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The framework has been introduced in previous works [7, 25, 26]. It is recapitulated here for convenience. In the RL approximation, the quark propagator is solved by the Gap equation [27, 28]:
$ S_f^{-1}(k) = Z_2 ({\rm i}\gamma\cdot k + Z_m m_f) + \frac{4}{3} (Z_2)^2 \int^\Lambda_{{\rm d} q} \tilde{D}^f_{\mu\nu}(l)\gamma_\mu S_{f}(q)\gamma_\nu, $
(1) where
$ f = \{u,d,s,c,b,t\} $ represents the quark flavor.$ S_{f}(k) $ is the quark propagator, which can be decomposed as$ S_{f}(k) = \dfrac{Z_f(k^2)}{{\rm i}{\not{k}} + M_f(k^2)} $ .$ Z_f(k^2) $ is the quark dressing function and$ M_f(k^2) $ the quark mass function.$ l = k-q $ .$ m_f $ is the current quark mass.$ \tilde{D}^f_{\mu\nu} $ represents the effective interaction for the self energy of the f-quark.$ Z_2 $ and$ Z_m $ are the renormalisation constants of the quark field and the quark mass, respectively.$ \int^\Lambda_{{\rm d} q} = \int ^{\Lambda} {\rm d}^{4} q/(2\pi)^{4} $ stands for a Poincar$ \acute{\text{e}} $ invariant regularized integration, with the regularization scale Λ .A meson is qualified by the Bethe-Salpeter amplitude (BSA),
$ \Gamma^{fg}(k;P) $ , with k and P being the relative and the total momentum of the meson, respectively.$ P^2 = -M^2_{fg} $ , and$ M_{fg} $ is the mass of the meson with quark flavor$ (f,g) $ . The BSA is solved by the Bethe-Salpeter equation (BSE) [29, 30]$ \Big{[} \Gamma^{fg}(k;P) \Big{]}^{\alpha}_{\beta} = - \int^\Lambda_{{\rm d} q} \frac{4}{3}(Z_{2})^{2} \tilde{D}^{fg}_{\mu\nu}(l) \Big[\gamma_{\mu}^{}\Big]^{\alpha}_\sigma \Big[\gamma_{\nu}\Big]^\delta_\beta \Big{[} \chi^{fg}(q;P) \Big{]}^{\sigma}_{\delta} , $
(2) where α, β, σ, and δ are the Dirac indexes.
$ \chi^{fg}(q;P) = S_{f}(q_{+}) \Gamma^{fg}(q;P) S_{g}(q_{-}) $ is the wave function,$ q_{+} = q + \eta P/2 $ ,$ q_{-} = q - (1-\eta) P/2 $ , η is the partitioning parameter describing the momentum partition between the quark and the antiquark and does not affect the physical observables.$ \tilde{D}^{fg}_{\mu\nu} $ represents the effective interaction between the quark and the antiquark. The leptonic decay constant of the pseudoscalar meson,$ f_{0^{-}} $ , is obtained by$ f_{0^-}P_{\mu} = Z_{2} N_{c} \;\text{tr} \int^{\Lambda}_{{\rm d} q} \gamma_{5}^{} \gamma_{\mu}^{} S_f(q_+)\Gamma^{fg}_{0^-}(q;P)S_g(q_-). $
(3) $ N_c = 3 $ , is the color number. “$ \text{tr} $ ” represents the trace of the Dirac index. The leptonic decay constant of the vector meson,$ f_{1^{-}} $ , is analogous to$ f_{1^-}M_{1^-} = Z_{2} N_{c} \;\text{tr} \int^{\Lambda}_{{\rm d} q} \gamma_{\mu}^{} S_f(q_+)\Gamma^{fg,\mu}_{1^-}(q;P)S_g(q_-). $
(4) $ \tilde{D}^{f}_{\mu\nu} $ and$ \tilde{D}^{fg}_{\mu\nu} $ are decomposed as$ \tilde{D}^{f}_{\mu\nu}(l) = \left(\delta_{\mu\nu}-\dfrac{l_{\mu}l_{\nu}}{l^{2}}\right) $ $ {\cal{G}}^f(l^2) $ and$ \tilde{D}^{fg}_{\mu\nu}(l) = \left(\delta_{\mu\nu}-\dfrac{l_{\mu}l_{\nu}}{l^{2}}\right){\cal{G}}^{fg}(l^2) $ . The dressing functions$ {\cal{G}}^f(l^2) $ and$ {\cal{G}}^{fg}(l^2) $ are modeled as$ {\cal{G}}^f(s) = {\cal{G}}^f_{IR}(s) + {\cal{G}}_{UV}(s), $
(5) $ {\cal{G}}^f_{IR}(s) = 8\pi^2\frac{D_f^2}{\omega_f^4} {\rm e}^{-s/\omega_f^2}, $
(6) $ {\cal{G}}^{fg}(s) = {\cal{G}}^{fg}_{IR}(s) + {\cal{G}}_{UV}(s), $
