Three-loop renormalization of the quantum action for a four-dimensional scalar model with quartic interaction with the usage of the background field method and a cutoff regularization

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The paper studies the quantum action for the four-dimensional real ϕ4-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization R-operation on the example of separate three-loop diagrams are also discussed. The explicit forms of the first three coefficients for the renormalization constants and for the β-function are presented. Consistency with previously known results is shown.