1. 1 SUSY mass measurements from invariant mass endpoints and boundary lines Konstantin Matchev Leptonic SUSY mini-team meeting April 21, 2009 In collaboration with: M. Burns, M. Park, arXiv:0903.4371 [hep-ph]

2. 2 Identify a sub-chain as shown Form all possible invariant mass distributions – M ll, M jll, M jl(lo), M jl(hi) Remove combinatorial background (OF and ME subtraction) Measure the endpoints and solve for the masses of A,B,C,D 4 measurements, 4 unknowns. Should be sufficient. Not so fast: –The measurements may not be independent – Piecewise defined functions -> multiple solutions? The classic endpoint method

3. 3 Combinatorics problems Lepton combinatorics Solution: OF subtraction Jet combinatorics Solution: Mixed Event subtraction

4. 4 MAMA MCMC B on-shell B off-shell Example: Dilepton invariant mass M LL M LL MBMBMBMB M C -M A

5. 5 Jet-lepton-lepton invariant mass M JLL There are 6 different cases to consider: (N jll,-) M JLL

6. 6 M JLL versus M LL scatter plot,, Bounded by a hyperbola OWS and a line UV Lester,Parker,White 06 Burns, KM, Park (2009)

7. 7 Where do the different M JLL cases come from? Regions (1,-) and (5,-): the upper boundary line is going down, the M JLL endpoint is at M LL =0 The remaining regions: the upper boundary is going up and the M JLL endpoint is at M LL >0. Where exactly? –Regions (2,-) and (3,-): the plot is cut before reaching the maximum allowed M JLL value –Regions (4,-) and (6,-): the plot is cut after reaching the maximum allowed M JLL value Region (1, - )Region (2, - ), (3, - )Region (4, - ) (5, 4 ) (6, 4 )

8. 8 “Low” jet-lepton pair invariant mass M JL(lo) Four additional cases: (-,N jl )

9. 9 “High” jet-lepton pair invariant mass The same 4 cases: (-,N jl ) M JL(hi)

10. 10 How many solutions? The endpoints are piecewise functions of the masses –11 cases altogether: (N jll,N jl ) It could have been even worse, but 3 cases are impossible –(2,1), (2,2), (3,3) Bad news: in (3,1), (3,2) and (2,3) the measured endpoints are not independent: (N jll,N jl ) regions

11. 11 The M JLL( Ѳ >π/2) invariant mass “threshold” Needed whenever M JLL in the rest frame of C L L M JLL(Ѳ>π/2)

12. 12 M JLL versus M LL scatter plot,, Bounded by a hyperbola OWS and a line UV Lester,Parker,White 06 The M JLL( Ѳ >π/2) invariant mass “threshold” Burns, KM, Park (2009)

13. 13 Posing the inverse problem Find the spectrum of A,B,C,D, given the 4 endpoints N jll not used: we have reduced the number of cases to four: –N jl =1, Region R 1 –N jl =2, Region R 2 –N jl =3, Region R 3 –N jl =4, Region R 4 May cross-check the solution with (N jll,N jl ) regions R3R3 R4R4 R2R2 R1R1

14. 14 The LHC inverse problem Solution: one for each region R 1, R 2, R 3 and R 4 Next: test for uniqueness Burns, KM, Park (2009)

15. 15 Multiple solutions? Not of the type previously discussed in the literature: Gjelsten, Miller, Osland (2005); Gjelsten, Miller, Osland, Raklev (2006)

16. 16 Mass ambiguities Exact spectrum duplication in (3,1), (3,2) and (2,3) Burns, KM, Park (2009)

17. 17 What have we learned so far? How the classic endpoint method works The inverse problem can be solved analytically 5 endpoint measurements may not be enough to uniquely determine 4 masses –Good news: at most 2-fold ambiguity –Bad news: will get even worse in the real world (with error bars) What can we do? –Improve precision at the LHC? Does not help. –Extra measurements from ILC? Expensive. –Longer decay chain? Not up to us. –Fresh new ideas? Yes!

18. 18 What is the fresh new idea? Pretty obvious: a two-dimensional (scatter) plot contains more information than the two individual one-dimensional histograms. Look at the scatter plot! –There is even more information in the 3D distribution Instead of looking for endpoints in 1D histograms, look at boundary lines in 2D scatter plots –For convenience, plot versus mass 2 instead of mass The shape of the scatter plot reveals the region R i Some special points provide additional measurements R1R1 R2R2 R3R3 Burns, KM, Park (2009)

19. 19 Understanding JL shapes Start with “near” versus “far” JL pairs (unobservable) The shape is a right-angle trapezoid ONPF Notice the correspondence between regions and point P R3R3 R4R4 R2R2 R1R1 Notice available measurements: n, f, p, perhaps also q Burns, KM, Park (2009)

20. 20 From “near-far” to “low-high” This reordering is simply origami: a 45 degree fold Burns, KM, Park (2009)

21. 21 The four basic JL shapes Burns, KM, Park (2009)

22. 22 JL scatter plots resolve the ambiguity (3,1) (2,3) (3,2)(2,3) R 1 versus R 3 R 2 versus R 3 “Drop” “Foot”

23. 23 Invariant mass summary Inverse LHC problem solved analytically Identified dangerous regions of parameter space with exact spectrum duplication Advertisement: look at scatter plots (in m 2 ) The shape of the scatter plots determines the type of region (N jll,N jl ), removes the ambiguity The boundaries of the scatter plots offer additional measurements, 11 altogether: as opposed to 5:

24. 24 Outlook What about 3D distributions? –Bounded by surfaces –Even more severe problem with statistics What about other 2D distributions? –M JL vs M LL –M JL vs M JLL What about cleverly chosen 1D distributions instead? How easy is it to see the kinematic boundary lines after backgrounds, detsim etc.? –Work in progress L. Pape (2006) Karapostoli,Sphicas,Pape (2008)

25. 25 BACKUPS

26. 26 Animation: Region R 1 Green dot: M jln endpoint Blue dot: M jlf endpoint Red dot: point P Endpoints given by (Low,High)=(Near,Far) M 2 jln M 2 jlf M 2 jl(lo) M 2 jl(hi) Region R 1

27. 27 Animation: Region R 2 M 2 jln M 2 jlf M 2 jl(lo) M 2 jl(hi) Region R 2 Green dot: M jln endpoint Blue dot: M jlf endpoint Red dot: point P Black dot: “Equal” endpoint Endpoints given by (Low,High)=(Equal,Far)

28. 28 Animation: Region R 3 M 2 jln M 2 jlf M 2 jl(lo) M 2 jl(hi) Region R 3 Green dot: M jln endpoint Blue dot: M jlf endpoint Red dot: point P Black dot: “Equal” endpoint Endpoints given by (Low,High)=(Equal,Near)

29. 29 Animation: Region R 4 (off-shell) M 2 jln M 2 jlf M 2 jl(lo) M 2 jl(hi) Region R 4 (off-shell) The shape is fixed: always a triangle “Low” and “High” endpoints are related:

30. 30 Animation: M JLL versus M LL scatter plot M 2 LL M 2 JLL (5,4) (6,4) (1,1) (3,2) (1,2) (2,3) (4,3) (4,2) (4,1) (3,1) (1,3) Region (1, - )Region (2, - ), (3, - )Region (4, - ) (5, 4 ) (6, 4 ) Several additional measurements besides the 1D endpoints:

31. 31 Mathematics of duplication Compose the two maps Apply to each pair of different regions –e.g. R 2 and R 1 This pair is safe! Only “boundary” effect due to the finite experimental precision

32. 32 Bad news! Examples of “real” duplication –Regions R 1 and R 3, namely (3,1) and (2,3) –Regions R 2 and R 3, namely (3,2) and (2,3) The extra measurement of M JLL does not help Part of region R 3 is safe