Study of the ${h}_{c}\left({1}^{1}{P}_{1}\right)$ meson via $\psi \left(2S\right)\to {\pi }^{0}{h}_{c}$ decays at BESIII

Using 448 million $\psi \left(2S\right)$ events, the spin-singlet $P$-wave charmonium state ${h}_{c}\left(1{}^{1}{P}_{1}\right)$ is studied via the $\psi \left(2S\right)\to {\pi }^{0}{h}_{c}$ decay followed by the ${h}_{c}\to \gamma {\eta }_{c}$ transition. The branching fractions are measured to be ${B}_{\text{Inc}}\left(\psi \left(2S\right)\to {\pi }^{0}{h}_{c}\right)×{B}_{\text{Tag}}\left({h}_{c}\to \gamma {\eta }_{c}\right)=\left(4.2{2}_{-0.26}^{+0.27}±0.19\right)×{10}^{-4}$, ${B}_{\text{Inc}}\left(\psi \left(2S\right)\to {\pi }^{0}{h}_{c}\right)=\left(7.32±0.34±\phantom{\rule{0ex}{0ex}}0.41\right)×{10}^{-4}$, and ${B}_{\text{Tag}}\left({h}_{c}\to \gamma {\eta }_{c}\right)=\left(57.6{6}_{-3.50}^{+3.62}±0.58\right)%$, where the uncertainties are statistical and systematic, respectively. The ${h}_{c}\left(1{}^{1}{P}_{1}\right)$ mass and width are determined to be $M=\left(3525.32±0.06±0.15\right)\text{}\text{}\mathrm{MeV}/{c}^{2}$ and $\Gamma =\left(0.7{8}_{-0.24}^{+0.27}±0.12\right)\text{}\text{}\mathrm{MeV}$. Using the center of gravity mass of the three ${\chi }_{cJ}\left({1}^{3}{P}_{J}\right)$ mesons [$M\left(c.o.g.\right)$], the $1P$ hyperfine mass splitting is estimated to be ${\Delta }_{\mathrm{hyp}}=M\left({h}_{c}\right)-M\left(c.o.g.\right)=\phantom{\rule{0ex}{0ex}}\left(0.03±0.06±0.15\right)\text{}\text{}\mathrm{MeV}/{c}^{2}$, which is consistent with the expectation that the $1P$ hyperfine splitting is zero at the lowest order.