Study of the $Y(4660)$ from a light-quark perspective

In this paper, we try to reveal the structure of the $Y(4660)$ from the light-quark perspective. We study the dipion invariant mass spectrum and the helicity angular distributions of the $e^+ e^- \to Y(4660) \to \psi(2S) \pi^+\pi^-$ process. In particular, we consider the effects of different light-quark SU(3) eigenstates inside the $Y(4660)$. The strong pion-pion final-state interactions as well as the $K\bar{K}$ coupled channel in the $S$-wave are taken into account model independently by using dispersion theory. We find that the light-quark SU(3) octet state plays a significant role in this transition, implying that the $Y(4660)$ contains a large light-quark component and thus might not be a pure conventional charmonium state. In the fit scheme considering both the SU(3) singlet and SU(3) octet states, two solutions are found, and both solutions reproduce the $\pi\pi$ invariant mass spectra well. New measurement data with higher statistics in the future will be helpful to better distinguish these two solutions.


I. INTRODUCTION
In recent years, a number of charmoniumlike states have been discovered and they challenge our current understanding of hadron spectroscopy. Among these states, the Y (4660) was first observed in the initial-state radiation process e + e − → γ ISR ψ(2S)π + π − by the Belle Collaboration [1].
In the present work, we will study the possible light-quark components of the Y (4660) to help reveal its internal structure. We will focus on the ππ invariant mass spectrum of the reaction e + e − → Y (4660) → ψ(2S)ππ, which was presented after applying an appropriate cut to the ψ(2S)ππ invariant mass in Ref. [33]. In this process, the ππ invariant mass can reaches above the KK threshold, and thus allows us to extract the information of the light-quark SU(3) flavorsinglet and flavor-octet components. If the Y (4660) contains no light quarks (as in the charmonium scenario), the light-quark source provided by the Y (4660) has to be in the form of an SU(3) singlet state. Therefore the determination of the contributions from different SU(3) eigenstate components is instructive to clarify the internal structure of the Y (4660), especially in the case if a nonzero SU(3) octet component is found to be indispensable to reproduce the experimental data. The similar strategy has been applied to study the nature of the Y (4260) state in our previous work [38].
The main difference is that in the present work we simultaneously fit to the experimental data of the ππ invariant mass distributions and the helicity angular distributions, while in Ref. [38] the helicity angular distribution data were not considered.
Parity and C-parity conservation require the dipion system in e + e − → Y (4660) → ψ(2S)ππ to be in even partial waves. The dipion invariant mass can reaches above the KK threshold, so the coupled-channel final-state interactions (FSIs) in the S-wave is strong and needs to be taken into account. Based on unitarity and analyticity, the modified Omnès representation is used in this study, where the left-hand-cut contribution is approximated by the sum of the Zexchange mechanism and the triangle diagrams Y (4660) →D * D * 1 (2600) →D * Dπ(D * D s K) → ψ(2S)ππ(ψ(2S)KK ). 1 2 At low energies, the amplitude should agree with the leading chiral contact results. For the leading contact couplings for Y (4660)ψ(2S)ππ and Y (4660)ψ(2S)KK , we construct the chiral Lagrangians in the spirit of the chiral effective field theory (χEFT) and the heavy-quark nonrelativistic expansion [39]. The parameters are then determined from fitting to the Belle data. The relevant Feynman diagrams considered are given in Fig. 1.
This paper is organized as follows. In Sec. II, we introduce the theoretical framework and elaborate on the calculation of the transition 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 (4660) and its structure. A summary is given in Sec. IV.

A. Lagrangians
In general, the light-quark sources provided by the Y (4660) in the Y (4660) → ψ(2S)π + π − transition may come from two ways, one is from the possible light-quark components contained in the Y (4660) (e.g., in the charm-and anticharm-mesons molecule or the four-quark scenarios), and the other way is that the light quarks are excited by the Y (4660) from vacuum (e.g., in the pure cc or the hybrid state scenarios). Here we do not distinguish these two types of possible light-quark sources, but take them into account in an unified scheme, since what is matter here is the relative strengths between the light-flavor SU(3) singlet part and SU(3) octet part acting in this transition.
Considering the light-quarks sources provided by the Y (4660), the Y (4660) can be decomposed into SU(3) singlet and octet components of light quarks, where Note that the heavy-quark (e.g., c quark) components are contained in the V heavy , and they are not distinguishable in |V 1 and |V 8 . Expressed in terms of a 3 × 3 matrix in the SU(3) flavor space, the Y (4660) is written as The effective Lagrangian for the Y (4660)ψ(2S)ππ and Y (4660)ψ(2S)KK contact couplings, at leading order in the chiral as well as the heavy-quark nonrelativistic expansion, reads [38,39] 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 can be filled nonlinearly into with the Goldstone fields Here F corresponds to the pion decay constant, F π = 92.1 MeV, in the chiral limit.
We need the Z c Y (4660)π and the Z c ψ(2S)π interacting Lagrangians to calculate the contri- In order to calculate the triangle diagrams Y (4660) 3 we need the Lagrangians for the coupling of the Y (4660) toD * D * 1 as well as the couplings of the D * 1 to Dπ and D s K, where P denotes the pseudoscalar meson π or K. We also need the Lagrangian for the ψ(2S)D * Dπ vertex, which at leading order in heavy-meson chiral perturbation theory is where the charm mesons are collected in The gauge-invariant γ * (µ) and Y (4660)(ν) interaction can be written as where p is the momentum of the virtual photon γ * .

