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Peter Athron David Miller In collaboration with Quantifying Fine Tuning (arXiv: , Phys.Rev.D76:075010, arXiv: [hep-ph], AIP Conf.Proc.903: ,2007. arXiv: [hep-ph] )

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Outline Motivations for supersymmetry Hierarchy problem Little Hierarchy Problem (of Susy) Traditional Tuning Measure New tuning measure Applications SM Toy model MSSM

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Supersymmetry Only possible extension to Poincare symmetry Unifies gauge couplings Provides Dark Matter candidates Leptogenesis in the early universe Elegant solution to the Hierarchy Problem! Essential ingredient for M-Theory

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Expect New Physics at Planck Energy (Mass) Hierarchy Problem Higgs mass sensitive to this scale Supersymmetry (SUSY) removes quadratic dependence Enormous Fine tuning! SUSY? Standard Model (SM) of particle physics Eliminates fine tuning Beautiful description of Electromagnetic, Weak and Strong forces Neglects gravitation, very weak at low energies (large distances)

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Little Hierarchy Problem Constrained Minimal Supersymmetric Standard Model (CMSSM) Z boson mass predicted from CMSSM parameters Fine tuning?

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Superymmetry Models with extended Higgs sectors NMSSM nMSSM E 6 SSM Supersymmetry Plus Little Higgs Twin Higgs Alternative solutions to the Hierarchy Problem Technicolor Large Extra Dimensions Little Higgs Twin Higgs Need a reliable, quantitative measure of fine tuning to judge the success of these approaches. Solutions?

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J.R. Ellis, K. Enqvist, D.V. Nanopoulas, & F.Zwirner (1986) R. Barbieri & G.F. Giudice, (1988) Define Tuning is fine tuned % change in from 1% change in Observable Parameter Traditional Measure J. A. Casas, J. R. Espinosa and I. Hidalgo (2004)

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Limitations of the Traditional Measure Considers each parameter separately Fine tuning is about cancellations between parameters. A good fine tuning measure considers all parameters together. Implicitly assumes a uniform distribution of parameters Parameters in L GUT may be different to those in L SUSY parameters drawn from a different probability distribution Takes infinitesimal variations in the parameters Observables may look stable (unstable) locally, but unstable (stable) over finite variations in the parameters. Considers only one observable Theories may contain tunings in several observables Global Sensitivity (discussed later)

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parameter space volume restricted by, Parameter space point, Unnormalised Tuning: New Measure `` Compare dimensionless variations in ALL parameters With dimensionless variations in ALL observables

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Global Sensitivity Consider: responds sensitively to All values of appear equally tuned! throughout the whole parameter space (globally) All are atypical? True tuning must be quantified with a normalised measure G. W. Anderson & D.J Castano (1995) Only relative sensitivity between different points indicates atypical values of

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parameter space volume restricted by, Parameter space point, Unnormalised Tunings New Measure Normalised Tunings mean value `` `` AND

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Probability of random point lying in : Probability of a point lying in a “typical” volume: New Measure Define: We can associate our tuning measure with relative improbability! volume with physical scenarios qualitatively “similar” to point P

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Standard Model Obtain over whole parameter range:

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Four observables, three parameters Large cancellations ) fine tuning

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Choose a point P in the parameter space at GUT scale Take random fluctuations about this point. Using a modified version of Softsusy (B.C. Allanach) Run to Electro-Weak Symmetry Breaking scale. Predict M z and sparticle masses Count how many points are in F and in G. Apply fine tuning measure Fine Tuning in the CMSSM

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Tuning in

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Tuning

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m 1/2 (GeV)

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“Natural” Point 1

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“Natural” Point 2

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If we normalise with NP1If we normalise with NP2 Tunings for the points shown in plots are:

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Naturalness comparisons of BSM models need a reliable tuning measure, but the traditional measure neglects: Many parameter nature of fine tuning; Tunings in other observables; Behaviour over finite variations; Probability dist. of parameters; Global Sensitivity. New measure addresses these issues and: Demonstrates and increase with. Naïve interpretation: tuning worse than thought. Normalisation may dramatically change this. If we can explain the Little hierarchy Problem. Alternatively a large may be reduced by changing parameterisation. Could provide a hint for a GUT. Fine Tuning Summary

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For our study of tuning in the CMSSM we chose a grid of points: Plots showing tuning variation in m 1/2 were obtained by taking the average tuning for each m 1/2 over all m 0. Plots showing tuning variation in m 0 were obtained by taking the average tuning for each m 0 over all m 1/2. Technical Aside To reduce statistical errors:

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