Electroweak Baryogenesis in the Next-to-MSSM · 2007. 3. 6. · Features of the NMSSM • 5 neutral and 1 charged scalars (3 scalar and 2 pseudoscalar when CP cons.) The lightest

Electroweak Baryogenesis in theNext-to-MSSM

SHUICHIRO TAO (KYUSHU UNIV.)

WITH KOICHI FUNAKUBO (SAGA UNIV.)

YITP - June 30, 2004 – p.1

Summary

• One-loop formulation with all CP phases

• Parameter search→ mass bounds for the charged Higgs boson

• EWPT in CP conserving case

• Higgs spectrum in CP violating case

Todo

• EWPT in CP violated case

YITP - June 30, 2004 – p.2

Features of the NMSSM

• 5 neutral and 1 charged scalars(3 scalar and 2 pseudoscalar when CP cons.)The lightest Higgs scalar H1 can escape from the bound

mHSM > 114GeV

because of small gH1ZZ , ⇒ HSM = H2 [Miller, et. al. NPB681]

M2N =

0@ M2

S M2SP

(M2SP )T M2

P

1A , M2

SP =

0BB@

0 32

sin β

0 32

cos β

− 12

− sin 2β

1CCA Ivnv

YITP - June 30, 2004 – p.3


• 5 neutral and 1 charged scalars

• CP violation at the tree level: I

I = Im(λκ∗ei(θ−2φ))

from the vacuum condition, I related to the other combinations

I ∼ Im(λAλei(θ+φ)) ∼ Im(κAκe3iφ)

• Only one combination of the phases is physical

• Our formalism is independent of prametrisation for the phases

• One can escape from the EDM constraint

YITP - June 30, 2004 – p.3


• 5 neutral and 1 charged scalars

• CP violation at the tree level: I• Pure NMSSM regime

vn → ∞ with λvn and κvn fixed ⇒ MSSM [Ellis, et. al. PRD39]

→ new features expected for vn = O(100)GeV

we focus on this regime

YITP - June 30, 2004 – p.3

NMSSM

Superpotential:

W = ydHdQDc − yuHuQU c − λNHdHu − κ

3N 3

λ〈N〉 ∼ µ in the MSSM

• Z3 symmetry← We assume it’s already broken by higher dimensional operatorIf one include tadpole term → nMSSM [Menon, et. al. hep-ph/0404184]

• All couplings are dimensionless

YITP - June 30, 2004 – p.4

NMSSM

Superpotential:

W = ydHdQDc − yuHuQU c − λNHdHu − κ

3N 3

λ〈N〉 ∼ µ in the MSSM

SUSY X soft terms:

Lsoft � −m2Nn∗n +

[λAλnΦdΦu +

κ

3Aκn

3 + h.c.]

λAλ〈n〉 ∼ µB in the MSSM

YITP - June 30, 2004 – p.4

NMSSM - Higgs sector

The tree-level Higgs potential:

V =m21Φ

†dΦd + m2

2Φ†uΦu + m2

Nn∗n − (λAλnΦdΦu +κ

3Aκn

3 + h.c.)

+g22 + g2

1

8(Φ†

dΦd − Φ†uΦu)

2 +g22

2(Φ†

dΦu)(Φ†uΦd)

+ |λ|2n∗n(Φ†dΦd + Φ†

uΦu) + |λΦdΦu + κn2|2

Order parameters:

〈Φd〉 =1√2

( vd

0

), 〈Φu〉 =

eiθ

√2

( 0

vu

), 〈n〉 =

eiφ

√2vn

YITP - June 30, 2004 – p.5

Tadpole conditions

m21 = Rλvn tan β − 1

2m2

Z cos(2β) − 12|λ|2(v2

u + v2n) − 1

2Rv2

n tanβ

m22 = Rλvn cot β +

12m2

Z cos(2β) − 12|λ|2(v2

d + v2n) − 1

2Rv2

n cotβ

m2N = Rλ

vdvu

vn+ Rκvn − 1

2|λ|2(v2

d + v2u) − |κ|2v2

n −Rvdvu

Iλ =12Ivn, Iκ = −3

2I vdvu

vn.

