Electroweak phase transitions in the MSSM with an extra $U(1)'$
Electroweak phase transitions in the MSSM with an
extra U (1)′
S.W. Ham(1), E.J. Yoo(2), and S.K. Oh(1,2)
(1) Center for High Energy Physics, Kyungpook National University,
Daegu 702-701, Korea
(2) Department of Physics, Konkuk University, Seoul 143-701, Korea
Abstract
We investigate the possibility of electroweak phase transition in the minimal
supersymmetric standard model (MSSM) with an extra U(1)′. This model has two
Higgs doublets and a singlet, in addition to a singlet exotic quark superfield. We
find that at the one-loop level this model may accommodate the electroweak phase
transitions that are strongly first-order in a reasonably large region of the parameter
space. In the parameter region where the phase transitions take place, we observe
that the lightest scalar Higgs boson has a smaller mass when the strength of the
phase transition becomes weaker. Also, the other three heavier neutral Higgs bosons
get more large masses when the strength of the phase transition becomes weaker.
http://arxiv.org/abs/0704.0328v1
I. INTRODUCTION
The baryon asymmetry of the universe can be dynamically generated during the evolution
of the universe, if the mechanism of baryogenesis satisfies the three Sakharov conditions
[1]. The three Sakharov conditions are: the presence of baryon number violation, the
violation of both C and CP, and a deviation from thermal equilibrium. It is known that
the universe can escape out of the thermal equilibrium by means of electroweak phase
transition, which should be strongly first-order in order to ensure sufficient deviation from
thermal equilibrium to generate the baryon asymmetry that is observed today. However,
it has been already recognized that the Standard Model (SM) has some difficulty to realize
the desired electroweak phase transition. The present experimental lower bound on the
mass of the SM Higgs boson does not allow the electroweak phase transition to be strongly
first-order [2, 3]. The electroweak phase transition is weakly first-order or higher order in
the SM. Thus, the SM is inadequate to generate sufficient baryon asymmetry. Moreover,
the amount CP violation in the Cabibbo-Kobayashi-Maskawa (CKM) matrix is too small
to account for the baryon asymmetry of the observed universe [4].
Consequently, new physical models beyond the SM have extensively been studied for
the possibility of reasonable explanation of the baryon asymmetry of the universe. Espe-
cially, the low energy supersymmetric models have been studied widely within the context
of electroweak baryogenesis [5-7]. The simplest supersymmetric model that includes the
SM is the minimal supersymmetric standard model (MSSM), which possesses in its su-
perpotential the µ term that accounts for the mixing between two Higgs doublets. The
µ parameter, which has the mass dimension, causes some problem with respect to its
energy scale [8]. Several possibilities have been investigated in the literature to solve the
so-called µ problem [9-12]. Introducing an additional U(1)′ to the MSSM is one of the
plausible explanations for the µ problem of the MSSM.
The MSSM with an extra U(1)′ can not only solve the µ problem but we will show
that it can also overcome the difficulties that the SM encounters when the SM tries to
satisfy the Sakharov conditions. This model can accommodate sufficient CP violation,
because it possesses other sources of CP violation besides the CKM matrix. It is possible
to realize the explicit CP violation in this model by means of complex CP phases arising
from the soft SUSY breaking terms [12].
Then, it is the purpose of this paper to show that this model indeed allows the strongly
first-order electroweak phase transitions such that it can successfully explain the baryo-
genesis. The characteristics of the electroweak phase transitions are determined essen-
tially by the temperature-dependent part of the Higgs potential. We construct the full
temperature-dependent Higgs potential at the one-loop level, and examine if the elec-
troweak phase transition may be strongly first-order. Two methods are employed for the
construction of the temperature-dependent Higgs potential. One method assumes that
the critical temperature at which the electroweak phase transition occurs is relatively
high, thus the temperature-dependent effective potential is approximated by retaining
only terms proportional to T 2, whereas the other method carries out numerically exact
integrations of the temperature-dependent effective potential. The thermal effects of par-
ticles whose masses are comparatively smaller than the critical temperature are included
at the one-loop level in the former method, whereas the particle content is different in the
latter method.
