The Fully-Differential Quark Beam Function at NNLO

TL;DR

This work delivers the first NNLO calculation of a fully-unintegrated parton distribution by computing the two-loop quark dBF–PDF matching coefficients ${\mathcal{I}}_{ij}^{(2)}$ within SCET. The authors derive a master formula for ${\mathcal{I}}_{ij}^{(2)}(t,z,\vec{k}_{\perp}^{\,2},\mu)$ containing $\mathcal{L}_n$ distributions in $t$ and introduce the novel $J^{(2)}_{ij}(t,z,\vec{k}_{\perp}^{\,2})$ functions that encode the transverse momentum dependence, which can only be fixed after integrating over $\vec{k}_{\perp}^{\,2}$. The calculation is cross-checked with two gauges and two diagram-discontinuity methods, and the resulting expressions enable NNLO singular contributions and NNLL$'$ / N$^3$LL resummations for observables sensitive to both virtuality and transverse momentum. The results have broad implications for precision predictions in processes with double-differential measurements and could inform improvements to initial-state radiation modeling in event generators. The quark dBF integrates to the virtuality-dependent beam function, validating the consistency of the factorization framework.

Abstract

We present the first calculation of a fully-unintegrated parton distribution (beam function) at next-to-next-to-leading order (NNLO). We obtain the fully-differential beam function for quark-initiated processes by matching it onto standard parton distribution functions (PDFs) at two loops. The fully-differential beam function is a universal ingredient in resummed predictions of observables probing both the virtuality as well as the transverse momentum of the incoming quark in addition to its usual longitudinal momentum fraction. For such double-differential observables our result provides the part of the NNLO singular cross section related to collinear initial-state radiation (ISR), and is important for the resummation of large logarithms through N3LL.

Figures (1)

• Figure 1: Examples of Feynman diagrams contributing to the calculation of the NNLO matching coefficients ${\mathcal{I}}_{q_iq_j}$, ${\mathcal{I}}_{q_i\bar{q}_j}$ and ${\mathcal{I}}_{q_ig}$, respectively. The complete list of relevant axial-gauge diagrams, when using dimensional regularization, is displayed in figure 2 of ref. Gaunt:2014xga.