Gauge Extensions of the MSSM - FermilabFigure 1: Fine tuning in the MSSM (measured by ! µ2) as a...

Gauge Extensions of the MSSM

P.B., A Delgado, D.E. Kaplan & T. Taithep-ph/0309149hep-ph/0404251

The Standard Model

The Minimal Supersymmetric Standard Model

Anagrams to“TeSted daMn hard”

(l,o)

The Standard Model


~2x as many particles


(l,o)

The Standard Model



~5x as many parameters (105 total)


(l,o)

The Standard Model



~5x as many parameters (105 total)


(l,o)


(l,o)+ 21 other letters

The Standard Model


Also Anagrams to“reSilient Theory-Scam

aMidst Madmen” +(plurmd)

The Standard Model


Well known tree-level constraint:

mh0 ≤ MZ

λ|H|4

λ is not constrained

V (φ)V (φ)

〈φ〉〈φ〉

λ ∝ g2

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160

mh°

(GeV/c2)

mA°

(GeV

/c2)

Excludedby LEP

TheoreticallyInaccessible

No Mixing

1

10

0 20 40 60 80 100 120 140

1

10

mh°

(GeV/c2)

tan!

Excludedby LEP


No Mixing

1

10

0 200 400

1

10

mA°

(GeV/c2)

tan!

Excludedby LEP

No Mixing

1

10

0 200 400

1

10

mH+-

(GeV/c2)

tan!

Excludedby LEP

No Mixing

Figure 5: MSSM parameter exclusions, at 95% c.l., for the no-mixing benchmark scenario, withmt = 179.3 GeV/c2. The figure shows the excluded and theoretically inaccessible regions asfunctions of the MSSM parameters in four projections: (mh, mA), (mh, tan β),(mA, tan β) and(mH+ , tanβ). The dashed lines indicate the boundaries of the regions expected to be excludedon the basis of Monte Carlo simulations with no signal.

20

LEP Working Group, Summer ‘04

LEP-Bounds on the MSSM

Large radiative corrections from the stop sector help the MSSM evade the direct search bounds on mh0

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160

mh°

(GeV/c2)

mA°

(GeV

/c2)

Excludedby LEP


mh°

-max

1

10

0 20 40 60 80 100 120 140

1

10

mh°

(GeV/c2)

tan!

Excludedby LEP


mh°

-max

1

10

0 200 400

1

10

mA°

(GeV/c2)

tan!

Excludedby LEP

mh°

-max

1

10

0 200 400

1

10

mH+-

(GeV/c2)

tan!

Excludedby LEP

mh°

-max

Figure 4: The MSSM exclusions, at 95% c.l., for the mh-max benchmark scenario, withmt = 179.3 GeV/c2. The figure shows the excluded and theoretically inaccessible regionsas functions of the MSSM parameters in four projections: (mh, mA), (mh, tanβ),(mA, tanβ)and (mH+ , tan β). The dashed lines indicate the boundaries of the regions expected to beexcluded on the basis of Monte Carlo simulations with no signal.

19

LEP Working Group, Summer ‘04

LEP-Bounds on the MSSM

More parameter space available in the Max-Mixing scenario

● Any increase in comes at a price Dimensional Analysis

λ ∝ ln

(m2

S

m2t

)

δm2

φ ∝ m2

S

mh0

MAX-mixing?

● Two-possibilities: (Kane, Nelson, L. Wang, T. Wang) the Higgs sector has escaped detection... fine-tuned at ~.1 %, difficult to explain SUSY sectorMax-Mixing scenario, CP-even Higgs soon...still fine-tuned, Max-mixing is hard to generate

100 110 120 130 1401

10

100

1000

mh (GeV)

!µ2

Figure 1: Fine tuning in the MSSM (measured by ∆µ2 ) as a function of the Higgs mass (in GeV)for tanβ = 10.

tan β = 10 (such large value of tan β minimizes the fine tuning, as discussed above).

We only include the dominant one-loop correction to mh, as shown in eq. (4), and make

the simplifying assumption that the soft parameters are universal at the GUT scale.

Although the fine tuning can be made smaller in non-universal cases, figure 1 shows the

typical size of ∆µ2 in the MSSM. As expected from the previous discussions, ∆µ2 grows

exponentially for increasing mh. The dependence of ∆µ2 with tanβ is shown in fig. 2

for mh at the LEP bound5, mh = 115 GeV (the optimal choice for the fine tuning).

The curve for ∆µ2 increases exponentially for decreasing cos2 2β, again as expected.

This curve can be interpreted as a LEP lower bound on the MSSM fine tuning.

Finally, fig. 3 shows contour lines of constant ∆µ2 in the (m, tanβ) plane, where m

is the universal soft mass at ΛUV . We also plot dashed contour lines of constant mh

and the LEP lower bound on mh. Again, it is clear how the fine tuning is greater for

smaller tanβ and how it grows, together with mh, for larger m. The upper horizontal

line and the mh = 115 GeV contour line correspond to figs. 1 and 2 respectively. It is

instructive to examine the behaviour of the lines of constant ∆µ2 along which m and5With our choice of universal soft masses, the mass of the pseudoscalar Higgs is generically large.

In that case the LEP bound reduces to that in the SM: mh>∼ 115 GeV.

