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

2. 2 R. Lafaye - Wednesday, 27 June 2007 What do we do? (1)  From a set of measurements: oLow scale indirect constraints  M top, BR(b  s  ), (g-2) , M W, sin 2  eff, BR(B s   +  - )  Or dark matter constraints  h 2 from WMAP oLHC measurements (di-lepton edges, sq R  1 0 mass difference, …) oOr ATLAS+CMS measurements (no need to be combined) oAnd then ILC measurements  Compare to Susy theoretical predictions: oSpectrum calculators:  Suspect[A. Djouadi, J.L. Kneur & G. Moultaka]  Softsusy[B.C. Allanach]  Isasusy[H. Baer, F.E. Paige, S.D. Protopopescu & X. Tata] oLHC Cross sections: Prospino2[T. Plehn et al.] oLC Cross sections: MsmLib[G. Ganis] oBranching ratios: Hdecay, Sdecay [A. Djouadi, M. Mühlleitner & M. Spira] oDark matter: Micromegas [G. Bélanger, F. Boudjema, A. Pukhov & A. Semenov]

3. 3 R. Lafaye - Wednesday, 27 June 2007 What do we do ? (2) Then, find best  2 using different fitting technics: oMinuit, Grid scan oMarkov chains (to be described here) Measurements defined as: x:= ( x exp   exp   syst and x th   th ) Where: o  exp is gaussian and might be correlated o  syst is gaussian and 99% correlated (LHC jet or lepton energy scale) o  th two possibilities (see next slides) Needs careful likelihood ( L ) study to extract parameter errors: o  2 = -2 ln( L ) oBest method is to smear input values according to uncertainties oAnd perform a set of toy-experiments oStart with a two constraints toy-model fit

4. 4 R. Lafaye - Wednesday, 27 June 2007 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 x exp -x th L max x exp -x th y exp -y th L

5. 5 R. Lafaye - Wednesday, 27 June 2007 Theoretical errors (2) Second case: Use Rfit scheme [A.Hoecker, H.Lacker, S.Laplace, F.Lediberder] if(|x exp -x th |<  th )  2 = 0 else  2 = [(|x exp -x th |-  th )/  exp ] 2 x exp -x th L max y exp -y th x exp -x th L There is no convolution between experimental and theoretical errors Just a theoretically allowed range for x th

6. 6 R. Lafaye - Wednesday, 27 June 2007 Markov Chains (1) Definition: The future state depends on the current but not on the past Principle: oStarts from a point in the parameter space. Compute  2 cur oPick a new point (using Monte-Carlo technics) and compute  2 new oSwitch to new point  if  2 new <  2 cur  or Random[0,1] <  2 cur /  2 new Advantages: oFaster than crude scan, for N parameters: c N  c 1 N (instead of c 1 N ) oMarkov chains do not rely on  2 shape in parameter space (no use of gradients) oAbility to find secondary minima (  to a grid scan) Drawbacks: oExact minimum not found (use gradient fit around minima to improve) oChoice of new point implies the use of priors (like for any Monte-Carlo) oBad choice of priors  Limited parameter space region scanned

7. 7 R. Lafaye - Wednesday, 27 June 2007 Markov Chains (2) Choosing the next point: oFlat: Pick parameter values evenly between the allowed range oBreit-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.

8. 8 R. Lafaye - Wednesday, 27 June 2007 mSUGRA at LHC mSUGRA SPS1a as a benchmark point: m 0 =100 GeV, m 1/2 =250 GeV, tan  =10, A 0 =-100 GeV,  >0 and m top =171.4 GeV The LHC “experimental” data from cascade decays: Theoretical errors: o3% for gluino and squark masses o1% for other sparticle masses q  q  0 2  0 2  l  l  R l  R  l   0 2 ~~ ~ ~ ~ ~

9. 9 R. Lafaye - Wednesday, 27 June 2007 mSUGRA Markov Chains Scan SPS1a benchmark point results from Markov Chains oSFitter output #1: fully inclusive likelihood map oSFitter output #2: ranked list of local maxima m0m0 m 1/2 z axis: minimum  2 found in all other directions (tan , A 0, , m top )

10. 10 R. Lafaye - Wednesday, 27 June 2007 Markov Chains priors 2 different priors for tan  : oRight: Flat prior (use the low scale parameter tan  ) oLeft: Use the high scale parameter B  1/ tan  2  Around tan  =10 the two priors picks an equivalent number of points  Choosing the B prior favors low values of tan  Number of points 1/  2 min

11. 11 R. Lafaye - Wednesday, 27 June 2007 Frequentist or Bayesian (1) SFitter provide a multi-dimensional likelihood map One can then perform his favorite analysis… Bottom: Frequentists look for lower  2 in all other directions Top: 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   2 min shape remains the same. But the true minimum might not be found  More points or use Minuit around minima 1/  2 marg 1/  2 min

12. 12 R. Lafaye - Wednesday, 27 June 2007 Frequentist or Bayesian (2) Left: flat (Rfit) theoretical errors Right: gaussian errors m top m 1/2 m0m0 A0A0 m0m0 A0A0  2 min

13. 13 R. Lafaye - Wednesday, 27 June 2007 Frequentist or Bayesian (3) Left: Frequentist:  2 min and flat (Rfit) theoretical errors Right: Bayesian: marginalize  2 and gaussian errors, B prior m top m 1/2 m0m0 A0A0 m0m0 A0A0 Bayesian preferred solutions:  m 0 = 50 GeV  m 1/2 = 250 GeV  A 0 = 1400 GeV  m top = 169 GeV 22

14. 14 R. Lafaye - Wednesday, 27 June 2007 mSUGRA Markov Chains SPS1a benchmark point results from Markov Chains oSFitter output #1: fully inclusive likelihood map oSFitter output #2: ranked list of local maxima sgn(  ) m 1/2

15. 15 R. Lafaye - Wednesday, 27 June 2007 mSUGRA Minuit Results Perform a Minuit fit around the best minimum  2 point: SPS1aΔLHC massesΔLHCedges m0m0 1003.91.2 m 1/2 2501.71.0 tanβ101.10.9 A0-1003320 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

16. 16 R. Lafaye - Wednesday, 27 June 2007 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 

17. 17 R. Lafaye - Wednesday, 27 June 2007 pMSSM up to GUT Scale Still need to include low scale parameter errors for completness [SFitter+J.L. Kneur] Testing unification

18. 18 R. Lafaye - Wednesday, 27 June 2007 Summary SFitter combines measurements from different sources into a determination of supersymmetric parameters  Markov Chains: oSpeed up the scan of the parameter space oEspecially useful for large number of parameters oProvide likelihood map and ranked list of minima  Frequentist or Bayesian: oBoth can be performed from the likelihood map oBayesian output greatly depends on the choice of priors oFull frequentist analysis needs a lot of “toys”  Work in progress: o Minuit fit around minima including all uncertainties o Full frequentist treatment including  2 min p-value and coverage determination