Electroweak Vacuum
Stability andRenormalized Vacuum
Field Fluctuation
Hiroki Matsui
KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, JapanThe Graduate University of Advanced Studies (Sokendai),Tsukuba, Ibaraki 305-0801,
Japan, matshiro@post.kek.jp
Based mainly on: K, Kohri and H, Mastui, Phys.Rev. D94 (2016) no.10, 103509,
arXiv:1607.08133 (to appear in JCAP), arXiv:1704.06884, arXiv:1708.?????

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 2017

1 / 24

Introduction

Electroweak Vacuum Metastability

Electroweak Vacuum Metastability
The recent LHC experiments of the Higgs boson mass
mh = 125.09 ± 0.21(stat) ± 0.11(syst) GeV and the top quark mass
mt = 172.44 ± 0.13(stat) ± 0.47(syst) GeV suggest that theelectroweak vacuum is metastable and finally cause a catastrophic
vacuum decay through quantum tunneling.

104
SM
Instability

6
0.4

180

6 8 10
200

104

10
14
86
12

Metastability

Stability

0.2

107

Instability

12
ility 10stab 8 9ta7Me65
LI =104 GeV

150

100

14

16

19

Stability

50

1011
1012
1013

176

1016

174

0.02

Higgs coupling ΛHMPl L

0.04

0.06

1,2,3 ΣMeta-stability

1019

172

1018
1014

0
0

50

109
1010

170

10 12 14 1618
0.0 8
-0.06 -0.04 -0.02 0.00

108

Instability
178
Top pole mass Mt in GeV

0.6

No EW vacuum
16
18

Top pole mass Mt in GeV

4
12
14
18
16
8610
10
410
14
12
1068

Non-perturbativity

Gauge coupling g2 HMPl L µ g1 HMPl L

0.8

100

150

Higgs pole mass Mh in GeV

200

168
120

17

10

Stability
122

124

126

128

130

132

Higgs pole mass Mh in GeV[D. Buttazzo, G. Degrassic, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, A. Strumia, JHEP 1312 (2013)]

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20172 / 24

Introduction

Electroweak Vacuum Metastability

Electroweak Vacuum Stability
during Large-scale Inflation
In de Sitter space or inflationary Universe, the vacuum field fluctuation
δφ2 enlarges in proportion to the Hubble scale H.
δφ2

1/2

≈ O (H) & ΛI ≈ 1011 GeV =⇒ CATASTROPHE !?

If the inflationary vacuum fluctuation
δφ2 overcomes the barrier of Veff (φ),
it can trigger off a false vacuum decay.
The preheating or reheating case arealso trouble.
[ J. R. Espinosa, G. F. Giudice, and A. Riotto, JCAP 0805, 002(2008), M. Herranen, T. Markkanen, S. Nurmi, and A.
Rajantie, Phys.Rev.Lett.113 211102 (2014), A. Hook, J.
Kearney, B. Shakya, and K. M. Zurek, JHEP 01,061 (2015) ]Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20173 / 24

Introduction

Electroweak Vacuum Metastability

Electroweak Vacuum Stability
around Evaporating Black HoleIn Schwarzschild space, δφ2 proportional to inverse of the black-holemass MBH and approximately approach the Hawking thermal fluctuationnear the black-hole event horizon.
δφ2

1/2

≈ O (1/MBH ) & ΛI ≈ 1011 GeV =⇒ CATASTROPHE !?

