electroweak baryogenesis in the nmssmsusy06.physics.uci.edu/talks/6/konstandin.pdf · 2006. 6....

Electroweak Baryogenesis in the nMSSM

Thomas Konstandin, KTH Stockholm

SUSY 06-

S.J. Huber, CERNT. Prokopec, U. Utrecht

M.G. Schmidt, U. Heidelberghep-ph/0410135, hep-ph/0505103, hep-ph/06xxxxx

The nMSSM Introduction Transport Theory The Phase Transition Electric Dipole Moments Conclusions

Why is the nMSSM interesting?

Panagiotakopoulos, Pilaftsis (’02)

The nearly Minimal Supersymmetric Standard Model has thefollowing effective superpotential

WnMSSM = λSH1 · H2 −m2

12

λS + WMSSM ,

and has the virtues to solve the µ-problem of the MSSM byintroducing a dynamical µ-term

µ = −λ 〈S〉 .

In this model singlet self-couplings are forbidden by a R’-symmetry.The resulting model has neither problems with the stability of thehierarchy nor with domain walls.


Higgs potential of the nMSSM

Including soft SUSY breaking terms, the Higgs potential reads

V0 = m21H

†1H1 + m2

2H†2H2 + m2

s |S |2 + λ2|H1 · H2|

2

+λ2|S |2(H†1H1 + H

†2H2) +

g2

8(H†

2H2 − H†1H1)

2 +g2

2|H†

1H2|2

+ ts(S + h.c .) + (aλSH1 · H2 + h.c .) − m212(H1 · H2 + h.c .).

Compared to the MSSM, the new singlet terms help to strengthenthe EWPT and introduce additional sources of CP violation.


Picture of Electroweak Baryogenesis

Kuzmin, Rubakov, Shaposhnikov (’87)

Higgs vev

symmetric phase broken phase

lw

vw




Higgs vev


CP-violating particle density

(charginos, neutralinos)

CP-violating interaction




Higgs vev



sphalerons active sphalerons not active



Cohen, Kaplan, Nelson (’94)

Higgs vev

symmetric phase broken phasesphalerons active sphalerons not active


transport


First Principle Transport Equations: A Simple Example

Kainulainen, Prokopec, Schmidt, Weinstock (’01)

Kadanoff-Baym equations (bosonic, without interactions):

e−i♦{(k2 − m2(z)),S<(kµ, z)} = 0

with 2♦{A,B} := ∂XµA ∂kµB − ∂kµA ∂XµB .

Free bosonic theory with one flavour and a constant mass inequilibrium:

The hermitian/antihermitian parts are the so calledconstraint/kinetic equations:

(k2 − m2)S< = 0, kµ∂XµS< = 0

Using the KMS condition and the correct normalization

iS< = 2π sign(k0) δ(k2 − m2) n(k0), n(k0) =1

exp(βk0) − 1


Gradient Expansion

If the background in the MSSM is weakly varying (lw Tc � 1) theMoyal star product can be simplified by the semi-classicalapproximation

∂k∂X ≈1

Tc lw� 1 → e−i♦ ≈ 1 − i♦ + · · ·

The simplest example for an transport equation in a varyingbackground is for one bosonic flavour with real mass

(k2 − m2)S< = 0

(kµ∂µ −1

2(∂zm

2) ∂kz)S< = 0.


Fermionic Systems

Kainulainen, Prokopec, Schmidt, Weinstock (’01)

T.K., Prokopec, Schmidt (’04)

After spin projection the fermionic system of equations reads

“

2i k0 −

k0∂t + ~k‖ · ∇‖

k0

”

Ss0 − (2iskz + s∂z ) S

s3 − 2imhe

i2

↼∂z

⇀∂kz S

s1 − 2imae

i2

↼∂z

⇀∂kz S

s2 = 0

“

2i k0 −

k0∂t + ~k‖ · ∇‖

k0

”

Ss1 − (2skz − is∂z ) S

s2 − 2imhe

i2

↼∂z

⇀∂kz S

s0 + 2mae

i2

↼∂z

⇀∂kz S

s3 = 0

“

2i k0 −

k0∂t + ~k‖ · ∇‖

k0

”

Ss2 + (2skz − is∂z ) S

s1 − 2mhe

i2

↼∂z

⇀∂kz S

s3 − 2imae

i2

↼∂z

⇀∂kz S

s0 = 0

“

2i k0 −

k0∂t + ~k‖ · ∇‖

k0

”

Ss3 − (2iskz + s∂z ) S

s0 + 2mhe

i2

↼∂z

⇀∂kz S

s2 − 2mae

i2

↼∂z

⇀∂kz S

s1 = 0 ,

where S0 . . . S3 are 2 × 2 matrices in flavour space and s denotesthe spin.


CP violation: Chargino masses in the nMSSM

T.K., Prokopec, Schmidt, Seco (’05)

The chargino transport equations for the left/right handeddeviations from equilibrium (δS< = S< − S<

eq) are of the form

kµ∂µδS< +i

2

[

m2, δS<]

+ Γ k0 δS< = Sources(S<eq)

with the following properties

full 2x2 flavour structure (flavor basis, CP violation explicit)

sources are without ambiguities from the gradient expansion

the second term induces flavor oscillations ω = ∆m2/k


Sources for EWBG

This approach resembles two mechanisms of EWBG from formerapproachesJoyce, Prokopec, Turok (’96)

Cline, Joyce, Kainulainen (’97,’00)

Fromme, Huber (’06)

The dispersion shift source from the WKB approach:

S(2) ∼{

m†′′m − m†m′′, ∂kzS<

}

.

Carena, Moreno, Quiros, Seco, Wagner (’00)

Carena, Quiros, Seco, Wagner (’02)

Cirigliano, Profumo, Ramsey-Musolf (’06)

Cirigliano, Ramsey-Musolf, Tulin, Lee (’06)

Sources from flavor mixing effects, e.g.

S(1) ∼[

m†′m − m†m′, ∂kzS<

]

.


Situation in the MSSM

In the MSSM case the higgsino/wino mass matrix is:

m =

(

M2 g 〈H2〉g 〈H1〉 µ

)

In the MSSM the second order source is two weak to explain theobserved BAU. The first order sources show resonant behavior dueto flavor oscillations. (M2 = 200 GeV in the left plot)

-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

100 200 300 400M2

100

200

300

400

µ c

mA=200


CP violation: Chargino masses in the nMSSM

Huber, Konstandin, Prokopec, Schmidt (in preparation)

In the nMSSM the µ term contains a z-dependent complex phase

µ(z) = −λ 〈S〉 = −λφs(z) e iqs (z)

In the nMSSM second order sources dominate

The dynamical parameter µ = λ 〈S〉 leads to novel anddominating sources

Charginos are generically non-degenerate (M2 & µ)

Thin wall profiles


Phase Transition due to Tree–Level Dynamics

In the MSSM, a first order phase transition requires a light stop.Menon, Morrissey, Wagner (’04)

In a simplified model without CP violation and the thermaleffective potential ∆V T = αT 2φ2, the criterion for a first orderphase transition is

m2s <

1

λ

∣

∣

∣

∣

λ2ts

ms

− ms a

∣

∣

∣

∣

.

This criterion seems to be decisive, even if one-loop effects and CPviolation is taken into account

0

1

2

3

4

5

0 1 2 3 4 5

lhs

rhs

spectrum+phase transitionEq. (27)

0

1

2

3

4

5

0 1 2 3 4 5

lhs

rhs

spectrumEq. (27)


Thin Bubble Walls

In the MSSM, the bubble wall is rather thick lwTc ∼ 20 − 30(Moreno, Quiros, Seco (’98)).In the nMSSM, bubble walls are comparably thin, what enhancesthe second order sources in the gradient expansion

0

20

40

60

80

100

0 2 4 6 8 10 12 14

#

lw / Tc-1

# of models

For numerical methods of multi dimensional phase transitions seeKonstandin, Huber (’06).


eEDM: One Loop Contributions

One-loop contributions to the EDMs are of the following form

dd

γ

dRdR

d d

dLdL

g g

As in the MSSM, these contributions have to be suppressed bylarge sfermion masses in the first two generations. This restrictioncan be relaxed, since in the nMSSM CP violation could be small inthe broken phase.

BAU ∝ ∆Arg(µ) in the nMSSM,

∝ Arg(M2µ) in the MSSM,

EDM ∝ Arg(M2µ).


eEDM: Two Loop Contributions

Chang, Chang, Keung (’02), Pilaftsis (’02)

Potentially dangerous diagrams are of the following form with achargino in the loop

↑ γ(k, µ)

ϕ (p)

l

l + ql + p

γ(q, ν)

fL fR(q) fR

However in the nMSSM, these contributions are not as severe as inthe MSSM, since usually tan(β) ∼ O(1).


Numerical Results

A numerical analysis of the BAU leads to the following result (setspassed LEP constraints and have a first order phase transition)

0

50

100

150

200

250

300

350

0 2 4 6 8 10

#

η10

# of models# of models

0

50

100

150

200

0 2 4 6 8 10

#

η10

# of models# of models

The left (right) plot shows the generated BAU for M2 = 1 TeV(M2 = 200 GeV). 50% (63%) of the models are in accordance withobservation. The lower models fulfill the the EDM bounds with 1TeV sfermion masses, 4.8 % (6.2 %).


The Landau Pole in λ

Constructing nMSSM models without Landau pole in theparameter λ requires some tuning. Choosing 〈S〉 = 250 GeV,λ = 0.55, mA = 500 GeV, tan(β) = 2.0, Arg(ts) = 0.3 and

t1/3s = 70 GeV one obtains the following result

80

90

100

110

120

130

140

150

160

172 173 174 175 176 177 178 179 180 181 182

GeV

aλ

mHTc

φbrok

The produced BAU is in this case η10 ≈ 2.7.


Conclusions

EWBG in the MSSM possible if ...

Strong first order phase transition in the light stop scenario

η10 & 1 only for almost mass degenerate charginos (±20 GeV)with masses χ+

1/2 . 400 GeV

Additional argument needed to suppress two loop EDMs

EWBG in the nMSSM is very promising

Strong first order phase transition due to tree-level dynamics

η10 & 1 for most of the parameter space

two loop EDMs small due to tan(β) ∼ O(1)


Open questions

Landau pole in λ

“Fat Higgs”

0.7 < λ < 0.8

One-loop EDMs

Heavy sfermions (split SUSY)

Suppressed CP violation in broken phase

electroweak baryogenesis in the nmssmsusy06.physics.uci.edu/talks/6/konstandin.pdf · 2006. 6....

Documents