# Joint Physics Analysis Center

This project is supported by NSF

# $\gamma p\to\eta^{(')} p$

We present the model published in [Mat17a] concerning th beam asymmetry of $\eta$ and $\eta'$ beam asymmetries.
We report here only the main features of the model.
The code can be downloaded in Resources section and simulated in the Simulation section.

Denoting $\eta$ and $\eta'$ quantities by bare and primed symbols respectively, the beam asymmetry is defined by \begin{align} \Sigma^{(')} & = \frac{d \sigma^{(')}_\perp - d \sigma^{(')}_\parallel} {d \sigma^{(')}_\perp + d \sigma^{(')}_\parallel}, \end{align} with $d \sigma_\perp$ and $d \sigma_\parallel$ denoting the differential cross section with a photon polarization parallel and perpendicular to the reaction plane. Natural exchanges $\rho,\omega$ and $\phi$ contribute to $d\sigma_\perp$ and unnatural exchanges $b,h$ and $h'$ contribute to $d\sigma_\parallel$. The other unnatural exchanges $\rho_2, \omega_2$ and $\phi_2$ also contribute to $d\sigma_\parallel$. We separate the contribution from natural and unnatural exchanges \begin{align}\label{eq:kNkU} k_N &= \frac{d\sigma'_\perp}{d\sigma_\perp}, & k_U &= \frac{d\sigma'_\parallel}{d\sigma_\parallel}. \end{align} and rewrite the ratio of $\eta'$ and $\eta$ beam asymmetries as \begin{align} \nonumber \frac{\Sigma'}{\Sigma} & = 1+ \frac{1-\Sigma^2}{\Sigma} \cdot \frac{k_N - k_U}{(1+\Sigma) k_N + (1-\Sigma) k_U}, \\ & \equiv 1+ \epsilon \label{eq:ratio} \end{align} We use the CGLN invariant amplitudes $A_i$ defined in [Chew57a].
The scalar amplitudes $A_i = \sum_{V,A,E} A_i^V + A_i^A + A_i^E$ receive contribution from $V = \rho, \omega, \phi$, $A = b, h, h'$ and $E = \rho_2, \omega_2, \phi_2$. For the natural Regge poles $V = \rho,\omega,\phi$ (with $s$ expressed in GeV$^2$): \begin{align} \nonumber A_{1}^{(')V}(s,t) & = t \beta^{(')V}_{1}(t) \frac{1- e^{-i\pi \alpha_V(t)}}{ \sin\pi \alpha_V(t)} s^{\alpha_V(t)-1} & A_2^{(') V}(s,t) & = (-1/t) A_{1}^{(')V}(s,t) \\ A_{4}^{(')V}(s,t) & = \phantom{t} \beta^{(')V}_{4}(t) \frac{1- e^{-i\pi \alpha_V(t)}}{ \sin\pi \alpha_V(t)} s^{\alpha_V(t)-1} & A_3^{(') V}(s,t) & = 0 \label{eq:V} \end{align} The factor $t$ in $A_{1}^{(')V}$ comes from the factorization of the Regge pole residues and conservation of angular momentum.
The unnatural exchange contribution are $A = b,h,h'$ and $E = \rho_2,\omega_2,\phi_2$ \begin{align} A_{2}^{(')A}(s,t) & = \beta^{(')A}_{2}(t) \frac{1- e^{-i\pi \alpha_A(t)}}{ \sin\pi \alpha_A(t)} s^{\alpha_A(t)-1} & A_{1}^{(')A}(s,t) & = A_{3}^{(')A}(s,t) = A_{4}^{(')A}(s,t) = 0 \\ A_{3}^{(')E}(s,t) & = \beta^{(')E}_{2}(t) \frac{1- e^{-i\pi \alpha_E(t)}}{ \sin\pi \alpha_E(t)} s^{\alpha_E(t)-1} & A_{1}^{(')E}(s,t) & = A_{2}^{(')E}(s,t) = A_{4}^{(')E}(s,t) = 0 \end{align}
In this webpage we propose the following flexible parametrization for the residues and trajectories (ommitting the index $V,A,E$) \begin{align} \label{eq:betas} \beta^{(')}_i(t) & = g^{(')}_{i\gamma} g_{i N} e^{b_i t} (1-\gamma_{i,1} t) (1-\gamma_{i,2} t) \\ \alpha(t) & = \alpha_{0} + \alpha_{1} t \end{align}
The observables are expressed with the scalar amplitudes (K is an irrelevant kinematical factor): \begin{align} \label{eq:cgln} d\sigma^{(')}_\perp(s,t) & = K \left[ |A^{(')}_1|^2 - t|A^{(')}_4|^2 \right], & d\sigma^{(')}_\parallel(s,t) & = K \left[ |A^{(')}_1 +t A^{(')}_2|^2 - t|A^{(')}_3|^2 \right] \end{align} so that the relevant quantities are \begin{align} k_N &= \frac{|A^{'}_1|^2 - t|A^{'}_4|^2}{|A_1|^2 - t|A_4|^2}, & k_U &= \frac{|A^{'}_1 +t A^{'}_2|^2 - t|A^{'}_3|^2}{|A_1 +t A_2|^2 - t|A_3|^2}. \end{align}

## References

[Chew57a]
G.F. Chew, M.L. Goldberger, F.E. Low and Y. Nambu,
Relativistic dispersion relation approach to photomeson production,'' Phys. Rev. 106, (1957) 1345.

