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

摘要

We propose the addition of a new "soft-collinear" mode to soft collineareffective theory (SCET) below the usual soft scale to factorize and resumlogarithms of jet radii $R$ in jet cross sections. We consider exclusive 2-jetcross sections in $e^+e^-$ collisions with an energy veto $\Lambda$ onadditional jets. The key observation is that there are actually two pairs ofenergy scales whose ratio is $R$: the transverse momentum $QR$ of the energeticparticles inside jets and their total energy $Q$, and the transverse momentum$\Lambda R$ of soft particles that are cut out of the jet cones and theirenergy $\Lambda$. The soft-collinear mode is necessary to factorize and resumlogarithms of the latter hierarchy. We show how this factorization occurs inthe jet thrust cross section for cone and $k_T$-type algorithms at$O(\alpha_s)$ and using the thrust cone algorithm at $O(\alpha_s^2)$. Weidentify the presence of hard-collinear, in-jet soft, global (veto) soft, andsoft-collinear modes in the jet thrust cross section. We also observe here thatthe in-jet soft modes measured with thrust are actually the "csoft" modes ofthe theory SCET$_+$. We dub the new theory with both csoft and soft-collinearmodes SCET$_{++}$. We go on to explain the relation between the "unmeasured"jet function appearing in total exclusive jet cross sections and thehard-collinear and csoft functions in measured jet thrust cross sections. We donot resum logs that are non-global in origin, arising from the ratio of thescales of soft radiation whose thrust is measured at $Q\tau/R$ and of thesoft-collinear radiation at $2\Lambda R$. Their resummation would require theintroduction of additional operators beyond those we consider here. The stepswe outline here are a necessary part of summing logs of $R$ that are global innature and have not been factorized and resummed beyond leading-log levelpreviously.