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I am studying this paper about the strong CP problem and axions by R.D. Peccei:
http://arxiv.org/abs/hep-ph/0607268
At the moment I read through the introduction and I have some questions about some points, so please could you help me clear out some things?. [In general I'll try to feed back this thread with more questions around the topic].
#1. It says that for massless quarks, and 2 light quarks [for QCD scale] we have an approximate global symmetry U[2]_V \times U[2]_A, so a vector and axial current symmetry. However if I look at the massless lagrangian [just the kinetic term] I have:
\mathcal{L}= i \bar{q} \gamma^\mu \partial_\mu q
What are the transformations done for the U[1] cases? I think the axial will have something like \propto \exp[ia \gamma_5]. In that case I don't see how the \exp[-ia \gamma_5] \gamma^\mu \exp[ia \gamma_5]= \gamma^\mu (it's not even a symmetry for the massless lagrangian)
#2. In Eq 1: I am having a problem in seeing that the triangle chiral anomaly gives rise to a divergence to the axial current of the form: \partial_\mu J^\mu_5 = g F^{\mu \nu}_a \tilde{F}_{a \mu \nu}. Do you have any source where I can see how does someone extract this result?
#3. In Eq 7: The gauge transformation of A goes like:
A^i \rightarrow \Omega A^i \Omega ^{-1} + \frac{i}{g} \nabla^i \Omega \Omega^{-1}
I think the second term is zero (\Omega \Omega^{-1}=1) or does it mean that (\nabla^i \Omega ) \Omega^{-1}. Then, also why does it give to the Omegas a spatial dependence (\textbf{r})?
#4 Finally for now, under Eq8, I don't understand what it means by the Jacobian of the mapping S_3 \rightarrow S_3 (what is S_3? the symmetry group of order 3!=6?)
http://arxiv.org/abs/hep-ph/0607268
At the moment I read through the introduction and I have some questions about some points, so please could you help me clear out some things?. [In general I'll try to feed back this thread with more questions around the topic].
#1. It says that for massless quarks, and 2 light quarks [for QCD scale] we have an approximate global symmetry U[2]_V \times U[2]_A, so a vector and axial current symmetry. However if I look at the massless lagrangian [just the kinetic term] I have:
\mathcal{L}= i \bar{q} \gamma^\mu \partial_\mu q
What are the transformations done for the U[1] cases? I think the axial will have something like \propto \exp[ia \gamma_5]. In that case I don't see how the \exp[-ia \gamma_5] \gamma^\mu \exp[ia \gamma_5]= \gamma^\mu (it's not even a symmetry for the massless lagrangian)
#2. In Eq 1: I am having a problem in seeing that the triangle chiral anomaly gives rise to a divergence to the axial current of the form: \partial_\mu J^\mu_5 = g F^{\mu \nu}_a \tilde{F}_{a \mu \nu}. Do you have any source where I can see how does someone extract this result?
#3. In Eq 7: The gauge transformation of A goes like:
A^i \rightarrow \Omega A^i \Omega ^{-1} + \frac{i}{g} \nabla^i \Omega \Omega^{-1}
I think the second term is zero (\Omega \Omega^{-1}=1) or does it mean that (\nabla^i \Omega ) \Omega^{-1}. Then, also why does it give to the Omegas a spatial dependence (\textbf{r})?
#4 Finally for now, under Eq8, I don't understand what it means by the Jacobian of the mapping S_3 \rightarrow S_3 (what is S_3? the symmetry group of order 3!=6?)
