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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., Comment: v1: LaTeX, 13 pages, 0 figures, 6 tables

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., Comment: v1: LaTeX, 13 pages, 0 figures, 6 tables

\date{\today} Within the diquark-antidiquark model the masses of the $0^{++}, cc\bar c\bar c$ resonances are calculated, using the expansion of the four-quark wave function in the set of the hyperspherical functions. The interaction is defined via a universal pair-wise potential, which does not contain fitting parameters. The resulting masses $M_4(nS)$ are shown to be very sensitive to the value of $c-$quark mass, chosen in relativistic string Hamiltonian, and $m_c=1.24, 1.30, 1.43$ (in GeV) are considered. The choice of $m_c$, equal to the current mass, $m_c=1.245$ GeV, yields three $nS~(n_r=0,1,2)$ states in a very good agreement with the masses of the $X(6550), X(6900), X(7287)$ resonances, if the gluon-exchange interaction is totally neglected. This fact indicates on a possible screening of the gluon-exchange interaction inhe $cc\bar c\bar c$ system. For $m_c=1.43$~GeV the ground state mass $M_4(1S)=6557$~MeV is obtained in agreement with experiment only if $\alpha_{\rm V}\cong 0.39(1)$ is used, however, in this case the masses of the $2S, 3S$ radial excitations exceed the masses of $X(6900), X(7280)$ by $\sim 100$~MeV., Comment: 16 pages

\date{\today} Within the diquark-antidiquark model the masses of the $0^{++}, cc\bar c\bar c$ resonances are calculated, using the expansion of the four-quark wave function in the set of the hyperspherical functions. The interaction is defined via a universal pair-wise potential, which does not contain fitting parameters. The resulting masses $M_4(nS)$ are shown to be very sensitive to the value of $c-$quark mass, chosen in relativistic string Hamiltonian, and $m_c=1.24, 1.30, 1.43$ (in GeV) are considered. The choice of $m_c$, equal to the current mass, $m_c=1.245$ GeV, yields three $nS~(n_r=0,1,2)$ states in a very good agreement with the masses of the $X(6550), X(6900), X(7287)$ resonances, if the gluon-exchange interaction is totally neglected. This fact indicates on a possible screening of the gluon-exchange interaction inhe $cc\bar c\bar c$ system. For $m_c=1.43$~GeV the ground state mass $M_4(1S)=6557$~MeV is obtained in agreement with experiment only if $\alpha_{\rm V}\cong 0.39(1)$ is used, however, in this case the masses of the $2S, 3S$ radial excitations exceed the masses of $X(6900), X(7280)$ by $\sim 100$~MeV., Comment: 16 pages

\date{\today} Within the diquark-antidiquark model the masses of the $0^{++}, cc\bar c\bar c$ resonances are calculated, using the expansion of the four-quark wave function in the set of the hyperspherical functions. The interaction is defined via a universal pair-wise potential, which does not contain fitting parameters. The resulting masses $M_4(nS)$ are shown to be very sensitive to the value of $c-$quark mass, chosen in relativistic string Hamiltonian, and $m_c=1.24, 1.30, 1.43$ (in GeV) are considered. The choice of $m_c$, equal to the current mass, $m_c=1.245$ GeV, yields three $nS~(n_r=0,1,2)$ states in a very good agreement with the masses of the $X(6550), X(6900), X(7287)$ resonances, if the gluon-exchange interaction is totally neglected. This fact indicates on a possible screening of the gluon-exchange interaction inhe $cc\bar c\bar c$ system. For $m_c=1.43$~GeV the ground state mass $M_4(1S)=6557$~MeV is obtained in agreement with experiment only if $\alpha_{\rm V}\cong 0.39(1)$ is used, however, in this case the masses of the $2S, 3S$ radial excitations exceed the masses of $X(6900), X(7280)$ by $\sim 100$~MeV., Comment: 16 pages

The scalar resonances $X(3915), X(3960), X(4140)$ are considered as exotic four-quark states: $cq\bar c \bar q, cs\bar c \bar s, cs\bar c\bar s$, while the $X(3863)$ is proved to be the $c\bar c, 2\,^3P_0$ state. The masses and the widths of these resonances are calculated in the framework of the Extended Recoupling Model, where a four-quark system is formed inside the bag and has relatively small size ($\la 1.0$~fm). Then the resonance $X(3915)$ appears due to the transitions: $J/\psi\omega$ into $D^{*+}D^{*-}$ (or $D^{*0}\bar D^{*0})$ and back, while the $X(3960)$ is created due to the transitions $D_s^+D_s^-$ into $J/\psi\phi$ and back, and the $X(4140)$ is formed in the transitions $J/\psi\phi$ into $D_s^{*+}D_s^{*-}$ and back. The characteristic feature of the recoupling mechanism is that this type of resonances can be predominantly in the $S$-wave decay channels and has $J^P=0^+$. In two-channel case the resonance occurs to be just near the lower threshold, while due to coupling to third channel (like the $c\bar c$ channel) it is shifted up and lies by (20--30)~MeV above the lower threshold. The following masses and widths are calculated: $M(X(3915))=3920$~MeV, $\Gamma(X(3915))=20$~MeV; $M(X(3960))=3970$~MeV, $\Gamma(X(3960)=45(5)$~MeV, $M(X(4140))= 4120(20)$~MeV, $\Gamma(X(4140))=100$~MeV, which are in good agreement with experiment., Comment: 16 pages

The resonances, containing $c\bar c$ plus $s\bar s$ (or light $q\bar q$) quarks in the mass region $(3900-4700)$ MeV, are analyzed in the relativistic strong coupling theory, with and without channel coupling phenomena. The conventional charmonium spectrum is presented, being calculated with the relativistic string Hamiltonian, which does not contain fitting parameters, while for high excitations the universal flattened confining potential is used. It is shown that $X(4274),X(4500),X(4700)$ can be identified as $3\,^3P_1, 4\,^3P_0, 5\,^3P_0$ states. The exotic states are considered using the coupled-channel (recoupling) theory, when two mesons $m_1,m_2$ transfer into another two mesons $m_3, m_4$ and back (infinite number of times), creating the four-quark systems. The resonances $X(3875), X(3915), Z_{cs}(3985), X(4140)$ can be explained in this way as the exotic four-quark states in the $S$-wave decay channels., Comment: 19 pages

