Global analysis of charge exchange meson production at high energies

Many experiments that are conducted to study the hadron spectrum rely on peripheral resonance production. Hereby, the rapidity gap allows the process to be viewed as an independent fragmen- tation of the beam and the target, with the beam fragmentation dominated by production and decays of meson resonances. We test this separation by determining the kinematic regimes that are dominated by factorizable contributions, indicating the most favorable regions to perform this kind of experiments. In doing so, we use a Regge model to analyze the available world data of charge exchange meson production with beam momentum above 5 GeV in the laboratory frame, that are not dominated by either pion or Pomeron exchanges. We determine the Regge residues and point out the kinematic regimes which are dominated by factorizable contributions.


I. INTRODUCTION
The new generation of high statistics experiments e.g. Belle II, BESIII, CLAS12, CMS, COMPASS, GlueX, J-PARC, LHCb, andPANDA, have dedicated programs to study the hadron spectrum, whose quantitative description is pivotal for a complete understanding of Quantum Chromodynamics (QCD). These experiments demand a high level of precision in the amplitude analysis [1] necessary to obtain reliable extractions of hadron properties from the data. In particular, the diffraction of photons or mesons on the nucleon target at high energies, as studied at GlueX, CLAS12 and COMPASS, is expected to provide information on hybrids, exotics and the gluonic degrees of freedom, via independent fragmentation of the beam and of the target (see Fig. 1), with the beam fragmentation dominated by production and decays of mesons.
Regge phenomenology underlies such processes and provides the theoretical framework for studying high energy scattering. In Regge theory, resonances in the exchanged channel are related to each other and are described by the Regge trajectories, also referred to as reggeons. Specifically, the pion diffractive dissociation at COMPASS is dominated by exchanges of reggeons with vacuum quantum numbers, including the Pomeron (P). Photon induced reactions at the Jefferson Lab (JLab) may also proceed by exchange of reggeons with nonvacuum quantum numbers. Regge trajectories provide * jannes.nys@ugent.be † hillerbl@uni-mainz.de specific information about the dynamics responsible for the formation of resonances [2][3][4] that can be used to constrain amplitudes of other reactions, e.g. production of light hadrons in heavy flavor hadron decays. In order to confidently separate the beam and target fragmentation in peripheral scattering, it is necessary to establish the validity of Regge pole factorization. Establishing the production mechanism in production of resonances is also the necessary first step in determination of their quantum numbers and other characteristics, as their quark model nature [5]. This is particularly relevant when searching for new states, e.g. hybrid mesons, which is one of the main goals of the spectroscopy program at JLab [6,7]. In the near future new data on peripheral resonance production will be coming primarily from JLab experiments, and therefore it is important to validate Regge mechanisms for beam energies of E γ ∼ O(10 GeV). Even though high energy peripheral processes are expected to be dominated by exchanges of leading Regge poles, there are sub-leading singularities, e.g. Regge cuts and/or poles in daughter trajectories, which have to be assessed [8]. With this goal in mind, we have recently studied πN scattering, and π 0 N , ηN , π∆ and neutral vector meson photoproduction [9][10][11][12][13][14][15], obtaining several results that we briefly summarize below. In π 0 photoproduction, we used finite energy sum rules (FESR's) [13] to demonstrate that there is a good agreement between the partial wave models (PWA) for low energy amplitudes [16][17][18][19][20][21][22][23], and the Regge parametrization of the highenergy data. In η photoproduction, the PWA models are less constrained by the available low-energy data and we have shown how the high-energy data can help reducing uncertainties, specifically those related to unnatural exchanges [11,13]. Our prediction for the π 0 photopro-duction cross section [10] compares favorably with the CLAS data [24], and our results on π 0 , η and η photoproduction beam asymmetries [14,15] are in agreement with the GlueX results [6,25]. The main conclusion one can draw from these comparisons is that, at forward scattering angles, the natural Regge poles dominate over the unnatural ones and over the non-pole contributions. In general, for natural exchanges we have found a good agreement with factorization. Specifically, a zero in the residue of the ω exchange in πN [9] implies a similar behavior for the photoproduction reactions with ω exchange. Indeed, a zero is found in the η photoproduction amplitude and a strong dip is present in the cross section for π 0 photoproduction [11]. Complementary to these analyses, we now study reactions with meson beams sharing the same nucleon residues. For these reactions, the amount of high-energy data is abundant, which allows for a detailed study of both the residue factorizability, and the energy dependence of the observables. Additionally, mesonic beams allow for less exchanges compared to photon beams, therefore allowing us to study the dominant natural exchanges in isolations. For those kinematics where Regge factorization holds, information about the residues can be applied to photoproduction reactions. Since the Regge picture has been well established for πN scattering and π and η ( ) photoproduction off the proton, in this work, we proceed to examine the Regge pole model in a global analysis of several quasi-two-body reactions of interest to peripheral resonance production. When sub-leading contributions, such as Regge cuts or daughters, are accounted for in the amplitudes, the factorization approximation is violated. We aim to identify the kinematics for which such violations can be expected, while for processes dominated by factorizable exchanges we provide amplitudes and residues that are compatible with the world data at high energies. These can be used to model the production mechanism in fragmentation experiments, allowing the isolation of the resonant part intended to search for hybrids.
In the Regge pole approximation, the amplitudes are well constrained by unitarity and analyticity and are specified by a small number of parameters. Thus, in principle, a large enough data set in principle makes it possible to test the Regge pole dominance hypothesis [26]. In this work we perform a global analysis of all available data on charge exchange (CEX) quasi-two-body reactions with meson beams, that are dominated by vector and tensor Regge trajectories. Except for the Pomeron exchange, which does not contribute to CEX reactions, vector and tensor exchanges are expected to dominate in the energy range of interest. Furthermore, we exclude processes in which pion exchange is possible. At high energies, the pion pole is close to the physical region and becomes more sensitive to the sub-leading Regge contributions. This, in general, requires a special treatment [12,27,28]. The data set considered in this paper includes 23 reactions and 1271 differential cross section data points, as described in Section III B, and summarized in Table II.
