# Tau lepton branching ratios

• Fek
In summary: 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 not taken into account Kaons with the Cabibo factors (cos(x)) for non mixed states, and sin(x) for mixed states?But sin and cos together should work, and give 20%, right. Okay, blame phase space for the rest.

## 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

[/B]
The W+ boson can decay to any combination of quarks with +1 charge, but the CKM matrix suppresses many of these heavily, so let's 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?

Fek said:
so let's 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.

Fek said:
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
mfb said:
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.

Fek said:
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
mfb said:
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.

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?

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.

## 1. What is a tau lepton branching ratio?

The branching ratio of a tau lepton refers to the likelihood that it will decay into a specific set of particles. It is represented as a percentage, with a total of 100% representing all possible decay modes.

## 2. How is the branching ratio of a tau lepton determined?

The branching ratio is determined through experimental measurements and theoretical calculations, which take into account the fundamental properties of the tau lepton and its interactions with other particles.

## 3. Why is the study of tau lepton branching ratios important?

The study of tau lepton branching ratios can provide valuable insights into the underlying physics of particle interactions and the fundamental properties of the tau lepton itself. It is also a crucial aspect of understanding and predicting the behavior of high-energy particle collisions.

## 4. How do branching ratios differ between different particles?

The branching ratios of particles can vary significantly depending on their mass, charge, and other properties. In general, heavier particles have more decay modes available to them, resulting in lower branching ratios.

## 5. Can branching ratios change over time?

Yes, branching ratios can change over time as new particles and decay modes are discovered through experimental research. Theoretical predictions of branching ratios are also constantly being refined and updated, leading to changes in the estimated values.