萝莉的烦躁期What if one uses other regularization schemes,such as the scheme with a momentum cut-o??In a cut-o?scheme,composite operators are?nite,as long as the cut-o?is kept ?nite.The operators are also free of any IR renormalon singularities.Their physical matrix elements depend on two scales:the non-perturbative scale generated from the breaking of the scale symmetry in the non-perturbative QCD[22]and the perturbative scale that is served as a cut-o?for both the log and power divergences.If the degree of divergence of an operator is d O,its matrix element goes like the d O power of the cut-o?when the cut-o?approaches in?nity.In light of dimensional regularization,it is tempting to de?ne a pure non-perturbative matrix element by subtracting o?the corresponding perturbative matrix element.Although the cut-o?dependence now disappears,the IR renormalon singularities are introduced through perturbative matrix elements.Thus a pure non-perturbative matrix element is again not well-de?ned.
To summarize the above discussion,operators with power divergence cannot be speci?ed uniquely after subtracting their logarithmic divergence in dimensional regularization.One must specify in addition how the IR renormalons are regularized,or,equivalently,how the limit?→0is taken.In a cut-o?regularization,the operators are cut-o?dependent, independent of how the logarithmic divergence is regulated.For example,β(g)/2gF2has
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