The resonances, containing $c\bar c$ plus $s\bar s$ (or light $q\bar q$) quarks in the mass region $(3900-4700)$ MeV, are analyzed in the relativistic strong coupling theory, with and without channel coupling phenomena. The conventional charmonium spectrum is presented, being calculated with the relativistic string Hamiltonian, which does not contain fitting parameters, while for high excitations the universal flattened confining potential is used. It is shown that $X(4274),X(4500),X(4700)$ can be identified as $3\,^3P_1, 4\,^3P_0, 5\,^3P_0$ states. The exotic states are considered using the coupled-channel (recoupling) theory, when two mesons $m_1,m_2$ transfer into another two mesons $m_3, m_4$ and back (infinite number of times), creating the four-quark systems. The resonances $X(3875), X(3915), Z_{cs}(3985), X(4140)$ can be explained in this way as the exotic four-quark states in the $S$-wave decay channels., Comment: 15 pages

The masses of excited charmed mesons are shown to decrease by $\sim (50-150)$~MeV due to a flattening of the confining potential at large distances, which effectively takes into account open decay channels. The scale of the mass shifts is similar to that in charmonium for $\psi(4660)$ and $\chi_{c0}(4700)$. The following masses of the first excitations: $M(2\,{}^3P_0)=2874$~MeV, $M(2\,{}^3P_2)=2968$~MeV, $M(2\,{}^3D_1)=3175$~MeV, and $M(2\,{}^3D_3)=3187$~MeV, and second excitations: $M(3\,{}^1S_0)=3008$~MeV, $M(3\,{}^3S_1)=3062$~MeV, $M(3\,{}^3P_0)=3229$~MeV, and $M(3\,{}^3P_2) =3264$~MeV, are predicted. The other states with $L=0,1,2$ and $n_r \geq 3$ have their masses in the region $M(nL)\geq 3.3$~GeV., Comment: 15 pages, 4 tables

The formalism of the coupled $q\bar q$ and the $\varphi\varphi ( \pi-\pi$, $K\bar K, \pi K,...$) scalar channels is formulated, taking into account the ground and radial excited $q\bar q$ poles. The basic role is shown to be played by the transition coefficients $k^{(I)} (q\bar q, |\varphi\varphi)$, which are calculated using the quark-chiral Lagrangian without free parameters. The resulting method, called the pole projection mechanism (PPM), ensures: 1) one resonance for each $\varphi\varphi$ channel from the basic $q\bar q$ pole, e.g. the $f_0 (500)$ resonance in the $\pi\pi$ channel; 2) a possibility to have two $\varphi\varphi$ resonances, coupled to the same $q\bar q$ state, when the channel coupling is taken into account in the meson-meson channels, which yields $f_0 (500)$ and $f_0(980)$ from the same $n\bar n$ pole around 1 GeV; 3) the strong pole shift down for special ($\pi\pi, \pi K)$ channels due to large transition coefficients $k^{(I)}$, computed in this formalism without free parameters. The parameters of calculated complex poles are in reasonable agreement with the experimental data of the resonances $f_0(500)$, $f_0(980)$, $a_0(980)$, $a_0(1450)$, $K^*_0(700)$, $K^*_0(1430)$, $f_0(1370)$ and $f_0(1710)$., Comment: v2: 22 pages, 5 figures; new explanations are added, several important insertions are made, the list of references is enlarged

The formalism of the coupled $q\bar q$ and the $\varphi\varphi ( \pi-\pi$, $K\bar K, \pi K,...$) scalar channels is formulated, taking into account the ground and radial excited $q\bar q$ poles. The basic role is shown to be played by the transition coefficients $k^{(I)} (q\bar q, |\varphi\varphi)$, which are calculated using the quark-chiral Lagrangian without free parameters. The resulting method, called the pole projection mechanism (PPM), ensures: 1) one resonance for each $\varphi\varphi$ channel from the basic $q\bar q$ pole, e.g. the $f_0 (500)$ resonance in the $\pi\pi$ channel; 2) a possibility to have two $\varphi\varphi$ resonances, coupled to the same $q\bar q$ state, when the channel coupling is taken into account in the meson-meson channels, which yields $f_0 (500)$ and $f_0(980)$ from the same $n\bar n$ pole around 1 GeV; 3) the strong pole shift down for special ($\pi\pi, \pi K)$ channels due to large transition coefficients $k^{(I)}$, computed in this formalism without free parameters. The parameters of calculated complex poles are in reasonable agreement with the experimental data of the resonances $f_0(500)$, $f_0(980)$, $a_0(980)$, $a_0(1450)$, $K^*_0(700)$, $K^*_0(1430)$, $f_0(1370)$ and $f_0(1710)$., Comment: v2: 22 pages, 5 figures; new explanations are added, several important insertions are made, the list of references is enlarged

The masses of excited charmed mesons are shown to decrease by $\sim (50-150)$~MeV due to a flattening of the confining potential at large distances, which effectively takes into account open decay channels. The scale of the mass shifts is similar to that in charmonium for $\psi(4660)$ and $\chi_{c0}(4700)$. The following masses of the first excitations: $M(2\,{}^3P_0)=2874$~MeV, $M(2\,{}^3P_2)=2968$~MeV, $M(2\,{}^3D_1)=3175$~MeV, and $M(2\,{}^3D_3)=3187$~MeV, and second excitations: $M(3\,{}^1S_0)=3008$~MeV, $M(3\,{}^3S_1)=3062$~MeV, $M(3\,{}^3P_0)=3229$~MeV, and $M(3\,{}^3P_2) =3264$~MeV, are predicted. The other states with $L=0,1,2$ and $n_r \geq 3$ have their masses in the region $M(nL)\geq 3.3$~GeV., Comment: 15 pages, 4 tables

The formalism of the coupled $q\bar q$ and the $\varphi\varphi ( \pi-\pi$, $K\bar K, \pi K,...$) scalar channels is formulated, taking into account the ground and radial excited $q\bar q$ poles. The basic role is shown to be played by the transition coefficients $k^{(I)} (q\bar q, |\varphi\varphi)$, which are calculated using the quark-chiral Lagrangian without free parameters. The resulting method, called the pole projection mechanism (PPM), ensures: 1) one resonance for each $\varphi\varphi$ channel from the basic $q\bar q$ pole, e.g. the $f_0 (500)$ resonance in the $\pi\pi$ channel; 2) a possibility to have two $\varphi\varphi$ resonances, coupled to the same $q\bar q$ state, when the channel coupling is taken into account in the meson-meson channels, which yields $f_0 (500)$ and $f_0(980)$ from the same $n\bar n$ pole around 1 GeV; 3) the strong pole shift down for special ($\pi\pi, \pi K)$ channels due to large transition coefficients $k^{(I)}$, computed in this formalism without free parameters. The parameters of calculated complex poles are in reasonable agreement with the experimental data of the resonances $f_0(500)$, $f_0(980)$, $a_0(980)$, $a_0(1450)$, $K^*_0(700)$, $K^*_0(1430)$, $f_0(1370)$ and $f_0(1710)$., Comment: v2: 22 pages, 5 figures; new explanations are added, several important insertions are made, the list of references is enlarged

