# Isospin violating decays of positive parity \(B_s\) mesons in HM\(\chi \)PT

## Abstract

Recent lattice QCD results suggest that the masses of the first two positive parity \(B_s\) mesons lie below the *BK* threshold, similar to the case of \(D^*_{s0}(2317)^+\) and \(D_{s1}(2460)^+\) mesons. The mass spectrum of \(B_s\) mesons seems to follow the pattern of a \(D_s\) mass spectrum. As in the case of charmed mesons, the structure of positive parity \(B_s\) mesons is very intriguing. To shed more light on this issue, we investigate the strong isospin violating decays \(B_{s0}^{*0} \rightarrow B_s^0 \pi ^0\), \(B_{s1}^{0} \rightarrow B_s^{*0} \pi ^0\), and \(B_{s1}^{0} \rightarrow B_s^0 \pi \pi \) within heavy meson chiral perturbation theory. The two-body decay amplitude arises at tree level and we show that the loop corrections give significant contributions. On the other hand, in the case of three-body decay \(B_{s1}^{0} \rightarrow B_s^0 \pi \pi \) the amplitude occurs only at loop level. We find that the decay widths for these decays are \(\Gamma (B_{s1}^{0} \rightarrow B_s^0 \pi \pi )\sim 10^{-3}\,\) keV, and \(\Gamma (B_{s0}^{*0} \rightarrow B_s^0 \pi ^0) \le 55\,\) keV, \(\Gamma (B_{s1}^{0} \rightarrow B_s^{*0} \pi ^0) \le 50\,\) keV. More precise knowledge of the coupling constant describing the interaction of positive and negative parity heavy mesons with light pseudo-scalar mesons would help to increase the accuracy of our calculation.

## 1 Introduction

Two positive parity mesons \(B_s\) states: the \(J^P=1^+\) state \(B_{s1}(5830)^0\) and the \(J^P=2^+\) state \(B^*_{s2}(5840)^0\) were observed by the CDF and LHCb collaborations [1, 2, 3, 4]. Recent lattice results [5, 6], as well as other work [7, 8, 9, 10, 11, 12, 13, 14, 15], have indicated that the observed states are most likely members of the (\(1^+,2^+\)) doublet. However, the positive parity doublet of the \(B_s\) states (\(0^+,1^+\)) is still unobserved. The above-mentioned studies also suggest that the (\(0^+,1^+\)) doublet of \(B_s\) states might have masses below the *BK* and \(B^*K\) thresholds. However, some relativistic quark models analysis [16, 17, 18] suggested that masses of the (\(0^+,1^+\)) doublet \(B_s\) states should be above the *BK* and \(B^*K\) thresholds. This reminds one strongly of the “history” of establishing the charm meson spectrum in which detection of positive parity states below *DK* threshold was not predicted by the quark models. After the observation of \(D_{s 0}^*(2317)\) and \(D_{s 1}(2460)\) charmed mesons, we faced a long-lasting dilemma on the structure of \(D_s\) (\(0^+,1^+\)). The issue is whether \(D_{s 0}^*(2317)\) and \(D_{s 1}(2460)\) are \(\bar{q}q\) states or more exotic compounds [14, 15, 19]. One of the first explanations of the low mass of \(D_{s 0}^*(2317)\) was offered by the authors of [13], based on the unitarized model of *DK* mesons scattering. Similarly, the work of [14] based on the scattering of Goldstone bosons off heavy-light pseudo-scalar and vector mesons with the use of the approximate crossing symmetry of the unitarized scattering amplitude, predicted the existence of positive parity B meson states. The masses of positive parity heavy mesons were studied within the heavy meson chiral Lagrangians approach by the authors of [15, 20]. The dilemmas on the structure of positive parity charm meson states are nicely summarized in the work of [19].

