I have a question that asks me to calculate the lepton current for the decay of a tau lepton in to 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:(adsbygoogle = window.adsbygoogle || []).push({});

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

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):

[tex]u_{r}(p) = \sqrt{E+m}(c, e^{i\phi}s, p/(E+m) * c, e^{i\phi}s * p/(E+m) )[/tex]

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

[/tex]

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

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# Tau Lepton Decay Question

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