di?erence of the dimension of the operator and its rank of spin in representations of the Lorentz group.Although the notion of twist arises from study of deep-inelastic scattering,no reference here is needed to this application.[All operators can be classi-?ed in terms of twist because of the Lorentz symmetry.]The twist-two and twist-three operators have just logarithmic divergence because their matrix elements in physical states are dimensionless quantities.They cannot mix under renormalization with any operators of lower dimensions.However,they can mix among themselves,yielding

a dimensionless mixing matrix.The renormalized operators depend on the genuine

renormalization scaleµ2.

?Operators with power divergence:Operators of twist-four and higher normally have power divergence.Exceptions include?nite operators discussed above and operators with special symmetry properties.The simplest example of operators with power divergence is F2.One might argue that since the energy-momentum tensor is a?nite operator and F2appears in its trace(trace anomaly),so(β(g)/2g)F2must be a?nite operator.This is true as long as one is talking about the di?erence of the matrix elements in the excited states of QCD and the vacuum.Since the vacuum energy-momentum density is not a physical observable,(β(g)/2g)F2in the vacuum needs not to be?nite(the proper normal ordering of the operator is always implied here.) Once an operator has quadratic or quartic or higher-order divergence,they can mix under renormalization with lower-twist operators.For instance,F2can mix with the trivial operator1,and¯ψ?Fµνγνγ5ψcan mix with¯ψγµψ,etc.As shall become clear soon,these mixings have important implications about the twist separation in the OPE.

Let me consider the renormalization of power-divergent operators in dimensional regu-larization.Although one cannot do non-perturbative calculations in this regularization,the coe?cient functions in the leading-twist are normally calculated in the scheme[21].Since perturbative QCD does not have any mass scale,the mixing of higher-twist operators with lower-twist ones vanishes identically.[Integrals of type d d k/k m are taken to be zero.]Only logarithmic divergence appears in the matrix elements of the operators,which can be renor-

malized in a standard way.Thus it seems that composite operators can be de?ned up to their logarithmic divergence and are devoid of any power-dependent perturbative contributions.

This standard treatment of composite operators in dimensional regularization is in fact deceptive.In principle,infrared and ultraviolet physics are entirely di?erent and shall not be

6