It was already suggested by the authors of [19] that a study of the strong and radiative decay modes of positive parity \(D_s\) states might help in differentiating between these scenarios. Since both systems of positive parity states \(D_s\) and \(B_s\) are rather similar, the systematic analyses of strong and radiative decay dynamics of \(B_s\) mesons would help in clarifying their structure. In our study we rely on the results of lattice calculations presented in Ref. [5]. These authors determined the spectrum of \(B_s\) 1*P* states and they found that the masses of \(B_{s1}(5830)^0\) and \(B^*_{s2}(5840)^0\) agree very well with the experimental results [1, 2, 3, 4]. They predicted also the existence of the spin zero positive parity state (\(J^P= 0^+\)) with the mass \(m_{B_{s0}} =5.711(13)(19)\) GeV and the state \(J^P= 1^+\) with the mass \(m_{B_{s0}} =5.750(17)(19)\) GeV. Both states have masses below the *BK* and \(B^*K\) threshold. This immediately indicates that both states can decay strongly if isospin is violated. Motivated by the result of lattice calculation and relying on our findings in the appropriate charm sector [21], we determine the partial decay widths of both meson states to the final state containing one or two pions: \(B_{s0}^{*0} \rightarrow B_s^0 \pi ^0\), \(B_{s1}^{0} \rightarrow B_s^{*0} \pi ^0\), and \(B_{s1}^{0} \rightarrow B_s^0 \pi \pi \).

Studies of these decays were performed already by [10, 11, 12, 22, 23, 24]. The authors of [23] assumed that the positive parity \(0^+\) and \(1^+\) \(B_s\) states have a structure of *BK* molecules, accounting for the similarity with \(D_s\), and they suggest that they are rather narrow states with partial decay widths about 50–\(60\,\) keV. On the other hand, authors of [10, 11, 12, 22] used different approach based on the assumption that the decays proceed through the channels \(B_{s0}^{*0} \rightarrow B_s^0 \eta \rightarrow B_s^0 \pi ^0\) and \(B_{s1}^{0} \rightarrow B_s^{*0} \eta \rightarrow B_s^{*0} \pi ^0\) with the help of \(\eta \)–\(\pi \) mixing and predicted the partial decay widths in the range of 10–40 GeV. As already discussed in [20, 21, 25, 26] chiral loop corrections play an important role in strong decays of \(D_s\) positive parity states and their contribution to the strong decay modes can be as large as the effect of \(\eta \)–\(\pi \) mixing. Since \(\pi \) and \(\pi \pi \) in the final state of these decays are having very small momenta, both decay modes are ideal to use heavy meson chiral perturbation theory (HM\(\chi \)PT).

In this paper, we determine the isospin violating decay amplitudes of positive parity \(B_s\) mesons, members of the (\(0^+,1^+\)) doublet, using HM\(\chi \)PT. For two-body decays, there is a tree-level contribution to decay amplitude arising from the \(\eta \)–\(\pi \) mixing and loop contribution which is the divergent. The divergent loop contribution requires the regularization by the counter-terms. On the other hand, in the isospin violating two-body decays of \(D_{s0}^*(2317)\) and \(D_{s1}(2460)\) mesons, chiral loops contribute significantly [21]. This was indicated already in Ref. [27] within a different framework in which only part of the loop contributions are included in the decay amplitudes of \(D_{s0}^*(2317)\) and \(D_{s1}(2460)\). As we pointed out in [21], the isospin violating three-body decay amplitude can arise at loop level only within HM\(\chi \)PT. These loop contributions are then finite. In the case of charm decays, the ratio of the decay widths for \(D_{s1}(2460)^+\rightarrow D_s^{*+}\pi ^0\) and \(D_{s1}(2460)^+\rightarrow D_s^+ \pi ^+ \pi ^-\) is known experimentally. From this ratio we were able to constrain the finite size of the counter-terms necessary to regularize the two-body decay amplitude \(D_{s1}(2460)^+ \rightarrow D_s^{*+} \pi ^0\). The heavy quark symmetry implies the same size of counter-term contributions for the \(B_s\) system as in the case of charm mesons. Therefore, by adopting the result of the lattice calculation showing that \(B_s\) mesons, part of the (\(0^+,1^+\)) doublet, have masses below *BK* and \(BK^*\), we are able to predict their partial decay widths.

The basic HM\(\chi \)PT formalism is introduced in Sect. 2. In Sect. Theampl we calculate the decay widths of the two-body strong decays of positive parity \(B_s\) doublet (\(0^+\), \(1^+\)). In Sect. 4, the calculation of the three-body decay width \(B_{s1}^{0} \rightarrow B_s^0 \pi \pi \) decay mode will be presented, while a short conclusion will be given in Sect. 5.

