1. 1 Efficiency calculation using the System8 method Benoit Clément, Thomas Delemontex, Fabienne Ledroit

2. 2 What is System8 ? System8 is a method to determine signal and background fractions, as well as selection efficiencies. First developed for b-tagging efficiencies measurements… and used sucessfully. (It as been adopted as one of the efficiency measurement by every b-tagging algorithms) DØ Note 4159 / NIM-A paper accepted (no number yet) First developed for b-tagging efficiencies measurements… and used sucessfully. (It as been adopted as one of the efficiency measurement by every b-tagging algorithms) DØ Note 4159 / NIM-A paper accepted (no number yet) It uses only data. Some correction factors may be added from MC. It allows to extract normalized distribution of sensitive variables in data.

3. 3 A bit of maths… Assuming s independent data samples containing b different signal or backgrounds, one can write for each sample j (j=1..s) an equation: with n j i : fraction of signal i in sample j. Assuming s independent data samples containing b different signal or backgrounds, one can write for each sample j (j=1..s) an equation: with n j i : fraction of signal i in sample j. Moreover supposing t different selection criteria (cuts on various variables), one can write for each sample j (j=1..s) and each criteria k (k=1..t) another equation: with ε k i : efficiency of cut k on signal i. q k j : fraction of remaining events in sample j after cut k. Moreover supposing t different selection criteria (cuts on various variables), one can write for each sample j (j=1..s) and each criteria k (k=1..t) another equation: with ε k i : efficiency of cut k on signal i. q k j : fraction of remaining events in sample j after cut k.

4. 4 A bit more… The efficiencies of applying x criteria k i,i=1..x can be written : If the selection criteria are independent, then the kappa factors are equal to 1. In general small correlation remains, the kappas are close to unity. And then, new equations applying several criteria : The efficiencies of applying x criteria k i,i=1..x can be written : If the selection criteria are independent, then the kappa factors are equal to 1. In general small correlation remains, the kappas are close to unity. And then, new equations applying several criteria :

5. 5 And finally… The various q j are easily mesured on data. Selections and samples bring : 2 t.s equations, b.(s+t) unkowns (fractions and efficiencies). You need as many equations as unknowns : 2 t.d = b.(d+t) Selections and samples bring : 2 t.s equations, b.(s+t) unkowns (fractions and efficiencies). You need as many equations as unknowns : 2 t.d = b.(d+t) Simplest solutions are : (a) s = 1, t = 3, b = 2 and (b) s = 2, t = 2, b = 2. => 8 equations, 8 unknowns : System8. Simplest solutions are : (a) s = 1, t = 3, b = 2 and (b) s = 2, t = 2, b = 2. => 8 equations, 8 unknowns : System8.

6. 6 The System

7. 7 Analytic solution (1) 1. Use equation (1)to get rid of f b

8. 8 Analytic solution (2) 2. Extract ε a = F(ε b, f b ) in equations (2-4)

9. 9 Analytic solution (3) 3. Substitute f b and the ε a s in equations (5-7) gives 3 polynomial equation for the ε b s

10. 10 Analytic solution (4) 4. The 3 previous equations gives ε b (U) as function of ε b (V) 5. By double substitution, we obtain an equation for ε b (X) that simplify into a 2 nd order polynomial : 2 solutions !

11. 11 Analytic solution (5) 6. For each solution of ε b (X) we substitute everything in the last equation. This gives a single equation of a single unknown f a that can be rewritten as a 43th order polynomial. 6. For each solution of ε b (X) we substitute everything in the last equation. This gives a single equation of a single unknown f a that can be rewritten as a 43th order polynomial.

12. 12 No longer analytic The last equation is solved numerically: - for both possible values of ε b (X) - with the constraint 0<f a <1 Then all 8 unknowns can be determined. There can be many solutions, the physical one(s) can be determined knowing that - All fraction an efficiency must be between 0 and 1 - We may have a priori knowledge of the expected value for a part of the solution. The last equation is solved numerically: - for both possible values of ε b (X) - with the constraint 0<f a <1 Then all 8 unknowns can be determined. There can be many solutions, the physical one(s) can be determined knowing that - All fraction an efficiency must be between 0 and 1 - We may have a priori knowledge of the expected value for a part of the solution.

