The analysis of the electroweak bubble wall and sphaleron decoupling condition in the …research.kek.jp/group/riron/workshop/KEKPH09/Mar.6/... · 2009-03-09 · The analysis of the

The analysis of the electroweak bubble wall and

sphaleron decoupling condition in the MSSM

Eibun Senaha (National Central U, Taiwan)

Mar. 6, 2009@KEKPH09

in collaboration withKoichi Funakubo (Saga U)

Outline

Motivation Baryon Asymmetry of the Universe (BAU)

Electroweak phase transition ■ Sphaleron decoupling ■ Critical bubble wall

Summary

Motivation■ Energy budget of the Universe

Baryons4.6%

Dark Matter23%

Dark Energy72%

■ 95% of the Universe is made of dark object.■ It should be stressed that there remains a mystery in the visible sector as well. Where did antibaryons go?

Dark EnergyDark MatterBaryons

Motivation■ Energy budget of the Universe

Baryons4.6%

Dark Matter23%

Dark Energy72%

■ 95% of the Universe is made of dark object.■ It should be stressed that there remains a mystery in the visible sector as well. Where did antibaryons go?

Dark EnergyDark MatterBaryons

BAU Baryon Asymmetry of the Universe (BAU)

KM phase is not sufficient

××

phase transition is not 1st order

(1) Baryon(B) violation(2) C violation

CP violation(3) out of equilibrium

Sakharov’s conditions (‘67)

○ sphaleron processSM case

○ chiral gauge interaction

nB

n!=

nb ! nb

n!= (6.21± 0.16)" 10!10

Extension of the SMMSSM, 2HDM, singlet-extended MSSM etc.

We discuss the MSSM baryogenesis (BG) in the following.

Tension in the MSSM BG

■ There is a tension between the LEP data and the sphaleron decoupling in the MSSM.

More precise analysis of the sphaleron decoupling is needed.

1

10

0 20 40 60 80 100 120 140

1

10

mh (GeV/c

2)

tan!

Excludedby LEP

TheoreticallyInaccessible

No Mixing, (a)

(b)

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160

mh (GeV/c

2)

mA

(G

eV/c

2)

Excludedby LEP

TheoreticallyInaccessible

No Mixing, (a)

(a)

■ The LEP data put a strong constrains on the light Higgs boson.

Baryogenesis mechanism

!

Z

brokenphase

symmetric phase

vwall

"sph(s) >> HB+L"sph < H

(b)

f, f f,fCP

• Asymmetry of the charge flow of theparticle ( due to CP violation)

!• Accumulation of the charge in thesymmetric phase

!• B generation via sphaleron process

!• Decoupling of sphaleron process in thebroken phase

• Strongly 1st order phase transition

" Decoupling of the sphaleron process at T <# Tc :

!(b)sph/T 3

c < H(Tc) =" !c

Tc># 1

6

EW Baryogenesis mechanism

D: diffusion constant ~ mean free path X a few, Γ: particle changing rate

Diffusion equation

n: particle # density,


!

Z

broken

phase

symmetric

phase

vwall

"sph

(s)>> H

B+L"sph < H(b)

f, f f,fCP

• Asymmetry of particle number densities atthe phase boundary due to CP violation (CPviolating current, source term)

• Left-handed particle number densities di!useinto the symmetric phase (coupled di!usioneqs. with source terms)

• B generation via sphaleron process

• Decoupling of sphaleron process in the brokenphase


! Decoupling of the sphaleron process at T <" Tc :

!(b)sph/T 3

c < H(Tc) =! !c

Tc>" 1

— Seminar@IOPAS, June 16, 2006 — 7/26

H:Hubble constant

Dn!!i + vwalln!

i +!

j !ijnj = [source term]

[Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ]


!

Z

brokenphase

symmetric phase

vwall


(b)

f, f f,fCP







!(b)sph/T 3

c < H(Tc) =" !c

Tc># 1

6



Diffusion equation



!

Z

broken

phase

symmetric

phase

vwall

"sph

(s)>> H

B+L"sph < H(b)

f, f f,fCP







!(b)sph/T 3

c < H(Tc) =! !c

Tc>" 1


H:Hubble constant

Dn!!i + vwalln!

i +!


Sakharov’s cond.



!

Z

brokenphase

symmetric phase

vwall


(b)

f, f f,fCP







!(b)sph/T 3

c < H(Tc) =" !c

Tc># 1

6



Diffusion equation



!

Z

broken

phase

symmetric

phase

vwall

"sph

(s)>> H

B+L"sph < H(b)

f, f f,fCP







!(b)sph/T 3

c < H(Tc) =! !c

Tc>" 1


H:Hubble constant

Dn!!i + vwalln!

i +!


Sakharov’s cond.


Effective potential

The critical bubbles in the MSSM!

Eibun Senaha

March 2, 2009

In order to evaluate the sphaleron decoupling precisely, we should also know the nucleationtemperature TN . For this purpose, we calculate the critical bubble numerically.

1 Gauge-Higgs system

We consider the gauge-Higgs system in the MSSM. The Lagrangian is reduced to

Lgauge+Higgs = "1

4F a

µ!Faµ! " 1

4Bµ!

µ! +!

i=d,u

(Dµ!i)†Dµ!i " Ve!(!d, !u), (1.1)

where

Dµ!d =

"!µ + ig2

"a

2Aa

µ " ig1

2Bµ

#!d, Dµ!u =

"!µ + ig2

"a

2Aa

µ + ig1

2Bµ

#!u. (1.2)

Ve! is composed of the tree-level and one-loop e"ective potentials:

Ve!(!d, !u) = V0(!d, !u) + #V (!d, !u; T ), (1.3)

where the tree-level part is

V0(!d, !u) = m21!

†d!d + m2

2!†u!u " (m2

3#ij!id!

ju + h.c.)

+g22 + g2

1

8(!†

d!d " !†u!u)

2 +g22

2(!†

d!u)(!†u!d), (1.4)

and one-loop part is

#V (!d, !u; T ) =!

A

cA

$F0(m

2A) +

T 4

2$2IB,F

"m2

A

T 2

#%. (1.5)

where

F0(m2) =

m4

64$2

"ln

m2

M2" 3

2

#, IB,F (a2) =

& !

0

dx x2 ln'1 # e"

#x2+a2

(. (1.6)

The classical Higgs fields are parameterized as

!d(x) =ei"d(x)

$2

"%d(x)

0

#, !u =

ei("u(x)+#)

$2

"0

%u(x)

#, (1.7)

As we will see in the following, & % &d + &u is the gauge-invariant physical direction. Inorthogonal to this, we can identify that & % &d " &u is the unphysical direction which isassociated with the NG modes.

!Since February 27, 2009

1


Eibun Senaha

March 2, 2009




Lgauge+Higgs = "1

4F a

µ!Faµ! " 1

4Bµ!

µ! +!

i=d,u

(Dµ!i)†Dµ!i " Ve!(!d, !u), (1.1)

where

Dµ!d =

"!µ + ig2

"a

2Aa

µ " ig1

2Bµ

#!d, Dµ!u =

"!µ + ig2

"a

2Aa

µ + ig1

2Bµ

#!u. (1.2)


Ve!(!d, !u) = V0(!d, !u) + #V (!d, !u; T ), (1.3)


V0(!d, !u) = m21!

†d!d + m2

2!†u!u " (m2

3#ij!id!

ju + h.c.)

+g22 + g2

1

8(!†

d!d " !†u!u)

2 +g22

2(!†

d!u)(!†u!d), (1.4)


#V (!d, !u; T ) =!

A

cA

$F0(m

2A) +

T 4

2$2IB,F

"m2

A

T 2

#%. (1.5)

where

F0(m2) =

m4

64$2

"ln

m2

M2" 3

2

#, IB,F (a2) =

& !

0

dx x2 ln'1 # e"

#x2+a2

(. (1.6)


!d(x) =ei"d(x)

$2

"%d(x)

0

#, !u =

ei("u(x)+#)

$2

"0

%u(x)

#, (1.7)



1


Eibun Senaha

March 2, 2009




Lgauge+Higgs = "1

4F a

µ!Faµ! " 1

4Bµ!

µ! +!

i=d,u

(Dµ!i)†Dµ!i " Ve!(!d, !u), (1.1)

where

Dµ!d =

"!µ + ig2

"a

2Aa

µ " ig1

2Bµ

#!d, Dµ!u =

"!µ + ig2

"a

2Aa

µ + ig1

2Bµ

#!u. (1.2)


Ve!(!d, !u) = V0(!d, !u) + #V (!d, !u; T ), (1.3)


V0(!d, !u) = m21!

†d!d + m2

2!†u!u " (m2

3#ij!id!

ju + h.c.)

+g22 + g2

1

8(!†

d!d " !†u!u)

2 +g22

2(!†

d!u)(!†u!d), (1.4)


#V (!d, !u; T ) =!

A

cA

$F0(m

2A) +

T 4

2$2IB,F

"m2

A

T 2

#%, (1.5)

where

F0(m2) =

m4

64$2

"ln

m2

M2" 3

2

#, IB,F (a2) =

& !

0

dx x2 ln'1 # e"

#x2+a2

(. (1.6)


!d(x) =ei"d(x)

$2

"%d(x)

0

#, !u =

ei("u(x)+#)

$2

"0

%u(x)

#, (1.7)



1


Eibun Senaha

March 2, 2009




Lgauge+Higgs = "1

4F a

µ!Faµ! " 1

4Bµ!