Recent discussion about the vacuum
stability around the evaporating(primordial) black hole which can beformulated by large density fluctuationsin the early universe, has been growing.
[ P. Burda, R. Gregory, and I. Moss, Phys.Rev.Lett.115,071303(2015), N. Tetradis, JCAP 1609,036 (2016), D. Gorbunov, D.
Levkov, and A. Panin, arXiv:1704.05399, K. Mukaida and M.
Yamada, arXiv:1706.04523 ]Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20174 / 24

Introduction

Electroweak Vacuum Metastability

Electroweak Vacuum Metastability
in Curved BackgroundWe assume the bare Higgs potential in the curved spacetime.
V (φ) ≡

λ1m2 + ξR φ2 + φ4 + · · ·24

where ξ is the non-minimal curvature coupling.
The Friedmann- Lemaitre-Robertson-Walker (FLRW) metric and the
Schwarzschild metric can be written by



dr 2
2
2
2
2
2
ds 2 = −dt 2 + a (t)
+
r
dθ
+
sin
θdϕ
1 − Kr 2



dr 2
2MBH
dt 2 ++ r 2 dθ2 + sin2 θdϕ2
ds 2 = − 1 −r
1 − 2MBH /r

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 2017

5 / 24

Introduction

Electroweak Vacuum Metastability

The Vacuum Field Fluctuation in
Curved Spacetime
In quantum field theory (QFT), the two-point correlation function δφ2corresponding to the vacuum field fluctuation δφ2 −→ ∞ becomes(quadratically or logarithmically) divergent.
Z
Z∞
dk 2
2
2
3
δφ = d k|δφk (t, x)| =∆ (k) −→ ∞.
k δφ0These UV divergences must be removed by using the regularization or the
renormalization method, and we take out the dynamical field fluctuation
being depend on the curved background, i.e. H, R, Rabcd R abcd .
In QFT, there are various regularization methods like cutoff regularization,
dimensional regularization, ζ-function regularization, lattice regularization,
point-splinting regularization (QFT in curved spacetime), adiabatic
regularization (QFT in curved spacetime) etc.

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20176 / 24

Introduction

Electroweak Vacuum Metastability

The Vacuum Field Fluctuation in
De-Sitter (Inflation) Spacetime
The vacuum field fluctuation δφ2 of the Higgs field can be written by
Z∞
1
2
δφ2 =
dkk 2 Ω−1
1 + 2|βk | + αk β∗k δϕ2k + α∗k βk δϕ∗k 2
k
4π2 a2 (η) 0
where αk (η) and βk (η) are the Bogoliubov coefficients.
δφ2 = δφ2

(s)

(s)

1

δφ2
δφ2

(d)

=
=

+ δφ2
Z∞

4π2 a2 (η)
1
4π2 a2 (η)

Z0∞
0

(d)

dkk 2 Ω−1kdkk 2 Ω−1
k {2nk + 2Rezk }2

where we introduce nk = |βk | which can be interpreted as the particle number
density and zk = αk β∗k δϕ2k .

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20177 / 24

Introduction

Electroweak Vacuum Metastability

The Vacuum Field Fluctuation in
De-Sitter (Inflation) Spacetime
The divergences of δφ2(s)correspond to the Minkowskian divergences#" M 2 (φ)3
M 2 (φ)
1
2 (s)
=
δφ
× ln
− − log 4π − γ −
16π2
µ2

2

M 2 (φ) = m2 + 3λφ2 + (ξ − 1/6) R
These divergences can be canceled by the counterterms δm2 , δξ and δλ.
  2


M (φ)
3
M 2 (φ)(s)
δφ2 ren =
ln
−
16π2
µ2
2
(s)

From the above renormalized vacuum field fluctuations δφ2 ren , we canconstruct the one-loop effective potential on the curved spacetime
  2


1 2 2 1
λ 4 M 2 (φ)
M (φ)
3
2
ln
−
Veff (φ) = m φ + ξRφ + φ +
2
2
4
16π2
µ2
2

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20178 / 24

Introduction

Electroweak Vacuum Metastability

The Vacuum Field Fluctuation in
De-Sitter (Inflation) Spacetime
The adiabatic regularization is the powerful method to obtain the dynamical
vacuum fluctuation.
Z∞
1
2 (s)
2
2 (d)
=
= δφ − δφδφ
dkk 2 Ω−1
k {2nk + 2Rezk }4π2 a2 (η) 0
Z ∞

