# Leading QCD-induced four-loop contributions to the *β*-function of the Higgs self-coupling in the SM and vacuum stability

- 62 Downloads
- 11 Citations

## Abstract

We present analytical results for the leading top-Yukawa and QCD contribution to the *β*-function for the Higgs self-coupling λ of the Standard Model at four-loop level, namely the part ∝ *y* _{ t } ^{4} *g* _{ s } ^{6} independently confirming a result given in [1]. We also give the contribution ∝ *y* _{ t } ^{2} *g* _{ s } ^{6} of the anomalous dimension of the Higgs field as well as the terms ∝ *y* _{ t } *g* _{ s } ^{8} to the top-Yukawa *β*-function which can also be derived from the anomalous dimension of the top quark mass. We compare the results with the RG functions of the correlators of two and four scalar currents in pure QCD and find a new relation between the anomalous dimension *γ* _{0} of the QCD vacuum energy and the anomalous dimension *γ* _{ m } ^{ SS } appearing in the RG equation of the correlator of two scalar currents. Together with the recently computed top-Yukawa and QCD contributions to *β* _{ gs } [2, 3] the *β*-functions presented here constitute the leading four-loop contributions to the evolution of the Higgs self-coupling. A numerical estimate of these terms at the scale of the top-quark mass is presented as well as an analysis of the impact on the evolution of *λ* up to the Planck scale and the vacuum stability problem.

## Keywords

Higgs Physics Perturbative QCD Renormalization Group## Notes

### **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.

## References

- [1]S.P. Martin,
*Four-loop standard model effective potential at leading order in QCD*,*Phys. Rev.***D 92**(2015) 054029 [arXiv:1508.00912] [INSPIRE].ADSGoogle Scholar - [2]A.V. Bednyakov and A.F. Pikelner,
*Four-loop strong coupling β-function in the standard model*, arXiv:1508.02680 [INSPIRE]. - [3]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].ADSCrossRefGoogle Scholar - [4]F. Bezrukov and M. Shaposhnikov,
*Standard model Higgs boson mass from inflation: two loop analysis*,*JHEP***07**(2009) 089 [arXiv:0904.1537] [INSPIRE].ADSCrossRefGoogle Scholar - [5]M. Holthausen, K.S. Lim and M. Lindner,
*Planck scale boundary conditions and the Higgs mass*,*JHEP***02**(2012) 037 [arXiv:1112.2415] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [6]J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori, A. Riotto and A. Strumia,
*Higgs mass implications on the stability of the electroweak vacuum*,*Phys. Lett.***B 709**(2012) 222 [arXiv:1112.3022] [INSPIRE].ADSCrossRefGoogle Scholar - [7]Z.-z. Xing, H. Zhang and S. Zhou,
*Impacts of the Higgs mass on vacuum stability, running fermion masses and two-body Higgs decays*,*Phys. Rev.***D 86**(2012) 013013 [arXiv:1112.3112] [INSPIRE].ADSGoogle Scholar - [8]F. Bezrukov, M.Yu. Kalmykov, B.A. Kniehl and M. Shaposhnikov,
*Higgs boson mass and new physics*,*JHEP***10**(2012) 140 [arXiv:1205.2893] [INSPIRE]. - [9]G. Degrassi et al.,
*Higgs mass and vacuum stability in the standard model at NNLO*,*JHEP***08**(2012) 098 [arXiv:1205.6497] [INSPIRE].ADSCrossRefGoogle Scholar - [10]K.G. Chetyrkin and M.F. Zoller,
*Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the standard model*,*JHEP***06**(2012) 033 [arXiv:1205.2892] [INSPIRE].ADSCrossRefGoogle Scholar - [11]M.F. Zoller,
*Vacuum stability in the SM and the three-loop β-function for the Higgs self-interaction*,*Subnucl. Ser.***50**(2014) 557 [arXiv:1209.5609] [INSPIRE].Google Scholar - [12]I. Masina,
*Higgs boson and top quark masses as tests of electroweak vacuum stability*,*Phys. Rev.***D 87**(2013) 053001 [arXiv:1209.0393] [INSPIRE].ADSGoogle Scholar - [13]M.F. Zoller,
*Standard model β-functions to three-loop order and vacuum stability*, arXiv:1411.2843 [INSPIRE]. - [14]M.F. Zoller,
*Three-loop β-function for the Higgs self-coupling*, PoS (LL2014) 014 [arXiv:1407.6608] [INSPIRE]. - [15]M. Zoller,
*β-function for the Higgs self-interaction in the Standard Model at three-loop level*, PoS (EPS-HEP 2013) 322 [arXiv:1311.5085] [INSPIRE]. - [16]D. Buttazzo et al.,
*Investigating the near-criticality of the Higgs boson*,*JHEP***12**(2013) 089 [arXiv:1307.3536] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [17]A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner and O.L. Veretin,
*Stability of the electroweak vacuum: gauge independence and advanced precision*,*Phys. Rev. Lett.***115**(2015) 201802 [arXiv:1507.08833] [INSPIRE].ADSCrossRefGoogle Scholar - [18]N.V. Krasnikov,
*Restriction of the fermion mass in gauge theories of weak and electromagnetic interactions*,*Yad. Fiz.***28**(1978) 549 [INSPIRE].Google Scholar - [19]H.D. Politzer and S. Wolfram,
*Bounds on particle masses in the Weinberg-Salam model*,*Phys. Lett.***B 82**(1979) 242 [*Erratum ibid.***B 83**(1979) 421] [INSPIRE]. - [20]P.Q. Hung,
*Vacuum instability and new constraints on fermion masses*,*Phys. Rev. Lett.***42**(1979) 873 [INSPIRE].ADSCrossRefGoogle Scholar - [21]M. Bobrowski, G. Chalons, W.G. Hollik and U. Nierste,
*Vacuum stability of the effective Higgs potential in the minimal supersymmetric standard model*,*Phys. Rev.***D 90**(2014) 035025 [arXiv:1407.2814] [INSPIRE].ADSGoogle Scholar - [22]S.R. Coleman and E.J. Weinberg,
*Radiative corrections as the origin of spontaneous symmetry breaking*,*Phys. Rev.***D 7**(1973) 1888 [INSPIRE].ADSGoogle Scholar - [23]N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio,
*Bounds on the fermions and Higgs boson masses in grand unified theories*,*Nucl. Phys.***B 158**(1979) 295 [INSPIRE].ADSCrossRefGoogle Scholar - [24]M. Sher,
*Electroweak Higgs potentials and vacuum stability*,*Phys. Rept.***179**(1989) 273 [INSPIRE].ADSCrossRefGoogle Scholar - [25]M. Lindner, M. Sher and H.W. Zaglauer,
*Probing vacuum stability bounds at the Fermilab collider*,*Phys. Lett.***B 228**(1989) 139 [INSPIRE].ADSCrossRefGoogle Scholar - [26]C. Ford, D.R.T. Jones, P.W. Stephenson and M.B. Einhorn,
