Can the nature of a_0(980) be tested in the D_s^{+}\to π^{+}π^0 η decay?
Xi-Zhe Ling, Ming-Zhu Liu, Jun-Xu Lu, Li-Sheng Geng, Ju-Jun Xie
aa r X i v : . [ h e p - ph ] F e b Can the nature of a (980) be tested in the D + s → π + π η decay? Xi-Zhe Ling, Ming-Zhu Liu,
2, 1
Jun-Xu Lu, Li-Sheng Geng,
1, 3, 4, ∗ and Ju-Jun Xie
5, 6, 4, † School of Physics, Beihang University, Beijing 100191, China School of Space and Environment, Beihang University, Beijing 100191, China Beijing Key Laboratory of Advanced Nuclear Materials and Physics,Beihang University, Beijing 100191, China School of Physics and Microelectronics,Zhengzhou University, Zhengzhou, Henan 450001, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Science and Technology,University of Chinese Academy of Sciences, Beijing 101408, China ( Dated: February 11, 2021)
Abstract
From the amplitude analysis of the D + s → π + π η decay, the BESIII Collaboration firstly observed the D + s → a (980) + π and D + s → a (980) π + decay modes, which are expected to occur through the pure W -annihilation processes. The measured branching fraction B [ D + s → a (980) +(0) π , a (980) +(0) → π +(0) η ] is, however, found to be larger than those of known W -annihilation decays by one order of magni-tude. This apparent contradiction can be reconciled if the two decays are induced by internal W -conversionor external W -emission mechanisms instead of W -annihilation mechanism. In this work, we propose thatthe D + s decay proceeds via both the external and internal W -emission instead of W -annihilation mech-anisms. In such a scenario, we perform a study of the D + s → π + π η decay by taking into account thecontributions from the tree diagram D + s → ρ + η → π + π η and the intermediate ρ + η and K ∗ ¯ K/K ¯ K ∗ triangle diagrams. The intermediate a (980) state can be dynamically generated from the final state inter-actions of coupled K ¯ K and πη channels, and it is shown that the experimental data can be described fairlywell, which supports the interpretation of a (980) as a molecular state. ∗ [email protected] † [email protected] . INTRODUCTION The charmed meson weak decays into light mesons provide a very good channel to studymeson-meson interactions at low energies and the nature of the low-lying scalar mesons [1–4].We refer to Ref. [5] for a review about the study on the interactions of light hadrons from theweak decays of the D mesons. Very recently, in Ref. [6], the BESIII Collaboration firstly re-ported on the observation of the decay modes of D + s → a (980) + π and D + s → a (980) π + in the amplitude analysis of the D + s → π + π η decay. The D + s → a (980) +(0) π decays areclaimed as W -annihilation dominant processes. However, the measured absolute branching frac-tion of B [ D + s → a (980) +(0) π , a (980) +(0) → π +(0) η ] = (1 . ± . sta . ± . sys . )% isfound to be larger than those of normal W -annihilation processes by at least one order of mag-nitude. In Ref. [7] it was proposed that the D + s → π + π η decay actually occurs via an internal W -conversion process, where πK ¯ K was produced at the first step, and then the πη state is gener-ated from final state interaction of K ¯ K in S -wave and isospin I = 1 , and the a (980) resonanceis dynamically generated via ¯ KK and πη coupled channel interactions as described in the unitarychiral theory [8, 9]. The main purpose of Ref. [7] is to get the a (980) signal, and thus, onlythe experimental data with the cut M π + π − > GeV were studied, for which the tree diagram D + s → ρ + η → π + π η does not contribute. On the other hand, in Ref. [11] it was shown that theexperimental measurements can also be described as an external W -emission process D + s → ρ + η .Then the ρ meson decays into a pair of ππ and the πη pair fuses into a (980) , which subsequentlydecays into πη again. As both the W -internal conversion and W -external emission processes arebelieved to be larger than the W -annihilation process, the puzzle seems to be resolved, thoughthe two theoretical studies seem to give conflicting results regarding the responsible weak decaymechanism.In the present work, we revisit this issue and argue that the two theoretical works are notnecessarily contradicting with each other. As a matter of fact, the triangle mechanism of Ref. [11]may offer a way to estimate the unknown weak decay coupling between D + s and ¯ KK ∗ ( ¯ K ∗ K ) ,i.e., the first vertex of the triangle diagrams for D + s → K + ¯ K ∗ /K ∗ + ¯ K → π + π η .At the quark level, the decay of D + s → ρ + η proceeds through external W -emission c → ¯ sW + as shown in Fig 1(a). According to the review of the Particle Data Group [12], the absolutebranching fractions of the decay modes D + s → K ∗ + ¯ K and D + s → K + ¯ K ∗ are (5 . ± . and (5 . ± . respectively, which are comparable to the absolute branching fraction of D + s → + η that is (8 . ± . . As a result, if D + s → ρ + η can contribute to the D + s → a +(0)0 π [ a ≡ a (980) ] process via the triangle diagrams as in Ref. [11], the D + s → K + ¯ K ∗ / ¯ K K ∗ + processes shown in Fig 1(b) can also contribute via the triangle diagrams, where the K ∗ decaysinto Kπ and K ¯ K produce the πη pair through final state interactions, from which the a (980) resonance can be produced. The latter has not been considered either in Ref. [7] or in Ref. [11]. As a result, in this work we will consider both D + s → ρ + η and D s → K + ¯ K ∗ /K ∗ + ¯ K inducedtriangle diagrams, which lead to π + a and π a +0 final states. FIG. 1. (a) External W -emission mechanism for D + s → ρ + η and (b) internal W -conversion mechanismsfor D + s → K + ¯ K ∗ /K ∗ + ¯ K . Compared to Ref. [11], we make a further improvement. It is well known that the a (980) state does not behave like a normal Breit-Wigner resonance, because of the closeness of the K ¯ K threshold. It can be dynamically generated as a molecular state from the K ¯ K and πη coupledchannel interactions in the chiral unitary approach [8, 13–16]. In this work, we then investigatewhether with the chiral unitary amplitudes one can describe the BESIII data [6].The article is organized as follows. In Sec. II, we lay out the theoretical formalism. In Sec. IIIwe show our theoretical results and discussions are also given comparing with the experimentaldata from Ref. [6]. We summarize in Sec. IV. As we will show later, the anomalously large a πη ′ coupling adopted in Ref. [11] helps to increase the branchingfraction to meet the experimental number. IG. 2. Tree-level diagram for D + s → ηρ + → ηπ π + . II. THEORETICAL FORMALISM
To calculate the decay width of D + s → ηπ π + , we consider the contribution from both thetree-level diagram of Fig. 2 and the triangle diagrams of Fig. 3 and Fig. 4. In the process de-scribed by the tree-level diagram, D + s first decays into ρ + and η , then ρ + decays to π + π as shownin Fig 2. As pointed out in Ref. [11], the π + / π meson can interact with the η meson to form a (980) + /a (980) via the triangle diagrams shown in Fig. 3. On the other hand, if the processesdepicted in Fig. 3 can occur, those depicted in Fig. 4 can also occur, because 1) the branching frac-tions of D s decaying into K ∗ ¯ K/K ¯ K ∗ are comparable to that of D s decaying into ρη and 2) K ¯ K is a dominant channel to which the a (980) state couples. As a result, their contributions cannot beneglected. In this work, therefore, we also consider the contribution from D + s → K ∗ + ¯ K /K + ¯ K ∗ ,which can proceed through the triangle mechanisms shown in Fig 4. FIG. 3. Triangle rescattering diagrams for D + s → ( ρ + η → ) π + π η . The follwing effective Hamiltonian is needed to describe the D + s → ρ + η and D + s → K ∗ + ¯ K /K + ¯ K ∗ IG. 4. Triangle rescattering diagrams for D + s → ( K ∗ + ¯ K → ) π + π η and D + s → ( K + ¯ K ∗ → ) π + π η . processes, H eff = G F √ V cs V ud [ c eff (¯ ud )(¯ sc ) + c eff (¯ sd )(¯ uc )] , (1)where G F = 1 . × − GeV − is the Fermi constant, V cs and V ud are the CKM matrix elements, c eff , are the effective Wilson coefficients, and ¯ q q stand for vector and axial vector currents, ¯ q γ µ (1 − γ ) q [17–21]. The amplitude of D + s → ρ + η and D + s → K ∗ + ¯ K /K + ¯ K ∗ can be writtenas the products of two hadronic matrix elements [22, 23] A (cid:0) D + s → ηρ + (cid:1) = G F √ V cs V ud a (cid:10) ρ + | (¯ ud ) | (cid:11) (cid:10) η | (¯ sc ) | D + s (cid:11) (2) A (cid:0) D + s → K ∗ + ¯ K (cid:1) = G F √ V cs V ud a (cid:10) ¯ K | (¯ sd ) | (cid:11) (cid:10) K ∗ + | (¯ uc ) | D + s (cid:11) , (3) A (cid:0) D + s → ¯ K ∗ K + (cid:1) = G F √ V cs V ud a ′ (cid:10) ¯ K ∗ | (¯ sd ) | (cid:11) (cid:10) K + | (¯ uc ) | D + s (cid:11) . (4)where a = c eff + c eff /N c and a = c eff /N c + c eff with N c the number of colors. It should benoted that a can be calculated in the naive factorization approach, but a and a ′ cannot be easilyobtained within the factorization approach [24, 25]. In the present work we will fix all of them byfitting directly to data.The current matrix elements between a pseudoscalar meson or vector meson and the vacuum5ave the following form: (cid:10) ρ + | (¯ ud ) | (cid:11) = m ρ f ρ ǫ ∗ µ , (cid:10) ¯ K | (¯ sd ) | (cid:11) = − f ¯ K p µ ¯ K , (cid:10) ¯ K ∗ | (¯ sd ) | (cid:11) = m ¯ K ∗ f ¯ K ∗ ǫ ∗ µ , (5)where f K , f K ∗ , and f ρ are the decay constants for K , K ∗ , and ρ mesons, respectively, and ǫ ∗ µ isthe polarization vector of ρ or K ∗ meson. In this work, we take f ρ = 210 MeV, f K = 158 MeV, f K ∗ = 214 MeV as in Refs. [22, 26, 27].The hadronic matrix elements can be written in terms of form factors as follows [28] (cid:10) η | (¯ sc ) | D + s (cid:11) = ( p D s + p η ) µ F + ( q ) + q µ F − ( q ) , (6) (cid:10) K ∗ + | (¯ uc ) | D + s (cid:11) = ǫ ∗ α m D s + m K ∗ + (cid:2) − g µα P · q ′ A (cid:0) q ′ (cid:1) + P µ P α A + (cid:0) q ′ (cid:1) (7) + q ′ µ P α A − (cid:0) q ′ (cid:1) + iε µαβγ P β q ′ γ V (cid:0) q ′ (cid:1)(cid:3) , (8) (cid:10) K + | (¯ uc ) | D + s (cid:11) = ( p D s + p K + ) µ F + ( q ′′ ) + q ′′ µ F − ( q ′′ ) , (9)where q µ ( ′ , ′′ ) represent the momentum of ρ , K , and K ∗ mesons, respectively, and P µ = ( p D s + p K ∗ ) µ . The form factors of F ± ( t ) , A ( t ) , A + ( t ) , A − ( t ) , and V ( t ) with t ≡ q ( ′ , ′′ )2 can be parame-terized as [28] X ( t ) = X (0)1 − a (cid:0) t/m D s (cid:1) + b (cid:0) t /m D s (cid:1) . (10)In this work, we take these form factors F ± , A , ± and V from Ref. [28]: ( F + (0) , a, b ) D s → η =(0 . , . , . , ( F + (0) , a, b ) D s → K = (0 . , . , . , ( A + (0) , a, b ) D s → K ∗ = (0 . , . , . , ( A (0) , a, b ) D s → K ∗ = (1 . , . , − . , and ( A − (0) , a, b ) D s → K ∗ = ( − . , . , . . Notethat the terms containing V ( q ′ ) and F − ( q ( q ′′ )) do not contribute to the processes we study here.For the strong decays ρ + → π + π and K ∗ + → Kπ ( ¯ K ∗ → ¯ Kπ ), the amplitudes are A (cid:0) ρ + → π + π (cid:1) = g ρππ ǫ · ( p π + − p π ) , A ( K ∗ → Kπ ) = g K ∗ Kπ ǫ · ( p π − p K ) , (11)where