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?

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/hep-ph/0607268 Nov 19 '17 at 12:17
• What is this $\psi$ term in the interaction part for the axions? Apr 25 '19 at 11:03
• @MichaelAngelo, the $\psi$ represents "any fields in the theory", i.e., $\cal{L}(\partial_\mu a; \psi)$ stands for arbitrary interaction terms between $\partial_\mu a$ and any of the other fields in the theory. (See eq. 28 of arxiv.org/abs/hep-ph/9606475) Aug 25 '20 at 10:39

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.
• Thank you for the answer. The kinetic term is clear and the interaction between the axion and fermion is there to obtain the most general form of the effective Lagrangian? Should I interpret the last term to absorb the axial anomaly in the shift of the Nambu-Goldstone-Boson under a $U(1)$-transformation $a \rightarrow a +\alpha F_a$? Nov 19 '17 at 12:08