[go: up one dir, main page]

Skip to main content
Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Renormalization-group equations of the LEFT at two loops: dimension-five effects

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 03 June 2025
  • Volume 2025, article number 7, (2025)
  • Cite this article
Download PDF

You have full access to this open access article

Journal of High Energy Physics Aims and scope Submit manuscript
Renormalization-group equations of the LEFT at two loops: dimension-five effects
Download PDF
  • Luca Naterop  ORCID: orcid.org/0000-0003-1586-15191,2 &
  • Peter Stoffer  ORCID: orcid.org/0000-0001-7966-26961,2 
  • 159 Accesses

  • 10 Citations

  • 1 Altmetric

  • Explore all metrics

A preprint version of the article is available at arXiv.

Abstract

We present the first part of a systematic calculation of the two-loop anomalous dimensions in the low-energy effective field theory (LEFT): the effects at dimension five in the power counting. Our calculation is performed in a basis with generic mass matrices. We employ the algebraically consistent ’t Hooft-Veltman scheme for γ5 and we correct for evanescent as well as chiral-symmetry-breaking effects by including the appropriate finite counterterms. We also provide results for the CP-even sector in a scheme that coincides with naive dimensional regularization. We discuss two methods to avoid the explicit construction of gauge-variant operators, which in principle are needed for the cancellation of sub-divergences, even in the background-field method. The two methods are consistent with each other and with existing partial results. Our work is a further step towards a complete EFT framework for physics beyond the Standard Model at next-to-leading-logarithmic accuracy.

Article PDF

Download to read the full article text

Similar content being viewed by others

Renormalization-group equations of the LEFT at two loops: dimension-six baryon-number-violating operators

Article Open access 23 July 2025

Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme

Article Open access 13 February 2024

Low-energy effective field theory below the electroweak scale: operators and matching

Article Open access 06 March 2018

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.
  • Computational Number Theory
  • Crystal Field Theory
  • Field Theory and Polynomials
  • Special Functions
  • String Theory
  • Quantum Electrodynamics, Relativistic and Many-body Calculations
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

References

  1. W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621 [INSPIRE].

  2. B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].

    Article  ADS  Google Scholar 

  3. F. Feruglio, The chiral approach to the electroweak interactions, Int. J. Mod. Phys. A 8 (1993) 4937 [hep-ph/9301281] [INSPIRE].

  4. B. Grinstein and M. Trott, A Higgs-Higgs bound state due to new physics at a TeV, Phys. Rev. D 76 (2007) 073002 [arXiv:0704.1505] [INSPIRE].

    Article  ADS  Google Scholar 

  5. R. Alonso et al., The Effective Chiral Lagrangian for a Light Dynamical “Higgs Particle”, Phys. Lett. B 722 (2013) 330 [Erratum ibid. 726 (2013) 926] [arXiv:1212.3305] [INSPIRE].

  6. G. Buchalla, O. Catà and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552 [arXiv:1307.5017] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  7. G. Buchalla, O. Catà and C. Krause, On the Power Counting in Effective Field Theories, Phys. Lett. B 731 (2014) 80 [arXiv:1312.5624] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  8. B.M. Gavela, E.E. Jenkins, A.V. Manohar and L. Merlo, Analysis of General Power Counting Rules in Effective Field Theory, Eur. Phys. J. C 76 (2016) 485 [arXiv:1601.07551] [INSPIRE].

    Article  ADS  Google Scholar 

  9. A. Pich, I. Rosell, J. Santos and J.J. Sanz-Cillero, Fingerprints of heavy scales in electroweak effective Lagrangians, JHEP 04 (2017) 012 [arXiv:1609.06659] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  10. S. Karmakar, A. Dighe and R.S. Gupta, SMEFT predictions for semileptonic processes, Phys. Rev. D 111 (2025) 055002 [arXiv:2404.10061] [INSPIRE].

    Article  Google Scholar 

  11. R. Bause, H. Gisbert, M. Golz and G. Hiller, Lepton universality and lepton flavor conservation tests with dineutrino modes, Eur. Phys. J. C 82 (2022) 164 [arXiv:2007.05001] [INSPIRE].

    Article  ADS  Google Scholar 

  12. R. Bause, H. Gisbert, M. Golz and G. Hiller, Interplay of dineutrino modes with semileptonic rare B-decays, JHEP 12 (2021) 061 [arXiv:2109.01675] [INSPIRE].

    Article  ADS  Google Scholar 

  13. E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016 [Erratum ibid. 12 (2023) 043] [arXiv:1709.04486] [INSPIRE].

