Towards a gaugeinvariant treatment of the electroweak phase transition
Eibun Senaha (ibsCTPU) Nov. 11, 2017@Natl Taiwan U
based on
[2] C.W. Chiang (Natl Taiwan U), M. RamseyMusolf (UMassAmherst), 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
•
Gaugedependence (ξ) of the effective potential
•
Impact of ξ on 1
^{st}order phase transition in classical scaleinv. U(1) models: T
C, T
N, GW
•
Gaugeinv. 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. RamseyMusolf, E.S, 1707.09960
Motivation

Standard perturbative treatment of v
C/T
Cis gaugedependent .
v
C: minimum of V
effat T
CT
C: T at which V
effhas degenerate minima
T=TC
vC
For successful EW baryogenesis: v
_{C}T
_{C}& 1 ^{1}
st
order EWPT

1
^{st}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
(NielsenFukudaKugo (NFK) identity)
JHEP07(2011)029
Φ V_{eff}
Ξ_{1} Ξ_{2}
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 gaugedependent.
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
_{C}.
Such eﬀects may be deduced directly from Nielsen’s identity. By expressing V
_{eﬀ}(φ) and C(φ, ξ) in (2.26 ) as a series in !,
V
_{eﬀ}(φ) = V
_{0}(φ) + ! V
_{1}(φ) + !
^{2}V
_{2}(φ) + . . . (2.27) C(φ, ξ) = c
_{0}+ ! c
_{1}(φ) + !
^{2}c
_{2}(φ) + . . . , (2.28) and retaining terms through O(!) in both sides of ( 2.26), we find
∂V
_{0}∂ξ + ! ∂V
_{1}∂ξ = −c
^{0}∂V
_{0}∂φ − !
!
c
_{0}∂V
_{1}∂φ + c
_{1}∂V
_{0}∂φ
"
. (2.29)
Since the treelevel potential V
_{0}is strictly gauge independent, setting O(!
^{0}) terms equal implies c
_{0}= 0. Upon setting O(!
^{1}) terms equal we have
∂V
_{1}∂ξ = −c
^{1}∂V
_{0}∂φ ; (2.30)
that is, the oneloop potential is gaugeindependent only where the treelevel potential is extremized, and not where the oneloop potential is extremized. Therefore, the critical temperature based on the oneloop extremum is gaugedependent. 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 gaugeindependence at each order in perturbation theory.
3 Gaugeindependent determination of T
^{C}In the following sections, we detail one method for performing a gaugeindependent per turbative analysis of the EWPT. As discussed above, exact expressions for the critical
– 15 –
Fig. taken from H. Patel and M. RamseyMusolf, JHEP,07(2011)029
Gauge dependence of V eff
@V
_{e↵}@⇠ = C(', ⇠) @V
_{e↵}@'
[N.K. Nielsen, NPB101 (’75) 173, R. Fukuda, T. Kugo, PRD13 (’76) 3469.]1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
gauge boson
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
gauge boson NG boson
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
1loop
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
1loop Tree
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
1loop effective potential
NFK identity at 1loop level:
ξ dependence disappears at
1loop Tree
gauge boson
NG boson ghost
µ^{✏}V_{1}^{A+G+c}(', ⇠) = i 2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln( k^{2} + ⇠m¯ ^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆
2 ln( k^{2} + ⇠m¯ ^{2}_{A})
= i
2µ^{✏}
Z d^{D}k (2⇡)^{D}
(D 1) ln( k^{2} + ¯m^{2}_{A}) + ln
✓
1 + m¯ ^{2}_{G}
k^{2} + ⇠m¯ ^{2}_{A}
◆ .
