- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Three-loop corrections to the Fermi decay constant in the MS¯ scheme

Stephen P. Martin

PDF

Phys. Rev. D 112, 073002 – Published 6 October, 2025

DOI: https://doi.org/10.1103/mpy1-jmzz

Abstract

I present the leading 3-loop contributions to the Fermi decay constant in the Standard Model, using the tadpole-free pure MS¯ scheme. The calculation is exact in the limit in which the quantum chromodynamics coupling, the top-quark Yukawa coupling, and the square root of the Higgs self-coupling are all treated as large compared to the electroweak gauge couplings. The effect of the 3-loop contribution is to decrease the estimate of the Higgs vacuum expectation value (VEV), for fixed on-shell inputs, by about 3.5 MeV. The renormalization scale dependence of the computed Fermi constant is greatly reduced compared to the previously known complete 2-loop order result, and as a fraction is now less than ±2×10−6 for renormalization scales between 90 and 250 GeV. This corresponds to a theoretical uncertainty in the VEV that is well under 1 MeV, and much smaller than the current parametric uncertainties coming mostly from the top-quark mass. I also comment on the impact on the precise prediction of the W-boson mass.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (109)

  1. T. Kinoshita and A. Sirlin, Radiative corrections to Fermi interactions, Phys. Rev. 113, 1652 (1959).
  2. Y. Nir, The mass ratio mc/mb in semileptonic B decays, Phys. Lett. B 221, 184 (1989).
  3. T. van Ritbergen and R. G. Stuart, On the precise determination of the Fermi coupling constant from the muon lifetime, Nucl. Phys. B564, 343 (2000).
  4. M. Steinhauser and T. Seidensticker, Second order corrections to the muon lifetime and the semileptonic B decay, Phys. Lett. B 467, 271 (1999).
  5. A. Pak and A. Czarnecki, Mass effects in muon and semileptonic b→c decays, Phys. Rev. Lett. 100, 241807 (2008).
  6. A. Ferroglia, G. Ossola, and A. Sirlin, Considerations concerning the radiative corrections to muon decay in the Fermi and standard theories, Nucl. Phys. B560, 23 (1999).
  7. M. Fael, K. Schönwald, and M. Steinhauser, Third order corrections to the semileptonic b→c and the muon decays, Phys. Rev. D 104, 016003 (2021).
  8. M. Czakon, A. Czarnecki, and M. Dowling, Three-loop corrections to the muon and heavy quark decay rates, Phys. Rev. D 103, L111301 (2021).
  9. D. M. Webber et al. (MuLan Collaboration), Measurement of the positive muon lifetime and determination of the Fermi constant to part-per-million precision, Phys. Rev. Lett. 106, 041803 (2011).
  10. V. Tishchenko et al. (MuLan Collaboration), Detailed report of the mulan measurement of the positive muon lifetime and determination of the Fermi constant, Phys. Rev. D 87, 052003 (2013).
  11. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  12. A. Sirlin, Radiative corrections in the SU(2)-L x U(1) theory: A simple renormalization framework, Phys. Rev. D 22, 971 (1980).
  13. W. J. Marciano and A. Sirlin, Radiative corrections to neutrino induced neutral current phenomena in the SU(2)-L x U(1) theory, Phys. Rev. D 22, 2695 (1980); 31, 213(E) (1985).
  14. A. Sirlin and W. J. Marciano, Radiative corrections to muon-neutrino Nμ+N→μ−+X and their effect on the Determination of ρ2 and sin2θW, Nucl. Phys. B189, 442 (1981).
  15. J. J. van der Bij and F. Hoogeveen, Two loop correction to weak interaction parameters due to a heavy fermion doublet, Nucl. Phys. B283, 477 (1987).
  16. A. Djouadi and C. Verzegnassi, Virtual very heavy top effects in LEP/SLC precision measurements, Phys. Lett. B 195, 265 (1987).
  17. A. Djouadi, O(ααs) vacuum polarization functions of the standard model gauge bosons, Nuovo Cimento Soc. Ital. Fis. 100A, 357 (1988).
  18. B. A. Kniehl, Two loop corrections to the vacuum polarizations in perturbative QCD, Nucl. Phys. B347, 86 (1990).
  19. F. Halzen and B. A. Kniehl, Δ r beyond one loop, Nucl. Phys. B353, 567 (1991).
  20. R. Barbieri, M. Beccaria, P. Ciafaloni, G. Curci, and A. Vicere, Radiative correction effects of a very heavy top, Phys. Lett. B 288, 95 (1992); 312, 511(E) (1993).
  21. R. Barbieri, M. Beccaria, P. Ciafaloni, G. Curci, and A. Vicere, Two loop heavy top effects in the standard model, Nucl. Phys. B409, 105 (1993).
  22. J. Fleischer, O. V. Tarasov, and F. Jegerlehner, Two loop heavy top corrections to the rho parameter: A simple formula valid for arbitrary Higgs mass, Phys. Lett. B 319, 249 (1993).
  23. A. Djouadi and P. Gambino, Electroweak gauge bosons selfenergies: Complete QCD corrections, Phys. Rev. D 49, 3499 (1994); 53, 4111(E) (1996).
  24. G. Degrassi, P. Gambino, and A. Vicini, Two loop heavy top effects on the mZ−mW interdependence, Phys. Lett. B 383, 219 (1996).
  25. G. Degrassi, P. Gambino, and A. Sirlin, Precise calculation of MW, sin2θW (MZ), and sin2θefflept, Phys. Lett. B 394, 188 (1997).
  26. P. Malde and R. G. Stuart, Complete O(Nfα2) weak contributions to the muon lifetime, Nucl. Phys. B552, 41 (1999).
  27. A. Freitas, W. Hollik, W. Walter, and G. Weiglein, Complete fermionic two loop results for the MW−MZ interdependence, Phys. Lett. B 495, 338 (2000); 570, 265(E) (2003).
  28. A. Freitas, W. Hollik, W. Walter, and G. Weiglein, Electroweak two loop corrections to the MW−MZ mass correlation in the standard model, Nucl. Phys. B632, 189 (2002); B666, 305(E) (2003).
  29. M. Awramik and M. Czakon, Complete two loop bosonic contributions to the muon lifetime in the standard model, Phys. Rev. Lett. 89, 241801 (2002).
  30. A. Onishchenko and O. Veretin, Two loop bosonic electroweak corrections to the muon lifetime and MZ−MW interdependence, Phys. Lett. B 551, 111 (2003).
  31. M. Awramik, M. Czakon, A. Onishchenko, and O. Veretin, Bosonic corrections to Δr at the two loop level, Phys. Rev. D 68, 053004 (2003).
  32. M. Awramik and M. Czakon, Complete two loop electroweak contributions to the muon lifetime in the standard model, Phys. Lett. B 568, 48 (2003).
  33. L. Avdeev, J. Fleischer, S. Mikhailov, and O. Tarasov, 0(ααs2) correction to the electroweak ρ parameter, Phys. Lett. B 336, 560 (1994); 349, 597(E) (1995).
  34. K. G. Chetyrkin, J. H. Kuhn, and M. Steinhauser, Corrections of order O(GFMt2αs2) to the ρ parameter, Phys. Lett. B 351, 331 (1995).
  35. K. G. Chetyrkin, J. H. Kuhn, and M. Steinhauser, QCD corrections from top quark to relations between electroweak parameters to order αs2, Phys. Rev. Lett. 75, 3394 (1995).
  36. J. J. van der Bij, K. G. Chetyrkin, M. Faisst, G. Jikia, and T. Seidensticker, Three loop leading top mass contributions to the ρ parameter, Phys. Lett. B 498, 156 (2001).
  37. M. Faisst, J. H. Kuhn, T. Seidensticker, and O. Veretin, Three loop top quark contributions to the ρ parameter, Nucl. Phys. B665, 649 (2003).
  38. Y. Schroder and M. Steinhauser, Four-loop singlet contribution to the ρ parameter, Phys. Lett. B 622, 124 (2005).
  39. K. G. Chetyrkin, M. Faisst, J. H. Kuhn, P. Maierhofer, and C. Sturm, Four-loop QCD corrections to the ρ parameter, Phys. Rev. Lett. 97, 102003 (2006).
  40. R. Boughezal and M. Czakon, Single scale tadpoles and O(GFmt2αs3) corrections to the ρ parameter, Nucl. Phys. B755, 221 (2006).
  41. M. Awramik, M. Czakon, A. Freitas, and G. Weiglein, Precise prediction for the W boson mass in the standard model, Phys. Rev. D 69, 053006 (2004).
  42. G. Degrassi, P. Gambino, and P. P. Giardino, The mW−mZ interdependence in the standard model: A new scrutiny, J. High Energy Phys. 05 (2015) 154.
  43. B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, Two-loop electroweak threshold corrections in the standard model, Nucl. Phys. B896, 19 (2015).
  44. B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, mr: A C++ library for the matching and running of the standard model parameters, Comput. Phys. Commun. 206, 84 (2016).
  45. S. P. Martin and D. G. Robertson, Standard model parameters in the tadpole-free pure MS¯ scheme, Phys. Rev. D 100, 073004 (2019); The latest version of the SMDR code can be obtained from https://github.com/davidgrobertson/SMDR or https://www.niu.edu/spmartin/SMDR/.
  46. C. Ford, I. Jack, and D. R. T. Jones, The standard model effective potential at two loops, Nucl. Phys. B387, 373 (1992); B504, 551(E) (1997).
  47. S. P. Martin, Three-loop standard model effective potential at leading order in strong and top Yukawa couplings, Phys. Rev. D 89, 013003 (2014).
  48. S. P. Martin, Effective potential at three loops, Phys. Rev. D 96, 096005 (2017).
  49. S. P. Martin, Two loop effective potential for a general renormalizable theory and softly broken supersymmetry, Phys. Rev. D 65, 116003 (2002).
  50. S. P. Martin, Three-loop effective potential for softly broken supersymmetry, Phys. Rev. D 109, 015019 (2024).
  51. S. P. Martin, Four-loop standard model effective potential at leading order in QCD, Phys. Rev. D 92, 054029 (2015).
  52. S. P. Martin, Pole mass of the W boson at two-loop order in the pure MS¯ scheme, Phys. Rev. D 91, 114003 (2015).
  53. S. P. Martin, Z-boson pole mass at two-loop order in the pure MS¯ scheme, Phys. Rev. D 92, 014026 (2015).
  54. S. P. Martin, Three-loop QCD corrections to the electroweak boson masses, Phys. Rev. D 106, 013007 (2022).
  55. S. P. Martin and D. G. Robertson, Higgs boson mass in the standard model at two-loop order and beyond, Phys. Rev. D 90, 073010 (2014).
  56. S. P. Martin, Top-quark pole mass in the tadpole-free MS¯ scheme, Phys. Rev. D 93, 094017 (2016).
  57. S. P. Martin, Matching relations for decoupling in the standard model at two loops and beyond, Phys. Rev. D 99, 033007 (2019).
  58. Z. Alam and S. P. Martin, Standard model at 200 GeV, Phys. Rev. D 107, 013010 (2023).
  59. S. P. Martin and H. H. Patel, Two-loop effective potential for generalized gauge fixing, Phys. Rev. D 98, 076008 (2018).
  60. S. P. Martin, Taming the Goldstone contributions to the effective potential, Phys. Rev. D 90, 016013 (2014).
  61. J. Elias-Miro, J. R. Espinosa, and T. Konstandin, Taming infrared divergences in the effective potential, J. High Energy Phys. 08 (2014) 034.
  62. P. Maierhöfer, J. Usovitsch, and P. Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun. 230, 99 (2018).
  63. P. Maierhöfer and J. Usovitsch, Kira 1.2 release notes, arXiv:1812.01491.
  64. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
  65. S. P. Martin and D. G. Robertson, Evaluation of the general 3-loop vacuum Feynman integral, Phys. Rev. D 95, 016008 (2017).
  66. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mpy1-jmzz for more details.
  67. K. G. Chetyrkin, M. Faisst, C. Sturm, and M. Tentyukov, ε-finite basis of master integrals for the integration-by-parts method, Nucl. Phys. B742, 208 (2006).
  68. D. J. Broadhurst, Three loop on-shell charge renormalization without integration: Lambda-MS (QED) to four loops, Z. Phys. C 54, 599 (1992).
  69. L. V. Avdeev, Recurrence relations for three loop prototypes of bubble diagrams with a mass, Comput. Phys. Commun. 98, 15 (1996).
  70. D. J. Broadhurst, Massive three-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur. Phys. J. C 8, 311 (1999).
  71. J. Fleischer and M. Y. Kalmykov, Single mass scale diagrams: Construction of a basis for the epsilon expansion, Phys. Lett. B 470, 168 (1999).
  72. M. Steinhauser, MATAD: A program package for the computation of MAssive TADpoles, Comput. Phys. Commun. 134, 335 (2001).
  73. Y. Schroder and A. Vuorinen, High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles, J. High Energy Phys. 06 (2005) 051.
  74. S. P. Martin, Renormalized ϵ-finite master integrals and their virtues: The three-loop self-energy case, Phys. Rev. D 105, 056014 (2022).
  75. A. I. Davydychev and M. Y. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B699, 3 (2004).
  76. M. Y. Kalmykov, About higher order epsilon-expansion of some massive two- and three-loop master-integrals, Nucl. Phys. B718, 276 (2005).
  77. S. Bekavac, A. G. Grozin, D. Seidel, and V. A. Smirnov, Three-loop on-shell Feynman integrals with two masses, Nucl. Phys. B819, 183 (2009).
  78. J. Grigo, J. Hoff, P. Marquard, and M. Steinhauser, Moments of heavy quark correlators with two masses: Exact mass dependence to three loops, Nucl. Phys. B864, 580 (2012).
  79. D. R. T. Jones, Two loop diagrams in Yang-Mills theory, Nucl. Phys. B75, 531 (1974).
  80. W. E. Caswell, Asymptotic behavior of nonAbelian gauge theories to two loop order, Phys. Rev. Lett. 33, 244 (1974).
  81. D. R. T. Jones, The two loop beta function for a G1×G2 gauge theory, Phys. Rev. D 25, 581 (1982).
  82. M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization, Nucl. Phys. B222, 83 (1983).
  83. M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 2. Yukawa couplings, Nucl. Phys. B236, 221 (1984).
  84. M. E. Machacek and M. T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 3. Scalar quartic couplings, Nucl. Phys. B249, 70 (1985).
  85. M. x. Luo and Y. Xiao, Two loop renormalization group equations in the standard model, Phys. Rev. Lett. 90, 011601 (2003).
  86. O. V. Tarasov, A. A. Vladimirov, and A. Y. Zharkov, The Gell-Mann-low function of QCD in the three loop approximation, Phys. Lett. 93B, 429 (1980).
  87. S. A. Larin and J. A. M. Vermaseren, The three loop QCD Beta function and anomalous dimensions, Phys. Lett. B 303, 334 (1993).
  88. L. N. Mihaila, J. Salomon, and M. Steinhauser, Gauge coupling beta functions in the standard model to three loops, Phys. Rev. Lett. 108, 151602 (2012).
  89. L. N. Mihaila, J. Salomon, and M. Steinhauser, Renormalization constants and beta functions for the gauge couplings of the standard model to three-loop order, Phys. Rev. D 86, 096008 (2012).
  90. A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Model at three loops, J. High Energy Phys. 01 (2013) 017.
  91. T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin, The four loop beta function in quantum chromodynamics, Phys. Lett. B 400, 379 (1997).
  92. M. Czakon, The four-loop QCD beta-function and anomalous dimensions, Nucl. Phys. B710, 485 (2005).
  93. A. V. Bednyakov and A. F. Pikelner, Four-loop strong coupling beta-function in the standard model, Phys. Lett. B 762, 151 (2016).
  94. A. V. Bednyakov and A. F. Pikelner, On the four-loop strong coupling beta-function in the SM, EPJ Web Conf. 125, 04008 (2016).
  95. M. F. Zoller, Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level, J. High Energy Phys. 02 (2016) 095.
  96. C. Poole and A. E. Thomsen, Weyl consistency conditions and γ5, Phys. Rev. Lett. 123, 041602 (2019).
  97. J. Davies, F. Herren, C. Poole, M. Steinhauser, and A. E. Thomsen, Gauge coupling β functions to four-loop order in the standard model, Phys. Rev. Lett. 124, 071803 (2020).
  98. P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118, 082002 (2017).
  99. F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, The five-loop beta function of Yang-Mills theory with fermions, J. High Energy Phys. 02 (2017) 090.
  100. T. Luthe, A. Maier, P. Marquard, and Y. Schröder, The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge, J. High Energy Phys. 10 (2017) 166.
  101. K. G. Chetyrkin and M. F. Zoller, Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the standard model, J. High Energy Phys. 06 (2012) 033.
  102. A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Yukawa coupling beta-functions in the standard model at three loops, Phys. Lett. B 722, 336 (2013).
  103. A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Three-loop SM beta-functions for matrix Yukawa couplings, Phys. Lett. B 737, 129 (2014).
  104. K. G. Chetyrkin, Quark mass anomalous dimension to O(αs4), Phys. Lett. B 404, 161 (1997).
  105. J. A. M. Vermaseren, S. A. Larin, and T. van Ritbergen, The 4-loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B 405, 327 (1997).
  106. P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Quark mass and field anomalous dimensions to O(αs5), J. High Energy Phys. 10 (2014) 076.
  107. K. G. Chetyrkin and M. F. Zoller, β-function for the Higgs self-interaction in the standard model at three-loop level, J. High Energy Phys. 04 (2013) 091; 09 (2013) 155(E).
  108. A. V. Bednyakov, A. F. Pikelner, and V. N. Velizhanin, Higgs self-coupling beta-function in the standard model at three loops, Nucl. Phys. B875, 552 (2013).
  109. K. G. Chetyrkin and M. F. Zoller, Leading QCD-induced four-loop contributions to the β-function of the Higgs self-coupling in the SM and vacuum stability, J. High Energy Phys. 06 (2016) 175.

Outline

Information

Phys. Rev. D 112, 073002– Published 6 October, 2025
  • Received 29 July 2025
  • Accepted 8 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation