Three-body unitary coupled-channel approach to radiative J/ψ decays and η (1405 / 1475)

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I. INTRODUCTION
Since the first observation in 1967 [1], the light isoscalar pseudoscalar states in 1.4-1.5 GeV region, named η(1405/1475), has invited lots of debates about its peculiar features in experimental data and about various theoretical interpretations.Two major questions on η(1405/1475), which are still open, are: (i) Are there one or two η excited states in this energy region ?; (ii) How is the internal structure of the excited state(s) like ?What makes η(1405/1475) difficult to understand is that η(1405/1475) could include various components such as a quark-antiquark pair, various hadronic coupled-channels, and a glueball, reflecting the complex nature of the Quantum Chromodynamics (QCD) in the low energy regime.Also, the mixing between (uū + d d)/ √ 2 and ss is significant only in the isoscalar pseudoscalar sector.Thus, understanding η(1405/1475) seems particularly important to deepen our understanding of QCD.
The conventional quark model expects radially excited η and η ′ states in this energy region, and they correspond to η(1295) and (one of) η(1405/1475) states, respectively [15,16].If η(1405/1475) includes two states, what is its nature ?A proposal was made to interpret η(1405) as a glueball [17].However, the isoscalar pseudoscalar glueball from lattice QCD (LQCD) predictions turned out to be significantly heavier [18][19][20][21][22].Meanwhile, a LQCD prediction from Ref. [23] indicated only two states in this region.However, the authors did not identify them with η(1295) and η(1405/1475) since the experimental situation is unclear.Thus, although the two-state solution for η(1405/1475) is not accommodated in the quark model, it is not forbidden by any strong theoretical arguments.
Another peculiar property of η(1405/1475) is its anomalously large isospin violation in η(1405/1475) → πππ decays, as found in radiative J/ψ decays by the BESIII collaboration in 2012 [24].The BESIII found that the decays mostly proceed as η(1405/1475) → f 0 (980)π → πππ, and that the rate is significantly larger than that expected from η(1405/1475) → a 0 (980)π followed by the a 0 (980)-f 0 (980) mixing.It was also found that the f 0 (980) width in the ππ invariant mass distribution is significantly narrower (∼ 10 MeV) than those seen in other processes (∼ 50 MeV) [15].A theoretical explanation for these experimental findings was made in Refs.[25][26][27][28].First the authors pointed out that a K * KK triangle loop from a η(1405/1475) decay can hit an on-shell kinematics, causing a triangle singularity (TS) that can significantly enhance the amplitude.At the same time, this triangle loop causes the isospin violation due to the mass difference between K ± and K 0 in the K * K − K + and K * K0 K 0 triangle loops.This mechanism can naturally explain the large isospin violation without any additional assumptions.
The discovery of the potentially important TS effects in the η(1405/1475) decays encouraged theorists to describe all η(1405/1475)-related data, including processdependent lineshapes, with one η(1405/1475) state, based on the "Occam's Razor" principle [25][26][27][28].Indeed, it was shown that the TS mechanisms can shift the resonant peak position somewhat, depending on K Kπ, ηππ, and πππ final states.However, the experimental data of K Kπ and ηππ were rather limited at this time, and these theoretical results were not sufficiently tested.Also, it has not been possible to discriminate one-and two-state solutions of η(1405/1475).
A significant advancement has been made recently by a BESIII analysis of J/ψ → γ(K S K S π 0 ) data from the high-statistics ∼ 10 10 J/ψ decay samples [29].They fitted the data with J P C = 0 −+ , 1 ++ , and 2 ++ partial wave amplitudes, and identified two η(1405/1475) states in the 0 −+ amplitude with a high statistical significance.
There are however theoretical issues in the BESIII analysis since they described the η(1405/1475) states with Breit-Wigner (BW) amplitudes.The BW amplitude is known to be unsuitable in cases when a resonance is close to its decay channel threshold and/or when more than one resonances are overlapping [30].This difficulty arises since the BW amplitude does not consider the unitary.In the present case, η(1405) is close to the K * K threshold, and η(1405) and η(1475) are overlapping.Furthermore, while coupling parameters in the BW formalism implicitly absorb loop contributions, they cannot simulate non-smooth behavior such as TS.Thus, it is highly desirable to develop an appropriate approach where the data are fitted with a unitary coupled-channel J/ψ decay amplitude, and η(1405/1475) poles are searched by analytically continuing the amplitude.The η(1405/1475) exists in a complicated coupledchannel system consisting of quasi two-body channels such as K * K and a 0 π and three-body channels such as K Kπ and ππη.The unitary coupled-channel approach seems the only possible option to describe such a system.Also in this approach, we automatically take account of the TS effects that are expected to play an important role, and thus taking over the sound physics in the previous models of Refs.[25][26][27][28].
In this work 1 , we develop a three-body unitary coupled-channel model for radiative J/ψ decays to threemeson final states of any J P C .Then we use the model to fit K S K S π 0 Dalitz plot pseudodata generated from the BESIII's 0 −+ amplitude for J/ψ → γ(K S K S π 0 ) [29].At the same time, the branching fractions of other final states such as ηπ + π − and ρ 0 γ relative to that of K Kπ are also fitted.Based on the obtained model, we examine the pole structure of η(1405/1475) in the complex energy plane to see if η(1405/1475) is one or two state(s).We also use the model to predict η(1405/1475) → ηππ, γππ, and πππ lineshapes and branchings.By examining the η(1405/1475) decay mechanisms for different final states, we identify dominant mechanisms and address major issues regarding η(1405/1475) how the process dependent lineshapes and large isospin violations come about.
Precise Dalitz plot data are a great target for a threebody unitary model.Single-channel three-body unitary frameworks based on the Khuri-Treiman equations have been used extensively to analyze Dalitz data in elastic kinematical regions: e.g., Refs.[32,33] for ω/ϕ → πππ.However, Dalitz-plot analyses covering inelastic kinematical regions with coupled-channel three-body unitary frameworks are very limited: e.g., Ref. [34] for D + → K − π + π + and the present analysis.Since more and more precise Dalitz data are expected from the contemporary experimental facilities, the importance of the three-body unitary coupled-channel analysis will be increasing.Thus, related theoretical developments have been made recently [35][36][37].
Three-body unitary analysis like the present work involves pole extractions.There are literatures [38][39][40] that discuss the pole extraction from three-body unitarity amplitudes.Practically, however, such a pole extraction from experimental three-body distributions had not been done until recently.The first case was made in Refs.[41,42] where a ρπ single-channel model was used to analyze m π + π − π − lineshape data for τ − → π + π − π − ν τ , extracting an a 1 (1260) pole.Reference [42] ( [41]) treated the three-body unitarity rigorously (partially).The two analyses highlighted the importance of the full threebody unitarity in the pole extraction since an additional spurious pole existed in Ref. [41].Our present analysis treats the three-body unitarity as rigorously as in Ref. [42].Furthermore, we improve the pole extraction method of Ref. [42] since we consider relevant coupledchannels and fit Dalitz plot distributions rather than the projected invariant mass distributions.
The organization of this paper is as follows.In Sec.II, we present formulas for the radiative J/ψ decay amplitude based on the three-body unitary coupled-channel model and the partial decay width.In Sec.III, we analyze Dalitz plot pseudodata from the BESIII 0 −+ amplitude for J/ψ → γ(K S K S π 0 ).The quality of the fits is shown, and the η(1405/1475) poles are extracted.In Sec.IV, we predict the lineshapes of ηππ, γπ + π − , πππ final states from the radiative J/ψ decays.The branching fractions for the πππ final states are also predicted.Finally, in Sec.V, we summarize the paper, and discuss the future prospects.

