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Peter Athron David Miller In collaboration with Measuring Fine Tuning

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+ physical mass = “bare mass” + “loop corrections” =+ divergent Cut off integral at Planck Scale Fine tuning Hierarchy Problem

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Beyond the Standard Model Physics Technicolour Large Extra Dimensions Little Higgs Twin Higgs Supersymmetry

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MSSM EWSB constraint (Tree Level) : If sparticle mass limits ) Parameters Need tuning for Little Hierarchy Problem

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Superymmetry Models with extended Higgs sectors NMSSM nMSSM ESSM 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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R. Barbieri & G.F. Giudice, (1988) Define Tuning is fine tuned % change in from 1% change in Observable Parameter Traditional Measure

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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 In some regions of parameter space 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

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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 Tuning New Measure Normalised Tuning mean value `` `` AND

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Toy SM

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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 often M z (and sparticle masses) is ok Apply fine tuning measure Fine Tuning in the CMSSM

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Tuning in

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Tuning

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Normal points spectrums “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. Conclusions

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Tuning in

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Tuning

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m 1/2 (GeV)

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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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For example... MSUGRA benchmark point SPS1a: ALL

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Supersymmetry The only possible extension to space-time 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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