- #1
QuantumCosmo
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Hi,
I was wondering about the U(1)_A problem. The Lagrangian exhibits a (in the limit of vanishing quark masses) U(1)_A symmetry but due to the chiral anomaly, the current [tex]J_5^{\mu}[/tex] is not conserved:
[tex]\partial_{\mu}J_5^{\mu} = G\tilde{G} + 2i\bar{u}\gamma_5 u +...[/tex]
The [tex]G\tilde{G}[/tex] term is itself the divergence of the (not gauge invariant) current [tex]K^{\mu}[/tex].
(I have left out constant factors etc)
So in the limit of vanishing quark masses, the current [tex]\tilde{J}_5^{\mu} = J_5^{\mu} - K^{\mu}[/tex] is conserved and so is the charge
[tex] \tilde{Q}_5 = \int \tilde{J}_5^{\mu} \d^3[/tex]
Now it seems that although there actually isn't a U(1)_A symmetry in my theory, I still get a Goldstone boson because [tex] \tilde{Q}_5[/tex] is conserved.
But I thought Goldstone bosons occurred because of spontenously broken continuous symmetries and not because of conserved charges?
Can anyone help me with that?
Thank you very much,
Quantum
I was wondering about the U(1)_A problem. The Lagrangian exhibits a (in the limit of vanishing quark masses) U(1)_A symmetry but due to the chiral anomaly, the current [tex]J_5^{\mu}[/tex] is not conserved:
[tex]\partial_{\mu}J_5^{\mu} = G\tilde{G} + 2i\bar{u}\gamma_5 u +...[/tex]
The [tex]G\tilde{G}[/tex] term is itself the divergence of the (not gauge invariant) current [tex]K^{\mu}[/tex].
(I have left out constant factors etc)
So in the limit of vanishing quark masses, the current [tex]\tilde{J}_5^{\mu} = J_5^{\mu} - K^{\mu}[/tex] is conserved and so is the charge
[tex] \tilde{Q}_5 = \int \tilde{J}_5^{\mu} \d^3[/tex]
Now it seems that although there actually isn't a U(1)_A symmetry in my theory, I still get a Goldstone boson because [tex] \tilde{Q}_5[/tex] is conserved.
But I thought Goldstone bosons occurred because of spontenously broken continuous symmetries and not because of conserved charges?
Can anyone help me with that?
Thank you very much,
Quantum