## 2 Framework

*H*, while the positive parity states are entering in the field

*S*:

*P*annihilate the vector and pseudo-scalar mesons, respectively, while \(P^*_{1\mu } \) and \(P_0\) annihilate the axial-vector and scalar mesons, respectively. Within chiral perturbation theory, the light pseudo-scalar mesons are accommodated into the octet \(\Sigma =\xi ^2=e^{(2i\Pi /f)}\) with

*B*states as well as a mass splitting between vector (axial-vector) and pseudo-scalar (scalar) fields. According to [33], the mass splitting between \(B_s\) and

*B*states is \(87\,\)MeV, while the splitting between vector and pseudo-scalar states is \(45\,\)MeV. Since these splittings are much smaller than \(\Delta _{SH}\), they can be safely neglected.

The coupling constants *g*, *h*, and \(\tilde{g}\) were already discussed by several authors and determined by several methods [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. We will use recent results of the lattice QCD: \(g=0.54(3)(^{+2}_{-4})\) [39], \(\tilde{g}=-0.122(8)(6)\), and \(h=0.84(3)(2)\) [45]. The lattice study for h coupling is derived from the non-strange scalar/axial states and then SU(3) symmetry has been used to relate non-strange/strange systems. However, based on the experimental and phenomenological studies of the B meson positive parity states, it was found that these systems with and without strangeness are rather different. That indicates the possibility that the nature of the positive parity \(B_s\) system is not necessarily the same as in the case of positive parity *B* states, which can cause a difference in a value of the coupling constant *h* for these two systems. Furthermore, existing phenomenological determinations of the coupling constant h, based on light-cone sum rules [34, 38, 49] or HM\(\chi \)PT approach [26], are also performed on the non-strange systems and predict smaller values of \(h \in (0.5\)–0.6). Due to luck of lattice studies of the coupling constant *h* in the strange systems, we vary *h* in the range (0.5–0.95) in order to investigate its influence on decay widths.

Lattice results will also be used for the \(B_{s0}^*\) and \(B_{s1}\) masses, as well as \(\Delta _{SH}\) [5]: \(m_{B_{s0}}=5,711(13)(19)\,\)GeV, \(m_{B_{s1}}=5.75(17)(19)\,\)GeV, and \(\Delta _{SH}=375(13)(19)\,\)MeV.

*h*. Parameters \(\lambda ^\prime _1\) and \(\tilde{\lambda ^\prime _1}\) can be absorbed into the definition of heavy meson masses by a phase redefinition of

*H*and

*S*, while \(\lambda _1\) and \(\tilde{\lambda _1}\) split the masses of SU(3) flavor triplets of \(H_a\) and \(S_a\) [25, 26]. Therefore, only contributions proportional to \(\kappa ^\prime _1\), \(\kappa ^\prime _9\), \(\kappa ^\prime _5\), \(\delta ^\prime _2\), and \(\delta ^\prime _3\) will be explicitly included in the amplitudes.

## 3 The amplitudes and the decay widths of two-body decay modes

*h*as already noticed in [21], for the charm meson decays. The wave-function renormalization factor is responsible for this behavior.

The dependence of the decay widths on the coupling constant *h* in the range (0.5–0.95) is shown in Fig. 5. As seen from Fig. 5, the decay widths are in the range of (0.1–55) keV for the range of coupling constant \(h=0.84(3)(2)\) as found by lattice calculation [45]. This value of *h* has been obtained in the non-strange system and one might expect that the value is not necessarily the same for positive parity mesons containing a strange quark. As can be seen from Fig. 5, if one would measure larger decay widths than the above stated ones, this would indicate lower values of the coupling constant *h* and possibly a more complicated structure of the positive parity \(B_s\) mesons. Nevertheless, due to the uncertainties in the values of the counter-terms, the allowed region for the decay widths grows rapidly if we lower the *h*. Note that we use the range of values for the counter-term (0.1–1.2) as found in [21]. Hopefully, measurements of the decay widths for positive parity \(D_s\) mesons, planned in the near future, would enable us to determine counter-terms more precisely and consequently to help in achieving a better precision of our predictions. For the central value \(h=0.84\), the range is 1 keV \(\le \Gamma (B^{0}_{s1} \rightarrow B^*_s \pi ^0) \le 30\,\)keV. The decay rates for \(B^{0}_{s1} \rightarrow B^*_s \pi ^0\) and \(B^{*0}_{s0} \rightarrow B_s \pi ^0\) are almost equal, with the small difference due to the different masses of the final and initial \(B_s\) states.