II. MODEL
A. Radiative J/ψ decay amplitudes within three-body unitary coupled-channel approach In constructing our three-body unitary coupledchannel model, we basically follow the formulation presented in Refs.[34,35].However, there is one noteworthy difference.While we specified a particle with its isospin state in Refs.[34,35], we now use its charge state.This is an important extension of the model to describe isospinviolating processes.In what follows, we sketch our model, putting an emphasis on the differences from Refs.[34,35].
A radiative J/ψ decay mechanism within our model is diagrammatically represented by Fig. 1(a).First, J/ψ radiatively couples, via a vertex Γ γM * j ,J/ψ , to a bare excited state (M * ) such as η(1405/1475) of J P C = 0 −+ and f 1 (1420) of J P C = 1 ++ ; we consider M * with I = 0 (I: isospin) in this work.Second, the bare M * nonperturbatively couples with quasi two-body Rc and threebody abc states to form a dressed M * propagator ḠM * [Fig.1(b)] that includes M * resonance pole(s).Here, abc are pseudoscalar mesons (π, K, η), and R is a bare twomeson resonance such as K * , a 0 (980), and f 0 (980).Particles R and ab also nonperturbatively couple through a vertex Γ ab,R , forming a dressed R propagator τ R,R ′ [Fig.1(e)] that includes R resonance pole(s).Third, M * decays to a final abc via a dressed M * i → Rc decay vertex ΓcR,M * i [Fig.1(c)] that includes nonperturbative final state interactions.The amplitude formula for the above radiative J/ψ decay process is given by 2 A γabc,J/ψ = with where cyclic permutations (abc), (cab), (bca) are indicated by cyclic abc ; the indices i and j specify one of bare M * states belonging to the same J P C ; E denotes the abc total energy in the abc center-of-mass (CM) frame.Below, we give a more detailed expression for each of the components in the amplitude.
The J/ψ → γM * j vertex is given in a general form as where g ℓs J/ψM * j γ is a coupling constant and m M * j being a bare M * j mass; Y ℓm (q) denotes the spherical harmonics with q ≡ q/|q|, and ℓ is restricted by the parityconservation.When M * belongs to J P C = 0 −+ , Eq. (3) reduces to (up to a constant overall factor) In our numerical analysis from Sec. III, we use the coupling g J/ψη * j γ defined in this reduced form.The R → ab vertex is given by 2 We denote a particle x's mass, momentum, energy, polarization, spin and its z-component in the abc center-of-mass frame by mx, px, Ex, ϵx, sx, and s z x , respectively; Ex = m 2 x + p 2 x with px = |px|.The mass values for pseudoscalar mesons (π, K, η) are taken from Ref. [15].Symbols with tilde such as px indicate quantities in the J/ψ-at-rest frame.
where the parentheses are Clebsch-Gordan coefficients, and t x and t z x are the isospin of a particle x and its zcomponent, respectively; p * a denotes a particle a's momentum in the ab CM frame.Since particles a and b are pseudoscalar in this paper, the total spin is S = 0 and the orbital angular momentum is L = s R .Thus we simplify the above notation for the R → ab vertex as with a vertex function and use this notation hereafter.The coupling g ab,R and cutoff c ab,R in Eq. ( 7), and the bare mass m R in Eq. ( 8) are determined by analyzing L-wave ab scattering data as detailed in Appendix A where the parameter values are presented.The dressed R propagator [Fig.1(e)] is given by with the R self-energy with M ab (q) = E a (q) + E b (q) and m R being a bare mass of R; R and R ′ in Eq. ( 9) have the same spin state (s R = s R ′ ).Due to the Bose symmetry, we have a factor B ab : B ab = 1/2 for identical particles a and b; B ab = 1 otherwise.In Eq. ( 9), the a 0 -f 0 mixing occurs (Σ a0,f0 ̸ = 0) due to the mass difference between ab = K + K − and K 0 K0 states.The dressed R propagators include R resonance poles as summarized in Tables III-V in Appendix A.
The dressed where l is the relative orbital angular momentum between R and c.The dressed where the first and second terms are direct decay and rescattering mechanisms, respectively.Common isobar models do not have the second term.We use a bare vertex function including a dipole form factor as where C cR) l are coupling and cutoff parameters, respectively.We also have introduced J P C partial wave amplitudes for cR → c ′ R ′ scatterings, , that is obtained by solving the scattering equation [Fig.1(d)]: with The driving term Z c,J P C (c ′ R ′ ) l ′ ,(cR) l , what we call the Zdiagram, is diagrammatically expressed in the first term of the r.h.s. of Fig. 1(d); c indicates an exchanged particle.Explicit formulas for the partial-wave-expanded Z-diagram can be found in Appendix C of Ref. [35].One important difference from Ref. [35] is that we here do not project the Z diagrams onto a definite total isospin state.As a result, an isospin-violating The second term in the r.h.s. of Eq. ( 14) is a vectormeson exchange mechanism based on the hidden local symmetry model [43].In the present case, this mechanism works for K * K ↔ K * K, K * K interactions.Formulas are presented in Appendix A of Ref. [34], but here we use a different form factor of (1 + p 2 /Λ 2 ) −2 (1 + p ′2 /Λ 2 ) −2 with Λ = 1 GeV, rather than Eq.(A15) of Ref. [34].
The dressed M * propagator [Fig.1(b)] is given by where the M * self energy in the second term is given by The above formulas show that the dressed M * propagator (M * pole structure) and the dressed M * i → Rc form factor (M * decay mechanism) are explicitly related by the common dynamics.This is a consequence of the three-body unitarity.
In the above formulas, we assumed that two-body ab → a ′ b ′ interactions occur via bare R-excitations, ab → R → a ′ b ′ .We can straightforwardly extend the formulas if two-body interactions are from bare Rexcitations and separable contact interactions, as detailed in Ref. [34].Also, the above formulas are valid when c is a pseudoscalar meson, and need to be slightly modified for Rc = ρρ channel.We consider the spectator ρ width in the first term of the r.h.s. of Eq. ( 8) by E − E R (p) → E − E R (p) + iΓ ρ /2; Γ ρ = 150 MeV and is constant.Also, the label in the bare form factor of Eq. ( 12) is extended to include the total spin of ρρ (s ρρ ) as (cR) l → (cR) lsρρ .

