Fate of Electroweak Vacuum during Preheatingppp.ws/PPP2016/slides/ema.pdf · Parametric resonance h...

Fate of Electroweak Vacuum during Preheating

Yohei Ema

Based on arXiv:1602.00483In collaboration with K. Mukaida and K. Nakayama

Progress in Particle Physics 2016.09.06

University of Tokyo

Introduction

Metastability

102 104 106 108 1010 1012 1014 1016 1018 1020

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

RGE scale m in GeV

Hig

gsqu

artic

coup

lingl

3s bands inMt = 173.3 ± 0.8 GeV HgrayLa3HMZL = 0.1184 ± 0.0007HredLMh = 125.1 ± 0.2 GeV HblueL

Mt = 171.1 GeV

asHMZL = 0.1163

asHMZL = 0.1205

Mt = 175.6 GeV

Electroweak (EW) vacuum may be metastable??[Buttazzo+ 13]

µ [GeV]

�

becomes negative for the center value of .� ⇠ 1010 GeV Mtop

Cosmology must be compatible with it.

hmax

⇠ 1010 GeV

hmax

Ve↵(h)

h

Metastability

107 108109

1010

1011

10121013

1014

1016

120 122 124 126 128 130 132168

170

172

174

176

178

180

Higgs pole mass Mh in GeVToppolemassM

tinGeV

1017

1018

1019

1,2,3 s

Instability

Stability

Meta-stability

102 104 106 108 1010 1012 1014 1016 1018 1020

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

RGE scale m in GeV

Hig

gsqu

artic

coup

lingl

3s bands inMt = 173.3 ± 0.8 GeV HgrayLa3HMZL = 0.1184 ± 0.0007HredLMh = 125.1 ± 0.2 GeV HblueL

Mt = 171.1 GeV

asHMZL = 0.1163

asHMZL = 0.1205

Mt = 175.6 GeV

Electroweak (EW) vacuum may be metastable??

Instability

Meta-stability

Stability

⌧EW > ⌧U

[Buttazzo+ 13]

µ [GeV]

� Mtop

Mhiggs

becomes negative for the center value of .� ⇠ 1010 GeV Mtop

Cosmology must be compatible with it.

hmax

⇠ 1010 GeV

Metastability vs. inflation• Light fields acquire fluctuations during inflation.

ph�h2i ' Hinf

2⇡

pN

[Starobinsky, Yokoyama 94]

(random walk with for each one e-folding)Hinf/2⇡

• Inflation scale must be low for EW vacuum to be stable.(if BSM does not affect much on the higgs potential)

[e.g. Hook+ 14; Espinosa+ 15]

(Potential and reheating dynamics are neglected.)

Hinf

. 4⇥ 108 GeV

✓60

N

◆✓hmax

1010 GeV

◆Roughly, or, P (|h| > h

max

) ' e�x

2

/p⇡x < e�3N , x =

p2⇡h

max

/pNH

inf

Stabilization during inflation

Induce effective mass for higgs during inflation.

• The following couplings are often considered:

[e.g. Espinosa+ 08; Lebedev+ 13]

• One way to avoid this constraint:

(BSM or low-scale inflation are of course other choices…)

m2e↵;h = c2�2, 12⇠H2

inf during inflation

c & O(Hinf/�inf), ⇠ & O(0.1)Suppress fluctuations if

�Lint =

(12c

2�2h2 · · · quartic12⇠Rh2 · · · curvature

Stabilization during inflation

Induce effective mass for higgs during inflation.

• The following couplings are often considered:

[e.g. Espinosa+ 08; Lebedev+ 13]

• One way to avoid this constraint:

(BSM or low-scale inflation are of course other choices…)

m2e↵;h = c2�2, 12⇠H2

inf during inflation

c & O(Hinf/�inf), ⇠ & O(0.1)Suppress fluctuations if

�Lint =

(12c

2�2h2 · · · quartic12⇠Rh2 · · · curvature

How about after inflation??

Motivation1. EW vacuum metastability (BSM negligible)

vs. high scale inflation



2. Stabilization mechanism during inflation:

�Lint =1

2c2�2h2,

1

2⇠Rh2.




�Lint =1

2c2�2h2,

1

2⇠Rh2.

3. How about after inflation?? [our work]- Mass term oscillates resonance.

- Even tachyonic during some period (curvature).

Outline

1.Introduction

2.Resonant particle production

3.Preheating dynamics of Higgs

4.Summary

Dynamics of inflaton

1. Slow-roll during inflation.

2. Oscillate after inflation.

3. Finally decay, and reheating completes.

= beginning of hot big bang

if exponential particle production

preheating epoch

= accelerated expansion

Parametric resonance

hh h

h

inflaton

q ⌘ c2�2

4m2�

� 1If , adiabaticity only at around the origin.