The paper is organized as follows. We discuss the main features of the formalism in Sections II and III A, leaving the technicalities to the Appendices. The results of the fits are discussed in Section III B and in Section V we summarize the main conclusions. The kinematics and our conventions are discussed in detail in Appendix A. Appendix B contains a summary of the effect of factorization of the Regge residues on the forward behavior of the helicity amplitudes. The interaction Lagrangians used in the fits are contained in Appendix C, with estimates of the corresponding coupling constants derived in Appendix D. Our method for building Regge amplitudes from singleparticle exchange amplitudes is discussed with an example in Appendix E. Finally, Appendix F provides the expressions that allow one to determine the t-channel helicity residues directly from the s-channel residues and vice versa, without the need for introducing Lagrangians.

II. FORMALISM
We consider reactions of the type (see Fig. 2) where the p i 's are the 4-momenta and the µ i 's are the helicities in the center of mass frame, referred to as the s-channel frame. The standard Mandelstam variables are We parametrize the high-energy s-channel helicity amplitudes following the analysis of Cohen-Tannoudji et al. [29] which leads to a factorized form as discussed in [26]. Specifically, in the large s limit the amplitude of a Regge pole (e) described by a trajectory α e (t), which in the schannel physical region is approximated by a linear function, α e (t) = α 0 e + α 1 e t, is given by were ζ e = ±1 is the reggeon signature and s 0 is a scale above which the pole approximation is expected to dominate, s s 0 . The latter is related to the range of the strong interaction. In the following we use s 0 = 1 GeV 2 . We refer to right (wrong) signature points as those kinematics for which the signature factor ζ e + e −iπαe(t) in Eq. (2) is finite (vanishes). Except for kinematic factors, discussed below, Regge theory does not fix the tdependence of the residues β e µ4µ3µ2µ1 (t). Unitarity in the t-channel requires that the trajectory and residues are real functions in the kinematic region of interest, and that the t-channel helicity residues are factorizable in a separate contribution coming from the meson vertex and a contribution from the baryon vertex. It was shown in [26] that in the high-energy limit, due to properties of the crossing matrix, the s-channel helicity residues must also be factorizable. Hence, the residues β e µ4µ3µ2µ1 (t) in Eq. (2) can be written in a product form [30] β e µ4µ3µ2µ1 (t) = β e24 In the following, the β e24 µ2µ4 (t) and β e13 µ1µ3 (t) will be referred to as the bottom and top or meson and nucleon residues respectively. The superscripts indicate which external particles the residue depends upon. This factorized form illustrates the role played by coupled channels, since the residues β eij µiµj (t) in Eq. (3) (where (i, j) = (1, 3) or (i, j) = (2, 4)) are shared among various reactions which involve the same particles.
The function sin πα e (t) in the denominator of Eq. (2) reflects the existence of particle poles in the t-channel with integer spins J given by the value of the trajectory at a pole, J = α e (t = m 2 e ). Regge amplitudes involve particles of definite parity and naturality η = P (−1) J = ±1, which is accounted for by the signature factor (ζ e + e −iπαe(t) )/2. For physical spins, it reduces to (ζ e + (−1) J )/2 and retains only the poles which satisfy ηP = ζ e . Since poles with negative spins are unphysical, the residues must contain additional zeros. For example, zeros in even (odd) signature residues must be present at values of t for which α e (t) is equal to an even (odd) negative integer. In addition α e = 0 in the even signature trajectory may occur at a value of t in the s-channel physical region. For some trajectories, this would correspond to an unphysical exchange of a ghost spin-0 particle. This happens in the a 2 trajectory 1 , where α a2 (t) = 0 corresponds to negative values of t, and therefore an additional zero must be included at α a2 (t) = 0 in the residue of the a 2 exchange. Hadron Regge trajectories are observed to satisfy the so-called exchange degeneracy (EXD), which can be understood in terms of local Regge-resonance duality and large-N c limit [5,31]. As a consequence, residues of the a 2 and ρ exchanges should have the same t-dependence 2 . Together with the requirement for zeros at the negative values of α e with the right signature, the residues of a 2 and ρ must contain zeros at all non-positive spins. While the above reasoning invokes EXD, which holds only approximately, the zeros at α e (t) = 0 are more general, as will be discussed at the end of this Section. Zeros at non-negative opposite signature spins, such as the zero at α ρ (t) = 0, are referred to as wrong signature zeros (WSZ). To include all the expected zeros, as discussed above we require [8,32,33] where l e is the lowest physical spin on the trajectory α e (t), or on the trajectory of its exchange-degenerate partner, i.e. l ρ = l a2 = 1. Since it is not known a priori which one of the two factorizable residues should the zeros be attributed to, we pull this factor out of the product of residues and absorb it in the definition of the remaining terms in Eq. (2). The trajectories are assumed to be linear functions and are given in Table I. Notice that we assume weak degeneracy for all exchanges: α ρ (t) = α a2 (t) and α K * (t) = α K * 2 (t). As discussed in Appendix B, for t → 0 the most singular behavior of residues corresponding to t-channel exchanges of definite parity is given by β eij µiµj (t) ∼ √ −t |µi−µj | . In order to make this kinematic tdependence explicit, we define the reduced residueŝ β µiµj (t), which are regular in t. The general form of the large s amplitude implied by Regge theory is therefore given by where F e (s, t) is defined below. In the phenomenological analysis of the data, we have found that one obtains better fits if the t-dependence implied by the exact angular behavior of the amplitude is used instead of its limit corresponding the high energy scattering in the forward direction, z s → 1 + 2t /s where z s is the cosine of the scattering angle in the s-channel frame, and t = t − t min with t min = t(z s = +1) = O(1/s) [33]. Specifically, from the overall angular-momentum conservation it follows that the s-channel helicity amplitude is proportional to the half-angle factor [34,35] ξ µµ (s, t) = 1 − z s 2 where µ = µ 1 − µ 2 and µ = µ 3 − µ 4 are the net helicity in the initial and final state, respectively. The half-angle factor incorporates the kinematic singularities in t of the s-channel helicity amplitude, and in the forward direction at high energies it reduces to By comparing with the asymptotic limit of the Regge pole expression given by Eq. (6), we obtain where the half-angle factor is fully taken into account. Alternatively we can restore the half angle factor by multiplying the Regge formula (Eq. 6) by a factor R where ν = (s − u)/2. It is worth noting that the small |t| behavior imposed by angular-momentum conservation troduced by the additional requirement of factorization of the residue in Eqs. (5) and (6). We have normalized R in such a way that R → 1 for s → ∞, such that the s-dependence of the helicity amplitude remains s αe(t) , as in Eq. (6). The final functional form for the amplitudes used for the fits reads Regge theory does not predict the full t-dependence and the single-particle model only approximates the tdependence close to the single particle pole. An exponential factor is introduced with a slope parameter and in the following we will implicitly assume the presence of an exponential factor in the residueŝ The function F e (s, t) is often referred to as the Regge propagator, and is given by In the t → m 2 e limit, where m e is the mass of the meson with lowest spin J e on the trajectory α e (t), one recovers the known single-particle propagator 3 This property illustrates that the Regge amplitude in Eq. (11) reduces to the single-particle exchange amplitude near the particle poles. This property constrains the residues near the pole to be numerically close to the phenomenological values from single-particle exchange models [33]. Note that in our approach the reggeons are used to directly construct the s-channel helicity amplitudes. A more natural approach would be to start in the rest frame of the reggeon, i.e. in the t-channel center-of-mass (c.m.) frame and apply crossing relations. In the t-channel frame the relation between the residue zeros and angular momentum conservation is more transparent. Consider for example, the t-channel amplitude ππ → N N with a ρ exchange. A spin-0 exchange is unphysical in the t-channel helicity flip component. Such a contribution is referred to as 'nonsense' [8]. Therefore, a nonsense wrong-signature zero (NWSZ) is often introduced at α ρ = 0 only in the t-channel helicity-flip component, in order to remove such a spurious contribution. For the t-channel non-flip component, the Finite-Energy Sum Rule analysis indeed show a finite residue at the wrongsignature point (α ρ = 0) [9]. However, starting from the t-channel, a global analysis is tedious and less straightforward, as discussed for example in [34]. Indeed, for each channel one must trace the kinematic singularities in t and remove them in order to construct a set of helicity amplitudes which contains only dynamic singularities in t. In contrast, the t-singularities of the s-channel helicity amplitudes are easy to find as they originate entirely from the half-angle factors defined in Eq. (7), see Appendix A. By merely dividing out the half-angle factors, one is able to write down s-channel helicity amplitudes that are free from kinematic singularities in t. For a global phenomenological analysis it is therefore preferential to deal directly with the s-channel amplitudes.

III. CONSTRAINED SU(3)-EXD FIT
The total number of parameters describing Regge residues is substantial. In order to obtain a robust estimate we first carry out a fit to the data in which we impose both SU(3) and exchange degeneracy (EXD). Furthermore, we fix the t-dependence of the reduced residueŝ β eij µiµj (see Eq. 11) using a single-particle exchange model obtained from the effective Lagrangians as discussed in Appendices C and E. We refer to this analysis as the 'global SU(3)-EXD fit', which is similar to the analyses of [32,33]. The results are shown in Section III B. We later relax all these assumptions for a second unconstrained fit, in Section IV.

A. Constraints
To derive the SU(3) couplings, we consider SU(3) relations based on the Lagrangians in Appendix C for the single-particle exchange model. Using the relation between the single-particle and the Regge residues one obtains the SU(3) constraints for the latter, using the residues of the ρ and K * exchanges 4 as input. Since we do not consider reactions dominated by P or π exchange, we do not fit N N cross section data. If we did, it would allow us to constrain the overall normalization of the fit, through the determination ofβ eN N . Instead, we extract the top vertices from resonance decay widths. We fixβ ρπ − π − = 8.4 using the Lagrangian coupling, g V P P = −4.2, as discussed in Appendix D. Note that the sign is chosen such that the contributions to the total cross sections match correctly. While SU(3) allows one to relate the various Regge residues of vector exchanges, it does not relate these to residues of tensor exchanges (a 2 and K * 2 ). The a 2 and K * 2 couplings for the global SU(3)-EXD fit are obtained by demanding EXD for the helicity residues. In summary, we use the following relations obtained from duality arguments [5] 4 We denote reggeons by their lowest spin particles. Also, K * ≡ K * (892) and K * 2 ≡ K * 2 (1430).
for any helicity combination. Since K + p → K + p and K + n → K + n are exotic in the s-channel, duality requires thatβ ρK + K +β ρpp = −β a2K + K +β a2pp , since the propagator is normalized according to Eq. (14) (and similarly for f and ω exchanges). Note that exact EXD is not necessarily fulfilled within the single-particle model. In particular, EXD requires the residues of two exchanges to be equal for any t, which might not be possible, especially when only a subset of the interaction Lagrangians is considered, as is often the case in the literature. Therefore, an 'EXD propagator' (which amounts to adding or subtracting the propagators of EXD contributions) is usually introduced to circumvent this issue, while strong EXD is in fact a property of the residues. In the latter approach, one effectively assumes that both exchanges have the single-particle residue of the lowest spin exchange. It is worth noting that EXD is sometimes used incorrectly. It is interpreted as a property of a single reggeon, while it is in fact a relation between different Regge pole contributions. Therefore, the EXD propagator can only be used in reactions where both EXD partners are allowed. For instance, there is no strong EXD for the residues β eγπ µγ µπ (t) of π photoproduction reactions, since there is no exotic s-channel reaction containing these residues.
In the global SU(3)-EXD fit, we opt to introduce a single exponential damping factor for each independent helicity configuration b nf , b sf , with nf and sf referring to helicity non-flip and single flip amplitudes. These are fitted to the data. In this fit we set the double-flip amplitudes to zero.

B. Results
In the following, we present results of the global analysis of the high-momentum (p lab ≥ 5 GeV) differential cross section data for a large number of reactions, including reactions with π and K beams that are dominated by ρ, a 2 , K * and K * 2 exchanges 5 . Hereby, we consider the channels with strangeness and charge exchange (CEX). The data used in this analysis are listed in Table II.
The Γ function in Eq. (4) introduces zeros in the cross sections for reactions dominated by the ρ exchange, e.g. π − p → π 0 n. In this case, daughter poles or other subleading singularities become relevant as they tend to fill in the zeros [33,37]. In the SU(3)-EXD we focus only on the leading natural Regge poles and do not include any subleading Regge contribution. Instead of introducing new amplitude components to fill in the dips, we prefer to work with the well defined Regge pole model. Therefore, we must reduce the contribution from dip regions in ρ-only dominated channels in Figs. 4 and 8a, by rescaling the data error bars using a factor f (t) = 1/|t − t 0 |,  (3)-EXD fit (α(1)) and as obtained from the unconstrained fit (α(2)). The Mandelstam variable t must be expressed in units GeV 2 .
where t 0 is defined such that α ρ (t 0 ) = 0. Such a rescaling is required in order for the fit to be less sensitive to kinematic regions dominated by components beyond the leading pole model. The goal of this work is to detect the regions where the Regge pole approximation holds best, by manually varying the kinematic regions. Additionally, the available data is subject to sizable and uncontrolled systematic errors. Therefore, we aim to provide a qualitative description, and do not report the χ 2 values and errors of the fits. The global SU(3)-EXD fit contains nine free parameters, which are given in Table III. The other three are the η − η mixing angle θ P and the single-and non-helicityflip exponential factors b sf and b nf , respectively. In total, N data = 1271 high-energy forward scattering data points are fitted, with p lab ≥ 5 GeV and 0 ≤ −t ≤ 0.8 GeV 2 . For the total cross sections, we consider data at slightly higher energies, p lab ≥ 10 GeV, since the Regge-based model in [38] matches best above this energy. The parameters of the natural Regge trajectories are fixed to the values given in Table I. Since there are too many amplitudes describing the production of spin 3/2 baryons compared to the available data, additional constraints are needed. Specifically, we keep only a single term in the interaction Lagrangian of vector-meson-octet-baryon-decupletbaryon couplings (VBD), i.e. we set g V BD = 0 in Eq. (C19) for all VBD combinations. More details are given in Appendix C. Note that this sets all double flip components to zero, in agreement with the data. Additionally, it is worth mentioning that the reduced residue of the non-flip component is proportional to t in the single-meson form in Table VII. Even though this is not required by factorization and angular-momentum conservation, the contribution is therefore required to vanish at t = 0 in the single-meson exchange approximation.
The comparison between the data and the model is shown in Figs. 3-13. The global SU(3)-EXD fit is stable and provides a remarkably good description of all the key features in the data. In Table IV we list all exchanges that contribute to the reactions we have analyzed and values of the residues derived from the fit. We do not take into account the exponential factor in Eq. (12) when we extrapolate the residues to the pole, as it is expected II: Available differential cross section data and references.
The contribution from the individual Regge poles to the elastic amplitudes in Eq. (16) is given by Hence, defining the following linear combinations, one can investigate the individual contributions from the ρ and a 2 exchanges. In the case of exact SU (3) and EXD, all three cross section combinations must be equal.
Inspecting the results shown in Fig. 3, one sees that σ ρ (KN ) and σ a2 (KN ) differ by a few mb only, indicating a small violation of exchange degeneracy. The compatibility of σ ρ (KN ) and σ ρ (πN ) illustrates that the SU (3) symmetry is well respected in KN and πN scattering. The Γ function in Eq. (13) is introduced to remove spurious spin exchange components. The spin 0 exchange term in the a 2 trajectory at α a2 (t) = 0 corresponds to an unphysical pole with negative mass squared. As discussed earlier, due to EXD, a zero in the residue for a 2 exchange forces a zero in the residue of the ρ exchange, even though the Regge propagator for the ρ does not contain a pole at α ρ (t) = 0. The residue zero results in the vanishing of the cross section, which can be observed in all reactions where ρ exchange dominates, i.e. in πN → πN and πN → π∆ in Fig. 4.
The SU(3) predictions for η ( ) production are depicted in Figs. 5a and 5b, and an overlay of both reactions for p lab = 40 GeV is given in Fig. 5c. The comparison between η and η fits can be used to extract information on the pseudoscalar mixing. Using SU(3) constraints, one estimates the relative couplings as (see Appendix C) As discussed in Appendix C, the Okubo-Zweig-Iizuka (OZI) ss suppression rule requires that the relative coupling of the singlet η 1 and octet η 8 components is given by S T = √ 2. Hence, for mixing angles θ P ≈ −0.17 and under the OZI assumption (S T = √ 2), one finds g a2πη ≈ g a2πη . The exact size of the pseudoscalar mixing angle θ P is unknown, but the various theoretical estimates suggest values in the range −0.38 θ P −0.17 [76][77][78][79][80][81].
The fit results in a good correspondence to KN → KN and KN → K∆ CEX in Figs. 6 and 7. In these reactions, both ρ and a 2 contribute. While the ρ dominates the very forward region, the a 2 exchange fills up the dip of the ρ in the neighborhood of α ρ (t) = 0.
For ωN production at very high energies in Fig. 8a, a fit with only ρ exchange shows the correct s-dependence over a wide energy range. The data also clearly show a dipping behavior near α ρ (t) = 0, as expected for a pure ρ contribution. The data on ω production in Fig. 8 are dominated by ρ exchange at very high energies p lab ≥ 100 GeV, since in the forward direction the b 1 exchange contribution is suppressed by a factor of s −1/2 relative to the natural exchange. In the fits, we wish to determine the residues of only the leading Regge poles that are constrained by multiple channels. Therefore, we consider only p lab ≥ 100 GeV to isolate the ρ component, and neglect the b 1 exchange. The data on ω∆ production are rather scarce and at energies sensitive to the b 1 exchange (see Fig. 9). Therefore, we do not consider them in the global SU(3)-CEX fit.
Finally we consider strangeness exchange Figs. 10-13. The effective trajectories 6 obtained in [70] from Λ and Σ 0 production data, are much flatter than the ones compatible with α K * (t = m 2 K * ) = 1 and α K * (t = m 2 K * 2 ) = 2 used in this work. They obtain α 0 eff = 0.32 and α 1 eff = 0.23 − 0.43 GeV −2 . This disagreement indicates that secondary contributions (such as additional poles) are present at higher −t. The global fit indeed does not reproduce the high-|t| region. However, a very good agreement is found in the very forward region. This kinematic domain follows the s-dependence compatible with our trajectories in Table I. For both Λ and Σ production, an inconsistency is observed for the time-reversal related reactions in Figs. 10 and 11 respectively. The mismatch is related to the time-reversal symmetry of the β K * (2) πK (t) vertex imposed in the fit [33]. The latter requires β K * (2) πK (t) = −β K * (2) Kπ (t), while the bottom vertex remains the same. Under the assumption of EXD, the cross section of the time-reversed reactions are therefore expected to be the same. The SU(3) relations force the Λ (and to a lesser extent Σ) production to be dominated by helicity nonflip contributions in the forward direction 7 . Note that the non-flip contributions respect the EXD and SU (3) relations very well. 6 The effective trajectories are obtained by fitting α eff (t) to the s-dependence at fixed t, where f (t) is a fitting parameter which may be different for all t.

IV. UNCONSTRAINED FIT
In the unconstrained fit, we relax the EXD and SU(3) constraint and only keep SU(2), isospin, as a good symmetry. We also fit the Regge trajectory parameters of the ρ and a 2 exchange. In this fit, we can keep the singleparticle approximation for the residues and fit the couplings in Table VII (without their SU(3) decomposition) for both vector and tensor exchanges. Indeed, when the EXD constraint is removed, the residues of the tensor exchanges must be determined independently. However, this approach is quite cumbersome. Additionally, this approach forces the residues to be restricted to the singleparticle exchange residue. The tensor poles are quite far away from the physical region and it can no longer be expected that this approximation is reliable in the physical region. Therefore, in our next fit the t-dependence of the residues is no longer constrained by the single-particle model and we use Eq. (20) instead. Hereby, the reduced residues are parametrized aŝ where the constants g eif µiµ f and b eif µiµ f are all fitted independently, unless stated otherwise.
Abandoning the strict connection with the particle exchange model also implies, for example, that the non-flip component of the V BD coupling is no longer required to vanish at t = 0 (seeβ V BD ∝ t for the single-particle residue in Table VII). All this significantly increases the number of parameters from 9 to 110, and we fit them in steps including a few reactions at the time. The tdependence in our fit is now entirely absorbed into the exponential factor. Next, we describe the step-wise fitting process. We take advantage of the fact that a given exchange is related to a limited set of reactions. In the first step, we determine the trajectory intercepts, α 0 ρ , α 0 a2 , and the parameters of the residues,β ρN N ++ ,β a2N N ++ ,β ρN N −+ ,β a2N N −+ ,β ρKK , β a2KK ,β ρπη ( ) andβ a2πη ( ) cf. Eq. (12). The slopes α 1 ρ (α 1 a2 ) follow from the requirement that α ρ (t = m 2 ρ ) = 1 (α a2 (t = m 2 a2 ) = 2). The intercepts are allowed to vary in the range 0.4 ≤ α 0 (ρ,a2) ≤ 0.55. The results of these fits are depicted in Figs. 3, 4a, 5, and 6 Assuming S T = √ 2, we obtain the mixing angle θ P = −0.33. This angle is compatible with the values found in the recent literature [80,81]. Inspecting the results in Fig. 5a one finds that the cross section rises rapidly for −t ≥ 0.8 GeV 2 and cannot be described within the pure Regge-pole picture. Therefore, this kinematic domain is excluded in the channels that follow.
With the couplings at the top vertex determined through the residues listed above, we proceed to determine the bottom couplings, i.e.β ρN ∆ µ N µ∆ andβ a2N ∆ µ N µ∆ , from a combined fit to πN → η∆, πN → π∆ and KN → K∆ cross sections. The large number of helicity couplings leads to a rather unconstrained fit and thus, based on the result of the SU(3)-EXD fit, we eliminate the double-flip components. Furthermore, we keep the ratio of the two single-flip components g eN ∆ (e = ρ, a 2 ) fixed to the value obtained from the SU(3)-EXD fit. Their exponential t-dependence is assumed to be the same and is fitted to the data. The t-dependence of the non-flip components is fixed to the SU(3)-EXD values. The results of the fit are depicted in Figs. 4b, and 7. Relaxing the conditionβ ρN ∆ + 1 2 + 1 2 (t = 0) = 0 imposed by the singleparticle exchange correspondence in Table VII seems to slightly improve the fit at forward angles. This effect is even clearer for Σ * production channels in Fig. 13.