Either way, we obtain almost the same physical results. Unlike the MSSM, this model
allows a strongly first-order electroweak phase transition in a wide region of the parame-
ter space, and the first-order electroweak phase transition can be strong enough without
requiring a light stop quark. An interesting behavior of this model with respect to the
strongly first-order electroweak phase transition is that the mass of the lightest neutral
Higgs boson becomes larger when the phase transition gets stronger. On the other hand,
the masses of the other three neutral Higgs bosons become smaller when the phase tran-
sition gets stronger.
II. ZERO TEMPERATURE
The MSSM with an extra U(1)′ accommodates in its Higgs sector two Higgs doublets
H1 = (H
1 , H
1 ), H2 = (H
2 , H
2 ), and one Higgs singlet, S. In terms of these Higgs fields,
the relevant part of the superpotential of this model may be written as
W ≈ htQH2t
R + hbQH1b
R + hkSDLD̄R − λSH
ǫH2 , (1)
where we take into account only the third generation: tcR and b
R are, respectively, the
right-handed singlet top and bottom quark superfields, DR is the right-handed singlet
exotic quark (a vector-like down quark) superfield, Q is the left-handed SU(2) doublet
quark superfield of the third generation, and DL is the left-handed singlet exotic quark
superfield. Further, ht, hb and hk are, respectively, the dimensionless Yukawa coupling
coefficients of top, bottom, and exotic quark superfields, and ǫ is an antisymmetric 2× 2
matrix with ǫ12 = 1.
From the superpotential, at zero temperature, we can construct the Higgs potential
at the tree level, which may be read as
V0 = VF + VD + VS , (2)
where
VF = |λ|
2[(|H1|
2 + |H2|
2)|S|2 + |HT
1~σH1 +H
2~σH2)
(|H1|
2 − |H2|
(Q̃1|H1|
2 + Q̃2|H2|
2 + Q̃3|S|
2)2 ,
VS = m
2 +m2
2 +m2
|S|2 − [λAλ(H
ǫH2)S +H.c.] , (3)
where ~σ denotes the three Pauli matrices, g1, g2, and g
are the U(1), SU(2), and U(1)′
gauge coupling constants, respectively, Q̃1, Q̃2, and Q̃3 are the U(1)
′ hypercharges of H1,
H2, and S, respectively, and m
i (i = 1, 2, 3) are the soft SUSY breaking masses. In the
Higgs potential, λ and Aλ may in general be complex numbers. However, they will be
assumed to be real in the subsequent discussions, as we do not consider CP violation in
the Higgs sector. The soft masses are also assumed to be real, without loss of generality,
and they are eventually eliminated by imposing minimum conditions with respect to the
neutral Higgs fields, The gauge invariance of the superpotential under of U(1)′ requires
that the three U(1)′ hypercharges should satisfy Q̃1 + Q̃2 + Q̃3 = 0.
The above Higgs potential at the tree level would allow the three neutral Higgs fields
, and S to develop the vacuum expectation values (VEVs) v1(0), v2(0), and s(0),
respectively. Remark that these VEVs are obtained at zero temperature. However, for
simplicity, we omit the temperature dependence of these VEVs until next section where
we take into account the finite temperature effect.
The tree-level Higgs potential should now be corrected by the radiative one-loop effects.
In SUSY models, the radiative corrections due to the top and stop quarks contribute
most dominantly to the tree-level Higgs sector. Besides, if tanβ = v2/v1 is very large,
the radiative corrections due to the bottom and sbottom quarks should also be included
since they become no longer negligible. Furthermore, the radiative corrections due to the
exotic quark and squark may be important since the Yukawa coupling of the exotic quark
to the singlet field S can be large at the electroweak scale [11]. Therefore, we take into
account all the contributions from the top, bottom, exotic quark sector to the tree-level
Higgs potential.
The one-loop radiative corrections are evaluated by the effective potential method [13].
We assume that the squark masses are degenerate. Ignoring the mixings in the masses of
the squarks [14], the one-loop effective potential is given by
l=t,b,k
+ log
m̃2 +M2l
, (4)
where t, b, and k, respectively are top, bottom, and exotic quark fields including the
corresponding squark fields, Mt = ht|H2|, Mb = hb|H1|, Mk = hk|S| are the field-
dependent quark masses, and m̃ is the soft SUSY breaking mass, which is assumed that
m̃ = 1000 GeV ≫ mq (q= t, b, or k).
