Variable Flavor Number Scheme for Final State Jets in Thrust

TL;DR

The paper develops a variable flavor number scheme for final-state jets to incorporate secondary heavy-quark mass effects in e+e- thrust within SCET. It introduces a four-scenario mass-mode framework, with explicit hard, jet, and soft function corrections, threshold matchings, and gap-subtraction to control renormalon ambiguities. Dispersion-relations techniques relate massive-quark corrections to massive-gluon results, enabling explicit O(α_s^2 C_F T_F) calculations that cover all relevant mass hierarchies and ensure proper massless and decoupling limits. The work provides universal threshold corrections and consistency relations across the RG evolution, and demonstrates the practical impact of heavy-quark masses on the thrust peak and tail, including notable bottom and top mass effects. This VFNS framework offers a systematic path to extend to primary heavy-quark production and other jet-related observables in collider phenomenology.

Abstract

We present results for mass effects coming from secondary radiation of heavy quark pairs related to gluon splitting in the thrust distribution for e+e- collisions. The results are given in the dijet limit where the hard interaction scale and the scales related to collinear and soft radiation are widely separated. We account for the corresponding fixed-order corrections at O(alpha_s^2) and the summation of all logarithmic terms related to the hard, collinear and soft scales as well as the quark mass at N3LL order. We also remove the O(Lambda_QCD) renormalon in the partonic soft function leading to an infrared evolution equation with a matching condition related to the massive quark threshold. The quark mass can be arbitrary, ranging from the infinitely heavy case, where decoupling takes place, down to the massless limit where the results smoothly merge into the well known predictions for massless quarks. Our results are formulated in the framework of factorization theorems for e+e- dijet production and provide universal threshold corrections for the renormalization group evolution of the hard current, the jet and soft functions at the scale where the massive quarks are integrated out. The results represent a first explicit realization of a variable flavor number scheme for final state jets along the lines of the well known flavor number dependent evolution of the strong coupling alpha_s and the parton distribution functions.

Figures (17)

• 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: Diagrams at $\mathcal{O}(\alpha_s)$ for virtual and real secondary radiation of gluons with mass $M$ in primary massless quark production.
• Figure 3: 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{['sec:dispersion']}.
• Figure 4: The different scenarios depending on the hierarchy between the mass scale $\mu_m$ and the hard, jet and ultrasoft scales. 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 (dashed box). This leads to a matching condition and to a change in the evolution factor.
• Figure 5: R-evolution of $\Omega_1(R,\mu=R)$ with a massive bottom quark at $\mathcal{O}(\alpha_s^3)$ as described in the text. The curves represent purely massless evolution (red, dashed), massive evolution including threshold matching at $\overline{m}_b(\overline{m}_b)$ (blue, solid) and massive evolution without threshold matching (green, dotted)