The higher $χ_{cJ}(nP)$, $h_c(nP)$ 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 $h_c(3P)$, $\chi_{c1}(3P)$, $\chi_{c0}(4P)$, $\chi_{c0}(5P)$, $\chi_{c1}(5P)$ 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 $V_{ge}(r)$ are considered. In first case the non-screened GE potential with large vector coupling at asymptotic, $\alpha_{\rm V}(\rm asym.)=0.635$, and the $c$-quark mass $m_c=1.430$ GeV are taken; in second case a screened $V_{ge}$ and $m_c=1.385$ are investigated, and in third case the GE potential is totally suppressed, $V_{ge}=0$, 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 $\cong 4.81$ 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.