6

.1% cancellation

10% cancellation

1% cancellation

(Casas, Espinosa, Hidalgo)

Can we raise ? mh0

● Need a new, dimensionless, tree-level parameter (Haber & Sher, Espinosa & Quiros, ...)

New Gauge Groups New Matter

A gluon loop diagram

1

g2

new, Wµ

new

∝ g2

new ∝ λ2

S

λSSHuHd

MSSM+λ

′H

4

MSSM

+ New StuffSUSY Breaking, . . .

Energy Weak TeV 10-100 TeV

Physics 170 Final Exam, December 9, 1999

λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1

Symmetry Breaking

〈Σ〉

●

. . .

. . .

Req R1 R2 RN

● mW ′ ∼ 〈Σ〉

SU(2)W

1

g2=

1

g21

+1

g22


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1

●

Symmetry Breaking

〈Σ〉

SU(2)W

●

. . .

. . .

Req R1 R2 RN

● Potentially large quartic as

Hu, Hd f

Hu, Hd f

mW ′ ∼ 〈Σ〉

1

g2=

1

g21

+1

g22

g1 → ∞, g → g2

Wµ→ A

µ

2

Non-decoupling D-terms

● Non-decoupling limit, must have stable ratio of scales, communicates with Higgs sector at two-loops only.D-flat breaking, so the Higgs sector is still light

m2

Σ > 〈Σ〉

● The low-energy theory ‘remembers’ its origins

●


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

1


1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4

(2)

mh0 ∼ λv

2(3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g

2+ g

′2

8→

∆g2+ g

′2

8(10)

H, H (11)

〈Σ〉 = u

( 1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W′, Z

′with m ∼ u

2 (g2

1 + g2

2) (14)

m2

Σ|Σ|2

(15)

λ ∼∆g

2+ g

′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2∼ g

2(18)

m2

Σ + 〈Σ〉 → ∆g2∼ g

2

1 (19)

H H Σ E + mΣ (20)

1

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV, MV = 4.5TeV (21)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (22)

∆ ≡ 7, mh0 ≤ 210 GeV (23)

2


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1no effect


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ* 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ* 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

1

big effect!


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ * 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ + 〈Σ〉 → ∆g2 ∼ g2

1 (19)

H H Σ E + mΣ (20)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

1

Σ3rd Gen.

1st Gen.

2nd Gen.

“Topflavor”

Asymptotic Freedom

● different flavor structure to prevent non-perturbative physics before the GUT scale

● Third generation must be included with the Higgs to keep the top heavyUVIR

g1

A New Fine-Tuning?

• Now that SU(2)1 is asymptotically free, we canraise g1 at u to its perturbative limit α1(u) ∼ 1.

• s2 " .05, u ≥ 3.1 TeV.

• This allows∆ " 20 ⇒ mh " 350 GeV!

• However, to have g1(u) maximal, we are effectivelytuning two independent parameters: u and the

confinement scale Λ1.

• A (heuristic) way to quantify the fine-tuning is to ask

how close g1(MG ) has been tuned to its criticalvalue in order to have a given Higgs mass:

0

50

100

150

200

250

300

350

400

450

500

10-2

10-1

MSSM Limit

!g(MG

) / gc(M

G)

mh

(GeV

)

Sample Point:H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≡ 7, mh0 ≤ 210 GeV (24)

2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≡ 7, mh0 ≤ 210 GeV (24)

2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)

2

A New Mass Bound for SUSY

● apparent SU(5) unification could be explained by SU(5) x SU(5)

Gauge Coupling Unification

Breaks to SU(3) x SU(2) x SU(2) x U(1) at a high-scale, leaves behind spectator fields to give the observed coupling constants

SU (3) × SU (2) × U (1) [SU (3)]3

ψ, hΣc

[SU(3)]3 [SU(3)]3Σ

Figure 2: A theory space diagram of the product unification model discussed in the text. Inaddition to the usual SU(3)×SU(2)×U(1), there is an SU(3)3 gauge group. The Σ, Σc fieldsare bi-fundamentals connecting these groups. The three generations of matter, denoted ψ,and the pair of Higgs doublets, h, are charged under SU(3) × SU(2) × U(1). We considera model where SU(3) × SU(2) × U(1) unifies into the trinified group SU(3)3 at the GUTscale.

TeV 710 TeV

MBreak

MSUSY

MGUT

G=[ SU(3) x SU(2) x U(1) ]SM 2 GUTMSSM

MSSM x MSSM

Figure 3: The minimal accelerated unification model, with N = 2, where two copies of theMSSM gauge group break down to the diagonal subgroup at the TeV scale.

reasons are two–fold. First, this representation is the smallest possible. In trinification,the Σ fields fall into representations of SU(3) that only contribute ∆b = 3 to each betafunction. In SU(5) and SO(10) unification, the link fields add 5 and 10 to ∆b, respectively.Thus trinification contributes the least possible amount to the gauge coupling beta functions,which helps keep the theory perturbative. Second, this model is closely related to the minimalaccelerated unification model, which we now discuss.

2.3.2 Accelerated Unification

In accelerated unification models [10], the Standard Model gauge group, GSM, is the remnantof an enlarged group, GN

SM, that breaks to the diagonal subgroup at the TeV scale. Thepresence of extra matter changes the gauge coupling beta functions, causing the theories tounify at a much lower scale (see Fig. 3).

