Resummation Improved Rapidity Spectrum for Gluon Fusion Higgs Production

TL;DR

The paper tackles large perturbative corrections in color-singlet production caused by timelike Sudakov logarithms in the gluon form factor. It develops a factorized resummation framework using a timelike-evolved hard function $H(q^2,\mu)$ with a complex scale $\mu_H=-iQ$, yielding the resummed cross section $\sigma_{\text{res}} = U_H(\mu_H,\mu_{\text{FO}})\,[H(\mu_H)/H(\mu_{\text{FO}})\,\sigma_{\text{FO}}]_{\text{FO}}$ and applying it to gluon-fusion Higgs production, Higgs rapidity, and related processes. The results show substantially improved perturbative convergence and reduced uncertainties, with total cross sections at $ ext{N3LO}+\text{N3LL}'_{\varphi}$ and rapidity spectra at $\text{NNLO}+\text{NNLL}'_{\varphi}$ approaching anticipated higher-order accuracy. The method is also extended to quark-induced channels (bottom-quark annihilation and Drell–Yan), yielding smaller improvements for $\text{bbH}$ and confirming small uncertainties for Drell–Yan, thereby demonstrating broad applicability and reliability of timelike-log resummation in initial- and final-state color-singlet production.

Abstract

Gluon-induced processes such as Higgs production typically exhibit large perturbative corrections. These partially arise from large virtual corrections to the gluon form factor, which at timelike momentum transfer contains Sudakov logarithms evaluated at negative arguments $\ln^2(-1) = -π^2$. It has been observed that resumming these terms in the timelike form factor leads to a much improved perturbative convergence for the total cross section. We discuss how to consistently incorporate the resummed form factor into the perturbative predictions for generic cross sections differential in the Born kinematics, including in particular the Higgs rapidity spectrum. We verify that this indeed improves the perturbative convergence, leading to smaller and more reliable perturbative uncertainties, and that this is not affected by cancellations between resummed and unresummed contributions. Combining both fixed-order and resummation uncertainties, the perturbative uncertainty for the total cross section at N$^3$LO$+$N$^3$LL$^\prime_\varphi$ is about a factor of two smaller than at N$^3$LO. The perturbative uncertainty of the rapidity spectrum at NNLO$+$NNLL$^\prime_\varphi$ is similarly reduced compared to NNLO. We also study the analogous resummation for quark-induced processes, namely Higgs production through bottom quark annihilation and the Drell-Yan rapidity spectrum. For the former the resummation leads to a small improvement, while for the latter it confirms the already small uncertainties of the fixed-order predictions.

Figures (10)

• Figure 1: Illustration of the scale variations used to estimate the perturbative uncertainties. Left: The overall variations of $\mu_\mathrm{FO}$, which determines $\Delta_{\mu}$ (in conjunction with the variation of $\kappa_F$, which is not shown). Right: The phase variation for $\mu_H$ for fixed $\mu_\mathrm{FO}$, which determines the resummation uncertainty $\Delta_\varphi$.
• Figure 2: Illustration of the fixed-order perturbative series for gg to X at $\mu_\mathrm{FO} = m_X$ for the inclusive $K$-factor (left), the hard function $H_{gg}$ at $\mu_H = m_X$ (center), and the normalized remainder $R/\sigma^{(0)}$ (right). The middle panel also shows the N3LO hard function $H_{gg}$ at $\mu_H = -\mathrm{i} m_X$ (black long dashed), for which it contains no timelike logarithms.
• Figure 3: The total cross section for gg to X at $E_\mathrm{cm} = 13 \,\mathrm{TeV}$ at fixed order (left) and including the resummation of timelike logarithms (right). All results are normalized to the central LO prediction at $\mu_\mathrm{FO} = m_X$.
• Figure 4: The gg to H cross section at $E_\mathrm{cm} = 13 \,\mathrm{TeV}$ and $m_H = 125\,\mathrm{GeV}$ in the rEFT scheme. Left: The cross section as a function of the resummation phase $\varphi$ of the hard scale $\mu_H = \mu_\mathrm{FO} \exp(-\mathrm{i} \varphi)$, with the uncertainty bands corresponding to $\Delta_{\mu}$ only. Right: Comparison of the fixed-order results for $\mu_\mathrm{FO} = m_H$ and $\mu_\mathrm{FO} = m_H/2$, and the resummed results with $\mu_\mathrm{FO} = \mathrm{i} \mu_H = m_H$. All results are given as the percent difference from the $\text{N3LO}+ N3LLphi$ central value. The uncertainty bars show $\Delta_{\mu}$ for the fixed-order results and $\Delta_{\mu} \oplus \Delta_\varphi$ for the resummed results (with the inner bars visible at the lower orders showing $\Delta_\varphi$ only). The fixed LO results are out of range.
• Figure 5: The perturbative remainder $R(Y)/\sigma^{(0)}(Y)$ as a function of the Higgs rapidity $Y$ normalized to the LO spectrum $\sigma^{(0)}(Y) \equiv \mathrm{d} \sigma^{(0)} / \mathrm{d} Y$ in the rEFT limit for $\mu_\mathrm{FO} = m_H$ (left) and $\mu_\mathrm{FO} = m_H/2$ (right).