The formalism of the coupled $q\bar q$ and the $\varphi\varphi ( \pi-\pi$, $K\bar K, \pi K,...$) scalar channels is formulated, taking into account the ground and radial excited $q\bar q$ poles. The basic role is shown to be played by the transition coefficients $k^{(I)} (q\bar q, |\varphi\varphi)$, which are calculated using the quark-chiral Lagrangian without free parameters. The resulting method, called the pole projection mechanism (PPM), ensures: 1) one resonance for each $\varphi\varphi$ channel from the basic $q\bar q$ pole, e.g. the $f_0 (500)$ resonance in the $\pi\pi$ channel; 2) a possibility to have two $\varphi\varphi$ resonances, coupled to the same $q\bar q$ state, when the channel coupling is taken into account in the meson-meson channels, which yields $f_0 (500)$ and $f_0(980)$ from the same $n\bar n$ pole around 1 GeV; 3) the strong pole shift down for special ($\pi\pi, \pi K)$ channels due to large transition coefficients $k^{(I)}$, computed in this formalism without free parameters. The parameters of calculated complex poles are in reasonable agreement with the experimental data of the resonances $f_0(500)$, $f_0(980)$, $a_0(980)$, $a_0(1450)$, $K^*_0(700)$, $K^*_0(1430)$, $f_0(1370)$ and $f_0(1710)$., Comment: v2: 22 pages, 5 figures; new explanations are added, several important insertions are made, the list of references is enlarged

The masses of excited charmed mesons are shown to decrease by $\sim (50-150)$~MeV due to a flattening of the confining potential at large distances, which effectively takes into account open decay channels. The scale of the mass shifts is similar to that in charmonium for $\psi(4660)$ and $\chi_{c0}(4700)$. The following masses of the first excitations: $M(2\,{}^3P_0)=2874$~MeV, $M(2\,{}^3P_2)=2968$~MeV, $M(2\,{}^3D_1)=3175$~MeV, and $M(2\,{}^3D_3)=3187$~MeV, and second excitations: $M(3\,{}^1S_0)=3008$~MeV, $M(3\,{}^3S_1)=3062$~MeV, $M(3\,{}^3P_0)=3229$~MeV, and $M(3\,{}^3P_2) =3264$~MeV, are predicted. The other states with $L=0,1,2$ and $n_r \geq 3$ have their masses in the region $M(nL)\geq 3.3$~GeV., Comment: 15 pages, 4 tables

The spectrum of the $s\bar s$ mesons is studied performing a phenomenological analysis of the Regge trajectories defined for the excitation energies. For the $\phi(3 ^3S_1)$ state the mass $M(\phi(3S))=2100(20)$ MeV and the leptonic width $\Gamma_{ee}(\phi(3S))=0.27(2)$ keV are obtained, while the mass of the $2 ^3D_1$ state, $M(\phi(2 ^3D_3))=2180(5)$ MeV, appears to be in agreement with the mass of the $\phi(2170)$ resonance, and its leptonic width, $\Gamma_{ee}(2 ^3D_1)=0.20\pm 0.10$ keV, has a large theoretical uncertainty, depending on the parameters of the flattened confining potential., Comment: 7 pages, no figures

The dynamical picture of a quark-antiquark interaction in light mesons, which provides linearity of radial and orbital Regge trajectories (RT), is studied with the use of the relativistic string Hamiltonian with flattened confining potential and taking into account the self-energy and string corrections. Due to the flattening effect both slopes, $\beta_n$ of the radial and $\beta_l$ of the orbital RT, decrease by $\sim 30\%$ with the value of $\beta_n=1.30(5)$~GeV$^2$ being larger than $ \beta_l=0.95(5)$~GeV. The self-energy correction provides the linearity of RT and remains important up to very high excitations; the string correction decreases the slope of the orbital RT, while the intercept $\beta_0=0.51(1)~ $GeV$^2$ is equal to the squared centroid mass of $\rho(1S)$. If the universal gluon-exchanged potential without fitting parameters and screening function, as in heavy quarkonia, is taken, then the slope of the radial RT decreases, $\beta_n=1.15(8)$~GeV$^2$, and its value coincides with the slope of the orbital RT, $\beta_l=1.08(8)$~GeV$^2$ within theoretical errors, producing the universal RT., Comment: 11 pages, two figures

The spectrum of the $s\bar s$ mesons is studied performing a phenomenological analysis of the Regge trajectories defined for the excitation energies. For the $\phi(3 ^3S_1)$ state the mass $M(\phi(3S))=2100(20)$ MeV and the leptonic width $\Gamma_{ee}(\phi(3S))=0.27(2)$ keV are obtained, while the mass of the $2 ^3D_1$ state, $M(\phi(2 ^3D_3))=2180(5)$ MeV, appears to be in agreement with the mass of the $\phi(2170)$ resonance, and its leptonic width, $\Gamma_{ee}(2 ^3D_1)=0.20\pm 0.10$ keV, has a large theoretical uncertainty, depending on the parameters of the flattened confining potential., Comment: 7 pages, no figures

The dynamical picture of a quark-antiquark interaction in light mesons, which provides linearity of radial and orbital Regge trajectories (RT), is studied with the use of the relativistic string Hamiltonian with flattened confining potential and taking into account the self-energy and string corrections. Due to the flattening effect both slopes, $\beta_n$ of the radial and $\beta_l$ of the orbital RT, decrease by $\sim 30\%$ with the value of $\beta_n=1.30(5)$~GeV$^2$ being larger than $ \beta_l=0.95(5)$~GeV. The self-energy correction provides the linearity of RT and remains important up to very high excitations; the string correction decreases the slope of the orbital RT, while the intercept $\beta_0=0.51(1)~ $GeV$^2$ is equal to the squared centroid mass of $\rho(1S)$. If the universal gluon-exchanged potential without fitting parameters and screening function, as in heavy quarkonia, is taken, then the slope of the radial RT decreases, $\beta_n=1.15(8)$~GeV$^2$, and its value coincides with the slope of the orbital RT, $\beta_l=1.08(8)$~GeV$^2$ within theoretical errors, producing the universal RT., Comment: 11 pages, two figures

