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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 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Outline Hierarchy problem of SM. Fine-tuning: SUSY Little Higgs Conclusions. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Hierarchy problem of SM SM as an effective theory valid up to a cut-off scale ΛSM → Radiative corrections to the Higgs mass: Veltman No fine-tuning between tree-level and 1 loop contributions to mh →ΛSM≤ few TeV ( “Big” hierarchy problem ) . E.g. mh =130 GeV THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Tension between these bounds in ΛSM and the experimental bounds on the effective scale of non-renormalizable operators (that parametrize new physics). 1 L LSM 2 O LH Typically LH > 10 TeV ~ “Little” hierarchy problem THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . 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. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . FINE-TUNING Barbieri & Giudice Standard definition of the fine-tuning parameters: v 2 (α1 , α 2 ,...) v 2 Δ Δ 2 , Total αi i = 10 10% fine-tuning = 100 1% fine-tuning with αi the independent parameters of the model. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . 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 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Top mass: mt = 178 ± 4.3 GeV with But mh has not been measured: aver SM < 2.5 TeV THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . 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: 2 0 2 0 2 1 2 2 0 2 0 2 V m H 2 m H2 (m H H h.c.) ( g g ' ) H1 H 2 2 0 0 1 1 2 2 3 1 2 8 with m12, 2 2 mH1, 2 2 m3 B 2 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Along the breaking direction in the H10, H20 space: 1 2 2 1 4 V m v v 2 4 where λ and m2 are functions of the soft masses and the μ-parameter at the initial scale. Minimization: Fine-tuning: m2 mi2 v2 2 v MSSM 1 2 1 tree ( g g ' ) cos 2 cos2 2 2 2 8 15 m m c m s m s 2 2 2 2 2 2 ~2 1.01 2.31m 2 1 2 3 2 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Contourplot of the fine-tuning in the MSSM THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . LOW SCALE SUSY 2 F2 F 2 msoft 2 msoft ~ 2 , SUSY ~ 4~ 2 M M M with F the SUSY breaking scale and M the messenger scale. - Gravity-mediated models: M~1019 GeV - Low scale SUSY models: F and M of similar order ~ TeV – Concrete example: W 2S T H1 H 2 l H1H 2 2 2M t K T H1 H 2 2 2 2 4 T 4M 2 1 M 2 T 2 H 1 2 H2 2 e 1 2 H1 H 2 2M 4 4 ~ where T is the singlet responsible for the breaking of SUSY and m = ΛS2 / M THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Integrating the singlet T out: 2HDM ~ m12 m2 2 1m 2 , m3 0 2 2 1 2 ~2 2 m 1 2 ( g g ' ) 21 2 2 4 M 1 2 2 ~ 3 ( g g ' ) 2 (12 m 2 e1 2 ) 2 4 M 1 2 12 2 4 g 2 e1 2 2 2 t M l 5 0 , 6 7 M ~ μ = 0.3 M , m =0.5 M , e1 = -2, αt = 1 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Little Higgs Models Stabilization of Higgs mass by making the Higgs a pseudo-Goldstone boson resulting from a spontaneously broken approximate symmetry. Spectrum: SM L. H. H.E. cut-off mh ~ 200 GeV m ~ f~ 1 TeV ~ 4 f ~ 10 TeV New particles at 1 TeV than cancel quadratic divergences in mh. THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . 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 : THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . The Littlest Higgs Tree-level Lagrangian: (g1, g2 , g1´, g2´) (1, 2) constrained by Radiative corrections: THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . The Littlest Higgs The operators O 1 and O2 already at tree-level: c and c’ unknown coefficients. 3 c ctree c1loop ctree 4 c' c'tree c'1loop c'tree 24 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . The Littlest Higgs Electroweak symmetry breaking. At energies beneath m , integrating out the triplet: with a c( g 2 g '2 ) c' 1 , b c( g12 g '1 ) 2 2 2 2 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Fine-tuning in the Littlest Parametrization of the amount of fine-tuning: Rough estimate: heavy top contribution t mh2 = 2 with t mh2 0.37 f 2 2 t2 e.g. for f = 1 TeV, mh = 150 GeV t mh2 / mh2 33 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . 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 a c( g 2 g '2 ) c' 1 2 2 2 b c( g12 g '1 ) 2 M 2 (a b ) f 2 1 / 1 / a 1 / b THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Fine-tuning in the Littlest Case A. f = 1TeV , g’12= g’22= 2 g’2 mh = 115 GeV mh = 250 GeV THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Fine-tuning in the Littlest Case A → c small → Implicit fine-tuning between ctree and c1-loop c tree instead of c cc c1loop ctree 3 tree 4 c' c'tree c'1loop c'tree 24 total with c total with c tree mh = 250 GeV THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Fine-tuning in the Littlest Case B. f = 1TeV , g’12= g’22= 2 g’2 mh = 115 GeV Fine-tuning larger than case A. total with c tree Delicately tuned THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Littlest with T-parity Extra symmetry: T-parity. Cheng & Low Coupling h2Φ is forbidden, and also direct couplings of SM fields to new gauge bosons . g1 g 2 2 g , g '1 g '2 2 g ' 1 (a b ) 4 Parameters : c, c´ , λ1 , λ2 Two cases: A) λ1 < λ2 B) λ2 < λ1 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . [SU(2)]2 x U(1)Y model Peskin et al. Differences from the Littlest: There is a quadratic divergence contribution to mh2 due to U(1)Y Case A mh = c 2 g´2 2 / 162 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. mh = 250 GeV THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Fine-tuning in the Simplest Global [SU(3)×U(1)]2 / [SU(2)×U(1)]2 f1 = f2 = 1 TeV Two scales: f1 , f2 . Radiatively induced δm2<0 : Add tree-level mass μ2 Parameters: f1 , f2 , μ2, λ 1 , λ 2 . THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Conclusions SM → hierarchy problem → Physics Beyond SM ~ few TeV. SUSY MSSM Logarithmic and finite contributions from sparticles Bounds on sparticles masses MSSM ~5 % fine-tuned → λtree is small Low scale SUSY λtree is larger → Improvement in the fine-tuning problem No big effects of running THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS . Conclusions “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 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 THE FINE-TUNING PROBLEM IN SUSY AND LITTLE HIGGS .