$\sigma$ exchange in the one-boson exchange model involving the ground state octet baryons

Based on the one-boson-exchange framework that the $\sigma$ meson serves as an effective parameterization for the correlated scalar-isoscalar $\pi\pi$ interaction, we calculate the coupling constants of the $\sigma$ to the $\frac{1}{2}^+$ ground state light baryon octet ${\mathbb B}$ by matching the amplitude of ${\mathbb B}\bar{{\mathbb B}}\to\pi\pi\to\bar{{\mathbb B}}{\mathbb B}$ to that of ${\mathbb B}\bar{{\mathbb B}}\to\sigma\to\bar{{\mathbb B}}{\mathbb B}$. The former is calculated using a dispersion relation, supplemented with chiral perturbation theory results for the ${\mathbb B}{\mathbb B}\pi\pi$ couplings and the Muskhelishvili-Omn\` es representation for the $\pi\pi$ rescattering. Explicitly, the coupling constants are obtained as $g_{NN\sigma}=8.7_{-1.9}^{+1.7}$, $g_{\Sigma\Sigma\sigma}=3.5_{-1.3}^{+1.8}$, $g_{\Xi\Xi\sigma}=2.5_{-1.4}^{+1.5}$, and $g_{\Lambda\Lambda\sigma}=6.8_{-1.7}^{+1.5}$. These coupling constants can be used in the one-boson-exchange model calculations of the interaction of light baryons with other hadrons.

Based on the one-boson-exchange framework that the σ meson serves as an effective parameterization for the correlated scalar-isoscalar ππ interaction, we calculate the coupling constants of the σ to the 1 2 + ground state light baryon octet B by matching the amplitude of B B → ππ → BB to that of B B → σ → BB.The former is calculated using a dispersion relation, supplemented with chiral perturbation theory results for the BBππ couplings and the Muskhelishvili-Omnès representation for the ππ rescattering.Explicitly, the coupling constants are obtained as g N N σ = 8.7 +1.7 −1.9 , g ΣΣσ = 3.5 +1.8 −1.3 , g ΞΞσ = 2.5 +1.5 −1.4 , and g ΛΛσ = 6.8 +1.5 −1.7 .These coupling constants can be used in the one-boson-exchange model calculations of the interaction of light baryons with other hadrons.
Hadronic molecules [7], one of the most promising candidates for exotic states, are loosely bound states of hadrons and a natural extension of the atomic nuclei (such as the deuteron as a protonneutron bound state) and offer an explanation of the many experimentally observed near-threshold structures, in particular in the heavy-flavor hadron mass region [15].
As a generalization of the one-pion-exchange potential [16], the one-boson-exchange (OBE) model has played a crucial role in studying composite systems of hadrons [10,11,[17][18][19][20][21][22][23][24][25].Taking the deuteron as an example, it is widely accepted in the OBE model that its formation involves the long-range interaction from the one-pion exchange and the middle-range interaction from the σ-meson exchange; see Ref. [18] for a detailed review.However, unlike narrow width particles that are associated with clear resonance peaks or dips observed in experiments, the scalar-isoscalar σ meson, which plays a crucial role in nuclear and hadron physics, had remained a subject of considerable debates for several decades until it was established as the lowest-lying hadronic resonance in quantum chromodynamics (QCD) in the past twenty years based on rigorous dispersive analyses of ππ scattering [26][27][28] (see, e.g., Refs.[29,30] for reviews).The dispersive techniques have recently been applied to determine the nature of the σ at unphysical pion masses [31,32].
The σ meson in the OBE model can be considered as approximating the correlated S-wave isoscalar ππ exchange in a few hundred MeV range [18,[33][34][35][36][37][38], and some modifications to its properties have been made in order to improve the accuracy of the approximation [18,33].However, the effective coupling constants between the σ and various hadrons remain highly uncertain.One example is the widely used nucleon-nucleon-σ coupling g N N σ , which ranges roughly from 8 to 14 [18,33,36].For its couplings to other ground state octet baryons, g ΣΣσ , g ΞΞσ , and g ΛΛσ , there are rare systematic discussions and error analyses.Most of them are estimated either by the quenched quark model or the SU(3) symmetry model assuming the σ to be a certain member of the light-flavor multiplet [23,24].The use of the one-σ exchange instead of the correlated ππ exchange may raise some questions: Is this approximation reasonable?How good is the approximation?In the present work, we try to address these questions by considering the scattering of the baryon and antibaryon in the ground state octet, B B → BB, through the intermediate state of the correlated IJ = 00 ππ pair or the σ meson and calculate the effective coupling constants of g BBσ .We will make use of dispersion relations following Ref.[39], and similar methods have been used in, e.g., Refs.[37,38] to derive the baryon-baryon-σ couplings, and Ref. [40] to derive the σ coupling to heavy mesons.Here, we will match the dispersive amplitudes of B B → ππ at low energies to the chiral amplitudes up to the next-to-leading order (NLO).This paper is structured as follows.The formalism is presented in Sec.II, including the calculation of the OBE amplitude in Sec.II A and the amplitude from the dispersion relation (DR) with a careful treatment of kinematical singularities in Sec.II B. In Sec.III, we conduct an analysis of the two amplitudes and present the scalar coupling constants g BBσ along with an error analysis.
This includes a comparison of the s-channel processes, as detailed in Sec.III A, and a comparison of the corresponding t/u-channel processes utilizing the crossing symmetry in Sec.III B. A brief summary is given in Sec.IV.The adopted conventions, the result of N N σ coupling in the SU (2) framework, and crossing symmetry relations are relegated to the appendices.