g ρππ and g K ∗ Kπ denote the ρ coupling to ππ and the K ∗ coupling to Kπ . With the massesof these particles and the partial decay widths of ρ → ππ and K ∗ → Kπ quoted in the PDG [12],we obtain g ρππ = 6 . , g K ∗ + Kπ = 3 . , and g K ∗ Kπ = 3 . . On the other hand, the couplings a = 0 . , a = 1 . and a ′ = 1 . are determined by fitting them to the experimental branchingfractions B ( D + s → ρ + η ) = (8 . ± . , B ( D + s → K ∗ + ¯ K ) = (5 . ± . and B ( D + s → K + ¯ K ∗ , ¯ K ∗ → K − π + ) = (2 . ± . quoted in the PDG [12]. It should be stressed that the partial decay widths determine only the absolute value of the corresponding couplingconstants, but not their phases. In this work, we assume that they are real and positive, which seems to be areasonable choice given the reasonable description of the experimental data as shown below. D + s → π + π η from thetree-level diagram of Fig. 2 as A tree1 = i A ( D + s → ηρ + ) A ( ρ + → π + π ) m − m ρ + + im ρ + Γ ρ + , (12)with m = ( p π + p π + ) the invariant mass squared of the π + π system.Next, we can write the total decay amplitude of D + s → π + π η for those triangle diagramsshown in Fig. 3, A ρη = A ρηa + A ρηb , (13) A ρηa = t πη → πη ( m ) Z d q (2 π ) i A ( D + s → ηρ + ) A ( ρ + → π + π ) (cid:16) q − m ρ + + im ρ + Γ ρ + (cid:17) (cid:0) q − m η + iǫ (cid:1) (cid:0) q − m π (cid:1) , (14) A ρηb = t πη → πη ( m ) Z d q (2 π ) i A ( D + s → ηρ + ) A ( ρ + → π π + ) (cid:16) q − m ρ + + im ρ + Γ ρ + (cid:17) (cid:0) q − m η + iǫ (cid:1) (cid:0) q − m π + (cid:1) , (15)where m = ( p π + p η ) , m = ( p π + + p η ) , and the momenta ( q , q , q ) are those of ( ρ + , η, π ) , respectively. The t πη → πη stands for the two-body πη → πη scattering amplitude,which depends on the invariant mass of the πη system.The decay amplitudes of D + s → π + π η via triangle diagrams shown in Fig. 4 are written as A K ∗ K = A K ∗ Ka + A K ∗ Kb + A K ∗ Kc + A K ∗ Kd , (16) A K ∗ Ka = i √ Z d q (2 π ) A (cid:0) D + s → K ∗ + ¯ K (cid:1) A ( K ∗ + → π + K ) t K ¯ K → πη ( m ) (cid:0) q − m K ∗ + + im K ∗ + Γ K ∗ + (cid:1) (cid:0) q − m K + iǫ (cid:1) (cid:0) q − m K (cid:1) , (17) A K ∗ Kb = − i Z d q (2 π ) A (cid:0) D + s → K ∗ + ¯ K (cid:1) A ( K ∗ + → π K + ) t K ¯ K → πη ( m ) (cid:0) q − m K ∗ + + im K ∗ + Γ K ∗ + (cid:1) (cid:0) q − m K + iǫ (cid:1) (cid:0) q − m K + (cid:1) , (18) A K ∗ Kc = − i √ Z d q (2 π ) A (cid:0) D + s → ¯ K ∗ K + (cid:1) A (cid:0) ¯ K ∗ → π + K − (cid:1) t K ¯ K → πη ( m ) (cid:0) q − m K ∗ + im K ∗ Γ K ∗ (cid:1) (cid:0) q − m K + + iǫ (cid:1) (cid:0) q − m K − (cid:1) , (19) A K ∗ Kd = − i Z d q (2 π ) A (cid:0) D + s → ¯ K ∗ K + (cid:1) A (cid:0) ¯ K ∗ → π ¯ K (cid:1) t K ¯ K → πη ( m ) (cid:0) q − m K ∗ + im K ∗ Γ K ∗ (cid:1) (cid:0) q − m K + + iǫ (cid:1) (cid:0) q − m K (cid:1) , (20)with momenta ( q , q , q ) for ( K ∗ ( ¯ K ∗ ) , ¯ K ( K ) , K ( ¯ K )) , respectively. It is worth mentioning thatone needs to include the isospin factor − q and q for Figs. 4 (a) and (c) and Figs. 4 (b) and(d), respectively. The t K ¯ K → πη stands for the two-body K ¯ K → πη scattering amplitude, whichdepends on the invariant mass of the πη system. It should be noted that in the present work, forthe ρ and K ∗ vector meson propagators, we take G µνV ( q V ) = i ( − g µν + q µV q νV /q V ) q V − m V + im V Γ V , (21)7here m V and Γ V are the mass and width of the vector mesons.The triangle loop integrals in these above amplitudes are ultraviolet divergent, in general oneneeds to include phenomenological form factors to prevent ultraviolet divergence, as shown inRefs. [29–35]. However, as discussed in Refs. [10, 11], the ultraviolet divergences