Infrared Renormalons and Power Corrections in Deep-Inelastic Sum Rules(5)

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

d 4ξ

e iq ·ξT J em µ(ξ)J em ν(0) =?2i?µναβq α

3C 2(αs (Q 2))J 5β(0)?8

π?3.58 αs (Q 2)

π 3+...,(6)

for three ?avors.Here the renormalization scale has been chosen to be µ2=Q 2,and so there are no logarithms of type ln k Q 2/µ2in c n .Notice that the coe?cient c n is growing rapidly with n .An estimate for the fourth-order term gives ?130(αs (Q 2)/π)4

[19].As a matter of fact,there are indications that c n may grow like n !for large n with a ?xed sign!

To see the rapid growth of c n ,consider the contribution from a n -loop bubble-chain dia-gram shown in ing dimensional regularization and modi?ed minimal subtraction (

4π

(β′0?+γ)k × 6Γ2(2??/2)Γ(?/2)

Γ(1+(n ?k )?/2)Γ(3?(n +2?k )?/2) ,(7)

where C F =4/3and β′0=5?2n f /3represents the contribution to the leading-order β-

function from the one-loop gluon self-energy (n f is the number of ?avors).The symbol Fn refers to taking ?nite part of the expression,which turns out to be non-trivial.With the help of the symbolic program MAPLE,I get the following result for the rational part,

n

Fn[...]20

2.5761918×101921

5.4100006×1020221.1902018×1022.(8)

The irrational part that involves Reiman ζ-functions and πis small compared with the rational part,at least at small orders.Eq.(8)indicates that the contribution from the bubble-chain diagram grows like ～?5×n !at large orders.

One might argue that the large coe?cients from this particular diagram might cancel with those from other diagrams at the same order.However,indications are that the large n behavior of c n for the sum of all n -loop diagrams is qualitatively similar to the bubble-chain diagram.To see that,consider the Borel-transformed series for C (αs ),

4

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