Exclusive $f_{1}(1285)$ meson production for energy ranges available at the GSI-FAIR with HADES and PANDA

We evaluate the cross section for the $p p \to p p f_{1}(1285)$ and $p \bar{p} \to p \bar{p} f_{1}(1285)$ reactions at near threshold energies relevant for the HADES and PANDA experiments at GSI-FAIR. We assume that at energies close to the threshold the $\omega \omega \to f_{1}(1285)$ and $\rho^{0} \rho^{0} \to f_{1}(1285)$ fusion processes are the dominant production mechanisms. The vertex for the $VV \to f_{1}$ coupling is derived from an effective coupling Lagrangian. The $g_{\rho \rho f_1}$ coupling constant is extracted from the decay rate of $f_{1}(1285) \to \rho^{0} \gamma$ using the vector-meson-dominance ansatz. We assume $g_{\omega \omega f_1} = g_{\rho \rho f_1}$, equality of these two coupling constants, based on arguments from the naive quark model and vector-meson dominance. The amplitude for the $VV \to f_{1}$ fusion, supplemented by phenomenological vertex form factors for the process, is given. The differential cross sections at energies close to the threshold are calculated. In order to determine the parameters of the model the $\gamma p \to f_{1}(1285) p$ reaction is discussed in addition and results are compared with the CLAS data. The possibility of a measurement by HADES@GSI is presented and discussed. We performed a Monte Carlo feasibility simulations of the $p p \to p p f_1$ reaction for $\sqrt{s}$ = 3.46 GeV in the $\pi^+ \pi^- \pi^+ \pi^-$ (not shown explicitly) and $\pi^+ \pi^- \pi^+ \pi^- \pi^0$ final states using the PLUTO generator. The latter one is especially promising as a peak in the $\pi^+ \pi^- \eta$ should be observable by HADES.


I. INTRODUCTION
The production of light axial-vector mesons with quantum numbers I G J PC = 0 + 1 ++ is very interesting and was discussed in a number of experimental and theoretical papers. For example, the f 1 (1285) meson was measured in two-photon interactions in the reaction e + e − → e + e − ηπ + π − (η → γγ) by the Mark II [1], the TPC/Two-Gamma [2,3], and, more recently, by the L3 [4] collaborations. In such a process the γ * γ * → f 1 (1285) vertex, associated with corresponding transition form factors, is the building block in calculating the amplitude. Different vector-vector-f 1 vertices and corresponding transition form factors were suggested in the literature [5][6][7][8][9][10][11][12][13][14]. It was suggested in [15] that a measurement of the e + e − → e + e − f 1 (1285) reaction with double tagging at Belle II at KEK could shed new light on the γ * γ * f 1 coupling with two virtual photons.
The f 1 (1285) meson was also measured in the photoproduction process γp → f 1 (1285)p by the CLAS Collaboration at JLAB [16]. The differential cross sections were measured from threshold up to a center-of-mass energy of W γp = 2.8 GeV in a wide range of production angles. The f 1 (1285) photoproduction was studied extensively from the theoretical point of view; see [17][18][19][20][21]. There, the t-channel ρ and ω exchange (either Regge trajectories or meson exchanges) is the dominant reaction mechanism for the small-t behaviour of the cross section, that is, in the forward scattering region. The contribution of the u-channel proton-exchange term with the coupling of f 1 (1285) to the nucleon is dominant at the backward angles [18][19][20]22]. In [20] the authors showed that also the s-channel nucleon resonance N(2300) with J P = 1/2 + may play an important role in the reaction of γp → f 1 (1285)p around √ s = 2.3 GeV. As was shown in [20] other contributions, the s-channel proton-exchange term, the u-channel N(2300)-exchange term, and the contact term, are very small and can be neglected in the analysis of the CLAS data. The Primakoff effect by the virtual photon exchange in the t-channel was discussed in [21]. This mechanism is especially important in the forward region and at higher W γp energies.
The pp → pp f 1 (1285) reaction was already measured by the WA102 Collaboration for center-of-mass energies √ s = 12.7 and 29.1 GeV [23][24][25][26]. There the dominant contribution at √ s = 29.1 GeV is most probably related to the double-pomeron-exchange (PP-fusion) mechanism; see [27]. In [27] the pp → pp f 1 (1285) and pp → pp f 1 (1420) reactions were considered in the tensor-pomeron approach [28]. A good description of the WA102 data [25] at √ s = 29.1 GeV was achieved. A study of central exclusive production (CEP) of the axial vector mesons f 1 at high energies (RHIC, LHC) could shed more light on the coupling of two pomerons to the f 1 meson [27]. As discussed in Appendix D of [27] at the lower energy √ s = 12.7 GeV the reggeized-vector-meson-exchange or reggeonreggeon-exchange contributions should be taken into account.
The ωω → f 1 and ρ 0 ρ 0 → f 1 fusion are the most probable low energy production processes. We know how the ω and ρ 0 couple to nucleons. However, the couplings of ωω → f 1 and ρ 0 ρ 0 → f 1 are less known. We note that future experiments at HADES and PANDA will provide new information there. The ρ 0 ρ 0 → f 1 (1285) coupling constant can be obtained from the decays: f 1 → ρ 0 γ and/or f 1 → π + π − π + π − .
In the present analysis we obtain the g ρρ f 1 coupling constant from the radiative decay process f 1 (1285) → γρ 0 → γπ + π − using the vector-meson-dominance (VMD) ansatz; see Appendices A and B. We discuss briefly our results for the γp → f 1 (1285)p reaction and compare with the CLAS data in Appendix C. From this comparison we estimate the form-factor cutoff parameters.
The PANDA experiment (antiProton ANnihilations at DArmstadt) [29] will be one of the key experiments at the Facility for Antiproton and Ion Research (FAIR) which is currently being constructed. At FAIR, a system of accelerators and storage rings will be used to generate a beam of antiprotons with a momentum between 1.5 and 15 GeV/c. The design maximum energy in the center-of-mass (c.m.) system for antiproton-proton collisions is √ s ≃ 5.5 GeV. The exclusive production of the f 0 (1500) meson in antiproton-proton collisions via the pion-pion fusion mechanism was discussed for the PANDA experiment in [30]; see also Fig. 3 of [31]. The pion-pion fusion contribution grows quickly from the threshold, has a maximum at √ s ≃ 6 GeV and then drops slowly with increasing energy. The predicted cross section for the pp f 0 (1500) final state is σ f 0 = 0.3 − 0.8 µb for √ s = 5.5 GeV; see Sec. III C of [30]. At intermediate energies (e.g. for the WA102 and COMPASS experiments) other exchange processes such as the reggeon-reggeon, reggeon-pomeron and pomeron-pomeron exchanges are very probable; see e.g. [31].
