Spacetime curvature and Higgs stability during and after inflation - PowerPoint PPT Presentation

Spacetime curvature and Higgs stability during and after inflation arXiv:1407.3141 (PRL 113, 211102) arXiv:1506.04065 Matti Herranen 3 Sami Nurmi 4 Tommi Markkanen 12 Arttu Rajantie 2 1 King’s College London 2 Imperial College London 3 Niels Bohr International Academy, Copenhagen 4 University of Jyväskylä Birmingham October 2015 Markkanen Higgs Stability 1 / 29

Outline Introduction 1 Higgs stability during inflation (QFT in Minkowski) 2 Higgs stability after inflation 3 Conclusions 4 Markkanen Higgs Stability 1 / 29

Introduction Introduction 1 Higgs stability during inflation (QFT in Minkowski) 2 Higgs stability after inflation 3 Conclusions 4 Markkanen Higgs Stability 2 / 29

Standard Model Higgs potential V ( ϕ ) V ( ϕ ) 0 0 v 0 v Behaviour very sensitive V ( φ ) has a minimum at to M h and M t φ = v A vacuum at φ � = v incompatible with observations New physics needed to stabilize the vacuum? Markkanen Higgs Stability 3 / 29

Current status Figure : Degrassi et al. (2013) 180 10 7 10 10 Instability Instability Meta � stability Pole top mass M t in GeV 175 1,2,3 Σ 170 10 12 Stability 165 115 120 125 130 135 Higgs mass M h in GeV Meta stable at 99% CL [1] Lifetime much longer than 13 . 8 · 10 9 years Is this also true for the early Universe ? [1] Buttazzo et al. (2013); Spencer-Smith (2014); Bednyakov, Kniehl, Pikelner, & Veretin (2015) Markkanen Higgs Stability 4 / 29

Inflation and the Standard Model We assume the SM to be valid at high energies Potential peaks at Λ max Assuming also an early stage of exponential cosmological expansion (inflation) with a scale H Important if Λ max � H State of the art calculations [2]: Λ max ∼ 10 11 GeV V � Φ � BICEP2/Keck/Planck H � 10 14 GeV V max 0 BICEP2: Λ max ≪ H v � max [2] Degrazzi et. al.(2013); Buttazzo et. al. (2013) Markkanen Higgs Stability 5 / 29

Outline Introduction 1 Higgs stability during inflation (QFT in Minkowski) 2 Higgs stability after inflation 3 Conclusions 4 Markkanen Higgs Stability 6 / 29

Higgs stability during inflation Inflation induces fluctuations to the Higgs field ∆ φ ∼ H Fluctuations may be treated as stochastic variables [3] ⇒ We can assign a probability density P ( φ ) to φ The essential input for P ( φ ) is ¯ V eff ( φ ) , the effective potential [3] Starobinsky (1986); Starobinsky & Yokoyama (1994) Markkanen Higgs Stability 7 / 29

1-loop Effective potential Derivation of V eff ( φ ) is a standard calculation [4] A theory with a massive self-interacting scalar field 2 m 2 φ 2 + λ V eff ( φ ) = 1 4 ! φ 4 � �� � classical effective mass � �� � � � M ( φ ) 2 � � + M ( φ ) 4 ; M ( φ ) 2 = m 2 + λ − 3 2 φ 2 log 64 π 2 µ 2 2 � �� � quantum µ is the renormalization scale Similarly one may derive the potential for the SM Higgs [4] Coleman & Weinberg (1972) Markkanen Higgs Stability 8 / 29

Effective potential for the SM Higgs 5 � � log M 2 i ( φ ) V eff ( φ ) = − 1 2 m 2 φ 2 + 1 n i � 4 λφ 4 + 64 π 2 M 4 i ( φ ) − c i µ 2 i = 1 i ( φ ) = κ i φ 2 − κ ′ ; M 2 i κ ′ Φ κ i i n i c i i W ± g 2 / 4 1 6 0 5 / 6 ( g 2 + g ′ 2 ) / 4 Z 0 2 3 0 5 / 6 y 2 t 3 − 12 t / 2 0 3 / 2 m 2 φ 4 1 3 λ 3 / 2 m 2 χ i λ 3 / 2 5 3 Explicit µ dependence? Markkanen Higgs Stability 9 / 29

