Decay of a D Meson to K[SUP]+[/SUP] and [tex]\Pi[/tex][SUP]-[/SUP]

[Ajihood]

Homework Statement

Hi guys, I have been given the problem of drawing the feyman diagrams and finding the decay rates for the two decays:

D⁰ $$\rightarrow$$ K⁺ + $$\pi$$^-

and

D⁰ $$\rightarrow$$ K^- + $$\pi$$⁺

D⁰ $$\rightarrow$$ $$\pi$$^- + $$\pi$$⁺

Homework Equations

The coupling constant g = 10^-5 for the weak force. But we have quarks changing so we have sin$$\vartheta$$_c and cos$$\vartheta$$_c facotrs associated with the cabbibo matrix. We take $$\vartheta$$_c = 12.7 degrees.

The Attempt at a Solution

I have no problem drawing the feynman diagram for the second decay, with c $$\rightarrow$$ s and the W⁺ decaying to u and anti d. This has a coupling constant

The problem is with the first equation. I can get it if I have c $$\rightarrow$$ d and then W⁺ decaying to u and anti s.

This is my problem. Can W⁺ decay to a u and anti-s. All the examples I find on the net have W boson decaying to a quark and anti-quark in the same generation... Can I do this?

I can get the third decay easy, but it is the same as the second with W⁺ decaying to u and anti-d. It just worries me it is so similar.

So am I on the right track?

Thanks in advanced.

 

[diazona]

Yep, that can happen. You can draw a vertex involving the W boson and any two quarks (subject to charge conservation, of course). It's just that the vertices involving two quarks in the same generation are overwhelmingly more likely; that's probably why people talk about them the most.

The specific probabilities of the different quark combinations at a weak interaction vertex are given by the CKM matrix.

 

[Ajihood]

Thanks Diazona. I just wanted to double check that, as I have never seen an example of that occurring!

Also, since I have a c $$\rightarrow$$ u and W⁺ $$\rightarrow$$ u + anti-s I have two vertices for the decay D⁰ $$\rightarrow$$ K⁺ + $$\pi$$^-. This means the total decay rate will depend on the square of the coupling constants g_cu and g_uanti-s multiplied togther.

But because of the CKM matrix g_cu = g_wsin$$\vartheta$$_c and g_uanti-s = g_wsin$$\vartheta$$_c. Where g_w is given to be 10^-5

So the total decay rate for D⁰ $$\rightarrow$$ K⁺ + $$\pi$$^- ~ g²_cug²_uanti-s = (g_wsin$$\vartheta$$_c)²

Is this right?

Thanks

 

[diazona]

Almost. Keep in mind that the charm quark can't decay to an up quark directly, because of charge conservation, so you should probably go back and take another look at your Feynman diagram. But other than that, I believe you have the right idea.

 

[Ajihood]

Sorry, I just want to get this clear, as I am pretty sure I have it right but I think I am misunderstanding what you just meant.

My Feynman diagram has a c $$\rightarrow$$ d and in that process it emits a W⁺. This W⁺ then decays to a u and anti-s. I thought charge is conseved overall in this. I wish I could show you my diagram! (also the anti-u is unaffected in the decay and continues across)

 

[diazona]

Yep, that sounds correct. The way your previous post was phrased, I thought you were talking about a charm quark decaying directly to an up quark and a W+. So your diagram sounds right after all.