Joint Physics Analysis Center

                                                                                         
This project is supported by NSF

$\gamma p\to \pi^0 \eta p$

kinematics We present the model published in [Mat19a] illustrating the polarized moments for the reaction \begin{align} \vec \gamma (p_\gamma) p(p_N) \to \eta(p_\eta) \pi^0(p_\pi) p(p'_N), \end{align} with a linearly polarized beam. The code can be downloaded in Resources section and simulated in the Simulation section.

Definitions

$\Omega = (\theta,\phi)$ are the angles of the $\eta$ in the $\eta\pi$ rest frame. $\Phi$ is the angle between the polarization and the reaction plane. The angles are illustrated on the figure in the helicity frame. The other kinematical variables are the total energy squared $s = (p_\gamma+p_N)^2$, the momentum transferred $t = (p_N- p'_N)^2$ and the $\eta\pi$ mass $m_{\eta\pi}^2 = (p_\eta+p_\pi)^2$.

With a linearly plarized beam, with polarization $P_\gamma$, the intensity is decomposed as \begin{align} I(\Omega,\Phi) & = I^0(\Omega) - P_\gamma I^1(\Omega) \cos 2 \Phi - P_\gamma I^2(\Omega) \sin 2 \Phi. \end{align} The moments are defined by \begin{align} H^0(LM) & =\phantom{- \frac{2}{P_\gamma}} \frac{1}{2\pi}\int I(\Omega,\Phi) d^L_{M0}(\theta) \cos M \phi \ d\Omega d\Phi ,\\ H^1(LM) & = - \frac{2}{P_\gamma}\frac{1}{2\pi}\int I(\Omega,\Phi) d^L_{M0}(\theta) \cos M \phi \cos 2\Phi\ d\Omega d\Phi ,\\ \text{Im }H^2(LM) & = \frac{2}{P_\gamma}\frac{1}{2\pi}\int I(\Omega,\Phi) d^L_{M0}(\theta) \cos M \phi \sin 2 \Phi\ d\Omega d\Phi. \end{align}

The $4\pi$ integrated beam asymmetry $\Sigma_{4\pi}$ and the beam asymmetry along the $y$ axis $\Sigma_y$ are defined by \begin{align} \Sigma_{4\pi} & = \frac{1}{P_\gamma} \frac{\int [I(\Omega,0) - I(\Omega,\frac{\pi}{2})] d\Omega} {\int [I(\Omega,0) + I(\Omega,\frac{\pi}{2})] d\Omega } \\ \Sigma_y & = \frac{1}{P_\gamma} \frac{[I(\Omega_y,0) - I(\Omega_y,\frac{\pi}{2})] } { [I(\Omega_y,0) + I(\Omega_y,\frac{\pi}{2})] }. \end{align} They can be expressed with the moments ($\Sigma_y$ truncated to $L=4$) \begin{align} \Sigma_{4\pi} & = \frac{H^1(00)}{H^0(00)} \\ \Sigma_y & = \frac{H^1(00) - \frac{5}{2} H^1(20) - 5 \sqrt{\frac{3}{2}} H^1(22) + \frac{27}{8} H^1(40) + \frac{9}{2}\sqrt{\frac{5}{2}} H^1(42) + \frac{9}{4}\sqrt{\frac{35}{2}} H^1(44)} {H^0(00) - \frac{5}{2} H^0(20) - 5 \sqrt{\frac{3}{2}} H^0(22) + \frac{27}{8} H^0(40) + \frac{9}{2}\sqrt{\frac{5}{2}} H^0(42) + \frac{9}{4}\sqrt{\frac{35}{2}} H^0(44)} \end{align}