Using theβ ρN N couplings extracted in the previous fitting steps, one can now determine theβ ρπω residue from a fit to the ωN production data at very high energies. At forward angles and low energies the b 1 exchange does not represent the full strength required to reproduce the cross section of ω production. Notice that the NWSZ at α b1 (t) = 0 would force the cross section to vanish near t ≈ 0. This mismatch is typically associated with the existence of a trajectory with quantum numbers I Gζη = 1 ++− , with the lowest spin meson located on the trajectory being the yet undiscovered ρ 2 (I G J P C = 1 + 2 −− ) [33,36] 8 Due to its positive signature, this contribution is not required to vanish at α ρ2 (t) = 0, and might be even more important at forward angles than the b 1 exchange, provided that α ρ2 (t) is similar to α b1 (t); note however that α ρ2 (t) is undetermined, as pointed out in [11]. This is because the NWSZ of the b 1 lies at t = 0 which forces the b 1 contribution to vanish in the forward direction, independent of the factors in Eq. (5). This lack of strength in our model in the forward direction hinders an unambiguous extraction of the b 1 couplings. While we do include a contribution from the b 1 exchange to absorb the different energy dependence at lower energies, we do not quote the results for the latter, since these couplings are unreliable.
For the strangeness exchange channels the fits are somewhat more difficult since one cannot separate the K * from K * 2 exchanges due to the lack of definite G parity. Additionally, there is less sensitivity to the trajectory parameters due to a limited energy range in the data. We therefore keep the trajectories fixed to the ones used in the global SU(3)-EXD fit. Additionally, the EXD constraint is imposed on the fits to reduce the number of free parameters. In the global SU(3)-EXD fits shown in Figs. 10a-11c, and Table VII We can therefore carry out two-step fits, where we first determine the non-flip coupling from these very forward data. In a second step, we fix the non-flip coupling and extract the flip contribution from a full t range fit. One observes that the helicity-flip couplings do not obey the SU(3) constraints well, in contrast to the non-flip contributions. Obtaining the helicity flip contributions from the fit turns out to be ambiguous. They depend strongly on the considered t range and the final results deviate heavily from the SU(3)-EXD predictions. This issue was anticipated in the previous section, where we commented (based on the s-dependence) that the large −t behavior of the cross section is dominated by contributions other than the Regge poles considered here.
Theβ K * (2) Kη ( ) couplings are determined in a separate fit using the η ( ) Λ production data in Figs. 12a-12b. As mentioned before, the SU(3) constraint does not hold well for these couplings.
For the Σ * production channels in Figs. 13a-13b, we fix theβ coupling constants and the exponential t-dependence to the ones obtained in the global SU(3)-EXD fit. EXD is not imposed for the remaining coupling constants. The SU(3)-EXD fit already provided a reliable representation of these channels. The main difference to the unconstrained fit is the forward behavior of the cross section. Indeed, as discussed before, the single-meson exchange approximation forces the non-flip component to vanish at t = 0. Therefore, no amplitude survives in the forward direction. In the unconstrained fit, the nonflip component is allowed to contribute in the forward direction. This feature seems to be favored by the data.

V. CONCLUSIONS
We assessed the applicability of the Regge pole model by performing a global fit to charge and strange exchange quasi-two-body reactions at large momenta p lab ≥ 5 GeV. We have found that the Regge pole model provides a good description of the data for a large amount of channels, while requiring only a small number of free SU(3) and EXD related parameters. It was shown that the inclusion of these constraints offers a solid way to reduce the number of free parameters of the fit. The large number of free parameters in the unconstrained fit allows for too much freedom compared with the number of available data to yield a unique result. SU(3) and EXD constraints are especially useful to determine the relevant regions of the vast parameter space of the residues. In kinematic domains where wrong-signature zeros can be expected (such as in channels dominated by ρ exchange), secondary contributions become relevant. For these channels, we find the single pole model and factorization to work well in the domain 0 ≤ −t ≤ 0.5 GeV 2 . The cross sections of reactions dominated by a 2 exchanges are well reproduced by the Regge pole models up to t ∼ −0.8 GeV 2 , due to the lack of a dip. Reactions dominated by strangeness ex- . 0 * 0 * changes follow the Regge pole model remarkably well in the forward region up to t ∼ −0.6 GeV 2 . Especially those channels dominated by non-flip baryon vertices are in good agreement with the constraints imposed by SU (3), which should therefore be considered in future fits. These are the kinematic domains were factorization can be used as a reliable approximation to model beam-target fragmentation. The presented model provides a solid description of the production mechanism needed to describe the production amplitude in peripheral resonance production, of relevance to hybrid searches. Our predictions and our model will be made available online on the JPAC website [86,87]. With the online version of the model, users have the possibility to vary the model parameters and generate the observables. . 0 * 0 *

ACKNOWLEDGMENTS
We dedicate this work to the memory of Mike Pennington. He was an example of humanity and scientific dedication whose efforts and support made the JPAC Collaboration possible. This work was supported by Re-            In the s-channel center-of-mass (c.m.) frame, the cosine of the scattering angle for 1 + 2 → 3 + 4 is given by The conventions are illustrated in Fig. 2. Introducing t min (s) ≡ t(z s = +1) and t = t − t min , the above can be simplified or in other words 1 − z s ∼ −t . At leading s, t = t . Furthermore, where (i, j) is a s-channel pair.
The expressions for the helicity amplitudes in terms of the covariant couplings are obtained by properly selecting the polarization angles. We use the 'particle 2' convention of Jacob and Wick [88], which requires that for the helicity states of the two particles in a two-particle s-channel helicity state lim p→0 − p, −µ| p, µ = 1. (A6) Since the baryons in our convention are a 'particle 2' in the s-channel helicity pairs, one must use the following conventions where χ (2) (θ, φ) is a 'particle 2' Pauli spinor, i.e. the fermion (in a two-particle pair) goes opposite to the direction determined by (θ, φ). For a 'particle 1' Pauli spinor we have Using the above definition, the 'particle 2' Pauli spinor is defined as where ω = µ 1 − µ 3 and ω = µ 2 − µ 4 . The above is the most singular kinematic behavior of the amplitude, and cannot be cast into the factorizable form in Eq. (3). One can show [89] that the simplest factorizable form consistent with a definite parity exchange is given by In Eq. (11), t is used explicitly for the factors stemming from the half-angle factors in Eq. (B1).

