Role of the scalar in the process

Based on BESIII measurements of the reaction , we investigate this process by considering the S-wave pseudoscalar-pseudoscalar interaction within the unitary chiral approach and the contributions from the intermediate resonances and . Our calculation can reasonably reproduce the experimental data, and our results imply that , which is dynamically generated from the S-wave pseudoscalar-pseudoscalar interaction, plays an important role in this process, and the contributions from the intermediate resonances and are also necessary. More precise measurements of this process in future can shed light on the nature of and .


I. INTRODUCTION
In the last few decades, the constituent quark model has successfully explained the composition of most mesons [1]. However, the properties of light scalar mesons are difficult to describe within the constituent quark model. For scalar mesons with masses less than 1 GeV ( , , , and ), there are different explanations, such as states, tetraquark states, or molecular states [2][3][4][5]. In addition, the nature of isospin-zero scalar mesons with larger masses, such as , , and , is also difficult to fully understand [1]. Recently, BESIII and Belle/ Belle II accumulated considerable experimental information about charmed hadrons decays, which provides an important platform to explore the internal structures of these scalar mesons [6][7][8][9]. S (1710) 1723 ± 11 ± 2 140 ± 14 ± 4 D + s → K 0 For instance, the BESIII Collaboration observed the scalar 1) with a mass of MeV and a width of MeV in the process [10]. Later, a similar structure was also observed in the process by the BESIII Collaboration, associated with a new scalar state with a mass of MeV and a width of MeV [11].
MeV mass discrepancy between and causes doubt as to whether they are isospin partners. However, the line-shapes of and in the and invariant mass distributions are so similar that the measured mass discrepancy of and may be due to the closeness of their peaks to the threshold of the mass spectrum. We previously investigated both reactions and found that the BESIII measurements of both reactions could be well reproduced by regarding the intermediate state as the molecular state [12,13], which is also supported by the study in Ref. [14].

KK
The BESIII Collaboration recently performed an amplitude analysis of the process and determined the branching fraction ( ) = (2.8 0.4 0.4) [15], which improves the precision by a factor of two compared to the results of the CLEO Collaboration [16]. The significant signal and an enhancement structure around 1300 MeV were observed in the invariant mass spectrum, and the latter structure could be associated with the intermediate resonances and . The scalar could be explained as the molecular state dynamically generated from the S-wave pseudoscalar-pseudoscalar interaction 1) It should be stressed that BESIII does not distinguish between the and in the , and denotes the combined state as [10].  [17,18], which is supported by many studies [6][7][8][9]. In addition, the resonance is a broad state, and its mass and width are not well established. There are various explanations for the nature of , such as the state, molecular state [19][20][21], [22], and glueball [23,24]. In Refs. [19][20][21], is also explained as a bound system of two vector mesons, which is challenged in Ref. [25]. Therefore, a study of the process should help deepen our understanding of the nature of , , and .
In this work, we investigate the process by considering the scalar generated from the S-wave pseudoscalar-pseudoscalar interaction within the chiral unitary approach and the contributions from the intermediate resonances and .
This paper is organized as follows: In Sec. II, we present the mechanism for the process . The results and discussions are given in Sec. III, followed by a short summary in the last section.

II. FORMALISM
In this section, we present the mechanism of the process . This process occurs in three steps, weak decay, hadronization, and final state interactions [7,9,12,13]. First, the c quark of the initial weakly decays into an s quark and a boson, and the boson subsequently decays into a quark pair. Then, all the quarks, along with the quark pair (= + + ) created from the vacuum with the quantum numbers , hadronize into hadrons, which can be classified as the internal emission of Figs hadronize into or , and the hadronization of the other quarks may be expressed as where correspond to the u, d, and s quarks, respectively, and M is the matrix (3)

SU(3)
Within flavor symmetry, the matrix M can be written in terms of pseudoscalar mesons as [7] Because has a large mass and does not play a role in the generation of , we ignore the component in this study. Then, Eqs. (1) and (2) can be rewritten as For the external emission of Fig. 1(c), the quarks of the decay hadronize into , and the pair, along with the created pair, hadronize into the states ηη.
For the external emission of Fig. 1(d), the pair may hadronize into the η meson, and the quarks of the decay, along with the created pair, hadronize into , which contributes to the process . Thus, we have Then, the processes involved in the decay of into all possible states can be expressed as and are the CKM matrix elements. Because the external emission of bosons is colorfavored relative to the internal emission, an extra color factor C can be introduced to account for the relative weight of the external emission with respect to the internal emission. For the external emission, the quark pair from the decay can form the color singlet , and u and have three choices of colors, whereas for the internal emission, u, , s, and quarks from the decay have fixed colors. Thus, the factor C is taken to be three in this study [26][27][28][29]. Now, we have all the possible components after hadronization, where denotes the factors of the production vertices containing all the dynamics. After the preliminary weak decay, the meson pairs of , , and may undergo the S-wave final state interaction to give rise to the final state, where the scalar meson can be dynamically generated, as shown in Fig. 2.
The amplitude of the S-wave pseudoscalar-pseudoscalar interaction, generating the scalar , can now be written as where is the loop function of the two-meson propagator, and is the transition amplitude of the i-channel to j-channel, both of which are functions of the invariant mass . The loop function is given by Role of the scalar in the process Chin. Phys. C 47, 043101 (2023) π 0K0 K +K0 K + π + π 0 1) Here we neglect the components . Indeed, the interaction of the to can be given by the intermediate resonances, which is shown very small by BESIII [15].
and are the masses of the two mesons in the loop of the i-channel, and P and q are the four-momenta of the two-meson system and second meson, respectively. The Mandelstam invariant . The loop function of Eq. (15) is logarithmically divergent, and there are two methods of solving this singular integral, either using the three-momentum cut-off method, or the dimensional regularization method. The choice of a particular regularization scheme does not, of course, affect our argumentation. In this study, we perform the integral for q in Eq. (15) with a cut-off MeV [30,31]. The transition amplitude can be obtained by solving the Bethe-Salpeter equation in coupled channels, where V is a matrix of the interaction kernel. We take five channels, , , , , and . The explicit expressions of the matrix elements in the Swave are given by [32][33] where MeV is the pion decay constant, s is invari- ant mass square of the meson-meson system, and , , and are the masses of the pion, kaon, and η mesons, respectively [1]. The unitary normalization and is easily considered to identify the particle when using the loop function G without an extra factor [30].
In addition to the scalar , BESIII has observed the enhancement structure around 1300 MeV in the invariant mass distribution, which could be associated with the resonances and . Hence, we also take into account contributions from the intermediate resonances and .
For the contribution of in the decay , we describe it using the Breit-Winger form, where α is the strength of . Considering that the mass and width of have large uncertainties [1], we fix its mass to be 1300 MeV, the center position of the enhancement structure in the invariant mass distribution of BESIII measurements [15], and take the width as a free parameter.
Considering that couples to in the Dwave, we can write the contribution of this resonance as is the momentum of in the rest frame, π 0 π + π 0 π 0 The parameter θ is the angle between the momentum of and in the rest frame of the system [34], where ( ) is the energy of ( ) in the rest frame, and is the ( ) momentum in the same frame. We give the explicit forms of these vari- ( π + π 0 π 0 π 0 where and are the energies of and in the rest frame, respectively,