Towards a gauge-invariant treatment of the electroweak phase transition

Towards a gauge-invariant treatment of the electroweak phase transition

Eibun Senaha (ibs-CTPU) Nov. 11, 2017@Natl Taiwan U

based on

[2] C.W. Chiang (Natl Taiwan U), M. Ramsey-Musolf (UMass-Amherst), E.S, arXiv: 1707.09960,

[1] C.W. Chiang (Natl Taiwan U), E.S., arXiv: 1707.06765 (PLB)

Workshop of Recent Developments in QCD and Quantum Field Theories

•

Introduction

•

Gauge-dependence (ξ) of the effective potential

•

Impact of ξ on 1

-order phase transition in classical scale-inv. U(1) models: T

, T

, GW

•

Gauge-inv. analysis on EW phase transition in SM w/

a complex scalar

•

Summary

Outline

C.W. Chiang, E.S., 1707.06765 (PLB)

C.W. Chiang, M. Ramsey-Musolf, E.S, 1707.09960

Motivation

-

Standard perturbative treatment of v

/T

is gauge-dependent .

v

: minimum of V

at T

T

: T at which V

has degenerate minima

T=TC

vC

For successful EW baryogenesis: v

T

& 1 ¹

st

-order EWPT

-

1

-order phase transition has interesting physical implications:

electroweak baryogenesis, gravitational waves, etc.

-

Effective potential is used for such calculations.

For example,

problem

How (numerically) serious? and how do we treat it?

Thorny problem

Effective potential is gauge dependent!!

V eff ∋ ∌

1PI diagrams only

Because

Leg corrections are needed to remove the ξ dependence.

ξ ξ

Jackiw, PRD9,1686 (1974)

- Energies at stationary points do not depend on ξ - Generally, VEV depends on a gauge parameter ξ

(Nielsen-Fukuda-Kugo (NFK) identity)

JHEP07(2011)029

Φ V_eff

Ξ₁ Ξ₂

Figure 2. A schematic illustration of the behavior of the exact effective potential as the gauge parameter ξ is varied according to Nielsen’s identity. The values of the potential at its extrema stay unchanged but the fields extremizing the potential are gauge-dependent.

artifact of an approximation scheme. In particular, na¨ıvely truncating the perturbative effective potential at a finite order in perturbation theory leads to apparent violations of Nielsen’s identity. In fact, we identify this na¨ıve truncation as the principal source of gauge dependence in the standard determination of T

.

Such effects may be deduced directly from Nielsen’s identity. By expressing V

(φ) and C(φ, ξ) in (2.26 ) as a series in !,

V

(φ) = V

(φ) + ! V

(φ) + !

V

(φ) + . . . (2.27) C(φ, ξ) = c

+ ! c

(φ) + !

c

(φ) + . . . , (2.28) and retaining terms through O(!) in both sides of ( 2.26), we find

∂V

∂ξ + ! ∂V

∂ξ = −c

∂V

∂φ − !

!

c

∂V

∂φ + c

∂V

∂φ

"

. (2.29)

Since the tree-level potential V

is strictly gauge independent, setting O(!

) terms equal implies c

= 0. Upon setting O(!

) terms equal we have

∂V

∂ξ = −c

∂V

∂φ ; (2.30)

that is, the one-loop potential is gauge-independent only where the tree-level potential is extremized, and not where the one-loop potential is extremized. Therefore, the critical temperature based on the one-loop extremum is gauge-dependent. We note that gauge dependence appears formally at a higher order than the order of approximation. Yet, numerically, there is no limit to its sensitivity, and one should necessarily strive to ob- tain a critical temperature that strictly maintains gauge-independence at each order in perturbation theory.

3 Gauge-independent determination of T

In the following sections, we detail one method for performing a gauge-independent per- turbative analysis of the EWPT. As discussed above, exact expressions for the critical

– 15 –

Fig. taken from H. Patel and M. Ramsey-Musolf, JHEP,07(2011)029

Gauge dependence of V eff

@V

@⇠ = C(', ⇠) @V

@'

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

gauge boson

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

gauge boson NG boson

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

1-loop

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

1-loop Tree

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

1-loop effective potential

NFK identity at 1-loop level:

ξ -dependence disappears at

1-loop Tree

gauge boson

NG boson ghost

µ^✏V₁^A+G+c(', ⇠) = i 2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln( k² + ⇠m¯ ²_A) + ln( k² + ⇠m¯ ²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆

2 ln( k² + ⇠m¯ ²_A)

= i

2µ^✏

Z d^Dk (2⇡)^D



(D 1) ln( k² + ¯m²_A) + ln

✓

1 + m¯ ²_G

k² + ⇠m¯ ²_A

◆ .

¯

m

(' = v) = 0, @V

@'

= 0

@V

(', ⇠)

@⇠ = C(', ⇠) @V

(')

@'

e.g., Abelian-Higgs model

[N.B.] commonly used vacuum condition:

@(V

+ V

)

@' = 0

1-loop effective potential T≠0

Using a high-T expansion, one gets

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

ξ terms disappear if m ¯

= 0

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

ξ terms disappear if m ¯

= 0

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

ξ terms disappear if m ¯

= 0

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

ξ terms disappear if m ¯

= 0

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

1-loop effective potential T≠0

Using a high-T expansion, one gets

“T

-terms” are gauge-independent.

V₁(', ⇠; T ) + V₁(', ⇠; T )

= T²

24 ( ¯m²_h + ¯m²_G + 3 ¯m²_A) T 12⇡

h( ¯m²_h)^3/2 + ( ¯m²_G + ⇠m¯ ²_A)^3/2 + (3 ⇠^3/2)( ¯m²_A)^3/2i

+ 1

64⇡²



¯

m⁴_h ln ↵_BT²

¯

µ² + ( ¯m²_G + ⇠m¯ ²_A)² ln ↵_BT²

¯

µ² + 3 ¯m⁴_A

✓

ln ↵_BT²

¯

µ² + 2 3

◆

(⇠m¯ ²_A)² ln ↵_BT²

¯

µ² .

gauge dependent!!

v

T

= 2E

T

ξ terms disappear if m ¯

= 0

V_e↵(', ⇠; T ) = V₀(') + V₁(', ⇠) + V₁(', ⇠; T )

= D(T² T₀²)'² ET ('²)^3/2 + ^T

4 '⁴

JHEP07(2011)029

20 10 0 10 20

70 75 80 100 140

Gauge Parameter, CriticalTemperature,T C(GeV)

0.035 (mH 65 GeV)

Landau:

78.0 GeV latt: 126.8 GeV

104.2 GeV

70.6 GeV 120

: :

Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.

Also included are lattice results, that yield a critical temperature of 126.8 GeV, in- dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of T

, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of T

as ob- tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).

We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.

While the gauge-independent perturbative estimation of T

falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop T

as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases T

in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.

We now turn our attention to the sphaleron scale, ¯ v(T ), which we plot in figure 7.

We observe that in the vicinity of the T

obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in T

. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop T

in the full theory,

– 33 –

[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]

SM case

JHEP07(2011)029

20 10 0 10 20

70 75 80 100 140

Gauge Parameter, CriticalTemperature,T C(GeV)

0.035 (mH 65 GeV)

Landau:

78.0 GeV latt: 126.8 GeV

104.2 GeV

70.6 GeV 120

: :

Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.

Also included are lattice results, that yield a critical temperature of 126.8 GeV, in- dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of T

, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of T

as ob- tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).

We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.

While the gauge-independent perturbative estimation of T

falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop T

as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases T

in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.

We now turn our attention to the sphaleron scale, ¯ v(T ), which we plot in figure 7.

We observe that in the vicinity of the T

obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in T

. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop T

in the full theory,

– 33 –

[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]

SM case

JHEP07(2011)029

20 10 0 10 20

70 75 80 100 140

Gauge Parameter, CriticalTemperature,T C(GeV)

0.035 (mH 65 GeV)

Landau:

78.0 GeV latt: 126.8 GeV

104.2 GeV

70.6 GeV 120

: :

Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.

Also included are lattice results, that yield a critical temperature of 126.8 GeV, in- dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of T

, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of T

as ob- tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).

We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.

While the gauge-independent perturbative estimation of T

falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop T

as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases T

in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.

We now turn our attention to the sphaleron scale, ¯ v(T ), which we plot in figure 7.

We observe that in the vicinity of the T

obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in T

. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop T

in the full theory,

– 33 –

[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]

SM case

Classical scale-inv. U(1) model

ξ dependence is different from the massive U(1) model case.

V

inherently depends on the ⇠ parameter. The one-loop e↵ective potential takes the form [13, 27]

V

('

) =

4 '

+ m ¯

64⇡

✓

ln m ¯

¯ µ

3 2

◆

+ 3 m ¯

64⇡

✓

ln m ¯

¯ µ

5 6

◆

+ m ¯

64⇡

✓

ln m ¯

¯ µ

3 2

◆ (⇠ ¯ m

)

64⇡

✓

ln ⇠ ¯ m

¯ µ

3 2

◆

, (6)

where the field-dependent masses of S, Z

, and G in the R

gauge are respectively given by

¯

m

= 3

'

, m ¯

= (g

Q

'

)

, m ¯

= ¯ m

+ ⇠ ¯ m

, (7) with ¯ m

=

'

. Even though the ⇠-dependent terms are partly cancelled among the gauge boson, the NG boson and the ghosts, the ⇠ dependence still remains at this stage.

Minimizing the one-loop e↵ective potential in Eq. (6) with respect to '

and evaluating it at '

= v

, one can solve for

iteratively and obtains to the leading order that

S

' 3m

16⇡

v

✓

ln m

¯ µ

1 3

◆

, (8)

where we have dropped terms of higher order in

. This result is in stark di↵erence from the corresponding one in U (1)

models without the scale symmetry. Putting

back to Eq. (6), we obtain

V

('

) ' 3 ¯ m

64⇡

✓

ln '

v

1 2

◆

, (9)

which shows no ⇠ dependence. It should be emphasized that in ordinary U (1) models without scale invariance, ¯ m

cannot be considered as a result of one-loop e↵ects as in the above case.

In that case, V

('

) depends on ⇠ except at the point where ¯ m

= 0, corresponding to the parameter set when the tree-level potential, rather than the one-loop potential, assumes its minimum. Albeit no gauge dependence shows up in Eq. (9), we will point out with an explicit example below that the ⇠ dependence cannot be relegated to the second order in perturbation at finite temperatures due to a thermal resummation.

It is well known that at high temperatures perturbative expansions break down and require thermal resummation, i.e., reorganizing the expansions in such a way that domi- nant thermal pieces are summed up to all orders. Following the resummation method for Abelian gauge theories presented in Refs. [28, 29], the thermal masses of the longitudinal and transverse parts ( m

) of the Z

boson as well as the thermal mass of S are added

5

Minimization condition -> λ

= O(g’

/16π

) One gets

ξ independent!!

- ξdependence will appear from 2-loop order.

V_e↵('_S) = ^S

4 '⁴_S + 3 m¯ ⁴_Z0

64⇡²

✓

ln m¯ ²_Z0

¯ µ²

5 6

◆

+ m¯ ⁴_G,⇠

64⇡² ln m¯ ²_G,⇠

¯ µ²

3 2

! (⇠ ¯m²_Z0)² 64⇡²

✓

ln ⇠ ¯m²_Z0

¯ µ²

3 2

◆ ,

- Finite-T 1-loop effective potential is also ξ independent.

¯

m

= (g

Q

'

)

, m ¯

=

'

+ ⇠ ¯ m

.

where

Classical scale-inv. U(1) model

ξ dependence is different from the massive U(1) model case.

V

inherently depends on the ⇠ parameter. The one-loop e↵ective potential takes the form [13, 27]

V

('

) =

4 '

+ m ¯

64⇡

✓

ln m ¯

¯ µ

3 2

◆

+ 3 m ¯

64⇡

✓

ln m ¯

¯ µ

5 6

◆

+ m ¯

64⇡

✓

ln m ¯

¯ µ

3 2

◆ (⇠ ¯ m

)

64⇡

✓

ln ⇠ ¯ m

¯ µ

3 2

◆

, (6)

where the field-dependent masses of S, Z

, and G in the R

gauge are respectively given by

¯

m

= 3

'

, m ¯

= (g

Q

'

)

, m ¯

= ¯ m

+ ⇠ ¯ m

, (7) with ¯ m

=

'

. Even though the ⇠-dependent terms are partly cancelled among the gauge boson, the NG boson and the ghosts, the ⇠ dependence still remains at this stage.

Minimizing the one-loop e↵ective potential in Eq. (6) with respect to '

and evaluating it at '

= v

, one can solve for

iteratively and obtains to the leading order that

S

' 3m

16⇡

v

✓

ln m

¯ µ

1 3

◆

, (8)

where we have dropped terms of higher order in

. This result is in stark di↵erence from the corresponding one in U (1)

models without the scale symmetry. Putting

back to Eq. (6), we obtain

V

('

) ' 3 ¯ m

64⇡

✓

ln '

v

1 2

◆

, (9)

which shows no ⇠ dependence. It should be emphasized that in ordinary U (1) models without scale invariance, ¯ m

cannot be considered as a result of one-loop e↵ects as in the above case.

In that case, V

('

) depends on ⇠ except at the point where ¯ m

= 0, corresponding to the parameter set when the tree-level potential, rather than the one-loop potential, assumes its minimum. Albeit no gauge dependence shows up in Eq. (9), we will point out with an explicit example below that the ⇠ dependence cannot be relegated to the second order in perturbation at finite temperatures due to a thermal resummation.

It is well known that at high temperatures perturbative expansions break down and require thermal resummation, i.e., reorganizing the expansions in such a way that domi- nant thermal pieces are summed up to all orders. Following the resummation method for Abelian gauge theories presented in Refs. [28, 29], the thermal masses of the longitudinal and transverse parts ( m

) of the Z

boson as well as the thermal mass of S are added

5

Minimization condition -> λ

= O(g’

/16π

) One gets

ξ independent!!

- ξdependence will appear from 2-loop order.

V_e↵('_S) = ^S

4 '⁴_S + 3 m¯ ⁴_Z0

64⇡²

✓

ln m¯ ²_Z0

¯ µ²

5 6

◆

+ m¯ ⁴_G,⇠

64⇡² ln m¯ ²_G,⇠

¯ µ²

3 2

! (⇠ ¯m²_Z0)² 64⇡²

✓

ln ⇠ ¯m²_Z0

¯ µ²

3 2

◆ ,

- Finite-T 1-loop effective potential is also ξ independent.

¯

m

= (g

Q

'

)

, m ¯

=

'

+ ⇠ ¯ m

.

where