(7) $ {\cal{G}}^{fg}_{IR}(s) = 8\pi^2\frac{D_f}{\omega_f^2}\frac{D_g}{\omega_g^2} {\rm e}^{-s/(\omega_f\omega_g)}, $
(8) $ {\cal{G}}_{UV}(s) = \frac{8\pi^{2} \gamma_{m}^{} {\cal{F}}(s)}{\text{ln}\big[\tau+(1+s/\Lambda^{2}_{\rm QCD})^2\big]}, $
(9) where
$ s = l^2 $ .$ {\cal{G}}^f_{IR}(s) $ and$ {\cal{G}}^{fg}_{IR}(s) $ are the infrared interactions responsible for the hadron properties.$ \omega_f $ represents the interaction width in momentum space, and$ D_f $ is the infrared strength.$ {\cal{G}}_{UV}(s) $ keeps the one-loop perturbative QCD limit in the ultraviolet.$ {\cal{F}}(s) = [1 - \exp(-s^2/ $ $ [4m_{t}^{4}])]/s $ ,$ \gamma_{m}^{} = 12/(33-2N_{f}) $ , with$ m_{t} = 1.0 \;{\rm{ GeV}}\, $ ,$ \tau = e^{10} - 1 $ ,$ N_f = 5 $ , and$ \Lambda_{\text{QCD}} = 0.21 \;{\rm{ GeV}}\, $ . This model turned out to be successful for all ground state pseudoscalar and vector mesons, from heavy, heavy-light to light mass scales. It has been extended to the scalar and axial vector mesons [26]. These successes support that Eq. (8) contains the proper flavor dependence. However, the same interaction should not apply to radial excited mesons if high precision is required, as stated in Section I.To mimic the interesting difference between the radial excited states and the ground states, Eq. (8) is changed into
$ {\cal{G}}^{fg}_{IR}(s) = 8\pi^2\frac{\eta_f D_f}{\omega_f^2}\frac{ \eta_ g D_g}{\omega_g^2} {\rm e}^{-s/(\alpha_f\omega_f \alpha_g\omega_g)}. $
(10) While
$ \omega_f $ and$ D_f $ ($ f = \{c,b\} $ ) are kept unchanged from those in Ref. [7],$ \alpha_f $ and$ \eta_f $ ($ f = \{c,b\} $ ) are free parameters.$ \alpha_f $ presents the changing of the interaction width and$ \eta_f $ the infrared strength. The four free parameters,$ \alpha_c $ ,$ \alpha_b $ ,$ \eta_c $ , and$ \eta_b $ , are fixed by fitting the masses and leptonic decay constants of$ \psi(2S) $ and$ \varUpsilon(2S) $ ($ M_{\psi(2S)} $ ,$ M_{\varUpsilon(2S)} $ ,$ f_{\psi(2S)} $ ,$ f_{\varUpsilon(2S)} $ ) to the experimental values. The exprimental values of$ f_{\psi(2S)} $ and$ f_{\varUpsilon(2S)} $ are extracted from the branch decay width of the vector meson to$ e^+ e^- $ . The values are$ f_{\psi(2S)} = -0.208\;{\rm{ GeV}} $ ,$ f_{\varUpsilon(2S)} = -0.352\;{\rm{ GeV}} $ [25]. Then, the masses and leptonic decay constants of$ \eta_c(2S) $ ,$ B_c(2S) $ ,$ B^*_c(2S) $ , and$ \eta_b(2S) $ are calculated with refixed interactions. -
Before discussing the results, we should first explain the process of solving the BSE, Eq. (2). Eq. (2) is solved as a
$ P^{2} $ -dependent eigenvalue problem,$ \lambda^{fg}(P^2) \Big{[} \Gamma^{fg}(k;P) \Big{]}^{\alpha}_{\beta} = - \int^\Lambda_{{\rm d} q} \frac{4}{3}(Z_{2})^{2} \tilde{D}^{fg}_{\mu\nu}(l) \Big[\gamma_{\mu}^{}\Big]^{\alpha}_\sigma \Big[\gamma_{\nu}\Big]^\delta_\beta \Big{[} \chi^{fg}(q;P) \Big{]}^{\sigma}_{\delta}. $
(11) The meson mass is determined by
$ \lambda^{fg}(P^2 = -M^2_{fg}) = 1 $ . However, due to the singularity of the quark propagators in the complex momentum plane, there is a lower bound value for$ P^2 $ in Eq. (11). Only when$ P^2 > -M_{\rm max}^2 $ , Eq. (11) is solvable.$ M_{\rm max}^2 $ defines the contour border of the calculable region of the quark propagators; see appendix of Ref. [31] for this problem.As the masses of the radial excited mesons are beyond the contour border, i.e.
