Introduction to electroweak
phase transition
Eibun Senaha (NCU)
March 12, 2009@NTHU
Outline
Overview
Effective potential
Sphaleron
Electroweak phase transition (EWPT)
SM, MSSM
Updated analysis of EWPT in the MSSM
Summary
Motivation
■ Energy budget of the Universe
Baryons
4.6%
Dark Energy
Dark Matter
Baryons
Dark Matter
23%
Dark Energy
72%
■ 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?
Motivation
■ Energy budget of the Universe
Baryons
4.6%
Dark Energy
Dark Matter
Baryons
Dark Matter
23%
Dark Energy
72%
■ 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?
BAU
Baryon Asymmetry of the Universe (BAU)
nB
nb − nb̄
η ≡
=
nγ
nγ
−10
= (4.7 − 6.5) × 10
(95%C.L.)
■ If the BAU is generated before
T=1 MeV, the light element
abundances can be explained
by the standard Big-Bang
cosmology.
Question:
How did the BAU arise
dynamically?
[PDG ’08]
Sakharov’s criteria
To get the BAU from initially baryon symmetric Universe,
the following conditions must be satisfied. [Sakharov, ’67]
(1) Baryon number (B) violation
(2) C and CP violation
(3) out of equilibrium
(1) is trivial.
w/o (2), namely, if C and CP symmetries exist,
∂ρ(t)
[ρ(t), O] = 0, (O = C, CP ), i!
+ [ρ(t), H] = 0,
∂t
−1
!nB " = tr(ρB nB ) = tr(ρB nB O O)
= tr(ρB OnB O−1 ) = −tr(ρB nB )
∴ !nB " = 0
(CBC −1 = −B, (CP )B(CP )−1 = −B)
B is vector-like
w/o (3), namely, if the B violating process is in
equilibrium, one would get
nb = nb̄
⇒
"nB # = 0
[N.B.] The masses of particle and antiparticle are
assumed to be the same. (∵ CPT theorem)
Two possibilities
(1) B-L generation above the electroweak (EW) scale
Leptogenesis, GUTs, Affleck-Dine etc
(2) B generation during the EW phase transition (PT)
EW baryogenesis (BG)
(2) is directly linked to EW Physics.
It is testable at colliders
EW baryogenesis
B violation: sphaleron process
C violation: chiral gauge interaction
CP violation: Kobayashi-Maskawa (KM) phase and
other sources in the BSM.
out of equilibrium: 1st order PT with expanding
bubble wall
The SM has the problems with the last 2 conditions:
■ The KM phase is too small to generate the BAU.
■ The PT is not 1st order for the viable Higgs mass.
(>114.4 GeV)
The SM failed to explain the BAU
Mechanism of EWBG
[Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ]
H:Hubble constant
Due to CP violation, asymmetry of
particle number densities at the
bubble wall occur.
they diffuse into symmetric phase.
Left-handed particle number
densities are converted into B via
sphaleron process.
Sphaleron process is decoupled
after the PT.
B is frozen.
Mechanism of EWBG
[Kuzmin, Rubakov, Shaposhnikov, PLB155,36 (‘85) ]
H:Hubble constant
Sakharov’s cond.
Due to CP violation, asymmetry of
particle number densities at the
bubble wall occur.
they diffuse into symmetric phase.
Left-handed particle number
densities are converted into B via
sphaleron process.
Sphaleron process is decoupled
after the PT.
B is frozen.
Effective potential
Effective potential T=0
The effective potential is defined by !
Veff (ϕ) = −Γ[ϕ(x) = ϕ]/ d4 x
1-loop effective potential:
i
V1 (ϕ) = −
2
!
"
#
d4 p
−1
ln
det
iG
(p; ϕ) .
4
(2π)
e.g. scalar loop: After a Wick rotation,
1 !
V1 (ϕ) = µ
2
!
#
dD pE " 2
2
ln
p
+
m
(ϕ) ,
E
D
(2π)
D =4−#
After performing the integration, we can get
4
m
V1 (ϕ) =
64π 2
!
MS-bar scheme:
2
"
2
m
3
− − ln 4π + γE + ln 2 − + O(#) + · · · .
#
µ
2
4
m
V1 (ϕ) =
64π 2
!
2
"
m
3
ln 2 −
.
