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, \Gamma[\itex], and Branching Ratios. I haven&#039;t found very clear information online so far. Here&#039;s just an example below that I&#039;d like help with. I&#039;m unsure of what formulas I&#039;d have to use, if someone could give me an indication about how I might start this, I&#039;d be most grateful. Thank for any help! <br /> <br /> <h2>Homework Statement </h2><br /> <br /> (i) Estimate the partial decay width and branching ratio (BR) for \phi → e^{+}e^{-}. Where \phi is s s-bar (a meson with strange and anti-strange).<br /> <br /> (ii) Make a rough estimate of \Gamma(\tau^{-} → K^{-}\nu_{\tau})/\Gamma(\tau^{-} → \pi^{-}\nu_{\tau})<br /> <br /> K^{-} is u-bar and s (i.e. a meson with anti-up and strange). <br /> <br /> <h2>Homework Equations</h2><br /> <br /> The CKM matrix is: <br /> <br /> 0.974***0.277***0.004<br /> 0.227***0.973***0.042<br /> 0.008***0.042***0.999<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> 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.