-
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
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 2 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 2 / 24
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Outline
1 Leptogenesis in the non-relativistic limit
2 Relativistic corrections
3 Radiative corrections
4 Summary
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 3 / 24
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Outline
1 Leptogenesis in the non-relativistic limit
2 Relativistic corrections
3 Radiative corrections
4 Summary
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 4 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 5 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 5 / 24
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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!
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 6 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 7 / 24
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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
therefor expand 1/EN
1EN≈ 1MN
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 8 / 24
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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
therefor expand 1/EN 1EN ≈1
MN
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 8 / 24
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Rate equations in the non-relativistic regime
(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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 9 / 24
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Rate equations in the non-relativistic regime
(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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 9 / 24
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Outline
1 Leptogenesis in the non-relativistic limit
2 Relativistic corrections
3 Radiative corrections
4 Summary
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 10 / 24
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Adding relativistic corrections
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
therefor expand 1/EN 1EN ≈1
MN+ ~p
2
2M3N
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 11 / 24
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Relativistic corrections
(ddt
+ 5H)
u = −Γu (u − ueq) u ≡ 1MN · 2∫ d3p
(2π)3~p2
2MNfN
(ddt
+ 3H)
nN = −ΓN(nN − n
eqN
)+ ΓN,u (u − ueq)
(ddt
+ 3H)
nB−L = ΓB−L,N(nN − n
eqN
)+ ΓB−L,u (u − ueq)− ΓB−L nB−L
Γu = Γ0 ΓN,u = Γ0 ΓB−L,u = e Γ0
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 12 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 13 / 24
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Outline
1 Leptogenesis in the non-relativistic limit
2 Relativistic corrections
3 Radiative corrections
4 Summary
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 14 / 24
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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]]
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 15 / 24
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Radiative corrections
(ddt
+ 5H)
u = −Γu (u − ueq)
(ddt
+ 3H)
nN = −ΓN(nN − n
eqN
)+ ΓN,u (u − ueq)
(ddt
+ 3H)
nB−L = ΓB−L,N(nN − n
eqN
)+ ΓB−L,u (u − ueq)− ΓB−L nB−L
NLO coefficients:
Γu = a Γ0 ΓN = a Γ0 ΓN,u = (a− 2b) Γ0
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 16 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 17 / 24
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Outline
1 Leptogenesis in the non-relativistic limit
2 Relativistic corrections
3 Radiative corrections
4 Summary
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 18 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 19 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 19 / 24
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Thank you for your attention!
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 20 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 21 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 22 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 23 / 24
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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
Mirco Wörmann (Universität Bielefeld) Non-relativistic leptogenesis Kosmologietag 2014 24 / 24
Leptogenesis in the non-relativistic limitRelativistic correctionsRadiative correctionsSummary