The strong CP problem and higher-dimensional gauge theories

We discuss a natural scenario to solve the strong CP problem in the framework of the higher-dimensional gauge theory. An axion-like field A(y) has been built in as the extra-space component of the higher-dimensional gauge field. The coupling of A(y) with gluons is attributed to the radiatively induced "Chern-Simons" (CS) term. We adopt a toy model with some unknown gauge symmetry U(1)x. The CS term is obtained in two ways: first by a concrete 1-loop calculation and next by use of Fujikawa's method to deal with the chiral anomaly in 4D space-time. The obtained results are identical, which implies that the radiative correction to the CS term is "1-loop exact" and is also free from UV-divergence even though the theory itself is non-renormalizable. As a novel feature of this scenario, the thus-obtained CS term is no longer linear in the field A(y) as in the usually discussed CS term in 5D space-time but is a periodic function of A(y), since A(y) has a physical meaning as the Wilson-loop phase. We argue how such a novel feature in this scenario causes modification of the ordinary solutions of the strong CP problem based on the axion fields.