Study of the $h_c$($1^1P_1$) meson via $ψ(2S) \rightarrow π^0 h_c$ decays at BESIII
Using 448 million $ψ(2S)$ events, the spin-singlet $P$-wave charmonium state $h_c$(1$^1P_1$) is studied via the $ψ(2S) \rightarrow π^0 h_c$ decay followed by the $h_c \to γη_c$ transition. The branching fractions are measured to be $\mathcal{B}_{\rm Inc}(ψ(2S) \rightarrow π^0h_c) \times \mathcal{B}_{\rm Tag}(h_c \rightarrow γη_c) = (4.22 ^{+0.27}_{-0.26} \pm 0.19) \times 10^{-4}$ , $\mathcal{B}_{\rm Inc}(ψ(2S) \rightarrow π^0h_c) = (7.32 \pm 0.34 \pm 0.41) \times 10^{-4}$, and $\mathcal{B}_{\rm Tag}(h_c \rightarrow γη_c) = (57.66 ^{+3.62}_{-3.50} \pm 0.58)\%$, where the uncertainties are statistical and systematic, respectively. The $h_c$(1$^1P_1$) mass and width are determined to be $M = (3525.32 \pm 0.06 \pm 0.15)~\rm{MeV}/c^{\rm{2}}$ and ${\mathit Γ} = (0.78 ^{+0.27}_{-0.24} \pm 0.12)~\rm{MeV}$. Using the center of gravity mass of the three $χ_{cJ}$(1$^3 P_J$) mesons ($M(\rm c.o.g.)$), the $1P$ hyperfine mass splitting is estimated to be $Δ_{\rm hyp} = M(h_c) - M(\rm c.o.g.) = (0.03 \pm 0.06 \pm 0.15) ~ \rm{MeV}/c^{\rm{2}}$, which is consistent with the expectation that the $1P$ hyperfine splitting is zero at the lowest-order.