Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On threshold resummation beyond leading 1-x order

G. Grunberg, V. Ravindran

TL;DR

This work tests the viability of threshold resummation in momentum space beyond the leading 1−x order by comparing simple and two-scale ansatze against exact three-loop results for non-singlet DIS structure functions F2 and F3. It finds that a straightforward single-scale (and even a two-scale) ansatz cannot reproduce all finite-β0 terms at next-to-leading order, indicating an obstruction to exact threshold resummation at this order. Nonetheless, the analysis preserves some predictive power: the leading logarithms for each color structure are recovered, and certain NNLL patterns appear for the CF^3 contribution, along with a universal behavior of the leading LL terms in the physical evolution kernels for F2 and F3. The results motivate a decomposition of the evolution kernel into an exponentiating part and a non-exponentiating remainder, with implications for extending threshold resummation to higher orders and for related processes like Drell–Yan.

Abstract

We check against exact finite order three-loop results for the non-singlet F_2 and F_3 structure functions the validity of a class of momentum space ansaetze for threshold resummation at the next-to-leading order in 1-x, which generalize results previously obtained in the large-β_0 limit. We find that the ansaetze do not work exactly, pointing towards an obstruction to threshold resummation at this order, but still yield correct results at the leading logarithmic level for each color structures, as well as at the next-to-next-to-leading logarithmic level for the specific C_F^3 color factor. A universality of the leading logarithm contributions to the physical evolution kernels of F_2 and F_3 at the next-to-leading order in 1-x is observed.

On threshold resummation beyond leading 1-x order

TL;DR

This work tests the viability of threshold resummation in momentum space beyond the leading 1−x order by comparing simple and two-scale ansatze against exact three-loop results for non-singlet DIS structure functions F2 and F3. It finds that a straightforward single-scale (and even a two-scale) ansatz cannot reproduce all finite-β0 terms at next-to-leading order, indicating an obstruction to exact threshold resummation at this order. Nonetheless, the analysis preserves some predictive power: the leading logarithms for each color structure are recovered, and certain NNLL patterns appear for the CF^3 contribution, along with a universal behavior of the leading LL terms in the physical evolution kernels for F2 and F3. The results motivate a decomposition of the evolution kernel into an exponentiating part and a non-exponentiating remainder, with implications for extending threshold resummation to higher orders and for related processes like Drell–Yan.

Abstract

We check against exact finite order three-loop results for the non-singlet F_2 and F_3 structure functions the validity of a class of momentum space ansaetze for threshold resummation at the next-to-leading order in 1-x, which generalize results previously obtained in the large-β_0 limit. We find that the ansaetze do not work exactly, pointing towards an obstruction to threshold resummation at this order, but still yield correct results at the leading logarithmic level for each color structures, as well as at the next-to-next-to-leading logarithmic level for the specific C_F^3 color factor. A universality of the leading logarithm contributions to the physical evolution kernels of F_2 and F_3 at the next-to-leading order in 1-x is observed.

Paper Structure

This paper contains 10 sections, 128 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The conjecture
  3. Checking the ansatz
  4. Two-scale ansatz
  5. Comment on reference Laenen:2008ux
  6. Concluding remarks
  7. Relevant terms in the expansions of $c_1^{\otimes 2}(x)$, $c_1^{\otimes 3}(x)$, and $c_2(x)\otimes c_1(x)$ in parametric form
  8. Relevant terms in the expansions of $c_2(x)$ and $c_1^{\otimes 2}(x)$ ($F_2$ structure function)
  9. Relevant terms in the expansions of $c_3(x)$, $c_2(x)\otimes c_1(x)$ and $c_1^{\otimes 3}(x)$ ($F_2$ structure function)
  10. Relevant terms in the expansions of $c_1(x)$, $c_2(x)$, $c_1^{\otimes 2}(x)$, $c_3(x)$, $c_2(x)\otimes c_1(x)$ and $c_1^{\otimes 3}(x)$ ($F_3$ structure function)