Vacuum stability and the mass of the Higgs boson. Holger Gies

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1 Vacuum stability and the mass of the Higgs boson Holger Gies Helmholtz Institute Jena & TPI, Friedrich-Schiller-Universität Jena & C. Gneiting, R. Sondenheimer, PRD 89 (2014) , [arxiv: ], & R. Sondenheimer, EPJ C75 (2015) 2, 68, [arxiv: ], & A. Eichhorn, J. Jaeckel, T. Plehn, M.M. Scherer, R. Sondenheimer, JHEP 1504 (2015) 022, [arxiv: ] Seminar, KFU Graz,

2 Standard model particle spectrum: pre-lhc

3 Standard model particle spectrum? (ANDERSON 62; BROUT,ENGLERT 64; HIGGS 64; GURALNIK,HAGEN,KIBBLE 64)

4 Search for the Higgs boson 4 Jul ATLAS & 14 Mar 2013, CERN press release:... the new particle is looking more and more like a Higgs boson... CMS 12 : ± 0.4(stat) ± 0.5(sys)GeV, ATLAS 12 : ± 0.4(stat) ± 0.4(sys)GeV

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6 Numbers matter standard model best before: Λ = M Planck?

7 Validity range of the standard model Λ: UV cutoff SM as effective theory scale of maximum UV extension Λ? scale of new physics: Λ NP Λ

8 Higgs boson mass and maximum validity scale (HAMBYE,RIESSELMANN 97) SM: e.g. (KRIVE,LINDE 76; MAIANI,PARISI,PETRONZIO 78; KRASNIKOV 78; POLITZER, WOLFRAM 78; HUNG 79; LINDNER 85; WETTERICH 87; SHER 88; FORT,JONES,STEPHENSON, EINHORN 93; ALTARELLI,ISIDORI 94; SCHREMPP,WIMMER 96;... ) BSM: e.g., (CABBIBO, MAIANI,PARISI,PETRONZIO 79; ESPINOSA,QUIROS 91;... )

9 Higgs boson mass and maximum validity scale upper bound lower bound (HAMBYE,RIESSELMANN 97) SM: e.g. (KRIVE,LINDE 76; MAIANI,PARISI,PETRONZIO 78; KRASNIKOV 78; POLITZER, WOLFRAM 78; HUNG 79; LINDNER 85; WETTERICH 87; SHER 88; FORT,JONES,STEPHENSON, EINHORN 93; ALTARELLI,ISIDORI 94; SCHREMPP,WIMMER 96;... ) BSM: e.g., (CABBIBO, MAIANI,PARISI,PETRONZIO 79; ESPINOSA,QUIROS 91;... )

10 Upper bound of Higgs boson mass triviality bound = perturbativity bound = unitarity bound Higgs boson mass mh 2 v 2 U eff (φ = v), U eff (φ = v) λ R = large Higgs boson masses: scalar interactions dominate (λ 1)

11 Upper bound of Higgs boson mass triviality: perturbation theory predicts its own failure 10 (LANDAU 55) (GELL-MANN, LOW 54) 1 λ R 1 λ Λ = β 0 ln Λ v, β 0 > λ R and v fixed: ( = Λ sing v exp ) 1 β 0 λ R Landau pole singularity in practice: perturbativity criterion, e.g. λ Λpert π 4 = Λ = Λ pert

12 Upper bound of Higgs boson mass triviality: perturbation theory predicts its own failure 10 (LANDAU 55) (GELL-MANN, LOW 54) 1 λ R 1 λ Λ = β 0 ln Λ v, β 0 > Triviality lim Λ v : = λ R 0 in simple models: nonperturbative evidence from lattice simulations (LUESCHER,WEISZ 88; HASENFRATZ,JANSEN,LANG,NEUHAUS,YONEYAMA 87; WOLFF 11; BUIVIDOVICH 11; WEISZ,WOLFF 12,... )

13 Lower bound of Higgs boson mass vacuum stability / meta-stability bound effective potential á la Coleman Weinberg: U eff (φ) = 1 2 µ2 φ λ(φ)φ4 e.g., λ(φ) from RG-improved perturbation theory: t λ = 3 4π 2 ( ht 4 + ht 2 λ + 1 [ 2g 4 + (g 2 + g 2 ) 2] 1 ) 16 4 λ(3g2 + g 2 ) + λ 2

14 Lower bound of Higgs boson mass effective potential á la Coleman Weinberg: U eff (φ) = 1 2 µ2 φ λ(φ)φ4 e.g., λ(φ) from RG-improved perturbation theory: t λ = 3 4π 2 ( ht 4 + ht 2 λ + 1 [ 2g 4 + (g 2 + g 2 ) 2] 1 ) 16 4 λ(3g2 + g 2 ) + λ 2 (GABRIELLI ET AL. 13)

15 Lower Bound of Higgs boson mass meta-stability: tunneling time > age of universe Near critical standard model: (BUTTAZZO ET AL. 13) NNLO calculation (DEGRASSI ET AL. 12) earlier calculations, e.g., (ISIDORI,RIDOLFI,STRUMIA 01)

16 Lower Bound of Higgs boson mass Near critical standard model: (BUTTAZZO ET AL. 13: UPDATE V4) Stability seems to prefer m H 130GeV or m top 171GeV

17 Lower Bound of Higgs boson mass Near critical standard model: (BUTTAZZO ET AL. 13: UPDATE V4) Stability seems to prefer m H 130GeV or m top 171GeV The Higgs potential has the worrisome feature that it might become metastable at energies above 100bn gig-electron-volts,

18 Lower bound of Higgs boson mass Reasons for concern? α s dependence top mass dependence (DEGRASSI ET AL. 12) larger error of top pole mass 3? (ALEKHIN,DJOUADI,MOCH 12) compared to the MC fit parameter determined by Tevatron Decay of the false vacuum warning time s... the expansion of the bubble is a clean sweep... (COLEMAN 77)

19 2nd thoughts on the lower bound True minimum of U eff (φ) at φ GeV > M Pl? UV IR RG flow?

20 2nd thoughts on the lower bound True minimum of U eff (φ) at φ GeV > M Pl? UV IR RG flow? simple top-higgs Yukawa model: lattice simulation vs. 1-loop PT with cutoff vs. 1-loop Λ-removed PT (HOLLAND,KUTI 03; HOLLAND 04) Criticism: too few scales? (EINHORN,JONES 07)

21 Top-Higgs Yukawa toy model (HOLLAND, KUTI 03; BRANCHINA,FAIVRE 05) Z 2 symmetric model 1 S = 2 ( φ)2 + U(φ) + ψi / ψ + ihφ ψψ includes relevant top quark + Higgs field (+ largest Yukawa coupling) discrete symmetry breaking no Goldstone bosons (as in SM) avoids intricate questions arising from gauge symmetry (FROHLICH,MORCHIO,STROCCHI 81; LANG,REBBI,VIRASORO 81; JERSAK,LANG,NEUHAUS,VONES 85; MAAS 12)

22 Top-Higgs Yukawa models generating functional: Z [J] = DφD ψdψe S[φ, ψ,ψ]+ Jφ Λ = Dφ e S B[φ] S F,Λ [φ]+ Jφ Λ (HOLLAND, KUTI 03; HOLLAND 04) top-induced effective potential U F (φ) = 1 2Ω ln det Λ( 2 + ht 2φ φ) det Λ ( 2, ) CAVE: cutoff Λ / regularization dependent

23 Top-induced effective potential exact results for fermion determinants for homogeneous φ e.g., sharp cutoff: U F,t (φ) = Λ2 8π 2 h2 t φ 2 }{{} <0 (mass-like term) π 2 [ ) )] ht 4 φ 4 ln (1 + Λ2 ht 2 + h 2 φ 2 t φ 2 Λ 2 Λ 4 ln (1 + h2 t φ 2 Λ 2 }{{} >0 (interaction part) mass-like term: contributes to χsb = v 246GeV interaction part: strictly positive = cannot induce instability for any finite Λ (HG,SONDENHEIMER 14)

24 Rederiving the instability try to send Λ : U F,t (φ) = Λ2 8π 2 h2 t φ ( 16π 2 h4 t φ [ln 4 Λ2 h 2 ht 2 + const. + O t φ 2 )] φ 2 Λ 2

25 Rederiving the instability try to send Λ : U F,t (φ) = Λ2 8π 2 h2 t φ ( 16π 2 h4 t φ [ln 4 Λ2 h 2 ht 2 + const. + O t φ 2 )] φ 2 Λ 2 renormalization: trade (Λ, m Λ, λ Λ ) for (µ, v, λ v )? U F,t (φ) 1 ) 16π 2 h4 t φ (ln 4 h2 t φ 2 µ 2 + const.

26 Rederiving the instability try to send Λ : U F,t (φ) = Λ2 8π 2 h2 t φ ( 16π 2 h4 t φ [ln 4 Λ2 h 2 ht 2 + const. + O t φ 2 )] φ 2 Λ 2 renormalization: trade (Λ, m Λ, λ Λ ) for (µ, v, λ v )? U F,t (φ) 1 ) 16π 2 h4 t φ (ln 4 h2 t φ 2 µ 2 + const. instability occurs beyond h2 t φ 2 Λ 2 > 1 (HG,SONDENHEIMER 14) similar problems for other reg s implicit renormalization conditions would violate unitarity (HOLLAND, KUTI 03; HOLLAND 04) (BRANCHINA,FAIVRE 05; GNEITING 05)

27 Contradiction? t λ = 3 4π 2 h4 t +... = λ < 0 interaction part of U F,t (φ) > 0 strictly positive

28 Contradiction? t λ = 3 4π 2 h4 t +... = λ < 0 U eff (φ) 1 2 λ(φ)φ4 interaction part of U F,t (φ) > 0 strictly positive

29 Contradiction? t λ = 3 4π 2 h4 t +... = λ < 0 U eff (φ) 1 2 λ(φ)φ4 interaction part of U F,t (φ) > 0 strictly positive ( h 2 t CAVE: O )-terms φ 2 Λ 2 for finite Λ

30 Summary, Part I no in-/meta-stability from top (fermion) fluctuations... if cutoff Λ is kept finite but arbitrary no in-/meta-stability at all? U eff (φ) = U Λ (φ) + U B (φ) + U F (φ)

31 Summary, Part I no in-/meta-stability from top (fermion) fluctuations... if cutoff Λ is kept finite but arbitrary no in-/meta-stability at all? U eff (φ) = U Λ (φ) + U B (φ) + U F (φ) }{{}}{{} arbitrary generically stable... in-/meta-stabilities from the bare action/uv completion lower Higgs mass bounds? = nonperturbative methods recommended if not needed extensive lattice simulations: (FODOR,HOLLAND,KUTI,NOGRADI,SCHROEDER 07) constraining 4th generations: implications for dark matter models: (GERHOLD,JANSEN ) (GERHOLD,JANSEN,KALLARACKAL 10; BULAVA,JANSEN,NAGY 13) (EICHHORN, SCHERER 14)

32 Higgs boson mass bounds as a UV to IR mapping Λ

33 Higgs boson mass bounds as a UV to IR mapping Λ

34 Higgs boson mass bounds as a UV to IR mapping Λ

35 Higgs boson mass bounds as a UV to IR mapping Λ

36 Higgs boson mass bounds as a UV to IR mapping Λ

37 Higgs boson mass bounds as a UV to IR mapping Λ QFT + RG

38 Higgs boson mass bounds as a UV to IR mapping Λ QFT + RG = consistency bounds

39 Higgs boson mass bounds as a UV to IR mapping microscopic action at cutoff Λ: S Λ = S Λ (m 2 Λ, λ Λ, λ 6,Λ,..., h Λ,... ) = RG: mapping to IR observables RG = v 246GeV, m top 173GeV, m H = m H [S Λ ] By contrast: RG-improved PT fix: v, m top m H upwards RG = U eff

40 Simple Example: mean-field theory MFT ˆ= large-n f limit ˆ= pure top loop: U MF (φ) = U Λ (φ) ln det reg (i / + ih Λ φ) (HG,GNEITING,SONDENHEIMER 13) UV bare potential, e.g, U Λ (φ) = 1 2 m2 Λφ λ Λφ 4, λ Λ 0 trade: m Λ, h Λ v, m top Higgs boson mass: m 2 H(Λ, λ Λ ) = m4 top 4π 2 v 2 [ 2 ln ( 1 + Λ2 m 2 top ) ] 3Λ4 + 2mtop 2 Λ2 (Λ 2 + mtop 2 + v 2 λ Λ )2

41 Simple Example: mean-field theory Higgs boson mass bound [ m 2 H(Λ, λ Λ ) = m4 top 4π 2 v 2 2 ln ( 1 + Λ2 m 2 top ) 3Λ4 + 2m 2 top Λ2 (Λ 2 + m 2 top )2 requirement: well defined path integral: λ Λ 0 for φ 4 theory (HG,GNEITING,SONDENHEIMER 13) ] + v 2 λ Λ cf. lattice (HOLLAND 04; FODOR,HOLLAND,KUTI,NOGRADI,SCHROEDER 07; GERHOLD,JANSEN ) extended mean field ˆ= NLO 1/N f expansion: 250 mhgev /6 λ Λ = 1/3 2/ GeV

42 Nonperturbative tool: functional RG RG flow equation: t Γ k k k Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 = (WETTERICH 93)

43 RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 RG trajectory: Γ k =Λ = S Λ = 1 2 ( φ)2 + U Λ (φ) +...

44 RG trajectory: RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1

45 RG trajectory: RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1

46 RG Flow Equation t Γ k = 1 2 Tr tr k (Γ (2) k + R k ) 1 RG trajectory: Γ k 0 = Γ = U eff +...

47 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ m H GeV GeV λ Λ = 0

48 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ m H GeV GeV λ Λ = 0 λ Λ = 0.1

49 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ 800 m H GeV λ Λ = 0 λ Λ = 0.1 λ Λ = 1 GeV

50 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ 800 m H GeV λ Λ = 0 λ Λ = 0.1 λ Λ = 1 λ Λ = 10 GeV

51 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ 800 m H GeV λ Λ = 0 λ Λ = 0.1 λ Λ = 1 λ Λ = 10 λ Λ = 100 GeV

52 Higgs boson mass bounds from functional RG Top-Higgs toy model (HG,GNEITING,SONDENHEIMER 13) Systematic derivative expansion: ( ) Γ k = d d Zφk x 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ihk φ ψψ 800 m H GeV λ Λ = 0 λ Λ = 0.1 λ Λ = 1 λ Λ = 10 λ Λ = 100 GeV = conventional lower bound for λ Λ = 0 ( mean-field result) agreement with lattice (HOLLAND 04; FODOR,HOLLAND,KUTI,NOGRADI,SCHROEDER 07; GERHOLD,JANSEN )

53 The IR window for the Higgs boson mass m H vλ R (WETTERICH 87) mapping: λ Λ λ R not surjective on R + e.g. for φ 4 bare potential, fix Λ = 10 7 GeV mhgev (HG,GNEITING,SONDENHEIMER 13) convergence check of derivative expansion NLO / LO strong coupling U eff solver (polynom. exp.) Λ

54 General microscopic actions S Λ is a priori unconstrained. Consider, e.g., U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48 φ6 (HG,GNEITING,SONDENHEIMER 13) for λ 3Λ > 0 we can choose λ 2Λ < 0:

55 General microscopic actions S Λ is a priori unconstrained. Consider, e.g., U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48 φ6 (HG,GNEITING,SONDENHEIMER 13) mhgev for λ 3Λ > 0 we can choose λ 2Λ < 0: GeV λ 3Λ = 0, λ 2Λ = 0 λ 3Λ = 3, λ 2Λ = 0.08 lower bound relaxed = consistency bound < conventional bound string models with λ < 0: (HEBECKER,KNOCHEL,WEIGAND 13)

56 Renormalizable field theories seeming contradiction with common wisdom...?... observables are determined by renormalizable operators... y-axis: m H observable, x-axis: Λ?

57 RG mechanism for lowering the lower bound g i g 2 Theory Space g 1

58 RG mechanism for lowering the lower bound g i Gaußian fixed point g 2 Theory Space g 1

59 RG mechanism for lowering the lower bound g i renormalizable operators span a hyperplane g 2 Theory Space g 1

60 RG mechanism for lowering the lower bound g i non-renormalizable operators are exponentially damped towards the IR flow g 2 Theory Space g 1

61 RG mechanism for lowering the lower bound g i non-renormalizable operators are exponentially damped towards the IR flow BUT: RG time (scales) is needed for this damping g 2 Theory Space g 1

62 Consistency bounds from generalized bare actions e.g., U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48 φ6 (HG,GNEITING,SONDENHEIMER 13) mhgev GeV = consistency bound shifted Λ axis λ 3Λ = 0, λ 2Λ = 0 λ 3Λ = 3, λ 2Λ = 0.08

63 chiral Yukawa model: Towards the standard model (HG,SONDENHEIMER 14) [ S = µ φ µ φ + U(φ φ) + ti / t + bi / b + ih b ( ψ L φb R + b R φ ψ L ) + ih t ( ψ ] L φ C t R + t R φ C ψ L) φ = ( ) φ1 + iφ 2 φ 4 + iφ 3 ψ L = ( tl b L ) enforce decoupling of Goldstone bosons (m G = 0) k 2 k 2 + m 2 G k 2 k 2 + m 2 G + gv 2 k choose gauge boson masses gv 2 k = (80.4 GeV)2 cf. lattice model (GERHOLD,JANSEN )

64 Conventional lower Higgs boson mass bound for φ 4 -type bare potentials: mhgev GeV (HG,SONDENHEIMER 14) FRG: NLO derivative expansion λ 2Λ = 0 λ 2Λ = 1 λ 2Λ = 10 λ 2Λ = 100 = lower bound close to simple toy model:... bottom quark has little quantitative influence

