Leptogenesis and the Non-Equilibrium Condition
Björn Garbrecht
Physik-Department T70
Technische Universität München
Baryons over antibaryons: the nuclear physics of Sakharov
ECT? Trento, 29/07/16
Outline
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Dynamical Generation of the BAU: Sakharov Conditions
What was known in 1966
1928: Dirac equation 1955: CP T theorem (Pauli) 1964: CP (Cronin & Fitch)
1932: positron
1956: parity violation (Wu)
1964: CMB (Penzias & Wilson)
Today: Baryon-to-photon ratio
ηB = (6.16 ± 0.15) × 10−10 [Planck, 68% confidence level],
i.e. in the early Universe, there was one extra quark
dark energy
per ten billion quark-antiquark pairs.
68.3 %
Creating the baryon (B) asymmetry of the Universe
(BAU) from symmetric initial conditions
(baryogenesis) requires [Sakharov (1966)]
1 B violation,
2 charge (C) and charge-parity (CP ) violation,
3 departure from equilibrium.
baryons 4.9 %
26.8 %
dark matter
[Planck (2013)]
Remark on 3rd Condition
CP T -invariance theorem is the QFT generalization of the time reversal invariance
well-known from classical mechanics and electrodynamics.
Need to break time-reversal (T ) invariance to allow for CP asymmetry.
Realize this thermodynamically (2nd law, non-equilibrium → irreversible processes).
Kinetic Equations in Early Universe Cosmology
Inferring the Asymmetry from Observations
Baryon-to-photon ratio ηB = (6.16 ± 0.15) × 10−10 can be inferred
from observations
light elements produced during Big Bang Nucleosnynthesis (BBN),
anisotropies of the cosmic microwave background (CMB), in particular
baryon acoustic oscillations (BAOs).
Theoretical predictions from Boltzmann equations that model network
of nuclear reactions or scatterings of photons, electrons and nuclei/ons.
µ
∇µ jX
=∂t nX − ∇ · jX + 3HnX = CX
Z Y
1
d 3 pi
CX =
δ 4 (pX + pA1 + · · · − pB1 − · · · )
3 )2E
2EX
(2π
i
i
× (1 ± fX )(1 ± fA1 ) · · · fB1 · · · |MB1 B2 ···→XA1 A2 ··· |2
− fX fA1 · · · (1 ± fB1 ) · · · |MXA1 A2 ···→B1 B2 ··· |2
nX :
jX :
jµ:
∇µ :
H:
CX :
fi :
number density
current density
four-current
cov. derivative
Hubble rate
collision term
distribution fn.
Agreement between BBN & and CMB values for BAU huge success
→ aim to repeat this e.g. for baryogenesis or dark matter abundance
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Non-Equilibrium and Neutrinos
Neutrinos Masses & Mixings
Mass Parameters for Active Neutrinos from Oscillation Experiments
∆m221 = 7.50 × 10−5 eV2 , ∆m231 = 2.457 × 10−3 eV2 (NH),
[Gonzalez-Garcia, Maltoni, Schwetz (2014)], upper bound from [Planck(2015)] @ 95% c.l.
P
mi < 0.23eV
i
Normal
Inverted
m23
m22
-5
2
solar: 7.5×10 eV
|νn i =
P
m21
∗
Uai
|νa i,
atmospheric:
a
a = e, µ, τ ; n = 1, 2, 3


m1
0
0
 0
m2
0  = U † mU ∗
0
0
m3
for Majorana neutrinos, for Dirac
neutrinos just like CKM mechanism.
U is the Pontecorvo-MakiNakagawa-Sakata (PMNS) matrix.
2.4×10-3 eV2
atmospheric:
m22
2.4×10-3 eV2
solar: 7.5×10-5 eV2
m21
νe
νµ
Active neutrino mass differences & mixing
[from JUNO physics paper (2015)]
m23
ντ
Why is the mass scale of neutrinos (several meV) much below that for other SM
particles, e.g. electron (511 keV), or top quark (173 GeV)?
Non-Equilibrium and Neutrinos
Seesaw Mechanism
Type I seesaw: Introduce nN hypothetical right-handed neutrinos N (RHNs) with
Majorana mass matrix M (can take it to be diagonal), couple these to to Standard
Model lepton ` = (ν, eL ) and Higgs doublets φ, h|φ|i = v = 174 GeV:
∗¯ †
`a φ Ni −Yia N̄i φ`a ; a = e, µ, τ
LSM → LSM + 21 N̄ic (i∂/ − Mij )Nj −Yia
c 0
mD
ν
(3 + nN ) × (3 + nN ) matrix: 12 (ν̄ N̄ c )
, mD = Y † v
mD M
N
|
{z
}
=:M
Assume
||M|| ||m||
block-diagonalization
m 0
1 − θθ†
θ
†
= Ũ MŨ , Ũ =
+ O(θ3 ) , θ = Y † vM −1
†
†
0 M
−θ
1−θ θ
Majorana mass matrix for light neutrinos: m = v 2Y † M −1 Y ∗ , diagonalised by U ,
for heavy neutrinos MN = M + 12 θ† θM + M θT θ∗ + O(θ3 ), diagonalised by UN
†
ν light = U † (1 − 12 θθ† )ν − θN c , ν heavy = UN
(1 − 21 θT θ∗ )N − θT ν c
¯ † and
RHNs are their only antiparticles → can decay either Ni → `φ or Ni → `φ
therefore violate lepton number L.
This talk
Comprehensive overview of leptogenesis in the type I seesaw mechanism throughout the
parameter space, in particular for all mass scales M .
Non-Equilibrium and Neutrinos
Out-of-Equilibrium Dynamics
Simplest Meaningful Network of Kinetic Equations Describing Leptogenesis:
|Yi |2
dnN i
Γi =
Mi
+ 3HnN i =Γi (nN i − neq
)
Ni
8π
dt
3
dnL
Γi M 2 − M
+ 3HnL =εΓi (nN i − neq
e T
W =
N i ) − W nL
dt
4
T
nN i : number density of Ni ; nL : lepton charge density; W : washout rate;
ε = (ΓN i→`H − ΓN i→`H
¯ ∗ )/(ΓN i→`H + ΓN i→`H
¯ ∗ ): decay asymmetry; T : temperature
Best compromise between large L violating rate∗ (1st S. condition) and and large
deviation from equilibrium (3rd S. condition):
Γ ∼ H for T ∼ M (i.e. at freezeout, when the RHNs become Maxwell suppressed).
2 2
Y 2 M/(8π) ∼ T 2 /mPl ∼ M 2 /mPl
m∼ Y Mv
⇒
m ∼ 8πv 2/mPl ∼ 0.1 meV
Light Neutrino Mass Scale points to Role of RHNs in Baryogenesis
Out-of-equilibrium property of the N independent of the mass scale.
Tendency of being somewhat close to equilibrium, i.e. Γ H around T ∼ M →
strong washout. However, ∃ a lot of parametric freedom.
∗
Asymmetry in lepton sector transferred to baryons by the sphaleron.
CP Violation & Quantum Interference
P
Example leptogenesis: Recall L ⊃ −
¯ † Ni )
(Yi N̄i φ` + Yi∗ `φ
i=1,2
Consider first N1 → `φ:
2
= Y1∗ |Atree | + Y1 Y2∗2 |Aloop |eiϕloop ⊃
×-term
(Y1∗2 Y22 e−iϕloop + Y12 Y2∗2 eiϕloop )|Atree Aloop |
¯ †:
Rate for CP conjugate process N1 → `φ
2
∗ 2
iϕ
2 ∗2 −iϕ
∗2 2 iϕ
Y1 |Atree | + Y1 Y2 |Aloop |e loop ⊃ (Y1 Y2 e loop + Y1 Y2 e loop )|Atree Aloop |
×-term
Difference: 4Im[eiϕloop ]Im[Y1∗2 Y22 ]|Atree Aloop |
CP Violation & Quantum Interference
P
Example leptogenesis: Recall L ⊃ −
¯ † Ni )
(Yi N̄i φ` + Yi∗ `φ
i=1,2
Consider first N1 → `φ:
2
= Y1∗ |Atree | + Y1 Y2∗2 |Aloop |eiϕloop ⊃
×-term
(Y1∗2 Y22 e−iϕloop + Y12 Y2∗2 eiϕloop )|Atree Aloop |
¯ †:
Rate for CP conjugate process N1 → `φ
2
∗ 2
iϕ
2 ∗2 −iϕ
∗2 2 iϕ
Y1 |Atree | + Y1 Y2 |Aloop |e loop ⊃ (Y1 Y2 e loop + Y1 Y2 e loop )|Atree Aloop |
×-term
Difference: 4Im[eiϕloop ]Im[Y1∗2 Y22 ]|Atree Aloop |
Im[eiϕloop ] comes precisely from contributions where the anti-Higgs and the
anti-lepton in the loops propagate on shell (i.e. they fulfill the energy-momentum
relation of real particles). → on-shell cuts
Quantum interference between the “direct” path and the case where antiparticles
are produced, then rescattered into particles.
Leptogenesis – Standard Approach
decay asymmetry:
εN i =
[Fukugita, Yanagida (1986);
Covi, Roulet, Vissani (1996)]
ΓN i→`H −ΓN i→`H
¯ ∗
ΓN i→`H +ΓN i→`H
¯ ∗
“Wave Function” & “Vertex” Contributions:
εwf
Ni =
1
8π
=
εvertex
Ni
P
j6=i
1
8π
† 2
Mi Mj Im[(Y Y )ij ]
2
2
†
(Y Y )ii
Mi −Mj
P
j6=i
Mj
Mi
1− 1+
Mj2
Mi2
log 1 +
Mi2
Mj2
Im[(Y Y † )2 ]
ij
(Y † Y )ii
Resonant enhancement for Mi → Mj .
Understand situation when |Mi2 − Mj2 | Mi,j ΓN i,j does not hold.
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Kinetic Equations – Strong Washout, Unflavoured
Reparametrization and Nonrelativistic Approximations
Kinetic Equations
dnN 1
+ 3HnN i =Γ1 (nN 1 − neq
N1)
dt
dnL
+ 3HnL =εΓ1 (nN 1 − neq
N 1 ) − W nL
dt
|Y1 |2
M1
8π
3
Γ1 M 2 − M
W =
e T
4
T
Γ1 =
2
3
Normalize number densities to entropy density s = g? 2π45T ∝ 1/V → Hubble
dilution term drops. (g? : number of relativistic dofs., g? = 106.75 in the SM.)
YN 1 = nN 1 /s, YL = nL /s.
Introduce dimensionless parameter z = M1 /T → d/dt = z/H(z = 1) d/dz.
H = aȧ = żz ⇔ dz = zHdt = z1 H(z = 1)dt
1
T∝a
H∝T 2
Kinetic Equations – Strong Washout, Unflavoured
Reparametrization and Nonrelativistic Approximations
Kinetic Equations
dYN 1
zΓ1
=
(YN 1 − YNeq1 )
dz
H(z = 1)
dYL
zΓ1
zW
=ε
(YN 1 − YNeq1 ) −
YL
dz
H(z = 1)
H(z = 1)
|Y1 |2
M1
8π
Γ1 3
W = z 2 e−z
4
Γ1 =
2
3
Normalize number densities to entropy density s = g? 2π45T ∝ 1/V → Hubble
dilution term drops. (g? : number of relativistic dofs., g? = 106.75 in the SM.)
YN 1 = nN 1 /s, YL = nL /s.
Introduce dimensionless parameter z = M1 /T → d/dt = z/H(z = 1) d/dz.
Strong washout (Γ1 H) → freezeout W ∼ H occurs when M1 T
→ non-relativistic √
approximations applicable:
R d3 p − p2 +M 2 /T
3
1
3
3
eq
1
nN 1 = (2π)3 2e
= Tπ2 z 2 K2 (z) ≈ 2− 2 π − 2 z 2 e−z T 3
Can further approximate
zΓ1
(YN 1
H(z=1)
− YNeq1 ) =
eq
dYN 1
.
dz
Result depends only on
ε (trivially),
washout strength K := Γ1 /H(z = 1) (NB K ∝ Y 2 /M ∝ m up to flavour mixing).
