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

KMI Topics, June 19, 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

Spectral Density Example: Superconductivity

V

I

Scanning
Tunneling
Microscopy

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

dI
d
∼
dV
dV

»Z ω=f +eV–Zdω Dos(ω) ∼ Dos(ω) ∼ ρ(ω, p) ,

f

(1)

p

The Spectral Density ρ(ω, p) is responsible fora PHISICAL CONTENTS of the system.
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

Decay of Right-Handed Neutrino: Typical Example

L
L

NR = Right-Handed Neutrino

NR

NR

Φ

Φ

LT = (ν, l)Φ = Higgs Doublet

Figure: Plumacher et.al. PRD (2010).

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

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Imaginary Self-Energy of Right-Handed Neutrino Comes into Play!

L

Im ΣNR = Im

NR = Right-Handed Neutrino
LT = (ν, l)NR
ΦK. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC

Φ = Higgs Doublet

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(2)

If the standard-model leptons have non-trivial spectral properties in EW scale
plasma, the lepton number creation via the NR decay will be significantly
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

Quasi-Particle
Anti-Plasmino

32
ω/mtherm

In QED and QCD at extremely high T ,the Hard Thermal-Loop (HTL) approx.
indicates that a probe fermion interactingwith thermally excited gauge bosons
and anti-fermions admits acollective excitation mode
(See, The text book by LeBellac).

1
0
-1
-2
-3
-4
0

1

2
|p|/mtherm

3

4

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 fixing (Boyanovsky(’05)).
The gauge invariance and the strength of the ultrasoft mode should be
investigated.

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

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

QGP Physics

When the probe massless fermion interacts with the massive mesonicmode in plasma (an effective discription of QGP), the spectral density of
the fermion shows 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

Goal: From QGP to Electroweak Plasma

Partly motivated by the low energy scale (resonant) leptogenesis, weinvestigate the spectral properties of the standard model neutrinos at finite
T around the electroweak scale.
We utilize the technology developed in the QGP physics (Kitazawa et.al.
(’06), Satow et.al. (’10)), which leads to the followings advantages:
the use of Rξ gauge (Fujikawa et.al. (’72)): Necessary to investigate thegauge invariance of the spectral density.
Without recourse to the HTL approximation: Necessary to evaluate thestrength of the collective modes.

We discuss the implications of the neutrino spectral properties to the
leptogenesis in the EW scale.

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

Lagrangian and Feyman Diagram

Massless Lepton Sector:
„ i«
i h
i
X h i i
ν
µ
µ†
µ
+JW
Wµ +Zµ JZµ +AEM
LL =(ν̄ , l̄L )i6 ∂+l̄Ri i6 ∂ lRi + Wµ† JWµ JEM ,ilL
i=e,µ,τ

(3)
Gauge Boson Propagator in Rξ Gauge:`´22gµν − qµ qν /MW,Z(T )qµ qν /MW,Z(T )Gµν (q, T ) = −
+ 2
,
2
2
2
q − MW,Z (T )q − ξMW,Z(T )K. MiuraA , Y. HidakaB , D. SatowB , and T. KunihiroC(4)

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Neutrino Self-Energy and Spectral Density
We evaluate the one-loop self-energy diagrams at finite T in the imaginary
time formalism: Matsubara Sum ⇒ Analytic Continuation (iωm → ω + iη)(ν)⇒ Retarded Self-Energy Σret (p, ω; T ).
For the massless left-handed neutrinos, the finite-T effects are solely
encoded in the coefficients in the decompositionXˆ˜(ν)Σret (p, ω; T ) =PR Λs,p γ 0 PL Σ(ν)
s (|p|, ω; T ) ,

(5)

s=±

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

(6)

For the spectral density, similarly,Xˆ˜
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

Higgs Effective Potential I

Higgs Effective Potential:
–
»
µ2
T2
λ
Veff = − 0 1 − 2v 2 (T ) + v 4 (T ) ,2
44µ0 /(2λ + 3g 2 /4 + g 0 2 /4)(7)

The potential has been derived in the Rξ gauge by taking account of theleading order of the high T expansion for thermal one-loop effects of the
Higgs and gauge bosons in addition to the tree-level Higgs potential.
The ξ dependences cancel out between the Nambu-Goldstone modes andthe ghost contributions.

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

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Higgs Effective Potential II

1.4

T/v0
v(T)/v0
MW(T)/v0
MZ(T)/v0

1.2
1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6
T/v0

0.8

1

1.2

The effective potential leads to the second-order phase transition. Note that inreality the possibility of the strong first-order transition has been ruled out
within the standard model.

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

Introduction
Preliminaries
Results
Summary

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
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 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 ,(11)

ρ+v0 at T/v0 = 0.8
100
80
60
40
20
0
0
0.05

0.2
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

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

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 the spacelike
region:
Z h
q
i
(ν)
2
δ ω + |k| − |p − k|2 + MW
ImΣ+
3
k
ˆ
˜ ˆ ˜
× NF (1 + NB ) + NB (1 − NF ) · · · · .
(12)

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

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

Introduction
Preliminaries
Results
Summary

Landau Damping Suppression
Incident Particle Induced Particlel¯ : NFk

W−µ : (1 + NB )

νp−kp = (ω, p)

For a small external momentum (ω, p) and a not small MW , Rthe Landau
Damping becomes kinematically limited in the loop integral k :Z ∞G (x) =

dx xNB (x) =x∞Xe −nx ˆ
n=1

n2

2
˜
ω 2 − |p|2 − MW
1 + nx , x => 0 . (13)2T (ω − |p|)Here, the HTL limit T  MW , ω, |p| leads to x → 0 and G (x) → ζ(2).
The condition T ∼ MW and the resultan finite x leads to G (x) < ζ(2) andthereby 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

(14)

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.

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

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

Introduction
Preliminaries
Results
Summary

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

Introduction
Preliminaries
Results
Summaryξ Depenence of Self-Energy in Three-Peak Region|p|= 0.02 .
v0T= 0.5 ,
v0

0.08
0

0.06
0.04
ReΣ+/v0ImΣ+/v0

-0.02
-0.04
-0.06

= (0.02, 0.5)
-0.1
-1-0.75 -0.5 -0.25

0
ω/v0

0.25

0.5

0.75

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

ξ=0.1
ξ=1
ξ=10
UGauge

0.02
0
-0.02
(p/v0, T/v0)= (0.02, 0.5)

-0.04

ξ=0.1
ξ=1
ξ=10
UGauge(p/v0, T/v0)

-0.08

(15)

-0.06
-0.08
1

-1

-0.75 -0.5 -0.25

0
ω/v0

0.25

0.5

0.75

1

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

(17)
(18)

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

(19)

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

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

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 can appear attemperature comparable to T∗ . Thus, the three-peak collective modes
could affect the leptogenesis at T & T∗ , and therefore, the baryogenesis.

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

Neutrino Spectral Density at Electroweak Scale Temperature

Introduction
Preliminaries
Results
Summary

Future Plan

It is desirable to estimate how large the effects of two-loop or higher-order
diagrams are on the neutrino spectral density.
The present formulation should be extended to include the bare fermion
mass effects.
It is interesting to evaluate the lepton number creation with respect to the
non-trivial spectral properties of the neutrinos as well as the charged
leptons by adopting the explicit model of the leptogenesis.
The spectral density of the right-handed neutrinos?

Thanks for Your Attention!

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

Neutrino Spectral Density at Electroweak Scale Temperature