Flavoured Leptogenesis from Nonequilibrium QFT · 2011. 5. 5. · Why baryogenesis? To explain theexcess of matter over antimatterin the universe: [WMAP 2003] n B n = 6:1+0:3 0:2

Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions

Flavoured Leptogenesis from Nonequilibrium QFT

Matti Herranena

in collaboration with

M. Benekea B. Garbrechta C. Fidlera P. Schwallerb

Institut für Theoretische Teilchenphysik und Kosmologie,RWTH Aachen Universitya

Institut für Theoretische Physik,Universität Zürichb

Bielefeld, 5.5.2011

M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 1 / 25


Outline

Introduction

Closed Time Path (CTP) Formalism of Noneq. QFT

CTP Approach to Flavoured Leptogenesis

Numerical Results

Conclusions and outlook



Why baryogenesis?

To explain the excess of matter over antimatter in the universe:[WMAP 2003]

nB

nγ=(6.1+0.3−0.2

)× 10−10

Why is there something rather than nothing?



Leptogenesis [Fukugita, Yanagida (1986)]

The Standard Model is extended by adding heavy right-handedMajorana neutrinos Ni, i = 1, 2, . . . (eg. see-saw models)

L ⊃ −h∗abφ ψ`bPRψRa − Y∗iaψà(εφ)†ψNi −12ψNiMiψNi + h.c.

Lepton asymmetry is generated through out-of-equilibrium L- andCP-violating Yukawa decays: N1 → `φ



Nonequilibrium QFT Approach to Leptogenesis1

1related / complementary aspects in:Buchmüller, Fredenhagen (2000); De Simone, Riotto (2007);Garny, Hohenegger, Kartavtsev, Lindner (2009 & 2010);Anisimov, Buchmüller, Drewes, Mendizabal (2010);Anisimov, Besak, Bödeker (2010)



Closed Time Path∗ (CTP) Formalism∗ a.k.a. Schwinger-Keldysh Formalism [Schwinger (1961); Keldysh (1964); Calzetta, Hu (1988)]

Usually in QFT matrix elements are computed within the in-outframework:

〈out|S|in〉 ←− time-ordered correlators: 〈T[ψ(x)ψ(y)]〉

In leptogenesis we want to calculate expectation values in a finitedensity medium, e.g.

j0(x) = 〈ψ(x)γ0ψ(x)〉 ≡ tr[ρ ψ(x)γ0ψ(x)

]ρ is an (unknown) quantum density operator



−→ In-In generating functional on a Closed Time Path:

Im t

Re t

Path-ordered correlators:

iSC(x, y) = 〈TC [ψ(x)ψ(y)]〉

Four propagators with respect to real time variable:

iS+−(u, v) = iS<(u, v) = −〈ψ(v)ψ(u)〉iS−+(u, v) = iS>(u, v) = 〈ψ(u)ψ(v)〉iS++(u, v) = iST(u, v) = 〈T(ψ(u)ψ(v))〉

iS−−(u, v) = iST(u, v) = 〈T(ψ(u)ψ(v))〉



Kadanoff-Baym Equations

Generic Schwinger-Dyson equations for (CTP) propagators:

= +G G0 Σ

Kadanoff-Baym equations are the <,> components:

(i∂/− m)S<,> − ΣH � S<,> − Σ<,> � SH =12(Σ> � S< − Σ< � S>

)(A� B)(u, v) ≡

∫d 4w A(u,w)B(w, v) denotes convolution

Renormalization, thermal corrections (thermal masses etc.)

Finite width effects

Collision term



Approximations

Wigner representation:

S(k, x) =

∫d 4r eik·r S

(x +

r2, x− r

2),

and gradient expansion to the lowest order in

x-derivatives: ∂xS(k, x), ∂xΣ(k, x), etc.

coupling constants in Σ(k, x)

−→ Constraint and Kinetic Equations:

2(k0 − k · γγ0)iγ0S<,>` −{

Σ/H` γ

0, iγ0S<,>`

}−{

iΣ/<,>` γ0, γ0SH`

}= −1

2

(iC` − iC†`

)i∂ηiγ0S<,>` −

[Σ/H` γ

0, iγ0S<,>`

]−[iΣ/<,>` γ0, γ0SH

`

]= −1

2

(iC` + iC†`

)η is conformal time variable: dη = dt/a(t)



The zeroth order solutions to Contraint Equation:

Flavour covariant free propagators for `:

iS<àb(k, η) = −2πδ(k2)k/[ϑ(k0)f+àb(k, η)− ϑ(−k0)(1ab − f−àb(−k, η))

]iS>àb(k, η) = −2πδ(k2)k/

[−ϑ(k0)(1ab − f+àb(k, η)) + ϑ(−k0)f−àb(−k, η)

]f±àb(k, η) are time-dependent distribution functions

a, b are flavour indices

Similar (unflavoured) free propagators for N1 and φ



Contributions to Lepton Collision term

Y-Yukawa interactions:

` `

φ

N

MSM h-Yukawa and gauge interactions:

` `

`

φ

` `

`

A



Kinetic Equation for Lepton Number Densities[Beneke, Garbrecht, Fidler, MH, Schwaller (2010)]

Substituting free propagators into Kinetic Equation

−→ First order equation for n±àb(η) =∫ d3k

(2π)3 f±àb(k, η):

∂δn±àb

∂η=∑

c

[Ξeffac δn±`cb − δn±àcΞ

effcb ]∓ i∆ωeff

àbδn±àb

−∑

c

[Wacδn±`cb + δn±∗`ca W∗bc]± Sab − Γbl(δn+àb + δn−àb)− Γ±fl

àb

Gradients of the mixing matrices: Ξeff ∼ U†∂ηU

Flavor oscillations by the thermal masses: ∆ωeff` ∼ h2T , Y2T

Collision terms: Washout, CP-violating source, Flavour-blinddamping, Flavour-sensitive damping



Fast Flavour-blind Gauge Interactions

` `

`

A

Blue cut←→ tree-level pair creation andannihilation and 1↔ 2 scatterings

Kinematically forbidden for on-shellexcitations −→ Γbl ∼ g4

2T

Force kinetic equilibrium with generalized chemical potentialsµ±ab(η):

f±àb(k, η) =

(1

eβ|k|−βµ± + 1

)ab

Flavour and momentum dependence factorizes to first order in µ±ab:

δn±àb = µ±abT2

12=⇒ δf±àb(k, η) = δn±àb

12β3eβ|k|

(eβ|k| + 1)2

−→ Simple interaction terms in the kinetic equationM. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 13 / 25


Suppression of Flavour Oscillations

Toy equations for flavour oscillations

∂δn±àb∂η

= ∓i∆ωeffàbδn±àb − Γbl(δn+

àb + δn−àb)

Parametrically Γbl � ∆ωeff`

−→ Two over-damped solutions with short and long time scales:

Short mode with τshort = 1/(2Γbl) forces an effective constraint:

δn−ab = −δn+ab

Flavour oscillations of δn+ab are over-damped with a long decay time:

τdamp ∼ 2Γbl/(∆ωeffàb)2



Suppression of Flavour Oscillations

Toy equations for flavour oscillations

∂δn±àb∂η

= ∓i∆ωeffàbδn±àb − Γbl(δn+

àb + δn−àb)

Parametrically Γbl � ∆ωeff`

−→ Two over-damped solutions with short and long time scales:

Short mode with τshort = 1/(2Γbl) forces an effective constraint:

δn−ab = −δn+ab

Flavour oscillations of δn+ab are over-damped with a long decay time:

τdamp ∼ 2Γbl/(∆ωeffàb)2



Final Kinetic Equations

Lepton asymmetries q ≡ δn+ − δn− (L- and R-handed):

∂qàb

∂η=∑

c

[Ξacq`cb − qàcΞcb −Wacq`cb − qàcWcb]− Γflàb + 2Sab

∂qRab

∂η= −Γfl

Rab

Majorana neutrino N1:

∂fN1(kcom)

∂η= −2|Y1|2

kcomµ

2kcom 0Σµ

N(kcom)[fN1(kcom)− f eq

N1(kcom)]

Higgs field φ remains in thermal equilibrium



Lepton Asymmetries: Yàb =n+àb−n−àb

s , Fully Flavoured case

hΤ=0.030

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

10-11

10-10

10-9

10-8

z

ÈYÈ

dark blue: Y`11

light blue: Y`22

brown dotted: Re[Y`12]

red dashed: Im[Y`12]

M1 = 1012 GeV, M2 = 1014 GeV

YYuk =

(1.4× 10−2 1× 10−2

i× 10−1 10−1

)

Y`12 = Y∗`21 strongly suppressed before freeze-out at z = M1T ≈ 10

−→ Flavour off-diagonals Y`12,21 can be negleted




s , Unflavoured case

hΤ=0.001

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

10-11

10-10

10-9

10-8

z

ÈYÈ

dark blue: Y`11

light blue: Y`22



M1 = 1012 GeV, M2 = 1014 GeV

YYuk =

(1.4× 10−2 1× 10−2

i× 10−1 10−1

)

Y`12 = Y∗`21 decay away long after freeze-out at z ≈ 10

−→ Flavour damping by Γfl` can be neglected




s , Intermediate regime

hΤ=0.007

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

10-11

10-10

10-9

10-8

z

ÈYÈ

dark blue: Y`11

light blue: Y`22



M1 = 1012 GeV, M2 = 1014 GeV

YYuk =

(1.4× 10−2 1× 10−2

i× 10−1 10−1

)

Full kinetic equations need to be solved!



Total Lepton Asymmetry as a function of Γfl`

Scale M1,2 and the couplings YYuk such that Γfl` is varying while the

washout and source terms remain constant

Scenario A

1010 1011 1012 1013 10140

1.´10-9

2.´10-9

3.´10-9

4.´10-9

5.´10-9

6.´10-9

M1@GeVD

ÈY11+

Y22È

fully flavoured

full kinetic equations

unflavoured



Conclusions

First principle description of leptogenesis within the CTPframework

RIS subtraction procedure not required

Simple kinetic equations for lepton number densities, including

Quantum statistical corrections in loops and external states

Sizable for weak washout

Flavour effects

Flavour oscillations are overdamped by fast gauge interactions

Full flavoured equations are needed between fully flavoured andunflavoured regimes



Outlook

Systematic inclusion of thermal effects

Finite widths

Thermal masses

Spectator processes

Resonant leptogenesis

Flavour coherence effects between Neutrinos Ni


Washout contribution

` `

φ

N

Blue cut←→ tree-level decays andinverse decays N1 ↔ `φ

12

tr∫

d3k(2π)3

∫ ∞0

dk0

2πCYàb = −

∑c

Wacδn+`cb

with

Wac =12

Y∗1aY1c

∫d3k

(2π)32|k|d3k′

(2π)32√

k′2 + (a(η)M1)2

d3k′′

(2π)32|k′′| (2π)4δ4(k′ − k − k′′)

× 2k · k′(fN1(k′) + fφ(k′′)

) 12β3 eβ|k|

(eβ|k| + 1)2


CP-violating Source Sab: Wave-function Contribution

Orange cut←→ Interference betweentwo s-channel scatterings

Blue cut←→ Interference betweenloop and tree-level decays

Swfab = i

∑c

[Y∗1aY∗1cY2cY2b − Y∗2aY∗2cY1cY1b]

×(−M1

M2

)∫d3k′

(2π)32√

k′2 + (a(η)M1)2

ΣNµ(k′)ΣµN(k′)gw

δfN1(k′)

where the thermal decay rate is

ΣµN(k) = gw

∫d3p

(2π)32|p|d3q

(2π)32|q| (2π)4δ4(k − p− q) pµ(

1− f eq` (p) + f eq

φ (q))


CP-violating Source Sab: Vertex Contribution

Orange cut←→ Interference betweens- and t-channel scatterings

Blue cut←→ Interference betweenloop and tree-level decays

In the strongly hierarchical case, M1 � M2:

SVab =

12

Swfab =⇒ Sab =

32

Swfab

ΣµN(k)

MN�T−−−−→ gwkµ

16π recover standard approximation


Flavour-sensitive MSM Yukawa Interactions

` `

`

φ

Blue cut←→ tree-level pair creation andannihilation and 1↔ 2 scatterings

Kinematically forbidden for on-shellexcitations

Γflàb =Γan

([h†h]acq`cb + q†àc[h

†h]cb − h†acqRcdhdb − h†adq†Rdchcb

)+Γsc

([h†h]acq`cb + q†àc[h

†h]cb − h†acqRcdhdb − h†adq†Rdchcb

)

Example: 2 flavours, charged lepton basis

Γfl` = (Γan + Γsc) h2

τ

[(1 00 0

)q` + q`

(1 00 0

)− 2

(qR11 0

0 0

)]

Γan,sc ∼ g22T need to be calculated including (thermal) finite width

corrections for ` and φ propagators


Documents

Flavoured Leptogenesis from Nonequilibrium QFT · 2011. 5. 5. · Why baryogenesis? To explain theexcess of matter over antimatterin the universe: [WMAP 2003] n B n = 6:1+0:3 0:2