A measurement at low energies, such as HADES@GSI would be interesting to impose constraints on the VV → f 1 (1285) vertices. In this paper we wish to make first estimates of the total and differential cross sections for the pp → pp f 1 (1285) and pp → pp f 1 (1285) reactions at energies relevant for the HADES and PANDA experiments. We shall present some differential distributions for the HADES energy at √ s = 3.46 GeV and for the future experiments with the PANDA detector at √ s = 5.0 GeV. The experimental possibilities of such measurements will be discussed in addition.
In [27] for the reaction (2.1) the pomeron-pomeron-fusion mechanism was considered which seems to dominate at the WA102 energy of √ s = 29.1 GeV. As discussed in Appendix D of [27] at lower energies other fusion mechanisms may be important. We shall take into account only the main processes at energies close to the threshold, the VV-fusion mechanism, shown by the diagrams in Fig. 1. There can be also the a 0 1 (1260)π 0 -fusion mechanism not discussed in the present paper. Note that due to the large width of the a 1 (1260) the decay f 1 (1285) → π ± a ∓ 1 can easily occur for off-shell a 1 (1260) and this is an important decay mode in the f 1 → 2π + 2π − channel as will be discussed in [32].
The VV-fusion mechanisms (VV stands for ωω or ρ 0 ρ 0 ) for f 1 production in proton-proton collisions.
The kinematic variables for the reaction (2.1) are For the kinematics see e.g. Appendix D of [31]. The amplitude for the reaction (2.1) includes two terms The VV-fusion (VV = ρ 0 ρ 0 or ωω) amplitude can be written as Here ǫ α (λ) is the polarisation vector of the f 1 meson, Γ are the V pp and VV f 1 vertex functions, respectively, and∆ (V) µν is the propagator for the reggeized vector meson V. At very low energies the latter must be replaced by ∆ (V) µν , the standard propagator for the vector meson V. We shall now discuss all these quantities in turn.
First we discuss the VV f 1 coupling. We start by considering the on shell process of two real vector particles V fusing to give an f 1 meson: The angular momentum analysis of such reactions was made in [31]. The spins of the two vectors can be combined to a total spin S = 0, 1, 2. Then S has to be combined with the orbital angular momentum l to give the spin J = 1 and parity +1 of the f 1 state. From Table 8 of [31] we find that there is here only one possible coupling, namely (l, S) = (2, 2). A convenient corresponding coupling Lagrangian, given in (D9) of [27], reads with M 0 ≡ 1 GeV and g VV f 1 a dimensionless coupling constant. U α (x) and V κ (x) are the fields of the f 1 meson and the vector meson V, respectively. For the Levi-Civita symbol we use the normalisation ε 0123 = +1. The expression for the VV f 1 vertex obtained from (2.6) is as follows [27]. Here the label "bare" is used for a vertex as derived from (2.6) without a form-factor function. The vertex function (2.8) satisfies the relations For realistic applications we should multiply the 'bare' vertex (2.8) by a phenomenological cutoff function (form factor) F VV f 1 which we take in the factorised ansatz We make the assumption thatF V (t) is parametrized as where the cutoff parameter Λ V , taken to be the same for both ρ 0 and ω, is a free parameter. For the on-shell V and f 1 mesons we have with the tensor-to-vector coupling ratio, κ V = f V NN /g V NN . We use the following values for these coupling constants: We give a short discussion of values for the ρpp and ωpp coupling constants found in the literature. For the the ρNN coupling constants one finds g ρpp = 2.63 − 3.36 [34,35] and κ ρ is expected to be κ ρ = 6.1 ± 0.2 [36]. There is a considerable uncertainty in the ωNN coupling constants. From Table 1 of [33] we see a broad range of values: g ω pp ≃ 10 to 21 and κ ω ≃ −0.16 to +0.14. For example, in [36] it was estimated g ω pp = 20.86 ± 0.25 and κ ω = −0.16 ± 0.01; see Table 3 of [36]. Within the (full) Bonn potential [37] values of g ω pp = 15.85 and κ ω = 0 are required for a best fit to NN scattering data. In [38] it was shown that such a fairly large value of g ω pp must be considered as an effective coupling strength rather than as the intrinsic ωNN coupling constant. They found that the additional repulsion provided by the correlated πρ exchange to the NN interaction allows g ωNN to be reduced by about a factor 2, leading to an "intrinsic" ωNN coupling constant which is more in line with the value one would obtain from the SU(3) flavour symmetry considerations, g ωNN = 3 g ρNN cos(∆θ V ) [35], where ∆θ V ≃ 3.7 • is the deviation from the ideal ω-φ mixing angle. The values of g ω pp = 7.0 − 10.5 and κ ω ≃ 0 were found to describe consistently the πN scattering and π photoproduction processes [39]. The values of g ω pp = 9.0 and κ ω = −0.5 have been used in the analysis of the pp → ppω reaction to reproduce the shape of the measured ω angular distribution; see Fig. 7 of [40]. It was shown [41] that the energy dependence of the total cross section and the angular distribution for pp → ppω can be described rather reasonably even with a vanishing κ ω (g ω pp = 9.0, κ ω = 0); see Fig. 4 of [41]. Finally we note that in [28] the couplings of the ω R and ρ R reggeons to the proton were estimated from high-energy scattering data and found as g ρ R pp = 2.02 and g ω R pp = 8.65 ; (2.14) see (3.60) and (3.62) of [28]. Taking all these informations into account we think that our choice (2.13) for the coupling constants is quite reasonable. The form factor F V NN (t) in (2.12), describing the t-dependence of the V-(anti)proton coupling, can be parametrized as where Λ V NN > m V and t < 0. Please note that the form factor F V NN (t) is normalized to unity at t = m 2 V . On the other hand, the reggeon-proton couplings (2.14) are defined for t = 0. Since F V NN (0) < 1 we expect that g ρ R pp < g ρpp and g ω R pp < g ω pp , which is indeed the case; see (2.13) and (2.14).
The coupling constant g VV f 1 and cutoff parameters Λ V and Λ V NN should be adjusted to experimental data. Examples are discussed in Appendices B and C. There, the form factor F VV f 1 (2.10) is used for F f 1 (m 2 f 1 ) = 1 and for different kinematic conditions ofF V (q 2 ) (2.11), that is, for spacelike (q 2 < 0) and timelike (q 2 > 0) momentum transfers of the V meson, and also at q 2 = 0. In Appendix B we discuss the radiative decays of the f 1 (1285) meson in two ways f 1 → ργ (B1) and f 1 → (ρ 0 → π + π − )γ (B2) where we have F ρρ f 1 (m 2 ρ , 0, m 2 f 1 ) and F ρρ f 1 (q 2 > 0, 0, m 2 f 1 ), respectively. In Table III in Appendix B we collect our results for g ρρ f 1 extracted from the decay rate of f 1 → ρ 0 γ using the VMD ansatz. The process f 1 → ρ 0 ρ 0 → 2π + 2π − , where both ρ 0 mesons carry timelike momentum transfers, will be studied in detail in [32]. For the γp → f 1 p reaction, discussed in Appendix C, we have F ρρ f 1 (0, q 2 < 0, m 2 f 1 ). This is closer to the VV → f 1 fusion mechanisms shown in Fig. 1 where both V mesons have spacelike momentum transfers. From comparison of the model to the f 1 -meson angular distributions of the CLAS experimental data [16] we shall extract the cutoff parameter Λ V NN in the V-proton vertex (2.15); see (C7)-(C12) and Fig. 14 in Appendix C.
In the following we shall use the VV f 1 coupling (2.6) and the corresponding vertex (2.8)-(2.11) for our VV → f 1 fusion processes of Fig. 1 for both: normal off-shell vector mesons V and reggeized vector mesons V R .
The standard form of the vector-meson propagator is given e.g. in (3.2) of [28] i∆ For higher values of s 1 and s 2 we must take into account reggeization. We do this, following (3.21), (3.24) of [42], by making in (2.16) the replacements for i = 1 or 2, and s thr is the lowest value of s i possible here: We use the standard linear form for the vector meson Regge trajectories (cf., e.g., [43]) Our reggeized vector meson propagator, denoted by∆ In the following we shall also consider the CEP of the f 1 (1285) with subsequent decay into ρ 0 γ: with p 34 = p 3 + p 4 . Here p 3 , p 4 and λ 3 = 0, ±1, λ 4 = ±1 denote the four-momenta and helicities of the ρ 0 meson and the photon, respectively.
The amplitude for the reaction (2.22) can be written as in (2.4) but with the replacements Here ǫ (ρ) and ǫ (γ) are the polarisation vectors of ρ 0 and γ, respectively, and ∆ is the transverse part of the f 1 propagator which has a structure analogous to (2.16). The factor e/γ ρ comes from the ρ-γ transition vertex; see (3.23)-(3.25) of [28].
In practical calculations we introduce in the ρρ f 1 vertex the form factor F f 1 (p 2 34 ) [see (2.10)] for the virtual f 1 meson In (2.23) we shall use a simple Breit-Wigner ansatz for the f 1 meson propagator (2.25) The mass and total width of f 1 meson from [44] are We note that the mass of 1281.0 ± 0.8 MeV measured in the CLAS experiment [16] is in very good agreement with the PDG average value (2.26). The total width measured by the CLAS Collaboration is however smaller than the value (2.27): Then the corresponding amplitudes are as in (2.4) but with the replacement The main decay modes of the f 1 (1285) are [44] 4π, ηππ, KKπ, and ρ 0 γ. If the f 1 is to be identified and measured in CEP in any one of these channels one will have to consider background processes giving the same final state, for instance, pp4π. Therefore, in this section we discuss two background reactions: CEP of 4π via ρ 0 ρ 0 in the continuum and CEP of ρ 0 γ in the continuum.
First we discuss the exclusive production of ρ 0 ρ 0 in proton-proton collisions, Diagrams for exclusive continuum ρ 0 ρ 0 production in proton-proton collisions. There are also the diagrams with p 3 ↔ p 4 .
There can also be ρρ fusion with exchange of an intermediate σ ≡ f 0 (500) meson and the σσ fusion with ρ 0 exchange. From the Bonn potential [37,45] we get for the squared coupling constant g 2 σpp /4π ≃ 6.0 which is smaller than g 2 π pp /4π ≃ 14.0. Moreover, we can expect that |g σρρ | ≪ |g ρωπ |. Due to large form-factor uncertainties and the poorly known σρρ coupling we neglect these contributions in our present study. Other contributions may be due to the exchanges of the f 2 (1270) meson ( f 2 -ρ 0 -f 2 or ρ 0 -f 2 -ρ 0 ) and the neutral a 2 (1320) meson (a 2 -ω-a 2 or ω-a 2 -ω). For the f 2 ρρ and a 2 ωρ couplings one could use the rather well known couplings from (3.55), (3.56), (7.29)-(7.34) and (3.57), (3.58), (7.38)-(7.43) of [28], respectively. Since the f 2 pp coupling, taking it equal to f 2R pp from (3.49), (3.50) of [28], is rather large, the f 2 -ρ 0 -f 2 fusion may give a large background contribution. Since g ω R pp > g a 2R pp , see (3.52) and (3.60) of [28], and the a 2 ωρ couplings have values similar to the f 2 ρρ couplings the ω-a 2 -ω contribution may also be potentially important. However, we expect that the tensor meson propagator(s) will reduce the cross section for these processes.
At higher energies the pomeron plus f 2 reggeon (P + f 2R ) fusion [(P + f 2R )-ρ 0 -(P + f 2R )] and ρ 0 fusion with P + f 2R exchange [ρ 0 -(P + f 2R )-ρ 0 ] will be important, probably the dominant processes; see [46]. We expect that these processes will give only a small contribution in the threshold region, of interest for us here. Therefore, we shall neglect also these mechanisms in the following.
With the assumption, motivated above, that the diagrams of Fig. 2 represent the dominant reaction mechanisms in the threshold region, the continuum amplitude for the reaction (2.32) can be written as The ωωand ππ-fusion amplitudes (2.33) are given by µ 's denote the polarisation vectors of the outgoing ρ 0 mesons. The standard pion prop- is used in the calculations. The reggeized vector meson propagator, denoted by∆ .17) and (2.18) and with the relevant s ij , s thr , and t i , the four-momentum transfer squared, in the pρ 0 and ρ 0 ρ 0 subsystems.
With k ′ , µ and k, ν the four-momentum and vector index of the outgoing ρ 0 and incoming ω meson, respectively, and k ′ − k the four-momentum of the pion the ρωπ vertex, including form factor, reads 1 where g ρωπ ≃ ±10 [33,35,41]. We note that the value of g ρωπ = +10, has been extracted in [35] from the measured ω → π 0 γ radiative decay rate and the positive sign from the analysis of pion photoproduction reaction in conjunction with the VMD assumption. In [41] it was found that the data for the reaction pp → ppω strongly favour a negative sign of the coupling constant g ρωπ . In our case, the sign of g ρωπ does not matter as this coupling occurs twice in the amplitudes (2.34) and (2.35). We use a factorized ansatz for the form factor The form factor (2.37) should be normalised as F(0, m 2 ω , m 2 π ) = 1, consistent with the kinematics at which the coupling constant g ρωπ is determined. This is the ω → π 0 γ reaction where ω and π 0 are on shell and the virtual ρ 0 which gives the γ has mass zero. Following [35] we take We assume for the cutoff parameters that they are equal to a common value Λ M ≡ Λ Mω = Λ Mρ = Λ Mπ . Following [35] we take Λ M = 1.45 GeV. Smaller values of the cutoff parameters, Λ Mρ = Λ Mπ = 1.0 GeV, were used in [40] (see Table I there). Also a dipole form factor F V (t) in (2.38) was considered; see [47,48] and Table II of [40].