µ! +!

i=d,u

(Dµ!i)†Dµ!i " Ve!(!d, !u), (1.1)

where

Dµ!d =

"!µ + ig2

"a

2Aa

µ " ig1

2Bµ

#!d, Dµ!u =

"!µ + ig2

"a

2Aa

µ + ig1

2Bµ

#!u. (1.2)


Ve!(!d, !u) = V0(!d, !u) + #V (!d, !u; T ), (1.3)


V0(!d, !u) = m21!

†d!d + m2

2!†u!u " (m2

3#ij!id!

ju + h.c.)

+g22 + g2

1

8(!†

d!d " !†u!u)

2 +g22

2(!†

d!u)(!†u!d), (1.4)


#V (!d, !u; T ) =!

A

cA

$F0(m

2A) +

T 4

2$2IB,F

"m2

A

T 2

#%, (1.5)

where

F0(m2) =

m4

64$2

"ln

m2

M2" 3

2

#, IB,F (a2) =

& !

0

dx x2 ln'1 # e"

#x2+a2

(. (1.6)


!d(x) =ei"d(x)

$2

"%d(x)

0

#, !u =

ei("u(x)+#)

$2

"0

%u(x)

#, (1.7)



1

1-loop:

Tree:

■ The gauge bosons and 3rd generation of quarks/squarks are taken into account.

Fitting function: [K. Funakubo]

■ The fitting function is used in our numerical analysis.

■ To discuss the phase transition(PT) we use the effective potential.

IB,F (a2) = e!aN!

n=0

cb,fn an, |IB,F ! IB,F | < 10!6 (N = 40).

Phase transition

0

0 50 100 150 200 250 300

v [GeV]

Veff

T=Tc

T>Tc

T<Tc

■ The PT must be 1st order to realize out of equilibrium.

■ At Tc, the potential has two degenerate minima.

e.g. 1dim

■ Light stop plays a crucial role in strengthening the 1st order PT.

Xt = At ! µ cot !m2q ! m2

tR, X2

t ,

[Carena, Quiros, Wagner,PLB380 (‘96) 81]

To be consistent with the LEP bound on mH and !-parameter, we shouldtake

m2t1

= m2tR

+y2

t sin2 !

2

!1! X2

t

m2q

"v2 +O(g2)

Et1 !y3

t sin3 !

4"

2"

!1# X2

t

m2q

"3/2c.f. High T expansion:

light stop mass ⇒

Ve! ! "(ESM + Et1)Tv3

■ Order parameters = Higgs VEVs

0

large effect!

(mt1 < mt)

Sphaleron

Sphaleron decoupling condition:1

B

dB

dt! " 13Nf

4 · 32!2

"!#3

W

$NtrNrote!Esph/T < H(T ) ! 1.66

#g"T

2/mP

Energy

sphaleron

instanton

Ncs0-1 1

Esph

!B = 3!NCS

NCS =g22

32!2

!d3x "ijkTr

"FijAk !

2

3g2AiAjAk

#

!B != 0Instanton: quantum tunnelingSphaleron: thermal fluctuation

■ A static saddle point solution w/ finite energy of the gauge-Higgs system. [N.S. Manton, PRD28 (’83) 2019]

Hubble constant

[K. Funakubo]10% corr.

v

T>

0.050

E

!43.17 + ln(!NtrNrot) + ln

""!mW

#! 2 ln

"T

100GeV

#$

" 0.025# (43.17 + 4.38 + 0.416) = 1.20

where E = Esph/(4!v/g2) = 2.00, NtrNrot = 80.13, "2! = 2.3m2

W , # = 1

Experimental constraints

■ ρ-parameter:

■ Higgs bounds@ [PLB565, 61 (2003)]

■ Lower bounds for SUSY particles:

10-2

10-1

1

20 40 60 80 100 120

mH

(GeV/c2)

95

% C

L l

imit

on

!2

LEP"s = 91-210 GeV

Observed

Expected for background

(a)

■ B physics observables:

In the LHS, H is mostly responsible for the electroweak symmetry breaking since !H" #

v0. Thus the Higgs potential that we examine may be approximately cast into the form

V (!H"T ).

In order to avoid the washout of the generated BAU after the phase transition, the

sphaleron process must be decoupled. This condition is reduced to

vC/TC >$ 1. (10)

From the argument by using the high temperature expansion of V , one can see that

vC/TC can be enhanced when the coe!cient of the cubic term with a negative coe!cient

(denoted "E) gets larger and/or the quartic term does smaller. The latter implies that

the light H is favored. The former can be realized by the additional contributions from the

bosons. In fact, the e#ect of the light stop t1 on "E is sizable due to the large top Yukawa

coupling and the number of degrees of freedom, i.e., 6. In the high temperature expansion,

it follows that

"Et1 #1

6!

NCm3t

v30

!

