Factorization and Resummation for Generic Hierarchies between Jets

TL;DR

This work develops ${\rm SCET}_+$, an extension of Soft-Collinear Effective Theory, to systematically resum large logarithms from generic hierarchies among jet kinematics in multijet processes. By introducing intermediate collinear-soft and soft-collinear modes, the framework derives factorization theorems for distinct regimes ($c+$, $s+$, $cs+$) and demonstrates their application to $e^+e^-\to 3$ jets and to $pp\to N$ jets, including color/spin correlations and regime overlaps. The results are stitched together with nonsingular fixed-order corrections and profile scales to provide a complete description across the full jet phase space, enabling improvements over parton showers and enabling NNLL-level resummation. This framework has practical implications for jet substructure, jet binning, and the integration with Monte Carlo tools (e.g., Geneva, MINLO), and offers a path to systematically account for nonglobal logarithms via controlled hierarchical multijet configurations.

Abstract

Jets are an important probe to identify the hard interaction of interest at the LHC. They are routinely used in Standard Model precision measurements as well as in searches for new heavy particles, including jet substructure methods. In processes with several jets, one typically encounters hierarchies in the jet transverse momenta and/or dijet invariant masses. Large logarithms of the ratios of these kinematic jet scales in the cross section are at present primarily described by parton showers. We present a general factorization framework called SCET$_+$, which is an extension of Soft-Collinear Effective Theory (SCET) and allows for a systematic higher-order resummation of such kinematic logarithms for generic jet hierarchies. In SCET$_+$ additional intermediate soft/collinear modes are used to resolve jets arising from additional soft and/or collinear QCD emissions. The resulting factorized cross sections utilize collinear splitting amplitudes and soft gluon currents and fully capture spin and color correlations. We discuss how to systematically combine the different kinematic regimes to obtain a complete description of the jet phase space. To present its application in a simple context, we use the case of $e^+e^- \to $3 jets. We then discuss in detail the application to N-jet processes at hadron colliders, considering representative classes of hierarchies from which the general case can be built. This includes in particular multiple hierarchies that are either strongly ordered in angle or energy or not.

Figures (6)

• Figure 1: Different hierarchies for three-jet events in $e^+e^-$ collisions.
• Figure 2: Illustration of the multistage matching procedure for the $c+$ regime of ${\rm SCET}_+$ with $t\ll u\sim Q^2$. The modes and their virtuality scale are indicated.
• Figure 3: Illustration of the multistage matching procedure for the $s+$ regime of ${\rm SCET}_+$ with $t\sim u\ll Q^2$. The modes and their virtuality scale are indicated.
• Figure 4: Illustration of the multistage matching procedure for the $cs+$ regime of ${\rm SCET}_+$ with $t \ll u \ll Q^2$. The parent SCET is matched onto an intermediate ${\rm SCET}_+$ with a single soft-collinear sector. In the final matching step, this is further matched onto separate soft-collinear and csoft modes.
• Figure 5: Schematic overview of the fixed-order content of the different theories discussed here. Approaching the $cs+$ regime in the center, more and more logarithms get large and are resummed, at the expense of additional expansions. The missing fixed-order corrections from these expansions can be incorporated by adding back the relevant nonlogarithmic fixed-order differences between the theories.