Peter Athron David Miller In collaboration with Measuring Fine Tuning

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

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

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

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?

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

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

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

Toy SM

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

Tuning in

Tuning

Normal points spectrums “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. Conclusions

Tuning in

Tuning

m 1/2 (GeV)

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:

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  Elegant solution to the Hierarchy Problem!  Essential ingredient for M-Theory