- #1
manfromearth
- 5
- 0
- Homework Statement
- Consider the hypothetical decay of the ##\phi(1020)## meson into ##\pi^{+}\pi^{-}\pi^{0}##. Is This decay possible?
- Relevant Equations
- $$\phi(1020) \rightarrow \pi^{+}\pi^{-}\pi^{0}$$
$$J^{PC}(\phi)=1^{--}, J^{P}(\pi^{\pm})=0^{-}, J^{PC}(\pi^{0})=0^{-+}$$
I started checking for angular momentum conservation.
The initial state has ##J_{in}=S_{\phi}=1##. The pions in the final state all have 0 spin, so the total angular momentum in the final state comes only from orbital momentum. Call ##L_{\pm}## the orbital momentum of the charged pions orbiting each other, ##L_{(\pm)0}## the orbital angular momentum of the system of charged pions rotating around the neutral pion. So, the final state has ##J_{fin} = L_{(\pm)0} + L_{\pm}##.
Angular momentum conservation requires ##J_{in}=J_{fin}##. So:
##1=J_{fin} = L_{(\pm)0} + L_{\pm}##
So i start combining integer angular momenta with the usual rules, and look what combinations give me a total angular momenta of ##J_{fin}=1##.
Adding the momenta ##L_{(\pm)0}## and ##L_{(\pm)}## i get possible ##J_{fin}## going from ##L_{(\pm)0}+L_{(\pm)}## to ##|L_{(\pm)0}-L_{(\pm)}|##. Doing this for some values i get the following table:
What I get from this is that, for example, looking at the 2nd row: i can satisfy angular momentum conservation with ##L_{\pm}## an even number and ##L_{(\pm)0}## odd. Looking at other rows in the table that have ##J_{fin}=1## as possible values, I see that in general momentum conservation can be satisfied with whatever parity for ##L_{\pm}## and ##L_{(\pm)0}##: One can be odd, the other even or also Even-Even or Odd-Odd.
Now i Apply Parity conservation and Charge conjugation conservation.
Parity conservation: $$P(\phi)=-1,\quad P(\pi^{+}\pi^{-}\pi^{0})=P(\pi^{+}\pi^{-})P(\pi^{0})=(-1)^{L_{\pm}}(+1)(-1)^{L_{(\pm)0}} $$
Charge conjugation conservation: $$C(\phi)=-1,\quad C(\pi^{+}\pi^{-}\pi^{0})=C(\pi^{+}\pi^{-})C(\pi^{0})=(-1)^{L_{\pm}}(+1)$$
From this i see that: to preserve parity i must have one of the L odd, the other even. To preserve Charge conjugation i must have ##L_{\pm}## odd. Since i can choose the evenness/oddness of Ls as i want and still conserve angular momentum, i can pick ##L_{\pm}## to be odd and ##L_{(\pm)0}## even. This will conserve P,C,J and i would conclude that:
The proposed decay does not violate P,C,J conservation and is therefore possible.
Is this the right way to proceed? Am i missing something? I have seen examples of a scalar particle decaying into 3 scalar products but never one with a spin 1 decaying into 3 spin 0.
The initial state has ##J_{in}=S_{\phi}=1##. The pions in the final state all have 0 spin, so the total angular momentum in the final state comes only from orbital momentum. Call ##L_{\pm}## the orbital momentum of the charged pions orbiting each other, ##L_{(\pm)0}## the orbital angular momentum of the system of charged pions rotating around the neutral pion. So, the final state has ##J_{fin} = L_{(\pm)0} + L_{\pm}##.
Angular momentum conservation requires ##J_{in}=J_{fin}##. So:
##1=J_{fin} = L_{(\pm)0} + L_{\pm}##
So i start combining integer angular momenta with the usual rules, and look what combinations give me a total angular momenta of ##J_{fin}=1##.
Adding the momenta ##L_{(\pm)0}## and ##L_{(\pm)}## i get possible ##J_{fin}## going from ##L_{(\pm)0}+L_{(\pm)}## to ##|L_{(\pm)0}-L_{(\pm)}|##. Doing this for some values i get the following table:
What I get from this is that, for example, looking at the 2nd row: i can satisfy angular momentum conservation with ##L_{\pm}## an even number and ##L_{(\pm)0}## odd. Looking at other rows in the table that have ##J_{fin}=1## as possible values, I see that in general momentum conservation can be satisfied with whatever parity for ##L_{\pm}## and ##L_{(\pm)0}##: One can be odd, the other even or also Even-Even or Odd-Odd.
Now i Apply Parity conservation and Charge conjugation conservation.
Parity conservation: $$P(\phi)=-1,\quad P(\pi^{+}\pi^{-}\pi^{0})=P(\pi^{+}\pi^{-})P(\pi^{0})=(-1)^{L_{\pm}}(+1)(-1)^{L_{(\pm)0}} $$
Charge conjugation conservation: $$C(\phi)=-1,\quad C(\pi^{+}\pi^{-}\pi^{0})=C(\pi^{+}\pi^{-})C(\pi^{0})=(-1)^{L_{\pm}}(+1)$$
From this i see that: to preserve parity i must have one of the L odd, the other even. To preserve Charge conjugation i must have ##L_{\pm}## odd. Since i can choose the evenness/oddness of Ls as i want and still conserve angular momentum, i can pick ##L_{\pm}## to be odd and ##L_{(\pm)0}## even. This will conserve P,C,J and i would conclude that:
The proposed decay does not violate P,C,J conservation and is therefore possible.
Is this the right way to proceed? Am i missing something? I have seen examples of a scalar particle decaying into 3 scalar products but never one with a spin 1 decaying into 3 spin 0.