65 Lowering the lower Higgs boson mass bound generalized bare potential with λ 6,Λ (φ φ) 3 interaction: (HG,SONDENHEIMER 14) mhgev λ 3Λ = 0, λ 2Λ = 0 λ 3Λ = 3, λ 2Λ = GeV = same RG mechanism at work

66 Lowering the lower Higgs boson mass bound generalized bare potential with λ 6,Λ (φ φ) 3 interaction comparison with lattice data: mhgev GeV (HG,SONDENHEIMER 13, 14) (HEDGE,JANSEN,LIN,NAGY 13) (CHU,JANSEN,KNIPPSCHILD,LIN,NAGY 15) = RG mechanism confirmed

67 A serious toy model (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) Standard Model vs. (BUTTAZZO ET AL. 13) = toy model satisfies SM constraints at low energies: m H 125GeV, m top (m top ) 164GeV, α s (M Z ) = naive instability scale: Λ I 10 10

68 A serious toy model (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) Standard Model vs. Z 2 model (BUTTAZZO ET AL. 13) = toy model satisfies SM constraints at low energies: m H 125GeV, m top (m top ) 164GeV, α s (M Z ) = naive instability scale: Λ I 10 10

69 A serious toy model (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) Standard Model vs. Z 2 SU(3) model (BUTTAZZO ET AL. 13) = toy model satisfies SM constraints at low energies: m H 125GeV, m top (m top ) 164GeV, α s (M Z ) = naive instability scale: Λ I 10 10

70 A serious toy model (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) Standard Model vs. Z 2 SU(3) + fiducial EW (BUTTAZZO ET AL. 13) = toy model satisfies SM constraints at low energies: m H 125GeV, m top (m top ) 164GeV, α s (M Z ) = naive instability scale: Λ I 10 10

71 Stable flows with higher dimensional operators (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) +φ 6 term: = cutoff scale Λ 10 2 Λ I > naive instability scale

72 Stable flows with higher dimensional operators (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) +φ 6 term: RG flow of the potential: = cutoff scale Λ 10 2 Λ I > naive instability scale = stable potential on all scales (UV IR)

73 Pseudo-stable flows with higher dimensional operators (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) +φ 6 term: = cutoff scale Λ Λ I naive instability scale

74 Pseudo-stable flows with higher dimensional operators (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) +φ 6 term: RG flow of the potential: = cutoff scale Λ Λ I naive instability scale = stable potential on UV and IR scales = meta-stable potential at intermediate scales PE artifact? convergence radius?

75 Stability analysis of the effective potential (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) function RG analysis (polynomially expanded potential) I stable on all scales II pseudo-stable (UV- and IR-stable, meta-stable inbetween) III meta-stable UV potential (IR fate?)

76 Higgs mass consistency bounds (EICHHORN,HG,JAECKEL,PLEHN,SCHERER,SONDENHEIMER 15) = m H 1 GeV (in stable region) = m H 5 GeV (in pseudo-stable region) measured Higgs mass could be within consistency bounds!

77 Summary, Part II Bounds on the Higgs boson mass (or any other physical IR observable) arise from a mapping S micro O phys... provided by the RG For effective quantum field theories (with a cutoff Λ): bounds on O phys = f [S Λ ]... full S Λ not just the renormalizable operators lowering the conventional lower Higgs boson mass bound is possible... without in-/meta-stable vacuum

78 TODO list Λ study of general S Λ evolution of several minima in-/meta-/pseudo-stabilities?

79 Implications if m H < conventional lower bound: new physics at lower scales first constraints on underlying UV completion if m H exactly on the convenional lower bound: (e.g. if m top 171GeV)... criticality underlying UV completion has to explain absence of higher dimensional operators flat potential

80 Candidates standard model + asymptotically safe gravity gravity fluctuations induces a UV fixed point λ 0 (WEINBERG 76; REUTER 96) (PERCACCI ET AL 03 09) = m H put onto conventional lower bound (WETTERICH,SHAPOSHNIKOV 10) (BEZRUKOV,KALMYKOV,KNIEHL,SHAPOSHNIKOV 12) Asymptotically safe gauged Higgs Yukawa model (HG,RECHENBERGER,SCHERER,ZAMBELLI 13) = line of fixed points approaching flat potential with v/k

81 Summary, Part III Numbers matter... m top, m H QFT is more than a collection of recipes... new insight from new tools vacuum stability: no reason for concern... so far...

82