Two- and three-body structure of the Y(4660)

We study general features of three-meson bound states using the Y(4660) as an example. Here the Y(4660) is assumed to be either a two-body bound state of the f_0(980), itself a bound state of K and Kbar, and the psi'= psi(2s), or a three-body bound state of psi', K, and Kbar. In particular, we investigate in detail the interplay of the various scales inherent in the problem, namely the f_0 binding energy, the Y binding energy, and the K-psi' scattering length. This allows us to understand under which circumstances the substructure of the f_0(980) can be neglected in the description of the Y(4660).

determine the structure of such states close to scattering thresholds. For twobody states located near a threshold there is a powerful method originally proposed by Weinberg for the deuteron [10]. This method was extended in Ref. [11] and allows one to quantify the two-body molecular component. A complementary formulation of this problem is given by effective range theory and the effective field theory (EFT) for large scattering length. The latter framework can also be extended to three-and four-body molecules [12,13]. Both methods effectively analyze the low-energy pole structure of the scattering amplitude in a model-independent way.
In this paper, we want to investigate to which extent it is possible to distinguish between two-and three-body molecules and which scales govern this distinction. In particular, we analyze the example of the Y (4660) in detail. This state was proposed to be a ψ ′ f 0 (980) molecule based on an analysis with the Weinberg method [14] while the f 0 (980) itself was proposed to be a molecule ofK and K mesons [6,7]. It is therefore important to elucidate whether the internal structure of the f 0 (980) can be neglected for the description of the Y (4660) or not. The answer to this question, of course, depends on the physical scales in this problem. For a deeply-bound f 0 (980), one expects that the internal structure can be neglected. Performing numerically exact three-body calculations, we determine for which parameter ranges the Y (4660) can be interpreted as a ψ ′ f 0 (980) molecule and for which ranges as a ψ ′ KK molecule. We stress that we only investigate the interplay of a possible two-body vs. three-body nature of the Y (4660) in order to understand the interplay of the various scales. No explicit compact component is included for the Y (4660) nor is the width of the f 0 (980) included. Thus, this work is mainly of theoretical interest and we are not yet in the position to compare with experimental line shapes. We start with a brief description of our formalism and some analytical considerations before we present and discuss the results of our three-body calculation. The paper ends with an outlook.

Formalism:
In this section, we briefly outline the approach used to derive the amplitude for ψ ′ f 0 (980)-scattering. For this purpose, we set up a non-relativistic effective field theory (EFT) for three distinguishable particles with different masses. 1 Note that the typical momentum of the kaons in this problem is of order 70 MeV or less such that they can still be treated non-  relativistically. The underlying Lagrangian reads [15] where the scalar functions ψ k and d k are particle-and dimer fields, respectively, and m k is the mass of particle k. The indices i, j, k in Eq. (1) are always different from each other such that the mass of particle k can also be labeled by m ij = m ji . This convention will be used below. Moreover, a two-body state of particles i and j can be labeled either by the indices ij or by the index k (i = j = k). The dimer fields are introduced for convenience and are not dynamical. An equivalent theory without dimer fields can be obtained by inserting the classical equation of motion for the non-dynamical dimer fields. Furthermore, g k and h are coupling-constants, and the factors c k sum up to one. From the Lagrangian (1), we can directly derive the Feynman-rules of our theory in momentum space and construct the general amplitude A ij for particle-dimer scattering in the channel i → j. The resulting equations are illustrated in Figs. 1 and 2.
After renormalizing the amplitude and performing a partial wave decomposition, the inhomogeneous integral equation for the l-th partial wave reads where p and k is the incoming and outgoing momentum, respectively. The kernel factorizes into the driving term and the full dimer propagator where are the reduced masses of the two-particle system labeled by ij or by k (i = j = k) and the particle-dimer system labeled by k, respectively. The Legendre functions of the second kind Q l and their arguments c ik (p, q) are defined by where P l denotes the l-th Legendre polynomial. We use standard solution techniques in order to solve the integral Eq. (2) numerically. The a priori undetermined, dimensionless function H(Λ), that by construction contributes only in the s-wave, is directly related to the parameter h in Eq. (1). The function H(Λ) plays the role of a running coupling constant and is determined by the renormalization procedure up to an unknown three-body parameter [16]. We fix its value by imposing that a three-particle bound-state exists at the binding energy E = −B. This requires that the amplitude A ij in Eq. (2) has a pole at this energy. However, any low-energy three-body observable can be chosen for this purpose. After fixing H, we are able to calculate the full particle-dimer scattering amplitude and therefrom, observables such as scattering lengths, cross sections and so forth can be predicted. The obtained results are independent of the value of the cutoff Λ, as long as Λ is large compared to all momentum scales in the problem.
