THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS...

THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS .

THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS

Irene Hidalgo.IFT, Madrid

Collaboration with: Pre-SUSY A. Casas 6 July 2005 J.R. Espinosa


OutlineOutline

Hierarchy problem of SM.

Fine-tuning:SUSY Little Higgs

Conclusions.


Hierarchy problem of SMHierarchy problem of SM

SM as an effective theory valid up to a cut-off scale ΛSM → Radiative corrections to the Higgs mass:

No fine-tuning between tree-level and 1 loop contributions to mh → ΛSM≤ few TeV ( “Big” hierarchy problem ) .

E.g. mh =130 GeV

Veltman


Tension between these bounds in ΛSM and the experimental bounds on the effective scale of non-renormalizable operators (that parametrize new physics).

Typically

“Little” hierarchy problem

LH10 TeV

~>

2

1SM

LH

L L O


Veltman´s condition (1-loop):

ΛSM could be larger than expected if Veltman´s condition is fulfilled.

At higher order this condition becomes cut-off dependent.

Kolda & Murayama


FINE-TUNINGFINE-TUNING

Standard definition of the fine-tuning parameters:

,

with αi the independent parameters of the model.

i

2

αTotal ΔΔ i

Barbieri & Giudice

= 10 10% fine-tuning = 100 1% fine-tuning

221

2 ,...)α,(α vv


SM: ΛSM as an indepedent parameter Veltman´s throat

Contourplot of ΔΛ

Other relevant parameters in the SM for the fine-tuninng: λ t and λ

Kolda & Murayama


Top mass: mt = 178 ± 4.3 GeV

with

But mh has not been measured:

< 2.5 TeVSMaver


SUSY modelsSUSY models

SUSY:

There are the same number of bosonic and fermionic degrees of freedom.The hierarchy problem is solved due to the cancellation of quadratic divergences of the Higgs mass.

The Minimal Supersymmetric extension of the SM: the MSSM

Higgs sector: 2 doublets, H1 and H2 . Tree-level scalar potential:

2202

201

2202

01

23

202

22

201

21 )'(

8

1.).(

HHggchHHmHmHmV

with 222

2,1 2,1Hmm Bm 23


Along the breaking direction in the H10, H2

0 space:

where λ and m2 are functions of the soft masses and the μ-parameter at the initial scale.

Minimization: Fine-tuning:

MSSM

422

4

1

2

1vvmV

22 m

v

2

2

v

mi

2cos15

12cos)'(

8

1 2222 ggtree

222

23

222

221

2 ~31.201.1 msmsmcmm


Contourplot of the fine-tuning in the MSSM


LOW SCALE SUSY

with the SUSY breaking scale and M the messenger scale.- Gravity-mediated models: M~1019 GeV- Low scale SUSY models: and M of similar order ~ TeV

– Concrete example:

where T is the singlet responsible for the breaking of SUSY and m = ΛS

2 / M

2

2

4

2

2

22 ~~,~

M

m

M

F

M

Fm soft

SUSYsoft

F

F

22121

2

2HH

M

lHHTW S

4

2

4

1212

2

2

1

2

21

4

2

2

2

2

1

2

2

4

HHM

eHHT

M

TM

HHTK t

~


Integrating the singlet T out: 2HDM

μ = 0.3 M , m =0.5 M , e1 = -2, αt = 1

0,~ 23

21

222

21 mmmm

M

l

Meg

emM

gg

M

mgg

t

765

2

221

12

4

21

2212

223

2

221

2221

,0

222

1

)~(2

)'(4

1

~2)'(

4

1

~


Little Higgs ModelsLittle Higgs Models

Stabilization of Higgs mass by making the Higgs a pseudo-Goldstone boson resulting from a spontaneously broken approximate symmetry. Spectrum:

New particles at 1 TeV than cancel quadratic divergences in mh.

SM L. H. H.E. cut-off

mh ~ 200 GeV ~ 4 f ~ 10 TeVm ~ f~ 1 TeV


The Littlest Higgs is a non-linear σ model based on a global SU(5) symmetry which is spontaneously broken to SO(5) at the scale f ~ 1 TeV.

An [SU(2)×U(1)]2 subgroup of SU(5) is gauged, and is spontaneously broken to the diagonal SU(2)×U(1) subgroup.

New states that cancel the quadratic divergences:

– Heavy top T :

– Extra gauge bosons W’ , B’ : ,

- Triplet :

The Littlest HiggsArkani-Hamed et al.


Tree-level Lagrangian:

Radiative corrections:

The Littlest Higgs

(g1, g2 , g1´, g2´) (1, 2)

constrained by


The operators O 1 and O2 already at tree-level:

c and c’ unknown coefficients.

The Littlest Higgs

244

3

1

1

treelooptree

treelooptree

c'c'c'c'

cccc


Electroweak symmetry breaking.At energies beneath m , integrating out the triplet:

The Littlest Higgs

with

)'(,')'( 21

21

21

22

22 ggccggc ba


Parametrization of the amount of fine-tuning:Rough estimate: heavy top contribution

t mh2 = 2

with

2 t2

t mh2 0.37 f 2

e.g. for f = 1 TeV, mh = 150 GeV

t mh2 / mh

2 33

Fine-tuning in the Littlest


But heavy top contribution is not all.

Using the standard definition of fine-tuning parameters.Parameters in Littlest: c, c´ , λ1 , λ2 , g1 , g2 , g´1 , g´2 . (Constraints between them)Two regions:

A) λ ≈ λb « λa ≈ M2Φ/f2

B) λ ≈ λa « λb ≈ M2Φ/f2


ba

ba

b

a

fM

ggc

cggc

/1/1/1

)(

)'(

')'(

22

21

21

21

22

22



Case A. f = 1TeV , g’12= g’2

2= 2 g’2

mh = 115 GeV mh = 250 GeV



Case A → c small → Implicit fine-tuning between ctree and c1-loop

c instead of c

mh = 250 GeV

tree

total with c total with ctree

24''''4

3

1

1

treelooptree

treelooptree

cccc

cccc



Case B. f = 1TeV , g’12= g’2

2= 2 g’2

mh = 115 GeV

total with ctree

Fine-tuning larger than case A.

Delicately tuned


Extra symmetry: T-parity.

Coupling h2Φ is forbidden, and also direct couplings of SM fields to new gauge bosons .

Parameters : c, c´ , λ1 , λ2

Two cases:

A) λ1 < λ2

B) λ2 < λ1

)(4

1

'2'',2 2121

ba

gggggg

Littlest with T-parity

Cheng & Low


Differences from the Littlest:

There is a quadratic divergence contribution to mh2 due to U(1)Y

Absence of the heavy B’ boson.

Two regions (A and B heirs of the Littlest):

Case A similar fine-tuning as Littlest.

Case B is worse in terms of fine-tuning.

[SU(2)]2 x U(1)Y model

mh2 = c g´2 2 / 162

mh = 250 GeV

Peskin et al.

Case A


Global [SU(3)×U(1)]2 / [SU(2)×U(1)]2

Two scales: f1 , f2 .

Radiatively induced δm2<0 :

Add tree-level mass μ2

Parameters: f1 , f2 , μ2, λ 1 , λ 2 .

Fine-tuning in the Simplest

f1 = f2 = 1 TeV


ConclusionsConclusions SM → hierarchy problem → Physics Beyond SM ~ few TeV.

SUSY

MSSM

Logarithmic and finite contributions from sparticles

Bounds on sparticles masses

λtree is small

Low scale SUSY

λtree is larger

No big effects of running

→ MSSM ~5 % fine-tuned

→ Improvement in the fine-tuning problem


“Little Higgs” models.

Rough estimate with the heavy top contribution : few % fine-tuned.

Taking into account the standard definition of fine-tuning and all the parameters in the 3 studied models:

More fine-tuned than the rough More fine-tuned than the rough estimate due to implicit tunings between estimate due to implicit tunings between the parameters of the models to work the parameters of the models to work properly and have the correct EW scale.properly and have the correct EW scale.

Conclusions

Minimum value of Δ accessible

by varying the parameters

THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS...

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