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  • Open Access

Second-order renormalized Hamiltonian of Yukawa theory

Kamil Serafin*, Carter M. Gustin, and Peter J. Love†

  • Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

  • *Contact author: kamil.serafin.physics@gmail.com
  • †Also at Brookhaven National Laboratory.
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Phys. Rev. D 112, 096003 – Published 4 November, 2025

DOI: https://doi.org/10.1103/63qr-r25y

Abstract

Using the renormalization group procedure for effective particles we calculate the effective Hamiltonians in the theory of a fermion field coupled to a scalar field via the Yukawa interaction. The theory is renormalized by the addition of counterterms. Necessary counterterms are determined by computing matrix elements of the effective Hamiltonian. All calculations are performed up to the second order in the expansion in powers of the coupling constant. Renormalized effective Hamiltonians are well-defined symmetric forms acting in the Fock space as opposed to the renormalized bare Hamiltonian, which is not well defined without regularization. We introduce computational techniques that should streamline higher-order calculations and may be of independent interest.

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References (72)

  1. S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 74, 2890 (2014).
  2. Z. Davoudi et al., Report of the Snowmass 2021 topical group on lattice gauge theory, in Snowmass 2021 (2022), arXiv:2209.10758.
  3. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  4. W. Kamleh, D. B. Leinweber, and A. Virgili, Numerical indication that center vortices drive dynamical mass generation in QCD, Phys. Rev. D 110, L051502 (2024).
  5. K. Cichy and M. Constantinou, A guide to light-cone PDFs from Lattice QCD: An overview of approaches, techniques and results, Adv. High Energy Phys. 2019, 3036904 (2019).
  6. M. C. Bañuls and K. Cichy, Review on novel methods for lattice gauge theories, Rep. Prog. Phys. 83, 024401 (2020).
  7. K. Nagata, Finite-density lattice QCD and sign problem: Current status and open problems, Prog. Part. Nucl. Phys. 127, 103991 (2022).
  8. C. W. Bauer et al., Quantum simulation for high-energy physics, PRX Quantum 4, 027001 (2023).
  9. A. Di Meglio et al., Quantum computing for high-energy physics: State of the art and challenges, PRX Quantum 5, 037001 (2024).
  10. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for quantum field theories, Science 336, 1130 (2012).
  11. E. A. Martinez et al., Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature (London) 534, 516 (2016).
  12. P. A. M. Dirac, Forms of relativistic dynamics, Rev. Mod. Phys. 21, 392 (1949).
  13. S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rep. 301, 299 (1998).
  14. S. D. Glazek, Perturbative formulae for relativistic interactions of effective particles, Acta Phys. Pol. B 43, 1843 (2012).
  15. S. D. Glazek, Similarity renormalization group approach to boost invariant Hamiltonian dynamics, Acta Phys. Pol. B 29, 1979 (1998).
  16. S. D. Glazek and K. G. Wilson, Renormalization of Hamiltonians, Phys. Rev. D 48, 5863 (1993).
  17. S. D. Glazek and K. G. Wilson, Perturbative renormalization group for Hamiltonians, Phys. Rev. D 49, 4214 (1994).
  18. H. Yukawa, On the interaction of elementary particles I, Proc. Phys. Math. Soc. Jap. 17, 48 (1935).
  19. R. Machleidt, Phenomenology and meson theory of nuclear forces, in Handbook of Nuclear Physics, edited by I. Tanihata, H. Toki, and T. Kajino (Springer, Singapore, 2022), pp. 1–53.
  20. K. G. Wilson, T. S. Walhout, A. Harindranath, W.-M. Zhang, R. J. Perry, and S. D. Glazek, Nonperturbative QCD: A weak coupling treatment on the light front, Phys. Rev. D 49, 6720 (1994).
  21. K. Serafin, M. Gómez-Rocha, J. More, and S. D. Głazek, Dynamics of heavy quarks in the Fock space, Phys. Rev. D 109, 016017 (2024).