B. Radiative J/ψ decay rate formula
The partial decay width for a radiative J/ψ decay, J/ψ → γ(abc), is given by where m ab and m ac are the invariant masses of the ab and ac subsystems, respectively; pγ denotes the photon momentum in the J/ψ-at-rest frame; M J/ψ→γ(abc) is the invariant amplitude that is related to Eq. ( 1) with an overall kinematical factor.A Bose factor B is: B = 1/3!for identical three particles abc; B = 1/2!for identical two particles among abc; B = 1 otherwise.The J/ψ spin state is implicitly averaged.Using our amplitude of Eq. ( 2) for the J/ψ radiative decays via M * -excitations, the decay formula of Eq. ( 17) can be written as with where The invariant amplitudes M M * i →abc and M J/ψ→M * j γ are related to components of the amplitude in Eq. ( 2) by with and For the case of i = j, Eqs. ( 20) and ( 21) reduce to the standard formulas of the M * -decay Dalitz plot distribution and J/ψ two-body decay width, respectively.Our decay formula of Eq. ( 19) can be made look similar to that of a Breit-Wigner model by a replacement: i being Breit-Wigner mass and width, respectively.

III. DATA ANALYSIS AND η(1405/1475) POLES
In this paper, we study radiative J/ψ decays via η(1405/1475) excitations with the unitary coupledchannel model described above.We thus consider only the J P C = 0 −+ partial wave contribution in the above formulas.In the following, we discuss our dataset, our default setup of the model, and analysis results.

A. Dataset
A main part of our dataset is K S K S π 0 Dalitz plot pseudodata.We generate the pseudodata using the E-dependent 0 −+ partial wave amplitude from the recent BESIII Monte Carlo (MC) analysis on J/ψ → γ(K S K S π 0 ) [29].We often denote this process by J/ψ → γ(0 −+ ) → γ(K S K S π 0 ).The pseudodata is therefore detection efficiency-corrected and background-free.
The pseudodata includes ∼ 1.23 × 10 5 events in total, being consistent with the BESIII data, and is binned as follows.The range of 1300 ≤ E ≤ 1600 MeV is divided into 30 E bins (10 MeV bin width; labeled by l).Furthermore, in each E bin, we equally divide (0.95 GeV) 2 ≤ m 2 K S K S ≤ (1.50 GeV) 2 and (0.60 GeV) 2 ≤ m 2 K S π 0 ≤ (1.15 GeV) 2 into 50 × 50 bins (labeled by m); m ab is the ab invariant mass.We denote the event numbers in {l, m} and l-th bins by N l,m and Nl (≡ m N l,m ), respectively; their statistical uncertainties are N l,m and Nl , respectively.We fit both {N l,m } and { Nl } pseudodata, since {N l,m } and { Nl } would efficiently constrain the detailed decay dynamics and the resonant behavior (pole structure) of η(1405/1475), respectively.We use the bootstrap method [45] to estimate the statistical uncertainty of the model, and we thus generate and fit 50 pseudodata samples.
Other final states from the radiative J/ψ decays are also considered in our analysis.We fit the model to a ratio of partial decay widths [15] and also another ratio [8,9] The partial widths Γ in the above ratios are calculated by integrating the E distributions [Eq.( 18)] for K Kπ, π + π − η, and ρ 0 γ final states over the range of 1350 MeV < E < 1550 MeV.The ratio of Eq. ( 25) is important to constrain the a 0 (980)π contributions since the relative coupling strengths of a 0 (980) → K K and a 0 (980) → ηπ are experimentally fixed in a certain range [46][47][48][49].Also, f 0 η and ρρ channels indirectly contribute to K S K S π 0 through loops, and therefore the K S K S π 0 data does not constrain their parameters well.Since these channels directly contribute to the ηπ + π − and ρ 0 γ final states, the above ratios will be a good constraint.The partial width for all K Kπ final states in Eqs. ( 25) and ( 26) is 12 times larger than that of K S K S π 0 , as determined by the isospin CG coefficients.The MC solution-based {N l,m }, { Nl }, R exp 1 , and R exp 2 are simultaneously fitted, with a χ 2 -minimization, by our default model described in the next subsection; the actual BESIII data are not directly fitted.We calculate χ 2 from {N l,m } by comparing N l,m to the differential decay width [dΓ J/ψ→γ(abc) /dEdm 2 ab dm 2 ac of Eqs. ( 18)-( 21)] evaluated at the bin center and multiplied by the bin volume.We omit N l,m on the phase-space boundary from the χ 2 calculation.This simplified procedure keeps the computation time reasonable.Also, if a bin has N l,m < 10, it is combined with neighboring bins to have more than 9 events for the χ 2 calculation.The number of bins for {N l,m } depends on the pseudodata samples, and is 4496-4575.χ 2 from { Nl }, R exp 1 , and R exp 2 are weighted appropriately to reasonably constrain the model.