1 % |Xt|2

m2q

"3/2

, (11)

for mq & m2tR

= 0, where Xt = At % µ/ tan ". The high temperature expansion makes it

easy to see how the first order phase transition is strengthen analytically. It is, however,

untrustworthy to use this approximation when the masses of the particles in loops are larger

than TC . Therefore we will adopt the alternative method. (funakubo-san’s method)

IV. THE EXPERIMENTAL CONSTRAINTS

Here we present the experimental constraints that we impose in the numerical calculation.

When the masses of the three neutral Higgs boson (Hi) are smaller than 114.4 GeV and/or

the sum of two of them are smaller than about 195 GeV, we require

g2HiZZ ' Br(Hi ( ff) < FHiZ(mHi),

g2HiHjZ ' Br(Hi ( ff) ' Br(Hj ( ff) < FHiHj(mHi + mHj), (12)

where f = b, # . FHiZ and FHiHj are the 95% C.L. upper limits from the LEP experiments [7,

8].

5

In the LHS, H is mostly responsible for the electroweak symmetry breaking since !H" #

v0. Thus the Higgs potential that we examine may be approximately cast into the form

V (!H"T ).

In order to avoid the washout of the generated BAU after the phase transition, the

sphaleron process must be decoupled. This condition is reduced to

vC/TC >$ 1. (10)

From the argument by using the high temperature expansion of V , one can see that

vC/TC can be enhanced when the coe!cient of the cubic term with a negative coe!cient

(denoted "E) gets larger and/or the quartic term does smaller. The latter implies that

the light H is favored. The former can be realized by the additional contributions from the

bosons. In fact, the e#ect of the light stop t1 on "E is sizable due to the large top Yukawa

coupling and the number of degrees of freedom, i.e., 6. In the high temperature expansion,

it follows that

"Et1 #1

6!

NCm3t

v30

!

1 % |Xt|2

m2q

"3/2

, (11)

for mq & m2tR

= 0, where Xt = At % µ/ tan ". The high temperature expansion makes it

easy to see how the first order phase transition is strengthen analytically. It is, however,

untrustworthy to use this approximation when the masses of the particles in loops are larger

than TC . Therefore we will adopt the alternative method. (funakubo-san’s method)

IV. THE EXPERIMENTAL CONSTRAINTS

Here we present the experimental constraints that we impose in the numerical calculation.

When the masses of the three neutral Higgs boson (Hi) are smaller than 114.4 GeV and/or

the sum of two of them are smaller than about 195 GeV, we require

g2HiZZ ' Br(Hi ( ff) < FHiZ(mHi),

g2HiHjZ ' Br(Hi ( ff) ' Br(Hj ( ff) < FHiHj(mHi + mHj), (12)

where f = b, # . FHiZ and FHiHj are the 95% C.L. upper limits from the LEP experiments [7,

8].

5

We also consider the constraint on the ! parameter which is a measure of the SU(2)

symmetry breaking. !! = !q! + !H! < 0.002 is imposed in our calculation, where !q!

and !H! are the contributions of squarks and the physical Higgs bosons, respectively. It is

turns out that as long as we take mtR ! mq, !q! would not exceed the current upper bound

even for mt1 ! mb1due to the suppressed couplings in the loop. In almost region in our

numerical calculation, the mass di"erence between one of the neutral Higgs bosons and the

charged Higgs bosons is small, which implies the custodial SU(2) symmetry approximately

exists, we then have !H! " 0.

The experimental lower bounds on the SUSY particles are taken into account, especially

for the lightest stop and the chargino ("±1 ), we impose mt1 > 95.7 GeV, m!±

1> 94 GeV [6].

The other SUSY particles are assumed to be heavy enough.

Currently, the enormous data of the B physics observables are available, especially Bu #

#$" , B # Xs% and Bs # µ+µ! are relevant for the LHS. The averaged experimental values

of those processes are reported by HFAG [10]:

Br(Bu # #$" )exp = 1.41+0.43!0.42 $ 10!4, (13)

Br(B # Xs%)exp = (3.52 ± 0.23 ± 0.09) $ 10!4, (14)

Br(Bs # µ+µ!)exp < 0.23 $ 10!7. (15)

As mentioned in Introduction, the first two modes can give the severe constraints for the light

charged Higgs bosons and last one could be important for the light neutral Higgs bosons.

At the tree level, the ratio of the two branching ratios of Br(Bu # #$" )MSSM/SM is given by

Br(Bu # #$" )MSSM

Br(Bu # #$" )SM=

!

1 % tan2 &m2

Bu

m2H±

"2

& rH . (16)

From the latest data: |Vub| = (3.95 ± 0.35) $ 10!3, fB = 200 ± 20 MeV which are involved

in Br(Bu # #$" )SM, it follows that rH = 1.28 ± 0.52. We will take the 95% C.L. interval

range of rH .

6

e.g. Higgsstrahlung

e.g. chargino mass > 94 GeV

!! ! "TZZ(0)

m2Z

" "TWW (0)

m2W

< 0.002

Allowed region■ The allowed region is highly constrained by the experimental data.

