1. Introduction
2. Chiral effective Lagrangian for excited heavy-light mesons
where $({P}_{{1}^{+}}^{\nu },{P}_{{2}^{+}}^{* \mu \nu })$ refer to JP = (1+, 2+) states, and $({P}_{{1}^{-}}^{\nu },{P}_{{2}^{-}}^{* \mu \nu })$ refer to JP = (1−, 2−) states, respectively. v$\mu$ is the velocity of an on-shell heavy quark, i.e. p$\mu$ = mQv$\mu$ with v2 = 1. Then the chiral effective Lagrangian for excited heavy-light mesons can be written as [33, 32]
where
The covariant derivative ∇$\mu$ = ∂$\mu$ − iV$\mu$ with ${V}_{\mu }=\tfrac{{\rm{i}}}{2}({{\rm{\Omega }}}^{\dagger }{\partial }_{\mu }$ Ω + Ω∂$\mu$Ω†), and ${A}_{\mu }=\tfrac{{\rm{i}}}{2}\left({{\rm{\Omega }}}^{\dagger }{\partial }_{\mu }{\rm{\Omega }}-{\rm{\Omega }}{\partial }_{\mu }{{\rm{\Omega }}}^{\dagger }\right)$. The Ω field is related to the chiral field $U(x)=\exp ({\rm{i}}\pi (x)/{f}_{\pi })$ through U = Ω2. The parameters ${g}_{T},{g}_{R},{g}_{{TR}},{m}_{T}$, and mR are the LECs of the Lagrangian. They are free ones at the level of the effective theory.
which yields the mass splitting
For the (1−, 2−) doublet, the situation is subtle since the quantum numbers of the possible candidates in PDG are not well determined. With respect to the mass splitting between chiral partners in the charmed meson sector, the masses of the states in the R$\mu$ doublet should have masses ∼6000 MeV. This means that it is reasonable to identify the quantum numbers of the state BJ(5970) as 1− or 2− or maybe the 1− and 2− states have nearly degenerate masses.
3. Chiral effective Lagrangian of excited heavy-light mesons from QCD
where q(x), Q(x) and G$\mu$(x) are the light-quark, heavy-quark and gluon fields, respectively.
where Π2 and ${\bar{{\rm{\Pi }}}}_{2}$ are the heavy-light meson field and its conjugate, respectively and, mQ is the mass of heavy quark Q. $\bar{{\rm{\Sigma }}}$ is the self-energy of the light quark propagator. I1 = diag(1, 1, 0) and I4 = diag(0, 0, 1) are matrices in the flavor space. JΩ is the chiral rotated external source.
• | When integrating in the heavy-light meson fields, we have taken the heavy quark limit which brings in an uncertainty of order 1/mQ. And, to integrate out the gluon and quark fields, we left infinitely many gluon Green functions in the action. |
• | Further approximations, i.e. the chiral limit, the large Nc limit and the leading order in dynamical perturbation, are made in order for the derived action could serve practical purposes. The chiral limit causes an uncertainty of order mu,d, and large Nc limit suffers from 1/Nc corrections. Taking the leading order in dynamical perturbation is a similar technique as the rainbow-ladder approximation in the Dyson-Schwinger-Bethe-Salpeter (DSBS) formulism. The errors brought in by this last approximation is hard to be estimated quantitatively, but its justification is supported by the relatively success of the rainbow-ladder approximation in the DSBS formulism. |
where
Here, the fields H and G refer to JP = (0−, 1−) states and JP = (0+, 1+) states, the fields T$\mu$ and R$\mu$ refer to JP = (1+, 2+) states and JP = (1−, 2−) states respectively. Since we are interested in the heavy-light meson doublets with ${j}_{l}^{P}={\tfrac{3}{2}}^{\pm }$ in this work, we focus on the chiral effective Lagrangian for ${{\rm{\Pi }}}_{2}^{\mu }$ alone. So the chiral effective action is reduced to
Taking the derivative expansion up to the first order, we obtain
In the calculation, we have used the relation ${T}^{\mu }{/}\!\!\!{v}=-{T}^{\mu }$. It is easy to identify the mass term and the kinetic term in equation (
Upto the first order of the derivative expansion, one obtains
with ZT being the wave function renormalization factor
We also obtain the coupling constant between the chiral partner fields T$\mu$ and R$\mu$ as
In the above expressions, Σ(−p2) stands for self-energy of light quarks to be calculated from QCD. This is an apparent indicator that we have established a connection between the low energy constants of the chiral effective theory and the Green functions of QCD.