This theory represents the leading order of an EFT around the limit of large pair scattering lengths a k [12,13]. Corrections are suppressed by powers of r 0 /a k and pr 0 , where r 0 is the range of the interaction and p is a typical momentum. If the scattering lengths are not sufficiently large, the theory can still be taken as a particular three-body model for hadronic molecules. As such, it can be used to test the limits of applicability of the Weinberg approach to systems like the Y (4660). By calculating the scattering length and the effective range, we are able to investigate the question, whether the three-particle bound-state can effectively be treated as a two-particle system, consisting of one particle and an elementary dimer. If the substructure of the dimer becomes relevant, the full three-particle picture is necessary. Transferred to our specific example Y (4660), this means that we want to distinguish between the two alternatives ψ ′ f 0 (980) and ψ ′ KK. In the Weinberg formalism [10], the quantity Z ∈ [0, 1] is defined as where Y denotes the wave function of the physical state. Thus, Z measures the "non-two-particle"-fraction of the wave function. From the definition (7), we directly conclude that Z → 0 implies that the Y (4660) is effectively a two-body system with an elementary f 0 (980)-dimer, whereas Z → 1 means that it has to be seen as something else, in our case a three-body molecule. This method of distinction between two-and three-particle molecules can of course also be applied to other candidates for hadronic molecules in order to reveal their few-particle nature. The quantity Z is directly related to the residue Z pole of the bound state pole in the two-particle scattering amplitude by Z = 1 − Z pole .
Performing a straightforward calculation within the Weinberg formalism, using the Lippmann-Schwinger equation and the effective range expansion, the quantity Z can be related to the effective range parameters in two different ways [10]. The two corresponding Z-factors, which we denote by should be (approximately) equal if the Weinberg formalism is applicable. The equations (8,9) receive corrections from the finite range of the interaction. They are exact in the limit of vanishing binding energy of the Y (4660) relative to the ψ ′ f 0 -threshold with Z = Z a = Z r kept fixed [10]. The quantities a ψ ′ f 0 and r ψ ′ f 0 are the scattering length and the effective range for ψ ′ f 0 -scattering, respectively. Moreover, is the scattering length within a pure two-particle picture.
The positive binding energies are defined via Thereby all quantities on the right sides of the equations in (8) can be calculated in the full three-body model. Combining the two foregoing conditions should hold. Below we will use the validity of Eq. (13) as a diagnostic tool to identify the range of applicability of the Weinberg method.
The full derivation of the equations (8) can be found in [10]. We omit it at this point, but outline its essential idea. Utilizing the binding energy between the particle ψ ′ and f 0 (980) as defined in Eq. (11), the Schrödinger equation can be applied in the form HereV describes the interaction andĤ 0 denotes the free-particle part of the Hamiltonian, such thatĤ holds, where E(p) = p 2 /(2µ ψ ′ f 0 ) and p is the relative momentum of the ψ ′ and f 0 in the center-of-mass. Employing the relations (14) and (15), the momentum-integral appearing in the definition of Z (7) assumes the form dp In order to deduce the formulas (8), the crucial requirement of the Weinberg method is that the numerator on the right-hand-side of Eq. (16) can be approximated by its value at zero momentum, dp For this to hold, the numerator has to vary slowly over the range of momenta contributing to the integral, i.e.
In other words, the range of the form factor ψ ′ f 0 (p)|V |Y has to be much larger than the range of the denominator, 2µ ψ ′ f 0 B ψ ′ f 0 .

Analytical Considerations:
Thus we have to understand the relevant momentum-scales of ψ ′ f 0 (p)|V |Y . In the general case, with an incoming particle and dimer of type i this form-factor ψ i d i (p) |V |Y is proportional to i Y (4660) Fig. 3. Feynman diagram for the form-factor d i ψ i (p)|V |Y with a particle and an dimer of type i in the incoming channel and Y (4660) as the outgoing bound state.
the bound state amplitude F i as it is depicted in Fig. 3. It can be calculated by solving the homogeneous analog of Eq. (2), where we have set l = 0 for an s-wave state. In the rest frame of the Y (4660), the total energy is simply E = −B Y . The quantity p is the relative momentum between the incoming particles. Solving Eq. (19) exactly requires the same effort as solving the scattering equation and can only be done numerically.
Such numerical solutions will be presented below.
