On the $\eta$ and $\eta'$ Photoproduction Beam Asymmetry at High Energies

We show that, in the Regge limit, beam asymmetries in $\eta$ and $\eta'$ photoproduction are sensitive to hidden strangeness components. Under reasonable assumptions about the couplings we estimate the contribution of the $\phi$ Regge pole, which is expected to be the dominant hidden strangeness contribution. The ratio of the asymmetries in $\eta'$ and $\eta$ production is estimated to be close to unity in the forward region $0<-t/\text{GeV}^2 \leq 1$ at the photon energy $E_\text{lab} = 9$~GeV, relevant for the upcoming measurements at Jefferson Lab.

Meson photoproduction plays an important role in studies of the hadron spectrum and searches for exotic mesons [1][2][3][4][5], in particular for hybrids. The latter contain a large gluon component and are predicted in phenomenological models of QCD and lattice simulations [6][7][8][9][10][11]. Identifying the nature of new resonances requires to establish their quantum numbers first, which constrain both decays and production mechanisms. At Jefferson Lab, with photon energies E lab ∼ 5 − 11 GeV, meson resonances are produced via beam frag-pected soon. Similar measurements will also be performed by the CLAS12 experiments [14] in the near future.
In this letter we give an estimate for the η photoproduction beam asymmetry at high energies. The beam asymmetry is defined as with dσ ⊥, ≡ dσ ⊥, dt (s, t) denoting the differential cross section for photons polarized perpendicular or parallel to the reaction plane, s and t are the usual Mandelstam variables. Unprimed and primed quantities refer to η and η , respectively. We wish to estimate the quantity Using the recent measurements of the η beam asymmetry [13,15], one can extract the quantities 2dσ ⊥ dσ ⊥ + dσ = 1 + Σ, 2dσ dσ ⊥ + dσ = 1 − Σ. ( It is convinient to rewrite the ratio under interest as In order to evaluate this ratio, one must determine the quantities In our evaluation of k V,A , we proceed as follows. We first identify the Regge poles. For the natural exchanges, we extract their residues at the photon vertex by considering the radiative decays and the residues at the nucleon vertex from nucleon-nucleon total cross section. We then estimate the residues of the unnatural exchanges. Finally we give our prediction for in Eq. (2) and list the possible deviations from our assumptions.
Regge poles are classified according to the internal quantum numbers of the particles with the lowest spin located on the corresponding trajectory. The natural exchanges dominating the η ( ) photoproduction are ρ, ω and φ, and the unnatural ones are b, h and h . 1 Asymptotically dσ ⊥ and dσ are dominated by natural and unnatural exchanges, respectively, and so are the corresponding cross section ratios k V and k A . In absence of hidden strangeness (ss) in the proton and for a vanishing contribution from the associated exchange of φ and h mesons, 2 one finds where ϕ is the η − η mixing angle in the flavor basis. Phenomenological analyses of radiative decays give tan ϕ ∼ 0.7 − 0.9 [17], consistent with predictions of chiral Lagrangians [18,19]. Eq. (6) implies Σ = Σ, so one concludes that a sizable deviation of the ratio of beam asymmetries from unity indicates either non-negligible contributions from hidden strangeness φ and h exchanges, and/or significant deviation from the quark model description of the η mesons or from the Regge pole dominance.
The leading contributions to the differential 1 The lowest spin exchange in the b trajectory is the isovector 1 +− state in the PDG [16], the b1(1235). Similarly, the lowest spin exchange on the h and h trajectory are the isoscalar 1 +− states, the h1(1170) and h1(1380). The former is observed decaying into ρπ, the latter intoKK * (892), which suggests ideal mixing. 2 We assume ideal ω/φ and h/h mixing.
where K is a kinematical factor that cancels out in the polarization observables. The A i , i = 1, . . . , 4 are the conventional CGLN invariant amplitudes [21]. At leading order in the energy, the scalar amplitudes A i are related to the s−channel helicity amplitudes A λ ,λλγ 4 by These combinations of (s−channel) helicity amplitudes are parity conserving in the t−channel. A 1 and A 4 involve natural exchanges and A 1 +tA 2 and A 3 involve unnatural exchanges. 5 Specifically, we write the scalar The factorization of Regge residues yields a simple form for the kinematical singularities in t [22] A λ ,λλγ ∝ √ −t |λγ |+|λ−λ | .
Comparing Eq. (8) and Eq. (10), we see that A 1 , hence the residue β 1 (t), needs to be proportional to t. In the (physical) region under consideration, i.e. the forward direction where t is small and negative, the residues β ( )V i (t), with kinematical singularities removed, are regular real functions of the momentum transfered t. A standard parametrization of the residues [15,23,24], β ∝ 1/Γ(α) describes both the exponential suppression seen in data and zeros at ghost poles. 6 In general, however, since all natural (unnatural) poles have approximatively the same trajectory α V (t) (α A (t)), the beam asymmetry depends weakly on the details of the t-dependence (c.f. Eq. (2)) and is primarily determined by the relative strengths of the various exchanges. Accordingly, for the evaluation of the ratio in Eq. (2), we use The additional factor of t in β V 1 (t) is required by factorization of Regge residues, as we noticed above, and can be understood using an effective Lagrangian to desribe the exchange of a ρ meson [20]. The g 1V and g 4V denote the nucleon couplings and the g ( ) V γ denotes the coupling at the γη ( ) vertex. In Eq. (11), we kept a universal exponential slope b. In the ratio of beam asymmetries in Eq. (2) the exponential factor cancels out. It is needed for the determination of the ρ nucleon helicity flip coupling g 1ρ when fitting π − p → π 0 n differential cross section, discussed below.