The dynamical picture of a quark-antiquark interaction in light mesons, which provides linearity of radial and orbital Regge trajectories (RT), is studied with the use of the relativistic string Hamiltonian with flattened confining potential and taking into account the self-energy and string corrections. Due to the flattening effect both slopes, $\beta_n$ of the radial and $\beta_l$ of the orbital RT, decrease by $\sim 30\%$ with the value of $\beta_n=1.30(5)$~GeV$^2$ being larger than $ \beta_l=0.95(5)$~GeV. The self-energy correction provides the linearity of RT and remains important up to very high excitations; the string correction decreases the slope of the orbital RT, while the intercept $\beta_0=0.51(1)~ $GeV$^2$ is equal to the squared centroid mass of $\rho(1S)$. If the universal gluon-exchanged potential without fitting parameters and screening function, as in heavy quarkonia, is taken, then the slope of the radial RT decreases, $\beta_n=1.15(8)$~GeV$^2$, and its value coincides with the slope of the orbital RT, $\beta_l=1.08(8)$~GeV$^2$ within theoretical errors, producing the universal RT., Comment: 11 pages, two figures

The spectrum of the $s\bar s$ mesons is studied performing a phenomenological analysis of the Regge trajectories defined for the excitation energies. For the $\phi(3 ^3S_1)$ state the mass $M(\phi(3S))=2100(20)$ MeV and the leptonic width $\Gamma_{ee}(\phi(3S))=0.27(2)$ keV are obtained, while the mass of the $2 ^3D_1$ state, $M(\phi(2 ^3D_3))=2180(5)$ MeV, appears to be in agreement with the mass of the $\phi(2170)$ resonance, and its leptonic width, $\Gamma_{ee}(2 ^3D_1)=0.20\pm 0.10$ keV, has a large theoretical uncertainty, depending on the parameters of the flattened confining potential., Comment: 7 pages, no figures

The leptonic widths of high $\psi$-resonances are calculated in a coupled-channel model with unitary inelasticity, where analytical expressions for mixing angles between $(n+1)\,^3S_1$ and $n\,^3D_1$ states and probabilities $Z_i$ of the $c\bar c$ component are derived. Since these factors depend on energy (mass), different values of mixing angles $\theta(\psi(4040))=27.7^\circ$ and $\theta(\psi(4160))=29.5^\circ$, $Z_1\,(\psi(4040))=0.76$, and $Z_2\,(\psi(4160))=0.62$ are obtained. It gives the leptonic widths $\Gamma_{ee}(\psi(4040))=Z_1\, 1.17=0.89$~keV, $\Gamma_{ee}(\psi(4160))=Z_2\, 0.76=0.47$~keV in good agreement with experiment. For $\psi(4415)$ the leptonic width $\Gamma_{ee}(\psi(4415))=~0.55$~keV is calculated, while for the missing resonance $\psi(4510)$ we predict $M(\psi(4500))=(4515\pm 5)$~MeV and $\Gamma_{ee}(\psi(4510)) \cong 0.50$~keV., Comment: 10 pages, 6 references corrected, some new material added

The leptonic widths of high $\psi$-resonances are calculated in a coupled-channel model with unitary inelasticity, where analytical expressions for mixing angles between $(n+1)\,^3S_1$ and $n\,^3D_1$ states and probabilities $Z_i$ of the $c\bar c$ component are derived. Since these factors depend on energy (mass), different values of mixing angles $\theta(\psi(4040))=27.7^\circ$ and $\theta(\psi(4160))=29.5^\circ$, $Z_1\,(\psi(4040))=0.76$, and $Z_2\,(\psi(4160))=0.62$ are obtained. It gives the leptonic widths $\Gamma_{ee}(\psi(4040))=Z_1\, 1.17=0.89$~keV, $\Gamma_{ee}(\psi(4160))=Z_2\, 0.76=0.47$~keV in good agreement with experiment. For $\psi(4415)$ the leptonic width $\Gamma_{ee}(\psi(4415))=~0.55$~keV is calculated, while for the missing resonance $\psi(4510)$ we predict $M(\psi(4500))=(4515\pm 5)$~MeV and $\Gamma_{ee}(\psi(4510)) \cong 0.50$~keV., Comment: 10 pages, 6 references corrected, some new material added

The leptonic widths of high $\psi$-resonances are calculated in a coupled-channel model with unitary inelasticity, where analytical expressions for mixing angles between $(n+1)\,^3S_1$ and $n\,^3D_1$ states and probabilities $Z_i$ of the $c\bar c$ component are derived. Since these factors depend on energy (mass), different values of mixing angles $\theta(\psi(4040))=27.7^\circ$ and $\theta(\psi(4160))=29.5^\circ$, $Z_1\,(\psi(4040))=0.76$, and $Z_2\,(\psi(4160))=0.62$ are obtained. It gives the leptonic widths $\Gamma_{ee}(\psi(4040))=Z_1\, 1.17=0.89$~keV, $\Gamma_{ee}(\psi(4160))=Z_2\, 0.76=0.47$~keV in good agreement with experiment. For $\psi(4415)$ the leptonic width $\Gamma_{ee}(\psi(4415))=~0.55$~keV is calculated, while for the missing resonance $\psi(4510)$ we predict $M(\psi(4500))=(4515\pm 5)$~MeV and $\Gamma_{ee}(\psi(4510)) \cong 0.50$~keV., Comment: 10 pages, 6 references corrected, some new material added

It is shown that two physical phenomena are important for high excitations: (i) the screening of the universal gluon-exchange potential and (ii) the flattening of the confining potential owing to creation of quark loops, and both effects are determined quantitatively. Taking the first effect into account, we predict the masses of the ground states with $l=0,1,2$ in agreement with experiment. The flattening effect ensures the observed linear behaviour of the radial Regge trajectories $M^2(n)=m_0^2 + n_r \mu^2$ GeV$^2$, where the slope $\mu^2$ is very sensitive to the parameter $\gamma$, which determines the weakening of the string tension $\sigma(r)$ at large distances. For the $\rho$-trajectory the linear behaviour starts with $n_r=1$ and the values $\mu^2=1.40(2)$~GeV$^2$ for $\gamma=0.40$ and $\mu^2=1.34(1)$~GeV$^2$ for $\gamma=0.45$ are obtained. For the excited states the leptonic widths: $\Gamma_{\rm ee}(\rho(775))=7.0(3)$~keV, $\Gamma_{\rm ee}(\rho(1450))=1.7(1)$~keV, $\Gamma_{\rm ee}(\rho(1900))=1.0(1)$~keV, $\Gamma_{\rm ee}(\rho(2150))=0.7(1)$~keV, and $\Gamma_{\rm ee}(1\,{}^3D_1)=0.26(5)$~keV are calculated, if these states are considered as purely $q\bar q$ states. The width $\Gamma_{\rm ee}(\rho(1700))$ increases if $\rho(1700)$ is mixed with the $2\,{}^3S_1$ state, giving for a mixing angle $\theta=21^\circ$ almost equal widths: $\Gamma_{\rm ee}(\rho(1700))=0.75(6)$~keV and $\Gamma_{\rm ee}(1450)=1.0(1)$~keV., Comment: 12 pages

It is shown that two physical phenomena are important for high excitations: (i) the screening of the universal gluon-exchange potential and (ii) the flattening of the confining potential owing to creation of quark loops, and both effects are determined quantitatively. Taking the first effect into account, we predict the masses of the ground states with $l=0,1,2$ in agreement with experiment. The flattening effect ensures the observed linear behaviour of the radial Regge trajectories $M^2(n)=m_0^2 + n_r \mu^2$ GeV$^2$, where the slope $\mu^2$ is very sensitive to the parameter $\gamma$, which determines the weakening of the string tension $\sigma(r)$ at large distances. For the $\rho$-trajectory the linear behaviour starts with $n_r=1$ and the values $\mu^2=1.40(2)$~GeV$^2$ for $\gamma=0.40$ and $\mu^2=1.34(1)$~GeV$^2$ for $\gamma=0.45$ are obtained. For the excited states the leptonic widths: $\Gamma_{\rm ee}(\rho(775))=7.0(3)$~keV, $\Gamma_{\rm ee}(\rho(1450))=1.7(1)$~keV, $\Gamma_{\rm ee}(\rho(1900))=1.0(1)$~keV, $\Gamma_{\rm ee}(\rho(2150))=0.7(1)$~keV, and $\Gamma_{\rm ee}(1\,{}^3D_1)=0.26(5)$~keV are calculated, if these states are considered as purely $q\bar q$ states. The width $\Gamma_{\rm ee}(\rho(1700))$ increases if $\rho(1700)$ is mixed with the $2\,{}^3S_1$ state, giving for a mixing angle $\theta=21^\circ$ almost equal widths: $\Gamma_{\rm ee}(\rho(1700))=0.75(6)$~keV and $\Gamma_{\rm ee}(1450)=1.0(1)$~keV., Comment: 12 pages

It is shown that two physical phenomena are important for high excitations: (i) the screening of the universal gluon-exchange potential and (ii) the flattening of the confining potential owing to creation of quark loops, and both effects are determined quantitatively. Taking the first effect into account, we predict the masses of the ground states with $l=0,1,2$ in agreement with experiment. The flattening effect ensures the observed linear behaviour of the radial Regge trajectories $M^2(n)=m_0^2 + n_r \mu^2$ GeV$^2$, where the slope $\mu^2$ is very sensitive to the parameter $\gamma$, which determines the weakening of the string tension $\sigma(r)$ at large distances. For the $\rho$-trajectory the linear behaviour starts with $n_r=1$ and the values $\mu^2=1.40(2)$~GeV$^2$ for $\gamma=0.40$ and $\mu^2=1.34(1)$~GeV$^2$ for $\gamma=0.45$ are obtained. For the excited states the leptonic widths: $\Gamma_{\rm ee}(\rho(775))=7.0(3)$~keV, $\Gamma_{\rm ee}(\rho(1450))=1.7(1)$~keV, $\Gamma_{\rm ee}(\rho(1900))=1.0(1)$~keV, $\Gamma_{\rm ee}(\rho(2150))=0.7(1)$~keV, and $\Gamma_{\rm ee}(1\,{}^3D_1)=0.26(5)$~keV are calculated, if these states are considered as purely $q\bar q$ states. The width $\Gamma_{\rm ee}(\rho(1700))$ increases if $\rho(1700)$ is mixed with the $2\,{}^3S_1$ state, giving for a mixing angle $\theta=21^\circ$ almost equal widths: $\Gamma_{\rm ee}(\rho(1700))=0.75(6)$~keV and $\Gamma_{\rm ee}(1450)=1.0(1)$~keV., Comment: 12 pages

Radiative decays of $X(3872)$ are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels $D\bar D^*$ are described with the string breaking mechanism. In SCA the transition rate $\tilde{\Gamma}_2=\Gamma(2\,{}^3P_1 \rightarrow \psi\gamma)=71.8$~keV and large $\tilde{\Gamma}_1=\Gamma(2\,{}^3P_1\rightarrow J/\psi\gamma)=85.4$~keV are obtained, giving for their ratio the value $\tilde{R_{\psi\gamma}}=\frac{\tilde{\Gamma}_2}{\tilde{\Gamma}_1}=0.84$. In the CC approach three factors are shown to be equally important. First, the admixture of the $1\,{}^3P_1$ component in the normalized wave function of $X(3872)$ due to the CC effects. Its weight $c_{\rm X}(E_{\rm R})=0.200\pm 0.015$ is calculated. Secondly, the use of the multipole function $g(r)$ instead of $r$ in the overlap integrals, determining the partial widths. Thirdly, the choice of the gluon-exchange interaction for $X(3872)$, as well as for other states above threshold. If for $X(3872)$ the gluon-exchange potential is taken the same as for low-lying charmonium states, then in the CC approach $\Gamma_1= \Gamma(X(3872)\rightarrow J/\psi\gamma) \sim 3$~keV is very small, giving the large ratio $R_{\psi\gamma}=\frac{\mathcal{B}(X(3872)\rightarrow \psi(2S)\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi\gamma)}\gg 1.0$. Arguments are presented why the gluon-exchange interaction may be suppressed for $X(3872)$ and in this case $\Gamma_1=42.7$~keV, $\Gamma_2= 70.5$~keV, and $R_{\psi\gamma}=1.65$ are predicted for the minimal value $c_{\rm X}({\rm min})=0.185$, while for the maximal value $c_{\rm X}=0.215$ we obtained $\Gamma_1=30.8$~keV, $\Gamma_2=73.2$~keV, and $R_{\psi\gamma}=2.38$, which agrees with the LHCb data., Comment: 12 pages, no figures

