The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region

Using dispersion theory, the electromagnetic Sigma-to-Lambda transition form factors are expressed as the product of the pion electromagnetic form factor and the $\Sigma\bar{\Lambda}\to\pi\pi$ scattering amplitudes with the latter estimated from SU(3) chiral perturbation theory including the baryon decuplet as explicit degrees of freedom. The contribution of the $K\bar{K}$ channel is also taken into account and the $\pi\pi$-$K\bar{K}$ coupled-channel effect is included by means of a two-channel Muskhelishvili-Omn\`{e}s representation. It is found that the electric transition form factor shows a significant shift after the inclusion of the $K\bar{K}$ channel, while the magnetic transition form factor is only weakly affected. However, the $K\bar{K}$ effect on the electric form factor is obscured by the undetermined coupling $h_A$ in the three-flavor chiral Lagrangian. The error bands of the Sigma-to-Lambda transition form factors from the uncertainties of the couplings and low-energy constant in three-flavor chiral perturbation theory are estimated by a bootstrap sampling method.


Introduction
Electromagnetic form factors (EMFFs) give access to the strong interaction, which provides one of the most notorious challenges in the Standard Model due to the nonperturbative nature of Quantum Chromodynamics (QCD) at the low energy scale. On the one hand, the EMFFs can be extracted from a variety of experimental processes, such as lepton-hadron scattering, lepton-antilepton annihilation or radiative hadron decays. These EMFFs can be measured over a large energy range. On the other hand, dispersion theory, which is a powerful nonperturbative approach, allows for a theoretical description of the EMFFs. Consequently, the EMFFs are an ideal bridge between experimental measurements and theoretical studies of the low-energy strong interaction.
In the last decade, much research effort both in experiment and theory was focused on the nucleon EMFFs, largely triggered by the so-called proton radius puzzle [1]. For recent reviews, see e.g. Refs. [2,3,4,5,6]. In the process of unravelling this puzzle, dispersion theory has played and is playing a crucial role in the theoretical description of the nucleon EMFFs [7,8,9,10]. The dispersion theo-retical parametrization of the nucleon EMFFs, first proposed in the early works [11,12,13] and further developed in Refs. [14,15,9], incorporates all constraints from unitarity, analyticity, and crossing symmetry, as well as the constraints on the asymptotic behavior of the form factors from perturbative QCD [16]. The state of the art of dispersive analyses of the nucleon EMFFs is reviewed in Ref. [17]. Very recently, all current measurements on electron-proton scattering, electron-positron annihilation, muonic hydrogen spectroscopy, and polarization measurements from Jefferson Laboratory could be consistently described in a dispersion theoretical analysis of the nucleon EMFFs [18].
The dispersive prescription of parameterizing the nucleon EMFFs can also be applied to other hadron states. The first two straightforward extensions concern the Delta baryon and the hyperon states, with the former obtained by flipping the spin of one of the quarks inside the nucleon and the latter by replacing one or several up or down quarks with one or more strange quarks. The EMFFs of the Delta and the hyperons provide complementary information about the intrinsic structure of the nucleon [19]. The electromagnetic properties of the Delta baryon have been studied in detail in Ref. [20]. Recent investigations of the hyperon EM structure are given in Refs. [21,22,19,23,24,25,26]. Ref. [19] considered once-subtracted dispersion relations for the electromagnetic Sigma-to-Lambda transition form factors (TFFs) and expressed these in terms of the pion EMFF and the two-pion-Sigma-Lambda scattering amplitudes. Using an Omnès representation, the pion EMFF could be expressed as the Omnès function of the pion P -wave phase shift which has been well determined from the Roy-type analyses of the pion-pion scattering amplitude [27]. An improved parameterization of the pion EMFF is also available, which includes further inelasticities and is applicable at higher energies [28]. Moreover, the two-pion-Sigma-Lambda scattering amplitudes could be calculated in a model-independent way by using three-flavor chiral perturbation theory (ChPT) [21]. Combining these studies and taking some reasonable values for couplings and the low-energy constants in three-flavor ChPT, the electromagnetic Sigma-to-Lambda TFFs were predicted in Ref. [19] where the pion rescattering and the role of the explicit inclusion of the decuplet baryons in three-flavor ChPT were also investigated.