  14. W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197 [Erratum ibid. 11 (2022) 148] [arXiv:1908.05295] [INSPIRE].

  15. L. Lehman, Extending the Standard Model Effective Field Theory with the Complete Set of Dimension-7 Operators, Phys. Rev. D 90 (2014) 125023 [arXiv:1410.4193] [INSPIRE].

    Article  ADS  Google Scholar 

  16. Y. Liao and X.-D. Ma, Renormalization Group Evolution of Dimension-seven Baryon- and Lepton-number-violating Operators, JHEP 11 (2016) 043 [arXiv:1607.07309] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  17. C.W. Murphy, Dimension-8 operators in the Standard Model Effective Field Theory, JHEP 10 (2020) 174 [arXiv:2005.00059] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  18. H.-L. Li et al., Complete set of dimension-eight operators in the standard model effective field theory, Phys. Rev. D 104 (2021) 015026 [arXiv:2005.00008] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  19. Y. Liao and X.-D. Ma, An explicit construction of the dimension-9 operator basis in the standard model effective field theory, JHEP 11 (2020) 152 [arXiv:2007.08125] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  20. R.V. Harlander, T. Kempkens and M.C. Schaaf, Standard model effective field theory up to mass dimension 12, Phys. Rev. D 108 (2023) 055020 [arXiv:2305.06832] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  21. Y. Liao, X.-D. Ma and Q.-Y. Wang, Extending low energy effective field theory with a complete set of dimension-7 operators, JHEP 08 (2020) 162 [arXiv:2005.08013] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  22. C.W. Murphy, Low-Energy Effective Field Theory below the Electroweak Scale: Dimension-8 Operators, JHEP 04 (2021) 101 [arXiv:2012.13291] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  23. H.-L. Li et al., Low energy effective field theory operator basis at d ≤ 9, JHEP 06 (2021) 138 [arXiv:2012.09188] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  24. E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10 (2013) 087 [arXiv:1308.2627] [INSPIRE].

    Article  ADS  Google Scholar 

  25. E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence, JHEP 01 (2014) 035 [arXiv:1310.4838] [INSPIRE].

    Article  ADS  Google Scholar 

  26. R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04 (2014) 159 [arXiv:1312.2014] [INSPIRE].

    Article  ADS  Google Scholar 

  27. E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions, JHEP 01 (2018) 084 [Erratum ibid. 12 (2023) 042] [arXiv:1711.05270] [INSPIRE].

  28. A.J. Buras and P.H. Weisz, QCD Nonleading Corrections to Weak Decays in Dimensional Regularization and ’t Hooft-Veltman Schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].

  29. A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Effective Hamiltonians for ∆S = 1 and ∆B = 1 nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B 370 (1992) 69 [Addendum ibid. 375 (1992) 501] [INSPIRE].

  30. A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Two loop anomalous dimension matrix for ∆S = 1 weak nonleptonic decays I: \( \mathcal{O}\left({\alpha}_s^2\right) \), Nucl. Phys. B 400 (1993) 37 [hep-ph/9211304] [INSPIRE].

  31. M. Ciuchini, E. Franco, G. Martinelli and L. Reina, The ∆S = 1 effective Hamiltonian including next-to-leading order QCD and QED corrections, Nucl. Phys. B 415 (1994) 403 [hep-ph/9304257] [INSPIRE].

  32. M. Ciuchini, E. Franco, L. Reina and L. Silvestrini, Leading order QCD corrections to b → sγ and b → sg decays in three regularization schemes, Nucl. Phys. B 421 (1994) 41 [hep-ph/9311357] [INSPIRE].

  33. G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].

  34. K.G. Chetyrkin, M. Misiak and M. Münz, |∆F| = 1 nonleptonic effective Hamiltonian in a simpler scheme, Nucl. Phys. B 520 (1998) 279 [hep-ph/9711280] [INSPIRE].

  35. A.J. Buras, M. Misiak and J. Urban, Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the standard model, Nucl. Phys. B 586 (2000) 397 [hep-ph/0005183] [INSPIRE].

  36. C. Bobeth, P. Gambino, M. Gorbahn and U. Haisch, Complete NNLO QCD Analysis of \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) and Higher Order Electroweak Effects, JHEP 04 (2004) 071 [hep-ph/0312090] [INSPIRE].

  37. M. Gorbahn and U. Haisch, Effective Hamiltonian for non-leptonic |∆F| = 1 decays at NNLO in QCD, Nucl. Phys. B 713 (2005) 291 [hep-ph/0411071] [INSPIRE].

  38. T. Huber, E. Lunghi, M. Misiak and D. Wyler, Electromagnetic logarithms in \( \overline{B}\to {X}_s{l}^{+}{l}^{-} \), Nucl. Phys. B 740 (2006) 105 [hep-ph/0512066] [INSPIRE].

  39. M. Gorbahn, U. Haisch and M. Misiak, Three-loop mixing of dipole operators, Phys. Rev. Lett. 95 (2005) 102004 [hep-ph/0504194] [INSPIRE].

  40. M. Czakon, U. Haisch and M. Misiak, Four-Loop Anomalous Dimensions for Radiative Flavour-Changing Decays, JHEP 03 (2007) 008 [hep-ph/0612329] [INSPIRE].

  41. J. Aebischer, M. Fael, C. Greub and J. Virto, B physics Beyond the Standard Model at One Loop: Complete Renormalization Group Evolution below the Electroweak Scale, JHEP 09 (2017) 158 [arXiv:1704.06639] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  42. G. Panico, A. Pomarol and M. Riembau, EFT approach to the electron Electric Dipole Moment at the two-loop level, JHEP 04 (2019) 090 [arXiv:1810.09413] [INSPIRE].

    Article  ADS  Google Scholar 

  43. P. Morell and J. Virto, On the two-loop penguin contributions to the Anomalous Dimensions of four-quark operators, JHEP 04 (2024) 105 [arXiv:2402.00249] [INSPIRE].

    Article  ADS  Google Scholar 

  44. J. Aebischer, P. Morell, M. Pesut and J. Virto, Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR, arXiv:2501.08384 [INSPIRE].

  45. Y. Liao and X.-D. Ma, Renormalization Group Evolution of Dimension-seven Operators in Standard Model Effective Field Theory and Relevant Phenomenology, JHEP 03 (2019) 179 [arXiv:1901.10302] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  46. M. Chala and A. Titov, Neutrino masses in the Standard Model effective field theory, Phys. Rev. D 104 (2021) 035002 [arXiv:2104.08248] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  47. M. Chala, G. Guedes, M. Ramos and J. Santiago, Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions I, SciPost Phys. 11 (2021) 065 [arXiv:2106.05291] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  48. M. Accettulli Huber and S. De Angelis, Standard Model EFTs via on-shell methods, JHEP 11 (2021) 221 [arXiv:2108.03669] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  49. S. Das Bakshi, M. Chala, Á. Díaz-Carmona and G. Guedes, Towards the renormalisation of the Standard Model effective field theory to dimension eight: bosonic interactions II, Eur. Phys. J. Plus 137 (2022) 973 [arXiv:2205.03301] [INSPIRE].

    Article  Google Scholar 

  50. A. Helset, E.E. Jenkins and A.V. Manohar, Renormalization of the Standard Model Effective Field Theory from geometry, JHEP 02 (2023) 063 [arXiv:2212.03253] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  51. S. Das Bakshi and Á. Díaz-Carmona, Renormalisation of SMEFT bosonic interactions up to dimension eight by LNV operators, JHEP 06 (2023) 123 [arXiv:2301.07151] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  52. D. Zhang, Renormalization group equations for the SMEFT operators up to dimension seven, JHEP 10 (2023) 148 [arXiv:2306.03008] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  53. M. Chala and X. Li, Positivity restrictions on the mixing of dimension-eight SMEFT operators, Phys. Rev. D 109 (2024) 065015 [arXiv:2309.16611] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  54. D. Zhang, Revisiting renormalization group equations of the SMEFT dimension-seven operators, JHEP 02 (2024) 133 [arXiv:2310.11055] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  55. R. Boughezal, Y. Huang and F. Petriello, Renormalization-group running of dimension-8 four-fermion operators in the SMEFT, Phys. Rev. D 110 (2024) 116015 [arXiv:2408.15378] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  56. S.D. Bakshi et al., Renormalization of the SMEFT to dimension eight: Fermionic interactions I, JHEP 12 (2025) 214 [arXiv:2409.15408] [INSPIRE].

