The Gluon Beam Function at Two Loops

TL;DR

The paper delivers the two-loop (NNLO) gluon beam-function matching coefficients within SCET, completing the NNLO set of virtuality-dependent beam functions when combined with prior quark results. By computing $\mathcal{I}_{gj}^{(2)}(t,z,\mu)$ analytically and validating against known anomalous dimensions and splitting functions, the work enables $N^3LL$ resummation for observables sensitive to parton virtuality, such as beam thrust and $N$-jettiness. Numerical studies with MSTW2008 PDFs show that NNLO corrections are significant, particularly for gluons, but that the residual matching-scale dependence is reduced by about a factor of two, signaling improved perturbative stability for resummed predictions. The analysis employs cross-checks across gauges and calculation methods, reinforcing the consistency of beam and jet function evolution at higher orders.

Abstract

The virtuality-dependent beam function is a universal ingredient in the resummation for observables probing the virtuality of incoming partons, including N-jettiness and beam thrust. We compute the gluon beam function at two-loop order. Together with our previous results for the two-loop quark beam function, this completes the full set of virtuality-dependent beam functions at next-to-next-to-leading order (NNLO). Our results are required to account for all collinear ISR effects to the N-jettiness event shape through N^3LL order. We present numerical results for both the quark and gluon beam functions up to NNLO and N^3LL order. Numerically, the NNLO matching corrections are important. They reduce the residual matching scale dependence in the resummed beam function by about a factor of two.

Figures (6)

• Figure 1: Diagrams contributing to the calculation of the NNLO matching coefficients ${\mathcal{I}}_{gg}$ (a-l) and ${\mathcal{I}}_{gq}$ (m-s) when using dimensional regularization. Left-right mirror graphs and graphs with reversed fermion flow in the loop are not displayed. The blob in diagrams (h,i,p) represents the full one-loop gluon self-energy. The graphs can be computed using either standard QCD Feynman rules or SCET Feynman rules with collinear quark and gluon lines. Using axial gauge and QCD Feynman rules, this is the complete set of nontrivial diagrams. Using SCET Feynman rules, it has to be supplemented by diagrams involving vertices of four collinear particles. In Feynman gauge, additional diagrams with Wilson line connections (see e.g. figure \ref{['fig:endpoint']}) or ghost loops contribute.
• Figure 2: Diagrammatic calculation of the endpoint ($z\to1$) contributions to the partonic beam function in Feynman gauge. The connections to the collinear Wilson lines in the beam function operator denoted as $\otimes$ in the left (example) diagram can also be drawn explicitly as connections to double lines that represent the Wilson lines along the $\bar{n}$ direction in the adjoint representation (middle diagram). In the limit $z\to1$ the incoming gluon lines can be replaced by (adjoint) Wilson lines along the $n$ direction as shown in the right diagram. The calculation of the quark-quark channel endpoint is analogous, but with all the Wilson lines in the fundamental color representation.
• Figure 3: The one-loop (blue) and two-loop (orange) corrections to the integrated beam function $\widetilde{B}_i$ in percent relative to the tree level result for $i=d$ (upper left), $i=u$ (upper right), $i=\bar{d}$ (lower left), and $i=g$ (lower right) as a function of the minus-momentum fraction $x$ carried by the parton $i$. Dotted lines show the contributions from off-diagonal channels ($q\to g$, $g \to q$), dashed lines the diagonal channels ($q\to q$, $g\to g$), as detailed in the text. The solid lines show the total result after summing over diagonal and off-diagonal contributions.
• Figure 4: Residual matching scale dependence in the resummed integrated beam function $\widetilde{B}_i\otimes U_B$ at $x = 0.01$ for $i=d$ (upper left), $i=u$ (upper right), $i=\bar{d}$ (lower left), and $i=g$ (lower right). In all cases we show the correction in percent relative to the fixed NNLO result at the central scale $\widetilde{B}_{i0} \equiv \widetilde{B}_i^{\rm NNLO}(t_{\rm max}, x, \mu_B = \sqrt{t_{\rm max}})$.
• Figure 5: The cut ladder diagram. The cut is denoted by a dashed line. The discussion in this section applies regardless of the species of partons in the diagram, so we just use straight lines to denote the particles in the diagram.