µ
2
μ: renormalization scale
Effective potential at finite T
Imaginary time formalism:
!
4
dk
→ iT
4
(2π)
∞ !
"
n=−∞
#
#
dk
#
(·
·
·
)
#0
(2π)3
k =iωn
3
ωn =
!
2nπT (boson)
(2n + 1)πT (fermion)
1-loop effective potential:
∞ "
!
T
d3 k
2
2
V1 (ϕ, T ) =
ln(wn + w ),
3
2 n=−∞ (2π)
Frequency sum:
∞
!
ω=
z
1
1
= + z
,
2
2
2
z + 4π n
2 e −1
n=−∞
∞
!
z
1
1
= − z
2
2
2
z + (2n + 1) π
2 e +1
n=−∞
#
2
k + m2
(boson)
(fermion)
!
"
3
$%
#
dk w
−w/T
V1 (ϕ, T ) =
+ T ln 1 ∓ e
3
(2π) 2
T4
= V1 (ϕ) + 2 IB,F (a2 ), a2 = m2 (ϕ)/T 2 .
2π
where
T=0
2
IB,F (a ) =
!
0
∞
2
"
√
− x2 +a2
dx x ln 1 ∓ e
#
,
[N.B.]
Since the divergences appear only in the 1st term (T=0
part), the counter terms at T=0 are enough.
High T expansion
For a=m/T<<1, IB,F can be expanded in powers of a.
boson:
4
2
4
!
2
π
π 2 π 2 3/2 a
a
3
IB (a ) = − + a − (a ) −
log
−
45 12
6
32
αB 2
2
fermion:
4
2
4
7π
π 2 a
IF (a ) =
− a −
360 24
32
2
!
2
a
3
log
−
αF
2
log αB = 2 log 4π − 2γE " 3.91,
"
"
+ O(a6 )
+ O(a6 )
log αF = 2 log π − 2γE " 1.14,
Euler’s constant: γE ! 0.577
The bosonic loop gives a cubic term a3 which
comes from a zero frequency mode.
Validity of HTE
Boson case
0.3
HTE 2
IB (a2 ) − IB
(a )
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
a
2
|IB (a ) −
HTE 2
IB (a )|
< 0.05 for a <
∼ 2.3
4
5
Validity of HTE
Fermion case
0.3
IF (a2 ) − IFHTE (a2 )
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
a
2
|IF (a ) −
HTE 2
IF (a )|
< 0.05 for a <
∼ 1.7
4
5
Sphaleron
Sphaleron
■ A static saddle point solution with finite energy of the
gauge-Higgs system.
[N.S. Manton, PRD28 (’83) 2019]
Energy
∆B != 0
Instanton: quantum tunneling
Sphaleron: thermal fluctuation
sphaleron
Esph
instanton
0
-1
Transition rates:
(b)
Γsph
(s)
Γsph
1
Ncs
NCS =
! (αW T )4 e−Esph /T ,
! (αW T )4 ,
∆B = 3∆NCS
g22
32π 2
!
#
"
2
3
d x "ijk Tr Fij Ak − g2 Ai Aj Ak
3
(broken phase)
(symmetric phase),
αW = g22 /4π
B violation is active at finite T but is suppressed at T=0.
no proton decay problem
Sphaleron decoupling condition:
To preserve the generated B after the PT,
(b)
Γsph
Tc3
< H(Tc )
⇒
vc
>
1.
Tc ∼
Hubble parameter:
√
H(T ) ! 1.66 g∗ T 2 /mP
Sphaleron solution for SU(2) gauge-Higgs system
i
Ai (µ, r, θ, φ) = − f (r)∂i U (µ, θ, φ)U −1 (µ, θ, φ),
g2
!
"
#
" #$
v
0
0
Φ(µ, r, θ, φ) = √ (1 − h(r))
+ h(r)U (µ, θ, φ)
.
−iµ
e cos µ
1
2
Ansatz:
Energy functional:
Esph
" # $
%
#
$
2
2
2
4πv ∞
df
8
ξ
dh
λ 2 2
2 2
2
2
dξ 4
=
+ 2 (f − f ) +
+ h (1 − f ) + 2 ξ (h − 1)2 .
g2 0
dξ
ξ
2 dξ
4g2
!
Equation of motion:
ξ = g2 vr
d2
2
1 2
f (ξ) = 2 f (ξ)(1 − f (ξ))(1 − 2f (ξ)) − h (ξ)(1 − f (ξ)),
2
dξ
ξ
4
!
"
d
λ 2
2 dh(ξ)
2
ξ
= 2h(ξ)(1 − f (ξ)) + 2 (h (ξ) − 1)h(ξ).
dξ
dξ
g2
PT in the SM
Order of the PT
This is what the 1st and 2nd order PTs look like.
■ order parameter
= Higgs VEV
■ EWBG requires
1st order PT
vC ≡ lim v "= 0
T ↑TC
Higgs potential
c.t.
Veff (ϕ) = V0 (ϕ) + ∆V (ϕ) + ∆V
c.t.
= V0 (ϕ) + ∆g V (ϕ) + ∆t V (ϕ) + ∆t V (ϕ, T ) + ∆V ,
Tree:
1-loop:
2
2
4
V0 (Φ) = −µ |Φ| + λ|Φ|
!
"
!
"
∆g V (ϕ) = 2 · 3F m2W (ϕ) + 3F m2Z (ϕ) ,
!
"
!
" m4 (ϕ) # m2 (ϕ) 3 $
2
ln
−
∆t V (ϕ) = −4 · 3F mt (ϕ) , F m2 (ϕ) =
2
2
64π
1-loop finite T:
Mren
#
! "
T
∆V (ϕ, T ) = 2
ni IB (a2i ) + nt IF (a2t ) ,
2π i=W,Z
4
Renormalization conditions:
!
∂(∆V (ϕ) + ∆V c.t. ) !!
= 0,
!
∂ϕ
ϕ=v0
!
2
c.t. !
∂ (∆V (ϕ) + ∆V ) !
= 0,
!
2
∂ϕ
ϕ=v0
(vacuum)
(Higgs mass)
2
Higgs potential (cont)
If we use the high T expansion,
2
Veff (ϕ, T ) ! D(T −
T02
1
=
D
B =
D =
E =
λT =
!
"
T02 )ϕ2
λT 4
− ET |ϕ| +
ϕ + ··· ,
4
3
1 2
mh − 2Bv02 ,
4
$
3 # 4
4
4
2m
+
m
−
4m
W
Z
t ,
4
2
64π v0
$
1 # 2
2
2
2m
+
m
+
2m
W
Z
t ,
2
8v0
$
1 # 3
3
−2
2m
+
m
∼
10
,
W
Z
3
4πv0
%
&
'(
2
2
2
2
mW
mh
3
mZ
mt
4
4
4
1 − 2 2 2 2mW log
+ mZ log
− 4mt log
,
2
2
2
2
2v0
8π v0 mh
αB T
αB T
αF T
The critical temperature Tc is given by
2
T
0
Tc2 =
.
2
1 − E /(λTc D)
At Tc
The potential has two degenerate
minima at
2ETc
ϕ = 0, ϕc =
.
λTc
Sphaleron decoupling condition:
Veff
2E
ϕc
>
=
1
∼
Tc
λTc
T>Tc
T=Tc
0
T<Tc
0
50
100
150
200
! (GeV)
250
Since λTc ∼ m2h /2v02 ,
mh <
48
GeV.
∼
300
This upper bound has been excluded by the LEP data.
[N.B.] Higgs mass (λ)
E
strength of the PT
strength of the PT
What is the minimally required value of E for mh=114.4 GeV?
Minimal value of E
ϕc
2E
From the sphaleron decoupling condition,
=
>1
Tc
λTc
Emin
m2h
> 2 ! 0.054,
4v0
for mh = 114.4 GeV
SM contributions:
ESM
!
"
1
3
3
=
2m
+
m
W
Z ! 0.01
3
4πv0
Note: The origin of E is the zero frequency modes of the
bosonic loops.
To have a strong 1st order PT, the extra bosonic degrees
of freedom are needed.
Caveat
“Bosons do not always play a role.”
Suppose that the bosonic particle whose mass is given by
2
2
2
2
2
M = m + g ϕ , m : gauge invariant mass
g : coupling constant
For m2 ! g 2 ϕ2
3
3
Veff ! −g ϕ T ⇒ strengthen the 1st order PT
The loop effect is large.
2
2
For m ! g ϕ
2
3
3
No (−g ϕ T ) term in Veff
The loop effect is vanishing.