Callan-Symanzik equation for massless λφ 4 theory The effective potential is renormalized at a scale µ λ 0 → λ R + δλ, φ → ( 1 + δ Z ) φ However, the physical result must not depend on µ We can impose this by demanding d d µ V eff ( φ ) = 0 This can be used to improve the perturbative result Leads to running parameters , e.g. λ ( µ ) Same can be done for the SM Markkanen Higgs Stability 10 / 29

SM running (1-loop) 1.2 g 3 1.0 y t SM couplings 0.8 g 0.6 g’ 0.4 0.2 Λ 0.0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 RGE scale Μ � GeV � For large φ , the potential is dominated by the quartic term λφ 4 V ( φ ) ∼ λ ( µ ) φ 4 4 Markkanen Higgs Stability 11 / 29

Scale independence V eff One can easily show that for the SM to 1-loop [5] d V eff = 0 + O ( � 2 ) ¯ d µ We must choose µ to make the higher order terms as small as possible [6] The optimal choice µ ∼ φ ⇒ No large logarithms Now we have a well-defined potential with no unknown parameters! [5] Casas et. al. (1994) [6] Ford et. al. (1993) Markkanen Higgs Stability 12 / 29

Generalization to curved space <2-> It is possible to include (classical) gravity in the quantum calculation, R = 12 H 2 ⇒ The SM includes a non-minimal ξ -term, ∼ ξ R φ 2 Always generated by running in curved space Virtually unbounded by the LHC, ξ EW < 10 15 [7] Curvature induces running of the constants [8] Leading potential contributions: Flat space, φ ≫ m Curved space, H ≫ φ ≫ m V eff ( φ ) ≈ λ ( φ ) V eff ( φ ) ≈ λ ( H ) φ 4 + ξ ( H ) φ 4 R φ 2 4 4 2 [7] Atkins & Calmet (2012) [8] Zurek, Kearney & Yoo (2015); TM (2014) Markkanen Higgs Stability 13 / 29

1-loop Effective potential in curved space V eff ( φ, R ) = − 1 2 m 2 ( t ) φ ( t ) 2 + 1 2 ξ ( t ) R φ ( t ) 2 + 1 4 λ ( t ) φ ( t ) 4 9 � � � � M 2 � i ( t ) n i � � 64 π 2 M 4 ; M 2 i ( t ) = κ i φ ( t ) 2 − κ ′ + i ( t ) i + θ i R log − c i µ 2 ( t ) i = 1 κ ′ Φ κ i θ i i n i c i i g 2 / 4 1 / 12 3 / 2 1 2 0 W ± g 2 / 4 − 1 / 6 5 / 6 2 6 0 g 2 / 4 − 1 / 6 3 / 2 3 − 2 0 ( g 2 + g ′ 2 ) / 4 0 4 1 1 / 12 3 / 2 ( g 2 + g ′ 2 ) / 4 0 Z 0 5 3 − 1 / 6 5 / 6 ( g 2 + g ′ 2 ) / 4 0 6 − 1 − 1 / 6 3 / 2 y 2 t / 2 1 / 12 3 / 2 t 7 − 12 0 m 2 φ 8 1 3 λ ξ − 1 / 6 3 / 2 m 2 χ i 9 3 λ ξ − 1 / 6 3 / 2 Markkanen Higgs Stability 14 / 29