    Google Scholar 

  57. I. Brivio and M. Trott, The Standard Model as an Effective Field Theory, Phys. Rept. 793 (2019) 1 [arXiv:1706.08945] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  58. G. Isidori, F. Wilsch and D. Wyler, The standard model effective field theory at work, Rev. Mod. Phys. 96 (2024) 015006 [arXiv:2303.16922] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  59. A. Carmona, A. Lazopoulos, P. Olgoso and J. Santiago, Matchmakereft: automated tree-level and one-loop matching, SciPost Phys. 12 (2022) 198 [arXiv:2112.10787] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  60. J. Fuentes-Martín et al., A proof of concept for matchete: an automated tool for matching effective theories, Eur. Phys. J. C 83 (2023) 662 [arXiv:2212.04510] [INSPIRE].

    Article  ADS  Google Scholar 

  61. J. Fuentes-Martín, A. Palavrić and A.E. Thomsen, Functional matching and renormalization group equations at two-loop order, Phys. Lett. B 851 (2024) 138557 [arXiv:2311.13630] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  62. L. Allwicher et al., Computing tools for effective field theories: SMEFT-Tools 2022 Workshop Report, 14–16th September 2022, Zürich, Eur. Phys. J. C 84 (2024) 170 [arXiv:2307.08745] [INSPIRE].

    Article  ADS  Google Scholar 

  63. A.E. Thomsen, A Partially Fixed Background Field Gauge, JHEP 12 (2024) 185 [arXiv:2404.11640] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  64. J. de Vries, G. Falcioni, F. Herzog and B. Ruijl, Two- and three-loop anomalous dimensions of Weinberg’s dimension-six CP-odd gluonic operator, Phys. Rev. D 102 (2020) 016010 [arXiv:1907.04923] [INSPIRE].

    Article  ADS  Google Scholar 

  65. Z. Bern, J. Parra-Martinez and E. Sawyer, Structure of two-loop SMEFT anomalous dimensions via on-shell methods, JHEP 10 (2020) 211 [arXiv:2005.12917] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. J. Aebischer, A.J. Buras and J. Kumar, NLO QCD renormalization group evolution for nonleptonic ∆F=2 transitions in the SMEFT, Phys. Rev. D 106 (2022) 035003 [arXiv:2203.11224] [INSPIRE].

    Article  ADS  Google Scholar 

  67. J. Fuentes-Martín et al., Evanescent operators in one-loop matching computations, JHEP 02 (2023) 031 [arXiv:2211.09144] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  68. J. Aebischer, M. Pesut and Z. Polonsky, Renormalization scheme factorization of one-loop Fierz identities, JHEP 01 (2024) 060 [arXiv:2306.16449] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  69. E.E. Jenkins, A.V. Manohar, L. Naterop and J. Pagès, An algebraic formula for two loop renormalization of scalar quantum field theory, JHEP 12 (2023) 165 [arXiv:2308.06315] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  70. E.E. Jenkins, A.V. Manohar, L. Naterop and J. Pagès, Two loop renormalization of scalar theories using a geometric approach, JHEP 02 (2024) 131 [arXiv:2310.19883] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  71. L. Naterop and P. Stoffer, Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme, JHEP 02 (2024) 068 [arXiv:2310.13051] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  72. J. Aebischer, M. Pesut and Z. Polonsky, A simple dirac prescription for two-loop anomalous dimension matrices, Eur. Phys. J. C 84 (2024) 750 [arXiv:2401.16904] [INSPIRE].

    Article  Google Scholar 

  73. A.V. Manohar, J. Pagès and J. Roosmale Nepveu, Field redefinitions and infinite field anomalous dimensions, JHEP 05 (2024) 018 [arXiv:2402.08715] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  74. S. Di Noi, R. Gröber and M.K. Mandal, Two-loop running effects in Higgs physics in Standard Model Effective Field Theory, JHEP 12 (2025) 220 [arXiv:2408.03252] [INSPIRE].

    Google Scholar 

  75. L. Born, J. Fuentes-Martín, S. Kvedaraite˙ and A.E. Thomsen, Two-Loop Running in the Bosonic SMEFT Using Functional Methods, arXiv:2410.07320 [INSPIRE].

  76. S. Di Noi et al., γ5 schemes and the interplay of SMEFT operators in the Higgs-gluon coupling, Phys. Rev. D 109 (2024) 095024 [arXiv:2310.18221] [INSPIRE].

    Article  ADS  Google Scholar 

  77. J. Fuentes-Martín, A. Moreno-Sánchez, A. Palavrić and A.E. Thomsen, A Guide to Functional Methods Beyond One-Loop Order, arXiv:2412.12270 [INSPIRE].

  78. G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

  79. P. Breitenlohner and D. Maison, Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 1, Commun. Math. Phys. 52 (1977) 39 [INSPIRE].

