1. Barbara Storaci, Nicola Serra, Niels Tuning Bs →μ+μ- Bfys-meeting, 15 th May 2009 1

2. Index Bs → μ + μ - analysis strategy Goal Toy Model description Comparison of analysis methods 1-bin approach Modified Frequentist Approach (MFA) Unbinned Approach Results and Conclusions 2 Talk given at the last Bs → μ + μ – working group meeting (7 th May 2009) : http://indico.cern.ch/contributionDisplay.py?contribId=2&confId=53090

3. Bs → μ + μ - 3 FCNC process: Sensitive to new physics Rare Decay [ SM: BR = (3.35 ± 0.32) 10 -9 ]: Precise prediction: small hadronic uncertainties thanks to ΔMs measurement. In all MSSMs BR(Bs → μ + μ - ) prop to tan 6 β sensitive to models with high tanβ In NUHM (which contains cMSSM) prediction of BR(Bs → μ + μ - ) ~ 10 -8 Beyond Tevatron discovery possibility, but early measurement for LHCb One of the key channels for (early) NP discovery

4. Analysis Strategy (1) Pre-Selection PV   2 <14 IPS <6 Invariant Mass Likelihood (IML)[± 600MeV] Z pos SV downstream respect to PV 4 Selection Invariant Mass Likelihood (IML) [± 60MeV] Particle Identification Likelihood (PIDL) Geometrical Likelihood (GL) > 0.5 Variables built and chosen to be uncorrelated (see roadmap!) BR extraction/ CL limit 3 possible methods 1-bin Approach Modified Frequentist Approach Unbinned Approach TOPIC of this TALK Only for completeness: GL built with:  (Bs) IPS (Bs) DOCA (distance of closest approach between the 2 μ) Isolation of the 2 μ

5. Analysis Strategy (2) Pre-Selection PV   2 <14 IPS <6 Invariant Mass Likelihood (IML)[± 600MeV] Z pos SV downstream respect to PV 5 Selection Invariant Mass Likelihood (IML) [± 60MeV] Particle Identification Likelihood (PIDL) Geometrical Likelihood (GL) > 0.5 Variables built and chosen to be uncorrelated BR extraction (CL limit) 3 possible methods 1-bin Approach Modified Frequentist Approach Unbinned Approach TOPIC of this TALK Only for completeness: GL built with:  (Bs) IPS (Bs) DOCA (distance of closest approach between the 2 μ) Isolation of the 2 μ

6. Goals Extraction of BR with different methods Study for different signal (BR) Assumptions: Input: generated with toy model All methods used same toy experiments Complete factorization of GL and Bs Mass Used large sample to describe signal (equivalent to control channel in real data) Side-Band perfectly describes peak region 6

7. Toy Model Description (1) Only tw0 variables: GL, Bs Mass Signal: Polynomial Background: Exponential e -11.7 x Signal: Gaussian Background: Polynomial Exp to match roadmap prediction in the sensitive region (GL>0.5) Bg = GL(exp)  Bs(pol) Sg = GL(pol)  Bs(gauss) Peak_pdf = Nb [GL(exp)  Bs(pol)] + Ns [GL(pol)  Bs(gauss)] 7

8. Toy Model Description (2) For each toy, we produce three statistically independent samples: E.g. 1 year, BR=3.5 10 -9 1. 20000 → “ Control Channel” 2. 70000 → “ Side Band” (e.g. [M Bs -120, M Bs -60]  [M Bs +60, M Bs +12o]) 3. 70042 → “ Peak Region” Extract BR (Ns) with different methods 8 N.B. all numbers Poison fluctuating

9. Analysis Methods 1-bin Approach (Simplest): Rectangular cuts on variables to select a bin (mass- window, cut on GL) Modified Frequentist Approach (Roadmap): 3 bins, GL[0.5 – 1.0], 5 bins, Bs Mass [5309.6 – 5429.6] Estimation of probability per each bin (for an assumed BR value) of the hypotheses signal + background or only background Un-binned Approach (This talk): “Generalization” of MFA in an unbinned way (see later) 9

10. 1-bin Approach Sensitivie region! GL Bs Mass 5429.6 5309.6 0.761.0 Mass window 5309.6 - 5429.6 GL cut at 0.76 GL cut optimization with Figure Of Merit (FOM) for a 3 sigma sensitivity: Real BR = 3.5 10 -9 (red line) 3 years of data taken N toys = 500 → The method is unbiased Measurement 3.44 +/- 0.05 Sensitivity 0.97 +/- 0.03 10

11. MFA method Box of two variables: Bs mass, GL 3 bins, GL[0.5 – 1.0] 5 bins, Bs Mass [5309.6 – 5429.6] For given BR, estimate probability that data in a bin are signal+background or only background Calculate: GL Bs Mass 5429.6 5309.6 0.51.0 Sensitive region! 11

12. Br extraction with MFA Calculating ΔΧ 2 for different BR hypotheses Probability of a certain amount of signal (assuming a Br) hypothesis with the expected background (from side-band) Probability of having only background Minimum from parabola interpolation → The method is unbiased Real BR = 3.5 10 -9 (red line) 3 years of data taken N toys = 500 Measurement 3.50 +/- 0.04 Sensitivity 0.92 +/- 0.03 12 Added by us!!!

13. Why an unbinned strategy? No bin size optimization necessary Gain sensitivity to BR? Used a flexible tool (RooFit) → easy to extend for more dimensions and more background sources 13

14. Unbinned method Fit: Background (i.e. Side-Band) (e.g. [M Bs -120, M Bs -60]  [M Bs +60, M Bs +12o]) pdf_bg’ = GL(RooKeysPdf)  Bs(Pol) (Extended pdf ) Signal (i.e. Control-Channel) pdf_sg’ = GL(RooKeysPdf)  Bs(gauss) (Extended pdf) Use the result to build: pdf_tot’ = Nb pdf_bg’ + Ns pdf_sg’ Fit data with pdf_tot’ to obtain Ns (for the moment parameter Nb fitted from Side Band, work in progress for a simultaneous fit) GL Bs Mass 5429.6 5309.6 0.51.0 Sensitive region! Use NO knowledge from generation:  GL: no assumption on signal and background shape  Bs Mass: reasonable assumption 14

15. RooKeysPdf: idea Technique to produce a continuous estimates of f(x) of the parent distribution without assume a model (non- parametric) (Direct generalization of the Averaged Shifted Histograms (ASH) techniques) Convolution of Gaussian with different sigmas drawn under each point of the dataset Kernel Estimation in High-Energy Physics (auth: Cranmer K.) http://cdsweb.cern.ch/record/477613?ln=en ONLY for illustrative purpose!!!  Accurate estimation of parent distribution with narrow kernels in high statistics region  Wide kernels to smooth out statistical fluctuations in low-density regions 15

16. Preliminary Results: SM 2 years1 year 3 years MFA and unbinned methods unbiased Unbinned method seems to give better RMS Both methods run on the SAME toys!!! 16

17. Preliminary Results: Br=10 -8 0.25 year 1 year 0.5 year Both methods run on the SAME toys!!! Both methods unbiased Unbinned method seems to be good also at low statistics 17

18. 3 methods comparison 3 years In ideal conditions all methods work well (Remember: cuts were optimized in 1-bin method) MFA lies in between the 1-bin and unbinned methods (no bin size optimization for MFA here) 18

19. Conclusions Implementation of BR extraction in the Modified Frequentist Approach First investigation of an alternative method (unbinned) to extract BR in the Bs → μ + μ - All methods unbiased in ideal conditions Unbinned approach good in extreme conditions: small BR, small amount of statistics (early measurement) 19