R = Re[λκ∗ei(θ−2ϕ)], I = Im[λκ∗ei(θ−2ϕ)],

Rλ =1√2Re[λAλei(θ+ϕ)], Iλ =

1√2Im[λAλei(θ+ϕ)],

Rκ =1√2Re[κAκei3ϕ], Iκ =

1√2Im[κAκei3ϕ].

YITP - June 30, 2004 – p.6

Constraints for the parameters

The parameters are restricted

The vacuum that break electroweak symmetryis the absolute minimum of the potential

• V (vac.) = min{V }• Positive mass2 of the all scalars

YITP - June 30, 2004 – p.7

Constrained parameter - example

Charged Higgs mass bound

• Positive mass2 of pseudo-scalar

detM2P > 0 =⇒ lower bound of mH±

mH± >3R2v2v2

n

4Rκ − 3Rv2 sin 2β+ m2

W − 12|λ|2v2

where

R = Re[λκ∗ei(θ−2ϕ)], Rκ =1√2Re[κAκei3ϕ]

� In the MSSM : m2H± = m2

W + m2A =⇒ m2

H± > m2W

YITP - June 30, 2004 – p.8




• Higgs potential at the vac. is lower than that at origin

V (vac.) < V (0) =⇒ upper bound of mH±

m2H± < 2|λ|2v2

n

1sin2 2β

+ 2|κ|2 v4n

v2

1sin2 2β

+ m2Z cot2 2β + m2

W

+ Rv2n

v2

1sin 2β

− 43Rκ

v3n

v2

1sin2 2β

.

� In the MSSM limit : this bound → ∞

YITP - June 30, 2004 – p.8





CP cons.

Pure NMSSM regime

YITP - June 30, 2004 – p.8




• Higgs potential at the vac. is lower than that at originGeVGeV

GeV

κ = −0.3 κ = 0.3

Aκ GeVAκ

tan β = 5, vn = 300GeV, ⇒ κAκ > 0 is favored

YITP - June 30, 2004 – p.8





In fact, we search the allowed parameternumerically and systematically at one-loop level

YITP - June 30, 2004 – p.8

Allowed parameters

1. The masses of the Higgs bosons are heavier than114GeV, or their coupling to Z boson is less than 0.1

2. The electroweak vacuum is the global minimum of theeffective potential

3. The mass-squared of the squarks are positive

tanβ = 2, 3, 5, 10, 20 and vn = 100, 150, 200, , , 1000 GeV

• heavy-squark scenario (mq̃, mt̃R) = (1000, 800)GeV

• light-squark scenario-1 (mq̃, mt̃R) = (1000, 10)GeV

• light-squark scenario-2 (mq̃, mt̃R) = (500, 10)GeV

YITP - June 30, 2004 – p.9

Allowed parametersA

κ(G

eV)

vn = 100GeV

mH±(GeV)

vn = 200GeV vn = 300GeV

vn = 400GeV vn = 500GeV vn = 600GeV

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

x

YITP - June 30, 2004 – p.9

Allowed parameters

crit

κ

λ

light Higgs

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

YITP - June 30, 2004 – p.9

Electroweak phase transition

• The effective potential at finite temperature

V (T ) = Vtree + ∆qV + ∆q̃V + ∆gV + ∆T V

• For the Baryogenesis

• Strong 1st order PT : vC > TC

• CPV (but now we investigate the PT in CP cons. case)

YITP - June 30, 2004 – p.10

Electroweak phase transition

tan β mH± Aκ vn λ κ mq̃ mt̃R

3 400GeV −200GeV 200GeV 0.9 −0.1 1000GeV 800GeV

H1 H2 H3 H4 H5

mass(GeV) 64.1156 145.332 187.338 436.500 5964.78

g2HiZZ 2.69889 × 10−3 0.996300 0 1.00092 × 10−3 0

T=0

v

vn

0

50

100

150

200

250

300

0 50 100 150 200 250 300

T=100GeV

0

50

100

150

200

250

300

0 50 100 150 200 250 300

T=155GeV TC=155.4GeV

0

50

100

150

200

250

300

0 50 100 150 200 250 3000

50

100

150

200

250

300

0 50 100 150 200 250 300

vC = 163.2GeV

TC = 155.4GeV

vC

example of strong 1st order in light-mass and small-coupling case

YITP - June 30, 2004 – p.10

CP violation

We used light-mass and small-coupling

• 3 scalar mixing in NMSSM [Miller, et. al. NPB681]

light Higgs mass ⇒ strong 1st order PTAnd more, CPV is nessesary for BaryogenesisBut now we weren’t include CP phase

Is there other mixing that related to CPV?

