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Global Fits: Making Sense out of What we Will See

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.

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Global Fits: Making Sense out of What we Will See

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  1. 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

  2. “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

  3. 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

  4. 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

  5. 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

  6. 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

  7. The Solution to the Problem – Global Fits Valid for all NP and SM parameters Carry out a simultaneous fit of all relevant NP and SM parameter to the experimental data/constraints. 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 well-defined answer. Joint HEP-APP IOP meeting on SUSY - March 24th 2010 5

  8. Wmap blobs today The Solution to the Problem – Global Fits Another Example: the so-called “WMAP strips” WMAP strips a few years ago 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

  9. Different Statistical Approaches Bayesian vs Frequentist Different answers to different questions ρbar = 0.139 [+0.025 -0.027] ηbar = 0.341[+0.016 -0.015] ρbar = 0.155 [+0.022 -0.022] ηbar = 0.342[+0.014 -0.014] 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

  10. Marginal Posterior vs Profile Likelihood Bayesian vs Frequentist Marginal posterior: Profile likelihood: Physical analogy: (thanks to Tom Loredo) Heat: Posterior: Relevant for underconstrained system In general different if: likelihood = “hottest” hypothesis Posterior = region with most heat θ2 (plot depicts likelihood contours - prior assumed flat over wide range) θ1 Joint HEP-APP IOP meeting on SUSY - March 24th 2010

  11. 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

  12. R. Lafaye , M. Rauch, T. Plehn, D. Zerwas  S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P. Hobson Global NP Fits in the LHC (Discovery) Era Sfitter P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J. Stelzer GFitter 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 MasterCode Superbayes Joint HEP-APP IOP meeting on SUSY - March 24th 2010 13

  13. R. Lafaye , M. Rauch, T. Plehn, D. Zerwas  S.S. AbdusSalam, B.C. Allanach, M.J. Dolan, F. Feroz, M.P. Hobson Global NP Fits in the LHC (Discovery) Era Sfitter P. Bechtle, K. Desch, M. Uhlenbrock, P. Wienemann H. Flächer, M. Goebel, J. Haller, A. Höcker, K. Mönig, J. Stelzer GFitter 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 MasterCode Superbayes Joint HEP-APP IOP meeting on SUSY - March 24th 2010 14

  14. 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 non-collider 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 Global NP Fit Methodology in a Nutshell Joint HEP-APP IOP meeting on SUSY - March 24th 2010 15

  15. Constrained MSSM Analysis Pipeline SCANNING ALGORITHM Likelihood = 0 4 CMSSM parameters θ = {m0, m1/2, A0, tanβ} (fixing sign(μ) > 0) NO Physically acceptable? EWSB, no tachyons, neutralino CDM Observable quantitiesfi(θ ,Ψ) CDM relic abundance BR’s EW observables g-2 Higgs masssparticle spectrum (gamma-ray, neutrino,antimatter flux, direct detection x-section) RGE Non-linear numerical function via SoftSusy 2.0.18 DarkSusy 4.1 MICROMEGAS 2.2 FeynHiggs 2.5.1 Hdecay 3.102 YES 4 SM “nuisance parameters” Ψ={mt, mb,αS, αEM } Joint likelihood function Data: Gaussian likelihood(CDM, EWO, g-2, b→sγ, ΔMBs) other observables have only lower/upper limits Data: Gaussian likelihoods for each of the Ψj (j=1...4) Joint HEP-APP IOP meeting on SUSY - March 24th 2010

  16. 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 Finding the Favoured Regions Joint HEP-APP IOP meeting on SUSY - March 24th 2010 16

  17. 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) Finding the Favoured Regions Joint HEP-APP IOP meeting on SUSY - March 24th 2010 16

  18. 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 Ωh2=0.1099 ± 0.0066 ± 0.012 Mw=80.399 ± 0.023 GeV Δaμ =(24.6 ± 8)E-10 Relic density Mw g-2 R(Bτν) = 1.43±0.43 R(BXsγ) = 1.117±0.12 R: Measurement/SM Joint HEP-APP IOP meeting on SUSY - March 24th 2010 21

  19. Where are we today?Details in Frederic Ronga’s and Roberto Ruiz de Austri’s talksHere just a few points...

  20. CMSSM Today: Frequentist vs. Bayesian 0809.3792 [hep-ph] 0705.0487 [hep-ph] “naturaless prior” “log prior” “flat 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? 0808.4128 [hep-ph] Joint HEP-APP IOP meeting on SUSY - March 24th 2010 33

  21. Prior Dependence 0809.3792 [hep-ph] 0705.0487 [hep-ph] “flat prior” “naturalness prior” “flat prior” “flat prior” “flat prior” “naturalness 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

  22. Prior Dependence due to Weak Constraints Global Fit to CMSSM with four parameters: M0, M1/2, A0, tanβ [μ>0] strong prior dependence 0705.0487 [hep-ph] “naturalness prior” “flat 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 dimensions (eg, pMSSM with ~ 20 free parameters) 0806.1184 [hep-ph] Global Fit to Large Volume String scenario with only two param.: M0, tanβ [μ>0] smaller prior dependence “naturalness prior” “flat prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 29

  23. LHC Data (Discoveries) will help a lot  M(l+l-) GeV 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” “log prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 32 0907.0594 [hep-ph]

  24. 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 8-parameters global CMSSM fit: full MultiNest fit • Standard MCMC(SuperBayeS v1.23, 2006) 500 CPU days • MultiNest algorithm(SuperBayeS v1.35, 2008)<3 CPU daysspeed-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

  25. Connection to Astro(particle)physics

  26. Direct Dark Matter Searches & LHC 0809.3792 [hep-ph] Sensitivity Plot: WIMP(LSP) Mass vs. σpSI σpSI: 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

  27. 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

  28. 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. Cumberbatch et al (in prep) Joint HEP-APP IOP meeting on SUSY - March 24th 2010

  29. 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

  30. Supplementary material after this slide Joint HEP-APP IOP meeting on SUSY - March 24th 2010

  31. 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 non-collider 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 Global NP Fit Methodology in a Nutshell Joint HEP-APP IOP meeting on SUSY - March 24th 2010 15

  32. Constrained MSSM Analysis Pipeline SCANNING ALGORITHM Likelihood = 0 4 CMSSM parameters θ = {m0, m1/2, A0, tanβ} (fixing sign(μ) > 0) NO Physically acceptable? EWSB, no tachyons, neutralino CDM Observable quantitiesfi(θ ,Ψ) CDM relic abundance BR’s EW observables g-2 Higgs masssparticle spectrum (gamma-ray, neutrino,antimatter flux, direct detection x-section) RGE Non-linear numerical function via SoftSusy 2.0.18 DarkSusy 4.1 MICROMEGAS 2.2 FeynHiggs 2.5.1 Hdecay 3.102 YES 4 SM “nuisance parameters” Ψ={mt, mb,αS, αEM } Joint likelihood function Data: Gaussian likelihood(CDM, EWO, g-2, b→sγ, ΔMBs) other observables have only lower/upper limits Data: Gaussian likelihoods for each of the Ψj (j=1...4) Joint HEP-APP IOP meeting on SUSY - March 24th 2010

  33. 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. “naturalness prior” “log prior” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 37

  34. Non-Universal Higgs Models 0903.1279 [hep-ph] “flat prior” “log prior” NUHM II M0, M1/2, A0, tanβ, MHu2, MHd2 0907.4468 [hep-ph] MasterCode NUHM I M0, M1/2, A0, tanβ, MH2 [MHu2 = MHd2] Also for NUHM I the frequentist approach clearly favors low mass SUSY! Joint HEP-APP IOP meeting on SUSY - March 24th 2010 38

  35. Closer to the Experiment“Soft Scale Models”

  36. 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

  37. “Soft SUSY breaking” MSSM L Joint HEP-APP IOP meeting on SUSY - March 24th 2010 41

  38. “Soft SUSY breaking” MSSM L Gaugino’s and their masses M3, M2, M1 Joint HEP-APP IOP meeting on SUSY - March 24th 2010 42

  39. “Soft SUSY breaking” MSSM L Squarks and sleptons, and their masses Joint HEP-APP IOP meeting on SUSY - March 24th 2010 43

  40. “Soft SUSY breaking” MSSM L Tri-linear couplings A Joint HEP-APP IOP meeting on SUSY - March 24th 2010 44

  41. “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

  42. 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

  43. “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

  44. “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

  45. “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 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? MSSM18 LHC 300/fb Joint HEP-APP IOP meeting on SUSY - March 24th 2010 49

  46. Global Fits and DiscoveriesA few Projections

  47. LHC Data (Discoveries) will help a lot  M(l+l-) GeV If we are really luck we might see these spectacular signatures already very early at the LHC! 0808.4128 [hep-ph] 0808.4128 [hep-ph] Including a ”edge discovery” Joint HEP-APP IOP meeting on SUSY - March 24th 2010 51 0907.0594 [hep-ph]

  48. 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

  49. Connection to Astro(particle)physics

  50. Direct Dark Matter Searches & LHC 0809.3792 [hep-ph] Sensitivity Plot: WIMP(LSP) Mass vs. σpSI σpSI: 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

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