  80. P. Breitenlohner and D. Maison, Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 2, Commun. Math. Phys. 52 (1977) 55 [INSPIRE].

  81. P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].

  82. F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].

    Article  ADS  Google Scholar 

  83. D. Kreimer, The γ5 Problem and Anomalies: A Clifford Algebra Approach, Phys. Lett. B 237 (1990) 59 [INSPIRE].

  84. J.G. Körner, D. Kreimer and K. Schilcher, A Practicable γ5 scheme in dimensional regularization, Z. Phys. C 54 (1992) 503 [INSPIRE].

  85. C. Schubert, The Yukawa Model as an Example for Dimensional Renormalization With γ5, Nucl. Phys. B 323 (1989) 478 [INSPIRE].

  86. R. Ferrari, A. Le Yaouanc, L. Oliver and J.C. Raynal, Gauge invariance and dimensional regularization with γ5 in flavor changing neutral processes, Phys. Rev. D 52 (1995) 3036 [INSPIRE].

  87. C. Cornella, F. Feruglio and L. Vecchi, Gauge invariance and finite counterterms in chiral gauge theories, JHEP 02 (2023) 244 [arXiv:2205.10381] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  88. H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ5 applied to chiral YM theories: full one-loop counterterm and RGE structure, JHEP 08 (2020) 024 [arXiv:2004.14398] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  89. H. Bélusca-Maïto et al., Two-loop application of the Breitenlohner-Maison/’t Hooft-Veltman scheme with non-anticommuting γ5: full renormalization and symmetry-restoring counterterms in an abelian chiral gauge theory, JHEP 11 (2021) 159 [arXiv:2109.11042] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  90. H. Bélusca-Maïto et al., Introduction to Renormalization Theory and Chiral Gauge Theories in Dimensional Regularization with Non-Anticommuting γ5, Symmetry 15 (2023) 622 [arXiv:2303.09120] [INSPIRE].

    Article  ADS  Google Scholar 

  91. D. Stöckinger and M. Weißwange, Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ5 in the BMHV scheme, JHEP 02 (2024) 139 [arXiv:2312.11291] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  92. P. Olgoso Ruiz and L. Vecchi, Spurious gauge-invariance and γ5 in dimensional regularization, JHEP 12 (2024) 080 [arXiv:2406.17013] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  93. P.L. Ebert, P. Kühler, D. Stöckinger and M. Weißwange, Shedding light on evanescent shadows — Exploration of non-anticommuting γ5 in Dimensional Regularisation, JHEP 01 (2025) 114 [arXiv:2411.02543] [INSPIRE].

    Article  Google Scholar 

  94. L.F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].

  95. L.F. Abbott, M.T. Grisaru and R.K. Schaefer, The Background Field Method and the S Matrix, Nucl. Phys. B 229 (1983) 372 [INSPIRE].

  96. J.A. Dixon and J.C. Taylor, Renormalization of wilson operators in gauge theories, Nucl. Phys. B 78 (1974) 552 [INSPIRE].

  97. H. Kluberg-Stern and J.B. Zuber, Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 2. Gauge Invariant Operators, Phys. Rev. D 12 (1975) 3159 [INSPIRE].

  98. S.D. Joglekar and B.W. Lee, General Theory of Renormalization of Gauge Invariant Operators, Annals Phys. 97 (1976) 160 [INSPIRE].

  99. W.S. Deans and J.A. Dixon, Theory of Gauge Invariant Operators: Their Renormalization and S Matrix Elements, Phys. Rev. D 18 (1978) 1113 [INSPIRE].

  100. G. Falcioni and F. Herzog, Renormalization of gluonic leading-twist operators in covariant gauges, JHEP 05 (2022) 177 [arXiv:2203.11181] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  101. M. Misiak and M. Münz, Two loop mixing of dimension five flavor changing operators, Phys. Lett. B 344 (1995) 308 [hep-ph/9409454] [INSPIRE].

  102. K.G. Chetyrkin, M. Misiak and M. Münz, Beta functions and anomalous dimensions up to three loops, Nucl. Phys. B 518 (1998) 473 [hep-ph/9711266] [INSPIRE].

  103. P. Gambino, M. Gorbahn and U. Haisch, Anomalous Dimension Matrix for Radiative and Rare Semileptonic B Decays Up to Three Loops, Nucl. Phys. B 673 (2003) 238 [hep-ph/0306079] [INSPIRE].