In the present work, we extend the theoretical framework used in Ref. [19] to explore the effect of the KK inelasticity on the electromagnetic Sigma-to-Lambda transition form factors. This is performed by considering the two-channel Muskhelishvili-Omnès representation when introducing the pion rescattering effects. In principle, one should include even more inelasticities when implementing the dispersion theoretical parameterization for the Sigmato-Lambda TFFs, as done in our previous work on the nucleon EMFFs [18]. However, it is difficult in the current case due to the poor data base which is required when constructing reliable inelasticities in the higher energy region, that is above the KK threshold at ∼ 1 GeV. Note that the four-pion channel has negligible effects in the energy region around 1 GeV, see Ref. [29]. It is also known that the contribution of the four-pion channel to the pion and kaon form factors below 1 GeV is a three-loop effect in ChPT [30] and thus is heavily suppressed.
We remark that the 4π channel was shown to play an important role starting from 1.4 GeV in the S-wave case [31]. This is caused by the presence of the nearby scalar resonances f 0 (1370) and f 0 (1500) which were both observed to have a sizable coupling to four-pion states [32,33,34]. There is no evidence, however, for the presence of corresponding 1 − isovector states in the energy region of 1...2 GeV in the P -wave case. Moreover, another experimental finding of these references is that the 4π system likes to cluster into two resonances in the energy region above 1 GeV [31]. The lowest candidate is supposed to be the ρρ channel for the P -wave isovector problem. From the phenomenological point of view, the inelasticity around 1 GeV should be saturated to a good approximation by the ππ and KK coupled-channel treatments in the Pwave case. Moreover, as we will show later, the effect of KK inelasticity is small. Thus, the relative ratio between the effects of the KK and four-pion channel could be enhanced. To investigate this relative ratio, a sophisticated calculation on the four-pion inelasticity is needed which goes beyond the present work.
In the present work, the KK inelasticity is implemented using SU(3) ChPT. The inclusion of the KK channel allows one to construct the Sigma-to-Lambda transition form factors up to 1 GeV precisely. In addition, the estimation of the theoretical uncertainties is improved by using the bootstrap approach [35].
The paper is organized as follows: In Sect. 2 we introduce the dispersion theoretical description of the electromagnetic Sigma-to-Lambda transition form factors and present the coupled-channel Muskhelishvili-Omnès representation for the inclusion of the KK inelasticity. Numerical results are collected in Sect. 3. The paper closes with a summary. Some technicalities are relegated to the appendix.

Formalism
Here, we discuss the basic formalism underlying our calculations. We first write down once-subtracted dispersion relations for the electric and magnetic Sigma-to-Lambda transition form factor and then discuss in detail their various ingredients, namely the vector form factor of the pion and the kaon and the amplitudes for Σ 0Λ → ππ and Σ 0Λ → KK, in order.

