Transverse momentum dependence of the angular distribution of the Drell-Yan process

TL;DR

This work analyzes the transverse-momentum dependence of the Drell–Yan dilepton angular distribution by examining the four helicity structure functions $W_T$, $W_L$, $W_Δ$, and $W_{ΔΔ}$ within collinear QCD. It identifies the leading small-$Q_\bot$ divergences arising at fixed order, constructs a current-conserving asymptotic tensor to relate these divergences to the angular-integrated cross section, and implements Collins–Soper–Sterman resummation to all orders, yielding finite predictions as $Q_\bot \to 0$. The authors show that the resummed results preserve the Lam–Tung relation $W_L=2W_{ΔΔ}$ for the leading divergent terms and provide a consistent framework for predicting the full $Q_\bot$-dependent angular distribution, including a distinct treatment for the single-spin-flip term $W_Δ$. The findings have practical implications for precision vector-boson polarization studies and SIDIS observables, with extensions to $W$, $Z$, and beyond.

Abstract

We calculate the transverse momentum Q_{\perp} dependence of the helicity structure functions for the hadroproduction of a massive pair of leptons with pair invariant mass Q. These structure functions determine the angular distribution of the leptons in the pair rest frame. Unphysical behavior in the region Q_{\perp} --> 0 is seen in the results of calculations done at fixed-order in QCD perturbation theory. We use current conservation to demonstrate that the unphysical inverse-power and \ln(Q/Q_{\perp}) logarithmic divergences in three of the four independent helicity structure functions share the same origin as the divergent terms in fixed-order calculations of the angular-integrated cross section. We show that the resummation of these divergences to all orders in the strong coupling strength α_s can be reduced to the solved problem of the resummation of the divergences in the angular-integrated cross section, resulting in well-behaved predictions in the small Q_{\perp} region. Among other results, we show the resummed part of the helicity structure functions preserves the Lam-Tung relation between the longitudinal and double spin-flip structure functions as a function of Q_{\perp} to all orders in α_s.

Figures (4)

• Figure 1: Diagrammatic representation of hadronic dilepton production via a virtual photon of four-momentum $q$.
• Figure 2: Feynman diagrams for quark-antiquark annihilation to a virtual photon plus a gluon.
• Figure 3: Feynman diagrams for quark-gluon scattering to produce a virtual photon plus a quark.
• Figure 4: The transverse momentum dependence of the angular-integrated Drell-Yan cross section, obtained from the contributions of the helicity structure functions, $W_T$ and $W_L$, in Eq. (\ref{['st-full-cs']}) is shown as a solid line and compared with data from Fermilab experiment E772 E772 for $Q$ in the interval $(8,9)$ GeV. The dashed and dot-dashed curves show our calculations for the contributions from $W_T$ and $W_L$. The inset shows the $W_L$ contribution on an expanded scale.