Higgs mass bounds from the functional renormalization group

Transcription

1 Higgs mass bounds from the functional renormalization group René Sondenheimer A. Eichhorn, H. Gies, C. Gneiting, J. Jäckel, T. Plehn, M. Scherer Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Schladming Winter School th of March

2 access:

3 access: th of July 2012: ATLAS: CMS: ± 0.4(stat) ± 0.5(sys) GeV ± 0.4(stat) ± 0.4(sys) GeV

4 access: th of July 2012: ATLAS: CMS: ± 0.4(stat) ± 0.5(sys) GeV ± 0.4(stat) ± 0.4(sys) GeV CERN press release (14th of March 2013): "...the new particle is looking more and more like a Higgs boson..."

5 rst Higgs mass bounds were derived and discussed in perturbation theory [e.g.: Krive, Linde '76; Lindner '85; Sher '89; Ford et al. '93; Casas et al. '96; Isidori et al. '01;...]

6 rst Higgs mass bounds were derived and discussed in perturbation theory [e.g.: Krive, Linde '76; Lindner '85; Sher '89; Ford et al. '93; Casas et al. '96; Isidori et al. '01;...] [Hagiwara et al. '02]

7 rst Higgs mass bounds were derived and discussed in perturbation theory [e.g.: Krive, Linde '76; Lindner '85; Sher '89; Ford et al. '93; Casas et al. '96; Isidori et al. '01;...] [Hagiwara et al. '02] vacuum stability? UΦ 246 GeV

8 rst Higgs mass bounds were derived and discussed in perturbation theory [e.g.: Krive, Linde '76; Lindner '85; Sher '89; Ford et al. '93; Casas et al. '96; Isidori et al. '01;...] [Hagiwara et al. '02] UΦ 246 GeV vacuum stability? Second minimum occurs at a trans-planckian scale? Convexity properties? discrepancy to lattice simulations [Holland, Kuti '03; Gerhold, Jansen '07]

9 the eective potential is essentially dominated by top uctuations

10 the eective potential is essentially dominated by top uctuations Higgs-Yukawa toy model with Z 2 symmetry [ ] 1 S = d d x 2 ( µφ) 2 + U(φ 2 ) + ψi / ψ + ihφ ψψ

11 the eective potential is essentially dominated by top uctuations Higgs-Yukawa toy model with Z 2 symmetry [ ] 1 S = d d x 2 ( µφ) 2 + U(φ 2 ) + ψi / ψ + ihφ ψψ 1-loop beta function for the quartic coupling λφ 4 : β λ = 1 4π 2 [ 3λ 2 + λh 2 h 4]

12 the eective potential is essentially dominated by top uctuations Higgs-Yukawa toy model with Z 2 symmetry [ ] 1 S = d d x 2 ( µφ) 2 + U(φ 2 ) + ψi / ψ + ihφ ψψ 1-loop beta function for the quartic coupling λφ 4 : β λ = 1 4π 2 [ 3λ 2 + λh 2 h 4] eective potential (Coleman-Weinberg): U e (φ) = m2 2 φ2 + λ(µ = φ) φ 4 8 U eff Φ

13 generating functional: Z(J) = DφD ψdψ e S[φ, ψ,ψ]+ Jφ = Dφ e S B [φ] S F,Λ [φ]+ Jφ Λ Λ

14 generating functional: Z(J) = DφD ψdψ e S[φ, ψ,ψ]+ Jφ = Λ top-induced eective potential U F (φ 2 ) = 1 [ 2Ω ln detλ ( 2 + h 2 φ 2 ] ) det Λ ( 2 ) Dφ e S B [φ] S F,Λ [φ]+ Jφ Λ

15 generating functional: Z(J) = DφD ψdψ e S[φ, ψ,ψ]+ Jφ = Λ top-induced eective potential U F (φ 2 ) = 1 [ 2Ω ln detλ ( 2 + h 2 φ 2 ] ) det Λ ( 2 ) Dφ e S B [φ] S F,Λ [φ]+ Jφ Λ sharp cuto U F (φ 2 ) = Λ2 8π 2 h2 φ [ 16π 2 h 4 φ 4 ln (1 + Λ2 h 2 φ 2 ) + h 2 φ 2 Λ 2 Λ 4 ln (1 + h2 φ 2 interaction part strictly positive cannot induce instability Λ 2 [Gies, RS: arxiv: ] )]

16 t Γ k = 1 2 STr [ Γ (2) k t R k + R k ] [Wetterich: Phys.Lett. B301 (1993) 90-94]

17 Systematic derivative expansion: Γ k = d d x ( ) Zφk 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ih k φ ψψ

18 Systematic derivative expansion: Γ k = d d x ( ) Zφk 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ih k φ ψψ β functions: t U k = β Uk η φ := t ln Z φk = β ηφ t h 2 k = β h 2 k η ψ := t ln Z ψk = β ηψ

19 Systematic derivative expansion: Γ k = d d x ( ) Zφk 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ih k φ ψψ β functions: t U k = β Uk η φ := t ln Z φk = β ηφ t h 2 k = β h 2 k η ψ := t ln Z ψk = β ηψ Initial conditions and ne tuning: U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 or U Λ = λ 2Λ 8 (φ2 v 2 Λ )2 Φ Φ

