The Standard model Higgs boson as the inflaton

Transcription

1 The Standard model Higgs boson as the inflaton F. Bezrukov M. Shaposhnikov EPFL, Lausanne, Suisse Institute for Nuclear Research, Moscow, Russia PONT d Avignon May 2008 Phys. Lett. B 659, 703 (2008) [arxiv: [hep-th]]

2 Outline Outline 1 Inflation virtues and problems Cosmological requirements Standard chaotic inflation Problems with using the SM Higgs for inflation 2 SM with non-minimal coupling to gravity The action Conformal transformation Inflation in the model Radiative corrections no danger 3 Predictions and expectations CMB parameters spectrum and tensor modes Higgs mass 4 Conclusions F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 2 / 22

3 Outline Inflation virtues and problems 1 Inflation virtues and problems Cosmological requirements Standard chaotic inflation Problems with using the SM Higgs for inflation 2 SM with non-minimal coupling to gravity The action Conformal transformation Inflation in the model Radiative corrections no danger 3 Predictions and expectations CMB parameters spectrum and tensor modes Higgs mass 4 Conclusions F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 3 / 22

4 Inflation virtues and problems Cosmological implications Cosmological requirements Problems in cosmology Flatness problem (at T M P density was tuned Ω 1 < ) Entropy of the Universe S Size of the Universe (at T M P size was M 1 P ) Horizon problem Solution Inflation! F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 4 / 22

5 Inflation virtues and problems Cosmological implications Cosmological requirements Problems in cosmology Flatness problem (at T M P density was tuned Ω 1 < ) Entropy of the Universe S Size of the Universe (at T M P size was M 1 P ) Horizon problem Solution Inflation! F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 4 / 22

6 CMB Temperature fluctuations Inflation virtues and problems Cosmological requirements CMB spectrum Polarization F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 5 / 22

7 Inflation virtues and problems Standard chaotic inflation Usually required for inflation Scalar field quartic coupling constant λ mass m GeV, Present in the Standard Model The Higgs boson λ 1 m H 100GeV Standard chaotic inflation Even if one writes a potential that flattens at large field values: Radiative corrections from t, W generate δv rad #h 4 logh V 4 M P 1/4 0 M P λ M P ϕ Solution: Non-minimal coupling to gravity F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 6 / 22

8 Inflation virtues and problems Scalar fields present in the SM Usually required for inflation Scalar field quartic coupling constant λ mass m GeV, Present in the Standard Model The Higgs boson λ 1 m H 100GeV Problems with using the SM Higgs for inflation Even if one writes a potential that flattens at large field values: Radiative corrections from t, W generate δv rad #h 4 logh V 4 M P 1/4 0 M P λ M P ϕ Solution: Non-minimal coupling to gravity F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 7 / 22

9 Inflation virtues and problems Scalar fields present in the SM Usually required for inflation Scalar field quartic coupling constant λ mass m GeV, Present in the Standard Model The Higgs boson λ 1 m H 100GeV Problems with using the SM Higgs for inflation Even if one writes a potential that flattens at large field values: Radiative corrections from t, W generate δv rad #h 4 logh V 4 M P 1/4 0 M P λ M P ϕ Solution: Non-minimal coupling to gravity F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 7 / 22

10 Outline SM with non-minimal coupling to gravity 1 Inflation virtues and problems Cosmological requirements Standard chaotic inflation Problems with using the SM Higgs for inflation 2 SM with non-minimal coupling to gravity The action Conformal transformation Inflation in the model Radiative corrections no danger 3 Predictions and expectations CMB parameters spectrum and tensor modes Higgs mass 4 Conclusions F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 8 / 22

11 SM with non-minimal coupling to gravity The action Possible operators in the SM (+gravity) Dimension 4 No new degrees of freedom (no higher derivatives) S = d 4 x [ g SM { Tr(Fµν F µν ) + D µh 2 M2 P 2 R ξ H HR 2 +R 2 + R µν R µν + R µνλρ R µνλρ + R V (H) + Ψ /DΨ + YH Ψ L Ψ R +m N c N ] F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 9 / 22

12 SM with non-minimal coupling to gravity The action Possible operators in the SM (+gravity) Dimension 4 No new degrees of freedom (no higher derivatives) S = d 4 x [ g SM { Tr(Fµν F µν ) + D µh 2 M2 P 2 R ξ H HR 2 +R 2 + R µν R µν + R µνλρ R µνλρ + R V (H) + Ψ /DΨ + YH Ψ L Ψ R +m N c N ] F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 9 / 22