B. Model setup
For the present analysis of the radiative J/ψ decays, we consider the following coupled-channels as a default in our model described in Sec.II.We include two bare M * of J P C = 0 −+ ; we refer to them as η * hereafter.The Rc channels are Rc = K * (892) K, κ K, a 0 (980)π, a 2 (1320)π, f 0 η, ρ(770)ρ(770), and f 0 π, where charge indices are suppressed 3 .To form positive Cparity states, K * (892)K and κK channels are implicitly included.A symbol R may refer to more than one bare state and/or contact interactions.For example, the f 0 π channel includes two bare states and one contact interaction that nonperturbatively couple with ππ − K K continuum states, forming f 0 (500), f 0 (980), and f 0 (1370) poles; see Appendix A for details.
We mention the channels considered in the BESIII amplitude analysis of J/ψ → γ(K S K S π 0 ) [29].In the 0 −+ partial wave, the BESIII considered η(1405) and η(1475) resonances that decay into K * (892) K, a 0 (980)π, and a 2 (1320)π.All resonances, except for a 0 (980), are described with Breit-Wigner amplitudes.No rescattering nor channel-coupling such as those in the second term of the r.h.s. of Fig. 1(c) is taken into account.In addition, a nonresonant Kπ p-wave amplitude supplements the K * (892) tail region.Clearly, our coupled-channel model includes more channels than the BESIII model does.This is to satisfy the coupled-channel three-body unitarity, and also to describe different final states in a unified manner.
3 κ is also referred to as K * 0 (700) in the literature.
We consider isospin-conserving η * → Rc decays in Eq. ( 12) for all bare η * and Rc states specified in the first paragraph of this subsection.One exception applies to the lighter bare η * → ρρ which is set to zero.This is because the lighter bare η * seems consistent with an excited ss state from the quark model [16] and LQCD prediction [23], and ss → ρρ should be small for the OZI rule.
We may add nonresonant (NR) amplitudes A J P C ,NR γabc,J/ψ , which does not involve M * excitations, to the resonant amplitudes A J P C γabc,J/ψ of Eq. ( 1).We can derive A J P C ,NR γabc,J/ψ and modify A J P C γabc,J/ψ so that their sum still maintains the three-body unitarity.However, this introduces too many fitting parameters to determine with the dataset in the present analysis.We thus use a simplified NR amplitude in this work [cf.Eq. ( 4)]: where c NR is a complex constant.Only when fitting the J/ψ → γ(0 −+ ) → γK S K S π 0 Dalitz plot pseudodata, this NR term is added to M J/ψ→γ(abc) in Eq. ( 17) and c NR is determined by the fit.
We summarize the parameters fitted to the dataset discussed in the previous subsection.We have two bare η * masses in Eq. ( 15), two complex coupling constants (g J/ψη * j γ ) in Eq. ( 4), and one complex constant c NR in Eq. (27).We also adjust real coupling parameters C in Eq. (12).While the cutoffs Λ M * i (cR) l in Eq. ( 12) are also adjustable, we fix them to 700 MeV in this work to reduce the number of fitting parameters and speed up the fitting procedure.Since the overall strength and phase of the full amplitude are arbitrary, we have 25 fitting parameters in total.The parameter values obtained from the fit are presented in Table IX All the radiative J/ψ decay processes included in our dataset for the fit are isospin-conserving.Since the isospin-violating effects are very small in these processes, we make the model isospin-symmetric for fitting and extracting poles and thus use the averaged mass for each isospin multiplet.The amplitude formulas in Sec.II A reduce to isospin symmetric ones given in Refs.[34,35].This simplification significantly speeds up the fitting and pole extraction procedures.When calculating the isospin-violating J/ψ → γ(0 −+ ) → γ(πππ) amplitude of Eq. ( 2), we still use the isospin symmetric Ḡη * (E) and parameters determined by fitting the dataset, and the pole positions stay the same; the isospin violations occur in τ R,R ′ and ΓcR,η * i due to the difference between m K ± and m K 0 .
C. Fits to KSKSπ 0 Dalitz plot pseudodata generated from BESIII 0 −+ amplitude By fitting the 50 bootstrap samples of the K S K S π 0 Dalitz plot pseudodata with the default dynamical con-tents described above, we obtain χ 2 /ndf = 1.40-1.54(ndf: number of degrees of freedom) by comparing with {N l,m }.The ratios of Eqs. ( 25) and ( 26) are also fitted simultaneously, obtaining R th 1 ∼ 7.5 and R th 2 ∼ 0.025, respectively.
The Dalitz plot distributions obtained from the fit are shown in Fig. 2 for representative E values, in comparison with one of the bootstrap samples 4 .The fit quality is reasonable overall.For E < ∼ 1400 MeV, there is a peak near the K S K S threshold.While this is seemingly the a 0 (980) contribution, it is actually due to a constructive interference between K * (892) and K * (892), as detailed later.For E > ∼ 1430 MeV, on the other hand, the main pattern is mostly understood as the K * (892) and K * (892) resonance contributions.The good fit quality can be seen more clearly in the K S K S and K S π 0 invariant mass distributions as shown in Figs. 3 and 4, respectively.The model is well-fitted to the K * peak (the sharp peak near the K S K S threshold) in the m 2 The E dependence of the radiative J/ψ decay to K S K S π 0 , obtained by integrating the Dalitz plots in Fig. 2, is shown in Fig. 5(a).The E-dependence would be largely determined by the pole structure of the η(1405/1475) resonances.The E distribution shows a broad peak with an almost flat top, and our model reasonably agrees with the pseudodata.We now study dynamical details.The η * decay mechanisms can be separated according to Rc states in Fig. 1(a) that directly couple to the final states.We will refer to these Rc states as final Rc states.Contributions from the final K * K, κ K, and a 0 (980)π states are shown separately in Fig. 5(a).The final K * K and κ K contributions are the first and second largest, while the final a 0 (980)π contribution is very small.The constant nonresonant contribution from Eq. ( 27) gives a small phase-space shape contribution.
The final K * K, κ K, and a 0 (980)π contributions are also shown separately in Figs.5(b), 5(c), and 5(d), respectively, and main contributions from the diagrams in Fig. 6 are also shown.The direct decays of Fig. 6(a) and single-rescattering mechanisms of Fig. 6(b) are obtained by perturbatively expanding the dressed η * decay vertex of Fig. 1(c) in terms of V in Eq. ( 14), and taking the first two terms.The final K * K contribution is mostly from the direct decay, while the final κ K and a 0 (980)π contributions are dominantly from the single-rescattering mechanism and therefore a coupled-channel effect.The K * KK triangle loop causes a triangle singularity (TS) in the the final a 0 (980)π contribution at E ∼ 1.4 GeV.However, we do not find a large contribution from the TS.The TS-induced enhancement may have been suppressed since the K * K pair is relatively p-wave.
Figure 7 illustrates the mechanism that creates the sharp a 0 (980)-like enhancement near the K S K S thresh- The BESIII model obtained from their amplitude analysis describes the data rather differently from ours (see Fig. 3 of Ref. [29]) such as: (i) The a 0 (980)π contribution is the largest overall; (ii) The K * K contribution is

This work
FIG. 4. The KSπ 0 invariant mass distributions.Other features are the same as those in Fig. 3.
comparable to a 0 (980)π only around E = 1.5 GeV; (iii) The κ K channel is not included.These differences come mainly from the fact that our model is fitted not only to the K S K S π 0 Dalitz plot pseudodata but also to the ratios of Eqs. ( 25) and ( 26); the BESIII model was fitted to the J/ψ → γ(K S K S π 0 ) data only.The ratio of Eq. ( 25) is, albeit a large uncertainty, an important constraint on the final a 0 (980)π contribution to η * → K Kπ, since the relative coupling of a 0 (980) → K K to a 0 (980) → πη is exper-imentally determined in a certain range [15,[46][47][48][49].The final a 0 (980)π contribution to K Kπ needs to be small as in our model in order to satisfy the ratio of Eq. ( 25).Furthermore, the κ K channel in our model gives a substantial contribution through the channel-coupling required by the unitarity.
Since the a 0 (980)π contribution is very different between our and the BESIII models, one may wonder how much our result depends on a particular a 0 (980) model.As we discussed in Sec.III B, our default a 0 (980) model is based on Ref. [46] and |g a0(980)→K K /g a0(980)→ηπ | ∼ 1.
D. Fit with one bare η * state It is important to examine if the BESIII data can also be fitted with a single bare η * model, since the η(1405/1475) was claimed to be a single state in the literature.We try to fit only the m K S K S π 0 (= E) distribution, but a reasonable fit is not achievable.The result is shown in Fig. 8 along with the final Rc contributions.The final κ K and a 0 π contributions have lineshapes expected from the η * pole position, 1416 − 61i MeV.The triangle singularity caused by the K * KK-loop does not noticeably shift the lineshape of the final a 0 (980)π contribution.The lineshape of the final K * K contribution has its peak at 30-40 MeV higher than the peak positions of the final κ K and a 0 (980)π contributions, since its threshold opens at E ∼ 1.4 GeV and the K * K pair is relatively p-wave.Still, the peak shift is not large enough to explain the significantly broader peak of the pseudodata.
Another possible single-state solution for η(1405/1475) describes the BESIII data by including an interference with η(1295).To examine this possibility, we include two bare η * states, and restrict one of the bare masses below 1.4 GeV, and the other around 1.6 GeV.We are not able to obtain a reasonable fit to the pseudodata with this model.We thus conclude that two bare η * for η(1405/1475) are necessary to reasonably fit the K S K S π 0 pseudodata generated from the BESIII 0 −+ amplitude.

