Study of the Magnetic Dipole Transition of $J\mathrm{/}\psi \to \gamma {\eta }_c$ via ${\eta }_c\to p\bar{p}$

Using $(10.087\pm 0.044)\times {10}^9$ $J\mathrm{/}\psi$ events collected with the BESIII detector at the $e^{+}{e}^{-}$ BEPCII collider, we present the first amplitude analysis of $J\mathrm{/}\psi \to \gamma p\bar{p}$ with the $p\bar{p}$ invariant mass in the ${\eta }_c$ mass region $[2.70,3.05]$~GeV/$c^2$. The product branching fraction $\mathcal{B}(J\mathrm{/}\psi \to \gamma {\eta }_c)\times \mathcal{B}({\eta }_c\to p\bar{p})$ is precisely determined to be $(2.11\pm {0.02}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.07}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-5}$. Combining with the product branching fractions $\mathcal{B}({\eta }_c\to p\bar{p})\times \mathcal{B}({\eta }_c\to \gamma \gamma )$ and $\mathcal{B}(J\mathrm{/}\psi \to \gamma {\eta }_c)\times \mathcal{B}({\eta }_c\to \gamma \gamma )$, the branching fractions of $\mathcal{B}(J\mathrm{/}\psi \to \gamma {\eta }_c)$ and $\mathcal{B}({\eta }_c\to \gamma \gamma )$ are calculated to be $(2.29\pm {0.01}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.04}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}\pm {0.18}_{\mathrm{o}\mathrm{p}\mathrm{b}\mathrm{f}})\mathrm{\%}$ and $(2.28\pm {0.01}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.04}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}\pm {0.18}_{\mathrm{o}\mathrm{p}\mathrm{b}\mathrm{f}})\times {10}^{-4}$, respectively, which are consistent with the latest lattice quantum chromodynamics calculations. Here, opbf is the uncertainty from the other product branching fractions used in the calculation.