HIGGS-CURVATURE COUPLING AND POST-INFLATIONARY VACUUM STABILITY - PowerPoint PPT Presentation

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

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 .

CERN 
 merchandising


 
 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 pl R + ξ R ϕ † ϕ + 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 particle- physics experiments are very weak: | ξ | . 2 . 6 × 10 15 (Atkins & Calmet 2012) LHC: But in the early universe , R ↑↑ , and its effects can be important.

+ ξ R φ 2 + ξ R φ 2 CERN 
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 + ξ R φ 2 in curved spacetime

2. SM HIGGS POTENTIAL & HIGGS INSTABILITY (RGI) SM Higgs potential: 1.0 V ( ϕ ) = λ ( ϕ ) 0.8 ϕ 4 λ ( φ )>0 0.6 4 V H j Lê V H j + L λ ( φ )<0 0.4 v ⇠ O (10 2 )GeV ⌧ ϕ • Potential has a maximum 
 0.2 j + j - j 0 (barrier) at φ +. 0.0 * * ô • For φ > φ + , we have λ ( φ )<0. - 0.2 - 0.4 • Higgs develops a second 
 2 ¥ 10 16 4 ¥ 10 16 6 ¥ 10 16 8 ¥ 10 16 1 ¥ 10 17 j @ GeV D vacuum at φ _>> φ 0 , φ +. Running of λ ( φ ) is very sensitive to top-quark mass . world average: m t = (173.34±0.76) GeV (Note: For m t <171.5GeV; φ + , φ 0 —> + ∞ ) G. Degrassi et al. (2012); Bezrukov et al. (2012)

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 
 Constraints from 
 INFLATION PREHEATING

2. SM HIGGS POTENTIAL & HIGGS INSTABILITY If the Higgs had decay to the high-energy vacuum in the If the Higgs had decay to this vacuum in the past, the past, the Universe would have immediately collapsed. 
 Universe would have immediately collapsed. 
 This imposes strong constraints to ξ . This imposes strong constraints to ξ . Constraints from 
 Constraints from 
 INFLATION PREHEATING • If ξ ~0, Higgs is effectively massless during inflation and fluctuates: H ∗  H (max) ' 8 . 4 ⇥ 10 13 GeV ∗ − 2 π 2 λϕ 4 ✓ ◆ Yokoyama, 
 ϕ ⇠ H ∗ � ϕ + , ϕ 0 P eq ( ϕ ) = N exp Starobinsky 3 H 4 (1994) ∗ the Higgs becomes unstable! • Introducing a small coupling ξ >0 saves the day: m 2 h, e ff = ξ R Herrannen, Markannen, Lower bound: ξ & 0 . 06 Nurmi & Rajantie (2014)

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

3. TACHYONIC RESONANCE • We consider a chaotic inflation model with quadratic potential : V ( φ ) = 1 m φ ≈ 6 × 10 − 6 m p 2 m 2 φ φ 2 • If , inflaton decays in a slow-roll regime, causing the 
 φ & O (10) m p exponential expansion of the Universe. • When , inflation ends, and the inflaton starts oscillating φ ∗ ≈ 2 m p around the minimum of its potential ( preheating ).

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

3. TACHYONIC RESONANCE R ( t ) φ ( t ) ( k/a ( t )) 2 + ξ R ( t ) p ω k ( t ) = h k ∼ e | ω k | t m 2 φ ( t ) ≈ 0 R ( t ) < 0 h, e ff ( t ) ≡ ξ R ( t ) < 0

3. TACHYONIC RESONANCE R ( t ) φ ( t ) ( k/a ( t )) 2 + ξ R ( t ) p ω k ( t ) = h k ∼ e | ω k | t m 2 φ ( t ) ≈ 0 R ( t ) < 0 h, e ff ( t ) ≡ ξ R ( t ) < 0 The presence of the negative effective mass induces a strong excitation of the Higgs field modes: tachyonic resonance.

3. TACHYONIC RESONANCE TWO REGIMES IN THE HIGGS TIME-EVOLUTION ( λ = 0): h ϕ ( t ) i / t − 2 / 3 h ϕ ( t ) i ⇠ e α t initial tachyonic resonance late-time dynamics (Obtained from lattice simulations)

4. HIGGS INSTABILITY We now introduce the Higgs potential V( φ ) = λ ( φ ) φ 4 :  k 2 � Higgs EOM: 
 ¨ a 2 + ξ R ( t ) + ∆ + λ ( ϕ ) h ϕ 2 i h k + h k = 0 (h = φ a 3/2 ) R(t)<0: Destabilizing effect 
 λ ( φ )<0: Destabilizing effect 
 R(t)>0: Stabilizing 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.)


 4. LATTICE SIMULATIONS Momenta captured: 1. N 3 =256 3 points, L ≈ (0.03 m ϕ ) -1 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) The Higgs goes to negative-energy ξ > ξ c : vacuum at time ti . We determine ξ c ξ < ξ c : The Higgs goes to EW vacuum. (Figueroa, Rajantie & F .T., arXiv:1612.xxxxx)

4. LATTICE SIMULATIONS: RESULTS m t ≈ 172 . 12GeV Negative-energy 
 vacuum Potential BARRIER λ ( φ )>0: the Higgs is safe! For ξ ≳ ξ c ≈ 12.1 , the Higgs field becomes unstable at a time t i ( ξ )

ξ =9, m t =173.34 GeV DONEC QUIS NUNC

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 >173.34 GeV , 
 lattice approach is not valid.

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 : ! 3 Z 1 + 1 g 0 2 Y µ ν Y µ ν + ( D µ Φ )( D µ Φ ) + λ ( Φ † Φ ) 2 X a 3 ( t ) d 4 x S = − W a µ ν W µ ν g 2 a a =1 WITH 
 WITHOUT 
 g. bosons: g. bosons: • 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 .

5. CONCLUSIONS • As the Ricci scalar was much greater in the past than now, early- universe cosmology can provide tight constraints for the Higgs- curvature coupling . • With lattice simulations, one can determine upper bounds for ξ : m 2 φ 2 inflation & 
 0 . 06 . ξ . 4 m t =173.34 GeV • Bounds are dependent on inflationary model and running of λ ( φ ) . Lower-energy models can widen this range.