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

Irene Hidalgo. IFT, Madrid Collaboration with: Pre-SUSY A. Casas July 2005 J.R. Espinosa

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**Outline Hierarchy problem of SM. Fine-tuning: Conclusions. SUSY**

Little Higgs Conclusions.

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**Hierarchy 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

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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 LH 10 TeV ~ >

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**Veltman´s condition (1-loop):**

Kolda & Murayama 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.

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**FINE-TUNING = 100 1% fine-tuning**

Barbieri & Giudice Standard definition of the fine-tuning parameters: , with αi the independent parameters of the model. = 10% fine-tuning = 100 1% fine-tuning

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**SM: ΛSM as an indepedent parameter Veltman´s throat Contourplot of ΔΛ**

Kolda & Murayama 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

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** < 2.5 TeV Top mass: mt = 178 ± 4.3 GeV with**

But mh has not been measured: < 2.5 TeV SM aver

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SUSY 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: with

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**Along the breaking direction in the H10, H20 space:**

where λ and m2 are functions of the soft masses and the μ-parameter at the initial scale. Minimization: Fine-tuning: MSSM

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**Contourplot of the fine-tuning in the MSSM**

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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 = ΛS2 / M ~

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**~ Integrating the singlet T out: 2HDM**

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

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**Little Higgs Models mh ~ 200 GeV SM L. H. H.E. cut-off m ~ f~ 1 TeV**

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 m ~ f~ 1 TeV ~ 4 f ~ 10 TeV

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The Littlest Higgs Arkani-Hamed et al. 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 :

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**The Littlest Higgs Tree-level Lagrangian: Radiative corrections:**

(g1, g2 , g1´, g2´) (1, 2) constrained by

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

c and c’ unknown coefficients.

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**The Littlest Higgs Electroweak symmetry breaking.**

At energies beneath m , integrating out the triplet: with

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**Fine-tuning in the Littlest**

Parametrization of the amount of fine-tuning: Rough estimate: heavy top contribution t mh2 = 2 with 2 t2 t mh2 f 2 e.g. for f = 1 TeV, mh = 150 GeV t mh2 / mh2 33

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**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

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**Fine-tuning in the Littlest**

Case A. f = 1TeV , g’12= g’22= 2 g’2 mh = 250 GeV mh = 115 GeV

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**Fine-tuning in the Littlest**

Case A → c small → Implicit fine-tuning between ctree and c1-loop c instead of c tree total with c total with c mh = 250 GeV tree

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**Fine-tuning in the Littlest**

mh = 115 GeV Case B. f = 1TeV , g’12= g’22= 2 g’2 Fine-tuning larger than case A. total with c Delicately tuned tree

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**Littlest with T-parity**

Cheng & Low 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

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** mh2 = c g´2 2 / 162 [SU(2)]2 x U(1)Y model mh = 250 GeV Case A**

Peskin et al. 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. Case A mh2 = c g´2 2 / 162 mh = 250 GeV

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**Fine-tuning in the Simplest**

f1 = f2 = 1 TeV 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 .

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**Conclusions → MSSM ~5 % fine-tuned**

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

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**Conclusions Minimum value of Δ accessible by varying the parameters**

“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 studied models: More fine-tuned than the rough estimate due to implicit tunings between the parameters of the models to work properly and have the correct EW scale. Minimum value of Δ accessible by varying the parameters

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