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.