Vacuum decay rate in the standard model and beyond Yutaro Shoji - - PowerPoint PPT Presentation

vacuum decay rate in the standard model and beyond
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Vacuum decay rate in the standard model and beyond Yutaro Shoji - - PowerPoint PPT Presentation

The 24th regular meeting of the NHWG, Dec 21-22 2018 Vacuum decay rate in the standard model and beyond Yutaro Shoji (KMI, Nagoya U.) Phys. Lett. B771(2017)281; M. Endo, T. Moroi, M. M. Nojiri, YS JHEP11(2017)074; M. Endo, T. Moroi, M. M.


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SLIDE 1

Vacuum decay rate in the standard model and beyond

Yutaro Shoji (KMI, Nagoya U.)

The 24th regular meeting of the NHWG, Dec 21-22 2018 https://github.com/YShoji-HEP/ELVAS

  • Phys. Lett. B771(2017)281; M. Endo, T. Moroi, M. M. Nojiri, YS

JHEP11(2017)074; M. Endo, T. Moroi, M. M. Nojiri, YS

  • Phys. Rev. Lett. 119 (2017) no.21, 211801; S. Chigusa, T. Moroi, YS
  • Phys. Rev. D97 (2018) no.11, 116012; S. Chigusa, T. Moroi, YS

Work in progress with S. Oda, D. Takahashi

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SLIDE 2

Introduction

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SLIDE 3

Vacuum decay in the standard model

Vacuum Energy Field value

V (Φ) = λ(Φ†Φ)2 − M 2(Φ†Φ)

102 GeV

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  • [ / ]

λ

Standard Model Planck EW

slide-4
SLIDE 4

Vacuum decay in the standard model

Vacuum Energy Field value

V (Φ) = λ(Φ†Φ)2 − M 2(Φ†Φ)

102 GeV

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1010 GeV

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1017 GeV

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Standard Model EW vacuum of the SM is meta-stable

Decay

slide-5
SLIDE 5

Vacuum decay

Vacuum Energy Field value Space

slide-6
SLIDE 6

Vacuum decay

Vacuum Energy Field value Space Bubble nucleation Quantum tunneling

slide-7
SLIDE 7

Vacuum decay

Bubble nucleation Bubble expansion

Vacuum Energy Field value Space

slide-8
SLIDE 8

Bubble nucleation rate

γ = Ae−B

B = SE(¯ φ)

Bounce solution (Fubini-Lipatov instanton)

B = 8π2 3|λ|

¯ φ(r) = s 8 |λ| R R2 + r2

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R :

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Size of the instanton

Independent of R (classical scale invariance)

A rate to have a bubble in unit volume for a negative λ

slide-9
SLIDE 9

Bubble nucleation rate

γ = Ae−B

A rate to have a bubble in unit volume

A ∼ ✓det S00

E|bounce

det S00

E|false

◆1/2

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+ + + + · · · + +

Higgs Top NG boson Gauge Ghost Mixed [C. G. Callan, S. R. Coleman, ’77]

Quantum correction to B

exp(

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)

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slide-10
SLIDE 10

Typical scale?

γ = Ae−B

A rate to have a bubble in unit volume

A ∼ M 4

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sha1_base64="0Qqndsvum3BJ+g+a06gw7AVwt6w=">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</latexit>

A ∼ ✓det S00

E|bounce

det S00

E|false

◆1/2

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Dimensional analysis Typical scale

What is the typical scale?

SM is classically scale invariant

slide-11
SLIDE 11

Renormalization scale?

γ = Ae−B

B = SE(¯ φ)

B = 8π2 3|λ|

A rate to have a bubble in unit volume

What is the renormalization scale for λ?

  • [ / ]

λ

Standard Model

slide-12
SLIDE 12

The pre-factor is important!

