Measuring Supersymmetry

Measuring Supersymmetry Michael Rauch ( in collaboration with Rémi Lafaye, Tilman Plehn, Dirk Zerwas) arxiv:79.3985 [hep-ph] Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Outline Supersymmetry Current Status Determining SUSY Parameters Kinematic Edges as Experimental Input Possible Collider Results Testing Unification Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 2

Supersymmetry Symmetry between bosons and fermions: Q boson = fermion ; Q fermion = boson Q: Supersymmetry Operator Simplest model: Minimal Supersymmetric Standard Model (MSSM) Supersymmetric partner to each Standard Model particle Two Higgs doublets 5 Higgs bosons (h,h,a,h ± ) Particles with same quantum numbers mix (e.g. Zino, Photino, 2 Higgsino 4 Neutralino) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 3

msugra Unification at the GUT scale ( 6 GeV ): plus Apparent unification of gauge couplings (general feature of MSSM) Common scalar mass: m Common sfermion mass: m /2 Common trilinear coupling: A ratio of the Higgs vacuum expectation values at the electro-weak scale: tan β = v 2 v one sign: sgn µ Evolve three parameters defined at the GUT scale via renormalisation group equations down to electro-weak scale: Weak-scale MSSM parameters Masses and couplings /M i..8.6.4 M M 2 M 3.2 2 4 6 8 2 4 6 8 log(q) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 4

Current Status Standard Model experimentally very well confirmed Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Current Status Standard Model experimentally very well confirmed Few experimental deviations Dark matter 23% Dark Matter content in the universe Possible candidate in the SM: Neutrinos neutrino mass limits prevent accounting for total content M W σ deviation LEP 2.3σ excess in Higgs-like events near 98 GeV g 2 of the Muon 3.4σ deviation Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Current Status Standard Model experimentally very well confirmed Few experimental deviations Dark matter 23% Dark Matter content in the universe Possible candidate in the SM: Neutrinos neutrino mass limits prevent accounting for total content M W σ deviation LEP 2.3σ excess in Higgs-like events near 98 GeV g 2 of the Muon 3.4σ deviation Some theoretical problems No gauge coupling unification Hierarchy problem (higher-order corrections to the Higgs-boson mass proportional to mass of the heaviest coupling particle GUT scale (?)) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Current Status Standard Model experimentally very well confirmed Few experimental deviations Dark matter 23% Dark Matter content in the universe Possible candidate in the SM: Neutrinos neutrino mass limits prevent accounting for total content M W σ deviation LEP 2.3σ excess in Higgs-like events near 98 GeV g 2 of the Muon 3.4σ deviation Some theoretical problems No gauge coupling unification Hierarchy problem (higher-order corrections to the Higgs-boson mass proportional to mass of the heaviest coupling particle GUT scale (?)) Look for possible ultra-violet completions Supersymmetry Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Determining SUSY parameters nowadays: Parameters in the Lagrangian m, µ, tan(β), M {,2,3},... Observables: Masses Feynman diagrams, RG evolution,... Kinematic endpoints Cross sections Branching ratios... after SUSY discovery: Observables m h, m gχ, three-particle edge(χ 4,ẽ L,χ ), BR,... Lagrangian parameters M ± GeV M 2 ± GeV M 3 ± GeV µ ± GeV tan β ±......? Tools to reconstruct SUSY parameters Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 6

Current Fits Fits of current data to supersymmetry (only msugra) [Allanach, Cranmer, Lester, Weber 25-7] [Roszkowski, Ruiz de Austra, Trotta 26/7] [Buchmüller, Cavanaugh, De Roeck, Heinemeyer, Isidori, Paradisi, Ronga, Weber, Weiglein 27] Observables: Dark Matter Ω DM h 2 g 2 µ M W sin 2 θ W BR(b sγ) BR(B s µ + µ )... Predictions for SUSY mass spectrum Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 7

Current Fits (2) Predictions for SUSY mass spectrum [plots by Allanach et al.] Bayesian: (flat tan β prior) (REWSB+same order prior) m (TeV) 4 3.5 3 2.5 2.5.5.5.5 2 M /2 (TeV) P/P(max).8.6.4.2 m (TeV) 4 3.5 3 2.5 2.5.5 2 3 4 5 6 7 tan β P/P(max).8.6.4.2 m (TeV) 4 3.5 3 2.5 2.5.5 2 3 4 5 6 7 tan β P/P(max).8.6.4.2 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 8

