$\eta$ photoproduction off the deuteron and low-energy $\eta$-nucleon interaction

We study $\eta$ photoproduction off the deuteron ($\gamma d\to\eta pn$) at a special kinematics: $\sim 0.94$ GeV of the photon beam energy and $\sim 0^\circ$ of the scattering angle of the proton. This kinematics is ideal to extract the low-energy $\eta$-nucleon scattering parameters such as $a_{\eta N}$ (scattering length) and $r_{\eta N}$ (effective range) because the $\eta$-nucleon elastic scattering is significantly enhanced. We show that if a ratio $R$, the $\gamma d\to\eta pn$ cross section divided by the $\gamma p\to\eta p$ cross section convoluted with the proton momentum distribution in the deuteron, is measured with 5% error, ${\rm Re}[a_{\eta N}]$ (${\rm Re}[r_{\eta N}]$) can be determined at the precision of $\sim\pm$0.1 fm ($\sim\pm$0.5 fm), significantly narrowing down the currently estimated range of the parameters. The measurement is ongoing at the Research Center for Electron Photon Science (ELPH), Tohoku University.


Introduction
The low-energy η-nucleon interaction can be characterized with two parameters, the scattering length a ηN and effective range r ηN . The existence of exotic η-mesic nuclei largely depends on a ηN that determines the attractive or repulsive nature of the low-energy ηN interaction [1]. However, a ηN has not been well determined yet. Previous works have attempted to extract a ηN and r ηN by analyzing the πN → πN, ηN and γN → πN, ηN reaction data [1], and also the pn → ηd reaction data [2]. These analyses gave fairly consistent results for the imaginary parts of a ηN and r ηN which are within Im[a ηN ] = 0.2-0.3 fm and Im[r ηN ] = −1-0 fm, respectively [1]. However, their real parts are not welldetermined: Re[a ηN ] = 0.2-0.9 fm and Re[r ηN ] = −6 to +1 fm. The large model-dependence in the real parts stems from the difficulty of isolating the ηN scattering amplitudes from other mechanisms involved in the reactions analyzed.
An ongoing η photoproduction experiment [3] at the Research Center for Electron Photon Science (ELPH), Tohoku University is designed to overcome the difficulty of determining a ηN by utilizing a special kinematics. In this experiment, a photon beam with E γ ∼ 0.94 GeV hits a deuteron target and the recoil proton from γd → ηpn is detected at θ p ∼ 0 • . At this kinematics, an η produced from a quasi-free proton is almost at rest, and thus it would interact strongly with the spectator neutron. On the other hand, the struck proton goes away with a large momentum, and thus it would not interact with the η and neutron. This seems an ideal kinematical condition, referred to as the ELPH kinematics, to determine the low-energy ηN scattering parameters. The present theoretical analysis [4] will show that a combined cross-section data for γd → ηpn and γp → ηp expected to be taken in the ELPH experiment would indeed lead to significant reduction of the current uncertainty of a ηN and r ηN .

Model
We study γd → ηpn relevant to the ELPH experiment with a model based on the impulse and the first-order rescattering mechanisms as illustrated in Fig and NN → NN processes with M (′) =π, η, as well as with a realistic deuteron wave function. By doing so, we can reliably isolate the amplitude for the ηN → ηN subprocess from data using well-predicted background contributions. Regarding γN → MN and MN → M ′ N amplitudes, we employ those generated with a dynamical coupled-channels (DCC) model [5,6]. The DCC model is a multichannel unitary model for the πN and γN reactions in the nucleon resonance region. It was constructed fitting ∼ 27, 000 data points, and successfully describes [5,6] πN → πN, ππN, ηN, KΛ, KΣ and γN → πN, ππN, ηN, KΛ, KΣ reactions over the energy region from the thresholds up to √ s 2.1 GeV. For example, the DCC model describes γp → ηp differential cross sections in a very good agreement with data over the energy region relevant to the following calculations of γd → ηpn. This confirms that the most important γp → ηp amplitudes among the elementary amplitudes for describing γd → ηpn have been well tested by the data. This DCC model predicts the ηN scattering parameters to be a ηN = 0.75 + 0.26i fm and r ηN = −1.6 − 0.6i fm, which are consistent with the previously estimated ranges. As for the deuteron wave function and the NN scattering amplitudes, we use the CD-Bonn potential [7] to generate them.