Our aim here is to investigate to what extent the underlying scales of the form factor ψ i d i (p) |V |Y can be understood in a simple way. For this purpose, we approximate the full amplitude [F k ] l (q) on the right-hand-side of Eq. (19) by a constant. In this point-like approximation, the p-dependence of where we have inserted the definitions (3) and (4). Using (6), we deduce that for E = −B Y the quantity c := c ik (p, q) is always smaller than −1 and approaches −∞ for p → 0. Hence a Taylor-expansion of Q 0 around 1/c = 0 can be performed and we approximate: Inserting this result into (20), the momentum dependence of ψ i d i (p) |V |Y for p → 0 has the form: This expression is nearly constant for momenta so that the range of the form-factor can be estimated by √ 2µ k B Y . Coming back to our specific situation with ψ ′ and the f 0 -dimer, the range of ψ ′ f 0 (p)|V |Y is estimated as 2µ ψ ′ K B Y . From this we are able to formulate the condition for the validity of Eq. (17). Since the reduced mass terms µ ψ ′ K and µ ψ ′ f 0 are both of the order m K and B ψ ′ f 0 = B Y − B f 0 , this is equivalent to: where we have introduced the dimensionless parameter b 0 := B f 0 /B Y ∈ (0, 1), which is the binding energy of the f 0 -dimer relative to the binding energy of the Y (4660). Thus in this approximation the only relevant scale that controls the applicability of the Weinberg-method is b 0 which has to be close to 1, meaning that f 0 has to be a deeply bound state or an elementary particle.
We stress that b 0 can not be the only relevant scale in this problem. In particular, the scale characterizing the Kψ ′ andKψ ′ interactions does not enter at all in this simple picture. In the full solution of the three-body problem where the Y (4660) emerges as a dynamically generated three-particle-state, the situation will be more complex. We will investigate this question below by solving Eq. (2) numerically.
4. Results and discussion: Using our formalism, we now present some results for ψ ′ f 0 (980)-scattering within our three-body approach for the Y (4660). In order to calculate observables, we first have to fix the 3 masses and scattering lengths, appearing in (3), (4) and (6). Ignoring the -for our purposeinsignificant widths and errors, we take PDG values for the 3 masses by setting m ψ ′ = 3686.1 MeV and m K = mK = (m K + + m K 0 )/2 = 495.6 MeV. The corresponding two-particle scattering lengths are currently unknown, but for the kaon-kaon-interaction we can determine a KK = 1/ 2µ KK B f 0 by employing the analogue of Eq. (9). This corresponds to treating the f 0 as shallow bound state of K andK. For a ψ ′ K and a ψ ′K , we have neither experimental data nor predictions from lattice calculations. Due to the absence of a two-particle bound-state and owing to symmetry reasons we can, however, conclude that a ψ ′ K = a ψ ′K < 0 should hold. Another restriction on this quantity's magnitude comes from the fact that in ψ ′ K-scattering no quark exchange is possible in contrast to DK-scattering, for instance. Therefore, its interaction should be suppressed compared to the one in the latter case, implying that |a ψ ′ K | should be smaller than typical, non-resonant scattering lengths for DK-processes.
In [17], DK scattering lengths were calculated, using unitarised chiral perturbation theory and in Ref. [18] they were extracted from lattice QCD simulations. Both analyses agree and provide values of the order of 0.1 fm. Thus we demand −0.1 fm a ψ ′ K = a ψ ′K < 0 fm.
We note that, given this assumption, a ψ ′ K is not large compared to the range of forces such that the EFT for large scattering lengths is strictly not applicable. In this study, however, we use the EFT as a specific three-body model that can be solved numerically in order to test the range of applicability of the Weinberg method. In relation to the corresponding thresholds in Eq. (10) and (12) the masses m Y = 4664 ± 11 ± 5 MeV and m f 0 = 980 ± 10 MeV have large errors respectively so that we are still allowed to vary the binding energies in a region 0 < B f 0 < B Y 20 MeV and be consistent with experiment.  (11)). Our numerical values within the three-particle model (3P) are compared to the ones in a simple two-particle model (2P). The unknown scattering length a ψ ′ K = a ψ ′K is set to −0.1 fm.
In Fig. 4, the numerically calculated ψ ′ f 0 (980)-scattering length within the three-particle model (3P) is depicted for several values of the total binding energy B Y . It is also compared to the one of Eq. (9) in a hypothetical twoparticle system (2P) with no substructure in the f 0 (980)-dimer. The variation in B f 0 is expressed in a dimensionless relative binding energy b 0 . The unknown quantity a ψ ′ K = a ψ ′K is set to −0.1 fm. Our numerical results for the ψ ′ f 0 (980)-scattering length confirm the behavior deduced from the analytical considerations above (cf. Eq. (25)). For a weakly bound f 0 (980)-dimer, that is b 0 → 0, we see deviations between the two models, whereas for a strongly bound dimer with b 0 → 1 the results nearly coincide. In the latter case a ψ ′ f 0 diverges to +∞. We also calculated the effective range in Fig. 5 for the same parameters. It diverges at values where the scattering length vanishes. Unfortunately, neither experimental data nor lattice calculations are available at present and no comparison to data for ψ ′ f 0 (980)-scattering can be made.
