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TMF, 2011, Volume 169, Number 1, Pages 100–111 (Mi tmf6712)  

This article is cited in 11 scientific papers (total in 11 papers)

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

L. Ts. Adzhemyan, M. V. Kompaniets

Saint Petersburg State University, St.~Petersburg, Russia

Abstract: 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

DOI: https://doi.org/10.4213/tmf6712

Full text: PDF file (505 kB)
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English version:
Theoretical and Mathematical Physics, 2011, 169:1, 1450–1459

Bibliographic databases:

Document Type: Article
Received: 20.10.2011

Citation: L. Ts. Adzhemyan, M. V. Kompaniets, “Renormalization group and the $\varepsilon$-expansion: Representation of the $\beta$-function and anomalous dimensions by nonsingular integrals”, TMF, 169:1 (2011), 100–111; Theoret. and Math. Phys., 169:1 (2011), 1450–1459

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v169/i1/p100

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. Ts. Adzhemyan, M. V. Kompaniets, S. V. Novikov, V. K. Sazonov, “Representation of the $\beta$-function and anomalous dimensions by nonsingular integrals: Proof of the main relation”, Theoret. and Math. Phys., 175:3 (2013), 717–726  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Kalagov G.A., Nalimov M.Yu., “Higher-Order Asymptotics and Critical Indexes in the Phi(3) Theory”, Nucl. Phys. B, 884 (2014), 672–683  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Adzhemyan L.Ts., Kompaniets M.V., “Five-Loop Numerical Evaluation of Critical Exponents of the Phi(4) Theory”, 15th International Workshop on Advanced Computing and Analysis Techniques in Physics Research, Journal of Physics Conference Series, 523, IOP Publishing Ltd, 2014, 012049  crossref  isi  scopus
    4. L. Ts. Adzhemyan, S. E. Vorobyeva, M. V. Kompaniets, “Representation of the $\beta$-function and anomalous dimensions by nonsingular integrals in models of critical dynamics”, Theoret. and Math. Phys., 185:1 (2015), 1361–1369  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. A. L. Pismenskii, “Calculation of the critical index $\eta$ for the $\varphi^3$ theory by the conformal bootstrap method”, Theoret. and Math. Phys., 185:1 (2015), 1516–1521  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. Pismensky A.L., “Calculation of Critical Index Eta of the Phi(3)-Theory in Four-Loop Approximation By the Conformal Bootstrap Technique”, Int. J. Mod. Phys. A, 30:24 (2015), 1550138  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Pismensky A.L. Pis'mak Yu.M., “Scaling Violation in Massless Scalar Quantum Field Models in Logarithmic Dimensions”, J. Phys. A-Math. Theor., 48:32, SI (2015), 325401  crossref  mathscinet  zmath  isi  elib  scopus
    8. Danco M. Hnatic M. Komarova M.V. Lucivjansky T. Nalimov M.Yu., “Superfluid Phase Transition With Activated Velocity Fluctuations: Renormalization Group Approach”, Phys. Rev. E, 93:1 (2016), 012109  crossref  mathscinet  adsnasa  isi  scopus
    9. Adzhemyan L.Ts., Hnatic M., Kompaniets M., Lucivjansky T., Mizisin L., “Numerical Calculation of Scaling Exponents of Percolation Process in the Framework of Renormalization Group Approach”, Mathematical Modeling and Computational Physics (MMCP 2015), EPJ Web of Conferences, 108, eds. Adam G., Busa J., Hnatic M., EDP Sciences, 2016, 02005  crossref  mathscinet  isi  scopus
    10. Adzhemyan L.Ts. Hnatic M. Kompaniets M.V. Luoivjansky T. Mizisin L., “Directed Percolation: Calculation of Feynman Diagrams in the Three-Loop Approximation”, Mathematical Modeling and Computational Physics 2017 (Mmcp 2017), Epj Web of Conferences, 173, ed. Adam G. Busa J. Hnatic M. Podgainy D., E D P Sciences, 2018, UNSP 02001  crossref  isi  scopus
    11. L. Ts. Adzhemyan, S. E. Vorob'eva, E. V. Ivanova, M. V. Kompaniets, “Representation of renormalization group functions by nonsingular integrals in a model of the critical dynamics of ferromagnets: The fourth order of the $\varepsilon$-expansion”, Theoret. and Math. Phys., 195:1 (2018), 584–594  mathnet  crossref  crossref  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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