S parameter at non-zero temperature and chemical potential Ulrik Ishøj Søndergaard
February 2012 Søndergaard, Pica, Sannino: Phys. Rev. D 84, 075022 (2011)
Electroweak Precision Observables
α
Electron anomalous magnetic moment
GF
Muon lifetime
�Z
Resonance
Classic review: M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992)
At Lagrangian level �
� = �(�� � � �)
2 �H
λ
Classic review: M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992)
At Lagrangian level �
Broken phase:
� = �(�� � � �)
� = �(�0 � �0 � �)
2 �H
λ
Experimental setup: External fermions are light
Experimental setup: External fermions are light =⇒ =⇒
µ ν
� � -term in propagators suppressed
‘direct’ BSM corrections suppressed
Experimental setup: External fermions are light =⇒ =⇒
µ ν
� � -term in propagators suppressed
‘direct’ BSM corrections suppressed
Oblique corrections dominate
Tree-Level expression Example: Charged Current
�W W =
2 �0 I 2 + 2 2�0 �
1 I − 2 − ���0
Tree-Level expression Example: Charged Current
�W W =
2 �0 I 2 + 2 2�0 �
1 I − 2 − ���0
All Orders Vacuum Polarization
*
�W W =
2 �∗ ZW �∗ I I + − 2 2 2 2�∗ � − ���∗ 2
‘ed quantities depend on q
Tree-Level expression Example: Charged Current
�W W =
2 �0 I 2 + 2 2�0 �
1 I − 2 − ���0
All Orders Vacuum Polarization
*
�W W =
2 �∗ ZW �∗ I I + − 2 2 2 2�∗ � − ���∗ 2
‘ed quantities depend on q
+ Most SM corrections can be absorbed into these variables (including direct corrections)
Choosing a scheme
�� ≡?
M. Awramik et al., Phys. Rev. D69, 053006 (2004), hep-ph/0311148 ← SM mW calc. + others
Choosing a scheme
�� ≡?
2 � 2 W sin θ = 1 − W ‘On shell’ �2Z
M. Awramik et al., Phys. Rev. D69, 053006 (2004), hep-ph/0311148 ← SM mW calc. + others
Choosing a scheme
�� ≡?
2 � 2 W sin θ = 1 − W ‘On shell’ �2Z
Better:
sin 2θ� =
�
4πα��∗ (�2Z ) √ 2GF �2Z
M. Awramik et al., Phys. Rev. D69, 053006 (2004), hep-ph/0311148 ← SM mW calc. + others
Choosing a scheme
�� ≡?
2 � 2 W sin θ = 1 − W ‘On shell’ �2Z
Better:
sin 2θ� =
�
4πα��∗ (�2Z ) √ 2GF �2Z
We can now compute any EW observable as an expression of �� � α� GF and �Z Up to corrections from new physics!
M. Awramik et al., Phys. Rev. D69, 053006 (2004), hep-ph/0311148 ← SM mW calc. + others
Choosing a scheme 2 � 2 W sin θ = 1 − W ‘On shell’ �2Z
�� ≡?
Better:
sin 2θ� =
�
4πα��∗ (�2Z ) √ 2GF �2Z
We can now compute any EW observable as an expression of �� � α� GF and �Z Up to corrections from new physics!
Ex:
�2W 2 − cos θ� 2 �Z α�2 = 2 � − �2
2 2 1 � � � 2 4π � � � � − � � � − 16π Π33 (0) − Π3Q (0) +� (Π11 (0) − Π33 (0)) + 16π Π11 (0) − Π33 (0) 2� 2 2 2 2 � � �� 4� � �� � �� � � �� �
≡S
≡T
≡U
M. Awramik et al., Phys. Rev. D69, 053006 (2004), hep-ph/0311148 ← SM mW calc. + others
T
0.5 B SM
Aug 10
0.4
G fitter preliminary
0.3 0.2 0.1 0 MH D [114,1000] GeV mt = 173.3! 1.1 GeV
-0.1 MH
-0.2 -0.3
68%, 95%, 99% CL fit contours (M =120 GeV, U=0) H
-0.4 -0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
S
GFITTER
Definitions Peskin-Takeuchi
S = 16π =
�
� Π33 (0)
−
� −16πΠ3Y (0)
� Π3Q (0)
�
Definitions Peskin-Takeuchi �2 → 0
He-Polonsky-Su
S = 16π =
�
� Π33 (0)
−
� −16πΠ3Y (0)
� Π3Q (0)
�
� Π3Y (� ) − Π3Y (0) � S = −16π � �2 =�2Z �2 2
We will not do this
Definitions S = 16π
Peskin-Takeuchi �2 → 0
He-Polonsky-Su T →0 µ→0
=
�
� Π33 (0)
−
� −16πΠ3Y (0)
� Π3Q (0)
� Π3Y (� ) − Π3Y (0) � S = −16π � �2 =�2Z �2 2
We will not do this
Π3Y (�2 � �� T � µ) − Π3Y (0� �� T � µ) S = −16π �2
�0 = �|q|�
�
2
� =
2 �0
−q
2
� = coth η ≥ 1
S at zero T and μ Degenerate technifermions: No Y dependence � � 6 �� 2 4 � 1 � 1 � � 2 2 S(� /�T F ) = 1+ + +� 2 4 6π 10 �T F 70 �T F �6T F
� = �[�]NT F /2
fund:
= NT C NT F /2
S at zero T and μ Degenerate technifermions: No Y dependence � � 6 �� 2 4 � 1 � 1 � � 2 2 S(� /�T F ) = 1+ + +� 2 4 6 6π 10 �T F 70 �TCP � 3 - Origins F TF
� = �[�]NT F /2
= NT C NT F /2
3.0 2.5 2.0 6ΠS��
fund:
Particle Physics & Origin of Mass
1.5 1.0 0.5 0.0 0
1
2
4
q2 �m2
8
16
�
FIG. 1: Real (blue-solid) and imaginary (red-dashed) parts for 6πS the normalized parameter as function of increasing q2 /m2 �
Bo no ke lim the be
S at zero T and μ Degenerate technifermions: No Y dependence � � 6 �� 2 4 � 1 � 1 � � 2 2 S(� /�T F ) = 1+ + +� 2 4 6 6π 10 �T F 70 �TCP � 3 - Origins F TF
� = �[�]NT F /2
= NT C NT F /2
� S(0) = 6π sxc.hu:1128191
3.0 2.5 2.0 6ΠS��
fund:
Particle Physics & Origin of Mass
1.5 1.0 0.5 0.0 0
1
2
4
q2 �m2
8
16
�
FIG. 1: Real (blue-solid) and imaginary (red-dashed) parts for 6πS the normalized parameter as function of increasing q2 /m2 �
Bo no ke lim the be
CP 3 - Origins Particle Physics & Origin of Mass
1.3
1
� >1.1 6π 6ΠS��
1.2
1.1
10
11
12
13
14
15
16
Nf (a) SU(3) with fundamental fermions.
1.2
1
1.3
1.3
1.2
1.2
6ΠS��
6ΠS��
2-loop perturbation
6ΠS��
1.3
1.1
1
7
8
(b) SU(2)
1.1
2.0
2.2
2.4
2.6
2.8
3.0
1
3.2
2.0
2.1
2.
Nf (c) SU(3) with 2-index symmetric fermions.
(d) SU
FIG. 2: Normalized conformal S-parameter near the perturbative upper bound of the co
to consider the 2-loops β-function to determine α at the
with
CP 3 - Origins Particle Physics & Origin of Mass
1.3
1.2
1.1
1
� >1.1 6π 6ΠS��
2-loop perturbation
6ΠS��
1.3
10
11
12
13
14
15
16
Nf (a) SU(3) with fundamental fermions.
1
1.3
1.3
1.2
1.2
� � 1.1 S(0) � 1�57 > 6π 6π 1
2.0
2.2
2.4
6ΠS��
WSR + vector dominance + Large N rescaling 6ΠS��
1.2
7
8
(b) SU(2)
1.1
2.6
2.8
3.0
1
3.2
2.0
2.1
2.
Nf
(c) SU(3) with 2-index symmetric fermions.
(d) SU
FIG. 2: Normalized conformal S-parameter near the perturbative upper bound of the co
Peskin, Takeuchi, Phys. Rev. D 46, 381 (1992)to consider the 2-loops β-function to determine α at the
with
CP 3 - Origins Particle Physics & Origin of Mass
1.3
1.2
1.1
1
� >1.1 6π 6ΠS��
2-loop perturbation
6ΠS��
1.3
10
11
12
13
14
15
16
Nf (a) SU(3) with fundamental fermions.
1
1.3
1.3
1.2
1.2
� � 1.1 S(0) � 1�57 > 6π 6π 1
2.0
2.2
2.4
6ΠS��
WSR + vector dominance + Large N rescaling 6ΠS��
1.2
7
8
(b) SU(2)
1.1
2.6
2.8
3.0
1
3.2
2.0
2.1
WSR + more sophisticated approx + Large N rescaling Nf
(c) SU(3) with 2-index symmetric fermions.
� � S(0) � 1�88 > 6π 6π Peskin, Takeuchi, Phys. Rev. D 46, 381 (1992)
2.
(d) SU
FIG. 2: Normalized conformal S-parameter near the perturbative upper bound of the co
to consider the 2-loops β-function to determine α at the
with
Calculation For a degenerate technifermion doublet: 1 loop µν ΠLH = T
∞ �
�=−∞
�
Π3Y
1 = ΠLR 2
� � �/ + � ν �/ + �/ + � �p µ Tr γ PL 2 γ PH 3 2 (2π) � −� (� + �)2 − �2 3
�0 = �(2� + 1)πT + µ
ΠLR (�� �2 � T � µ) − ΠLR (�� 0� T � µ) S = −8π �2
Results Cold (βm >> 1):
√ � � �� � cosh(βµ) sech (η) 3 2π −β� 1 S+ = − � · 1+� � �2 3/2 6π (β�) β� �/� 1 − 2 cosh η 4
Results Cold (βm >> 1):
√ � � �� � cosh(βµ) sech (η) 3 2π −β� 1 S+ = − � · 1+� � �2 3/2 6π (β�) β� �/� 1 − 2 cosh η 4
Hot (βm > 1):
√ � � �� � cosh(βµ) sech (η) 3 2π −β� 1 S+ = − � · 1+� � �2 3/2 6π (β�) β� �/� 1 − 2 cosh η 4
Hot (βm