Current Fits (2) P.3.25.2.5. w= w=2 profile P.35.3.25.2.5. w= w=2 profile χ 2 4 3.5 3 2.5 2.5.5.5.5.5 2 2.5 3 3.5 4 m ql (TeV).2.4.6.8 m χ (TeV).5 LEP excluded 4 2 m h [right-hand plot by Buchmüller et al.] Low-energy TeV-scale SUSY fits data very well Mass ranges for SUSY particles Mass of the lightest Higgs boson compatible with LEP limit Discovery of SUSY at the LHC Additional observables from collider data SFitter (or Fittino) [Bechtle, Desch, Wienemann] Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 9

What SFitter does Set of measurements LHC measurements: kinematic edges, thresholds, masses, mass differences cross sections, branching ratios ILC measurements Indirect Constraints electro-weak: M W, sin 2 θ W ; (g 2) µ flavour: BR(b sγ), BR(B s µ + µ ); dark matter: Ωh 2 or even ATLAS and CMS measurements separately Compare to theoretical predictions Spectrum calculators: SoftSUSY, SuSPECT, ISASUSY LHC cross sections: Prospino2 LC cross sections: MsmLib Branching Ratios: SUSYHit (HDecay + SDecay) micromegas [Allanach; Djouadi, Kneur, Moultaka; Baer, Paige, Protopopescu, Tata] [Plehn et al.] [Ganis] [Djouadi, Mühlleitner, Spira] [Bélanger, Boudjema, Pukhov, Semenov] g-2 [Alexander, Stöckinger] Using as glue: SLHAio [Kreiß] Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Parameter Scans MSSM parameter space is high-dimensional: SM: 3+ parameters (m t, α s, α,... ) msugra: 5 parameters (m,m /2, A,tan(β),sgn(µ)) General MSSM: 5 parameters On loop-level observables depend on every parameter Simple inversion of the relations not possible Parameter scans Error estimates on parameters in the minimum Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Parameter Scans MSSM parameter space is high-dimensional: SM: 3+ parameters (m t, α s, α,... ) msugra: 5 parameters (m,m /2, A,tan(β),sgn(µ)) General MSSM: 5 parameters On loop-level observables depend on every parameter Simple inversion of the relations not possible Parameter scans Error estimates on parameters in the minimum Find best points (best χ 2 ) using different fitting techniques: Gradient search (Minuit) + Reasonably fast Limited convergence, only best fit 8 6 4.9.8.7.6.5.4.3.2. 8 6 4.9.8.7.6.5.4.3.2. 2 2 2 4 6 8 2 4 6 8 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Parameter Scans MSSM parameter space is high-dimensional: SM: 3+ parameters (m t, α s, α,... ) msugra: 5 parameters (m,m /2, A,tan(β),sgn(µ)) General MSSM: 5 parameters On loop-level observables depend on every parameter Simple inversion of the relations not possible Parameter scans Error estimates on parameters in the minimum Find best points (best χ 2 ) using different fitting techniques: + Reasonably fast Gradient search (Minuit) fixed Grid scan Limited convergence, only best fit + scans complete parameter space many points needed (O(e N )) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Parameter Scans MSSM parameter space is high-dimensional: SM: 3+ parameters (m t, α s, α,... ) msugra: 5 parameters (m,m /2, A,tan(β),sgn(µ)) General MSSM: 5 parameters On loop-level observables depend on every parameter Simple inversion of the relations not possible Parameter scans Error estimates on parameters in the minimum Find best points (best χ 2 ) using different fitting techniques: + Reasonably fast Gradient search (Minuit) fixed Grid scan Weighted Markov Chains Limited convergence, only best fit + scans complete parameter space many points needed (O(e N )) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p.

Markov Chains Markov Chain (MC): Sequence of points, chosen by an algorithm (Metropolis-Hastings), only depending on its direct predecessor Picks a set of average points according to a potential V (e.g. inverse log-likelihood, /χ 2 ) Point density resembles the value of V (i.e. more points in region with high V ) Scans high dimensional parameter spaces efficiently [Baltz, Gondolo 24] msugra MC scans with current exp. limits [Allanach, Cranmer, Lester, Weber 25-7; Roszkowski, Ruiz de Austra, Trotta 26/7] Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 2

Weighted Markov Chains Weighted Markov Chains: Improved evaluation algorithm for binning: [Plehn, MR] Weight points with value of V : Take care of Overcounting because point density is already weighted ( number of points ) /V (point) Ppoints [based on Ferrenberg, Swendsen 988] Correct account for regions with zero probability (maintain additional chain which stores points rejected because V (point) = ) + Fast scans of high-dimensional spaces O(N) + Does not rely on shape of χ 2 (no derivatives used) + Can find secondary distinct solutions Exact minimum difficult to find Additional gradient fit Bad choice of proposal function for next point leads to bad coverage of the space Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 3

msugra as a Toy Model msugra with LHC measurements (SPSa kinematic edges): pick one set of measurements, randomly smeared from the true values Free parameters: m, m /2, tan(β), A, sgn(µ), m t SFitter output : Fully-dimensional exclusive likelihood map (colour: minimum χ 2 over all unseen parameters) SFitter output 2: Ranked list of minima: m /2 8 6 4 2 2 4 6 8 m e+7 χ 2 m m /2 tan(β) A µ m t SPSa. 25.. -. + 7.4 ).32.4 25.2 2.7-7.7 + 7.9 2) 7.8 6.3 243.6 4.3-3.3 7.7 3) 3.9 3.5 258.2 2.2 848.4 + 74.4 4) 75. 7.3 25.4 5. 778.8 73.6 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 4

Bayesian or Frequentist? SFitter provides full-dimensional log-likelihood map project onto plotable - or 2-dimensional spaces Bayesian: m /2 8 6 4 2 2 4 6 8 m e+7 /χ 2 3.5e-5 3e-5 2.5e-5 2e-5.5e-5 e-5 5e-6 2 4 6 8 m /2 Marginalisation of χ 2 in all other directions Frequentist: m /2 8 6 4 2 2 4 6 8 m e+7 /χ 2 9 8 7 6 5 4 3 2 2 4 6 8 Profile likelihood: Value of bin is value of smallest χ 2 occuring in this bin m /2 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Bayesian or Frequentist? SFitter provides full-dimensional log-likelihood map project onto plotable - or 2-dimensional spaces Bayesian: m /2 8 6 4 2 2 4 6 8 m e+7 /χ 2 3.5e-5 3e-5 2.5e-5 2e-5.5e-5 e-5 5e-6 2 4 6 8 m /2 Marginalisation of χ 2 in all other directions Frequentist: m /2 8 6 4 2 2 4 6 8 m e+7 /χ 2 9 8 7 6 5 4 3 2 2 4 6 8 m /2 Different methods answer different questions. Bayesian and Frequentist! Everybody can choose his/her favourite analysis... Profile likelihood: Value of bin is value of smallest χ 2 occuring in this bin Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 5

Purely high-scale model m, m /2, A defined at the GUT-scale tan(β) defined at the weak scale Replace tan(β) with high-scale quantity B Flat prior in B yields prior tan(β) 2 SPSa with LHC kinematic edges (tan(β) vs. /χ 2 ): /χ 2 /χ 2 9 8 7 6 5 4 3 2 7e-6 6e-6 5e-6 4e-6 3e-6 2e-6 e-6 flat B prior 5 5 2 25 3 tan(β) 5 5 2 25 3 tan(β) /χ 2 /χ 2.3.25.2.5..5 flat tan(β) prior 2.5e-6 2e-6.5e-6 e-6 5e-7 5 5 2 25 3 tan(β) 5 5 2 25 3 tan(β) Bayesian: Large influence of choice of prior Choosing flat B prior strongly favours low values of tan(β). Frequentist: Two plots should be identical (no prior in χ 2 calculation) Indirect influence via Markov Chain proposal function Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 6

Error determination Treatment of errors: All experimental errors are Gaussian σ 2 exp = σ2 stat + σ2 syst(j) + σ2 syst(l) Systematic errors from jet (σ syst(j) ) and lepton energy scale (σ syst(l) ) assumed 99% correlated each.8.6.4.2.8.6.4.2 -.2 -σ theo -σ exp Theory error added as box-shaped ( (RFit scheme [Hoecker, Lacker, Laplace, Lediberder]) 2log L χ 2 = P measurements Parameter errors: SPSa theo exp zero theo exp zero xdata x pred σ theo -σ theo for x data x pred < σ theo «2 for x data x pred σ theo σexp theo exp gauss theo exp flat LHC masses LHC edges m 4..5 2.97 2.7 m /2 25 2..73 2.99 2.64 tan β.42.65 3.36 2.45 A - 35.4 2.2 5.5 49.6 m t 7.4.93.26.89.97 Use kinematic edges for parameter determination instead of masses +σ theo χ 2 +σ theo +σ exp Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 7