Quartic coupling: �Lint =1

2c2�2h2

+ Bose enhancement [many times of oscillation].

Parametric resonance occurs.

µqtc ' O(0.1)nh ⇠ p3⇤ e2µqtcm�t,

where the typical momentum is p⇤ ⌘p

cm�� ⇠ m�q1/4.

[Kofman+ 94; 97]⇠m2e↵;h

m2�

Tachyonic resonanceCurvature coupling:�Lint =

1

2⇠Rh2

R =1

M2P

h4V (�)� �2

ifrom Friedmann equations.

Higgs is tachyonic for some region at around the origin.

inflaton

h h h

µcrv ' 2

* particle production is dominated by the first a few oscillations.

[Bassett+ 97; Tujikawa+ 99; Dufaux+ 06]

where the typical momentum is p(tac)⇤ ' m�q1/4, q ⌘ 3

4

✓⇠ � 1

4

◆�2

M2P

.

Tachyonic resonance occurs for q � 1 :

nh ⇠ p(tac)3

⇤ eµcrvp⇠

�iniMP ,

Outline

1.Introduction



4.Summary

Set up

� hc2�2h2, ⇠Rh2

Main idea: resonant Higgs production induces tachyonic mass.

• Consider inflaton oscillation epoch

• Inflaton potential: V (�) =1

2m2

��2

�m2self;h ' �3|�|hh2i

It may force Higgs to roll down to the true vacuum.

From now,

• Analytically derive the condition for EW vacuum decay. • Verify it by numerical simulations.

Higgs-inflaton coupling :Lint = �1

2c2�2h2 nh ⇠ 1

32⇡2

r⇡

2µqtcm�te2µqtcm�tp3⇤ µqtc ' O(0.1)with

• Cosmic expansion should kill resonance before that time.

at the end of the resonance, or��m2

self;h(t)��⇠0

. p2⇤

c . 10�4

0.1

µqtc

� h m�

1013 GeV

i

• Higgs tachyonic mass grows:

That time interval �t : c2�2(m��t)2 ⇠��m2

self;h

��⇠0

�m2self;h(t) ' 3�hh2(t)i ⇠ �3|�| nh(t)

!k⇤;h(t)

• dominates at around the origin.�m2self;h c2�2

EW vacuum decays.• If the growth rate exceeds unity, |�me↵;h|�t ⇠ |�m2self;h|/p2⇤ > 1,

Quartic: lattice simulationWe have verified our condition by classical lattice simulations.

(added 6th term for large field for convergence )

c = 1⇥ 10�4 c = 2⇥ 10�4

�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 10/m�, dt = 10�3/m�

10-8

10-6

10-4

10-2

100

102

0 20 40 60 80 100

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-8

10-6

10-4

10-2

100

102

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>


(added 6th term for large field for convergence ) EW vacuum decays!!

c = 1⇥ 10�4 c = 2⇥ 10�4


10-8

10-6

10-4

10-2

100

102

0 20 40 60 80 100

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-8

10-6

10-4

10-2

100

102

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

Higgs-curvature coupling :Lint = �1

2⇠Rh2 nh ⇠ 1

16⇡2

r⇡

2eneffµcrv

p⇠�ini/Mplp(tac)

3

⇤ with µcrv ' 2, ne↵ & 1

⇠ . 10⇥

2

ne↵µcrv

�2 "p2Mpl

�ini

#2

We require at the end of the resonance, or|�m2self;h|⇠R⇠0 . ⇠R

• : effective number of times of oscillationne↵

ne↵ ' 1 for �ini =p2MP , ne↵ ' 1.5-2 for �ini =

p0.2MP

• Bound depends on the initial inflaton amplitude.

c . 10�4

0.1

µqtc

� h m�

1013 GeV

icf. : independent of the initial inflaton amplitude

We have verified our condition by classical lattice simulations.


⇠ = 10 ⇠ = 20

Curvature: lattice simulation

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>


We have verified our condition by classical lattice simulations.


⇠ = 10 ⇠ = 20


10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

EW vacuum decays!!�ini =

p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 20/m�, dt = 10�3/m�

Outline

1.Introduction



4.Summary




�Lint =1

2c2�2h2,

1

2⇠Rh2.

3. How about during preheating?? [our work]- Mass term oscillates resonance.

- Even tachyonic during some period (curvature).

