On the nature of the $Y(4260)$: a light-quark perspective

The $Y(4260)$ has been one of the most puzzling pieces among the so-called $XYZ$ states. In this paper, we try to gain insights into the structure of the $Y(4260)$ from the light-quark perspective. We study the dipion invariant mass spectrum of the $e^+ e^- \to Y(4260) \to J/\psi \pi^+\pi^-$ process and the ratio of the cross sections ${\sigma(e^+e^- \to J/\psi K^+ K^-)}/{\sigma(e^+e^- \to J/\psi \pi^+\pi^-)}$. In particular, we consider the effects of different light-quark SU(3) eigenstates inside the $Y(4260)$. The strong pion-pion final-state interactions as well as the $K\bar{K}$ coupled channel in the $S$-wave are taken into account in a model-independent way using dispersion theory. We find that the SU(3) octet state plays a significant role in these transitions, implying that the $Y(4260)$ contains a large light-quark component. Our findings suggest that the $Y(4260)$ is neither a hybrid nor a conventional charmonium state, and they are consistent with the $Y(4260)$ having a sizeable $\bar D D_1$ component which, however, is not completely dominant.

In this work, we will study the possible light-quark components of the Y (4260) to help reveal its internal structure. We will focus on the ππ invariant mass spectrum of the reaction e + e − → Y (4260) → J/ψππ, which is one of the most accurately measured channels and is the discovery channel of the Y (4260). In this process, the dipion invariant mass reaches above the KK threshold, and thus allows us to extract the information of the light-quark SU(3) flavor-singlet and flavoroctet components. The ratio of the cross sections σ(e + e − → J/ψK + K − )/σ(e + e − → J/ψπ + π − ) is relevant to the strange-quark component, and will also be taken into account. If the Y (4260) contains no light quarks (as in the hybrid state or the charmonium scenarios), the light-quark source provided by the Y (4260) has to be in the form of an SU(3) singlet state. Thus the determination of the contributions from different SU(3) eigenstate components is instructive to clarify the structure of the Y (4260), especially in the case if a nonzero SU(3) octet component is found to be indispensable to reproduce the experimental data.
The conservation of parity and C-parity constrains the dipion system in e + e − → Y (4260) → J/ψππ to be in even partial waves. The dipion invariant mass m ππ goes up to more than 1.1 GeV. In this energy region, there are strong coupled-channel final-state interactions (FSIs) in the S-wave, which include the scalar resonances f 0 (500) and f 0 (980) and can be taken into account modelindependently using dispersion theory. Based on unitarity and analyticity, the modified Omnès representation is used in this study, where the left-hand-cut contributions are approximated by the sum of the Z c (3900)-exchange mechanism and the triangle diagrams Y (4260) →DD 1 (2420) → DD * π(DD * s K) → J/ψππ(J/ψKK) [29,40,41]. 1 At low energies, the amplitude should agree with the leading chiral results, so the subtraction terms in the dispersion relations can be determined by matching to the chiral contact terms. For the leading contact couplings for Y (4260)J/ψππ and Y (4260)J/ψKK , we construct the chiral Lagrangians in the spirit of the chiral effective field theory (χEFT) and the heavy-quark nonrelativistic expansion [42]. The parameters are then fixed from fitting to the BESIII data. A diagrammatic representation of all contributions is given in Fig. 1.
This paper is organized as follows. In Sec. II, we describe the theoretical framework and elaborate on the calculation of the amplitudes as well as the dispersive treatment of the FSI. In Sec. III, we present the fit results and discuss the light-quark components of the Y (4260) and its structure. A brief summary is given in Sec. IV. 1 We also need to take account of the Y (4260) → J/ψKK process in the coupled-channel FSI.

A. Lagrangians
In general, the Y (4260) can be decomposed into SU(3) singlet and octet components of light quarks, where and the ratio of the component strengths r ≡ b/a can be determined through fitting to the data. Expressed in terms of a 3 × 3 matrix in the SU(3) flavor space, it is written as The effective Lagrangian for the Y (4260)J/ψππ and Y (4260)J/ψKK contact couplings, at leading order in the chiral expansion and respecting the heavy-quark spin symmetry, reads [42][43][44] where . . . denotes the trace in the SU(3) flavor space, J = (ψ/ √ 3) · ½, and v µ = (1, 0) is the velocity of the heavy quark. The lightest pseudoscalar mesons, being the pseudo-Goldstone bosons from the spontaneous breaking of chiral symmetry, can be filled nonlinearly into with the Goldstone fields Here F is the pion decay constant in the chiral limit, and we take the physical value 92.1 MeV for it.
We need to define the Z c Y (4260)π and the Z c J/ψπ interacting Lagrangians to calculate the contribution of the intermediate Z c states, namely Y (4260) → Z c π → J/ψππ. Note that there is no hint so far for the existence of a hidden-charm strange partner of the Z c state [45]. We thus parametrize the Z c states in a matrix as The leading-order Lagrangians are [46] which give the S-wave pionic vertices proportional to the pion energy. Note that the SU(3) singlet and octet components of the Y (4260) are not distinguishable in the Z c Y (4260)π interaction, as the strange-quark component is irrelevant here.
In order to calculate the triangle diagrams Y (4260) →DD 1 (2420) →DD * π(DD * s K) → J/ψππ(J/ψKK), 2 we need the Lagrangians for the coupling of the Y (4260) toDD 1 as well as the couplings of the D 1 to D * π and D * s K [27,47,48], where P denotes the pseudoscalar meson π or K. We also need the Lagrangian for the J/ψD * Dπ and J/ψD * s DK vertices, which at leading order in heavy-meson chiral perturbation theory is [49] where the charm mesons are collected in H a = V a · σ + P a with P a (V a ) = (D ( * )0 , D ( * )+ , D The gauge-invariant γ * (µ) and Y (4260)(ν) two-point coupling is given by where p is the momentum of the virtual photon γ * . 2 Here and in the following,DD1 always means the negative C-parity combination ofDD1 and DD1.

B. Amplitudes of Y (4260) → J/ψP P processes
First we consider the decay amplitude of Y (4260)(p a ) → J/ψ(p b )P (p c )P (p d ), which is described in terms of the Mandelstam variables The variables t P and u P can be expressed in terms of s and the scattering angle θ according to where θ is defined as the angle between the positive pseudoscalar meson and the Y (4260) in the rest frame of the P P system, and λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc) is the Källén triangle function. We define q as the 3-momentum of final J/ψ in the rest frame of the Y (4260) with For the Y (4260) → J/ψπ + π − process, since the crossed-channel exchanged Z c and DD * can be on-shell, the left-hand cut (l.h.c.) produced intersects and overlaps with the right-hand cut (r.h.c.).
Implementing the modified Omnès solution method to obtain the amplitude including FSI relies on the ability to separate the amplitude into two parts having either l.h.c. or r.h.c. only. A way of separating the two has been proposed in Ref. [51], using the spectral representation of the resonance propagator as well as a consistent application of the iǫ prescription for the energy variables. 3 Similarly we use the spectral representations of the Z c propagator and the D 1 propagator [51], where The off-shell-width effects of the broad intermediate resonances could play a role in the process discussed [29,39], and we construct the energy-dependent widths for the broad vector resonances. Taking into account that the Z c J/ψπ vertex is in an S-wave and proportional to the energy of the pion, and the D 1 → D * π decays in a D-wave, the energy-dependent widths of Z c and D 1 read where k QP (s) = λ 1/2 (M 2 Q , m 2 P , s)/(2 √ s) is the magnitude of the three-vector momentum of the pion, and E QP (s) = m 2 π + k 2 QP (s). The thresholds in Eq. (14) are x thr D 1 = (M D + m π ) 2 and x thr Zc = (M ψ + m π ) 2 , respectively. 4 Notice that the integration convolves with other parts of the amplitude. Now the Z c -exchange amplitude readŝ where C Zc Y ψ ≡ C ZcY π C Zcψπ is the product of the coupling constants for the exchange of the Z c . The amplitude has been partial-wave decomposed, and P l (cos θ) are the standard Legendre polynomials.