The Higgs sector of the present model consists of six physical Higgs bosons: a pair
of charged Higgs boson, one neutral pseudoscalar Higgs boson, and three neutral scalar
Higgs bosons. The tree-level mass of the charged Higgs boson is given by
m2C± = m
W − λ
2v2 +
2λAλs
sin 2β
, (5)
where v =
v21 + v
2 = 175 GeV and m
W = g
2/2 is the squared mass of the W boson.
At the tree level, the mass of the charged Higgs boson might be either smaller or larger
than the W boson mass.
The tree-level mass of the neutral pseudoscalar Higgs boson is given by
m2A =
2λAλv
sin 2α
, (6)
where tanα = (v/2s) sin 2β implies the splitting between the electroweak symmetry break-
ing scale and the extra U(1)′ symmetry breaking scale. Note that these tree-level masses
of both the neutral pseudoscalar and the charged Higgs bosons do not receive any radiative
corrections, because the squark masses are degenerate.
The tree-level squared masses of the three neutral scalar Higgs bosons are considerably
affected by the radiative corrections. Their squared masses at the one-loop level are given
as the eigenvalues of the 3×3 one-loop level mass matrix, whose elements may be written
M11 = m
Z cos
2 β + 2g
v2 cos2 β +m2A sin
2 β cos2 α + fa(m
M22 = m
Z sin
2 β + 2g
v2 sin2 β +m2A cos
2 β cos2 α + fa(m
t ) ,
M33 = 2g
2 +m2A sin
2 α + fa(m
M12 = g
Q̃1Q̃2v
2 sin 2β + (λ2v2 −m2Z/2) sin 2β −m
A cos β sin β cos
2 α ,
M13 = 2g
1 Q̃1Q̃3vs cos β + 2λ
2vs cosβ −m2A sin β cosα sinα ,
M23 = 2g
Q̃2Q̃3vs sin β + 2λ
2vs sinβ −m2A cos β cosα sinα , (7)
where m2Z = (g
)v2/2 is the squared mass of the Z boson, and the function fa(m
is defined as
3h2qm
m̃2 +m2q
4h2qm
m̃2 +m2q
(m̃2 +m2q)
. (8)
We assume that the masses of three scalar Higgs bosons Si are sorted such that mS1 ≤
mS2 ≤ mS3 .
III. FINITE TEMPERATURE
Now, let us study the temperature dependence of the Higgs potential in order to inves-
tigate the nature of the electroweak phase transition in the MSSM with an extra U(1)′.
We evaluate VT , the temperature-dependent part of the Higgs potential at the one-loop
level, using the effective potential method. It is given as [15]
l=B,F
dx x2 log
1± exp
x2 +m2l (φi)/T
, (9)
where B and F stand for bosons (t̃, b̃, and k̃) and fermions (t, b, and k), and nt = nb =
nk = −12 and nt̃ = nb̃ = nk̃ = 12. The negative sign is for bosons and the positive sign
is for fermions. Thus, the full Higgs potential at finite temperature at the one-loop level
is given by
V (T ) = V0 + V1 + VT (10)
For numerical analysis, we need to set the values of the relevant parameters of the
model. As in the previous section, the soft SUSY breaking mass is set as m̃ = 1000 GeV.
The quark masses are set as mt = 175 GeV, mb = 4 GeV, and mk = 400 GeV. From
these values, mq̃ =
m̃2 +m2q (q = t, b, k) yield the squark masses as mt̃ = 1015 GeV,
= 1000 GeV, and m
= 1077 GeV.
Some caution should be taken for setting the values of Q̃i (i=1, 2, 3), the U(1)
hypercharges of the Higgs doublets and the Higgs singlet. In the MSSM with an extra
U(1)′, the extra neutral gauge boson mass (mZ′) and the mixing angle (αZZ′) between the
two neutral gauge bosons (Z,Z ′) may impose strong constraints on the parameter values.
For our numerical analysis, mZ′ is estimated to be larger than 600 GeV, and αZZ′ smaller
than 2 × 10−3, for tan β = 3 and s(T = 0) = 500 GeV. Besides, as recent research has
suggested [10], we impose the constraint of Q̃1Q̃2 > 0. Further, the U(1)
′ gauge invariance
condition requires that Q̃3 = −(Q̃1 + Q̃2).
In this paper, we define new charges Qi = g
1Q̃i since Q̃i appear always together with
. Then, one may establish the allowed area in the (Q1, Q2)-plane by imposing the above
constraints. For tanβ = 3 and s(T = 0) = 500 GeV, the result is shown in Fig. 1, where
the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner of Fig. 1
are the allowed areas. The hatched region is the excluded area. There are two specific
points in Fig. 1, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at
the star-marked point correspond to the ν-model of E6 gauge group realizations [11]. We
would take the values of Q1 and Q2 at the cross-marked point, namely, (Q1, Q2) = (-1,
-0.1), and hence Q3 =1.1.
With these parameter values at hand, we would investigate the possibility of the
strongly first-order electroweak phase transition by using two different ways. The first
method is to retain only the dominant T 2-proportional part from the high-temperature
approximation of VT , and to take account only those particles whose masses are relatively
small [6]. The second method is to perform the integration in VT in numerically exact way,
and to consider only the contributions of top, bottom, and exotic quarks and squarks.
1. Method A
Let us start with the high temperature approximation of VT , which is expressed as [3]
VT ≈ −
i=t,b,k
T 2m2i (φi)
m4i (φi)
m2i (φi)
cFT 2
i=t̃,b̃,k̃
T 2m2i (φi)
Tm3i (φi)
m4i (φi)
m2i (φi)
cBT 2
, (11)
where log cF = 2.64 and log cB = 5.41. It is known that in the SM the high temperature
approximation is consistent with the exact integration of VT within 5 % at temperature
T for mF/T < 1.6 and mB/T < 2.2, where mF and mB are respectively the fermion mass
and the boson mass that participate in the potential.
We select those terms that are proportional to T 2 in the above expression, which
become most dominant at high temperature. Thus, we assume that the temperature at
which the electroweak phase transition takes place is sufficiently high. We also assume
that the U(1) and SU(2) gaugino masses M1 and M2 in the chargino and neutralino
sectors are very much larger than the other mass parameters. We take into account the
thermal effects due to the Higgs bosons, W , Z, and the extra U(1) gauge boson in the
boson sector, and t, b, k quarks, the lighter chargino, and the three light neutralinos in the
fermion sector, because their masses are relatively small as compared with temperature,
similarly to the analyses of previous articles [6]. Explicitly, the T 2 terms in the high
temperature approximation of VT can be expressed as
+ 4m2
+ 2m2
+ (2g2
+ 6g2
+ 6λ2)(|H1|
2 + |H2|
2) + 12λ2|S|2
+ 12g
2 + Q̃2
2 + Q̃2
|S|2) + 2g
Q̃1Q̃2(|H1|
2 + |H2|
Q̃2Q̃3(|H2|
2 + |S|2) + 2g
Q̃1Q̃3(|H1|
2 + |S|2)
1 (Q̃1 + Q̃2)(Q̃1|H1|
2 + Q̃2|H2|
2 + Q̃3|S|
+6(h2t |H2|
2 + h2b |H1|
2 + h2k|S|
. (12)
Now, the neutral scalar Higgs fields develop the temperature-dependent VEVs, v1(T ),
v2(T ), and s(T ), which we will simply denote v1, v2, and s, respectively. In terms of
these temperature-dependent VEVs, the vacuum at finite temperature is defined as the
minimum of V (T ) as
〈V (v1, v2, s, T )〉 = 〈V0〉+ 〈V1〉+ 〈VT 〉 , (13)
where
〈V0〉 = m
g21 + g
)2 + λ2(v2
s2 + v2
− 2λAλv1v2s+
(Q̃1v
+ Q̃2v
+ Q̃3s
2)2 ,
〈V1〉 = fb(m
t ) + fb(m
b) + fb(m
〈VT 〉 =
+ 4m2
+ 2m2
+ (2g2
+ 6g2
+ 6λ2)(v2
) + 12λ2s2
+ 12g
+ Q̃2
+ Q̃2
s2) + 2g
Q̃1Q̃2(v
1 Q̃2Q̃3(v
2 + s
2) + 2g
1 Q̃1Q̃3(v
1 + s
1 (Q̃1 + Q̃2)(Q̃1v
1 + Q̃2v
2 + Q̃3s
2) + 6(h2tv
2 + h
1 + k
. (14)
In the above expressions, the function fb is defined as
+ log
m̃2 +m2q
, (15)
and the soft SUSY breaking masses at the one-loop level are given as
cos 2β − λ2(s(0)2 + v(0)2 sin2 β) + λAλs(0) tanβ
Q̃1(Q̃1v(0)
2 cos2 β + Q̃2v(0)
2 sin2 β + Q̃3s(0)
2)− fc(m
b(0))
cos 2β − λ2(s(0)2 + v(0)2 cos2 β) + λAλs(0) cotβ
Q̃2(Q̃1v(0)
2 cos2 β + Q̃2v(0)
2 sin2 β + Q̃3s(0)
2)− fc(m
t (0))
= − λ2v(0)2 +
2s(0)
v(0)2Aλ sin 2β
Q̃3(Q̃1v(0)
2 cos2 β + Q̃2v(0)
2 sin2 β + Q̃3s(0)
2)− fc(m
k(0)) , (16)
where v1(0), v2(0), and s(0) are the VEVs evaluated at zero temperature in the preceding
section, tan β = v2(0)/v1(0), v(0) =
v1(0)2 + v2(0)2 = 175 GeV, and the function fc is
defined as
3h2qm
2 + 2 log
m̃2 +m2q
m̃2 +m2q
. (17)
Now, let us determine the critical temperature at which the electroweak phase tran-
sition takes place. In our analysis, the critical temperature is defined by a temperature
at which 〈V (T )〉 has two distinct minima with equal value, that is, a pair of degenerate
vacua. In order to have a pair of degenerate vacua, the potential 〈V (T )〉 should satisfy
the minimum condition of
0 = 2m2
s− 2λAλv1v2 + 2λ
1 Q̃3s(Q̃1v
1 + Q̃2v
2 + Q̃3s
2) + 2h2kmkfc(m
s[24λ2 + 24g
+ 20g
Q̃3(Q̃1 + Q̃2) + 12k
2] , (18)
which is obtained by calculating the first derivative of the full effective potential at the
finite temperature with respect to s.
For given parameter values at given temperature, one may solve the above minimum
condition to express s in terms of the other two VEVs, v1 and v2. Then, by substituting
s into 〈V (v1, v2, s, T )〉, one may obtain 〈V (v1, v2, T )〉 which depends only on v1 and v2.
By inspecting the shape of 〈V (v1, v2, T )〉 on the (v1, v2)-plane for given parameter values
at given temperature, we may determine whether it possess a pair of degenerate vacua or
In Fig. 2, the equipotential contours of 〈V (v1, v2, T )〉 are plotted on the (v1, v2)-
plane, where the parameter values are set as tanβ = 3, λ = 0.8, s(0) = 500 GeV,
mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the
critical temperature Tc. One can easily spot two distinct minima of 〈V (v1, v2, T )〉 on
the (v1, v2)-plane, namely, one at (0, 0) and the other at (275, 640) GeV. The phase of
the state is symmetric at the minimum point (0, 0) on the (v1, v2)-plane, whereas it is
broken at (275, 640) GeV. The electroweak phase transition may take place from (0, 0) to
(275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is
first-order.
The distance on the (v1, v2)-plane between the two minima of 〈V (v1, v2, T )〉, defined
as vc, determines the strength of the electroweak phase transition. The electroweak phase
transition is said to be strong if vc/Tc > 1, and weak otherwise. In Fig. 2, the distance is
calculated to be
(275− 0)2 + (640− 0)2 = 696 (GeV) . (19)
In Fig. 2, the strength of the electroweak phase transition is about vc/Tc = 6.9, which
definitely tells that the electroweak phase transition is a strong one. Therefore, the
particular parameter values set for Fig. 2 yields an electroweak phase transition which
is first-order as well as strong. Note that vc does not depend on s, that is, we need not
to know the values of s at the two minima to calculate vc. Actually, vc is the VEV at
the broken phase. The masses of the neutral scalar Higgs bosons at zero temperature
for the parameter values of Fig. 2 are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and
mS3 = 1827 GeV.
We repeat the above job of analysis, varying the values of the relevant parameters. We
find that there are a large number of sets of parameter values that allow strongly first-order
electroweak phase transitions. Thus, the MSSM with an extra U(1)′ may accommodate
TABLE 1: Some sets of λ and mA that allow strongly first-order electroweak phase
transitions in the MSSM with an extra U(1)′, obtained by Method A. The values of other
parameters are fixed as tanβ = 3, s(0) = 500 GeV, m̃ = 1000 GeV, and Tc = 100
GeV. The pair of numbers in the third column are the coordinates of the broken-phase
minimum of 〈V (v1, v2, T )〉. The coordinates of its symmetric-phase minimum is (0, 0)
for all sets. The three numbers in the fourth column are the masses of S1, S2, and S3,
respectively. The number in the last column is the strength of the first-order electroweak
phase transition.
λ mA (GeV) (v1, v2) (GeV) mS1 , mS2 , mS3 (GeV) vc/Tc
0.1 478 (1750, 1650) 120, 524, 792 26
0.2 675 (1400, 1500) 118, 674, 796 23
0.3 900 (1200, 1400) 112, 786, 908 18
0.4 1109 (870, 1200) 104, 792, 1112 15
0.5 1306 (600, 1000) 93, 796, 1307 12
0.6 1486 (430, 850) 82, 800, 1485 8
0.7 1660 (340, 700) 70, 803, 1658 7
0.8 1830 (275, 640) 56, 807, 1827 6.9
the desired phase transitions for a wide region in its parameter space. Some of the results
are listed in Table 1, where tanβ = 3, s(0) = 500 GeV, and T = 100 GeV are fixed as
the values set in Fig. 2, whereas λ and mA have different values. The set of numbers in
the last row of Table 1 is the numerical result of Fig. 2.
Every set of numbers in each row of Table 1 gives 〈V (v1, v2, T )〉 a pair of degenerate
minima, the minimum of symmetric phase at (0, 0) on the (v1, v2)-plane, and the one of
broken phase at a different point on the (v1, v2)-plane as given in Table 1. The electroweak
phase transition is strongly first-order. One may easily observe in Table 1 that, as the
value of λ increases, a larger value of mA allow desired phase transitions. On the other
hand, the strength of the phase transition is reinforced if the value of λ decreases.
The masses of the neutral scalar Higgs bosons exhibit some interesting behavior. For a
larger value of mA, both S2 and S3 have also larger masses whereas S1 has a smaller mass.
The tendency is that the strength of the phase transition is reinforced if mS1 increases
and if mA, mS2 , and mS3 decrease. In the SM, the strength of the first order electroweak
phase transition decreases if its single Higgs boson mass is increased. Also, in the MSSM,
we have a weaker phase transition if the lighter one of its two scalar Higgs bosons has a
larger mass. In this regard, the tendency of our model is opposite to those of the SM or
the MSSM. One can see that this strange behavior also occurs in some parameter region
of a non-minimal SUSY model, as shown in Fig. 3 of Ref. [7].
2. Method B
The second method evaluates VT by exact integration to obtain the temperature-dependent
full potential V (T ) at one-loop level, where the thermal effects of top, bottom, and exotic
quarks and squarks are taken into account. The thermal effects of the gauge bosons can
be a help for strengthening the first-order electroweak phase transition, but we would
omit them, since the strength of the phase transition is already strong enough.
This method starts with the exact integral expression for 〈VT 〉 after replacing the
neutral Higgs fields by their VEVs as
〈VT 〉 = −
l=t,b,k
dx x2 log
1− exp
m2l (v1, v2, s)
l=t̃,b̃,k̃
dx x2 log
1 + exp
m̃2 +m2l (v1, v2, s)
 ,(20)
which is different from 〈VT 〉 of Method A, while 〈V0〉 and 〈V1〉 are the same as those of
Method A. From the full 〈V (T )〉 = 〈V0〉 + 〈V1〉 + 〈VT 〉, we obtain a minimum condition
for degenerate vacua as
0 = 2m2
s− 2λAλv1v2 + 2λ
)s+ 2g
Q3s(Q̃1v
+ Q̃2v
+ Q̃3s
+ 2h2kmkfc(m
dx x2
2h2ks exp(−
x2 +m2k/T
x2 +m2k/T
1 + exp(−
x2 +m2k/T
dx x2
2h2ks exp(−
x2 + (m̃2 +m2k)/T
x2 + (m̃2 +m2k)/T
1 + exp(−
x2 + (m̃2 +m2k)/T
] , (21)
where mk depends only on s and is independent from v1 and v2.
Solving the above minimum condition is harder than solving the corresponding mini-
mum condition of Method A. Nevertheless, we can solve it by using the bisection method
to express s in terms of the other parameters. Then, eliminating s from 〈V (T )〉, we can
obtain the expression for 〈V (v1, v2, T )〉 which depends only on v1 and v2. Subsequent
steps of numerical analysis are the same as the previous method.