Solutions for the Freeze-Out Asymmetry
Varying Washout Strength
YNeq[z_] = 2 ^ (-1 / 2) * Pi ^ (-3 / 2) * z ^ (3 / 2) * Exp[-z]; ϵ = 1;
S = TableNDSolve[
{YL '[z] == ϵ * D[YNeq[z], z] - Kws / 4 * z ^ (5 / 2) * Exp[-z] * YL[z], YL[0.1] ⩵ 0},
YL, {z, 0.5, 100}], {Kws, {10, 15, 20, 25, 30}};
0.008
Out[4]=
YL /ϵ
0.006
0.004
0.002
5
10
15
20
z
Analytic Approximation:
ηB ' 0.96 × 10−2 εκ,
2
1
where κ = zB (K)K , zB (K) = 1 + 2 log 1 +
[Buchmüller, Di Bari, Plümacher (2004)]
h
i5 3125πK 2
1024 log 1024
πK 2
Solutions for the Freeze-Out Asymmetry
Varying Initial Conditions
YNeq[z_] = 2 ^ (-1 / 2) * Pi ^ (-3 / 2) * z ^ (3 / 2) * Exp[-z]; ϵ = 1;
With{Kws = 15},
S = TableNDSolve[
{YL '[z] == ϵ * D[YNeq[z], z] - Kws / 4 * z ^ (5 / 2) * Exp[-z] * YL[z], YL[0.1] ⩵ IC},
YL, {z, 0.5, 100}], {IC, {-1, -0.5, 0, 0.5, 1}};
;
0.0060
0.0055
Strong washout → freeze-out
asymmetry approximately
independent from initial asymmetry.
Out[7]=
YL /ϵ
0.0050
0.0045
0.0040
0.0035
5
10
15
20
z
NB The evolution of the asymmetries in the crossover between relativistic and
nonrelativistic regimes for z . 1 has not yet been studied in all details but is irrelevant for
the freeze-out asmmetry in the strong washout regime.
Mass Scale of RHNs
Unless there is resonant enhancement, leptogenesis in the strong
washout regime requires M & 1010 GeV because ε ∼ Y 2 [Davidson, Ibarra
(2002)].
Leaving leptogenesis aside, RHNs are allowed throughout the mass
range because they can always be decoupled.
Barring strong decoupling, BBN yields strong constraints for masses
. 100 MeV, oscillation experiments for masses . eV.
In absence of SUSY or other cancellation mechanism, the RHNs will
contribute to the Higgs mass. In order to avoid destabilization, require
P †
mM 3
M
2
7
∆m2φ = i [Y4πY2]ii Mi2 log Mµi ∼ 4π
2 v 2 log µ . mφ → MN . 10 GeV
[Vissani (1997)]
In presence of SUSY, there is a slight tension of the high temperatures
with gravitino production that can either lead to overclosure (i.e. too
much dark matter, if stable) or a conflict with BBN (if unstable).
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Need for Going beyond the Standard Approach
Real Intermediate State (RIS) Problem
Interference of tree & loop amplitudes → CP violation.
(∗)
CP violating contributions from discontinuities
→ loop momenta where cut particles are on shell.
Is
an extra process or is it already accounted for by
and
?
Including (∗) only → CP asymmetry is already generated in
equilibrium. CP T theorem.
Need for Going beyond the Standard Approach
(Inverse) Decays & CP Asymmetry
Consider the squared matrix elements, ε being the decay asymmetry.
Naive multiplication∗ suggests that an asymmetry is generated already
in equilibrium: Γ`φ
¯ ∗ →`φ ∼ 1 + 2ε, Γ`φ→`φ
¯ ∗ ∼ 1 − 2ε
Ad hoc fix: Subtract real intermediate states (RIS) from
[Kolb, Wolfram (1980)].
Better Way Out
Compute the real time (time dependent perturbation theory), non-equilibrium
(statistical physics) evolution of the quantum field theory states of interest.
Since charges and currents are e.g. given by j µ (x) = hψ̄(x)γ µ ψ(x)i, calculate in
particular evolution of two-point functions.
∗
Do not try this at home: The unstable N are not asymptotic states of a unitary S matrix
→ conflict with the CP T theorem.
Subtraction of Real Intermediate States (RIS)
Kinetic equations (with washout term not shown)
zΓ1
dY`
=
{YN 1 (1 + ε) − YNeq1 (1 − ε)}
dz
2Hz=1
dY`¯
zΓ1
=
{YN 1 (1 − ε) − YNeq1 (1 + ε)}
dz
2Hz=1
YL = Y` − Y`¯
Oh no!
zΓ1
dYL
=
ε(YN 1 + YNeq1 )
dz
Hz=1
[Kolb, Wolfram (1980)]
Subtraction of Real Intermediate States (RIS)
[Kolb, Wolfram (1980)]
Kinetic equations (with washout term not shown)
zΓ1
dY`
RIS
RIS
=
{YN 1 (1 + ε) − YNeq1 (1 − ε)} −γ`H→`H
¯ ∗ +γ`H
¯ ∗ →`H +γ`H→`H
¯ ∗ −γ`H
¯ ∗ →`H
dz
2Hz=1
dY`¯
zΓ1
RIS
RIS
=
{YN 1 (1 − ε) − YNeq1 (1 + ε)} +γ`H→`H
¯ ∗ −γ`H
¯ ∗ →`H −γ`H→`H
¯ ∗ +γ`H
¯ ∗ →`H
dz
2Hz=1
YL = Y` − Y`¯
Oh no!
zΓ1
dYL
RIS
RIS
=
ε(YN 1 + YNeq1 ) − 2γ`H→`H
¯ ∗ + 2γ`H
¯ ∗ →`H + 2γ`H→`H
¯ ∗ − 2γ`H
¯ ∗ →`H
dz
Hz=1
zΓ1
1−ε
zΓ1 eq 1
RIS
γ`H→
Y eq (1 − ε) ×
≈
YN 1 (1 − 2ε)
¯ ∗ =
`H
2Hz=1 N 1
2
H
4
z=1
| {z }
|
{z
}
resonant
production of N1
RIS
γ`H
¯ ∗ →`H ≈
branching ratio
for decay
zΓ1 eq 1
Y
(1 + 2ε)
Hz=1 N 1 4
dYL
zΓ1
=
ε(YN 1 − YNeq1 ) − 2γ`H→`H
¯ ∗ + 2γ`H
¯ ∗ →`H
dz
Hz=1
Subtraction of Real Intermediate States (RIS)
[Kolb, Wolfram (1980)]
Kinetic equations (with washout term not shown)
zΓ1
dY`
RIS
RIS
=
{YN 1 (1 + ε) − YNeq1 (1 − ε)} −γ`H→`H
¯ ∗ +γ`H
¯ ∗ →`H +γ`H→`H
¯ ∗ −γ`H
¯ ∗ →`H
dz
2Hz=1
dY`¯
zΓ1
RIS
RIS
=
{YN 1 (1 − ε) − YNeq1 (1 + ε)} +γ`H→`H
¯ ∗ −γ`H
¯ ∗ →`H −γ`H→`H
¯ ∗ +γ`H
¯ ∗ →`H
dz
2Hz=1
YL = Y` − Y`¯
Oh no!
