Matching next-to-leading-order and high-energy-resummed calculations of heavy-quarkonium-hadroproduction cross sections

TL;DR

The paper tackles the instability of heavy-quarkonium hadroproduction cross sections in collinear-factorisation at high energies caused by large small-$z$ corrections. It implements High-Energy Factorisation with a leading-logarithmic $\ln(1/z)$ resummation, formulated directly in $z$-space, and matches it to NLO CF corrections using two approaches, including a novel Inverse-Error-Weighting scheme. The authors derive DLAs for the resummation, connect them to the DGLAP evolution to preserve PDF consistency, and provide NNLO predictions for the CF coefficient function’s small-$z$ behavior. The work demonstrates that proper resummation stabilizes predictions and offers a path toward constraining small-$x$ PDFs, while outlining the need for full NLO+NLL($\ln(1/z)$) accuracy in future implementations.

Abstract

The energy dependence of the total hadroproduction cross section of pseudo-scalar quarkonia is computed via matching Next-to-Leading Order (NLO) Collinear-Factorisation (CF) results with resummed higher-order corrections, proportional to $α_s^{n}\ln^{n-1}(1/z)$, to the CF hard-scattering coefficient, where $z=M^2/\hat{s}$ with $M$ and $\hat{s}$ being the quarkonium mass and the partonic center-of-mass energy squared. The resummation is performed using High-Energy Factorisation (HEF) in the Doubly-Logarithmic (DL) approximation, which is a subset of the leading logarithmic $\ln (1/z)$ approximation. Doing so, one remains strictly consistent with the NLO and NNLO DGLAP evolution of the PDFs. By improving the treatment of the small-$z$ asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in the $z$-space and, for the first time, by using the Inverse-Error Weighting (InEW) matching procedure. As a by-product of the calculation, the NNLO term of the CF hard-scattering coefficient proportional to $α_s^2\ln(1/z)$ is predicted from HEF.

Figures (13)

• Figure 1: Typical diagrams contributing to the high-energy factorisation amplitude in the $gq$-channel. The dashed lines denote Reggeised gluon exchanges, while solid circles denote Lipatov's vertices.
• Figure 2: Dimensionless integrand function (\ref{['eq:fm-def']}) averaged over $\phi$, at ${\bf n}_{T2}^2=0$, as a function of ${\bf n}_{T1}^2={\bf q}_{T1}^2/(4m_Q^2)$, normalised on its value at ${\bf n}_{T1}^2=0$. The three dashed curves correspond to the coefficient functions for production of $Q\bar{Q}[m]$-states with $m={}^1S_0,$${}^3P_0$ and ${}^3P_2$ (section \ref{['sec:H-HEF']}). The solid curve depicts the coefficient function for the unbound $Q\bar{Q}$-pair production Ellis:1990hw, integrated over the phase space of the $Q\bar{Q}$ pair. The short-dashed line represents the $\overline{\text{MS}}$ subtraction term $\theta(1-{\bf n}_{T1}^2)$.
• Figure 3: Comparison of the energy dependence of the hadroproduction cross sections of a $Q\bar{Q}[{}^1S_0^{(1)} ]$ state with $M=3$ GeV ( left panel) and $M=9.4$ GeV ( right panel) in CF at LO (grey curve) and NLO(blue curve). The central member of the CT18NLO PDF set Hou:2019efy has been used. The dashed line depicts the central prediction of the NLO calculation using the $\hat{\mu}_F$ prescription (Equation \ref{['eq:muF-hat']}) Lansberg:2020ejc. The LDME $\langle {\cal O} [ {}^1S_0^{(1)} ] \rangle$ was set to 1 GeV$^3$ in both cases for illustration purposes.
• Figure 4: Comparison of the energy dependence of the hadroproduction cross sections of a $Q\bar{Q}[{}^1S_0^{[1]} ]$ state with $M=3$ GeV ( left panel) and $M=9.4$ GeV ( right panel) for the matched NLO CF + DLA HEF calculations using the subtractive (orange curve and right-shaded band) and InEW matchings(red curve and left-shaded band). The dashed line depicts the central prediction of the NLO calculation using the $\hat{\mu}_F$ prescription (Equation \ref{['eq:muF-hat']}) Lansberg:2020ejc. The LDME $\langle {\cal O} [ {}^1S_0^{[1]} ] \rangle$ was set to 1 GeV$^3$.
• Figure 5: Plots of the InEW weights of the CF contribution for different channels as function of $z$ with $\mu_F=\mu_R=M$.