The sphaleron process is not decoupled at Tc.

Loophole:⇒ The PT begins to proceed with bubble wall at below Tc.

We need to know the dynamics of bubble wall.

vC

TC=

107.10 GeV

116.27 GeV= 0.92

supercooling

Maximal v/T:mq = 1200 GeV, mtR ! 0, At = Ab = "300 GeV.

tan ! = 10.1, mH± = 127.4 GeV

Critical bubble

! 1

!2

!!2dh2

d!

"+ h2

!h2

1 cos2 "

h21 cos2 " + h2

2 sin2 "

d#

d!

"2

+1

v3 sin "

$Ve!

$%u= 0, (1.28)

! 1

!2

d

d!

!!2h2

1h22

h21 cos2 " + h2

2 sin2 "

d#

d!

"+

1

v4 sin2 " cos2 "

$Ve!

$#= 0, (1.29)

with the boundary conditions:

lim!!"

h1(!) = 0, lim!!"

h2(!) = 0, lim!!"

#(!) = 0, (1.30)

and

dh1(!)

d!

####!=0

= 0,dh2(!)

d!

####!=0

= 0,d#(!)

d!

####!=0

= 0. (1.31)

2 CP-conserving case

In the CP-coserving case the energy functional is reduced to

E = 4&

$ "

0

dr r2

%1

2

&!d%d

dr

"2

+

!d%u

dr

"2'+ Ve!(%d, %u; T )

(. (2.1)

The EOM for %d and %u are given by

! 1

r2

d

dr

!r2d%d

dr

"+

$Ve!

$%d= 0, (2.2)

! 1

r2

d

dr

!r2d%u

dr

"+

$Ve!

$%u= 0. (2.3)

The boundary conditions for EOM are imposed in the symmetric phase as

limr!"

%d(r) = 0, limr!"

%u(r) = 0, (2.4)

and in the broken phase as

d%d(r)

dr

####r=0

= 0,d%u(r)

dr

####r=0

= 0. (2.5)

It is convenient to parameterize the Higgs profiles (%d, %u) in terms of the dimensionlessquantities. We thus change variables as

! = vr, h1(!) =%d(r)

v cos ", h2(!) =

%u(r)

v sin ". (2.6)

In this case, E takes the form

E = 4&v

$ "

0

d! !2

%1

2

&!dh1

d!

"2

cos2 " +

!dh2

d!

"2

sin2 "

'+ Ve!(h1, h2; T )

(, (2.7)

where Ve! = Ve!/v4. Correspondingly, the EOM are rewritten as

! 1

!2

!!2dh1

d!

"+

1

v3 cos "

$Ve!

$%d= 0, (2.8)

4

! 1

!2

!!2dh2

d!

"+ h2

!h2

1 cos2 "

h21 cos2 " + h2

2 sin2 "

d#

d!

"2

+1

v3 sin "

$Ve!

$%u= 0, (1.28)

! 1

!2

d

d!

!!2h2

1h22

h21 cos2 " + h2

2 sin2 "

d#

d!

"+

1

v4 sin2 " cos2 "

$Ve!

$#= 0, (1.29)


lim!!"

h1(!) = 0, lim!!"

h2(!) = 0, lim!!"

#(!) = 0, (1.30)

and

dh1(!)

d!

####!=0

= 0,dh2(!)

d!

####!=0

= 0,d#(!)

d!

####!=0

= 0. (1.31)



E = 4&

$ "

0

dr r2

%1

2

&!d%d

dr

"2

+

!d%u

dr

"2'+ Ve!(%d, %u; T )

(. (2.1)


! 1

r2

d

dr

!r2d%d

dr

"+

$Ve!

$%d= 0, (2.2)

! 1

r2

d

dr

!r2d%u

dr

"+

$Ve!

$%u= 0. (2.3)


limr!"

%d(r) = 0, limr!"

%u(r) = 0, (2.4)


d%d(r)

dr

####r=0

= 0,d%u(r)

dr

####r=0

= 0. (2.5)


! = vr, h1(!) =%d(r)

v cos ", h2(!) =

%u(r)

v sin ". (2.6)


E = 4&v

$ "

0

d! !2

%1

2

&!dh1

d!

"2

cos2 " +

!dh2

d!

"2

sin2 "

'+ Ve!(h1, h2; T )

(, (2.7)


! 1

!2

!!2dh1

d!

"+

1

v3 cos "

$Ve!

$%d= 0, (2.8)

4

! 1

!2

!!2dh2

d!

"+ h2

!h2

1 cos2 "

h21 cos2 " + h2

2 sin2 "

d#

d!

"2

+1

v3 sin "

$Ve!

$%u= 0, (1.28)

! 1

!2

d

d!

!!2h2

1h22

h21 cos2 " + h2

2 sin2 "

d#

d!

"+

1

v4 sin2 " cos2 "

$Ve!