Likewise, the monopole form factor (2.15) in the V pp vertex (2.12) is assumed with the cutoff parameter Λ V NN . We take Λ V NN = 0.9 GeV and 1.35 GeV in accordance with (C10) and (C7), respectively.
Taking into account the statistical factor 1 2 due to the identity of the two ρ 0 mesons in (2.32) we get for the amplitude squared respectively. Using (2.30) and (2.42) we obtain M (ωω fusion) As will be discussed in the following, from the π + π − π + π − channel it may be rather difficult to extract the f 1 (1285) signal. Another decay channel worth considering is ρ 0 γ.
Therefore, now we discuss the exclusive production of the ρ 0 γ continuum in protonproton collisions, with p 4 and λ 4 = ±1 the four-momentum and helicities of the photon.
(a) In order to calculate the amplitude for the reaction (2.45) we use the standard VMD model with the γV couplings as given in (3.23)-(3.25) of [28]. We shall consider the dia-grams shown in Fig. 3. The result is as follows: We could also have πη and πσ fusion contributions. For these we have to replace in the left (right) diagram in Fig. 3(c) the lower (upper) particles (π 0 , ρ 0 ) by (η, ω) or (σ, ω), respectively. Discussing first πη fusion we note that the couplings η pp and ωωη are smaller than those of π 0 pp and ρωπ [33]. In addition, the η exchange is suppressed relative to the π 0 exchange because of the heavier mass occurring in the propagator. Another mechanism is the πσ fusion involving the σpp and σωω vertices. However, here g σωω ∼ 0.5 [33] is extremely small. Moreover, the ω → γ transition coupling is much smaller than the ρ → γ one; see (A5). Therefore, we neglect the πη and πσ contributions in our considerations. Thus, we are left with the (ω + ρ 0 )-π 0 -ω, ω-π 0 -(ω + ρ 0 ), and π 0 -ω-π 0 contributions, which we shall treat in a way similar to (2.34) and (2.35). As an example, the M (ωω fusion) pp→ppρ 0 γ amplitude can be written as in (2.34) with the following replacement: In the case of the diagrams with the ω → γ transition, the outgoing ω has fourmomentum squared p 2 = 0. Since nothing is known about the form factor at the ρωπ vertex where both the π 0 and ρ 0 are off their mass-shell, we assume in (2.36) the form factor (2.37) as F(m 2 ρ , 0, m 2 π ) = 1 which is consistent with (2.38) and (2.39). The ργ-continuum processes in proton-antiproton collisions can be treated in a completely analogous way to the ργ-continuum processes in proton-proton collisions but with the appropriate replacements given by (2.30) and (2.42).

III. NUMERICAL RESULTS
We start by showing the integrated cross section for the exclusive reaction pp → pp f 1 (1285) as a function of collision energy √ s from threshold to 8 GeV. Note that due to (2.31) the cross sections and distributions for the VV-fusion mechanism are equal for pp and pp scattering for the same kinematical values.
In Fig. 4 we show results for for the VV-fusion contributions (V = ρ, ω) for different parameters given by (C7), (C9) and (C10) in Appendix C. We assume g ωω f 1 = g ρρ f 1 ≡ g VV f 1 ; see (A9). The cross section first rises from the threshold √ s thr = 2m p + m f 1 to √ s ≈ 5 GeV (PANDA energy range), where it starts to decrease towards higher energies. The region of fast growth of the cross section is related to the fast opening of the phase space, while the reggeization is responsible for the decreasing part. Without the reggeization the cross section would continue to grow. The reggeization, calculated according to (2.17)-(2.19), reduces the cross section by a factor of 1.8 already for the HADES c.m. energy √ s = 3.46 GeV. For comparison we also show the high-energy contribution of the PP → f 1 (1285) fusion (see the red dashed line) with parameters fixed in [27]; see Eq. (3.7) there.
At near-threshold energies one should consider final state interactions (FSI) between the two produced protons; see e.g. [33,48]. But the effect is sizeable only for extremely small excess energies of tens of MeV: Q exc = √ s − √ s thr . In our case, we have Q exc > 300 MeV and this FSI effect can be neglected.
We remind the reader that our calculation of the VV-fusion processes should only be applied at energies √ s 8 GeV. In the intermediate energy range also other processes like f 2R f 2R fusion must be considered; see the discussion in Appendix D of [27].
The salient feature of the results shown in Fig. 4 is the high sensitivity of the VVfusion cross section to the different sets of parameters. In our procedure of extracting the coupling constant g VV f 1 and the form-factor cutoff parameters from the CLAS data [see Appendices B and C] the dominant sensitivity is on g VV f 1 , not on the form factors. Also the form of reggeization used in our model, according to (2.17)-(2.21), affects the size of the cross section. With the parameter values of (C10) we get With the parameter values of (C7) we get As mentioned above, the different numbers in (3.1) and (3.2) compared to (3.3) and (3.4) reflect mainly the different couplings g VV f 1 . Indeed, from (3.3) and (3.1) we get for the cross section ratio 3.8, from (3.4) and (3.2) we get 5.0, and from (C7) and (C10) we get for the ratio of the coupling constants squared 5.6, not far from the two numbers above. . We show also the pomeron-pomeron fusion mechanism (red dashed line). In the right panel, the solid line is for the parameters of (C10) and the reggeized propagators∆ 5.0 GeV (PANDA). One can observe that dσ/dt decreases rapidly at forward scattering |t| → |t| min , where |t| min ≃ 0.3 GeV 2 at √ s = 3.46 GeV. At near threshold energy the values of small |t 1 | and |t 2 | are not accessible kinematically. The maximum of dσ/dt appears at −t 1,2 ≃ 0.65 GeV 2 for the parameter values of (C10) and at −t 1,2 ≃ 0.77 GeV 2 for those of (C11). The close-to-threshold production of the f 1 meson, therefore, probes corresponding form factors, (2.10), (2.11) and (2.15), at relatively large values of |t 1 | and |t 2 |, far from their on mass-shell values at t 1,2 = m 2 V where they were normalised. Thus, the VV-fusion cross section is very sensitive to the choice of the form factors. Therefore the HADES and PANDA experiments have a good opportunity to study physics of large four-momentum transfer squared. In Fig. 6 we present the contributions for the ωωand ρρ-fusion processes separately and their coherent sum (total). The interference term is shown also (see the green solid line). Both processes play roughly similar role. For large values of |t 1 | and |t 2 |, in spite of g ρpp < g ω pp (2.13), the spin-flip term of the ρ 0 -proton coupling is important. For √ s = 5 GeV the ωω-fusion contribution is the dominant process for |t 1,2 | 0.5 GeV 2 . There one can see also a large constructive interference effect.
