T [GeV]

FIG. 7: Esph(T )/T as a function of T , where Esph(T ) is
cal-culated using the high-T eﬀective potential in Eq. (10). From
right to left, the dots mark for Esph(T_{C}^{LO})/T_{C}^{LO} = 61.31,
Esph(TN)/TN = 74.23 and Esph(TC)/TC = 78.00, where
T_{C}^{LO} = 90.4 GeV, TN = 84.9 GeV and TC = 83.1 GeV.

S1. The dotted line satisfies the condition in Eq. (19),
from which we obtain S_{3}(T_{N})/T_{N} = 152.01 and T_{N} =
84.9 GeV, which is closer to T_{C} = 83.1 GeV obtained
in the PRM scheme. The degrees of supercooling is
thus (T_{C}^{LO} − T^{N})/T_{C}^{LO} ≃ 6.1%, where T_{C}^{LO} = 90.4 GeV
and is one order of magnitude larger than that in the
minimal supersymmetric SM (MSSM) case (see, e.g.,
Refs. [23, 24]).

It is found that the supercooling becomes larger as
α decreases, and eventually the condition of Eq. (19)
cannot be fulfilled for α ! −21.4^{◦}, rendering a stronger
lower bound on α than the vacuum condition mentioned
above. For the critical α, we obtain T_{C}^{LO} = 78.1 GeV and
T_{N} = 47.3 GeV, leading to (T_{C}^{LO} − T^{N})/T_{C}^{LO} ≃ 39.4%.

In such a large supercooling case, it is worth studying how the EWPT develops. If it mostly proceeds via the bubble nucleations rather than expansions, the EWBG may not work since the baryon asymmetry is normally facilitated with the help of the bubble expansions.

Fig. 7 displays E_{sph}(T )/T against T for α = −20.5^{◦}
in S1, where E_{sph}(T ) is estimate based on the high-T
eﬀective potential in Eq. (10). From right to left, the
three dots mark the results for E_{sph}(T_{C}^{LO})/T_{C}^{LO} = 61.31,
E_{sph}(T_{N})/T_{N} = 74.23 and E_{sph}(T_{C})/T_{C} = 78.00 using
the values of T_{C}^{LO}, T_{N} and T_{C} given above.

In Fig. 8, the dimensionless sphaleron energy E(T ) is
plotted as a function of T . Apparently, E(T ) decreases as
T increases, showing that the temperature dependence of
E_{sph}(T ) is not fully embodied in Ω(T ), and a na¨ıve
scal-ing formula E_{sph}(T ) = E_{sph}(0)v(T )/v_{0} is no longer valid,
especially when T approaches T_{C} (for earlier studies, see
Refs. [24–26]). As in Fig. 7, the three dots correspond
to E(T_{C}^{LO}) = 1.82, E(T^{N}) = 1.86 and E(T^{C}) = 1.87 from
right to left.

**E** **sph** **(T)/T in cxSM**

### E

_{sph}

### (T

_{C}

### )

### T

_{C}

### = 78.00, E

_{sph}

### (T

_{N}

### )

### T

_{N}

### = 74.23, E

_{sph}

### (T

_{C}

^{HT}

### )

### T

_{C}

^{HT}

### = 61.31,

### Is this scaling law valid?

### If T-dependence comes from v(T) only, one has

### E

_{sph}

### (T ) = 4⇡¯ v(T )

### g

_{2}

### E(T )

### E

_{sph}

### (T ) = E

_{sph}

### (0) v(T ) ¯ v

_{0}

**T-dependence of E** **sph** **(T)**

8

1.75 1.8 1.85 1.9 1.95 2

0 20 40 60 80 100

T [GeV]

E(T)

FIG. 8: E(T ) = g^{2}Esph(T )/(4π¯v) as a function of T . The
three dots correspond to E(TC^{LO}) = 1.82, E(T^{N}) = 1.86 and
E(T^{C}) = 1.87 from right to left.

0 50 100 150 200 250

-0.4 -0.2 0 0.2 0.4 0.6

δ2

[GeV]

T_{C}
v¯_{C}
T_{C}^{LO}

v¯^{LO}

C

FIG. 9: TC and VEV’s as functions of δ2. For δ2 ! −0.5, we have √

λd2 < −δ^{2}, which makes the Higgs potential
un-bounded from below. For −0.5 ! δ^{2} ! 0.34, the phase A is
absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77,
phase A turns into the global minimum at T = 0.

