Particle Physics: Partial Decay Widths and Branching Ratios

[Collisionman]

Hello there,

This isn't specifically homework, it is study. I'm having a difficult time trying to understand how to calculate/estimate partial decay widths, [itex]\Gamma[\itex], and Branching Ratios. I haven't found very clear information online so far. Here's just an example below that I'd like help with. I'm unsure of what formulas I'd have to use, if someone could give me an indication about how I might start this, I'd be most grateful. Thank for any help!

Homework Statement

(i) Estimate the partial decay width and branching ratio (BR) for [itex]\phi → e^{+}e^{-}[/itex]. Where [itex]\phi[/itex] is s s-bar (a meson with strange and anti-strange).

(ii) Make a rough estimate of [itex]\Gamma(\tau^{-} → K^{-}\nu_{\tau})[/itex]/[itex]\Gamma(\tau^{-} → \pi^{-}\nu_{\tau})[/itex]

[itex]K^{-}[/itex] is u-bar and s (i.e. a meson with anti-up and strange).

Homework Equations

The CKM matrix is:

0.974***0.277***0.004
0.227***0.973***0.042
0.008***0.042***0.999

The Attempt at a Solution

I am unsure about how to tackle these questions.

 

[RobyVonRintein]

How is this going to go? Strong processes always dominate, and we can imagine a lot of strong processes here; ss --> cc for example, mediated by a gluon. Getting an EM process like this to go is going to be at least 1/1000 times smaller. So that's the branching ratio (very roughly). The full width is just hbar/lifetime, so you can determine the full width and multiply by 1/1000 to get the partial width. As far as I know, there's no way to get a "good' estimate for either of these without doing Feynman rules for a bunch of diagrams, will be complicated because of the form factor in the initial state.

(ii) is a little easier; the tau throws a W-, which can decay into either su or ud. ud has a CKM element of ~1, whereas us has ~.22. So, the K is about four times less likely -- and we have to square amplitude to get probability. So I'd guess about 1/16 = 0.0625. Sure enough, the actual ratios are K 0.7% of the time while pi is 10% of the time, so the ratio is .7/10 = 0.07, darn close.

 

[mfb]

ss -> cc is not possible as charm quarks are too heavy. ss->uu or ss->dd are possible.

 

[RobyVonRintein]

Good call. To clarify for the OP: if you had s and sbar crashing into one another (as at a collider), then you could get ss --> cc, because the relative kinetic energy between them would make up for the smaller mass energy. But in this case you have s and sbar bound together, so there is a frame in which both particles are at rest, and the total E_initial is just m_s^2. As a result, there's not enough energy to form two charms, which will have E_final m_c^2 + kinetic (even if kinetic is 0).

 

[bipin sonawane]

what is physical interpretation of branching ratio?

 

[mfb]

The fraction of particles that decays to some specific set of other particles. As an example, 57% of Higgs bosons decay to a pair of b-quarks (one quark and one antiquark). The branching ratio to b-quarks is 57%.

This thread is from 2013.