YITP - June 30, 2004 – p.11

CP violation



• The squark phase, also in MSSM [Carena, et. al. NPB586]

Scalar-pseudoscalar mixing at one-loop levelWe investigated the effect of this phase in MSSM

Im(µAt)

on the PT in previous paper [Funakubo, et. al. PTP109]

We gain light Higgs, but 1st order PT is weakened by this phaseIs there other phase?

YITP - June 30, 2004 – p.11

CP violation




• The tree level CP phase in NMSSMScalar-pseudoscalar mixing at tree levelOne can escape EDM constraint

We expect good result

YITP - June 30, 2004 – p.11

CP violation




• The tree level CP phase in NMSSM

-15000

-10000

-5000

0

5000

10000

15000

20000

0 0.2 0.4 0.6 0.8 1

m_H1^2

m_H2^2

m_H3^2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

g_H1ZZ

g_H2ZZ

g_H3ZZ

g_H4ZZ

g_H5ZZ

del_kappa/pi

YITP - June 30, 2004 – p.11

Summary

• One-loop formulation with all CP phases

• Parameter search→ mass bounds for the charged Higgs boson

• EWPT in CP conserving case

• Higgs spectrum in CP violating case

Todo

• EWPT in CP violated case

YITP - June 30, 2004 – p.12

EWPT in the NMSSM

order parameters :

vd = v cos β = y cosα cosβ

vu = v sinβ = y cos α sinβ

vn = y sinα

−→ y3-term, even at the tree level [Pietroni, NPB402]

0

50

100

150

200

250

0

50

100

150

200

-5·107

0

5·107

1·108

50

100

150

200

250

0

50

100

150

200 vn

v

V

VEV

0 50 100 150 200 250 3000

50

100

150

200

250

v

vn

YITP - June 30, 2004 – p.13

CP phase and the EDM

The phases apear in following 3 combinations

I = Im[λκ∗ei(θ−2ϕ)], Iλ =1√2Im[λAλei(θ+ϕ)], Iκ =

1√2Im[κAκei3ϕ]

from vacuum condition, these are related

Iλ =1

2Ivn, Iκ = −3

2I vdvu

vn.

EDM related phases that appear in Higgs sector are

δEDM ⊃ δλ + ϕ + θ

YITP - June 30, 2004 – p.14

CP phase and the EDM

define

δ′λ ≡ δλ + θ + ϕ, δ′κ ≡ δκ + 3ϕ

so, describe the phases explicitly

Arg(λκ∗ei(θ−2ϕ)) = δλ − δκ + θ − 2ϕ = δ′λ − δ′κ,

Arg(λAλei(θ+ϕ)) = δλ + δAλ+ θ + ϕ = δ′λ + δAλ

,

Arg(κAκei3ϕ) = δκ + δAκ + 3ϕ = δ′κ + δAκ

and

δEDM ⊃ δλ + ϕ + θ = δ′λ

Although one choose the phase δ′λ = 0, the vacuum condition can be satisfied.

YITP - June 30, 2004 – p.14

Maximal effect of the CP phase

now we have

R = Re[λκ∗], I = Im[λκ∗],

Rλ =1√2Re[λAλ], Iλ =

1√2Im[λAλ],

Rκ =1√2Re[κAκ], Iκ =

1√2Im[κAκ].

from the vacuum condition, Iλ and Iκ are described by the function of I. The inputparameters are λ, κ, δκ, mH± , δAκ = 0, π. The input mH± gives Rλ, and the inputδAκ = 0, π gives following simple relation,

Aκ = ±3λvdvu√2vn

,

so the parameter Aκ becomes dependent one. Then Rκ determined.

Note : in this situation, we can see the effect of the CP phase δκ maximally.YITP - June 30, 2004 – p.15

Documents

Electroweak Baryogenesis in the Next-to-MSSM · 2007. 3. 6. · Features of the NMSSM • 5 neutral and 1 charged scalars (3 scalar and 2 pseudoscalar when CP cons.) The lightest