Soft-collinear mode for jet cross sections in soft collinear effective theory

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 ${e}^{+}{e}^{\ensuremath{-}}$ collisions with an energy veto $\mathrm{\ensuremath{\Lambda}}$ 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 $\mathrm{\ensuremath{\Lambda}}R$ of soft particles that are cut out of the jet cones and their energy $\mathrm{\ensuremath{\Lambda}}$. 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 $\mathcal{O}({\ensuremath{\alpha}}_{s})$ and using the thrust cone algorithm at $\mathcal{O}({\ensuremath{\alpha}}_{s}^{2})$. 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 ${\mathrm{SCET}}_{+}$. We dub the new theory with both csoft and soft-collinear modes ``${\mathrm{SCET}}_{++}$.'' 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 nonglobal in origin, arising from the ratio of the scales of soft radiation whose thrust is measured at $Q\ensuremath{\tau}/R$ and of the soft-collinear radiation at $2\mathrm{\ensuremath{\Lambda}}R$. 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.