Constant terms in threshold resummation and the quark form factor

TL;DR

The paper investigates how N-independent constant terms in threshold resummation for DIS and DY relate to the second logarithmic derivative of the massless quark form factor in four dimensions. By combining standard DIS/DY resummation formalisms with a dispersive (large-$\beta_0$) approach, it verifies conjectured relations to ${\mathcal O}(\alpha_s^4)$ and provides an all-orders check in the large-$\beta_0$ limit, yielding a dispersive representation of the quark form factor. It derives explicit expressions for the three-loop coefficients $B_3$ and $D_3$ in terms of diagonal splitting function and form-factor contributions, and demonstrates that these relations persist across general resummation schemes. The results enhance understanding of how purely virtual, form-factor-like pieces determine constant terms in Sudakov resummation and suggest a universal IR structure across related inclusive processes.

Abstract

We verify to order alpha_s^4 two previously conjectured relations, valid in four dimensions, between constant terms in threshold resummation (for Deep Inelastic Scattering and the Drell-Yan process) and the second logarithmic derivative of the massless quark form factor. The same relations are checked to all orders in the large beta_0 limit; as a byproduct a dispersive representation of the form factor is obtained. These relations allow to compute in a symmetrical way the three-loop resummation coefficients B_3 and D_3 in terms of the three-loop contributions to the virtual diagonal splitting function and to the quark form factor, confirming results obtained in the literature.