AbstractThe decays B+→J/ψπ+π−K+are studied using a data set corresponding to an integrated luminosity of 9 fb−1collected with the LHCb detector in proton-proton collisions between 2011 and 2018. Precise measurements of the ratios of branching fractions with the intermediate ψ2(3823), χc1(3872) and ψ(2S) states are reported. The values are$$ {\displaystyle \begin{array}{c}\frac{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\uppsi}_2(3823){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi_2(3823)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\upchi}_{\mathrm{c}1}(3872){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\upchi_{\mathrm{c}1}(3872)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(3.56\pm 0.67\pm 0.11\right)\times {10}^{-2},\\ {}\frac{{\mathcal{B}}_{{\mathrm{B}}^{+}\to {\uppsi}_2(3823){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi_2(3823)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to \uppsi \left(2\mathrm{S}\right){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi \left(2\mathrm{S}\right)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(1.31\pm 0.25\pm 0.04\right)\times {10}^{-3},\\ {}\frac{{\mathcal{B}}_{\mathrm{B}+\to {\upchi}_{\mathrm{c}1}(3872){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\upchi_{\mathrm{c}1}(3872)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}{{\mathcal{B}}_{{\mathrm{B}}^{+}\to \uppsi \left(2\mathrm{S}\right){\mathrm{K}}^{+}}\times {\mathcal{B}}_{\uppsi \left(2\mathrm{S}\right)\to \mathrm{J}/{\uppsi \uppi}^{+}{\uppi}^{-}}}=\left(3.69\pm 0.07\pm 0.06\right)\times {10}^{-2},\end{array}} $$BB+→ψ23823K+×Bψ23823→J/ψπ+π−BB+→χc13872K+×Bχc13872→J/ψπ+π−=3.56±0.67±0.11×10−2,BB+→ψ23823K+×Bψ23823→J/ψπ+π−BB+→ψ2SK+×Bψ2S→J/ψπ+π−=1.31±0.25±0.04×10−3,BB+→χc13872K+×Bχc13872→J/ψπ+π−BB+→ψ2SK+×Bψ2S→J/ψπ+π−=3.69±0.07±0.06×10−2,where the first uncertainty is statistical and the second is systematic. The decay of B+→ψ2(3823)K+with ψ2(3823)→J/ψπ+π−is observed for the first time with a significance of 5.1 standard deviations. The mass differences between the ψ2(3823), χc1(3872) and ψ(2S) states are measured to be$$ {\displaystyle \begin{array}{c}{m}_{\upchi_{\mathrm{c}1}(3872)}-{m}_{\uppsi_2(3823)}=47.50\pm 0.53\pm 0.13\;\mathrm{MeV}/{c}^2,\\ {}{m}_{\uppsi_2(3823)}-{m}_{\uppsi_2\left(2\mathrm{S}\right)}=137.98\pm 0.53\pm 0.14\;\mathrm{MeV}/{c}^2,\\ {}{m}_{\upchi_{\mathrm{c}1}(3872)}-{m}_{\uppsi_2\left(2\mathrm{S}\right)}=185.49\pm 0.06\pm 0.03\;\mathrm{MeV}/{c}^2,\end{array}} $$mχc13872−mψ23823=47.50±0.53±0.13MeV/c2,mψ23823−mψ22S=137.98±0.53±0.14MeV/c2,mχc13872−mψ22S=185.49±0.06±0.03MeV/c2,resulting in the most precise determination of the χc1(3872) mass. The width of the ψ2(3823) state is found to be below 5.2 MeV at 90% confidence level. The Breit-Wigner width of the χc1(3872) state is measured to be$$ {\Gamma}_{\upchi_{\mathrm{c}1}(3872)}^{\mathrm{BW}}={0.96}_{-0.18}^{+0.19}\pm 0.21\;\mathrm{MeV} $$Γχc13872BW=0.96−0.18+0.19±0.21MeVwhich is inconsistent with zero by 5.5 standard deviations.