B. Amplitudes of Y (4660) → ψ(2S)P P processes
The decay amplitude of Y (4660)(p a ) → ψ ′ (p b )P (p c )P (p d ) can be described in terms of the Mandelstam variables We define q as the three-momentum of the final ψ(2S) in the rest frame of the Y (4660) with where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc) is the Källén triangle function.
Using the Lagrangians in Eq. (3), the projections of the S-and D-waves of the chiral contact terms are obtained as where the kaon decay constant F K = 0.113 GeV has been employed, and σ P ≡ 1 − 4m 2 P /s. For the D-wave, the single-channel FSI will be taken into account and we only give the amplitude of the process involving pions.
For the Y (4660) → ψ(2S)π + π − process, since the crossed-channel exchanged Z 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.) and requires special treatment. As discussed in Ref. [42], the l.h.c. is in fact in the unphysical Riemann sheet. The proper analytical continuation for the energy variable q 2 → q 2 + iǫ helps to locate the l.h.c. in the right position so that it does not overlap with the r.h.c. in the physical Riemann sheet. Also we will take into account the finite width in the D * 1 propagator. Using the Lagrangians in Eq. (6), the projections of S-and D-waves of the Z c -exchange amplitude are obtained aŝ where Zc + 2iM Zc Γ Zc )/κ π (s), and Q 0 (y) is the Legendre function of the second kind, In 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 [43]. Explicitly, the partial-wave where is the product of the coupling constants for the triangle diagrams.

C. Final-state interactions with a dispersive approach, Omnès solution
There are strong FSIs in the ππ system, which can be taken into account model-independently using dispersion theory. Based on unitarity and analyticity, the Omnès solutions will be used in this study. Similar methods to consider the FSI have been applied previously e.g. in Refs. [38,[43][44][45][46][47][48][49][50][51][52]. Because the invariant mass of the pion pair reaches above the KK threshold, we will take account of the coupled-channel (ππ and KK) FSIs for the dominant S-wave component, while for the D-wave only the single-channel (ππ) FSI will be considered.
For Y (4660) → ψ(2S)π + π − , the partial-wave decomposition of the amplitude including the s-channel FSI reads where M π l (s) includes the r.h.c. part and accounts for the s-channel rescattering, and the "hat function"M π l (s) contains the l.h.c., contributed by the possible crossed-channel pole terms or the open-flavor 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). θ is the angle between the positive pseudoscalar meson and the Y (4660) in the rest frame of the P P system.
For the D-wave, we will take account of the single-channel FSI. 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 [59,60]. The modified Omnès solution of Eq. (25) is [51,61] M 2 (s) = Ω 0 2 (s) P n−1 where the polynomial P n−1 2 (s) is a subtraction function, and the Omnès function is defined as [62] Ω 0 2 (s) = exp We will use the Madrid-Kraków group [63] result for δ 0 2 (s), which is smoothly continued to π for s → ∞.
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 results given in Eq. (13). Therefore, for the S-wave, the amplitude 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 polarization-averaged modulus square of the e + e − → Y (4660) → ψ(2S)π + 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  [36]. We also have tried to allow the mass and width of the Y (4660) to float freely, and found that the fit quality changes only slightly. At last, the ππ invariant mass spectra and the helicity angular distribution for e + e − → ψ(2S)π + π − can be calculated using where the limits of integration are chosen to be identical to the cuts used to get the experimental rate [33], N is the normalization factor, k 1 and k 5 represent the three-momenta of e ± and Φ in the center of mass frame, respectively, and k * 3 denotes the three-momenta of π ± in the rest frame of the ππ system. They are given as

A. Characteristics of singlet and octet contributions
In this work we perform fits simultaneously taking into account the experimental data of the ππ invariant mass distributions and the helicity angular distributions collected in the Y (4660) region of the e + e − → ψ(2S)ππ process [33]. Using the constraint h 8 = h 1 g 8 /g 1 , 4 there are five free parameters in our fits:  Table I.
It is obvious that in Fit I the peak around 1 GeV in the ππ mass spectrum is not reproduced, although the angular distribution can be described. In contrast, in Fits IIa and IIb, including the SU(3) octet terms, the fit qualities are improved significantly. The fit quality of Fit IIb is a little better than that of Fit IIa. singlet and octet components. The experimental data are taken from Ref. [33].  solutions reproduce the ππ invariant mass spectra well. Notice that the present data is limited in statistics, and new measurement data with higher statistics in the future will be helpful to distinguish between these two solutions.