in the triangleloop diagrams integrals cancel out (for more details see Ref. [36]), thus we do not need to introducethese form factors in this work.In Ref. [11] the two-body scattering amplitude t πη → πη is parameterized with the Breit-Wignerform. In this work, we describe the final state interaction between π and η as well as the interactionbetween K and ¯ K with the chiral unitary approach. The scattering amplitudes t πη → πη and t K ¯ K → πη can be obtained by solving the following Bethe-Salpeter equation t = [1 − V G ] − V, (22)where G is the loop function of two mesons and V is the transition potential. The loop functioncan be regularized by either the dimensional regularization scheme or the cutoff regularizationscheme. In this work we employ the dimensional regularization scheme. The potential V is a × matrix of coupled channels K ¯ K and πη . At the leading chiral order, the transition potential V can be explicitly written as [8, 37–41] V K ¯ K → K ¯ K = − f s, V K ¯ K → πη = √ f (cid:18) s − m K − m π − m η (cid:19) , V πη → πη = − f m π , where we take the isospin multiplets as K = ( K + , K ) , ¯ K = ( ¯ K , − K − ) , and π = ( − π + , π , π ) .Then, solving the Bethe-Salpeter equation with µ = 1 GeV, a ( µ ) πη = − . , a ( µ ) ¯ KK = − . [9], we obtain a resonance with mass m = 983 . MeV and width
Γ = 105 . MeV.This can be associated with the a (980) state.With the so-obtained decay amplitudes, one can calculate the invariant mass distributions of D + s → ηπ + π as a function of m and m [12]: d Γ = 1(2 π ) |A| m D s dm dm , (23)where A is the total decay amplitude, which is A = A tree1 + A ρη + A K ∗ K .One can easily obtain the single differential invariant mass distribution d Γ /dm ππ and d Γ /dm πη by integrating over m πη and m ππ with the limits of the Dalitz Plot, respectively.8 II. NUMERICAL RESULTS AND DISCUSSION
We first show the theoretical results for the π + π invariant mass distribution in Fig. 5 in com-parison with the BESIII data [6]. The solid curve stands for the total contributions from the treediagram and the triangle diagrams, while the dashed curve stands for the contribution from onlythe tree diagram. The solid curve has been adjusted to the strength of the experimental data ofBESIII at its peak [6]. Since we have considered the tree diagram contribution from the ρ + meson,one can see that the ρ + peak can be well reproduced. Furthermore, the high energy points for the π + π invariant mass distributions can also be well reproduced by including the contributions fromthe triangle diagrams. It is interesting to mention that the interference between the tree diagramand triangle diagrams is destructive below m π + π = 0 . GeV, while above that energy point, theinterference is constructive. Besides, with a , a , and a ′ determined as specified above, we ob-tain an absolute branching ratio of B (cid:16) D + s → π +(0) (cid:16) a → (cid:17) π η (cid:17) = 1 . . This is in niceagreement with the BESIII measurement. Total Tree
FIG. 5. Invariant mass distribution of π + π for the D + s → η ( ρ + → ) π π + decay, in comparison with theexperimental data taken from Ref. [6]. In Fig. 6 and Fig. 7, we show the π η and π + η invariant mass distributions of the decay D + s → π + π η without and with the cut of m π + π > GeV, respectively. From Fig. 6, one can see thatthe contribution from the tree diagram is predominant. The theoretical results can describe theexperimental data rather well, particularly the shoulder around m πη ∼ . GeV. In addition, ourtheoretical results do not show a pronounced asymmetric peak around m πη ∼ . GeV as in9ef. [11] (see Fig. 4 of that reference).
