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Global Fits: Making Sense out of What we Will See Oliver Buchmüller Roberto Trotta Imperial College London •Global Fits – Motivation & History • “SUSY Fits” – where are we today? •The future Joint HEP-APP IOP meeting on SUSY - March 24th 2010 “Random Scans”: Why we Need Better Tools Gogoladze et al (2008) [one of many such studies] • Points accepted/rejected in a in/out fashion (e.g., 2-sigma cuts) • No statistical measure attached to density of points: no probabilistic interpretation of results possible • Inefficient/Unfeasible in high dimensional parameters spaces (N>3) • Explores only a very limited portion of the parameter space! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Statistics in High-D spaces • “Random scans” of a high-dimensional parameter space only probe a very limited sub-volume: this is the concentration of measure. • Statistical fact: the norm of d draws from U[0,1] concentrates around (d/3)1/2 with constant variance Tom Loredo (cosmostats09 talk) Damien Francois (2005) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Geometry in High-D Spaces • Geometrical fact: in d dimensions, most of the volume is near the boundary. The volume inside the spherical core of d-dimensional cube is negligible. Volume of cube 1 Volume of sphere 1 Ratio Sphere/Cube Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Scanning the New Physics Parameter Space Determining the preferred NP parameter space is a multi-dimensional problem. Even in one of the simplest cases, the CMSSM, there are four NP parameters (M0, M1/2, A0, tanβ) as well as SM parameters like mtop or mb. The conventional (now historical) strategy in the field was to carry out “2D scans” by fixing the other relevant parameters to certain values. An arbitrary example: 0106334 [hep-ph] tanβ=10 tanβ=50 2D scans strongly depend on the “fixed NP parameters” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 2 Scanning the New Physics Parameter Space Mtop=170 GeV Mtop=180 GeV There is also a strong dependence on the important SM parameters! (which are known only with a limited accuracy) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 3 The Solution to the Problem – Global Fits Carry out a simultaneous fit of all relevant NP and SM parameter to the experimental data/constraints. Valid for all NP and SM parameters Marginalize (= integrate) or maximise along the hidden dimensions so as to obtain results that account for the multi-dimensional nature of the problem. This gives a statistically welldefined answer. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 5 The Solution to the Problem – Global Fits Another Example: the so-called “WMAP strips” WMAP strips a few years ago Wmap blobs today In 2D scans, enforcing the cosmological relic abundance results in narrow “allowed regions” (the “WMAP strips”), whose location changes with the value of the fixed parameters. Once fixed parameters are included and hidden dimensions accounted for, WMAP strips widen to become “WMAP blobs” 0306219 [hep-ph] Joint HEP-APP IOP meeting on SUSY - March 24th 2010 6 Different Statistical Approaches Bayesian vs Frequentist Different answers to different questions ρbar = 0.155 [+0.022 -0.022] ηbar = 0.342[+0.014 -0.014] ρbar = 0.139 [+0.025 -0.027] ηbar = 0.341[+0.016 -0.015] CKM triangle Fits Today: Overall good agreement between the two approaches but still differences in the details. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Marginal Posterior vs Profile Likelihood Bayesian vs Marginal posterior: Frequentist Profile likelihood: Physical analogy: (thanks to Tom Loredo) Heat: Posterior: Relevant for underconstrained system θ2 Posterior = region with most heat likelihood = “hottest” hypothesis (plot depicts likelihood contours - prior assumed flat over wide range) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 θ1 Bayesian Statistic on the Rise? As far as I can think back, the frequentist approach has dominated the field of particle physics – it still does but bayesian methodology becomes increasingly more important it seems. In particular there is strong influence from the field of astrophysics: Joint HEP-APP IOP meeting on SUSY - March 24th 2010 11 Global NP Fits in the LHC (Discovery) Era R. Lafaye , M. Rauch, T. Plehn, D. Zerwas H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J. Stelzer