What is the Lepton Current for Tau Lepton Decay in the Rest Frame?

[Willa]

I have a question that asks me to calculate the lepton current for the decay of a tau lepton into the tau neutrino and a pion. It asks me to work in the rest frame of the tau lepton, taking the spin to be fully polarized in the +z direction. The lepton current is given by:

$$j^{u} = u_{bar}(p_3)\gamma^{u} (1/2)(1-\gamma^{5})u(p_1)$$

where p1 is the tau lepton 4 momentum and p3 is the neutrino 4 momentum. u-bar should be the standard adjoint spinor, I just suck at latex.

I was given the right and left handed helicity spinors as well (c = cos theta/2, s = sin theta/2):

$$u_{r}(p) = \sqrt{E+m}(c, e^{i\phi}s, p/(E+m) * c, e^{i\phi}s * p/(E+m) )$$
$$u_{l}(p) = \sqrt{E+m}(-s, e^{i\phi}c, p/(E+m) * s, -e^{i\phi}c * p/(E+m) )$$

The solution by the way is quoted as being of the form ~(-s, -c, -ic, s)

So what I did was say that only Left handed chiral states are involved in the weak interaction, so I said the neutrino must be in the left handed helicity state. But to conserve helicity I said that the tau lepton must also start in left handed helicity state (since the pion has no spin so no helicity).
But that resulted in the wrong answer!?
I found that if I used the right handed expression for the tau lepton (with theta=0), then I obtained the given answer. But I don't understand why that works - surely that violates helicity conservation.

I hope someone out there understands my problem, I can try and give more detail if needed.

thanks in advance for the help

 

[Willa]

ok so since I posted this problem I've realized that I can't say the tau is in a left helicity state because helicity is only conserved in the relativistic limit, and we are working in the tau rest frame.

So basically the problem I think boils down to "How do you determine the spinor for the tau lepton at rest, with it's spin polarized in the +z direction?". In particular, why does it work when the spinor is proportional to (1,0,0,0)?

 

[Timo]

I don't understand the original question you were given and I cannot tell things you were given from things you assumed yourself in your original post. But since you weren't given any reply till now, I can at least say what I would usually do in respect to this question:

How do you determine the spinor for the tau lepton at rest, with it's spin polarized in the +z direction?

I'd use the projection operator Pz(+) projecting on a polarisation in positive z direction and get my state as $$u_{pol}(p,+) = \sum_{\lambda=\pm} P_z(+) u_{unpol}(p,\lambda)$$.
Sidenote: In Bjorken-Drell convention, $$P_s(\lambda) = \frac{1 + \lambda \gamma^5 \gamma^\mu s_\mu}{2}$$ with s=(0,0,0,1), the z-axis, in your case.

EDIT: Like said, I am not completely sure if that help you - I don't understand the original problem. But the projectors give you handle on polarized spinors, so it might at least be a way to tackle the problem.

 

[Willa]

unfortunately that doesn't help since I don't know what the spinor is for an unpolarized state is.

I think that the two helicity eigenstate spinors I gave in my first post form a complete set, if that's true then any state must be a linear combination of those two states. The question is then why is a state, with 0 momentum and spin fully polarized in the +z direction, represented by a pure right handed helicity state with p=0 and theta=0 (even though it's not moving anywhere).

Personally I would have been tempted in some way to say the state was a linear combination of the right handed state with theta=0 and the left handed state with theta=pi, since the left handed state is where the spin is in the opposite direction to the momentum. But since the momentum is 0, it's hard to say what the "direction" of the momentum is.

Basically I'm still stuck, but thanks for trying.