  22. S. D. Głazek, Elementary example of exact effective-Hamiltonian computation, Phys. Rev. D 103, 014021 (2021).
  23. S. D. Glazek, Renormalization group procedure for effective particles: Elementary example of exact solution with finite mass corrections and no involvement of vacuum, Phys. Rev. D 85, 125018 (2012).
  24. S. D. Glazek, Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles, Phys. Rev. D 87, 125032 (2013).
  25. T. Masłowski, The exact solution of the Wegner flow equation with the Mielke generator for 3×3 Hermitian matrices, Phys. Scripta 99, 105282 (2024).
  26. H. C. Pauli and S. J. Brodsky, Solving field theory in one space one time dimension, Phys. Rev. D 32, 1993 (1985).
  27. H. C. Pauli and S. J. Brodsky, Discretized light cone quantization: Solution to a field theory in one space one time dimensions, Phys. Rev. D 32, 2001 (1985).
  28. T. Eller, H. C. Pauli, and S. J. Brodsky, Discretized light cone quantization: The massless and the massive Schwinger model, Phys. Rev. D 35, 1493 (1987).
  29. A. Harindranath and J. P. Vary, Solving two-dimensional φ4 theory by discretized light-front quantization, Phys. Rev. D 36, 1141 (1987).
  30. K. Hornbostel, S. J. Brodsky, and H. C. Pauli, Light cone quantized QCD in (1+1)-dimensions, Phys. Rev. D 41, 3814 (1990).
  31. K. Hornbostel, The application of light cone quantization to quantum chromodynamics in (1+1)-dimensions, Other thesis, Stanford University, 1988.
  32. J. P. Vary, M. Huang, S. Jawadekar, M. Sharaf, A. Harindranath, and D. Chakrabarti, Critical coupling for two-dimensional ϕ4 theory in discretized light-cone quantization, Phys. Rev. D 105, 016020 (2022).
  33. J. P. Vary, H. Honkanen, J. Li, P. Maris, S. J. Brodsky, A. Harindranath, G. F. de Teramond, P. Sternberg, E. G. Ng, and C. Yang, Hamiltonian light-front field theory in a basis function approach, Phys. Rev. C 81, 035205 (2010).
  34. P. Wiecki, Y. Li, X. Zhao, P. Maris, and J. P. Vary, Basis light-front quantization approach to positronium, Phys. Rev. D 91, 105009 (2015).
  35. Y. Li, P. Maris, and J. P. Vary, Quarkonium as a relativistic bound state on the light front, Phys. Rev. D 96, 016022 (2017).
  36. S. Jia and J. P. Vary, Basis light front quantization for the charged light mesons with color singlet Nambu–Jona-Lasinio interactions, Phys. Rev. C 99, 035206 (2019).
  37. J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Parton distribution functions from a light front Hamiltonian and QCD evolution for light mesons, Phys. Rev. Lett. 122, 172001 (2019).
  38. Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary (BLFQ Collaboration), Transverse structure of electron in momentum space in basis light-front quantization, Phys. Rev. D 103, 036005 (2021).
  39. J. Lan, K. Fu, C. Mondal, X. Zhao, and J. P. Vary (BLFQ Collaboration), Light mesons with one dynamical gluon on the light front, Phys. Lett. B 825, 136890 (2022).
  40. S. Xu, C. Mondal, J. Lan, X. Zhao, Y. Li, and J. P. Vary (BLFQ Collaboration), Nucleon structure from basis light-front quantization, Phys. Rev. D 104, 094036 (2021).
  41. S. Nair, C. Mondal, X. Zhao, A. Mukherjee, and J. P. Vary (BLFQ Collaboration), Basis light-front quantization approach to photon, Phys. Lett. B 827, 137005 (2022).
  42. Z. Kuang, K. Serafin, X. Zhao, and J. P. Vary (BLFQ Collaboration), All-charm tetraquark in front form dynamics, Phys. Rev. D 105, 094028 (2022).
  43. B. Lin, S. Nair, C. Mondal, S. Xu, Z. Hu, P. Zhang, X. Zhao, and J. P. Vary (BLFQ Collaboration), Chiral-odd gluon generalized parton distributions in the proton: A light-front quantization approach, Phys. Lett. B 860, 139153 (2025).
  44. M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, Quantum simulation of quantum field theory in the light-front formulation, Phys. Rev. A 105, 032418 (2022).
  45. M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Light-front field theory on current quantum computers, Entropy 23, 597 (2021).
