Pattens of PT

3

(a) (b)

(c) (d)

FIG. 1: Patterns of symmetry breaking at finite temperature.

where ' and '_S denote the neutral doublet and singlet background fields, respectively, and

⌃_H =



8 + ²

24 + 3g₂² + g₁²

16 + y_t²

4 T² ⌘ ¯⌃_HT² , (11)

⌃_S = ( ₂ + d₂)T²

12 ⌘ ¯⌃_ST² . (12)

For m₂ < 0 and b₂ < 0, T = 0 minima will exist for non-zero v₀ and v_S0. In the limit of vanishing a₁, at suffi-ciently high-T the only minimum of the theory occurs at the origin, denoted by “O”. As T decreases, one generi-cally expects that a secondary minimum at '_s ⌘ ˜v^S 6= 0 will first appear, since ¯⌃_S < ¯⌃_H. At a temperature T₁, the minimum at ˜v_S will become the global minimum in-dicated by “A” in Fig. 1. As T further decreases, an ad-ditional minimum at (' ⌘ v 6= 0, '^S ⌘ v^S 6= 0) develops, becoming the global minimum at temperature T₂ < T₁, corresponding to point “B” in Fig. 1. The universe will then follow a two-step symmetry-breaking trajectory in field space indicated Fig. 1, where one may have either

˜

v_S(T₂) v_S(T₂) > 0 or < 0. We will henceforth denote T₂ as the EWSB critical temperature, T_C and the value of ' at this temperature as v_C(T_C). After a straightforward calculation, one finds

¯

v_C '

r2 ₂v˜_S(T_C)

˜

v_S(T_C) v_S(T_C) , (13) T_C '

s 1

2 ¯⌃_H

✓

m² v˜_S²(T_C)

2 ²

◆

. (14)

Here, the bar over v_C indicates that it has been com-puted using the high-T e↵ective potential (10). T_C and

¯

v_C obtained in this way are the leading-order (LO) gauge-invariant results. For positive (negative) ₂ one has

˜

v_S(T_C) larger (smaller) than v_S(T_C). In addition, for positive ₂, the potential will have a barrier between the

minima at A and B. In this case, the EWPT at T = T_C will be first order. Note, however, that an oversized pos-itive ₂ may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup-posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of ₂, as will be discussed below. For negative ₂, in contrast, T_C is always raised which may prevent ¯v_C/T_C becoming a sufficient size. In fact, SFOEWPT cannot be found for

2 < 0 in our numerical analysis.

The situation can be more complex in the presence of non-vanishing a₁ and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential:

V_e↵('; T ) = V₀(') + V₁('; T ) (15) where ' = (', '_S);

V₁('; T ) = X

j

n_j



V_CW( ¯m²_j) + T⁴

2⇡² I_B,F m¯ ²_j T²

! , (16) and n_j count the degrees of freedom for particle species j, and ¯m_j are '-dependent masses. The Coleman-Weinberg potential V_CW and I_B,F (a²) are respectively given by [16, 17]

V_CW(m²) = m⁴ 64⇡²

✓

ln m²

µ² c

◆

, (17)

I_B,F (a²) =

Z ₁

0

dx x² ln ⇣

1 ⌥ e ^px2+a2⌘

, (18) where c = 3/2 for scalars and fermions and 5/6 for gauge bosons, and µ is the renormalization scale.

For a₁ 6= 0, the high-T minimum will no longer lie at the origin but will be shifted by a₁ along the '_S direction. Alternately, EWSB may occur directly from the origin to point B, as Type (d) in Fig. 1. However, this phase transition is not first order since ˜v_S is zero as seen from Eq. (13). In what follows, we exclusively explore two cases: Type-(a) EWPT with a₁ 6= 0 and Type-(c) EWPT with a₁ = 0, i.e.,

S1: m_H2 = 230 GeV, v_S0 = 40 GeV, a₁ = (110 GeV)³,

S2: m_H2 = 150 GeV, v_S0 = 0 GeV, ₂ = 2, d₂ = 1,

and m_A = m_H1/2 ' 62.5 GeV in both cases. In Fig. 2, temperature evolutions of the VEVs are plotted in S1 (Left) and S2 (Right). For the former, it is found that A ! B transition is first-order, and T² = T_C = 90.4 GeV and ¯v_C = 158.2 GeV. For the latter, on the other hand, one can see that O ! A transition is second order while A ! B transition is first order. We obtain T¹ = 176.8 GeV for the former transition and T₂ = T_C = 85.8 GeV,

¯

v_C = 181.4 GeV, ¯v_SC = 0 GeV and ˜v_SC = 153.8 GeV for the latter transition. Fig. 3 shows contours of the high-T e↵ective potential at high-T = 180 GeV (Upper Left), 100 GeV (Upper Right), T_C (Lower Left) and 0 GeV (Lower Right) in the case of S2.

Because of 2 fields, there are many patterns of phase transitions.

3

(a) (b)

(c) (d)

FIG. 1: Patterns of symmetry breaking at finite temperature.

where ' and '_S denote the neutral doublet and singlet background fields, respectively, and

⌃_H =



8 + ²

24 + 3g₂² + g₁²

16 + y_t²

4 T² ⌘ ¯⌃_HT² , (11)

⌃_S = ( ₂ + d₂)T²

12 ⌘ ¯⌃_ST² . (12)

For m₂ < 0 and b₂ < 0, T = 0 minima will exist for non-zero v₀ and v_S0. In the limit of vanishing a₁, at suffi-ciently high-T the only minimum of the theory occurs at the origin, denoted by “O”. As T decreases, one generi-cally expects that a secondary minimum at '_s ⌘ ˜v^S 6= 0 will first appear, since ¯⌃_S < ¯⌃_H. At a temperature T₁, the minimum at ˜v_S will become the global minimum in-dicated by “A” in Fig. 1. As T further decreases, an ad-ditional minimum at (' ⌘ v 6= 0, '^S ⌘ v^S 6= 0) develops, becoming the global minimum at temperature T₂ < T₁, corresponding to point “B” in Fig. 1. The universe will then follow a two-step symmetry-breaking trajectory in field space indicated Fig. 1, where one may have either

˜

v_S(T₂) v_S(T₂) > 0 or < 0. We will henceforth denote T₂ as the EWSB critical temperature, T_C and the value of ' at this temperature as v_C(T_C). After a straightforward calculation, one finds

¯

v_C '

r2 ₂v˜_S(T_C)

˜

v_S(T_C) v_S(T_C) , (13) T_C '

s 1

2 ¯⌃_H

✓

m² v˜_S²(T_C)

2 ²

◆

. (14)

Here, the bar over v_C indicates that it has been com-puted using the high-T e↵ective potential (10). T_C and

¯

v_C obtained in this way are the leading-order (LO) gauge-invariant results. For positive (negative) ₂ one has

˜

v_S(T_C) larger (smaller) than v_S(T_C). In addition, for positive ₂, the potential will have a barrier between the

minima at A and B. In this case, the EWPT at T = T_C will be first order. Note, however, that an oversized pos-itive ₂ may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup-posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of ₂, as will be discussed below. For negative ₂, in contrast, T_C is always raised which may prevent ¯v_C/T_C becoming a sufficient size. In fact, SFOEWPT cannot be found for

2 < 0 in our numerical analysis.

The situation can be more complex in the presence of non-vanishing a₁ and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential:

V_e↵('; T ) = V₀(') + V₁('; T ) (15) where ' = (', '_S);

V₁('; T ) = X

j

n_j



V_CW( ¯m²_j) + T⁴

2⇡² I_B,F m¯ ²_j T²

! , (16) and n_j count the degrees of freedom for particle species j, and ¯m_j are '-dependent masses. The Coleman-Weinberg potential V_CW and I_B,F (a²) are respectively given by [16, 17]

V_CW(m²) = m⁴ 64⇡²

✓

ln m²

µ² c

◆

, (17)

I_B,F (a²) =

Z ₁

0

dx x² ln ⇣

1 ⌥ e ^px2+a2⌘

, (18) where c = 3/2 for scalars and fermions and 5/6 for gauge bosons, and µ is the renormalization scale.

For a₁ 6= 0, the high-T minimum will no longer lie at the origin but will be shifted by a₁ along the '_S direction. Alternately, EWSB may occur directly from the origin to point B, as Type (d) in Fig. 1. However, this phase transition is not first order since ˜v_S is zero as seen from Eq. (13). In what follows, we exclusively explore two cases: Type-(a) EWPT with a₁ 6= 0 and Type-(c) EWPT with a₁ = 0, i.e.,

S1: m_H2 = 230 GeV, v_S0 = 40 GeV, a₁ = (110 GeV)³,

S2: m_H2 = 150 GeV, v_S0 = 0 GeV, ₂ = 2, d₂ = 1,

and m_A = m_H1/2 ' 62.5 GeV in both cases. In Fig. 2, temperature evolutions of the VEVs are plotted in S1 (Left) and S2 (Right). For the former, it is found that A ! B transition is first-order, and T² = T_C = 90.4 GeV and ¯v_C = 158.2 GeV. For the latter, on the other hand, one can see that O ! A transition is second order while A ! B transition is first order. We obtain T¹ = 176.8 GeV for the former transition and T₂ = T_C = 85.8 GeV,

¯

v_C = 181.4 GeV, ¯v_SC = 0 GeV and ˜v_SC = 153.8 GeV for the latter transition. Fig. 3 shows contours of the high-T e↵ective potential at high-T = 180 GeV (Upper Left), 100 GeV (Upper Right), T_C (Lower Left) and 0 GeV (Lower Right) in the case of S2.

Because of 2 fields, there are many patterns of phase transitions.

