Next-to-Next-to-Leading Order QCD Corrections to $\eta_t \to HZ$

Abstract

Motivated by recent LHC observations of a threshold enhancement in the $t\bar t$ system consistent with toponium-like dynamics, we compute the next-to-next-to-leading order (NNLO) QCD correction to the hard short-distance coefficient for $\eta_t\to HZ$, where $\eta_t$ denotes the pseudoscalar color-singlet configuration $t\bar t({}^1S_0^{[1]})$. Within the NRQCD factorization framework, we retain the full dependence on the Higgs- and $Z$-boson masses and include both the two-loop virtual corrections and the real double-gluon-emission channel. At next-to-leading order (NLO), the finite-mass result remains close to its large-mass limit. The NNLO coefficient develops logarithms of both the renormalization scale $\mu_R$ and the NRQCD factorization scale $\mu_\Lambda$. For $\mu_R=\mu_\Lambda=m_t$, the NLO and NNLO terms suppress the leading-order width by about $29\%$ and $19\%$, respectively; the two-loop contribution is about two thirds of the one-loop effect and of the same sign, so the perturbative series converges slowly and the NNLO prediction amounts to about $52\%$ of the leading-order width. The resulting coefficient provides a necessary ingredient for future threshold studies of $\eta_t\to HZ$ and for assessing the sensitivity of this channel to the top-Higgs interaction. The accuracy of the result is further supported by three checks: reproduction of the known large-mass limit at one loop, independence of the extracted coefficient on the $\gamma_5$ prescription, and consistency of its scale logarithms with the renormalization-group structure.