Dispersion relations for the Sigma-to-Lambda TFFs
The electromagnetic Sigma-to-Lambda TFFs are defined as in Refs. [21,19], with t = (p − p) 2 = q 2 the four-momentum transfer squared. The scalar functions F 1 (t) and F 2 (t) are called the Dirac and Pauli transition form factors, respectively. One also writes the electric and magnetic Sachs transition form factors, given by the following linear combinations, with the normalizations F 1 (0) = G E (0) = 0 and F 2 (0) = G M (0) = κ ≈ 1.98. Here, κ is estimated from the experimental width of the decay Σ 0 → Λγ, see Ref. [19] for details. Unlike the nucleon case where one constructs dispersion relations for F 1 and F 2 [17], we work with the electric and magnetic Sachs form factors, i.e. G E and G M , for the Sigma-to-Lambda TFFs as in Ref. [19] since the Sigma-to-Lambda TFFs are of pure isovector type and the helicity decomposition used in Ref. [19] can easier be applied to the Sachs FFs. In order to apply the spectral decomposition to estimate the imaginary part Im G E/M , we consider the matrix element of the electromagnetic current Eq. (1) in the time-like region (t > 0), which is obtained via crossing symmetry, where p 3 and p 4 are the momenta of the Σ 0 andΛ created by the electromagnetic current, respectively. The fourmomentum transfer squared in the time-like region is then t = (p 3 + p 4 ) 2 . With the ππ and KK inelasticities taken into account as depicted in Fig. 1, the unitarity relations for the Sigma-to-Lambda TFFs read [36,37,19,24], where q π/K (t) = is the center-of-mass momentum of the ππ/KK two-body continuum with λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + zx) the Källén function. F V π/K (t) is the vector-isovector form factor (J = I = 1) of the pion/kaon and T ππ E (t) and T ππ M (t) are two independent reduced P -wave Σ 0Λ → π + π − amplitudes in the helicity basis. Similarly, T KK E (t) and T KK M (t) denote the corresponding reduced amplitudes for Σ 0Λ → K + K − (K 0K 0 ). Then the once-subtracted dispersion relations for the Sigma-to-Lambda TFFs are written as, where the last term G anom E/M (t) denotes the contribution of the anomalous cut which is non-zero when there exists an anomalous threshold in the involved processes [38,39,40,41,24]. This does happen when the KK channel is taken into account, see Appendix A for detailed discussions and the explicit expressions of the anomalous part. Next, we need to consider the various factors contributing to Eq. (6).

The P -wave Omnès matrix and π, K vector form factors
In this subsection, we derive the P -wave isovector Omnès matrix and solve for the pion and kaon EMFFs in the coupled-channel formalism. Ω satisfies the unitarity rela- where with is the diagonal phase space matrix. The J = I = 1 ππ-KK coupled-channel T -matrix t 1 1 is parameterized as where g and ψ are the modulus and phase of the P -wave isovector ππ → KK scattering amplitude, respectively. The inelasticity η is defined by Then we can write the dispersion relation for the Omnès matrix Ω as The analytic solution of the integral equation Eq. (23) was given in Ref. [47] for the single-channel problem. However, there are no known analytic solutions for two or more channel cases where one has to construct the solutions numerically, either by an iterative procedure [48] or a discretization method [31]. Here, we adopt the iterative approach to solve the P -wave ππ-KK coupled-channel Omnès matrix. Substituting Eq. (18) into Eq. (23), one obtains a two-dimensional system of integral equations Re where and P denotes the principal value. Searching for solutions of Ω(t) is equivalent to searching for two independent solutions of the integral equation set for the two-dimensional array (χ 1 , χ 2 ) T . Using the iterative procedure, one can obtain a series of solutions (χ λ 1 , χ λ 2 ) T starting with various initial inputs χ 1 (t) = 1, χ 2 (t) = λ, where λ is a real parameter. Note that the iterative process is linear and the results of the iteration is therefore a linear function of λ [48]. Then the solution family {(χ λ 1 , χ λ 2 ) T } contains only two linearly independent members. Here, we take the same convention as Ref. [46] to construct two independent solutions, (Ω 11 , Ω 21 ) T and (Ω 12 , Ω 22 ) T , that satisfy the normalizations Ω 11 (0) = Ω 22 (0) = 1 and Ω 12 (0) = Ω 21 (0) = 0 , from two arbitrary solutions (χ λ1 1 , χ λ1 2 ) T and (χ λ2 1 , χ λ2 2 ) T . With the two-channel Muskhelishvili-Omnès representation, the binary function composed of the vector FFs of the pion and the kaon fulfills the same unitarity relation Eq. (18). Then one can solve the pion and kaon vector form factors which are normalized as F V π (0) = 1 and F V K (0) = 1/2. To solve the J = I = 1 ππ-KK Omnès matrix, the required input is the P -wave isovector ππ-KK scattering matrix t 1 1 , i.e. Eq. (21), that is constructed from the ππ Pwave isovector phase shift δ 1 1 , the modulus g and phase ψ of the P -wave isovector ππ → KK amplitude. The phase shift δ 1 1 up to 1.4 GeV was extracted precisely from the Roy-type analyses of the pion-pion scattering amplitude in Ref. [27]. We take the same prescription as in Ref. [28] to extrapolate it smoothly to reach π at infinity. Then δ 1 1 (t) is given by where Here, t 0 = (1.4 GeV) 2 , t 1 = (1.05 GeV) 2 and t 2 = (10 GeV) 2 . The P -wave ππ → KK amplitude up to √ t 3 = 1.57 GeV is taken from Ref. [49] where the modulus g in the region of 4M 2 π ...4M 2 K was solved from the Roy-Steiner equation with the experimental data of P -wave ππ → KK scattering [50,51] above the KK threshold as input, while the phase ψ was fitted to experimental data [50,51]. Note that the two-channel Muskhelishvili-Omnès representation in terms of ππ and KK intermediate states should only work well in the lower energy region [46]. Further, the asymptotic values of phase shifts in the coupled-channel systems have to satisfy to ensure that the system of integral equations, Eq. (24), has a unique solution [31,52]. n is the number of channels that are considered in the formalism. It requires ψ = δ 1 1,ππ + δ 1 1,KK ≥ 2π in Eq. (21). g and ψ are extrapolated smoothly to 0 and 2π by means of [31] where the extrapolation point t 4 of ψ should be far away from 1.5 GeV since there is a structure located around 1.5 GeV in the phase of the P -wave ππ → KK amplitude.
Here we take the value √ t 4 = 5 GeV for ψ. Such a structure should also leave trails in the modulus g. However, only g up to √ 2 GeV is estimated in Ref. [49] and a small bump around 1.5 GeV in g is only reflected roughly by several data points above 1.4 GeV measured by Ref. [50], see Fig. 9 in Ref. [49]. The modulus used in our work is presented in Fig. 3, while the δ 1 1 and ψ are presented when we show the solved pion and kaon vector form factors. The obtained Ω matrix elements are presented in Fig. 4. The pion and kaon vector form factors calculated from Eq. (26) are then given in Fig. 5. Clearly, one can see from Fig. 5 that the phase of F V π and F V K are consistent with the input ππ phase shift δ 1 1 and the phase ψ of the Pwave ππ → KK scattering amplitude respectively, which is similar with the finding for the S-wave case by Ref. [46].