13 SM with non-minimal coupling to gravity The action Possible operators in the SM (+gravity) Dimension 4 No new degrees of freedom (no higher derivatives) S = d 4 x [ g SM { Tr(Fµν F µν ) + D µh 2 M2 P 2 R ξ H HR 2 +R 2 + R µν R µν + R µνλρ R µνλρ + R V (H) + Ψ /DΨ + YH Ψ L Ψ R +m N c N ] F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon 08 9 / 22

14 SM with non-minimal coupling to gravity The action Non-minimally coupled scalar field inflation Quite an old idea Add φ 2 R term to/instead of the usual M P R term in the gravitational action A.Zee 78, L.Smolin 79, B.Spokoiny 84 D.Salopek J.Bond J.Bardeen 89 S J = d 4 x g { } M2 + ξ h 2 µ h ν h R + g µν λ (h 2 v 2) h is the Higgs field M v ξ so M M P = 1 = GeV 8πGN F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

15 SM with non-minimal coupling to gravity The action Non-minimally coupled scalar field inflation Quite an old idea Add φ 2 R term to/instead of the usual M P R term in the gravitational action A.Zee 78, L.Smolin 79, B.Spokoiny 84 D.Salopek J.Bond J.Bardeen 89 S J = d 4 x g { } M2 + ξ h 2 µ h ν h R + g µν λ (h 2 v 2) h is the Higgs field M v ξ so M M P = 1 = GeV 8πGN F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

16 SM with non-minimal coupling to gravity Conformal transformation Conformal transformation It is possible to get rid of the non-minimal coupling by the conformal transformation (field redefinition) ĝ µν = Ω 2 g µν, Ω 2 = 1 + ξ h2 M 2 P and also redefinition of the Higgs field to make canonical kinetic term { dχ dh = Ω 2 + 6ξ 2 h 2 /MP 2 h χ for h < MP /ξ Ω 4 = ( ) Ω 2 exp 2χ 6MP for h > M P /ξ Resulting action (Einstein frame action) { } S E = d 4 x ĝ M2 P 2 ˆR + µ χ µ χ 1 λ (h(χ) 2 2 Ω(χ) 4 v 2) 2 4 F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

17 SM with non-minimal coupling to gravity Inflationary potential Inflation in the model U(χ) λm 4 /ξ 2 /4 Slow roll inflation λm 4 /ξ 2 / χ end χ COBE χ For χ > M P : U(χ) λm4 P 4ξ 2 ( 1 exp ( 2χ )) 2 6MP F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

18 Slow roll stage SM with non-minimal coupling to gravity Inflation in the model ε = M2 P 2 Slow roll ends at χ end M P ( du/dχ ) 2 4 ( 3 exp U η = MP 2 d 2 U/dχ 2 4 ( U 3 exp 4χ ) 6MP 2χ ) 6MP Number of e-folds of inflation at the moment h N is N 6 8 χ 60 5M P COBE normalization U/ε = (0.027M P ) 4 gives λ N ξ COBE λ = m H 2v Connection of the parameter ξ and the Higgs mass! h 2 N h2 end M 2 P /ξ F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

19 SM with non-minimal coupling to gravity After inflation reheating Inflation in the model U(χ) λm 4 /ξ 2 /4 λm 4 /ξ 2 /16 Reheating 0 0 χ end χ COBE χ M P ξ < χ < M P : U(χ) λm4 P 4ξ 2 ( 1 exp ( 2χ )) 2 6MP F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

20 SM with non-minimal coupling to gravity After inflation reheating Inflation in the model U(χ) λm 4 /ξ 2 /4 λm 4 /ξ 2 /16 Reheating 0 0 χ end χ COBE χ M P ξ < χ < M P : U λm2 P 6ξ 2 χ2, Ω 1, χ 3 ξ h 2 2 M P, T reh GeV F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

21 SM with non-minimal coupling to gravity After inflation back to the SM Inflation in the model U(χ) λm 4 /ξ 2 /4 Reheating Standard Model λm 4 /ξ 2 λ v 4 /4 / v 0 0 χ end χ COBE χ For χ < M P /ξ : the Standard Model F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