  46. M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Simulating hadronic physics on noisy intermediate-scale quantum devices using basis light-front quantization, Phys. Rev. A 103, 062601 (2021).
  47. W. M. Kirby, S. Hadi, M. Kreshchuk, and P. J. Love, Quantum simulation of second-quantized Hamiltonians in compact encoding, Phys. Rev. A 104, 042607 (2021).
  48. M. Kreshchuk, J. P. Vary, and P. J. Love, Simulating scattering of composite particles, arXiv:2310.13742.
  49. W. Qian, M. Li, C. A. Salgado, and M. Kreshchuk, Efficient quantum simulation of QCD jets on the light front, Phys. Rev. D 111, 096001 (2025).
  50. W. A. Simon, C. M. Gustin, K. Serafin, A. Ralli, G. R. Goldstein, and P. J. Love, Ladder operator block-encoding, arXiv:2503.11641.
  51. B. H. Allen and R. J. Perry, Systematic renormalization in Hamiltonian light front field theory, Phys. Rev. D 58, 125017 (1998).
  52. R. D. Kylin, B. H. Allen, and R. J. Perry, Systematic renormalization in Hamiltonian light front field theory: The massive generalization, Phys. Rev. D 60, 067704 (1999).
  53. B. H. Allen and R. J. Perry, Glueballs in a Hamiltonian light front approach to pure glue QCD, Phys. Rev. D 62, 025005 (2000).
  54. S. D. Glazek and T. Maslowski, Renormalized Poincare algebra for effective particles in quantum field theory, Phys. Rev. D 65, 065011 (2002).
  55. S. D. Glazek, Non-local structure of renormalized Hamiltonian densities on the light-front hyperplane in space-time, Acta Phys. Pol. B 41, 1937 (2010).
  56. A. Mielke, Flow equations for band-matrices, Eur. Phys. J. B 5, 605 (1998).
  57. S. D. Glazek and J. Mlynik, Optimization of perturbative similarity renormalization group for Hamiltonians with asymptotic freedom and bound states, Phys. Rev. D 67, 045001 (2003).
  58. S. D. Glazek and J. Mlynik, Accuracy estimate for a relativistic Hamiltonian approach to bound state problems in theories with asymptotic freedom, Acta Phys. Pol. B 35, 723 (2004).
  59. F. Wegner, Flow equations and normal ordering: A survey, J. Phys. A 39, 8221 (2006).
  60. S. D. Glazek and R. J. Perry, The impact of bound states on similarity renormalization group transformations, Phys. Rev. D 78, 045011 (2008).
  61. E. Anderson, S. K. Bogner, R. J. Furnstahl, E. D. Jurgenson, R. J. Perry, and A. Schwenk, Block diagonalization using similarity renormalization group flow equations, Phys. Rev. C 77, 037001 (2008).
  62. K. G. Wilson, A model of coupling constant renormalization, Phys. Rev. D 2, 1438 (1970).
  63. T. Maslowski and M. Wieckowski, Fourth order similarity renormalization of a model Hamiltonian, Phys. Rev. D 57, 4976 (1998).
  64. K. Serafin, Relativistic model of Hamiltonian renormalization for bound states and scattering amplitudes, Few Body Syst. 58, 125 (2017).
  65. R. J. Perry and K. G. Wilson, Perturbative renormalizability with an infinite number of relevant and marginal operators, Nucl. Phys. B403, 587 (1993).
  66. R. J. Perry, Using Wilson’s renormalization group to repair symmetries, Phys. Rep. 348, 33 (2001).
  67. A. Houriet and A. Kind, Invariant classification of the terms of the matrix S, Helv. Phys. Acta (Switzerland) 22, 319 (1949).
  68. G. C. Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268 (1950).
  69. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995).
  70. G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publications (American Mathematical Society, 1975).
  71. M. Reed and B. Simon, Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness (Academic Press, New York, 1975).
  72. S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, edited by B. G.-G. Chen, D. Derbes, D. Griffiths, B. Hill, R. Sohn, and Y.-S. Ting (World Scientific Publishing, Hackensack, 2018).

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Phys. Rev. D 112, 096003– Published 4 November, 2025
  • Received 6 August 2025
  • Accepted 23 September 2025
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