Summary• Studied EW vacuum stability during the preheating epoch with:

Lint = �1

2c2�2h2, � 1

2⇠Rh2

• Obtained upper bounds on the couplings:

Unstable during inflation

Dependent on reheating

Unstable during preheating

�1

2c2�2h2 :

�1

2⇠Rh2 :

c�ini

⇠⇠ 0.2

⇠ Hinf ⇠ 10Hinf

[high scale inflation]

⇠ 10M2pl/�

2ini

Back up

Adiabaticity• Mode equation of Higgs:

• Number density of Higgs:

nk;h(t) =1

2!k;h(t)

��hk;h

��2+ !2

k;h |hk;h|2�� 1

2

: [violation of adiabaticity] or,: [tachyonic] is important for particle production.

Its time evolution is nk;h = O(|!k;h/!2k;h|)!k;hnk;h

hk +⇥!2k;h(t) +�2(t)

⇤hk = 0,

�2 = �9H2/4� 3H/2, !2k;h(t) = m2

e↵;h(t) + k2/a2 + �m2self;h.where

� = �(t) sinm�t, �(t) ' �i

a(t)3/2m2

e↵;h(t) = c2�2, or

⇠

M2P

h4V � ˙�2

i,

R =1

M2P

h4V � �2

i

(|!k;h/!2

k;h| & 1

!2k;h < 0

Mathieu equationAk

0 5 10

5

10

-2q/3

Ak= 2q/3

Ak= 2q

q

zeroth

first

second

third

Ak=

[Tsujikawa+ 99]

• Quartic coupling

• Curvature coupling

Ak =k2

a2m2�

+ 2q, q =c2�2

4m2�

Ak =k2

a2m2�

+⇠�2

2M2pl

,

q =3

4

✓⇠ � 1

4

◆�2

M2pl

expansionexpansion

[quartic]

[curvature]

d2hk

dz2+ (Ak � 2q cos z)hk = 0

z = m�t+ const.


Classical lattice simulation:

(1) Divide space coordinates into meshes.

(2) Introduce gaussian fluctuations (quantum fluctuation).

(3) Solve the discretized classical equations of motion.

[Khlebnikov, Tkachev 96]

�i,j �i+1,j · · ·

· · ·

�i,j�1

[Polarski, Starobinsky 96]

* Classical approximation is valid in the large occupation number limit.

Quartic: spectrumThe number density of higgs for each momentum:


c = 1⇥ 10�4 c = 2⇥ 10�4


10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;

h +

1/2

Comoving momentum k/mφ

mφt = 05

1015202530

10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;

h +

1/2


mφt = 0102030405060708090

Curvature: spectrumThe number density of higgs for each momentum:



⇠ = 10 ⇠ = 20

10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;

h +

1/2


mφt/π = 012

10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;

h +

1/2


mφt/π = 0123456789

Coupling with SM particles

� hc2�2h2, ⇠Rh2So far…


� h

�

c2�2h2, ⇠Rh2

� : top quarks, EW gauge bosons

In reality…


� h

�

top Yukawa gauge

c2�2h2, ⇠Rh2


In reality…


� h

�

top Yukawa gauge

c2�2h2, ⇠Rh2

inflaton decay (reheating)

In reality…



� h

�

top Yukawa gauge

c2�2h2, ⇠Rh2In reality…



� h

�

top Yukawa gauge

c2�2h2, ⇠Rh2


• Higgs decays into top quarks via top Yukawa coupling.

• Higgs annihilates into gauge boson pairs.

Reduce Higgs resonance efficiency (instant preheating)

Induce effective potential for Higgs

We will see they are less significant during preheating.

Instant preheatingHiggs is massive for the large field value region of inflaton.

Higgs can decay via Yukawa at that region. [Felder+ 98]

There are two cases:

(1) [Resonant production rate] [Decay rate] ⌧

�(2) [Resonant production rate] [Decay rate]

• Early stage of quartic coupling

• Late stage of quartic coupling

• Curvature coupling

m2H;h = c2�2, ⇠R

* For curvature coupling, growth/decay rate , and hence (2) always holds./p

⇠

Instant preheating

In this case, the decay just slightly reduces the production rate.

(1) [Resonant production rate] [Decay rate] ⌧

(2) [Resonant production rate] [Decay rate] �

• Higgs decay rate: �h!tt ⇠ ↵tc�

Higgs completely decays for ↵tc� � m�/⇡.

• Inflaton decay rate at that epoch: �inst ⇠c2

4⇡4p3↵t/2

m�

For c & O(0.1), inflaton completely decays via this process.

Otherwise, (1) is eventually violated and the situation reduces to (2).

↵t = y2t /4⇡

AnnihilationConsider the following simplified Lagrangian:

�Lint =1

2g2h�h

2�2 +1

4g2��

4,

with � : light scalar field (mimicking gauge bosons).

