Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(20)

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

c1=?1

9πQ2

)(26)

The power term here is quite close to the power term in the cut-o?scheme in the previous section.I emphasize again that this coe?cient function must be used together with the speci?c twist-four operator in Eq.(23).

Another way to de?ne the twist-four operator is to let the non-locality of the operator in Eq.(23)Λbe the same as the lattice spacing a,approaching zero in the continuum limit. Then the lattice matrix element diverges like1/a2.A?nite twist-four operator can be de?ned by subtracting from the original operator its perturbative matrix element on the lattice,with a low momentum cut-o?in loop integrals.Thus the twist-four contribution is again free of the IR renormalons but still depends on the infrared cut-o?.Unfortunately,it is di?cult to do perturbative calculations with a nucleon state.However,the idea is clearly applicable to vacuum condensates in the QCD sum rules[9,30].

MENTS AND CONCLUSIONS

According to the discussions in the previous sections,the leading-twist contribution to the Bjorken sum rule can still be regarded as a power series inαs(Q2).However,the coe?cient in each order is now not just a numerical number;it contains the subtraction of the soft contributions that are represented by a power series inΛ2/Q2.The most important soft-subtractions are those nominally suppressed by one power of1/Q2.As the order of the perturbation n→∞,the subtraction grows like n!,independent of the value of Q2and the subtraction scaleΛ2.Therefore,the series inαs would converge if other renormalons singularities and sources of divergent factors did not exit.[It is known that ultraviolet renormalons and the singularities in the Borel plane caused by instanton-anti-instanton pairs still make the series asymtotic,but I ignore them here.]

Thus to the accuracy of1/Q2,one can successively take into account the QCD corrections in the following way.First calculate the coe?cient function C2(αs,Λ2/Q2)in powers ofαs. At each other,the1/Q2subtraction must be calculated explicitly.At lower orders,the subtraction is smaller than the pure loop contributions.However,the former must be kept to maintain the accuracy.As n increases,both the loop contributions and subtractions decrease initially,and their di?erence also decreases.If the di?erence becomes the size of the non-perturbative1/Q2terms,the latter must be included.With the further increase of n,both the perturbative contributions and their subtractions start to grow.However, the cancellation now becomes more complete and the residual remains small.The?rst few terms of the Bjorken sum rule is,

10(g p1(x,Q2)?g n1(x,Q2))dx=g Aπ(1?16Λ2π 2(1?ηΛ2

27

Oµ4(Λ2)

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