Quantum and Medium Effects in Resonant Leptogenesis Andreas - - PowerPoint PPT Presentation

Andreas Hohenegger, Mérida 2012

Quantum and Medium Effects in Resonant Leptogenesis

Andreas Hohenegger

with Tibor Frossard Mathias Garny Alexander Kartavtsev

(arXiv: 0909.1559, 0911.4122, 1002.0331, 1112.642, in preparation) École Polytechnique Fédérale de Lausanne

PASCOS Mérida, 2012-6-7

Andreas Hohenegger, Mérida 2012

Leptogenesis

CP CP CP CP CP CP CP

L = LSM+ 1

2 ¯

Ni

• i /

∂ − Mi

• Ni

−hαi ¯ ℓα ˜ φPRNi − h†

iα ¯

Ni ˜ φ†PLℓα

bL bL tL sL sL cL dL dL uL νe νµ ντ

Andreas Hohenegger, Mérida 2012

Leptogenesis

kinetic equations needed, usually generalized Boltzmann equations (BEs) kµDµf ℓ (X, k) =

• interactions of ℓ

C ℓ +.↔i+.

k

[.f ℓ .](X, k) 2 ↔ 1 collision term

C ℓ φ ↔Ni (k) = 1 2

• dΠ φ

p dΠNi q (2π)4δ(k + p − q)

• −
• where

=

• 2

(1 ± f ℓ

k )(1 ± f φ p )f Ni q

=

• 2

f ℓ

k f φ p (1 ± f Ni q )

2 ↔ 1 collision term antiparticles

C ℓ φ ↔Ni (k) = 1 2

• dΠ φ

p dΠNi q (2π)4δ(k + p − q)

• −

Andreas Hohenegger, Mérida 2012

Leptogenesis

• ne-loop vertex and self-energy contribution

=

Ni φ ℓ

+

Ni Nj

+

Ni Nj

parametrize matrix elements by CP-violating parameter

• 2

= 1 2(1 + ǫi) |M|2

Ni ,

• 2

= 1 2(1 − ǫi) |M|2

Ni

ǫi = |M|2

Ni→ℓφ − |M|2 Ni→¯ ℓ ¯ φ

|M|2

Ni→ℓφ + |M|2 Ni→¯ ℓ ¯ φ

ǫvac

i

= 2 Im

• (h†h)2

ij

• (h†h)iiMi
• Im
• ∗

×

• + 2Im
• ∗

×

• self-energy contrib. completed by finite width (Paschos et. al., Pilaftsis et. al.,

Plümacher et. al., Covi et. al. . . . )

ǫvac

i

= − Im

• (h†h)2

ij

• (h†h)ii(h†h)jj

R R2 + A2 , with R ≡ M2

j − M2 i

MiΓj

Andreas Hohenegger, Mérida 2012

Resonant Leptogenesis

self-energy contribution is resonantly enhanced

R R/(R2 + A2) A = M1/M2, A = Γ1/Γ2 − M2/M1 A = Γ1/Γ2 + M2/M1 0.01 0.1 1 0.01 0.1 1 10 102

resonance conditions: ǫvac

i

∼ 1 if Im

• (h†h)2

ij

• (h†h)ii(h†h)jj ∼ 1 and |R| ≃ A

require |Mj − Mi| ∼ Γi,j

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

repeat nonequilibrium quantum field analysis of resonant “leptogenesis” within toy-model 0911.4122 for SM+3νR

• ψi → b b , ψi → b b
• =

⇒

• Ni → ℓ φ , Ni → ℓ φ
• see also: Buchmüller, Anisimov, Drewes, Mendizabal and Garbrecht, Herannen, Schwaller, et. al.

divergence of lepton current (flavour independent) ∂µjµ

L (x) = dnL

dt + 3 HnL jµ

L (x) =

• α,a

¯ ℓa

α(x)γµℓa α(x)

• = −
• α,a

Tr [γµSℓ

αα aa (x, x)]

Kadanoff-Baym equations

i / ∂xSℓ

αβ F (x, y) =

x0 d4z Σℓ

αγ ρ (x, z)Sℓ γβ F (z, y) −

y0 d4z Σℓ

αγ F (x, z)Sℓ γβ ρ (z, y)

i / ∂xSℓ

αβ ρ (x, y) =

x0

y0 d4z Σℓ αγ ρ (x, z)Sℓ γβ ρ (z, y)

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

1-loop self-energy

SN ∆ S

− → gradient expansion, Wigner transformation . . .

∂µjµ

L (t) = 2 ∞

• dk 0

2π

• d3k

(2π)3 Tr

• Σℓ

αβ < (t, /

k)Sℓ

βα > (t, /

k) − Σℓ

αβ > (t, /

k)Sℓ

βα < (t, /

k)

• −

¯ Σℓ

βα < (t, /

k) ¯ Sℓ

αβ > (t, /

k) − ¯ Σℓ

βα > (t, /

k) ¯ Sℓ

αβ < (t, /

k)

• compare with conventional Boltzmann equations

∂µjµ

L (x) =

• d3k

(2π)3E ℓ

k

• C ℓ φ ↔Ni (k) − C ℓ φ ↔Ni (k)

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

Kadanoff-Baym ansatz and quasi-particle approximation for leptons and Higgs (lepton flavour-diagonal) S ℓ

αβ ρ (t, /

k) = δαβ PL/ kPR (2π) sign(k0) δ(k2) ∆ φ ρ(t, p) = (2π) sign(p0) δ(p2) quasi-particle approximation insufficient for Majorana neutrinos

• verlap due to finite width neglected

diagonal approximation neglects cross-correlations SN

ij ρ(t, q) =

δij (/ q + Mi) (2π) sign(q0) δ(q2 − M2

i )

extended QP-approximation required

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

equilibrium solution for ˆ S with off-diagonal components elegant analytic solution with off-shell dynamics possible (neglecting back-reaction and expansion) 1112.6428 rewrite full ˆ S in terms of diagonal ˆ S and off-diagonal ˆ Σ′ ˆ S−1(x, y) = ˆ S−1(x, y) − ˆ Σ′(x, y) solve for ˆ S ˆ SF(ρ) = ˆ ΘR ˆ SF(ρ) − ˆ SR ˆ Σ′

F(ρ) ˆ

SA ˆ ΘA where ˆ ΘA =

• I + ˆ

Σ′

A ˆ

SA −1 and ˆ ΘR =

• I + ˆ

SR ˆ Σ′

R

−1 insert ˆ S in self-energy analyse the pole structure of ˆ S (extended) quasi-particle approximation for diagonal propagator

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

spectral functions for R = 100, T = 0.1 M1, M1

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

consider 2RHN, (M1, M2 ≪ M3)

M1 = 1 · 103 GeV h†h =

• (h†h)11

(h†h)12 (h†h)21 (h†h)22

• = 10−12
• 1.7

0.14 + 0.2i 0.14 − 0.2i 0.33

• M2 > M1, (h†h)11 > (h†h)22

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

spectral functions for R = 100, T = 0.1 M1, M1

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

spectral functions for R = 10, T = 0.1 M1, M1

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

spectral functions for R = 10, T = 0.1 M1, M1

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

q0

I

q0

J

S11

ρ

S22

ρ

S11

ρ

S22

ρ

S12

ρ

N1, N2 undergo level-crossing

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

quantum-corrected rate equation

∂µjµ

L =

• i
• dΠ ℓ

k dΠ φ p dΠNi q (2π)4δ(k + p − q)

• |M|2

Ni→ℓφ

• −
• − |M|2

Ni→¯ ℓ ¯ φ

• −

corrected CP-violating parameter in R ≫ 1 limit

ǫi = Im

• (h†h)2

ij

• (h†h)ii(h†h)jj

(M2

i − M2 j )MiΓj

• M2

i − Mj

2 +

• Γj/Mj q · Lρ

2 k · Lρ k · q

medium effects

Lρ(t, / q) = 16π

• dΠ ℓ

k dΠ φ p (2π)4δ(q − p − k) /

k(1 + f φ − f ℓ )

retrieve conventional amplitudes in R → ∞, T → 0 limit

Andreas Hohenegger, Mérida 2012

Nonequilibrium Quantum Field Theory

Lh(ρ) capture medium effects (two components for each sufficient)

mℓ = mφ = 0, T = M1

q M1 (−)Lρ(h) q0

L0

ρ

L3

ρ

L0

h

L3

h

0.5 1 1.5 2 2.5 3 3.5 4 5 10 15 20

Andreas Hohenegger, Mérida 2012

Rate Equations

rate equations dYNi dz = −κi z γD

Ni

2Γvac

i

neq

Ni

(YNi − Y eq

Ni )

dYL dz =

• i

κi z ǫiγD

Ni

• 2Γvac

i

neq

Ni

(YNi − Y eq

Ni ) − (1 + cφℓ)cℓ

γW

Ni

2Γvac

i

2neq

ℓ

YL

• medium corrected reaction densities (compare Garbrecht et. al., Wong et. al., Hannestad
• et. al., Pastor et. al.)

γD

Ni ≡

• dΠNi

q dΠ ℓ k dΠ φ p (2π)4δ(q − k − k)

× |M|2

Ni (1 + f φ − f ℓ )fNi

• ǫiγD

Ni

• ≡
• dΠNi

q dΠ ℓ k dΠ φ p (2π)4δ(q − k − k)

×ǫi |M|2

Ni (1 + f φ − f ℓ )fNi

γW

Ni ≡

• dΠNi

q dΠ ℓ k dΠ φ p (2π)4 δ(q − p − k)

× |M|2

Ni (1 + f φ − f ℓ )(1 − fNi )fNi

Andreas Hohenegger, Mérida 2012

Preliminary Results

averaged CP-violating parameter

z ǫ1 R = 1 R = 10

• ǫvac

1

R = 100 R = 1 R = 10

• ǫ1

R = 100 |R| < 10 −1 −0.1 −0.01 −10−3 −10−4 −10−5 −10−6 0.1 1 10

Andreas Hohenegger, Mérida 2012

Thermal Masses and Modified Dispersion Relations

Lh(ρ) depend on the kinematics in the decay thermal masses of leptons and Higgs m ℓ ∼ 0.2T, m φ ∼ 0.4T,

φ → ℓ + Ni for m φ > m ℓ + Mi

= + + ǫi for Higgs (1 + f φ − f ℓ ) − → (f φ + f ℓ ) leptons have modified dispersion relations (Weldon; Giudice et.al.; Plümacher, Kießig) dΠ ℓ

q =

• d3k dk0

(2π)3 δ(k2 − m2

ℓ ) −

→

• d3k dk0

(2π)3 δ(δ ℓ (k)) collinear enhancement can enable Majorana decay with enhanced rate at high T (Bödecker et. al., Laine et. al.)

Andreas Hohenegger, Mérida 2012

Conclusions

Nonequilibrium quantum field theory can put Leptogenesis on solid ground improved rate equations for quasi-degenerate case quantitative corrections can be significant need quantum analysis with off-shell dynamics in degenerate case need to include further medium effects