# Main difference between P, S & D Partial Waves in Decays?

1. Mar 25, 2015

### Naeem Anwar

I am a little confused about; how to identify that Hadron/Meson may have S, P or D wave contribution to its decay to other hadrons. e.g in case of light meson decay

a1→ ρπ

this decay channel have two partial waves S & D, so my question is that from where I can guess that this channel have two or more partial wave contributions to total decay width?

2. Mar 26, 2015

### Einj

You need to consider the quantum numbers of the particles involved. In this case the $a_1$ has $J^{P}=1^+$, the $\rho$ has $J^P=1^-$ and the $\pi$ has $J^P=0^-$. Your decay is therefore of the kind $1^+\to1^-0^-$.
Since this is a strong decay you need to conserve both total angular momentum and parity. The total parity of the final system is given by $P_{\rho\pi}=P_\rho P_\pi (-1)^L$, where L is the relative angular momentum. The possible candidates to conserve you parity are therefore L=0,2,4,6,... Now with L=0 and L=2 you are fine and you can conserve both parity and total angular momentum. With L>2 though, even if parity is conserved, you can't conserve momentum anymore since $J=1\otimes 0\otimes L$ for L>2 always gives total angular momentum greater that 1 (the J of the $a_1$ meson).

Is this what you were looking for?

3. Mar 26, 2015

### Naeem Anwar

Got it! Yes I was exactly looking this. Thanks for this brief & exact answer, I am on the track now. Let me please correct here once more, for the channel $a_2\to\rho\pi$ with quantum numbers $2^+\to1^-0^-$, by using relation $P_{\rho\pi}=P_\rho P_\pi (-1)^L$, now possible value of L=0,2,4,6... but allowed value is only 2 (D wave). To conserve parity allowed values are 0 & 2 but to conserve $J$ only 2, so my question is that which total angular momentum I will keep in mind to conserve here? The $J$ of $a_2$ the $J$ of $\rho$ or their relative difference?

4. Mar 26, 2015

### Einj

The total angular momentum of the final system, which in your case is $J_\rho\otimes J_\pi\otimes L$ must be equal to the total angular momentum of the initial one, which in your case is $J_{a_2}=2$.

5. Apr 14, 2015

### Naeem Anwar

Dear Sir Einj..!
Thanks for the valuable comments. Could you kindly recommend some basic literature on the fundamentals of hadron decays. I am quite new in this field, just started the graduation course work with little research related the strong decays of hadrons (mesons).

Thanks! Looking forward to listen from you.

6. Apr 14, 2015

### Einj

For this advanced topics is usually quite hard to find decent understandable literature. I have few suggestions but you might need to spend some time on them since they are not really straightforward:

1) Wise - "Heavy Quark Physics"
2) Donoghue - "Dynamics of the Standard Model"

These two are real books. However I always found the best reference to be a review from Casalbuoni et al.: http://arxiv.org/abs/hep-ph/9605342 . I think so far is the most complete review we have and it's quite understandable.

I hope this is useful!