Radiative decays of $X(3872)$ are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels $D\bar D^*$ are described with the string breaking mechanism. In SCA the transition rate $\tilde{\Gamma}_2=\Gamma(2\,{}^3P_1 \rightarrow \psi\gamma)=71.8$~keV and large $\tilde{\Gamma}_1=\Gamma(2\,{}^3P_1\rightarrow J/\psi\gamma)=85.4$~keV are obtained, giving for their ratio the value $\tilde{R_{\psi\gamma}}=\frac{\tilde{\Gamma}_2}{\tilde{\Gamma}_1}=0.84$. In the CC approach three factors are shown to be equally important. First, the admixture of the $1\,{}^3P_1$ component in the normalized wave function of $X(3872)$ due to the CC effects. Its weight $c_{\rm X}(E_{\rm R})=0.200\pm 0.015$ is calculated. Secondly, the use of the multipole function $g(r)$ instead of $r$ in the overlap integrals, determining the partial widths. Thirdly, the choice of the gluon-exchange interaction for $X(3872)$, as well as for other states above threshold. If for $X(3872)$ the gluon-exchange potential is taken the same as for low-lying charmonium states, then in the CC approach $\Gamma_1= \Gamma(X(3872)\rightarrow J/\psi\gamma) \sim 3$~keV is very small, giving the large ratio $R_{\psi\gamma}=\frac{\mathcal{B}(X(3872)\rightarrow \psi(2S)\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi\gamma)}\gg 1.0$. Arguments are presented why the gluon-exchange interaction may be suppressed for $X(3872)$ and in this case $\Gamma_1=42.7$~keV, $\Gamma_2= 70.5$~keV, and $R_{\psi\gamma}=1.65$ are predicted for the minimal value $c_{\rm X}({\rm min})=0.185$, while for the maximal value $c_{\rm X}=0.215$ we obtained $\Gamma_1=30.8$~keV, $\Gamma_2=73.2$~keV, and $R_{\psi\gamma}=2.38$, which agrees with the LHCb data., Comment: 12 pages, no figures

Radiative decays of $X(3872)$ are studied in single-channel approximation (SCA) and in the coupled-channel (CC) approach, where the decay channels $D\bar D^*$ are described with the string breaking mechanism. In SCA the transition rate $\tilde{\Gamma}_2=\Gamma(2\,{}^3P_1 \rightarrow \psi\gamma)=71.8$~keV and large $\tilde{\Gamma}_1=\Gamma(2\,{}^3P_1\rightarrow J/\psi\gamma)=85.4$~keV are obtained, giving for their ratio the value $\tilde{R_{\psi\gamma}}=\frac{\tilde{\Gamma}_2}{\tilde{\Gamma}_1}=0.84$. In the CC approach three factors are shown to be equally important. First, the admixture of the $1\,{}^3P_1$ component in the normalized wave function of $X(3872)$ due to the CC effects. Its weight $c_{\rm X}(E_{\rm R})=0.200\pm 0.015$ is calculated. Secondly, the use of the multipole function $g(r)$ instead of $r$ in the overlap integrals, determining the partial widths. Thirdly, the choice of the gluon-exchange interaction for $X(3872)$, as well as for other states above threshold. If for $X(3872)$ the gluon-exchange potential is taken the same as for low-lying charmonium states, then in the CC approach $\Gamma_1= \Gamma(X(3872)\rightarrow J/\psi\gamma) \sim 3$~keV is very small, giving the large ratio $R_{\psi\gamma}=\frac{\mathcal{B}(X(3872)\rightarrow \psi(2S)\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi\gamma)}\gg 1.0$. Arguments are presented why the gluon-exchange interaction may be suppressed for $X(3872)$ and in this case $\Gamma_1=42.7$~keV, $\Gamma_2= 70.5$~keV, and $R_{\psi\gamma}=1.65$ are predicted for the minimal value $c_{\rm X}({\rm min})=0.185$, while for the maximal value $c_{\rm X}=0.215$ we obtained $\Gamma_1=30.8$~keV, $\Gamma_2=73.2$~keV, and $R_{\psi\gamma}=2.38$, which agrees with the LHCb data., Comment: 12 pages, no figures

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted., Comment: 21 pages, 5 figures, revtex4

Radiative decays of X(3872) with $J^{PC}=1^{++}$ are studied in the coupled-channel approach, where the $c\bar c$ states are described by relativistic string Hamiltonian, while for the decay channels $DD^*$ a string breaking mechanism is used. Within this method a sharp peak and correct mass shift of the $2 {}^3P_1$ charmonium state just to the $D^0D^{*0}$ threshold was already obtained for a prescribed channel coupling to the $DD^*$ decay channels. For the same value of coupling the normalized wave function (w.f.) of X(3872) acquires admixture of the $1 {}^3P_1$ component with the w.f. fraction $c_1=0.153 (\theta=8.8^\circ$), which increases the transition rate $\Gamma(X(3872)\rightarrow J/\psi\gamma)$ up to 50-70 keV, making the ratio $R=\frac{\mathcal{B}(X(3872)\rightarrow \psi^{\prime}\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi \gamma)}=0.8\pm 0.20 (th)$ significantly smaller, as compared to $R\simeq 5$ for X(3872) as a purely $2 {}^3P_1$ state., Comment: 14 pages,2 Tables

Radiative decays of X(3872) with $J^{PC}=1^{++}$ are studied in the coupled-channel approach, where the $c\bar c$ states are described by relativistic string Hamiltonian, while for the decay channels $DD^*$ a string breaking mechanism is used. Within this method a sharp peak and correct mass shift of the $2 {}^3P_1$ charmonium state just to the $D^0D^{*0}$ threshold was already obtained for a prescribed channel coupling to the $DD^*$ decay channels. For the same value of coupling the normalized wave function (w.f.) of X(3872) acquires admixture of the $1 {}^3P_1$ component with the w.f. fraction $c_1=0.153 (\theta=8.8^\circ$), which increases the transition rate $\Gamma(X(3872)\rightarrow J/\psi\gamma)$ up to 50-70 keV, making the ratio $R=\frac{\mathcal{B}(X(3872)\rightarrow \psi^{\prime}\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi \gamma)}=0.8\pm 0.20 (th)$ significantly smaller, as compared to $R\simeq 5$ for X(3872) as a purely $2 {}^3P_1$ state., Comment: 14 pages,2 Tables

