Measuring Fine Tuning Peter Athron In collaboration with David Miller

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

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

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

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

Traditional Measure is fine tuned Define Tuning Observable R. Barbieri & G.F. Giudice, (1988) Define Tuning Parameter % change in from 1% change in is fine tuned

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. Considers only one observable Theories may contain tunings in several observables 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. Implicitly assumes a uniform distribution of parameters Parameters in LGUT may be different to those in LSUSY parameters drawn from a different probability distribution

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

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

Toy SM

Fine Tuning in the CMSSM 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 Mz and sparticle masses Count how often Mz (and sparticle masses) is ok Apply fine tuning measure

Tuning in

Tuning

Normal points spectrums “Natural” Point 1 Normal points spectrums

“Natural” Point 2

Tunings for the points shown in plots are: If we normalise with NP1 If we normalise with NP2 Tunings for the points shown in plots are:

Conclusions 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.

Tuning in

Tuning

m1/2(GeV)

m1/2(GeV)

For our study of tuning in the CMSSM we chose a grid of points: Technical Aside For our study of tuning in the CMSSM we chose a grid of points: To reduce statistical errors: Plots showing tuning variation in m1/2 were obtained by taking the average tuning for each m1/2 over all m0. Plots showing tuning variation in m0 were obtained by taking the average tuning for each m0 over all m1/2.

MSUGRA benchmark point SPS1a: For example . . . MSUGRA benchmark point SPS1a: ALL

Supersymmetry The only possible extension to space-time Unifies gauge couplings Provides Dark Matter candidates Leptogenesis in the early universe Essential ingredient for M-Theory Elegant solution to the Hierarchy Problem!