Fermionic Glauber Operators and Quark Reggeization

TL;DR

This work develops an SCET-based framework for forward scattering that includes t-channel Glauber quark exchange, constructing a nonlocal fermionic potential operator that captures subleading power corrections in the Regge limit. Through tree-level matching and a rapidity renormalization group analysis, it demonstrates LL Reggeization of the quark at the amplitude level and derives the LL BFKL equation at the cross-section level, extending the known Regge behavior beyond gluon exchanges. The study also shows how simultaneous Glauber quark and gluon exchanges reproduce the bar{6} and 15 color-channel structures, and applies the formalism to q\bar{q}→γγ to confirm BFKL evolution in a quark-initiated process. Collectively, the results establish a systematic operator-based EFT approach to Regge limit amplitudes in QCD and lay groundwork for higher-order analyses and broader Reggeon structures.

Abstract

We derive, in the framework of soft-collinear effective field theory (SCET), a Lagrangian describing the $t$-channel exchange of Glauber quarks in the Regge limit. The Glauber quarks are not dynamical, but are incorporated through non-local fermionic potential operators. These operators are power suppressed in $|t|/s$ relative to those describing Glauber gluon exchange, but give the first non-vanishing contributions in the Regge limit to processes such as $q\bar q \to gg$ and $q\bar q \to γγ$. They therefore represent an interesting subset of power corrections to study. The structure of the operators, which describe certain soft and collinear emissions to all orders through Wilson lines, is derived from the symmetries of the effective theory combined with constraints from power and mass dimension counting, as well as through explicit matching calculations. Lightcone singularities in the fermionic potentials are regulated using a rapidity regulator, whose corresponding renormalization group evolution gives rise to the Reggeization of the quark at the amplitude level and the BFKL equation at the cross section level. We verify this at one-loop, deriving the Regge trajectory of the quark in the $3$ color channel, as well as the leading logarithmic BFKL equation. Results in the $\bar 6$ and $15$ color channels are obtained by the simultaneous exchange of a Glauber quark and a Glauber gluon. SCET with quark and gluon Glauber operators therefore provides a framework to systematically study the structure of QCD amplitudes in the Regge limit, and derive constraints on higher order amplitudes.

Figures (7)

• Figure 1: Feynman rules for tree level $qg$ forward scattering with zero, one and two soft gluon emissions, generated by the soft operator $\mathcal{O}_s$. Soft emissions at higher orders in $\alpha_s$ are also produced by $\mathcal{O}_s$.
• Figure 2: (a) Full theory and (b) effective theory graphs with a single soft emission. We refer to the effective theory vertex as the Fadin-Sherman vertex since it first appeared in Fadin:1976nwFadin:1977jr.
• Figure 3: (a) Full theory and (b) effective theory graphs with two soft emissions. In the effective theory, the first three graphs are $T$-product contributions, and the fourth graph is the two emission Feynman rule from the Fadin-Sherman vertex.
• Figure 4: One-loop virtual contributions to the renormalization of the collinear operator $\mathcal{O}_n$. The V graphs are labeled a) and b), and the Wilson line graphs are labeled c) and d).
• Figure 5: One-loop virtual contributions to the renormalization of the soft operator $\mathcal{O}_s$. The flower graph is labeled a) and the eye graph is labeled b).