$\gamma p\to V p$

We present the model published in [Mathieu:2018xyc] concerning the spin density matrix elements of light neutral vector meson ($\omega$, $\rho^0$, $\phi$) photoproduction.
We report here only the main features of the model.
The code can be downloaded in Resources section and simulated in the Simulation section.

Our model for the helicity amplitudes of vector meson photoproduction involves the $\pi$ and $\eta$ meson unnatural exchanges, Pomeron, $f_2$ and $a_2$ natural exchanges: \begin{align} {\cal M}_{\substack{\lambda_V, \lambda_\gamma \\ \lambda', \lambda}}(s,t) & = \sum_{E = \pi,\eta, {\mathbb P}, f_2, a_2} {\cal M}^{E}_{\substack{\lambda_V, \lambda_\gamma \\ \lambda', \lambda}}(s,t). \end{align} The helicities are defined in the center-of-mass frame of the reaction $\gamma(k,\lambda_\gamma) N(p,\lambda)\to V(q,\lambda_V) N'(p',\lambda')$.
The Mandelstam variables are $s = (k+p)^2$ and $t = (k-q)^2$.
At high energy, the amplitudes for the $\pi$ and $\eta$ meson exchange reads \begin{align} \label{eq:pion} {\cal M}^P_{\substack{\lambda_V, \lambda_\gamma \\ \lambda', \lambda}}(s,t) & = \beta^P_0 \lambda_\gamma \delta_{\lambda,-\lambda'} \sqrt{-t} \left[ \delta_{\lambda_\omega,\lambda_\gamma} - \sqrt{2} \lambda_\gamma \frac{ \sqrt{-t} }{m_V}\delta_{\lambda_\omega,0}+ \frac{-t}{m_V^2} \delta_{\lambda_\omega,-\lambda_\gamma} \right] \\ &\times\frac{1+e^{-i\pi\alpha_\pi(t)}}{2 \sin \alpha_\pi(t)}\left(\frac{s}{s_0} \right)^{\alpha_\pi(t)} \end{align} with $\beta^P_0 = -m_V^2 \pi \alpha_1 g_{V P\gamma}g_{P NN}e^{b_P t}/2$ and $P = \pi, \eta$.
We will assume factorization into a top and a bottom vertex together with a Regge factor: \begin{align} \label{eq:naturalM} {\cal M}^R_{\substack{\lambda_V, \lambda_\gamma \\ \lambda', \lambda}}(s,t) & = \beta_R^{\gamma V} e^{b_Rt} \ V^{\gamma V}_{\lambda_V \lambda_\gamma}(t) R(s,t) V^{pp}_{\lambda' \lambda}(t). \end{align} The helicity structure at the photon vertex is: \begin{align} \label{eq:topV} V^{\gamma V}_{\lambda_V \lambda_\gamma}(t) & = \lambda_\gamma^2 \left(\delta_{\lambda_V,\lambda_\gamma} + \beta^R_1 \frac{\sqrt{-t}}{m_V} \frac{\lambda_\gamma}{\sqrt{2}} \delta_{\lambda_V,0} +\beta^R_2 \frac{-t}{m^2_V} \delta^{\lambda_V,-\lambda_\gamma} \right). \end{align} The helicity structure at the nucleon vertex is: \begin{align} V^{pp}_{\lambda' \lambda}(t) & = \delta_{\lambda,\lambda'} + 2 \lambda \kappa_R \frac{\sqrt{-t}}{2m} \delta_{\lambda,-\lambda'}. \end{align} The Regge factor is, with the trajectory $\alpha(t) = \alpha_0 + \alpha_1 t$: \begin{align} R(s,t) & = \frac{\alpha_R(t)}{\alpha_R(0)} \frac{1+e^{-i\pi\alpha_t(t)}}{\sin\pi\alpha_R(t)} \left(\frac{s}{s_0}\right)^{\alpha_R(t)}. \end{align}

The spin density matrix elements (SDME) computed in the $s$-channel frame are equivallent to the SDME in the helicity frame.
The relation between the helicity frame and the Gottfried-Jackson frame is \begin{align} \rho^i_{MM'}|_{GJ} & = \sum_{\lambda \lambda'} d^1_{M\lambda}(\theta) \rho^i_{\lambda \lambda'}|_{H} d^1_{M'\lambda'}(\theta) \end{align} The definition of the SDME $\rho^i_{MM'}$ and the frames are given in Ref. [Mathieu:2018xyc].
At high energy, the rotation angle $\theta$ is simply \begin{align} \cos\theta & = \frac{m_V^2+t}{m_V^2-t}. \end{align} The code uses the leading $s$ expression for the rotatio angles to be consistent with the helicity amplitudes.
One can check that with only the unnatural exchanges (set the natural exchanges couplings to 0), the SDME in the Gottfried-Jackson frame take the simple form $\rho^0_{00} = \text{Re }\rho^0_{10}= \rho^0_{1-1}=0$.

The parameters are determined on the SLAC data Ref. [Ballam:1972eq]. We refer to the publication [Mathieu:2018xyc] for the procedure. In the Simulation section, the default values of the parameters are listed in Table. The scale factor in the Regge factor is set to $s_0 = 1$ GeV$^2$.

