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An isotopic effect in phi photoproduction at a few GeV

A distinct isotopic effect in phi photoproduction at 2-5 GeV region is identified by examining the production amplitudes due to Pomeron-exchange and meson-exchange mechanisms. This effect is mainly caused by the pi-eta interference constrained by SU(3) symmetry and the isotopic structure of the gamma NN coupling in the direct phi-radiation amplitude. It can be tested experimentally by measuring differences in the polarization observables between the gamma-p and gamma-n reactions.

that the data at high energies can be described by a Pomeron trajectory with an intercept α(0) ∼ 1.08. This trajectory will be called the P 1 trajectory in this paper. Within Regge theory, it is easy to see that the P 1 trajectory can only be identified with objects with spin J > 0. It was suggested [7] that a different Pomeron trajectory, as inspired by the glueball (J π = 0 + , M 2 gb ≃ 3 GeV) predicted by Lattice QCD calculations and a QCD-based Dual Landau-Ginsburg model [13], could dominate the cross section at low energies by having a negative intercept α(0). This Pomeron trajectory, called P 2 , will be also considered in this work.
The DL-model was examined in Ref. [2] by using an Euclidean QCD model of mesons to evaluate exactly the quark-loop integration in Fig.2. It was shown that the interpretation of the Pomeron as gluonic object is consistent with the data up to very large Q 2 and W . For the Q 2 = 0 and W < 5 GeV region considered in this work, the exact loop integration, as done in Ref. [2], can be approximated by the factorized form employed by Donnachie and Landshoff. We therefore follow Donnachie and Landshoff and use this simplified version of the model to define the Pomeron-exchange amplitudes. We will not consider two-gluon-exchange models of Pomeron as proposed in Refs. [4,5,14].
Within the DL-model, the invariant amplitude can be written as where ε γν and ε φµ are respectively the polarization vectors of the photon and φ meson, F Pn α describes the Pomeron-nucleon vertex, and Γ Pnα,µν is associated with the Pomeron-vectormeson coupling which is related to the γ → qq vertex Γ ν and the qq → φ vertex V µ , as shown in Fig. 2. We follow the DL-model and assume that the P 1 -nucleon coupling is photon-like. For the P 2 -nucleon vertex, we assume a scalar coupling. Accordingly, we have where u(p) is the nucleon spinor with momentum p. By using the factorization approximation of Donnachie and Landshoff, an explicit evulation [9] of the Dirac algebra associated with the quark-loop shown in Fig.2 leads to the following form of the Pomeron-vector meson coupling in Eq.(1) where the transversality conditions M P 1,2 µν ·q µ = M P 1,2 µν ·k ν = 0 are fullfilled. The strength factors M P 1,2 0 in Eq.(1) take the conventional form of Regge parameterization where the trajectories are taken to be α 1 (t) = α 1 (0) + α ′ 1 t = 1.08 + 0.25 t, α 2 (t) = −0.75 + 0.25 t, s 0 = α ′ −1 .
As discussed above, α 1 (t) is chosen to fit the data at large s and α 2 (t) is inspired by the (J π = 0 + ,M 2 gb ∼ 3 GeV 2 ) glueball, predicted by Lattice QCD calculations and a QCD-based Dual Landau-Ginsbug model [13]. We follow Ref. [6] to define the overall form factor in Eq.(4) as F (t) = F φ (t) · F N (t) [6] with The correcting functions F 1,2 (s, t) for Eq.(4) are The strength factor C 1 in Eq.(4) is chosen to reproduce dσ/dt| θ=0 at large s, where the cross section is entirely due to P 1 trajectory. The strength factor C 2 is chosen to reproduce dσ/dt| θ=0 at low energies where all mechanisms shown in Fig.1 are important. The slope B is chosen to reproduce the available data of angular distributions, dσ/dt. Note that the phase of the Pomeron exchange amplitudes is fixed and controlled by the exponential factors in (4). At t = 0, P 1 amplitude is almost purely imaginary and negative, while the P 2 amplitude is complex. Pseudoscalar exchange amplitudes. We calculate the (π, η)-exchange by using the following effective Lagrangians where ϕ = π 0 , η, and A β is the photon field. For the πNN coupling, we use the standard value g πN N = (4π · 14) 1/2 . The status of the ηNN coupling is not so clear. The value of g 2 ηN N /4π reported in the literatures varies from 0 to about 7. The largest value comes from fits to NN scattering data, using one boson exchange potential models [15]. But many other studies [16][17][18][19] favor a smaller value, g 2 ηN N /4π ∼ 0.3 − 1.0. In our calculation we accept the SU(3) symmetry prediction [20] based on the most recent value of F/D = 0.575 [21]. This leads to g ηN N = (0.99 · 4π) 1/2 which is close to its "overall" averaged value. Later, we will discuss how our prediction of polarization observables can be used to pin down the value of g ηN N .
