The 1-loop effective potential for the Standard Model in curved spacetime

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1 The 1-loop effective potential for the Standard Model in curved spacetime arxiv: (JHEP) The 1-loop effective potential for the SM in curved spacetime arxiv: (Review) Cosmological Aspects of Higgs Vacuum Metastability Tommi Markkanen 1 In collaboration with: Arttu Rajantie 1, Stephen Stopyra 1 and Sami Nurmi 2 1 Imperial College London, 2 University of Jyväskylä September-2018, Paris Analytical Methods

2 Outline 1 Motivation 2 Effective potential in curved space 3 Applications

3 Motivation 1 Motivation 2 Effective potential in curved space 3 Applications

4 Standard Model Higgs potential Minimum at φ = v Sensitive to M h and M t Effective potential for φ m h V(φ) λ(φ) 4 φ4 New minimum λ(µ) < 0 [1] A true vacuum at φ v decay! new physics! [1] Buttazzo et al. (13) ; Degrassi et al. (12)

5 Current status With central values: λ(µ Λ ) = 0 µ Λ GeV

6 Implications from metastability V (φ) Fluctuations 0 Tunneling Metastable Minimum φ True Minimum

7 Current status 180 Instability region Meta-stability region Stability region Metastabiliy favoured over stability!

8 Implications from metastability II Lifetime of the Universe today (much) longer than years [2] In the early Universe vacuum decay can easily be triggered! In the early Universe one must include spacetime curvature in the effective potential [2] Andreassen et al. (18) ; Chigusa et al. (18)

9 Outline 1 Motivation 2 Effective potential in curved space 3 Applications

10 1-loop Effective potential Derivation of V eff (φ) is a standard calculation [3] A theory with a massive self-interacting scalar field V eff (φ) = 1 2 m2 φ 2 + λ 4! φ4 }{{} classical effective mass [ { ( }} ) { + M(φ)4 M(φ) 2 64π 2 log µ 2 3 ] 2 }{{} quantum ; M(φ) 2 = m 2 + λ 2 φ2 µ is the renormalization scale Large log s problematic [3] Coleman & Weinberg (72)

11 RG improvement Physics must be µ-independent: { d dµ V eff(φ) = 0 µ µ + β λ λ + γ φφ Running parameters ( λ λ(µ), etc. ) φ } V eff (φ) = 0 We are free to choose µ Renormalization group improvement (RGI) optimization of µ the usual choice for φ m µ φ V RGI (φ) = λ(φ) 4 φ4 + loops A better approximation if loop correction is fixed to vanish The (SM) metastability comes from fermions

12 Yukawa theory: 1 2 ( φ)2 + ψ /ψ λ 4 φ4 gφ ψψ (1-loop) (λ(φ)/4)φ 4 V RGI (φ) V eff (φ) V (φ) µ The critical threshold log(φ/µ 0 ) λ(µ 0 ) 0.73g 2 (µ 0 ) V(φ ).

13 RG improvement for λφ 4 theory in curved space I Background strictly classical! V(φ) = 1 2 m2 φ 2 + λ 4! φ4 ; 1-loop [ m 2 + λ 2 φ2 1 6 R] 2 64π 2 [ log ( m 2 + λ 2 φ2 1 6 R µ 2 ) 3 ] 2 A new operator is generated! Should be included from the start Curvature mass L L + ξrφ 2 The SM in principle includes a non-minimal ξ-term Always generated by running in curved space, β ξ (ξ 1 6 ) Virtually unbounded by the LHC, ξ EW < [4] [4] Atkins & Calmet (12)

14 RG improvement for λφ 4 theory in curved space II log s contain curvature dependence m 2 + λ 2 φ2 1 6 R /µ2 RG improvement Curvature induced running [5] running scale, φ m µ 2 φ 2 + R Leading approximations: Flat space, φ m V RGI (φ) λ(φ) 4 φ4 Curved space, H φ m V RGI (φ) λ(h) 4 φ4 + ξ(h) 2 Rφ2 This is not the whole story! [5] East et. al. (16); Zurek, Kearney & Yoo (15); TM et. al. (14); TM (14)

15 RG improvement for λφ 4 theory in curved space III The full 1-loop result: + [ m 2 + λ 2 φ2 1 6 R] 2 64π 2 ( Rµνδη R µνδη R µν R µν) 64π 2 [ log ( m 2 + λ 2 φ2 1 6 R )] [ log µ 2 ( m 2 + λ 2 φ2 1 6 R )] µ 2 All radiatively generated operators are: V Λ, R, Rφ 2, R 2, R αβ R αβ, R αβγδ R αβγδ O(R 2 )-type operators couple to the Higgs logarithmically β functions depend on the particle content

16 β s for the SM gravitational operators Operators coupling to gravity for the SM L g = ξ 2 Rφ2 V Λ + κr α 1R 2 α 2R µνr µν α 3R µνδη R µνδη 16π 2 β ξ = ( ξ 1 ] )[12λ + 2Y 2 3(g ) 2 9g π 2 β VΛ = 2m 4 ( 16π 2 β κ = 4m 2 ξ 1 ) 6 16π 2 β α1 = 2ξ 2 2ξ π 2 β α2 = π 2 β α3 = O(R 2 ) operators and quantum gravity?! [6] [6] Salvio (18)

17 Full 1-loop result, de Sitter (arxiv: ) { [ V (1) eff,sm (φ) = 1 12 M 2 n im 4 i 64π 2 i log d µ 2 i + n ih 4 log M2 i Table 1. Contributions to the effective potential (4.14), where Ψ stands for W ±, Z 0, top quark t, Higgs h, the Goldstone bosons χ µ 2 W and χ Z and the ghosts c W and c Z. i=1 ] }, Ψ i n i d i n i M 2 i 1 2 3/2 34/15 m 2 W + H2 W ± 2 6 5/6 34/5 m 2 W + H /2 4/15 m 2 W 2H /2 17/15 m 2 Z + H2 Z /6 17/5 m 2 Z + H /2 2/15 m 2 Z 2H2 t /2 38/5 m 2 t + H 2 h 8 1 3/2 2/15 m 2 h + 12(ξ 1/6)H2 χ W 9 2 3/2 4/15 m 2 χ + ζm 2 W + 12(ξ 1/6)H2 χ Z /2 2/15 m 2 χ + ζm 2 Z + 12(ξ 1/6)H2 c W /2 4/15 ζm 2 W 2H2 c Z /2 2/15 ζm 2 Z 2H2

18 SM in de Sitter, ξ EW = 0 [V(ϕcl) V(0)]/µ 4 Λ H 0 = µ Λ H 0 = µ Λ H 0 = µ Λ H 0 = µ Λ H 0 = µ Λ H 0 = µ Λ ϕ cl /µ Λ Change mostly due to the generation of a negative ξ(µ)

19 Jordan Einstein (classical background) The Jordan frame action S = d 4 x [ R g 2 M2 pl 1 2 ( µ φ) 2 1 ] 2 ξrφ2 V(φ) Einstein frame via a scaling ḡ µν = Ω 2 g µν, Ω 2 = 1 ξφ2 Mpl 2, V(φ) V(φ) (Partial) quantization in the new frame Einstein frame The ξrφ 2 is removed, the potential changes Debate over equivalence/inequivalence after quantization Crucial for Higgs inflation For a spectator field identical results up to φ/m pl 1 [7] [7] TM, Rajantie & Nurmi (17)

20 Metric quantization For the metric fluctuations, use the ADM form ds 2 = N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) Tedious calculation No change to a fixed background in the Jordan frame [8] All new interactions suppressed. Semi-classical error ( ) φ O M pl For spectator Higgs dynamics [8] TM, Rajantie & Nurmi (17) µ Λ /M pl 10 8

21 Comments on the derivation Calculation based on the resummed Heat Kernel [9] Essentially an expansion around the UV State independent Gives all β functions What about the IR? Needs to be accounted for For inflation use V RGI (φ) as input for Stochastic approach [10] Hawking-Moss instanton [11] Applications beyond SM metastability [9] Parker & Toms (85) [10] Starobinsky & Yokoyama (94) [11] Hawking & Moss (82)

22 Stochastic approach with RG effects IR fluctuations as stochastic variables, noise from UV [12] Talks by: Burgess, Silverstein, Vennin, Renaux-Petel Classical statistics with the probability density P(t, φ) Ṗ(t, φ) = 1 [ P(t, φ)v (φ) ] + H3 2 3H φ 8π 2 P(t, φ) φ2 (Fokker-Planck) For vacuum stability correct treatment of UV essential (Otherwise, needs to be investigated case-by-case) V(φ) V RGI (φ) Is it possible to maintain ξ = 0 for 60 e-folds? [12] Starobinsky (1986); Starobinsky & Yokoyama (1994)

23 Applications 1 Motivation 2 Effective potential in curved space 3 Applications

24 Decay of the Higgs vacuum during inflation For large H metastability problematic [13] The Higgs is light during inflation (in de Sitter) Including curvature in the effective potential essential [14] Can (de)stabilize the flat space prediction Vacuum decay rate from the Hawking-Moss instanton Similar to the stochastic approach Decay rate ( ) Γ exp 8π2 V RGI 3H 4 [13] Espinosa, Giudice & Riotto (08); Kobakhidze & Spencer-Smith (14); Hook et. al. (14); Fairbairn & Hogan (14); Enqvist, Meriniemi & Nurmi (14); Zurek, Kearney & Yoo (15) [14] Herranen et al. (14) & (15); Markkanen et al. (18)

25 Current status inflation Stability tight constraints for ξ 1 Low scale inflation healthy

26 Decay of the Higgs vacuum after inflation After inflation ξrφ 2 < 0 explosive (tachyonic) particle creation! R = 1 M 2 P [4U(Φ) Φ 2 ] 2 U(Φ) = m2 Φ 2 Φ2 U(Φ) = λ Φ 4 Φ4 U(Φ) = 0 Φ M P R M 2 P

27 Current status reheating Hubble rate H/µΛ 10 1 Non-minimal coupling ξ φ > µλ φ > 10µΛ φ > 102 µλ Hubble rate H [GeV] Curvature induced running effects backreaction Stability tight constraints for ξ & 1

28 Despicable Dark Relics, (arxiv: ) ξrχ 2 < 0 gives particle creation for any decoupled singlet U(Φ) = m2 Φ 2 Φ2 Ω DM < Ω χ Ω DM > Ω χ H inf = GeV H inf = GeV 10 8 λ = λ = λ = 10 7 m GeV m GeV ξ

29 For a large H, curvature significantly effects the effective potential for the SM Running of couplings from H Rφ 2, R 2, R αβ R αβ, R αβγδ R αβγδ are always generated and they couple to the Higgs Constraints from stability during inflation and reheating Potent gravitational particle creation Dark Matter Thank You!

30 Sensitivity to the choice of µ A loop calculation is never fully scale invariant How dependent is the result on the choice µ 2 =φ 2 +R? µ 2 = αφ 2 + βr α, β {0.1 10} Veff Φ max Φ max

31 Potential during inflation II Quantum field theory on a curved background Example: self-interacting scalar field in curved space S = d 4 x [ 1 g 2 ( µφ) m2 φ 2 + ξ 2 Rφ2 + λ ] 4! φ4 ; R = 12H 2 For a constant field δs φ δφ = 0 V (φ) = 0 V(φ) = V (φ) Quantum correction to LO in fluctuations, ˆφ ˆφ + ˆφ φ + ˆφ V (φ) = m 2 φ + ξrφ + λ 6 φ3 + λ }{{} 2 φ ˆφ 2 }{{} classical quantum Same calculation in flat and curved space The only difference is the form of the mode

32 Flat space ds 2 = dt 2 + dx 2 [ + M(φ) 2 ] ˆφ = 0 Curved space (FLRW) ds 2 = dt 2 + a 2 (t)dx 2 [ ] + M(φ) 2 + ξr ˆφ = 0 f k = 1 ω e iωt ω 2 = k 2 + M(φ) 2 [ V Q (φ) = M(φ)4 64π 2 log ( M(φ) 2 µ 2 W 2 = f k = 1 W e i t W ( k2 a(t) 2 + M(φ)2 + ξ 1 ) R 6 ) 3 ] ; M(φ) 2 = m 2 + λ 2 2 φ2 V Q (φ) = [ M(φ) 2 + ( ξ 1 6) R ] 2 64π 2 [ ( M(φ) 2 + ( ξ 1 6 log µ 2 ) R ) 3 ] 2