Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(9)

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

mixed.However,the statement d d k/k2=0,for instance,has a ridiculous implication that the ultraviolet divergence is cancelled by the integration in the infrared region!Not only the integration is not reliable in the low k2region due to non-perturbative physics,but also is?nite.Thus the implied cancellation can never occur in reality.Actually,the ultraviolet power divergence should not be thrown away like this because they dictate the correct operator mixing.Furthermore,a perturbative calculation in the low k2region generates IR renormalons just like the coe?cient functions discussed in the last section.Thus d d k/k2= 0is mathematically convenient,but physically unwanted.

Indeed,in a study of the two-dimensional O(N)non-linear sigma model,David found that the composite operators with power divergence are not well-de?ned in dimensional regularization[15].He calculated the spin-wave condensate in the physical vacuum using the regularization(which is possible in1/N expansion)and found that the result is not unique.His argument for the presence of the ambiguity in the composite operator is subtle and roughly goes like this:When?=4?d is kept?nit in dimensional regularization,the higher-twist matrix element has a series of poles on the positive real-axis of the?plane starting from?=the degree of divergence of the operator.The poles are present because at these?,one cannot distinguish a higher-twist contribution from a lower-twist one.These poles make the limit of?→0ambiguous,depending on whether one takes Im?>or<0.

The interpretation of David’s ambiguity is that composite operators in dimensional regu-larization have IR renormalons.They were not there originally,but appear when the scheme insists to treat the contribution in the soft momentum region in perturbation theory.Those perturbative soft contributions are needed to cancel the ultraviolet power divergence be-cause the relations like d d k/k m=0are used.Consequently the power divergence is killed with a price:emergence of IR renormalon singularities.It is these singularities that cause the ambiguity in David’s calculation.[An explicit example of the renormalons in matrix elements is provided in the next section in Eq.(20)]Thus in dimensional regularization,in addition to subtract the logarithmic divergence of an operator,one needs to specify how the IR renormalons are regulated.