Complete measurement of $\mathrm{\Lambda }$ electromagnetic form factors

The exclusive process $e^{+}{e}^{-}\to \mathrm{\Lambda }\bar{\mathrm{\Lambda }}$, with $\Lambda \to p\pi^-$and$\bar{\Lambda} \to \bar{p}\pi^+$,hasbeenstudiedat$\sqrt{s} =$2.396GeVformeasurementofthe$\Lambda$ electric and magnetic form factors, $G_E$ and $G_M$. A data sample, corresponding to an integrated luminosity of 66.9 pb${}^{-1}$, was collected with the BESIII detector for this purpose. A multi-dimensional analysis with a complete decomposition of the spin structure of the reaction enables a determination of the modulus of the ratio $R=\mathrm{\mid }{G}_E\mathrm{/}{G}_M\mathrm{\mid }$ and, for the first time for any baryon, the relative phase $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_E-{\mathrm{\Phi }}_M$. The resulting values are obtained using the recent and most precise measured value of the asymmetry parameter ${\alpha }_{\mathrm{\Lambda }}$ = $0.750\pm 0.010$ to be $R=0.96\pm 0.14(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.02(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.})$ and $\Delta\Phi=37^{$\mathrm{o}$}\pm~12^{$\mathrm{o}$}~(\rm stat.)\pm~6^{$\mathrm{o}$}~(\rm sys.)$, respectively. In addition, the cross section is measured with unprecedented precision to be $\sigma = 118.7~\pm~5.3~(\rm stat.)\pm~5.1~(\rm sys.)$ pb, which corresponds to an effective form factor of $\mathrm{\mid }G\mathrm{\mid }=0.123\pm 0.003(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.003(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.})$. The contribution from two-photon exchange is found to be negligible. Our result enables the first complete determination of baryon time-like electromagnetic form factors.