Events m p + h (GeV) Exp. Total Tree Events m p h (GeV) Exp. Total Tree
FIG. 6. Invariant mass distributions of πη for the D + s → π ( a (980) → ) π π + η decay, in comparison withthe BESIII data [6]. In Fig. 7, the dashed curves represent the contributions from the ρη triangle diagrams shown inFig. 3, while the blue-dashed curves represent the contributions from the ¯ KK ∗ triangle diagramsshown in Fig. 4, and the red-solid curves stand for the sum of the two contributions. From Fig. 7,one can see that after the m π + π > GeV cut, the a (980) signal is well reproduced, where it isdynamically generated from the πη and K ¯ K coupled channel interactions. However, our resultsat the a (980) peak position are somehow larger than the experimental data, especially for thecase of a (980) + . It should be noted that in Ref. [7], the πK ¯ K final states were produced at thefirst step with the internal W -emission mechanism, and then the final state interaction of K ¯ K produces a (980) , which then decays to πη . Clearly, Ref. [7] and the present work share the samemechanism for the final state interactions. As a result, both can describe the πη line shapes, butthe present work also determines the global strength of the D s decay. In principle, both weakmechanisms may play a role. However, a quantitative consideration of the mechanism of Ref. [7]inevitably introduces additional free parameters for the weak interaction, which cannot yet bedetermined. Hence, we will leave such a study to a future work when more precise experimentaldata become available.It is worthwhile mentioning that in our framework that the K ¯ K ∗ contribution is larger than the ρη contribution, while the former was not considered in Ref. [11], where the ρη ′ channel plays animportant role and a large coupling for a (980) to πη ′ is used. However, both in the unitary chiralapproach and from the experimental information, it is known that the πη ′ coupling to the a (980) .6 0.8 1.0 1.2 1.4 1.6 1.80102030 Events m p + h (GeV) Exp. Total KK * rh Events m p h (GeV) Exp. Total KK * rh FIG. 7. Invariant mass distributions of πη with the cut of m π + π > GeV for the decay D + s → π ( a (980) → ) π π + η , in comparison with the BESIII data [6]. resonance is small. d G /dm K + K (10 -15 ) m K + K (GeV) FIG. 8. Invariant K + ¯ K mass distribution for the D + s → π K + ¯ K decay. In addition, we study the a (980) state in the K ¯ K channel from the D + s → πK ¯ K decay byincluding the contributions from the triangle diagrams, which can be easily obtained with thereplacement of the πη final state by K ¯ K in Figs. 3 and 4. The resulting predictions for the K + ¯ K invariant mass distributions are shown in Fig. 8, which can serve as a highly non-trivial check ofthe mechanism proposed in this work. 11 V. SUMMARY
We studied the D + s → π + π η decay recently analysed by the BESIII Collaboration, wherethe D + s → π a (980) + and D + s → π + a (980) decay modes are claimed as the W -annihilationdominant processes observed for the first time, and their branching fractions, however, are oneorder of magnitude larger than those of known W -annihilation decays. Inspired by Ref. [11], weproposed that the anomalously large branching ratios of these decay modes can be understood viatriangle diagrams. At first, the D + s meson decays weakly into either ρ + η or K ¯ K ∗ /K ∗ ¯ K . Thevector mesons then decay into a pair of pseudoscalar mesons, ππ or Kπ . One of them interactswith the pseudoscalar meson from the weak decay of D + s , and generates dynamically the a (980) state. With the weak decay couplings determined by fitting to the experimental branching frac-tions, our method predicted both the absolute branching ratio of D s → a π and the πη invariantmass distributions, which are in nice agreement with the BESIII data. For the D + s → π + π η de-cay, the contribution from the tree diagram as shown in Fig. 2 is the most dominant. After the cutof M π + π > GeV, the contributions of the triangle diagrams are crucial to produce the a (980) resonance. In addition, we predicted the K ¯ K invariant mass distributions of the D s → πK ¯ K decay, which can be checked by future experimental measurements.Furthermore, the present work provides a way to estimate the effective weak coupling appearingin the D s → K ∗ ¯ K → K ¯ Kπ vertex. The same mechanism might also be relevant to those ofsimilar processes, such as the D + → π + K ¯ K and D + → π + π η reactions [41, 42]. ACKNOWLEDGMENTS
This work is partly supported by the National Natural Science Foundation of China underGrants No. 11735003, No. 11975041, No. 11775148, No. 12075288, and No. 11961141004. It isalso supported by the Youth Innovation Promotion Association CAS (2016367). [1] E. Klempt, M. Matveev and A. V. Sarantsev, Eur. Phys. J. C (2008) 39.[2] B. Bhattacharya, C. W. Chiang and J. L. Rosner, Phys. Rev. D (2010) 096008.[3] W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C (2016), 700.[4] V. R. Debastiani, W. H. Liang, J. J. Xie and E. Oset, Phys. Lett. B (2017) 59.
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