E. Pole positions for η(1405) and η(1475)
The properties of a resonance are characterized by its pole position and residue of the (scattering or decay) amplitude.In the present unitary coupled-channel framework, a pole position corresponds to a complex energy E that satisfies det [ Ḡ−1 (E)] = 0, where Ḡ−1 (E) has been defined in Eq. ( 15) and is analytically continued to the complex E-plane.The analytic continuation involves deformations of the integral paths in Eqs. ( 9), ( 11), ( 13  and ( 16).Otherwise, singularities on the complex momentum planes cross the real momentum paths as E goes to complex values, invalidating the analytic continuation.The driving term Z c,J P C (c ′ R ′ ) l ′ ,(cR) l in Eq. ( 14) and τ R,R ′ in Eqs.(11), (13), and ( 16) cause such singularities.To avoid these singularities, a possible deformed path to be used in Eqs.(11), (13), and ( 16) can be found in Fig. 7 of Ref. [42].The energy denominator in Eq. ( 9) also causes a singularity and, for a complex E, we need to avoid it by choosing a deformed path as found in Fig. 3 of Ref. [42].Our procedure of the analytic continuation is very similar to those discussed in detail in Ref. [42], and we do not go into it further.
We search for poles in the region of Re Thus we specify the pole's RS of these channels in Table I; the relevant RS of the other channels should be clear 5 .The locations of the poles and branch points are also shown in Fig. 9.
The BESIII analysis result (Breit-Wigner parameters) is also shown for comparison.A noticeable difference is that our model describes η(1405) with two poles (α = 1, 2).The two pole structure does not mean two physical states but is simply due to the fact that a pole coupled to a channel is split into two poles on different RS of this channel.The mass and width values are fairly similar between our and the BESIII results.However, the use of the Breit-Wigner amplitude could cause an artifact due to the issues discussed in the introduction and below, which might explain the difference between the two analysis results.In Ref. [30], a unitary coupled-channel model and an isobar (Breit-Wigner) model were fitted to the same pseudo-data.Resonance poles from the two models can be significantly different, particularly when two resonances are overlapping.Also, if the pole is located near a threshold, the lineshape (E dependence) caused by the pole can be distorted by the branch cut.In the present case, η(1405) and η(1475) are fairly overlapping and η(1405) is located near the K * K threshold.Our three-body unitary coupled-channel analysis fully considers these issues and is a more appropriate pole-extraction method.
We examine the resonance pole contributions to the E distribution.For this purpose, we expand the dressed η * propagator of Eq. ( 15) around the resonance pole at  I. The nonresonant (NR) contribution is from Eq. ( 27).
M Rα as [50], with ) and γ(π + π − π 0 ), respectively, obtained with various choices of g J/ψη * j γ in Eq. ( 4).The red triangles in (a) and (b) are the same as those in Fig. 5(a ∆(E) and ∆ ′ (M Rα ) = d∆(E)/dE| E=M Rα .Then we replace Ḡij (E) in the full amplitude of Eq. ( 2) with the above expanded form, and calculate the m K S K S π 0 distribution.In Fig. 10, we show each of the pole contributions and their coherent sum, in comparison with the full calculation.The α = 2 pole contribution is not included in the figure since the K * K branch cut mostly screens this pole contribution to the amplitude on the physical real E axis.The contributions from the α = 1 and 3 poles are dominant, and the lineshape of the full calculation is mostly formed by the the pole contributions.The nonresonant term in Eq. ( 27) enhances the spectrum overall through the interference.Still, the branch cuts and non-pole contribution are missing in the pole approximation of Eq. ( 28), and their effects should explain the difference between the red triangles and the magenta squares in the figure.
The resonance amplitude of Eq. ( 28) suggests that one of the pole contributions can be eliminated from our full model by adjusting the coupling g J/ψη * j γ in the initial vertex of Eq. ( 4).Specifically, we can eliminate the contribution of the pole α by setting as demonstrated in Fig. 11(a).The figure shows a full calculation without the pole approximation of Eq. ( 28).
Eliminating the initial radiative transition of J/ψ → [α = 1], we obtain the magenta squares (g J/ψ[α=1]γ = 0) showing a single peak from the α = 3 pole.Similarly, a calculation with g J/ψ[α=3]γ = 0 gives the green diamonds that have a single peak from the α = 1 pole.Among various processes that include η(1405/1475)decay into K Kπ final states, some of them show a single peak from either of η(1405) or η(1475), and others have a broad peak from a coherent sum of them.Figure 11(a) indicates that our coupled-channel model can describe both cases by appropriately adjusting the couplings of initial vertices.
In the presented analysis, two bare states are required for reasonably fitting the dataset.The lighter bare mass is determined to be ∼ 1.6 GeV, while the heavier one being ∼ 2.3 GeV, as listed in Table IX of Appendix.The heavier bare mass is not tightly constrained by the fit, and those in the range of 2-2.4 GeV can give comparable fits.Within our coupled-channel model, the bare states are mixed and dressed by meson-meson continuum states, forming the resonance states.In concept, the bare states are similar to states from a quark model or LQCD without two-hadron operators.The lighter bare state seems compatible with the excited ss [15,16,23].The heavier bare state could be either of a second radial excitation of η (′) , a hybrid [23], a glueball [18][19][20][21][22], or a mixture of these states.
In this section, we present E dependences of various final states from the radiative J/ψ decays via η(1405/1475), using the three-body unitary coupledchannel model developed in the previous section.The model has been fitted to the K S K S π 0 Dalitz plot pseudodata (Fig. 2) and the ratios of Eqs. ( 25) and (26).
A. π + π − η and π 0 π 0 η final states We show in Fig. 12(a) the m ππη (= E) distributions for the π + π − η final state; the π 0 π 0 η distribution is smaller by a factor of 1/2.The lineshape is qualitatively consistent with the MARK III analysis [3].The final a 0 (980)π and f 0 η states have comparable contributions.On the other hand, the K Kπ final state are mainly from the final K * K and κ K contributions, as seen in Fig. 5(a).Since different Rc final states couple with η(1405) and η(1475) differently, the K Kπ and ππη final states have different E dependences.The ππη final states give a single peak at m ππη ∼ 1.4 GeV, while the K Kπ distribution has a flat peak.The process-dependent lineshape of the η(1405/1475) decays can thus be understood.
In Figs.12(b) and 12(c), we decompose the final a 0 (980)π and f 0 η contributions into direct decays [Fig.6(a)] and single-rescattering mechanisms [Fig.6(b)].The final a 0 (980)π state is mostly from the singlerescattering mechanisms and the direct decays are minor.On the other hand, a completely opposite trend applies to the final f 0 η state.In more detail, the K * K K, κK K, and f 0 πη triangle mechanisms contribute to the final a 0 (980)π state.We find that the three loops give comparable contributions, even though only the K * K K loop causes a triangle singularity.This is perhaps because the K * K pair is relatively p-wave, suppressing the triangle singularity.We also present in Fig. 13 a prediction for the m 2 πη distribution from the default model.Clear a 0 (980) peaks are predicted, which is qualitatively consistent with the data [4].This prediction should be confronted with the future data from the BESIII.As already discussed, the final a 0 (980)π contribution to the K Kπ and ππη final states are related by the relative coupling of a 0 (980) → K K to a 0 (980) → πη determined experimentally [15,[46][47][48][49].As we have seen in Fig. 5(a), the final a 0 (980)π contribution to K Kπ is very small to satisfy the ratio of Eq. ( 25).If the final a 0 (980)π contribution to K Kπ were as large as that of the BESIII amplitude model, then Eq. ( 25) requires that the final a 0 (980)π → ππη amplitude has to be drastically canceled by destructively interfering with the final f 0 η → ππη amplitude.Such a large cancellation seems unlikely since there is no symmetry behind.Also, the large cancellation makes the a 0 (980) peak in the m πη distribution rather unclear, but the data [4] shows a clear a 0 (980) peak.As shown in Fig. 13, our default model creates a clear a 0 (980) peak.
C. π + π − π 0 and π 0 π 0 π 0 final states Our default model makes predictions for the isospinviolating J/ψ → γ(0 −+ ) → γ(πππ); the model has not been constrained by any data of the πππ final states.These isospin-violating processes are mainly from the mechanisms of Fig. 15 that are not completely canceled due to the small difference between the charged and neutral K masses.In particular, the isospin-violating mechanisms in Figs.15(b) and 15(c) are called the a 0 -f 0 mixing.The m πππ distributions are shown in Fig. 16(a).The π + π − π 0 distribution is almost twice as large as the π 0 π 0 π 0 distribution.The m πππ distributions have a single peak at ∼ 1.4 GeV.
Contributions from the diagrams of Figs.loop diagram of Fig. 15(a) generates a clear peak.As has been discussed in the literature, this K * KK triangle loop is significantly enhanced by a TS occurring at E ∼ 1.40 GeV.The κ KK triangle loop without TS gives a smaller contribution.The TS-enhancement is larger around the higher end of the TS energy range since the p-wave K * K pair suppresses the TS-enhancement around the lower end.This explains the peak position in Fig. 16.
The a 0 -f 0 mixing contribution is very small.This is because η(1405/1475) → a 0 (980)π is very little as seen in Fig. 5(a).This small branching is required by the experimental ratio of Eq. (25).The two-loop mechanisms of Fig. 15(d) are sizable; the second loop involves a TS.
A part of the two-loop contribution is from mechanisms where the two loops are mediated by v HLS in Eq. ( 14).The coherent sum of the mechanisms in Fig. 15 (green diamonds in Fig. 16) mostly explain the full calculation (red triangles).
We confront our predictions for the π + π − π 0 and π 0 π 0 π 0 lineshapes with the BESIII data [24] in Figs.17(a) and 17(b), respectively.Our model correctly predicts the peak position.This remarkable agreement suggests that the peak position is determined by a kinematical effect (triangle singularity) that does not depend on dynamical details.However, the peak width from our calculation seems somewhat broader than the data; we Events/(0.025GeV) where the π + π − pair is from ρ 0 decay.The default model predicts the lineshape of the red curve which has been smeared with the experimental bin width, scaled by a factor, and augmented by a linear background (BG) to fit the data [9].
will come back to this point later.
