Study the molecular nature of $\sigma$, $f_{0}(980)$, and $a_{0}(980)$ states

We investigate the characteristics of $\sigma$, $f_{0}(980)$, and $a_{0}(980)$ with the formalism of chiral unitary approach. With the dynamical generation of them, we make a further study of their properties by evaluating the couplings, the compositeness, the wave functions and the radii. We also research their properties in the single channel interactions, where the $a_{0}(980)$ can not be reproduced in the $K\bar{K}$ interactions with isospin $I=1$ since the potential is too weak. In our results, the states of $\sigma$ and $f_{0}(980)$ can be dynamically reproduced stably with varying cutoffs both in the coupled channel and the single channel cases. We find that the $\pi\eta$ components is much important in the coupled channel interactions to dynamically reproduce the $a_{0}(980)$ state, which means that $a_{0}(980)$ state can not be a pure $K\bar{K}$ molecular state. We obtain their radii as: $|\langle r^2 \rangle|_{f_0(980)} = 1.80 \pm 0.35$ fm, $|\langle r^2 \rangle|_{\sigma} = 0.68 \pm 0.05$ fm and $|\langle r^2 \rangle|_{a_0(980)} = 0.94 \pm 0.09$ fm. Based on our investigation results, we conclude that the $f_{0}(980)$ state is mainly a $K\bar{K}$ bound state, the $\sigma$ state a resonance of $\pi\pi$ and the $a_{0}(980)$ state a loose $K\bar{K}$ bound state. From the results of the compositeness, they are not pure molecular states and have something non-molecular components, especially for the $\sigma$ state.


I. INTRODUCTION
Even though Quantum Chromodynamics (QCD) is the fundamental theory of strong interaction and governs the high energy region, the nature and the structure of the lowest scalar mesons still problematic and under debate. One of the main topics of the high energy physics is to comprehend the properties of the hadronic resonances. The conventional picture of the hadrons based on the quark model is the baryon made of qqq and the meson qq.
However, that is not the whole picture of the observed hadrons, with the development of the experiments, many resonances have been found, which may have complex structures since their nature cannot be interpreted by the conventional ways, such as tetraquarks [1], hybrids [2], and glueballs [3] for mesons, and pentaquarks and heptaquarks for baryons, or molecular states. These exotic states have drawn much attention both in theories and experiments to understand their structure and decay properties, see more details in the reviews [4][5][6][7][8][9][10][11][12][13][14]. In the low energy region the perturbative QCD failed because of the confinement, so we need to explore a non-perturbative QCD, such as Lattice QCD [15][16][17], QCD sum rules [18][19][20][21][22][23][24], Effective Field Theory [25][26][27], Chiral Unitary Approach (ChUA) [28][29][30][31][32], and so on. In case of meson-meson and meson-baryon interaction, chiral dynamics is crucial in understanding the structure and nature of the resonances, and it has shown that many known resonances are dynamically generated as an outcome of the hadron-hadron interaction [33].
Following the work of Ref. [34], we continue to study the properties of the σ [or f 0 (500)], f 0 (980) [35], and a 0 (980) [36] states. Although the states of f 0 (980) and a 0 (980) are nearly degenerated, they have different isospin and other properties. Several proposals were made about the nature of these scalar particles, such as qq state [37][38][39][40], multiquark states [1, 22-24, 41, 42], or KK molecules [43][44][45][46]. The Evidence of four-quark nature for the f 0 (980) and a 0 (980) states are found in the φ meson radiative decay [42] where more experimental informations and discussions can be referred to Refs. [47][48][49]. The nature of the σ resonance is different from the other two. The masses of the f 0 (980) and a 0 (980) are close to the KK threshold, conversely σ is far above ππ threshold. Moreover the decay width of the sigma is very large, which does not behave like an ordinary Breit-Wigner resonance [50].
