Fock–Schwinger method in the case of different masses

The result of removing heavy particles with different mass from the theory spectrum can be described at low energy by the effective action, which is an asymptotic series in inverse powers of the mass. It is demonstrated how the leading terms of the asymptotic series can be calculated on the basis of the Fock–Schwinger proper time method. Theory with broken $U(3) \times U(3)$ chiral symmetry is considered to illustrate this approach. It is shown that different masses lead to the formation of a set of vertices describing the effects of the explicit breaking of flavor symmetry, which are usually not revealed in a less rigorous consideration of the problem. The result can be used in the study of theories with spontaneous symmetry breaking and in the development of effective theories of the Standard Model.