The gauge and matter content of the N = 2 trinified model is summarized in Fig. 4.There are two copies of the low energy gauge group, which we denote [SU(3)C × SU(2)L ×

7

● Accelerated Unification or Product Unification models can also increase the Higgs mass (40-50 GeV at Tree-level) without requiring ‘Unifons’.

Maloney, Pierce, Wacker


1


1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1

3rd Gen.

Σ1st & 2nd Gen.

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)

W ⊃ µ′H ′H ′ + λcQccH ′ + λfHΣH ′ + λfH′ΣH (26)

W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)

2

● oblique corrections to

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)

W Z MW MZ Z ′, W ′ Z ′ W ′ W, Z 〈H〉 MW , MZ (31)

W Z MW MZ Z ′, W ′ Z ′ W ′ W, Z 〈H〉 MW , MZ (32)

2

● non-oblique corrections to

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)

W Z MW MZ Z ′, W ′ Z ′ W ′ W, Z 〈H〉 MW , MZ GF fL (31)


2

● non-universal 3rd Generation couplings!

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

2

● vertex corrections

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)



2

Precision Corrections

Precision Electroweak Constraints

• The dominant corrections are:– Oblique corrections to W and Z masses.

– Vertex corrections to fL .

– Non-oblique contributions to GF .

– Small triplet VEV in Σ as before.

– Non-universal corrections to 3rd family couplings!

• Strongest constraints are from τ and b couplings.

• Bounds are a function of s2:

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MV

u

c2

M (T

eV

)

●


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

m2

Σ|Σ|2 (15)

λ ∼∆g2 + g′2

8(16)

g1 g2 (17)

m2

Σ* 〈Σ〉 → ∆g2 ∼ g2 (18)

m2

Σ+ 〈Σ〉 → ∆g2 ∼ g2

1(19)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1

LargeSmall


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1

LargeSmall


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1

Lower Bounds as a function of


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1

Precision Corrections

Phenomenology● Lightest Higgs mass which violates the MSSM bound (and LEP-II bound) for a range of Tan (β)● Consistently altered mass sum rules (usual VVh couplings):

m2

h = ∆′m2

Z cos2(2β)m2

H = m2

A + ∆′m2

Z sin2(2β)m2

H+ = m2

A + ∆m2

W

● WW and ZZ Higgs decay modes can dominate

● Heavy Vector bosons which couple preferentially to the third generation

300 fb−1100 fb−130 fb−110 fb−1

perturb. limit

top-flavor see-saw

0.05

orbifolded L-R

perturbative limit

top-flavor

MW ′ (TeV)

g′/g

SM

10987654321

10

1

0.1 300 fb−1100 fb−130 fb−110 fb−1

0.05

f <4f <2f <1 TeV

MW ′ (TeV)

g′/g

SM

10987654321

10

1

0.1

(a) (b)

Figure 2: 95% confidence-level exclusion reach as a function of W ′ mass at the LHC for arbitrary g′/gSM

.Superimposed are the predictions of (a) various classes of perturbative models, and (b) Little Higgs. The shortdot-dashed contours denote the maximally allowed parameter space for a given f . The solid contours denote theperturbative parameter space (αi = g2

i /(4π) < 1/π ≈ 0.32).

space shown by the triple-dashed contours of Fig. 2b.When examining Fig. 2b it should be questioned whether the theory is really perturbative

if one of the couplings is√

4π. A more reasonable perturbative bound of αi = g2i /(4π) < 1/π ≈

0.32 is shown via solid contours in Fig. 2b. The figure stops for f = 4 TeV, since it becomesincreasing unnatural for f to be larger than 1 TeV in Little Higgs scenarios. 2 However, even fas large as 6–8 TeV can be mostly covered in the central perturbative region (g′/g

SM∼ 1)which

is favored for more complicated models. The conclusion to be drawn is that the W ′ bosonsappearing in Little Higgs models should either be seen or excluded in the first year of runningat the LHC.

Acknowledgments

This work is supported by the U. S. Department of Energy under contract No. DE-AC02-76CH03000.

References

1. M. Schmaltz, Nucl. Phys. Proc. Suppl. 117, 40 (2003); preprints on new variations on theLittle Higgs models continue to appear monthly.

2. J. L. Hewett, F. J. Petriello and T. G. Rizzo, arXiv:hep-ph/0211218.3. T. Han, H. E. Logan, B. McElrath and L. T. Wang, Phys. Rev. D 67, 095004 (2003).4. Zack Sullivan, in production.5. Zack Sullivan, Phys. Rev. D 66, 075011 (2002).6. D. Acosta et al. [CDF Collaboration], Phys. Rev. Lett. 90, 081802 (2003).7. T. Affolder et al. [CDF Collaboration], Phys. Rev. Lett. 87, 231803 (2001).8. T. Sjostrand, L. Lonnblad and S. Mrenna, arXiv:hep-ph/0108264.9. J. S. Conway et al., in Proceedings of the Workshop on Physics at Run II – Supersymme-

try/Higgs, Fermilab, 1998, hep-ph/0010338, p. 39.10. ATLAS Collaboration, “ATLAS Detector and Physics Performance Technical Design Re-

port, Volume I,” CERN-LHCC-99-14, ATLAS-TDR-14, May 25, 1999.11. T. Stelzer, Z. Sullivan and S. Willenbrock, Phys. Rev. D 58, 094021 (1998).12. B. W. Harris, E. Laenen, L. Phaf, Z. Sullivan and S. Weinzierl, Phys. Rev. D 66, 054024

(2002).13. D. J. Muller and S. Nandi, Phys. Lett. B 383, 345 (1996).14. E. Malkawi, T. Tait and C. P. Yuan, Phys. Lett. B 385, 304 (1996).15. H. J. He, T. Tait and C. P. Yuan, Phys. Rev. D 62, 011702 (2000).16. Y. Mimura and S. Nandi, Phys. Lett. B 538, 406 (2002).

Z. Sullivan

+ Singlet

Singlet + Gauge Extensions

New Matter


1

∝ λ2

S

λSSHuHd

MSSM+λ

′H

4

MSSMSUSY Breaking, . . .

Energy Weak TeV 10-100 TeV

● No SM field will do● Prefers Tan (β) ~ 1● Bounded by perturbativity

UVIR

λS

+ Singlet

← 135 GeV Espinosa & Quiros

Figure 1: Theoretical upper limits on the lightest Higgs-boson mass as a function of the cutoffscale Λ of the NMSSM Higgs sector.

λ and κ that the value of λ always decreases when it is run down from a high scale:

dλ

d lnµ=

λ

16π2

(4λ2 + 2κ2 + 3y2

t + 3y2b + y2

τ − 3g2 − g′2), (3)

dκ

d lnµ=

6κ

16π2

(λ2 + κ2

), (4)

where yb and yτ are the bottom and tau Yukawa couplings, and g and g′ are the SU(2)L and

U(1)Y gauge couplings. For instance, if we assume that λ becomes non-perturbative at a scale Λ

above the weak scale but below ΛGUT, i.e. λ(Λ) = 2π, then a lower value of Λ results in a higher

value of λ when it is run down to the weak scale. This, therefore, results in a higher upper

limit on the mass of the lightest Higgs boson. We have illustrated this in Fig. 1, where we have

taken κ(Λ) = 0 and λ(Λ) = 2π, which maximizes the Higgs-boson mass, and chosen the value of

tan β such that the largest Higgs mass is obtained for each value of Λ. The improvement gained

by lowering the scale at which the Higgs-singlet sector becomes non-perturbative is clear. A

scale of Λ = 104 GeV, for example, results in an upper Higgs mass bound of approximately

340 GeV, and lower values of Λ give even larger Higgs-boson masses. Although there are many

uncertainties in the Higgs mass values obtained in this way, for example those arising from

effects of non-zero values for κ, one-loop effects involving λ, higher-order effects and so on,

we expect that we can still obtain the lightest Higgs-boson mass as large as about 300 GeV,

4

Birkedal, Chacko, Nomura

Mass Bounds

● λS ≤ 2π

Intermediate Physics

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

104

106

108

1010

1012

1014

1016

g1

g3

tan ! = 10

yt

"S

u = 3 TeV

µ (GeV)

g

Figure 2: Renormalization group flow of couplings for the model, with g1(3 TeV) = 2,λS(200 GeV) = 0.8 and tanβ = 10.

messengers M,M via the superpotential coupling SMM . The result is a one-loop (instead

of two loop) squared soft-mass for S which is roughly 4π/α larger than the squared MSSM

soft-masses, as desired.

Another option which makes use of extra dimensions is gaugino mediation [18]: SUSY

breaks on a sequestered brane, couples directly to all bulk fields (including the gauginos),

and then communicates to the MSSM matter on a visible brane through the gauginos.

Gaugino masses arise from a superpotential term of the form XWαWα, while bulk scalar

fields receive squared masses from the Kahler term X†XS†S. Therefore, if we put the singlet

in the bulk, its mass receives an enhancement of√

ML relative to the gaugino masses. Here,

L is the length of the extra-dimension, while M is some fundamental scale. Rough bounds

from flavor constraints and naive dimensional analysis predict 10 ! ML ! 100, resulting

in the proper enhancement for the soft-mass of S.

3. Perturbativity Constraints

The enhanced quartic effect in the non-decoupling limit is strongly limited by a desire for

– 5 –

〈Σ〉

Harnik, Kribs, Larson, Murayama“Fat Higgs”

Confining Dynamics or

or

Chang, Kilic, Mahbubani

Product Groups

AdS 5-d Structure Birkedal, Chacko, Nomura

Figure 1: Theoretical upper limits on the lightest Higgs-boson mass as a function of the cutoffscale Λ of the NMSSM Higgs sector.

λ and κ that the value of λ always decreases when it is run down from a high scale:

dλ

d lnµ=

λ

16π2

(4λ2 + 2κ2 + 3y2

t + 3y2b + y2

τ − 3g2 − g′2), (3)

dκ

d lnµ=

6κ

16π2

(λ2 + κ2

), (4)

where yb and yτ are the bottom and tau Yukawa couplings, and g and g′ are the SU(2)L and

U(1)Y gauge couplings. For instance, if we assume that λ becomes non-perturbative at a scale Λ

above the weak scale but below ΛGUT, i.e. λ(Λ) = 2π, then a lower value of Λ results in a higher

value of λ when it is run down to the weak scale. This, therefore, results in a higher upper

limit on the mass of the lightest Higgs boson. We have illustrated this in Fig. 1, where we have

taken κ(Λ) = 0 and λ(Λ) = 2π, which maximizes the Higgs-boson mass, and chosen the value of

tan β such that the largest Higgs mass is obtained for each value of Λ. The improvement gained

by lowering the scale at which the Higgs-singlet sector becomes non-perturbative is clear. A

scale of Λ = 104 GeV, for example, results in an upper Higgs mass bound of approximately

