Investigating $\eta^\prime_{1}(1855)$ exotic states in $J/\psi\to\eta^\prime_{1}(1855)\eta^{(\prime)}$ decays

An analysis of the $J/\psi\to \eta\eta^\prime\gamma$ decay by the BESIII collaboration claims the observation of an exotic state $\eta_1(1855)$ with $I^GJ^{PC}=0^+1^{-+}$. To establish its C-parity partner $\eta^\prime_{1}(1855)$ in the picture of the $K \bar{K}_1(1400)$ molecular state, we propose that $J/\psi\to\eta^\prime_{1}(1855)\eta^{(\prime)}$ receives the main contributions from the final state interaction of $KK^*$($K^+ K^{*-}$, $K^- K^{*+}$, $K^0 \bar{K}^{*0}$, and $\bar{K}^0 K^{*0}$). Specifically, $K$ and $K^*$ in $J/\psi\to KK^*$ decays transform as $\eta^\prime_{1}(1855)\eta^{(\prime)}$, by exchanging a $K_1(1400)$. We predict ${\cal B}(J/\psi\to \eta^{\prime}_1(1855)\eta)=(6.3^{+12.6}_{-3.5})\times 10^{-6}$, and ${\cal B}(J/\psi\to \eta^{\prime}_1(1855)\eta^{\prime}) =(6.5^{+6.6}_{-4.6})\times 10^{-6}$, which can be studied in the $J/\psi\to K^*\bar{K}^*\eta^{(\prime)}$ decays.


I. INTRODUCTION
Although most conventional hadrons are mesons or baryon, Quantum Chromodynamics actually allows the existence of other types of states, called exotic states as long as the color confinement is satisfied. One decisive way to judge whether a meson is exotic states or not is to examine its J P C , for which conventional mesons can't have quantum numbers J P C = 0 −− , (even) +− , and (odd) −+ . The BESIII Collaboration has recently observed a new state η 1 (1855) ≡ η 1 with quantum numbers J P C = 1 −+ on the ηη ′ invariant mass spectrum of the J/ψ → ηη ′ γ decay [1,2] and determined the mass and width to be The J P C of η 1 unambiguously indicates it is an exotic state. However, it deserves more efforts to further determine which type of exotic states the η 1 is.
Many theoretical hypotheses interpreting the nature of η 1 have been proposed immediately after its observation, such as anssg isoscalar hybrid meson [3][4][5][6][7] or a tetraquark state [8], but the mass' being around the threshold of total mass of K andK 1 (1400) ≡K 1 makes the η 1 more naturally to be interpreted as a KK 1 +c.c. molecular state [9][10][11]. (KK 1 denotes the various combinations K + K − 1 , K − K + 1 , K 0K 0 1 , andK 0 K 0 1 in the following.) Reference [10] has showed the binding energies of the isoscalar KK 1 (1400) are all negative in various situations in its Fig. 2 and proved the attractive force between K andK 1 , by exchanging mesons, is strong enough to form a bound state using the one-boson-exchange model. This shows the newly discovered η 1 could be the candidate of a KK 1 molecular state with J P C = 1 −+ . At the same time, the molecular model uniquely predicted that η 1 (1855) should have a C-parity partner with [10,11]. Hence, examining existence of the η ′ 1 is very important to decide whether the η (′) 1 are molecular states or not.
In this paper, we analyze the productions of η (′) 1 , assuming they are KK 1 bound states, in the J/Ψ → η ′ 1 η (′) decays. In principle, all the possible bases that can connect the initial state J/Ψ and final state η ′ 1 η (′) should be considered. The direct estimations of these production process at the quark level are difficult, but the loops composed by hadrons can be regarded as the major contributions as indicated in Refs. [12,13]. The dominant diagrams contributing to J/Ψ → η ′ 1 η (′) , as depicted in Fig. 1, can give large enough branching ratios to be observed. The η ′ 1 could be produced through the final state interaction in the J/ψ → KK * decay (KK * = K + K * − , K − K * + , K 0K * 0 , andK 0 K * 0 ), followed by the K-K * rescattering. The K and K * then transform to η ′ 1 η (′) with the K 1 exchange in the triangle-rescattering process. The size of the contribution from this triangle-rescattering effect highly depends on the couplings of involved intermediate interactions, which are crucial and required in calculation of the triangle loop. Fortunately, the branching fractions of the J/ψ → KK * and K 1 → K * π decays have been measured to be at 10 −2 level and almost 100%, respectively [14], implying a strong coupling constant g K 1 K * η (′) with the helping of the SU(3) flavour symmetry. In addition, the η ′ 1 , as a candidate of a KK 1 molecular, should couple to KK 1 strongly [10]. Therefore, we investigate the J/ψ → η ′ 1 η (′) decays in the molecular model in this work and show that they are anticipated to be accessible in the BESIII experiment.

