Search for the $a_0(980)-f_0(980)$ mixing in weak decays of $D_s/B_s$ mesons

Scalar mesons $a^0_0(980)$ and $f_0(980)$ can mix with each other through isospin violating effects, and the mixing intensity has been predicted at the percent level in various theoretical models. However the mixing has not been firmed established on the experimental side to date. In this work we explore the possibility to extract the $a_0-f_0$ mixing intensity using weak decays of heavy mesons: $D_s\to [\pi^0\eta, \pi\pi] e^+\nu$, $B_s\to [\pi^0\eta, \pi\pi]\ell^+\ell^-$ and the $B_s\to J/\psi [\pi^0\eta,\pi^+\pi^-]$ decays. Based on the large amount of data accumulated by various experimental facilities including BEPC-II, LHC, Super KEKB and the future colliders, we find that the $a_0-f_0$ mixing intensity might be determined to a high precision, which will lead to a better understanding of the nature of scalar mesons.


I. INTRODUCTION
Light scalar mesons below 1GeV play an important role in understanding the QCD vacuum since they share the same quantum numbers J P C . But due to the nonperturbative nature of QCD at low energy the internal structure of scalar mesons is extremely complicated and still under controversy. They have been interpreted as quark-antiquark, tetra-quarks, hadronic molecule, quark-antiquark-gluon hybrid, and etc [1].
Among various phenomena, it is anticipated that the mixing between the a 0 0 (980) and f 0 (980) resonances may shed light on the nature of these two resonances, and therefore has been studied extensively on different aspects and in various processes. For an incomplete list of discussions in the literature, please see Refs.  and references therein. To date no firm experimental determination on this quantity is available yet. The possibility of extracting the a 0 0 (980)-f 0 (980) mixing from the J/ψ → φa 0 0 (980) → φηπ 0 reaction has been explored in Refs. [17,18]. This reaction is an isospin breaking process with the initial state of isospin 0 and the final state of isospin 1. BES-III collaboration has used this process to determine the mixing [26]: where the uncertainties are statistical, systematics due to the measurement and the parametrization, respectively. As one can see, the statistical significance is only about 3.4σ. To more precisely determine the mixing intensity, two parallel researches can be conducted in the future. On the one hand, one may collect more data on the J/ψ (and ψ ′ ) and accordingly the errors in this quantity can be reduced significantly. On the other side, one may look for new channels that can be used to determine the mixing parameter. This will also provide a cross-check of the results derived from the J/ψ decays. In this work, we will focus on the latter category. Weak decays of heavy mesons are not only of great value to determine the standard model parameters (see Ref. [27] for a recent review), but can also provide an ideal platform to study hadron structures [28]. In the following, we will examine the possibility to extract the mixing intensity from the rare decays of D s and B s : D s → [π 0 η, ππ]e + ν, B s → [π 0 η, ππ]ℓ + ℓ − and the B s → J/ψ[π 0 η, π + π − ] decays. An advantage in these modes is that the lepton (or the J/ψ) is an iso-singlet system and thus there is a natural isospin filter. At the quark level, the intermediate state has I = 0. It should be noticed that the semileptonic D s and B s decays into the π + π − via the f 0 (980) have already been observed by CLEO-c [29][30][31] and LHCb collaboration [32], respectively. The branching fraction of the B s → J/ψf 0 (980) → J/ψπ + π − is also measured in Refs. [33][34][35][36][37][38][39][40][41].
The rest of this paper is organized as follows. In Sec.II, we will give a brief overview of the a 0 − f 0 mixing mechanism. We will discuss the mixing effects in B s and D s decays in Sec. III. A short summary is presented in the last section.
II. THE f 0 (980) − a 0 (980) MIXING MECHANISM For the nearly degenerate a 0 0 (980) with isospin 1 and f 0 (980) with isospin 0, both can couple to the KK state, but the charged and neutral kaon thresholds are different by about 8 MeV. This difference leads to the a 0 0 (980)-f 0 (980) mixing. In the following we will use the abbreviation a 0 and f 0 to denote the a 0 0 (980) and f 0 (980) for simplicity. For illustration, we consider the propagation of the f 0 (980) and include the loop corrections through two pseudo-scalars M 1 and M 2 . The one-loop corrections are shown in Fig 1. If one sums these loop corrections in the chain approximation, the f 0 (980) propagator will become: with the loop corrections Here the g f 0 M 1 M 2 denotes the coupling of the f 0 with the M 1 , M 2 . The real part of the M 2 will renormalize the bare mass, leading to the pole in the propagator as the physical mass. The remanent multiplicative constant in the real part is absorbed by the field strength renormalization factor. The imaginary part of the M 2 will result in a nonzero mass-dependent decay width: With the incorporation of the mixing effects, we have the a 0 /f 0 propagator: where D a and D f are the denominators of the resummed propagators for the a 0 0 (980) and f 0 (980), respectively: Since the mixing term is already small at leading order, it is not necessary to sum all order corrections. We have the expression for the D af : The relation between the D af and the mass-dependent f 0 → a 0 mixing parameter ξ is given as: As one can see, the mixing parameter arises due to the different masses of the charged and neutral Kaon. The results also rely on the couplings g a 0 0 (980)K + K − , g f 0 (980)K + K − and the mass pole position in the propagator. Various theoretical models predict different values for these quantities, and a thorough discussion has been presented in Refs. [17,18].