340 GeV, and lower values of Λ give even larger Higgs-boson masses. Although there are many

uncertainties in the Higgs mass values obtained in this way, for example those arising from

effects of non-zero values for κ, one-loop effects involving λ, higher-order effects and so on,

we expect that we can still obtain the lightest Higgs-boson mass as large as about 300 GeV,

4

● λS ≤ 2π

← 135 GeV Espinosa & Quiros

▲

✖

♥

♥ P.B, Delgado, Kaplan, Tait

✖ Harnik, Kribs, Larson, Murayama ▲ Birkedal, Chacko, Nomura (AdS)

Pure D-terms →

Mass Bounds

← Triplet (Y=-1)

Phenomenology● Higgs mass at odds with the MSSM, altered mass-sum rules, preference for low Tan (β)

● Smaller values of Tan (β) allowed by perturbativity (enhanced top couplings!)

0.5

0.6

0.70.80.9

1

2

3

4

5

6

789

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.5

0.6

0.70.80.9

1

2

3

4

5

6

789

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

g1 = 1.2

g1 = 1.5

g1 = 2

g1 = 3

Not Gauge

Extended

!S (200 GeV)

tan

"

Figure 3: Allowed regions of tanβ and λS(200 GeV) for given values of the new SU(2) couplingg1(3 TeV). For the indicated regions, all couplings in the model remain perturbative during 1-loopRG evolution up to the GUT scale. From left to right, the regions are: the non-gauge extendedmodel of [6], and our gauge extension with g1(u) =1.2, 1.5, 2, and 3.

perturbative unification. Both couplings λS and yt feed into each other’s renormalization

group equations (RGE’s) with positive coefficients. If either λS or yt is large at a low scale

(required for mh0 > MZ , or low tan β, respectively), non-perturbative physics is reached

long before MGUT .

Both of these problems are largely ameliorated by the presence of new, relatively strong,

gauge interactions, which drive both yt and λS down at large scales, owing to the Higgs and

top quark participation in the stronger group. The dominant terms in the renormalization

group equations at one loop (including the spectator matter described above, necessary for

gauge coupling unification and Yukawa interactions for the first two generations) are1,

dg1

dt= −2

g31

16π2

1These RGE’s are valid above the SU(2) × SU(2) breaking scale, u. Below u, we use the RGE’s

appropriate for the broken phase.

– 6 –

Light Charged Higgs

100

120

140

160

180

200

220

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

tan ! = 0.8

tan ! = 1

tan ! = 2

"S

MH± (G

eV

)

Figure 5: The charged Higgs mass versus λS for different values of tanβ. The area below each linerepresents the parameter space in which the charged Higgs is the lightest Higgs in the spectrum.The lines themselves represent the charged Higgs masses for which the charged and lightest CP-evenHiggs bosons are degenerate. The soft parameters were chosen as mQ = mU = µ = 200 GeV andAt = 200 GeV, and there is moderate dependence on them. For example, if the above parametersare set at 500 GeV, the values of the lines at λ = 1.2 are (from lowest to highest tanβ) 162, 210,and 145 GeV.

with masses up to about 400 GeV in the low tan β region [28] with 100 fb−1. Another

production mechanism is through an off-shell W boson, qq′ → W ∗ → H+A0 [29], leading

to final states with tbbb. Two of the bottoms reconstruct mA0, and thus typically have

much higher energies than bb from gluon splitting.

Note that despite the absence of additional weak scale Higgs bosons, this theory mod-

ifies the usual MSSM m2H± −m2

A = m2W mass relation. Even when the spectrum is roughly

consistent with the MSSM (say, for modest λS and large mA such that mh0 has mass just

above the LEP bound and h0 has largely SM-like couplings), this fact can be tested at the

LHC through the associated production of H±A0 provided the masses are less than about

300 GeV [29]. As with any extension of the MSSM that affects the Higgs quartic, one could

also combine a measurement of the light CP even Higgs mass with precision measurements

– 10 –

● Spectrum where the charged Higgs is the lightest state in the Higgs spectrum

m2

H+ = m2

A + ∆m2

W − λSv2

Light Charged Higgs

● Possibility of rare top decays

3 Assorted Tree-Level Diagrams

four photon interaction

2 fermions emit a scalar

1 fermion decays to a scalar and a fermion

6

tb

H+

tanβ < 1, mH+ < 120 GeV

mH+ < 160 GeV

Tevatron (2 ) can exclude (Run II Higgs Report, Carena et al.)

fb−1

LHC (100 ) can exclude (CERN Top Yellow Book, Beneke et al.)

fb−1

● Modifies branching ratios to jets vs. leptons compared to the usual top decay

Conclusions

● ‘The MSSM’=’Resilient-scam amidst theory madmen’● Max-Mixing and %-ish fine-tuning are needed to evade LEP-II bounds on the MSSM!

● Signals of SUSY will likely include low-energy evidence of additional structure beyond the MSSM--

massive vectors bosonsaltered mass relations in the Higgs Sectorlight, charged Higgs fields

● Will some other successes of the MSSM (radiative EWSB, simple GUT unification) be lost?

Supplemental Slides

0

20

40

60

80

100

120

140

160

180

200

60 70 80 90 100 110 1200

20

40

60

80

100

120

140

160

180

200

mh°

(GeV/c2)

mA°

(GeV

/c2)

!1!

!2!

<1!


mh°

-max

Figure 3: Contours of the p-value 1−CLb for the mh-max benchmark scenario, in the (mh, mA)projection of the MSSM parameter space. The theoretical limits are indicated. In the whiteregions labeled < 1σ the data deviate by less than one standard deviation from the expectationbased on Monte Carlo simulation with no signal. Similarly, in the dark-grey (green) regionslabeled ≥ 1σ the deviation is between one and two standard deviations and in the light-grey(blue) regions labeled ≥ 2σ it is between two and three standard deviations. The dashed linerepresents the upper edge of the region excluded at 95% c.l. by this search (compare to Figure 4,upper left).

18

0

0.5

1

1.5

2

2.5

3

103

104

105

106

107

108

10910

1010

1110

1210

1310

1410

15

gY

gC

g1

g2

yt

µ (GeV)

g

Gauge Coupling Unification

Figure 12: The radiatively corrected light CP-even Higgs mass is plotted (a) as a function of Xt, where Xt ≡ At−µ cotβ,for Mt = 174.3 GeV and two choices of tan β = 3 and 30, and (b) as a function of tanβ, for the maximal mixing [upperband] and minimal mixing [lower band] benchmark cases. In (b), the central value of the shaded bands corresponds toMt = 175 GeV, while the upper [lower] edge of the bands correspond to increasing [decreasing] Mt by 5 GeV. In both(a) and (b), mA = 1 TeV and the diagonal soft squark squared-masses are assumed to be degenerate: MSUSY ≡ MQ =MU = MD = 1 TeV.

prediction for mh corresponds to its theoretical upper bound, mh = mZ . Including radiative corrections,the theoretical upper bound is increased. The dominant effect arises from an incomplete cancellation12

of the top-quark and top-squark loops (these effects cancel in the exact supersymmetric limit). Thequalitative behavior of the radiative corrections can be most easily seen in the large top squark masslimit, where in addition, the splitting of the two diagonal entries and the off-diagonal entry of thetop-squark squared-mass matrix are both small in comparison to the average of the two top-squarksquared-masses:

M2S ≡ 1

2(M2t1

+ M2t2

) . (38)

In this case, the upper bound on the lightest CP-even Higgs mass is approximately given by

m2h

<∼ m2Z +

3g2m4t

8π2m2W

[ln

(M2

S

m2t

)+

X2t

M2S

(1 − X2

t

12M2S

)]. (39)

12In certain regions of parameter space (corresponding to large tanβ and large values of µ), the incomplete cancellationof the bottom-quark and bottom-squark loops can be as important as the corresponding top sector contributions. Forsimplicity, we ignore this contribution in eq. (39).

31

Max vs. Min mixing

80

90

100

110

120

130

140

501052

mh

Tan( !)

Tree-level mass

80

90

100

110

120

130

140

501052

mh

Tan( !)

Min mixing with 500 GeV stop

80

90

100

110

120

130

140

501052

mh

Tan( !)

Max mixing with 500 GeV stop

Figure 12: The radiatively corrected light CP-even Higgs mass is plotted (a) as a function of Xt, where Xt ≡ At−µ cotβ,for Mt = 174.3 GeV and two choices of tan β = 3 and 30, and (b) as a function of tanβ, for the maximal mixing [upperband] and minimal mixing [lower band] benchmark cases. In (b), the central value of the shaded bands corresponds toMt = 175 GeV, while the upper [lower] edge of the bands correspond to increasing [decreasing] Mt by 5 GeV. In both(a) and (b), mA = 1 TeV and the diagonal soft squark squared-masses are assumed to be degenerate: MSUSY ≡ MQ =MU = MD = 1 TeV.

prediction for mh corresponds to its theoretical upper bound, mh = mZ . Including radiative corrections,the theoretical upper bound is increased. The dominant effect arises from an incomplete cancellation12

of the top-quark and top-squark loops (these effects cancel in the exact supersymmetric limit). Thequalitative behavior of the radiative corrections can be most easily seen in the large top squark masslimit, where in addition, the splitting of the two diagonal entries and the off-diagonal entry of thetop-squark squared-mass matrix are both small in comparison to the average of the two top-squarksquared-masses:

M2S ≡ 1

2(M2t1

+ M2t2

) . (38)

In this case, the upper bound on the lightest CP-even Higgs mass is approximately given by

m2h

<∼ m2Z +

3g2m4t

8π2m2W

[ln

(M2

S

m2t

)+

X2t

M2S

(1 − X2

t

12M2S

)]. (39)

12In certain regions of parameter space (corresponding to large tanβ and large values of µ), the incomplete cancellationof the bottom-quark and bottom-squark loops can be as important as the corresponding top sector contributions. Forsimplicity, we ignore this contribution in eq. (39).

