Peter Athron

David Miller

In collaboration with

Quantifying Fine Tuning (arXiv:0705.2241, Phys.Rev.D76:075010, 2007.

arXiv:0707.1255 [hep-ph], AIP Conf.Proc.903:373-376,2007.

arXiv:0710.2486 [hep-ph] )

Outline

Motivations for supersymmetry Hierarchy problem

Little Hierarchy Problem (of Susy) Traditional Tuning Measure New tuning measure Applications

SM Toy model MSSM

Supersymmetry

Only possible extension to Poincare symmetry

Unifies gauge couplings

Provides Dark Matter candidates

Leptogenesis in the early universe

Elegant solution to the Hierarchy Problem!

Essential ingredient for M-Theory

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?

Superymmetry Models with extended Higgs sectors NMSSM nMSSM E6SSM

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?

J.R. Ellis, K. Enqvist, D.V. Nanopoulas, & F.Zwirner (1986)

R. Barbieri & G.F. Giudice, (1988)

Define Tuning

is fine tuned

% change in from 1% change in

Observable

Parameter

Traditional Measure

J. A. Casas, J. R. Espinosa and I. Hidalgo (2004)

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

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 (discussed later)

parameter space volume restricted by,

Parameter space point,

Unnormalised Tuning:

New Measure

`` ``

Compare dimensionless variations in ALL parameters

With dimensionless variations in ALL 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 Tunings

New Measure

Normalised Tunings

mean value

`` ``

`` `` AND

Probability of random point lying in :

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

New Measure

Define:

We can associate our tuning measure with relative improbability!

volume with physical scenarios qualitatively “similar” to point P

Standard Model

Obtain over whole parameter range:

Four observables, three parameters

Large cancellations ) fine tuning

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 many points are in F and in G. Apply fine tuning measure

Fine Tuning in the CMSSM

Tuning in

Tuning in

Tuning

Tuning

m1/2(GeV)

m1/2(GeV)

“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.

Fine Tuning Summary

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: