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

[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?

 

[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?

 

[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?

 

[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 $J_{a_2}=2$.

 

[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.

 

[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!