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

2. physical mass = “bare mass” + “loop corrections”
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
Need a reliable, quantitative measure of fine tuning to judge the success of these approaches.

6. 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

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. 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

9. parameter space volume restricted by,
New Measure
Parameter space point,
parameter space volume 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. Normal points spectrums
“Natural” Point 1
Normal points spectrums

16. “Natural” Point 2

17. 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:

18. 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.

19. Tuning in

20. Tuning

21. m1/2(GeV)

22. m1/2(GeV)

23. 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.

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

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!