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] )

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

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

 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)

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

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?

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)

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)

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

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

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

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

Standard Model Obtain over whole parameter range:

Four observables, three parameters Large cancellations ) fine tuning

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

Tuning in

Tuning

m 1/2 (GeV)

“Natural” Point 1

“Natural” Point 2

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

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

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: