Coulomb Gas and Sine-Gordon Model in Arbitrary Dimension

Higgs, inflaton and axion physics are examples where scalar fields naturally appear, in the two former cases, usually with a polynomial type self-interaction. In this work possible applications of periodic scalar potentials are discussed which belong to the sine-Gordon universality class. For example, a periodic self-interaction is proposed here as a possible extension, UV completion of the standard model Higgs potential. Using the usual parametrisation of the field around the first minimum, one recovers the Lagrangian of the Higgs sector but with a periodic interaction term for a single-component real scalar field. Another possible application is inflationary cosmology where the so called natural inflation i.e., the periodic PNGB potential has already been used as a competing inflationary model. Finally, one has to mention the periodic axion potential which was proposed to retain the CP conserving nature of QCD where, in principle, gauge symmetry and renormalizability allow the inclusion of CP violating terms but experimental data do not favour such an extension. The purpose of this work is to map out the exact phase structure of the sine-Gordon model by the functional renormalization group (RG) method in arbitrary dimension and to study the consequences of its renormalization on Higgs, inflaton and axion physics. It is shown that the pure sine-Gordon model has a single (broken) phase for d>2. Results are used to study the phase structure of the equivalent neutral Coulomb gas and to demonstrate that the isotropic XY spin model belongs to a different universality class for d>2. The findings (i) open a new platform to perform stability studies for models with periodic type Higgs potential which can be used as a possible UV completion, (ii) clarify the role of the RG running of the frequency parameter in the PNGB inflationary model, (iii) confirm the flattening of the axion potential.