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Renormalization, cutoff, and gluing for a quartic model with boundary

A. V. Ivanov*

  • *Contact author: regul1@mail.ru
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Phys. Rev. D 114, 105009 – Published 6 November, 2026

DOI: https://doi.org/10.1103/ssdk-cw96

Abstract

In this paper, we study the renormalization of a three-dimensional quartic model on a compact connected Riemannian manifold with a boundary. We show that, in addition to the standard shift of the mass parameter, it is necessary to take into account boundary contributions involving singular coefficients. This indicates the need to extend the classical action within the framework of standard renormalization theory. The requirements of consistency regarding the gluing of manifolds lead to renormalization of an operator on the boundary.

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References (43)

  1. L. D. Faddeev and A. A. Slavnov, Gauge Fields: An Introduction to Quantum Theory, Frontiers in Physics Vol. 83 (Addison-Wesley, Reading, MA, 1991), pp, 1–236.
  2. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, MA, 1995), pp. 1–868.
  3. J. C. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion (Cambridge University Press, Cambridge, England, 1984), pp. 1–380.
  4. I. M. Gel’fand and G. E. Shilov, Generalized Functions, Volume 1: Properties and Operations (AMS Chelsea Publishing, New York, 1964), Vol. 377, pp. 1–423.
  5. V. S. Vladimirov, Methods of the Theory of Generalized Functions (CRC Press, London, 2002), pp. 1–328.
  6. O. I. Zavialov, Renormalized Quantum Field Theory (Kluwer Academic Publishers, Dodrecht, Boston, 1990), pp. 1–524.
  7. D. I. Kazakov, Radiative Corrections, Divergences, Regularization, Renormalization, Renormalization Group and All That in Examples in Quantum Field Theory, JINR, UC-2008-34 (2009), 1–91, arXiv:0901.2208.
  8. V. S. Vladimirov and V. V. Zharinov, Equations of Mathematical Physics (Fizmatlit, Moscow, 2004), pp. 1–400.
  9. K. Symanzik, Schrodinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B190, 1 (1981).
  10. D. M. McAvity and H. Osborn, Quantum field theories on manifolds with curved boundaries: Scalar fields, Nucl. Phys. B394, 728 (1993).
  11. M. Atiyah, Topological quantum field theories, Publ. Math. l’Inst. Hautes Sci. 68, 175 (1988).
  12. G. Segal, The definition of conformal field theory, Topology, Geometry and Quantum Field Theory, London Mathematical Society Lecture Note Series Vol. 308 (Cambridge University Press, Cambridge, England, 2004), pp. 421–577.
  13. N. Reshetikhin, Lectures on quantization of gauge systems, New Paths Towards Quantum Gravity (Springer, Berlin Heidelberg, 2010), pp. 125–190.
  14. A. S. Cattaneo, P. Mnev, and N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, Commun. Math. Phys. 357, 631 (2018).
  15. S. Kandel, Functorial quantum field theory in the Riemannian setting, Adv. Theor. Math. Phys. 20, 1443 (2016).
  16. S. Kandel, P. Mnev, and K. Wernli, Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing, Adv. Theor. Math. Phys. 25, 1847 (2021).
  17. C. G. Bollini and J. J. Giambiagi, Dimensional renormalization: The number of dimensions as a regularizing parameter, Nuovo Cimento B 12, 20 (1972).
  18. G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B44, 189 (1972).
  19. T. Bakeyev and A. Slavnov, Higher covariant derivative regularization revisited, Mod. Phys. Lett. A 11, 1539 (1996).
  20. K. V. Stepanyantz, The higher covariant derivative regularization as a tool for revealing the structure of quantum corrections in supersymmetric gauge theories, Proc Steklov Inst Math / Trudy Matematicheskogo instituta imeni VA Steklova 309, 284 (2020).
  21. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, Berlin, 1987), pp. 1–535.
  22. A. N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Chapman and Hall/CRC, Boca Raton, 2004), pp. 1–681.
  23. N. V. Kharuk, Three-loop renormalization with a cutoff in a sextic model, J. Math. Sci. 299, 878 (2026).
  24. L. L. Salcedo, Linearized renormalization, Eur. Phys. J. C 85, 945 (2025).
  25. Rongchan Zhu and Xiangchan Zhu, Lattice approximation to the dynamical Φ34-model, Ann. Probab. 46, 397 (2018).
  26. S. Albeverio and S. Kusuoka, The invariant measure and the flow associated to the Φ34-quantum field model, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20, 1359 (2020).
  27. I. Bailleul, N. V. Dang, L. Ferdinand, and T. D. Tô, Φ34-measures on compact Riemannian 3-manifolds, arXiv:2304.10185.
  28. M. V. Kraskov, Yu., and M. Pis’mak, Casimir energy for the interaction of n δ-plates with a massless real scalar field, Theor. Math. Phys. 227, 1026 (2026).
  29. A. V. Ivanov, Explicit cutoff regularization in coordinate representation, J. Phys. A 55, 495401 (2022).
  30. A. V. Ivanov, An applicability condition of a cutoff regularization in the coordinate representation, Funct. Anal. Appl. 59, 1 (2025).
  31. A. V. Ivanov and I. V. Korenev, Quasi-local probability averaging in the context of cutoff regularization, arXiv:2603.28235.
  32. A. V. Ivanov, Effective actions, cutoff regularization, quasi-locality, and gluing of partition functions, J. Phys. A 58, 135401 (2025).
  33. M. Nakahara, Geometry, Topology and Physics (IoP, Bristol, 2003), pp. 1–573.
  34. J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics Vol. 218 (Springer, New York, 2012), pp. 1–708.
  35. M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, 2nd ed., Applied Mathematical Sciences (Springer, New York, 2011), pp. 1–637.
  36. N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Willey, New York, 1980), p. 1.
  37. V. I. Lebedev and V. I. Agoshkov, Poincaré-Steklov Operators and their Applications in Analysis (Academy of Sciences, Moskva, USSR, 1983), pp. 1–184.
  38. M. Lüscher, Dimensional regularisation in the presence of large background fields, Ann. Phys. (N.Y.) 142, 359 (1982).
  39. A. V. Ivanov and N. V. Kharuk, Special functions for heat kernel expansion, Eur. Phys. J. Plus 137, 1060 (2022).
  40. I. Chavel, Riemannian Geometry: A Modern Introduction (Cambridge University Press, Cambridge, England, 2006), pp. 1–471.
  41. O. I. Zavialov, Bogolyubov’s R-operation and the Bogolyubov–Parasyuk theorem, Russ. Math. Surv. 49, 67 (1994).
  42. N. V. Kharuk, Four-loop renormalization with a cutoff in a sextic model, J. Phys. A 58, 395401 (2025).
  43. 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, Nucl. Phys. B1006, 116647 (2024).

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Phys. Rev. D 114, 105009– Published 6 November, 2026
  • Received 30 August 2026
  • Accepted 19 October 2026
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