Parity and C-parity conservation (or isospin conservation combined with Bose symmetry) require the pion pair to be in even angular momentum partial waves. We only take into account the Sand D-wave components in this study, neglecting the effects of higher partial waves. Explicitly, the projections of S-and D-waves of the Z c -exchange amplitude read and respectively, where y(s, x ′ ) ≡ (3s 0 − s − 2x ′ )/κ π (s), and Q 0 (y) is the Legendre function of the second kind, Notice that the analytic continuation of Q 0 (y) should be taken into account since the Z c can be on-shell in the physical region. There are two finite branch points in Q 0 (y(s, x ′ )), In the range of s − < s < s + , the argument of the logarithm in Eq. (19) becomes negative, and the continuation reads [56][57][58] Q 0 (y) = Now we briefly discuss the calculation of the triangle diagrams. We only keep the terms proportional to ǫ Y · ǫ ψ , and omit the remaining terms proportional to contractions of momenta with the polarization vectors, which are suppressed in the heavy-quark nonrelativistic expansion [44]. Explicitly, the partial-wave projections of the triangle amplitude for the Y (4260) → J/ψππ(J/ψKK) where C loop Y ψ ≡ yh ′ g ψP is the product of the coupling constants for the triangle diagrams. For the chiral contact terms, using the Lagrangians in Eq. (3), we have The projections of the S-and D-waves of the chiral contact terms are given by For the D-wave, where the ππ scattering is almost elastic in the energy range considered here, we only give the amplitude of the process involving pions.

C. Final-state interactions with a dispersive approach, Omnès solution
There are strong FSI in the ππ system in particular in the isospin-0 S-wave, which can be taken into account model-independently using dispersion theory. Since the invariant mass of the pion pair reaches above the KK threshold, we will consider the coupled-channel (ππ and KK) FSI for the dominant S-wave component, while for the D-wave only the single-channel (ππ) FSI will be considered.
For Y (4260) → J/ψπ + π − , the partial-wave expansion of the amplitude including FSI reads where M π l (s) contains the r.h.c. part and accounts for the s-channel rescattering, and the "hat function"M π l (s) represents the l.h.c., contributed by the crossed-channel pole terms or the openflavor loop effects. In this study, we approximate the l.h.c. by the sum of the Z c -exchange diagram and the triangle diagrams, i.e.,M π l (s) =M Zc,π l (s) +M loop,π l (s). The method of approximating the l.h.c. in dispersion relations by including the most relevant resonance exchanges (in the case of no loops) has been applied previously e.g. in Refs. [43,[59][60][61][62][63][64][65].
For the S-wave, we will take into account the two-channel rescattering effects. The functionŝ M l (s) do not have a r.h.c., so the two-channel unitarity condition leads to the discontinuity of the production amplitudes as where the two-dimensional vectors M 0 (s) andM 0 (s) stand for the r.h.c. and the l.h.c. parts of both the ππ and the KK final states, respectively, The two-dimensional matrices T 0 0 (s) and Σ(s) are given by and Σ(s) ≡ diag σ π (s)θ(s − 4m 2 π ), σ K (s)θ(s − 4m 2 K ) . Three input functions enter the T 0 0 (s) matrix: the ππ S-wave isoscalar phase shift δ 0 0 (s), and the modulus and phase of the ππ → KK S-wave amplitude g 0 0 (s) = |g 0 0 (s)|e iψ 0 0 (s) . To estimate the uncertainty due to the dispersive input for the ππ/KK rescattering, we will use two different T 0 0 (s) matrices, the Dai-Pennington (DP) [63][64][65] and the Bern/Orsay (BO) [66,67] parametrizations, and compare the results. Note that the inelasticity parameter η 0 0 (s) in Eq. (28) is related to the modulus |g 0 0 (s)| by These inputs are used up to √ s 0 = 1.3 GeV, below the onset of further inelasticities from the 4π intermediate states, where the f 0 (1370) and f 0 (1500) resonances become important that couple strongly to 4π [68,69]. Above s 0 , the phases δ 0 0 (s) and ψ 0 0 are guided smoothly to 2π by means of [70] The solution of the inhomogeneous coupled-channel unitarity condition in Eq. (26) is given by where Ω(s) satisfies the the homogeneous coupled-channel unitarity relation Im Ω(s) = T 0 * 0 (s)Σ(s)Ω(s), and its numerical results have been computed, e.g., in Refs. [70][71][72][73].
