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 LGUT may be different to those in LSUSY
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 Mz and sparticle masses
Count how often Mz (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 NP1 If 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
m1/2(GeV)
m1/2(GeV)
For our study of tuning in the CMSSM we chose a grid of points:
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.
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