*The Effective potential and the renormalization group*,*Nucl. Phys.***B 395**(1993) 17 [hep-lat/9210033] [INSPIRE]. - [27]G. Altarelli and G. Isidori,
*Lower limit on the higgs mass in the standard model: An update*,*Phys. Lett.***B 337**(1994) 141.ADSCrossRefGoogle Scholar - [28]B.A. Kniehl, A.F. Pikelner and O.L. Veretin,
*Two-loop electroweak threshold corrections in the standard model*,*Nucl. Phys.***B 896**(2015) 19 [arXiv:1503.02138] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [29]L.N. Mihaila, J. Salomon and M. Steinhauser,
*Gauge coupling β-functions in the standard model to three loops*,*Phys. Rev. Lett.***108**(2012) 151602 [arXiv:1201.5868] [INSPIRE].ADSCrossRefGoogle Scholar - [30]L.N. Mihaila, J. Salomon and M. Steinhauser,
*Renormalization constants and β-functions for the gauge couplings of the Standard Model to three-loop order*,*Phys. Rev.***D 86**(2012) 096008 [arXiv:1208.3357] [INSPIRE].ADSGoogle Scholar - [31]A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin,
*Anomalous dimensions of gauge fields and gauge coupling β-functions in the standard model at three loops*,*JHEP***01**(2013) 017 [arXiv:1210.6873] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [32]K.G. Chetyrkin and M.F. Zoller,
*β-function for the Higgs self-interaction in the Standard Model at three-loop level*,*JHEP***04**(2013) 091 [arXiv:1303.2890] [INSPIRE]. - [33]A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin,
*Yukawa coupling β-functions in the standard model at three loops*,*Phys. Lett.***B 722**(2013) 336 [arXiv:1212.6829] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [34]A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin,
*Higgs self-coupling β-function in the standard model at three loops*,*Nucl. Phys.***B 875**(2013) 552 [arXiv:1303.4364] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [35]A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin,
*Three-loop Higgs self-coupling β-function in the standard model with complex Yukawa matrices*,*Nucl. Phys.***B 879**(2014) 256 [arXiv:1310.3806] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [36]T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin,
*The four loop β-function in quantum chromodynamics*,*Phys. Lett.***B 400**(1997) 379 [hep-ph/9701390] [INSPIRE]. - [37]M. Czakon,
*The four-loop QCD β-function and anomalous dimensions*,*Nucl. Phys.***B 710**(2005) 485 [hep-ph/0411261] [INSPIRE]. - [38]P. Nogueira,
*Automatic Feynman graph generation*,*J. Comput. Phys.***105**(1993) 279.ADSMathSciNetCrossRefzbMATHGoogle Scholar - [39]T. Seidensticker,
*Automatic application of successive asymptotic expansions of Feynman diagrams*, in 6^{th}*International Workshop on New Computing Techniques in Physics Research: Software Engineering, Artificial Intelligence Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation (AIHENP 99)*, April 12-16, Heraklion, Crete, Greece (1999), hep-ph/9905298 [INSPIRE]. - [40]R. Harlander, T. Seidensticker and M. Steinhauser,
*Complete corrections of order αα*_{s}*to the decay of the Z boson into bottom quarks*,*Phys. Lett.***B 426**(1998) 125 [hep-ph/9712228] [INSPIRE]. - [41]
- [42]M. Tentyukov and J.A.M. Vermaseren,
*The multithreaded version of FORM*,*Comput. Phys. Commun.***181**(2010) 1419 [hep-ph/0702279] [INSPIRE]. - [43]T. Van Ritbergen, A. Schellekens and J. Vermaseren,
*Group theory factors for Feynman diagrams*,*Int. J. Mod. Phys.***A 14**(1999) 41.ADSMathSciNetCrossRefzbMATHGoogle Scholar - [44]M. Misiak and M. Münz,
*Two loop mixing of dimension five flavor changing operators*,*Phys. Lett.***B 344**(1995) 308 [hep-ph/9409454] [INSPIRE]. - [45]K.G. Chetyrkin, M. Misiak and M. Münz,
*β-functions and anomalous dimensions up to three loops*,*Nucl. Phys.***B 518**(1998) 473 [hep-ph/9711266] [INSPIRE]. - [46]M. Steinhauser,
*MATAD: a program package for the computation of MAssive TADpoles*,*Comput. Phys. Commun.***134**(2001) 335 [hep-ph/0009029] [INSPIRE]. - [47]A.V. Smirnov,
*Algorithm FIRE — Feynman Integral REduction*,*JHEP***10**(2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [48]A.V. Smirnov,
*FIRE5: a C++ implementation of Feynman Integral REduction*,*Comput. Phys. Commun.***189**(2015) 182 [arXiv:1408.2372] [INSPIRE].ADSCrossRefGoogle Scholar - [49]J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen,
*The four loop quark mass anomalous dimension and the invariant quark mass*,*Phys. Lett.***B 405**(1997) 327 [hep-ph/9703284] [INSPIRE]. - [50]K.G. Chetyrkin,