In Fig. 18, we also compare the m π + π − distribution from our full calculation with the BESIII data [24].Again, the agreement is reasonable, showing the sound predictive power of the coupled-channel model that appropriately account for the relevant kinematical effect for the isospin violation.The f 0 (980)-like peak width (∼ 10 MeV) is much narrower than the world average (∼ 50 MeV) [15].This occurs because the (K * )K + K − and (K * )K 0 K0 loops in Fig. 15 almost exactly cancel with each other due to the isospin symmetry, except in a small window (∼8 MeV) of 2m K ± < m ππ < 2m K 0 where the two loops are rather different and the cancellation is incomplete.Furthermore, the TS enhances the f 0 (980)like peak.Therefore, the f 0 (980) pole plays a minor role in developing the peak in Fig. 18.
How do η(1405) and η(1475) resonances work in J/ψ → γ(πππ) ?We address this question by using the models shown in Fig. 11(a).In the figure, the models labeled by g J/ψ[α=1]γ = 0 and g J/ψ[α=3]γ = 0 do not have J/ψ → γη(1405) and J/ψ → γη(1475) couplings, respectively, and they are normalized to have the same peak height in the m K S K S π 0 distribution.Then, we use them to calculate J/ψ → γ(π + π − π 0 ) as shown in Fig. 11(b).For the model of g J/ψ[α=3]γ = 0, the peak positions are almost the same for K Kπ and πππ final states.This is because η(1405) → πππ is dominant and the η(1405) mass and the TS region overlap well.However, the peak width is narrower for πππ because the TS region is narrower than the η(1405) width.On the other hand, the model of g J/ψ[α=1]γ = 0 gives a significantly suppressed m πππ distribution in comparison with the model of g J/ψ[α=3]γ = 0.This is because the η(1475) mass is outside of the TS region and η(1475) → πππ is not enhanced.In this way, we understand how the different K Kπ and πππ lineshapes in Fig. 11 are caused.
Finally, we compare ratios of K Kπ and πππ branching fractions from our model with the experimental counterpart.Using the K Kπ and πππ branching ratios in Refs.[15,24], we have experimental ratios: Our coupled-channel model predicts: R th 3 = 0.0020 − 0.0021, R th 4 = 0.0010 − 0.0011, (34) which is significantly smaller than the data.A possible reason for the deficit is that we do not consider a contribution from the J P C = 1 ++ partial wave that includes f 1 (1285) and f 1 (1420).The BESIII analysis [29] found that 20-30% of J/ψ → γ(K S K S π 0 ) is from the 1 ++ contribution in which f 1 (1420) → K * K is a dominant mechanism.Considering the consistency with J/ψ → γ(K Kπ), J/ψ → γ(πππ) should come not only from the mechanisms of Fig. 15 but also from similar mechanisms that originate from f 1 decays.In particular, the triangle diagram from the f 1 (1420) decay similar to Fig. 15(a) would be significantly enhanced by the TS, since the f 1 (1420) mass and width have a good overlap with the TS region.Furthermore, f 1 (1420) creates an s-wave K * K pair while η(1405) creates a p-wave pair.Thus the triangle mechanism from f 1 (1420) is more enhanced by the TS than that from η(1405).This 1 ++ contribution might explain the difference between our prediction of Eq. ( 34) and the experimental ratios of Eqs.(32) and (33).We also note that the BESIII [24] did not separate out a possible f 1 (1420) contribution from Γ[J/ψ → γη(1405/1475) → γ(πππ)] in Eqs.(32) and (33).The stronger TS enhancement would create a sharper peak in the m πππ lineshape.In Fig. 17, our 0 −+ model shows a peak somewhat broader than the data.By adding a sharper 1 ++ peak, the data might be better fitted.
channel analysis is desirable.Thus, we developed a model for radiative J/ψ decays to three pseudoscalar-meson final states of any partial wave (J P C ).Also, a slight extension was made to include γρ(ρ → π + π − ) final state.The main components of the model are two-body πK, ππ, K K, and πη scattering models that generate K * 0 (700)(= κ), K * (892), f 0 (500)(= σ), f 0 (980), a 0 (980), and a 2 (1320) resonance poles in the scattering amplitudes.The two-body scattering models as well as bare resonance states were implemented into the three-body coupled-channel scattering equation (Faddeev equation).By solving the equation, we obtained the three-body unitary amplitudes with which we described the final-state interactions in the radiative J/ψ decays.
Using the BESIII's J P C = 0 −+ amplitude for J/ψ → γK S K S π 0 , we generated K S K S π 0 Dalitz plot pseudo data for 30 energy bins in 1.3 GeV ≤ m K S K S π 0 ≤ 1.6 GeV.Then the pseudo data were fitted with the coupled-channel model.The experimental branching ratios of η(1405/1475) → ηππ and η(1405/1475) → γρ relative to that of η(1405/1475) → K Kπ were simulta- ).The BESIII data and the background polynomial (BG) are from Ref. [24].Our full calculation has been smeared with the bin width, scaled to fit the data, and augmented by the background.
neously fitted.We obtained a reasonable fit with two bare η * states while, with one bare η * state, we did not find a reasonable solution.A noteworthy difference from the BESIII amplitude model is that the a 0 (980)π contribution is dominant (very small) in the BESIII (our) model.The small a 0 (980)π contribution is required by the empirical branching ratio of η(1405/1475) → ηππ that was not considered in the BESIII analysis.
Our 0 −+ amplitude was analytically continued to reach three poles in the η(1405/1475) region.Two poles corresponding to η(1405) were found near the K * K threshold, and are located on different Riemann sheet of the K * K channel.Another pole is η(1475).We made 50 bootstrap fits, and estimated statistical uncertainties of the pole positions (Table I).This is the first pole determination of η(1405/1475) and, furthermore, the first-ever pole determination from analyzing experimental Dalitz plot distributions with a manifestly three-body unitary coupled-channel framework.
The obtained model was used to predict the ηππ and γπ + π − lineshapes of J/ψ → γ(0 −+ ) → γ(ηππ) and γ(γρ) processes.The predicted lineshapes are processdependent and reasonably consistent with the existing data.We also applied the model to the isospin-violating J/ψ → γ(0 −+ ) → γ(πππ).The importance of the triangle singularity from the K * KK loop was clarified, while the a 0 (980)-f 0 (980) mixing gave a tiny contribution.Furthermore, the two-loop contribution was calculated for the first time, and this contribution was shown to significantly enhance the isospin violation.The predicted πππ and π + π − lineshapes agree well with the BESIII data.Although the predicted branching fraction underestimates the data, we may expect the 1 ++ partial wave including f 1 (1420) to fill the deficiency.
Here, we stress that all of the above conclusions are based on the Dalitz plot pseudo data including only the 0 −+ contribution, and on the current branching ratios of η(1405/1475) → ηππ and η(1405/1475) → γρ relative to that of η(1405/1475) → K Kπ.Since all of this experimental information was extracted with simpler Breit-Wigner models, our results might be biased.This situation encourages further studies.
In the next step, we will extend the present analysis by including more partial waves such as 1 ++ and 2 ++ , and directly analyze the BESIII data of J/ψ → γK S K S π 0 .Then we can perform the partial wave decomposition with our unitary coupled-channel framework by ourselves.With the 0 −+ amplitude obtained in this way, the two-pole solution of η(1405/1475) needs to be reexamined.Also, we can study the relevant resonances such as η(1405/1475) and f 1 (1420) with the unitary coupledchannel framework consistently.
with b LI ab being a cutoff; B ab is a factor associated with the Bose symmetry: B ab = 1/2 for identical particles a and b, and B ab = 1 otherwise.The partial wave amplitude is then given by Next, we also include bare R-excitation mechanisms in the interaction as with m R being the bare R mass.A bare R → ab vertex function is denoted by f LI ab,R (q) and f LI R,ab (q) = f LI ab,R (q); an explicit form has been given in Eq. (7).With the interaction of Eq. (A6), the resulting scattering amplitude is given by The second term has been given in Eq. (A3).The dressed R → ab vertex, denoted by fab,R , is given by The dressed Green function for R, τ LI R ′ ,R (p, E), in Eq. (A7) has been given in Eqs. ( 8) and ( 9) with f ab,R ′ being replaced by fab,R ′ .
The partial wave amplitude, T LI a ′ b ′ ,ab in Eq. (A7), is related to the S-matrix by s LI ab,ab (E) = η LI e 2iδ LI = 1 − 2πiρ ab B ab T LI ab,ab (q o , q o ; E) ,(A10) where δ LI and η LI are the phase shift and inelasticity, respectively; q o is the on-shell momentum (E = E a (q o ) + E b (q o )); ρ ab = q o E a (q o )E b (q o )/E is the phasespace factor.
2. Fits to ππ, πK, and πη scattering data In our unitary coupled-channel model for describing the radiative J/ψ decays in the η(1405/1475) region, ππ − K K, πK, and πη − K K coupled-channel scattering amplitudes of E < ∼ 1.2 GeV are the major components.Our choices for the scattering models such as the number of R and contact interactions are specified in Table II.We determine the parameters in the two-meson scattering models such as h LI a ′ b ′ ,ab , b LI ab , m R , g ab,R , and c ab,R in Eqs.(A1), (A6), and (7) using experimental information.For the ππ−K K and πK s-and p-wave scattering amplitudes, we fit empirical scattering amplitudes by adjusting the model parameters, and obtain reasonable fits as seen in Fig. 19(a-e).
Regarding the πη − K K s-wave scattering amplitude that includes the a 0 (980) pole, we consider two experimental inputs.First, our a 0 (980) propagator (τ LI R ′ ,R in Eq. (A7)) is fitted to the denominator of the a 0 (980) amplitude [Eq.(4) of Ref. [60]] from the BESIII amplitude analysis on χ c1 → ηπ + π − .Second, the ratio of coupling strengths (including the form factor) between the a 0 (980) → πη and a 0 (980) → K K is fitted to an empirical value of 1.03 from Ref. [46].Furthermore, the relative phase between the πη → πη and πη → K K amplitudes is chosen to be consistent with those from the chiral unitary model [61].In Fig. 19(f), we show our πη → πη and πη → K K scattering amplitudes defined by Finally, we obtain the πη − K K − ρπ d-wave scattering amplitude with the a 2 (1320) pole by adjusting the model parameters so that the mass and width of a 2 (1320), and branching fractions of a 2 (1320) → πη and a 2 (1320) → K K are reproduced; all of the fitted a 2 (1320) properties are from the PDG listing [15].
From the obtained partial wave amplitudes, resonance poles are extracted and presented in Tables III -V.Overall, the pole locations are consistent with those listed in the PDG [15].Numerical values of the fitting parameters  Table IX presents model parameters determined by fitting J/ψ → γ(0 −+ ) → γ(K S K S π 0 ) Dalitz plot pseudodata and the branching fractions of η(1405/1475) → ηπ + π − and η(1405/1475) → ρ 0 γ relative to that of  (c(ab) LI ) l are coupling and cutoff parameters, respectively.This bare vertex function is used in a dressed vertex and a self energy in a similar manner as the bare vertex F (cR) l ,M * i in Eq. ( 12) is used in Eqs.(10), (11), and ( 16).   4), the bare mass m η * i in Eq. ( 15), and bare couplings C i cR n in Eq. ( 12) and C i c(ab) LI in Eq. (B1); the subscripts l are suppressed.R n stands for n-th bare R state, while (ab)LI is a direct decay into two pseudoscalar mesons (ab) with the orbital angular momentum L and total isospin I.The nonresonant amplitude parameter, cNR, has been introduced in Eq. (27).Since the overall normalization of the full amplitude is arbitrary in our model, a common scaling factor can be multiplied to g J/ψη * i γ and cNR.All cutoffs [Λ i cR n in Eq. ( 12) and Λ i c(ab) LI in Eq. (B1)] are fixed to 700 MeV.