Finally, we have checked the sensitivity of our results on the choice of the \(B_{s0}^*\) and \(B_{s1}\) masses as well as \(\Delta _{SH}\). If we use the lattice results for the masses from Ref. [6] instead of [5], our results are modified by about 10 %.

## 4 The three-body decays: amplitudes and decay widths

*q*is the momentum of \(B_s^+\), then \(\mathrm{d}m^2_{12}=(p_++p_-)^2\) and \(\mathrm{d}m^2_{23}=(p_-+q)^2\). In the heavy quark limit \(P^\mu =M_i v^\mu \), \(q^\mu =M_fv^\mu \), and \(\epsilon \cdot v=0\), the amplitude is simplified to the following form:

## 5 Comments and discussion

Systematically using HM\(\chi \)PT, we determine the decay widths of the isospin violating decay modes of positive parity \(B_s\) mesons: \(B_{s0}^{*0} \rightarrow B_s^0 \pi ^0\), \(B_{s1}^{0} \rightarrow B_s^{*0} \pi ^0\), and \(B_{s1}^{0} \rightarrow B_s^0 \pi \pi \). The masses of the decaying particles and the values of the coupling constants are taken from the lattice studies.

Predictions of the \(B^{*0}_{s0} \rightarrow B_s \pi ^0\) and \(B^{0}_{s1} \rightarrow B^*_s \pi ^0\) decay widths

Method | \(\Gamma (B^{*0}_{s0} \rightarrow B_s \pi ^0\)) (keV) | \(\Gamma (B^{0}_{s1} \rightarrow B^*_s \pi ^0)\) (keV) | \(\Gamma (B^{0}_{s1} \rightarrow B_s \pi \pi \)) (keV) |
---|---|---|---|

Molecule picture [23] | 46.7 | 50.1 | |

Molecule picture [27] | \(0.8\pm 0.8\) | \(1.8 \pm 1.8\) | |

Heavy quark and chiral symmetry [12] | 21.5 | 21.5 | \(\approx \) 0.05 |

7.92 | 10.36 | ||

\(^3P_0\) model [22] | 35 | 38 | |

Light-cone sum rules [24] | 7.92 | 10.36 | |

Chiral loop corrections (this work) | |||

\(h=0.79-0.89\) | <55 | <50 | \(\approx \)0.001 |

The two-body decays of \(B_{s0}^{*0}\) and \(B_{s1}^{0}\) occur at three level through \(\eta \)–\(\pi \) mixing. We find that the chiral loop corrections can significantly enhance or suppress the decay amplitudes being almost of the same order of magnitude as the tree-level contribution. We can only give a range of values for the decay widths. Namely, the decay widths are very sensitive to the value of the coupling constant *h* and change significantly if the coupling constant *h* is varied within the error bars determined by the lattice studies [45]. Also, the counter-terms are known only within a range of values in [21]. Let us note that the experimental measurement of the decay widths of the positive parity mesons \(D^*_{s0}(2317)^\pm \) and \(D_{s1}(2460)^\pm \), planned in the near future by various experiments, can lead to a more precise determination of the counter-terms [21]. Together with a detection and a measurement of the decay widths and the branching ratios of the strong decays of the positive parity \((0^+,1^+)\) \(B_s\) meson doublet, it will in principle allow one to extract the value of the coupling constant *h* exclusively from the \(D_s\) and \(B_s\) positive parity \((0^+,1^+)\) doublet system. If this value would then deviate significantly from what is found by assuming the validity of *SU*(3) symmetry or quark model considerations, it would point at an unusual nature of these states.

In Table 1, we give the results of other existing studies. The authors of [23] find higher values of the decay widths in the molecular picture of positive parity \(B_s\) states. In their approach, however, wave-function renormalization, which in our case tends to lower the decay widths significantly, is not taken into consideration. Note also that the contributions of \(K^*\) loops, present in [23], are a higher order correction in the HM\(\chi \)PT approach and therefore are not included in our analysis.

It will be interesting if current experimental searches at LHCb and planned studies at Belle II would lead to the discovery of both states \(B_{s0}^{*0}\) and \(B_{s1}^{0}\). We hope that our study might shed more light on this issue.

## Notes

### Acknowledgments

The work of SF and APB was supported in part by the Slovenian Research Agency.

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