B. S2 Case

In this case, there is no constraint among b_{1}, b_{2}, δ_{2} and
d_{2} from the tadpole condition of Eq. (6), as opposed to
the S1 case. Note that the nontrivial vacuum phase A in
ϕ_{S} exists if

˜

v_{S}^{2}(T ) = − 2
d_{2}

!b_{1} + b_{2} + 2Σ_{S}(T )"

(26)
is positive, which implies that b_{1} + b_{2} must be negative
as long as Σ_{S}(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ_{2}. For

T [GeV] TC = 84.3 T_{C}^{LO} = 99.8 TN = 96.6

¯

v(T ) [GeV] 193.0 167.0 173.5 Esph(T )/T 84.36 61.67 66.02

E(T ) 1.92 1.92 1.92

TABLE I: VEVs and sphaleron energies at the diﬀerent
tem-peratures, TC, T_{C}^{LO} and TN for δ2 = 0.55 in S2, where the
last two are calculated by use of the high-T eﬀective
poten-tial (10).

δ_{2} ! −0.5, the Higgs potential is unbounded from below,
i.e., √

λd_{2} < −δ^{2} (λ = 0.52, d_{2} = 0.5). For −0.5 !
δ_{2} ! 0.34, there is no phase A since T^{C} is too large to
satisfy Eq. (26). For δ_{2} " 0.77, phase A becomes the
global minimum at T = 0. In this case, the EWPT never
happens.

As in the S1 case, we also evaluate S_{3}(T ), E_{sph}(T ) and
E, fixing δ^{2} = 0.55. The results are summarized in Table
I. One can see that as in the S1 case, the degree of the
su-percooling is one order of magnitude larger than the
typ-ical MSSM value [23, 24], i.e., (T_{C}^{LO} − T^{N})/T_{C}^{LO} ≃ 3.2%.

However, one distinctive feature of S2 is that E is
inde-pendent of T , implying that E_{sph}(T ) = E_{sph}(0)¯v(T )/v_{0}
as in the SM. This is due to the fact that there is no
singlet Higgs contribution to E_{sph} since ¯v_{S} = 0.

Before closing this section, we make a comment on the
DM mass m_{A} dependence in SFOEWPT. In both S1 and
S2 cases, the vacuum energy of phase B increases as m_{A}
increases and surpasses the vacuum energy of phase A
at some critical value, and hence T_{C} cannot be defined
in the NLO calculation. As will be discussed in the next
section, a relatively large m_{A} would be required to obtain
the correct DM abundance, which could be in conflict
with the realization of SFOEWPT.

VII. DARK MATTER

In this model, the pseudoscalar particle A can be a can-didate for DM. We will study its properties in S1 and S2.

We use micrOMEGAs [27, 28] to calculate the relic
den-sity of A, Ω_{A}, and its spin-independent scattering cross
section with nucleon N , σ_{SI}^{N}.

The observed DM relic density is [29]

Ω_{DM}h^{2} = 0.1186 ± 0.0020 . (27)
We will use the central value as a constraint on Ω_{A}h^{2}.

If the relic abundance of A is less than the observed
one, σ_{SI}^{N} should be scaled as

˜

σ_{SI}^{N} = σ_{SI}^{N}

# Ω_{A}
Ω_{DM}

$

. (28)

Recently, the LUX Collaboration has updated their
experimental results [30]. After combining the previous
LUX data, the minimum exclusion limit becomes σ_{SI}^{N} ≃
1.1 × 10^{−46} cm^{2} at around 50 GeV of the DM mass.

### Is this scaling law valid?

### If T-dependence comes from v(T) only, one has

### E

_{sph}

### (T ) = 4⇡¯ v(T )

### g

_{2}

### E(T )

### E

_{sph}

### (T ) = E

_{sph}

### (0) v(T ) ¯ v

_{0}

**T-dependence of E** **sph** **(T)**

8

1.75 1.8 1.85 1.9 1.95 2

0 20 40 60 80 100

T [GeV]

E(T)

FIG. 8: E(T ) = g^{2}Esph(T )/(4π¯v) as a function of T . The
three dots correspond to E(TC^{LO}) = 1.82, E(T^{N}) = 1.86 and
E(T^{C}) = 1.87 from right to left.

0 50 100 150 200 250

-0.4 -0.2 0 0.2 0.4 0.6

δ2

[GeV]

T_{C}
v¯_{C}
T_{C}^{LO}

v¯^{LO}

C

FIG. 9: TC and VEV’s as functions of δ2. For δ2 ! −0.5, we have √

λd2 < −δ^{2}, which makes the Higgs potential
un-bounded from below. For −0.5 ! δ^{2} ! 0.34, the phase A is
absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77,
phase A turns into the global minimum at T = 0.

B. S2 Case

In this case, there is no constraint among b_{1}, b_{2}, δ_{2} and
d_{2} from the tadpole condition of Eq. (6), as opposed to
the S1 case. Note that the nontrivial vacuum phase A in
ϕ_{S} exists if

˜

v_{S}^{2}(T ) = − 2
d_{2}

!b_{1} + b_{2} + 2Σ_{S}(T )"

(26)
is positive, which implies that b_{1} + b_{2} must be negative
as long as Σ_{S}(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ_{2}. For

T [GeV] TC = 84.3 T_{C}^{LO} = 99.8 TN = 96.6

¯

v(T ) [GeV] 193.0 167.0 173.5 Esph(T )/T 84.36 61.67 66.02

E(T ) 1.92 1.92 1.92

TABLE I: VEVs and sphaleron energies at the diﬀerent
tem-peratures, TC, T_{C}^{LO} and TN for δ2 = 0.55 in S2, where the
last two are calculated by use of the high-T eﬀective
poten-tial (10).

δ_{2} ! −0.5, the Higgs potential is unbounded from below,
i.e., √

λd_{2} < −δ^{2} (λ = 0.52, d_{2} = 0.5). For −0.5 !
δ_{2} ! 0.34, there is no phase A since T^{C} is too large to
satisfy Eq. (26). For δ_{2} " 0.77, phase A becomes the
global minimum at T = 0. In this case, the EWPT never
happens.

As in the S1 case, we also evaluate S_{3}(T ), E_{sph}(T ) and
E, fixing δ^{2} = 0.55. The results are summarized in Table
I. One can see that as in the S1 case, the degree of the
su-percooling is one order of magnitude larger than the
typ-ical MSSM value [23, 24], i.e., (T_{C}^{LO} − T^{N})/T_{C}^{LO} ≃ 3.2%.

However, one distinctive feature of S2 is that E is
inde-pendent of T , implying that E_{sph}(T ) = E_{sph}(0)¯v(T )/v_{0}
as in the SM. This is due to the fact that there is no
singlet Higgs contribution to E_{sph} since ¯v_{S} = 0.

Before closing this section, we make a comment on the
DM mass m_{A} dependence in SFOEWPT. In both S1 and
S2 cases, the vacuum energy of phase B increases as m_{A}
increases and surpasses the vacuum energy of phase A
at some critical value, and hence T_{C} cannot be defined
in the NLO calculation. As will be discussed in the next
section, a relatively large m_{A} would be required to obtain
the correct DM abundance, which could be in conflict
with the realization of SFOEWPT.

VII. DARK MATTER

In this model, the pseudoscalar particle A can be a can-didate for DM. We will study its properties in S1 and S2.

We use micrOMEGAs [27, 28] to calculate the relic
den-sity of A, Ω_{A}, and its spin-independent scattering cross
section with nucleon N , σ_{SI}^{N}.

The observed DM relic density is [29]

Ω_{DM}h^{2} = 0.1186 ± 0.0020 . (27)
We will use the central value as a constraint on Ω_{A}h^{2}.

If the relic abundance of A is less than the observed
one, σ_{SI}^{N} should be scaled as

˜

σ_{SI}^{N} = σ_{SI}^{N}

# Ω_{A}
Ω_{DM}

$

. (28)

Recently, the LUX Collaboration has updated their
experimental results [30]. After combining the previous
LUX data, the minimum exclusion limit becomes σ_{SI}^{N} ≃
1.1 × 10^{−46} cm^{2} at around 50 GeV of the DM mass.

### Is this scaling law valid?

### If T-dependence comes from v(T) only, one has

### No, it breaks down especially when T approaches T

C### . E

_{sph}

### (T ) = 4⇡¯ v(T )

### g

_{2}

### E(T )

### E

_{sph}

### (T ) = E

_{sph}

### (0) v(T ) ¯ v

_{0}

**T-dependence of E** **sph** **(T)**

8

1.75 1.8 1.85 1.9 1.95 2

0 20 40 60 80 100

T [GeV]

E(T)

FIG. 8: E(T ) = g^{2}Esph(T )/(4π¯v) as a function of T . The
three dots correspond to E(TC^{LO}) = 1.82, E(T^{N}) = 1.86 and
E(T^{C}) = 1.87 from right to left.

0 50 100 150 200 250

-0.4 -0.2 0 0.2 0.4 0.6

δ2

[GeV]

T_{C}
v¯_{C}
T_{C}^{LO}

v¯^{LO}

C

FIG. 9: TC and VEV’s as functions of δ2. For δ2 ! −0.5, we have √

λd2 < −δ^{2}, which makes the Higgs potential
un-bounded from below. For −0.5 ! δ^{2} ! 0.34, the phase A is
absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77,
phase A turns into the global minimum at T = 0.

B. S2 Case

In this case, there is no constraint among b_{1}, b_{2}, δ_{2} and
d_{2} from the tadpole condition of Eq. (6), as opposed to
the S1 case. Note that the nontrivial vacuum phase A in
ϕ_{S} exists if

˜

v_{S}^{2}(T ) = − 2
d_{2}

!b_{1} + b_{2} + 2Σ_{S}(T )"

(26)
is positive, which implies that b_{1} + b_{2} must be negative
as long as Σ_{S}(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ_{2}. For

T [GeV] TC = 84.3 T_{C}^{LO} = 99.8 TN = 96.6

¯

v(T ) [GeV] 193.0 167.0 173.5 Esph(T )/T 84.36 61.67 66.02

E(T ) 1.92 1.92 1.92

TABLE I: VEVs and sphaleron energies at the diﬀerent
tem-peratures, TC, T_{C}^{LO} and TN for δ2 = 0.55 in S2, where the
last two are calculated by use of the high-T eﬀective
poten-tial (10).

δ_{2} ! −0.5, the Higgs potential is unbounded from below,
i.e., √

λd_{2} < −δ^{2} (λ = 0.52, d_{2} = 0.5). For −0.5 !
δ_{2} ! 0.34, there is no phase A since T^{C} is too large to
satisfy Eq. (26). For δ_{2} " 0.77, phase A becomes the
global minimum at T = 0. In this case, the EWPT never
happens.

As in the S1 case, we also evaluate S_{3}(T ), E_{sph}(T ) and
E, fixing δ^{2} = 0.55. The results are summarized in Table
I. One can see that as in the S1 case, the degree of the
su-percooling is one order of magnitude larger than the
typ-ical MSSM value [23, 24], i.e., (T_{C}^{LO} − T^{N})/T_{C}^{LO} ≃ 3.2%.

However, one distinctive feature of S2 is that E is
inde-pendent of T , implying that E_{sph}(T ) = E_{sph}(0)¯v(T )/v_{0}
as in the SM. This is due to the fact that there is no
singlet Higgs contribution to E_{sph} since ¯v_{S} = 0.

Before closing this section, we make a comment on the
DM mass m_{A} dependence in SFOEWPT. In both S1 and
S2 cases, the vacuum energy of phase B increases as m_{A}
increases and surpasses the vacuum energy of phase A
at some critical value, and hence T_{C} cannot be defined
in the NLO calculation. As will be discussed in the next
section, a relatively large m_{A} would be required to obtain
the correct DM abundance, which could be in conflict
with the realization of SFOEWPT.

VII. DARK MATTER

In this model, the pseudoscalar particle A can be a can-didate for DM. We will study its properties in S1 and S2.

We use micrOMEGAs [27, 28] to calculate the relic
den-sity of A, Ω_{A}, and its spin-independent scattering cross
section with nucleon N , σ_{SI}^{N}.

The observed DM relic density is [29]

Ω_{DM}h^{2} = 0.1186 ± 0.0020 . (27)
We will use the central value as a constraint on Ω_{A}h^{2}.

If the relic abundance of A is less than the observed
one, σ_{SI}^{N} should be scaled as

˜

σ_{SI}^{N} = σ_{SI}^{N}

# Ω_{A}
Ω_{DM}

$

. (28)

Recently, the LUX Collaboration has updated their
experimental results [30]. After combining the previous
LUX data, the minimum exclusion limit becomes σ_{SI}^{N} ≃
1.1 × 10^{−46} cm^{2} at around 50 GeV of the DM mass.

### Is this scaling law valid?

### If T-dependence comes from v(T) only, one has

### No, it breaks down especially when T approaches T

C### .

### ∵ presence of v

S### (T).

### E

_{sph}

### (T ) = 4⇡¯ v(T )

### g

_{2}

### E(T )

### E

_{sph}

### (T ) = E

_{sph}

### (0) v(T ) ¯ v

_{0}