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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: LaTeX, 18 pages, 0 figures, 7 tables; v2: new Eq.(2) and Appendices A and B are added

\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 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 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 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

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

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

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 $\psi((n+1) {}^3S_1)$ and $\psi(n {}^3D_1)$ are calculated using the relativistic string Hamiltonian with "linear+gluon-exchange" potential. They occur in the range 4.5-5.8 GeV, in particular, $M(3D)=4.54$ GeV, $M(5S)=4.79$ GeV, $M(4D)=4.85$ GeV are calculated with accuracy $\sim 50$ MeV. Linear Regge trajectories: $M^2(nS)=M^2(\psi(4.42))+ 2.91$ GeV$^2 (n-4)$ $(n\geq 4) $ and $M^2(nD)=(4.54^2+ 2.88 (n-3))$ GeV$^2$ ($n\geq 3$) are obtained only for higher charmonium excitations. They have slopes two times larger than those of light mesons and give good description of calculated masses. These masses are compared with enhancements in some recent $e^+e^-$ experiments., Comment: LaTeX, 15 pages.References and short comments added. Invited paper to the special issue of Yadernaya Fizika, dedicated to the centennial anniversary of the birthday of A.B.Migdal

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

A universal description of the hyperfine splittings (HFS) in bottomonium and the $B_q (q=n,s,c)$ mesons is obtained with a universal strong coupling constant $\alpha_s(\mu)=0.305(2)$ in a spin-spin potential. Other characteristics are calculated within the Field Correlator Method, taking the freezing value of the strong coupling independent of $n_f$. The HFS $M(B^*)- M(B)=45.3(3)$ MeV, $M(B_s^*) - M(B_s)=46.5(3)$ MeV are obtained in full agreement with experiment both for $n_f=3$ and $n_f=4$. In bottomonium, $M(\Upsilon(9460))- M(\eta_b)=70.0(4)$ MeV for $n_f=5$ agrees with the BaBar data, while a smaller HFS, equal to 64(1) MeV, is obtained for $n_f=4$. We predict HFS $M(\Upsilon(2S))-M(\eta_b(2S))=36(1)$ MeV, $M(\Upsilon(3S))- M(\eta(3S))=27(1)$ MeV, and $M(B_c^*) - M(B_c)= 57.5(10)$ MeV, which gives $M(B_c^*)=6334(1)$ MeV, $M(B_c(2 {}^1S_0))=6865(5)$ MeV, and $M(B_c^*(2S {}^3S_1))=6901(5)$ MeV., Comment: 5 pages revtex4

The dielectron widths of $\Upsilon(nS) (n=1,...,7)$ and vector decay constants are calculated using the Relativistic String Hamiltonian with a universal interaction. For $\Upsilon(nS) (n=1,2,3)$ the dielectron widths and their ratios are obtained in full agreement with the latest CLEO data. For $\Upsilon(10580)$ and $\Upsilon(11020)$ a good agreement with experiment is reached only if the 4S--3D mixing (with a mixing angle $\theta=27^\circ\pm 4^\circ$) and 6S--5D mixing (with $\theta=40^\circ\pm 5^\circ$) are taken into account. The possibility to observe higher "mixed $D$-wave" resonances, $\tilde\Upsilon(n {}^3D_1)$ with $n=3,4,5$ is discussed. In particular, $\tilde\Upsilon(\approx 11120)$, originating from the pure $5 {}^3D_1$ state, can acquire a rather large dielectron width, $\sim 130$ eV, so that this resonance may become manifest in the $e^+e^-$ experiments. On the contrary, the widths of pure $D$-wave states are very small, $\Gamma_{ee}(n{}^3 D_1) \leq 2$ eV., Comment: 13 pages, no figures

The possibility to observe new bottomonium states with $J^{PC}= 1^{--}$ in the region $10.7-11.1$ GeV is discussed. The analysis of the di-electron widths shows that the $(n+1)^3S_1$ and $n^3D_1$ states ($n\geq 3$) may be mixed with a rather large mixing angle, $\theta\sim30^\circ$ and this effect provides the correct values of $\Gamma_{ee}(\Upsilon(10580))$ and $\Gamma_{ee}(\Upsilon(11020))$. On the other hand, the $S-D$ mixing gives rise to an increase by two orders of magnitude of the di-electron widths of the mixed $\tilde\Upsilon(n^3D_1$) resonances ($n=3,4,5$), which originate from pure $D-$wave states. The value $\Gamma_{ee}(\tilde\Upsilon(3D))=0.095^{+0.028}_{-0.025}$ keV is obtained, being only $\sim 3$ times smaller than the di-electron width of $\Upsilon(10580)$, while $\Gamma_{ee}(\tilde\Upsilon(5D))\sim 135$ eV appears to be close to $\Gamma_{ee}(\Upsilon(11020))$ and therefore this resonance may become manifest in the $e^+e^-$ experiments. The mass differences between $M(nD)$ and $M((n+1)S) (n=4,5)$ are shown to be rather small, $50\pm 10$ MeV., Comment: 5 pages, no figures

The spin-dependent interactions in mesons are considered in detail in the framework of the Field Correlator Method. Analytic expressions for the spin-dependent potentials in heavy and light quarkonia are derived with the QCD string moment of inertia taken into account. Recent lattice data are analysed using these formulae and the data are shown to be consistent with very small values of the gluonic correlation length, <0.1 fm. The Field Correlator and the Eichten-Feinberg definitions of the spin-dependent potentials in the lowest, Gaussian approximation for the QCD vacuum are compared to one another and the two approaches are shown to be equivalent in the limit of a vanishing vacuum correlation length, whereas for finite values of the latter the difference between these two approaches can be explained by the contribution of the higher-order field corelators, starting from the quartic one., Comment: RevTeX4, 37 pages, to appear in Phys.Rev.D

The mass shifts of the $P$-wave $D_s$ and $B_s$ mesons due to coupling to $DK$ and $BK$ channels are calculated in the coupling channel model without fitting parameters. The strong mass shifts down for $0^+$ and ${1^+}'$ states have been obtained, while ${1^+}"$ and $2^+$ states remain almost in situ. The masses of $0^+$ and ${1^+}'$ states of $B_s$ mesons have been predicted., Comment: to be published in the Proceedings of the 14th International QCD Conference, 7th-12th July 2008, Montpellier, France

In the framework of the coupled channel model the mass shifts of the $P$--wave excitations of $D_s$ and $B_s$ mesons have been calculated. The corresponding coupling to $DK$ and $BK$ channels is provided by the effective chiral Lagrangian which is deduced from QCD and does not contain fitting parameters. The strong mass shifts down for $0^+$ and ${1^+}'$ states have been obtained, while ${1^+}"$ and $2^+$ states remain almost at rest. Two factors are essential for large mass shifts: strong coupling of the $0^+$ and $1^{+'}$ states to the $S$-wave decay channel, containing a Nambu-Goldstone meson, and the chiral flip transitions due to the bispinor structure of both heavy-light mesons. The masses $M(B^*_s(0^+))=5710(15)$ MeV and $M(B_s(1^{+'}))=5730(15)$ MeV are predicted. Experimental limit on the width $\Gamma(D_{s1}(2536))<2.3$ MeV puts strong restrictions on admittable mixing angle between the $1^+$ and $1^{+'}$ states., Comment: to be published in the Proceedings of the Conference QUARKS-2008, Sergiev Posad, Russia, 23rd-29th May 2008

The vacuum gluonic correlation length is extracted from recent lattice data on spin-dependent interquark potentials in heavy quarkonia. It is shown that the data are consistent with extremely small values of the correlation length, Tg<0.1 fm., Comment: LaTeX2e, 6 pages, uses jetpl.cls (included), version to appear in JETP Lett

Spin-dependent interaction in hadrons is considered in the approach of the QCD string. The string moment of inertia, which ensures the correct (inverse) Regge slope 2*pi*sigma, is found to suppress the spin-orbit interaction. For light quarks and moderate angular momenta the suppression constitutes around 25% whereas for large angular momenta the spin-orbit interaction is suppressed by the factor L^(-2/5). For heavy quarks the effect manifests itself as a string correction for the spin-dependent potential., Comment: LaTeX2e, 6 pages, uses jetpl.cls, version to appear in JETP Lett

The di-electron widths of $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$, and their ratios are shown to be in good agreement with experiment, if in all cases the $S-D$ mixing with a large mixing angle $\theta\approx 34^\circ$ is taken. Arguments are presented why continuum states give small contributions to the wave functions at the origin. We find that the Y(4360) resonance, considered as a pure $3 {}^3D_1$ state, would have very small di-electron width, $\Gamma_{ee}(Y(4360))=0.060$ keV. On the contrary, for large mixing between the $4 {}^3S_1$ and $3 {}^3D_1$ states with the mixing angle $\theta=34.8^\circ$, $\Gamma_{ee}(\psi(4415))=0.57$ keV coincides with the experimental number, while a second physical resonance, probably Y(4360), has also a rather large $\Gamma_{ee} (Y(\sim 4400))=0.61$ keV. For the higher resonance Y(4660), considered as a pure $5 {}^3S_1$ state, we predict the di-electron width $\Gamma_{ee}(Y(4660))=0.70$ keV, but it becomes significantly smaller, namely 0.31 keV, if the mixing angle between the $5 {}^3S_1$ and $4 {}^3D_1$ states $\theta=34^\circ$. The mass and di-electron width of the $6 {}^3S_1$ charmonium state are calculated., Comment: 19 pages, no figures

The vector and pseudoscalar decay constants are calculated in the framework of the Field Correlator Method. Di-electron widths: $\Gamma_{ee}(J/\psi)=5.41$ keV, $\Gamma_{ee}(\psi'(3686))=2.47$ keV, $\Gamma_{ee}(\psi''(3770))=0.248$ keV, in good agreement with experiment, are obtained with the same coupling, $\alpha_s=0.165$, in QCD radiative corrections. We show that the larger $\alpha_s=0.191\pm 0.004$ is needed to reach agreement with experiment for $\Gamma_{\gamma\gamma}(\eta_c)=7.22$ keV, $\Gamma_{\gamma\gamma} (\chi(^3P_0))=3.3$ keV, $\Gamma_{\gamma\gamma}(\chi(^3P_2))= 0.54$ keV, and also for $\Gamma(J/\psi\to 3g)=59.5$ keV, $\Gamma(J/\psi\to \gamma 2g)=5.7$ keV. Meanwhile even larger $\alpha_s=0.238$ gives rise to good description of $\Gamma(\psi'\to 3g)=52.7$ keV, $\Gamma(\psi'\to \gamma 2g)= 3.5$ keV, and provides correct ratio of the branching fractions: $\frac{\mathcal{B}(J/\psi\to light hadrons)}{\mathcal{B}(\psi'\to light hadrons)}=0.24.$, Comment: 8 pages, no figures

The mass shifts of the $P$-wave $D_s$ and $B_s$ mesons due to coupling to $DK$, $D^*K$ and $BK$, $B^*K$ channels are studied using the chiral quark-pion Lagrangian without fitting parameters. The strong mass shifts down $\sim 140$ MeV and $\sim 100$ MeV for $D^*_s(0^+)$ and $D_s(1^{+'})$ and $\sim 100$ MeV for $B^*_s(0^+)$ and $B_s(1^{+'})$ are calculated. Two factors are essential for large mass shifts: strong coupling of the $0^+$ and $1^{+'}$ states to the $S$-wave decay channel, containing a Nambu-Goldstone meson, and the chiral flip transitions due to the bispinor structure of both heavy-light mesons. The masses $M(B^*_s(0^+))=5695(10)$ MeV and $M(B_s(1^{+'}))=5730(15)$ MeV,very close to $M(B(0^+))$ and $M(B(1^{+'}))$, are predicted. Experimental limit on the width $\Gamma(D_{s1}(2536))<2.3$ MeV puts strong restrictions on admittable mixing angle between the $1^+$ and $1^{+'}$ states, $|\phi|<6^{\circ}$, which corresponds to the mixing angle $\theta$ between the $^3P_1$ and $^1P_1$ states, $29^{\circ}<\theta< 41^{\circ}$., Comment: 30 pages, 11 tables, 7 figures; ReVTeX 4; submitted to Phys. Rev. D

It is shown that a 25(20)% difference between the decay constants $f_{D_s}(f_{B_s})$ and $f_D(f_B)$ occurs due to large differences in the pole masses of the $s$ and $d(u)$ quarks. The values $\eta_D =f_{D_s}/f_D\approx 1.23(15)$, recently observed in the CLEO experiment, and $\eta_B=f_{B_s}/f_B\approx 1.20$, obtained in unquenched lattice QCD, can be reached only if the running mass $m_s$ at low scale is $m_s(\sim 0.5 $ GeV)$= 170 - 200$ MeV. Our results follow from the analytical expression for the pseudoscalar decay constant $f_{\rm P}$ based on the path-integral representation of the meson Green's function., Comment: 6 pages, no figures; revtex4

Meson Green's functions and decay constants $f_{\Gamma}$ in different channels $\Gamma$ are calculated using the Field Correlator Method. Both, spectrum and $f_\Gamma$, appear to be expressed only through universal constants: the string tension $\sigma$, $\alpha_s$, and the pole quark masses. For the $S$-wave states the calculated masses agree with the experimental numbers within $\pm 5$ MeV. For the $D$ and $D_s$ mesons the values of $f_{\rm P} (1S)$ are equal to 210(10) and 260(10) MeV, respectively, and their ratio $f_{D_s}/f_D$=1.24(3) agrees with recent CLEO experiment. The values $f_{\rm P}(1S)=182, 216, 438$ MeV are obtained for the $B$, $B_s$, and $B_c$ mesons with the ratio $f_{B_s}/f_B$=1.19(2) and $f_D/f_B$=1.14(2). The decay constants $f_{\rm P}(2S)$ for the first radial excitations as well as the decay constants $f_{\rm V}(1S)$ in the vector channel are also calculated. The difference of about 20% between $f_{D_s}$ and $f_D$, $f_{B_s}$ and $f_B$ directly follows from our analytical formulas., Comment: 37 pages, 10 tables, RevTeX4

The relativistic string Hamiltonian is used to calculate the masses and decay constants of $B_q$ mesons: they appear to be expressed through onlythree fundamental values: the string tension $\sigma $, $\alpha_s$, and the quark pole masses. The values $f_B =186 $ MeV, $f_{B_s}= 222$ MeV are calculated while $f_{B_c}$ depends on the $c$-quark pole mass used, namely $f_{B_c}=440 (424)$ MeV for $m_c =1.40 (1.35)$ GeV. For the $1P$ states we predict the spin-averaged masses: $\bar M(B_J)=5730$ MeV and $\bar M(B_{sJ})=5830$ MeV which are in good agreement with the recent data of the D0 and CDF Collaborations, at the same time owning to the string correction being by $\sim 50$ MeV smaller than in other calculations., Comment: 9 pages, ICHEP 2006

The hyperfine splittings in heavy quarkonia are studied using new experimental data on the di-electron widths. The smearing of the spin-spin interaction is taken into account, while the radius of smearing is fixed by the known $J/\psi-\eta_c(1S)$ and $\psi(2S)-\eta'_c(2S)$ splittings and appears to be small, $r_{ss} \approx 0.06$ fm. Nevertheless, even with such a small radius an essential suppression of the hyperfine splittings ($\sim 50%)$ is observed in bottomonium. For the $nS b\bar b$ states $(n=1,2,...6)$ the values we predict (in MeV) are 28, 12, 10, 6, 6, and 3, respectively. In single-channel approximation for the $3S$ and $4S$ charmonium states the splittings 16(2) MeV and 12(4) MeV are obtained., Comment: 13 pages, no figures

The hyperfine splittings in heavy quarkonia are studied in a model-independent way using the experimental data on di-electron widths. Relativistic correlations are taken into account together with the smearing of the spin-spin interaction. The radius of smearing is fixed by the known $J/\psi-\eta_c(1S)$ and $\psi(2S)-\eta'_c(2S)$ splittings and appears to be small, $r_{ss} \cong 0.06$ fm. Nevertheless, even with such a small radius an essential suppression of the hyperfine splittings ($\sim 50%)$ is observed in bottomonium. For the $nS~ b\bar b$ states $(n=1,2,...,6)$ we predict the values (in MeV) 28, 12, 10, 6, 6, and 3, respectively. For the $3S$ and $4S$ charmonium states the splittings 16(2) MeV and 12(4) MeV are obtained., Comment: 23 pages

The precision measurement of the hyperfine splitting $\Delta_{\rm HF} (1P, c\bar c)=M_{\rm cog} (\chi_{cJ}) - M(h_c) = -0.5 \pm 0.4$ MeV in the Fermilab--E835 experiment allows to determine the gluonic condensate $G_2$ with high accuracy if the gluonic correlation length $T_g$ is fixed. In our calculations the negative value of $\Delta_{\rm HF} = -0.3 \pm 0.4$ MeV is obtained only if the relatively small $T_g = 0.16$ fm and $G_2 = 0.065 (3)$ GeV${}^4$ are taken. These values correspond to the ``physical'' string tension $(\sigma \approx 0.18 $ GeV$^2$). For $T_g \ge 0.2$ fm the hyperfine splitting is positive and grows for increasing $T_g$. In particular for $T_g = 0.2$ fm and $G_2 = 0.041 (2)$ GeV${}^4$ the splitting $\Delta_{\rm HF} = 1.4 (2)$ MeV is obtained, which is in accord with the recent CLEO result., Comment: 9 pages revtex 4, no figures

Agreement with the measured electronic widths of the $\psi(4040)$, $\psi(4415)$, and $\Upsilon (11019)$ resonances is shown to be reached if two effects are taken into account: a flattening of the confining potential at large distances and a total screening of the gluon-exchange interaction at $r\ga 1.2$ fm. The leptonic widths of the unobserved $\Upsilon(7S)$ and $\psi(5S)$ resonances: $\Gamma_{e^+e^-}(\Upsilon (7S))=0.11$ keV and $\Gamma(\psi(5S))\approx 0.54$ keV are predicted., Comment: 11 pages revtex4

The splittings $1D-1P, 1F-1P, 2D-2P$ are shown to be the most convenient characteristics to determine the vector coupling in IR region. In background perturbation theory the splitting $\Delta =1D -1P$ appears to be in agreement with $\Delta (\exp) =261.1 \pm 2.2$ MeV only for large freezing value, $\alpha_{crit} \cong 0.60$, which corresponds to $\Lambda_{\bar{MS}} (2$-loop, $n_f=5)\cong 240$ MeV $(\alpha_s(M_Z) =0.1193(2)$. The masses $M_{cog} (1F)=10362\pm 3$ MeV and $M_{cog} (2D) =10452\pm 4 $ MeV are predicted., Comment: 8 pages. The talk presented at the workshop "QUARKS-2004"

