Update on strong and radiative decays of the $D_{s0}^*(2317)$ and $D_{s1}(2460)$ and their bottom cousins

Abstract

The isospin breaking and radiative decay widths of the positive-parity charm-strange mesons, $D^{*}_{s0}$ and $D_{s1}$, and their predicted bottom-strange counterparts, $B^{*}_{s0}$ and $B_{s1}$, as hadronic molecules are revisited. This is necessary, since the $B^{*}_{s0}$ and $B_{s1}$ masses used in (Cleven et al., Eur Phys J A 50:149, 2014) were too small, in conflict with the heavy quark flavour symmetry. Furthermore, not all isospin breaking contributions were considered. We here present a method to restore heavy quark flavour symmetry, correcting the masses of $B^{*}_{s0}$ and $B_{s1}$, and include the complete isospin breaking contributions up to next-to-leading order. With this we provide updated hadronic decay widths for all of $D^{*}_{s0}$, $D_{s1}$, $B^{*}_{s0}$ and $B_{s1}$. Results for the partial widths of the radiative deays of $D_{s0}^*(2317)$ and $D_{s1}(2460)$ are also renewed in light of the much more precisely measured $D^{*+}$ width. We find that $B_s\pi ^0$ and $B_s\gamma$ are the preferred channels for searching for $B_{s0}^*$ and $B_{s1}$, respectively.

Appendix

Table 6 The S-wave scattering lengths $a_0$ and effective ranges $r_{0}$ for the scattering of a psuedo-Goldstone boson off a ground state heavy meson. For the latter, we only show the values for those channels with the lowest threshold in each coupled-channel system. The uncertainties are propagated from those of the low-energy constants in the fit of Ref. [26]

For reference, in this appendix we provide the numerical results of the values of the S-wave scattering lengths $a_0$ and effective ranges $r_{0}$ for the charm-light and bottom-light meson scattering. We use the following convention for the effective range expansion of the S-wave amplitude:

$$\begin{aligned} T_{L=0}(s) = - \frac{8\pi \sqrt{s}}{\frac{1}{a_{0}}+\frac{1}{2}r_{0} k^2 -i\,k+\mathcal {O}(k^{4})}, \end{aligned}$$

(10)

where $\sqrt{s}$ and k are the total energy and the size of three-momentum in the center-of-mass frame, respectively. In particular, the effective range, for which we only show the values for those with the lowest threshold in each coupled-channel system, predictions have been not been presented before. The results are shown in Table 6. The $B_s\pi$ and $D_s\pi$ effective ranges, $-1284_{-1336}^{+446}\,$fm and $201_{-270}^{+100}\,$fm, respectively, are not listed in the table since they are unreliable. The reason is that the leading order chiral amplitudes for these elastic scattering vanish and they receive contributions only from higher orders; consequently the amplitude has a zero close to threshold from higher orders, to which the effective range is sensitive.

Not all of the values for the charm-light scattering lengths completely agree with those given in Ref. [26]. The reason is that in Ref. [26], the charmed mesons and kaons are heavier than their physical masses when extrapolated to the physical pion mass, $M_D^\text {lat.}\simeq 1947$ MeV and $M_K^\text {lat.}\simeq 560$ MeV. Here, we use the physical masses for all mesons instead. The large values of the scattering lengths for the isoscalar $D\bar{K}$ and $\bar{B} \bar{K}$ are due to the near-threshold virtual states predicted in Ref. [9], which are about 20 MeV below the $D\bar{K}$ threshold and almost at the $\bar{B} \bar{K}$ threshold, respectively. Thus, the systems are fine tuned, and it is this difference that also makes the effective ranges of these two channels have opposite signs. The existence of an isoscalar $D\bar{K}$ virtual state received support from lattice QCD calculations in Ref. [33].

Using Weinberg’s expressions to relate the scattering length and effective range to the probability of finding the two-hadron component in the physical wave function of a state, the so-called compositeness $1-Z$ [34],

$$\begin{aligned} a_0 \approx -2\left( \frac{1-Z}{2-Z}\right) \frac{1}{\gamma } ,\quad r_{0} \approx -\frac{Z}{1-Z} \frac{1}{\gamma }, \end{aligned}$$

(11)

where $\gamma =\sqrt{2 \mu \epsilon }$ is the binding momentum, with $\epsilon$ the binding energy and $\mu$ the reduced mass, we find that the compositeness for the pole in the isoscalar $\bar{B}^{(*)} K$ system,

$$\begin{aligned} (1-Z)_{\bar{B}^{(*)} K} = 0.7\pm 0.2\pm 0.1, \end{aligned}$$

(12)

is in the same ballpark as that of the $D_{s0}^*$ found in Refs. [26, 33, 35,36,37]. Here, the first uncertainty is propagated from those of the parameters and the second one is an additional intrinsic uncertainty in the above relations. It is usually estimated as $\mathcal {O}(\gamma /\beta )$, where $\beta$ is a hard scale set by the neglected heavier degrees of freedom, since Ref. [34] used a constant form factor approximation for the coupling of the composite system to its constituents. However, as shown in Ref. [38], in case of a negative effective range, as is the case here, the above relations hold even when the constant form factor approximation is lifted. Consequently, the intrinsic uncertainty should be $\mathcal {O}(\gamma ^2/\beta ^2)$; see also Ref. [39]. With $\beta \sim m_\rho$, and $\gamma \approx 0.2$ GeV, one has $\gamma ^2/\beta ^2\approx 0.07$.