Joint resummation in electroweak boson production

TL;DR

Joint resummation combines threshold and transverse-momentum resummation in electroweak boson production by performing a double inverse Mellin/Fourier transform with a carefully defined chi function that couples energy and recoil logarithms. The approach yields a perturbatively well-defined cross section at all nonzero $Q_T$, complemented by a Gaussian nonperturbative smearing term whose strength is fitted to data; matching to fixed-order results preserves correct high-$Q_T$ behavior and positivity. Applied to Z production at the Tevatron, the method describes the observed $Q_T$ distribution well when nonperturbative effects are included, with a single parameter $g\approx 0.8\,\text{GeV}^2$ capturing low-$Q_T$ enhancement. The work provides a unified framework for perturbative and nonperturbative contributions in hadronic reactions and points to extensions to Higgs production and semi-inclusive processes at the LHC.

Abstract

We present a phenomenological application of the joint resummation formalism to electroweak annihilation processes at measured boson momentum Q_T. This formalism simultaneously resums at next-to-leading logarithmic accuracy large threshold and recoil corrections to partonic scattering. We invert the impact parameter transform using a previously described analytic continuation procedure. This leads to a well-defined, resummed perturbative cross section for all nonzero Q_T, which can be compared to resummation carried out directly in Q_T space. From the structure of the resummed expressions, we also determine the form of nonperturbative corrections to the cross section and implement these into our analysis. We obtain a good description of the transverse momentum distribution of Z bosons produced at the Tevatron collider.

Figures (6)

• Figure 1: (a) Fractional deviation $\Delta$ (as defined in Eq. (\ref{['frdev']})) between the "exact" ${\cal O} (\alpha s{$α_s${ }})$ result and the ${\cal O} (\alpha s{$α_s${ }})$ expansion of the jointly resummed cross section. We consider here Z boson production at the Tevatron; the cross section has been integrated over $66<Q<116$ GeV. (b) Comparison of $d \sigma^{\rm fixed(1)} /d Q_T$ (solid) and $d\sigma^{\rm exp(1)}/d Q_T$ (dashed) on absolute scale.
• Figure 2: Choice of contour for Mellin inversion.
• Figure 3: Choice of contour for $b$ integration (thick solid lines). The straight sections of the contour from $0$ to $b_c$ are to be interpreted as on the positive real axis. The remaining curves represent lines of singularity discussed in the text.
• Figure 4: $Q_T$ distribution for Z production at $\sqrt s=1.8$ TeV calculated using the narrow width approximation ($Q=M_Z$). The $Q_T$ space method result and the joint resummation method result are matched to the cross section at ${\cal O}(\alpha_s)$. The CTEQ5M cteq parton distributions have been used.
• Figure 5: CDF data cdf on Z production compared to joint resummation predictions (matched to the ${\cal O} (\alpha s{$α_s${ }})$ result according to Eq. (\ref{['joint:match']})) without nonperturbative smearing (dashed) and with Gaussian smearing using the nonperturbative parameter $g=0.8$ GeV$^2$ (solid). The normalizations of the curves have been adjusted in order to give an optimal description; see text. The dotted and dash-dotted lines show the fixed-order results at ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha_s^2)$, respectively. The lower plot makes the large $Q_T$ region more visible.