Three-loop jet function for boosted top quarks

TL;DR

This work tackles precision top-quark mass determination in boosted-top production by computing the inclusive bHQET jet function to three loops in the resonance region $M^2 - m^2 \ll m^2$. The authors implement a full three-loop calculation, verify non-Abelian exponentiation, reproduce the cusp and non-cusp anomalous dimensions up to three loops, and confirm the $n_\ell^2 \alpha_s^3$ renormalon prediction. They present the explicit three-loop exponent coefficient $\tilde{b}_{30}$ for $N_c=3$ with its color decomposition and discuss relations among jet-mass schemes and a renormalon-based estimate of the four-loop coefficient $\tilde{b}_{40}(n_\ell)$. The results complete the set of ingredients for N$^3$LL$^\prime$ self-normalized thrust predictions, enabling more precise top-mass calibrations in parton-shower MCs and potentially improved indirect top-quark mass determinations at future lepton colliders.

Abstract

We present the calculation of the inclusive jet function for highly energetic heavy quarks at order $\mathcal{O}(α_s^3)$ using boosted Heavy-Quark Effective Theory (bHQET). This jet function describes the effect of collinear radiation emitted by energetic heavy quarks on observables dependent on the jet invariant mass $M$. In particular, we focus on the regime $M^2 - m^2 \ll m^2$, which is relevant for boosted top quark production at high-energy colliders in the resonance region. Our results are consistent with non-Abelian exponentiation and reproduce the known cusp and non-cusp anomalous dimensions up to three loops. We also verify that the $n_\ell^2 α_s^3$ contribution, with $n_\ell$ denoting the number of light quark flavors, agrees with predictions from renormalon calculus. This calculation completes the list of ingredients required for the N$^3$LL$^\prime$ resummed (self-normalized) thrust distribution, an essential component for calibrating the top quark mass parameter in parton-shower Monte Carlo generators. It likewise contributes to the invariant-mass distribution of reconstructed top quarks, enabling precise mass determinations at future lepton colliders. Finally, we determine the relation between the pole and two short-distance jet-mass schemes at $\mathcal{O}(α_s^3)$ and provide an estimate of the non-logarithmic part of the four-loop jet function based on renormalon dominance.

Figures (3)

• Figure 1: Sample of Feynman diagrams contributing to the jet function at three loops. Double, curly, dashed and solid lines represent heavy-quarks, gluons, light-like Wilson lines and light quarks, respectively.
• Figure 2: Difference between the heavy quark MSR and jet masses in the derivative (left) and non-derivative (right) schemes in absolute value. In the plots we use $n_\ell=5$ for the number of active light flavors and $\alpha_s^{(n_\ell=5)}(m_Z)=0.1180$. For both masses we expand $\alpha_s(R)$ in terms of $\alpha_s(\mu)$, and generate uncertainty bands varying $\mu$ in the range $R/2$ to $2R$.
• Figure 3: Estimates of $\tilde{b}_{40}$ for various numbers of light flavors $n_\ell$ with $N_c=3$. In the left panel, we show the estimates when including lower-order information up to one (blue), two (green), and three (red) loops. In the right panel, we use three-loop input to make predictions for a large set of $n_\ell$ values (in blue), and show as a red band the predictions obtained using the results for the various flavor coefficients. For details in the uncertainty calculation, we refer to our paper Ref. Clavero:2024yav.