II. FORMALISM
To determine the scalar coupling, g BBσ , between the baryon B in the1 2 + ground baryon octet and the σ meson, we first utilize the DR and chiral perturbation theory (ChPT) to calculate the amplitude of B(p 1 , λ 1 )+ B(p 2 , λ 2 ) → B(p 3 , λ 3 )+B(p 4 , λ 4 ) with a correlated ππ intermediate state, as depicted in Fig. 1(a).Here, p i and λ i represent the four momentum and the helicity of particle B or B, respectively.Additionally, we restrict the quantum numbers of the two-body intermediate state, ππ, to be IJ = 00.This S-wave amplitude can be denoted as M DR B(λ 1 )+ B(λ 2 )→ B(λ 3 )+B(λ 4 ),0 (s), where the subscript 0 indicates the S-wave, s represents the square of the total energy of the system in the center-of-mass (c.m.) frame, 1 and the superscript DR indicates the result obtained from the DR.
Next, we proceed to calculate the same amplitude, with the intermediate σ meson, as depicted in Fig. 1(b).In this case, we utilize the OBE model, and the corresponding amplitude can be expressed as M OBE B(λ 1 )+ B(λ 2 )→ B(λ 3 )+B(λ 4 ) (s).Finally, we compare the aforementioned amplitudes to extract the coupling constant g BBσ in the phenomenological baryon-baryon-σ coupling Lagrangian.

A. The OBE amplitude
According to the following effective Lagrangian coupling the σ meson to the baryons in the SU(3) flavor octet [18,41], the OBE amplitude for the Feynman diagram depicted in Fig. 1(b) reads where C B is a flavor factor, For simplicity, we choose λ i = 1/2 (i = 1, . . ., 4) throughout the paper. 2With this choice we have where m B is the isospin averaged mass of the baryon B.
Here a once-subtracted dispersive integral is employed to facilitate the convergence of the dispersive integral.The threshold s = 4m 2 B is chosen as the subtraction point, and we set the subtraction constant M DR (s = 4m 2 B ) = 0 as Eq. ( 3) since the two amplitudes will be matched later.In order to capture the ππ rescattering in the σ region and avoid the interference from other resonances, e.g., the f 0 (980), the upper limit of the dispersive integral in Eq. ( 4) is set to s 0 = (0.8 GeV) 2 , as in Ref. [39] (see also Ref. [42]).We will investigate the impact of varying the upper limit of the integration on the final result and regard it as a part of the uncertainty of the coupling constants.
Next, let us discuss the discontinuity.Taking into account the unitary relation that is fulfilled by the partial-wave T -matrix elements, we can express the discontinuity of the S-wave amplitude (here the partial wave refers to that between the pions) along the cut s ∈ [4M 2 π , +∞) in terms of T B B→ππ,0 (s) and T ππ→ BB,0 (s).However, it is crucial to notice that when dealing with systems that involve spins, particularly those containing fermions, kinematical singularities arise [43].These singularities stem from the definition of the wave functions for the initial and final states.Following Ref. [43], we introduce the kinematical-singularity-free amplitudes, Then the unitary relation for the S-wave T -matrix elements is given by disc M B B→ BB,0 (s) = 2iρ π (s) where is the two-body phase space factor.Moreover, as we will discuss in detail in Sec.II B 4, the treatment of kinematical singularity plays a vital role in guaranteeing the self-consistency of the theory.
Furthermore, with the phase conventions outlined in Appendix A and considering the isospin scalar ππ system, we obtain the following relations Then, we obtain the following discontinuities, disc disc

The SU(3) ChPT framework
To obtain the low-energy S-wave amplitudes T B B→ππ,0 (s), we need the corresponding chiral baryon-meson Lagrangian.The leading-order (LO) chiral Lagrangian is given by [44], L which contains three low-energy constants (LECs), m 0 , D and F .Here, ⟨•⟩ means trace in the flavor space, the baryon octet are collected in the matrix, and the chiral vielbein and covariant derivative are given by with Notice that the chiral connection Γ µ containing two pions is a vector, the two pions from that term cannot be in the S-wave.As a result, only the t-and u-channel exchanges depicted in Fig. 2 (a) and (b), which contribute to the LHC part of T B B→ππ,0 (s), will be present in the LO calculation.
In addition, the LO Lagrangian contains the ΣΛπ coupling terms of the form Λγ µ γ 5 ∂ µ πΣ and

The partial-wave amplitudes
Using the LO and NLO Lagrangians given in Eqs.(13,17), we can calculate the tree-level amplitude for the process of B B → ππ as depicted in Fig. 2.However, in order to determine the final state ππ with IJ = 00, we need to perform a partial-wave (PW) expansion.The generalized PW expansion of the helicity amplitude for arbitrary spin can be found in Ref. [49].The final PW amplitude for B B → ππ reads where L is the relative orbital angular momentum of the pions, λ 1 and λ 2 are the third components of the helicities of B and B. The basis is such that B B is expanded in terms of |θ 0 , ϕ 0 ; λ 1 , λ 2 ⟩, and the B B relative momentum is chosen to be along the z axis so that θ 0 = ϕ 0 = 0; (θ, ϕ) are the polar and azimuthal angles of the ππ relative momentum.
For the tree-level S-wave amplitude for B B → ππ, the LHC part from the t-and u-channel where The contact terms, which are from the NLO Lagrangian and contribute to the RHC part of T B B→ππ,0 (s) after taking into account the ππ rescattering, read 5 Here we consider only the baryon exchanges such that the two mesons emitted are two pions since we focus on the correlated S-wave two-pion exchange.That is, although we use an SU(3) chiral Lagrangian, the exchanged baryon has the same strangeness as the external ones.The framework may be understood as an SU(2) one for each of the baryons, but with the LECs matched to those in the SU(3) Lagrangian.
where the parameter F π is the decay constant of the π in the chiral limit.Since we use the LECs determined in Ref. [48], we adopt the same value F π = 87.1 MeV [50] for consistency.
Moreover, employing Eq. ( 5), the tree-level S-wave amplitudes for B B → ππ after eliminating the kinematical singularities read, for the LHC part, and for the contact term part,