$ M_{fg} > M_{\rm max} $ , the following form is used to fit$ \lambda^{fg}(P^2) $ :$ \frac{1}{\lambda^{fg}(P^2)} = \frac{1 + \displaystyle\sum^{N_{o}}_{n = 1}\, a_{n} (P^{2} + s_{0}^{})^n}{1 + \displaystyle\sum^{N_{o}}_{n = 1}\, b_{n} (P^{2} + s_{0}^{})^n} \, , $
(12) where
$ N_{o} $ is the order of the series$ s_{0}^{} $ , and$ a_{n} $ and$ b_{n} $ are the parameters to be determined by the least square method.$ f_{fg}(P^2) $ is fitted by$ f_{fg}(P^2) = \frac{f_0 + \displaystyle\sum^{N_o}_{n = 1}\, c_n (P^2+M^2)^n}{1 + \displaystyle\sum^{N_o}_{n = 1}\, d_n (P^2+M^2)^n}, $
(13) where
$ f_{0}^{} $ ,$ c_{n} $ , and$ d_{n} $ are parameters, and$ M^{2} $ is the square of the mass. The physical value of the decay constant is$ f_{fg}(-M^2) = f_{0}^{} $ . This extrapolation scheme has been used in Refs. [25, 26]. The error is controllable as long as the physical mass of the meson is not considerably larger than$ M_{\rm max} $ . The extrapolation results are illustrated by$ B_c(2S) $ in the two upper figures of Fig. 1.$ N_o = 1, 2, 3 $ are considered in practice, and the differences between the three cases are estimated to be the error from extrapolation.Figure 1. (color online) Extrapolation of the eigenvalue,
$\lambda(P^2)$ , and the leptonic decay constant,$f(P^2)$ , of$B_c(2S)$ . The upper two figures show the extrapolation with$N_o = 1, 2, 3$ using Eq. (12) and Eq. (13). The lower two figures show the results of using the three different groups of parameters that are listed in Table A1 in the appendix. The vertical dot-dashed orange line is the contour border on the right of which the direct calculation can be applied. The green stars present the location of the meson masses.Three groups of
$ \omega_f $ and$ D_f $ are used, which are the same as in Ref. [7]. Different groups of parameters correspond to different interaction widths. In each case,$ \alpha_f $ and$ \eta_f $ ($ f = \{c,b\} $ ), are fixed by fitting the masses and leptonic decay constants of$ \psi(2S) $ and$ \varUpsilon(2S) $ to the experimental values. Then, masses and leptonic decay constants of$ \eta_c(2S) $ ,$ B_c(2S) $ ,$ B^*_c(2S) $ , and$ \eta_b(2S) $ are outputted. The differences in the outputs are considered to be the errors due to varying the parameters. This is illustrated by$ B_c(2S) $ , the two lower figures of Fig. 1.The masses and leptonic decay constants of
$ \eta_c(2S) $ ,$ \psi(2S) $ ,$ B_c(2S) $ ,$ B^*_c(2S) $ ,$ \eta_b(2S) $ , and$ \varUpsilon(2S) $ herein (the DSE results) and the available experimental values (the expt. results) are listed in Table 1. For the DSE results, the first error is from the extrapolation, and the second is from varying the parameters. The extrapolation errors of the charmonium masses are larger than those of the$ B_c $ mesons and the bottomonium, because heavier mesons are less affected by the higher orders in Eq. (12) in the extrapolation process. Comparing the DSE results with the experimental values, the largest mass deviation is$ 20\;{\rm{ MeV}} $ , in the case of$ \eta_c(2S) $ . Therefore, the systematic error of the DSE calculated spectrum is estimated to be$ 20\;{\rm{ MeV}} $ . The mass of$ B^*_c(2S) $ is not well determined experimentally, due to a lack of knowledge of the$ B^*_c(1S) $ mass. Herein, I provide the most precise prediction of the$ B^*_c(2S) $ mass to date. Note that my prediction is also consistent with the potential model results, for example, see Table IV in Ref. [37].meson $J^{\text{PC}}$ $M^{\text{DSE}}_{c\bar{c}}$ $M^{\text{expt.