Peter Athron

David Miller

In collaboration with

Fine Tuning

Expect New Physics at Planck Energy (Mass)

Hierarchy Problem

Higgs mass sensitive to this scale

Supersymmetry (SUSY) removes quadratic dependence

Enormous Fine tuning!

SUSY?

Standard Model (SM) of particle physics

Eliminates fine tuning

Beautiful description of Electromagnetic, Weak and Strong forces

Neglects gravitation, very weak at low energies (large distances)

Little Hierarchy Problem

Constrained Minimal Supersymmetric Standard Model (CMSSM)

Z boson mass predicted from CMSSM parameters

Fine tuning?

Only low mass SUSY avoids fine tuning

SM masses sensitive to SUSY masses

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.

Considers only one observable

Theories may contain tunings in several observables

Global Sensitivity G. W. Anderson & D.J Castano (1995)

Consider:

All points are tuned? All points are special, atypical scenarios?

True tuning must be quantified with a normalised measure

No unnatural cancellation!

parameter space volume restricted by,

Parameter space point,

Unnormalised Tuning:

New Measure

`` ``

Compare dimensionless variations in ALL parameters

With dimensionless variations in ALL observables

parameter space volume restricted by,

Parameter space point,

Unnormalised Tuning:

New Measure

Tuning:

mean value

`` ``

Compare dimensionless variations in ALL parameters

With dimensionless variations in ALL observables

Remove Global Sensitivity

Probability of random point lying in :

Probability of a point lying in a “typical” volume:

New Measure

Define:

We measure the relative improbability!

volume with physical scenarios qualitatively “similar” to point P

Standard Model

Obtain over whole parameter range:

Large numbers of observables and parameters Numerical Approach

Choose a point P in the parameter space. Take random fluctuations about this point. Count how many points are in and Apply tuning measure

Fine Tuning in the CMSSM

Tuning

Tuning in

“Natural” Point 1

“Natural” Point 2

If we normalise with NP1 If we normalise with NP2

Tunings for the points shown in plots are:

Fine Tuning in the SM SUSY

CMSSM appears fine tuned in Little Hierarchy Problem

New measure considers how:all observables restrict space formed by all parameters in comparison to “typical” (global sensitivity)

CMSSM may not be fine tuned

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

Beyond the Standard Model Physics

Technicolor

Large Extra Dimensions

Little Higgs

Twin Higgs

Supersymmetry

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?

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