In Fig. 3, equipotential contours of 〈V (v1, v2, T )〉 obtained by the present method is
plotted on the (v1, v2)-plane, where the parameter values are set slightly different from
the previous method: tan β = 3, λ = 0.8, s(0) = 500 GeV, mA = 1780 GeV, and T = 100
GeV. The shape of the equipotential contours of Fig. 3 is almost the same as that of
Fig. 2. One can see that there are two distinct minima in Fig. 3, just like Fig. 2:
one at (0, 0), and the other at (165, 440) GeV on the (v1, v2)-plane, indicating that the
phase transition is first order. The strength of the first-order phase transition is strong,
since vc/Tc = 4.7. The masses of the three scalar Higgs bosons are evaluated at zero
temperature as mS1 = 82 GeV, mS2 = 804 GeV, and mS3 = 1777 GeV.
TABLE 2: Some sets of λ and mA that allow strongly first-order electroweak phase
transitions in the MSSM with an extra U(1)′, obtained by Method B. Other descriptions
are the same as Table 1.
λ mA GeV (v1B, v2B) GeV mSi GeV vc/Tc
0.1 462 (1600, 1600) 121, 468, 791 22
0.2 663 (1400, 1400) 118, 662, 795 19
0.3 885 (1100, 1100) 113, 785, 894 15
0.4 1095 (800, 1200) 106, 792, 1098 14
0.5 1287 (680, 990) 97, 796, 1288 12
0.6 1457 (400, 750) 91, 799, 1456 8
0.7 1620 (300, 600) 86, 801, 1618 6
0.8 1780 (165, 440) 82, 804, 1777 4.7
Comparing Fig. 3 with Fig. 2, one may safely remark that Method A and Method
B lead qualitatively the same results. Either method, whether 〈VT 〉 is calculated by
direct integration or is simplified by high-temperature approximation, and whether the
participating particles at the one-loop level are somewhat exhaustive or selective, we
find that the MSSM with and extra U(1)′ allows strongly first-order electroweak phase
transitions for certain region in its parameter space.
We repeat the numerical analysis by varying the parameter values. and some of the
results are listed in Table 2. Like in Table 1, tan β = 3, s(0) = 500 GeV, and T = 100
GeV are fixed, whereas λ and mA are varied. The set of numbers in the last row of Table
2 is the numerical result of Fig. 3. Comparing Table 2 with Table 1, one may notice that
the numbers are slightly different from each other but the general behavior of the two
tables is exactly the same.
IV. DISCUSSIONS AND CONCLUSIONS
We investigate the MSSM with an extra U(1)′ if it could accommodate strongly first-
order electroweak phase transitions to provide sufficient baryon asymmetry, for reasonable
masses of scalar Higgs bosons. To do so, we need the temperature-dependent part of
the Higgs potential at the one-loop level. Explicitly, its expression is obtained by two
complementary methods: Method A employs high-temperature approximation and retains
only the most dominant T 2 terms, and takes into account the thermal effects at the one-
loop level of various participating particles. On the other hand, method B performs
numerical integrations, and the thermal effects of top, bottom, and exotic quarks and
squarks are accounted for.
Both methods lead us to essentially the same conclusion: the strongly first-order
electroweak phase transition is possible in the MSSM with an extra U(1)′, for a wide
region in its parameter space. The masses of the scalar Higgs bosons are obtained within
reasonably acceptable ranges. Accordingly, we may expect that the MSSM with an extra
U(1)′ can explain the baryon asymmetry of the universe.
We remark that the MSSM with an extra U(1)′ exhibits an interesting behavior with
respect to the correlation between the strength of the phase transition and the Higgs
boson masses. The MSSM with an extra U(1)′ is opposite to the SM or to the MSSM
in the sense that the mass of the lightest scalar Higgs boson increases when the strength
of the strongly first-order electroweak phase transition becomes stronger. In the SM, its
single Higgs boson has a larger mass when the strength of the first order electroweak
phase transition decreases. In the MSSM, we also have a larger mass for the lighter one
of its two scalar Higgs bosons when the phase transition becomes weaker.
ACKNOWLEDGMENTS
This research is supported by KOSEF through CHEP. The authors would like to
acknowledge the support from KISTI (Korea Institute of Science and Technology Infor-
mation) under ”The Strategic Supercomputing Support Program” with Dr. Kihyeon Cho
as the technical supporter. The use of the computing system of the Supercomputing
Center is also greatly appreciated.
