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

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 $\Lambda$ 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 $\Lambda R$ of soft particles that are cut out of the jet cones and their energy $\Lambda$. 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 $O(\alpha_s)$ and using the thrust cone algorithm at $O(\alpha_s^2)$. 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 $Q\tau/R$ and of the soft-collinear radiation at $2\Lambda R$. 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.