*Quark mass anomalous dimension to O*(*α*_{S}^{4}),*Phys. Lett.***B 404**(1997) 161 [hep-ph/9703278] [INSPIRE]. - [51]K.G. Chetyrkin,
*Correlator of the quark scalar currents and*Γ_{tot}(*H*→*hadrons*)*at O*(*α*_{S}^{3})*in pQCD*,*Phys. Lett.***B 390**(1997) 309 [hep-ph/9608318] [INSPIRE]. - [52]P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,
*Quark mass and field anomalous dimensions to*\( \mathcal{O}\left({\alpha}_s^5\right) \),*JHEP***10**(2014) 076 [arXiv:1402.6611] [INSPIRE].ADSCrossRefGoogle Scholar - [53]P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,
*Scalar correlator at O*(*α*_{s}^{4})*, Higgs decay into b-quarks and bounds on the light quark masses*,*Phys. Rev. Lett.***96**(2006) 012003 [hep-ph/0511063] [INSPIRE]. - [54]J.R. Ellis, M.K. Gaillard and D.V. Nanopoulos,
*A phenomenological profile of the Higgs boson*,*Nucl. Phys.***B 106**(1976) 292 [INSPIRE].ADSCrossRefGoogle Scholar - [55]M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,
*Remarks on Higgs boson interactions with nucleons*,*Phys. Lett.***B 78**(1978) 443 [INSPIRE].ADSCrossRefGoogle Scholar - [56]B.A. Kniehl and M. Spira,
*Low-energy theorems in Higgs physics*,*Z. Phys.***C 69**(1995) 77 [hep-ph/9505225] [INSPIRE]. - [57]K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser,
*Decoupling relations to O*(*α*_{S}^{3})*and their connection to low-energy theorems*,*Nucl. Phys.***B 510**(1998) 61 [hep-ph/9708255] [INSPIRE]. - [58]J.H. Lowenstein,
*Differential vertex operations in Lagrangian field theory*,*Commun. Math. Phys.***24**(1971) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [59]Y.-M.P. Lam,
*Perturbation Lagrangian theory for scalar fields: Ward-Takahasi identity and current algebra*,*Phys. Rev.***D 6**(1972) 2145 [INSPIRE].ADSGoogle Scholar - [60]P. Breitenlohner and D. Maison,
*Dimensional renormalization and the action principle*,*Commun. Math. Phys.***52**(1977) 11 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [61]V.P. Spiridonov and K.G. Chetyrkin,
*Nonleading mass corrections and renormalization of the operators mp*\( \overline{s} \)*iψ and g*2(*μν*),*Sov. J. Nucl. Phys.***47**(1988) 522 [INSPIRE].Google Scholar - [62]V. P. Spiridonov,
*Anomalous dimension of G*_{μν}^{2}*and β function*, preprint IYaI-P-0378 (1984).Google Scholar - [63]K.G. Chetyrkin and J.H. Kuhn,
*Quartic mass corrections to R*_{had},*Nucl. Phys.***B 432**(1994) 337 [hep-ph/9406299] [INSPIRE]. - [64]Y. Schröder,
*Automatic reduction of four loop bubbles*,*Nucl. Phys. Proc. Suppl.***116**(2003) 402 [hep-ph/0211288] [INSPIRE]. - [65]F. Di Renzo, A. Mantovi, V. Miccio and Y. Schröder, 3
*D lattice QCD free energy to four loops*,*JHEP***05**(2004) 006 [hep-lat/0404003] [INSPIRE]. - [66]K.G. Chetyrkin, unpublished (1998).Google Scholar
- [67]K.G. Chetyrkin, R.V. Harlander and J.H. Kuhn,
*Quartic mass corrections to R*_{had}*at*\( \mathcal{O}\left({\alpha}_s^3\right) \),*Nucl. Phys.***B 586**(2000) 56 [*Erratum ibid.***B 634**(2002) 413] [hep-ph/0005139] [INSPIRE]. - [68]C. Sturm,
*Moments of heavy quark current correlators at four-loop order in perturbative QCD*,*JHEP***09**(2008) 075 [arXiv:0805.3358] [INSPIRE].ADSCrossRefGoogle Scholar - [69]A. Maier, P. Maierhofer, P. Marquard and A.V. Smirnov,
*Low energy moments of heavy quark current correlators at four loops*,*Nucl. Phys.***B 824**(2010) 1 [arXiv:0907.2117] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [70]ATLAS, CMS collaboration,
*Combined measurement of the Higgs boson mass in pp collisions at*\( \sqrt{s}=7 \)*and*8*TeV with the ATLAS and CMS experiments*,*Phys. Rev. Lett.***114**(2015) 191803 [arXiv:1503.07589] [INSPIRE]. - [71]Particle Data Group collaboration, K.A. Olive et al.,
*Review of particle physics*,*Chin. Phys.***C 38**(2014) 090001 [INSPIRE]. - [72]ATLAS, CDF, CMS, D0 collaboration,
*First combination of Tevatron and LHC measurements of the top-quark mass*, arXiv:1403.4427 [INSPIRE]. - [73]I.I.Y. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein,
*The pole mass of the heavy quark. Perturbation theory and beyond*,*Phys. Rev.***D 50**(1994) 2234 [hep-ph/9402360] [INSPIRE]. - [74]
- [75]A.H. Hoang, A. Jain, I. Scimemi and I.W. Stewart,
*Infrared renormalization group flow for heavy quark masses*,*Phys. Rev. Lett.***101**(2008) 151602 [arXiv:0803.4214] [INSPIRE].ADSCrossRefGoogle Scholar - [76]
- [77]
- [78]K.G. Chetyrkin and M.F. Zoller,
*Erratum: β-function for the Higgs self-interaction in the Standard Model at three-loop level*,*JHEP***09**(2013) 155.ADSCrossRefGoogle Scholar