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http://arxiv.org/abs/hep-ph/0703041
FIGURE CAPTION
FIG. 1. : The allowed area in the (Q1, Q2)-plane. For tanβ = 3 and s(T = 0) = 500
GeV, the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner are the
allowed areas, whereas the hatched region is the excluded area. There are two specific
points, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at the star-marked
point correspond to the ν-model of E6 gauge group realizations. The values of Q1 and Q2
at the cross-marked point are (Q1, Q2) = (-1, -0.1), and hence Q3 =1.1. In our discussions,
we choose this point.
FIG. 2. : The plot of the equipotential contours of 〈V (v1, v2, T )〉 on the (v1, v2)-plane,
obtained by Method A. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500
GeV, mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the
critical temperature Tc. Notice two distinct minima of 〈V (v1, v2, T )〉 on the (v1, v2)-plane:
(0, 0) where the phase of the state is symmetric, and (275, 640) GeV, where the phase
of the state is broken. The electroweak phase transition may take place from (0, 0) to
(275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is
first order. The distance between the two minima is vc = 696 GeV, indicating that the
strength of the first-order phase transition is strong (vc/Tc > 1). The masses of the three
scalar Higgs bosons are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and mS3 = 1827
FIG. 3. : The plot of the equipotential contours of 〈V (v1, v2, T )〉 on (v1, v2)-plane, ob-
tained by Method B. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500
GeV, mA = 1780 GeV, and Tc = 100 GeV. The coordinates of two minima are: (0, 0)
and (165, 440) GeV. The distance between the two minima is vc = 470 GeV, thus the
electroweak phase transition between the two minima is strongly first-order. The masses
of the three scalar Higgs bosons are obtained as mS1 = 82 GeV, mS2 = 804 GeV, and
mS3 = 1777 GeV.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
FIG. 1: The allowed area in the (Q1, Q2)-plane. For tanβ = 3 and s(T = 0) = 500
GeV, the small area near the point (Q1, Q2) = (-1, 0) and the upper right corner are the
allowed areas, whereas the hatched region is the excluded area. There are two specific
points, marked by a star (∗) and a cross (+). The values of Q1 and Q2 at the star-marked
point correspond to the ν-model of E6 gauge group realizations. The values of Q1 and Q2
at the cross-marked point are (Q1, Q2) = (-1, -0.1), and hence Q3 =1.1. In our discussions,
we choose this point.
0 50 100 150 200 250 300 350 400
V1 (GeV)
V2 (GeV)
FIG. 2: The plot of the equipotential contours of 〈V (v1, v2, T )〉 on the (v1, v2)-plane,
obtained by Method A. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500
GeV, mA = 1830 GeV, and the temperature is set as T = 100 GeV, which is actually the
critical temperature Tc. Notice two distinct minima of 〈V (v1, v2, T )〉 on the (v1, v2)-plane:
(0, 0) where the phase of the state is symmetric, and (275, 640) GeV, where the phase
of the state is broken. The electroweak phase transition may take place from (0, 0) to
(275, 640) GeV on the (v1, v2)-plane, which is evidently discontinuous and therefore it is
first order. The distance between the two minima is vc = 696 GeV, indicating that the
strength of the first-order phase transition is strong (vc/Tc > 1). The masses of the three
scalar Higgs bosons are obtained as mS1 = 56 GeV, mS2 = 807 GeV, and mS3 = 1827
0 25 50 75 100 125 150 175 200 225 250
V1 (GeV)
V2 (GeV)
FIG. 3: The plot of the equipotential contours of 〈V (v1, v2, T )〉 on (v1, v2)-plane, obtained
by Method B. The parameter values are set as tan β = 3, λ = 0.8, s(0) = 500 GeV, mA =
1780 GeV, and Tc = 100 GeV. The coordinates of two minima are: (0, 0) and (165, 440)
GeV. The distance between the two minima is vc = 470 GeV, thus the electroweak phase
transition between the two minima is strongly first-order. The masses of the three scalar
Higgs bosons are obtained as mS1 = 82 GeV, mS2 = 804 GeV, and mS3 = 1777 GeV.
	INTRODUCTION
	ZERO TEMPERATURE
	FINITE TEMPERATURE
	Method A
	Method B
	DISCUSSIONS AND CONCLUSIONS