20 Systematic derivative expansion: Γ k = d d x ( ) Zφk 2 µφ µ φ + U k (φ 2 ) + Z ψk ψi / ψ + ih k φ ψψ β functions: t U k = β Uk η φ := t ln Z φk = β ηφ t h 2 k = β h 2 k η ψ := t ln Z ψk = β ηψ Initial conditions and ne tuning: U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 or U Λ = λ 2Λ 8 (φ2 v 2 Λ )2 Φ Φ λ 1Λ (or v Λ ) v 0 = 246 GeV h Λ m top = 173 GeV

21 λ 2Λ = 0 m H GeV GeV

22 800 m H GeV λ 2Λ = 0 λ 2Λ = GeV

23 800 m H GeV λ 2Λ = 0 λ 2Λ = 0.1 λ 2Λ = GeV

24 800 m H GeV λ 2Λ = 0 λ 2Λ = 0.1 λ 2Λ = 1 λ 2Λ = 10 GeV

25 800 m H GeV λ 2Λ = 0 λ 2Λ = 0.1 λ 2Λ = 1 λ 2Λ = 10 λ 2Λ = 100 GeV

26 800 m H GeV λ 2Λ = 0 λ 2Λ = 0.1 λ 2Λ = 1 λ 2Λ = 10 λ 2Λ = 100 GeV Higgs mass is a monotonically increasing function of λ 2Λ! natural lower bound for λ 2Λ = 0 for a quartic UV potential cf. lattice [Holland and Kuti '04], [Jansen et al. '12]

27 generalised bare potentials, e.g.: U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48Λ 2 φ 6

28 generalised bare potentials, e.g.: U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48Λ 2 φ 6 for λ 3Λ > 0 we can choose λ 2Λ < 0

29 generalised bare potentials, e.g.: U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48Λ 2 φ 6 for λ 3Λ > 0 we can choose λ 2Λ < 0 mhgev GeV λ 3Λ = 0, λ 2Λ = 0 λ 3Λ = 3, λ 2Λ = 0.08

30 Extension of the simple Higgs-Yukawa model: running of the Yukawa couplings mainly inuenced by the gauge sectors [ ] t h = π 2 2 h3 8g 2 s h 9 4 g 2 h g 2 h

31 Extension of the simple Higgs-Yukawa model: running of the Yukawa couplings mainly inuenced by the gauge sectors [ ] t h = π 2 2 h3 8g 2 s h 9 4 g 2 h g 2 h Higgs-top-QCD model [ ] 1 S = d d x + U(φ 2 ) + ψi Dψ / + ihφ ψψ ( µφ)2 4 G µν i G µν i + S gf + S gh

32 Extension of the simple Higgs-Yukawa model: running of the Yukawa couplings mainly inuenced by the gauge sectors [ ] t h = π 2 2 h3 8g 2 s h 9 4 g 2 h g 2 h Higgs-top-QCD model [ ] 1 S = d d x + U(φ 2 ) + ψi Dψ / + ihφ ψψ ( µφ)2 4 G µν i G µν i + S gf + S gh ow equations t U k = β non-pert, U k t h 2 k = β non-pert, t g 2 k = β pert h 2 g 2 k k

33 Extension of the simple Higgs-Yukawa model: running of the Yukawa couplings mainly inuenced by the gauge sectors [ ] t h = π 2 2 h3 8g 2 s h 9 4 g 2 h g 2 h Higgs-top-QCD model [ ] 1 S = d d x + U(φ 2 ) + ψi Dψ / + ihφ ψψ ( µφ)2 4 G µν i G µν i + S gf + S gh ow equations t U k = β non-pert, U k t h 2 k = β non-pert, t g 2 k = β pert h 2 g 2 k k Furthermore, we modied these ow equations to model eectively the contributions from the electroweak gauge bosons by a ducial gauge coupling.

34 U Λ = λ 1Λ 2 φ2 + λ 2Λ 8 φ4 + λ 3Λ 48Λ 2 φ III GeV II Λ 6 I [Eichhorn,Gies,Jäckel,Plehn,Scherer,RS: arxiv: ]

35 Conclusions We found natural bounds for the Higgs mass in the framework of the functional RG for quartic UV potentials. The form of the UV potential can exert a signicant inuence on the mass bounds.

36 Conclusions We found natural bounds for the Higgs mass in the framework of the functional RG for quartic UV potentials. The form of the UV potential can exert a signicant inuence on the mass bounds. Outlook Extend the model to the full Standard Model. Study the inuence of other higher order operators. Solve the full ow equation for the potential with a global solver to get insights into the 'pseudo'-stable region.

37 Thanks for your attention!

38 Thanks for your attention! Thanks to Prof. Glozman, Prof Plessas and Dr. Pak for organising the Schladming Winter School 2015!!

39 Thanks for your attention! Thanks to Prof. Glozman, Prof Plessas and Dr. Pak for organising the Schladming Winter School 2015!! Thanks to the students from Graz for doing all the stu behind the scenes!!

40 Thanks for your attention! Thanks to Prof. Glozman, Prof Plessas and Dr. Pak for organising the Schladming Winter School 2015!! Thanks to the students from Graz for doing all the stu behind the scenes!! And many THANKS to Prof. Dunne, Prof. Kharzeev, Prof. Shifman, Prof. Son, Prof. Wiese and Prof. Zoller for the nice and enlightening lectures!!!