XYZ spectroscopy at electron-hadron facilities. II. Semi-inclusive processes with pion exchange

XYZ spectroscopy at electron-hadron facilities. II. Semi-inclusive processes with pion exchange D. Winney,1,2,* A. Pilloni ,3,4,†V. Mathieu,5,‡A. N. Hiller Blin,6,7 M. Albaladejo,8 W. A. Smith,9,10 and A. Szczepaniak9,10,11 (Joint Physics Analysis Center) 1Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China 2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China 3Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit`a degli Studi di Messina, I-98122 Messina, Italy 4INFN Sezione di Catania, I-95123 Catania, Italy 5Departament de Física Qu`antica i Astrofísica and Institut de Ci`encies del Cosmos, Universitat de Barcelona, E-08028, Spain 6Institute for Theoretical Physics, Regensburg University, D-93040 Regensburg, Germany 7Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle 14, 72076 Tübingen, Germany 8Instituto de Física Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, E-46071 Valencia, Spain 9Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47403, USA 10Department of Physics, Indiana University, Bloomington, Indiana 47405, USA 11Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA (Received 21 September 2022; accepted 12 October 2022; published 7 November 2022) Semi-inclusive processes are very promising to investigate XYZ hadrons at the next generation of electron-hadron facilities, because they generally boast higher cross sections. We extend our formalism of exclusive photoproduction to semi-inclusive final states. The inclusive production cross sections for charged axial-vector Zstates from pion exchange are predicted. We isolate the contribution of Δresonances at small missing mass. Production near threshold is shown to be enhanced roughly by a factor of two compared to the exclusive reaction. We benchmark the model with data of semi-inclusive b 1production. DOI: 10.1103/PhysRevD.106.094009 I. INTRODUCTION Appearance of the exotic XYZ states in the spectrum of heavy quarkonia is widely recognized as one of the most intriguing puzzles with potentially high impact on our understanding of QCD [1,2]. Most of these states have been observed only in specific channels, most notably in heavy hadron decays and via direct production in eþe− collisions [3]. Exploring alternative production processes, such as electroor photoproduction can provide complementary information on the nature of these states, while probing if they are real resonances or mere kinematic effects [4]. In a previous paper [5], we calculated production rates for several of these states in exclusive photoand electroproduction, at energies that have been proposed, both for the future Electron Ion Collider (EIC) [6] and a new facility that could take advantage of an energy upgrade of the CEBAF accelerator [7]. While exclusive reactions benefit from constrained kinematics, complementary information can be obtained from inclusive reactions. For example, there is ample literature on inclusive Xð3872Þproduction, e.g., in heavy-ion collisions and how it is relevant in unraveling its composition [6,8–11]. Hard events can be studied with perturbative QCD and effective field theories, and one can perform global fits of the long-distance matrix *[email protected] †[email protected] ‡[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 094009 (2022) 2470-0010=2022=106(9)=094009(13) 094009-1 Published by the American Physical Society elements from electron-hadron and hadron-hadron collisions, to be compared with model predictions [9]. Soft processes are dominated by specific kinematic configurations. Compared to exclusive production, inclusive reactions benefit from larger cross sections, and often rely less on model assumptions. In this paper, we focus on inclusive production of states that can process via one pion exchange. Modulo final state interactions, pion exchange is a rather well tested