Study of the isospin breaking decay at BESIII
Using measured branching fraction of the decay from the BESIII experiment, we estimate branching fraction of decay, which proceeds via the - mixing and the - mixing. The branching fraction is predicted to be about , which can be accessed with events collected at the BESIII. The decay is dominated by the contribution from - mixing. We find that the interference between the amplitudes due to - mixing and that due to - mixing is destructive. The branching fraction can be decreased by about owing to the interference effect. We also study the mass squared spectrum, and find that a narrow peak due to the - mixing in the mass squared spectrum should be observed. The observation of this decay in experiment will be helpful to determine the - mixing intensity and get information about the structures of the light scalar mesons.
The nature of the light scalar mesons and is still a hot topic in hadronic physics. Several models about the structure of the scalar mesons have been proposed, such as states, glueball, hybrid states, molecule states, tetra-quark states and the superpositions of these contents [1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11]. Due to the absence of convincing evidence, a final consensus has not been reached so far. Therefore, more researches both in theory and experiment are still needed.
The structure of and is closely related to the mixing of them, which was first suggested theoretically in Ref. . Its mixing intensity has been studied extensively on its different aspects and possible manifestations in various processes [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Recently, BESIII Collaboration has reported the first observation of mixing in the decays of and . In their work, the values of the mixing intensity for the transition was obtained
Here, is defined as
The theoretical calculation prefers to the solution-1 result of BESIII . Here, more works are needed to determine the final solution of .
The resonance, which deays dominantly via a intermediate state, is a vector meson, its . This resonance was first observed by BABAR Collaboration  and then confirmed by BESIII Collaboration  and Belle Collaboration . Recent result on resonance in decay from BESIII Collaboration is obtained as 
2 Two mechanisms of the decay
The isospin breaking decay can proceed via the - mixing and the - mixing. The amplitude can be written as
With , one can obtain
while, with , one can get
In BESIII analysis for the decays of and , they only assumed contribution from or mixing, which causes the isospin breaking decays. In fact, the final states of could be also induced by via - mixing. If it is the case that the isospin breaking decay is due to both and - mixings, actually the BESIII measured values for the mixing intensity in Eq. (1) has already included both effects. Therefore the results given in Eqs. (6) and (7) includes both the contributions of - mixing and - mixing physically.
As for the sole contribution of - mixing, the relative ratio of ) to ) is
where are the masses of , and , respectively. The function is
where , is the phase-space integral. While is the relevant phase-space integral that changes to in . The relative ratio of has been measured by BESIII  and CLEO Collaboration . The recent value measured by BESIII Collaboration is [42, 43]. Employing the relation , combining Eqs.(3), (8), (10) and using the particle masses taken from Table 1 and Table 2, one can obtain
3 The branching fraction
As mentioned in Eq.(2), both the - mixing and the - mixing can contribute to the decay of . The most characteristic feature of the first contribution is the narrow peak in the mass spectrum, which is due to the property of the - mixing amplitude [12, 16, 29]. As far as - mixing is concerned, however, the width in the mass spectrum should be the natural width of state, which is broad. Fortunately, the contribution from the - mixing is much smaller than that from the - mixing, so the narrow structure caused by the - mixing is expected to be observed, while the broad width from the effect of - mixing is negligibly small. The corresponding decay amplitude contributed by - mixing is [29, 45]
where , and is the coupling of to . is the invariant amplitude for the decay , which can be used to calculate the branching fraction
where is the decay width of . , and are the masses of the resonances , and , respectively. is the - mixing amplitude, and defined as
where for , , while for , , here . in Eq.(3) is the denominator for the propagator of the resonance ,
for , we have
where is the coupling of resonance to final states , , and is
As for the contribution of mixing, the transition amplitude is
where and are the couplings of to and to , respectively, which can be extracted from
where is the phase-space factor of the involved decays
here is the total decay width of . and in Eq. (3) denote the contributions from the - mixing and the - mixing, respectively, which are given in the following
Using the input parameters listed in Table 1 and Table 2, we obtain the result for the distribution curve of the mass squared spectrum for decay, which is shown in Fig. 3. In this figure, the narrow peak due to the - mixing can be clearly observed.
|[50, 51]||[50, 51]|
|[1, 49]||[1, 49]|
Furthermore, the branching fraction of the decay is obtained by performing the integration in the effective region , and the result is
where we have considered the errors of the mass and width of and , the errors from the branching fractions of the decays and as well as uncertainty from the branching fraction of the decay . The contribution from the - mixing dominates the predicted branching fraction. In additional, the interference of the amplitudes from the - mixing and mixing is destructive, the branching fraction is decreased by about owing to the interference effect.
4 Prospects for the measurement at BESIII
The final states , and in the cascade decay process are reconstructed through the decays , and . By employing the data reported by the Particle Data Group , we obtain
Because of the narrow peak near the thresholds in the invariant mass spectrum, the event selection criteria for the candidates has high efficiency. In addition, the final states contain six photons and two charged tracks, the detection efficiency for decay can be as large as after the final selection [37, 52, 42, 53]. The BESIII experiment will accumulate huge data sample of decays by the end of 2019 [53, 54, 55]. Therefore about 80 events for the decay of are expected in the decay sample at the BESIII. Therefore, the isospin breaking decay will be helpful to determine the final value of in addition to the process .
Basing on the branching fraction of the decay and the - mixing intensity measured recently by the BESIII, we study the isospin violation decay , which proceeds via the - mixing and the - mixing. It is found that the decay can reach a branching fraction of the order of , which can be accessed with events collected at BESIII by the end of 2019. The contribution from the - mixing dominates the decay. The interference between the amplitude caused by the - mixing and the amplitude caused by the - mixing is destructive, the branching fraction will be decreased by about because of the interference effect between the two mixings. In the distribution of the mass square spectrum, we find that the narrow peak due to - mixing should be expected, and the effect on the peak from mixing is negligibly small. This decay will be complementary to the decay , which will be helpful to determine the final solution of the - mixing intensity and understand the nature of the light scalar mesons.
This work is supported in part by the National Natural Science Foundation of China under Contracts Nos. 11335009, 11125525,11675137, 11875054, the Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1532257, CAS under Contract No. QYZDJ-SSW-SLH003, and the National Key Basic Research Program of China under Contract No. 2015CB856700. X. Cheng and R. Wang are supported by Nanhu Scholars Program of XYNU.
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