In Figs. 7 and 8 we show several differential distributions for the reaction pp → pp f 1 (1285) for √ s = 3.46 GeV relevant for the HADES experiment and for the reaction pp → pp f 1 (1285) for √ s = 5.0 GeV relevant for the PANDA experiment, respectively. We show the distributions in the transverse momentum of the f 1 (1285) meson, in x F,M , the Feynman variable of the meson, in the cos θ M where θ M is the angle between k and p a in the c.m. frame, and in φ pp , the azimuthal angle between the transverse momentum vectors p t,1 , p t,2 of the outgoing nucleons in the c.m. frame. We predict a strong preference for the outgoing nucleons to be produced with their transverse momenta being back-to-back (φ pp ≈ π). The distributions in cos θ M for the energies √ s = 3.46 GeV and √ s = 5.0 GeV have a different shape. This is explained in Fig. 9. One can observe from Figs. 6 and 9 that the ωωand ρρ-fusion processes have different kinematic depen- dences. With increasing energy √ s the averages of |t 1 | and |t 2 | decrease (damping by form factors), hence the ωω contribution becomes more important. Now we turn to the pp → pp( f 1 (1285) → ρ 0 γ) reaction and the discussion of background processes.
In Fig. 10  (2.28). For the set of parameters (C10) the VV-continuum contribution, due to the small value of Λ V NN , turns out to be negligible. The situation changes when we use the parameter set of (C7). But still the ππ-continuum contribution is larger than the VV-continuum contribution. In both cases the f 1 (1285) resonance is clearly visible, even without the reggeization effects in the continuum processes. This result makes us rather optimistic that an experimental study of the f 1 in the ρ 0 γ decay channel should be possible.
In our calculations we find practically no interference effects between the ππ and VV fusion contributions in the continuum. For our exploratory study we have neglected interference effects between the background ρ 0 γ and the signal f 1 → ρ 0 γ processes. We have also neglected the background processes due to bremsstrahlung of γ and ρ 0 from the nucleon lines. For an analysis of real data these effects should be included or at least estimated. But this goes beyond the scope of our present paper. Now we wish to discuss the integrated cross sections for the reactions pp → pp( f 1 → ρ 0 γ) and pp → pp( f 1 → ρ 0 γ) treated with exact 2 → 4 kinematics. In our calculation we took into account the reggeization effects according to (2.17)-(2.21) and the replacements given in (2.23). We consider two sets of parameters, (C10) and (C7), extracted from the CLAS data. With the parameter values of (C10) we get for √ s = 3.46 GeV : σ pp→pp( f 1 →ρ 0 γ) = 1.26 nb , for the exclusive axial-vector f 1 (1285) production compared to the continuum processes considered in the ρ 0 γ channel.
In Table I we have collected integrated cross sections in nb for the continuum processes considered. These numbers were obtained for g ρωπ = 10.0, Λ M = 1.45 GeV in (2.36)-(2.39), Λ V NN = 1.35 GeV in (2.15), and Λ πNN = 1.0 GeV in (2.40). The reggeization effects were included. We can observe very small numbers for the production of ρ 0 ρ 0 at √ s = 3.46 GeV which is caused by the threshold behaviour of the process (the assumption of a fixed ρ 0 -meson mass of m ρ = 0.775 GeV in the calculation) and limited phase space. TABLE I. The integrated cross sections in nb for the continuum processes in proton-(anti)proton collisions. We show results for the VVand ππ-fusion contributions separately and for their coherent sum ("total"). Now we compare the cross section for the ρρ continuum from Table I to the cross section for the f 1 (1285) signal according to with σ pp→pp f 1 from Fig. 4 and a branching ratio from [44]. Taking  These roughly estimated results show that, for the cases treated here, the background processes considered in the ρ 0 ρ 0 channel (see Table I) can be important only for √ s = 5.0 GeV in the pp case.
The reaction pp → ppρ 0 ρ 0 is treated technically as a 2 → 4 process. A better approach would be to consider the pp → ppπ + π − π + π − reaction, as a 2 → 6 process. This is however beyond the scope of the present study. In addition, as will be discussed in the following, the background for the pp → ppπ + π − π + π − reaction measured long ago by the bubble chamber experiment [50] was found to be much larger than the result for the continuum terms ("total") presented in Table I.

IV. HADES AND PANDA EXPERIMENTS
The HADES (High Acceptance Dielectron Spectrometer) is a magnetic spectrometer located at the SIS18 accelerator in the Facility for Antiproton and Ion Research (FAIR) in Darmstadt (Germany) [51]. It is a versatile detector allowing measurement of charged hadrons (pions, kaons and protons), leptons (electrons and positrons) originating from various reactions on fixed proton or nuclear targets in the energy regime of a few A · GeV. The spectrometer covers the polar angle region 18 • < θ < 80 • and features almost complete azimuthal coverage w.r.t. the beam axis. The detector has been recently upgraded by a large area electromagnetic calorimeter and a forward detector (for a recent review see [52]) extending the coverage to very forward region (0.5 • < θ < 7.5 • ). These upgrades allow to measure hadron decays involving photons and significantly improve acceptance for protons and hyperons which at these energies are emitted to large extent in forward directions.
The spectrometer is specialized for electron-positron pair detection but it also provides excellent hadron (pion, kaon, proton)-identification capabilities. It has a low material budget and consequently features an excellent invariant mass resolution for electronpositron pairs of ∆M/M ≈ 2.5 % in the ρ/ω/φ vector meson mass region.
The PANDA (antiProton ANnihilations at DArmstadt) detector is currently under construction at FAIR. PANDA will utilise a beam of antiprotons, provided by the High Energy Storage Ring (HESR), and with its almost full solid-angle coverage will be a detector for precise measurements in hadron physics. HESR will deliver antiprotons with momenta from 1.5 GeV/c up to 15 GeV/c (which corresponds to √ s ≃ 2.25 − 5.47 GeV) impinging on a cluster jet or pallet proton target placed in PANDA. The scientific programme of PANDA is very broad and includes charmonium and hyperon spectroscopy, elastic proton form-factor measurements, searches of exotic states and studies of in-medium hadron properties (for a recent review of stage-one experiments see [53]).
The luminosity of both detectors are comparable and are at the level of L = 10 31 cm −2 s −1 (after first years of operation and completion of the detector PANDA will increase it by one order of magnitude).