The magnitude of the coupling constant g φγϕ in Eq.(8) can be estimated through the decay widths [22] of φ → γπ and φ → γη. We obtain |g φγπ | = 0.14 and |g φγη | = 0.707. Consistent with our choice of ηNN coupling constant discussed above, we also employ the SU(3) symmetry to determine their signs. We find that their signs are the same but are opposite to the sign of g ωγπ , which is found to be positive in the pion photoproduction [23,24]. Since g ηpp and g π 0 pp (here we include the sign due to the matrix element of the isospin factor τ 3 ) have the same sign, we expect a constructive interference between π-and η-exchange amplitudes in the γp → φp reaction. The situation changes in γn → φn reaction. In this case the isospin property determines that g π 0 nn (= −g π 0 pp ) is of oppisite sign of g ηnn (= g ηpp ) and hence the corresponding interference becomes destructive.
The vertices in Fig.1b must be dressed by form factors. We choose the standard form Here we assume that the range of the π−exchange is identical to that determined by Friman and Soyeur [25] in their study of ρ and ω photoproduction. This leads us to choose Λ π = 0.7 GeV and Λ π φγπ = 0.77 GeV. There is no reliable source to fix the cutoff parameters of the η-exchange amplitude. For definitness, we set Λ η = 1 GeV and Λ φγη = 0.9 GeV which are the values for the σ−exchange determined by Friman and Soyeur [25] in their fit to the ρ photoproduction. These choices of cutoff parameters represent the main uncertainty of our predictions. Nevertheless, they are comparable to the values used in the literature and are sufficient for the present investigation. We find that our main conclusion concerning the isotopic effect is not changed unless very hard form factors( Λ α > about 1.5 GeV) are used. The hard form factors are in general not favored in reproducing the total cross sections at high energies. Direct φ − radiation amplitude. We evaluate this amplitude (Fig.1c) by using the following interaction Lagrangians with κ p(n) = 1.79(−1.91). The resulting direct and crossed amplitudes in Fig.1c are real and their signs are controlled by the constants g φN N and κ φ . Here we choose g φN N = −0.24 and κ φ = 0.2, which are obtained from an estimate [10] of the OZI-violation contributions due to φπρ couplings and the interaction of φ with the kaon cloud of the nucleon. The calculation of Ref. [10] is also constrainted by SU(3) sysmmetry, consistent with our evaluation of the (π, η)−exchange amplitudes discussed above. The φNN vertices in Fig.1c may be dressed by form factors. Namely, the direct and crossed terms must be dressed by F φ s and F φ u respectively. A possible form is [11,26] F φ a (a) = Λ 4 φ /(Λ 4 φ + (a − M 2 N ) 2 ), a = s, u. But this results in strong violation of gauge invariance with respect to the photon and φ meson fields. One can use the gauge invariant prescription of [27], where a universal form factor F φ (s, u) = 1 2 (F φ s (s) + F φ u (u)) is used to regularize both φNN vertices. But our predictions would then have an additional uncertainty [27] depending on the cutoff parameter Λ φ . To avoid this problem we use the prescription of [28] and simply set F φ s = F φ u = 1. Our prediction of the contribution from the direct φ−radiation (Fig.1c) may therefore be considered as an upper limit. Results. With the (π, η)-exchange and direct φ-radition amplitudes defined above, our first task is to determine the parameters of Pomeron-exchange amplitudes. The parameters s i and C i of Eq.(4) are not independent in the fit to the data. For example, a smaller value of s 2 will lead to a larger C 2 in the fit. For simplicity we set s 2 = 0. To account for the threshold behavior of the P 1 amplitude, we set s 1 = M N + M φ . We then find that the dσ/dt(θ = 0) data of γp → φp reaction can be reproduced with ξ = C 2 /C 1 = −0.55 and C 1 = 2.34. Our result is the thick solid curve in the left panel of Fig.3. The slope B = 1.7 GeV −2 of the form factor Eq.(6) is needed to reproduce the angular distribution, as shown in the right panel of Fig.3. Here we also display the relative importance between different amplitudes. We note that the P 1 contribution drops sharply as energy approaches the production threshold. On the other hand, the P 2 contribution increases very quickly as energy decreases. This striking difference in energy-dependence between P 1 and P 2 is due to the different signs of their intercepts: α 1 (0) = 1.08 and α 2 (0) = −0.75(see Eq. (5)). By comparing the π-curve and (π, η)-curve in Fig.3, we see that the π and η amplitudes interfere constructively. This is exactly what we expected in the discussion given above. The direct φ-radiation clearly dominates the cross sections at large angles.