}}_{c\bar{c}}$ $f^{\text{DSE}}_{c\bar{c}}$ $f^{\text{expt.}}_{c\bar{c}}$ $|f|$ [33]$|f|$ [36]$\eta_c(2S)$ $0^{-+}$ 3.618(25)(3) 3.638 −0.158(8)(4) − 0.170~0.172 0.197 $\psi(2S)$ $1^{--}$ 3.686(21)(0) 3.686 −0.208(5)(0) −0.208 0.207~0.216 0.182 meson $J^{\text{P}}$ $M^{\text{DSE}}_{c\bar{b}}$ $M^{\text{expt.}}_{c\bar{b}}$ $f^{\text{DSE}}_{c\bar{b}}$ $f^{\text{expt.}}_{c\bar{b}}$ $|f|$ [34]$|f|$ [35]$|f|$ [36]$B_c(2S)$ $0^{-}$ 6.874(9)(6) 6.872 −0.174(5)(4) − $0.198(35)$ 0.304(14) 0.251 $B^*_c(2S)$ $1^{-}$ 6.926(12)(6) − −0.216(9)(4) − − 0.325(14) 0.252 meson $J^{\text{PC}}$ $M^{\text{DSE}}_{b\bar{b}}$ $M^{\text{expt.}}_{b\bar{b}}$ $f^{\text{DSE}}_{b\bar{b}}$ $f^{\text{expt.}}_{b\bar{b}}$ $|f|$ [33]$|f|$ [36]$\eta_b(2S)$ $0^{-+}$ 9.989(13)(3) 9.999 −0.345(6)(1) − 0.291~0.299 0.367 $\varUpsilon(2S)$ $1^{--}$ 10.023(11)(0) 10.023 -0.352(4)(0) −0.352 0.336~0.350 0.367 Table 1. Masses and leptonic decay constants of the first radial excited heavy pseudoscalar and vector mesons (in GeV). The normalization convention
$f_\pi = 0.093\;{\text{GeV}}$ is used for the leptonic decay constants.$M^{\text{DSE}}_{q\bar{q}'}$ and$f^{\text{DSE}}_{q\bar{q}'}$ are the Dyson-Schwinger equation results, whereby the first error is from the extrapolation, and the second is from varying the parameters. The underlined values are used to fit the parameters$\alpha_{f,g}$ and$\eta_{f,g}$ in Eq. (10).$M^{\text{expt.}}_{q\bar{q}'}$ are the experimental values [32]. See the text for the experimental values of$f_{\psi(2S)}$ and$f_{\varUpsilon(2S)}$ . The results in Refs. [33-36] have to be divided by$\sqrt{2}$ for comparison with those of this study. Ref. [33] presents the nonrelativistic quark model results, cited therein for the case of “BGS log” with$\Lambda = 0.25\;{\text{GeV}}$ and$\Lambda = 0.40\;{\text{GeV}}$ . Ref. [34] presents the static potential model results. Ref. [35] presents a QCD sum rule estimation and Ref. [36] a nonrelativistic Cornell potential model result.There are no experimental values for
$ f_{\eta_c(2S)} $ ,$ f_{B_c(2S)} $ ,$ f_{B^*_c(2S)} $ , or$ f_{\eta_b(2S)} $ as of now. I list the leptonic decay constants from other models in Table 1. In this article, the normalization convention$ f_\pi = 0.093\;{\rm{ GeV}} \approx 0.131/\sqrt{2} $ $ \;{\rm{ GeV}} $ is used for the leptonic decay constants, so the results in Refs. [33-35] have to be divided by$ \sqrt{2} $ for comparison with those of this study. Ref. [33] includes the nonrelativistic quark model results, cited therein for the case of “BGS log” with$ \Lambda = 0.25\;{\rm{ GeV}} $ and$ \Lambda = 0.40\;{\rm{ GeV}} $ . Ref. [34] describes the static potential model results, and Ref. [35] presents a QCD sum rule estimation. Recently, some other model estimations have been made of the leptonic decay constants of$ \eta_c(2S) $ ,$ \psi(2S) $ ,$ \eta_b(2S) $ , and$ \varUpsilon(2S) $ [38, 39]. However, Ref. [33] is the most consistent one compared with this study. There are fewer studies on the$ B_c(2S) $ and$ B^*_c(2S) $ leptonic deacy constants. Results from Ref. [34] are consistent with those of this study. However, results in Ref. [35] are incompatible with the viewpoint herein. Ref. [36] calculated the entire S-wave heavy meson spectrum and the leptonic decay constants via the nonrelativistic Cornell potential model. The bottomonium decay constants therein are consistent with those from this study, while there are deviations for the charmonium and the$ B_c $ mesons.Regardless, the reasonableness of my results can be justified with the following three facts:
