How large are hadronic contributions to $h \to γγ$?

TL;DR

This work quantifies non-perturbative hadronic contributions to the SM Higgs decay $h \to \gamma \gamma$ using a dispersive, analyticity- and unitarity–constrained framework. By expressing the hadronic correction as a dispersion integral over two-body intermediate states with $S$-wave $X \to \gamma \gamma$ cross sections and hadronic form factors, the authors show these effects are quadratically suppressed with the light-quark mass and numerically tiny, $|C_{\gamma \gamma}^{\mathrm{had}}| \sim 4.1\times 10^{-8}$, leading to a width shift $|\delta_{\mathrm{had}}| \approx 0.0044\%$. They obtain a robust, conservative upper bound and discuss cross-checks, such as a Lowest-Meson-Dominance estimate that yields only a fraction of the full result, confirming the practical negligibility of hadronic uncertainties in $h \to \gamma \gamma$ and in $gg \to h$. The methodology reinforces the reliability of SM predictions for precision Higgs physics and can be extended to related processes like $h \to \gamma Z$, $h \to gg$, and $gg \to h$, where non-perturbative effects are likewise expected to remain subdominant.

Abstract

The decay of the Higgs boson into two photons, $h \to γγ$, is a loop-induced process within the Standard Model, predominantly mediated by loops of $W$ bosons and top quarks. While these leading contributions are well understood, the role of hadronic effects, which arise from non-perturbative QCD dynamics, has received less attention, with recent studies reporting puzzling and contradictory results. In this work, we present a systematic evaluation of the hadronic contributions to the $h \to γγ$ decay width using dispersion relations. Our analysis shows that these contributions are exceedingly small, as expected, altering the decay width by about $0.004\%$ under conservative assumptions. Therefore, hadronic effects can be safely neglected even in the context of future high-precision Higgs measurements at current and next-generation colliders. As an aside, we also estimate the possible size of hadronic contributions to Higgs production in gluon-gluon fusion.

Figures (6)

• Figure 1: Graphical illustration of the optical theorem applied to the hadronic contribution of the $h \to \gamma \gamma$ amplitude, as given in (\ref{['eq:unitarity']}). The ellipses in the second line denote all hadronic intermediate states $X$ other than $\pi^+ \pi^-$. Also shown are diagrammatic representations of the form factors $F_i^{\pi^+ \pi^-} (s)$ and the $S$-wave cross section $\sigma_{\text{$S$-wave}} \left ( \pi^+ \pi^- \to \gamma \gamma \right )$.
• Figure 2: Magnitudes of the form factors $F_i^X(s)$ corresponding to the operators $Q_i$ introduced in (\ref{['eq:Qi']}), shown for $X = \pi \pi, K K$. Further details are provided in the main text.
• Figure 3: $S$-wave contributions to the $X \to \gamma \gamma$ cross sections for $X = \pi^+ \pi^-$, $\pi^0 \pi^0$, $\pi^0 \eta$, $K^+ K^-$, and $K_S K_S$. See the main text for further explanations.
• Figure 4: $S$-wave contributions to the $X \to \gamma \gamma$ cross sections for $X = \pi^+ \pi^-$ and $K^+ K^-$ are shown. The figure compares the full results from Danilkin:2018qfnDeineka:2024mzt with the Born approximation given in (\ref{['eq:S-wave_Born']}).
• Figure 5: Functions $\kappa_i^X (s)$ introduced in (\ref{['eq:kappaiX']}) for the operators $Q_i$ in (\ref{['eq:Qi']}) and $X = \pi^+ \pi^-$, $\pi^0 \pi^0$, $K^+ K^-$, and $K_S K_S$. Up to $s = 2 \, \text{GeV}^2$, the results for charged pions and kaons are based on the full $S$-wave cross sections, whereas for $s > 2 \, \text{GeV}^2$ the Born approximation (\ref{['eq:S-wave_Born']}) is used. Consult the main text for further details.