We show that there are two reasons why the partial width for the transition $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))$ is suppressed. Firstly, the spin-averaged matrix element (m.e.) $\bar{I(3S|r|1P_J)}$ is small, being equal to 0.023 GeV$^{-1}$ in our relativistic calculations. Secondly, the spin-orbit splittings produce relatively large contributions, giving $I(3S|r|1P_2)=0.066$ GeV$^{-1}$, while due to large cancellation the m.e. $I(3S|r|1P_1)=-0.020$ GeV$^{-1}$ is small and negative; at the same time the magnitude of $I(3S|r|1P_0)=-0.063$ GeV$^{-1}$ is relatively large. These m.e. give rise to the partial widths: $\Gamma_2(\Upsilon(3S)\rightarrow \gamma\chi_{b2}(1P))=212$ eV, $\Gamma_0(\Upsilon(3S)\rightarrow \gamma\chi_{b0}(1P))=54$ eV, which are in good agreement with the CLEO and BaBar data, and also to $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))=13$ eV, which satisfies the BaBar limit, $\Gamma_1(exp.) < 22$ eV., Comment: 8 pages

Magnetic moments of charged and neutral mesons are calculated with the use of the relativistic Hamiltonian derived from the path integral form of the $q_1\bar q_2$ Green's function. The magnetic moments are shown to be expressed via the average quark energies which are defined by the fundamental quantities: the string tension $\sigma$, the current quark masses, and the strong coupling constant $\alpha_s$. Resulting values for vector, axial, and tensor light and $K$ mesons agree well with all available lattice data., Comment: 17 pages,5 tables

Theory of strong decays defines in addition to decay widths, also the channel coupling and the mass shifts of the levels above the decay thresholds. In the standard decay models of the 3P0 type the decay vertex is taken to be a phenomenological constant "gamma" and such a choice leads to large mass shifts of all meson levels due to real and virtual decays, the latter giving a divergent contribution. Here we show that taking the microscopic details of decay vertex into account, one obtains new string width coefficient, which strongly suppresses virtual decay contribution. In addition for a realistic space structure of the decay vertex of highly excited states, the decay matrix elements appear to be strongly different from those, where the constant "gamma" is used. From our analysis also follows that so-called flattening potential can imitate the effects of intermediate decay channels., Comment: 21 pages, 8 figures, typos corrected, to be published in Phys.Atom.Nucl

Radiative decays of X(3872) with $J^{PC}=1^{++}$ are studied in the coupled-channel approach, where the $c\bar c$ states are described by relativistic string Hamiltonian, while for the decay channels $DD^*$ a string breaking mechanism is used. Within this method a sharp peak and correct mass shift of the $2 {}^3P_1$ charmonium state just to the $D^0D^{*0}$ threshold was already obtained for a prescribed channel coupling to the $DD^*$ decay channels. For the same value of coupling the normalized wave function (w.f.) of X(3872) acquires admixture of the $1 {}^3P_1$ component with the w.f. fraction $c_1=0.153 (\theta=8.8^\circ$), which increases the transition rate $\Gamma(X(3872)\rightarrow J/\psi\gamma)$ up to 50-70 keV, making the ratio $R=\frac{\mathcal{B}(X(3872)\rightarrow \psi^{\prime}\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi \gamma)}=0.8\pm 0.20 (th)$ significantly smaller, as compared to $R\simeq 5$ for X(3872) as a purely $2 {}^3P_1$ state., Comment: 14 pages,2 Tables

Theory of strong decays defines in addition to decay widths, also the channel coupling and the mass shifts of the levels above the decay thresholds. In the standard decay models of the 3P0 type the decay vertex is taken to be a phenomenological constant "gamma" and such a choice leads to large mass shifts of all meson levels due to real and virtual decays, the latter giving a divergent contribution. Here we show that taking the microscopic details of decay vertex into account, one obtains new string width coefficient, which strongly suppresses virtual decay contribution. In addition for a realistic space structure of the decay vertex of highly excited states, the decay matrix elements appear to be strongly different from those, where the constant "gamma" is used. From our analysis also follows that so-called flattening potential can imitate the effects of intermediate decay channels., Comment: 21 pages, 8 figures, typos corrected, to be published in Phys.Atom.Nucl

Magnetic moments of charged and neutral mesons are calculated with the use of the relativistic Hamiltonian derived from the path integral form of the $q_1\bar q_2$ Green's function. The magnetic moments are shown to be expressed via the average quark energies which are defined by the fundamental quantities: the string tension $\sigma$, the current quark masses, and the strong coupling constant $\alpha_s$. Resulting values for vector, axial, and tensor light and $K$ mesons agree well with all available lattice data., Comment: 17 pages,5 tables

We show that there are two reasons why the partial width for the transition $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))$ is suppressed. Firstly, the spin-averaged matrix element (m.e.) $\bar{I(3S|r|1P_J)}$ is small, being equal to 0.023 GeV$^{-1}$ in our relativistic calculations. Secondly, the spin-orbit splittings produce relatively large contributions, giving $I(3S|r|1P_2)=0.066$ GeV$^{-1}$, while due to large cancellation the m.e. $I(3S|r|1P_1)=-0.020$ GeV$^{-1}$ is small and negative; at the same time the magnitude of $I(3S|r|1P_0)=-0.063$ GeV$^{-1}$ is relatively large. These m.e. give rise to the partial widths: $\Gamma_2(\Upsilon(3S)\rightarrow \gamma\chi_{b2}(1P))=212$ eV, $\Gamma_0(\Upsilon(3S)\rightarrow \gamma\chi_{b0}(1P))=54$ eV, which are in good agreement with the CLEO and BaBar data, and also to $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))=13$ eV, which satisfies the BaBar limit, $\Gamma_1(exp.) < 22$ eV., Comment: 8 pages

