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Why is 3 pion decay of eta (η) strong forbidden? 
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#1
Dec1411, 07:42 AM

P: 5

I have read that 3 pion decay of the eta is not allowed as a strong reaction, but proceeds as an electromagnetic interaction. I do not see why it is strong forbidden.



#2
Dec1411, 11:38 AM

Sci Advisor
P: 1,256

That is easiest to see using G parity, which is [tex]C e^{i\pi I_y}[/tex].
One or three pions have negative G, while the eta has positive G. 


#3
Dec1411, 11:41 AM

P: 5

How do you find the result of the I2 rotation operation?



#4
Dec1411, 04:07 PM

Sci Advisor
P: 1,256

Why is 3 pion decay of eta (η) strong forbidden?
Put "G Parity" into google.



#5
Dec1411, 08:27 PM

P: 518

From Gparity  Wikipedia
In general, P(G) = P(C)*(1)^I For fermionantifermion systems, P(G) = (1)^(L+S+I) For bosonantiboson systems, P(G) = (1)^(L+I) To calculate exp(i*pi*I2), use Wigner Dmatrix  Wikipedia for rotation matrices between quantummechanical angularmomentum states. One can carry angularmomentum features over to isospin without much trouble. For angular momentum j and state m to state m', find D(j,m,m',0,pi,0) = d(j,m,m',pi) It is only nonzero if m' =  m: d(j,m,m,pi) = (1)^(j+m) One can do the calculation more directly, by using isospin conservation and pion spinstatistics. Pions have spin 0, making them bosons, with their combined wavefunction always being symmetric. It also simplifies the treatment of their spins, since their combined spin is always 0, with a symmetric combined wave function. Orbital angular momentum is usually handled by setting it to 0 (swave), giving the same simplification and symmetry. Pions have isospin 1, while eta and eta' mesons have isospin 0. To find the total isospin of 3 pions, let's start with finding it for 2 pions. One gets these isospins and wavefunction symmetries: Symmetric: 0, 2 Antisymmetric: 1 One can prove this alternation of symmetry more generally, I think. Since their combined wavefunction must be symmetric, 2 swave pions must have isospin 0 or 2. Combining the isospin of the third pion gives possibilities 3, 1, and 1. Using my SemisimpleLieAlgebras package, I find: Symmetric: 3, 1 Mixed: 2, 1 Antisymmetric: 0 Thus, 3 swave pions cannot have zero isospin, and thus, an eta cannot decay into them without violating isospin. That can be done with the electromagnetic interaction, however. Let's depart from the swave hypothesis, while using the smallest possible orbital angular momenta. The sum of these values must always be even from parity conservation, and we have two possibilities: 1 dwave and 2 swave, and 1 swave and 2 pwave. The first one cannot have zero isospin, while the second one can, if the 2 pwave pions have total angular momentum 1. That makes antisymmetry, and that can be compensated for with isospin antisymmetry and total isospin 1. The third pion's isospin can then combine with it to make isospin 0. Thus, an eta can decay into 3 pions without violating isospin if 2 of them have pwave wavefunctions. 


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