We will focus on type (a) PT.

Pattens of PT

7

0 50 100 150 200 250

-22 -21 -20 -19 -18 -17 -16 -15 α [^◦]

[GeV]

T_C v¯_C T^LO

C

v¯^LO

C

FIG. 5: TC and ¯vC as a function of α in S1. The solid curves are obtained by the PRM scheme with RG improvement while the dashed ones by the high-T effective potential in Eq. (10).

0 100 200 300 400 500

80 82 84 86 88 90

S 3(T)/T

T [GeV]

FIG. 6: S3(T )/T vs. T for α = −20.5^◦ in S1. S3(T ) is evaluated by use of the high-T effective potential in Eq. (10).

The dotted horizontal line satisfies the condition in Eq. (19).

In this case, we have T_C^LO = 90.4 GeV and TN = 84.9 GeV, with the latter closer to TC calculated in the PRM scheme.

dropped here may not be neglected in the small α re-gion. As emphasized in Ref. [20], such a fake SFOEWPT to O(!) calculation can also happen in the SM. Even though a quantitative statement awaits a more precise analysis in that region, the negatively large α (positively large δ) does show the generic feature of SFOEWPT in this model.

As discussed in Sec. IV, the actual beginning of the EWPT is somewhat below the temperature at which the effective potential has two degenerate minima. Here, we calculate S₃(T ) to find T_N using the high-T effec-tive potential in Eq. (10). In Fig. 6, the solid curve shows S₃(T )/T as a function of T for α = −20.5^◦ in

10¹ 10² 10³ 10⁴ 10⁵

0 20 40 60 80 100

E sph(T)/T

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 TC^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.

benchmark point:

T

_N

= 84.9 GeV

S

₃

(T

_N

)

T

_N

= 152.01 GeV

cf., MSSM: O(0.1)%

T

C

= 83.1 GeV; T

N

(HT) = 84.9 GeV, T

C

(HT) = 90.4 GeV

m

_H2

= 230 GeV, v

_S0

= 40 GeV,

↵ = 20.5 , a

₁

= (110 GeV)

³

84.9 GeV 152.01 GeV

- HT case -

S 3 (T)/T

T

_C^HT

T

_N

T

_C^HT

' 6.1%

scale-inv. U(1): O(70)%

7

0 50 100 150 200 250

-22 -21 -20 -19 -18 -17 -16 -15 α [^◦]

[GeV]

T_C v¯_C T^LO

C

v¯^LO

C

FIG. 5: TC and ¯vC as a function of α in S1. The solid curves are obtained by the PRM scheme with RG improvement while the dashed ones by the high-T effective potential in Eq. (10).

0 100 200 300 400 500

80 82 84 86 88 90

S 3(T)/T

T [GeV]

FIG. 6: S3(T )/T vs. T for α = −20.5^◦ in S1. S3(T ) is evaluated by use of the high-T effective potential in Eq. (10).

The dotted horizontal line satisfies the condition in Eq. (19).

In this case, we have T_C^LO = 90.4 GeV and TN = 84.9 GeV, with the latter closer to TC calculated in the PRM scheme.

dropped here may not be neglected in the small α re-gion. As emphasized in Ref. [20], such a fake SFOEWPT to O(!) calculation can also happen in the SM. Even though a quantitative statement awaits a more precise analysis in that region, the negatively large α (positively large δ) does show the generic feature of SFOEWPT in this model.

As discussed in Sec. IV, the actual beginning of the EWPT is somewhat below the temperature at which the effective potential has two degenerate minima. Here, we calculate S₃(T ) to find T_N using the high-T effec-tive potential in Eq. (10). In Fig. 6, the solid curve shows S₃(T )/T as a function of T for α = −20.5^◦ in

10¹ 10² 10³ 10⁴ 10⁵

0 20 40 60 80 100

E sph(T)/T

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 TC^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.

benchmark point:

T

_N

= 84.9 GeV

S

₃

(T

_N

)

T

_N

= 152.01 GeV

cf., MSSM: O(0.1)%

T

C

= 83.1 GeV; T

N

(HT) = 84.9 GeV, T

C

(HT) = 90.4 GeV

m

_H2

= 230 GeV, v

_S0

= 40 GeV,

↵ = 20.5 , a

₁

= (110 GeV)

³

84.9 GeV 152.01 GeV

- HT case -

S 3 (T)/T

T

_C^HT

T

_N

T

_C^HT

' 6.1%

scale-inv. U(1): O(70)%

-

V(T

C

)/T

C

= (209.1GeV)/(65.52GeV)

= 3.2

-

Too strong 1

^st

-order EWPT may not be consistent!!

10² 10³ 10⁴ 10⁵ 10⁶

0 10 20 30 40 50 60

-

α =-22.0

^o

-

No nucleation forα<-21.4

^o

在文檔中 Towards a gauge-invariant treatment of the electroweak phase transition (頁 94-98)