Inclusive weak decays of the $J/ψ$: single-charm decay and weak annihilation
Conventional estimates of $J/ψ$ weak decays, based on independent single-charm decays, suggest a branching fraction of order $10^{-8}$. Searches with samples of approximately $10^{10}$ $J/ψ$ events have so far yielded upper limits on exclusive modes. We calculate the inclusive weak branching fraction using the optical theorem, including single-charm decay, charged- and neutral-current annihilation, and their interference. The heavy-quark expansion identifies the leading pair-annihilation structure and motivates vacuum saturation for the leading color-singlet component. With tree-level hard kernels and leading-logarithmic Wilson coefficients at $μ_0=M_{J/ψ}$, we obtain $B_{\rm weak}=(7.96^{+3.43}_{-0.43}\pm1.24)\times10^{-8}$. The contribution mediated by the $W$ boson alone, i.e. single-charm decays and $W$-exchange annihilation without the $W$-$Z$ interference, has a branching fraction of $(2.37^{+1.64}_{-0.42}\pm0.13)\times10^{-8}$. For the flavor sectors $(ΔC,|ΔS|)=(0,1)$ and $|ΔC|=1$, the branching fractions are $(4.97^{+16.44}_{-4.97}\pm0.99)\times10^{-10}$ and $(1.86^{+0.20}_{-0.06}\pm0.02)\times10^{-8}$, respectively, including charge-conjugate channels. The first uncertainty estimates missing perturbative corrections through scale variation over $1~\mathrm{GeV}\leqμ\leq3M_{J/ψ}$. The second is a hadronic uncertainty, obtained by jointly varying each vacuum-saturation and kinetic/chromomagnetic input within relative ranges of $Λ_{\rm QCD}/m_c\simeq20\%$. Pair annihilation dominates, with a substantial neutral-current contribution and scale-sensitive $W$-$Z$ interference.