The static force $F_B(r)$ and the strong coupling $\alpha_F(r)$, which defines the gluon-exchange part of $F_B(r)$, are studied in QCD background perturbation theory (BPT). In the region $r\la 0.6 $ fm $\alpha_F(r)$ turns out to be essentially smaller than the coupling $\alpha_B(r)$ in the static potential. For the dimensionless function $\Phi_B(r) = r^2 F_B(r)$ the characteristic values $\Phi_B(r_1) =1.0$ and $\Phi_B(r_0)=1.65$ are shown to be reached at the following $Q\bar Q$ separations: $r_1\sqrt{\sigma} =0.77, r_0\sqrt{\sigma} =1.09$ in quenched approximation and $r_1\sqrt{\sigma}=0.72, r_0\sqrt{\sigma}=1.04$ for $n_f =3$. The numbers obtained appear to be by only 8% smaller than those calculated in lattice QCD while the values of the couplings $\alpha_F(r_1)$ and $\alpha_F(r_0)$ in BPT are by $\sim 30% (n_f =3)$ and $50% (n_f=0)$ larger than corresponding lattice couplings. With the use of the BPT potential good description of the bottomonium spectrum is obtained., Comment: 21 pages, 1 figure. To be publised in Phys. Atom. Nucl. dedicated to the 70-th birthday of Yu.A. Simonov

The $b\bar b$ spectrum is calculated with the use of a relativistic Hamiltonian where the gluon-exchange between a quark and an antiquark is taken as in background perturbation theory. We observed that the splittings $\Delta_1= \Upsilon({\rm 1D})-\chi_b({\rm 1P})$ and other splittings are very sensitive to the QCD constant $\Lambda_V(n_f)$ which occurs in the Vector scheme, and good agreement with the experimental data is obtained for $\Lambda_V(2$-loop, $n_f=5)= 325\pm 10$ MeV which corresponds to the conventional $\Lambda_{\bar{MS}} (2-$loop, $n_f=5)= 238\pm 7$ MeV, $\alpha_s(2-$loop, $M_Z)=0.1189\pm 0.0005,$ and a large freezing value of the background coupling: $\alpha_{\rm crit} (2$-loop, $q^2=0)=\alpha_{\rm crit} (2$-loop, $r\to \infty)=0.58\pm 0.02$. If the asymptotic freedom behavior of the coupling is neglected and an effective freezing coupling $\alpha_{\rm static}=const$ is introduced, as in the Cornell potential, then precise agreement with $\Delta_1({\rm exp})$ and $\Delta_2({\rm exp})$ can be reached for the rather large value $\alpha_{\rm static} =0.43\pm 0.02$. We predict a value for the mass M(2D) = 10451\pm2 MeV., Comment: 17 pages, no figures, 6 tables

From the fact that the nonperturbative self-energy contribution $C_{\rm SE}$ to the heavy meson mass is small: $C_{\rm SE}(b\bar{b})=0$; $C_{\rm SE}(c\bar{c})\cong -40$ MeV \cite{ref.01}, strong restrictions on the pole masses $m_b$ and $m_c$ are obtained. The analysis of the $b\bar{b}$ and the $c\bar{c}$ spectra with the use of relativistic (string) Hamiltonian gives $m_b$(2-loop)$=4.78\pm 0.05$ GeV and $m_c$(2-loop)$=1.39 \pm 0.06$ GeV which correspond to the $\bar{\rm MS}$ running mass $\bar{m}_b(\bar{m}_b)=4.19\pm 0.04$ GeV and $\bar{m}_c(\bar{m}_c)=1.10\pm 0.05$ GeV. The masses $\omega_c$ and $\omega_b$, which define the heavy quarkonia spin structure, are shown to be by $\sim 200$ MeV larger than the pole ones., Comment: 18 pages, no figures, 8 tables

The hyperfine splitting for the 2S charmonium state is calculated and the predicted number is $\Delta_{\rm HF}(2S) = 57 \pm 8$ MeV, being by derivation the lower bound of this splitting. It results in $M(\eta_c(2S))= 3630 \pm 8$ MeV, which is smaller by two standard deviations than found in the Belle experiment \cite{ref.01}, but close to the $\eta_c(2S)$ mass observed by the same group in the experiment $e^+e^- \to J/\psi \eta_c$ \cite{ref.06} where $M(\eta_c(2S)) = 3622 \pm 12$ MeV was found., Comment: 8 pages RevTex

