Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density
at Electroweak Scale Temperature
K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC
KMIA , LNF-INFNA , RIKENB , Kyoto Univ.C

TQFT & Their Applications, August 26, 2013

References
K. Miura, Y. Hidaka, D. Satow, and T. Kunihiro, arXiv:1306.1701 [hep-ph].

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Leptogenesis, Particularly in Low Energy Scale

L
NR

NR = Right-Handed Neutrino
LT = (ν, l)ΦΦ = Higgs Doublet

The standard model (SM) seems to fail to explain the observed baryon
asymmetry of the universe (BAU): ηB = (nB − n̄B )/nγ ' 6.1 × 10−10 .
An extension of the SM by adding right-handed Majorana neutrinos (NR )may have a chance to account for the BAU (Fukugita et.al. (’86)): A decay ofNR (e.g. Fig.) generates a net lepton number, which are partiallyconverted into the baryon number via sphaleron process in the electroweak(EW) phase trans. (Kuzmin et.al. (’85), Klinkhamer et.al.(’84), Arnold et.al. (’87)).
If the mass difference between two NR s is in the order of their CP-violating
decay width, the CP asymmetry is dynamically enhanced (Pilaftsis (’97)), andthe leptogenesis in the EW scale can be relevant (Pilaftsis et.al. (’05)).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Leptogenesis, Particularly in Low Energy Scale

L
NR

NR = Right-Handed Neutrino
LT = (ν, l)ΦΦ = Higgs Doublet

The standard model (SM) seems to fail to explain the observed baryon
asymmetry of the universe (BAU): ηB = (nB − n̄B )/nγ ' 6.1 × 10−10 .
An extension of the SM by adding right-handed Majorana neutrinos (NR )may have a chance to account for the BAU (Fukugita et.al. (’86)): A decay ofNR (e.g. Fig.) generates a net lepton number, which are partiallyconverted into the baryon number via sphaleron process in the electroweak(EW) phase trans. (Kuzmin et.al. (’85), Klinkhamer et.al.(’84), Arnold et.al. (’87)).
If the mass difference between two NR s is in the order of their CP-violating
decay width, the CP asymmetry is dynamically enhanced (Pilaftsis (’97)), andthe leptogenesis in the EW scale can be relevant (Pilaftsis et.al. (’05)).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Leptogenesis, Particularly in Low Energy Scale

L
NR

NR = Right-Handed Neutrino
LT = (ν, l)ΦΦ = Higgs Doublet

The standard model (SM) seems to fail to explain the observed baryon
asymmetry of the universe (BAU): ηB = (nB − n̄B )/nγ ' 6.1 × 10−10 .
An extension of the SM by adding right-handed Majorana neutrinos (NR )may have a chance to account for the BAU (Fukugita et.al. (’86)): A decay ofNR (e.g. Fig.) generates a net lepton number, which are partiallyconverted into the baryon number via sphaleron process in the electroweak(EW) phase trans. (Kuzmin et.al. (’85), Klinkhamer et.al.(’84), Arnold et.al. (’87)).
If the mass difference between two NR s is in the order of their CP-violating
decay width, the CP asymmetry is dynamically enhanced (Pilaftsis (’97)), andthe leptogenesis in the EW scale can be relevant (Pilaftsis et.al. (’05)).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Spectral Density of Leptons in Leptogenesis

L

Im ΣNR = Im

NR = Right-Handed Neutrino
LT = (ν, l)NR

Φ = Higgs Doublet

Φ
(ν,l)

Z ∞
G̃Lepton (k, iωn ; T ) =dω−∞ρLeft (k, ω; T ).
ω + iωn(1)

If the standard-model leptons have non-trivial spectral properties in EW scale
plasma, the lepton number creation via the NR decay may be significantly
modified (c.f. Kiessig et.al., PRD. 2010).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Spectral Density of Leptons in LeptogenesisRR

ρ(ν,lL ) (ω)ω ω+iωn

Φ

R

Lk

Lk
NRi

ρ(ν,lL ) (ω)ω ω+iωnNRiΦρ(ν,lL ) (ω)ω ω+iωn

Ll
NRj

Φ

Lk
NRi

NRj
Φ

If the standard-model leptons have non-trivial spectral properties in EW scale
plasma, the CP asymmetry would be modified.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Spectral Property of Finite T Gauge Theory

4
3

Quasi-Particle
Anti-Plasmino2
ω/mtherm

There is a growing interestin the collective nature of the fermionsin the scenario of thermal leptogenesis(Drewes, arXiv:1303.6912).

1
0
-1
-2

In QED and QCD at extremely high T ,-3
the Hard Thermal-Loop (HTL) approx.
-4
0
1
2
3
4
indicates that a probe fermion interacting|p|/mwith thermally excited gauge bosons and anti-fermions admits a collective
excitation mode (See, The text book by LeBellac).
therm

In the neutrino dispersion relation in the electroweak scale plasma, theexistence of a novel branch in the ultrasoft-energy region has been
indicated by using the HTL and the unitary gauge (Boyanovsky, PRD. 2005).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Spectral Property of Finite T Gauge Theory

4
3

Quasi-Particle
Anti-Plasmino2
ω/mtherm

There is a growing interestin the collective nature of the fermionsin the scenario of thermal leptogenesis(Drewes, arXiv:1303.6912).

1
0
-1
-2

In QED and QCD at extremely high T ,-3
the Hard Thermal-Loop (HTL) approx.
-4
0
1
2
3
4
indicates that a probe fermion interacting|p|/mwith thermally excited gauge bosons and anti-fermions admits a collective
excitation mode (See, The text book by LeBellac).
therm

In the neutrino dispersion relation in the electroweak scale plasma, theexistence of a novel branch in the ultrasoft-energy region has been
indicated by using the HTL and the unitary gauge (Boyanovsky, PRD. 2005).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Spectral Property of Finite T Gauge Theory

4
3

Quasi-Particle
Anti-Plasmino2
ω/mtherm

There is a growing interestin the collective nature of the fermionsin the scenario of thermal leptogenesis(Drewes, arXiv:1303.6912).

1
0
-1
-2

In QED and QCD at extremely high T ,-3
the Hard Thermal-Loop (HTL) approx.
-4
0
1
2
3
4
indicates that a probe fermion interacting|p|/mwith thermally excited gauge bosons and anti-fermions admits a collective
excitation mode (See, The text book by LeBellac).
therm

In the neutrino dispersion relation in the electroweak scale plasma, theexistence of a novel branch in the ultrasoft-energy region has been
indicated by using the HTL and the unitary gauge (Boyanovsky, PRD. 2005).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

From QGP to Particle Cosmology

Goal: We investigate Neutrino Spectral Density at T & MW,Z
1

Without restricting ourselves to the dispersion,

2

In Rξ Gauge (Fujikawa et.al. PRD. 1972),

3

And discuss a possible implication to the leptogenesis.

Hints in QGP Physics
The Spectral Density of massless fermion coupled with the massive
mesonic mode in plasma (an effective discription of QGP) has been
investigated and shown to have a Three-Peak Structure with a Ultrasoft
Mode. (Kitazawa et.al. (’05-’06), Harada et.al. (’08), w.o. HTL).
In particular, when the fermion is coupled with the massive vectorial
meson, the Gauge Independent Nature of the three-peak structure has
been confirmed (Satow et.al. (’10)) by using the Stueckelberg formalism.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

From QGP to Particle Cosmology

Goal: We investigate Neutrino Spectral Density at T & MW,Z
1

Without restricting ourselves to the dispersion,

2

In Rξ Gauge (Fujikawa et.al. PRD. 1972),

3

And discuss a possible implication to the leptogenesis.

Hints in QGP Physics
The Spectral Density of massless fermion coupled with the massive
mesonic mode in plasma (an effective discription of QGP) has been
investigated and shown to have a Three-Peak Structure with a Ultrasoft
Mode. (Kitazawa et.al. (’05-’06), Harada et.al. (’08), w.o. HTL).
In particular, when the fermion is coupled with the massive vectorial
meson, the Gauge Independent Nature of the three-peak structure has
been confirmed (Satow et.al. (’10)) by using the Stueckelberg formalism.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Table of Contents

1

Introduction

2

Preliminaries

3

Results
Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

4

Summary

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Table of Contents

1

Introduction

2

Preliminaries

3

Results
Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

4

Summary

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Setups

Massless Lepton Sector:
LL

=

X h

(ν̄ i , l̄Li )i6 ∂

i=e,µ,τ

 iiν+ l̄Ri i6 ∂ lRi
i
lL

h
i
µ
µ†
µ
+ Wµ† JW
+ JW
Wµ + Zµ JZµ + AEMµ JEM ,

(2)

Weak Bosons in Rξ Gauge:22gµν − qµ qν /MW,Z(T )qµ qν /MW,Z(T )Gµν (q, T ) = −+.
22
q 2 − MW,Z (T )q 2 − ξMW,Z
(T )

(3)

Weak Boson Masses T  v (T ) (c.f. Manuel, PRD. 1998):p
g 2 + g 02 v (T )gv (T )
MW (T ) =+O(gT ) , MZ (T ) =+O(gT ) . (4)22

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Higgs Effective Potential (Rξ Gauge)

Veff = −
T 2 (2λ + 3g 2 /4 + g 0 2 /4) 2µ20
λ1−v (T ) + v 4 (T ) ,
2
4
4µ20

(5)

The ξ dependences cancel out between the Nambu-Goldstone modes andthe ghost contributions (Text Book by Kapusta).
The effective potential leads to the second-order phase transition. Notethat in reality the possibility of the strong first-order transition has been
ruled out within the standard model (Kajantie et.al.(’96), Y.Aoki
et.al.(’99), Csikor et.al.(’99, ’00)).
The temperature region satisfying MW,Z (T ) . T  v (T ) should exist,
and a non-trivial spectral property is anticipated there.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Higgs Effective Potential (Rξ Gauge)

Veff = −
T 2 (2λ + 3g 2 /4 + g 0 2 /4) 2µ20
λ1−v (T ) + v 4 (T ) ,
2
4
4µ20

(5)

The ξ dependences cancel out between the Nambu-Goldstone modes andthe ghost contributions (Text Book by Kapusta).
The effective potential leads to the second-order phase transition. Notethat in reality the possibility of the strong first-order transition has been
ruled out within the standard model (Kajantie et.al.(’96), Y.Aoki
et.al.(’99), Csikor et.al.(’99, ’00)).
The temperature region satisfying MW,Z (T ) . T  v (T ) should exist,
and a non-trivial spectral property is anticipated there.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Self-Energy and Spectral Density

For the massless left-handed neutrinos, the finite-T effects are solely
encoded in the coefficients in the decompositionX(ν)Σret (p, ω; T ) =PR Λs,p γ 0 PL Σ(ν)
s (|p|, ω; T ) ,

(6)

s=±

PL/R = (1 ∓ γ5 )/2 ,Λ±,p = (1 ± γ 0 γ · p/|p|)/2 .

(7)

For the spectral density, similarly,Xρ(ν) (p, ω; T ) =PR Λs,p γ 0 PL ρ(ν)
s (|p|, ω; T )s=±(ν)−Im Σ± (|p|, ω; T )/π(ν)

ρ± (|p|, ω; T ) =

(ν)

(ν)

{ω − |p| ∓ ReΣ± (|p|, ω; T )}2 + {ImΣ± (|p|, ω; T )}2

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature.

Introduction
Preliminaries
Results
Summary

Neutrino Self-Energy and Spectral Density

For the massless left-handed neutrinos, the finite-T effects are solely
encoded in the coefficients in the decompositionX(ν)Σret (p, ω; T ) =PR Λs,p γ 0 PL Σ(ν)
s (|p|, ω; T ) ,

(6)

s=±

PL/R = (1 ∓ γ5 )/2 ,Λ±,p = (1 ± γ 0 γ · p/|p|)/2 .

(7)

For the spectral density, similarly,Xρ(ν) (p, ω; T ) =PR Λs,p γ 0 PL ρ(ν)
s (|p|, ω; T )s=±(ν)−Im Σ± (|p|, ω; T )/π(ν)

ρ± (|p|, ω; T ) =

(ν)

(ν)

{ω − |p| ∓ ReΣ± (|p|, ω; T )}2 + {ImΣ± (|p|, ω; T )}2

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature.

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Table of Contents

1

Introduction

2

Preliminaries

3

Results
Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

4

Summary

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Low Temperature Region

T /v0 = 0.2 ,T /MW (T ) ' 0.63 ,(8)

ρ+v0 at T/v0 = 0.2
100
80
60
40
20
0
0
0.05

0.15
0.1

ω/v0

0.05
0
-0.05
-0.1
T/v0 = 0.2

-0.15
0

0.02

0.04

0.06

0.08

0.1

|p|/v0p/v0 0.1
0.15
0.2 -0.3 -0.2

-0.1

0

0.1
ω/v0

0.2 0.3

ω − |p| − Re Σ+ (ω, |p|, T ) = 0

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Intermediate Temperature Region I

T /v0 = 0.42 ,T /MW (T ) ' 1.45 ,ρ+v0 at T/v0 = 0.42

0.15

30
25
20
15
10
5
0
0
0.025
p/v00.05
0.075

0.1
0.05
ω/v0

(9)

0
-0.05
-0.1
T/v0 = 0.42
-0.15
0

0.02

0.04

0.06

0.08

0.1

|p|/v00.1-0.3-0.2 -0.1

0

0.1 0.2
ω/v0

0.3

ω − |p| − Re Σ+ (ω, |p|, T ) = 0

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Intermediate Temperature Region II

T /v0 = 0.5 ,T /MW (T ) ' 1.83 ,(10)

ρ+v0 at T/v0 = 0.5
30
25
20
15
10
5
0
0
0.025
p/v00.05
0.075

0.15
0.1

ω/v0

0.05
0
-0.05
-0.1
T/v0 = 0.5

-0.15
0

0.02

0.04

0.06

0.08

0.1

|p|/v00.1-0.1-0.3 -0.2

0

0.1
ω/v0

0.2 0.3

ω − |p| − Re Σ+ (ω, |p|, T ) = 0

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

High Temperature Region
T /v0 = 0.8 ,T /MW (T ) ' 4.9 .

ρ+v0 at T/v0 = 0.8
0.2

100
80
60
40
20
0
0
0.05

0.15
T/v0 = 0.8

0.1
ω/v0

0.05
0
-0.05
-0.1
-0.15
-0.2
0

0.02

0.04

0.06

0.08

0.1

|p|/v0p/v0 0.1
0.15
0.2 -0.3-0.2 -0.1

0

0.1
ω/v0

0.2

0.3

The spectral property becomes closer to the HTL result.
T /v (T ) ' 1.59 > 1: The additional thermal-loop corrections may modify
the spectral property (Hidaka-Satow-Kunihiro, Ncul.Phys.A, 2012).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

High Temperature Region
T /v0 = 0.8 ,T /MW (T ) ' 4.9 .

ρ+v0 at T/v0 = 0.8
0.2

100
80
60
40
20
0
0
0.05

0.15
T/v0 = 0.8

0.1
ω/v0

0.05
0
-0.05
-0.1
-0.15
-0.2
0

0.02

0.04

0.06

0.08

0.1

|p|/v0p/v0 0.1
0.15
0.2 -0.3-0.2 -0.1

0

0.1
ω/v0

0.2

0.3

The spectral property becomes closer to the HTL result.
T /v (T ) ' 1.59 > 1: The additional thermal-loop corrections may modify
the spectral property (Hidaka-Satow-Kunihiro, Ncul.Phys.A, 2012).

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Spectral Density at (|p|/v0 , T /v0 ) = (0.02, 0.5)

ρ+v0 at T/v0 = 0.5
12

30
25
20
15
10
5
0

(p/v0, T/v0) = (0.02, 0.5)
10

ρ+v0

8
6

0
0.025
p/v00.05
0.075

4
2
0
-0.2

-0.1

0
ω/v0

0.1

0.2

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

0.1 -0.3 -0.2

-0.1

0

0.1
ω/v0

0.2 0.3

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Landau Damping
Incident Particle Induced Particlel¯ : NFkν
p = (ω, p)

W−µ : (1 + NB )p−kLandau Damping (Fig.) makes the imaginary part being finite in thespacelike region:
Z h
q
i
(ν)
2
ImΣ+
3
δ ω + |k| − |p − k|2 + MW,Z
k

  
× NF (1 + NB ) + NB (1 − NF ) · · · · .
(11)For a small
R external momentum (ω, p) and a not small MW,Z , the phasespace in k admitting the Landau Damping will be restricted.
K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Landau Damping
Incident Particle Induced Particlel¯ : NFkν
p = (ω, p)

W−µ : (1 + NB )p−kLandau Damping (Fig.) makes the imaginary part being finite in thespacelike region:
Z h
q
i
(ν)
2
ImΣ+
3
δ ω + |k| − |p − k|2 + MW,Z
k

  
× NF (1 + NB ) + NB (1 − NF ) · · · · .
(11)For a small
R external momentum (ω, p) and a not small MW,Z , the phasespace in k admitting the Landau Damping will be restricted.
K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Landau Damping Suppression

Incident Particle Induced Particlel¯ : NFkν
p = (ω, p)

Z ∞G (x0 ) =dx xNB (x) =x0∞Xe −nx0 n=1n2

W−µ : (1 + NB )p−k2ω 2 − |p|2 − MW,Z
1 + nx0 , x0 =>0.
2T (ω − |p|)c.f. HTL Limit T  MW,Z , ω, |p|: x0 → 0 and G (x0 ) → ζ(2)  0.
The condition T ∼ MW,Z and the resultant finite x0 leads to G (x)  ζ(2)→ the suppression of the Landau damping.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Landau Damping Suppression

Incident Particle Induced Particlel¯ : NFkν
p = (ω, p)

Z ∞G (x0 ) =dx xNB (x) =x0∞Xe −nx0 n=1n2

W−µ : (1 + NB )p−k2ω 2 − |p|2 − MW,Z
1 + nx0 , x0 =>0.
2T (ω − |p|)c.f. HTL Limit T  MW,Z , ω, |p|: x0 → 0 and G (x0 ) → ζ(2)  0.
The condition T ∼ MW,Z and the resultant finite x0 leads to G (x)  ζ(2)→ the suppression of the Landau damping.

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Self-Energy at Three-Peak RegionT= 0.5 ,v0|p|
= 0.02 .
v0

0

0.15
0.1

(12)

ReΣ+/v0y = (ω - |p|)/v0-0.05
ReΣ+/v0ImΣ+/v0

0.05
-0.1

0
-0.05

-0.15

(|p|/v0, T/v0)-0.1(p/v0, T/v0)=(0.02, 0.5)=(0.02, 0.5)
-0.2

-0.15
-0.2

-0.1

0
ω/v0

0.1

0.2

-0.2

-0.1

0
ω/v0

0.1

0.2

In the right panel, the crossing points corresponds to the solutions of
ω − |p| − Re Σ+ (ω, |p|, T ) = 0(c.f. Kitazawa-Kunihiro-Nemoto, PTP 2007).
K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Spectral Density at (|p|/v0 , T /v0 ) = (0.02, 0.5)

12(p/v0, T/v0) = (0.02, 0.5)
10

ρ+v0

8
6
4
2
0
-0.2

-0.1

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

0
ω/v0

0.1

0.2

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

ξ Depenence of Three-Peak Spectral Density

12
10

ρ+v0

8

ξ=0.1
ξ=1
ξ=10
U-Gauge(p/v0, T/v0) = (0.02, 0.5)

6
4
2
0
-0.2

-0.1

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

0
ω/v0

0.1

0.2

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Sphaleron Freeze-out Temperature

The net baryon number Nb is produced in the sphaleron process when thechanging rate of Nb is larger than the expanding rate of the universe,
1 dNb≥ H(T ) ,
Nb dt

(13)

where,p
T2
H(T ) = 1.66 Ndof
' T 2 × 1.41 × 10−18 (GeV) ,MPL
4πv (T )1 dNb= −1023 · g 7 v (T ) exp −1.89
,
Nb dt
gT

(14)
(15)

and we obtainT ≥ T∗ ' 160 GeV ,K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroCT∗ /v0 ' 0.65 .

(16)

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Sphaleron Freeze-out Temperature

The net baryon number Nb is produced in the sphaleron process when thechanging rate of Nb is larger than the expanding rate of the universe,
1 dNb≥ H(T ) ,
Nb dt

(13)

where,p
T2' T 2 × 1.41 × 10−18 (GeV) ,H(T ) = 1.66 NdofMPL
4πv (T )1 dNb= −1023 · g 7 v (T ) exp −1.89
,
Nb dt
gT

(14)
(15)

and we obtainT ≥ T∗ ' 160 GeV ,K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroCT∗ /v0 ' 0.65 .

(16)

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Sphaleron Freeze-out Temperature

The net baryon number Nb is produced in the sphaleron process when thechanging rate of Nb is larger than the expanding rate of the universe,
1 dNb≥ H(T ) ,
Nb dt

(13)

where,p
T2' T 2 × 1.41 × 10−18 (GeV) ,H(T ) = 1.66 NdofMPL
4πv (T )1 dNb= −1023 · g 7 v (T ) exp −1.89
,
Nb dt
gT

(14)
(15)

and we obtainT ≥ T∗ ' 160 GeV ,K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroCT∗ /v0 ' 0.65 .

(16)

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

Neutrino Spectral Density around T = T∗ρ+v0 at p/v0 = 0.007

30
25
20
15
10
5
0.6
T/v0 0.65
0.7

-0.2

-0.1

T∗
' 0.65 ,v0

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

0
ω/v0

0.1

T∗
∼1.
v (T )

0.2(17)

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Table of Contents

1

Introduction

2

Preliminaries

3

Results
Neutrino Spectral Density: Overview
Three Peak Structure: In Details
Gauge Parameter ξ Dependence of Spectral Property
Implication to Low Energy Scale Leptogenesis

4

Summary

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Summary

We have investigated the spectral properties of standard-model
left-handed neutrinos at finite T around the electroweak scale in a way
where the gauge invariance is manifestly checked (Rξ gauge).
The spectral density of SM neutrino has the three-peak structure with theultrasoft mode with a physical significance when T /MW,Z & 1.
The collective excitation which involves the ultrasoft mode appears attemperature comparable to T∗ within the present approximation. The
three-peak collective modes could affect the leptogenesis at T & T∗ .
Future Work: It is desirable to estimate how large the effects of two-loop
or higher-order diagrams are on the neutrino spectral density.

Thanks for Your Attention!

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Summary

We have investigated the spectral properties of standard-model
left-handed neutrinos at finite T around the electroweak scale in a way
where the gauge invariance is manifestly checked (Rξ gauge).
The spectral density of SM neutrino has the three-peak structure with theultrasoft mode with a physical significance when T /MW,Z & 1.
The collective excitation which involves the ultrasoft mode appears attemperature comparable to T∗ within the present approximation. The
three-peak collective modes could affect the leptogenesis at T & T∗ .
Future Work: It is desirable to estimate how large the effects of two-loop
or higher-order diagrams are on the neutrino spectral density.

Thanks for Your Attention!

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale TemperatureBuckups

Table of Contents

5

Buckups

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Neutrino Spectral Density at Electroweak Scale TemperatureBuckups

Spectral Density Example: Superconductivity

V

I

Scanning
Tunneling
Microscopy

Slightly Underdoped Bi2212
STM Superconductor Scan
Renner et. al.,Phys.Rev.Lett., 80
149, (2008).

d
dI
∼
dV
dV

Z ω=f +eVZdω Dos(ω) ∼ Dos(ω) ∼ ρ(ω, p) ,f

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

(18)

p

Neutrino Spectral Density at Electroweak Scale TemperatureBuckups

Spectral Density Example: Superconductivity

V

I

Scanning
Tunneling
Microscopy

Slightly Underdoped Bi2212
STM Superconductor Scan
Renner et. al.,Phys.Rev.Lett., 80
149, (2008).

d
dI
∼
dV
dV

Z ω=f +eVZdω Dos(ω) ∼ Dos(ω) ∼ ρ(ω, p) ,f

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

(18)

p

Neutrino Spectral Density at Electroweak Scale TemperatureBuckups

Spectral Density Example: Superconductivity

V

I

Scanning
Tunneling
Microscopy

Slightly Underdoped Bi2212
STM Superconductor Scan
Renner et. al.,Phys.Rev.Lett., 80
149, (2008).

d
dI
∼
dV
dV

Z ω=f +eVZdω Dos(ω) ∼ Dos(ω) ∼ ρ(ω, p) ,f

K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

(18)

p

Neutrino Spectral Density at Electroweak Scale Temperature