Next-to-Leading Order QCD Corrections to the Decay Width $H \to Z γ$

TL;DR

This work provides a complete analytic computation of the next-to-leading order QCD corrections to the decay width $H\to Z\gamma$ in the Standard Model. The authors employ integration-by-parts to reduce the two-loop amplitudes to a set of master integrals, which are solved via differential equations in a canonical, uniform-weight basis, yielding results expressed in terms of $\log$-type functions and polylogarithms up to $\text{Li}_4$ and $\text{Li}_{2,2}$. An alternative basis based on logs and Li-functions is also developed to improve numerical stability across the full physical region. For $m_H=125.1$ GeV, the computed two-loop QCD corrections amount to about $0.22\%$ of the leading-order width, with the top-loop contributing $\approx+0.30\%$ and the bottom-loop about $-0.08\%$, underscoring a small but non-negligible QCD effect and suggesting that two-loop electroweak corrections may be more significant in precision tests.

Abstract

We present the analytic calculation of the two-loop QCD corrections to the decay width of a Higgs boson into a photon and a $Z$ boson. The calculation is carried out using integration-by-parts identities for the reduction to master integrals of the scalar integrals, in terms of which we express the amplitude. The calculation of the master integrals is performed using differential equations applied to a set of functions suitably chosen to be of uniform weight. The final result is expressed in terms of logarithms and polylogarithmic functions $\text{Li}_2$, $\text{Li}_3$, $\text{Li}_4$ and $\text{Li}_{2,2}$.

Figures (6)

• Figure 1: Feynman diagrams for the decay process $H \to Z \gamma$ at leading order.
• Figure 2: Feynman diagrams for the NLO QCD corrections to the decay process $H \to Z \gamma$. Diagrams with the reversed direction of the fermionic arrow are not shown. We can easily consider the $N_h^2$ contribution (one top-quark loop coupled to the Higgs and the other correcting the $Z$ or $\gamma$ propagator).
• Figure 3: Counterterm diagrams involved in the heavy-quark mass renormalization.
• Figure 4: Set of master integrals for the NLO QCD corrections to the decay $H\to Z \gamma$. The master integrals $M_2$, $M_4$, $M_6$, $M_9$ are functions of $p^2=-m_H^2$, while the master integrals $M_3$, $M_5$, $M_7$, $M_{10}$ are functions of $p^2=-m_Z^2$. $M_{15}$ and $M_{18}$ have the numerator explicitly written. A dot on a propagator indicates that the propagator is raised to power 2. Two dots means that the propagator is raised to power 3. See Appendix B.
• Figure 5: $\Gamma_{H \to Z \gamma}$ including LO and NLO QCD contributions.