E sph (T)/T in cxSM

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 S₃(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dependence of E_sph(T ) is not fully embodied in Ω(T ), and a na¨ıve scal-ing formula E_sph(T ) = E_sph(0)v(T )/v₀ is no longer valid, especially when T approaches T_C (for earlier studies, see Refs.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 ) = g²Esph(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 < −δ², which makes the Higgs potential un-bounded from below. For −0.5 ! δ² ! 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 b₁, b₂, δ₂ and d₂ 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_S²(T ) = − 2 d₂

!b₁ + b₂ + 2Σ_S(T )"

(26) is positive, which implies that b₁ + b₂ must be negative as long as Σ_S(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ₂. 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).

δ₂ ! −0.5, the Higgs potential is unbounded from below, i.e., √

λd₂ < −δ² (λ = 0.52, d₂ = 0.5). For −0.5 ! δ₂ ! 0.34, there is no phase A since T^C is too large to satisfy Eq. (26). For δ₂ " 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 S₃(T ), E_sph(T ) and E, fixing δ² = 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 )/v₀ 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_DMh² = 0.1186 ± 0.0020 . (27) We will use the central value as a constraint on Ω_Ah².

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 cm² 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 ) = g²Esph(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 < −δ², which makes the Higgs potential un-bounded from below. For −0.5 ! δ² ! 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 b₁, b₂, δ₂ and d₂ 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_S²(T ) = − 2 d₂

!b₁ + b₂ + 2Σ_S(T )"

(26) is positive, which implies that b₁ + b₂ must be negative as long as Σ_S(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ₂. 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).

δ₂ ! −0.5, the Higgs potential is unbounded from below, i.e., √

λd₂ < −δ² (λ = 0.52, d₂ = 0.5). For −0.5 ! δ₂ ! 0.34, there is no phase A since T^C is too large to satisfy Eq. (26). For δ₂ " 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 S₃(T ), E_sph(T ) and E, fixing δ² = 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 )/v₀ 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_DMh² = 0.1186 ± 0.0020 . (27) We will use the central value as a constraint on Ω_Ah².

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 cm² 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 ) = g²Esph(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 < −δ², which makes the Higgs potential un-bounded from below. For −0.5 ! δ² ! 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 b₁, b₂, δ₂ and d₂ 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_S²(T ) = − 2 d₂

!b₁ + b₂ + 2Σ_S(T )"

(26) is positive, which implies that b₁ + b₂ must be negative as long as Σ_S(T ) is positive.

Fig. 9 shows the dependence of EWPT on δ₂. 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).

δ₂ ! −0.5, the Higgs potential is unbounded from below, i.e., √

λd₂ < −δ² (λ = 0.52, d₂ = 0.5). For −0.5 ! δ₂ ! 0.34, there is no phase A since T^C is too large to satisfy Eq. (26). For δ₂ " 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 S₃(T ), E_sph(T ) and E, fixing δ² = 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 )/v₀ 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_DMh² = 0.1186 ± 0.0020 . (27) We will use the central value as a constraint on Ω_Ah².

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 cm² 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

T-dependence of E sph (T)

在文檔中 Towards a gauge-invariant treatment of the electroweak phase transition (頁 107-112)