sfitter: determining susy parameters rémi lafaye, tilman plehn, michael rauch, dirk zerwas
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R. Lafaye - Wednesday, 27 June 20072
What do we do? (1)
From a set of measurements:o Low scale indirect constraints
Mtop, BR(bs), (g-2), MW, sin2eff, BR(Bs+-)
Or dark matter constraints h2 from WMAP
o LHC measurements (di-lepton edges, sqR 10 mass difference, …)
o Or ATLAS+CMS measurements (no need to be combined)o And then ILC measurements
Compare to Susy theoretical predictions:o Spectrum calculators:
Suspect [A. Djouadi, J.L. Kneur & G. Moultaka] Softsusy [B.C. Allanach] Isasusy [H. Baer, F.E. Paige, S.D. Protopopescu & X. Tata]
o LHC Cross sections: Prospino2 [T. Plehn et al.]o LC Cross sections: MsmLib [G. Ganis]o Branching ratios: Hdecay, Sdecay [A. Djouadi, M. Mühlleitner & M. Spira] o Dark matter: Micromegas [G. Bélanger, F. Boudjema, A. Pukhov
& A. Semenov]
R. Lafaye - Wednesday, 27 June 20073
What do we do ? (2)
Then, find best 2 using different fitting technics:o Minuit, Grid scan
o Markov chains (to be described here)
Measurements defined as:
x:= ( xexp exp syst and xth th )
Where: exp is gaussian and might be correlated
syst is gaussian and 99% correlated (LHC jet or lepton energy scale)
th two possibilities (see next slides)
Needs careful likelihood (L) study to extract parameter errors: 2 = -2 ln(L)
o Best method is to smear input values according to uncertainties
o And perform a set of toy-experiments
o Start with a two constraints toy-model fit
R. Lafaye - Wednesday, 27 June 20074
Theoretical errors (1)
First case:
Treat theoretical errors as gaussian
Bayesian: assumes a pdf for theoretical predictions
Correlation between experimental uncertainties is smeared out
The gaussian tails extend far away from the usually admitted theoretical range
xexp-xth
Lm
ax
xexp-xth
yex
p-yth
LL
R. Lafaye - Wednesday, 27 June 20075
Theoretical errors (2)
Second case: Use Rfit scheme [A.Hoecker, H.Lacker, S.Laplace, F.Lediberder]
if(|xexp-xth|< th) 2 = 0
else 2 = [(|xexp-xth|- th)/exp] 2
xexp-xth
Lm
axy
exp-y
th
xexp-xth
LLThere is no convolution between experimental and theoretical errors
Just a theoretically allowed range for xth
R. Lafaye - Wednesday, 27 June 20076
Markov Chains (1)
Definition: The future state depends on the current but not on the past
Principle:o Starts from a point in the parameter space. Compute 2
cur
o Pick a new point (using Monte-Carlo technics) and compute 2new
o Switch to new point
if 2new < 2
cur
or Random[0,1] < 2cur /2
new
Advantages:o Faster than crude scan, for N parameters: cN c1 N (instead of c1
N)
o Markov chains do not rely on 2 shape in parameter space (no use of gradients)
o Ability to find secondary minima ( to a grid scan)
Drawbacks:o Exact minimum not found (use gradient fit around minima to improve)
o Choice of new point implies the use of priors (like for any Monte-Carlo)
o Bad choice of priors Limited parameter space region scanned
R. Lafaye - Wednesday, 27 June 20077
Markov Chains (2)
Choosing the next point:o Flat: Pick parameter values evenly between the allowed range
o Breit-Wigner: Has higher tails than gaussian, can go to zero at bounds Using BW can speed up the convergence if the 2 gradients are “nice”
Choosing the “right” priors:Problem: for a parameter “x” should we use a prior as a function of x, 1/x, ln(x) ?
Any choice will bias the fit toward a given parameter space region
Theoretically increasing the number of points in the chains can overcome this
Or alternatively, one can try different priors and make sure the whole parameter space is correctly scanned
Note that this last problem is not specific to Markov Chains as long as the 2 distribution has secondary minima. At least Markov Chains offer a solution.
