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--> --> -->My purpose herein is to study the radial excited states via a continuum QCD approach, the Dyson-Schwinger equation and the Bethe-Salpeter equation (DSBSE) approach. Many interesting properties of the excited states have been obtained using this approach, for example, the radial excited pseudoscalar mesons decouple from the axial-vector current in the chiral limit [14]. The electromagnetic properties and chiral symmetry restoration of the radial excited states were also studied [15, 16]. It is shown qualitatively that radial excited mesons are more sensitive to interactions at a large distance, though a realistic prediction for the excited mesons from the rainbow ladder (RL) approximation failed [17, 18].
In this paper, I focus on the the first radial excited heavy pseudoscalar and vector mesons, i.e.
I and my collaborators studied these excited mesons in the RL approximation in Ref. [25]. Therein, one gluon exchange effective interaction between a quark and an antiquark is fixed by the ground state mesons [7]. The effective interaction in Ref. [7] is universal for all light, heavy-light, and heavy ground pseudoscalar and vector mesons. It could be extended to ground state scalar and axial vector mesons [26]. When applying the same interaction to radial excited heavy mesons, we found that the spectrum is 1% lower than the available experimental value. The leptonic decay constant is lower by 12% for
What should an effective interaction be like? A proper interaction should not only produce the right spectrum, but also the right wave function for the mesons. The leptonic decay constant (see Eq. (3) and Eq. (4)) describes quark antiquark annihilation inside a meson and thus, is the simplest quantity related to the meson wave function. Therefore, herein I refix the effective interaction with the masses and leptonic decay constants of
$ 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) |
A meson is qualified by the Bethe-Salpeter amplitude (BSA),
$ \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) |
$ 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) |
$ 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) |
$ {\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) |
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) |
$ \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) |
As the masses of the radial excited mesons are beyond the contour border, i.e.
$ \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) |
$ 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) |

Three groups of
The masses and leptonic decay constants of
meson | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | |
![]() ![]() | ![]() ![]() | 3.618(25)(3) | 3.638 | ?0.158(8)(4) | ? | 0.170~0.172 | 0.197 | |
![]() ![]() | ![]() ![]() | 3.686(21)(0) | 3.686 | ?0.208(5)(0) | ?0.208 | 0.207~0.216 | 0.182 | |
meson | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() |
![]() ![]() | ![]() ![]() | 6.874(9)(6) | 6.872 | ?0.174(5)(4) | ? | ![]() ![]() | 0.304(14) | 0.251 |
![]() ![]() | ![]() ![]() | 6.926(12)(6) | ? | ?0.216(9)(4) | ? | ? | 0.325(14) | 0.252 |
meson | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | |
![]() ![]() | ![]() ![]() | 9.989(13)(3) | 9.999 | ?0.345(6)(1) | ? | 0.291~0.299 | 0.367 | |
![]() ![]() | ![]() ![]() | 10.023(11)(0) | 10.023 | -0.352(4)(0) | ?0.352 | 0.336~0.350 | 0.367 |
Table1.Masses and leptonic decay constants of the first radial excited heavy pseudoscalar and vector mesons (in GeV). The normalization convention
There are no experimental values for
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
3. The interaction is refixed by the experimental value of the masses and leptonic decay constants of
Finally, let us discuss the effective interaction between a quark and an antiquark in radial excited mesons. This is characterized by the dressing function

flavor | ![]() ![]() | Parameter-1 | |||
![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ||
c | 1.17 | 0.690 | 0.645 | 1.360 | 0.755 |
b | 4.97 | 0.722 | 0.258 | 1.323 | 0.671 |
flavor | ![]() ![]() | Parameter-2 | |||
![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ||
c | 1.17 | 0.730 | 0.599 | 1.304 | 0.817 |
b | 4.97 | 0.766 | 0.241 | 1.265 | 0.730 |
flavor | ![]() ![]() | Parameter-3 | |||
![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ||
c | 1.17 | 0.760 | 0.570 | 1.265 | 0.865 |
b | 4.97 | 0.792 | 0.231 | 1.225 | 0.766 |
TableA1.Three groups of parameters correspond to
$\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}), $ ![]() |