Past exam of the theoretical physics major option course of the University of Oxford / 2016 / 6 / a

The continuous spacetime symmetries are translations, spatial rotations and Lorentz boosts, together the connected Poincaré group. The theory also has local electromagnetic U(1) gauge symmetry, which is an internal symmetry rather than a spacetime transformation. A nonzero mass and a dimensionful interaction do not give an additional exact scale invariance.

In four dimensions the action is dimensionless and $[\partial ]=1$. The two kinetic terms giveThe axion-photon operator has mass dimension five, so this is an effective field theory. Its dimensionless interaction strength grows with energy as $|g_a|E$. Without information about an ultraviolet completion, a conservative parametric validity condition is $E\ll \Lambda$ with $\Lambda \sim |g_a{|}^{-1}$, and $m_a$ must also lie below that scale for the axion to be a degree of freedom in the effective theory. Numerical factors such as $4\pi$ depend on the ultraviolet dynamics and on the chosen perturbative criterion; the Lagrangian alone does not fix them.