# Tau lepton branching ratios

## Homework Statement

Branching ratios of tau lepton decay:
muon 17%
electron 17%

Use your knowledge of the decay of W bosons to justify these rates.

## The Attempt at a Solution

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The W+ boson can decay to any combination of quarks with +1 charge, but the CKM matrix suppresses many of these heavily, so lets just consider ud, us, cd, cs (where d/s are anti).

BR (electron) = g^2(w) / (3*((2g^2(w) cos^2x + 2g^2(w)sin^2x) + 2g^2(w)
= 1( 6 + 2) = 1/8

where g^2(w) cos^2x is the matrix element squared for ud and and cs
g^2(w)sin^2x is the matrix element squared for us and cd
g^2(w) is the matrix element squared for any lepton.

This doesn't give the right answer. Furthermore shouldn't density of states be very important given the fact that the lepton channel is 3 body, and in real life the tau decays into 3 or 4 particles most commonly?

mfb
Mentor
so lets just consider ud, us, cd, cs (where d/s are anti).
You ignored tb for a good reason. There is something else you should ignore for the same reason.

This doesn't give the right answer. Furthermore shouldn't density of states be very important given the fact that the lepton channel is 3 body, and in real life the tau decays into 3 or 4 particles most commonly?
That is a smaller effect from hadronization.

I'm afraid I just can't see it. I'v
You ignored tb for a good reason. There is something else you should ignore for the same reason.

Many thanks for your reply. I can see I shouldn't have include the charm containing mesons as they are slightly to heavy. This brings the branching ratios to 50% hadrons and 25% for each lepton. Have I missed something else or is the remaining discrepancy justified by a density of states argument? Many thanks.

mfb
Mentor
This brings the branching ratios to 50% hadrons and 25% for each lepton.
What happened to the quark colors now? The previous calculation seemed to have those.

Sorry made a silly mistake. So the BR should be 60% for hadrons and 20% for the electron
What happened to the quark colors now? The previous calculation seemed to have those.

Sorry made a silly mistake. BR should now be 60% for hadrons (3 * cos^2(x) + sin^2(x)) / (3 * (cos^2(x) + sin^2(x)) + 1 + 1 ) , and 20% for each lepton. Many thanks.

mfb
Mentor
Right, and that is close to the actual values. You can get an even better approximation if you take kaons into account (with up, not with charm).

Have I not taken into account Kaons with the Cabibo factors (cos(x)) for non mixed states, and sin(x) for mixed states?

mfb
Mentor
Well, I don't understand your numerator. Why is there no 3 for the sin term?
But sin and cos together should work, and give 20%, right. Okay, blame phase space for the rest.