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

TL;DR

<3-5 sentence high-level summary> Massive quark effects in exclusive Drell-Yan spectra are addressed using a soft-collinear effective theory (SCET) framework that systematically incorporates both primary and secondary heavy-quark contributions across all relevant mass-scale hierarchies. The authors derive NNLL$'$-accurate resummed predictions for the $q_T$ and beam-thrust observables, including mass-dependent beam, soft, and hard functions and their rapidity evolution, and provide explicit results for bottom-quark corrections. They discuss smooth merging across hierarchies via a variable-flavor matching scheme and illustrate potential implications for precision $W$-boson mass measurements at the LHC. The work also outlines how massive-quark effects enter the rapidity structure and furnishes a comprehensive set of ingredients for Monte Carlo implementations such as the Geneva framework.</p>

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.

Figures (21)

• Figure 1: Primary (a) and secondary (b) heavy-quark mass effects for $Z$-boson production.
• Figure 2: Effective theory modes for the $q_T$ spectrum with massive quarks for $q_T \ll Q$ and $m \gg \Lambda_{\rm QCD}$.
• Figure 3: Illustration of the renormalization group evolution for $q_T$ of the hard, beam, soft, and parton distribution functions in invariant mass and rapidity. The anomalous dimensions for each evolution step involve the displayed number of active quark flavors. The label $m$ indicates that the corresponding evolution is mass dependent.
• Figure 4: Relevant modes for the $q_T$ spectrum with $q_T \ll Q$ for different hierarchies between the quark mass $m$ and the scales $q_T$ and $Q$. The arrows indicate the relations between the modes and their associated contributions.
• Figure 5: Effective theory modes for the beam thrust spectrum with massive quarks for $m^2/Q \lesssim {\mathcal{T}} \ll Q$ and $m \gg \Lambda_{\rm QCD}$.