Axial current matrix element for pseudoscalar mesons

[Nixom]

I have some questions about the matrix
$$<0|j^{\mu}_5|\pi(p)>=if_{\pi}p^{\mu}$$
1.Pi is a psuedoscalar particle, but the current between vacuum and pi state is an axial one, why not a psuedoscalar one?
And how do we determine the current appearing in the similar matrix like $$<0|j|meson>$$

2.The r.h.s is propotional to the momentum of the meson pi, but in the vector current case it is propotional to the polarization and mass
$$<0|j^{\mu}|Vector(p)>=if_{V}m_{V}e^{\mu}$$, why?

 

[The_Duck]

I have some questions about the matrix
$$<0|j^{\mu}_5|\pi(p)>=if_{\pi}p^{\mu}$$
1.Pi is a psuedoscalar particle, but the current between vacuum and pi state is an axial one, why not a psuedoscalar one?

That matrix element would also be nonzero. When we use an axial-vector current, the LHS is scalar * axial-vector * pseudoscalar and so the RHS is a vector (proportional to p). If we used a pseudoscalar operator, the LHS would be scalar * pseudoscalar * pseudoscalar and the RHS would be some scalar. Since the axial-vector matrix element is (part of) what determines the rate of weak decays of mesons, it seems appropriate to define the "decay constant" via the axial-vector matrix element.

And how do we determine the current appearing in the similar matrix like $$<0|j|meson>$$

To define the decay constant of a given meson, you use the axial current with the same flavor quantum numbers as the meson. For example, for the decay constant of the $K^+$, you use the current $\bar{s}\gamma^\mu\gamma_5 u$.