Higher-Order Corrections in Threshold Resummation

TL;DR

This work pushes threshold resummation in Mellin-N space to N^3LL accuracy by organizing the exponent as $G^N$ with four logarithmic orders, enabling all-orders control of terms ${\rm O}(\alpha_s^2 (\alpha_s \ln N)^n)$. By matching to existing three-loop DIS results, the authors determine the universal jet-function coefficients $B_q$ and $B_g$ up to ${\rm O}(\alpha_s^3)$, uncovering a nontrivial relation with splitting functions and large-angle soft emissions in DY-like processes. The study includes both the fully exponentiated form and a tower expansion to assess numerical impact, finding that N^3LL corrections are typically small at moderate scales but essential for reliable predictions near threshold. Their tower expansion up to seven logarithmic towers stabilizes well and yields four-loop DIS $F_2$ predictions, while the full exponentiation confirms robust behavior under the minimal prescription. Overall, the results enhance precision for high-x observables and provide a framework for extending NNLL resummations to a broader class of processes.

Abstract

We extend the threshold resummation exponents G^N in Mellin-N space to the fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln N)^n to all orders in the strong coupling constant as. Comparing the results to our previous three-loop calculations for deep-inelastic scattering (DIS), we derive the universal coefficients B_q and B_g governing the final-state jet functions to order alpha_s^3, extending the previous quark and gluon results by one and two orders. A curious relation is found at second order between these quantities, the splitting functions and the large-angle soft emissions in Drell-Yan type processes. We study the numerical effect of the N^3LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms.

Figures (6)

• Figure 1: Left: the LL, NLL, N$^2$LL and N$^3$LL approximations for the resummation exponent for standard DIS. Right: the convolutions of the exponentiated results with a typical input shape.
• Figure 2: As the right part of the previous figure, but for a different value of ${n^{}_{\! f}}$ (left) and $\alpha_{\rm s}$ (right).
• Figure 3: As Fig. \ref{['pic:fig1']}, but for inclusive DIS by exchange of a scalar $\phi$ directly coupling to gluons.
• Figure 4: Convolution of the two-loop (left) and three-loop (right) contributions to the DIS coefficient functions $C_{2,\rm q}$ with a typical input shape. Shown are the full results vanNeerven:1991nnMoch:1999ebVermaseren:2005qc and the large-$x$ expansion by successively including the +-distributions ${\cal D}_k$, respectively starting with ${\cal D}_3$ and ${\cal D}_5$.
• Figure 5: As the previous figure, but for large-$N$ expansion in terms of decreasing powers of $\ln\, N\,$.