Renormalization group and the~$\varepsilon$-expansion: Representation of the~$\beta$-function and anomalous dimensions by nonsingular integrals
Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 1, pp. 100-111
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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 $\varepsilon^4$.
Keywords:
renormalization group, $\varepsilon$-expansion, multiloop diagrams, critical exponents.
@article{TMF_2011_169_1_a9,
author = {L. Ts. Adzhemyan and M. V. Kompaniets},
title = {Renormalization group and the~$\varepsilon$-expansion: {Representation} of the~$\beta$-function and anomalous dimensions by nonsingular integrals},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {100--111},
publisher = {mathdoc},
volume = {169},
number = {1},
year = {2011},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a9/}
}
TY - JOUR AU - L. Ts. Adzhemyan AU - M. V. Kompaniets TI - Renormalization group and the~$\varepsilon$-expansion: Representation of the~$\beta$-function and anomalous dimensions by nonsingular integrals JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2011 SP - 100 EP - 111 VL - 169 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a9/ LA - ru ID - TMF_2011_169_1_a9 ER -
%0 Journal Article %A L. Ts. Adzhemyan %A M. V. Kompaniets %T Renormalization group and the~$\varepsilon$-expansion: Representation of the~$\beta$-function and anomalous dimensions by nonsingular integrals %J Teoretičeskaâ i matematičeskaâ fizika %D 2011 %P 100-111 %V 169 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a9/ %G ru %F TMF_2011_169_1_a9
L. Ts. Adzhemyan; M. V. Kompaniets. Renormalization group and the~$\varepsilon$-expansion: Representation of the~$\beta$-function and anomalous dimensions by nonsingular integrals. Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 1, pp. 100-111. http://geodesic.mathdoc.fr/item/TMF_2011_169_1_a9/