The Muskhelishvili-Omnès representation
We now incorporate the ππ rescattering based on the tree-level amplitude, into the Muskhelishvili-Omnès representation.For the B B → ππ process, we partition the total S-wave kinematicalsingularity-free amplitude into the LHC and the RHC parts, Utilizing the ππ amplitude in the scalar-isoscalar channel T ππ→ππ,0 (s) = e iδ 0 (s) sin δ 0 (s)/ρ π (s), where δ 0 (s) is the S-wave isoscalar phase shift, and since there is no overlap between the LHC and RHC for kinematic-singularity-free amplitudes, 6 the unitary relation implies, To solve this equation, we first define the Omnès function [51], By using Ω 0 (s ± iϵ) = |Ω 0 (s)|e ±iδ 0 (s) , we further derive disc Therefore, we can derive a DR with n subtractions, where P n−1 (s) is an arbitrary polynomial of order n − 1.Finally, we obtain a DR for T new B B→ππ,0 (s) as For the phase shift δ 0 (s), we take the parametrization in Ref. [52].For the Ω 0 (s) Omnès function, we take the Ω 11 (s) matrix element of the coupled-channel Omnès matrix for the ππ-K K S-wave interaction obtained in Ref. [53]. 6The RHC is chosen to be along the positive s axis in the interval m 2 0 for the t-or u-channel process of B B → ππ, where m0 represents the mass of the exchanged particle.It can be easily proven that 4m FIG. 3. Real (left panel) and imaginary (right panel) parts of the tree-level t-and u-channel exchange amplitude for the process of Σ Σ → ππ projected to the ππ S-wave as given in Eq. ( 20).The branch cut of the square root function is chosen to be along the positive real s axis.
The above equation provides a reasonable form that incorporates the ππ rescattering.The LHC part L new B,0 (s) and the polynomial P n−1 (s) may be determined by matching at low energies to the chiral amplitudes as done in Refs.[54][55][56][57][58].We perform the matching when the ππ rescattering is switched off, i.e., δ 0 (s) = 0, which leads to Ω 0 (s) = 1.Consequently, for the B B → ππ process, we can approximate L new B,0 (s) ∼ ÂB new 0 (s) and Moreover, there is a polynomial ambiguity as discussed in Refs.[59,60].If the asymptotic value of the phase shift δ 0 (s) is not 0 but nπ as s → ∞, the corresponding Omnès function will approach 1/s n asymptotically.In our case, the phase shift δ 0 (s) s→∞ → π implies Ω 0 (s) s→∞ → 1/s, thus the general solution of the unitarity condition (36) contains 3 free parameters [59,60] (assuming that T new B B→ππ,0 is asymptotically bounded by s).However, although the standard twice subtracted DR via Eq.( 40) indeed grows like s (notice that n = 2), it contains only 2 free parameters in the polynomial, i.e., one parameter less than the general solution.Hence we propose an oversubtracted DR (twice subtracted DR with an order-2 polynomial matching to the ChPT amplitudes) that can be solved uniquely.In summary, the final DR is given as From the above derivation, it is important to note that Eq. ( 41) can only be applied when the singularity of the LHC is exclusively included in ÂB new 0 (s), and there is no overlap between the LHC and RHC.The original t-and u-channel exchange amplitudes Eqs.(19)(20)(21)(22) do not satisfy this condition due to the factor s − 4m 2 B .Let us take Σ Σ → ππ as an example.From Fig. 3, it becomes apparent that the amplitude in Eq. ( 20) includes the LHC −∞, 4M 2 π − M 4 π /m 2 Σ derived from the particle exchanging in the crossed channel, as well as a kinematical cut in the physical region.Therefore, directly substituting Eq. ( 20) into Eq.( 41) is invalid and disrupts the selfconsistency of the theory.By employing the method described in Sec.II B 1 to eliminate the kinematical singularities, the kinematical-singularity-free S-wave amplitude ÂΣ new 0 (s) in Eq. ( 28) has only the LHC and satisfies the condition for Eq. ( 41), as demonstrated in Fig. 4.

III. DETERMINATION OF COUPLING CONSTANTS
Now we compare the two amplitudes, M OBE in Eq. ( 3) and M DR and Eq. ( 4), to determine the coupling constant g BBσ .

A. Matching s-channel amplitudes
Let us first compare the two amplitudes in Eqs.(3,4) in the s-channel physical region, specifically s ≥ 4m 2 B .Since the amplitudes from exchanging σ and from exchanging the correlated S-wave ππ have the same Lorentz structure, we can compare the two amplitudes at large s values so that the pion masses and the σ mass in the OBE amplitude play little role.A comparison of M OBE and M DR in the physical region of s ≥ 4m 2 B is shown in Fig. 5, where g BBσ has been adjusted so that the two amplitudes coincide in the physical region and m σ = 0.5 GeV is taken.In fact, matching Eqs.(3,4) at s ≥ 4m 2 B , one gets Since s is much larger than both m 2 σ or z ≤ s 0 ≃ (0.8 GeV) 2 , one obtains the following sum rule: The numerical results of the scalar coupling constants are presented in Table I, where the uncertainties in the second to fourth columns arise from the error propagated from those of the NLO LECs and the choice of the upper limit for the dispersive integral (see below), corresponding to Eqs. (4,41).In addition to the results obtained in the SU(3) framework, we also investigate g N N σ in the SU(2) framework.The details are presented in Appendix B, and the results are listed in the last row in Table I, labeled as g SU N N σ .Moreover, remarks are made on the difference in g N N σ under the SU(2) and SU(3) frameworks in Appendix B.
Let us comment on the calculation of the two dispersive integrals.The first one, given by Eq. ( 41), is computed over the integration range of [4M 2π , ( ].The second one, given by Eq. ( 4), is integrated over [(2M π +ϵ) 2 , s 0 ]. 7Note that the range of the second integral is completely covered by that of the first one to avoid unphysical singularities.
The central values in Table I are obtained by setting √ s 0 to 0.8 GeV as in Ref. [39] and utilizing the central values of the NLO LECs provided in Ref. [48].The uncertainties of the NLO LECs as determined in Ref. [48] are propagated to the coupling constants by using the bootstrap method.The resulting average values and corresponding standard deviations introduce the first source of errors in the third and fourth columns in Table I (the b i LECs appear only in the RHC contributions, and we have fixed the pion decay constant; thus the second column does not have errors from LECs).Furthermore, we vary √ s 0 from 0.7 GeV to 0.9 GeV, which constitute the errors in the second column and the second source of errors in the third and fourth columns.
Results from other studies on these scalar couplings are also listed in Table I.For g N N σ that has been estimated in many works, we find a good agreement with existing results, which supports the validity of our framework.Here we briefly discuss the methods used in the literature.In Ref. [33], the authors investigated the S-wave N N → ππ amplitudes with the ππ rescattering and the results revealed that the intertwined contribution from the ππ S-wave can be elegantly TABLE I.The coupling constants g BBσ as given by the sum rule in Eq. ( 43). a The second, third and fourth columns list the results when only the LHC part shown in Fig. 2 (a) and (b), only the RHC part shown in Fig. 2 (c) and both of them are considered, respectively.The fifth to eleventh columns list the coupling constants from other references.For the seventh column, the values outside and inside the brackets represent the results calculated using different models in Ref. [34].The last column lists the mass (in MeV) of the σ determined in the t/u-channel amplitude matching, as detailed in III B. The last row for g SU( 2) N N σ lists the results obtained in the SU (2)  The numerical results show that the total coupling g total BBσ does not align with the mere addition of the LHC and RHC couplings, g LHC BBσ + g RHC BBσ .This difference stems from the fact that both the LHC and RHC terms in Eq. ( 41) share the same phase factor, specifically e iδ 0 (s) .Consequently, we anticipate the emergence of constructive and destructive interference effects in the subsequent computations involving the squared amplitude, as detailed in Eqs.(9)(10)(11)(12), as well as during the integration procedures outlined in Eq. (43).
described as a broad σ-meson with a mass of approximately m σ ∼ 4.8 M π and a coupling strength of g N N σ ∼ 12.78.In Ref. [18], displaying the outcomes derived from the Bonn meson-exchange model, they found that the correlated S-wave ππ exchange can be further approximated by a zero width scalar exchange, with the corresponding mass and coupling constant readjusted to 550 MeV and 8.46, respectively.In Ref. [34], the authors also considered the σ exchange as an effective parameterization for the correlated S-wave ππ exchange contribution.They utilized the result from the full Bonn meson-exchange model [18] for the nucleon, i.e., the value in the sixth column of Table I, and g ΛΛσ and g ΣΣσ are determined by a fit to the empirical hyperon-nucleon data using two different models, with the distinction lying in whether higher-order processes involving a spin-3 2 baryon in the intermediate state were considered in the hyperon-nucleon interaction.In Ref. [37], the authors calculated the B B′ → ππ and B B′ → K K amplitudes in the light of hadron-exchange picture.Based on an ansatz for Lagrangian, various symmetries and assumptions, they reduced the number of free parameters as many as possible, and then the parameters were fixed by adjusting the N N → ππ amplitudes to the quasi-empirical data.With these parameters and the existing ππ scattering phase shifts they got the B B′ → ππ and B B′ → K K amplitudes in the pseudo- MeV, where the uncertainties correspond to those of the couplings added in quadrature. 9These values are listed in the last column of Table I.This echoes previous attempts to modify the mass of σ, a broad resonance with a mass approximately equal to 4.8 M π [33], to a mass of 550 MeV with a zero width [18], which is within all the above ranges.The goal of such modification was to allow a single σ exchange to more accurately replicate the results of a correlated ππ exchange with IJ = 00.

