Higher-order power corrections in a transverse-momentum cut for colour-singlet production at NLO

TL;DR

This paper analyzes the dependence of colorless final-state production cross sections on a transverse-momentum cutoff within q_T-subtraction at NLO, deriving power corrections up to (q_T^{cut})^4. It develops a process-independent approach to compute all-order power corrections by extending the z-integration to 1 and introducing hatR_{ab}(z) and hatG_{ab}(z), separating universal soft/collinear contributions from non-universal ones. Analytic results are provided for Drell–Yan Z production and Higgs production in gluon fusion, with numerical studies at the LHC showing substantial reductions in cutoff dependence when higher-order power terms are included and clarifying the roles of universal vs non-universal contributions. The work enhances understanding of phase-space boundary behavior and subleading resummation structures, offering a practical framework for improving q_T subtraction at higher orders.

Abstract

We consider the production of a colourless system at next-to-leading order in the strong coupling constant $α_s$. We impose a transverse-momentum cutoff, qtcut, on the colourless final state and we compute the power corrections for the inclusive cross section in the cutoff, up to the fourth power. The study of the dependence of the cross section on qtcut allows for an understanding of its behaviour at the boundaries of the phase space, giving hints on the structure at all orders in $α_s$ and on the identification of universal patterns. The knowledge of such power corrections is also a required ingredient in order to reduce the dependence on the transverse-momentum cutoff of the QCD cross sections at higher orders, when the qt-subtraction method is applied. We present analytic results for both Drell--Yan vector boson and Higgs boson production in gluon fusion and we illustrate a process-independent procedure for the calculation of the all-order power corrections in the cutoff. In order to show the impact of the power-correction terms, we present selected numerical results and discuss how the residual dependence on qtcut affects the total cross section for Drell--Yan Z production and Higgs boson production via gluon fusion at the LHC.

Figures (6)

• Figure 1: Difference of the total cross sections $$σ^ > (1) - σ$$ as a function of $q_{ \rm T}^{\rm cut}$, for $Z$ boson production, in the $qg \rightarrow Z q$ (left pane) and in the $q\bar{q} \rightarrow Z g$ channel (right pane). The three curves correspond to the three possible choices of $\widetilde{\sigma}$: results for $\widetilde{\sigma} = \sigma^{ \rm LT }$ are displayed in blue, for $\widetilde{\sigma} = \sigma^{ \rm LT }+ \sigma^{ \rm NLT }$ are displayed in black and for $\widetilde{\sigma} = \sigma^{ \rm LT }+ \sigma^{ \rm NLT } + \sigma^{ \rm N^2LT }$ are displayed in red. The corresponding values of $a=$q_ T^ cut/m_ Z$^2$ are displayed on the top of the figure. The statistical errors of the integration are also shown, but they are totally negligible on the scale of the figure.
• Figure 2: Difference of the total cross sections $$σ^ > (1) - σ$$ as a function of $q_{ \rm T}^{\rm cut}$, for $H$ boson production, in the $qg \rightarrow H q$ (left pane) and in the $gg \rightarrow H g$ channel (right pane). Same legend as in fig. \ref{['fig:Z_qg_qq']}. The corresponding values of $a=$q_ T^ cut/m_ H$^2$ are displayed on the top of the figure.
• Figure 3: Results for $1 -$σ^ > (1) - σ$/\sigma_{\rm NLO}$ as a function of $q_{ \rm T}^{\rm cut}$, for $Z$ boson production, in $pp\rightarrow Zj$. Same legend as in fig. \ref{['fig:Z_qg_qq']}. In the left pane, the low-$q_{ \rm T}^{\rm cut}$ region is displayed, while, in the right pane, a larger region in $q_{ \rm T}^{\rm cut}$ is shown. The total cross section at NLO for $Z$ production, $\sigma_{\rm NLO}$, has been taken equal to 55668.1 pb.
• Figure 4: Results for $1 -$σ^ > (1) - σ$/\sigma_{\rm NLO}$ as a function of $q_{ \rm T}^{\rm cut}$, for $H$ boson production, in $pp\rightarrow Hj$. Same legend as in fig. \ref{['fig:Z_qg_qq']}. In the left pane, the low-$q_{ \rm T}^{\rm cut}$ region is displayed, while, in the right pane, a larger region in $q_{ \rm T}^{\rm cut}$ is shown. The total cross section at NLO for $H$ production, $\sigma_{\rm NLO}$, has been taken equal to 31.52 pb.
• Figure 5: Same as fig. \ref{['fig:Z_norm']}, but using only the universal part of $\hat{G}_{ab}^{(1,n,m)}(z)$ in computing the cross sections of eqs. (\ref{['eq:sigma_LT']})--(\ref{['eq:sigma_N2LT']}).