For the count rate estimates and signal to background considerations for the f 1 meson production we will use the properties of the HADES detector. This presents a "worse case" scenario. As it was shown in previous sections cross sections for the meson production in proton-proton interactions are about a factor 10 lower than for the protonantiproton case. Furthermore, the PANDA detector features also larger acceptance for the reaction of multi-particle finals and presents better opportunities for the studies discussed in this work. On the other hand HADES will measure proton-proton reactions at the c.m. energy √ s = 3.46 GeV (proton beam energy E kin = 4.5 GeV) already in 2021. Hence it will provide first valuable experimental results to verify our model predictions.
A. Simulation for 2π + 2π − and π + π − η decay channels We have considered production of the f 1 (1285) meson in proton-proton reactions and its decay into final states with four charged pions reconstructed in the HADES detector. For the f 1 production cross section we have assumed σ f 1 = 150 nb [estimate using the C7 parameter set; see (C7) and (3.3)].
Two reaction channels were simulated: In the second case the η meson is reconstructed via the η → π + π − π 0 decay channel, hence the final state has also four charged pions. The neutral pion from the η decay can be reconstructed via missing mass technique or via two photon decay. However, the latter case has smaller total reconstruction efficiency (see below for details). The f 1 (1285) meson decay into four charged pions has been simulated using the PLUTO event generator [54][55][56]. For the meson reconstruction four pions from the decay and at least one final state proton have been demanded in the analysis to establish exclusive channel identification. The HADES acceptance and reconstruction efficiencies for protons and pions have been parametrized as a function of the polar and azimuthal angles and the momentum. Furthermore, a momentum resolution ∆p/p = 2 % of the spectrometer for charged tracks has been taken into account in the simulation, as described in [51].
For the pp → pp2π + 2π − reaction a total cross section σ back = (227 ± 23) µb has been measured; see Table I of [50]. This reaction was measured in [50] at slightly higher energies E kin = 4.64 GeV (corresponding to proton beam momentum P = 5.5 GeV/c or √ s ≃ 3.5 GeV). 2 We tried to understand the large background in the π + π − π + π − channel. We analysed a few contributions due to double nucleon excitations. We considered the following processes: Both resonances have considerable branching fraction to the Nππ channel and the N(1535) to the Nη channel; see PDG [44]. In our evaluation (estimation) we used effective Lagrangians and relevant parameters from [57]. These parameters were found in [57] to describe the total cross section for the the pp → pnπ + reaction measured in the close-to-threshold region. The coupling constants and the cutoff parameters in the monopole form factors used in the calculation are the following ones: Similar values were also taken in [58] for the pn → dφ reaction. To describe the total cross sections of the pN → NNππ andpN →NNππ reactions measured in the near-threshold region the cutoff parameters Λ N * NM = 1.0 GeV were assumed in [59,60]. Therefore, our estimates for the reactions (4.3)-(4.5) with the parameters given in (4.6) should be treated rather as an upper limit.
There is a question about the role of the η ′ exchange in the reaction (4.5). For example, in [61] sub-threshold resonance-dominance of the N(1535) was assumed with g 2 N(1535)Nη ′ /4π = 1.1 to describe both the πN → η ′ N and NN → NNη ′ cross section data. However, it was shown in [62] that the N(1535) contribution is not necessary in these processes (see Figs. 9-14 of [62]) or, at least, its significant role (significant coupling strength of N(1535) → η ′ N used in [61,63]) was precluded.
For energy √ s = 3.5 GeV we get the cross section for the pp → N(1440)N(1440) reaction of the order of 0.8 mb. With the input from [64][65][66], g 2 N(1440)Nσ /4π = 1.33 and Λ N(1440)Nσ = 1.7 GeV, we get even smaller cross section by about 30 %. For the pp → N(1440)N(1535) reaction we get the cross section of 10 µb and for the pp → N(1535)N(1535) reaction about 7 µb. So we conclude that the double excitation of the N(1440) resonances via the σ-meson exchange is probably the dominant mechanism of this type in the pp → pp2π + 2π − reaction. This is due to the large N(1440)Nσ coupling. Taking BR(N(1440) → pπ + π − ) = 0.1 we get σ pp→N * N * →pp2π + 2π − ≃ 80 µb. This background is much higher than that for the ωωand ππ-fusion mechanisms considered in Sec. III; see Table I.
So far we have considered the 5-pion background with all components (1, 2, 3) listed in Table II. The contribution (1) can be, in principle, eliminated by using side-band subtraction method. We wish to discuss now separately the contribution (2), in the π + π − η mesonic state, to proof feasibility of the f 1 (1285) measurement. In Fig. 12 we make such a comparison. The nonreducible background contribution from double excitation of N * resonances has a broader distribution than the VV → f 1 signal. With our estimate of the cross section for the pp → pp f 1 (1285) reaction (see Table II) we expect that the f 1 (1285) could be observed in the π + π − η (→ π + π − π 0 ) channel. Invariant mass distribution of π + π − η observed in the ppπ + π − π + π − π 0 final state corresponding to the measurement with the p + p reactions at E kin = 4.5 GeV ( √ s = 3.46 GeV) with the HADES apparatus. Here, the two contributions (2) and (4) of Table II were included. The result includes the cut on the η meson mass 0.54 GeV < M π + π − π 0 < 0.56 GeV.

V. CONCLUSIONS
In the present paper we have discussed the possibility to observe the f 1 (1285) in the pp → pp f 1 (1285) reaction at energies close to the threshold where the pomeron-pomeron fusion, known to be the dominant mechanism at high energies, is expected to give only a very small contribution. Two different mechanisms have been considered: (a) ωω → f 1 (1285) fusion and (b) ρ 0 ρ 0 → f 1 (1285) fusion. We have estimated the cross section for √ s = 3.46 GeV for which a measurement will soon be possible for HADES@GSI. We have presented our method for the derivation of the VV → f 1 (1285) vertex for V = ρ 0 , ω. The coupling constant g ρρ f 1 has been extracted from the decay rate of f 1 → ρ 0 γ using the VMD ansatz. From naive quark model and VMD relations we have obtained equality of the g ρρ f 1 and the g ωω f 1 coupling constants; see Appendix A. In reality this relation can be expected to hold at the 20 % level. Then, we have fixed the cutoff parameters in the form factors and the corresponding coupling constants by fits to the CLAS experimental data for the process γp → f 1 (1285)p. There, the ρand ω-exchange contributions play a crucial role in reproducing the forward-peaked angular distributions, especially at higher energies, W γp > 2.55 GeV.
The corresponding ρρ and ωω fusion amplitudes have been written out explicitly. The two amplitudes have been used to estimate the total and differential cross sections for c.m. energy √ s = 3.46 GeV. The energy dependence close to the threshold has been discussed. The distributions in t (see Fig. 6) and the distributions in cos θ M (see Fig. 9) seem particularly interesting. The shape of these distributions gives information on the role of the individual fusion processes.