Result
We can make a parameter-free prediction for the γd → ηpn cross sections using the model described above. We can thus assess the validity of the model by confronting our model predictions with existing data. In Fig. 2(left), we show dσ/dΩ η at E γ = 775 MeV from our DCC-based model with and without the rescattering contributions along with the data. Our parameter-free prediction is found to be in an excellent agreement with the data. The ηN → ηN rescattering gives a slight enhancement in the backward direction, which is important for this nice agreement. A similar DCCbased model for γd → πNN [9,10] also gives predictions that agree well with data by taking account of significant rescattering effects.
We now move on to the γd → ηpn reaction at the ELPH kinematics (E γ = 0.94 GeV and θ p = 0 • ). In Fig. 2(right-top), the predicted threefold differential cross section, d 3 σ/dM ηn dΩ p , are presented as a function of the η-neutron invariant mass M ηn . The impulse mechanism [ Fig. 1(a)] including the γp → ηp (γn → ηn) amplitudes gives the dominant (negligible) contribution. A substantial contribution is from the η-exchange mechanism [ Fig. 1(b)], and the cross sections including the impulse mechanisms only are changed by −40 to +20% [difference between the dashed and dotted This feature is what we expect to find in this special kinematics, and indicates that the proton essentially does not interact with the ηn system. Thus multiple rescatterings beyond the first-order rescattering [Figs. 1(b)-1(d)] should be safely neglected for M ηn 1.5 GeV. We have also examined an off-shell momentum effect associated with the ηn → ηn scattering amplitude and found it very small. Because we are interested in a M ηn region close to the threshold, higher partial waves for the ηn → ηn amplitudes are negligible. Therefore, we modify the full γd → ηpn model by replacing the ηn scattering amplitude with the S -wave one parametrized with a ηN and r ηN . These scattering parameters are determined by analyzing the forthcoming ELPH data.
The ELPH data will be given in a form of the ratio, denoted by R expt , of the measured cross sections for γd → ηpn divided by those for γp → ηp convoluted with the proton momentum distribution in the deuteron. This is for removing systematic uncertainties of the acceptance from the detector coverage. Therefore, from the theoretical side, we need to calculate the corresponding quantity given by where σ full (σ imp ) is calculated with the full model (the impulse term only). Now the question is how sensitive R th is against changing a ηN and r ηN . Also, we are interested in what is the required precision of R expt for significantly reducing the current uncertainties of a ηN and r ηN . First Re[a ηN ] is changed over 0.2 -1.0 fm, with Im[a ηN ] = 0.25 fm and r ηN = 0 fm being fixed. The resulting cross sections cover the red striped region as shown in Fig. 3(top), within the considered ELPH kinematics and M ηn ≤ 1.505 GeV. The ratio R th also changes accordingly as shown in Fig. 3(bottom); R th shows more clearly the sensitivity to the variation of Re[a ηN ]. The cross section and thus R th changes by ∼25% at the quasi-free (QF) peak position of M ηn ∼ 1.488 GeV, as indicated by the width of the striped band. We also show the green solid bands that have the widths of ∼5% at the QF peak. This green band is covered by our model when Re[a ηN ] is varied by ±0.1 fm from 0.6 fm. This means that R expt data of 5% error per MeV bin can determine Re[a ηN ] at the precision of ∼ ±0.1 fm, significantly reducing the currently estimated range. Data of R expt with this precision is expected to be taken in the ongoing ELPH experiment [3]. Next Re[r ηN ] is varied over −6 -0 fm which is the currently estimated range, while the scattering length being fixed at the value from the latest DCC analysis [6], a ηn = 0.75+0.26i fm; Im[r ηN ] = 0 fm. Accordingly, the cross section and R th change over the red striped region in Fig. 4. The effect of changing r ηN is visible at ∼5 MeV above the ηN threshold. The ratio R th at M ηn = 1.5 GeV changes by ∼30% (∼5%) when Re[r ηN ] is changed over −6 -0 fm (−3.5 to −2.5 fm) as indicated by the red striped (green solid) band. Therefore, Re[r ηN ] at the precision of ±0.5 fm, which is significantly improved precision over the current estimates, can be obtained by measuring R expt data of 5% error per MeV bin.