 # Joint Physics Analysis Center This project is supported by NSF

# $\gamma p\to a_2 p$ and $\gamma p\to f_2 p$ We present the model published in [Mat19a] concerning tensor meson photoproduction \begin{align} \vec \gamma (p_\gamma, \lambda_\gamma) p(p_N, \lambda_p) \to T(p_T, \lambda_T) p(p'_N, \lambda_p'), \end{align} with a linearly polarized beam. The code can be downloaded in Resources section and simulated in the Simulation section.

## Models

For each exchange $E = \rho, \omega, b_1, h_1$ the amplitudes takes the form \begin{align} {\cal M}_{\lambda_\gamma \lambda_T ; \lambda_p \lambda'_p} &= -\sum_E T^{E}_{\lambda_\gamma \lambda_T}(t) R^E(s,t) B^E_{\lambda_p \lambda'_p}(t), \end{align} The Regge propagator is given by: \begin{equation}\label{eq:ReggePropa} R(s,t) = \frac{\tau + e^{-i\pi\alpha(t)}}{2} \, (-)^{\ell} \Gamma\left[\ell-\alpha(t)\right] (\alpha' s)^{\alpha(t)} . \end{equation} The Regge trajectories for the vector $V = \rho, \omega$ and the axial-vector $A = h_1,b_1$ are \begin{align*} \alpha_V(t) & = \alpha'_V t + 0.5 & \alpha_A(t) & = \alpha'_A (t-m_\pi^2) \end{align*} The bottom vertex depends on two helicity couplings for the vector exchange and one coupling for the axial-vector exchange. The top vertex depends on five couplings \begin{align} B^{V}_{\lambda_p \lambda'_p}(t) &= \left( \frac{-t'}{4m_p^2} \right)^{\frac{1}{2}|\lambda_p - \lambda'_p|} \left[ G^V_1 \delta_{\lambda_p, \lambda'_p} + 2 \lambda_p G^V_2 \delta_{\lambda_p, -\lambda'_p} \right] \\ B^A_{\lambda_p \lambda_p}(t) &= \left( \frac{-t'}{4m_p^2} \right) G_2^A \delta_{\lambda_p, -\lambda'_p} \\ T^{E}_{\lambda_\gamma \lambda_T}(t) & = \beta^{\gamma T}_E \left(\frac{-t'}{m_T^2}\right)^{\frac{1}{2}|\lambda_\gamma-\lambda_T|} \beta_{\lambda_\gamma \lambda_T}(t). \end{align} The nucleon couplings are \begin{align} G_1^\rho & = 1.63 & G_2^\rho &= 13.01& G_1^\omega &= 8.13& G_2^\omega & = 1.86 \end{align} The fitted overall photon couplings are \begin{align} \text{minimal: }&& \beta^V & = 0.251 \pm 0.053 & \beta^A & = 0.821 \pm 0.023 \\ \text{TMD: }&& \beta^V & = 1.060 \pm 0.073 & \beta^A & = 0.581 \pm 0.053 \\ \end{align} The model includes only isoscalar axial-vector exchanges $\beta^{\gamma T}_{b_1} = 0$. The $a_2$ and $f_2$ couplings are related by isospin relations \begin{align} \beta^V & = \beta^{\gamma a_2}_\omega = 3\beta^{\gamma a_2}_\rho = \beta^{\gamma f_2}_\rho = 3\beta^{\gamma f_2}_\omega , & \beta^A & = \beta^{\gamma a_2}_{h_1} = 3\beta^{\gamma f_2}_{h_1} \end{align} The helicity structures $\beta_{\lambda,\lambda_T}$ for the two models are

 $\beta_{1,2}$ $\beta_{1,1}$ $\beta_{1,0}$ $\beta_{1,-1}$ $\beta_{1,-2}$ Minimal $0$ $1/2$ $-1/\sqrt{6}$ $0$ $0$ TMD $-1/2$ $-t\big/2m_T^2$ $t\big/2\sqrt{6} m_T^2$ $0$ $0$ M1 $0$ $1/4$ $-1/\sqrt{6}$ $1/4$ 0

## References

[Mat20a] V. Mathieu, et al (JPAC), Exclusive tensor meson photoptoduction,'' arXiv:2005.01617 [hep-ph]

## Resources

The file main.cpp contains all the routines
To compile: gcc main.cpp

1. par_simu.txt:
The simulation parameters are $E_\gamma$, $\beta_V$, $\beta_A$, iso, $isGJ$.
iso = 0,1 for $f_2$ / $a_2$ photoproduction.
2. sdme0.txt and sdme1.txt:
$-t$ (GeV$^2$), $\rho^{0,1}_{00}$, $\rho^{0,1}_{11}$, $\rho^{0,1}_{22}$, $\rho^{0,1}_{10}$, Re $\rho^{0,1}_{1-1}$, Re $\rho^{0,1}_{20}$, Re $\rho^{0,1}_{21}$, Re $\rho^{0,1}_{2-1}$, Re$\rho^{0,1}_{2-2}$
3. sdme2.txt:
$-t$ (GeV$^2$), Im $\rho^2_{10}$, Im $\rho^2_{1-1}$, Im $\rho^2_{20}$, Im $\rho^2_{21}$, Im $\rho^2_{2-1}$, Im $\rho^2_{2-2}$
4. sdme_nat.txt and sdme_unn.txt:
$-t$, $\rho^{(\pm)}_{00}$, $\rho^{(\pm)}_{11}$, $\rho^{(\pm)}_{22}$, $\rho^{(\pm)}_{10}$, Re $\rho^{(\pm)}_{1-1}$, Re $\rho^{(\pm)}_{20}$, Re $\rho^{(\pm)}_{21}$, Re $\rho^{(\pm)}_{2-1}$, Re$\rho^{(\pm)}_{2-2}$
5. obs.txt:
$-t$ (GeV$^2$), $d\sigma/dt$ ($\mu$b/GeV$^2$), $\Sigma_{4\pi}$, $P_\sigma$

## Simulation

The user can choose the beam energy (in the target rest frame), the tensor meson $a_2(1320)$ or $f_2(1270)$ and the model.
The differential cross section, the integrated beam asymmetry and the parity asymmetry are frame independent.
The SDME are computed in the helicity frame.

Simulation specification:
 $E_\gamma$ in GeV Tensor meson: $a_2(1320)$ $f_2(1270)$ Model: Minimal TMD

Model parameters:
 Resonance $\beta_V$ $\beta_A$ Minimal TMD