zΓ1
dYL
RIS
RIS
=
ε(YN 1 + YNeq1 ) − 2γ`H→`H
¯ ∗ + 2γ`H
¯ ∗ →`H + 2γ`H→`H
¯ ∗ − 2γ`H
¯ ∗ →`H
dz
Hz=1
zΓ1
1−ε
zΓ1 eq 1
RIS
γ`H→
Y eq (1 − ε) ×
≈
YN 1 (1 − 2ε)
¯ ∗ =
`H
2Hz=1 N 1
2
H
4
z=1
| {z }
|
{z
}
resonant
production of N1
RIS
γ`H
¯ ∗ →`H ≈
branching ratio
for decay
zΓ1 eq 1
Y
(1 + 2ε)
Hz=1 N 1 4
dYL
zΓ1
=
ε(YN 1 − YNeq1 ) − 2γ`H→`H
¯ ∗ + 2γ`H
¯ ∗ →`H
dz
Hz=1
Wrong sign swept under rug
Closed-Time-Path Approach
In-in generating functional (in contrast to “in-out” for S matrix
elements): [SchwingerZ(1961); Keldysh (1965); Calzetta & Hu (1988)]
+ −
+
−
+
Dφ(τ )Dφ−
in Dφin hφin |φ(τ )ihφ(τ )|φin ihφin |%|φin i
Z[J+ , J− ] =
Z
=
Dφ− Dφ+ ei
R
d4 x{L(φ+ )−L(φ− )+J+ φ+ −J− φ− }
The Closed Time Path:
Aim: Calculate
hin|O(t)|ini
Path-ordered Green functions:
δ2
i∆ab (u, v) = − δJa (u)δJ
log
Z[J
,
J
]
+
−
b (v)
J± =0
= ihC[φa (u)φb (v)]i
e.g. j µ (x) = tr[γ µ hC[ψ − (x1 )ψ̄ + (x2 )]i]x1 =x2 =x
Wigner Transformation of Two-Point Functions (Green Function or Self Energy)
R
A(k, x) = d4 r eik·r A (x + r/2, x − r/2) →∼distribution function
x: average coordinate – macroscopic evolution
r → k: relative coordinate – microscopic (quantum)
Schwinger-Dyson & Kadanoff-Baym Equations
Feynman Rules
Vertices either + or −.
Connect vertices a = ± and b = ± with i∆ab .
Factor −1 for each − vertex.
Schwinger-Dyson equations →
These describe in principle the full
time evolution. However, truncations,
e.g. perturbation theory, are needed.
The <, >≡ +−, −+ parts of the Schwinger-Dyson equations are the celebrated
Kadanoff-Baym equation:
1
(−∂ 2 − m2 )∆<,> − ΠH ∆<,> − Π<,> ∆H =
Π> ∆< − Π< ∆>
|2
{z
}
collision term
Remaining linear combination gives pole-mass equation:
(−∂ 2 − m2 )i∆R,A − ΠR,A i∆R,A = iδ 4 , R, A: retarded, advanced,
ΠH , ∆H = Re[ΠR , ∆R ]
First principle derivation of Boltzmann-like kinetic equations.
[Keldysh (1965); Calzetta & Hu (1988)]
Path Ordered Green Functions
Four propagators (two of which are linearly
independent):
<
+−
i∆ (u, v) = i∆ (u, v) = hφ(v)φ(u)i
Wightman functions
i∆> (u, v) = i∆−+ (u, v) = hφ(u)φ(v)i
i∆T (u, v) = i∆++ (u, v) = hT [φ(u)φ(v)]i Feynman propagator
i∆T̄ (u, v) = i∆−− (u, v) = hT̄ [φ(u)φ(v)]i Dyson propagator
Perturbation theory can be formulated in terms of tree-level
expressions:
i∆< (p) =2πδ(p2 + m2 ) ϑ(p0 )f (p) + ϑ(−p0 )(1 + f¯(−p))
i∆> (p) =2πδ(p2 + m2 ) ϑ(p0 )(1 + f (p)) + ϑ(−p0 )f¯(−p)
0
i
2
2
0 ¯
i∆T (p) = 2
+
2πδ(p
+
m
)
ϑ(p
)f
(p)
+
ϑ(−p
)
f
(−p)
p − m2 + iε
0
−i
2
2
0 ¯
+
2πδ(p
+
m
)
ϑ(p
)f
(p)
+
ϑ(−p
)
f
(−p)
i∆T̄ (p) = 2
p − m2 − iε
As anticipated, can use these in order to compute charge and number
densities.
Kinetic Equations
The <, >≡ +−, −+ parts of the Schwinger-Dyson equations are the
celebrated Kadanoff-Baym equation:
1 >
(−∂ 2 −m2 )∆<,> −ΠH ∆<,> −Π<,> ∆H =
Π ∆ < − Π< ∆ >
{z
}
|2
collision term
Remaining linear combination gives pole-mass equation:
(−∂ 2 − m2 )i∆R,A − ΠR,A i∆R,A = iδ 4 , R, A: retarded, advanced
Hermitian/antihermitian decompostion of Kadanoff-Baym equations
→ kinetic equations:
k 0 ∂t i∆<,> = − 12 cos() ({iΠ> }{i∆< } − {iΠ< }{i∆> })
First principle derivation of Boltzmann-like equations.
[Keldysh (1965); Calzetta & Hu (1988)]
Gradient Expansion
Gradient Expansion
Leptogenesis in the CTP Approach
Schwinger-Dyson equations relevant for leptogenesis:
[Beneke, BG, Herranen, Schwaller (2010)]
Truncation that yields leading asymmetry in non-degenerate regime
Mi Γi , Mj Γj |Mi2 − Mj2 |:
Non-minimal truncations → e.g. systematic inclusion of thermal
corrections.
Unitarity Restored (without RIS)
CTP approach readily yields inclusive
rates for the creation of the charge
asymmetry.
No need to separately remove
unwanted contribution. No hand
waving explanation needed why these
are unphysical.
Note: Statistical Factors now on External and Internal Lines
Asymmetry ∝ [1 − f` (p) + fφ (k)] × [1 − f` (p0 ) + fφ (k0 )].
[Beneke, BG, Herranen, Schwaller (2010)]
CTP Methods and Baryogenesis
Non-equilibrium condition leads to technical problems in approach
based on S-matrix
CTP: Green’s function based inclusive approach as a cure of this
trouble
Leptogenesis: Clean example for out-of equilibrium decay scenarios
that is phenomenologically relevant and that works
CTP goes beyond Boltzmann calculations whenever virtual particles
are close to mass shell: CP cuts, exchange of soft degrees of freedom,
collinear radiation (long tem-scales of individual reactions), resonant
effects, flavour correlations
CTP can also be applied to Electroweak Baryogenesis → Spatial &
temporal inhomogeneity leads to complications that bar application to
fully phenomenological scenarios so far, cf. e.g. [Konstandin, Prokopec, Schmidt, Seco
(2005); Cirigliano, Lee, Ramsey-Musolf, Tulin (2009); Cirigliano, Lee, Tulin (2011)].
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Flavoured Leptogenesis
Neutrino Yukawa couplings: Yia N̄i `a φ
†
Wheninsensitive to lepton
flavour, can `a → Uab `b , Yia → Yib Uba such that
Y11
0
0
N1 decays only produce (U `)1 , which is a linear
0 
Y =  Y21 Y22
combination of `e , `µ , `τ .
Y31 Y32 Y33


qee qeµ qeτ
Back to flavour basis – quantum correlations 
qµe qµµ qµτ 
of charge densities:
qτ e qτ µ qτ τ
2
When H ∼ mTPl . Γτ,µ ∼ h2τ,µ T , i.e. T . 1012 GeV (for τ ) or T . 109 GeV (for
µ), the
 is destroyed by the SM lepton Yukawa couplings hτ,µ .
 flavour coherence
The resulting linear combination is only partly
qee
0
0
 0
aligned with the one that is produced in the deqµµ
0 
cays of N1 .
O(1) suppression of washout.
0
0
qτ τ
[Abada, Davidson, Josse-Michaux, Losada, Riotto (2006); Nardi, Nir, Roulet, Racker (2006)]
For initially lepton flavour violating contributions with vanishing L = Le + Lµ + Lτ :
washout of different flavours → lepton number violating asymmetry at freeze out
that receives contribution from the PMNS phase δ – but from other phases as well.
[Pascoli, Petcov, Riotto (2006)]
12
Even for T & 10 GeV, the asymmetries from the decays of the heavier RHNs
correspond to linear combinations of flavour that are in general misaligned with the
one washed out by the lightest of the RHNs [Di Bari (2005)]. → Richer phenomenology,
less predictivity.
Flavoured Leptogenesis
Partial Flavour (De)Coherence
Kinetic equations for number densities & flavour correlations of (anti-)leptons δn±
` :
∂δn±
fl
±
th
±
bl
+
− `ab
−Γ
=
±S
−
[W,
δn
∓i∆ω
δn
]
−
γ
δn
+
δn
ab
ab
ab
`ab
`ab
∂t
| {zab} |
{z `ab} | {z` } |
{z
} | {z }
CP oscillations
violating induced by
source
thermal masses
Γfl ∝ h2τ
1
0
0
0
washout
flavour-blind
pair-creation &
annihiliation
flavoursensitive
damping
, δn±
− 2h2τ δn±
R → directly damps off-diagonal correlations
`
Γdirect ∼ (h2aa + h2bb )T
(h2 −h2 )2
Γoverdamped ∼ aa g4 bb T
2
where a, b = e, µ, τ
Can interpolate between fully flavoured and
unflavoured regimes.
[Beneke, BG, Herranen, Fidler, Schwaller (2010)]
Spectator Effects
Rate for SM processes
mediated by Yukawa coupling
h and gauge coupling g in the
relativistic limit
ΓX ∼ T g 2 h2 log g, Hubble
rate H ∼ T 2 .
→ As the Universe cools, more
and more SM processes come
into equilibrium.
Besides Yukawa mediated
processes, there are also the
chiral anomalies (strong and
weak sphalerons).
down
charm
Weak Sph
1014
electron
strange
bottom
strong Sph
muon
tau
1012
up
1010
108
106
T @GeVD
Temperature bands TX ≤ T ≤ 20TX where TX is the
equilibration temperature ΓX (TX ) = H(TX ).
Asymmetries are transferred partly to all SM degrees of freedom → suppression of
washout. [Barbieri, Creminelli, Strumia, Tetradis (2000)]
Rephrase kinetic equations in terms of conserved SM charges, ∆a = B/3 − La .
dY∆a
zΓ1
zW
= εa H(z=1)
(YN 1 − YNeq1 ) − H(z=1)
(YLa + 12 YH )
dz
where YLa = Aab ∆b and YH = Ca ∆a → another O(1) effect.
E.g. for reaction X + Y ↔ Z, impose µX + µY − µZ = 0. → linear maps Aab , Ca .
This procedure holds for fully equilibrated spectator fields.
Partially Equilibrated Spectators
104
Fully equilibrated spectators are not a
realistic assumption in most regions of
parameter space.
1.5
10
1.4
0.8
1.3
102
These asymmetries are transferred to
the spectators.
For partially equilibrated spectators,
these asymmetries are also partially
protected from washout.
Moderate enhancement effect for
thermal initial conditions of the
RHNs. Vanishing initial conditions
(where the effect may be larger) are
yet to be explored.
0.7
3
K1
At early times, the deviation of the
RHNs from equilibrium is large →
large asymmetries present.
1.6
1.2
1.1
10
0.6
1
0.9
1011
1012
1013
M1 @GeVD
1014
Freeze out asymmetry for partially
equilibrated τ -Yukawa coupling hτ divided
by the limit τ → ∞. [BG, Schwaller (2014)]
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Resonant Leptogenesis
Regulator for Decay Asymmetry
Consider the mass-degenerate regime M1 → M2 and M1,2 & T (strong
washout).
Decay Asymmetry
Wave-function contribution:
† 2
2
2
1 P Mi Mj (Mi +Mj ) Im[(Y Y )ij ]
εwf
2
2
†
2
N i = 8π
(Y Y )
(M −M ) +R
j6=i
i
j
j
ii
In the degenerate limit, this will dominate over the vertex contribution.
