Strong coupling constant from bottomonium fine structure

From a fit to the experimental data on the $b\bar{b}$ fine structure, the two-loop coupling constant is extracted. For the 1P state the fitted value is $\alpha_s(\mu_1) = 0.33 \pm 0.01(exp)\pm 0.02 (th)$ at the scale $\mu_1 = 1.8 \pm 0.1$ GeV, which corresponds to the QCD constant $\Lambda^{(4)}(2-loop) = 338 \pm 30$ MeV (n_f = 4) and $\alpha_s(M_Z) = 0.119 \pm 0.002. For the 2P state the value$\alpha_s(\mu_2) = 0.40 \pm 0.02(exp)\pm 0.02(th)$at the scale$\mu_2 = 1.02 \pm 0.2$GeV is extracted, which is significantly larger than in the previous analysis of Fulcher (1991) and Halzen (1993), but about 30smaller than the value given by standard perturbation theory. This value$\alpha_s(1.0) \approx 0.40$can be obtained in the framework of the background perturbation theory, thus demonstrating the freezing of$\alpha_s$. The relativistic corrections to$\alpha_s$ are found to be about 15%.Comment: 18 pages LaTe

Higher excitations of the DD and DsD_s mesons

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 adde

The leptonic widths of high ψ\psi-resonances in unitary coupled-channel model

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 adde

Dielectron widths of the S-, D-vector bottomonium states

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 figure

The Hyperfine Splittings in Heavy-Light Mesons and Quarkonia

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 page

The gluonic condensate from the hyperfine splitting Mcog(χcJ)−M(hc)M_{\rm cog}(\chi_{cJ})-M(h_c) in charmonium

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 figure

The ccˉc\bar c interaction above threshold and the radiative decay X(3872)→J/ψγX(3872)\rightarrow J/\psi\gamma

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 figure

Light meson radial Regge trajectories

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 figur

The Hyperfine Splittings in Bottomonium and the Bq(q=n,s,c)B_q (q=n,s,c) Mesons

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 revtex

The heavy-quark pole masses in the Hamiltonian approach

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 table