Radiative decays of X(3872) with $J^{PC}=1^{++}$ are studied in the coupled-channel approach, where the $c\bar c$ states are described by relativistic string Hamiltonian, while for the decay channels $DD^*$ a string breaking mechanism is used. Within this method a sharp peak and correct mass shift of the $2 {}^3P_1$ charmonium state just to the $D^0D^{*0}$ threshold was already obtained for a prescribed channel coupling to the $DD^*$ decay channels. For the same value of coupling the normalized wave function (w.f.) of X(3872) acquires admixture of the $1 {}^3P_1$ component with the w.f. fraction $c_1=0.153 (\theta=8.8^\circ$), which increases the transition rate $\Gamma(X(3872)\rightarrow J/\psi\gamma)$ up to 50-70 keV, making the ratio $R=\frac{\mathcal{B}(X(3872)\rightarrow \psi^{\prime}\gamma)}{\mathcal{B}(X(3872)\rightarrow J/\psi \gamma)}=0.8\pm 0.20 (th)$ significantly smaller, as compared to $R\simeq 5$ for X(3872) as a purely $2 {}^3P_1$ state., Comment: 14 pages,2 Tables

The masses of higher $D(nL)$ and $D_s(nL)$ excitations are shown to decrease due to the string contribution, originating from the rotation of the QCD string itself: it lowers the masses by 45 MeV for $L=2 (n=1)$ and by 65 MeV for $L=3 (n=1)$. An additional decrease $\sim 100$ MeV takes place if the current mass of the light (strange) quark is used in a relativistic model. For $D_s(1\,{}^3D_3)$ and $D_s(2P_1^H)$ the calculated masses agree with the experimental values for $D_s(2860)$ and $D_s(3040)$, and the masses of $D(2\,{}^1S_0)$, $D(2\,{}^3S_1)$, $D(1\,{}^3D_3)$, and $D(1D_2)$ are in agreement with the new BaBar data. For the yet undiscovered resonances we predict the masses $M(D(2\,{}^3P_2))=2965$ MeV, $M(D(2\,{}^3P_0))=2880$ MeV, $M(D(1\,{}^3F_4))=3030$ MeV, and $M(D_s(1\,{}^3F_2))=3090$ MeV. We show that for $L=2,3$ the states with $j_q=l+1/2$ and $j_q=l-1/2$ ($J=l$) are almost completely unmixed ($\phi\simeq -1^\circ$), which implies that the mixing angles $\theta$ between the states with S=1 and S=0 ($J=L$) are $\theta\approx 40^\circ$ for L=2 and $\approx 42^\circ$ for L=3., Comment: 22 pages, no figures, 4 tables Two references and corresponding discussion added

The masses of higher $D(nL)$ and $D_s(nL)$ excitations are shown to decrease due to the string contribution, originating from the rotation of the QCD string itself: it lowers the masses by 45 MeV for $L=2 (n=1)$ and by 65 MeV for $L=3 (n=1)$. An additional decrease $\sim 100$ MeV takes place if the current mass of the light (strange) quark is used in a relativistic model. For $D_s(1\,{}^3D_3)$ and $D_s(2P_1^H)$ the calculated masses agree with the experimental values for $D_s(2860)$ and $D_s(3040)$, and the masses of $D(2\,{}^1S_0)$, $D(2\,{}^3S_1)$, $D(1\,{}^3D_3)$, and $D(1D_2)$ are in agreement with the new BaBar data. For the yet undiscovered resonances we predict the masses $M(D(2\,{}^3P_2))=2965$ MeV, $M(D(2\,{}^3P_0))=2880$ MeV, $M(D(1\,{}^3F_4))=3030$ MeV, and $M(D_s(1\,{}^3F_2))=3090$ MeV. We show that for $L=2,3$ the states with $j_q=l+1/2$ and $j_q=l-1/2$ ($J=l$) are almost completely unmixed ($\phi\simeq -1^\circ$), which implies that the mixing angles $\theta$ between the states with S=1 and S=0 ($J=L$) are $\theta\approx 40^\circ$ for L=2 and $\approx 42^\circ$ for L=3., Comment: 22 pages, no figures, 4 tables Two references and corresponding discussion added

The masses of higher $D(nL)$ and $D_s(nL)$ excitations are shown to decrease due to the string contribution, originating from the rotation of the QCD string itself: it lowers the masses by 45 MeV for $L=2 (n=1)$ and by 65 MeV for $L=3 (n=1)$. An additional decrease $\sim 100$ MeV takes place if the current mass of the light (strange) quark is used in a relativistic model. For $D_s(1\,{}^3D_3)$ and $D_s(2P_1^H)$ the calculated masses agree with the experimental values for $D_s(2860)$ and $D_s(3040)$, and the masses of $D(2\,{}^1S_0)$, $D(2\,{}^3S_1)$, $D(1\,{}^3D_3)$, and $D(1D_2)$ are in agreement with the new BaBar data. For the yet undiscovered resonances we predict the masses $M(D(2\,{}^3P_2))=2965$ MeV, $M(D(2\,{}^3P_0))=2880$ MeV, $M(D(1\,{}^3F_4))=3030$ MeV, and $M(D_s(1\,{}^3F_2))=3090$ MeV. We show that for $L=2,3$ the states with $j_q=l+1/2$ and $j_q=l-1/2$ ($J=l$) are almost completely unmixed ($\phi\simeq -1^\circ$), which implies that the mixing angles $\theta$ between the states with S=1 and S=0 ($J=L$) are $\theta\approx 40^\circ$ for L=2 and $\approx 42^\circ$ for L=3., Comment: 22 pages, no figures, 4 tables Two references and corresponding discussion added

Hyperfine splittings (HFS) are calculated within the Field Correlator Method, taking into account relativistic corrections. The HFS in bottomonium and the $B_q$ (q=n,s) mesons are shown to be in full agreement with experiment if a universal coupling $\alpha_{HF}=0.310$ is taken in perturbative spin-spin potential. It gives $M(B^*)-M(B)=45.7(3)$ MeV, $M(B_s^*)-M(B_s)=46.7(3)$ MeV ($n_f=4$), while in bottomonium $\Delta_{HF}(b\bar b)=M(\Upsilon(9460))-M(\eta_b(1S))=63.4$ MeV for $n_f=4$ and 71.1 MeV for $n_f=5$ are obtained; just latter agrees with recent BaBar data. For unobserved excited states we predict $M(\Upsilon(2S))-M(\eta_b(2S))=36(2)$ MeV, $M(\Upsilon(3S))-M(\eta(3S))=28(2)$ MeV, and also $M(B_c^*)=6334(4)$ MeV, $M(B_c(2S))=6868(4)$ MeV, $M(B_c^*(2S))=6905(4)$ MeV. The mass splittings between $D(2^3S_1)-D(2^1S_0)$, $D_s(2^3S_1)-D_s(2^1S_0)$ are predicted to be $\sim 70$ MeV, which are significantly smaller than in several other studies., Comment: 13 pages