Factorization and resummation for massive quark effects in exclusive Drell-Yan

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, T , as examples of exclusive observables. The theoretical description depends on the hierarchy between the hard, mass, and the q T (or T ) scales, ranging from the decoupling limit q T m to the massless limit m q T . The phenomenologically relevant intermediate regime m ∼ 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 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.

Two-loop massive quark jet functions in SCET

“…The mass mode jet functions are analogous to a corresponding definition for invariant mass dependent beam functions introduced in ref. [32]. Both types of jet functions are useful to gain a transparent view on the treatment of secondary massive quark effects and can be employed in practical applications.…”

“…To illustrate the application of the universal and the mass mode jet functions we also discuss their role in the equivalent factorization approaches of refs. [22,26] and [32] using as an example the double differential hemisphere mass distribution in e + e − annihilation (for the hard production of a boosted massive quarkantiquark pair). This discussion also clarifies the relation between both approaches and furthermore emphasizes that a smooth dependence on the value of the quark mass m (in the sense that assumptions on the hierarchy of m with respect to the other physical scales are not mandatory) can be achieved with them.…”

“…On the other hand, in ref. [32] a more systematic representation of the building blocks to sum virtuality and rapidity logarithms on an equal footing was provided.…”

“…It has also been advocated for this scale hierarchy in ref. [32] (originally developed for the massless primary quark and invariant mass dependent beam function [38]). In the factorization approach of ref.…”

“…In the factorization approach of ref. [32] separate factorization theorems are formulated for each possible hierarchy of the mass scale m with respect to the other kinematic scales. The approach is not designed to provide smooth dependence on m, but is more economical (in the sense of being minimalistic) concerning the appearance of collinear and soft mass mode contributions so that no cancellations arise between them in the context of a factorization theorem.…”