Analytic Next-To-Leading Order Calculation of Energy-Energy Correlation in Gluon-Initiated Higgs Decays

TL;DR

This paper provides the first fully analytic NLO calculation of the Energy-Energy Correlation (EEC) for gluon-initiated Higgs decays within the Higgs Effective Field Theory. The authors systematically map the 4-particle phase space to loop integrals via reverse unitarity, reduce to master integrals with a two-step IBP approach that accommodates nonlinear propagators, and solve the master integrals with canonical differential equations, fixing boundary conditions from collinear and back-to-back limits and matching to inclusive phase-space results. The final result expresses the Higgs EEC in terms of LO and NLO coefficients A_H(z) and B_H(z), decomposed into color components and built from harmonic polylogarithms up to weight 3, with detailed asymptotics in the $z\to0$ and $z\to1$ limits. The work also discusses nonperturbative effects using Pythia as a toy model and highlights the observable’s potential for probing QCD alongside Higgs physics at future $e^+e^-$ colliders, including prospects for determining $\alpha_s$ and directions toward NNLO.

Abstract

The energy-energy correlation (EEC) function in $e^+e^-$ annihilation is currently the only QCD event shape observable for which we know the full analytic result at the next-to-leading order (NLO). In this work we calculate the EEC observable for gluon initiated Higgs decay analytically at NLO in the Higgs Effective Field Theory (HEFT) framework and provide the full results expressed in terms of classical polylogarithms, including the asymptotic behavior in the collinear and back-to-back limits. This observable can be, in principle, measured at the future $e^+e^-$ colliders such as CEPC, ILC, FCC-ee or CLIC. It provides an interesting opportunity to simultaneously probe our understanding of the strong and Higgs sectors and can be used for the determinations of the strong coupling.

Figures (7)

• Figure 1: Some of the representative cut diagrams for real corrections to the Higgs EEC at NLO. For simplicity we do not show diagrams arising from the Higgs couplings to triple and quartic gluon vertices.
• Figure 2: Analytic fixed-order results for Higgs EEC at LO (lower curve) and NLO (upper curve) in the Higgs EFT. In both cases the solid black curves correspond to the central values, while the colored bands give the uncertainties from varying the renormalization scale $\mu$ between $m_H/2$ and $2 m_H$. We use $\mu = m_H$ as the central value. The number of flavors $N_f$ is set to 5 and the number of colors $N_c$ to 3.
• Figure 3: LO coefficient $A_H$ and its color components $A_{H,\textrm{lc}}$ and $A_{H,N_f}$ for $N_f =5$ and $N_c = 3$. Both components give positive contributions.
• Figure 4: NLO coefficient $B_H$ and its color components $B_{H,\textrm{lc}}$, $B_{H,\textrm{nlc}}$, $B_{H,\textrm{nnlc}}$ and $B_{H,N_f^2}$ for $N_f =5$ and $N_c = 3$. Notice that only the contribution of $B_{H,\textrm{lc}}$ is positive, while the three other components contribute negatively.
• Figure 5: Comparison of a Pythia simulation for Higgs EEC to the analytic LO and NLO results from eq. \ref{['eq:eecnlo']}. Both Pythia curves contain contributions from self-correlations, which are not included in the analytic result. The area under both Pythia curves is unity. Omitting the self-correlations decreases the area under the Pythia curve with haronization to $0.96$, while the area under the curve without hadronization becomes $0.88$. Adding self-correlations only increases the number of entries in the very last bin in the collinear ($\cos \chi \approx 1$) region, while the rest of the curve remains unchanged.