10-10

10-8

10-6

10-4

10-2

100

102

104

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

10-10

10-8

10-6

10-4

10-2

100

102

104

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

gh� = g�� = 0.5

Quartic Curvature


�Lint =1

2g2h�h

2�2 +1

4g2��

4,


10-10

10-8

10-6

10-4

10-2

100

102

104

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

10-10

10-8

10-6

10-4

10-2

100

102

104

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

gh� = g�� = 0.5

Quartic Curvature

10-8

10-6

10-4

10-2

100

102

0 10 20 30 40 50 60N

orm

aliz

ed b

y th

e in

itial

infla

ton

ampl

itude

Φin

i

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

Quartic


�Lint =1

2g2h�h

2�2 +1

4g2��

4,


10-10

10-8

10-6

10-4

10-2

100

102

104

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

10-10

10-8

10-6

10-4

10-2

100

102

104

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

gh� = g�� = 0.5

Quartic Curvature

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

Curvature

Annihilation

10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;

h +

1/2


mφt = 05

1015202530

10-1

100

101

102

103

104

1 10

Com

ovin

g nu

mbe

r den

sity

nk;χ

+ 1/

2


mφt = 05

1015202530

Higgs is resonantly produced, while is not efficiently produced.�

Spectrum of Higgs Spectrum of �

(Effective mass might suppress the production.) m2e↵;� ' g2h�hh2i

Cosmic expansion• Assume EW vacuum survives the preheating epoch.

• For illustration, neglect top Yukawa and gauge couplings.

Just after preheating, Higgs is stabilized by m2H;h = c2�2, ⇠R.

However, while m2H;h / a�3, �m2

self;h / hh2i / a�2.

�m2self;h eventually dominates over m2

H;h.

For before ,hh2i < h2

max

m2H;h < |�m2

self;h|

m� = 1.5⇥ 1013 GeV, hmax

= 1010 GeV

c . 3⇥ 10�5

0.1

µqtc

�, ⇠ . 0.5

2

ne↵µcrv

�2 "p2MP

�ini

#2

.

* It means most parameters for stabilization during inflation cause catastrophe.







H;h.


max

m2H;h < |�m2

self;h|

m� = 1.5⇥ 1013 GeV, hmax

= 1010 GeV

c . 3⇥ 10�5

0.1

µqtc

�, ⇠ . 0.5

2

ne↵µcrv

�2 "p2MP

�ini

#2

.Dynamics after preheating

is also non-trivial.

* It means most parameters for stabilization during inflation cause catastrophe.







H;h.


max

m2H;h < |�m2

self;h|

m� = 1.5⇥ 1013 GeV, hmax

= 1010 GeV

c . 3⇥ 10�5

0.1

µqtc

�, ⇠ . 0.5

2

ne↵µcrv

�2 "p2MP

�ini

#2

.Dynamics after preheating

is also non-trivial.

Thermalization is crucial

for EW vacuum stability.* It means most parameters for stabilization during inflation cause catastrophe.

Complete reheating

� h

�

top Yukawa gauge

c2�2h2, ⇠Rh2

inflaton decay (reheating)


Inflaton must decay to complete the reheating process.

Characterized by the reheating temperature .TRH

Complete reheating

10-10

10-8

10-6

10-4

10-2

100

102

104

0 20 40 60 80 100

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>a3<χ2>

prelim

inary

• For 105 GeV . TRH . 1010 GeV,

radiation from inflaton decay may stabilize Higgs after preheating.

Thermal bounce is valid after that.

TRH � 1010 GeV,• For

It might also induce EW vacuum decay??

�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283,

L = 10/m�, dt = 10�3/m�, g�� = 5⇥ 10�4

parametric resonance might occur

in other sectors during preheating.

Quartic: lattice simulation


c = 1⇥ 10�4 c = 2⇥ 10�4

We have changed the initial inflaton amplitude as .�ini =p0.2Mpl

10-8

10-6

10-4

10-2

100

102

0 20 40 60 80 100

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-8

10-6

10-4

10-2

100

102

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

�ini =p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�

Quartic: lattice simulation


c = 1⇥ 10�4 c = 2⇥ 10�4


10-8

10-6

10-4

10-2

100

102

0 20 40 60 80 100

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-8

10-6

10-4

10-2

100

102

0 10 20 30 40 50 60

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

EW vacuum decays!!�ini =

p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�




⇠ = 30


⇠ = 20

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>




⇠ = 30


⇠ = 20

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10 15 20 25 30

Nor

mal

ized

by

the

initi

al in

flato

n am

plitu

de Φ

ini

mφt

a3<φ>2

a3(<φ2> - <φ>2)a3<h2>

EW vacuum decays!!

Fate of Electroweak Vacuum during Preheatingppp.ws/PPP2016/slides/ema.pdf · Parametric resonance h...

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