Z∞
1
2
2
2 −1
=
dk2k |δχk | −dkk Ωk4π2 a2 (η) 00where we must determine exactly δχ (η). In de Sitter spacetime, the dynamical(d)(renormalized) vacuum field fluctuation δφ232H t/4π2 (d)4δφ' 3H /8π2 M 2 (φ)
 2
H /24π2

Hiroki Matsui

can be summarized as(M (φ) = 0)
(M (φ)  H)(M (φ) & H)

(KEK Theory
ElectroweakCenter, IPNS,
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 20179 / 24

Introduction

Electroweak Vacuum Metastability

The Electroweak Vacuum Stability
in Inflationary Universe
In curved spacetime, the effective Higgs potential must include the dynamical(d)vacuum field fluctuation δφ2
1 2 2 1
λλ(d) 2m φ + ξRφ2 + φ4 +
δφ2
φ
2
2
4
2


9
X
Mi2 (φ)
ni
4
(φ)
M
log
− Ci
+
i
64π2
µ2

Veff (φ) =i=1Mi2 (φ) = κi φ2 + κi

δφ2

(d)

+ κi0 + θi R

We can take the renormalization scale µ2 ≈ φ2 + R + δφ2µ ≈ (R + δφ2(d) 1/2)(d).
λ(µ)φ4& ΛI ≈ 1011 GeV =⇒<04[ M. Herranen, T. Markkanen, S. Nurmi, and A. Rajantie, Phys.Rev.Lett. 113,211102 (2014), K, Kohri and H, Mastui,
arXiv:1607.08133 (to appear in JCAP), arXiv:1704.06884 ]Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201710 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe
The inflationary vacuum fluctuation of the Higgs field δφ2 destabilize theeffective Higgs potential Veff (φ).

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201711 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe(d) 1/2
For µ ≈ R + δφ2& ΛI , the destabilization of Veff (φ) can be
determined by
1
ξ(µ)Rφ2 ≷
2

λ(µ)
(d) 2
δφ2
φ =⇒ Veff (φ) ≷ 0
2

In de-Sitter spacetime, the destabilization condition of Veff (φ) can be given by(R + δφ2(d) 1/2)& ΛI ,

ξ(µ)R < |λ(µ)| δφ2(d)=⇒ CATASTROPHE

Thus, we can obtain the constraint of the non-minimal coupling ξ(µ)

Hinf & ΛI and ξ(µ) . O 10−3 =⇒ Destabilized

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201712 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe
From the vacuum Higgs field fluctuations δφ2 , the catastrophic Higgs Anti-deSitter (AdS) domains or bubbles can be formed.

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201713 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe


(d)
(d)
as follows:with δφ2
The probability function P φ, δφ2




2
1
φ
(d)

=q
P φ, δφ2
exp −
2 (d)
2 (d)
2 δφ
2π δφ


Z φmax


φmax
(d)

P (φ < φmax ) ≡P φ, δφ2
dφ = erf  q
2 (d)
−φmax
2 δφ
The probability to generate Higgs AdS domains or bubbles can be given by

 q


2 (d)
2
2
δφ
φ
φ
max
max
'

exp −P (φ > φmax ) = 1 − erf  q
(d)
πφmax
2 (d)
2 δφ2
2 δφ
[ A. D. Linde, Nucl.Phys.B 372,421 (1992), J. R. Espinosa, G. F. Giudice, E. Morgante, A. Riotto, L. Senatore, A. Strumia, and
N. Tetradis, JHEP 09,174 (2015) ]Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201714 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe
At the end of the inflation, there are huge no correlation patches
being equivalent to e 3Nhor

