Factorization and resummation for transverse thrust

TL;DR

This work develops a Soft Collinear Effective Theory (SCET) framework for transverse thrust $T_ot$ in hadron collisions, deriving a factorization formula in the dijet limit $ au_ot=1-T_ot o0$ that enables all-order resummation of enhanced logarithms. It distinguishes leptonic (SCET$_{ m I}$) and hadronic (SCET$_{ m II}$) regimes, unveiling a collinear anomaly in the hadronic case that ties soft and beam functions together via an anomaly exponent $F_ot$ and a regulator-dependent yet universal structure. The authors compute all one-loop ingredients—jet, beam, soft, and hard functions—and outline how two-loop anomalous dimensions can be extracted numerically from fixed-order codes (e.g., EVENT2 for leptonic, NNLO tools for hadronic) to achieve NNLL and ultimately N$^2$LL resummation. They present a complete resummation framework in Laplace space and show how to translate it to momentum space, setting the stage for phenomenological studies and potential applications to other hadronic event shapes. The work also discusses regulatory subtleties, the role of Glauber gluons, and directions for automation and numerical implementation of SCET-based resummations in collider physics.

Abstract

We analyze transverse thrust in the framework of Soft Collinear Effective Theory and obtain a factorized expression for the cross section that permits resummation of terms enhanced in the dijet limit to arbitrary accuracy. The factorization theorem for this hadron-collider event-shape variable involves collinear emissions at different virtualities and suffers from a collinear anomaly. We compute all its ingredients at the one-loop order, and show that the two-loop input for next-to-next-to-leading logarithmic accuracy can be extracted numerically, from existing fixed-order codes.

Figures (12)

• Figure 1: Schematic representation of a dijet event in the $T_{\perp}\to 1$ limit for leptonic (left panel) and hadronic (right panel) collisions. The soft radiation $s$ and the collinear emissions $c_1$, $c_2$, $c_a$, $c_b$ are represented by different fields in the effective theory. The typical virtuality of the fields $c_a$, $c_b$ and $s$ is the same, and is lower than the virtuality of $c_1$ and $c_2$.
• Figure 2: Left: A two-jet configuration with low $\tau_{\perp}$. The figure shows the thrust axis $\vec{n}$ (green), the transverse thrust axis $\vec{n}_{\perp}$ (red), and the beam as a dashed line in the $z$-direction (blue). Right: A planar three-jet configuration with $\tau_{\perp}=0$. Any distribution of particles that is restricted to a plane which contains the beam has $\tau_{\perp}=0$.
• Figure 3: Virtualities of the different modes present in the hadron-collider case.
• Figure 4: Feynman diagram for the Born level $q\bar{q}\to q'\bar{q}'$ channel.
• Figure 5: Next-to-leading order real-emission diagrams for the quark jet function (first line), and for the gluon jet function (second and third lines). The red vertical lines represent the final-state cut. The crosses indicate the collinear Wilson lines.