HIGGS-CURVATURE COUPLING AND POST-INFLATIONARY VACUUM STABILITY

Transcription

1 HIGGS-CURVATURE COUPLING AND POST-INFLATIONARY VACUUM STABILITY Francisco Torrentí, IFT UAM/CSIC with Daniel G. Figueroa and Arttu Rajantie (arxiv:1612.xxxxx) V Postgraduate Meeting on Theoretical Physics, Oviedo, 18th November 2016

2 1. THE HIGGS-CURVATURE COUPLING The Standard Model Lagrangian in Minkowski spacetime (6 quarks +6 leptons+gauge bosons+higgs) possesses 19 free parameters. Since the discovery of the Higgs in the LHC (2012), we have determined all of them with great precision.

3 CERN merchandising

4 1. THE HIGGS-CURVATURE COUPLING In curved spacetime, there is one more possible term, required for the renormalisability of the theory: L = 1 2 M 2 plr + R' ' + L SM The Higgs-curvature coupling ξ runs with energy, and cannot be set to 0. As the Ricci scalar R is very small today, constraints from particlephysics experiments are very weak: LHC: (Atkins & Calmet 2012) But in the early universe, R, and its effects can be important.

5 + R 2 + R 2 + R 2 CERN merchandising in curved spacetime

6 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY (RGI) SM Higgs potential: V (') = (') 4 '4 v O(10 2 )GeV ' Potential has a maximum (barrier) at φ +. For φ>φ +, we have λ(φ)<0. Higgs develops a second vacuum at φ_>>φ 0,φ +. VHjLêVHj+L λ(φ)> λ(φ)<0 j + j 0 j - * * ô Running of λ(φ) is very sensitive to top-quark mass. world average: mt = (173.34±0.76) GeV G. Degrassi et al. (2012); Bezrukov et al. (2012) (Note: For m t <171.5GeV; φ +,φ 0 > + )

7 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY If the Higgs had decay to the high-energy vacuum in the past, the Universe would have immediately collapsed. This imposes strong constraints to ξ. Constraints from INFLATION Constraints from PREHEATING

8 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY If If the the Higgs had decay to to the this high-energy vacuum in the vacuum past, in the the past, Universe the Universe would would have have immediately collapsed. collapsed. This imposes strong constraints to ξ. Constraints from INFLATION Constraints from PREHEATING If ξ~0, Higgs is effectively massless during inflation and fluctuates: Yokoyama, Starobinsky (1994) P eq (') =N exp ' 4 H 4 H apple H (max) ' GeV ' H ' +, ' 0 the Higgs becomes unstable! Introducing a small coupling ξ>0 saves the day: m 2 h,e = R Herrannen, Markannen, Nurmi & Rajantie (2014) Lower bound: & 0.06

9 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY If If the the Higgs had decay to to the this high-energy vacuum in the vacuum past, in the the past, Universe the Universe would would have have immediately collapsed. collapsed. This imposes strong constraints to ξ. Constraints from INFLATION Constraints from PREHEATING SM Higgs is excited due to tachyonic resonance Upper bound (let s see how it works!)

10 3. TACHYONIC RESONANCE We consider a chaotic inflation model with quadratic potential: V ( )= 1 2 m2 2 m m p & O(10)m p If, inflaton decays in a slow-roll regime, causing the exponential expansion of the Universe. When 2m p, inflation ends, and the inflaton starts oscillating around the minimum of its potential (preheating).