2
Requirements: 1. large coupling g, 2. small m .
PT in the MSSM
Stop loop effect
[Carena, Quiros, Wagner, PLB380 (‘96) 81]
The LEP bounds on mH and ρ-parameter constraints demand
mq̃2 ! m2t̃R , Xt2 , Xt = At − µ/ tan β.
■ Stop masses:
m̄2t̃1
m̄2t̃2
yt2
2
!
2
|Xt |
sin β
2
v
,
=
+
+
1−
2
2
mq̃
!
"
2
2
2
yt sin β
|Xt |
2
2
2
2
= mq̃ + Dt̃L +
1+
v
"
m
q̃ ,
2
2
mq̃
m2t̃R
Dt̃2R
soft SUSY breaking masses: m2q̃ , m2t̃R ,
To have a large loop effect,
2
mt̃R
"
= 0 gives
Furthermore
2
mt̃R
Dt̃2L,R ∼ O(g 2 )
should be small.
mt̃1 < mt
Xt = 0 (no-mixing) maximizes the loop effect
Stop loop effect (cont)
Using the High T expansion,
■ Coefficient of cubic term in Veff(T)
Veff ! −(ESM + Et̃1 )T v
3
yt3
3
sin β
Et̃1 ! + √
4 2π
Et̃1 ! 0.054, for Xt = 0
!
2
|Xt |
1−
m2q̃
"3/2
Therefore E = (ESM + Et̃1 ) > Emin = 0.054
Such a light stop can play a role in strengthening the 1st
order PT.
As I wrote before, to have a strong 1st order PT
2
Requirements: 1. large coupling g, 2. small m .
→ top Yukawa yt
→
2
mt̃R
and large statistical factor nt̃ = NC × 2 = 6
=0
.
Updated analysis
[K. Funakubo (Saga U.), E.S.]
Tension in the MSSM BG
tan!
■ The LEP data put a strong constrains on the light Higgs boson.
160
mA (GeV/c2)
(b)
No Mixing, (a)
(a)
No Mixing, (a)
140
120
10
Theoretically
Inaccessible
100
Excluded
by LEP
80
60
40
Theoretically
Inaccessible
1
Excluded
by LEP
20
0
20
40
60
80
100 120 140
2
mh (GeV/c )
0
0
20
40
60
80
100 120 140
mh (GeV/c2)
■ There is a tension between the LEP data and the sphaleron
decoupling in the MSSM.
More precise analysis of the sphaleron decoupling is needed.
Allowed region
■ The allowed region is highly constrained by the experimental
data.
mq̃ = 1200 GeV, mt̃R ! 0, At = Ab = −300 GeV.
Maximal v/T:
tan β = 10.1, mH ± = 127.4 GeV
107.10 GeV
vC
=
= 0.92
TC
116.27 GeV
The sphaleron process is not
decoupled at Tc.
Loophole: supercooling
⇒ The PT begins to proceed
with bubble wall at below Tc.
We need to know the dynamics of bubble wall.
s:
##
#
#
#
#
dh
(ξ)
dh
(ξ)
dθ(ξ)
1
2
dh1 (ξ) ## # = 0, dh2 (ξ) # = 0, dθ(ξ)# ## = 0.
# θ(ξ)
#
0,
=
0,
= 0.
dξ
dξ
lim h1 (ξ) = dξ
0,dξ lim
(ξ)
=
0,
lim
=
0,
# #ξ=0h=
#
#
2
ξ=0
ξ=0
dξ
dξ
Critical
bubble
CP-conserving case
ξ→∞
ξ→∞
ξ=0
ξ→∞ξ=0
(ξ) = 0, lim h2 (ξ) = 0, lim θ(ξ) = 0,
ξ→∞
2
2
ξ=0
(1.30)
ξ→∞
(1.30)
(1.(
CP-conserving case
#
#
#
#
dh1 (ξ) ##= static solution
dh2 (ξ) ##
dθ(ξ)
criticalInbubble
which
is
unstable
against variation
#
# the energy =
the# CP-coserving
case
functional
is
reduced
to
=
0,
0,
=
0.
(1.31)
#
#
#
#
#
dhthe
In
case #the
functional
is reduced
to
dξ ξ=0 dθ(ξ)
dξenergy
dξ!
2 (ξ)CP-coserving
ξ=0
ξ=0
#
%
&!
