A Soft-Collinear Mode for Jet Cross Sections in Soft Collinear Effective Theory

TL;DR

The paper introduces a soft-collinear mode, enabling SCET++ to factorize and resum logs of jet radius R in exclusive 2-jet cross sections. It presents a detailed factorization into hard, jet, global soft, soft-collinear, and csoft sectors, derives two-loop anomalous dimensions, and relates measured and unmeasured jet/soft functions, validating against fixed-order QCD and providing a NNLL-resummed jet thrust formula. While non-global logarithms are not fully resummed here, the framework establishes the all-orders structure for global logs in jet cross sections and clarifies connections to prior work on unmeasured jet functions. Overall, this work lays critical groundwork for a complete resummation of jet-radius logarithms and their interplay with jet veto scales.

Abstract

We propose the addition of a new "soft-collinear" mode to soft collinear effective theory (SCET) below the usual soft scale to factorize and resum logarithms of jet radii $R$ in jet cross sections. We consider exclusive 2-jet cross sections in $e^+e^-$ collisions with an energy veto $Λ$ on additional jets. The key observation is that there are actually two pairs of energy scales whose ratio is $R$: the transverse momentum $QR$ of the energetic particles inside jets and their total energy $Q$, and the transverse momentum $ΛR$ of soft particles that are cut out of the jet cones and their energy $Λ$. The soft-collinear mode is necessary to factorize and resum logarithms of the latter hierarchy. We show how this factorization occurs in the jet thrust cross section for cone and $k_T$-type algorithms at $O(α_s)$ and using the thrust cone algorithm at $O(α_s^2)$. We identify the presence of hard-collinear, in-jet soft, global (veto) soft, and soft-collinear modes in the jet thrust cross section. We also observe here that the in-jet soft modes measured with thrust are actually the "csoft" modes of the theory SCET$_+$. We dub the new theory with both csoft and soft-collinear modes SCET$_{++}$. We go on to explain the relation between the "unmeasured" jet function appearing in total exclusive jet cross sections and the hard-collinear and csoft functions in measured jet thrust cross sections. We do not resum logs that are non-global in origin, arising from the ratio of the scales of soft radiation whose thrust is measured at $Qτ/R$ and of the soft-collinear radiation at $2ΛR$. Their resummation would require the introduction of additional operators beyond those we consider here. The steps we outline here are a necessary part of summing logs of $R$ that are global in nature and have not been factorized and resummed beyond leading-log level previously.

Figures (6)

• Figure 1: Phase space for cone and k$_T$/Sterman-Weinberg jets. The phase space for a collinear (blue) or soft (red) gluon emitted from a quark and antiquark in the computation of the $\mathcal{O}(\alpha_s)$ 2-jet rate in the cone and k$_T$-type/Sterman-Weinberg algorithms are shown.
• Figure 2: Soft phase space for one gluon. The phase space for one soft gluon emission is the same for both cone and k$_T$-type/Sterman-Weinberg algorithms. The original soft phase space on the left covers the region outside both jets where radiation is vetoed by the energy cut $E_g<\Lambda$. This region is actually sensitive to two distinct physical scales. On the right-hand side, this region is re-expressed in terms of region sensitive to one physical scale at a time. The purely soft (or "global") region covers all angles and is sensitive to the scale $2\Lambda$. The "soft-collinear" regions cover gluons of energy $\Lambda$ within the jet cones of angle $R$ and are sensitive to the scale $2\Lambda R$.
• Figure 3: Scales in ${\rm SCET}_{++}$ for the jet thrust cross section. The scaling of the light-cone components of momentum $(p^\pm,p^\mp,p_\perp)$ for each mode is shown. Hard virtual modes of scale $Q$ are integrated out. The jet scale is the same as the global thrust distribution; the usual soft scale is increased by $1/R$ due to the restriction of measured soft radiation to a cone of radius $R$, like the csoft mode of Bauer:2011uc. The soft veto on the energy $\Lambda$ of additional jets induces a global soft veto mode that cannot resolve the angle $R$ as well as the soft-collinear modes that can, see Eq. \ref{['eq:SCETR']}. The csoft scale here could also be below the soft veto and/or soft-collinear scales.
• Figure 4: Comparison between Eq. \ref{['eq:conerate2']} (solid lines) and the EERAD3 output (dots) for $\sigma_{N_c}$, $\sigma_{1/N_c}$ and $\sigma_{n_f}$, plotted (upper) as functions of $-\ln \Lambda/Q$ with fixed $R$. The linear dependence on $-\ln\Lambda/Q$ is shown as a check for the $\ln R$-enhanced terms predicted by our factorization theorem. Note the numerical sensitivity for small values of $R$ and $\Lambda/Q$. The differences (dots, with error bars) between Eq. \ref{['eq:conerate2']} and the EERAD3 output are plotted (lower) as functions of $-\ln \Lambda/Q$ with $R =$ 0.32. Performing $\chi^2$ fits within the fit regions (red), we extract the coefficients $C_{N_c}=-76\pm 8$, $C_{1/N_c}=-1.8\pm0.3$ and $C_{n_f}=10.0\pm0.1$.
• Figure 5: Comparison between Eq. (\ref{['eq:conerate2']}) (solid lines) and the output of EERAD3 (dots) for $\sigma_{N_c}$, $\sigma_{1/N_c}$ and $\sigma_{n_f}$, plotted as functions of $-\ln R$ with fixed $\Lambda/Q$. The quadratic dependence on $-\ln R$ is shown again as a check for the $\ln R$-enhanced terms predicted by our factorization theorem.