The fits to the γp → φp data completely fix the model and we can now apply the model to predict the φ photoproduction on neutron. Our results for γn → φn are shown in Fig.4. For comparison, we also display the corresponding γp → φp results. Here we observe that π− and η−exchange interfere destructively(comparing the π-curve and (π, η)-curve). The direct φ-radiation contribution(dash-dotted curves) is drastically different from that in Fig.3. This is due to the large difference between the γpp and γnn vertices. However, the change in the cross sections due to this isotopic effect is very small except at large scattering angles (comparing the top two curves in Fig.4) where the cross sections are very small. Experimental verification of this prediction is very difficult.
We now point out that the isotopic effect discussed above can be verified by examining polarization observables. As an example, we show in Fig.5 our prediction of the φ decay asymmetry [29] where σ (σ ⊥ ) are the cross sections induced by photons polarized in the direction parallel (perpendicular) to the γ − φ plane. These two cross sections refer to measurements of the symmetric K + K − pairs produced in the φ decay. Our prediction of Σ φ at E γ = 2 GeV are shown in Fig.5. We find that the predicted isotopic effect at forward angles is largely due to the π − η interference. This can be understood by comparing the (π, η)−curve and φNNcurve in the right panels of Figs. 3 and 4. Consequently, the predicted energy-dependence at θ = 0 shown in the left panel of Fig.6 can be used to explore the relatively undetermined ηNN coupling constant. To facilitate such a study, we show the dependence of Σ φ (θ = 0) on g ηN N in the right panel of Fig.6. In this calculation, the strength C 2 of the P 2 Pomeron is adjusted to maintain the same fit to the γp → φp data as g ηN N is varied in the calculation. At large angles, the cross sections are dominanted by the direct φ−radiation amplitude. Hence a measurement of Σ φ at large angles can be used to study the φNN coupling. However, such a measurement may be very difficult because the cross sections become very small at large angles.
In summary, we have developed a model to describe the φ photoproduction at low energies, √ s ∼ 2 − 5 GeV. In addition to the traditional Pomeron trajectory with an intercept of α 1 (0) = 1.08, the model contains a second Pomeron trajectory with a negative intercept α 2 (0) = −0.75 inspired by the (J π = 0 + , M 2 gb ∼ 3 GeV) glueball predicted by Lattice QCD calculations and a QCD-based Dual Landau-Ginsburg model [13]. The model also includes the contributions from (π, η)-exchange and direct φ-radiation from the nucleon. We have found a strong isotopic effect in the predicted φ decay asymmetry (Figs.5, 6), indicating the possibility of using the φ photoproduction polarization observables to explore the ηNN and φNN couplings. Experiments for measuring such observables at LEPS of SPring-8 (Japan) and TJNAF (USA) are highly desirable.
Physics of Osaka University. This work is also partially supported by the U.S. Department of Energy, Nuclear Physics Division, under Contract NO. W-31-109-ENG-38.