MircoWörmann - uni-bielefeld.deNon-relativisticleptogenesis MircoWörmann in collaboration with Dietrich Bödeker JCAP 02 (2014) 016 ArXiv ePrint: 1311.2593 Bielefeld,08.05.2014 Mirco

Non-relativistic leptogenesis

Mirco Wörmann

in collaboration with Dietrich Bödeker

JCAP 02 (2014) 016ArXiv ePrint: 1311.2593

Bielefeld, 08.05.2014

Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 1 / 24

Leptogenesis - Basics

Theory to explain the baryon asymmetry of the universe

Proposed by Yanagida and Fukugita in 1986

Standard Model extended by heavy right-handed Majorana-neutrinos

Their decays N ϕ + ` and N ϕ + ` can fulfillthe Sakharov conditions

L violation, cf. B − L violationCP violationDeparture from thermal equilibrium

Sphaleron processes convert B − L into B - asymmetry


Outline

1 Leptogenesis in the non-relativistic limit

2 Relativistic corrections

3 Radiative corrections

4 Summary


Outline




4 Summary


Motivation

Measure for the washout strength: K = Γ0/H(T = MN)K � 1: “Strong washout”K � 1: “Weak washout”

In the strong washout regime, asymmetry that was created atT > MN does not play a role

∆m2atm and ∆m2sol imply: 7 . K . 46

⇒ Final asymmetrie was created at T < MN ,i.e. in the non-relativistic regime


Rate equations in the non-relativistic regime

(ddt

+ 3H)

nN = −ΓN (nN − neqN )

(ddt

+ 3H)

nB−L = ΓB−L,N (nN − neqN )− ΓB−L nB−L

These equations are valid to all orders in the SM couplings!


Determining ΓN at leading order

Boltzmann equation

(∂t −Hp∂p) fN =MNΓ0EN

(e−EN/T − fN

)Yukawa interaction

LNYuk = hijNRi ϕ̃†`Lj + h.c. ⇒ Γ0 =|h11|2MN

8π

integrate Boltzmann equation over ~p



Boltzmann equation


(e−EN/T − fN

)Yukawa interaction


8π


therefor expand 1/EN

1EN≈ 1MN



Boltzmann equation


(e−EN/T − fN

)Yukawa interaction


8π


therefor expand 1/EN 1EN ≈1

MN



(ddt

+ 3H)

nN = −ΓN(nN − n

eqN

)(

ddt

+ 3H)

nB−L = ΓB−L,N(nN − n

eqN

)− ΓB−L nB−L

LO coefficients:

ΓN = Γ0

ΓB−L,N = e Γ0

ΓB−L =3

π2

(c` +

cϕ2

)z2K1(z)Γ0



(ddt

+ 3H)


eqN

)(

ddt

+ 3H)


eqN

)− ΓB−L nB−L

LO coefficients:

ΓN = Γ0 ΓB−L,N = e Γ0

ΓB−L =3

π2

(c` +

cϕ2

)z2K1(z)Γ0


Outline




4 Summary


Adding relativistic corrections

Boltzmann equation


(e−EN/T − fN

)Yukawa interaction


8π


therefor expand 1/EN 1EN ≈1

MN+ ~p

2

2M3N


Relativistic corrections

(ddt

+ 5H)

u = −Γu (u − ueq) u ≡ 1MN · 2∫ d3p

(2π)3~p2

2MNfN

(ddt

+ 3H)


eqN

)+ ΓN,u (u − ueq)

(ddt

+ 3H)


eqN

)+ ΓB−L,u (u − ueq)− ΓB−L nB−L

Γu = Γ0 ΓN,u = Γ0 ΓB−L,u = e Γ0


Relativistic corrections - Numerical results

1 10 K

0

0.1(κ

− κ

NR)

/ κ

T > 1013

GeV

T ~ 1013

GeV

T = 1012

- 1013

GeV

T = 1011

- 1012

GeV

T = 108 - 10

11 GeV


Outline




4 Summary


Radiative corrections

Before: LODecays (1 → 2)

Now: NLO2 → 21 → 31 → 2 virtual corrections

∂fN∂t

∣∣∣∣nN=0

= fF(EN) Γ0MNEN

{a +

~p2

M2Nb + O

(g4,

g3T 2

M2N,g2T 6

M6N

)}

a = 1− λT2M2N − |ht |2[

212(4π)2 +

7π260

T4M4N

]+ (g21 + 3g

22 )

[29

8(4π)2 −π2

80T4M4N

]b = −

[|ht |2 7π

2

45 + (g21 + 3g

22 )

π2

60

]T4M4N

A. Salvio, P. Lodone and A. Strumia, Towards leptogenesis at NLO: the right-handed neutrino interaction rate,JHEP 1108 (2011) 116 [arXiv:1106.2814 [hep-ph]]

M. Laine and Y. Schröder, Thermal right-handed neutrino production rate in the non-relativistic regime,JHEP 1202 (2012) 068 [arXiv:1112.1205 [hep-ph]]


Radiative corrections

(ddt

+ 5H)

u = −Γu (u − ueq)

(ddt

+ 3H)


eqN

)+ ΓN,u (u − ueq)

(ddt

+ 3H)


eqN

)+ ΓB−L,u (u − ueq)− ΓB−L nB−L

NLO coefficients:

Γu = a Γ0 ΓN = a Γ0 ΓN,u = (a− 2b) Γ0


Radiative corrections - Numerical results

1 10 K

1

1.1

1.2

κ/κ

LO

tree level

+ O(g2 )

+ O( λ v4 )

+ O(g2 v

8 )

MN = 10

10GeV

1 10K

MN = 10

8GeV


Outline




4 Summary


Summary

The non-relativistic expansion is a convenient tool for computingthe lepton asymmetry!

Relativistic corrections are smallAccuracy can be easily controlledRate equations are simpleRadiative corrections can be includedOnly works in the (favoured) strong washout regime

More work to doInclude radiative corrections in the B − L rate equationInclude flavor effects


Thank you for your attention!


Washout strength and effective light neutrino mass

K =Γ0

H(T = MN)=

|h11|2MN8π√

8π3g∗90

M2NMPl

=MPl

8π · 1.66√g∗ · v2· |h11|

2v2

MN

= const. · m̃1


Dependence on initial conditions

1 10K

0.01

0.1

κ

NR, thermal initial conditions

NR + O(v2), thermal initial cond’s

NR, zero initial densities

NR + O(v2), zero initial densities


Dependence on statistics

1 10

K

0.7

0.8

0.9

1

1.1κ

/ κ

cla

sssic

al

thermal intial nN

zero initial nN


Ratio of reaction rates and the Hubble rate

1 10 z

0.01

1

100

K -

1 Γ

/H

Γ N

ΓB-L

, T >1013

GeV

ΓB-L

, 1011

GeV> T >108GeV


Leptogenesis in the non-relativistic limitRelativistic correctionsRadiative correctionsSummary

Documents

MircoWörmann - uni-bielefeld.deNon-relativisticleptogenesis MircoWörmann in collaboration with Dietrich Bödeker JCAP 02 (2014) 016 ArXiv ePrint: 1311.2593 Bielefeld,08.05.2014 Mirco