# Vector Mesons decay constant

1. Jun 16, 2013

### GWSguy

I'm studying the $\tau \rightarrow \rho \; \nu_{\tau}$ decay. I'm asked to calculate the decay width, using a parameterization of the matrix element of the hadronic current. I actually find a matrix element of the form:
$$\left<\rho |\; \bar{u} \gamma^{\mu} \left( 1-\gamma^{5} \right ) d \; | 0\right>$$
in which I have both the vector and the axial current (u and d are up and down quarks). The $\rho$ meson is a spin 1 vector meson, so I expect that only the term from the vector current survives. I've infact verified this statement in many articles which report:
$$\left<\rho |\; \bar{u} \gamma^{\mu} d \; | 0\right> = f_{\rho} m_{\rho} \epsilon ^{\mu}$$
with $\epsilon ^{\mu}$ the polarization vector of the meson.

The problem is that I'm not able to demonstrate it. How can I formally demonstrate it? Is there any parity argument which allows me to exclude the axial term?

2. Jun 16, 2013

### GWSguy

Last edited by a moderator: May 6, 2017
3. Jun 21, 2013

### GWSguy

I know understand! The vector current is even under a G-parity transformation. I recall that G-parity is a multiplicative quantum number used for hadrons classification. It is defined as a rotation about the 2-axis, followed by charge conjugation, i.e. $$C \mathrm{e}^{i\pi I_{2}}$$ where $$I_{2}$$ is an isospin generator. The $$\rho$$ is even to. The axial current is odd under such a transformation.
For more details, I suggest Quarks and Leptons by Lev Okun.