"
"
'
(
of
radius.
$
2
2
= 0,
=
0,
=
0.
(1.31)
∞
#
#
dρu "2 '
!
$
2% 1&! dρd "2
dξ
dξ
∞
=0
ξ=0
! r2 1 " dρd + dρu! +"Veff (ρd , ρu ; T ) .(
E = 4π ξ=0dr
(2
dr
dr
1
0 + Veff (ρd , ρu ; T ) .
01 dr r ρd 2
E
=
4π
+
Higgs
fields:
CP-conserving Φcase
,
2 , drΦu = √ dr
d = 0√
0 by
EOM for ρd and ρu are
2 given
g caseTheTheEOM
for
ρdenergy
and ρu functional
are given by
!
"to
he CP-coserving
case
the
is reduced
Energy functional:
1 d ! 2 dρd "
2
ρu
∂Veff
% to
&!
"2− 12!d r"dρ
'd + ∂Veff = 0,(
energy functional$is∞reduced
2
∂ρd
2 dr
dρd − r drdρ
u
2 1
r
+
=; T )0, .
!
"
2
dr
r
+
+
V
(ρ
,
ρ
% &! E ="4π
!
"
'
(
eff
d
u
r1 dr
dru
∂ρeffd
2
2
dρ
∂V
d
2
2
dr
dr
dρd
"+
0 dρu
− 2 !r
= 0.
2 1
r
+
+ Veff (ρd , ρu1r; Tdr
)
.
(2.1)
dρ
∂V
d
∂ρueff
2 dru
Veff
dr
EOM2 for ρdr
d and ρu are given by
−
r2
r
+
r=
√
(2
x
(2.1)
(2
= 0.
dr are dr
∂ρ
uthe symmetric phase as
The
boundary
conditions
for
EOM
imposed
in
Equation
of
motion:
!
"
given by
1 d
dρd EOM∂V
eff imposed in the symmetric
2 for
e.g.
1dim
The boundary −
conditions
are
phase
as(2.2)
r
+
=
0,
lim
ρ
(r)
=
0,
lim
ρ
(r)
=
0,
!
"
d
u
2 dr
b.c.
r→∞
r→∞
r
dr
∂ρ
d
1 d
dρd
∂Veff
! lim"ρd (r) = 0, lim ρu (r)
= 0,
=
0,
(2.2)
2
1
dρ
∂V
d
r→∞
r→∞
r dr anddrin the broken
∂ρd −phase asr2 u + eff = 0.
!
"
2 dr
# ∂ρu
#
r
dr
1 d and2 dρ
∂V
#
in uthe broken
eff phase as
dρd (r) #
dρu (r) ##
− 2
r
+
= 0.
(2.3)
= 0,
= 0.
#
#
#
#
r dr
dr
∂ρfor
dr
dr(r) #r=0 phase as
The boundary
conditions
EOM aredρimposed
in
the
symmetric
u
#
r=0
(r)
dρ
d
u
−
r2
(2
+
(2.3)
(2
T=Tc
#
#
= 0,
= 0.0
#
#
dr phase
dr r=0
It is convenient
parameterize
the
Higgs
profiles
(ρd , ρu ) in termsT=T
of Nthe (2.4)
dimensionl
for EOM are imposed
in lim
the to
as
r=0
ρsymmetric
(r)
=
0,
lim
ρ
(r)
=
0,
d
u
r→∞ change variables
r→∞ as
quantities.
We be
thus
Bubbles
can
nucleated
at
below
Tc.
It
is
convenient
to
parameterize
the
Higgs
profiles
(ρ , ρ ) in terms of the dimensio
lim ρ (r) = 0, lim ρ (r) = 0,
(2.4)
d
inr→∞
the broken
phase
as u
r→∞
ρd (r)
d
0
u
ρu (r)
50
100
150
200
v [GeV]
250
300
Ecb (TBubble
) is the energy
of the critical bubble at temperature T 2 . Note that this is
Nucleation
per unit volume. We define the nucleation temperature TN as the temperature
to of
[3],nucleation
the bubbleofnucleation
per unit
time
per unitvolume
hccording
the rate
a criticalrate
bubble
within
a horizon
volumeisisgiven
equalbyto
! Since "
bble parameter at that temperature.
the
3/2 horizon scale is roughly given by
4 Ecb (T3)
, the
nucleation rate:
temperature
by
ΓN (T ) is
% defined
T
e−Ecb (T )/T ,
(4.