4. Numerical results and discussions
where the boundary conditions are
with $\alpha$s being the running coupling constant of QCD. ${\rm{\Lambda }}^{\prime} $ is an ultraviolet cutoff regularizing the integral, which is taken ${{\rm{\Lambda }}}^{{\prime} }\to \infty $ eventually. To solve the equation (
For convenience, we call it Gribov-Zwanziger (G-Z) Model. The parameters are M2 = 4.303 GeV2, (M2 + m2) = 0.526 GeV2 and λ4 = 0.4929 GeV4 [35]. $\alpha$0 is a model parameter to be determined. Although the UV behavior of G-Z formalism is inconsistent with QCD, it should not have a sizable impact on our results because the LECs are mostly controlled by its low energy behavior.
Figure 1. Running coupling constant of the G-Z model with a0 = 0.52. |
Figure 2. The lattice fittings of M(−p2) and Z(−p2) given in [36] and Σ(−p2) from the gap equation with G-Z Model. |
Table 1. The heavy-light meson masses and the coupling constants calculated from the G-Z Model with Λc = 1.6 GeV. The empirical values are give for comparison. |
a0 | Σ(0) (GeV) | fπ(GeV) | $-\langle \bar{\psi }\psi \rangle $ (GeV3) | gR | gT | gTR | mR (GeV) | mT (GeV) | Δm (GeV) |
0.50 | 0.158 | 0.073 | (0.328)3 | 0.882 | −0.479 | −0.990 | 1.365 | 1.104 | 0.261 |
0.51 | 0.193 | 0.084 | (0.350)3 | 0.936 | −0.442 | −0.987 | 1.411 | 1.091 | 0.321 |
0.52 | 0.226 | 0.093 | (0.368)3 | 0.991 | −0.409 | −0.984 | 1.459 | 1.080 | 0.379 |
0.53 | 0.259 | 0.102 | (0.385)3 | 1.047 | −0.377 | −0.981 | 1.510 | 1.071 | 0.439 |
0.54 | 0.291 | 0.109 | (0.399)3 | 1.107 | −0.347 | −0.979 | 1.564 | 1.064 | 0.500 |
Exp. | — | 0.093 [34] | — | — | — | — | 1.433a | 1.178 | 0.255 |
aThe experimental data of mR, mT and Δm are obtained by using the spin-averaged masses in the D meson sector with mc subtracted. |
Table 2. LECs calculated from lattice fittings given in [36]. The empirical values are give for comparison. |
Λc(GeV) | M(0)(GeV) | fπ (GeV) | $-\langle \bar{\psi }\psi \rangle $ (GeV3) | gR | gT | gTR | mR (GeV) | mT (GeV) | Δm (GeV) |
1.4 | 0.344 | 0.095 | (0.341)3 | 1.101 | −0.398 | −1.031 | 1.349 | 0.948 | 0.401 |
1.5 | 0.344 | 0.096 | (0.351)3 | 1.054 | −0.432 | −1.041 | 1.405 | 1.021 | 0.384 |
1.6 | 0.344 | 0.096 | (0.359)3 | 1.015 | −0.462 | −1.050 | 1.462 | 1.095 | 0.368 |
1.7 | 0.344 | 0.096 | (0.367)3 | 0.983 | −0.488 | −1.058 | 1.522 | 1.169 | 0.352 |
1.8 | 0.344 | 0.096 | (0.375)3 | 0.956 | −0.511 | −1.064 | 1.582 | 1.245 | 0.337 |
Exp. | — | 0.093 [34] | — | — | — | — | 1.433a | 1.178 | 0.255 |
aThe experimental data of mR, mT and Δm are obtained by using the spin-averaged masses in the D meson sector with mc subtracted. |
where mi is the mass of the decaying meson; mf and Ef are the mass and the energy of the heavy-light meson in the final states respectively; Eπ and pπ are the energy and the magnitude of the momentum of the pion, respectively.
where (G-Z) denotes results from the G-Z model with a0 = 0.52; (Lat.) denotes results from the lattice-fittings with Λc = 1.6 GeV. For the B mesons, the assignments of R doublet is yet unclear. If BJ(5970) is the 1− state in the R doublet, then we have
If BJ(5970) is the 2− state in the R doublet, then we have
In the future, when the corresponding experimental results are available, our theoretical predictions could help testing the assignments of these excited mesons.
5. Conclusions
Appendix. Formula for lecs with Z(−p2)
where Z(−p2) = 1/A(−p2) stands for the quark wave function renormalization and M(−p2) = B(−p2)/A(−p2) is the renormalization group invariant running quark mass. After a series of calculations, we can get the LECs with Z(−p2) as follows: