$\gamma p\to\eta^{(\prime)} p$

High energy model for $\eta^{(\prime)}$ beam asymmetry photoproduction

We present the model published in [Mathieu:2017jjs] 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^{(\prime)} & = \frac{d \sigma^{(\prime)}_\perp - d \sigma^{(\prime)}_\parallel} {d \sigma^{(\prime)}_\perp + d \sigma^{(\prime)}_\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 [Chew:1957tf].
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}^{(\prime)V}(s,t) & = t \beta^{(\prime)V}_{1}(t) \frac{1- e^{-i\pi \alpha_V(t)}}{ \sin\pi \alpha_V(t)} s^{\alpha_V(t)-1} & A_2^{(\prime) V}(s,t) & = (-1/t) A_{1}^{(\prime)V}(s,t) \\ A_{4}^{(\prime)V}(s,t) & = \phantom{t} \beta^{(\prime)V}_{4}(t) \frac{1- e^{-i\pi \alpha_V(t)}}{ \sin\pi \alpha_V(t)} s^{\alpha_V(t)-1} & A_3^{(\prime) V}(s,t) & = 0 \label{eq:V} \end{align} The factor $t$ in $A_{1}^{(\prime)V}$ 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}^{(\prime)A}(s,t) & = \beta^{(\prime)A}_{2}(t) \frac{1- e^{-i\pi \alpha_A(t)}}{ \sin\pi \alpha_A(t)} s^{\alpha_A(t)-1} & A_{1}^{(\prime)A}(s,t) & = A_{3}^{(\prime)A}(s,t) = A_{4}^{(\prime)A}(s,t) = 0 \\ A_{3}^{(\prime)E}(s,t) & = \beta^{(\prime)E}_{2}(t) \frac{1- e^{-i\pi \alpha_E(t)}}{ \sin\pi \alpha_E(t)} s^{\alpha_E(t)-1} & A_{1}^{(\prime)E}(s,t) & = A_{2}^{(\prime)E}(s,t) = A_{4}^{(\prime)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^{(\prime)}_i(t) & = g^{(\prime)}_{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^{(\prime)}_\perp(s,t) & = K \left[ |A^{(\prime)}_1|^2 - t|A^{(\prime)}_4|^2 \right], & d\sigma^{(\prime)}_\parallel(s,t) & = K \left[ |A^{(\prime)}_1 +t A^{(\prime)}_2|^2 - t|A^{(\prime)}_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}



Format of the input and output files: [show/hide]

  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*}


For each exchange, the user can supply parameters (residues and trajectories).
The parameters from [Mathieu:2017jjs] 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):

Natural exchanges (vector exchanges): [show/hide]

$\rho$ $g_{\rho \eta \gamma}$ $g_{\rho \eta' \gamma}$ $\alpha_{0,\rho}$ $\alpha_{1,\rho}$
$g_{1\rho}$ $b_{1\rho}$ $g_{4\rho}$ $b_{4\rho}$
$\gamma_{1,1}^\rho$ $\gamma_{1,2}^\rho$ $\gamma_{4,1}^\rho$ $\gamma_{4,2}^\rho$
$\omega$ $g_{\omega \eta \gamma}$ $g_{\omega \eta' \gamma}$ $\alpha_{0,\omega}$ $\alpha_{1,\omega}$
$g_{1\omega}$ $b_{1\omega}$ $g_{4\omega}$ $b_{4\omega}$
$\gamma_{1,1}^\omega$ $\gamma_{1,2}^\omega$ $\gamma_{4,1}^\omega$ $\gamma_{4,2}^\omega$
$\phi$ $g_{\phi \eta \gamma}$ $g_{\phi \eta' \gamma}$ $\alpha_{0,\phi}$ $\alpha_{1,\phi}$
$g_{1\phi}$ $b_{1\phi}$ $g_{4\phi}$ $b_{4\phi}$
$\gamma_{1,1}^\phi$ $\gamma_{1,2}^\phi$ $\gamma_{4,1}^\phi$ $\gamma_{4,2}^\phi$

Unnatural exchanges (vector exchanges): [show/hide]

$b$ $g_{b \eta \gamma}$ $g_{b \eta' \gamma}$ $\alpha_{0,b}$ $\alpha_{1,b}$
$g_{2b}$ $b_{2b}$ $\gamma_{2,1}^b$ $\gamma_{2,2}^b$
$h$ $g_{h \eta \gamma}$ $g_{h \eta' \gamma}$ $\alpha_{0,h}$ $\alpha_{1,h}$
$g_{2h}$ $b_{2h}$ $\gamma_{2,1}^h$ $\gamma_{2,2}^h$
$h'$ $g_{h' \eta \gamma}$ $g_{h' \eta' \gamma}$ $\alpha_{0,h'}$ $\alpha_{1,h'}$
$g_{2h'}$ $b_{2h'}$ $\gamma_{2,1}^{h'}$ $\gamma_{2,2}^{h'}$

Unnatural exchanges (pseudo-tensor exchanges): [show/hide]

$\rho_2$ $g_{\rho_2 \eta \gamma}$ $g_{\rho_2 \eta' \gamma}$ $\alpha_{0,\rho_2}$ $\alpha_{1,\rho_2}$
$g_{2\rho_2}$ $b_{2\rho_2}$ $\gamma_{2,1}^{\rho_2}$ $\gamma_{2,2}^{\rho_2}$
$\omega_2$ $g_{\omega_2 \eta \gamma}$ $g_{\omega_2 \eta' \gamma}$ $\alpha_{0,\omega_2}$ $\alpha_{1,\omega_2}$
$g_{2\omega_2}$ $b_{2\omega_2}$ $\gamma_{2,1}^{\omega_2}$ $\gamma_{2,2}^{\omega_2}$
$\phi_2$ $g_{\phi_2 \eta \gamma}$ $g_{\phi_2 \eta' \gamma}$ $\alpha_{0,\phi_2}$ $\alpha_{1,\phi_2}$
$g_{2\phi_2}$ $b_{2\phi_2}$ $\gamma_{2,1}^{\phi_2}$ $\gamma_{2,2}^{\phi_2}$