  104. J.-N. Lang, S. Pozzorini, H. Zhang and M.F. Zoller, Two-Loop Rational Terms in Yang-Mills Theories, JHEP 10 (2020) 016 [arXiv:2007.03713] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  105. K.G. Chetyrkin and F.V. Tkachov, Infrared r operation and ultraviolet counterterms in the MS scheme, Phys. Lett. B 114 (1982) 340 [INSPIRE].

  106. K.G. Chetyrkin and V.A. Smirnov, R* operation corrected, Phys. Lett. B 144 (1984) 419 [INSPIRE].

  107. V.A. Smirnov and K.G. Chetyrkin, R* Operation in the Minimal Subtraction Scheme, Theor. Math. Phys. 63 (1985) 462 [INSPIRE].

  108. F. Herzog and B. Ruijl, The R∗-operation for Feynman graphs with generic numerators, JHEP 05 (2017) 037 [arXiv:1703.03776] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  109. A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863 [https://doi.org/10.1093/oso/9780198855743.003.0002] [INSPIRE].

  110. J.C. Criado and M. Pérez-Victoria, Field redefinitions in effective theories at higher orders, JHEP 03 (2019) 038 [arXiv:1811.09413] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  111. V. Cirigliano, E. Mereghetti and P. Stoffer, Non-perturbative renormalization scheme for the CP-odd three-gluon operator, JHEP 09 (2020) 094 [arXiv:2004.03576] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  112. J.C. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge University Press, Cambridge (1984) [https://doi.org/10.1017/9781009401807] [INSPIRE].

  113. L. Naterop and P. Stoffer, Renormalization-group equations of the LEFT at two loops: dimension-six baryon-number-violating operators, arXiv:2505.03871 [INSPIRE].

  114. P. Nogueira, Automatic Feynman Graph Generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].

  115. J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].

  116. B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].

  117. H. Georgi, T. Tomaras and A. Pais, Strong CP violation without instantons, Phys. Rev. D 23 (1981) 469 [INSPIRE].

  118. R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without Supersymmetry in the Standard Model Effective Field Theory, Phys. Lett. B 739 (2014) 95 [arXiv:1409.0868] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  119. L. Naterop and P. Stoffer, in preparation.

  120. C. Abel et al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys. Rev. Lett. 124 (2020) 081803 [arXiv:2001.11966] [INSPIRE].

    Article  ADS  Google Scholar 

  121. E. Mereghetti et al., One-loop matching for quark dipole operators in a gradient-flow scheme, JHEP 04 (2022) 050 [Erratum ibid. 03 (2025) 101] [arXiv:2111.11449] [INSPIRE].

  122. J. Bühler and P. Stoffer, One-loop matching of CP-odd four-quark operators to the gradient-flow scheme, JHEP 08 (2023) 194 [arXiv:2304.00985] [INSPIRE].

    Article  ADS  Google Scholar 

  123. Ò.L. Crosas et al., One-loop matching of the CP-odd three-gluon operator to the gradient flow, Phys. Lett. B 847 (2023) 138301 [arXiv:2308.16221] [INSPIRE].

    Article  Google Scholar 

  124. Ò. L. Crosas and P. Stoffer, in preparation.

Download references

Acknowledgments

We thank B. Grinstein, A. V. Manohar, B. Ruijl, C.-H. Shen, A. Signer, D. Stockinger, A. E. Thomsen, and M. Zoller for useful discussions. Financial support by the Swiss National Science Foundation (Project No. PCEFP2_194272) is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Luca Naterop & Peter Stoffer

  2. PSI Center for Neutron and Muon Sciences, 5232, Villigen PSI, Switzerland

    Luca Naterop & Peter Stoffer

Authors
  1. Luca Naterop
    View author publications

    Search author on:PubMed Google Scholar

  2. Peter Stoffer
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Luca Naterop.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2412.13251

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naterop, L., Stoffer, P. Renormalization-group equations of the LEFT at two loops: dimension-five effects. J. High Energ. Phys. 2025, 7 (2025). https://doi.org/10.1007/JHEP06(2025)007

Download citation

  • Received: 21 January 2025

  • Accepted: 10 April 2025

  • Published: 03 June 2025

  • DOI: https://doi.org/10.1007/JHEP06(2025)007

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Effective Field Theories
  • Renormalization and Regularization
  • Renormalization Group
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Language editing
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our brands

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Discover
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Legal notice
  • Cancel contracts here

Not affiliated

Springer Nature

© 2025 Springer Nature