supersymmetric electroweak baryogenesis
TRANSCRIPT
Supersymmetric ElectroweakBaryogenesis
and Mixing in Transport EquationsThomas Konstandin
in collaboration with T. Prokopec and M.G. Schmidt
hep-ph/0410135
Stockholm, 18/02/05 – p.1/30
Outline
• Introduction to Electroweak Baryogenesis• EWB in the MSSM• Former Approaches• Transport Theory and EWB• Diffusion to the Quark Sector• Numerical Results• Conclusions
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IntroductionBaryogenesis is one of the cornerstones of theCosmological Standard Model.
The celebrated Sakharov conditions state the necessaryingredients for baryogenesis:
• C and CP violation• non-equilibrium• B-violation
B-violation is in the hot universe present due tosphaleron precesses.
C is violated in the electroweak sector of the SM
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Introduction
The sphaleron process:
• ∆B = 3, ∆L = 3, ∆NCS = 1
• B + L violating• B − L conserving• Exponentially suppressed
by the W mass• Topological effect of the
SU(2) gauge sector
Sphaleron bL
bL
tLsL
sL
cL
dL
dL
uLνe
νµ
ντ
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IntroductionCross-over versus electroweak phase transition:
<h>
VH<h>L at T >> 100 GeV
<h>
VH<h>L at T ~ 100 GeV
<h>
VH<h>L at T ~ 100 GeV
Phase transition Cross-over
<h>
VH<h>L at T = 0 GeV
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IntroductionElectroweak Baryogenesis:SHAPOSHNIKOV (’87)
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EWB in the MSSM
EWB in the SM excluded:• CP violation too small• No strong first order phase transition
EWB in the MSSM not excluded/disproved:• CP violation in the Higgsino/chargino sector• Strong first order phase transition in case of a light
stop
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EWB in the MSSM
In the MSSM case the Higgsino/chargino mass matrix is:
m =
(M2 g h2(z)
g h1(z) µc
)
with M2 and µc complex and
∂zm =
(0 g ∂zh2(z)
g ∂zh1(z) 0
)
and Tr(∂zMM † −M∂zM
†) ∼ Im(M2µ∗c) 6= 0 . CP-violation
is present on the tree level while in the SM ∂zM ∼M .
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EWB in the MSSM
Connection between macroscopic and microscopicscales:
• CP violation is a microscopic effect produced byinteraction of single quanta in the plasma.
• Transport is a macroscompic effect based onstatistical physics and particle densities.
Filling this gap means deriving semiclassical Boltzmannequations from first principles.
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Former Approaches
A) Via dispersion relations:CLINE, JOYCE, KAINULAINEN (’97, ’00)
The CP violating dispersion relation E(p, z) for a varyingmass m(z) = |m(z)|eiθ(z) is determined by the WKBmethod and put into the classical Boltzmann equation forthe particle density f :
E2 = ~p2 +m2 ± m2∂zθ
2pz(∂t + ∂pzE ∂z − ∂zE ∂pz)f = Coll.
This procedure does not contain mixing effects.
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Former ApproachesB) Via source terms:CARENA, MORENO, QUIROS, SECO, WAGNER (’00, ’02)
The additional term in the Schwinger-Dyson equation isinterpreted as an interaction term
The CP violating source is calculated from
jµ(Xµ) =1
τ
∫d4p
(2π)4pµδg(pµ, Xµ)
and put ’by hand’ into diffusion equations.
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Transport Theory and EWBC) Kadanoff-Baym Equations:
Statistical analogon to Schwinger-Dyson equations
(S−10 − ΣR)S< − Σ<SR =
1
2Σ<S> − 1
2Σ>S< = Coll.,
(S−10 − ΣR)A− ΣASR = 0,
and all functions depend on xµ and yµ. Used definitions:
S< := S+−, S> := S−+,
A := i2(S< − S>), SR := S++ − i
2(S< + S>),
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Transport Theory and EWB
Wigner space: Using
X = (x+ y)/2, r = (x− y)/2
and performing a Fourier transformation with respect to r
e−i♦{S−10 − ΣR, S
<} − e−i♦{Σ<, SR} = Coll.,
e−i♦{S−10 − ΣR,A} − e−i♦{ΣA, SR} = 0,
with2♦{A,B} := ∂XµA∂kµB − ∂kµA∂XµB,
all quantities become functions of Xµ and kµ.
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Transport Theory and EWBFree bosonic theory with a constant mass in equilibrium:• The constraint and kinetic equations are (KMS):
(k2 −m2)S< = 0, kµ∂µS< = 0
• And the solution reads:
iS< = 2π sign(k0) δ(k2 −m2) n(k0), n(k0) =1
exp(βk0)− 1
where we can read off the particle density∫
k0>0
d4k
(2π)42ik0 S
< =
∫d3k
(2π)3n(
√~k2 +m2)
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Transport Theory and EWBGradient expansion:
For space-time dependent mass matrices, theKadanoff-Baym equations can be expanded in gradientsas long as the background is weakly varying
∂k∂X ≈1
Tclw� 1 → e−i♦ ≈ 1− i♦
what is true in the MSSM (lw ≈ 20/Tc).
The simplest example for an transport equation in avarying background is for one flavour with real mass
(k2 −m2)S< = 0, (kµ∂µ −1
2(∂zm
2) ∂kz)S< = Coll..
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Transport Theory and EWB
After spin projection and in the wall frame:
2ik0gs0 − (2iskz + s∂z) g
s3 − 2imhEg
s1 − 2imaEg
s2 = 0
2ik0gs1 − (2skz − is∂z) gs2 − 2imhEg
s0 + 2maEg
s3 = 0
2ik0gs2 + (2skz − is∂z) gs1 − 2mhEg
s3 − 2imaEg
s0 = 0
2ik0gs3 − (2iskz + s∂z) g
s0 + 2mhEg
s2 − 2maEg
s1 = 0
with
E ≡ exp( i
2
←∂ z→∂ kz
), E† ≡ exp
(− i
2
←∂ kz→∂ z
).
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Transport Theory and EWB
In first order the kinetic equation is: g = geq + δg
kz∂zδgR +i
2
[m†m, δgR
]− 1
4
{(m†m)′, ∂kzδgR
}D
+k0ΓδgR − kz[V †′V, δgR
]= SR[geqR , g
eqL ]
• The source S depends on the lowest order solutionsgeqR and geqL
• We introduced a damping term in the time-likedirection to fulfill the boundary conditions at z → ±∞simultaneously
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Transport Theory and EWB
Sources:
SR[geqR , geqL ] = −kz
[V †′V, geq0 − g
eq3
]+
1
4
{(m†m)′, ∂kz(g
eq0 − g
eq3 )}T
− s
4k0
[m†′m−m†m′, g(0)
0
]T− s
4k0
{(m†m)′, g(0)
0
}T
where
geq3 ≡ V †geq3dV = (g(0)R − V †Ug
eqL U
†V )/2
geq0 ≡ V †geq0dV = (g(0)R + V †UgeqL U
†V )/2
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Transport Theory and EWB
Comments:
• The constraint equation is non-algebraic.• It is possible to decouple the kinetic equations, such
that it does not contain k0 explicitely and noinformation about the spectral function is needed.
• The kinetic equation for the deviation from thermalequilibrium in the mass eigenbasis reads g = geq + δg
kz∂zδg +i
2
[m2, δg
]+ · · · = S(geq)
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Transport Theory and EWBkz∂zδg + i
2
[m2, δg
]+ k0Γδg + · · · = S(geq)
• The term i2
[m2, δgR
]will lead to an oscillatory
behaviour of the off-diagonal particle densities,similar to neutrino oscillations with frequency∼ (m2
1 −m22)/kz. This can be seen using
i [σ3, σ1] = σ2, i [σ3, σ2] = −σ1
• Without this oscillation term CP would be conservedup to first order in gradients
• The phenomenological damping term k0Γδg has beenintroduced after identification of CP violating sources
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Transport Theory and EWB
Parameters chosen: M2 = 200 GeV, µc = 220 Gev
-2e-06
-1e-06
0
1e-06
2e-06
3e-06
4e-06
5e-06
6e-06
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
j/Tc3
z in GeV-1
j5j
j5(2)
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Transport Theory and EWB
Concluding remarks:
The first principle derivation of the transport equationsshows:
• Oscillations in the CP violating densities areimportant for the dynamics
• CP violation on the microscopic level has to beidentified before time-invariance is broken byphenomenological damping terms
and both effects suppress CP-violating effects relative tofromer approaches.
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Diffusion and the Sphaleron
Missing parts to determine the baryon asymmetry of theuniverse:
h~
q~
q
Y
and
Sphaleron bL
bL
tLsL
sL
cL
dL
dL
uLνe
νµ
ντ
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Diffusion and the Sphaleron
Diffusion equations:HUET,NELSON (’95)
vw n′Q = Dq n
′′Q − ΓY
[nQkQ− nTkT− nH + nh
kH
]− Γm
[nQkQ− nTkT
]
−6 Γss
[2nQkQ− nTkT
+ 9nQ + nTkB
]
vw n′T = Dq n
′′T + ΓY
[nQkQ− nTkT− nH + nh
kH
]+ Γm
[nQkQ− nTkT
]
+3 Γss
[2nQkQ− nTkT
+ 9nQ + nTkB
]
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Diffusion and the Sphaleron
Weak sphaleron:
nB = −3 Γwsvw
∫ 0
−∞dz nL(z) exp
(z
15Γws4 vw
)
and finally the baryon-to-entropy ratio is determined via
η =nBs, s ≈ 51.1 T 3
c .
and can be compared with
ηBBN ≈ 0.9× 10−10
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Numerical ResultsParameters chosen: M2 = 200 GeV, vw = 0.05,lw = 20/Tc, mA = 200 GeV, sin(φ)=1.0
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250 300 350 400 450 500
η / η
BB
N
µc
Stotal
Sa
Sb
Sc
Sd
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Numerical ResultsParameters chosen: vw = 0.05, lw = 20/Tc, mA = 200 GeV
100 200 300 400M2
100
200
300
400µ c
mA=200
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Pessimistic Conclusion
Supersymmetric baryogenesis is a rather unlikelyscenario based on• A light stop to acquire a strong first order phase
transition• The condition µc ≈M2 � 400 GeV of the a priori
unrelated parameters M2 and µc• A large CP-violating phase that hardly satisfies
experimental EDM bounds
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Optimistic Conclusion
We have been able to derive a formalism that reproducesthe qualitative behaviour of two existing methods in theirrange of applicability. In addition we found two additionalfeatures in the transport equations not taken into accountbefore.
Based on our analysis, supersymmetric baryogenesis isa scenario that• Can explain the BAU• Predicts µc ≈M2 � 400 GeV• Will be testable soon
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The End
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