# E+e- -> gamma f0 -> gamma pi0 pi0 cross section with VMD

1. Jun 30, 2013

### liberulo

e+e- --> gamma f0 --> gamma pi0 pi0 cross section with VMD

1. The problem statement, all variables and given/known data

Find the cross-section of $e^+e^- \to \gamma f_0(980) \to \gamma \pi^0 \pi^0$ using the vector meson dominance model.

2. Relevant equations

Some Feynman's rules:

The photon propagator is $-i \frac{g_{\mu\nu}}{q^2}$.
The propagator of $\varphi$-meson is $-i \frac{g_{\mu\nu} - \frac{q_\mu q_\nu}{m_\varphi^2}}{q^2 - m_\varphi^2 + i m_\phi \Gamma_\varphi}$, $\Gamma_\varphi$ - the particle width .
The $\gamma \varphi$-vertex is $-i e \frac{m_\varphi^2}{g_\varphi}$.
$g_{\varphi \omega f_0}$ is the $\varphi \omega f_0$-vertex constant.

3. The attempt at a solution
The effective Lagrangian is
$$\mathcal{L} = \mathcal{L}_{QED} + \mathcal{L}_{em} + \mathcal{L}_{str},$$
where
$$\mathcal{L}_{str} = g_{\varphi \omega f_0} {F_\varphi}^{\alpha \beta} {F_\omega}^{\mu \nu} \varepsilon_{\alpha \beta \mu \nu} f_0 + g_{f_0 \pi^0 \pi^0} f_0 \pi \pi,$$
$$\mathcal{L}_{em} = -e \frac{{m_\varphi}^2}{g_\varphi} \Phi^\mu A_\mu -e \frac{{m_\omega}^2}{g_\omega} \Omega^\mu A_\mu.$$
$\Phi^\mu, \Omega^\mu, A_\mu, f_0, \pi$ - $\varphi$, $\omega$, photon, $f_0$, $\pi^0$ fields.

After that I try to write the matrix element for the $e^+e^- \to \gamma f_0(980) \to \gamma \pi^0 \pi^0$ diagram. There is my trouble.
$$i M = \bar{v} (-i e \gamma_\mu ) u \cdot \left(-i \frac{g^{\mu \nu}}{q^2} \right) \left( -ie \frac{m_\varphi^2}{g_\varphi}\right) \left( -i \right) \frac{g_{\nu\alpha} - \frac{q_\nu q_\alpha}{m_\varphi^2}}{q^2 - m_\varphi^2 + i m_\varphi \Gamma_\varphi} g_{\varphi \omega f_0} \left( -i \right) \frac{g^{\alpha \beta} - \frac{k^\alpha k^\beta}{m_\omega^2}}{k^2 - m_\omega^2 + i m_\omega \Gamma_\omega} \left( -ie \frac{m_\omega^2}{g_\omega}\right) \cdot \\ \cdot ( k^\tau {\epsilon_{\gamma}}^\sigma - k^\sigma {\epsilon_{\gamma}}^\tau ) \varepsilon_{\tau \sigma ? ?} \cdot \frac{-i}{r^2 - m_{f_0}^2 + i m_{f_0} \Gamma_{f_0}} g_{f_0 \pi^0 \pi^0} .$$
k - the radiative photon four-momentum, $\epsilon_{\gamma}$ - the photon polarization, r - $f_0$ four-momentum.

Last edited: Jun 30, 2013