III. MIXING EFFECTS IN THE B s AND D s DECAYS
In this section, we will analyze the mixing intensity in the semileptonic decays of B s and D s mesons. More explicitly, the considered decay processes include B s → π 0 ηJ/ψ, B s → ππJ/ψ.
We will take the D s decay as the example, whose Feynman diagram is shown in the panel (a) of Fig. 2. After emitting the off-shell W -boson, the hadronic sector is thess which will couple to the iso-singlet component f 0 (980). Then the decay amplitudes for the D s → ππe + ν ≡ D s → f 0 e + ν → ππe + ν and D s → π 0 ηe + ν ≡ D s → f 0 e + ν → a 0 0 e + ν → π 0 ηe + ν are given as where the amplitudeÂ can be expressed in terms of the transition form factors: The double differential decay width is then derived as dΓ(D s → π 0 ηe + ν) where q 2 is the invariant mass of the lepton pair, and the s is the invariant mass square of the two pseudo-scalars. Here G F is the Fermi constant, V cs is the CKM matrix element, and the Källen function λ is: λ = m 4 Ds + s 2 + (q 2 ) 2 − 2(m 2 Ds q 2 + m 2 Ds s + sq 2 ). Since in this work we are interested in the mixing intensity in the a 0 (980) − f 0 (980) resonance region, one may integrate out the q 2 first, leading to where the coefficient C is obtained via the integration over q 2 . The mass-dependent mixing intensity can be defined as while in experiments one can directly measure the integrated mixing intensity: Here the s max denotes the lower and upper invariant mass cuts. In the previous BES-III analysis of the mixing intensity using the J/ψ decays [26], the mass of the mixing signal is set to 991.3 MeV at the center of charged and neutral kaon thresholds, and the width of the mixing signal is set to 8 MeV. It corresponds to For the f 0 (980), one may follow the BES-III analysis of the J/ψ → φπ + π − [42]: With the meson masses (in units of MeV) taken from Particle Data Group [1] m K + = 493.677, m K 0 = 497.614, m π 0 = 134.9766, m η = 547.862, we update the predictions for the mixing intensity ξ f a (s) at √ s = 991.3 MeV and give the results for the integrated quantity ξ f a with the kinematics in Eqs. (21) and (22) in table I. In the calculation, the isospin symmetry has been used for the ππ sytem. Results for the ξ f a are consistent with Refs. [17,18]. As one can see from this table, most predictions for the integrated mixing intensity are at the percent level.
The CLEO collaboration has firstly measured the branching fraction [30]: but a recent analysis based on the CLEO-c data gives a similar result with a smaller central value [31]: In near future the BES-III collaboration will collect about 3f b −1 data in e + e − collision at the energy around 4.18GeV [55]. This corresponds to a few times 10 6 events of the D s mesons and accordingly 3 MeV, which is at the center the K + K − and K 0K 0 threshold. The integrated mixing intensitȳ ξ f a (in unit of %) is evaluated by Eq. (20) with the kinematics in Eqs. (21) and (22). a few thousand events for the D s → π + π − e + ν decay before any kinematics cut. As we can see if the mixing intensity is at the percent level, there is a promising prospect to measure/constrain the mixing by BES-III collaboration using the D s → [π 0 η, π + π − ]e + ν.
Since much more data will be collected by experimental facilities including the LHCb detector [61] the Super-B factory at the KEK [62], it is likely to precisely derive the a 0 (980) and f 0 (980) mixing from these weak decays of heavy mesons.

IV. SUMMARY
To understand the internal structure of light scalar mesons is a long-standing problem in hadron physics. It is expected that some aspects can be unraveled by the study of a 0 0 (980)−f 0 (980) mixing. The two scalar mesons can couple to the K −K and will mix with each other due to the different masses for the charged and neutral kaons. The mixing intensity has been predicted at the percent level in various theoretical models. A number of processes have been proposed to study the mixing, but to date there is no firm evidence on the experimental side.
In this work we have proposed to use the weak decays of the B s and D s mesons to study the a 0 − f 0 mixing. We have studied the semileptonic decays of heavy mesons, D s → [π 0 η, ππ]e + ν, B s → [π 0 η, ππ]ℓ + ℓ − and the B s → J/ψ[π 0 η, π + π − ] decays. Based on the large amount of data accumulated by various experimental facilities including BEPC-II, Super KEKB, LHC and the future colliders like the High Intensity Electron Positron Accelerator (HIEPA) expected running at 2 − 7 GeV with the designed luminosity of 10 35 cm −2 s −1 , the Z-factory running at Z-pole and the circular electron-positron collider (CEPC), it is very likely that the a 0 − f 0 mixing intensity can be determined to a high precision, which will lead to a better understanding of the nature of scalar mesons.