P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann GFitter Sfitter Fittino L. Roszkowski, R. Ruiz de Austri, R. Trotta O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R. Ellis, H.Flacher, S. Heinemeyer, G. Isidori, K.A. Olive, F.J. Ronga, G. Weiglein S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P. Hobson MasterCode Superbayes Joint HEP-APP IOP meeting on SUSY - March 24th 2010 13 Global NP Fits in the LHC (Discovery) Era R. Lafaye , M. Rauch, T. Plehn, D. Zerwas H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J. Stelzer P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann GFitter Sfitter Fittino Many different groups of theorists and experimentalists and several different approaches! L. Roszkowski, R. Ruiz de Austri, R. Trotta O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R. Ellis, H.Flacher, S. Heinemeyer, G. Isidori, K.A. Olive, F.J. Ronga, G. Weiglein S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P. Hobson MasterCode Superbayes Joint HEP-APP IOP meeting on SUSY - March 24th 2010 14 Global NP Fit Methodology in a Nutshell • Chose a NP model • SUSY as show-case – so far considered models: • • GUT Scale model: CMSSM, NUHM, AMSB Soft Scale model: pMSSM • Chose the measurements • Considered (indirect) constraints are collider (EWK, flavour) and noncollider data (g-2, relic density) • It is crucial to have a consistent set of calculations for these constraints! • Chose your preferred statistical approach to confront model prediction Pi with measurements Mi with cov(Mi,Mi) • “χ2 “ as an example: (M – P)T cov-1 (M – P) [+ limits] • Find set of Pi that minimize χ2 Joint HEP-APP IOP meeting on SUSY - March 24th 2010 15 Constrained MSSM Analysis Pipeline SCANNING ALGORITHM Likelihood = 0 4 CMSSM parameters θ = {m0, m1/2, A0, tanβ} (fixing sign(μ) > 0) NO RGE 4 SM “nuisance parameters” Ψ={mt, mb,αS, αEM } Data: Gaussian likelihoods for each of the Ψj (j=1...4) Non-linear numerical function via SoftSusy 2.0.18 DarkSusy 4.1 MICROMEGAS 2.2 FeynHiggs 2.5.1 Hdecay 3.102 Observable quantities fi(θ ,Ψ) CDM relic abundance BR’s EW observables g-2 Higgs mass sparticle spectrum (gamma-ray, neutrino, antimatter flux, direct detection x-section) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Physically acceptable? EWSB, no tachyons, neutralino CDM YES Joint likelihood function Data: Gaussian likelihood (CDM, EWO, g-2, b→sγ, ΔMBs) other observables have only lower/upper limits Finding the Favoured Regions • Due to the weak nature of constraints, different scanning techniques and statistical methods will generally give different answers (also because the questions being asked are different!) • Traditional method: determine best fit parameter (find minimum) • • • • Markove Chain Monte Carlos (MCMC) MCMC and Minuit as “afterburner” Simulated annealing Genetic algorithm • Determine errors: Local Δ(LogLikelihood) • Intelligent sampling of parameter space with MCMC • Pseudo Experiment/Toy Experiment sampling Joint HEP-APP IOP meeting on SUSY - March 24th 2010 16 Finding the Favoured Regions • Alternatively, one might focus on the probability mass instead (Bayesian) • Best fit has no special status: look for the bulk of the posterior probability mass instead • Markov Chain Monte Carlo techniques (MCMC) • Nested sampling • Hamiltonian MC • Determine errors: region of parameter space containing e.g. 95% of samples • Might depend on measure chosen (prior) • Represents degree of knowledge (rather than coverage) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 16 Today: “Weak Constraints” only Biggest problem today – no significant deviation from the SM! Only ~2 to ~3σ evidence and relic density that cannot be explained in the SM. only weakly constraint NP parameter space The main players today Δaμ =(24.6 ± 8)E-10 Mw =80.399 ± 0.023 GeV Mw Ωh2 =0.1099 ± 0.0066 ± 0.012 Relic density g-2 R(BXsγ) = 1.117±0.12 R(Bτν) = 1.43±0.43 R: Measurement/SM Joint HEP-APP IOP meeting on SUSY - March 24th 2010 21 Where are we today? Details in Frederic Ronga’s and Roberto Ruiz de Austri’s talks Here just a few points... CMSSM Today: Frequentist vs. Bayesian 0809.3792 [hep-ph] 0705.0487 [hep-ph] “naturaless prior” “flat prior” 0808.4128 [hep-ph] “log prior” All priors find a “low mass SUSY” solution similar to the prior independent frequentist result. The key question is how constrained is actually this region? Has the frequentist approach missed “high mass” regions? Joint HEP-APP IOP meeting on SUSY - March 24th 2010 33 Prior Dependence 0809.3792 [hep-ph] “flat prior” 0705.0487 [hep-ph] “flat prior” “naturalness “flat prior” “naturalness prior” “flat prior” prior” “log prior” “log prior” Prior dependence of the Bayesian fits results from weak constraints on parameter space. Stronger assumptions (e.g. naturalness priors) lead to posteriors dominated by prior information (rather than data). Joint HEP-APP IOP meeting on SUSY - March 24th 2010 28 Prior Dependence due to Weak Constraints 0705.0487 [hep-ph] Global Fit to CMSSM with four parameters: M0, M1/2, A0, tanβ [μ>0] strong prior dependence “naturalness “flat prior” prior” Four (CMSSM) vs. two (LVS) free parameters. “Volume” effects coming from large-D parameter spaces are presently important. The question will remain in the future: i.e., once we’ll have LHC data, the frontier will shift to models in higher 0806.1184 [hep-ph] Global Fit to Large dimensions (eg, pMSSM with ~ 20 free parameters) Volume String scenario with only two param.: M0, tanβ [μ>0] smaller prior dependence “flat prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 “naturalness prior” 29 LHC Data (Discoveries) will help a lot If we are really luck we might see these spectacular signatures already very early at the LHC! Assumed ATLAS covariance matrix for the SU3 benchmark Point at 1/fb Assuming such a spectacular discovery as input reduces the prior dependence of the global bayesian fits to almost negligible levels. “flat prior” Joint HEP-APP IOP meeting on SUSY - March0907.0594 24th 2010 [hep-ph] M(l+l-) GeV “log prior” 32 Accelerated Inference from Neural Networks We will need faster methods to deal with more complex models and higher dimensional parameter spaces, independently of the statistics used. Neural networks provide a way forward for ultra-fast inference. Simulated ATLAS data with 1/fb luminosity Computational effort for an 8parameters global CMSSM fit: full MultiNest fit • Standard MCMC (SuperBayeS v1.23, 2006) 500 CPU days • MultiNest algorithm (SuperBayeS v1.35, 2008) <3 CPU days speed-up factor: 200 Neural Network • SuperBayeS+Neural Networks (Bridges, RT et al, upcoming) 15 CPU minutes speed-up factor: 50’000 Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Connection to Astro(particle)physics Direct Dark Matter Searches & LHC 0809.3792 [hep-ph] Sensitivity Plot: WIMP(LSP) Mass vs. pSI pSI: spin-independent dark matter WIMP elastic scattering cross section on a free proton. A convenient way to illustrate direct and indirect WIMP searches. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 0907.4468 [hep-ph] 54 Direct Detection Prospects in the CMSSM Prospects for WIMP discovery in the CMSSM framework for upcming 1t scale detectors are robust, independently of the choice of statistics. Bayesian posterior Profile likelihood 68% 68% 95% 95% Predicted reach with 1 tonne detectors Joint HEP-APP IOP meeting on SUSY - March 24th 2010 The Need for a Multiple Probes Approach • No single probe can cover the whole favoured parameter space, not even the LHC. • Astroparticle probes (direct and indirect detection) can increase the coverage of the favoured parameter space, and deliver increased statistical robustness. • High complementarity with direct detection methods. Need to establish a common language (and approach) to Use all this information consistently Global Fits are one possible tool for this. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Cumberbatch et al (in prep) Conclusions •Global fits are needed as tool to interpret new physics discoveries from multiple probes. •Can be used to explore connections between particle and astroparticle physics as well as cosmology. •Two independent statistical approaches: ought to give the same result (eventually, when data are constraining enough!) •In the meantime, different questions could give different answer if problem is underconstrained (as it is today) •The statistical frontier is a moving target and as we will make progress in understanding the first discoveries our questions will again be more complex! •More details now in the next two talks! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Supplementary material after this slide Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Global NP Fit Methodology in a Nutshell • Chose a NP model • SUSY as show-case – so far considered models: • • GUT Scale model: CMSSM, NUHM, AMSB Soft Scale model: pMSSM • Chose the measurements • Considered (indirect) constraints are collider (EWK, flavour) and noncollider data (g-2, relic density) • It is crucial to have a consistent set of calculations for these constraints! • Chose your preferred statistical approach to confront model prediction Pi with measurements Mi with cov(Mi,Mi) • “χ2 “ as an example: (M – P)T cov-1 (M – P) [+ limits] • Find set of Pi that minimize χ2 Joint HEP-APP IOP meeting on SUSY - March 24th 2010 15 Constrained MSSM Analysis Pipeline SCANNING ALGORITHM Likelihood = 0 4 CMSSM parameters θ = {m0, m1/2, A0, tanβ} (fixing sign(μ) > 0) NO RGE 4 SM “nuisance parameters” Ψ={mt, mb,αS, αEM } Data: Gaussian likelihoods for each of the Ψj (j=1...4) Non-linear numerical function via SoftSusy 2.0.18 DarkSusy 4.1 MICROMEGAS 2.2 FeynHiggs 2.5.1 Hdecay 3.102 Observable quantities fi(θ ,Ψ) CDM relic abundance BR’s EW observables g-2 Higgs mass sparticle spectrum (gamma-ray, neutrino, antimatter flux, direct detection x-section) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 Physically acceptable? EWSB, no tachyons, neutralino CDM YES Joint likelihood function Data: Gaussian likelihood (CDM, EWO, g-2, b→sγ, ΔMBs) other observables have only lower/upper limits AMSB & GMSB 0906.0957 [hep-ph] AMSB M0, M3/2, tanβ “naturalness prior” “flat prior” GMSB Mmess, Λ, tanβ If Ωh2 is only used as upper bound AMSB seems favored. “log prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 “naturalness prior” 37 Non-Universal Higgs Models 0903.1279 [hep-ph] “flat prior” NUHM II M0, M1/2, A0, tanβ, MHu2, MHd2 NUHM I M0, M1/2, A0, tanβ, MH2 [MHu2 = MHd2] “log prior” 0907.4468 [hep-ph] MasterCode Also for NUHM I the frequentist approach clearly favors low mass SUSY! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 38 Closer to the Experiment “Soft Scale Models” A Bottom-up Approach So far we have been far up at the GUT scale but for the first interpretation of the hopefully soon forthcoming LHC discoveries it will be beneficial to also consider “soft scale” models with parameters much closer to the experimental measurements/sensitivities (e.g. masses). Joint HEP-APP IOP meeting on SUSY - March 24th 2010 40 “Soft SUSY breaking” MSSM L Joint HEP-APP IOP meeting on SUSY - March 24th 2010 41 “Soft SUSY breaking” MSSM L Gaugino’s and their masses M3, M2, M1 Joint HEP-APP IOP meeting on SUSY - March 24th 2010 42 “Soft SUSY breaking” MSSM L Squarks and sleptons, and their masses Joint HEP-APP IOP meeting on SUSY - March 24th 2010 43 “Soft SUSY breaking” MSSM L Tri-linear couplings A Joint HEP-APP IOP meeting on SUSY - March 24th 2010 44 “Soft SUSY breaking” MSSM L Higgs sector: 2 complex doublets (1 for u-type, 1 for d-type) Joint HEP-APP IOP meeting on SUSY - March 24th 2010 45 How Much Parameters are Reasonable? 0903.1279 [hep-ph] pMSSM: 20 NP parameter Wow! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 46 “Fitting the Soft Scale” 0903.1279 [hep-ph] Bayesian fit In general very strong prior dependence – not surprising with 20+ parameters and only indirect constraints Joint HEP-APP IOP meeting on SUSY - March 24th 2010 47 “Fitting the Soft Scale” Frequentist fit to 18+ parameters but now assuming LHC input from 300/fb of data. 0907.2589 [hep-ph] MSSM18 LHC 300/fb Joint HEP-APP IOP meeting on SUSY - March 24th 2010 48 “Fitting the Soft Scale” Frequentist fit to 18+ parameters but now assuming LHC input from 300/fb of data. 0907.2589 [hep-ph] Fitting “soft scale” parameters that are much closer to the experimental MSSM18 LHC 300/fb measurements is an interesting approach but at least initially we need to reduce the set of parameters. Question: Can we define a meaningful set Of 4 to 5 NP “soft scale” parameters? Joint HEP-APP IOP meeting on SUSY - March 24th 2010 49 Global Fits and Discoveries A few Projections LHC Data (Discoveries) will help a lot If we are really luck we might see these spectacular signatures already very early at the LHC! M(l+l-) GeV 0808.4128 [hep-ph] 0808.4128 [hep-ph] Including a ”edge discovery” Joint HEP-APP IOP meeting on SUSY - March0907.0594 24th 2010 [hep-ph] 51 More Detailed Studies Use MCMC for likelihood maps Note SPS1a is close to the minima found by global fits (at lease MasterCode and Fittino) SPS1a Test running! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 52 Connection to Astro(particle)physics Direct Dark Matter Searches & LHC 0809.3792 [hep-ph] Sensitivity Plot: WIMP(LSP) Mass vs. pSI pSI: spin-independent dark matter WIMP elastic scattering cross section on a free proton. A convenient way to illustrate direct and indirect WIMP searches. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 0907.4468 [hep-ph] 54 Conclusion • Global NP Fits: Still in a tool and methodology development phase – getting ready for the LHC • Bayesian vs. Frequentist: Two competing schools? • Weakly constraint NP parameter space leads to strong prior dependences of the bayesian results. • The frequentist fits yield consistent results favoring light mass SUSY. • Fair question – implies the prior dependence of the bayesian fits that the frequentist results are not conclusive too? I don’t think so! • In any case the LHC will takes us to “statistic nirvana” (Bob Cousin) • “Soft Scale” Fits: Botton-up approach is a very interesting idea. Certainly closer to the experimental measurements! • How many parameters can we use and which ones? Can simplified (OSET) models be a guidance? Conclusion • Global NP Fits @ LHC: Like in the past (e.g. EW fit, CKM fit), global fits will be an important tool for the interpretation of (LHC) discoveries – hence crucial for establishing the true underlying theory of NP. • SUSY is a show-case and we eventually will also have to consider other NP models – BUT will we have a consistent set of calculations for the various collider and non-collider observable in these NP models? • I would like to thank B. Allanach, R.Trotta, A. De Roeck, F. Ronga, the Fittion, MasterCode, Superbayes groups (and many others) for input and very useful discussions! • MasterCode:O. Buchmueller, R. Cavanaugh, A. De Roeck, J.R. Ellis, H. Flacher, S. Heinemeyer, G. Isidori, K.A. Olive, F.J. Ronga, G. Weiglein • Fittino: P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann • Sfitter: Remi Lafaye , Michael Rauch, Tilman Plehn, Dirk Zerwas • S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P. Hobson • Gfitter: H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J. Stelzer • Superbayes: L. Roszkowski, R. Ruiz de Austri, R. Trotta Joint HEP-APP IOP meeting on SUSY - March 24th 2010 57 Bayesian vs. Frequentist – no math. Bayesian approach. This perspective on probabilities, says that a probability is a measure of a person’s degree of belief in an event, given the information available. Thus, probabilities refer to a state of knowledge held by an individual, rather than to the properties of a sequence of events. The use of subjective probability in calculations of the expected value of actions is called subjective expected utility. Frequentist approach. They understand probability as a long-run frequency of a ‘repeatable’ event and developed a notion of confidence intervals. Probability would be a measurable frequency of events determined from repeated experiments. Reichenbach, Giere or Mayo have defended that approach from a philosophical point of view, referred to by Mayo (1997) as the ‘error statistical view” (as opposed to the Bayesian or “evidential-relation view”). Joint HEP-APP IOP meeting on SUSY - March 24th 2010 58 Posterior & Profile Likelihood Slide from R. Trotta Joint HEP-APP IOP meeting on SUSY - March 24th 2010 59 Constraints Joint HEP-APP IOP meeting on SUSY - March 24th 2010 60 Posterior vs. Profile Likelihood Posterior 0809.3792 [hep-ph] Profile Likelihood “flat prior” “log prior” Profile Likelihood (PL) should not depend on prior assumptions (in contrast to the Posterior pdf) but when translating the prior dependent MCMC sampling into PLs the used sampling statistic is often not sufficient – thus resulting in a also “prior dependent” PL result – this should vanish with a complete sampling Joint HEP-APP IOP meeting on SUSY - March 24th 2010 61 Naturalness Prior See e.g. 075.0487 [hep-ph] Usually Higgs potential parameter B and μ are treated for Mz and tanβ using: Thus leading to the following “flat” prior assumption” Not using the EWK symmetry breaking conditions enables uniform prior assumptions in the more fundamental parameters B and μ: “Naturalness prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 62 Higgs Sector Joint HEP-APP IOP meeting on SUSY - March 24th 2010 63 Higgs Sector Joint HEP-APP IOP meeting on SUSY - March 24th 2010 64 Higgs Sector Joint HEP-APP IOP meeting on SUSY - March 24th 2010 65