Stability (Flat) 4 � 10 � 4 V eff � Φ � � � max 4 2 � 10 � 4 0 � 2 � 10 � 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Φ � � max For large H ( ∼ 10 3 Λ max ) , the SM is not stable [9] Coupling the Higgs to an inflaton ∼ Φ 2 φ 2 ⇒ stable [10] How does including curvature change this? [9] Kobakhidze & Spencer-Smith (2014); Hook et. al. (2014); Fairbairn & Hogan (2014); Enqvist, Meriniemi & Nurmi (2014); Zurek, Kearney & Yoo (2015) [10] Lebedev (2012); Lebedev & Westphal (2013) Markkanen Higgs Stability 15 / 29

Stability (curved) I First attempt, set ξ EW = 0 and H ∼ 10 3 Λ max V eff ( φ ) ≈ λ ( µ ) φ 4 + ξ ( µ ) 2 R φ 2 4 4 � 10 3 2 � 10 3 V eff � Φ � � � max 0 4 � 2 � 10 3 � 4 � 10 3 � 6 � 10 3 � 8 � 10 3 0.0 0.5 1.0 1.5 Φ � � max For large H one has λ ( µ ) < 0 , since µ 2 = φ 2 + R ξ Can become positive or negative depending on ξ EW Markkanen Higgs Stability 16 / 29

Stability results (curved space) II For large H one has λ ( µ ) < 0 , since µ 2 = φ 2 + R ξ Can become positive or negative depending on ξ EW ξ EW Ξ � Μ � 0 , 0 . 05 , 0 . 12 , 1 / 6 , 1 � 6 0 . 22 , 0 . 28 , 0 . 33 0 10 2 10 5 10 5 10 8 10 11 10 14 RGE scale Μ � GeV � Markkanen Higgs Stability 17 / 29

Stability results (curved space) III Now choosing ξ EW = 0 . 1 [11] 6 � 10 9 V eff � Φ � � � max 4 4 � 10 9 2 � 10 9 0 � 2 � 10 9 0 500 1000 1500 Φ � � max V max ( curved ) ≫ V max ( flat ) (and at a higher scale) � − 8 π 2 ( V max / 3 H 4 ) � P ∼ exp ⇒ Stable! [11] Espinosa, Giudice & Riotto (2008) Markkanen Higgs Stability 18 / 29

Stability results (curved space) IV The (in)stability of the potential is determined by ξ EW 10 1 � 4 � 10H V max 1 � 4 � 5H 1 V max I: Stability 10 � 1 Ξ EW 1 � 4 � H V max 10 � 2 II: Instability 10 � 3 10 9 10 10 10 11 10 12 10 13 10 14 H � GeV � Markkanen Higgs Stability 19 / 29

Outline Introduction 1 Higgs stability during inflation (QFT in Minkowski) 2 Higgs stability after inflation 3 Conclusions 4 Markkanen Higgs Stability 20 / 29

Markkanen Higgs Stability 21 / 29

Reheating Equation of state w = p /ρ changes, w inf = − 1 → w reh Energy of inflation is transferred to SM degrees of freedom, which (eventually) thermalize T = 0 → T reh The crucial moment is right after inflation, but before thermalization A very complicated and dynamical process [12] Reheating ⇔ Pre heating The Higgs always feels the dynamics of reheating (even without a direct coupling to the inflaton) [12] Kofman, Linde & Starobinsky (1997) Markkanen Higgs Stability 22 / 29

Reheating During reheating the inflaton oscillates ( p = w ρ ) Φ w reh w  - 1 ϵ  1 ϵ  1 The inflaton influences the Higgs via gravity ⇒ New stability constraints ! Two effects: A rapid drop in w , on average Oscillations in the complete solution Markkanen Higgs Stability 23 / 29

Oscillating mass (example) For example for a coupling L int ∝ g Φ 2 φ 2 Oscillating mass for Higgs 2 m 2 eff ∼ g Φ 2 0 cos 2 ( t M inf ) m eff 0 Parametric resonance via the Mathieu equation � � d 2 f ( z ) + A k − 2 q cos ( 2 z ) f ( z ) = 0 , z = t M inf dz 2 ⇒ Exponential amplification May result in a very large fluctuation [13] [13] Kofman, Linde & Starobinsky (1997) Markkanen Higgs Stability 24 / 29