I'm currently studying the pion leptonic decay and I'm getting a bit confused about some factors. Firstly, le correct lagrangian that describe the pion decay [itex]\pi^+\rightarrow l^+ + \nu_l[/itex] is:(adsbygoogle = window.adsbygoogle || []).push({});

$$L=\frac{4G_F}{\sqrt{2}}V_{ud}^* \bar{d}_L\gamma^\mu u_L \bar{\nu}_L \gamma_\mu l_L$$

We can't operate with the quark current and so we have to use an effective current [itex]J^\mu_L=1/2(V^\mu-A^\mu)[/itex] where V and A are the vector and axial curret. The vector current gives no contribution because the pion is pseudo scalar.

So we need to calculate the matrix element of the axial current. I'm a bit confused about the correct use of form factor. My professor wrote down the following matrix element:

$$\langle 0 |A_\mu|\pi^+\rangle = ip^{\pi}_\mu f_\pi$$

while in some books I found the form factor written as [itex]F_\pi[/itex] and realted to [itex]f_\pi[/itex] by some 2 or √2 factors.

Can some one tell me the exact relation between F and f?

Thanks

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# Π leptonic decay (current matrix element)

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