31

Carena, Haber

Higher order max vs min mixing

Figure 13: The radiatively corrected light CP-even Higgs mass is plotted as a function of MSUSY ≡ MQ = MU = MD,for Mt = 174.3 GeV, mA = 1 TeV and two choices of tanβ = 3 and tanβ = 30. Maximal mixing and minimal mixingare defined according to the value of Xt that yields the maximal and minimal Higgs mass as shown in fig. 12(a).

radiative corrections. The dominant corrections to M2, coming from the one-loop top and bottomquark and top and bottom squark contributions plus the two-loop leading logarithmic contributions,are given to O(h4

t , h4b) by [149,150,152]

δM211 " −µ2x2

t

h4t v

2

32π2s2

β

[1 + c11 ln

(M2

S

m2t

)]− µ2a2

b

h4bv

2

32π2s2

β

[1 + c12 ln

(M2

S

m2t

)], (41)

δM212 " −µxt

h4t v

2

32π2(6 − xtat)s

2β

[1 + c31 ln

(M2

S

m2t

)]+ µ3ab

h4bv

2

32π2s2

β

[1 + c32 ln

(M2

S

m2t

)], (42)

δM222 "

3h4tv

2

8π2s2

β ln

(M2

S

m2t

) [1 + 1

2c21 ln

(M2

S

m2t

)]

+h4

tv2

32π2s2

βxtat(12 − xtat)

[1 + c21 ln

(M2

S

m2t

)]− µ4 h4

bv2

32π2s2

β

[1 + c22 ln

(M2

S

m2t

)], (43)

where sβ ≡ sin β, cβ ≡ cos β, and the coefficients cij are:

cij ≡ tijh2t + bijh2

b − 32g23

32π2, (44)

33

Carena, Haber (2002)

Dependence on soft masses

0

0.5

1

1.5

2

2.5

3

103

104

105

106

107

108

10910

1010

1110

1210

1310

1410

15

gY

gC

g1

g2

yt

µ (GeV)

g

Unification

● Diagonal SU(5) Unification

Espinosa, Quiros (1998)

“MSSM+S” Limits

0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38

10 10 10 10 10 103 6 9 12 15 17

100

200

300

400

500

600

700

! [GeV]

m[G

eV

]

sin2"W

(!)

NMSSM

# S = 500 GeV

tan

tan

tan

tan

tan

$

$

$

$

$

= 1

= 2

= 3

= 5

= 30

h

FIG. 4: The five lines are for the same values of tan β in the NMSSM. The λs coupling of the

superpotential λsSHuHd term is assumed to be at its maximum allowed value without blowing up

before the scale Λ (λs < 5.1). Since sin2 θW (Λ) correlates directly with Λ we provide the sin2 θW (Λ)

values on the upper axis.

very different. The result, with ∆S = 500 GeV is

sin2 θW = 1/4 =⇒ Λ = 37 TeV, mh < 350 GeV,

sin2 θW = 3/8 =⇒ Λ " 2 × 1016 GeV, mh < 120 GeV.

We plot in figs. 5 and 6 the lightest Higgs mass in the NMSSM as a function of ∆S

confining ourselves to the two scenarios sin2 θW = 1/4 (Λ ∼ 8− 110 TeV) and sin2 θW = 3/8

(Λ = 2×1016 GeV). For low values of tan β the Higgs mass prediction is very well separated

between the two theories because the λs contribution is not suppressed much by sin2 2β

and the difference between the λs(∆S) allowed such that λs is still perturbative at Λ is

dramatically different for Λ ∼ 10 TeV (λs(∆S = 500 GeV) ∼ 2 allowed) and Λ = 2×1016 GeV

(λs(∆S = 500 GeV) ∼ 0.7 allowed). However, as we go to higher values of tanβ the Higgs

mass has very little dependence on the λ2s sin2 2β term since it is suppressed by 1/ tanβ at

high tan β. For that reason, the two tan β = 30 lines are very nearly on top of each other in

14

Tobe, Wells 2002

“MSSM+S” Limits

● add an extra pair of Higgs-like doublets

● include mixing terms in the Superpotential

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

2

Large mass term

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

2

‘Yukawa’ Feeding terms

● at low energies:

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

2

Yukawa’s with natural suppression

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

2


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

1

3rd Gen.

Σ1st & 2nd Gen.

H H Σ E < mΣ (20)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (21)

MV = 4.5TeV (22)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (23)

∆ ≈ 7, mh0 ≤ 210 GeV (24)

mh0 < ∼ 350 GeV (25)


W ⊃λfλc〈Σ〉

µ′QccH (27)

H ′ H ′ (28)

SU(2)2 SU(2)W (29)

H ′, H ′ (30)

2

Yukawas

Recall, we require

m2

Σ! 〈Σ〉 → ∆g2 ∼ g2 (19)

m2

Σ & 〈Σ〉 → ∆g2 ∼ g2

1 (20)

H H Σ E < mΣ (21)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (22)

MV = 4.5TeV (23)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (24)

∆ ≈ 7, mh0 ≤ 210 GeV (25)

mh0 < ∼ 350 GeV (26)


W ⊃λfλc〈Σ〉

µ′QccH (28)

H ′ H ′ (29)

SU(2)2 SU(2)W (30)

H ′, H ′ (31)



mΣ & 〈Σ〉 (34)

m2

H , m2

H(35)

L ⊃ Da(Tr

(Σ†σaΣ

)+ H†σaH − HσaH

†)

(36)

Tr σa = 0 (37)

2

1 TeV10 TeV

Two sources of fine-tuning:

● Below , a hard breaking of SUSY in the gauge sector from ● UV sensitive logarithmic contributions of into

m2

Σ! 〈Σ〉 → ∆g2 ∼ g2 (19)

m2

Σ & 〈Σ〉 → ∆g2 ∼ g2

1 (20)

H H Σ E < mΣ (21)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (22)

MV = 4.5TeV (23)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (24)

∆ ≈ 7, mh0 ≤ 210 GeV (25)

mh0 < ∼ 350 GeV (26)


W ⊃λfλc〈Σ〉

µ′QccH (28)

H ′ H ′ (29)

SU(2)2 SU(2)W (30)

H ′, H ′ (31)



mΣ & 〈Σ〉 (34)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (35)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (36)

L ⊃ Da(Tr

(Σ†σaΣ


†)

(37)

2

m2

Σ! 〈Σ〉 → ∆g2 ∼ g2 (19)

m2

Σ & 〈Σ〉 → ∆g2 ∼ g2

1 (20)

H H Σ E < mΣ (21)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (22)

MV = 4.5TeV (23)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (24)

∆ ≈ 7, mh0 ≤ 210 GeV (25)

mh0 < ∼ 350 GeV (26)


W ⊃λfλc〈Σ〉

µ′QccH (28)

H ′ H ′ (29)

SU(2)2 SU(2)W (30)

H ′, H ′ (31)



mΣ & 〈Σ〉 (34)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (35)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (36)

L ⊃ Da(Tr

(Σ†σaΣ


†)

(37)

2

m2

Σ! 〈Σ〉 → ∆g2 ∼ g2 (19)

m2

Σ & 〈Σ〉 → ∆g2 ∼ g2

1 (20)

H H Σ E < mΣ (21)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (22)

MV = 4.5TeV (23)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (24)

∆ ≈ 7, mh0 ≤ 210 GeV (25)

mh0 < ∼ 350 GeV (26)


W ⊃λfλc〈Σ〉

µ′QccH (28)

H ′ H ′ (29)

SU(2)2 SU(2)W (30)

H ′, H ′ (31)



mΣ & 〈Σ〉 (34)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (35)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (36)

L ⊃ Da(Tr

(Σ†σaΣ


†)

(37)

2

m2

Σ! 〈Σ〉 → ∆g2 ∼ g2 (19)

m2

Σ & 〈Σ〉 → ∆g2 ∼ g2

1 (20)

H H Σ E < mΣ (21)

〈Σ〉 = 2.5 TeV, mΣ ∼ 10 TeV (22)

MV = 4.5TeV (23)

g1(〈Σ〉) = 1.8, g2(〈Σ〉) = .70 (24)

∆ ≈ 7, mh0 ≤ 210 GeV (25)

mh0 < ∼ 350 GeV (26)


W ⊃λfλc〈Σ〉

µ′QccH (28)

H ′ H ′ (29)

SU(2)2 SU(2)W (30)

H ′, H ′ (31)



mΣ & 〈Σ〉 (34)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (35)

m2

H , m2

Hm2

Σ〈Σ〉 ∆ (36)

L ⊃ Da(Tr

(Σ†σaΣ


†)

(37)

2

only enters at two-loops!

L ⊃ Da(Tr

(Σ†σaΣ


†)

(38)

Tr σa = 0 (39)

Da (40)

#= 0 mΣ (41)

3

2

2

L ⊃ Da(Tr

(Σ†σaΣ


†)

(38)

∝ Tr σa = 0 (39)

Da (40)

$= 0 mΣ (41)

3

L ⊃ Da(Tr

(Σ†σaΣ


†)

(38)

∝ Tr σa = 0 (39)

Da (40)

$= 0 mΣ (41)

3

independent of

L ⊃ Da(Tr

(Σ†σaΣ


†)

(38)

∝ Tr σa = 0 (39)

Da (40)

$= 0 mΣ (41)

3

L ⊃ Da(Tr

(Σ†σaΣ


†)

(38)

∝ Tr σa = 0 (39)

Da (40)

$= 0 mΣ (41)

3


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1


λ ≤ 2503ξ

νε

(1)

VH ⊃ λH4 (2)

mh0 ∼ λv2 (3)

λSSHH (4)

λ → λ + λS (5)

SU(2)W (6)

SU(2)1 (7)

SU(2)2 (8)

→← (9)

λ ∼g2 + g′2

8→

∆g2 + g′2

8(10)

H, H (11)

〈Σ〉 = u

(1 00 1

)(12)

SU(2)1 × SU(2)2 → SU(2)D ≡ SU(2)L (13)

W ′, Z ′ with m ∼ u2(g2

1+ g2

2

)(14)

MW ′,Z′ ∼ u2(g2

1+ g2

2

)g1 ≡

g

sin φg2 ≡

g

cos φ(15)

m2

Σ|Σ|2 (16)

λ ∼∆g2 + g′2

8(17)

g1 g2 (18)

1

Fine-tuning

∆ =1 +

2m2

Σ

u2

1

g2

2

1 +2m2

Σ

u2

1

g2

1+g2

2

Delta, Delta, Delta

Gauge Extensions of the MSSM - FermilabFigure 1: Fine tuning in the MSSM (measured by ! µ2) as a...

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Transcript of Gauge Extensions of the MSSM - FermilabFigure 1: Fine tuning in the MSSM (measured by ! µ2) as a...