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201715 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe
The vacuum decay probability of the Universe can be expressed as
(d)

e

3Nhor

P (φ > φmax ) < 1 =⇒

δφ2
φ2max

<

1
6Nhor

where we can take the e-folding number Nhor ' NCMB ' 60. During inflation,
we can obtain the restriction of the non-minimal coupling ξ(µ) & O 10−2 notto generate the unwanted AdS domains or bubbles.
Hinf & ΛI and ξ(µ) . O 10−3 =⇒ DestabilizedHinf & ΛI and O 10−3 . ξ(µ) . O 10−2 =⇒ AdS domains
Hinf . ΛI or O 10−2 . ξ(µ) =⇒ Stable

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201716 / 24

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability in Inflationary Universe

The Electroweak Vacuum Stability
in Inflationary Universe
So far we consider the vacuum field fluctuation of the Higgs field, but the vacuum field
fluctuation of theW and Z bosons and top quark enlarge in proportional to the Hubble scale
Hinf and contribute to the effective Higgs potential as the effective masses
λλ1
1 2 2 1(d) 2(d)m φ + ξRφ2 + φ4 +
δφ2
φ + g 2 δW 2
2
2
4
2
4
#
"
9
X

Mi2 (φ)
1
ni
1 2
(d)
(d)
4
(φ)
g + g 02 δZ 2
+ yt2 δt 2
+
M
log
−
C
+
i .
4
2
64π2 iµ2
i=1

Veff (φ) =If the vacuum field fluctuations of W and Z bosons and the top quark are larger than the Higgs
one, the effective Higgs potential can be stabilized
δφ2

(d)

. δW 2

(d)

,

δZ 2

(d)

ξ(µ) & ξW (µ), ξZ (µ), ξt (µ)

,

δt 2

(d)

=⇒ Stabilized

non − catastrophe

Since these fluctuations of W and Z bosons and top quark depend on their mass as well as
Higgs field, they can be suppressed by their non-minimal coupling ξW ,Z ,t (µ).

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
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KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201717 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Vacuum Field Fluctuation in
Schwarzschild Spacetime



2MBH
dr 2
ds 2 = − 1 −dt 2 ++ r 2 dθ2 + sin2 θdϕ2
r
1 − 2MBH /r
r=0

i+

r

J+

=

i+

III

B

2M

r

B
H

i0

=

H

II

Hiroki Matsui

r

i−

i0

I

H

J−

J+

2M
H+

=

2M

r

B

IV

r=0

=

H−

2M

J−

B

H

i−

(KEK Theory
ElectroweakCenter, IPNS,
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201718 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Vacuum Field Fluctuation in
Schwarzschild Spacetime
The renormalized vacuum field fluctuation in the Boulware vacuum |0B i (vacuum polarization
around a static star) and the Unruh vacuum |0U i (evaporating black hole) via thepoint-splitting regularization can be give by
"∞
Z∞
h
2
dω X
1
(2l + 1) Rlin (r ; ω)
0B |δφ2 (x)|0B ren =2
16π 0 ω l=0
#
i
2
MBH
4ω2
2
out
−
+ Rl (r ; ω)−2
4
1 − 2MBH /r
48π r (1 − 2MBH /r )
"∞
Z∞
h
2
1
dω X
(2l + 1) Rlin (r ; ω)
0U |δφ2 (x)|0U ren =2
16π 0 ω l=0# πω 
i
2
MBH
4ω2
2
out
Rl (r ; ω)−
+ coth−κ1 − 2MBH /r
48π2 r 4 (1 − 2MBH /r )[P. Candelas, Phys. Rev. D21, 2185 (1980)]
where we introduceκ = (4MBH )−1 which is the surface gravity of the black hole and the factor
of coth πω
originates from the thermal features of the outgoing modes.
κ

