Neutrino Spectral Density at Electroweak Scale Temperaturesoken.editorial/sokendenshi/... ·...

IntroductionPreliminaries

ResultsSummary

Neutrino Spectral Densityat 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


ResultsSummary

Leptogenesis, Particularly in Low Energy Scale

NR

L

Φ

LT = (ν, l)

Φ = Higgs Doublet

NR = Right-Handed Neutrino

The standard model (SM) seems to fail to explain the observed baryonasymmetry 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 NRs is in the order of their CP-violatingdecay width, the CP asymmetry is dynamically enhanced (Pilaftsis (’97)), andthe leptogenesis in the EW scale can be relevant (Pilaftsis et.al. (’05)).



ResultsSummary


NR

L

Φ

LT = (ν, l)

Φ = Higgs Doublet







ResultsSummary


NR

L

Φ

LT = (ν, l)

Φ = Higgs Doublet







ResultsSummary

Spectral Density of Leptons in Leptogenesis

NR

L

Φ

Im ΣNR=Im LT = (ν, l)

Φ = Higgs Doublet


G̃Lepton(k, iωn;T ) =

∫ ∞−∞

dωρ(ν,l)Left(k, ω;T )

ω + iωn. (1)

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

modified (c.f. Kiessig et.al., PRD. 2010).



ResultsSummary

Spectral Density of Leptons in Leptogenesis

N iR

Lk

Φ N jR

Ll

ΦN i

R

Lk

Φ

N jR

N iR

Lk

Φ

∫ω

ρ(ν,l )(ω)ω+iωn

L

∫ω


L

∫ω


L

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



ResultsSummary

Spectral Property of Finite T Gauge Theory

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4

ω/m

ther

m

|p|/mtherm

Quasi-ParticleAnti-PlasminoThere is a growing interest

in the collective nature of the fermionsin the scenario of thermal leptogenesis(Drewes, arXiv:1303.6912).

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 a collectiveexcitation mode (See, The text book by LeBellac).

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



ResultsSummary


-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4

ω/m

ther

m

|p|/mtherm







ResultsSummary


-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4

ω/m

ther

m

|p|/mtherm







ResultsSummary

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 massivemesonic mode in plasma (an effective discription of QGP) has beeninvestigated and shown to have a Three-Peak Structure with a UltrasoftMode. (Kitazawa et.al. (’05-’06), Harada et.al. (’08), w.o. HTL).

In particular, when the fermion is coupled with the massive vectorialmeson, the Gauge Independent Nature of the three-peak structure hasbeen confirmed (Satow et.al. (’10)) by using the Stueckelberg formalism.



ResultsSummary

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 massivemesonic mode in plasma (an effective discription of QGP) has beeninvestigated and shown to have a Three-Peak Structure with a UltrasoftMode. (Kitazawa et.al. (’05-’06), Harada et.al. (’08), w.o. HTL).

In particular, when the fermion is coupled with the massive vectorialmeson, the Gauge Independent Nature of the three-peak structure hasbeen confirmed (Satow et.al. (’10)) by using the Stueckelberg formalism.



ResultsSummary

Table of Contents

1 Introduction

2 Preliminaries

3 ResultsNeutrino Spectral Density: OverviewThree Peak Structure: In DetailsGauge Parameter ξ Dependence of Spectral PropertyImplication to Low Energy Scale Leptogenesis

4 Summary



ResultsSummary

Table of Contents

1 Introduction

2 Preliminaries


4 Summary



ResultsSummary

Setups

Massless Lepton Sector:

LL =∑

i=e,µ,τ

[(ν̄ i , l̄ iL)i 6∂

(ν i

l iL

)+ l̄ iRi 6∂ l iR

]+[W †µJ

µW + Jµ†W Wµ + ZµJ

µZ + AEM

µ JµEM

], (2)

Weak Bosons in Rξ Gauge:

Gµν(q,T ) = −(gµν − qµqν/M

2W,Z(T )

)q2 −M2

W,Z(T )+

qµqν/M2W,Z(T )

q2 − ξM2W,Z(T )

. (3)

Weak Boson Masses T � v(T ) (c.f. Manuel, PRD. 1998):

MW(T ) =gv(T )

2+O(gT ) , MZ(T ) =

√g 2 + g ′2 v(T )

2+O(gT ) . (4)



ResultsSummary

Higgs Effective Potential (Rξ Gauge)

Veff = −µ20

2

[1− T 2(2λ+ 3g 2/4 + g ′ 2/4)

4µ20

]v 2(T ) +

λ

4v 4(T ) , (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 beenruled out within the standard model (Kajantie et.al.(’96), Y.Aokiet.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.



ResultsSummary

Higgs Effective Potential (Rξ Gauge)

Veff = −µ20

2

[1− T 2(2λ+ 3g 2/4 + g ′ 2/4)

4µ20

]v 2(T ) +

λ

4v 4(T ) , (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 beenruled out within the standard model (Kajantie et.al.(’96), Y.Aokiet.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.



ResultsSummary

Neutrino Self-Energy and Spectral Density

For the massless left-handed neutrinos, the finite-T effects are solelyencoded in the coefficients in the decomposition

Σ(ν)ret(p, ω;T ) =

∑s=±

[PRΛs,pγ

0PL

]Σ(ν)

s (|p|, ω;T ) , (6)

PL/R = (1∓ γ5)/2 , Λ±,p = (1± γ0γ · p/|p|)/2 . (7)

For the spectral density, similarly,

ρ(ν)(p, ω;T ) =∑s=±

[PRΛs,pγ

0PL

]ρ(ν)s (|p|, ω;T )

ρ(ν)± (|p|, ω;T ) =

−Im Σ(ν)± (|p|, ω;T )/π

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

(ν)± (|p|, ω;T )}2

.



ResultsSummary

Neutrino Self-Energy and Spectral Density

For the massless left-handed neutrinos, the finite-T effects are solelyencoded in the coefficients in the decomposition

Σ(ν)ret(p, ω;T ) =

∑s=±

[PRΛs,pγ

0PL

]Σ(ν)

s (|p|, ω;T ) , (6)

PL/R = (1∓ γ5)/2 , Λ±,p = (1± γ0γ · p/|p|)/2 . (7)

For the spectral density, similarly,

ρ(ν)(p, ω;T ) =∑s=±

[PRΛs,pγ

0PL

]ρ(ν)s (|p|, ω;T )

ρ(ν)± (|p|, ω;T ) =

−Im Σ(ν)± (|p|, ω;T )/π

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

(ν)± (|p|, ω;T )}2

.



ResultsSummary

Neutrino Spectral Density: OverviewThree Peak Structure: In DetailsGauge Parameter ξ Dependence of Spectral PropertyImplication to Low Energy Scale Leptogenesis

Table of Contents

1 Introduction

2 Preliminaries


4 Summary



ResultsSummary


Low Temperature Region

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

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.02 0.04 0.06 0.08 0.1

ω/v

0

|p|/v0

T/v0 = 0.2

0

0.05

0.1

0.15

0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

20

40

60

80

100

ρ+v0 at T/v0 = 0.2

p/v0

ω/v0

ω − |p| − Re Σ+(ω, |p|,T ) = 0



ResultsSummary


Intermediate Temperature Region I

T/v0 = 0.42 , T/MW (T ) ' 1.45 , (9)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.02 0.04 0.06 0.08 0.1

ω/v

0

|p|/v0

T/v0 = 0.42

0

0.025

0.05

0.075

0.1-0.3

-0.2-0.1

0 0.1

0.2 0.3

0 5

10 15 20 25 30

ρ+v0 at T/v0 = 0.42

p/v0

ω/v0

ω − |p| − Re Σ+(ω, |p|,T ) = 0



ResultsSummary


Intermediate Temperature Region II

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

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.02 0.04 0.06 0.08 0.1

ω/v

0

|p|/v0

T/v0 = 0.5

0

0.025

0.05

0.075

0.1-0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

5

10

15

20

25

30

ρ+v0 at T/v0 = 0.5

p/v0

ω/v0

ω − |p| − Re Σ+(ω, |p|,T ) = 0



ResultsSummary


High Temperature Region

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

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.02 0.04 0.06 0.08 0.1

ω/v

0

|p|/v0

T/v0 = 0.8

0

0.05

0.1

0.15

0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

20

40

60

80

100

ρ+v0 at T/v0 = 0.8

p/v0

ω/v0

The spectral property becomes closer to the HTL result.

