Fate of global symmetries in the Universe: QCD axion, quintessential axion and trans-Planckian inflaton decay-constant

Pseudoscalars appearing in particle physics are reviewd systematically. From the fundamental point of view at an ultra-violat completed theory, they can be light if they are realized as pseudo-Goldstone bosons of some spontaneously broken global symmetries. The spontaneous breaking scale is parametrized by the decay constant $f$. The global symmetry is defined by the lowest order terms allowed in the effective theory consistent with the gauge symmetry in question. Since any global symmetry is known to be broken at least by quantum gravitational effects, all pseudoscalars should be massive. The mass scale is determined by $f$ and the explicit breaking terms $\Delta V$ in the effective potential and also anomaly terms $\Delta\Lambda^4_G$ for some non-Abelian gauge groups $G$. The well-known example by non-Abelian gauge group breaking is the potential for the "invisible" QCD axion, via the Peccei-Quinn symmetry, which constitutes a major part of this review. Even if there is no breaking terms from gauge anomalies, there can be explicit breaking terms $\Delta V$ in the potential in which case the leading term suppressed by $f$ determines the pseudoscalar mass scale. If the breaking term is extremely small and the decay constant is trans-Planckian, the corresponding pseudoscalar can be a candidate for a `quintessential axion'. In general, $(\Delta V)^{1/4}$ is considered to be smaller than $f$, and hence the pseudo-Goldstone boson mass scales are considered to be smaller than the decay constants. In such a case, the potential of the pseudo-Goldstone boson at the grand unification scale is sufficiently flat near the top of the potential that it can be a good candidate for an inflationary model, which is known as `natural inflation'. We review all these ideas in the bosonic collective motion framework.