Calculating Decay Widths & Estimating Partial Width of τ− Lepton

[Sonko]

i was'nt sure whether to put this here in the homework questions because this is'nt really homework as its for my understanding... However my particle physics exam is tommoro and I'm having some problems understanding how to calculate decay width's of decays. i have the following from a past paper which i could use some help on (as I'm sure a similar quation will be asked)

A2. A τ− lepton has the following 4 decay channels:
(i) τ− → ν_ e− ¯νe;
(ii) τ− → ν_μ− ¯ ν μ;
(iii) τ− → ν_ ¯u d;
(iv) τ− → ν_ ¯u s.

(a) For each of the 4 decays, draw the lowest order Feynman diagrams, labelling
all external lines, internal lines, and vertices.

(b) Give simplified expressions for the decay width of each decay.
Estimate branching fractions for each decay mode, given that |Vud| = 0.97 and
|Vus| = 0.22.

(c) Estimate the partial width of the decay of the τ− lepton into a muon, given
that its lifetime is τ_ = 3 × 10−13 s [¯h = 6.588 × 10−25 GeV s.]


i can construct the feynman diagrams but i really don't know what is meant by "simplified expressions for the decay width of each decay" or how to estimate the partial width of the τ− lepton into a muon?

Thanks for any help you can give me

 

[anathema]

Thank you for reaching out for help with your particle physics exam. I am happy to assist you in understanding how to calculate decay widths of decays.

A decay width is a measure of the probability of a particle decaying into a specific final state. It is denoted by the symbol Γ and is typically measured in units of energy. In order to calculate the decay width, we need to use the Feynman diagrams you have constructed and apply the rules of Feynman diagrams to determine the amplitude for each decay process. The amplitude is then squared and integrated over all possible final states to obtain the decay width.

To simplify the expressions for the decay width, we can use the Fermi's Golden Rule, which states that the decay width is proportional to the square of the coupling constant (g) and the phase space factor (PSF). The PSF takes into account the available energy and momenta of the particles involved in the decay process. So, the simplified expression for the decay width of each decay will be:

Γ = g^2 x PSF

To estimate the branching fractions, we can use the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |Vud| and |Vus|, which describe the mixing between different quark flavors. The branching fraction is the ratio of the decay width of a particular decay channel to the total decay width of the particle. So, the branching fraction for each decay mode will be:

Branching fraction = Γ_i / ΣΓ

Where Γ_i is the decay width of the specific decay channel and ΣΓ is the total decay width of the particle.

To estimate the partial width of the τ− lepton into a muon, we can use the formula:

Γ(τ− → μ−ν_μν_τ) = |Vus|^2 x Γ_0

Where Γ_0 is the partial width of the τ− lepton into a muon in the absence of mixing. This can be calculated using the simplified expression for the decay width mentioned earlier.

I hope this helps in your understanding of calculating decay widths and estimating branching fractions and partial widths. Good luck with your exam tomorrow!