11 3. TACHYONIC RESONANCE To obtain the post-inflationary dynamics of the system, we solve the field and Friedmann equations: ȧ 2 H 2 (t) = ( 2 + m 2 2 ) +3H(t) + m 2 =0 a The inflaton solution is: 6m 2 p (t) ' (t)sin(m t) decaying amplitude (t) = r 8 3 m p m t And the Ricci scalar and scale factor: " ȧ # 2 R(t) 6 + ä a a ε(t): small oscillating function = 1 m 2 p (2m ) a(t) =t 2/3(1+ (t))

12 3. TACHYONIC RESONANCE R(t) (t)! k (t) = p (k/a(t)) 2 + R(t) (t) 0 R(t) < 0 m 2 h,e (t) R(t) < 0 h k e! k t

13 3. TACHYONIC RESONANCE R(t) (t)! k (t) = p (k/a(t)) 2 + R(t) (t) 0 R(t) < 0 m 2 h,e (t) R(t) < 0 h k e! k t The presence of the negative effective mass induces a strong excitation of the Higgs field modes: tachyonic resonance.

14 3. TACHYONIC RESONANCE TWO REGIMES IN THE HIGGS TIME-EVOLUTION (λ = 0): h'(t)i e t initial tachyonic resonance h'(t)i /t 2/3 late-time dynamics (Obtained from lattice simulations)

15 4. HIGGS INSTABILITY We now introduce the Higgs potential V(φ) = λ(φ)φ 4 : Higgs EOM: (h =φa 3/2 ) ḧ k + apple k 2 a 2 + R(t)+ + (')h'2 i h k =0 R(t)<0: Destabilizing effect R(t)>0: Stabilizing effect λ(φ)<0: Destabilizing effect λ(φ)>0: Stabilizing effect Analytically and/or numerically, it is difficult to determine the values of ξ for which the Higgs becomes unstable. (Herrannen et al., 2015) (Kohri & Matsui, 2016) Tachyonic resonance is a non-perturbative process, which must be studied with lattice simulations (i.e. solving the differential equations of motion in a discrete finite box). (Ema, Mukaida & Nakayama, 2016) (Figueroa, Rajantie & F.T., t.b.p.)

16 4. LATTICE SIMULATIONS 1. N 3 =256 3 points, L (0.03 m ϕ ) -1 Momenta captured: 0.18 m ϕ < p < 40 m ϕ 2. The running of λ(φ) is introduced in the lattice as a local function of the lattice point (not a constant). 3. We consider different runnings of λ(φ), corresponding to different values of the top-quark mass. (Note: We modify the running at high energies for numerical stability) We determine ξ c ξ>ξ c : ξ<ξ c : The Higgs goes to negative-energy vacuum at time ti. The Higgs goes to EW vacuum. (Figueroa, Rajantie & F.T., arxiv:1612.xxxxx)

17 4. LATTICE SIMULATIONS: RESULTS m t GeV Negative-energy vacuum Potential BARRIER λ(φ)>0: the Higgs is safe! For ξ ξc 12.1, the Higgs field becomes unstable at a time ti(ξ)

18 DONEC QUIS NUNC ξ=9, m t = GeV

19 4. LATTICE SIMULATIONS: RESULTS Instability time t i depends strongly on ξ Dependence on initial (random) quantum fluctuations Note: For values ξ<4, m t > GeV, lattice approach is not valid.

20 S = Z 4. LATTICE SIMULATIONS: RESULTS The SM Higgs is coupled to other SM particles: gauge bosons and fermions. They may affect the post-inflationary Higgs dynamics. Dominant decay products: electroweak gauge bosons: a 3 (t)d 4 x 1 g 2 3X a=1! Wµ W a a µ + 1 g 02 Y µ Y µ +(D µ )(D µ )+ ( ) 2 We introduce an Abelian- Higgs model in the lattice, mimicking the full non- Abelian structure of the Standard Model. Their effect is not very relevant. WITH g. bosons: WITHOUT g. bosons:

21 5. CONCLUSIONS As the Ricci scalar was much greater in the past than now, earlyuniverse cosmology can provide tight constraints for the Higgscurvature coupling. With lattice simulations, one can determine upper bounds for ξ: m 2 φ 2 inflation & m t = GeV Bounds are dependent on inflationary model and running of λ(φ). Lower-energy models can widen this range.