The new generation of high statistics experiments e.g., Belle II, BESIII, CLAS12, CMS, COMPASS, GlueX, J-PARC, LHCb, and P¯ANDA, 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. Other high-energy pQCD frameworks used in exclusive reactions, such as GPDs, meson DAs, …, require a hard interaction scale for factorization to hold [2]. Regge theory, on the other hand, is applicable for large center-of-mass energies (s), forward scattering angles (small -t), small meson masses, and, most importantly, the lack of a hard scale. 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 specific information about the dynamics responsible for the formation of resonances [3–5] 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 the production of resonances is also a necessary first step in the determination of their quantum numbers and other characteristics, as their quark model nature [6]. 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 [7,8].

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 Eγ∼O(10  GeV). Even though high energy peripheral processes are expected to be dominated by exchanges of leading Regge poles, there are subleading singularities, e.g., Regge cuts and/or poles in daughter trajectories, which have to be assessed [9]. With this goal in mind, we have recently studied πN scattering and π0N, ηN, πΔ, and neutral vector meson photoproduction [10–16], obtaining several results that we briefly summarize below. In π0 photoproduction, we used finite energy sum rules (FESR’s) [14] to demonstrate that there is a good agreement between the partial wave models (PWA) for low energy amplitudes [17–24] and the Regge parametrization of the high-energy 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 reduce uncertainties, specifically those related to unnatural exchanges [12,14]. Our prediction for the π0 photoproduction cross section [11] compares favorably with the CLAS data [25], and our results on π0, η, and η′ photoproduction beam asymmetries [15,16] are in agreement with the GlueX results [7,26]. 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 nonpole 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 [10] 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 π0 photoproduction [12]. 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 subleading 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 [27]. 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 subleading Regge contributions. This, in general, requires special treatment [13,28,29]. The data set considered in this paper includes 23 reactions and 1271 differential cross section data points, as described in Sec. III B, and summarized in Table II.

The paper is organized as follows. We discuss the main features of the formalism in Sec. II and Sec. III A, leaving the technicalities to the Appendices. The results of the fits are discussed in Sec. III B, and in Sec. 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 single-particle 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.

We consider reactions of the type (see Fig. 2) 1(p1,μ1)+2(p2,μ2)→3(p3,μ3)+4(p4,μ4),(1)where the pi’s are the 4-momenta, and the μi’s are the helicities in the center of mass frame, referred to as the s-channel frame. The standard Mandelstam variables are s=(p1+p2)2, t=(p1-p3)2, and u=(p1-p4)2. We parametrize the high-energy s-channel helicity amplitudes following the analysis of Cohen-Tannoudji et al. [30], which leads to a factorized form as discussed in [27]. Specifically, in the large s limit the amplitude of a Regge pole (e) described by a trajectory αe(t), which in the s-channel physical region is approximated by a linear function, αe(t)=αe0+αe1t, is given by Aμ4μ3μ2μ1(s,t)=-παe12βμ4μ3μ2μ1e(t)×ζe+e-iπαe(t)sinπαe(t)(ss0)αe(t),(2)where ζe=±1 is the reggeon signature, and s0 is a scale above which the pole approximation is expected to dominate, s≳s0. The latter is related to the range of the strong interaction. In the following, we use s0=1  GeV2. Notice that fixing s0 is a convention. Changing the value of s0 introduces solely a different t-dependence, which we absorb into the residue. We refer to right (wrong) signature points as those kinematics for which the signature factor ζe+e-iπαe(t) in Eq. (2) is finite (vanishes). Except for kinematic factors, discussed below, Regge theory does not fix the t-dependence of the residues βμ4μ3μ2μ1e(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 [27] 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 βμ4μ3μ2μ1e(t) in Eq. (2) can be written in a product form [31] βμ4μ3μ2μ1e(t)=βμ2μ4e24(t)βμ1μ3e13(t).(3)In the following, the βμ2μ4e24(t) and βμ1μ3e13(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 βμiμjeij(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 sinπαe(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=αe(t=me2). Regge amplitudes involve particles of definite parity and naturality η=P(-1)J=±1, which is accounted for by the signature factor (ζe+e-iπαe(t))/2. For physical spins, it reduces to (ζe+(-1)J)/2 and retains only the poles which satisfy ηP=ζe. 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 αe(t) is equal to an even (odd) negative integer. In addition, αe=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 a2 trajectory,¹

We denote reggeons by their lowest spin particles. Also, K*≡K*(892) and K2*≡K2*(1430).

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 βμiμjeij(t)∼-t|μi-μj|. In order to make this kinematic t-dependence explicit, we define the reduced residues β^μiμj(t), βμiμjeij(t)=-t|μi-μj|β^μiμjeij(t),(5)which are regular in t. The general form of the large s amplitude implied by Regge theory is therefore given by Aμ4μ3μ2μ1(s≫s0,t)=-t|μ1-μ3|-t|μ2-μ4|×β^μ1μ3e13(t)β^μ2μ4e24(t)Fe(s,t),(6)where Fe(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 to the high-energy scattering in the forward direction, zs→1+2t′/s, where zs is the cosine of the scattering angle in the s-channel frame, and t′=t-tmin with tmin=t(zs=+1)=O(1/s) [34]. Specifically, from the overall angular-momentum conservation it follows that the s-channel helicity amplitude is proportional to the half-angle factor [35,36] ξμμ′(s,t)=(1-zs2)|μ-μ′|/2(1+zs2)|μ+μ′|/2,(7)where μ=μ1-μ2 and μ′=μ3-μ4 are the net helicity in the initial and final states, 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 ξμμ′(s,t)→s→∞-ts|μ-μ′|.(8)By comparing with the asymptotic limit of the Regge pole expression given by Eq. (6), we obtain Aμ4μ3μ2μ1=ξμμ′(s,t)-ts|μ-μ′|×[-t|μ1-μ3|-t|μ2-μ4|β^μ1μ3e13(t)β^μ2μ4e24(t)Fe(s,t)],(9)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 R(s,t)≡(1-zs2ν-t)12|μ-μ′|(1+zs2)12|μ+μ′|,(10)where ν=(s-u)/2. It is worth noting that the small |t| behavior imposed by angular-momentum conservation βμ4μ3μ2μ1e(t)∝-t|(μ1-μ3)-(μ2-μ4)| in Eq. (8) is weaker than the behavior βμ4μ3μ2μ1e(t)∝-t|μ1-μ3|+|μ2-μ4| introduced 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 sαe(t), as in Eq. (6). The final functional form for the amplitudes used for the fits reads Aμ4μ3μ2μ1(s,t)=R(s,t)-t|μ1-μ3|-t|μ2-μ4|×β^μ1μ3e13(t)β^μ2μ4e24(t)Fe(s,t).(11)Regge theory does not predict the full t-dependence, and the single-particle model only approximates the t-dependence 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 residues β^μiμjeij(t)∝ebμiμjeijt.(12)

The function Fe(s,t) is often referred to as the Regge propagator and is given by Fe(s,t)=-ζeπαe1Γ(αe(t)-le+1)1+ζee-iπαe(t)2sinπαe(t)(ss0)αe(t).(13)In the t→me2 limit, where me is the mass of the meson with lowest spin Je on the trajectory αe(t), one recovers the known single-particle propagator³

Note that our convention is different from the one in [37] in the ζe prefactor. Our convention is correctly normalized to the single-particle pole as in Eq. (14), while the definition of [37] flips the sign according to the signature of the exchange. Naturally, this normalization affects the sign of the residues of the negative signature exchanges (ζe=-1). We absorb the sign difference into the top vertices of those exchanges.

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” [9]. Therefore, a nonsense wrong-signature zero (NWSZ) is often introduced at αρ=0 only in the t-channel helicity-flip component, in order to remove such a spurious contribution. For the t-channel nonflip component, the finite-energy sum rule analysis indeed shows a finite residue at the wrong signature point (αρ=0) [10]. However, starting from the t-channel, a global analysis is tedious and less straightforward, as discussed, for example, in [35]. 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 contain 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.

In this work, we will focus on the total and differential cross sections only. For most reactions, the amount of these type of data outweighs the polarization data count. Additionally, polarization observables are often measured at energies that are not high enough for our analysis. We restrict our model to the leading Regge pole contributions only. Polarization observables at high energies often function as a filter of specific t-channel quantum numbers and are sensitive to subtle interference effects in the amplitudes. Their interpretation is a study on its own. Since our goal is to obtain a qualitative description, we do not include such polarization data.

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 residues β^μiμjeij (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 [33,34]. The results are shown in Sec. III B. We later relax all these assumptions for a second unconstrained fit in Sec. IV.

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 a2 exchange. In this fit, we can keep the single-particle 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 single-particle 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 as β^μiμfeif(t)=gμiμfeifebμiμfeift,(20)where the constants gμiμfeif and bμiμfeif are all fitted independently, unless stated otherwise. The fitted residue parameters are tabulated in Tables V and VI.

Abandoning the strict connection with the particle exchange model also implies, for example, that the nonflip component of the VBD coupling is no longer required to vanish at t=0 (see β^+12+12VBD∝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 t-dependence 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, αρ0, αa20, and the parameters of the residues, β^++ρNN, β^++a2NN, β^-+ρNN, β^-+a2NN, β^ρKK, β^a2KK, β^ρπη(′), and β^a2πη(′) cf. Eq. (12). The slopes αρ1 (αa21) follow from the requirement that αρ(t=mρ2)=1 (αa2(t=ma22)=2). The intercepts are allowed to vary in the range 0.4≤α(ρ,a2)0≤0.55. The results of these fits are depicted in Figs. 3, 4a, 5, and 6 Assuming ST=2, we obtain the mixing angle θP=-0.33. This angle is compatible with the values found in the recent literature [81,82]. Inspecting the results in Fig. 5a one finds that the cross section rises rapidly for -t≥0.8  GeV2 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^μNμΔρNΔ and β^μNμΔa2NΔ, from a combined fit to πN→ηΔ, πN→πΔ, and KN→KΔ 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+12+32eNΔ/g-12+12eNΔ (e=ρ,a2) 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 nonflip components is fixed to the SU(3)-EXD values. The results of the fit are depicted in Figs. 4b, and 7. Relaxing the condition β^+12+12ρNΔ(t=0)=0 imposed by the single-particle 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 the β^ρNN couplings extracted in the previous fitting steps, one can now determine the β^ρπω residue from a fit to the ωN production data at very high energies. At forward angles and low energies, the b1 exchange does not represent the full strength required to reproduce the cross section of ω production. Notice that the NWSZ at αb1(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 IGζη=1++-, with the lowest spin meson located on the trajectory being the yet undiscovered ρ2 (IGJPC=1+2--) [34,37].⁸

These quantum numbers are not exotic (only the 0-- is), and both the quark model and lattice QCD results predict the existence of such states [83,84]. There are some experimental indications of the existence of ρ2 and ω2 mesons [85,86]. However, these states have been observed by a single group and are poorly established, thus needing confirmation [38].

For the strangeness exchange channels, the fits are somewhat more difficult since one cannot separate the K* from K2* 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, it appears that the 0-12+ production channels are dominated by Regge-pole nonflip contributions in the domain 0≤-t≤0.25  GeV2. We can therefore carry out two-step fits, where we first determine the nonflip coupling from these very forward data. In a second step, we fix the nonflip 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 nonflip 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.

The β^K(2)*Kη(′) couplings are determined in a separate fit using the η(′)Λ production data in Figs. 12a and 12b. As mentioned before, the SU(3) constraint does not hold well for these couplings.

For the Σ* production channels in Figs. 13a and 13b, we fix the β^+12+12K(2)*NΣ* 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 nonflip 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.

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 plab≥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  GeV2. The cross sections of reactions dominated by a2 exchanges are well reproduced by the Regge pole models up to t∼-0.8  GeV2, due to the lack of a dip. Reactions dominated by strangeness exchanges follow the Regge pole model remarkably well in the forward region up to t∼-0.6  GeV2. Especially those channels dominated by nonflip 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 where 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 [87,88]. With the online version of the model, users have the possibility to vary the model parameters and generate the observables.