Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(16)

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

O cut4β= k21,k22<Λ2d4k1(2π)4d4ξ1d4ξ2e ik1·ξ1e ik2·ξ2

ׯψ(0) γβγ5iD⊥(ξ1)·iD⊥(ξ2)+i?αηγδpγnδγβiDα(ξ1)iDη(ξ2) ψ(0)(18) where p and n are light-like vectors that are introduced as a part of the coordinate basis. Clearly,asΛ2→∞,O cut4β→O4β,the local twist-four operator appearing in Eq.(5).For a ?xedΛ2,O cut4βis a non-local operator.It is simple to show that at the one-loop order the perturbative calculation of the twist-four matrix element reproduces the soft perturbative contribution to the coe?cient function in Eq.(14).

The above result tells us several things about the operator product expansion in Eq.

(5).First,the higher-twist operator has an upper cut-o?in the Fourier components for the gluon?elds,and thus is?nite.To fully include the non-perturbative e?ect,the cut-o?Λ2 must be larger than any non-perturbative scale,such asΛQCD.Second,when quarks and gluons have virtuality greater thanΛ2,their e?ects are perturbative and are included in the coe?cient functions.The separation of the two e?ects are scale-dependent,but the sum is stly,although I discussed only one-loop diagrams with one soft-gluon,the non-perturbative twist-four contribution takes care of all Feynman diagrams with one soft-gluon. Diagrams with one soft-quark line do not contribute at the twist-four level[29].Diagrams with two soft-gluon or quark lines are related to the twist-six terms,which are beyond the scope of this paper.

Clearly,when taking O cut4βas a de?nition of higher-twist operator,the corresponding coe?cient function in Eq.(5)shall be modi?ed,

C2(αs)=C2pert(αs)+8

Q2

(19)

where C2pert(αs)the usual perturbation series in Eq.(2),C4(αs)is the coe?cient function for the twist-four operator.The perturbative matrix element of O cut4µis to be evaluated in a zero-momentum quark state and is expected to have IR renormalons.Since soft contributions are subtracted in Eq.(19),we expect that C2(αs)is free of the leading IR renormalon singularity.However,the renormalon poles at4πn/β0with n≥2are still present.To subtract them,one must consider the two-loop contributions to C2pert(αs)and the twist-six operators.

To see that the perturbative matrix element of O cut4βhas IR renormalons which cancels the leading IR renormalon in C2pert(αs),I consider again the bubble-chain diagram shown in Fig.1a.The corresponding contribution to the matrix element is shown in Fig.1b.A simple calculation shows,

O cut4β pert(Fig.1b)=αs(Q2)

4π

n

Fn 3Λ2n k=0C k n1

Γ(4??)

n?k

?

2

Λ2

?(n?k)/2 (20)

Thus the subtraction is bothΛ2and Q2dependent.If I?xΛ2at1GeV2,the subtraction is maximum at Q2=1GeV2.As Q2increase,the subtraction becomes smaller,indicating

11