It is well known that the ρ and ω trajectories are almost degenerate α ω (t) = α ρ (t) = 0.9t + 0.5 (with t expressed in GeV 2 ). For the φ Reggeon we assume the same slope (α ), but take into account the difference between the masses that determine the intercepts and, with the amplitudes described above, one obtains, Factorization of Regge residues allowed us to write the residues as a product of γη ( ) coupling g ( ) V γ and two nucleon couplings (g 1V and g 4V ). The photon couplings in η and η photoproduction can be estimated from radiative decays using The extracted couplings are summarized in Table 1. At leading order in the energy, g 1V and g 4V correspond to the s-channel helicity flip and non-flip couplings at the nucleon vertex respectively [20]. By denoting the flip and non-flip amplitudes as A V +,±1 , in the high-energy limit one obtains  (4) The nucleon non-flip couplings g 4V are determined by fitting the pp,pp, pn andpn total cross sections. At high energies the relevant exchanges contributing to these are the Pomeron P and the Regge poles f 2 , a 2 , ρ, ω and φ. From the optical theorem it follows that the total cross section is related to imaginary part of the forward scattering amplitude. We denote The axial exchanges vanish in the forward direction because of charge conjugation invariance, and do not contribute to the total cross section. The relative contribution of individual poles to the total cross section is given by where the top (bottom) sign corresponds to a (anti-)proton beam. We use the intercept values α ρ 0 = α ω 0 = α a 2 0 = α f 2 0 = 0.5, α φ 0 = 0.0 and α P 0 = 1.08. Only the data with p lab ≥ 15 GeV are included in the fit. The relevant couplings resulting from the fit are summarized in Table 2. The comparison to the data is presented on Fig. 1. We note that our value for the ratio of the nucleon couplings g 4φ /g 4ω = 1.29 (9) is significantly bigger than the same ratio extracted in other analyses. For instance in [25] it was found g 4φ /g 4ω = 0.34. This is because we neglected a large systematic uncertainty on g 4φ . We indeed note that g 4φ depends on the data selected for the fit. For instance if instead of selecting data with p lab > 15 GeV, we choose p lab > 30 GeV we obtain g 4φ = 4.20 (1.72) where the other nucleon couplings g 4ρ and g 4ω remain unchanged. In our approach the influence of the φ is nevertheless suppressed compared to the ω exchange by the difference in their intercept. This is evident on Fig. 2 where we illustrate the relative strength of the hidden strangeness exchange by the ratio T φ (s)/(T ω (s) + T φ (s)). Our value for this ratio lies in the range 0.1 − 0.3 in the region p lab = 10 − 100 GeV (see Fig 1). It is worth noticing that the coupling of the nucleon to the φ meson can be related to the strange electromagnetic form factors G s E and G s M [26][27][28]. These can either extracted from lattice simulations [29,30], or inferred by measurement of low-energy parity violation in ep scattering [31][32][33][34][35][36][37][38][39][40]. This might be an example of how high energy measurements can help in constraining the low energy information. The isoscalar exchanges are empirically found to be dominated by non-flip at the nucleon vertex [23]. Accordingly, we set g 1φ = g 1ω = 0. We therefore only need to determine the nucleon helicity-flip coupling of the ρ ex- change.
To this end, we analyze high-energy π − p → π 0 n data (which contains contribution from the ρ pole only) using We fit the high-energy data of [41] in the forward direction |t| ≤ 0.2 GeV 2 . The differential cross section is given by We do not attempt to fit larger values of |t| since our model has a very simple t dependence.
The main purpose of this fit is to estimate the nucleon helicity-flip coupling g 1ρ . The normalization of the ratio of the amplitudes in Eqs. (17) has been chosen to be in agreement with Eq. (15).
The fit yields b = 2.97(7) GeV −2 and g 1ρ /g 4ρ = 5.91(7) GeV −1 (in agreement with the standard value g 1ρ /g 4ρ 6 GeV −1 [23]) from which one obtains g 1ρ = 13.59(45) GeV −3 . The comparison of the model with data is shown in Fig. 3. One can now evaluate k V in Eq. (13).  Figure 3: π − p → π 0 n differential cross section at high energies in the forward direction. Comparison of the model with data from [41].
The results are given in Table 3 for 11 values of |t| below 1 GeV 2 at E lab = 9 GeV. We now turn our attention to the unnatural exchanges. The exchanges of b, h and h contribute only to A 2 with We consider only the b and h Regge poles in A 2 . By neglecting the hidden strangeness exchange h , the deviation of Σ /Σ from unity will be due to the φ exchange. It is empirically difficult to distinguish between b and h exchange. We will assume that they couple identically to the nucleon g 2b = g 2h and have degenerate trajectories α b (t) = α h (t). The Regge poles on the J P C = (2, 4, . . .) −− 13.49(45) 2.30(7) ω 0 7.28(10) φ 0 9.38(56) trajectory contribute to A 3 only. This amplitude, which is determines the difference between target and recoil asymmetries at high energies, is known to be small for the similar reaction γp → π 0 p [20]. Furthermore, the recent beam asymmetry measurements [13] showed that Σ ≈ 1, setting an upper limit to the A 3 contribution to η photoproduction [15]. Without any indication of significant A 3 contribution in γp → η ( ) p, we ignore it. These assumptions can be relaxed in the more flexible parametrization available online [42]. Under the above assumptions, one obtains Ideally, one would determine the couplings g ( ) Aγ following the procedure used in determination of the vector couplings. Unfortunately, there is no data on axial-vector radiative decays b, h → η ( ) γ. We can, however, estimate k A assuming that the axial-vector exchanges b and h follow the same pattern as the ρ exchange: Alternatively one could use the value obtained from ω decay or a combination of the two. Based on vectormeson dominance (VMD) and SU(3) flavor symmetry, one can estimate the b and h relative couplings to η ( ) γ. Overall the isoscalar contribution is found to be suppressed by a factor of three relative to the isovector contributions due to the γη ( ) vertex. We therefore assume that b is the dominant unnatural exchange. Using VMD and SU(3) flavor symmetry, it is therefore natural to assume that k A = k ρ . Without more information about axial-vector mesons, we consider k A constant in the forward direction. As an estimation of the systematic error, we investigated the effect on when k A = k ω is used.
Because of this, the hidden strangeness exchange is given by the φ only. One has Since k V > k A = k ρ , one expects Σ > Σ and hence > 0. Similarly, one expects < 0 for k A = k ω . The only remaining unknown quantity needed to estimate the ratio Σ /Σ in Eq. (4) is the η beam asymmetry. One could use the recent GlueX data [13] as input. However, the analysis contains measurements at only four values of t. We opt instead to use the predictions from [15], which allows us to evaluate Σ and in a variable t range. This approach is justified by the observation that the prediction is in agreement with the GlueX measurements in [13] and consistent with our hypotheses (negligible h pole, couplings proportional to decay widths and factorization of Regge poles). For completeness, we list the η beam asymmetry from [15] in Table 3 for 11 values of t in the range 0 ≤ −t/GeV 2 ≤ 1. Our final result for Σ /Σ at E lab = 9 GeV is presented in Fig. 4. Note that we applied firstorder error propagation to estimate the statistical error on the relevant quantities. The systematic errors are, however, expected to be larger than the statistical errors. Table 3: List of results for kV and , where the latter is provided for both assumption kA = kρ and kA = kω. We also provide the input for the η beam asymmetry from [15]. t is expressed in GeV 2 .  (34) -47 (51) From Table 3 and Fig. 4, one observes that the quantity = Σ /Σ − 1 is predicted to be small, of the order 10 −3 − 10 −4 for k U = k ρ . This is expected due to the presence of the factor 1 − Σ 2 in Eq. (4). The prediction with k U = k ω in green on Fig. 4 is indicated only to illustrate sensitivity to model assumptions. As we argued above, the value k U = k ρ is favored by SU (3) and the quark model. A sizable deviation of from unity would indicate one or more of the following possibilities: i) the axial-vector meson radiative decay is significantly different from the quark model predictions; ii) the hidden strangeness h Regge pole is not negligible; iii) the φ exchange has been incorrectly estimated; iv) the contribution of exchanges in the A ( ) 3 amplitudes are significant.
Even though the level of precision to resolve a difference between η and η photoproduction beam asymmetries might not be achieved with the GlueX or CLAS12 detectors, we remark that Σ decreases as |t| increases, resulting in increasing with |t|. This trend might nevertheless be observable at the JLab facil- ity. When the measurements will be available, the reader can test all these different hypotheses independently by playing with a flexible parametrization on the JPAC website [42,43]. However, given that the estimate is based on rather reasonable assumptions, a significant deviation from our prediction, if observed at these experiments, would indicate potential for "new physics" needed to describe meson photoproduction.
In our calculation, we proceeded by a separate evaluation of the natural (k V ) and unnatural (k A ) exchanges to photoproduction. Note that the quantity Σ /Σ is only mildly sensitive to the precise value of k A , since the dynamics are dominated by natural exchanges. This is illustrated in Fig. 4 where we plot the result for k A = k ω . Separate measurements of k V and k A , given by Eq. (5), would provide useful information to identify the source of deviations. In particular, it would provide us with more detailed information on the coupling of b and h to η γ relative to their coupling to ηγ (see the discussion related to Eq. (21)). These couplings are experimentally unconstrained at the moment.