Model

We use the reflectivity basis. Our toy model include only the following waves \begin{align} [\ell]^{(\epsilon)}_m = S_0^{(+)}, P_{0,1}^{(+)}, D_{0,1,2}^{(+)}. \end{align} $\epsilon$ is the reflectivity of the partial wave.
One can show that $\text{Im }H^2(LM) = -H^1(LM)$ with a basis icluding only waves $[\ell]^{(\epsilon)}_m$ with positive projection $m\ge 0$.
We thus will not display the moments $\text{Im} H^2(LM)$.
The wave are parametrized in the $s$-channel frame by \begin{align} [\ell]^{(\epsilon)}_m & = N_0 N_R \left( \delta_R \frac{\sqrt{-t}}{m_R} \right)^{|m-1|} \frac{m_R \Gamma_R}{m_R^2-m_{\eta\pi}^2-i m_R \Gamma_R}\ \Gamma[1-\alpha(t)](1-e^{-i\pi\alpha(t)}) s^{\alpha(t)} \end{align} where $R = a_0(980), \pi_1(1600), a_2(1320), a_2(1700) $ are the resonances in the partial wave $\ell=0,1,2$.
$N_0 = 20000$ is an overal normalization. The vector trajectory is $\alpha(t) = 0.5 + 0.9t$. $s$ and $t$ are expressed in GeV$^2$ in these expressions. The resonance parameters are indicated in the Table.

Model parameters. $m_R$ and $\Gamma_R$ are in GeV.
$R$ | $m_R$ $\Gamma_R$ | $\phantom{-}$$N_R$ $\phantom{-}$$\delta_R$
$a_0(980)$ | 0.980 0.075 | $\phantom{-}$ 1.00 $\phantom{-}$1.0
$\pi_1(1600)$ | 1.564 0.492 | $-0.03$ $-5.0$
$a_2(1320)$ | 1.318 0.107 | $-0.109$ $-2.0$
$a_2(1700)$ | 1.722 0.247 | $-0.036$ $-2.0$

They correspond to the values used in the publication [Mat19a] .
The parameters can be changed in the section Simulation

References

[Mat19a]
V. Mathieu, et al (JPAC),
``Moments of angular distribution and beam asymmetries in $\eta\pi^0$ photoproduction at GlueX,'' arXiv:1906.04841 [hep-ph],

Resources

The zip file contains the file main.c, main_GJ.c, modules.c and their header files.
The file main_GJ.c contains the same function as main.c but rotate the SDME to the GJ frame.
To compile: gcc main.c main_GJ.c modules.c

  1. par_simu.txt:
    The simulation parameters are $E_\gamma$, $t$, $dm$, $isGJ$
  2. par_model.txt:
    The model parameters are $m_R$, $\Gamma_R$, $x_R=1$, $N_r$, $\delta_R$.
    The three are lines for the three resonances: $a_0(980)$, $\pi_1(1600)$, $a_2(1320)$.
  3. momentX.txt:
    The first column is the $\eta\pi$ mass in GeV. The other columns are $LM=$00, 10,11, 20,21,22, 30,31,32,33, 40,41,42,43,44.
  4. BA.txt:
    The first column is the $\eta\pi$ mass in GeV. The second column is the 4 $\pi$ beam asymmetry. The third column is the beam asymmetry along the $y$ axis.
Funtion in main.c
  1. Moments: Compute the moments $H^{\alpha}(LM)$ from the SDME
  2. sdme: Compute the SDME $\rho^{\alpha, \ell \ell'}_{mm'}$ from the partial waves.
  3. partialwaves: Compute the partial waves for our toy model.
  4. printMoments: Print the moments in files.
  5. intH and intMoments: integrate the moments in $t$


A general code to reconstruct moments from partial waves is available: C/C++ moments
The zip file contains the files main.cpp, tools.cpp, moments.cpp and their hearders.
The file moments.cpp contains the routines returning all moments for given S, P, D waves (all spin projections).
The file tools.cpp contains the Wigner-d functions and other accessory functions.
The file main.cpp generates random waves and call the moments routines.

Simulation

The user can choose the beam energy (in the target rest frame) and the momentum transferred $t$.
The moments are computed in the interval $m_\eta + m_\pi \le m_{\eta\pi} \le 2.1$ GeV. The binning in $m_{\eta\pi}$$dm$ is specified by $dm$.
The moments are computed in the helicity frame but can be displayed in the GJ frame. The simulation takes a few second to rotate the moments the GJ frame.

Kinematics:
$t$ in GeV$^2$
$dm$ in GeV
Frame:

Model parameters:
Resonance $a_0(980)$ $\pi_1(1600)$ $a_2(1320)$ $a_2(1700)$
mass ($m_R$) in GeV
width ($\Gamma_R$) in GeV
normalization ($N_R$)
spin-flip coupling ($\delta_R$)