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Peter Athron
David Miller
In collaboration with
Quantifying Fine Tuning (arXiv:0705.2241)
Outline
Motivations for supersymmetry Hierarchy problem
Little Hierarchy Problem Traditional Tuning Measure New tuning measure ESSM
EWSB in the ESSM
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
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
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:
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
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:
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
Electroweak Symmetry Breaking in the E6SSM
Peter AthronIn collaboration with
S.F. King, D.J. Miller, S. Moretti & R. Nevzorov.
Exceptional Supersymmetric Standard Model (Phys.Rev. D73 (2006) 035009 arXiv:hep-ph/0510419 , Phys.Lett. B634 (2006)
278-284 arXiv:hep-ph/0511256 S.F.King, S.Moretti & R. Nevzorov)
Exotic coloured matter
3 generations of Singlet fields
Ordinary matter
E6 inspired model with an extra gauged U(1) symmetry
Matter content based on 3 generations of complete 27plet representations of E6 ) anomalies automatically cancelled
Provides a low energy alternative to the MSSM and NMSSM
3 generations of Higgs like fields
Extra SU(2) doublets(for gauge coupling unification)
E6SSM Superpotential
S = S3 develops vev, hSi = s, giving mass to exotic coloured fields
Hu = H2,3 & Hd = H1,3 develop vevs, hH0ui=vu &
hH0di=vd Give mass to ordinary matter via Higgs
Mechanism
generates an effective term
Solves -problem of MSSM as in NMSSM without tadpoles/domain walls problems
ESSMNMSSMMSSM
Two Loop Upper Bounds on the Light Higgs
NUHESSM GUT-scale universality assumptions
Highscale mass for all other scalar fields
Highscale mass for three generations of Singlet fields
Highscale mass for three generations of H2 fields
Highscale mass for three generations of H1 fields
Universal Gaugino MassUniversal trilinear soft mass
Minimal supergravity inspired GUT-scale constraints on parameters
EWSB constraints
Scalar masses
Gaugino massesTrilinear soft masses
Solutions must possess symmetry
Obtain RGE solutions for soft masses at EW scale
Fix v2 = vu2 + vd
2 = (174 GeV)2
Choose Yukawas 3=0.6; 1,2=0.46; 1,2,3=0.162
Choose tan = vu/vd = 10, s = 3 TeV
EWSB constraintsAllowed
EW tachyons
GUT tachyons Experimentally ruled out
Sample Spectrum
Conclusions Solutions with all universal masses are hard to find. Allow non universal Higgs masses. Dramatic improvement! Many spectrums which could be seen at the LHC. RGE solutions ) Gluino often lighter than the squarks Further work:
Include two loop RGEs for gaugino masses Look for solutions with stronger universality assumptions
1 loop RGE solutions for fully universal benchmark point
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