Appendix C: Interaction Lagrangians
In order to relate the helicity couplings to those of modern literature, we start from the set of effective Lagrangians given below. We consider interactions of pseudoscalars (P), vectors (V), axial vectors (A), tensors (T), octet baryons (B) and decuplet baryons (D).
We use the following conventions in the global SU(3) fit. The explicit form of the matrices for pseudoscalar, vector and tensor mesons is Here, θ T and θ P are the tensor and pseudoscalar mixing angles, respectively, between singlet and octet. Furthermore, we call θ I P the ideal mixing angle with sin θ I P = 1/ √ 3 and cos θ I P = 2/3. In the case of ideal mixing for vector mesons, the following relations hold: Listing of the reduced s-channel residues and their expressions in the single-particle exchange formalism. The expressions are given up to the flavor trace. We assumed M S = (m i + m f )/2. In practice, these couplings are evaluated on the mass pole of the exchanged particle. Only the lowest order tdependence, which is compatible with factorization, is used in the residue. Every residue contains an additional √ s 0 Je , where J e is the spin of the lightest particle on the trajectory.
β T P P 00 gT P P /2 β T BB Note that we have implicitly assumed a nonet symmetry for the pseudoscalar mesons in Eq. (C1), where the singlet/octet coupling ratio is S T = √ 2 [32]. The latter corresponds to neglecting the coupling to ss content. In the following, the trace is taken over flavor space, corresponding to the isospin couplings of the different channels in SU(3) symmetry.
The three-meson interaction Lagrangians consist only of a symmetric coupling due to G-parity conservation. All the couplings appearing in the following are fitting parameters, obtaining values as explained in App. D. The Lagrangians describing the couplings of tensor mesons to pseudoscalar and vector mesons are given by [90][91][92] L T P P = c 8 Concerning the couplings of vector to pseudoscalar mesons, the Lagrangians read The Lagrangian that couples vector, pseudoscalar and axial mesons b 1 is given by (we do not consider SU(3) relations for the b 1 exchange, since it is the only axial exchange considered here) [93,94].
Since one can couple two octets to both a symmetric and an antisymmetric octet, each meson-B-B vertex must be decomposed into two irreducible structures, with independent couplings. The couplings of the vectormeson fields V µ with momentum q to the octet baryons are described by the following Lagrangian [95] where M S = (m B1 + m B2 )/2, and m B1 and m B2 are the masses of the incoming and the outgoing baryon, respectively. The octet-baryon matrix is explicitly given as In the following, for simplicity we drop the notation with traces over commutators/anticommutators in flavor space. The couplings of tensor mesons f µν to the baryons are given in Refs. [96][97][98][99][100]. We follow the general definitions of the Lagrangian where the SU(3) couplings are given by the usual traces in flavor space, analogously to those in Eq. (C15). An axial vector couples to octet baryons via The explicit expressions for the flavor couplings are anal-ogous to those in Eq. (C15). Care must be taken in Eq. (C18), since only one of the couplings is allowed for a definite G-parity state of the axial meson A. Because of G-parity considerations, the vector coupling g v ABB does not contribute to b 1 exchanges, and the tensor coupling g t ABB is blocked for a 1 exchanges. The Lagrangians that describe the octet-to-decuplet transitions via a vector-meson and a tensor-meson emission are [101,102] where the explicit form of the fully symmetric decuplet matrix elements is Note that in the literature one often does not consider the minimal complete set of interaction Lagrangians when spin-3/2 baryons or tensor mesons are involved. In these analyses, one rather selects a single interaction term, which is usually unjustified. When consideringβ V BD Table VII, one observes that for g (2) V BD = g V BD = 0 and m 2 = m 4 , and neglecting t-dependent terms in β V BD µ V µ D (t), we reproduce the quark-model results in [33]: 3 and β V BD Based on this correspondence, we set g V BD = g V BD = 0 in the global SU(3)-EXD fit.
Appendix D: Estimating coupling constants from decay ratios The coupling constants that appear in the meson Lagrangians can be estimated based on measured decay ratios. These couplings serve as starting values for the fits.
From the data on decay widths of tensor mesons into pseudoscalars, one can extract the numerical values for the couplings c 8 T P P and c 1 T P P . When doing so in a global fit, Giacosa et al. [92] obtained 9 c 8 T P P ≈ 11.7 GeV −1 , c 1 T P P ≈ 13.6 GeV −1 .
The decay width of the ρ into two pions is Γ ρ→ππ = (149.1 ± 0.8) MeV [38]. Comparing this with the expression for the decay width in terms of the Lagrangian couplings, one finds g V P P = ±4.2. For completeness, it is worth mentioning that g V P P = ±4.5, when extracted from the K * → Kπ decay width.
Estimating the coupling g V V P is less straightforward. The decay ratio of the φ meson into the ωπ 0 channel is consistent with 0, both from theory and from experiment, while the only other channel measured so far is the decay of the φ into 3 pions, which can occur via the intermediate channel φ → ρπ. If one were to assume ideal mixing for the vector mesons, this coupling would vanish as well in the OZI limit. Therefore, for this particular estimate, we use the vector-meson mixing angle θ V = 39 o [92]. Assuming that all 3-pion decays of the φ happen via the πρ intermediate channel, one then obtains g V V P = 13.2 GeV −1 . Note that this is to be seen only as a starting value for the fits, since for this particular value many assumptions had to be made, which can give only a rough estimate of the coupling value.
To estimate g T V P , we use the information that the decay a 2 → 3π with decay width 105 MeV occurs dominantly through the ρπ intermediate state. Assuming only this intermediate state, one finds a partial width of 73.5 MeV. This leads to g T V P ≈ 6.8 GeV −2 . In fact, the coupling ranges between 6.4 GeV −2 and 6.8 GeV −2 , when the decay fraction into a ρπ intermediate state is varied between 88% and 100%. This is consistent with the results from K * 2 → πK * , Kρ and Kω, where the estimated g T V P ranges between 6.4 GeV −2 and 7.5 GeV −2 .
In order to estimate g V P A , we use the total decay width (142 ± 9) MeV of the b 1 . It dominantly decays into ωπ, leading to g V P A ≈ 4.0 GeV. All of the above-mentioned estimates are in good agreement with the quark-model predictions from [82].
One can extract the meson-baryon couplings using Eq. (C15) and (i = v, t) and relating them to the empirical couplings from nucleon-nucleon scattering data. The empirical results of the nucleon-nucleon Bonn potential from Refs. [103,104] read Furthermore, from [105], one finds Finally, one obtains The coupling g (2) T BB in Eq. (C17) is often estimated to be compatible with 0 when tensor-meson dominance (TMD) is assumed. One might question the validity of the TMD approach. Indeed, in [96] the authors state that g (1) T BB ≈ −g (2) T BB .

Appendix E: Covariant and helicity amplitudes
From the interaction Lagrangians in Appendix C, one determines the helicity amplitudes by choosing the appropriate polarization angles. Since we consider highenergy scattering, the amplitudes are expanded in powers of s and only the leading term is used. The s-channel helicity amplitudes are then cast onto the factorized form which coincides with the limit of our high-energy amplitudes in Eq. (6) for t → m 2 e . We introduced the convenient notation for the meson propagator From the analysis of kinematical singularities, we showed that the s-channel residues β eij µiµj (t) of an evasive reggeon must at least go as β eij µiµj (t) ∼ √ −t |µi−µj | . In order to unambiguously factorize our amplitudes, we first consider ππ, πN and N N scattering with a t-channel vector exchange V . One obtains the asymptotic expressions (up to isospin factors) From the above, we obtain A similar approach is followed for all reactions under consideration in this work. All residues considered in this work are listed in Table VII. The SU(3) flavor traces and commutators appear as factors then multiplying these residues.
As an example for how to build an amplitude inspired by single-particle-exchange, one can consider the helicityflip contribution to the process π − p → π 0 n. The full SU(3)-constrained amplitude at high energies is given by Here,β where the factor in front of the brackets is obtained from Table VII. The terms between brackets are obtained by working out the flavor traces.
In the unconstrained fit, we have since we have made the arbitrary choice of fitting the coupling constants g ρπ + π + and β ρpp −+ and relating all other charge states to these couplings.
The differential cross section is computed using Appendix F: t-channel decay couplings In various single channel analyses, one starts from the t-channel to model the reggeon contributions. Hereby, tchannel helicity couplings (denoted by γ λiλ k (t)) are used. The s-channel residues can be related to the t-channel residues by analytically continuing the s-channel amplitudes to the t-channel. To distinguish between the he-licity amplitudes in both channels we explicitly mention the channel in a subscript. Additionally, we use µ and λ for s-and t-channel helicities, respectively.
Consider the s-channel amplitude for an exchange e in Eq. (6). Near the lowest mass pole we have the simple form for the s-channel amplitude Note that in the above, the residues include the √ −t factors, which must also be evaluated at t = m 2 e . In the t-channel a + c → b + d, the helicity amplitude can be expanded in partial waves through A t λ4λ2λ3λ1 (s, t) = 16π where λ = λ 13 = λ 1 − λ 3 and λ = λ 24 = λ 2 − λ 4 and M t = max{|λ|, |λ |}. The z t is the cosine of the t-channel c.m. scattering angle. For a resonance e with spin J e , the corresponding partial-wave amplitude is parametrized as In the following, we relate the s-channel residues β in Eq. (F1) to the t-channel residues γ in Eq. (F3), without invoking Lagrangians to carry out the crossing (which is the inverse direction of the derivation by Fox and Hey [32]). We continue the s-channel helicity amplitudes into the t-channel and project them onto the t-channel helicity basis [106] A t λ4λ2λ3λ1 (s, t) = −i where the t → s-channel rotation angles are given by Here, φ is the Kibble function, c i = 1 (= 0) if particle i is (not) crossed, j is the s-channel and k the t-channel pair particle of i. Note that, following the path of Trueman and Wick [107], the square roots in the s-channel residues must be evaluated at t = t − i , since we cross the real s, t plane at negative t. This means that √ −t = i √ t. Note that we do not include a helicity dependent phase in Eq. (F4). This is in agreement with Trueman and Wick [107]. At leading s, we find that .
The rotation matrix has the property of being factorizable in a top-vertex and bottom-vertex rotation. If we assume for the moment that the left-hand side of Eq. (F4) can also be factorized then all of the above can be written in a factorized form where (i, k) = (1, 3) or (2,4). An additional factor of i must be included for the fermion vertex. In order to write Eq. (F2) in a factorized form, we realize that we are working in the high s limit, where z t is large.
The d-functions become factorizable when only their leading order in z t is considered A t λ4λ2λ3λ1 (s, t) = 16π where d J λ (z) ≡ e +iπλ/2 z 2 J Γ(2J + 1) Γ(J + |λ| + 1)Γ(J − |λ| + 1) Note that the above form is not fully factorized in the sense that the z t depends on both top and bottom par-ticle masses. We therefore consider the leading s form of z t at constant t z t = 2ts T 13 (t)T 24 (t) , and introduce the functions h J λ ik (z t ) ≡ e +iπλ ik /2 Γ(2J + 1) Γ(J + |λ ik | + 1)Γ(J − |λ ik | + 1) Close to the m e pole in t, the partial-wave expansion is dominated by the J e partial wave. In Eq. (F3) we have assumed a factorizable form for the t-channel partial-wave amplitude. Therefore, we can rewrite the partial-wave amplitude as where A t λ k λi,Je (t) = γ λiλ k (t) 1 t − m 2 e . (F17) The above can then be written in the factorized form where ξ(λ ik ) = e −iπλ ik for a bottom vertex, else ξ(λ ik ) = 1.
In summary, Eqs. (F22) and (F23) relate the s-channel residues at the pole directly to the t-channel residues.
With these expressions, one can compare the results obtained in this work directly to the decay couplings of various processes.