Three-loop renormalization with a cutoff in a sextic model

N. V. Kharuk

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 273-286

Résumé

The paper studies the quantum action for the three-dimensional real theory $\phi^6$ using the background field method. Three-loop renormalization is performed with a cutoff regularization in the coordinate representation. An explicit form of the first three coefficients for the renormalization constants is presented. The absence of non-local singular contributions is discussed.

Bibliographie

[1] C. Itzykson, J. B. Zuber, Quantum Field Theory, Mcgraw-hill, New York, 1980 | MR

[2] M. E. Peskin, D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995 | MR

[3] J. C. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge University Press, 1984 | MR | Zbl

[4] O. I. Zavialov, Renormalized quantum field theory, Kluwer Academic Publishers, Dodrecht–Boston, 1990 | MR | Zbl

[5] J. Math. Sci., 257:4 (2021), 526–536 | DOI | MR | Zbl

[6] A. V. Ivanov, N. V. Kharuk, “Two-loop Cutoff renormalization of 4-D Yang–Mills effective action”, J. Phys. G: Nucl. Part. Phys., 48 (2020), 015002 | DOI

[7] A. V. Ivanov, N. V. Kharuk, “Formula for two-loop divergent part of 4-D Yang–Mills effective action”, Eur. Phys. J. C, 82 (2022), 997 | DOI

[8] A. V. Ivanov, “Explicit cutoff regularization in coordinate representation”, J. Phys. A: Math. Theor., 55 (2022), 495401 | DOI | MR | Zbl

[9] A. V. Ivanov, N. V. Kharuk, “Three-loop divergences in effective action of 4-dimensional Yang–Mills theory with cutoff regularization: $\Gamma_4^2$-contribution”, Zap. Nauchn. Sem. POMI, 520, 2023, 162–188 | MR

[10] P. V. Akacevich, A. V. Ivanov, “On two-loop effective action of 2D sigma model”, Eur. Phys. J. C, 83 (2023), 653 | DOI

[11] A. V. Ivanov, 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, 2024, arXiv: ; 2402.14549https://www.pdmi.ras.ru/preprint/2024/24-02.html | MR

[12] A. V. Ivanov, N. V. Kharuk, Three-loop renormalization of the quantum action for a five-dimensional scalar cubic model with the usage of the background field method and a cutoff regularization, 2024, arXiv: ; 2404.07513https://www.pdmi.ras.ru/preprint/2024/24-05.html | MR

[13] L. N. Lipatov, “Calculation of the Gell–Mann–Low function in scalar theory with strong nonlinearity”, Sov. Phys. JETP, 44 (1976), 1055–1062

[14] R. D. Pisarski, “Fixed points of $(\phi^6)_3$ and $(\phi^4)_4$ theories”, Phys. Rev. D, 28 (1983), 1554–1556 | DOI

[15] W. A. Bardeen, M. Moshe, M. Bander, “Spontaneous breaking of scale invariance and the ultraviolet fixed point in $O(N)$-symmetric $(\phi^6_3)$ theory”, Phys. Rev. Lett., 52 (1984), 1188–1191 | DOI

[16] R. Gudmundsdottir, G. Rydnell, P. Salomonson, “More on $O(N)$-symmetric $\phi^6_3$ theory”, Phys. Rev. Lett., 53 (1984), 2529–2531 | DOI

[17] J. S. Hager, “Six-loop renormalization group functions of $O(n)$-symmetric $\phi^6$-theory and $\epsilon$-expansions of tricritical exponents up to $\epsilon^3$”, J. Phys. A, 35 (2002), 2703–2711 | DOI | MR | Zbl

[18] J. A. Gracey, “Renormalization of scalar field theories in rational spacetime dimensions”, Eur. Phys. J. C, 80 (2020), 604 | DOI

[19] I. M. Gel'fand, G. E. Shilov, Generalized Functions, v. 1, AMS Chelsea Publishing, 377, Properties and Operations, AMS, 1964 | MR

[20] D. I. Kazakov, Radiative Corrections, Divergences, Regularization, Renormalization, Renormalization Group and All That in Examples in Quantum Field Theory, 2009, arXiv: 0901.2208 [hep-ph]

[21] B. S. DeWitt, “Quantum theory of gravity. 2. The manifestly covariant theory”, Phys. Rev., 162 (1967), 1195–1239 | DOI | Zbl

[22] B. S. DeWitt, “Quantum theory of gravity. 3. Applications of the covariant theory”, Phys. Rev., 162 (1967), 1239–1256 | DOI | Zbl

[23] G. 't Hooft, “The background field method in gauge field theories”, Proceedings Acta Universitatis Wratislaviensis (Karpacz, 1975), v. 1, Wroclaw, 1976, 345–369

[24] L. F. Abbott, “Introduction to the background field method”, Acta Phys. Polon. B, 13:1–2 (1982), 33–50 | MR

[25] I. Ya. Aref'eva, A. A. Slavnov, L. D. Faddeev, “Generating functional for the S-matrix in gauge-invariant theories”, TMF, 21:3 (1974), 311–321

[26] M. Lüscher, “Dimensional regularisation in the presence of large background fields”, Annals Phys., 142 (1982), 359–392 | DOI | MR

[27] A. V. Ivanov, N. V. Kharuk, “Special functions for heat kernel expansion”, Eur. Phys. J. Plus, 137 (2022), 1060 | DOI | MR

[28] A. N. Vasil'ev, Functional methods in quantum field theory and statistics, Gordon Breach, London, 1998

[29] A. V. Ivanov, On a criterion for a cutoff regularization in the coordinate representation, 2024 , arXiv: https://www.pdmi.ras.ru/preprint/2024/24-04.html2403.09218