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201719 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Vacuum Field Fluctuation in
Evaporating Black HoleThe renormalized vacuum fluctuation δφ2 ren around the evaporating black hole can be
written as follows:
2(r → 2MBH )O TH
2δφ ren '(r → ∞)
0where the Hawking temperature TH = 1/8πMBH . The vacuum field fluctuation of the the
Higgs, W and Z bosons and the top quark approximately approach the Hawking thermal
fluctuations near the event horizon
2
δφ2 ren ' δW 2 ren ' δZ 2 ren ' δt 2 ren ' O TH
Thus, the maximal field value φmax being consistent with the hill of the effective Higgs
potential can be given by



1 2
1 2
φ2max ' O 102 ·g (µ) δW 2 ren +g (µ) + g 02 (µ) δZ 2 ren
4
4

1
1
+ yt2 (µ) δt 2 ren + λ(µ) δφ2 ren
2
2

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201720 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Electroweak Vacuum Stability
in Evaporating Black HoleLet us consider a huge number of the evaporating or evaporated
black holes NEBH where the large vacuum fluctuation exists onlyaround the event horizon.

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201721 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Electroweak Vacuum Stability
in Evaporating Black HoleBy applying the probability function P (φ > φmax )q!
NEBH 2 δφ2 ren
φ2max
NEBH · P (φ > φmax ) '
exp −
πφmax
2 δφ2 ren
≈ NEBH · e −O(100) . 1
The constraint of the number of the evaporating black holes can be given by mPBH 3/2
ρPBH
43
−21
NEBH . O 10 , β ≡. O 10ρtot formation
109 g
where ρPBH and ρtot are the energy density of the PBHs and the total energy
density of the Universe. This bound can be stronger than the known one for
mPBH . 109 g [ B. J. Carr, K, Kohri, Y. Sendouda, and J. Yokoyama, Phys. Rev. D81, 104019 (2010) ]Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
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Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201722 / 24

The Electroweak Vacuum Stability in Schwarzschild Spacetime

The Vacuum Field Fluctuation in Schwarzschild Spacetime

The Electroweak Vacuum Stability
in Evaporating Black HoleHowever, at the final stage of the evaporation of the black hole, the black-holemass MBH becomes extremely small and the Hawking temperature reaches the
Planck scale where TH → O (MPl ).
λeff (φ) 4 δλbsm 4 λ6 φ6
λ8 φ8
φ +
φ +
+
2
4 + ···
4
4
6 MPl
8 MPl


1
λ6 TH4
λ8 TH6=⇒ Veff (φ) =λeff TH2 + κ2 TH2 +
+
+
·
·
·
φ2
2
4
2
MPl
MPl


1
λ6 TH2
λ8 TH4+λeff + δλbsm +φ4 + · · ·2 + M4 + · · ·
4
MPl
Pl

Veff (φ) =where δλbsm express the running corrections from the BSM, λ6 and λ8
dimensionless coupling constants. Therefore, even a single evaporating black holecan be catastrophic for the vacuum stability through TH → O (MPl ) although
this possibility strongly depends on the Planck scale physics.

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201723 / 24

Conclusion and Summary

Conclusion and Outlook
In the curved background, the vacuum field fluctuations δφ2 grow inproportional to the curvature scale.
The large vacuum fluctuation of the Higgs field can destabilize the effective
Higgs potential, or generate the Higgs AdS domains or bubbles.These
unwanted phenomena could cause a catastrophic collapse of the Universe.
In the de-Sitter spacetime, the non-minimal coupling ξ(µ) sway thecosmological destiny of the Higgs vacuum.
In the Schwarzschild black-hole spacetime, the vacuum field fluctuation
δφ2 approaches approximately the Hawking thermal fluctuation near theblack-hole horizon. By incorporating the back-reaction effects of δφ2 andanalyzing the stability of the vacuum, we show that one evaporating blackhole does not cause serious problems in standard model vacuum and obtainan upper bound on the evaporating PBH abundance not to induce any
catastrophe.

Hiroki Matsui(KEK Theory
ElectroweakCenter, IPNS,
Vacuum
KEK,Stability
Tsukuba, Ibaraki 305-0801, Japan)
August 2, 201724 / 24