$#= 0, (1.29)


lim!!"

h1(!) = 0, lim!!"

h2(!) = 0, lim!!"

#(!) = 0, (1.30)

and

dh1(!)

d!

####!=0

= 0,dh2(!)

d!

####!=0

= 0,d#(!)

d!

####!=0

= 0. (1.31)



E = 4&

$ "

0

dr r2

%1

2

&!d%d

dr

"2

+

!d%u

dr

"2'+ Ve!(%d, %u; T )

(. (2.1)


! 1

r2

d

dr

!r2d%d

dr

"+

$Ve!

$%d= 0, (2.2)

! 1

r2

d

dr

!r2d%u

dr

"+

$Ve!

$%u= 0. (2.3)


limr!"

%d(r) = 0, limr!"

%u(r) = 0, (2.4)


d%d(r)

dr

####r=0

= 0,d%u(r)

dr

####r=0

= 0. (2.5)


! = vr, h1(!) =%d(r)

v cos ", h2(!) =

%u(r)

v sin ". (2.6)


E = 4&v

$ "

0

d! !2

%1

2

&!dh1

d!

"2

cos2 " +

!dh2

d!

"2

sin2 "

'+ Ve!(h1, h2; T )

(, (2.7)


! 1

!2

!!2dh1

d!

"+

1

v3 cos "

$Ve!

$%d= 0, (2.8)

4

b.c.

Higgs fields:

Energy functional:

!d =1!2

!!d

0

", !u =

1!2

!0!u

",

Equation of motion:

critical bubble = static solution which is unstable against variation of radius.

! 1

!2

!!2dh2

d!

"+ h2

!h2

1 cos2 "

h21 cos2 " + h2

2 sin2 "

d#

d!

"2

+1

v3 sin "

$Ve!

$%u= 0, (1.28)

! 1

!2

d

d!

!!2h2

1h22

h21 cos2 " + h2

2 sin2 "

d#

d!

"+

1

v4 sin2 " cos2 "

$Ve!

$#= 0, (1.29)


lim!!"

h1(!) = 0, lim!!"

h2(!) = 0, lim!!"

#(!) = 0, (1.30)

and

dh1(!)

d!

####!=0

= 0,dh2(!)

d!

####!=0

= 0,d#(!)

d!

####!=0

= 0. (1.31)



E = 4&

$ "

0

dr r2

%1

2

&!d%d

dr

"2

+

!d%u

dr

"2'+ Ve!(%d, %u; T )

(. (2.1)


! 1

r2

d

dr

!r2d%d

dr

"+

$Ve!

$%d= 0, (2.2)

! 1

r2

d

dr

!r2d%u

dr

"+

$Ve!

$%u= 0. (2.3)


limr!"

%d(r) = 0, limr!"

%u(r) = 0, (2.4)


d%d(r)

dr

####r=0

= 0,d%u(r)

dr

####r=0

= 0. (2.5)


! = vr, h1(!) =%d(r)

v cos ", h2(!) =

%u(r)

v sin ". (2.6)


E = 4&v

$ "

0

d! !2

%1

2

&!dh1

d!

"2

cos2 " +

!dh2

d!

"2

sin2 "

'+ Ve!(h1, h2; T )

(, (2.7)


! 1

!2

!!2dh1

d!

"+

1

v3 cos "

$Ve!

$%d= 0, (2.8)

4

r =!

x

Bubbles can be nucleated at below Tc.

e.g. 1dim

0

0 50 100 150 200 250 300

v [GeV]

Veff

T=Tc

T=TN

Bubble nucleation

vN

TN=

116.73

115.59= 1.01

■ The sphaleron process is not decoupled at TN either.

10% enhancement! But,

v

T> 1.35

(preliminary)

where Ecb(T ) is the energy of the critical bubble at temperature T 2. Note that this isa rate per unit volume. We define the nucleation temperature TN as the temperatureat which the rate of nucleation of a critical bubble within a horizon volume is equal tothe Hubble parameter at that temperature. Since the horizon scale is roughly given byH(T )!1, the nucleation temperature is defined by3

!N(TN)H(TN)!3 = H(TN). (4.2)

Since the Hubble parameter at temperature T is

H(T ) =

!8!G

3"(T ) =

"8!

3m2P

!2

30g"(T )T 4

#1/2

! 1.66 g"(T )1/2 T 2

mP, (4.3)

where g"(T ) is the e"ective massless degrees of freedom at T defined by

g"(T ) =$

B

#(T " mB(T ))gB +7

8

$

F

#(T " mF (T ))gF , (4.4)

with gB and gF being intrinsic degrees of freedom of boson B and fermion F , respectively,the definition of TN (4.2) is reduced to

"Ecb(TN)

2!TN

#3/2

e!Ecb(TN )/TN = 7.59 g"(TN)2 T 4N

m4P

, (4.5)

orEcb(TN)

TN" 3

2log

Ecb(TN)

TN= 152.59 " 2 log g"(TN) " 4 log

TN

100GeV. (4.6)

A Derivatives of the e!ective potential

Here we summarize the first and second derivatives of the e"ective potential, which areused in the tadpole conditions, the equations of motion and matrix elements in the relax-ation algorithm.