Weak-scale MSSM No need to assume specific SUSY-breaking scenario SUSY-breaking mechanism should be induced from data Use of Markov Chains makes scanning the 9-dimensional parameter space feasible Lack of sensitivity on one parameter does not slow down the scan (no need to fix parameters) Same SFitter output as before: Minima list and Likelihood map MSSM using SPSa spectrum and LHC kinematic edges: (Bayesian, full parameter space) M 2 8 6 4 2 2 4 6 8 M... e-4 e-5 % of points 9 8 7 6 5 4 3 2 2 4 6 8 M Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 8

Search Strategy () Full scan of 9D parameter space challenging Four-step procedure yields better and faster results: Weighted-Markov-Chain run with flat pdf over full parameter space 5 best points additionally minimised (full scan, no bias on starting point) M 2 8 6 4 2 2 4 6 8 M... e-4 e-5 % of points 9 8 7 6 5 4 3 2 2 4 6 8 M.6 e L 8 6 4... /χ 2.4.2. 8e-5 6e-5 2 4e-5 2e-5 2 4 6 8 M 2 2 3 4 5 6 7 8 9 M Correlations not aligned with parameters washed out in plots Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 9

Search Strategy (2) Full scan of 9D parameter space challenging Four-step procedure yields better and faster results: Weighted-Markov-Chain run with flat pdf over full parameter space 5 best points additionally minimised (full scan, no bias on starting point) Weighted-Markov Chain with flat pdf on Gaugino-Higgsino subspace: M, M 2, M 3, µ, tan β, m t Additional Minuit run with 5 best solutions Step Step 2 M 2 8 6 4... e-4 e-5 M 2 8 6 4 e+9 e+7 2 2 2 4 6 8 2 4 6 8 M M Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 2

Search Strategy (2) - results Only three neutralinos (χ, χ 2, χ 4 ) with masses (97.2 GeV, 8.5 GeV, 375.6 GeV ) and no charginos observable at the LHC in SPSa Mapping (M, M 2, µ) (χ, χ 2, χ 4 ) not unique sgn µ basically undetermined by collider data 8-fold solution.6 5 e+8 e+6.5.4 µ -5 /χ 2.3.2. - 2 4 6 8 M 2 4 6 8 M Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 2

Search Strategy (2) - results Only three neutralinos (χ, χ 2, χ 4 ) with masses (97.2 GeV, 8.5 GeV, 375.6 GeV ) and no charginos observable at the LHC in SPSa Mapping (M, M 2, µ) (χ, χ 2, χ 4 ) not unique sgn µ basically undetermined by collider data 8-fold solution µ < µ > SPSa M 96.6 75. 3.5 365.8 98.3 76.4 5.9 365.3 M 2 8.2 98.4 35. 3.9 87.5 3.9 348.4 37.8 µ -354. -357.6-77.7-59.9 347.8 352.6 78. 6.5 tan β 4.6 4.5 29. 32. 5. 4.8 29.2 32. M 3 583.2 583.3 583.3 583.5 583. 583. 583.3 583.4 m t 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 2

Search Strategy (3) Full scan of 9D parameter space challenging Four-step procedure yields better and faster results: Weighted-Markov-Chain run with flat pdf over full parameter space 5 best points additionally minimised (full scan, no bias on starting point) Weighted-Markov Chain with flat pdf on Gaugino-Higgsino subspace: M, M 2, M 3, µ, tan β, m t Additional Minuit run with 5 best solutions Weighted-Markov Chain with Breit-Wigner-shaped pdf on remaining parameters for all solutions of previous step Minimisation for best 5 points /χ 2 4.5e-6 4e-6 3.5e-6 3e-6 2.5e-6 2e-6.5e-6 e-6 5e-7 2 4 6 8 /χ 2.25.2.5..5 2 4 6 8 q L q L Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 22

Search Strategy (4) Full scan of 9D parameter space challenging Four-step procedure yields better and faster results: Weighted-Markov-Chain run with flat pdf over full parameter space 5 best points additionally minimised (full scan, no bias on starting point) Weighted-Markov Chain with flat pdf on Gaugino-Higgsino subspace: M, M 2, M 3, µ, tan β, m t Additional Minuit run with 5 best solutions Weighted-Markov Chain with Breit-Wigner-shaped pdf on remaining parameters for all solutions of previous step Minimisation for best 5 points Minuit run for best points of last step keeping all parameters variable Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 23

