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

HIGGS-CURVATURE COUPLING AND POST-INFLATIONARY VACUUM STABILITY

Francisco Torrentí, IFT UAM/CSIC

V Postgraduate Meeting on Theoretical Physics, Oviedo, 18th November 2016

with Daniel G. Figueroa and Arttu Rajantie 
 (arXiv:1612.xxxxx)

• 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

• In curved spacetime, there is one more possible term, required

for the renormalisability of the theory:
 
 


• 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:

• 1. THE HIGGS-CURVATURE COUPLING

|ξ| . 2.6 × 1015 (Atkins & Calmet 2012)

LHC:

L = 1 2M 2

plR + ξRϕ†ϕ + LSM

But in the early universe, R↑↑, and its effects can be important.

+ξRφ2

CERN
 merchandising
 in curved spacetime

+ξRφ2 +ξRφ2

• 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY

V (ϕ) = λ(ϕ) 4 ϕ4

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0.0 0.2 0.4 0.6 0.8 1.0 j @GeVD VHjLêVHj+L

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λ(φ)<0 λ(φ)>0

(RGI) SM Higgs potential:

• Potential has a maximum 


(barrier) at φ+.

• For φ>φ+, we have λ(φ)<0.
• Higgs develops a second


vacuum at φ_>>φ0,φ+.

• G. Degrassi et al. (2012); Bezrukov et al. (2012)

world average: mt = (173.34±0.76) GeV

Running of λ(φ) is very sensitive to top-quark mass.

(Note: For mt<171.5GeV; φ+,φ0 —> +∞)

v ⇠ O(102)GeV ⌧ ϕ

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

• 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY

• 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY

H∗  H(max)

∗

' 8.4 ⇥ 1013GeV

Yokoyama, 
 Starobinsky (1994)

• If ξ~0, Higgs is effectively massless during inflation and fluctuates:

ϕ ⇠ H∗ ϕ+, ϕ0

the Higgs becomes unstable!

If the Higgs had decay to this vacuum in the past, the Universe would have immediately collapsed.
 This imposes strong constraints to ξ.

Constraints from 
 INFLATION

Peq(ϕ) = N exp ✓ −2π2 3 λϕ4 H4

∗

◆

Constraints from 
 PREHEATING

ξ & 0.06

Herrannen, Markannen, Nurmi & Rajantie (2014)

• Introducing a small coupling ξ>0 saves the day:

m2

h,eff = ξR

Lower bound:

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.

If the Higgs had decay to this vacuum in the past, the Universe would have immediately collapsed.
 This imposes strong constraints to ξ.

Constraints from 
 INFLATION Constraints from 
 PREHEATING

• 2. SM HIGGS POTENTIAL & HIGGS INSTABILITY

SM Higgs is excited
 due to tachyonic resonance

Upper bound


(let’s see how it works!)

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.

• 3. TACHYONIC RESONANCE

V (φ) = 1 2m2

φφ2

• We consider a chaotic inflation model with quadratic potential:

mφ ≈ 6 × 10−6mp

• If , inflaton decays in a slow-roll regime, causing the


exponential expansion of the Universe.

φ & O(10)mp

• When , inflation ends, and the inflaton starts oscillating

around the minimum of its potential (preheating).

φ∗ ≈ 2mp

• 3. TACHYONIC RESONANCE

H2(t) ≡ ✓ ˙ a a ◆2 = ( ˙ φ2 + m2

φφ2)

6m2

p

¨ φ + 3H(t) ˙ φ + m2

φφ = 0

• To obtain the post-inflationary dynamics of the system, we solve the


field and Friedmann equations:

Φ(t) = r 8 3 mp mφt

φ(t) ' Φ(t) sin(mφt)

• The inflaton solution is:

decaying amplitude

And the Ricci scalar and scale factor:

R(t) ≡ 6 "✓ ˙ a a ◆2 + ¨ a a # = 1 m2

p

(2m2

φφ2 − ˙

φ2)

a(t) = t2/3(1+✏(t))

ε(t): small oscillating function

• 3. TACHYONIC RESONANCE

φ(t) R(t)

ωk(t) = p (k/a(t))2 + ξR(t)

φ(t) ≈ 0 R(t) < 0

m2

h,eff(t) ≡ ξR(t) < 0

hk ∼ e|ωk|t

• 3. TACHYONIC RESONANCE

φ(t) R(t)

ωk(t) = p (k/a(t))2 + ξR(t)

φ(t) ≈ 0 R(t) < 0

m2

h,eff(t) ≡ ξR(t) < 0

hk ∼ e|ωk|t

The presence of the negative effective mass induces a strong excitation of the Higgs field modes: tachyonic resonance.

hϕ(t)i ⇠ eαt

• 3. TACHYONIC RESONANCE

hϕ(t)i / t−2/3

TWO REGIMES IN THE HIGGS TIME-EVOLUTION (λ = 0):

late-time dynamics initial tachyonic resonance

(Obtained from lattice simulations)

• 4. HIGGS INSTABILITY

We now introduce the Higgs potential V(φ) = λ(φ)φ4:

¨ hk + k2 a2 + ξR(t) + ∆ + λ(ϕ)hϕ2i

• hk = 0

R(t)<0: Destabilizing effect
 R(t)>0: Stabilizing effect λ(φ)<0: Destabilizing effect
 λ(φ)>0: Stabilizing effect Higgs EOM:
 (h =φa3/2)

• 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.)

• 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)

• 1. N3=2563 points, L≈(0.03 mϕ)-1
• 2. The running of λ(φ) is introduced in the lattice as a local function
• f 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)

0.18 mϕ< p < 40 mϕ

Momenta captured:

We determine ξc ξ>ξc: ξ<ξc:

The Higgs goes to negative-energy vacuum at time ti. The Higgs goes to EW vacuum.

• 4. LATTICE SIMULATIONS

(Figueroa, Rajantie & F .T., arXiv:1612.xxxxx)

• 4. LATTICE SIMULATIONS: RESULTS

mt ≈ 172.12GeV

For ξ ≳ ξc ≈12.1, the Higgs field becomes unstable at a time ti(ξ)

λ(φ)>0: the Higgs is safe! Negative-energy
 vacuum Potential BARRIER

DONEC QUIS NUNC

ξ=9, mt=173.34 GeV

Instability
 time ti depends 
 strongly on ξ

• 4. LATTICE SIMULATIONS: RESULTS

Note: For values ξ<4, mt>173.34 GeV, 
 lattice approach is not valid.

Dependence on initial
 (random) quantum 
 fluctuations

S = − Z a3(t)d4x 1 g2

3

X

a=1

W a

µνW µν a

+ 1 g02 YµνY µν + (DµΦ)(DµΦ) + λ(Φ†Φ)2 !

• Dominant decay products: electroweak gauge bosons:
• The SM Higgs is coupled to other SM particles: gauge bosons and
• fermions. They may affect the post-inflationary Higgs dynamics.
• 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:
• 4. LATTICE SIMULATIONS: RESULTS

• 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:

0.06 . ξ . 4

• Bounds are dependent on inflationary model and running of

λ(φ). Lower-energy models can widen this range.

m2φ2 inflation &
 mt=173.34 GeV