Natural and unnatural exchange parameters

	$\mathbb P$	$f_2$	$a_2$		$\pi$	$\eta$
$\beta^{\gamma \omega}$	$0.739$	$0.730$	$1.256$	$g_{\omega P \gamma }$	$0.696$	$0.479$
$\beta^{\gamma \rho}$	$2.506$	$2.476$	$0.370$	$g_{\rho P \gamma }$	$0.252$	$0.136$
$\beta^{\gamma \phi}$	$0.932$	$0.000$	$0.000$	$g_{\phi P \gamma }$	$0.040$	$0.210$
$b$	$3.6$	$0.55$	$0.53$	$b $	$0.0$	$0.0$
$\beta_1$	$0.00$	$0.95$	$0.83$	$g_{P NN }$	$13.26$	$2.24$
$\beta_2$	$0.00$	$-$0.56	$0.00$
$\kappa$	$0.0$	$0.0$	$8.0$
$\alpha_0$	$1.08$	$0.5$	$0.5$	$\alpha_0$	$-0.013$	$-0.013$
$\alpha_1$	$0.2$	$0.9$	$0.9$	$\alpha_1$	$0.7$	$0.7$

They correspond to the values determined in the publication [Mathieu:2018xyc].

References

• [Ballam:1972eq]
  J. Ballam, et al.
  Vector Meson Production by Polarized Photons at 2.8 GeV, 4.7 GeV, and 9.3 GeV
  Phys. Rev. D 7 (1973), 3150; published on June 1, 1973

  Cite DOI

• [Mathieu:2018xyc]
  V. Mathieu, J. Nys, C. Fernández-Ramírez, A. Jackura, A. Pilloni, N. Sherrill, A. Szczepaniak, G. Fox
  Vector Meson Photoproduction with a Linearly Polarized Beam
  Phys. Rev. D 97 (2018), 094003; published on May 9, 2018

  Cite DOI ArXiv

Resources

• Publications: [Mathieu:2018xyc]
• C/C++: C/C++ file
• Input file: param.txt , simu.txt .
• Output files: resultsH.txt , resultsGJ.txt
• gnuplot files: gnuplotH.txt , gnuplotGJ.txt
• Contact person: Vincent Mathieu
• Last update: December 2017

Simulation

For each exchange, the user can supply parameters (couplings and trajectories).
The parameters from [Mathieu:2018xyc] are the default values.
The simulation displays the nine spin density matrix elements in both the helicity and the Gottfried-Jackson frame.

Beam energy in the lab frame (target rest frame):
$E_\gamma$ in GeV
Vector meson:   $\omega$   $\rho^0$   $\phi$

Natural exchanges (Pomeron and tensor exchanges): [show/hide]

$\mathbb P$		$\beta_{\mathbb P}^{\gamma\omega}$ :	$\beta_{\mathbb P}^{\gamma\rho}$ :	$\beta_{\mathbb P}^{\gamma\phi}$ :
		$\beta^{\mathbb P \phantom{\gamma}}_1$:	$\beta^{\mathbb P\phantom{\rho}\,}_2$:	$b_{\mathbb P}^{\phantom{\gamma\phi}\,}$:
		$\alpha_{0}^{\mathbb P\phantom{\gamma}}$:	$\alpha_{1}^{\mathbb P\phantom{\rho}\,}$:	$\kappa_{\mathbb P}^{\phantom{\gamma\phi}\,}$:
$\phantom{t}$
$f_2$		$\beta_{f_2}^{\gamma\omega}$ :	$\beta_{f_2}^{\gamma\rho}$ :	$\beta_{f_2}^{\gamma\phi}$ :
		$\beta^{f_2\phantom{\gamma}}_1$:	$\beta^{f_2\phantom{\gamma}}_2$:	$b_{f_2\phantom{\phi}}$:
		$\alpha_{0}^{f_2\phantom{\gamma}}$:	$\alpha_{1}^{f_2\phantom{\gamma}}$:	$\kappa_{f_2\phantom{\phi}}$:
$\phantom{t}$
$a_2$		$\beta_{a_2}^{\gamma\omega}$ :	$\beta_{a_2}^{\gamma\rho}$ :	$\beta_{a_2}^{\gamma\phi}$ :
		$\beta^{a_2\phantom{\gamma}}_1$:	$\beta^{a_2\phantom{\gamma}}_2$:	$b_{a_2\phantom{\phi}}$:
		$\alpha_{0}^{a_2\phantom{\gamma}}$:	$\alpha_{1}^{a_2\phantom{\gamma}}$:	$\kappa_{a_2\phantom{\phi}}$:
$\phantom{t}$

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

$\pi$		$\beta_{\pi}^{\gamma\omega}$ :	$\beta_{\pi}^{\gamma\rho}$ :	$\beta_{\pi}^{\gamma\phi}$ :
		$\alpha_{0}^{\pi \phantom{\gamma}}$:	$\alpha_{1}^{\pi\phantom{\gamma}}$:	$b_{\pi\phantom{\gamma\phi}}$:	$g_{\pi NN}$:
$\phantom{t}$
$\eta$		$\beta_{\eta}^{\gamma\omega}$ :	$\beta_{\eta}^{\gamma\rho}$ :	$\beta_{\eta}^{\gamma\phi}$ :
		$\alpha_{0}^{\eta \phantom{\gamma}}$:	$\alpha_{1}^{\eta\phantom{\gamma}}$:	$b_{\eta\phantom{\gamma\phi}}$:	$g_{\eta NN}$:
$\phantom{t}$