By applying the Weinberg-formalism, described in the previous section, we are also able to investigate the question at which scales Y (4660) can effectively be seen as a two-body molecule consisting of elementary particles f 0 (980) and ψ ′ and where the f 0 (980)-substructure has to be taken into account. Since the only undetermined scales in our system are B Y , B f 0 and a ψ ′ K , we expect the dimensionless quantities Z a and Z r , as defined in the equations (8), only to depend on 2 dimensionless ratios of these 3 parameters. We have already shown above that the relative binding energy of the f 0 and Y (4660), b 0 , plays an important role in determining the structure of the Y (4660). For the second ratio it is natural to choose b 1 : Since the scattering length a ψ ′ K is negative, the energy scale B 1 characterizes a virtual state in the ψ ′ K-channel but not a bound state. The functions Z a (b 0 ) for discrete values of b 1 , varied over a wide range, are displayed in Fig. 6. We find that Z a strongly depends on b 0 and b 1 . For b 0 → 0, Z a reaches the value of 1 so that Y (4660) has to be considered a three-body system. In the limit b 0 → 1, Z a approaches 0 as the f 0 (980)-dimer gets maximally bound. In this case, the Y (4660) can be seen as a two-particle system consisting of the elementary particles ψ ′ and f 0 (980).
Another feature shown in Fig. (6) is the fact that for certain parameters Z a exceeds its allowed interval [0, 1]. This leads us back to question at which scales the initial approximation (17) is valid. The approximate condition for b 0 (25) and the exact condition for b 0 and b 1 determined using Eq. (13) as a consistency-check for the theory, are summarized in Fig. 7. We show the consistency conditions for the Weinberg method in the (b 0 /b 1 )-(1/b 0 − 1)plane. The shaded grey area indicates the allowed region with 0 < Z a < 1 while the area to the left of the dotted horizontal line satisfies Z a ≈ Z r . The area to the left of the dashed horizontal line satisfies the approximate condition (25) for 1/b 0 − 1 ≪ 1. Within the lower left corner both our approximate and numerically exact calculations indicate the applicability of the Weinberg method. Outside of this area it leads to inconsistent results and can not be applied. For the case of the Y (4660) this area is also mostly the region that is  of physical interest. This is due to the fact that the experimental/theoretical constraints B Y 20 MeV and |a ψ ′ K | 0.1 fm restrict the parameter b 1 = 1/(2µ ψ ′ K a 2 ψ ′ K B Y ) to be greater than approximately 225.

Summary and Outlook:
The work presented can not yet be used to compare to experimental line shapes, especially since we assumed the f 0 (980) as a stable particle. In reality, due the decay f 0 (980) → ππ the Y (4660) becomes observable in ψ ′ ππ invariant mass distributions and the resulting line shapes are believed to contain important information on the nature of the Y [14]. Within the formalism presented, this channel could be included via a complex scattering length of theKK system -the impact of unstable constituents on the line shapes of particles is, e.g., discussed in Ref. [19]. In addition, the Y (4660) may also decay into other channels. Currently there is a discussion, if the signal seen in Λ cΛc , baptized X(4630), has its origin in the Y (4660) [20,21,22] -this channel could be parametrized by an imaginary part of the three-body interaction.
However, we do not expect that the possible extensions mentioned will distort significantly the conclusions on the applicability of the Weinberg method. Our analytical analysis summarized in Eq. (25) shows that the binding energy of the Y (4660) relative to the three-body threshold has to be large compared to the molecular binding energy relative to the ψ ′ f 0 threshold. Moreover, we found that for systems where the scattering length in the other subsystems is large, this kind of analysis can not be used and should either be replaced by a more complex, coupled channel analysis or abandoned all together. However, in a significant part of the parameter space allowed for the f 0 (980) and the Y (4660) the Weinberg method can be used to quantify the two-body molecular component, once better data is available.
In principle, we could also integrate spin-effects, higher derivatives in the fields and, via photon-coupling, even charge-dependent interactions into the Lagrangian (1). Clearly all these extensions imply additional parameters in our theory which would have to be determined. Furthermore, we could scan the field of possible hadronic molecules for dimer-and three-particle-candidates. An interesting application is the X(3872)-meson as a DDπ system. Since the Dπ-dimer could only appear in p-wave scattering, higher derivatives in the dimer fields would have to be included.