MSSM Parameter Reconstruction at the LHC

Transcription

1 MSSM Parameter Reconstruction at the LHC Michael Rauch Rémi Lafaye, Tilman Plehn, Dirk Zerwas Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 1

2 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 [Stöckinger] Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 2

3 Parameter Scans MSSM parameter space is high-dimensional: SM: 3+ parameters (m t, α s, α,... ) msugra: 5 parameters (m,m 1/2, A,tan(β),sgn(µ)) General MSSM: 15 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: + scans complete parameter space fixed Grid scan many points needed (O(e N )) Gradient search (Minuit) Weighted Markov Chains + Reasonably fast Limited convergence, only best fit Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 3

4 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, 1/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 Weighted Markov Chains: Improved evaluation algorithm for binning: Weight points with value of V : ( [Allanach, Lester, Weber 25-7; Roszkowski, Ruiz de Austra, Trotta 26/7] [Plehn, MR] number of points ) [based on Ferrenberg, Swendsen 1988] Ppoints 1/V (point) 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 not found Additional gradient fit Bad choice of proposal function for next point leads to bad coverage of the space Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 4

5 msugra as a Toy Model msugra with LHC measurements (SPS1a kinematic edges): pick one set of measurements, randomly smeared from the true values Free parameters: m, m 1/2, tan(β), A, sgn(µ), m t 1 SFitter output 1: Fully-dimensional exclusive likelihood map projected onto lower-dimensional space using marginalisation (Bayesian) profile likelihood (Frequentist) SFitter output 2: Ranked list of minima: m 1/ m 1e+7 1e χ 2 m m 1/2 tan(β) A µ m t SPS1a ) ) ) ) Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 5

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 σ theo -σ exp Theory error added as box-shaped ( (RFit scheme [Hoecker, Lacker, Laplace, Lediberder]) 2log L χ 2 = P measurements Parameter errors: SPS1a theo exp flat LHC masses 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 LHC edges theo exp flat m m 1/ tan β A m t σ theo χ 2 +σ theo +σ exp Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 6

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 19-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 SPS1a spectrum and LHC kinematic edges: (Bayesian, full parameter space) M M e-4 1e-5 % of points M 1 Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 7

8 Search Strategy (1) Full scan of 19D 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 1, M 2, M 3, µ, tan β, m t Additional Minuit run with 15 best solutions Step 1 Step M e-4 1e-5 M e+9 1e M 1 M 1 Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 8

9 Search Strategy (1) - results Only three neutralinos (χ 1, χ 2, χ 4 ) with masses (97.2 GeV, 18.5 GeV, GeV ) and no charginos observable at the LHC in SPS1a Mapping (M 1, M 2, µ) (χ 1, χ 2, χ 4 ) not unique sgn µ basically undetermined by collider data 8-fold solution e+8 1e µ /χ M M 1 Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 9

10 Search Strategy (1) - results Only three neutralinos (χ 1, χ 2, χ 4 ) with masses (97.2 GeV, 18.5 GeV, GeV ) and no charginos observable at the LHC in SPS1a Mapping (M 1, M 2, µ) (χ 1, χ 2, χ 4 ) not unique sgn µ basically undetermined by collider data 8-fold solution µ < µ > SPS1a M M µ tan β M m t Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 9

11 Search Strategy (2) Full scan of 19D 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 1, M 2, M 3, µ, tan β, m t Additional Minuit run with 15 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 Lack of measurements leads to underdeterminedness of parameter space: tan β, m A, M t R, A t, one of M τl or M τr not well constrained 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: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 1

12 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:.1.8 M 1 M 2 M 3 M τr,m τl,m t R,M br,m q3 L ; M 3 = 1 GeV 7 6 1/M i log(q) log(q) Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 11

13 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: 1/M i log(q) M 1 M 2 M 3 M τr,m τl,m t R,M br,m q3 L ; M 3 = 1 GeV For Demonstration Only log(q) Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 11

14 Summary & Outlook 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 Weak-scale MSSM successfully reconstructed from (simulated) LHC data Eight-fold degeneracy in the gaugino/higgsino sector Some parameters only very badly determined SFitter (despite its name) not tied to SUSY extend to other models/problems Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 12

15 Backup Slides Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 13

16 Experimental Input (edges) [Rémi Lafaye] Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 14

17 Experimental Input (edges) (Obs) = (meas) ± (exp) ± (theo) m h = ±.25 ± 2. m t = ± 1. ±. m µl,χ 1 = ± 1.6 ±.1 m g,χ 1 = ± 2.3 ± 6. m cr,χ 1 = ± 1. ± 4.2 m g, b1 = ± 1.5 ± 1. m g, b2 = ± 2.5 ±.7 Edge(χ 2, µ R,χ 1 ) = ±.3 ±.8 (mmax Edge( c L,χ 2,χ 1 ) = ± 1.4 ± 4.3 (mmax llq ) Edge( c L,χ 2, µ R) = ±.9 ± 3. (m low lq ) Edge( c L,χ 2, µ R,χ 1 ) = ± 1. ± 3.8 (mhigh lq ) Edge(χ 4, µ R,χ 1 ) = ± 2.3 ±.3 (mmax ll (χ 4 )) Edge(χ 4, τ L,χ 1 ) = ± 5. ±.8 (mmax ττ ) ll ) Threshold( c L,χ 2, µ R,χ 1 ) = ± 1.6 ± 2. (mmin Threshold( b 1,χ 2, µ R,χ 1 ) = ± 1.6 ± 2. (mmin llb ) llq ) Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 15

18 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 1, toy experiments with measurements smeared around correct value Minimise each toy experiment Plot resulting distribution of parameter points and fit with Gaussian m 1/ χ 2 Flat theory errors m 1/ χ 2 Gaussian theory errors Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 16

19 Example Test function (5-dim): Small Hypersphere r = 1, V max = (65, 25,35, 35, 35) Cuboid d = (173,12, 2,2, 2), V max = (85,225, 65, 65,65) Cube d = (1,1, 3,3, 3), V max = (75,75, 45,45, 45) Gaussian σ = (5, 15,15, 15, 15), V max = (25,25, 55,55, 55) Big Hypersphere r = 3, V max = (35,65, 65, 65,65) Background V = x 2 1 x2 2 x2 3 x2 4 x2 5 x x 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=58.219@(849.97,225.1,587.8,65.1,65.2) 4. V=25.11@(75.,749.99,45.,45.1,45.1) 5. V=16.42@(245.45,253.44,552.51,542.58,544.75) 6. V=12.116@(35.7,65.4,65.36,65.4,65.38) Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 17

20 Plot Details Parameters: x 1,...,x 5 [, 1] Bins: 5 5 PDF: Breit-Wigner ( 1 1+ x 2 i /σ2 ) with σ = 1 Number of Markov chains: 9 Number of points per chain: 1 7 Number of function evaluations: 33, 797, 153 Acceptance ratio:.19 Final r (measure of convergence): CPU time (3 GHz): 15 min Michael Rauch SFitter: MSSM Parameter Reconstruction at the LHC SUSY-GDR 7 p. 18