Furthermore, from the large N c limit calculations [51,52] and Regge theory [53] are confirmed that σ is not an ordinary qq structure. In the work of Ref. [54], using the ChUA, the potential of the pseudoscalars calculated from the chiral Lagrangians [55][56][57][58][59], and then by applying the unitarity in coupled channel scattering amplitudes, the σ, f 0 (980) and a 0 (980) are dynamically generated. Along the line of Ref. [54], we make a further investigation of the properties of the σ, f 0 (980), and a 0 (980) states by evaluating their compositeness, the wave functions and the radii both in the coupled channel and the single channel interactions.
In the present work, we will firstly introduce the formalism of the interactions of KK and its coupled channels. Then, we discuss the definition of the couplings and how to calculate the compositeness, the wave functions and the radii for a resonance in ChUA. Following, we show our results in details for the cases of the coupled channel and the single channel, respectively. Finally, we close with our conclusions.

II. FORMALISM
In this section, we firstly revisit the formalism of Ref. [54], where the interaction potentials for the coupled channels are derived from the lowest order chiral Lagrangian, and then performing the S-wave projection, the scattering amplitudes are evaluate with a set of on-shell Bethe-Salpeter equations. Next, we introduce the definitions of the couplings in the coupled channel, the wave functions, compositeness and radii of the generated resonances.
A. S-wave scattering amplitude in the coupled channels and single channel The most general chiral Lagrangian can be written in a perturbative manner according to the powers of the momenta of the pseudoscalar mesons [55][56][57][58], where the lowest order chiral Lagrangian L 2 contains the most general low energy interactions of the pseudoscalar meson octet, which is given by, where f is the pion decay constant, the value of which is taken as 92.4 MeV [60], stands for the trace of matrices, and Φ is the pseudo Goldstone boson fields, defined as Besides, the pseudoscalar meson mass matrix M is given by where we have taken the isospin limit (m u = m d ).
From this Lagrangian, Eq. (2), we can derive the tree level amplitudes for KK , ππ and πη channels, which will be used in the coupled channel Bethe-Salpeter equations. After performing the S-wave projection, the interaction potentials in the isospin basises are given by [54], where we specify the KK and ππ channels with the labels 1 and 2, respectively, for the case of isospin I = 0, and the KK and πη channels for the case of I = 1. For the on shell amplitudes, one can take p 2 i = m 2 i . For the scattering amplitudes of the coupled channels, one can solve the Bethe-Salpeter equations factorized on shell [54], It is worth to note that in the present case T , V , and G are 2 × 2 matrices. The element of the diagonal G matrix is the loop function of two intermediate mesons in the i-th channel, given by where p 1 and p 2 are the four-momenta of the two initial particles, respectively, and m 1 , m 2 are the masses of the two intermediate particles appearing in the loop. Note that the G function is logarithmically divergent. There are two methods to solve this singular integral, either using the three-momentum cut-off method [54], where the analytic expression is given (b) Imaginary part of G 11 by Ref [61], or the dimensional regularization method [62]. Using the cut-off method we can rewrite Eq. (8) as where q = | q|, ω i = ( q 2 + m 2 i ) 1/2 and s = (p 1 + p 2 ) 2 , and the cutoff, q max , is the only one free parameter. We show our results of the real part and the imaginary part of the G functions in the isospin I = 0 case in Fig. 1 with two different cutoffs (about their values see the discussions at the beginning of next section), where one can see that the imaginary part of the loop function is independent with the the cutoff, which leads to extrapolate to the Second Riemann sheet easily, see the discussions below.
Using ChUA, one also can easily determine the masses and the decay widths of the resonances produced in the coupled channel interactions just by looking for the poles in the second Riemann sheets. Thus, one need to extrapolate the analytical structure of the scattering amplitudes in the complex s plane. To fulfil these, one can extrapolate the G(s) function into the second Riemann sheet by where the three momentum in center-of-mass (CM) frame is given by with the usual Källen triangle function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc), see more details in Ref. [54].
