Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(3)

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

matrix elements vanish identically,and the non-perturbative higher-twist matrix elements are known to be calculable,for instance,in lattice QCD by subtracting the corresponding perturbative matrix elements[9],or in the QCD sum rule method[10].It turns out,however, this traditional view on the twist-separation is at best formal,and one in practice is forced to provide more details on the individual contribution.Eventually,the twist expansion has the similar degree of arbitrariness as the factorization of the coe?cient functions from the operator matrix elements in individual term of the OPE[11].This surprising feature of the twist-expansion in deep-inelastic sum rules was?rst recognized and studied,to the au-thor’s knowledge,by A.Mueller[12],although similar studies in vacuum current correlation functions had begun much early and the results are much more well-known[13–16].

This paper grows out of attempts to understand the1/Q2e?ects in the deep-inelastic sum rules[17]and Mueller’s paper on ambiguity of the higher-twist contribution[12].Since I believe most of the physicists working in deep-inelastic scattering are unaware of this issue, I intend to be pedagogical in my presentation.Therefore,the discussions in the paper are often mixed with results that are already known in the case of vacuum correlation functions.

I hope the references are su?ciently clear so the reader has no confusion about what is new and what is not.

In section II,I discuss the infrared(IR)renormalons in the coe?cient functions of the Bjorken sum rule.I start with an empirical observation and a calculation of a bubble-chain diagram to convince the reader that the perturbation series in fact diverges.Then I state some general results known about the properties of the renormalon singularities. In section III,I consider the renormalization of operators with power-like divergence and explain why the multiplicative renormalization of log divergence cannot de?ne the operators uniquely.The conclusion is that pure non-perturbative operators in general do not exist and higher-twist operators depend on handling of the perturbative contributions.In section IV,I derive a cut-o?version of the OPE by separating the soft and hard contributions to Feynman diagrams.The result shows that the coe?cient functions can be de?ned without IR renormalons,but depending on a momentum cut-o?,and the higher-twist operators depend on the same cut-o?due to perturbative power contributions.However,the sum of the twist expansion is cut-o?-independent.In section V,I argue that there exists a large degree of freedom in choosing the higher-twist operators.According to this,I de?ne a twist-four operator for the Bjorken sum rule suitable for lattice calculations.The corresponding twist-two coe?cient function is subsequently evaluated.The?nal section contains the discussion on the practical signi?cance of the1/Q2corrections in the Bjorken sum rule,and summary and conclusion of this paper.

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