Best points µ < µ > SPSa M 96.6 75. 3.5 365.8 98.3 76.4 5.9 365.3 M 2 8.2 98.4 35. 3.9 87.5 3.9 348.4 37.8 µ -354. -357.6-77.7-59.9 347.8 352.6 78. 6.5 tan β 4.6 4.5 29. 32. 5. 4.8 29.2 32. M 3 583.2 583.3 583.3 583.5 583. 583. 583.3 583.4 M τl 4.9 274.3 28.3 4794.2 28. 229.9 3269.3 8.6 M τr 348.8 29.9 292.7 3. 2266.5 38.5 29.9 255. M µl 92.7 92.7 92.7 92.9 92.6 92.6 92.7 92.8 M µr 3. 3. 3. 3.3 3. 3. 3. 3.2 MẽL 86.3 86.4 86.4 86.5 86.2 86.2 86.4 86.4 MẽR 3.5 3.5 3.6 3.7 3.4 3.4 3.5 3.6 M q3l 497. 497.2 494. 494. 495.6 495.6 495.8 495. M t R 73.9 92.3 547.9 95.8 547.9 46.5 978.2 52. M br 497.3 497.3 5.4 5.9 498.5 498.5 498.7 499.6 M ql 525. 525.2 525.3 525.5 525. 525. 525.2 525.3 M qr 5.3 5.3 5.4 5.5 5.2 5.2 5.4 5.5 A t ( ) -252.3-348.4-477. -259. -47. -484.3-243.4-465.7 A t (+) 384.9 48.8 64.5 432.5 739.2 774.7 44.5 656.9 m A 35.3 725.8 263. 2. 7.6 56.5 897.6 256. m t 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 24

Degenerate Solutions In total 9 parameters constrained by 22 measurements Measurements constructed from only 5 underlying masses Complete determination of parameter set not possible Five parameters not well constrained m A no heavy Higgses measurable M t R A t stop sector parameters do not enter edge measurements M τl or M τr only the lighter stau measured tan β change can always be accomodated by rotating M, M 2, M q3,... Single common link: m h 4-dimensional hyperplane in parameter space undetermined Can still assign errors to some of the badly determined parameters Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 25

Error analysis Technical procedure as in msugra case: smeared data sets Minimum determined for each data set individually Error determined from fit with Gaussian Most constrained parameters determinable with 5% accuracy Inclusion of theory errors leads to an increase of factor 2 on the parameter errors ILC data complementary to LHC Combination of the two experiments allows for precise determination of all parameters LHC ILC LHC+ILC SPSa M 2.± 7.8 3.±. 3.±.84 3. MẽR 35.± 8.3 35.8±.8 35.9±.77 35.8 m A 46.3±O( 3 ) 393.8±.6 393.9±.6 394.9 M t R 45.8±O( 2 ) 44.±O(4 2 ) 4.7±48.4 48.3 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 26

Testing Unification Apparent unification of gauge coupling parameters in the MSSM Question arises: Do other parameters unify as well? Should be tested by bottom-up running from weak scale to Planck scale Can give hints about supersymmetry breaking (e.g. test scalar-mass sum rules with a sliding scale) [Schmaltz et al.] Bottom-up running of gaugino masses and 3rd-generation sfermion masses:..8 M M 2 M 3 M τr,m τl,m t R,M br,m q3 L ; M 3 = GeV 7 6 /M i.6.4.2 5 4 3 2 2 4 6 8 2 4 6 8 log(q) 2 4 6 8 2 4 6 8 log(q) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 27

Testing Unification Apparent unification of gauge coupling parameters in the MSSM Question arises: Do other parameters unify as well? Should be tested by bottom-up running from weak scale to Planck scale Can give hints about supersymmetry breaking (e.g. test scalar-mass sum rules with a sliding scale) [Schmaltz et al.] Bottom-up running of gaugino masses and 3rd-generation sfermion masses: /M i..8.6.4.2 2 4 6 8 2 4 6 8 log(q) M M 2 M 3 M τr,m τl,m t R,M br,m q3 L ; M 3 = GeV 7 6 5 4 3 2 2 4 6 8 2 4 6 8 For Demonstration Only log(q) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 27

Summary & Outlook Current status: Low-energy supersymmetry fits precision data very well Parameter scans important to determine Lagrangian parameters from observables Problem of high-dimensional parameter spaces Markov Chains can do this effectively Improved algorithm developed Two types of output: Likelihood map and list of best points Both Bayesian and Frequentist from likelihood map Bayesian output significantly dependent on priors Tested with msugra SPSa: can reconstruct SPSa from (simulated) LHC data Repeated procedure with weak-scale MSSM: reconstruction works as well SFitter (despite its name) not tied to SUSY extend to other models/problems Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 28

Backup Slides Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 29

Metropolis-Hastings Algorithm start point suggested point probability density function (PDF) V (suggested point) V (start point) yes? > random number [, ) no replace add point to chain Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 3

Experimental Input (Edges) msugra SPSa as a benchmark point: m = GeV, m /2 = 25 GeV, A = GeV, tan β =, sgn µ = +, m t = 7.4 GeV LHC experimental data from cascade decays (best precision obtainable) q q l ± l χ 2 l χ Measurement Value Errors ( GeV ) ( GeV ) (stat) (syst) (m max llq ):Edge( q L,χ 2,χ ) 446.44.4 4.3 (m min llq ) :Thres( q L,χ 2, µ R,χ ) 2.95.6 2. (m low lq ) :Edge( c L,χ 2, µ R) 36.5.9 3. (m high lq ):Edge( c L,χ 2, µ R,χ ) 392.8. 3.8......... Theoretical Errors: msugra: 3% for gluino and squark masses, % for all other sparticle masses MSSM: % for gluino and squark masses,.5% for all other sparticle masses m h : 2 GeV (unknown higher order terms) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 3

Experimental Input (Edges) (Obs) = (meas) ± (exp) ± (theo) m h = 9.53 ±.25 ± 2. m t = 7.4 ±. ±. m µl,χ = 6.26 ±.6 ±. m g,χ = 59.96 ± 2.3 ± 6. m cr,χ = 45.52 ±. ± 4.2 m g, b = 98.97 ±.5 ±. m g, b2 = 64.6 ± 2.5 ±.7 Edge(χ 2, µ R,χ ) = 79.757 ±.3 ±.8 (mmax Edge( c L,χ 2,χ ) = 446.44 ±.4 ± 4.3 (mmax llq ) Edge( c L,χ 2, µ R) = 36.5 ±.9 ± 3. (m low lq ) Edge( c L,χ 2, µ R,χ ) = 392.8 ±. ± 3.8 (mhigh lq ) Edge(χ 4, µ R,χ ) = 257.4 ± 2.3 ±.3 (mmax ll (χ 4 )) Edge(χ 4, τ L,χ ) = 82.993 ± 5. ±.8 (mmax ττ ) ll ) Threshold( c L,χ 2, µ R,χ ) = 2.95 ±.6 ± 2. (mmin Threshold( b,χ 2, µ R,χ ) = 2.95 ±.6 ± 2. (mmin llb ) llq ) Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 32

msugra around Minima positive µ Bayesian A m t m t 2 9 8 7 6 - -5 5 5 2 A /χ 2.6.5.4.3.2. - -5 5 5 2 /χ 2.9.8.7.6.5.4.3.2. 6 7 8 9 2 A m t Frequentist A m t m t 2 9 8 7 6 - -5 5 5 2 A /χ 2 6 5 4 3 2 - -5 5 5 2 /χ 2 6 5 4 3 2 6 7 8 9 2 A m t Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 33

msugra around Minima negative µ Bayesian A m t m t 2 9 8 7 6 - -5 5 5 2 A /χ 2.5.45.4.35.3.25.2.5. 5e-5 - -5 5 5 /χ 2.8.7.6.5.4.3.2. 6 7 8 9 2 A m t Frequentist A m t m t 2 9 8 7 6 - -5 5 5 2 A /χ 2.2.8.6.4.2 - -5 5 5 /χ 2.2.8.6.4.2 6 7 8 9 2 A m t Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 34

Error determination Minuit output not usable for flat theory errors: Migrad function depends on parabolic approximation Cannot determine χ 2 for Minos to yield 68% CL intervals Need more general approach Perform, toy experiments with measurements smeared around correct value Minimise each toy experiment Plot resulting distribution of parameter points and fit with Gaussian.25.2.5..5 9 95 5 m.8.6.4.2..8.6.4.2 2 3 4 5 χ 2 Flat theory errors.4.2..8.6.4.2 9 95 5 m.8.7.6.5.4.3.2. 2 3 4 5 χ 2 Gaussian theory errors Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 35

MSSM errors LHC ILC LHC+ILC SPSa tan β.± 4.5 3.4± 6.8 2.3± 5.3. M 2.± 7.8 3.±. 3.±.84 3. M 2 93.3± 7.8 93.4± 3. 93.2± 2.3 92.9 M 3 577.2± 4.5 fixed 5 579.7± 2.8 577.9 M τl 227.8±O( 3 ) 83.8± 6.6 87.3± 2.9 93.6 M τr 64.±O( 3 ) 43.9± 7.9 4.± 4. 33.4 M µl 93.2± 8.8 94.4±. 94.5±. 94.4 M µr 35.± 8.3 35.9±. 36.±.89 35.8 MẽL 93.3± 8.8 94.4±.89 94.4±.84 94.4 MẽR 35.± 8.3 35.8±.8 35.9±.77 35.8 M q3l 48.4± 22. 57.2±O(4 2 ) 486.6± 9.5 48.8 M t R 45.8±O( 2 ) 44.±O(4 2 ) 4.7± 48.4 48.3 5.7± 7.9 fixed 5 54.± 7.4 52.9 M br M ql 524.6± 4.5 fixed 5 526.± 7.2 526.6 M qr 57.3± 7.5 fixed 5 58.4± 6.7 58. A τ fixed 633.2± O( 4 ) 39.6±O( 4 ) -249.4 A t -59.± 86.7-56.± O( 3 ) -5.± 43.4-49.9 A b fixed fixed -686.2±O( 4 ) -763.4 m A 46.3±O( 3 ) 393.8±.6 393.9±.6 394.9 µ 35.5± 4.5 343.7± 3. 354.8± 2.8 353.7 m t 7.4±. 7.4±.2 7.4±.2 7.4 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 36

Dark Matter Content of the universe: 73% Dark energy 4% Ordinary matter MSSM: χ 23% Dark matter [ NASA/WMAP Science Team as LSP ideal candidate for cold dark matter (CDM): massive, weakly interacting ] SFitter: Determine Lagrangian parameters Spectrum and couplings e.g. micromegas: Calculate relic density Ω CDM h 2 = n LSP m LSP [Bélanger et al.] Prediction of Ω CDM h 2 LHC : Ω CDM h 2 =.96 ±.33 LHC+ILC: Ω CDM h 2 =.9 ±.3 (improvement by one order of magnitude) Compare with experiment (Measurement of the fluctuations of the cosmic microwave background): WMAP: Ω CDM h 2 =.277 ±.8 [astro-ph/63449] Planck: Ω CDM h 2 =? ±.6 Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 37

Example Test function (5-dim): Small Hypersphere r =, V max = 75 @ (65, 25,35, 35, 35) Cuboid d = (73,2, 2,2, 2), V max = 6 @ (85,225, 65, 65,65) Cube d = (,, 3,3, 3), V max = 25 @ (75,75, 45,45, 45) Gaussian σ = (5, 5,5, 5, 5), V max = 6 @ (25,25, 55,55, 55) Big Hypersphere r = 3, V max = 2 @ (35,65, 65, 65,65) Background V =. + 4 3 x 2 x2 2 x2 3 x2 4 x2 5 x 2 8 6 4 2 2 4 6 8 x.9.8.7.6.5.4.3.2.. V=74.929@(655.,253.72,347.83,348.57,349.59) 2. V=59.972@(85.4,224.99,65.,649.99,654.56) 3. V=58.29@(849.97,225.,587.8,65.,65.2) 4. V=25.@(75.,749.99,45.,45.,45.) 5. V=6.42@(245.45,253.44,552.5,542.58,544.75) 6. V=2.6@(35.7,65.4,65.36,65.4,65.38) 7.... Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 38

Plot Details Parameters: x,...,x 5 [, ] Bins: 5 5 PDF: Breit-Wigner ( + x 2 i /σ2 ) with σ = Number of Markov chains: 9 Number of points per chain: 7 Number of function evaluations: 33, 797, 53 Acceptance ratio:.9 Final r (measure of convergence):.85 CPU time (3 GHz): 5 min Michael Rauch SFitter: Measuring Supersymmetry Edinburgh, Dec 7 p. 39