In the present case, only two coupled channels, the elements of the scattering amplitudes T matrix can be written as [54], where one defines For the single channel cases, we have the KK and ππ (πη) interaction channel separately for I = 0 (I = 1). Just by taking V 21 = 0, one can easily reduce Eq. (12) to, B. The couplings and the wave functions By applying the Laurent expansion of the amplitude close to the pole, the scattering amplitudes can be written as [63,64] where g i and g j are the coupling constants of the i-th and j-th channels, which can be calculated from the residue of the pole [65,66] Using Cauchy Integral formula, we can evaluate the residue as a loop integral in the complex s plane, where the integral is over a closed path in the complex s plane around the pole s = s pole .
Furthermore, with the couplings of the corresponding poles, one can generalize Weinberg's rule [67] for bound state or resonance to the ChUA [68] where an alternative derivation of this relationship can be found in Ref. [69]. This equation refer as the sum rule for the bound states or the resonances dynamically generated by the coupled channel interactions. More discussions and applications of this rule can be found in Refs. [70][71][72][73][74]. This equation holds for the resonance or the bound state which is a pure molecular state. However, in some cases, if a physical state couples not only to hadronhadron pairs, but also to a different component of non-molecular type, this relation becomes for the composite states where Z represents the probability that the system is not a molecule components but something else. As discussed in Ref. [73], the interpretation of Z as a probability non-molecular (meson-meson or meson-baryon state in ChUA) component is strict for bound states, which is related to the genuine component in the wave function of the state omitted from the coupled channels. Note that for a specified channel the G i function should be extrapolated to the right Riemann sheet for a corresponding pole of the state.
To understand more about the sources of the resonances, we study the wave function of the resonance at small distances. Once we have the wave function of a resonance, one can also investigate its form factor, which response the state to external sources. Following the formalism of Ref. [75], the wave function of a resonance in coordinate space is given by After performing the angle integration of the momentum, we obtained [65] where C is the normalization constant, and E ≡ √ s pole , thus, which is real for a pure bound state with zero width and otherwise complex for the general cases in ChUA. Note that here we put an extra form factor to regulate the scale of the wave function, and our conclusions do not change if we remove it. Using the wave functions that we have, one can evaluate the form factor of the states with its definition [75], with a normalization to keep F (q = 0) ≡ 1. For a generated state in ChUA, a pole with its width, which is complex, the form factor is complex too, see the results below. Finally, the radii of the states (or mean square distance) can be evaluated from the form factor, Note that a soft step function needed to make the form factor converge in this case. On the other hand, for the case of a weakly bound state, the radii of the state can also be obtained from the tail of the wave functions as done in Ref. [70] where the binding energy B E,i = m i + m i − M B , and the reduced mass µ i = m i m i m i +m i . Conceptually, r 2 i is the mean-squared distance of the bound state in the i-th channel.

III. RESULTS
We first revisit the KK interactions with its coupled channels of ππ or πη, where the states of σ, f 0 (980), and a 0 (980) are dynamically generated in the coupled channel approach as done in Ref. [54]. But, we make a further study of the couplings, the compositeness, the wave functions and the radii for these states to investigate more details on their properties, as show the results as below. To find more information about the structure of the poles corresponding these states, we examine the single channel interactions. Note that, for the only one free parameter in our approach, what we used below for the value of the cutoff is the one determined in Ref. [34] by dong a combined fit of the experimental data, q max = 931 MeV, which is a bit different with the ones used in Ref. [54]. To see the uncertainties of our calculations, we also show the results with the one of about 15% division to the upper limits, q max = 1080 MeV and varying the values between 15% division in some cases.

A. Coupled channel approach
We first calculate the phase shifts and the inelasticities. As done in Ref. [54], the twochannels S-matrix are used, where the observables of δ 1 , δ 2 correspond to the phase shifts of the channel 1, 2, respectively, and the one of η is the inelasticity. These observables can be calculated from the relationship between S-matrix and the scattering amplitude T -matrix, having where p cmi is the corresponding three momentum in the CM frame as discussed above. The sector because of the lake of experimental data for phase shifts and inelasticities, we make some predictions for them, where the structure of a 0 (980) can be clearly seen in the phase shifts.
Next, we show our results for the invariant mass distributions. As done in Ref [54], we compare our results with the data of πη invariant mass distribution from the reaction K − p → Σ(1385)π − η and the ones of KK from the reaction    where T ii is the scattering amplitude of the KK or πη channel, q cmi is three momentum in CM frame and C the normalization factor. To see more clearly the resonances dynamically produced in the coupled channel interactions, we plot the modulus squared of the scattering amplitudes in I = 0 and I = 1 sectors as shown in Figs. 5 and 6. From Figs. 5a and 5b of |T 11 | 2 and |T 12 | 2 for I = 0, the peak of f 0 (980) state is clearly seen. In Fig. 5c, the broad structure of T 22 are σ resonance, where the dip is the signal of f 0 (980) state closed to the KK threshold and the structure of the amplitudes are consistent with the ones calculated with dispersion method [76]. Likewise, the a 0 (980) resonance can be clearly seen in |T 11 | 2 , |T 12 | 2 , and |T 22 | 2 in I = 1 sectors in Fig. 6. In spite of showing the a 0 (980) resonance in the results of |T 22 | 2 , see Fig. 6c, there is an extra feature, which is called threshold effect [54,77], of which more details can be seen a recent review [78]. This feature is due to the strong coupling of the resonance a 0 (980) to the KK channel which cause to dwindle the width of the scattering amplitude and change the location of the maximum. This effect is           the KK channel, which means that the pole of a 0 is dominated by the KK channel but the contributions of πη is significant too.
Using the sum rule of Eq. (19), the compositeness can be calculated from the couplings of the dynamically generated resonances, where one can check whether f 0 and a 0 are a pure molecular state or have something else. Our results are given in Tables III and IV. From the results of Table III, once again we can conclude that the structure of f 0 is highly dominated by the KK molecular components, which is up to 80% with the central value of q max = 931 MeV, and has very small parts of the ππ components even though the coupling to the ππ channel is not so small, which is more than 1/3 of the one to the KK channel, see Table. I. By contrast, the σ state has large part components of ππ about 40% and quite tiny parts of KK, where one can find that this state still has much large parts of non-molecular components. Our resuts of the compositeness in Table. III for the states of σ and f 0 are consistert with the ones obtained in Ref. [74] with the inverse amplitude method. The a 0 state has a main components of KK and some contributions from the πη component, see Table IV, but it stil has something else about 30%. These results are comparable with the work of Ref. [70] where the properties of these resonances are investigated with the    [79] also conclude that the f 0 (980) and a 0 (980) states are not elementary states based with a Flatté parameterization analysis.
