Renormalization and Factorization Scale-Invariant Predictions for the Higgs Rare Decay $H\to J\mathrm{/}\psi +\gamma$ via the Principle of Maximum Conformality

We investigate the $J\mathrm{/}\psi$ direct production mechanism in the rare exclusive Higgs decay $H\to J\mathrm{/}\psi +\gamma$ within nonrelativistic QCD (NRQCD), which provides a clean probe for extracting the charm-quark Yukawa coupling to the Higgs boson. The Principle of Maximum Conformality (PMC) is used to remove conventional renormalization-scheme and scale ambiguities in the next-to-next-to-leading-order (N${}^2$LO) perturbative QCD series. Large logarithmic contributions arising from Yukawa coupling renormalization are resummed, providing a reliable foundation for subsequent analyses. Using the experimentally measured leptonic decay width of $J\mathrm{/}\psi$ and the N${}^2$LO perturbative result, we extract the factorization-scale-dependent long-distance matrix element $⟨J\mathrm{/}\psi (\mathit{\epsilon })\mathrm{\mid }{\psi }^{†}\mathit{\sigma }\cdot \mathit{\epsilon }\chi ({\mu }_{\mathrm{\Lambda }})\mathrm{\mid }0⟩$. Combining this with the factorization-scale-dependent short-distance coefficient, we obtain a factorization-scale-invariant decay width for the channel. Compared with earlier predictions in the literature, our fixed-order result for $\mathrm{\Gamma }(H\to J\mathrm{/}\psi +\gamma )$ is more robust and precise, with good convergence and no renormalization- or factorization-scale dependence. We find $\mathrm{\Gamma }(H\to J\mathrm{/}\psi +\gamma )=(6.4574_{-0.3995}^{+0.3995})\times {10}^{-11}$ GeV, where the uncertainty is the quadratic sum of contributions from $\mathrm{\Delta }{\alpha }_s(m_Z)=\pm 0.0009$, $\mathrm{\Delta }{\mathrm{\Gamma }}_{J\mathrm{/}\psi \to e^{+}{e}^{-}}=\pm 0.10\text{ GeV}$, $\mathrm{\Delta }{\bar{m}}_c({\bar{m}}_c)=\pm 0.0046\text{ GeV}$, and the estimated magnitude of N${}^3$LO contributions from Bayesian analysis. This work demonstrates for the first time how the PMC can be applied to obtain fixed-order perturbative predictions that are invariant under both renormalization and factorization scale variations.