13. 13 Results exemple

14. Single Top meeting - B. Clément - 20 nov 2003 14 System8 to estimate W/QCD background Use System8 instead of Matrix Method. QCD and W fractions in loose samples, as well as W and QCD efficiencies are output of the method. Sample : preselected loose sample in electron channel (p13) (Emmanuel’s preselection cuts DØ note 4274). Selection 1 : Electron likelihood LH > 0.4 (loose -> tight). Selection 2 : MET>35 GeV. Selection 3 : M TW > 40 GeV. 2 backgrounds : QCD and W+signal. Sample : preselected loose sample in electron channel (p13) (Emmanuel’s preselection cuts DØ note 4274). Selection 1 : Electron likelihood LH > 0.4 (loose -> tight). Selection 2 : MET>35 GeV. Selection 3 : M TW > 40 GeV. 2 backgrounds : QCD and W+signal. Proposed solution : 1 sample, 3 criteria.

15. Single Top meeting - B. Clément - 20 nov 2003 15 System to solve

16. Single Top meeting - B. Clément - 20 nov 2003 16 Results Problem : MET and MTW are not independent.  Use MC W+jets to estimate correlation factors : Problem : MET and MTW are not independent.  Use MC W+jets to estimate correlation factors : System 8 output : Efficiencies : ε LH W = (71.6 +/- 2.0) % ε LH QCD = (19.2+/- 1.8) % W and QCD fractions (loose sample) : n W = 0.53 +/- 0.03 n QCD = 0.47 +/- 0.02 W and QCD fractions (tight sample) : n W = 0.81 +/- 0.04 n QCD = 0.20+/- 0.02 System 8 output : Efficiencies : ε LH W = (71.6 +/- 2.0) % ε LH QCD = (19.2+/- 1.8) % W and QCD fractions (loose sample) : n W = 0.53 +/- 0.03 n QCD = 0.47 +/- 0.02 W and QCD fractions (tight sample) : n W = 0.81 +/- 0.04 n QCD = 0.20+/- 0.02 For proposed cuts : C MET,MTW = 1.032 +/- 0.01 C MET,LH = 0.997 +/- 0.01 C LH,MTW = 1.002 +/- 0.01 For proposed cuts : C MET,MTW = 1.032 +/- 0.01 C MET,LH = 0.997 +/- 0.01 C LH,MTW = 1.002 +/- 0.01

17. Single Top meeting - B. Clément - 20 nov 2003 17 Normalized distributions of sensitive variables from data For each bin : - Evaluate MC correlation factors (ALPGEN W+jets p14.02). - Solve the System8. => Normalized distributions of the variable. For each bin : - Evaluate MC correlation factors (ALPGEN W+jets p14.02). - Solve the System8. => Normalized distributions of the variable. Instead of a simple cut, use a bin for one of the criteria : for example, a < MET < b. Then the efficiencies for the new criteria is the fraction of W and QCD in that interval. Instead of a simple cut, use a bin for one of the criteria : for example, a < MET < b. Then the efficiencies for the new criteria is the fraction of W and QCD in that interval.

18. Single Top meeting - B. Clément - 20 nov 2003 18 M TW : correlation factors M TW bins : x<M TW <x+10, x = 10,20…120 LH > 0.4, MET > 35 GeV. M TW bins : x<M TW <x+10, x = 10,20…120 LH > 0.4, MET > 35 GeV. M TW - LH MET - M TW

19. Single Top meeting - B. Clément - 20 nov 2003 19 M TW distributions -W distribution is compared to MC W+jets. -System8 shows some instability when W and QCD efficiencies are close (crossing point ). -W distribution is compared to MC W+jets. -System8 shows some instability when W and QCD efficiencies are close (crossing point ). Crossing point

20. Single Top meeting - B. Clément - 20 nov 2003 20 M TW : other crosschecks Look at the stability of W, QCD fractions and efficiencies. Fit : 77.5 +/- 6.9 % Fit : 73.1 +/- 2.7 % Fit : 21.5 +/- 3.5 % Fit : 21.03 +/- 2.9 %

21. Single Top meeting - B. Clément - 20 nov 2003 21 Correlation Factors without correlation factors with correlation factors

22. 22 Summary