Results
Using the reduced amplitudes P  [21]. In SU(3) ChPT, F Φ can take three different values at LO, namely F π = 92.4 MeV, F K = 113.0 MeV and F η = 120.1 MeV [53]. Often, one chooses the average of these, that is, F Φ = (F π + F K + F η )/3. Here, we take F Φ = 100 ± 10 MeV to cover mainly the π and K contributions. h A can be determined from the experimental widths of either Σ * → Λπ or Σ * → Σπ. We take the value h A = 2.3 ± 0.3 [19], here an additional 10% error is added to account for the SU(3) flavor symmetry breaking effect when applied to the vertices involving a Ξ * . The low-energy constant b 10 was estimated in Ref. [54] based on the resonance saturation hypothesis as b 10 = 0.95 GeV −1 . A larger value b 10 = 1.24 GeV −1 is used in Ref. [21]. A very recent determination based on the ChPT fits to lattice data of the axial-vector currents of the octet baryons gives b 10 = 0.76 GeV −1 [55]. Taking all these determinations into account, b 10 = (1.0±0.3) GeV −1 is used here. Second, we introduce an energy cutoff Λ in the integration along the unitarity cut in Eq. (6) and Eq. (13). We consider two values for the cutoff, Λ = 1.5 and 2.0 GeV, to check the sensitivity of our results to it. Now we are in the position to present our numerical results for the electromagnetic Sigma-to-Lambda transition form factors. First, we present the electric transition form factor G E obtained with the radius-adjusted parameters given in Ref. [19], i.e. F Φ = 100 MeV, b 10 = 1.06 GeV −1 and h A = 2.22 where the radius is adjusted to the fourthorder ChPT result from Ref. [21], in Fig. 6. Note that Λ = 1.5 GeV is used in these calculations. The result from the single ππ channel consideration is also plotted for an intuitive comparison. Taking the same parameter values, we find good agreement with Ref. [19]. After the inclusion of the KK inelasticity, a logarithmic singularity located at the anomalous threshold t − = 0.935 GeV in the unphysical area of the time-like region is introduced into the TFF G E . Moreover, additional nonzero imaginary parts along the anomalous cut are produced for the TFFs by Eq. (42) and Eq. (43). This is similar to the triangle singularity mechanism that leads to a quasi-state phenomenon in the physical observables [56], except the anomalous threshold here can not be accessed directly by the experiments. The imaginary parts of G E in the space-like region, however, are still zero since the nonzero contributions from Eq. (42) are exactly canceled by those from the unitarity integral of Eq. (43). A similar plot for the magnetic  TFF G M is shown in Fig. 7 where there is a cusp-like structure rather than a logarithmic singularity in G E located at the anomalous threshold since the coefficient f in Eq. (39) which is proportional to (Y 2 − κ 2 ) does vanish at the anomalous threshold for G M . Note that such cusp-like structure is almost invisible due to the large scale variation of the magnitude of G M . With that set of parameters, a 52% decrease is produced by the KK channel for G E at t = −1 GeV 2 , 2 while only a 3% decrease happens for G M . One should be aware, however, of the large difference between the effects of KK channel in G E and G M is the result of the much larger magnitude that G M has overall than G E . The absolute effect of the KK inelasticity in G M is actually of compatible size as in G E (sometimes even larger). In Fig. 8, we show the electric transition form factor G E between the estimation including only the ππ intermediate state and the ππ-KK coupled-channel determination with errors. Note that the TFFs are real-valued in the space-like region. The solid curves are calculated again with the radius-adjusted parameters. The error bands in Fig. 8 are estimated by the bootstrap sampling over the three-dimensional parameter space that is spanned by F Φ , b 10 and h A . Note that the electric form factor is independent of the low-energy constant b 10 , see the expressions in Appendix B. As in Ref. [19], the uncertainty in h A gives the dominant contribution. The effect on G E introduced by the inclusion of the KK inelasticity is heav-ily intertwined with the large uncertainties from the variation of h A and Λ. Overall, the role of the cutoff is a bit more complicated than in the single ππ channel case. The situation is different for G M which is displayed in Fig. 9. The magnetic Sigma-to-Lambda transition form factor G M is almost unchanged after including the KK inelasticity. Moreover, G M has much larger absolute errors from the bootstrap method. At t = −1 GeV 2 , the bootstrap uncertainty from F Φ , h A and b 10 is already of order ±1, dominated by the uncertainty in b 10 . As in Ref. [19], we find a very small sensitivity of G M to the variation of the cutoff Λ. In addition to providing valuable insights into the electromagnetic structure of hyperons, experimental data for the transition form factors may thus also help to constrain these parameters.