¯
m
^{2}_{G}(' = v) = 0, @V
_{0}@'
_{'=v}= 0
@V
_{1}(', ⇠)
@⇠ = C(', ⇠) @V
_{0}(')
@'
e.g., AbelianHiggs model
[N.B.] commonly used vacuum condition:
@(V
_{0}+ V
_{1})
@' = 0
1loop effective potential T≠0
Using a highT expansion, one gets
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
ξ terms disappear if m ¯
^{2}_{G}= 0
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
ξ terms disappear if m ¯
^{2}_{G}= 0
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
ξ terms disappear if m ¯
^{2}_{G}= 0
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
ξ terms disappear if m ¯
^{2}_{G}= 0
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{2} ET ('^{2})^{3/2} + ^{T}
4 '^{4}
1loop effective potential T≠0
Using a highT expansion, one gets
“T
^{2}terms” are gaugeindependent.
V_{1}(', ⇠; T ) + V_{1}(', ⇠; T )
= T^{2}
24 ( ¯m^{2}_{h} + ¯m^{2}_{G} + 3 ¯m^{2}_{A}) T 12⇡
h( ¯m^{2}_{h})^{3/2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{3/2} + (3 ⇠^{3/2})( ¯m^{2}_{A})^{3/2}i
+ 1
64⇡^{2}
¯
m^{4}_{h} ln ↵_{B}T^{2}
¯
µ^{2} + ( ¯m^{2}_{G} + ⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} + 3 ¯m^{4}_{A}
✓
ln ↵_{B}T^{2}
¯
µ^{2} + 2 3
◆
(⇠m¯ ^{2}_{A})^{2} ln ↵_{B}T^{2}
¯
µ^{2} .
gauge dependent!!
v
_{C}T
_{C}= 2E
T
ξ terms disappear if m ¯
^{2}_{G}= 0
V_{e↵}(', ⇠; T ) = V_{0}(') + V_{1}(', ⇠) + V_{1}(', ⇠; T )
= D(T^{2} T_{0}^{2})'^{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 gaugeindependent 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 higherorder contributions included in (5.15) leads to a substantially larger value of T
_{C}, suggesting that the diﬀerence between the nonperturbative and O(!) perturbative results arises in part from the omission of higherorder contributions. In addition, we note that the precise definition of T
_{C}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 gaugeindependent perturbative estimation of T
_{C}falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in nonperturbative studies. To illustrate, we plot the oneloop T
_{C}as a function of the Higgs quartic selfcoupling λ in figure 6. We observe that increasing λ increases T
_{C}in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the eﬃciency of sphaleroninduced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in nonperturbative studies as well as in earlier gaugedependent 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
_{C}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
_{C}. 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 oneloop T
_{C}in the full theory,
– 33 –
[H.Patel, M.RamseyMusolf, 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 gaugeindependent 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 higherorder contributions included in (5.15) leads to a substantially larger value of T
_{C}, suggesting that the diﬀerence between the nonperturbative and O(!) perturbative results arises in part from the omission of higherorder contributions. In addition, we note that the precise definition of T
_{C}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 gaugeindependent perturbative estimation of T
_{C}falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in nonperturbative studies. To illustrate, we plot the oneloop T
_{C}as a function of the Higgs quartic selfcoupling λ in figure 6. We observe that increasing λ increases T
_{C}in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the eﬃciency of sphaleroninduced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in nonperturbative studies as well as in earlier gaugedependent 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
_{C}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
_{C}. 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 oneloop T
_{C}in the full theory,
– 33 –
[H.Patel, M.RamseyMusolf, 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 gaugeindependent 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 higherorder contributions included in (5.15) leads to a substantially larger value of T
_{C}, suggesting that the diﬀerence between the nonperturbative and O(!) perturbative results arises in part from the omission of higherorder contributions. In addition, we note that the precise definition of T
_{C}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 gaugeindependent perturbative estimation of T
_{C}falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in nonperturbative studies. To illustrate, we plot the oneloop T
_{C}as a function of the Higgs quartic selfcoupling λ in figure 6. We observe that increasing λ increases T
_{C}in agreement with our qualitative expectations in ( 3.11). As we discuss shortly, this trend implies that the eﬃciency of sphaleroninduced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in nonperturbative studies as well as in earlier gaugedependent 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
_{C}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
_{C}. 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 oneloop T
_{C}in the full theory,
– 33 –
[H.Patel, M.RamseyMusolf, JHEP,07(2011)029]
SM case
Classical scaleinv. U(1) model
ξ dependence is different from the massive U(1) model case.
V
_{CW}inherently depends on the ⇠ parameter. The oneloop e↵ective potential takes the form [13, 27]
V
_{e↵}('
_{S}) =
^{S}4 '
^{4}_{S}+ m ¯
^{4}_{S}64⇡
^{2}✓
ln m ¯
^{2}_{S}¯ µ
^{2}3 2
◆
+ 3 m ¯
^{4}_{Z}_{0}64⇡
^{2}✓
ln m ¯
^{2}_{Z}_{0}¯ µ
^{2}5 6
◆
+ m ¯
^{4}_{G,⇠}64⇡
^{2}✓
ln m ¯
^{2}_{G,⇠}¯ µ
^{2}3 2
◆ (⇠ ¯ m
^{2}_{Z}0)
^{2}64⇡
^{2}✓
ln ⇠ ¯ m
^{2}_{Z}0¯ µ
^{2}3 2
◆
, (6)
where the fielddependent masses of S, Z
^{0}, and G in the R
_{⇠}gauge are respectively given by
¯
m
^{2}_{S}= 3
_{S}'
^{2}_{S}, m ¯
^{2}_{Z}0= (g
^{0}Q
^{0}_{S}'
_{S})
^{2}, m ¯
^{2}_{G,⇠}= ¯ m
^{2}_{G}+ ⇠ ¯ m
^{2}_{Z}0, (7) with ¯ m
^{2}_{G}=
_{S}'
^{2}_{S}. 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 oneloop e↵ective potential in Eq. (6) with respect to '
_{S}and evaluating it at '
_{S}= v
_{S}, one can solve for
_{S}iteratively and obtains to the leading order that
S
' 3m
^{4}_{Z}016⇡
^{2}v
_{S}^{4}✓
ln m
^{2}_{Z}0¯ µ
^{2}1 3
◆
, (8)
where we have dropped terms of higher order in
_{S}. This result is in stark di↵erence from the corresponding one in U (1)
^{0}models without the scale symmetry. Putting
_{S}back to Eq. (6), we obtain
V
_{e↵}('
_{S}) ' 3 ¯ m
^{4}_{Z}_{0}64⇡
^{2}✓
ln '
^{2}_{S}v
_{S}^{2}1 2
◆
, (9)
which shows no ⇠ dependence. It should be emphasized that in ordinary U (1) models without scale invariance, ¯ m
^{2}_{G}cannot be considered as a result of oneloop e↵ects as in the above case.
In that case, V
_{e↵}('
_{S}) depends on ⇠ except at the point where ¯ m
^{2}_{G}= 0, corresponding to the parameter set when the treelevel potential, rather than the oneloop 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
_{L,T}) of the Z
^{0}boson as well as the thermal mass of S are added
5
Minimization condition > λ
S= O(g’
^{4}/16π
^{2}) One gets
ξ independent!!
 ξdependence will appear from 2loop order.
V_{e↵}('_{S}) = ^{S}
4 '^{4}_{S} + 3 m¯ ^{4}_{Z}0
64⇡^{2}
✓
ln m¯ ^{2}_{Z}0
¯ µ^{2}
5 6
◆
+ m¯ ^{4}_{G,⇠}
64⇡^{2} ln m¯ ^{2}_{G,⇠}
¯ µ^{2}
3 2
! (⇠ ¯m^{2}_{Z}0)^{2} 64⇡^{2}
✓
ln ⇠ ¯m^{2}_{Z}0
¯ µ^{2}
3 2
◆ ,
 FiniteT 1loop effective potential is also ξ independent.
¯
m
^{2}_{Z}0= (g
^{0}Q
^{0}_{S}'
_{S})
^{2}, m ¯
^{2}_{G,⇠}=
_{S}'
^{2}_{S}+ ⇠ ¯ m
^{2}_{Z}0.
where
Classical scaleinv. U(1) model
ξ dependence is different from the massive U(1) model case.
V
_{CW}inherently depends on the ⇠ parameter. The oneloop e↵ective potential takes the form [13, 27]
V
_{e↵}('
_{S}) =
^{S}4 '
^{4}_{S}+ m ¯
^{4}_{S}64⇡
^{2}✓
ln m ¯
^{2}_{S}¯ µ
^{2}3 2
◆
+ 3 m ¯
^{4}_{Z}_{0}64⇡
^{2}✓
ln m ¯
^{2}_{Z}_{0}¯ µ
^{2}5 6
◆
+ m ¯
^{4}_{G,⇠}64⇡
^{2}✓
ln m ¯
^{2}_{G,⇠}¯ µ
^{2}3 2
◆ (⇠ ¯ m
^{2}_{Z}0)
^{2}64⇡
^{2}✓
ln ⇠ ¯ m
^{2}_{Z}0¯ µ
^{2}3 2
◆
, (6)
where the fielddependent masses of S, Z
^{0}, and G in the R
_{⇠}gauge are respectively given by
¯
m
^{2}_{S}= 3
_{S}'
^{2}_{S}, m ¯
^{2}_{Z}0= (g
^{0}Q
^{0}_{S}'
_{S})
^{2}, m ¯
^{2}_{G,⇠}= ¯ m
^{2}_{G}+ ⇠ ¯ m
^{2}_{Z}0, (7) with ¯ m
^{2}_{G}=
_{S}'
^{2}_{S}. 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 oneloop e↵ective potential in Eq. (6) with respect to '
_{S}and evaluating it at '
_{S}= v
_{S}, one can solve for
_{S}iteratively and obtains to the leading order that
S
' 3m
^{4}_{Z}016⇡
^{2}v
_{S}^{4}✓
ln m
^{2}_{Z}0¯ µ
^{2}1 3
◆
, (8)
where we have dropped terms of higher order in
_{S}. This result is in stark di↵erence from the corresponding one in U (1)
^{0}models without the scale symmetry. Putting
_{S}back to Eq. (6), we obtain
V
_{e↵}('
_{S}) ' 3 ¯ m
^{4}_{Z}_{0}64⇡
^{2}✓
ln '
^{2}_{S}v
_{S}^{2}1 2
◆
, (9)
which shows no ⇠ dependence. It should be emphasized that in ordinary U (1) models without scale invariance, ¯ m
^{2}_{G}cannot be considered as a result of oneloop e↵ects as in the above case.
In that case, V
_{e↵}('
_{S}) depends on ⇠ except at the point where ¯ m
^{2}_{G}= 0, corresponding to the parameter set when the treelevel potential, rather than the oneloop 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
_{L,T}) of the Z
^{0}boson as well as the thermal mass of S are added
5
Minimization condition > λ
S= O(g’
^{4}/16π
^{2}) One gets
ξ independent!!
 ξdependence will appear from 2loop order.
V_{e↵}('_{S}) = ^{S}
4 '^{4}_{S} + 3 m¯ ^{4}_{Z}0
64⇡^{2}
✓
ln m¯ ^{2}_{Z}0
¯ µ^{2}
5 6
◆
+ m¯ ^{4}_{G,⇠}
64⇡^{2} ln m¯ ^{2}_{G,⇠}
¯ µ^{2}
3 2
! (⇠ ¯m^{2}_{Z}0)^{2} 64⇡^{2}
✓
ln ⇠ ¯m^{2}_{Z}0
¯ µ^{2}
3 2
◆ ,