22 SM with non-minimal coupling to gravity Radiative corrections Radiative corrections no danger Could be a problem In the ordinary situation effective potential is generated U(h) m4 (h) 64π 2 log m2 (h) µ 2 +AΛ 2 + BΛ 4 We suppose that quadratic divergences are dealt with (eg. in dimensional regularization) F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

23 SM with non-minimal coupling to gravity Radiative corrections Radiative corrections no danger Could be a problem In the ordinary situation effective potential is generated U(h) m4 (h) 64π 2 log m2 (h) µ 2 standard Yukawa interaction m = y h U y 4 h 4 log h2 µ 2 Spoils flatness of the potential (for top quark y 1!) F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

24 SM with non-minimal coupling to gravity Radiative corrections Radiative corrections no danger This is also cured by non-minimal coupling! Effective potential is still generated U(χ) m4 (χ) 64π 2 log m2 (χ) µ 2 Conformal transformation: fermions S J = d 4 x { } g ψ / ψ + yh ψψ S E = ˆψ = Ω 3/2 ψ { } d 4 x ĝ ˆψ / ˆψ + y h(χ) Ω(χ) ˆψ ˆψ F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

25 SM with non-minimal coupling to gravity Radiative corrections Radiative corrections no danger This is also cured by non-minimal coupling! Effective potential is still generated U(χ) m4 (χ) 64π 2 log m2 (χ) µ 2 The interactions are suppressed now! (where Ω(χ) h(χ) for large χ) = U(χ) y 4 M4 P ξ 2 m(χ) = y h(χ) χ const Ω(χ) ) (1 e 2χ 2 ) 6MP log( m 2 (χ) const µ 2 F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

26 SM with non-minimal coupling to gravity Radiative corrections Radiative corrections no danger This is also cured by non-minimal coupling! Effective potential is still generated U(χ) m4 (χ) 64π 2 log m2 (χ) µ 2 The same for self interactions m 2 (χ) = U (χ) = λm2 P 3ξ 2 (2e 2χ 6MP 1 )e 2χ 6MP = U(χ) 0 χ 0 F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

27 Outline Predictions and expectations 1 Inflation virtues and problems Cosmological requirements Standard chaotic inflation Problems with using the SM Higgs for inflation 2 SM with non-minimal coupling to gravity The action Conformal transformation Inflation in the model Radiative corrections no danger 3 Predictions and expectations CMB parameters spectrum and tensor modes Higgs mass 4 Conclusions F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

28 Predictions and expectations CMB parameters spectrum and tensor modes CMB parameters spectrum and tensor modes WMAP SM+ ξh 2 R (4N + 9) n = 1 6ε + 2η 1 (4N + 3) r = 16ε (4N + 3) F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

29 Predictions and expectations Higgs mass Expected window for the Higgs mass Standard Model should remain applicable up to M P /ξ GeV We expect the Higgs mass 130GeV < M H < 190GeV Discovery of the Higgs with different mass will close the model! M, GeV H 2 loop 1 loop M =180 GeV t M =175 GeV t M =170 GeV t M t =175 GeV Strong coupling Vacuum is unstable log( Λ/GeV) Yu.Pirogov O.Zenin 98 F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

30 Outline Conclusions 1 Inflation virtues and problems Cosmological requirements Standard chaotic inflation Problems with using the SM Higgs for inflation 2 SM with non-minimal coupling to gravity The action Conformal transformation Inflation in the model Radiative corrections no danger 3 Predictions and expectations CMB parameters spectrum and tensor modes Higgs mass 4 Conclusions F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

31 Conclusions Conclusions Adding non-minimal coupling ξ H HR of the Higgs field to the gravity makes inflation possible without introduction of new fields The new parameter of the model, non-minimal coupling ξ, relates the normalization of CMB fluctuations and the Higgs mass ξ m H/ 2v Predicted for CMB n s 0.97 r Expected for LHC Higgs mass 130GeV < M H < 190GeV F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

32 Appendix Outline Appendix 5 Appendix F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22

33 Note about fourth family Appendix Disallowed by vacuum metastability or strong coupling before M P. Pirogov, Zenin, 98 F. Bezrukov, M. Shaposhnikov (EPFL&INR) The Standard model Higgs boson as the inflaton PONT d Avignon / 22