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