We have discussed the possibility of a measurement of the pp → pp f 1 (1285) reaction by the HADES collaboration at GSI. For this, the π + π − π + π − , ρ 0 γ, and π + π − η channels, have been considered. For the four-pion channel we have estimated the background using the cross section from an old bubble chamber experiment [50]. We have found that the double excitation of the N(1440) resonances via the σ-meson exchange is probably the dominant mechanism in the pp → pp2π + 2π − reaction. The mechanisms considered by us: π 0 -ω-π 0 and ω-π 0 -ω exchanges give much smaller background cross sections. We conclude that it may be difficult to identify the f 1 (1285) meson in this channel. The ρ 0 γ channel should be much better suited as far as signal-to-background ratio is considered. There, however, dominant background channel ppπ + π − π 0 is of the order of 2 mb [50] and ρ 0 is so broad that it will not provide sufficient reductions (as it is the case in η decay channel). In our opinion the π + π − η(→ π + π − π 0 ) channel is especially promising. We have performed feasibility studies and estimated that a 30-days measurement with HADES should allow to identify the f 1 (1285) meson in the ppπ + π − η final state. No simulation of the π + π − η(→ π + π − π 0 ) channel has been done for PANDA energies.
In [68] the f 1 (1285) decays into a 0 (980)π 0 , f 0 (980)π 0 and isospin breaking were studied. An interesting proposal was also discussed in [69,70]: to study the anomalous isospin breaking decay f 1 (1285) → π + π − π 0 in central exclusive production of the f 1 . There is another important decay channel, KKπ, with branching fraction 9% [44] which can be used for f 1 meson studies in CEP. See also [71] for a discussion of the KKπ decay and the nature of the f 1 (1285) meson.
Predictions for the PANDA experiment at FAIR, for the pp → pp f 1 (1285) reaction, have also been presented. The possibility to study the underlying reaction mechanisms have been discussed. For the VV → f 1 (1285) fusion processes for √ s = 5.0 GeV we have obtained about 10 times larger cross sections than for √ s = 3.46 GeV. Thus we predict a large cross section for the exclusive axial-vector f 1 (1285) production, compared to the background continuum processes via VV and ππ fusion, in the ρ 0 γ channel. The ρ 0 γ channel seems, therefore, also promising for identifying the f 1 (1285) meson.
To conclude: we have shown that the study of f 1 (1285) production at HADES and PANDA should be feasible. From such experiments we will learn more on the nature of the f 1 . For instance, is it a normal qq state orKK * molecule [71,72]? Can it be described in holographic QCD [18]? In particular, we shall learn from f 1 CEP at low energies about the ρρ f 1 and ωω f 1 coupling strengths. These in turn are very interesting parameters for the calculations of light-by-light contributions to the anomalous magnetic moment of the muon [10][11][12][13][14][15]. The final aim for studies of f 1 CEP in proton-proton collisions should be to have a good understanding of this reaction, both from theory and from experiment, in the near threshold region, in the intermediate energy region 8 GeV √ s 30 GeV, and up to high energies available at the LHC as discussed in [27].
following quark content Consider now a radiative decay of the f 1 . After the emission of the photon by the f 1 the quark state should have the structure, with the quark charges e u = 2/3, e d = −1/3, e s = −1/3: Therefore, this simple argument suggests for the f 1 Vγ coupling constants the relation This is the relation suggested, e.g., in [17]. Now we can combine this with VMD which allows to relate the g VV f 1 and g f 1 Vγ by the standard Vγ transition vertices; see e.g. (3.23)-(3.25) of [28]. This gives, with e = √ 4πα em , where γ ρ > 0, γ ω > 0, and In the naive quark model plus VMD the hadronic light-quark-electromagnetic current is written as follows: J em µ = e e uū γ µ u + e dd γ µ d + e ss γ µ s Assuming m 2 ρ = m 2 ω (which is quite good) and m 2 ρ = m 2 φ (which is less good) we find from (A6) the following "ideal mixing" coupling ratios: From (A4) and (A7) we obtain with the "ideal" γV couplings With (A3) plus (A8) we obtain, thus, the simple estimate based on naive quark-model relations plus the simplest VMD ansatz.
That is, ideal mixing (A7) gives only an approximation, valid to within 15 %, compared to the experimental value (A11). We can, therefore, expect that also the relation (A9) may be violated in the real world by 15 to 20 %. We emphasize that the arguments presented in this Appendix depend crucially on the assumption made in (A1) that the f 1 (1285) is a normal qq state. The relation (A3) in particular could be quite different if this assumption is violated and the f 1 (1285) has another structure. In [71,72], for instance, the f 1 (1285) is described as a K * K molecule, not as a qq state. From (B1) and (B2) we will estimate the g ρρ f 1 coupling constant and the cutoff parameter Λ V in the form factor F ρρ f 1 from experiment.
Then, the coupling constant g ρρ f 1 , occurring in Γ (ρρ f 1 ) in the amplitudes above, can be adjusted to the experimental decay width Γ( f 1 (1285) → γρ 0 ). For the 1 → 2 decay process (B1) this is straightforward. For the 1 → 3 decay process (B2) this will be done with the help of a new Monte Carlo generator DECAY [73] designed for a general decay of the 1 → n type.
Unfortunately the partial decay width Γ( f 1 (1285) → γρ 0 ) appears to be not well known in the literature, see also the discussion in Sec. VII C and Table IV in [16], from PDG [44] : from CLAS [16] : Using the values of total widths accordingly from PDG (2.27) and the CLAS experiment (2.28) we get from PDG [44] : from CLAS [16] : We note that the CLAS result is in agreement with that found in [74], where the decay f 1 (1285) → ρ 0 γ was studied in the reaction π − N → π − f 1 N. Theoretical estimates based on the QCD inspired models such as the covariant oscillator quark model [75] and the Nambu-Jona-Lasinio model [8], which assume that the f 1 (1285) has a quarkantiquark nature, suggest (B8) rather than (B7). We hope that the future experimental measurements can clarify this issue. In the following we shall use both values, (B7) and (B8), to highlight the problem.
In Table III we collect our results for the two processes (B1) and (B2) obtained from (B7) and (B8). In the calculations we take m ρ = 775 MeV. We show results for the cutoff parameter from Λ ρ = 0.65 GeV to 2 GeV in (2.11). We expect the upper limit of the ρρ f 1 coupling constant to be not much larger than |g ρρ f 1 | ≃ 20. Otherwise one gets a nonrealistically large cutoff parameter Λ V NN in the V NN vertex (see the discussion in Appendix C). It is also interesting to compare our results with those of [72]. In [72] the radiative decays f 1 (1285) → γV were evaluated with the assumption that the f 1 (1285) is dynamically generated from the K * K interaction. In this model the partial decay widths strongly depend on the cutoff parameter Λ, for instance, Γ( f 1 (1285) → γρ 0 ) = 560 keV, or 1360 keV, for Λ = 1.0 GeV, or 2.5 GeV, respectively; see Table I of [72]. Moreover, there were also determined the ratios The dependence of both ratios on the cutoff parameter is rather weak. In the model of [72] the partial decay width of Γ( f 1 → γρ 0 ) is much larger than the ones of the γω and γφ channels due to constructive (destructive) interference of the triangle loop diagrams for the ρ 0 (ω and φ) production. Now we consider the decay f 1 → ωγ in our approach. We use the formula of (B3) with the replacements ρ → ω [γ ρ → γ ω (A5), g f 1 ρρ → g f 1 ωω , m ρ → m ω ]. In the calculation we take m ω = 783 MeV. We assume g ωω f 1 = g ρρ f 1 (A9) and take g ρρ f 1 corresponding to Λ ρ = 0.65 GeV and 2.0 GeV from Table III With Λ ρ = 0.65 GeV (first line in Table III), we obtain Γ( f 1 → γω) = 106.61 keV for |g f 1 ωω | = 27.37 and Γ( f 1 → γω) = 34.90 keV for |g f 1 ωω | = 15.66. Using the central values of (B7) and (B8) these correspond to the ratios of R 2 = 12.98 and R 2 = 12.99, respectively. With Λ ρ = 2.0 GeV (fourth line in Table III), we obtain Γ( f 1 → γω) = 112.61 keV for |g f 1 ωω | = 9.27 and Γ( f 1 → γω) = 36.81 keV for |g f 1 ωω | = 5.30. With the central values of (B7) and (B8) we obtain the ratios R 2 = 12.31 and R 2 = 12.30, respectively. These values for R 2 are about 2 times smaller than (B13) estimated in [72].
The recent average for R 1 given by PDG [44] is R 1 = 82.4 +11. 4 −23.8 . This is about 1 s.d. away from the theoretical result (B12) of [72]. But we have to keep in mind the differences in the width of f 1 → γρ 0 given by PDG and CLAS; see (B7) and (B8). There are currently no experimental data available for f 1 (1285) → γω decay. Further experiments will hopefully clarify the situation.

Appendix C: Photoproduction of the f 1 (1285) meson and comparison with the CLAS experimental data
Here we discuss the photoproduction of the f 1 (1285) meson. Using VMD and the g VV f 1 coupling constants introduced in (2.8) we have to calculate the diagram shown in Fig. 13. The differential cross section for the reaction γp → f 1 (1285)p will be compared with the CLAS data [16]. From this we will estimate the form factor and cutoff parameters of the model. . 13. Photoproduction of an f 1 meson via vector-meson exchanges.
The unpolarized differential cross section for the reaction γp → f 1 (1285)p is given by Here we work in the center-of-mass (c.m.) frame, s is the invariant mass squared of the γp system, and q and k are the c.m. three-momenta of the initial photon and final f 1 (1285), respectively. Taking the direction of q as a z axis we denote the polar and azimuthal angles of k by θ and φ.
We use standard kinematic variables The amplitude for the γp → f 1 (1284)p reaction via the vector-meson exchange includes two terms The generic amplitude with V = ρ 0 , ω, for the diagram in Fig. 13, can be written as where p b , p 2 and λ b , λ 2 = ± 1 2 denote the four-momenta and helicities of the incoming and outgoing protons.
We use the relations for the γ-V couplings (V = ρ 0 , ω) from (A4) and (A5). For the other building blocks of the amplitude (C4) see (2.8)-(2.21) in Sec. II. We can then write We perform the calculation of the total and differential cross sections with the cutoff parameter Λ ρ and corresponding VV f 1 coupling constant g VV f 1 from Table III. We choose the values from the last column (CLAS). For instance, |g ρρ f 1 | = 8.49 corresponds to Λ ρ = 1.0 GeV and |g ρρ f 1 | = 20.03 corresponds to Λ ρ = 0.65 GeV. We assume g ωω f 1 = g ρρ f 1 ≡ g VV f 1 ; see (A9). For the V pp coupling constants we take (2.13). For the V-proton form factor F V NN (t) we take the monopole form as in (2.15) with the parameter Λ V NN to be extracted from the CLAS data.
In the bottom right panel of Fig. 14 the individual ρand ω-exchange contributions at W γp = 2.75 GeV are shown. Here we use the parameters given in (C10). The ρ-exchange term is larger than the ω-exchange term due to larger coupling constants both in the γ → V transition vertex (A5), (A11) and for the tensor coupling in the V-proton vertex (2.12), (2.13). The differential distribution at W γp = 2.75 GeV peaks for cos θ = 0.7 corresponding to −t = 0.66 GeV 2 . The tensor coupling in the ρ-proton vertex with parameters κ ρ F ρNN (t) plays the most important role there. One can observe also an interference effect between the ρ and ω exchange terms. 3 In Fig. 15 we show the integrated cross sections for the reaction γp → f 1 (1285)p together with the CLAS data. Results for −0.8 < cos θ < 0.9 are presented. In the calculation we take (C10). In the left panel we show the respective contributions of ρ and ω exchanges and their coherent sum with the same V-proton coupling parameters as in the bottom right panel of Fig. 14. There, for W γp ≃ 2.7 GeV, a large interference between the ρ exchange term and the ω exchange term can be observed. In the right panel we compare our reggeized model results with those of the model without this effect. We note that the form of reggeization used in our model, calculated according to (2.17)-(2.21), affects both, the t-dependence of the V exchanges, and the size of the cross section. FIG. 14. The differential cross sections for the reaction γp → f 1 (1285)p → ηπ + π − p. Data are taken from Table V of [16]. The vertical error bars are the statistical and systematic uncertainties. Our results are scaled by a factor of 0.35 to account for the branching fraction from f 1 (1285) → ηπ + π − (C6). We take the V pp coupling constants from (2.13) and the different values of g VV f 1 corresponding to Λ V from the column "CLAS" of Table III. In the bottom right panel we show the individual contributions of ρ and ω exchanges and their coherent sum (total) at W γp = 2.75 GeV. For the ρ-exchange contribution also the results for only one type of coupling, tensor or vector, in the ρ-proton vertex (2.12) are shown. . The elastic f 1 (1285) photoproduction cross section as a function of the center-of-mass energy W γp . Five data points are obtained by integrating out the differential cross sections given in Table V of [16]. The experimental results have been scaled by the branching fraction BR( f 1 (1285) → ηπ + π − ) = 0.35; see (C6). We take the coupling parameters the same as in the bottom right panel of Fig. 14. We integrate for −0.8 < cos θ < 0.9. In the left panel the reggeized contributions of ρ and ω exchanges, their coherent sum (total), and the interference term are shown. In the right panel the solid line is the result from the reggeized model, the dotted line indicates the result without the reggeization.