Proposed forms for regulator R (M̄ = (Mi + Mj )/2):
Rj =
Rj =
Rj =
M̄ 4
† 2
2 2
64π 2 (Y Y )jj = M̄ Γj [Pilaftsis (1997); Pilaftsis, Underwood (2003)]
2
M̄ 4
†
†
64π 2 [Y Y ]ii − [Y Y ]jj [Anisimov, Broncano, Plümacher (2005)]
2
M̄ 4
†
†
[Garny, Hohenegger, Kartavtsev (2011)]
64π 2 [Y Y ]ii + [Y Y ]jj
Problem: When M̄ Γ |Mi2 − Mj2 | = ∆M 2 not satisfied, cannot expand
the RHN propagator as
Resonant Leptogenesis
Back to Schwinger-Dyson form
Evolution for Matrix-Valued RHN Distributions δfN h
A
A
2
eq0
0 + a (η) i[M 2 , δf
∗ t k·Σ̂N
∗ t k̃·Σ̂N
δfN
N h ] + fN = −2 Re[Y Y ] k0 − ihIm[Y Y ] k0 , δfN h
h
2k0
Σ̂A
N : spectral (cut part) self energy; a(η), η : scale factor and conformal time; 0 ≡ d/dη;
h: helicity.
i[M 2 , δfN h ]ij = i(Mi2 − Mj2 )δfN hij for diagonal M 2 → RHN “flavour” oscillations
Off-diagonal entries correspond to interference between the different Ni that give
rise to CP violation.
Evolution equations are well behaved for ∆M 2 → 0. Solutions δfN h
enter into the resummed RHN propagators. [BG, Herranen (2010)]
Resonant Leptogenesis in the Strong Washout Regime
Regulator:
R=
†
†
2
M̄ 4 ([Y Y ]11 +[Y Y ]22 )
64π 2
[Y Y † ]11 [Y Y † ]22
×((Im[Y Y † ]12 )2 +det Y Y † )
Applies in strong washout regime, i.e. a
large portion of parameter space.
blue: full result; red: result using effective decay
asymmetry ε; Yaa = nLaa /s. Here, M̄ Γ ∆M 2 .
|Yee |
εµµ
|Yµµ |
[BG, Gautier, Klaric (2014);
ετ τ
equations for δfN h .
Iso, Shimada (2014)].
|Yτ τ |
Neglect derivative term provided the
eigenvalues (that originate from the mass
and the damping terms) in the kinetic
equation for are larger than the Hubble
rate H → obtain linear system of
εee
Quasistatic Approximation
z=M̄ /T
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Leptogenesis from Oscillations
RHNs perform first oscillation at temperature Tosc ∼ (|Mi2 − Mj2 |mPl )1/3
→ Off-diagonal correlations for the RHNs
→ Sizeable asymmetries generated around Tosc (see plots on next slide)
Since Tosc Mi , the RHNs are typically relativistic and thermal effects are of
leading importance → calls for CTP techniques.
Typically, “early asymmetries” are washed out by the time T ∼ M . Only asymmetry
produced around T ∼ M survives (strong washout).
Loopholes to Preserve Early Asymmetries
GeV-scale RHNs (→ Tosc ∼ 105 GeV) typically do not equilibrate prior to the
electroweak phase transition where B settles to final value (sphaleron freezeout).
[Akhmedov, Rubakov, Smironov (1998); Asaka, Shaposhnikov (2005); BG, Drewes (2012)].
→Early asymmetry preserved until baryon number freezes in.
One individual active flavour (typically (νe , eL )) is only weakly washed out, such
that early asymmetries survive [BG (2014)].
Computations are numerically challenging and analytic estimates rough because of
oscillation scales, damping scales and levels of mass degeneracy that can be very
different through parameter space.
Leptogenesis from Oscillation – Dynamics
Generation of the BAU
T Γ ∆M 2
Re[δn12 odd ]/s
6. × 10-5
4. × 10-5
2. × 10-5
0.
-2. × 10-5
-4. × 10-5
-6. × 10-5
RHNs perform first oscillation at
Tosc ∼ (|Mi2 − Mj2 |mPl )1/3 .
Off-diagonal correlations lead to CP
violating source for lepton flavour
asymmetries (purely flavoured).
Δa/s
4. × 10-9
3. × 10-9
2. × 10-9
1. × 10-9
0.
-1. × 10-9
-2. × 10-9
-3. × 10-9
-4. × 10-9
Transfer of asymmetries into helicity
asymmetries of RHNs leads to YL 6= 0.
-10
Asymmetry frozen in at TEW , where
sphalerons are quenched by the
developing Higgs vev.
|YB |
10
Contributions from subsequent
oscillations average out.
10-11
10-2
10-1
100
z
z = TEW /T ; ∆a : lepton asymmetry in flavour a = e, µ, τ ; δn: number density of RHNs;
YB : entropy-normalized baryon asymmetry. [with Marco Drewes, Dario Gueter, Juraj Klarić (in preparation)]
Leptogenesis from Oscillation – Dynamics
Oscillatory vs Overdamped Regime
T Γ ∆M 2
T Γ ∆M 2
Re[δn12 odd ]/s
Re[δn12 odd ]/s
6. × 10-5
4. × 10-5
2. × 10-5
0.
-2. × 10-5
-4. × 10-5
-6. × 10-5
2. × 10-5
1. × 10-5
0.
-1. × 10-5
-2. × 10-5
4. × 10-9
3. × 10-9
2. × 10-9
1. × 10-9
0.
-1. × 10-9
-2. × 10-9
-3. × 10-9
-4. × 10-9
1. × 10-7
Δa/s
Δa/s
5. × 10-8
0.
-5. × 10-8
-1. × 10-7
10-8
10-10
|YB |
|YB |
10-9
10-10
10-11
10-11
10-2
z
10-1
100
10-1
z
100
z = TEW /T ; ∆a : lepton asymmetry in flavour a = e, µ, τ ; δn: number density of RHNs;
YB : entropy-normalized baryon asymmetry. [with Marco Drewes, Dario Gueter, Juraj Klarić (in preparation)]
Leptogenesis from Oscillation of GeV-scale RHNs
Asymmetries
Source term for flavoured early asymmetries around Tosc :
Sab =
32i
M 2 −M 2
ii
jj
c,i,j
i6=j
P
R
(2π)3 2
d3 k
√
2
k +M 2
ii
†
h
√ 2 2 A0 i Ai i
†
2 +2k2
2
Ai 2
× (Yai
Yic Ycj
Yjb ) (Mii
k +Mii Σ̂N k̂ Σ̂N
)(Σ̂A0
N +Σ̂N )−4|k|
† ∗ t
Aµ
+(Yai
Yic Ycj Yjb )Mii Mjj Σ̂A
×δfN hii (k) .
N µ Σ̂N
Σ̂A
N =
[Besak, Bödeker (2012);
BG, Glowna, Schwaller (2013)]
RHNs relativistic → Σ̂A
N dominated by thermal effects
Lepton number violating contribution ∼ M 2 /∆M 2 (Majorana mass insertion)
requires ∆M 2 /M 2 → 0 for resonant enhancement.
Lepton flavour violating (lepton number conserving) contribution ∼ T 2 /∆M 2 →
large enhancement for ∆M 2 T 2 → no/less pronounced mass degeneracy needed.
Leptogenesis is viable with non-degenerate (in mass) RHNs of the GeV scale [Drewes,
3
BG (2012)] and for masses & 5 × 10 GeV [BG (2014)].
Enhanced early production of RHNs is favourable for leptogenesis. This may happen
when extra degrees of freedom in scenarios beyond the Standard Model can be
radiated in the scattering processes at high temperatures.
Direct & Indirect Searches for GeV-scale RHNs
Direct searches:
Mixing between active neutrinos and RHNs:
†
Uai ≈ θai = vYai
/Mi
GeV-scale RHNs can be produced in heavy
meson (D, B) decays in B factories or beam
dump experiments.
Sensitivity of B factories is limited because RHNs can decay outside of the detector.
SHiP – Search for Hidden Particles: Proposed beam dump facility @ CERN:
Indirect signals:
Rate for 0νββ decay in type I seesaw mechanism can be enhanced for GeV-scale
RHNs compared to other mass regions.
Lepton universality in meson decays.
Charged lepton flavour violation.
Searches for GeV-scale RHNs and Leptogenesis
10-5
past experiments
10-6
BAU (upper bound)
U2
10-7
10-8
BBN
SHiP sensitivity
10-9
10-10
seesaw
BAU (lower bound)
10-11
0.1
0.5
1
5
10
[Canetti, Drewes, BG (2014)]
M [GeV]
Goal: Find regions where leptogenesis is viable (and understand these
in the high-dimensional parameter space).
Need better numerical performance, precision & better analytical
understanding. [Drewes, BG, Gueter, Klarić (in progress)]
Table of Contents
1. Matter-Antimatter Asymmetry
2. Neutrinos and the Baryon Asymmetry of the Universe
3. Vanilla Leptogenesis in the Strong Washout Regime
4. Leptogenesis & Methods from Non-Equilibrium Field Theory
5. Flavour and Spectator Effects
6. Resonant Leptogenesis
7. Leptogenesis from Oscillations
8. Conclusions
Conclusions
The observed neutrino mass scale suggests that the coupling strengths
of RHNs in the type I seesaw mechanism are favourable for
leptogenesis – independent of the scale of the seesaw.
“Vanilla” leptogenesis has very simple and robust predictions due to
strong washout. Flavour and spectator effects typically lead to a more
complicated picture – O(1) effects.
Standard leptogenesis requires seesaw scale above ∼ 1010 GeV, in
moderate tension with field theory arguments with or without SUSY.
Leptogenesis from oscillations operates at lower seesaw scales and
temperatures. RHNs at the GeV scale may be observed at the
intensity frontier if mixing angles are somewhat large.
Closed-Time-Path methods lead to a solid theoretical formulation of
baryogenesis – in particular concerning the description of
non-equilibrium required according to Sakharov.
→ Leptogenesis well motivated scenario for developing these
techniques.
50 Years of Sakharov’s Puzzle
A challenge to experiment & theory, QFT (QCD, EFT, lattice, ...),
hadrons & nuclear theory, atoms & molecules, BSM, non-equilibrium
physics
50 Years of Sakharov’s Puzzle
A challenge to experiment & theory, QFT (QCD, EFT, lattice, ...),
hadrons & nuclear theory, atoms & molecules, BSM, non-equilibrium
physics
When will we know the answer (if ever)?
50 Years of Sakharov’s Puzzle
A challenge to experiment & theory, QFT (QCD, EFT, lattice, ...),
hadrons & nuclear theory, atoms & molecules, BSM, non-equilibrium
physics
When will we know the answer (if ever)?
*
Probably before
The Netherlands win
the World Cup.
∗ Thanks
for trying. We do appreciate your effort!
Backup Slides
Situation in the Standard Model (SM)
Check Sakharov conditions:
1
B violated due to anomalous B + L violation
via sphaleron processes.
2
CP violation in CKM matrix, C and P violation
in weak interactions.
3
Deviation from equilibrium due to expansion of
the Universe.
Situation in the Standard Model (SM)
Check Sakharov conditions:
1
B violated due to anomalous B + L violation
via sphaleron processes.
2
CP violation in CKM matrix, C and P violation
in weak interactions.
3
Deviation from equilibrium due to expansion of
the Universe.
Situation in the Standard Model (SM)
Check Sakharov conditions:
1
B violated due to anomalous B + L violation
via sphaleron processes.
2
CP violation in CKM matrix, C and P violation
in weak interactions.
3
Deviation from equilibrium due to expansion of
the Universe.
However, conditions for baryogenesis not met quantitatively:
CP rephasing invariant normalised to electroweak scale is tiny:
h
i
Im det[mu m†u , md m†d ] ≈ −2Jm4t m4b m2c m2s , J ≈ 3 × 10−5 ,
m4 m4 m2 m2
2J t Tb12 c s ≈ 3 × 10−19 for T = 100 GeV.
Deviation from equilibrium H/Γ ∼ (T 2 /mPl )/(g 4 T ) = g −4 T /mPl is
tiny unless T is very high (mPl = 1.2 × 1019 GeV).
Situation in the Standard Model (SM)
Check Sakharov conditions:
1
B violated due to anomalous B + L violation
via sphaleron processes.
2
CP violation in CKM matrix, C and P violation
in weak interactions.
3
Deviation from equilibrium due to expansion of
the Universe.
However, conditions for baryogenesis not met quantitatively:
CP rephasing invariant normalised to electroweak scale is tiny:
h
i
Im det[mu m†u , md m†d ] ≈ −2Jm4t m4b m2c m2s , J ≈ 3 × 10−5 ,
m4 m4 m2 m2
2J t Tb12 c s ≈ 3 × 10−19 for T = 100 GeV.
Deviation from equilibrium H/Γ ∼ (T 2 /mPl )/(g 4 T ) = g −4 T /mPl is
tiny unless T is very high (mPl = 1.2 × 1019 GeV).
Baryogenesis requires physics beyond the Standard Model.
Baryogenesis Puzzle in New Physics Scenarios
Leptogenesis (this talk)
Electroweak baryogenesis – requires extension of the SM to provide strong first order
phase transition, extra CP violation → perhaps the best prospects for testability
Decay of scalar condensates or Q-balls, e.g. from SUSY flat directions
(Affleck-Dine)
Connection with asymmetric dark matter
Other paradigms less connected to specific models
Baryogenesis Puzzle in New Physics Scenarios
Leptogenesis (this talk)
Electroweak baryogenesis – requires extension of the SM to provide strong first order
phase transition, extra CP violation → perhaps the best prospects for testability
Decay of scalar condensates or Q-balls, e.g. from SUSY flat directions
(Affleck-Dine)
Connection with asymmetric dark matter
Other paradigms less connected to specific models
Hard to solve because
we do not know whether we
have all pieces,
we do not know whether the
pieces we have are part of the
puzzle,
some of the pieces probably
not even exist.
Matter Antimatter Asymmetry
Quantum Field Theory predicts matter and antimatter.
CP T theorem −→ These have very similar properties, in particular
same masses, same total decay rates.
Visible Universe almost entirely made of matter, almost no antimatter.
Aesthetics as well as modern Cosmology (inflation): Asymmetry
cannot be explained as an initial condition.
Kinetic Equations
The <, >≡ +−, −+ parts of the Schwinger-Dyson equations are the
celebrated Kadanoff-Baym equation:
1 >
(−∂ 2 −m2 )∆<,> −ΠH ∆<,> −Π<,> ∆H =
Π ∆ < − Π< ∆ >
{z
}
|2
collision term
Remaining linear combination gives pole-mass equation:
(−∂ 2 − m2 )i∆R,A − ΠR,A i∆R,A = iδ 4 , R, A: retarded, advanced
Hermitian/antihermitian decompostion of Kadanoff-Baym equations
→ kinetic equations:
k 0 ∂t i∆<,> = − 12 cos() ({iΠ> }{i∆< } − {iΠ< }{i∆> })
First principle derivation of Boltzmann-like equations.
[Keldysh (1965); Calzetta & Hu (1988)]
Path Ordered Green Functions
Four propagators (two of which are linearly
independent):
<
+−
i∆ (u, v) = i∆ (u, v) = hφ(v)φ(u)i
Wightman functions
i∆> (u, v) = i∆−+ (u, v) = hφ(u)φ(v)i
i∆T (u, v) = i∆++ (u, v) = hT [φ(u)φ(v)]i Feynman propagator
i∆T̄ (u, v) = i∆−− (u, v) = hT̄ [φ(u)φ(v)]i Dyson propagator
Perturbation theory can be formulated in terms of tree-level
expressions:
i∆< (p) =2πδ(p2 + m2 ) ϑ(p0 )f (p) + ϑ(−p0 )(1 + f¯(−p))
i∆> (p) =2πδ(p2 + m2 ) ϑ(p0 )(1 + f (p)) + ϑ(−p0 )f¯(−p)
0
i
2
2
0 ¯
i∆T (p) = 2
+
2πδ(p
+
m
)
ϑ(p
)f
(p)
+
ϑ(−p
)
f
(−p)
p − m2 + iε
0
−i
2
2
0 ¯
+
2πδ(p
+
m
)
ϑ(p
)f
(p)
+
ϑ(−p
)
f
(−p)
i∆T̄ (p) = 2
p − m2 − iε
As anticipated, can use these in order to compute charge and number
densities.
Gradient Expansion
Gradient Expansion
Unitarity in Soft Leptogenesis
Important new source of CP violation in
SUSY models: Relative phase between soft
masses and µ term.
→ Electric dipole moment (EDM) signals.
Does this work for leptogenesis?
Clearly not at T = 0. Asymmetry in leptons cancelled by sleptons.
Finite temperature: Sleptons follow Bose-Einstein statistics and leptons
Fermi-Dirac. → Does this bias a net asymmetry? [Fong, Gonzalez-Garcia (2009)]
No. Proper inclusion of quantum statistical effects on internal lines
shows that cancellation persists. [BG, Ramsey-Musolf (2013)]
Check all possible asymmetries for soft leptogenesis. Cancellation in
inclusive asymmetries always holds. ⇒ Comprehensive study of all
possible one-loop topologies that is of relevance for models other than
soft leptogenesis.
CTP Approach to Leptogenesis – Applications
Projects of the group in connection with non-equilibrium field theory
approach to leptogenesis:
Flavoured leptogenesis, dynamics of the decoherence of active lepton
flavours (importance of gauge interactions in that process) [Beneke, BG, Fidler,
Herranen, Schwaller (2010)]
Soft leptogenesis (this talk, previous slide)
[BG, Ramsey-Musolf (2013)]
Resonant leptogenesis from mass-degenerate RHNs → this talk
[BG, Garny
(2011); BG, Herranen (2011); BG, Gautier Klarić (2014)]
Baryogenesis from mixing of charged particles (i.e. Higgs and lepton
doublets) [BG (2012); BG (2012); BG, Izaguirre (2014)]
Thermal effects for scatterings among relativistic particles
[BG, Glowna,
Schwaller (2013); BG, Glowna, Herranen (2013)]
Leptogenesis with (non-degenerate) GeV or TeV scale RHNs → this
talk [Drewes, BG (2012); BG (2014); Canetti, Drewes, BG (2014); Drewes, Gueter, Klarić, BG (in progress)]
Spectator effects
[BG, Schwaller (2014)]
Asymmetry from Mixing of Charged Particles
Wave-function corrections lead to
off-diagonal correlations, that in turn
induce asymmetries in diagonal charge
densities.
Off-diagonal correlations are a non-equilibrium effect can occur for
SM leptons from loop that propagates nonequilibrium RHNs
[BG (2012); BG, Izaguirre (2014)].
Source overlooked so far
in type I seesaw. Can also
generate asymmetries
when using mixing Higgs
doublets [BG (2012)] or other
charged states that occur
replicatedly →
opportunities for model
building.
blue: asymmetry from lepton mixing
red: from standard leptogenesis.
Searches for GeV-Scale RHNs and Leptogenesis
Direct search bounds:
[from SHiP physics case paper (2015)]