For the $a_0(1450)$, considered as the $q\bar{q} 1^3P_0$ state, "experimental" tensor splitting, $c_{exp}=(-150\pm 40$) MeV, appears to be in contradiction with conventional theory of fine structure. There is no such discrepancy if the $a_0(980)$ belongs to the $1^3P_J q\bar{q}$ multiplet. The hadronic shift of the $a_0(980)$ is shown to be strongly dependent on the value of the strong coupling in spin-dependent interaction., Comment: 13 pages

Leading terms of the static quark-antiquark potential in the background perturbation theory are reviewed, including perturbative, nonperturbative and interference ones. The potential is shown to describe lattice data at short quark-antiquark separations with a good accuracy., Comment: 4 pages, 2 figures, talk at the NPD-2002 Conference, December 2-6, ITEP, Moscow, references updated

A new physical mechanism is suggested to explain the universal depletion of high meson excitations. It takes into account the appearance of holes inside the string world sheet due to $q\bar{q}$ pair creation when the length of the string exceeds the critical value $R_1 \simeq 1.4$ fm. It is argued that a delicate balance between large $N_c$ loop suppression and a favorable gain in the action, produced by holes, creates a new metastable (predecay) stage with a renormalized string tension which now depends on the separation r. This results in smaller values of the slope of the radial Regge trajectories, in good agreement with the analysis of experimental data in [Ref.3], Comment: 25 pages, 1 figure

In the framework of the QCD string approach it is shown that the spin-averaged masses $\bar{M}(nL)$ of all low-lying light mesons are well described using the string tension $\sigma$ as the only parameter. The Regge slope $\alpha'_L$ and the intercept $\alpha_L(0)$ of the Regge L-trajectory for $\bar{M}(nL)$ are calculated analytically and turn out to be $\alpha'_L = 0.80$ GeV${}^{-2}$ (for $L \leq 4$) and $\alpha_L(0) = -0.34$, in good agreement with the experimental data: $\alpha'_{L {\rm exp}} = 0.81 \pm 0.01$ GeV${}^{-2}$, $\alpha_{L {\rm exp}}(0) = -0.30 \pm 0.02$. To obtain this strong agreement with the data the nonperturbative quark self-energy contributions to the meson masses must be taken into account, which appeared to be large and negative for small values of L, and are even for larger values of L important for a close fit. From the present analysis of the meson spectra the restriction $\alpha_s \leq 0.40$ on the strong coupling is required., Comment: 18 pages RevTeX, 1 figure

Due to the nonperturbative contribution to the hyperfine splitting the mass of the $n^1P_1$ state is strongly correlated with the center of gravity $M_{\rm cog}(n^3P_J)$ of the $n^3P_J$ multiplet: $M(n^1P_1)$ is less than $M_{\rm cog}(n^3P_J)$ by about 40 MeV (20 MeV) for the 1P (2P) state. For $b_1(1235)$ the agreement with experiment is reached only if $a_0(980)$ belongs to the $1^3P_J$ multiplet. The predicted mass of $b_1(2^1P_1)$ is $\approx 1620$ MeV. For the isoscalar meson a correlation between the mass of $h_1$(1170) $(h_1(1380))$ and $M_{cog}(1^3P_J)$ composed from light (strange) quarks also takes place., Comment: 22 pages RevTeX

A striking contradiction between lattice short range static potential (nf=0) and standard perturbative potential, observed by Bali G.S., is investigated in the framework of the background perturbation theory. With the use of the background coupling which contains the only background parameter - mass mB, fixed by fine structure fit in bottomonium, the lattice data are nicely explained without introduction of exotic short range linear potential with large "string tension" approx. 1 GeV squared. A significant difference between the background coupling and standard perturbative coupling is found in the range 0.05 fm < r<0.15 fm, while at larger distances, r > 0.3 fm the background coupling fast approaches the freezing value. Some problems concerning the strong coupling properties at short and long distances are discussed and solutions are suggested., Comment: LaTeX2e, 17 pages, 6 PostScript figures, 1 GIF figure, final version to appear in Phys. Rev. D