FIG. 1 .
FIG. 1.(a) Diagrammatic representation for radiative J/ψ decay amplitude of Eq. (2).The dashed lines represent pseudoscalar mesons while the solid lines are bare two-meson resonances R. The double lines with M * i(j) represent bare states for M * such as η(1405/1475).(b) The dressed M * propagator: the first [second] diagram on the r.h.s. is a bare M * propagator [self energy].(c) The dressed M * decay vertex: the first [second] diagram is a bare vertex [rescattering term].The ellipse stands for the scattering amplitude X (d) Lippmann-Schwinger-like equation for the amplitude X. (e) The dressed R propagator: the first [second] diagram is a bare R propagator [self energy].

2 FIG. 2 .
FIG. 2. The KSKSπ 0 Dalitz plot distributions for J/ψ → γ(0 −+ ) → γ(KSKSπ 0 ).Our fit result and pseudodata (MC) are shown.The E values used in our calculation (the central values of the E bins of the pseudodata) are indicated in each panel.The distributions are shown, in the descending order, by the red, yellow, green, and blue colors.Depending on E, the same color means different absolute values.
), TABLE I. Locations of poles (Eη * ); each pole is labeled by α.The mass, width and Eη * are related by M = Re[Eη * ] and Γ = −2Im[Eη * ].Each pole is located on the Riemann sheet (RS) specified by (s K * K , s a 2 (1320)π ); sx = p(u) indicates the physical (unphysical) sheet of a channel x.Breit-Wigner parameters from the BESIII analysis are also shown.Errors are statistical only.
Appendix B: Parameters fitted to radiative J/ψ decay data

TABLE II .
Description of two-meson scattering models.Partial waves are specified by the orbital angular momentum L and the isospin I.

TABLE III .
Pole positions (M pole ) in our ππ scattering amplitudes.The Riemann sheets (RS) of the pole positions are specified by (sππ, s K K ) where sx = p(u) indicates that a pole is on the physical (unphysical) sheet of the channel x; "−" (hyphen) indicates no coupling to the channel.

TABLE IV .
Pole positions (M pole ) in our πK scattering amplitudes.The Riemann sheets (RS) of the pole positions are specified by (sπK ).
η(1405/1475) → K Kπ.When a two-meson scattering model includes contact interactions, we consider a direct bare M * → abc decay where two pseudoscalar mesons (ab) have an orbital angular momentum L and a total isospin I.We describe this bare vertex function with [cf.Eq.(12)]

TABLE V .
Pole positions (M pole ) in our πη scattering amplitudes.The Riemann sheets (RS) of the pole positions are specified by (sπη, s K K , sρπ).

TABLE VI .
(7)ameter values for the πK partial wave scattering models.The i-th bare R states (Ri) has a mass of mR i , and it decays into h1 and h2 particles with couplings (g h 1 h 2 ,R i ) and cutoffs (c h 1 h 2 ,R i ).Couplings and cutoffs for contact interactions are denoted by h h 1 h 2 ,h 1 h 2 and b h 1 h 2 , respectively.The parameters have been defined in Eqs.(A1), (A2), (A6), and(7).For simplicity, we suppress the superscripts, LI, of the parameters.The mass and cutoff values are given in the unit of MeV, and the couplings are dimensionless.

TABLE VII .
Parameter values for the ππ partial wave scattering models.See Table VI for the description.

TABLE VIII .
Parameter values for the πη partial wave scattering models.See TableVIfor the description.

TABLE IX .
Parameter values for i-th bare η * state obtained from one of the bootstrap fits.The symbols are the J/ψη * i γ coupling constant g J/ψη * i γ in Eq. (