The isospin for two quarks u,d, it is chosen that
u=|1/2,1/2>,d=|1/2,−1/2>,
in fundamental Rep of
SU(2).
the anti-quarks have anti-fundamental Rep of SU(2) [thus same as the fundamental Rep of SU(2)],
ˉd=−|1/2,1/2>,ˉu=|1/2,−1/2>,
be aware that there is a minus sign in front of the ˉd=−|1/2,1/2>.
Somehow the minus sign is crucial, to get the triplet and singlet state wavefunction correct. namely, we have
2⊗2=3⊕1
where the 3 is the triplet (3 pions) and 1 is the singlet (another meson), for the pseudo-scalar mesons.
Why do we know that we should choose ˉd=−|1/2,1/2>? In p.169 of Griffiths's book, he said the minus sign "is a technical detail, but it does not affect the result essentially."
I find that the minus sign is very crucial to get the correct wavefunction of pion π0 to be one of the triplet
uˉu−dˉd
=|1,0>=|1/2,1/2>|1/2,−1/2>+|1/2,−1/2>|1/2,1/2>
instead of being a singlet
uˉu+dˉd
=|0,0>
so why do we know the minus sign is there and is it key to the physics or not? I am asking a deeper reason behind it, because I know the minus sign there makes 100% sense for the pion wavefunctions.
This post imported from StackExchange Physics at 2020-10-29 20:02 (UTC), posted by SE-user annie marie heart