Secondary Heavy Quark Production in Jets through Mass Modes

TL;DR

This paper develops a mass-mode SCET framework to coherently describe secondary heavy-quark production in jet observables, focusing on e+e- thrust in the dijet limit. By introducing collinear and soft mass modes and employing a dispersion-based approach, it achieves a continuous description across mass hierarchies (Q, M, Qλ, Qλ^2) and derives four EFT scenarios that interpolate between decoupling and massless limits. The authors provide explicit one-loop results for hard, jet, and soft functions and introduce rapidity-log exponentiation to sum large logs, ensuring smooth transitions between scenarios. They also outline the dispersion method for secondary massive fermions, connect to ACOT-like flavor-number schemes, and demonstrate that the framework reproduces correct massless limits while preserving full mass dependence of the singular terms. The work lays groundwork for higher-order calculations and broad applicability to jet physics and hadron collisions.

Abstract

We present an effective field theory method to determine secondary massive quark effects in jet production taking the thrust distribution for e+ e- collisions in the dijet limit as a concrete example. The method is based on the field theoretic treatment of collinear and soft mass modes which have to be separated coherently from the collinear and ultrasoft modes related to massless quarks and gluons. For thrust the structure of the conceptual setup is closely related to the production of massive gauge bosons and involves four different effective field theories to describe all possible kinematic situations. The effective field theories merge into each other continuously and thus allow for a continuous description from infinitely heavy to arbitrarily small masses keeping the exact mass dependence of the most singular terms treated through factorization. The mass mode field theory method we present here is in the spirit of the variable fermion number scheme originally proposed by Aivazis, Collins, Olness and Tung and can also be applied in hadron collisions.

Figures (12)

• Figure 1: Diagrams at $\mathcal{O}(\alpha_s^2)$ for virtual and real secondary radiation of massive quark pairs in primary massless quark production.
• Figure 2: Figure illustrating the dispersion method for the vacuum polarization correction to the gluon propagator in the subtracted version with $\Pi(q^2=0)=0$ suitable for situations where the massive quark is not contributing to the renormalization group evolution. The explicit analytic form of the dispersion relations is discussed in Sec. \ref{['sect:outlook']}.
• Figure 3: Diagrams at $\mathcal{O}(\alpha_s^2)$ for virtual and real secondary radiation of gluons with mass $M$ in primary massless quark production.
• Figure 4: Localization of massless (ML) and massive (M) modes together with their mass-shell fluctuations (MM) in the $p^{+}$-$p^{-}$phase space according to their generic scaling for different hierarchies between $\lambda$ and $\lambda_M$. Modes with the same invariant mass are located on the same mass hyperbola. This is always the case for the collinear and soft mass-shell fluctuations.
• Figure 5: The different scenarios depending on the hierarchy between the mass scale $M$ and the hard, jet and ultrasoft scale. MM indicates mass-shell scaling, ML the massless one. With M we denote modes that have a mass $M$ but scale as their massless counterparts. The renormalization group evolution is also shown in the top-down evolution from the hard scale $\mu_H$ down to $\mu=\mu_S$. When the mass scale is crossed the mass shell fluctuations are integrated out. This leads to a matching condition and to a change in the evolution factor. In the case of secondary quark pairs production we evolve with $n_{\rm light}+n_{\rm mass}$ flavors above the mass scale and with $n_{\rm light}$ below ($n_{\rm mass}$ is the number of quark flavors with mass $M$).