Peter Athron

1. Measuring Fine Tuning Peter Athron In collaboration with David Miller

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

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

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

5. 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 • Little Higgs • Twin Higgs • Need a reliable, quantitative measure of fine tuning to judge the success of these approaches.

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

7. 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

8. G. W. Anderson & D.J Castano (1995) Consider: • Global Sensitivity 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

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

10. Toy SM

12. 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

13. Tuning in

14. Tuning

15. “Natural” Point 1 • Normal points spectrums

16. “Natural” Point 2

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

18. Conclusions • Probability dist. of parameters; • 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; • 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.

19. Tuning in

20. Tuning

21. m1/2(GeV)

22. m1/2(GeV)

23. 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.

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

27. 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!