\rho \to \pi \pi decay rate

[maelle]

Hello everyone,

I'm using the chiral perturbation theory for mesons to calculate Vector into two Pseudoscalars decay rates - hopefully to be able to calculate Tensor into two pseudoscalars decay rates later on.

I've got the lagrangien for the \rho \to \pi \pi decay rate :

L= f_{\rho \pi \pi} \epsilon_{ijk} \rho_i^\mu \pi_j D_\mu \pi_k

and I've got to end up with

\Gamma (\rho \to \pi \pi) = f_{\rho \pi \pi}^2 / (48 \pi) m_\rho [1- 4 m_\pi^2/m_\rho^2]^3/2.

I'm at loss at how I'm supposed to find that the matrix element (squared) is

M^2 = 4/3 f_{\rho \pi \pi} p_\pi^2

where p_\pi^2 = (m_\rho^2 - 4 m_\pi^2)/4 - but that last part I found out.

I've been checking textbooks to find a nice way to express a matrix element using a Lagrangian but all I can find is that same thing only for QED, and using the Feynman rules for QED. I'd love to compute my Feynman rules for my chiral perturbation theory!

Please help me ;)

 

[Hepth]

cleaned up

Hello everyone,

I'm using the chiral perturbation theory for mesons to calculate Vector into two Pseudoscalars decay rates - hopefully to be able to calculate Tensor into two pseudoscalars decay rates later on.

I've got the lagrangien for the \rho \to \pi \pi decay rate :

$$L= f_{\rho \pi \pi} \epsilon_{ijk} \rho_i^\mu \pi_j D_\mu \pi_k$$

and I've got to end up with

$$\Gamma (\rho \to \pi \pi) = f_{\rho \pi \pi}^2 / (48 \pi) m_\rho [1- 4 m_\pi^2/m_\rho^2]^3/2.$$

I'm at loss at how I'm supposed to find that the matrix element (squared) is

$$M^2 = 4/3 f_{\rho \pi \pi} p_\pi^2$$

where $$p_\pi^2 = (m_\rho^2 - 4 m_\pi^2)/4$$ - but that last part I found out.

I've been checking textbooks to find a nice way to express a matrix element using a Lagrangian but all I can find is that same thing only for QED, and using the Feynman rules for QED. I'd love to compute my Feynman rules for my chiral perturbation theory!

Please help me ;)

 

[maelle]

Way better, thanks! Didn't know there was a Latex interface here... Although for the decay rate it's

$$\Gamma (\rho \to \pi \pi) = \frac{f_{\rho \pi \pi}^2 }{ 48 \pi } m_\rho [1- 4 m_\pi^2/m_\rho^2]^{3/2}.$$

sorry for the misunderstanding.

Any ideas?