Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering

TL;DR

The paper develops a momentum-space threshold resummation for deep-inelastic scattering using soft-collinear effective theory (SCET), factorizing the non-singlet structure function into a hard coefficient, a jet function, and a PDF. By solving the RG equations directly in momentum space with well-chosen scales, it delivers analytic, Landau-pole–free expressions and establishes formal equivalence with the traditional moment-space approach. The work provides explicit perturbative inputs up to NNLO and outlines the endpoint evolution of PDFs, offering a robust framework applicable to other near-threshold processes such as Drell-Yan and inclusive B decays. The approach circumvents ambiguities inherent in Mellin-space resummation and clarifies the role of soft-collinear modes in the endpoint region.

Abstract

Renormalization-group methods in soft-collinear effective theory are used to perform the resummation of large perturbative logarithms for deep-inelastic scattering in the threshold region x->1. The factorization theorem for the structure function F_2(x,Q^2) for x->1 is rederived in the effective theory, whereby contributions from the hard scale Q^2 and the jet scale Q^2(1-x) are encoded in Wilson coefficients of effective-theory operators. Resummation is achieved by solving the evolution equations for these operators. Simple analytic results for the resummed expressions are obtained directly in momentum space, and are free of the Landau-pole singularities inherent to the traditional moment-space results. We show analytically that the two methods are nonetheless equivalent order by order in the perturbative expansion, and perform a numerical comparison up to next-to-next-to-leading order in renormalization-group improved perturbation theory.

Figures (9)

• Figure 1: Kinematics of DIS.
• Figure 2: Examples of diagrams involving anti-collinear gluon exchange. The dashed (dotted) lines represent anti-collinear (hard-collinear) quark lines. The wavy lines represent the electromagnetic currents. In graph (a) the anti-collinear gluon is part of the initial-state (nucleon) jet and the final-state propagator is hard-collinear, as required in the effective theory. In graphs (b) and (c) the anti-collinear gluon is part of the final-state jet, whose invariant mass then becomes hard. Graphs (b) and (c) are therefore not part of the effective-theory representation of the hadronic tensor as $x\to 1$.
• Figure 3: Soft-collinear Wilson line $W_C$.
• Figure 4: Examples of subleading time-ordered products in SCET which give rise to power corrections. Graph (a) leads to a perturbatively calculable power-suppressed jet function, while (b) and (c) lead to subleading parton distribution functions depending on three light-cone variables.
• Figure 5: Comparison between fixed-order (dashed) and resummed results (solid) for the $K$ factor. The green curves are the LO result, red NLO, black NNLO. For the resummed result, we set $\mu_h=Q$, $\mu_i=M_X$, $\mu_f=Q$, and $b(\mu_f)=4$. The fixed-order result is obtained by setting all scales equal to $\mu_f$.