Renormalization group and the \(\varepsilon\)-expansion: representation of the \(\beta\)-function and anomalous dimensions by nonsingular integrals. (English. Russian original) Zbl 1274.81167
Theor. Math. Phys. 169, No. 1, 1450-1459 (2011); translation from Teor. Mat. Fiz. 169, No. 1, 100-111 (2011).
Summary: 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 \(\varepsilon^4\).
MSC:
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
| 81T80 | Simulation and numerical modelling (quantum field theory) (MSC2010) |
| 82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |
| 81T18 | Feynman diagrams |
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