IV. SUMMARY
In this work, we evaluate the couplings of the σ meson to the 1  I, to the DR amplitudes using the central values of the LECs provided in Ref. [48] and setting √ s 0 to 0.8 GeV as in Ref. [39].
utilize DR and incorporate the ππ rescattering by Muskhelishvili-Omnès representation to obtain the DR amplitude.Considering the phenomenological σ exchange as an effective parameterization for the correlated ππ exchange contribution in the IJ = 00 channel, we determine the scalar coupling constants g BBσ from the s-channel matching, as listed in Table I.Specifically, g ΣΣσ = 3.5 +1.8 −1.3 , g ΞΞσ = 2.5 +1.5 −1.4 , g ΛΛσ = 6.8 +1.5 −1.7 , and g N N σ = 8.7 +1.7 −1.9 , where the errors are obtained by adding the corresponding ones in Table I in quadrature.This is achieved by comparing the DR amplitude and OBE amplitude in the physical region of the s-channel process, specifically, s ≥ 4m 2 B .Concurrently, we estimate the uncertainties of the scalar coupling constants arising from the NLO LECs [48] and variation of the upper limit for the dispersive integral.Moreover, by extending the analysis to the physical region of the corresponding t/u-channel process via the crossing relation, we obtain the σ mass to be used together with the determined BBσ coupling constant.The value depends on the process but is always around 550 MeV.We also compute the N N σ coupling by matching to the SU(2) CHPT amplitude with the LECs determined in Refs.[63,64], and the result is g SU(2) N N σ = 12.2 +1.9 −2.3 .The effective coupling constants obtained here can be used to describe the interaction between light hadrons and other hadrons through the σ exchange.The same method can be applied to the determination of the coupling constants of σ and other hadrons, such as heavy mesons and baryons, the interactions between which are crucial to understand the abundance of exotic hadron candidates observed at various experiments in last two decades.

FIG. 1 .
FIG. 1. Feynman diagrams for the s-channel process of B B → BB with the intermediate state of ππ (a) or σ (b).In (a), the black dots imply that the ππ rescattering is included.

B. The dispersive representation 1 .
The DR and the kinematical singularity One can write down a dispersive representation of the B B scattering amplitude corresponding to Fig. 1(a) as

FIG. 2 .
FIG.2.The tree-level Feynman diagrams for the process of B B → ππ.

FIG. 4 .
FIG.4.Real (left panel) and imaginary (right panel) parts of the tree-level t-and u-channel exchange amplitude as given in Eq. (28) for the process of Σ Σ → ππ projected to the ππ S-wave.The amplitude is free of kinematical singularities and has only the desired LHC.

FIG. 5 .
FIG. 5. Comparison of the OBE amplitudes with different coupling constants obtained from s-channel σ exchange and the DR amplitudes for different cases by using the central values of the LECs provided in Ref. [48] and setting √ s 0 to 0.8 GeV as in Ref. [39].The subscripts RHC, LHC and Total in M DR represent that the corresponding amplitudes consider only the RHC part shown in Fig. 2 (c), only the LHC part shown in Fig. 2 (a) and (b), and both contributions combined, respectively.

FIG. 6 .
FIG. 6.The Feynman diagram for the t-channel process of BB → BB with the intermediate state of ππ (a) or σ (b).In (a), the black dots imply ππ interaction.

2 +
FIG. 7. Comparison of the OBE amplitude, with the coupling constant taking the central value listed in Table I and different m Ξ σ values in the process of Ξ Ξ → ΞΞ, and the DR amplitude using the central values of the LECs provided in Ref. [48] and setting √ s 0 to 0.8 GeV as in Ref. [39].

4 (α 4 )λ 1 λ ′ 1 (α 1 )d 1 2 λ 2 λ ′ 2 (α 2 )d 1 2 λ 3 λ ′ 3 (α 3 )d 1 2 λ 4 λ ′ 4 (
M s-channel B(λ ′ 1 ) B(λ ′ 2 )→ B(λ ′ 3 )B(λ ′ 4 ),0 (s), (C1) where α i represents the Wigner rotation angles corresponding to the Lorentz transformation from the s-channel c.m. frame to the t-channel c.m. frame, and the subscript 0 signifies that the ππ of the t-channel process or the s-channel process forms an isoscalar S-wave.Considering that the crossing relation, Eq. (C1), is solely dependent on the particles of the external lines, the same relation is applicable regardless of whether there is a σ exchange or a correlated ππ exchange, is to ensure that the amplitude of the correlated ππ exchange with IJ = 00 and that of the σ exchange are approximately the same for the t-channel process of BB → BB within the t-channel physical region, i.e., t ≥ 4m 2 B , we require the corresponding s-channel amplitudes of B B → BB to approximate each other as well as possible, i.e., M OBE B B→ BB (s) ≃ M DR B B→ BB,0 (s) (C5) framework, as detailed in Appendix B.