hypothesis and given its proximity to the physical threshold it usually results in large cross sections. We test our model by comparing with data on the b1photoproduction. We find a good agreement with the cross section measured by the OmegaPhoton collaboration [12]. As in our previous work [5], in predicting the photoproduction cross section for the Zð0Þ c;b states we rely on the measured branching fractions and infer other properties from the wellestablished quarkonium phenomenology. This makes our predictions as agnostic as possible as far as the nature of these states. The paper is organized as follows. The following section, Sec. II outlines the formalism for single meson semiinclusive production. It includes discussion of the virtual pion-nucleon cross section, which we study in regimes of both small and large missing mass. Section III contains numerical results for the inclusive cross sections of axialvector mesons, in particular the b1ð1250Þand Zð0Þ c;b states. Finally, concluding remarks and summary of our results are given in Sec. IV. For reference we provide a summary of kinematic expressions relevant to inclusive processes in Appendix Aas well as formulas connecting SAID partialwaves to the total pion-nucleon cross section in Appendix B. II. FORMALISM We consider the process γp→QX, where Qis an axial-vector (quarkoniumlike) meson with mass mQ, and X collectively refers to unobserved particles with total invariant mass MX, also known as “missing mass.”We do not consider production of neutral Q0as it contains Pomeron exchange which, given the limited available data for the Z states, will depend more on model assumption. We note that Xhas baryon quantum numbers and a minimum mass Mmin which, for Qþ, is the nucleon mass and corresponds to the exclusive, γp→Qþnreaction. Since this process was already studied in Ref. [5], in the following we fix the minimum mass to be equal to the first inelastic threshold, i.e., Mmin ¼mπ0þmn(and for the Q−,Mmin ¼mπþþmp). The process is represented schematically in Fig. 1, and the relevant kinematics is summarized in Appendix A. A. Generalized optical theorem The extension of the single-particle exchange mechanism of the exclusive reactions to semi-inclusive production is given by the generalized optical theorem [13].A sketch of the derivation is provided below for illustration purposes and for a more detailed discussion we refer to [14]. The Lorentz-invariant differential cross section for the reaction γp→QX, is given by: EQ d3σ d3qf¼1 16π3 1 4Eγffiffiffi s p1 4X fλgX X d3pn ð2πÞ32EnjAγN→QX fλgj2 ×ð2πÞ4δ4qþp−q0−X n pn;ð1Þ where fλg¼λγ;λN;λQcollectively denotes particle helicities, and the sum over Xruns over all possible final states containing nunobserved particles. By analytical continuation, crossing symmetry relates the amplitude for γN→ QX to that of the reaction γN¯ Q→X. By summing over all intermediate states X, using unitarity for a 3→3amplitude one can express Eq. (1) in terms of the discontinuity of the forward elastic 3→3amplitude across the M2 Xcut1: DiscAγN¯ Q fλg¼1 2i½AγN¯ Q fλgðs;t;M2 XþiεÞ−AγN¯ Q fλgðs;t;M2 X−iεÞ ¼1 2X XZY n d3pn ð2πÞ32EnAγN→QX fλg 2ð2πÞ4δ4 ×qþp−q0−X n pn:ð2Þ Comparing with Eq. (1) one can write, FIG. 1. Semi-inclusive photoproduction of an axial-vector Q via pion exchange in the t-channel. The bottom vertex Bis generalized to consider the production of arbitrary final state X. 1We note that the discontinuity of a 3→3amplitude depends on eight kinematic variables. What enters the generalized optical theorem of Eq. (2) is the forward amplitude, with the final state particles having the same momenta as the initial state particles. In this kinematics there are only three independent variables, denoted, by s,t, and M2 Xas shown in Fig. 2. D. WINNEY et al. PHYS. REV. D 106, 094009 (2022) 094009-2 EQ d3σ d3qf¼1 16π3 1 2Eγffiffiffi s p1 4X fλg DiscAγN¯ Q λγλNλQ;ð3Þ or, in term of Mandelstam variables, d2σ dtdM2 X¼1 16π2 1 4E2 γs 1 4X fλg DiscAγN¯ Q λγλNλQ:ð4Þ The semi-inclusive cross section is therefore given by the discontinuity of the 3→3amplitude containing both t and M2 Xdependence as depicted in Fig. 2. As a check on normalization in Eq. (3) it is usefully to derive the exclusive formulas from it. Writing explicitly the one-body phase space in Eq. (2), one obtains, DiscAγN¯ Q λγλNλQ¼X λ0 NAγN→QN0 λγλNλQλ0 N 2πδðM2 X−m2 NÞ;ð5Þ whence dσ dt¼ZdM2 X d2σ dtdMX ¼1 16π 1 ð2Eγffiffiffi s pÞ2 1 4X fλgAγN→QN0 λγλNλQλ0 N 2;ð6Þ which agrees with the standard expression for the exclusive differential cross section [15]. Furthermore, using the t-channel pion exchange model for exclusive Zproduction previously considered in work [5] into Eq. (5), one obtains, DiscAγN¯ Q λγλNλQ¼jTλγλQPπj2X λ0 N jBλNλ0 Nj2πδðM2 X−m2 NÞ:ð7Þ Here, Tand Bare the πγQand πNN vertex functions respectively and Pπis the pion propagator. The second line is identified with the nucleon pole contribution to the elastic scattering of an off-shell pion (π) off the nucleon. We may thus write, DiscAγN¯ Q λγλNλQ¼jTλγλQPπj2DiscAπN λN;ð8Þ which shows how the M2 Xdependence of the exclusive amplitude generalizes to the inclusive case through the forward (virtual) πNscattering amplitude. Inserting Eq. (8) into Eq. (3) gives EQ d3σ d3qf¼K 16π3 1 2X λγλQjTλγλQPπj2σπN tot ;ð9Þ where we also use the optical theorem to express the forward πNelastic amplitude in terms of the total cross section, 1 2X λN ImAπN λN¼λ1=2ðM2 X;t;m 2 NÞσπN tot ;ð10Þ with σπN tot ≡σπN tot ðt; M2 XÞand the flux factor K≡Kðs; t; M2 XÞ¼λ1=2ðM2 X;t;m 2 NÞ 2Eγffiffiffi s p:ð11Þ In the approach of [5], the top vertex is approximated by an effective γQπLagrangian and thus in the t-channel frame the spin-flip interactions between the photon and Q vanish. The sum over helicities in the top vertex then reduces to: EQ d3σ d3qf¼K 16π3jTπðtÞPπj2σπN tot ;ð12Þ where the residue function TπðtÞis related to a dimensionless πγQcoupling constant: TπðtÞ¼gγQπ λ1=2ðt; 0;m 2 QÞ 2mQ et0=Λ2 π:ð13Þ Here we also include the exponential form factor with pion cutoff, Λπ¼900 MeV, to account for the observed momentum-transfer dependence of the OPE cross sections. For the ease of comparison with the exclusive kinematics and to avoid any spurious dependence on MX, we fix t0 FIG. 2. Diagrammatic representation of the generalized optical theorem. The semi-inclusive amplitude squared summed over all possible final states is related to the discontinuity of the 3→3forward elastic scattering amplitude. XYZ SPECTROSCOPY AT ELECTRON-HADRON FACILITIES. II. …PHYS. REV. D 106, 094009 (2022) 094009-3 to the value it takes for the exclusive reaction, t0≡t−tminðs; MX¼mNÞ. With an appropriately chosen pion propagator and parametrization for the (off-shell) πNcross section, Eq. (12) becomes the semi-inclusive generalization of the pion-exchange model. The form of the propagator depends on the energy range of interest for the production reaction. For instance, a Feynman diagram-inspired model of fixed-spin pion exchange with a scalar propagator Pπ¼1 m2 π−t;ð14Þ is expected to be reliable at energies near threshold. In the high energy limit, to leading order in sthe amplitude is given by Reggeized pion exchanges. In particular, we focus on the so-called triple Regge region, where s≫MX≫jtj in which the amplitude is simply obtained by replacing the pion propagator with Pπ→α0ξðtÞΓð−αðtÞÞs M2 XαðtÞ;ð15Þ in terms of a Regge pole [14]. The signature factor ξis given by ξðtÞ¼1 2½1þτe−iπαðtÞ;ð16Þ with τπ¼þ1and pion trajectory απðtÞ¼α0 πðt−m2 πÞwith α0 π¼0.7GeV2:ð17Þ Here we note that compared to the usual Regge pole form, M2 Xappears in the denominator of Eq. (15) instead of the constant mass scale related to the masses of particles involved, s0. This is because s≫M2 X≫s0≳jtj, thus cos θt→s=M2 X, with M2 Xsetting the dimensional scale. Examining the asymptotic behavior of Eq. (15), we see that at very large t(outside the