R. Lafaye - Wednesday, 27 June 20078
mSUGRA at LHC
mSUGRA SPS1a as a benchmark point:m0=100 GeV, m1/2=250 GeV, tan=10, A0=-100 GeV, >0 and mtop=171.4 GeV
The LHC “experimental” data from cascade decays:
Theoretical errors:o 3% for gluino and squark masses
o 1% for other sparticle masses
q q 02
02 l lR
lR l 02
~ ~
~ ~
~ ~
R. Lafaye - Wednesday, 27 June 20079
mSUGRA Markov Chains Scan
SPS1a benchmark point results from Markov Chainso SFitter output #1: fully inclusive likelihood mapo SFitter output #2: ranked list of local maxima
m0
m1/
2
z axis: minimum 2 found in all other directions (tan, A0, , mtop)
R. Lafaye - Wednesday, 27 June 200710
Markov Chains priors
2 different priors for tan:o Right: Flat prior (use the low scale parameter tan)o Left: Use the high scale parameter B 1/ tan2
Around tan=10 the two priors picks an equivalent number of points
Choosing the B prior favors low
values of tan
Num
ber
of
poin
ts1/
2m
in
R. Lafaye - Wednesday, 27 June 200711
Frequentist or Bayesian (1)SFitter provide a multi-dimensional likelihood mapOne can then perform his favorite analysis…
Bottom: Frequentists look for lower 2 in all other directionsTop: Bayesians marginalize the 2 in all other directions
B prior tan prior Choice of the prior
has a lot of influence on the marginalized plots
2min shape
remains the same. But the true minimum might not be found
More points or use Minuit around minima
1/2
mar
g1/
2m
in
R. Lafaye - Wednesday, 27 June 200712
Frequentist or Bayesian (2)
Left: flat (Rfit) theoretical errors
Right: gaussian errors
mto
pm
1/2
m0
A0
m0
A0
2min
2min
R. Lafaye - Wednesday, 27 June 200713
Frequentist or Bayesian (3)
Left: Frequentist: 2min and flat (Rfit) theoretical errors
Right: Bayesian: marginalize 2 and gaussian errors, B prior
mto
pm
1/2
m0
A0
m0
A0
Bayesian preferred solutions:
m0 = 50 GeV
m1/2 = 250 GeV
A0 = 1400 GeV
mtop = 169 GeV
2
R. Lafaye - Wednesday, 27 June 200714
mSUGRA Markov Chains
SPS1a benchmark point results from Markov Chainso SFitter output #1: fully inclusive likelihood mapo SFitter output #2: ranked list of local maxima
sgn(
)
m1/2
R. Lafaye - Wednesday, 27 June 200715
mSUGRA Minuit Results
Perform a Minuit fit around the best minimum 2 point:
SPS1a ΔLHC masses ΔLHCedges
m0 100 3.9 1.2
m1/2 250 1.7 1.0
tanβ 10 1.1 0.9
A0 -100 33 20
No correlations, no theoretical errors, Sign(μ) fixed
To be done:
o Include correlations and theoretical errors
o Proper CL coverage using Rfit scheme (non gaussian case)
o Define 2 p-value: mSUGRA & measurements agreement
using Monte-Carlo toys
R. Lafaye - Wednesday, 27 June 200716
pMSSM Fit
mSUGRA: 6D parameter space [including sgn()]
pMSSM: 15D parameter space
Use of Markov Chains makes a scan possible
Lack of sensitivity on one parameter does not slow down the scan (no need to use fixed parameters)
Low sensitivity on tan
R. Lafaye - Wednesday, 27 June 200717
pMSSM up to GUT Scale
Still need to include low scale parameter errors for completness
[SFitter+J.L. Kneur]Testing unification
R. Lafaye - Wednesday, 27 June 200718
Summary
SFitter combines measurements from different sources into a determination of supersymmetric
parameters
Markov Chains:o Speed up the scan of the parameter spaceo Especially useful for large number of parameterso Provide likelihood map and ranked list of minima
Frequentist or Bayesian:o Both can be performed from the likelihood mapo Bayesian output greatly depends on the choice of priorso Full frequentist analysis needs a lot of “toys”
Work in progress:o Minuit fit around minima including all uncertaintieso Full frequentist treatment including 2
min p-value and coverage determination
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