γ = Ae−B

B = SE(¯ φ)

B = 8π2 3|λ|

A rate to have a bubble in unit volume

A ∼ ✓det S00

E|bounce

det S00

E|false

◆1/2

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The renormalization scale uncertainty is canceled @1-loop

If we calculate A, we do not have such problems

We do not need “typical scale”

slide-13
SLIDE 13

Standard Model @ one-loop

Instability 107 109 1010 1012 115 120 125 130 135 165 170 175 180 Higgs mass Mh in GeV Pole top mass Mt in GeV 1,2,3 s Instability Stability Meta-stability

  • G. Ishidori, G. Ridolfi, A. Strumia; ‘01

However, There were no established ways to subtract dilatational and gauge zero modes They use an approx. for the gauge zero mode subtraction, which is gauge dependent

slide-14
SLIDE 14

However, There were no established ways to subtract dilatational and gauge zero modes They use an approx. for the gauge zero mode subtraction, which is gauge dependent

Gauge zero mode

JHEP 1711 (2017) 074, M. Endo, T. Moroi, M. M. Nojiri, YS We have found the correct treatment of the gauge zero mode

Flat direction Bounce

det S00

E|bounce = 0

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Higgs SU(2) [A. Kusenko, K. M. Lee, E. J. Weinberg; ’97]

slide-15
SLIDE 15

Contents

  • Introduction
  • Gauge invariance and analytic formulas
  • Models beyond the standard model
  • Summary
slide-16
SLIDE 16

Gauge invariance and analytic formulas

slide-17
SLIDE 17

Gauge fixing

LGF = 1 ξ (∂µAµ)2

NG boson

LGF = (∂µAµ − g ¯ φχ)2

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Numerical Evaluation Treatment of gauge zero modes Semi-analytical Evaluation

  • Phys. Lett. B771(2017)281

JHEP11(2017)074 Bounce solution

slide-18
SLIDE 18

Gauge fixing

LGF = 1 ξ (∂µAµ)2

NG boson

LGF = (∂µAµ − g ¯ φχ)2

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Numerical Evaluation Treatment of gauge zero modes Semi-analytical Evaluation

  • Phys. Lett. B771(2017)281

JHEP11(2017)074

Standard Model

  • A. Andreassen, W. Frost, M. D. Schwartz; ‘17
  • S. Chigusa, T. Moroi, YS; ‘ 17

Bounce solution

slide-19
SLIDE 19

Gauge contribution

M(S,L,ϕ)

J

≡ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −∆J + 3 r2 + g2 ¯ φ2 −2L r2 g ¯ φ′ − g ¯ φ∂r −2L r2 −∆J − 1 r2 + g2 ¯ φ2 −L r g ¯ φ 2g ¯ φ′ + g ¯ φ∂r + 3 rg ¯ φ −L r g ¯ φ −∆J + (∆0 ¯ φ) ¯ φ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ +

  • 1 − 1

ξ

⎜ ⎜ ⎜ ⎜ ⎝ ∂2

r + 3

r∂r − 3 r2 −L 1 r∂r − 1 r2

  • L

1 r∂r + 3 r2

  • −L2

r2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ,

A(Aµ,ϕ) =

Y

J=0

det M(S,L,ϕ)

J

det c M(S,L,ϕ)

J

!−(2J+1)2/2 det M(T )

J

det c M(T )

J

!−(2J+1)2

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M(T )

J

≡ −∆J + g2 ¯ φ2

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∆J = ∂2

r + 3

r ∂r − L2 r2

L = p 4J(J + 1) hatted operator: ¯

φ → 0

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Transverse modes are gauge invariant How about the others?

slide-20
SLIDE 20

Theorem

det M det c M = lim

r→∞

det[ψ1(r) · · · ψn(r)] det[ ˆ ψ1(r) · · · ˆ ψn(r)] ! lim

r→0

det[ψ1(r) · · · ψn(r)] det[ ˆ ψ1(r) · · · ˆ ψn(r)] !−1

M, c M

: (n x n) radial fluctuation operators

Mψi = 0, c M ˆ ψi = 0 : independent solutions (regular at r=0)