validity range of the model), the Γfunction grows faster than exponentially, exceeding the suppression from the form factor: et0=Λ2Γð−αðtÞÞ →et½Λ−2þα0−α0logðα0jtjÞ:ð18Þ Thus, in order to avoid unphysical contributions when integrating over the whole semi-inclusive phase-space, in the numerical studies below we impose a cutoff jtcutj≲1 α0e1=Λ2α0∼8.3GeV2;ð19Þ such that the amplitude is exponentially dampened in the entire trange considered. B. The πNtotal cross section The aspect of the model in Eq. (12) not present in the previous analysis of exclusive reactions is the generalized “bottom vertex”which is given by the total πNcross section. Since virtual pion-nucleon scattering cross sections are not known we will relate this as much as possible to the usual on-shell pion-nucleon scattering process. Additionally, as before, it is important to consider the different kinematic regimes of the variable M2 Xsince the near-threshold πNspectrum is dominated by nucleon resonances which may dramatically affect the inclusive production. In fact, the resonance region shows vastly different behavior between the different isospin states which means it is now important to clearly denote the charge of the channel considered. We fix the initial nucleon to a proton target such that Qproduction involves π∓p scattering in the bottom vertex. To this end we use the formalism of Ref. [16], which describes the total πpcross section at all energies. At small M2 Xthe cross section is dominated by the Δand N resonances. Because the intrinsic properties of these resonances are not the focus of our study, a sophisticated analytic model is not required. Instead, we interpolate the SAID partial wave amplitudes based on the T-matrix analysis of the GW-PWA group [17]. Partial waves for both parities and s-channel isospin projections are given up to orbital angular momentum L¼7. These may be related to t-channel isospin amplitudes by considering isospin crossing combinations as in Ref. [16]. We provide a brief summary of these relations in Appendix B. Importantly, the decomposition is in the form σπp tot ðM2 XÞ¼X L σπp LðM2 XÞ;ð20Þ where the right-hand side is calculated from the SAID partial waves at fixed L. The SAID partial waves are provided up to M2 X∼4.5GeV. At higher M2 Xthese are matched to a Regge-motivated parametrization which incorporates exchange physics relevant for the high energy regime. From Eq. (10), the cross section is calculated from t-channel isospin amplitudes: σπp tot ðM2 XÞ¼Im½CðþÞðM2 X;0Þ∓Cð−ÞðM2 X;0Þ plabðM2 XÞffiffiffiffiffi s0 p;ð21Þ where CðÞðsπp;t πpÞare the invariant, t-channel isoscalar and isovector πp→πpamplitudes, with energy squared sπp¼M2 X, momentum transfer tπp, and plabðM2 XÞ¼ λ1=2ðM2 X;m 2 p;m 2 πÞ=2mpthe pion momentum in the proton rest frame. At larger M2 X, the amplitudes Care taken to be the sum of Regge exchanges in the t-channel with the isoscalar corresponding to Pomeron and f2exchange while the isovector is dominated by ρexchange: D. WINNEY et al. PHYS. REV. D 106, 094009 (2022) 094009-4 CðþÞðM2 X;0Þ¼APðM2 X;0ÞþAfðM2 X;0Þ; Cð−ÞðM2 X;0Þ¼AρðM2 X;0Þ:ð22Þ Here each scalar amplitude Akðsπp;t πpÞtakes a simple Regge pole form which in the forward direction is given by: AkðM2 X;0Þ¼−ck 0ξkð0ÞΓðnk−αkð0ÞÞˆ ν ffiffiffiffiffi s0 pαkð0Þ;ð23Þ with the (reduced) crossing-symmetric variable ˆ ν¼νðM2 X;0Þ¼M2 X−m2 p−m2 π 2mp ;ð24Þ which is the energy of the pion in the nucleon rest frame. The f2trajectory is assumed to be degenerate with the ρ one with opposite signature. We take a simple linear form for all trajectories such that at tπp¼0we only require the intercept αð0Þ. The factor of ðn−αð0ÞÞ appearing inside the Gamma function is implemented with n¼ðτþ1Þ=2to remove the ghost pole at αðtπpÞ¼0for the f2and Pomeron exchanges in the more general expression. The parameters used are summarized in Table I. The total cross section as a function of M2 Xis shown in Fig. 3. As a comparison we additionally plot the simpler phenomenological parametrization from the PDG [18,19] the so-called HPR1R2model, which takes the form: σπp tot ðM2 XÞ¼PþHlog2M2 X M2 μ þR1M2 X M2 μ−η1 ∓R2M2 X M2 μ−η2 :ð25Þ Here M2 μ¼ðmπþmpþμÞ2and the parameters H¼0.272 mb, μ¼2.1206 GeV, η1¼0.4473, and η2¼ 0.5486 are process independent. For πpscattering the remaining parameters are P¼18.75,R1¼9.56, and R2¼1.767, all in units of mb. We consider this model to examine the impact of the low-energy resonance region in the production rates. In order to incorporate the off-shell pion in a minimal way we will assume that the dependence on the virtuality enters only through a kinematic change of phase space factors. For the low-energy regime we modify Eq. (20) with a factorized form: σπp tot ðt; M2 XÞ¼X L Rπ Lðt; M2 XÞσπp LðM2 XÞ;ð26Þ where Rπ Lðt; M2 XÞ¼pπ pπ2L ¼λðM2 X;t;m 2 pÞ λðM2 X;m 2 π;m 2 pÞL ð27Þ is the ratio of barrier factors of individual partial waves. We note that in the limit M2 X≫jtjthe rescaling ratio tends to unity and there is no need to apply it to the high-energy, Regge part in Eq. (22). The rescaling in Eq. (26) numerically involves large cancellations when evaluated very close to threshold, in particular for high-Lwaves. In the numerical studies below, we replace the value L→minðL; LmaxÞin the rescaling factor Rπ L, with the value of Lmax chosen to provide the appropriate rescaling to the dominant waves while keeping the higher waves numerically stable near threshold. Because the resonance peaks appear mainly in the Sand P-waves, with higher waves contributing primarily to the intermediate M2 Xregion near the matching point, we find that Lmax ¼3is sufficient for all processes considered. C. Relation to triple Regge amplitude The model in Eq. (12) has been considered in the past in the context of the triple Regge formula (see for example [20,21]), which takes the form: TABLE I. Parameters for Reggeon contributions to σπp tot in Eq. (23) and taken from [16]. ταð0Þc0 Pþ1.075 23.89 fþ0.49 71.35 ρ−5.01 FIG. 3. Low-energy behavior of the pion-nucleon elastic scattering as a function of the πpinvariant mass, Wπp¼ffiffiffiffiffiffiffi sπp p. Solid lines are calculated with Eq. (21) with the amplitudes from [16], while dashed are the high-energy parametrization Eq. (25) [18,19]. XYZ SPECTROSCOPY AT ELECTRON-HADRON FACILITIES. II. …PHYS. REV. D 106, 094009 (2022) 094009-5 EQ d3σ d3qfðγp→QXÞ ¼X k G∓ ππkðtÞ πs0ss M2 X2απðtÞM2 X s0αkð0Þ;ð28Þ schematically represented in Fig. 4. Here klabels the possible Regge exchanges in the bottom vertex, and Gππkis a triple Regge coupling, often parametrized with a phenomenological exponential form. The scale s0is customarily taken to 1GeV2. This form is expected to be reliable in the triple Regge kinematic region s≫MX≫jtj, where the particle Qcarries large momentum in the near-forward direction, i.e., when the fraction of longitudinal momentum x∼1. Since most of the literature focuses on this limit, it is worth considering how Eq. (12) compares to the phenomenological formula in Eq. (28). If we consider Eq. (12) in the triple Regge kinematics, we first note that the flux ratio K→ðM2 X=sÞand we may use the Reggeized form in Eq. (15) for the pion propagator. We must then consider σπp tot ðt; M2 XÞin this kinematics. From Eq. (23),asM2 X→∞we have Rπ L→1and the Reggeon pole form Eq. (23) reduces to: ImAkðM2 X;0Þ→γkM2 X s0αkð0Þ;ð29Þ where k¼P;f2;ρand the coupling constants are given by γk¼π 2 ck 0 Γðαkð0Þ−nkþ1Þffiffiffiffiffi s0 p 2mpαkð0Þ:ð30Þ Then, with Eq. (29), we may write Eq. (12) as: σπN tot ðM2 XÞ→X k 2mp ffiffiffiffiffi s0 pγkf k s0M2 X s0αkð0Þ−1 :ð31Þ where f ρ¼∓1and f f2;P¼þ1depending on the pion charge. Putting it all together, we can see that Eq. (21) in the triple Regge limit reduces precisely the form of Eq. (28) with the triple Regge vertex function given by G ππkðtÞ¼f k mpffiffiffiffiffi s0 pγk 8π2jα0TπðtÞξπðtÞΓð−απðtÞÞj2:ð32Þ Seeing the emergence of triple Regge behavior in the appropriate limit is reassuring, as the formula Eq. (12) generalizes the semi-inclusive cross section by loosening the requirement of large M2 Xand allows us to consider the fixed spin analog relevant for near threshold production, i.e., with small sand M2 X. D. Exclusive Q−Δ++ and Q+Δ0production Another consistency test for the semi-inclusive cross section formula in Eq. (12) is the opposite limit, MX∼ Mmin. We may consider the production of a meson Qtogether with a Δbaryon, which dominates the low-energy πN spectrum. This is particularly true for the πþpscattering, where the Δþþ is the only resonance in the mass region MX≲1.5GeV. Additionally, in this πNisospin-channel there is no analogous exclusive reaction, meaning the γp→Q−Δþþ reaction is already contained within the cross section, Eq. (12). Calculating the exclusive reaction with effective Lagrangian methods as in Ref. [5] should lie strictly below the prediction for the inclusive cross section and saturate the low energy regime. Although we will focus specifically on the Q−case, the discussion in this section is directly applicable for the opposite charge with a relative factor associated with isospin-projection: σðγp→QþΔ0Þ σðγp→Q−ΔþþÞ¼1 3:ð33Þ Because P-wave πþpscattering is dominated by the Δþþ, a straightforward way to consider Q−Δþþ production is to restrict the total cross section in Eq. (12) to only the P-wave component by replacing σπþp tot ðt; M2 XÞ→Rπ L¼1ðt; M2 XÞσπþp L¼1ðM2 XÞ:ð34Þ Alternatively, in analogy to the exclusive formalism of Ref. [5], we can include the spin-3=2baryon by changing FIG. 4. Diagrammatic representation of the triple Regge formula of Eq. (28). D. WINNEY et al. PHYS. REV. D 106, 094009 (2022) 094009-6 the bottom, πΔNvertex for which we take a simple effective Lagrangian as in [22]: LπNΔ¼gπNΔ mπ ¯ Δμ∂μπNþH:c:; ð35Þ or equivalently the bottom vertex: BλN;λΔ¼igπNΔ mπ ¯ uμðp0;λΔÞkμuðp; λNÞ;ð36Þ where kis the pion momentum, k¼p−p0and uμis the Rarita-Schwinger spinor: uμðp; λÞ¼X m1;m21;m 1;1 2;m 2 3 2;λϵμðp; m1Þuðp; m2Þ: ð37Þ The coupling gπNΔis calculated assuming the width to be saturated by the πNfinal state. With ΓΔ¼120 MeV and mΔ¼1.23 GeV, this leads to gπNΔ¼2.10. We may thus calculate the γp→QΔamplitude assuming a stable Δin the final state. To directly compare to the cross section in Eq. (12) we incorporate the Δ→πp lineshape via σðγp→Q−πþpÞ¼Z∞ M2 min dM2σðγp→Q−ΔþþÞdΔ→πpðM2Þ; ð38Þ where the QΔcross section is calculated at fixed sas a function of M2. Because of the proximity of the πp threshold we use a simple Breit-Wigner-like distribution proposed in Ref. [23] which is shown to provide a good description of the πpmass distribution in the Δmass region: dΔ→πpðM2Þ¼1 π ρðM2Þ˜ ΓΔ ½M2−m2 Δ2þ½ρðM2Þ˜ ΓΔ2;ð39Þ with ρðM2Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2−M2 min pand ˜ ΓΔ¼ΓΔmΔ=ρðm2 ΔÞ. Interestingly, this function is normalized across the mass distribution, obeying Z∞ M2 min dM2dΔ→πpðM2Þ¼1:ð40Þ Both Eqs. (34) and (38) are expected to yield similar results with minor differences related to the lineshape assumed for the Δ. III. NUMERICAL RESULTS Experimental data on the photoproduction of exotic quarkoniumlike states is virtually nonexistent. In order to benchmark the predictions for Zstates, we first consider the axial-vector analog in the light sector, the charged b1ð1235Þwhich has been looked at in the kinematic region of interest by the OmegaPhoton collaboration [24]. The applicability of the same amplitudes for both light and heavy meson production constitutes the primary model assumption, i.e., through the use of VMD which has recently been criticized [25]. As noted in Ref. [2], however, VMD predicts roughly the correct size of the ratio Γðχc2→γJ=ψÞ=Γðχc2→γγÞ, so it seems an appropriate method to obtain at least order-of-magnitude estimates. In any case, we note this assumption affects only the top vertex which is the same as in the exclusive analysis. Thus considering the b1still serves as a test of the inclusive extension of the pion exchange process which in principle is independent of the microscopic nature of Q. We begin with the comparison with the differential cross section data in Ref. [24]. The energy range covered by the measurement is Eγ¼25–55 GeV and for simplicity we take the midpoint center-of-mass energy, Wγp∼8.7GeV. At these energies with respect to the b1Nthreshold, the triple Regge behavior is expected to be dominant and we use exclusively the Reggeized form of the pion propagator, Eq. (15), with the high energy approximated kinematics as explained in Appendix A. The only undetermined parameter is the gγb1πcoupling governing the strength of the top vertex. We use the effective Lagrangian formalism previously considered in Ref. [5] for the γπb1vertex to extract this coupling from its decay widths. Luckily the radiative decay width for the b1is known and we may extract the coupling without relying on VMD. For Γðb1→πγÞ¼230 keV [26] we have gγb1π¼0.24. For comparison purposes, we calculate the inclusive cross section with the pion-nucleon interaction described with the full πNcross section Eq. (21), as well as with the Regge-only parametrization of Eq. (25). These are shown in Fig. 5compared to the data points in the highest xbins. We see a good agreement of both the models in the highest bin. To make the comparison more quantitative we may calculate the average cross section from each curve in this bin, yielding 0.69 and 0.56 μb for the model including nucleon resonances and the Regge-only parametrization respectively, both consistent with OmegaPhoton within uncertainties. The agreement between the two values is expected, because at this relatively large center-of-mass energy the nucleon resonances are squeezed in a small portion of the phase space. Away from the highest bin we note that single pion exchange severely underestimates the production. Indeed, since at high energies the upper limit of integration tis proportional to 1−xand the pion exchange is exponentially suppressed with jtj, the integral is also exponentially suppressed in 1−x. This suggests that the triple Regge contribution becomes quickly irrelevant already at x<0.9, although other top exchanges than the pion should be XYZ SPECTROSCOPY AT ELECTRON-HADRON FACILITIES. II. …PHYS. REV. D 106, 094009 (2022) 094009-7 added. Summarizing, we find good agreement with data in the region of validity of the model, and a trend toward zero away from this region. We may thus consider the cross section calculated in this formalism as a conservative lower bound of the total inclusive production rate. By integrating over x, we may also investigate the nearthreshold behavior as a function of invariant mass. For energies Wγp≲5GeV we use instead the fixed-spin pion exchange model of Eq. (14). We compare the inclusive prediction for b1ð1235Þ−production to the explicit b− 1Δþþ using the formalism in Sec. II D. The comparisons between the exclusive Δþþ prediction with and without inclusion of the subsequent Δ→πNdecay are shown in Fig. 6compared to the inclusive production. We see that the unstable Δcurve saturates the inclusive production up to the nominal b1Δ threshold, after which the contribution of other resonances become relevant. We also see good agreement between the two methods of incorporating the Δþþ decay. To this end in all subsequent numerical studies we consider the Δ→πp decay by restricting the total cross section in Eq. (34) to the SAID I¼3=2,L¼1partial wave, since it incorporates more accurately the Δþþ lineshape. The inclusive cross sections of both charged b 1are shown in Fig. 7. We now turn to the quarkoniumlike states in the hidden charm and bottom sectors. We use the couplings that were previously calculated from the observed hadronic decays of the Z-states within the VMD model, gγZπ×102¼5.17, 5.8, and 2.9 for Zc,Zband Z0 brespectively [5]. The total near-threshold production cross sections for the two charged Zstates are shown in Fig. 8. Examining the Z−inclusive cross sections, we again note that it is dominated by the Δþþ resonance. On the other hand, FIG. 5. Inclusive b1ð1235Þþproduction at Wγp¼8.7GeV using the Reggeized pion exchange. The solid curves are the inclusive cross section using the parametrization in Eq. (26) including nucleon resonances, while the dashed curves calculate the same process using the Regge-only parametrization of Eq. (25). When averaged over the highest bin, both curves are consistent with the experimental point within errors. Data from [24]. FIG. 6. Inclusive b1ð1235Þ−production as compared to exclusive b− 1pπþproduction through an intermediate Δþþ as a function of center-of-mass energy. The Δ→πpdecay is incorporated via a BW shape using Eq. (38) or from restricting the SAID PW sum in Eq. (34). The dashed line is the exclusive reaction calculated with the effective Lagrangian assuming a stable Δbaryon. FIG. 7. Total b1ð1235Þproduction as a function of Wγpwith a fixed-spin pion exchange. The bþ 1curve includes the sum of the inclusive contribution Eq. (12) and the exclusive process (dashed) as calculated in [5] which lies below the πNthreshold. The b− 1 production does not have a corresponding exclusive analog. D. WINNEY et al. PHYS. REV. D 106, 094009 (2022) 094009-8 the contributions from inelastic channels to Zþproduction is roughly the same size as the exclusive Zþnreaction enhancing the total production rate by a factor of 2 close to threshold. Figure 9provides a more detailed picture of the contributing processes. We see that, unlike the Z−case, the Δ0does not dominate the inclusive cross-section with contributions from other resonances being equally important. Further we see the asymptotic behavior of the inclusive cross section falls much slower than the Zþn final state as the center-of-mass energy grows. In fact, from the comparison with the asymptotic triple Regge formula in Sec. II C, we expect the curves to flatten out and grow slowly with energy, while the exclusive cross section decreases [5]. Explicit comparison of inclusive production compared to the respective exclusive reaction at large energies is shown in Table II. In Fig. 10 we show the transverse momentum distribution of the cross section. The diffractive production mechanism contributes primarily to the small-qTregion, meaning it may be expected to be predominant at nearthreshold energies where the possible produced inclusive final states have small invariant mass. FIG. 8. Total cross section predictions for charged, charmonium-like Z-states near-threshold via fixed-spin pion exchange. Left panel: Total inclusive Z−cross sections (solid lines) as compared with the exclusive γp→Z−Δþþ →Z−πþpcross section (dashed lines) as described in Sec. II D. Right panel: Total Zþcross sections (solid lines), which include the sum of the inclusive cross section and the exclusive nucleon pole contribution. This latter contribution is calculated as in Ref. [5] and is shown explicitly in dashed lines. FIG. 9. Contributions to the total cross section of near-threshold Zcð3900Þþproduction. The total curve represents the sum of the full inclusive contribution and the exclusive reaction. The dashed line corresponds to the sum of only Zþ cnand Zþ cΔ0contributions. TABLE II. High-energy production cross sections of the total inclusive process γp→QX as a function of Wγp, compared to the exclusive process. The exclusive cross sections fall asymptotically to zero while the inclusive approaches a constant as described in the text. σðγp→QXÞ[pb] σðγp→QþnÞ[pb] Q30 GeV 60 GeV 90 GeV 30 GeV 60 GeV 90 GeV b1ð1235Þ60 ×10360 ×10361 ×10343 2.3 <10−8 Zcð3900Þ187 146 140 19 1.0 <10−8 Zbð10610Þ163 15 5 150 10 <10−8 Zbð10650Þ40 4 1 37 2.4 <10−8 XYZ SPECTROSCOPY AT ELECTRON-HADRON FACILITIES. II. …PHYS. REV. D 106, 094009 (2022) 094009-9