To study the response of these states to the external sources, one need to know the form factor of these states. Thus, we evaluate the wave functions for them, and then, we can calculate the observables of the radii once we have their form factor. The wave functions of these state for all distances are shown in Fig. 9, where the real parts and imaginary parts of the wave functions for the f 0 , σ and a 0 states are given since the poles corresponding to these states are complex. From Fig. 9, one can see that, up to about 4 fm, the wave functions for them become zero. Once we have the wave functions, we can investigate the radii of these states with Eq. (23) which relate the wave functions at the origin, see Table. V with two cutoffs as above. As discuss before, we also can calculate the radii from the tail of the wave functions using Eq. (24), as shown in Table VI. From the results of Tables V and VI, we can clearly see that the radii of the f 0 and a 0 states in two approaches of Eqs. (23) and (24) are larger than the typical hadronic scale 0.8 fm [70], whereas the one of the σ state keeps in the the typical hadronic scale 0.8 fm. But, in Table VI, we find that the   become unstable, as shown in Fig. 10. In Fig. 10, we can see that the results with the seconde method are much stable and the ones with the first method have singularities when the cutoff move the pole near to the threshold where the binding energy becomes zero. As discussed in Ref. [80], the mean-squared radius is well defined with Eq. (23) both for the bound states and the resonance states. Thus, at the end, we obtain | r 2 | f 0 (980) = 1.80 ± 0.35 fm, | r 2 | σ = 0.68 ± 0.05 fm and | r 2 | a 0 (980) = 0.94 ± 0.09 fm, where we take the central value of the cutoff q max = 931 MeV within 15% uncertainties. |T | 2 , in Fig. 11, where one can see the sharp peak with nearly zero width in the KK channel on the left and the wide bump structure in the ππ channel on the right. Next, we search for the corresponding poles in second Riemann sheets. For the ππ channel interaction, as shown in Fig. 12 where we vary the cutoffs, we always find the pole in the second Riemann sheet above the threshold of which the mass changes weakly and the width varies not so much as the case of the coupled channel interactions. For the case of the KK channel, now the pole keeps below the threshold, and thus, has no width as a pure bound state since there is no decay channel, see Fig. 12 (c), which are more bound compared with the results of   coupled channel cases in Fig. 8. Therefore, we can conclude that the σ state is a resonance mainly formed by the ππ interaction and the one of the f 0 state is a bound state of the KK component as found in the coupled channel interactions above. To reveal more details, see Fig. 13 for the real and imaginary parts of the ππ scattering amplitudes in the coupled and the single channels, one can see that in the region of the σ state appeared, 400-700 MeV, the amplitudes are not affected so much by the coupled channel of KK, which is a bit far away from the threshold of KK. Indeed, the structure of the f 0 (980) state can be clearly seen closed to the threshold of KK, as shown in Fig. 13. However, in the isospin I = 1 sector, the potential of the KK channel is too weak to create a pole in the second Riemann sheet when it decouples to the πη channel, of which the potential is independent with the energy. This means that the coupled channel effects play much important role in the dynamical production of the a 0 state. As in case of the coupled channel interactions, we make a further studies of the compositeness, the wave functions and the radii. The results of the couplings are given in Table   VII, even though the strengths of the couplings have lost the relative meanings in the case of the single channel interaction. But, from the results of the compositeness, see Table VIII,  with the couplings obtained, the compositeness for the f 0 (980) state is a bit smaller than the ones of the coupled channel cases, which is consistent with the results of the coupled channel cases in Table III.
The wave functions of the σ and f 0 (980) states are shown in Figs. 14 and 15, respectively.    Fig. 16 with the sub-figure of Fig. 10 (a).

IV. CONCLUSIONS
In the present work, we investigate the properties of the σ, f 0 (980), and a 0 (980) states with the chiral unitary approach, where we use the formalisms of the coupled channels and   mean-squared distance of these dynamically generated states in both the coupled channels and the single channel formalisms. From the results of the couplings and the compositeness, we conclude that the f 0 (980) state is essentially made by the KK component, which is about 80%, and has very small parts of ππ. However, the σ state has the main contributions from the ππ channel, of which the component amounts to about 40%, and has quite small quantity of the KK component. Thus, the σ resonance has a large parts of something else except for the molecular components. For the case of the a 0 (980) state, the πη channel has important contributions to its generations in the coupled channel interactions. Even though it is dominated by the KK component with 55%, it also has large contributions of about 16% from the πη component. With the wave functions obtained, we calculate the radii of these states and get | r 2 | f 0 (980) = 1.80 ± 0.35 fm, | r 2 | σ = 0.68 ± 0.05 fm and | r 2 | a 0 (980) = 0.94 ± 0.09 fm, which can be tested in the future experiments. Finally, from our results of the couplings, the compositeness, the wave functions and the radii, we can conclude that the f 0 (980) state is mainly a KK bound state, the σ state a resonance of ππ and the a 0 (980) state a loose KK bound state.