Strong QCD Problem

The most general gauge invariant QCD Lagrangian upto dimension four reads

${\mathcal{L}}_{QCD}=-\frac{1}{4}{G}_{\mu \nu }^a{G}^{a,\mu \nu }+\bar{q}(i{\gamma }_{\mu }{D}^{\mu }-{\mathcal{M}}_q)q-\frac{{\alpha }_S}{8\pi }\theta {\tilde{G}}_{\mu \nu }^a{\tilde{G}}^{a,\mu \nu }$

The zero temperature mass of the axion field is determined by non-perturbative QCD effects, related to the axion scale $f_a$ and can be computed as

$m_a(T=0)=5.7\times 10^{-5}\frac{10^{-11}}{f_a}$.

The axion potential is periodic, but we can adopt a harmonic approximation as

$V(\theta )=m_a^2(T)(1-cos\theta )\approx \frac{1}{2}{m}_a^2(T){\theta }^2$.