A.1 tadpole conditions

The tadpole conditions are imposed in the vacuum at zero temperature.

##d$0 =1%2

%v0d

0

&, ##u$0 =

ei!0

%2

%0

v0u

&(A.1)

2In [3], the author only evaluated the contribution from translational zero modes which is proportionalto E3/2

cb , but not that from rotational zero modes and those from nonzero modes. He multiplied T 4 with theprefactor on the dimensional ground. We expect a factor of T 3 comes from the 3-dimensional translationalzero modes, while the remaining factor of T may have its origin in the negative mode corresponding tothe instability of the critical bubble. Any way, this uncertainty in the prefactor will have small e!ect onthe estimation of the nucleation temperature, which is mainly determined by the exponent.

3One may think that only one bubble nucleated within a horizon volume cannot convert the wholeregion into the broken phase. In this sense, this definition of nucleation temperature will give an upperbound of temperature at which the phase transition begins to proceed.

6

The boundary value !s is determined by finiteness of the energy functional[1]. At r = 0,the functions satisfy the Neumann-type boundary condition because of spherical symme-try,

"!h1(#) = 0, "!h2(#) = 0, "!! = 0, at # = 0. (2.31)

3 Numerical Analysis

To numerically study solutions to the equations of motion, we change the variable # withinfinite range to some variable with a finite range. We adopt x defined as

1 ! x = e!a!, or # = !1

alog(1 ! x), (3.1)

where a is some real number which will be chosen for later convenience. This relationmaps # " [0,#) to x " [0, 1] With this new variable, the equations of motion are rewrittenas

d2h1

dx2=

1

1 ! x

!

1 +2

log(1 ! x)

"dh1

dx+ h1

!h2

2 sin2 $

H

d!

dx

"2

+1

a2v3 cos $(1 ! x)2

"Ve!

"%d,(3.2)

d2h2

dx2=

1

1 ! x

!

1 +2

log(1 ! x)

"dh2

dx+ h2

!h2

1 cos2 $

H

d!

dx

"2

+1

a2v3 sin $(1 ! x)2

"Ve!

"%u,(3.3)

d2!2

dx2=

#1

1 ! x

!

1 +2

log(1 ! x)

"

! H

h21h

22

d

dx

!h2

1h22

H

"$d!

dx

+1

a2v4 cos2 $ sin2 $(1 ! x)2

H

h21h

22

"Ve!

"!

=

#1

1 ! x

!

1 +2

log(1 ! x)

"

! 2

!h"

1

h1+

h"2

h2! h1h"

1 cos2 $ + h2h"2 sin2 $

H

"$d!

dx

+1

a2v4(1 ! x)2

!1

h21 cos2 $

+1

h22 sin2 $

""Ve!

"!, (3.4)

where H(x) $ h1(x)2 cos2 $ + h2(x)2 sin2 $ and the prime denotes derivative with respectto x. The energy functional is expressed as

E = 4&v% 1

0dx log2(1 ! x)

&1 ! x

2a

'

(!

dh1

dx

"2

cos2 $ +

!dh2

dx

"2

sin2 $ +h2

1h22 cos2 $ sin2 $

H

!d!

dx

"2)

*

+1

a3(1 ! x)Ve!(h1, h2, !; T )

+

. (3.5)

4 Bubble Nucleation

According to [3], the bubble nucleation rate per unit time per unitvolume is given by

!N(T ) % T 4

!Ecb(T )

2&T

"3/2

e!Ecb(T )/T , (4.1)

5Nucleation T:

Nucleation rate:[A.D. Linde, NPB216 (’82) 421]

Numerical results:

E = 1.77, Ntr = 6.65, Nrot = 12.27

Sphaleron decoupling cond.@TN :

! = vr, h1(!) ="d(r)

v cos #, h2(!) =

"u(r)

v sin #

More examplesTable 1: Examples of the 1st order PT. |Ab| = |At| = 300 GeV, mtR = 10!4 GeV, mbR

= 1000GeV, |µ| = 100 GeV, M2 = 500 GeV, !A ! !At = !Ab

= ", !µ = 0.

mq (GeV) 1200 1300 1400 1500

tan # 10.11 9.87 9.75 9.57mH± (GeV) 127.30 127.40 127.40 127.40

vC/TC107.095

116.274= 0.921

107.512

116.496= 0.923

107.768

116.770= 0.923

107.914

117.045= 0.922

tan #C 13.812 13.640 13.606 13.465

vN/TN116.726

115.585= 1.010

117.155

115.798= 1.012

117.403

116.068= 1.012

117.530

116.340= 1.010

tan #N 13.684 13.503 13.462 13.317Ecb/(4"v0) 5.623 5.633 5.646 5.659

Ecb/TN 150.386 150.379 150.369 150.360

Esph/(4"v0/g2) 1.7686 1.7695 1.7704 1.7711Ntr 6.6522 6.6576 6.6623 6.6666Nrot 12.266 12.253 12.241 12.230

vN/TN > 1.345 1.344 1.344 1.343

1

Typically, vN/TN > 1.34 is needed for sphaleron decoupling.

Ab = At = !300 GeV, mtR = 10!4 GeV, mbR= 1000 GeV,

µ = 100 GeV, M2 = 500 GeV,



= ", !µ = 0.

mq (GeV) 1200 1300 1400 1500

tan # 10.11 9.87 9.75 9.57mH± (GeV) 127.30 127.40 127.40 127.40

vC/TC107.095

116.274= 0.921

107.512

116.496= 0.923

107.768

116.770= 0.923

107.914

117.045= 0.922

tan #C 13.812 13.640 13.606 13.465

vN/TN116.726

115.585= 1.010

117.155

115.798= 1.012

117.403

116.068= 1.012

117.530

116.340= 1.010

tan #N 13.684 13.503 13.462 13.317Ecb/(4"v0) 5.623 5.633 5.646 5.659

Ecb/TN 150.386 150.379 150.369 150.360

Esph/(4"v0/g2) 1.7686 1.7695 1.7704 1.7711Ntr 6.6522 6.6576 6.6623 6.6666Nrot 12.266 12.253 12.241 12.230

vN/TN > 1.345 1.344 1.344 1.343

1



µ = 100 GeV, M2 = 500 GeV,



= ", !µ = 0.

mq (GeV) 1200 1300 1400 1500

tan # 10.11 9.87 9.75 9.57mH± (GeV) 127.30 127.40 127.40 127.40

vC/TC107.095

116.274= 0.921

107.512

116.496= 0.923

107.768

116.770= 0.923

107.914

117.045= 0.922

tan #C 13.812 13.640 13.606 13.465

vN/TN116.726

115.585= 1.010

117.155

115.798= 1.012

117.403

116.068= 1.012

117.530

116.340= 1.010

tan #N 13.684 13.503 13.462 13.317Ecb/(4"v0) 5.623 5.633 5.646 5.659

Ecb/TN 150.386 150.379 150.369 150.360

Esph/(4"v0/g2) 1.7686 1.7695 1.7704 1.7711Ntr 6.6522 6.6576 6.6623 6.6666Nrot 12.266 12.253 12.241 12.230

vN/TN > 1.345 1.344 1.344 1.343

1



µ = 100 GeV, M2 = 500 GeV,



= ", !µ = 0.

mq (GeV) 1200 1300 1400 1500

tan # 10.11 9.87 9.75 9.57mH± (GeV) 127.30 127.40 127.40 127.40

vC/TC107.095

116.274= 0.921

107.512

116.496= 0.923

107.768

116.770= 0.923

107.914

117.045= 0.922

tan #C 13.812 13.640 13.606 13.465

vN/TN116.726

115.585= 1.010

117.155

115.798= 1.012

117.403

116.068= 1.012

117.530

116.340= 1.010

tan #N 13.684 13.503 13.462 13.317Ecb/(4"v0) 5.623 5.633 5.646 5.659

Ecb/TN 150.386 150.379 150.369 150.360

Esph/(4"v0/g2) 1.7686 1.7695 1.7704 1.7711Ntr 6.6522 6.6576 6.6623 6.6666Nrot 12.266 12.253 12.241 12.230

vN/TN > 1.345 1.344 1.344 1.343

1



µ = 100 GeV, M2 = 500 GeV,

Is MSSM BG dead?

TN ⇒ onset of the PT. We should know a temperature at which the PT ends. The sphaleron decoupling condition should be imposed at such a temperature.

Higher order (2-loop) contributions must be taken into account. [J.R. Espinosa, NPB475, (’06) 273]

It looks almost dead, but there might be a way out.

⇒ The sphaleron decoupling cond. might be relaxed. The potential can be extended in such a way that stop also has

a nontrivial VEV. (Color-Charge-Breaking vacuum)⇒ MSSM BG is viable. [Canena et al, NPB812,(‘09) 243] [N.B.] EW vacuum: metastable, CCB vacuum: global minimum

If the refined sphaleron decoupling cond. is used, is it still viable?

Summary We have studied the strength of the 1st order phase

transition in the MSSM.

v/T at TN can be enhanced by about 10% compared to that at Tc.

The sphaleron decoupling condition at TN is typically given by v/T>1.34.

The sphaleron process is not decoupled at both Tc and TN.

More refined analysis is needed to reach a convincing conclusion for successful EWBG in the MSSM.

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