[Mat17a]
V. Mathieu, J. Nys, C. Fernandez-Ramirez, A. Jackura, M. Mikhasenko, A. Pilloni, A. P. Szczepaniak and G. Fox (JPAC),
On the $\eta$ and $\eta'$ Photoproduction Beam Asymmetries,'' arXiv:1704.07684 [hep-ph],

## Resources

1. param.txt: The first line is the beam energy (in the lab frame) in GeV
The next 3x3 lines corresponds to the $\rho, \omega$ and $\phi$ exhchanges.
There are 3 lines for each exchange with the format:
• $g_{\eta \gamma}$ $g_{\eta' \gamma}$ $\alpha_0$ $\alpha_1$
• $g_{1}$ $b_{1}$ $g_{4}$ $b_{4}$
• $\gamma_{1,1}$ $\gamma_{1,2}$ $\gamma_{4,1}$ $\gamma_{4,2}$
The next 6x2 lines corresponds to the $b,h,h'$ and $\rho_2,\omega_2,\phi_2$ exhchanges.
There are 2 lines for each exchanges with the format:
• $g_{\eta \gamma}$ $g_{\eta' \gamma}$ $\alpha_0$ $\alpha_1$
• $g_{2(3)}$ $b_{2(3)}$ $\gamma_{2(3),1}$ $\gamma_{2(3),2}$
2. EtaBA.txt: The data for $\gamma p \to \eta p$ \begin{align*} t (\text{GeV}^2) \quad \cos\theta \quad \frac{d\sigma}{dt} (\mu\text{b/GeV}^2) \quad \frac{d\sigma}{d\Omega} (\mu\text{b}) \quad \Sigma \end{align*} The total cross sections $\sigma(\pi^\pm p)$ are in milli barns.
3. EtaP-BA.txt: The results of the simulations in the format \begin{align*} t(\text{GeV}^2) \quad \Sigma(\eta) \quad k_V \quad k_A \quad 10^4*\epsilon \quad \Sigma(\eta') \quad 1+\epsilon \end{align*}

## Simulation

For each exchange, the user can supply parameters (residues and trajectories).
The parameters from [Mat17a] are the default values.
The simulation displays the beam asymmetries, their ratio $\Sigma'/\Sigma$ and, $k_V$ and $k_A$.

Beam energy in the lab frame (target rest frame):

 $\rho$ $g_{\rho \eta \gamma}$ : $g_{\rho \eta' \gamma}$: $\alpha_{0,\rho}$: $\alpha_{1,\rho}$: $g_{1\rho}$ : $b_{1\rho}$: $g_{4\rho}$ : $b_{4\rho}$: $\gamma_{1,1}^\rho$ : $\gamma_{1,2}^\rho$: $\gamma_{4,1}^\rho$ : $\gamma_{4,2}^\rho$: $\phantom{t}$ $\omega$ $g_{\omega \eta \gamma}$ : $g_{\omega \eta' \gamma}$: $\alpha_{0,\omega}$: $\alpha_{1,\omega}$: $g_{1\omega}$ : $b_{1\omega}$: $g_{4\omega}$ : $b_{4\omega}$: $\gamma_{1,1}^\omega$ : $\gamma_{1,2}^\omega$: $\gamma_{4,1}^\omega$ : $\gamma_{4,2}^\omega$: $\phantom{t}$ $\phi$ $g_{\phi \eta \gamma}$ : $g_{\phi \eta' \gamma}$: $\alpha_{0,\phi}$: $\alpha_{1,\phi}$: $g_{1\phi}$ : $b_{1\phi}$: $g_{4\phi}$ : $b_{4\phi}$: $\gamma_{1,1}^\phi$ : $\gamma_{1,2}^\phi$: $\gamma_{4,1}^\phi$ : $\gamma_{4,2}^\phi$:

 $b$ $g_{b \eta \gamma}$ : $g_{b \eta' \gamma}$: $\alpha_{0,b}$: $\alpha_{1,b}$: $g_{2b}$ : $b_{2b}$: $\gamma_{2,1}^b$ : $\gamma_{2,2}^b$: $\phantom{t}$ $h$ $g_{h \eta \gamma}$ : $g_{h \eta' \gamma}$: $\alpha_{0,h}$: $\alpha_{1,h}$: $g_{2h}$ : $b_{2h}$: $\gamma_{2,1}^h$ : $\gamma_{2,2}^h$: $\phantom{t}$ $h'$ $g_{h' \eta \gamma}$ : $g_{h' \eta' \gamma}$: $\alpha_{0,h'}$: $\alpha_{1,h'}$: $g_{2h'}$ : $b_{2h'}$: $\gamma_{2,1}^{h'}$ : $\gamma_{2,2}^{h'}$:

 $\rho_2$ $g_{\rho_2 \eta \gamma}$ : $g_{\rho_2 \eta' \gamma}$: $\alpha_{0,\rho_2}$: $\alpha_{1,\rho_2}$: $g_{2\rho_2}$ : $b_{2\rho_2}$: $\gamma_{2,1}^{\rho_2}$ : $\gamma_{2,2}^{\rho_2}$: $\phantom{t}$ $\omega_2$ $g_{\omega_2 \eta \gamma}$ : $g_{\omega_2 \eta' \gamma}$: $\alpha_{0,\omega_2}$: $\alpha_{1,\omega_2}$: $g_{2\omega_2}$ : $b_{2\omega_2}$: $\gamma_{2,1}^{\omega_2}$ : $\gamma_{2,2}^{\omega_2}$: $\phantom{t}$ $\phi_2$ $g_{\phi_2 \eta \gamma}$ : $g_{\phi_2 \eta' \gamma}$: $\alpha_{0,\phi_2}$: $\alpha_{1,\phi_2}$: $g_{2\phi_2}$ : $b_{2\phi_2}$: $\gamma_{2,1}^{\phi_2}$ : $\gamma_{2,2}^{\phi_2}$: