# Low-energy effective field theory below the electroweak scale: matching at one loop

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## Abstract

We compute the one-loop matching between the Standard Model Effective Field Theory and the low-energy effective field theory below the electroweak scale, where the heavy gauge bosons, the Higgs particle, and the top quark are integrated out. The complete set of matching equations is derived including effects up to dimension six in the power counting of both theories. We present the results for general flavor structures and include both the *C P* -even and *C P* -odd sectors. The matching equations express the masses, gauge couplings, as well as the coefficients of dipole, three-gluon, and four-fermion operators in the low-energy theory in terms of the parameters of the Standard Model Effective Field Theory. Using momentum insertion, we also obtain the matching for the *C P* -violating theta angles. Our results provide an ingredient for a model-independent analysis of constraints on physics beyond the Standard Model. They can be used for fixed- order calculations at one-loop accuracy and represent a first step towards a systematic next-to-leading-log analysis.

## Keywords

Effective Field Theories Renormalization Group## Notes

### **Open Access**

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## Supplementary material

## References

- [1]A.V. Manohar,
*Effective theories for precision Higgs and flavor physics*, talk given at*Higgs and Effective Field Theory (HEFT 2019)*, April 15–18, Louvain, Belgium (2019).Google Scholar - [2]J.D. Jackson, S.B. Treiman and H.W. Wyld,
*Possible tests of time reversal invariance in Beta decay*,*Phys. Rev.***106**(1957) 517 [INSPIRE]. - [3]
- [4]F. Wilczek and A. Zee,
*Operator analysis of nucleon decay*,*Phys. Rev. Lett.***43**(1979) 1571 [INSPIRE]. - [5]
- [6]W. Buchmüller and D. Wyler,
*Effective Lagrangian analysis of new interactions and flavor conservation*,*Nucl. Phys.***B 268**(1986) 621 [INSPIRE]. - [7]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]. - [8]I. Brivio and M. Trott,
*The standard model as an effective field theory*,*Phys. Rept.***793**(2019) 1 [arXiv:1706.08945] [INSPIRE]. - [9]
- [10]E.E. Jenkins, A.V. Manohar and M. Trott,
*Renormalization group evolution of the standard model dimension six operators I: formalism and λ dependence*,*JHEP***10**(2013) 087 [arXiv:1308.2627] [INSPIRE]. - [11]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]. - [12]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]. - [13]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]. - [14]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 [arXiv:1709.04486] [INSPIRE]. - [15]E.E. Jenkins, A.V. Manohar and P. Stoffer,
*Low-energy effective field theory below the electroweak scale: anomalous dimensions*,*JHEP***01**(2018) 084 [arXiv:1711.05270] [INSPIRE]. - [16]
- [17]J. Gasser and H. Leutwyler,
*Chiral perturbation theory to one loop*,*Annals Phys.***158**(1984) 142 [INSPIRE]. - [18]J. Gasser and H. Leutwyler,
*Chiral perturbation theory: expansions in the mass of the strange quark*,*Nucl. Phys.***B 250**(1985) 465 [INSPIRE]. - [19]W. Dekens, E.E. Jenkins, A.V. Manohar and P. Stoffer,
*Non-perturbative effects in μ → eγ*,*JHEP***01**(2019) 088 [arXiv:1810.05675] [INSPIRE]. - [20]A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto,
*DsixTools: the standard model effective field theory toolkit*,*Eur. Phys. J.***C 77**(2017) 405 [arXiv:1704.04504] [INSPIRE]. - [21]J. Aebischer, J. Kumar and D.M. Straub,
*Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale*,*Eur. Phys. J.***C 78**(2018) 1026 [arXiv:1804.05033] [INSPIRE]. - [22]D.M. Straub,
*flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond*, arXiv:1810.08132 [INSPIRE]. - [23]I. Brivio, Y. Jiang and M. Trott,
*The SMEFTsim package, theory and tools*,*JHEP***12**(2017) 070 [arXiv:1709.06492] [INSPIRE]. - [24]A. Dedes et al.,
*SmeftFR — Feynman rules generator for the Standard Model Effective Field Theory*, arXiv:1904.03204 [INSPIRE]. - [25]MEG collaboration,
*New constraint on the existence of the μ*^{+}*→ e*^{+}*γ decay*,*Phys. Rev. Lett.***110**(2013) 201801 [arXiv:1303.0754] [INSPIRE]. - [26]MEG collaboration,
*Search for the lepton flavour violating decay μ*^{+}*→*e^{+}*γ with the full dataset of the MEG experiment*,*Eur. Phys. J.***C 76**(2016) 434 [arXiv:1605.05081] [INSPIRE]. - [27]SINDRUM II collaboration,
*A search for muon to electron conversion in muonic gold*,*Eur. Phys. J.***C 47**(2006) 337 [INSPIRE]. - [28]COMET collaboration,
*COMET conceptual design report*, KEK-2009-10 (2009).Google Scholar - [29]R.K. Kutschke,
*The Mu2e Experiment at Fermilab*, talk given at the 31^{st}*International Conference on Physics in collisions (PIC 2011)*, August 28–September 1, Vancouver, Canada (2011), arXiv:1112.0242 [INSPIRE]. - [30]COMET collaboration,
*A search for muon-to-electron conversion at J-PARC: the COMET experiment*,*PTEP***022**(2013) C01.Google Scholar - [31]ACME collaboration,
*Order of magnitude smaller limit on the electric dipole moment of the electron*,*Science***343**(2014) 269 [arXiv:1310.7534] [INSPIRE]. - [32]ACME collaboration,
*Improved limit on the electric dipole moment of the electron*,*Nature***562**(2018) 355 [INSPIRE]. - [33]G.M. Pruna and A. Signer,
*The μ → eγ decay in a systematic effective field theory approach with dimension 6 operators*,*JHEP***10**(2014) 014 [arXiv:1408.3565] [INSPIRE]. - [34]A. Crivellin, S. Davidson, G.M. Pruna and A. Signer,
*Renormalisation-group improved analysis of μ → e processes in a systematic effective-field-theory approach*,*JHEP***05**(2017) 117 [arXiv:1702.03020] [INSPIRE]. - [35]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]. - [36]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*, arXiv:1907.04923 [INSPIRE]. - [37]J. Aebischer, A. Crivellin, M. Fael and C. Greub,
*Matching of gauge invariant dimension-six operators for b → s and b → c transitions*,*JHEP***05**(2016) 037 [arXiv:1512.02830] [INSPIRE]. - [38]T. Hurth, S. Renner and W. Shepherd,
*Matching for FCNC effects in the flavour-symmetric SMEFT*,*JHEP***06**(2019) 029 [arXiv:1903.00500] [INSPIRE]. - [39]A. Dedes et al.,
*Feynman rules for the standard model effective field theory in R*_{ξ}*-gauges*,*JHEP***06**(2017) 143 [arXiv:1704.03888] [INSPIRE]. - [40]B. Grinstein and M.B. Wise,
*Operator analysis for precision electroweak physics*,*Phys. Lett.***B 265**(1991) 326 [INSPIRE]. - [41]
- [42]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]. - [43]A. Denner, G. Weiglein and S. Dittmaier,
*Application of the background field method to the electroweak standard model*,*Nucl. Phys.***B 440**(1995) 95 [hep-ph/9410338] [INSPIRE]. - [44]A. Denner, S. Dittmaier and G. Weiglein,
*The background field formulation of the electroweak standard model*,*Acta Phys. Polon.***B 27**(1996) 3645 [hep-ph/9609422] [INSPIRE]. - [45]A. Helset, M. Paraskevas and M. Trott,
*Gauge fixing the standard model effective field theory*,*Phys. Rev. Lett.***120**(2018) 251801 [arXiv:1803.08001] [INSPIRE]. - [46]T. Kugo and I. Ojima,
*Local covariant operator formalism of nonabelian gauge theories and quark confinement problem*,*Prog. Theor. Phys. Suppl.***66**(1979) 1 [INSPIRE]. - [47]S. Descotes-Genon et al.,
*The CKM parameters in the SMEFT*,*JHEP***05**(2019) 172 [arXiv:1812.08163] [INSPIRE]. - [48]A. Denner, H. Eck, O. Hahn and J. Kublbeck,
*Feynman rules for fermion number violating interactions*,*Nucl. Phys.***B 387**(1992) 467 [INSPIRE]. - [49]
- [50]R. Boughezal, C.-Y. Chen, F. Petriello and D. Wiegand,
*Top quark decay at next-to-leading order in the Standard Model Effective Field Theory*,*Phys. Rev.***D 100**(2019) 056023 [arXiv:1907.00997] [INSPIRE]. - [51]S.L. Adler,
*Perturbation theory anomalies*, in*Lectures on elementary particles and quantum field theory*, S. Deser et al. eds., MIT Press, Cambridge U.S.A. (1970).Google Scholar - [52]G. ’t Hooft and M.J.G. Veltman,
*Regularization and renormalization of gauge fields*,*Nucl. Phys.***B 44**(1972) 189 [INSPIRE]. - [53]P. Breitenlohner and D. Maison,
*Dimensional renormalization and the action principle*,*Commun. Math. Phys.***52**(1977) 11 [INSPIRE]. - [54]R. Ferrari, A. Le Yaouanc, L. Oliver and J.C. Raynal,
*Gauge invariance and dimensional regularization with γ*_{5}*in flavor changing neutral processes*,*Phys. Rev.***D 52**(1995) 3036 [INSPIRE]. - [55]P.A. Grassi, T. Hurth and M. Steinhauser,
*Practical algebraic renormalization*,*Annals Phys.***288**(2001) 197 [hep-ph/9907426] [INSPIRE]. - [56]T.L. Trueman,
*Spurious anomalies in dimensional renormalization*,*Z. Phys.***C 69**(1996) 525 [hep-ph/9504315] [INSPIRE]. - [57]J.G. Korner, N. Nasrallah and K. Schilcher,
*Evaluation of the flavor changing vertex b → sH using the Breitenlohner-Maison-’t Hooft-Veltman γ*_{5}*scheme*,*Phys. Rev.***D 41**(1990) 888 [INSPIRE]. - [58]S.A. Larin,
*The renormalization of the axial anomaly in dimensional regularization*,*Phys. Lett.***B 303**(1993) 113 [hep-ph/9302240] [INSPIRE]. - [59]A.V. Bednyakov and A.F. Pikelner,
*Four-loop strong coupling β-function in the Standard Model*,*Phys. Lett.***B 762**(2016) 151 [arXiv:1508.02680] [INSPIRE]. - [60]M.F. Zoller,
*Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level*,*JHEP***02**(2016) 095 [arXiv:1508.03624] [INSPIRE]. - [61]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]. - [62]M.J. Dugan and B. Grinstein,
*On the vanishing of evanescent operators*,*Phys. Lett.***B 256**(1991) 239 [INSPIRE]. - [63]S. Herrlich and U. Nierste,
*Evanescent operators, scheme dependences and double insertions*,*Nucl. Phys.***B 455**(1995) 39 [hep-ph/9412375] [INSPIRE]. - [64]N. Tracas and N. Vlachos,
*Two-loop calculations in QCD and the*∆*I*= \( \frac{1}{2} \)*rule in non-leptonic weak decays*,*Phys. Lett.***B 115**(1982) 419 [INSPIRE]. - [65]S. Herrlich and U. Nierste,
*The complete |*∆*S|*= 2*-Hamiltonian in the next-to-leading order*,*Nucl. Phys.***B 476**(1996) 27 [hep-ph/9604330] [INSPIRE]. - [66]A. Denner, L. Jenniches, J.-N. Lang and C. Sturm,
*Gauge-independent*\( \overline{MS} \)*renormalization in the 2HDM*,*JHEP***09**(2016) 115 [arXiv:1607.07352] [INSPIRE]. - [67]A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner and O.L. Veretin,
*On the b-quark running mass in QCD and the SM*,*Nucl. Phys.***B 916**(2017) 463 [arXiv:1612.00660] [INSPIRE]. - [68]J.M. Cullen, B.D. Pecjak and D.J. Scott,
*NLO corrections to h → b*\( \overline{b} \)*decay in SMEFT*,*JHEP***08**(2019) 173 [arXiv:1904.06358] [INSPIRE]. - [69]C. Hartmann and M. Trott,
*On one-loop corrections in the standard model effective field theory; the*Γ(*h → γ γ*)*case*,*JHEP***07**(2015) 151 [arXiv:1505.02646] [INSPIRE]. - [70]N.D. Christensen and C. Duhr,
*FeynRules — Feynman rules made easy*,*Comput. Phys. Commun.***180**(2009) 1614 [arXiv:0806.4194] [INSPIRE]. - [71]A. Alloul et al.,
*FeynRules 2.0 — A complete toolbox for tree-level phenomenology*,*Comput. Phys. Commun.***185**(2014) 2250 [arXiv:1310.1921] [INSPIRE]. - [72]T. Hahn,
*Generating Feynman diagrams and amplitudes with FeynArts 3*,*Comput. Phys. Commun.***140**(2001) 418 [hep-ph/0012260] [INSPIRE]. - [73]R. Mertig, M. Böhm and A. Denner,
*FEYN CALC: computer algebraic calculation of Feynman amplitudes*,*Comput. Phys. Commun.***64**(1991) 345 [INSPIRE]. - [74]V. Shtabovenko, R. Mertig and F. Orellana,
*New developments in FeynCalc 9.0*,*Comput. Phys. Commun.***207**(2016) 432 [arXiv:1601.01167] [INSPIRE]. - [75]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]. - [76]
- [77]A.V. Manohar,
*The HQET/NRQCD Lagrangian to order α/m*^{3},*Phys. Rev.***D 56**(1997) 230 [hep-ph/9701294] [INSPIRE]. - [78]A.V. Manohar,
*Introduction to effective field theories*, in the proceedings of the*Les Houches summer school: EFT in Particle Physics and Cosmology*, July 3–28, Les Houches, France (2018), arXiv:1804.05863 [INSPIRE]. - [79]A. Broncano, M.B. Gavela and E.E. Jenkins,
*Renormalization of lepton mixing for Majorana neutrinos*,*Nucl. Phys.***B 705**(2005) 269 [hep-ph/0406019] [INSPIRE]. - [80]S. Davidson, M. Gorbahn and M. Leak,
*Majorana neutrino masses in the renormalization group equations for lepton flavor violation*,*Phys. Rev.***D 98**(2018) 095014 [arXiv:1807.04283] [INSPIRE]. - [81]D. Espriu, J. Manzano and P. Talavera,
*Flavor mixing, gauge invariance and wave function renormalization*,*Phys. Rev.***D 66**(2002) 076002 [hep-ph/0204085] [INSPIRE]. - [82]B.A. Kniehl and A. Sirlin,
*Simple on-shell renormalization framework for the Cabibbo-Kobayashi-Maskawa matrix*,*Phys. Rev.***D 74**(2006) 116003 [hep-th/0612033] [INSPIRE]. - [83]B.A. Kniehl and A. Sirlin,
*A novel formulation of Cabibbo-Kobayashi-Maskawa matrix renormalization*,*Phys. Lett.***B 673**(2009) 208 [arXiv:0901.0114] [INSPIRE]. - [84]K.I. Aoki et al.,
*Electroweak theory. Framework of on-shell renormalization and study of higher order effects*,*Prog. Theor. Phys. Suppl.***73**(1982) 1 [INSPIRE]. - [85]A. Denner and T. Sack,
*Renormalization of the quark mixing matrix*,*Nucl. Phys.***B 347**(1990) 203 [INSPIRE]. - [86]B.A. Kniehl and A. Pilaftsis,
*Mixing renormalization in Majorana neutrino theories*,*Nucl. Phys.***B 474**(1996) 286 [hep-ph/9601390] [INSPIRE]. - [87]A. Pilaftsis,
*Gauge and scheme dependence of mixing matrix renormalization*,*Phys. Rev.***D 65**(2002) 115013 [hep-ph/0203210] [INSPIRE]. - [88]H. Lehmann, K. Symanzik and W. Zimmermann,
*On the formulation of quantized field theories*,*Nuovo Cim.***1**(1955) 205 [INSPIRE]. - [89]C. Balzereit, T. Mannel and B. Plumper,
*The renormalization group evolution of the CKM matrix*,*Eur. Phys. J.***C 9**(1999) 197 [hep-ph/9810350] [INSPIRE]. - [90]A. Denner, E. Kraus and M. Roth,
*Physical renormalization condition for the quark mixing matrix*,*Phys. Rev.***D 70**(2004) 033002 [hep-ph/0402130] [INSPIRE]. - [91]A. Sirlin and A. Ferroglia,
