SFitter: Determining SUSY Parameters Rémi Lafaye, Tilman Plehn, Michael Rauch, Dirk Zerwas

SFitter: Determining SUSY

Parameters

Rémi Lafaye, Tilman Plehn, Michael Rauch, Dirk Zerwas

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]


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


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


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


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


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.


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

~ ~

~ ~

~ ~


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)


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


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


Frequentist or Bayesian (2)

Left: flat (Rfit) theoretical errors

Right: gaussian errors

mto

pm

1/2

m0

A0

m0

A0

2min

2min


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


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


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


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


pMSSM up to GUT Scale

Still need to include low scale parameter errors for completness

[SFitter+J.L. Kneur]Testing unification


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

Documents

SFitter: Determining SUSY Parameters Rémi Lafaye, Tilman Plehn, Michael Rauch, Dirk Zerwas