Summary
In this paper, we extended the dispersion theoretical determination of the electromagnetic Sigma-to-Lambda transition form factors presented in Ref. [19] from the ππ intermediate state to the ππ-KK coupled-channel configuration within the SU(3) ChPT framework. After in-cluding the KK channel, a shift of the electric Sigma-to-Lambda transition form factor G E is presented, while the magnetic form factor G M stays essentially unchanged. At present, the dispersion theoretical determination of electromagnetic Sigma-to-Lambda transition form factors suffers from sizeable uncertainties due to the poor knowledge of the LEC b 10 and coupling h A . The precise determination of this three-flavor LEC from the future experiments will be helpful to pin down the hyperon TFFs. In a next step, it will be of interest to explore the elastic hyperon electromagnetic form factors based on the theoretical framework that combines dispersion theory and three-flavor chiral perturbation theory.

A Unitarity relations and the anomalous pieces
Let us start from the single channel case. The unitarity relations for the Σ-to-Λ TFFs G E/M (in the followings we drop the index E/M ) within the single ππ channel assumption read [19,24] where Σ π = σ π q 2 π with σ and q defined by Eq. (20) and Eq. (5) respectively, and q = √ tσ/2. Moving to the ππ-KK coupled-channel case, one first considers the vector pion and kaon form factors; they satisfy the unitarity relations [46,57], Similarly, the Σ 0Λ → ππ and Σ 0Λ → KK P -wave amplitudes fulfill the unitarity relations The key information that the above two equations provide us is the relative ratio between the ππ and KK channels in the J = I = 1 coupled-channel problem. Then with the single-ππ unitarity relations at hand already, that is, Eq. (31), one can easily extend to the two-channel case: That becomes Eq. (4) after substituting the identity q = √ tσ/2. Recalling that all the left-hand cut (LHC) part of T is included in K, then T − K only contains the right-hand cut (RHC) and its unitarity relation is given by Eq. (33) for the two-channel assumption. One can also write [46] 1 2i disc which leads to Eq. (13).
, the LHC and RHC will overlap, leading to the non-zero anomalous terms G anom and T anom in Eq. (4) and Eq. (13), respectively [38,39,40,41,24]. This indeed happens in the proton exchange diagram for the process ΣΛ → KK. Such anomalous contributions are estimated by the dispersive integrals of the discontinuity along the cut that connects the anomalous threshold to the starting point of the RHC (the physical threshold of the two-body intermediate state). The anomalous threshold t − is defined by [39] Numerically, t − = 0.935 GeV located at the real axis of t just below the KK threshold. To go further, one first has to derive the discontinuity along the anomalous cut for the TFFs G and the scattering amplitudes T . After implementing the partial-wave projection, namely the integration in Eq. (14) and Eq. (16), one obtains where f is the coefficient of the logarithm which is a smooth function over the transferred momentum square t without any cut. The anomalous threshold is generated by the logarithm function. As illustrated in Refs. [39,24], the discontinuity of K N along the anomalous cut reads Note that the argument of √ z is defined in the range of [0, π) in the present work. Regarding T , one can rewrite Eq. (35) into where we replace (− Im Ω −1 ) with ( Ω −1 t 1 * 1 Σ) in the second line since ImΩ −1 12 = ImΩ −1 22 = 0 below the KK threshold. 3 Finally, the discontinuity of the TFFs G along the anomalous cut can be read off straightforwardly in terms of that of T , Substituting Eq. (39) into the above equation, one obtains Then we arrive at the expressions for G anom and T anom . They are 3 This replacement is necessary since Im Ω −1 is solved numerically in our calculation and ImΩ −1 12 = ImΩ −1 22 = 0 always holds in the unphysical region. The combined quantity Ω −1 t 1 * 1 Σ can be simplified analytically when multiplied to discanom K. Then it turns out that the products Ω −1 t 1 * 1 and Σ discanom K, respectively, are finite along the anomalous cut. Moreover, the identity − Im Ω −1 = Ω −1 t 1 * 1 Σ is checked numerically and does hold near the KK threshold.