*Radiative corrections in precision electroweak physics: a historical perspective*,*Rev. Mod. Phys.***85**(2013) 263 [arXiv:1210.5296] [INSPIRE]. - [92]A. Denner, S. Dittmaier and J.-N. Lang,
*Renormalization of mixing angles*,*JHEP***11**(2018) 104 [arXiv:1808.03466] [INSPIRE]. - [93]J. Fleischer and F. Jegerlehner,
*Radiative corrections to Higgs decays in the extended Weinberg-Salam model*,*Phys. Rev.***D 23**(1981) 2001 [INSPIRE]. - [94]L. Baulieu and R. Coquereaux,
*Photon-Z mixing in the Weinberg-Salam model: effective charges and the a*=*−*3*gauge*,*Annals Phys.***140**(1982) 163 [INSPIRE]. - [95]M. Böhm, A. Denner and H. Joos,
*Gauge theories of the strong and electroweak interaction*, B.G. Teubner, Stuttgart, Germany (2001).Google Scholar - [96]H. Georgi, T. Tomaras and A. Pais,
*Strong CP-violation without instantons*,*Phys. Rev.***D 23**(1981) 469 [INSPIRE]. - [97]B. Grinstein, R.P. Springer and M.B. Wise,
*Effective Hamiltonian for weak radiative B meson decay*,*Phys. Lett.***B 202**(1988) 138 [INSPIRE]. - [98]T. Inami and C.S. Lim,
*Effects of superheavy quarks and leptons in low-energy weak processes KL → μ*\( \overline{\mu} \)*, K*^{+}*→ π*^{+}*ν*\( \overline{\nu} \)*and K*^{0}↔ \( \overline{K} \)^{0},*Prog. Theor. Phys.***65**(1981) 297 [*Erratum ibid.***65**(1981) 1772] [INSPIRE]. - [99]P.L. Cho and M. Misiak,
*b → sγ decay in*SU(2)_{L}*×*SU(2)_{R}*×*U(1)*extensions of the Standard Model*,*Phys. Rev.***D 49**(1994) 5894 [hep-ph/9310332] [INSPIRE]. - [100]X.-G. He and B. McKellar,
*Constraints on the anomalous W W γ couplings from b → sγ*,*Phys. Lett.***B 320**(1994) 165 [hep-ph/9309228] [INSPIRE]. - [101]B. Grzadkowski and M. Misiak,
*Anomalous W tb coupling effects in the weak radiative B-meson decay*,*Phys. Rev.***D 78**(2008) 077501 [*Erratum ibid.***D 84**(2011) 059903] [arXiv:0802.1413] [INSPIRE]. - [102]S. Alioli et al.,
*Right-handed charged currents in the era of the Large Hadron Collider*,*JHEP***05**(2017) 086 [arXiv:1703.04751] [INSPIRE]. - [103]A. De Rujula, M.B. Gavela, O. Pene and F.J. Vegas,
*Signets of CP-violation*,*Nucl. Phys.***B 357**(1991) 311 [INSPIRE]. - [104]J. Fan and M. Reece,
*Probing charged matter through Higgs diphoton decay, gamma ray lines and EDMs*,*JHEP***06**(2013) 004 [arXiv:1301.2597] [INSPIRE]. - [105]W. Dekens and J. de Vries,
*Renormalization group running of dimension-six sources of parity and time-reversal violation*,*JHEP***05**(2013) 149 [arXiv:1303.3156] [INSPIRE]. - [106]V. Cirigliano et al.,
*CP violation in Higgs-gauge interactions: from tabletop experiments to the LHC*,*Phys. Rev. Lett.***123**(2019) 051801 [arXiv:1903.03625] [INSPIRE]. - [107]F. Boudjema, K. Hagiwara, C. Hamzaoui and K. Numata,
*Anomalous moments of quarks and leptons from nonstandard W W gamma couplings*,*Phys. Rev.***D 43**(1991) 2223 [INSPIRE]. - [108]B. Gripaios and D. Sutherland,
*Searches for C P -violating dimension-6 electroweak gauge boson operators*,*Phys. Rev.***D 89**(2014) 076004 [arXiv:1309.7822] [INSPIRE]. - [109]G.J. Gounaris and C.G. Papadopoulos,
*Studying trilinear gauge couplings at next linear collider*,*Eur. Phys. J.***C 2**(1998) 365 [hep-ph/9612378] [INSPIRE]. - [110]E. Braaten, C.-S. Li and T.-C. Yuan,
*The evolution of Weinberg’s gluonic CP violation operator*,*Phys. Rev. Lett.***64**(1990) 1709 [INSPIRE]. - [111]G. Buchalla, A.J. Buras and M.E. Lautenbacher,
*Weak decays beyond leading logarithms*,*Rev. Mod. Phys.***68**(1996) 1125 [hep-ph/9512380] [INSPIRE].