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
- Publications: [Mat17a]
- C/C++: C/C++ file
- Input file: param.txt , EtaBA.txt .
- Output files: EtaP-BA.txt .
- Contact person: Vincent Mathieu
- Last update: May 2017
- 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:- $g_{\eta \gamma}$ $g_{\eta' \gamma}$ $\alpha_0$ $\alpha_1$
- $g_{1}$ $b_{1}$ $g_{4}$ $b_{4}$
- $\gamma_{1,1}$ $\gamma_{1,2}$ $\gamma_{4,1}$ $\gamma_{4,2}$
There are 2 lines for each exchanges with the format:- $g_{\eta \gamma}$ $g_{\eta' \gamma}$ $\alpha_0$ $\alpha_1$
- $g_{2(3)}$ $b_{2(3)}$ $\gamma_{2(3),1}$ $\gamma_{2(3),2}$
- 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.
- 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*}