Nucleation
Bubble nucleation
2πT
Nucleation T:
[A.D. Linde, NPB216 (’82) 421]
ΓN (TN )H(TN )−3 = H(TN ).
(4.2)
5
he Hubble parameter at temperature T is
H(T ) =
∗ (T )
!
8πG
ρ(T ) =
3
"
Numerical results: (preliminary)
2
#1/2
2
8π π
T
116.73
v
4
1/2
N
g
(T
)T
!
1.66
g
(T
)
,
∗
∗ 1.01
=
=
2
3mP 30
mP
TN
115.59
is the effective massless degrees of freedom
at T defined by
10% enhancement!
(4.3)
But,
$
7$
g∗ (T ) =
θ(T − mB (T ))gB Sphaleron
+
θ(T −decoupling
mF (T ))gF , cond.@T(4.4)
N:
8 F
B
E = 1.77, Ntr = 6.65, Nrot = 12.27
and gF being intrinsic degrees of freedom of boson B and
v fermion F , respectively,
>
1.35
nition of TN (4.2)ρd is
reduced
to
(r)
ρu (r)
ξ = vr, h1 (ξ) =
, h2 (ξ) =
" v cos β #3/2 v sin β
■ The
T
4
Ecb (TN )
T
sphaleron process
isN not
decoupled
e−Ecb (TN )/T
= 7.59
g∗ (TN )2 N4 ,at
2πT
m
TN either.
(4.5)
More examples
−4
−4
Table 1: Examples
of
the
1st
order
PT.
|A
|
=
|A
|
=
300
GeV,
m
=
10
b
t
Ab = At = −300 GeV, mt̃R = 10 GeV, mb̃Rt̃R= 1000 GeV,
GeV,mb̃R = 1000
GeV, |µ| = µ
100=GeV,
2 = 500 GeV, δA ≡ δAt = δAb = π, δµ = 0.
100 M
GeV,
M = 500 GeV,
2
mq̃ (GeV)
tan β
mH ± (GeV)
vC /TC
tan βC
vN /TN
tan βN
Ecb /(4πv0 )
Ecb /TN
Esph /(4πv0 /g2 )
Ntr
Nrot
vN /TN >
1200
10.11
127.30
107.095
= 0.921
116.274
13.812
116.726
= 1.010
115.585
13.684
5.623
150.386
1.7686
6.6522
12.266
1.345
1300
9.87
127.40
107.512
= 0.923
116.496
13.640
117.155
= 1.012
115.798
13.503
5.633
150.379
1.7695
6.6576
12.253
1.344
1400
9.75
127.40
107.768
= 0.923
116.770
13.606
117.403
= 1.012
116.068
13.462
5.646
150.369
1.7704
6.6623
12.241
1.344
1500
9.57
127.40
107.914
= 0.922
117.045
13.465
117.530
= 1.010
116.340
13.317
5.659
150.360
1.7711
6.6666
12.230
1.343
Typically, vN /TN > 1.34 is needed for sphaleron decoupling.
More examples
−4
−4
Table 1: Examples
of
the
1st
order
PT.
|A
|
=
|A
|
=
300
GeV,
m
=
10
b
t
Ab = At = −300 GeV, mt̃R = 10 GeV, mb̃Rt̃R= 1000 GeV,
GeV,mb̃R = 1000
GeV, |µ| = µ
100=GeV,
2 = 500 GeV, δA ≡ δAt = δAb = π, δµ = 0.
100 M
GeV,
M = 500 GeV,
2
mq̃ (GeV)
tan β
mH ± (GeV)
vC /TC
tan βC
vN /TN
tan βN
Ecb /(4πv0 )
Ecb /TN
Esph /(4πv0 /g2 )
Ntr
Nrot
vN /TN >
1200
10.11
127.30
107.095
= 0.921
116.274
13.812
116.726
= 1.010
115.585
13.684
5.623
150.386
1.7686
6.6522
12.266
1.345
1300
9.87
127.40
107.512
= 0.923
116.496
13.640
117.155
= 1.012
115.798
13.503
5.633
150.379
1.7695
6.6576
12.253
1.344
1400
9.75
127.40
107.768
= 0.923
116.770
13.606
117.403
= 1.012
116.068
13.462
5.646
150.369
1.7704
6.6623
12.241
1.344
1500
9.57
127.40
107.914
= 0.922
117.045
13.465
117.530
= 1.010
116.340
13.317
5.659
150.360
1.7711
6.6666
12.230
1.343
Typically, vN /TN > 1.34 is needed for sphaleron decoupling.
More examples
−4
−4
Table 1: Examples
of
the
1st
order
PT.
|A
|
=
|A
|
=
300
GeV,
m
=
10
b
t
Ab = At = −300 GeV, mt̃R = 10 GeV, mb̃Rt̃R= 1000 GeV,
GeV,mb̃R = 1000
GeV, |µ| = µ
100=GeV,
2 = 500 GeV, δA ≡ δAt = δAb = π, δµ = 0.
100 M
GeV,
M = 500 GeV,
2
mq̃ (GeV)
tan β
mH ± (GeV)
vC /TC
tan βC
vN /TN
tan βN
Ecb /(4πv0 )
Ecb /TN
Esph /(4πv0 /g2 )
Ntr
Nrot
vN /TN >
1200
10.11
127.30
107.095
= 0.921
116.274
13.812
116.726
= 1.010
115.585
13.684
5.623
150.386
1.7686
6.6522
12.266
1.345
1300
9.87
127.40
107.512
= 0.923
116.496
13.640
117.155
= 1.012
115.798
13.503
5.633
150.379
1.7695
6.6576
12.253
1.344
1400
9.75
127.40
107.768
= 0.923
116.770
13.606
117.403
= 1.012
116.068
13.462
5.646
150.369
1.7704
6.6623
12.241
1.344
1500
9.57
127.40
107.914
= 0.922
117.045
13.465
117.530
= 1.010
116.340
13.317
5.659
150.360
1.7711
6.6666
12.230
1.343
Typically, vN /TN > 1.34 is needed for sphaleron decoupling.
More examples
−4
−4
Table 1: Examples
of
the
1st
order
PT.
|A
|
=
|A
|
=
300
GeV,
m
=
10
b
t
Ab = At = −300 GeV, mt̃R = 10 GeV, mb̃Rt̃R= 1000 GeV,
GeV,mb̃R = 1000
GeV, |µ| = µ
100=GeV,
2 = 500 GeV, δA ≡ δAt = δAb = π, δµ = 0.
100 M
GeV,
M = 500 GeV,
2
mq̃ (GeV)
tan β
mH ± (GeV)
vC /TC
tan βC
vN /TN
tan βN
Ecb /(4πv0 )
Ecb /TN
Esph /(4πv0 /g2 )
Ntr
Nrot
vN /TN >
1200
10.11
127.30
107.095
= 0.921
116.274
13.812
116.726
= 1.010
115.585
13.684
5.623
150.386
1.7686
6.6522
12.266
1.345
1300
9.87
127.40
107.512
= 0.923
116.496
13.640
117.155
= 1.012
115.798
13.503
5.633
150.379
1.7695
6.6576
12.253
1.344
1400
9.75
127.40
107.768
= 0.923
116.770
13.606
117.403
= 1.012
116.068
13.462
5.646
150.369
1.7704
6.6623
12.241
1.344
1500
9.57
127.40
107.914
= 0.922
117.045
13.465
117.530
= 1.010
116.340
13.317
5.659
150.360
1.7711
6.6666
12.230
1.343
Typically, vN /TN > 1.34 is needed for sphaleron decoupling.
Is MSSM BG dead?
It looks almost dead, but there might be a way out.


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]
⇒ 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 EW bayogenesis focusing on the
EWPT.
In the SM, the PT is not 1st order for the viable
Higgs mass. mH > 114.4 GeV
In the MSSM, the 1st order PT can be strengthen
by the light stop. mt̃1 < mt
However, it is found that 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.
Review papers
A.G. Cohen, D.B. Kaplan, A.E. Nelson, hep-ph/9302210
M. Quiros, Helv.Phys.Acta 67 (’94)
V.A. Rubakov, M.E. Shaposhnikov, hep-ph/9603208
K. Funakubo, hep-ph/9608358
M. Trodden, hep-ph/9803479
A. Riotto, hep-ph/9807454
W. Bernreuther, hep-ph/0205279