Zero-mode product expansions and higher order couplings in gauge backgrounds

We study a four-dimensional low-energy effective field theory derived from extra dimensional field theories with general gauge backgrounds. We find that products among fermionic zero-modes and lightest scalar modes are expanded by other fermionic zero-modes and lightest scalar modes. Using this aspect, we show that higher-order couplings among the fermionic zero-modes and the lightest scalar mode can be decomposed into three-point couplings. This selection rule originating from a structure of the Dirac-type operators indicates the operator product expansion the underlying conformal field theory.