Threshold Resummation of the Structure Function F_L

Abstract

The behaviour of the quark coefficient function for the longitudinal structure function F_L in deep-inelastic scattering is investigated for large values of the Bjorken variable x. We combine a highly plausible conjecture on the large-x limit of the physical evolution kernel for this quantity with our explicit three-loop results to derive the coefficients of the three leading large-x logarithms, alpha_s^n ln^(2n-1-k) (1-x), k = 1, 2, 3, to all orders in the strong coupling constant alpha_s. Corresponding results are derived for the non-C_F part of the gluon coefficient function suppressed by a factor 1-x, and for the analogous subleading (1-x) ln^k (1-x) contributions in the quark case. Our results appear to indicate an obstacle for an exponentiation with a higher logarithmic accuracy.

Figures (1)

• Figure 1: Successive large-$N$ approximations by the leading 1, 2, 3 and (left) 4 large-$N$ logarithms $\ln^{\,k}\widetilde{N} \equiv (\ln N + \gamma_{\rm e})^k$ for the third- and fourth-order quark coefficient function of ${F_{\:\! L}}$ for four flavours. Also shown is the complete third-order all-$N$ result computed in Refs Moch:2004xuVermaseren:2005qc. The curves have been scaled to correspond to the expansion parameter $\alpha_{\rm s}$ instead of $a_{\rm s} = \alpha_{\rm s}/(4\pi)$ used in our formulae.