Study of the $\psi_2(3823)$ and $\chi_{c1}(3872)$ states in $B^+ \rightarrow \left( J\psi\pi^+\pi^-\right)K^+$ decays

The decays B$^{+}$→ J/ψπ$^{+}$π$^{−}$K$^{+}$ are studied using a data set corresponding to an integrated luminosity of 9 fb$^{−1}$ collected with the LHCb detector in proton-proton collisions between 2011 and 2018. Precise measurements of the ratios of branching fractions with the intermediate ψ$_{2}$(3823), χ$_{c1}$(3872) and ψ(2S) states are reported. The values are$ {\displaystyle \begin{array}{c}\frac{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\uppsi}_2(3823){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi_2(3823)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\upchi}_{\mathrm{c}1}(3872){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\upchi_{\mathrm{c}1}(3872)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(3.56\pm 0.67\pm 0.11\right)\times {10}^{-2},\\ {}\frac{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\uppsi}_2(3823){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi_2(3823)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to \uppsi \left(2\mathrm{S}\right){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi \left(2\mathrm{S}\right)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(1.31\pm 0.25\pm 0.04\right)\times {10}^{-3},\\ {}\frac{{\mathcal{B}}_{\mathrm{B}+\to {\upchi}_{\mathrm{c}1}(3872){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\upchi_{\mathrm{c}1}(3872)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to \uppsi \left(2\mathrm{S}\right){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi \left(2\mathrm{S}\right)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(3.69\pm 0.07\pm 0.06\right)\times {10}^{-2},\end{array}} $where the first uncertainty is statistical and the second is systematic. The decay of B$^{+}$→ ψ$_{2}$(3823)K$^{+}$ with ψ$_{2}$(3823) → J/ψπ$^{+}$π$^{−}$ is observed for the first time with a significance of 5.1 standard deviations. The mass differences between the ψ$_{2}$(3823), χ$_{c1}$(3872) and ψ(2S) states are measured to be$ {\displaystyle \begin{array}{c}{m}_{\upchi_{\mathrm{c}1}(3872)}-{m}_{\uppsi_2(3823)}=47.50\pm 0.53\pm 0.13\;\mathrm{MeV}/{c}^2,\\ {}{m}_{\uppsi_2(3823)}-{m}_{\uppsi_2\left(2\mathrm{S}\right)}=137.98\pm 0.53\pm 0.14\;\mathrm{MeV}/{c}^2,\\ {}{m}_{\upchi_{\mathrm{c}1}(3872)}-{m}_{\uppsi_2\left(2\mathrm{S}\right)}=185.49\pm 0.06\pm 0.03\;\mathrm{MeV}/{c}^2,\end{array}} $resulting in the most precise determination of the χ$_{c1}$(3872) mass. The width of the ψ$_{2}$(3823) state is found to be below 5.2 MeV at 90% confidence level. The Breit-Wigner width of