Renormalization group and the $\varepsilon$-expansion: Representation of the $\beta$-function and anomalous dimensions by nonsingular integrals

In the framework of the renormalization group and the $\varepsilon$-expansion, we propose expressions for the $\beta$-function and anomalous dimensions in terms of renormalized one-irreducible functions. These expressions are convenient for numerical calculations. We choose the renormalization scheme in which the quantities calculated using $R$ operations are represented by integrals that do not contain singularities in $\varepsilon$. We develop a completely automated calculation system starting from constructing diagrams, determining relevant subgraphs, combinatorial coefficients, etc., up to determining critical exponents. As an example, we calculate the critical exponents of the $\varphi^3$ model in the order $\varepsilon^4$.