Higher $$\chi_{{cJ}}{(nP)}$$, $${h}_{{c}}{(nP)}$$ States and the Role of the Gluon-Exchange Potential

The masses, the fine-structure splitting, and two-photon decay widths of the higher $$nP$$ -charmonium states are calculated in the relativistic string model, reduced to the spinless Salpeter equation, where the static potential has no fitting parameters and the c-quark mass has the physical value. The resulting masses of $$h_{c}\left(3P\right)$$ , $$\chi_{c1}\left(3P\right)$$ , $$\chi_{c0}\left(4P\right)$$ , $$\chi_{c0}\left(5P\right)$$ , $$\chi_{c1}\left(5P\right)$$ are obtained in a good agreement with the experimental masses of the LHCb resonances: $$h_{c}$$ (4300), $$\chi_{c1}$$ (4274), $$X$$ (4500), $$X$$ (4700), $$X$$ (4685). To test sensitivity of results to a chosen gluon-exchange (GE) potential, three types of $$V_{\textrm{ ge}}(r)$$ are considered. In first case the non-screened GE potential with large vector coupling at asymptotic, $$\alpha_{V}(\textrm{asym.})=0.635$$ , and the c-quark mass $$m_{c}=1.430$$ GeV are taken; in second case a screened $$V_{\textrm{ge}}$$ and $$m_{c}=1.385$$ GeV are investigated, and in third case the GE potential is totally suppressed, $$V_{\textrm{ge}}=0$$ , and $$m_{c}=1.32$$ GeV. The agreement with experiment is reached only if the same (universal) flattened confining potential, introduced in the analysis of the radial Regge trajectories of light mesons, is used. The unobserved $$6P$$ , $$0^{+}$$ resonance with the mass $${\cong}4.826(13)$$ GeV, near $$J/\psi\phi$$ (1680) threshold, is predicted. Our analysis shows that the screening of the GE potential is possible but weakly affects the physical results obtained. The calculated two-photon decay widths weakly differ in the three cases but may become an important factor, which distinguishes $$c\bar{c}$$ and four-quark states.