For the D-wave, the single-channel FSI will be taken into account. In the elastic ππ rescattering region, the partial-wave unitarity condition reads where the phase of the isoscalar D-wave amplitude δ 0 2 coincides with the ππ elastic phase shift, as required by Watson's theorem [74,75]. The modified Omnès solution of Eq. (33) can be obtained as [43,76] where the polynomial P n−1 2 (s) is a subtraction function, and the Omnès function is defined as [77] Ω 0 2 (s) = exp We will use the result given by the Madrid-Kraków group [78] for δ 0 2 (s), which is smoothly continued to π for s → ∞.
In order to determine the necessary number of subtractions that guarantees the convergence of the dispersive integrals in Eqs. (31) and (34), we need to investigate the high-energy behavior of the integrands. First, it is known that for a phase shift δ I l (s) approaching k π at high energies, the corresponding single-channel Omnès function falls asymptotically as s −k . As a consequence, we have Ω 0 2 (s) ∼ 1/s at large s. Furthermore, the coupled-channel Omnès function Ω I l (s) is found to fall asymptotically as 1/s for large s [70], provided the asymptotic condition δ I l (s) ≥ 2π for s → ∞, where δ I l (s) is the sum of the eigen-phase shifts. Second, we have checked that in the intermediate energy region of 1 GeV 2 s ≪ M 2 Y (4260) , the inhomogeneity contributed by the Z c -exchange and the triangle diagrams grows at most linearly in s. So we conclude that in the dispersive representations of Eqs. (31) and (34), three subtractions for each of them are sufficient to make the dispersive integrals convergent. On the other hand, at low energies the amplitudes M 0 (s) and M 2 (s) should match to those from χEFT. Namely, in the limit of switching off the FSI at s = 0, Ω(0) = ½ and Ω 0 2 (0) = 1, the subtraction terms should agree well with the low-energy chiral amplitudes given in Eq. (24). Therefore, for the S-wave, the integral equation takes the form where M χ 0 (s) = M χ,π 0 (s), 2/ √ 3 M χ,K 0 (s) T , while for the D-wave, it can be written as The amplitude for Y (4260) → J/ψπ + π − can be expressed in terms of the ingredients discussed above as M decay (s, cos θ) = M π 0 (s) +M π 0 (s) + M π 2 (s) +M π 2 (s) P 2 (cos θ) .
The polarization-averaged modulus-square of the e + e − → Y (4260) → J/ψπ + π − amplitude can be written as where E is the center-of-mass energy of the e + e − collisions, and we set the γ * Y (4260) coupling constant c γ to 1 since it can be absorbed into the overall normalization when we fit to the event distributions. Here we use the energy-independent width for the Y (4260), and the values of the Y (4260) mass and width are taken as 4222 MeV and 44.1 MeV, respectively, which are the central values of the BESIII fit in Ref. [33]. We also have tried to allow the mass and width to float freely, and found that the fit quality changes only slightly. At last, the ππ invariant mass distribution of where k 1 and k 5 denote the 3-momenta of e ± and J/ψ in the center-of-mass frame, respectively, and k * 3 is the 3-momenta of π ± in the rest frame of the ππ system. They are given as For e + e − → Y (4260) → J/ψK + K − , the relevant Feynman diagrams can be obtained by replacing all external pions by kaons in Fig. 1 (for (c1), the exchanged D * needs to be replaced by D * s ), but without diagram (b1) due to the absence of the Z c ψK vertex. Most ingredients of the amplitude of e + e − → Y (4260) → J/ψK + K − have been given in the above.