II. FORMALISM
In this section, we analyze the J/ψ → η ′ 1 η (′) decay in the molecular model. Its trianglerescattering process, as depicted in Fig. 1, can be separated into three parts:J/ψ → KK * , The first part is J/ψ → KK * . The relevant Lagrangian term is [15,16] where g ψKK * is the coupling constant for the J/ψ → KK * decay and ǫ ν ψ (ǫ ν K * ) is the polarization four-vector of the J/ψ(K * ). We derive this amplitude to be The relations between various g ψKK * can be given by SU (3) flavour symmetry [15,16]: where g 8 , g M 88 , g E 88 and δ E are the coupling constants of the octet term, the mass-breaking term, and the electromagnetic-breaking term, and the phase angle between electromagnetic and strong interaction, respectively.
The second part is η ′ 1 → KK 1 . The corresponding Lagrangian term is and the amplitude is derived to be where p E is the Euclidean Jacobi momentum and g η ′ can be parameterized as a Gaussian form vertex function [17,18] or a pole form vertex function [19][20][21]: with Λ 1 is the size parameter. In this paper, the Gaussian form will cause the integration divergence and, alternatively, we choose the pole vertex form correlation function. We determine the coupling constants between the hadronic molecule and its components using the pole vertex form by the consequence of the Källén-Lehmann representation (see Eqs. [22][23][24][25][26][27] and the triangle diagram can be calculate by the ′ tHooft-Veltman technique [22] (see Note that using the Gaussian form is more common than the Pole form to determine the coupling constant by the compositeness condition in an one loop diagram [23]. However, calculating triangle diagram with the Gaussian form may have singularities, called anomalous thresholds [24]. A parameter z loc has been proposed [25] to prejudge whether there are anomalous thresholds. The parameter z loc describes the triangle diagram in which A particle decay into B and C particles with a, b, and c particles as propagators. Take Fig. 1 for example, A/B/C = J/Ψ, η ′ 1 , η (′) and a/b/c = K/K 1 /K * . Then, z loc is given by with α a + α b + α c = 1 and α i ≥ 0 (i = a, b, c). There will be no anomalous thresholds if z loc is always positive. 1 In this paper, the set of masses (M A , M B , M C , m a , m b .m c ) = (m J/Ψ , m η ′ 1 , m η (′) , m K , m K 1 , m K * ) make z loc not always positive, implying divergence will happen if the Gaussian form is used.
The third part is K 1 → K * η (′) , whose Lagrangian term and amplitude are written as and where and where (a, b) are the parameters for coupling constants and (θ, φ) are the mixing angles of [26]. Eventually, the amplitude of the triangle-rescattering process for the J/ψ → η ′ 1 η (′) decay is obtained by where 4 i=1 sums over all possible Feynman diagrams for K is the monopole form factor [27][28][29], which can be adopted to represent the off-shell effect by exchanging K 1 mesons and also plays the role of avoiding integration divergences. Besides, q 2 = p η ′ 1 − q 1 and q 3 = p ψ − q 1 correspond to the momentum flows in Fig. 1. In the general form, one can expresses the amplitude as To obtain g ψη ′ 1 η (′) , one needs to integrate over the variables of the triangle loop in Eq. 13, which gives The propagators of K * and K 1 are supposed to contain vector and tensor structures, but contributions from the tensor structures will be zero due to the term ǫ µνρσ in Eq. 13. Hence, we only consider the vector structures here. The vector four-point function is written as such that one obtains with the linear combination of scalar point functions The D 2 term doesn't contributed to M(J/ψ → η ′ 1 η (′) ) due to ǫ µναβ p µ ψ p ν ψ = 0. The one loop scalar 3-and 4-point functions can be calculated with the ′ tHooft-Veltman technique [22,[30][31][32][33][34] and are given by As for the term D ′ρ , one can obtain D ′ρ = p ρ η ′ 1 D ′ 1 +p ρ ψ D ′ 2 by replacing m K 1 in Eqs. (17)(18)(19) with Λ 2 . Next, one can derive g ψη ′ 1 η (′) by comparing Eqs. 14 and 15 and definingD 1 = D 1 − D ′ 1 : At this point, all parameters relevant to M(J/ψ → η ′ 1 η (′) ) are given except g η ′ 1 KK 1 , which can be determined by a consequence of the Källén-Lehmann representation (see the discussion below). Upon resummation of the one-loop contributions, as in Fig. 2, the propagator of η ′ 1 takes the form The metric tensor term, g µν , already provides enough information to determine the coupling g η ′ 1 KK 1 and the rest can be ignored (denoted as [...]). The spectral function of the state η ′ 1 in the Källén-Lehmann representation can be obtained as the imaginary part of the propagator [19,20]: and the normalization is required to be satisfied: In the above equation, we define Σ µν (p 2 ) = −g µν Σ(p 2 ) + ... , and can give where and The normalization of the spectral function of the η ′ 1 , Eq. 23, causes a constraint in the Källén-Lehmann representation. With this constraint, the g η ′ 1 KK 1 term in Eq. 26 can be evaluated analytically with the ′ tHooft-Veltman technique. More specifically, one can obtain an analytical expression for d η ′ 1 (p 2 ) by substituting Eqs. 24-27 into Eq. 22. After integrating out the momentum p 2 in Eq. 23, one can determine the relationship between g η ′ 1 KK 1 and Λ 1 , as shown in Fig. 3. In this relationship, the masses of the relevant mesons are the only input parameters.
Note that I 2a , I 2b , or I 2c itself possesses logarithmic divergence. However, Eq. 25 doesn't diverge because the divergence will cancel by I 2a − I 2c and I 2b − I 2c .
2 Parameters (g 8 , g M 88 , g E 88 ), and δ E in SU(3) flavor symmetry theory describe the J/ψ to pseudoscalarvector decays, e.g. J/ψ → ρπ, J/ψ → ωπ, and J/ψ → K * K. They are extracted from fitting to measured branching ratios. Approximately, we present the resonant branching fractions as and, with B(η ′ 1 → K * K * ) ≃ 22% [10], predict The uncertainty is evaluated by repeating the calculation after varying the parameters according to their uncertainties. The uncertainty associated with Λ 1 dominates. The larger Λ 1 yields the lower predicted branching fractions, but when Λ 1 is set to be in the typical region, the predicted branching fractions remains being of O(10 −6 ).
The BESIII collaboration has collected about 10 billion J/ψ events at √ s = 3.097 GeV and can reach the sensitivity of 10 −7 -10 −6 level for branching fractions of J/ψ decays. Our proposal is shown to be accessible by the BESIII experiment.