Singlet night in Feynman-ville: one-loop matching of a real scalar

An Erratum to this article was published on 10 July 2020

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A complete one-loop matching calculation for real singlet scalar extensions of the Standard Model to the Standard Model effective field theory (SMEFT) of dimension- six operators is presented. We compare our analytic results obtained by using Feynman diagrams to the expressions derived in the literature by a combination of the universal one-loop effective action (UOLEA) approach and Feynman calculus. After identifying contributions that have been overlooked in the existing calculations, we find that the pure diagrammatic approach and the mixed method lead to identical results. We highlight some of the subtleties involved in computing one-loop matching corrections in SMEFT.

A preprint version of the article is available at ArXiv.

Change history


  1. [1]

    B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP10 (2010) 085 [arXiv:1008.4884] [INSPIRE].

  2. [2]

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

  3. [3]

    E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP10 (2013) 087 [arXiv:1308.2627] [INSPIRE].

  4. [4]

    E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence, JHEP01 (2014) 035 [arXiv:1310.4838] [INSPIRE].

  5. [5]

    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, JHEP04 (2014) 159 [arXiv:1312.2014] [INSPIRE].

  6. [6]

    R. Alonso, H.-M. Chang, E.E. Jenkins, A.V. Manohar and B. Shotwell, Renormalization group evolution of dimension-six baryon number violating operators, Phys. Lett.B 734 (2014) 302 [arXiv:1405.0486] [INSPIRE].

  7. [7]

    C. Grojean, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Scaling of Higgs Operators and Γ(h → γγ), JHEP04 (2013) 016 [arXiv:1301.2588] [INSPIRE].

  8. [8]

    J. Elias-Miró, J.R. Espinosa, E. Masso and A. Pomarol, Higgs windows to new physics through d = 6 operators: constraints and one-loop anomalous dimensions, JHEP11 (2013) 066 [arXiv:1308.1879] [INSPIRE].

  9. [9]

    C. Zhang, Effective field theory approach to top-quark decay at next-to-leading order in QCD, Phys. Rev.D 90 (2014) 014008 [arXiv:1404.1264] [INSPIRE].

  10. [10]

    G.M. Pruna and A. Signer, The μ → eγ decay in a systematic effective field theory approach with dimension 6 operators, JHEP10 (2014) 014 [arXiv:1408.3565] [INSPIRE].

  11. [11]

    J. Brod, A. Greljo, E. Stamou and P. Uttayarat, Probing anomalous \( t\overline{t}Z \)ttZ interactions with rare meson decays, JHEP02 (2015) 141 [arXiv:1408.0792] [INSPIRE].

  12. [12]

    C. Cheung and C.-H. Shen, Nonrenormalization Theorems without Supersymmetry, Phys. Rev. Lett.115 (2015) 071601 [arXiv:1505.01844] [INSPIRE].

  13. [13]

    M. Gorbahn and U. Haisch, Indirect probes of the trilinear Higgs coupling: gg → h and h → γγ, JHEP10 (2016) 094 [arXiv:1607.03773] [INSPIRE].

  14. [14]

    J. de Blas, J.C. Criado, M. Pérez-Victoria and J. Santiago, Effective description of general extensions of the Standard Model: the complete tree-level dictionary, JHEP03 (2018) 109 [arXiv:1711.10391] [INSPIRE].

  15. [15]

    A. Drozd, J. Ellis, J. Quevillon and T. You, The Universal One-Loop Effective Action, JHEP03 (2016) 180 [arXiv:1512.03003] [INSPIRE].

  16. [16]

    B. Henning, X. Lu and H. Murayama, How to use the Standard Model effective field theory, JHEP01 (2016) 023 [arXiv:1412.1837] [INSPIRE].

  17. [17]

    M.K. Gaillard, The Effective One Loop Lagrangian With Derivative Couplings, Nucl. Phys.B 268 (1986) 669 [INSPIRE].

  18. [18]

    L.-H. Chan, Derivative Expansion for the One Loop Effective Actions With Internal Symmetry, Phys. Rev. Lett.57 (1986) 1199 [INSPIRE].

  19. [19]

    O. Cheyette, Effective Action for the Standard Model With Large Higgs Mass, Nucl. Phys.B 297 (1988) 183 [INSPIRE].

  20. [20]

    F. del Aguila, Z. Kunszt and J. Santiago, One-loop effective lagrangians after matching, Eur. Phys. J.C 76 (2016) 244 [arXiv:1602.00126] [INSPIRE].

  21. [21]

    M. Boggia, R. Gomez-Ambrosio and G. Passarino, Low energy behaviour of standard model extensions, JHEP05 (2016) 162 [arXiv:1603.03660] [INSPIRE].

  22. [22]

    B. Henning, X. Lu and H. Murayama, One-loop Matching and Running with Covariant Derivative Expansion, JHEP01 (2018) 123 [arXiv:1604.01019] [INSPIRE].

  23. [23]

    S.A.R. Ellis, J. Quevillon, T. You and Z. Zhang, Mixed heavy-light matching in the Universal One-Loop Effective Action, Phys. Lett.B 762 (2016) 166 [arXiv:1604.02445] [INSPIRE].

  24. [24]

    J. Fuentes-Martin, J. Portoles and P. Ruiz-Femenia, Integrating out heavy particles with functional methods: a simplified framework, JHEP09 (2016) 156 [arXiv:1607.02142] [INSPIRE].

  25. [25]

    Z. Zhang, Covariant diagrams for one-loop matching, JHEP05 (2017) 152 [arXiv:1610.00710] [INSPIRE].

  26. [26]

    S.A.R. Ellis, J. Quevillon, T. You and Z. Zhang, Extending the Universal One-Loop Effective Action: Heavy-Light Coefficients, JHEP08 (2017) 054 [arXiv:1706.07765] [INSPIRE].

  27. [27]

    M. Krämer, B. Summ and A. Voigt, Completing the scalar and fermionic Universal One-Loop Effective Action, JHEP01 (2020) 079 [arXiv:1908.04798] [INSPIRE].

  28. [28]

    T. Cohen, M. Freytsis and X. Lu, Functional Methods for Heavy Quark Effective Theory, arXiv:1912.08814 [INSPIRE].

  29. [29]

    M. Jiang, N. Craig, Y.-Y. Li and D. Sutherland, Complete One-Loop Matching for a Singlet Scalar in the Standard Model EFT, JHEP02 (2019) 031 [arXiv:1811.08878] [INSPIRE].

  30. [30]

    C. Anastasiou, A. Carmona, A. Lazopoulos and J. Santiago, Match Maker, talk given at SMEFT-TOOLS 2019, Durham U.K. (2019), JSantiagoSMEFT-TOOLS19.pdf.

  31. [31]

    U. Haisch, M. Ruhdorfer, E. Salvioni, E. Venturini and A. Weiler, in preparation.

  32. [32]

    M. Frigerio, A. Pomarol, F. Riva and A. Urbano, Composite Scalar Dark Matter, JHEP07 (2012) 015 [arXiv:1204.2808] [INSPIRE].

  33. [33]

    R. Balkin, M. Ruhdorfer, E. Salvioni and A. Weiler, Dark matter shifts away from direct detection, JCAP11 (2018) 050 [arXiv:1809.09106] [INSPIRE].

  34. [34]

    V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf and G. Shaughnessy, Complex Singlet Extension of the Standard Model, Phys. Rev.D 79 (2009) 015018 [arXiv:0811.0393] [INSPIRE].

  35. [35]

    C. Gross, O. Lebedev and T. Toma, Cancellation Mechanism for Dark-Matter-Nucleon Interaction, Phys. Rev. Lett.119 (2017) 191801 [arXiv:1708.02253] [INSPIRE].

  36. [36]

    M. Ruhdorfer, E. Salvioni and A. Weiler, A Global View of the Off-Shell Higgs Portal, SciPost Phys.8 (2020) 027 [arXiv:1910.04170] [INSPIRE].

  37. [37]

    G.F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP06 (2007) 045 [hep-ph/0703164] [INSPIRE].

  38. [38]

    C. Bobeth, M. Misiak and J. Urban, Photonic penguins at two loops and mt dependence of BR[B → Xsl+l ], Nucl. Phys.B 574 (2000) 291 [hep-ph/9910220] [INSPIRE].

  39. [39]

    P. Gambino and U. Haisch, Complete electroweak matching for radiative B decays, JHEP10 (2001) 020 [hep-ph/0109058] [INSPIRE].

  40. [40]

    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun.140 (2001) 418 [hep-ph/0012260] [INSPIRE].

  41. [41]

    A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 — A complete toolbox for tree-level phenomenology, Comput. Phys. Commun.185 (2014) 2250 [arXiv:1310.1921] [INSPIRE].

  42. [42]

    T. Hahn and M. Ṕerez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput. Phys. Commun.118 (1999) 153 [hep-ph/9807565] [INSPIRE].

  43. [43]

    H.H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun.197 (2015) 276 [arXiv:1503.01469] [INSPIRE].

  44. [44]

    F. Jegerlehner, M. Yu. Kalmykov and O. Veretin, MS versus pole masses of gauge bosons: Electroweak bosonic two loop corrections, Nucl. Phys.B 641 (2002) 285 [hep-ph/0105304] [INSPIRE].

  45. [45]

    F. Jegerlehner, M. Yu. Kalmykov and O. Veretin, MS-bar versus pole masses of gauge bosons. 2. Two loop electroweak fermion corrections, Nucl. Phys.B 658 (2003) 49 [hep-ph/0212319] [INSPIRE].

  46. [46]

    A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortsch. Phys.41 (1993) 307 [arXiv:0709.1075] [INSPIRE].

  47. [47]

    J. Fleischer and F. Jegerlehner, Radiative Corrections to Higgs Decays in the Extended Weinberg-Salam Model, Phys. Rev.D 23 (1981) 2001 [INSPIRE].

  48. [48]

    A. Denner, L. Jenniches, J.-N. Lang and C. Sturm, Gauge-independent \( \overline{MS} \)renormalization in the 2HDM, JHEP09 (2016) 115 [arXiv:1607.07352] [INSPIRE].

  49. [49]

    J.M. Cullen, B.D. Pecjak and D.J. Scott, NLO corrections to h → \( b\overline{b} \)decay in SMEFT, JHEP08 (2019) 173 [arXiv:1904.06358] [INSPIRE].

  50. [50]

    A.J. Buras, Weak Hamiltonian, CP-violation and rare decays, in Probing the standard model of particle interactions. Proceedings of Summer School in Theoretical Physics, NATO Advanced Study Institute, 68th session, Les Houches France (1997), pg. 281 [hep-ph/9806471] [INSPIRE].

  51. [51]

    P. Gambino, A. Kwiatkowski and N. Pott, Electroweak effects in the B0− B0mixing, Nucl. Phys.B 544 (1999) 532 [hep-ph/9810400] [INSPIRE].

  52. [52]

    A.J. Buras, P. Gambino and U.A. Haisch, Electroweak penguin contributions to nonleptonicF = 1 decays at NNLO, Nucl. Phys.B 570 (2000) 117 [hep-ph/9911250] [INSPIRE].

  53. [53]

    T. Cohen, As Scales Become Separated: Lectures on Effective Field Theory, PoS(TASI2018)011 [arXiv:1903.03622] [INSPIRE].

  54. [54]

    J.D. Wells and Z. Zhang, Effective theories of universal theories, JHEP01 (2016) 123 [arXiv:1510.08462] [INSPIRE].

  55. [55]

    D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun.180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].

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Correspondence to Ennio Salvioni.

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ArXiv ePrint: 2003.05936

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Haisch, U., Ruhdorfer, M., Salvioni, E. et al. Singlet night in Feynman-ville: one-loop matching of a real scalar. J. High Energ. Phys. 2020, 164 (2020).

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  • Beyond Standard Model
  • Effective Field Theories