Peccei-Quinn-symmetry and effective Lagrangian for the Axion field

To solve the strong CP-problem Peccei and Quinn suggested the use of a new $U(1)$-symmetry called the PQ-symmetry. For this symmetry they constructed an effective Lagrangian involving the Nambu-Goldstone-boson of the spontanoues broken PQ-symmetry:

$$\mathcal{L}_a := -\frac{1}{2}\partial_\mu a\partial^\mu a -\mathcal{L}_{int}(\partial_\mu a; \psi)+ (\frac{a}{F_a}\xi+\overline{\theta})\frac{g^2}{32\pi^2}F_{\mu\nu}^a\tilde{F}^{\mu\nu}_a $$

In his paper R. Peccei (https://arxiv.org/pdf/hep-ph/0607268.pdf) said the first and second term are needed to make the whole standard model Lagrangian invariant under $U(1)$ and the last term ensures that $U_{PQ}(1)$ has the right axial anomaly.

I have serveral questions to this arguments:

1. 
   

   Why do we need the first and the second term to make the Lagrangian invariant under $U(1)$? Shouldn't it be invariant without that terms (beside from the anomaloues breaking)?

   
   


2. 
   

   The standart model Lagrangian has already an anomaloues axial current from the QCD-sector. Why is it neccessary to implement the last term for this?

   
   


3. 
   

   In which step of the solution to the CP-problem do we make use of the $U(1)$-symmetry? The mechanism is based on this effective Lagrangian couldn't it be constructed from another theory?

   
   
   

Answer

1. 
   

   You don't need the first and second terms for invariance as such. But you should recognize that these are simply the kinetic and interaction terms for the axion field. Are you proposing to include a field without kinetic term into the Lagrangian?

   
   


2. 
   

   You want an additional anomaly coming from the axion term here, so that the total anomaly in the Peccei-Quinn model cancels.

   
   



3. 
   

   You "use" the Peccei-Quinn $\mathrm{U}_\text{PC}(1)$ symmetry when you spontaneously break it under the influence of the periodic effective potential generated by the anomalous currents. The resulting physical axion is the Goldstone boson of the $\mathrm{U}_\text{PC}(1)$ broken by the VEV taken by the axion field w.r.t. to this potential.