[J. H. van Vleck, ’28; R. H. Cameron and W. T. Martin, ’45; …] We give a proof for our case in JHEP11(2017)074

slide-21
SLIDE 21

Decomposition of simultaneous differential equations

⎞ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ ∂rχ L r χ g ¯ φχ ⎞ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 rg2 ¯ φ2η 1 Lr2g2 ¯ φ2∂r(r2η) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2 ¯ φ′ g2 ¯ φ3ζ 1 g ¯ φζ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,

M(S,L,ϕ)

J

Ψi = 0

∆Jχ = 2¯ φ0 rg2 ¯ φ3 η + 2 r3 ∂r ✓ r3 ¯ φ0 g2 ¯ φ3 ζ ◆ − ξζ ✓ ∆J − g2 ¯ φ2 − 2 ¯ φ0 ¯ φ 1 r2 ∂rr2 ◆ η = −2L2 ¯ φ0 r ¯ φ ζ ∆Jζ = 0

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Ψi

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[JHEP11(2017)074; M. Endo, T. Moroi, YS]

slide-22
SLIDE 22

Solutions

∆Jχ = 2¯ φ0 rg2 ¯ φ3 η + 2 r3 ∂r ✓ r3 ¯ φ0 g2 ¯ φ3 ζ ◆ − ξζ ✓ ∆J − g2 ¯ φ2 − 2 ¯ φ0 ¯ φ 1 r2 ∂rr2 ◆ η = −2L2 ¯ φ0 r ¯ φ ζ ∆Jζ = 0

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ζ = 0, η = 0, χ = r2J

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ζ = r2J

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(A) (B) (C) => Solve (A) => Solve (A) and (B)

  • Sol. 1
  • Sol. 2
  • Sol. 3

Physical fluctuations (gauge invariant)

✓ ∆J − g2 ¯ φ2 − 2 ¯ φ0 ¯ φ 1 r2 ∂rr2 ◆ f (η)

J

= 0

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f (η)

J

:

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ζ = 0, η = f (η)

J

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slide-23
SLIDE 23

Result

¯ c) = lim r→∞ 2π

⇣ 2πmφ ¯ φ(0)r3 ¯ φ(r) ˆ f (σ) (r) ⌘1/2

Y

J=1/2

B @ J ¯ φ(0)f (η)

J (r)

⇣ f (T )

J

(r) ⌘2 mφr ¯ φ(r) ˆ f (σ)

J

(r) ⇣ ˆ f (T )

J

(r) ⌘2 1 C A

−(2J+1)2/2

A(Aµ,ϕ) =

Y

J=0

det M(S,L,ϕ)

J

det c M(S,L,ϕ)

J

!−(2J+1)2/2 det M(T )

J

det c M(T )

J

!−(2J+1)2

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∆J ˆ f (T )

J

= 0

⇥ ∆J − m2

φ

⇤ ˆ f (σ)

J

= 0

 ∆J − g2 ¯ φ2 − 2 ¯ φ0 ¯ φ 1 r2 ∂rr2

  • f (η)

J

= 0

[∆J − g2 ¯ φ2]f (T )

J

= 0

lim

r→0

f (T )

J

(r) r2J = lim

r→0

ˆ f (T )

J

(r) r2J = 1 lim

r→0

f (η)

J (r)

r2J = lim

r→0

ˆ f (σ)

J

(r) r2J = 1

Transverse modes Longitudinal / NG modes Gauge independent!

slide-24
SLIDE 24

Result

¯ c) = lim r→∞ 2π

⇣ 2πmφ ¯ φ(0)r3 ¯ φ(r) ˆ f (σ) (r) ⌘1/2

Y

J=1/2

B @ J ¯ φ(0)f (η)

J (r)

⇣ f (T )

J

(r) ⌘2 mφr ¯ φ(r) ˆ f (σ)

J

(r) ⇣ ˆ f (T )

J

(r) ⌘2 1 C A

−(2J+1)2/2

A(Aµ,ϕ) =

Y

J=0

det M(S,L,ϕ)

J

det c M(S,L,ϕ)

J

!−(2J+1)2/2 det M(T )

