PRM:

0 50 100 150 200 250

0 50 100 150 200

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

0 50 100 150 200 250

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

FIG. 3: Evolution of VEV’s as a function of T using the high-T e↵ective potential in Type-(a) EWPT with a1 6= 0 (Left) and Type-(c) EWPT with a₁ = 0 (Right). For the former, the A ! B transition is first-order, with T² = T_C = 90.4 GeV and

v(T_C) = 158.2 GeV. For the latter, the O ! A transition is second-order while the A ! B transition is first-order. It is found that T₁ = 224.6 GeV, T₂ = T_C = 99.8 GeV and ¯v(T_C) = 167.0 GeV.

where n_B(t) is the baryon number density at a time t af-ter the onset of the transition, n_B(0) is the initial baryon number density, and t_EW is the duration of the EWPT.

Requiring that S > exp( X), one obtains the baryon number preservation criterion, or BNPC [34]:

E_sph(T_C)

T_C 7 ln v(T¯ _C) T_C >

ln X ln ⇣ t_EW t_H

⌘ + ln QF + ln  , (23)

where t_H is the Hubble time,  is the fluctuation deter-minant about the classical sphaleron solution, and Q and F encode the e↵ects of rotational and translational zero modes as well as the unstable mode about the sphaleron⁴. It is convenient to express E_sph in terms of an energy scale ⌦(T ) associated with the EWSB that is typically of order T_C. To this end, we write

E_sph(T ) = 4⇡⌦(T )

g₂ E(T ) . (24)

When the only scalar fields in the theory are SU(2)_L doublets, the natural choice for ⌦(T ) is

v(T ). For the cxSM, there exists a second possibil-ity:

v²(T ) + (¯v_S^A(T ) v¯_S^B(T ))². Either choice is ac-ceptable, as the BNPC depends on E_sph and the dif-ference in the choice of ⌦(T ) will be compensated by

4 The quantity X parametrizes the degree to which the initial baryon asymmetry may be diluted by sphaleron processes. Its value will depend on the initial value of the asymmetry obtained from a computation of the CPV transport dynamics in a given model.

the corresponding convention for E(T ). Here we follow Refs. [19, 43] where it is argued that ⌦(T ) = ¯v(T ) en-capsulates the primary T -dependence of the sphaleron energy. Nevertheless, we find that the residual T -dependence of E can be non-negligible in some cases. The detailed calculation of E_sph(T ) is given in Appendix A.

From these considerations, one obtains from the BNPC (23) a requirement on the ratio ¯v(T_C)/T_C:

v(T_C)

T_C & ⇣^sph(T_C) . (25) In the literature, one often finds this requirement quoted as v(T_C)/T_C & 1, where v(T^C) is computed using the full one-loop e↵ective potential V_e↵ rather than V ^high-T [44–

47]. As discussed in Ref. [34], this procedure, as well as the conventional method for computing T_C, introduces an unphysical gauge dependence. In what follows, we will perform a gauge-invariant computation. We also address the impact of the µ-dependence by implementing a RG-improved analysis. These and other theoretical issues associated with the BNPC and _N are discussed below.

V. GAUGE-INVARIANT METHOD BEYOND

THE LEADING ORDER

Here, we delineate the gauge-invariant treatment for EWPT and sphaleron rate. Determination of T_C and ¯v_C using the high-T e↵ective potential is obviously gauge independent. Beyond this order, however, the potential barrier inherently depends on the gauge fixing parameter, which may lead to the gauge-dependent T_C and v_C as in the ordinary method. Nevertheless, the gauge-invariant T_C can still be obtained by use of a method advocated in

HT:

0 50 100 150 200 250

0 50 100 150 200

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

0 50 100 150 200 250

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

v(T_C) = 158.2 GeV. For the latter, the O ! A transition is second-order while the A ! B transition is first-order. It is found that T₁ = 224.6 GeV, T₂ = T_C = 99.8 GeV and ¯v(T_C) = 167.0 GeV.

where n_B(t) is the baryon number density at a time t af-ter the onset of the transition, n_B(0) is the initial baryon number density, and t_EW is the duration of the EWPT.

Requiring that S > exp( X), one obtains the baryon number preservation criterion, or BNPC [34]:

E_sph(T_C)

T_C 7 ln v(T¯ _C) T_C >

ln X ln ⇣ t_EW t_H

⌘ + ln QF + ln  , (23)

E_sph(T ) = 4⇡⌦(T )

g₂ E(T ) . (24)

When the only scalar fields in the theory are SU(2)_L doublets, the natural choice for ⌦(T ) is

v(T ). For the cxSM, there exists a second possibil-ity:

v²(T ) + (¯v_S^A(T ) v¯_S^B(T ))². Either choice is ac-ceptable, as the BNPC depends on E_sph and the dif-ference in the choice of ⌦(T ) will be compensated by

From these considerations, one obtains from the BNPC (23) a requirement on the ratio ¯v(T_C)/T_C:

v(T_C)

V. GAUGE-INVARIANT METHOD BEYOND

THE LEADING ORDER

HT:

benchmark point:

HT PRM

m

_H₂

= 230 GeV, v

_S0

= 40 GeV,

↵ = 20.5 , a

₁

= (110 GeV)

minima of ^V

^{high T}

^('

ⁱ

^{; T )}

HT vs. PRM

¯

v(T

)

T

> v(T ¯

_C^HT

) T

_C^HT

T_C^HT = 90.4 GeV, ¯v(T_C^HT) = 158.2 GeV

T_C = 83.1 GeV, ¯v(T_C) = 180.2 GeV

0 50 100 150 200 250

0 50 100 150 200

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

0 50 100 150 200 250

T [GeV]

¯v(T),¯v

A S(T)and¯v

B S(T)[GeV]

¯ v(T )

v_S^B(T )

v_S^A(T )

v(T_C) = 158.2 GeV. For the latter, the O ! A transition is second-order while the A ! B transition is first-order. It is found that T₁ = 224.6 GeV, T₂ = T_C = 99.8 GeV and ¯v(T_C) = 167.0 GeV.

where n_B(t) is the baryon number density at a time t af-ter the onset of the transition, n_B(0) is the initial baryon number density, and t_EW is the duration of the EWPT.

Requiring that S > exp( X), one obtains the baryon number preservation criterion, or BNPC [34]:

E_sph(T_C)

T_C 7 ln v(T¯ _C) T_C >

ln X ln ⇣ t_EW t_H

⌘ + ln QF + ln  , (23)

E_sph(T ) = 4⇡⌦(T )

g₂ E(T ) . (24)

When the only scalar fields in the theory are SU(2)_L doublets, the natural choice for ⌦(T ) is

v(T ). For the cxSM, there exists a second possibil-ity:

v²(T ) + (¯v_S^A(T ) v¯_S^B(T ))². Either choice is ac-ceptable, as the BNPC depends on E_sph and the dif-ference in the choice of ⌦(T ) will be compensated by

From these considerations, one obtains from the BNPC (23) a requirement on the ratio ¯v(T_C)/T_C:

v(T_C)

V. GAUGE-INVARIANT METHOD BEYOND

THE LEADING ORDER

HT:

PRM:

benchmark point:

HT PRM

Caution: v(T

) is subject to uncertainties of T

.

m

_H₂

= 230 GeV, v

_S0

= 40 GeV,

↵ = 20.5 , a

₁

= (110 GeV)

minima of ^V

^{high T}

^('

ⁱ

^{; T )}

HT vs. PRM

¯

v(T

)

T

> v(T ¯

_C^HT

) T

_C^HT

T_C^HT = 90.4 GeV, ¯v(T_C^HT) = 158.2 GeV

T_C = 83.1 GeV, ¯v(T_C) = 180.2 GeV

在文檔中 Towards a gauge-invariant treatment of the electroweak phase transition (頁 69-72)

PRM:

HT:

HT:

PRM:

benchmark point:

m

= 230 GeV, v

= 40 GeV,

↵ = 20.5 , a

= (110 GeV)

minima of V

('

; T )

HT vs. PRM

¯

v(T

)

T

> v(T ¯

) T

HT:

PRM:

benchmark point:

Caution: v(T

) is subject to uncertainties of T

.

m

= 230 GeV, v

= 40 GeV,

↵ = 20.5 , a

= (110 GeV)

minima of V

('

; T )

HT vs. PRM

¯

v(T

)

T

> v(T ¯

) T

minima of ^V

^('

^{; T )}

minima of ^V

^('

^{; T )}