T/v(T ) ' 1.59 > 1: The additional thermal-loop corrections may modifythe spectral property (Hidaka-Satow-Kunihiro, Ncul.Phys.A, 2012).



ResultsSummary


High Temperature Region

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

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.02 0.04 0.06 0.08 0.1

ω/v

0

|p|/v0

T/v0 = 0.8

0

0.05

0.1

0.15

0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

20

40

60

80

100

ρ+v0 at T/v0 = 0.8

p/v0

ω/v0

The spectral property becomes closer to the HTL result.

T/v(T ) ' 1.59 > 1: The additional thermal-loop corrections may modifythe spectral property (Hidaka-Satow-Kunihiro, Ncul.Phys.A, 2012).



ResultsSummary


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

0

2

4

6

8

10

12

-0.2 -0.1 0 0.1 0.2

ρ+v

0

ω/v0

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

0 0.025

0.05 0.075

0.1-0.3

-0.2-0.1

0 0.1

0.2 0.3

0

5

10

15

20

25

30

ρ+v0 at T/v0 = 0.5

p/v0

ω/v0



ResultsSummary


Landau Damping

ν

l̄ :

W µ− :

NF

(1 + NB)

Incident Particle Induced Particle

p = (ω,p) p− k

k

Landau Damping (Fig.) makes the imaginary part being finite in thespacelike region:

ImΣ(ν)+ 3

∫k

δ[ω + |k| −

√|p− k|2 + M2

W,Z

]×[NF(1 + NB) + NB(1− NF)

]·[· · ·]. (11)

For a small external momentum (ω, p) and a not small MW,Z, the phasespace in

∫k

admitting the Landau Damping will be restricted.



ResultsSummary


Landau Damping

ν

l̄ :

W µ− :

NF

(1 + NB)


p = (ω,p) p− k

k

Landau Damping (Fig.) makes the imaginary part being finite in thespacelike region:

ImΣ(ν)+ 3

∫k

δ[ω + |k| −

√|p− k|2 + M2

W,Z

]×[NF(1 + NB) + NB(1− NF)

]·[· · ·]. (11)

For a small external momentum (ω, p) and a not small MW,Z, the phasespace in

∫k

admitting the Landau Damping will be restricted.



ResultsSummary


Landau Damping Suppression

ν

l̄ :

W µ− :

NF

(1 + NB)


p = (ω,p) p− k

k

G(x0) =

∫ ∞x0

dx xNB(x) =∞∑n=1

e−nx0

n2

[1 + nx0

], x0 =

ω2 − |p|2 −M2W,Z

2T (ω − |p|) > 0 .

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.



ResultsSummary


Landau Damping Suppression

ν

l̄ :

W µ− :

NF

(1 + NB)


p = (ω,p) p− k

k

G(x0) =

∫ ∞x0

dx xNB(x) =∞∑n=1

e−nx0

n2

[1 + nx0

], x0 =

ω2 − |p|2 −M2W,Z

2T (ω − |p|) > 0 .

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.



ResultsSummary


Self-Energy at Three-Peak Region

T

v0= 0.5 ,

|p|v0

= 0.02 . (12)

-0.2

-0.15

-0.1

-0.05

0

-0.2 -0.1 0 0.1 0.2

ImΣ

+/v

0

ω/v0

(|p|/v0, T/v0)

=(0.02, 0.5)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.2 -0.1 0 0.1 0.2R

eΣ+/v

0

ω/v0

(p/v0, T/v0)

=(0.02, 0.5)

ReΣ+/v0y = (ω - |p|)/v0

In the right panel, the crossing points corresponds to the solutions ofω − |p| − Re Σ+(ω, |p|,T ) = 0

(c.f. Kitazawa-Kunihiro-Nemoto, PTP 2007).



ResultsSummary


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

0

2

4

6

8

10

12

-0.2 -0.1 0 0.1 0.2

ρ+v

0

ω/v0

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



ResultsSummary


ξ Depenence of Three-Peak Spectral Density

0

2

4

6

8

10

12

-0.2 -0.1 0 0.1 0.2

ρ+v

0

ω/v0

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

ξ=0.1ξ=1

ξ=10U-Gauge



ResultsSummary


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

Nb

dNb

dt

∣∣∣ ≥ H(T ) , (13)

where,

H(T ) = 1.66√

NdofT 2

MPL' T 2 × 1.41× 10−18 (GeV) , (14)

1

Nb

dNb

dt= −1023 · g 7v(T ) exp

[−1.89

4πv(T )

gT

], (15)

and we obtainT ≥ T∗ ' 160 GeV , T∗/v0 ' 0.65 . (16)



ResultsSummary




Nb

dNb

dt

∣∣∣ ≥ H(T ) , (13)

where,

H(T ) = 1.66√

NdofT 2

MPL' T 2 × 1.41× 10−18 (GeV) , (14)

1

Nb

dNb

dt= −1023 · g 7v(T ) exp

[−1.89

4πv(T )

gT

], (15)




ResultsSummary




Nb

dNb

dt

∣∣∣ ≥ H(T ) , (13)

where,

H(T ) = 1.66√

NdofT 2

MPL' T 2 × 1.41× 10−18 (GeV) , (14)

1

Nb

dNb

dt= −1023 · g 7v(T ) exp

[−1.89

4πv(T )

gT

], (15)




ResultsSummary


Neutrino Spectral Density around T = T∗

0.6

0.65

0.7-0.2 -0.1 0 0.1 0.2

5

10

15

20

25

30ρ+v0 at p/v0 = 0.007

T/v0

ω/v0

T∗v0' 0.65 ,

T∗v(T )

∼ 1 . (17)



ResultsSummary

Table of Contents

1 Introduction

2 Preliminaries


4 Summary



ResultsSummary

Summary

We have investigated the spectral properties of standard-modelleft-handed neutrinos at finite T around the electroweak scale in a waywhere 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. Thethree-peak collective modes could affect the leptogenesis at T & T∗.

Future Work: It is desirable to estimate how large the effects of two-loopor higher-order diagrams are on the neutrino spectral density.

Thanks for Your Attention!



ResultsSummary

Summary

We have investigated the spectral properties of standard-modelleft-handed neutrinos at finite T around the electroweak scale in a waywhere 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. Thethree-peak collective modes could affect the leptogenesis at T & T∗.

Future Work: It is desirable to estimate how large the effects of two-loopor higher-order diagrams are on the neutrino spectral density.

Thanks for Your Attention!


Buckups

Table of Contents

5 Buckups


Buckups

Spectral Density Example: Superconductivity

V

I

Scanning

Tunneling

Microscopy

Renner et. al.,

Phys.Rev.Lett., 80

149, (2008).

Slightly Underdoped Bi2212

STM Superconductor Scan

dI

dV∼ d

dV

[∫ ω=εf +eV

εf

dω Dos(ω)

]∼ Dos(ω) ∼

∫p

ρ(ω, p) , (18)


Buckups


V

I

Scanning

Tunneling

Microscopy

Renner et. al.,

Phys.Rev.Lett., 80

149, (2008).



dI

dV∼ d

dV

[∫ ω=εf +eV

εf

dω Dos(ω)

]∼ Dos(ω) ∼

∫p

ρ(ω, p) , (18)


Buckups


V

I

Scanning

Tunneling

Microscopy

Renner et. al.,

Phys.Rev.Lett., 80

149, (2008).



dI

dV∼ d

dV

[∫ ω=εf +eV

εf

dω Dos(ω)

]∼ Dos(ω) ∼

∫p

ρ(ω, p) , (18)


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