# Factorization and Resummation for Massive Quark Effects in Exclusive Drell-Yan

## TL;DR

This paper develops an effective-theory framework using soft-collinear effective theory to incorporate massive quark effects into resummed differential distributions for exclusive processes like Drell-Yan, covering various mass and scale hierarchies.

## Contribution

It introduces a comprehensive calculation of quark-mass effects at NNLL' order for $q_T$ and beam thrust, including primary and secondary contributions across different hierarchies.

## Key findings

- Calculated all ingredients for primary and secondary mass effects at NNLL' order.
- Analyzed rapidity divergences and evolution in the massive quark case.
- Enabled detailed studies of quark-mass effects in $W$ and $Z$ boson spectra.

## Abstract

Exclusive differential spectra in color-singlet processes at hadron colliders are benchmark observables that have been studied to high precision in theory and experiment. We present an effective-theory framework utilizing soft-collinear effective theory to incorporate massive (bottom) quark effects into resummed differential distributions, accounting for both heavy-quark initiated primary contributions to the hard scattering process as well as secondary effects from gluons splitting into heavy-quark pairs. To be specific, we focus on the Drell-Yan process and consider the vector-boson transverse momentum, $q_T$, and beam thrust, $\mathcal T$, as examples of exclusive observables. The theoretical description depends on the hierarchy between the hard, mass, and the $q_T$ (or $\mathcal T$) scales, ranging from the decoupling limit $q_T \ll m$ to the massless limit $m \ll q_T$. The phenomenologically relevant intermediate regime $m \sim q_T$ requires in particular quark-mass dependent beam and soft functions. We calculate all ingredients for the description of primary and secondary mass effects required at NNLL$'$ resummation order (combining NNLL evolution with NNLO boundary conditions) for $q_T$ and $\mathcal T$ in all relevant hierarchies. For the $q_T$ distribution the rapidity divergences are different from the massless case and we discuss features of the resulting rapidity evolution. Our results will allow for a detailed investigation of quark-mass effects in the ratio of $W$ and $Z$ boson spectra at small $q_T$, which is important for the precision measurement of the $W$-boson mass at the LHC.