J

det c M(T )

J

!−(2J+1)2

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  • | |
  • ln A(Aµ,ϕ)

MS = ln VG +

  • ln A′(Aµ,ϕ)

MS ,

  • ln A′(Aµ,ϕ)

MS =

1 3 + 2g2 |λ| + g4 |λ|2 1 + γE + ln µR 2

  • − g4

|λ|2 31 3 − π2

  • − 1

4 − 1 3γE − 2 ln AG − 1 2 ln |λ| 8 + 1 2 ln π − 3 2Sσ(zg) − 3 2 ln Γ(1 − zg)Γ(2 + zg). (

We can calculate it analytically with the MS-bar renormalization! [A. Andreassen, W. Frost, M. D. Schwartz; ’17, S. Chigusa, T. Moroi, YS; ’18]

slide-25
SLIDE 25

Standard Model

  • !"#

!"" !"$ !"% !"& !'# !(# !(" !($ !(% !(& !&# !! / )*+ !! / !"#

  • S. Chigusa, T. Moroi, YS; ‘ 18

cf.) A. Andreassen, W. Frost, M. D. Schwartz; ‘17

  • S. Chigusa, T. Moroi, YS; ‘ 17

log10

  • γPl × Gyr Gpc3
slide-26
SLIDE 26

How about

  • ther models?
slide-27
SLIDE 27

Bubble nucleation rate (decay rate of a vacuum) [A. Andreassen, W. Frost, M. D. Schwartz; ’17, S. Chigusa, T. Moroi, YS; ’18]

B = 8π2 3|λ|

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  • ln A′(h)

MS = −3

4 − 6 ln AG + 5 2 ln π 3 − 5 2 ln |λ| 8 + 3 ln µR 2 ,

  • ln A(σ)

MS = −1

2Sσ(zκ) + κ |λ| + κ2 3|λ|2

  • 1 + γE + ln µR

2

  • ,
  • ln A(ψ)

MS = − y4

3|λ|2

  • 1 + γE + ln µR

2

  • − 2y2

3|λ| 25 4 + γE + ln µR 2

  • + Sψ(zy),
  • .

. .

γ =

  • d ln R 1

R4

  • A′(h)A(σ)A(ψ)A(Aµ,ϕ)e−B

MS, µ

Manifestly gauge invariant analytic formulas

Tree 1-loop

slide-28
SLIDE 28

ELVAS

https://github.com/YShoji-HEP/ELVAS c++ package for ELectroweak VAcuum Stability

slide-29
SLIDE 29

BSM fermions

!"#$ !"#% !&#$ !&#% !'#$ !'#% !%#$ $#$ $#" $#' $#( $#) !#$ *+,!"[ !! / -./ ] !! ! ! " #$ #% #& $'$ $'% $'& $'! $'" #'$ ()*!"[ !! / +,- ] !! ! " # $% $& $! %'% %'& %'! %'" %'# $'% ()*!"[ !! / +,- ] !!

Right handed neutrino Vector-like quarks Vector-like leptons

  • S. Chigusa, T. Moroi, YS; ‘ 18

Fermions make our vacuum unstable

slide-30
SLIDE 30

BSM scalars

Two Higgs Doublet Model (in the context of the DFSZ axion model)

* Charge breaking vacuum may appear * Bottom and Tau Yukawa couplings can become large * Bounce is multi-field working with S. Oda, D. Takahashi at OIST * Additional quartic couplings We need to include quantum corrections from photon as well Large bottom and tau couplings with a large tanβ Vacuum structure becomes more non-trivial First example of one-loop calculation for multi-field bounce solution (but it can be reduced to single-field) Scalars are more interesting!

slide-31
SLIDE 31

Summary

  • We found a way to treat the gauge zero modes gauge

invariantly and the final result is manifestly gauge invariant.

  • We applied it to the standard model and confirmed that

the lifetime of the EW vacuum is much longer than the age of the Universe.

  • We provide a c++ package that can calculate decay rates

for generic models with approximate scale invariance.