Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(6)

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

C (b )=

∞ n =0c n +1

4π n ,(11)

where γis related to β2,the two-loop beta function,and γ1,the one-loop anomalous dimen-sion of the twist-four operator

[12].The existence of the IR renormalons destroys the Borel summability of the perturbation series since the integral in Eq.(10)is now ill-de?ned.Unlike Borel-summable series,the IR-renormalon-plagued series re?ects the truly divergent nature of the expanding quantity in perturbation theory;the divergence has nothing to do with the choice of the expansion parameter.Since at large Q 2higher-twist terms are exponentially small compared with the terms in the leading-twist,and since di?erent ways of regularizing the Borel integral in Eq.

(10)produce an uncertainty of order of 1/Q 2,the the nature of the OPE clearly deserves a more careful examination [15].

POSITE OPERATORS WITH POWER DIVERGENCE

Composite operators in QCD have a special importance:The chiral symmetry breaking is characterized by a non-vanishing expectation value of ¯ψψin the vacuum.The electro-weak couplings of quarks are through vector and axial-vector currents ¯ψγ

µψand ¯ψγµγ5ψ.The trace anomaly of the energy-momentum tensor are proportional to F 2=F αβF αβ,and the axial anomaly is proportional to F αβ?F

αβ.Finally,Wilson’s operator product expansion employs myriad composite operators.

Composite operators are badly divergent in ?eld theory,and are not made ?nite through the standard renormalization of Green’s functions [18].Rather,they are renormalized indi-vidually and have their own renormalization scale dependence.For these reasons,composite operators are normally not physical observables.When they do appear as a part of a physical quantity,the renormalization scale is determined either by physics consideration or cancelled by the scale-dependence of other quantities.

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