1. The RL approximation is suffcient for the pseudoscalar and vector mesons;
2. The interaction patterns in Eq. (7), Eq. (10) and Eq. (9) contain the proper flavor dependence, so the
$ B_c $ mesons share this universal interaction form with the charmonium and the bottomonium;3. The interaction is refixed by the experimental value of the masses and leptonic decay constants of
$ \varUpsilon(2S) $ and$ \psi(2S) $ , so the interactions for the radial excited mesons are realistic.Finally, let us discuss the effective interaction between a quark and an antiquark in radial excited mesons. This is characterized by the dressing function
$ {\cal{G}}^{fg} $ . The dressing functions,$ {\cal{G}}^{c\bar{c}}(k^2) $ ,$ {\cal{G}}^{c\bar{b}}(k^2) $ and$ {\cal{G}}^{b\bar{b}}(k^2) $ , for the ground states and the first radial excited states are depicted in Fig. 2. We can see that the dressing functions of the radial excited mesons are smaller in the region$ k^2 \lesssim 0.7 \sim 1.1\;{\rm{ GeV}}^2 $ and larger in the region$ k^2 \gtrsim 0.7 \sim 1.1\;{\rm{ GeV}}^2 $ . This can be understood as follows: the energy of quarks in the radial excited mesons is larger, so the soft interaction declines, and the hard interaction increases. This feature also applys to light mesons. As the light excited meson mass is farther from the contour border, the extrapolation of Eq. (12) and Eq. (13) has a larger error. This problem is reserved for future studies.Figure 2. (color online) Effective interaction dressing functions of the ground states (Eqs. (7)-(9)) and the first radial excited states (Eq. (7), Eq.(10) and Eq. (9)). The region boundary is defined by Parameter-1 and Parameter-3 in Table A1 in the appendix.
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The three groups of parameters corresponding to
$ \omega_u = 0.45, 0.50, 0.55 \;{\rm{ GeV}} $ are listed in Table A1. The quark mass$ \bar{m}_f^{\zeta} $ is defined byflavor $\bar{m}_f^{\zeta=2{\text{GeV}}}$ Parameter-1 $w_f $ $D_f^2$ $\alpha_f$ $\eta_f$ c 1.17 0.690 0.645 1.360 0.755 b 4.97 0.722 0.258 1.323 0.671 flavor $\bar{m}_f^{\zeta=2{\text{GeV}}}$ Parameter-2 $w_f $ $D_f^2$ $\alpha_f$ $\eta_f$ c 1.17 0.730 0.599 1.304 0.817 b 4.97 0.766 0.241 1.265 0.730 flavor $\bar{m}_f^{\zeta=2{\text{GeV}}}$ Parameter-3 $w_f $ $D_f^2$ $\alpha_f$ $\eta_f$ c 1.17 0.760 0.570 1.265 0.865 b 4.97 0.792 0.231 1.225 0.766 Table A1. Three groups of parameters correspond to
$\omega_{u/d} = 0.45, 0.50, 0.55 \;{\text{GeV}}$ .$\bar{m}_f^{\zeta=2{\text{GeV}}}$ ,$\omega_f$ and$D_f$ are measured in GeV. α and η are of unit 1.$\tag{A1} \bar{m}_f^{\zeta} = \hat{m}_f\Big /\left[\frac{1}{2}{\rm{ln}}\frac{\zeta^2}{\Lambda^2_{\rm{QCD}}}\right]^{\gamma_m}, $
$\tag{A2} {\hat m_f} = \mathop {\lim }\limits_{{p^2} \to \infty } {\left[ {\frac{1}{2}{\rm{ln}}\frac{{{p^2}}}{{\Lambda _{{\rm{QCD}}}^2}}} \right]^{{\gamma _m}}}{M_f}({p^2}), $
with ζ being the renormalisation scale,
$ \hat{m}_f $ the renormalisation-group invariant current-quark mass, and$ M_f(p^2) $ the quark mass function.
Radial excited heavy mesons
- Received Date: 2021-08-24
- Available Online: 2021-12-15
Abstract: In this study, the first radial excited heavy pseudoscalar and vector mesons (