A. Characteristics of singlet and octet contributions
The two pions in the final state must come from light-flavor sources. It is instructive to discuss what would be expected for the dipion invariant mass distributions produced from pure SU (3) flavor singlet and octet sources, which are proportional to (ūu +dd +ss)/ √ 3 and (ūu +dd − 2ss)/ √ 6, respectively, without considering the left-hand-cut contribution. It is well-known that the nonstrange and strange scalar pion form factors, 0|(ūu +dd)|π + π − and 0|ss|π + π − , behave very differently. The former has a broad bump around 0.5 GeV, and has a narrow dip at around 1 GeV, while the latter has a narrow peak at around 1 GeV. The narrow structures are manifestations of the scalar meson f 0 (980), which couples differently to the nonstrange and strange sources [73,79].
It is therefore natural to expect that the SU(3) singlet and octet pion scalar form factors should also be dramatically different.
To demonstrate the characteristic structures in the dipion mass spectrum from the singlet and octet sources for the current problem, we need to take into account the energy dependence in the chiral contact terms. Their contributions are separately shown with varying h i /g i in Fig. 2. and octet spectra are stable against the variation of h i /g i : the singlet spectra display a broad bump below 1 GeV, and around 1 GeV there is a dip for h 1 /g 1 1; the octet spectra have little contribution below 0.9 GeV, and show a sharp peak around 1 GeV, corresponding to the f 0 (980).
It is also worthwhile to notice that both of them have different behaviors from both the nonstrange and the strange pion scalar form factors. Therefore, one expects that precise measurements of the dipion invariant mass distributions can provide valuable information about the light-quark content of the source, considering the J/ψ to be a SU(3) flavor singlet.

B. Fitting to the BESIII data
In this work we perform fits taking into account the experimental data sets of the ππ invariant mass distributions of e + e − → J/ψπ + π − and the ratios of the cross sections σ(e + e − → J/ψK + K − )/σ(e + e − → J/ψπ + π − ) measured at two energy points E = 4.23 GeV and E = 4.26 GeV by the BESIII Collaboration [80,81]. As in Refs. [55,80], we regard the measurements at E = 4.23 GeV and E = 4.26 GeV as independent, and thus the coupling constants are allowed to be different in the fits of these two data sets. For the normalization factor for each data set, we choose to absorb it into the coupling constants. There are six free parameters in our fits:   I:  Experimental  and  theoretical  values  for  the  cross  sections  ratios σ(e + e − → J/ψK + K − )/σ(e + e − → J/ψπ + π − ) × 10 2 . The experimental data are taken from Ref. [81]. considered in all the fits. 5 The parameter C Zc Y ψ , as a product of the Y Zcπ and Zcψπ couplings, is related to the partial widths of the Y → Zcπ and Zc → J/ψπ. In principle, it can be determined from a thorough analysis of the Zc and Y line shapes; such an analysis that takes into account the ππ FSI is not available yet. Thus, here we make a compromise by focusing on the ππ distribution and taking C Zc Y ψ as a free parameter. The uncertainty due to the dispersive input for the ππ/KK rescattering is estimated by comparing the fits with the two different T 0 0 (s) matrices (DP [63][64][65] vs. BO [66,67]). In Fig. 3, the best fit results of the ππ mass spectrum in e + e − → J/ψπ + π − are shown, where the borders of the bands represent the fit results using these two different T 0 0 (s) matrix parametrizations. The fit results of the ratios of the cross sections σ(e + e − → J/ψK + K − )/σ(e + e − → J/ψπ + π − ) are given in Table I. The fitted parameters as well as the χ 2 /d.o.f. are shown in Tables II and III for the DP and BO parametrizations, respectively. As can be seen from Fig. 3 as well as Tables II and III, the fit quality to the data set at E = 4.23 GeV is worse than that at E = 4.26 GeV, in particular in the region close to the lower kinematical boundary and for the highest data point. Notice that by using the inputs from known scattering observables in the dispersion relations, the effects of resonances in the considered partial waves, i.e. the f 0 (500), f 0 (980), and f 2 (1270), are included automatically. Since the data set at E = 4.26 GeV has a larger phase space to reveal the nontrivial structure and the fits are better, we discuss the fit results of this data set in more details.
It is interesting to compare Fits IIa and IIb. In Fit IIa, the SU(3) octet chiral contact terms are not included. The experimental data, especially the broad peak in the region lower than 0.6 GeV, cannot be described well. In contrast, in Fit IIb, including the SU(3) octet chiral contact terms, the fit quality is improved significantly. A similar improvement is also observed comparing Fits Ib and Ia. We also perform two further Fits IIc and IId for the E = 4.26 GeV data set, considering only the contact terms and switching off the left-hand cuts: in Fit IIc we only retain the SU(3) The borders of the bands represent our best fit results using two different T 0 0 (s) matrices. The backgroundsubtracted and efficiency-corrected experimental data are taken from Ref. [80]. singlet component, while in Fit IId, both the SU(3) singlet and octet components are taken into account. The result is shown in Fig. 4, and the fit couplings are also listed in Table II Tables II and III, we have g 8 /g 1 = 1.2 ± 0.2 and h 8 /h 1 = 57 ± 76 in the DP parametrization and g 8 /g 1 = 1.1 ± 0.1 and h 8 /h 1 = 102 ± 152 in the BO one, which agree well with each other within errors. Note that h 8 /h 1 is not as stable as g 8 /g 1 : the reason is that h 1 is small in most fits. In theDD 1 hadronic molecule scenario of Y (4260), one has from which the light-quark component reads |uū we can estimate the ratio of β/α = −0.30 ± 0.05 based on our results of g 8 /g 1 . Thus we conclude that there is a large light-quark SU(3) octet component in the Y (4260), and scenarios of a hybrid or conventional charmonium are disfavored since the light quarks have to be produced in the SU (3) singlet state in such states. Also our study shows that theDD 1 component of the Y (4260) may not be completely dominant. This is not unnatural, as the Y (4260) mass, being around 4.22 GeV, is about 70 MeV below theDD 1 threshold.
In Fig. 5, we plot the moduli of the S-and D-wave amplitudes from the chiral contact terms, the Z c -exchange terms, and the triangle diagrams for Fit IIb. An interesting feature is that the 6 Notice that any isoscalar pair of nonstrange charm and anti-charm mesons has the same SU(3) structure.
D-wave contribution is comparable to the S-wave contribution in almost the whole phase space.
Such a large D-wave contribution in the Y ψΦΦ transition again indicates that the Y (4260) cannot be a conventional charmonium state, for which the ππ S-wave should be dominant. Notice that in theDD 1 hadronic molecule interpretation [29,82], the ππ D-wave emerges naturally since the D 1 decays dominantly into D-wave D * π. Also one observes that the contributions from the chiral contact terms and the l.h.c. contributions are of the same order. Amongst the l.h.c. contributions, both the Z c term and the triangle diagrams appear far from negligible. A better distinction of the effects of the Z c and the open-charm loops requires a detailed analysis of the J/ψπ distribution and is beyond the scope of the present paper.

IV. CONCLUSIONS
We have used dispersion theory to study the processes e + e − → Y (4260) → J/ψππ(KK).
In particular, we have analyzed the roles of the light-quark SU (3)  when the latter is constrained in the Y (4260) region [83].