To cross-check whether this prescription is correct, we present the calculation of a scalar triangle loop function Fig. 10. The exact agreement is achieved only when the anomalous contribution is taken into account. B The reduced amplitudes K π/K and P 0,π/K The four-point amplitudes M ΣΛ→ππ/KK (t, θ) are calculated up to next-to leading order within the framework of SU(3) chiral perturbation theory. It turns out that the explicit inclusion of the decuplet baryon in the three-flavor ChPT Lagrangian is important to reproduce the correct G E/M (0) 4 and reasonable electric and magnetic transition radii, r 2 E and r 2 M [19]. We use the same Lagragians as in Ref. [19]. To be specific, the relevant interaction part of the leading order (LO) chiral Lagrangian that contains both the octet and decuplet states as active degrees of freedom for the reactions of interest is given by [21,58] L (1) (44) and the relevant NLO Lagrangian reads [59,60] L (2) where · · · denotes a flavor trace. The chirally covariant derivatives are defined by with Here, v and a are external sources and u 2 = U = exp(iΦ/F Φ ) with the Goldstone bosons encoded in the matrix The octet baryons also make up a 3×3 matrix in the flavor space that is given by Finally, T abc is a totally symmetric flavor tensor that denotes the decuplet baryons, The amplitudes M ΣΛ→ππ/KK are described as a Born term in the LO plus a contact term in the NLO within the three-flavor ChPT, see Fig. 11 and Fig. 12.  From above Lagrangians, one obtains the Σ-exchange Born term for Σ 0 (p 1 ) +Λ(p 2 ) → π − (p 3 ) + π + (p 4 ), Σ ) the propagator of the exchanged Σ in the t-and u-channel respectively. And the Σ * -exchange Born term, with the spin-3/2 Rarita-Schwinger propagator [61] i∆ µν (p) = −i 3 m p 2 (g µρ g νβ g αλ + g µα g νλ g βρ )σ αβ p ρ p λ , and t = (p 1 − p 4 ) 2 , u = (p 1 − p 3 ) 2 . Here, m denotes the mass of the exchanged spin-3/2 resonance. The NLO contact term for the reaction Σ 0 (p 1 ) +Λ(p 2 ) → π − (p 3 ) + π + (p 4 ) is given by The corresponding expressions for the Σ 0 (p 1 ) +Λ( Ξ ) the propagator of the exchanged proton and Ξ baryon, respectively. To proceed, it is helpful to introduce the following equivalents, where s = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 is the center-of-mass energy. p z and p c.m. denote the modulus of the threedimensional center-of-mass momenta of the ΣΛ and ππ/KK two-body systems, respectively, i.e. p c.m. = q π/K . The equations (55) are calculated in the center-of-mass frame with the p z the modulus of the three-momentum along the direction of the z-axis and θ is the scattering angle of π or K. Substitute Eqs. (51), (52), (53), (54) into Eqs. (14), (15), (16), (17), we obtain P E 0,π , P M 0,π , K E π and K M π for the ππ inelasticity, Note that we subtract a term K M Σ * ,low in the polynomial part of the magnetic amplitude P M 0,π , which denotes the low-energy limit of the LHC contribution of the decupletexchanged magnetic amplitude. It is proposed to remove the doubly counted decuplet baryon contribution caused by the using of the resonance saturation assumption for the estimation of b 10 in the present ChPT framework. A similar term K E Σ * ,low should be subtracted in P E 0,π . However, it belongs to a higher chiral order and is dropped here. Note that P E NLO belongs to P 1 (s) that is beyond the accuracy of Eq. (13) and is also dropped. Taking the same convention with Ref. [19], K M Σ * ,low is given by .
Note that the Pascalutsa prescription of the spin-3/2 particle will bring an ambiguity in the P E Σ * and P E Ξ * while it keeps K E Σ * and K E Ξ * consistent with the interaction between the decuplet and octet states listed in Eq. (44), see Ref. [19] for the details. The uncertainties on the TFFs originating from such ambiguity, however, are negligible when compared with the parameter errors. And we take he same convention with Ref. [19] where the O(M 2 π , s) and O(M 2 K , s) terms are dropped in the P E Σ * and P E Ξ * . Further,