1. 1 The latest and greatest tricks in studying missing energy events Konstantin Matchev With: M. Burns, P. Konar, K. Kong, F. Moortgat, L. Pape, M. Park arXiv:0808.2472 [hep-ph], arXiv:0810.5576 [hep-ph], arXiv:0812.1042 [hep-ph], arXiv:0903.4371 [hep-ph], arXiv:0906.2417 [hep-ph], arXiv:090?.???? [hep-ph] Fermilab, LPC August 10-14, 2009

2. 2 These slides cover: “A general method for model-independent measurements of particle spins, couplings and mixing angles in cascade decays with missing energy at hadron colliders”, JHEP (2008) –Burns, Kong, KM, Park “Using subsystem M T2 for complete mass determinations in decay chains with missing energy at hadron colliders”, JHEP (2009) –Burns, Kong, KM, Park “s 1/2 min – a global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders”, JHEP (2009). –Konar, Kong, KM “Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy”, JHEP (2009) –Burns, KM, Park “Precise reconstruction of sparticle masses without ambiguities”, JHEP (200?) –KM, Moortgat, Pape, Park 67 pp 46 pp 32 pp 47 pp Total No of pages : 229 pp 37 pp

3. 3 MET events: experimentalist’s view What is going on here? This is why I am interested in MET!

4. 4 Q: What do we do for a living? A: Hunt for new particles. How? First make it, then detect it. Suppose it is: –Unstable, decays visibly to SM particles Resonant mass peak. Example: Z’. EASY –Unstable, decays semi-visibly to SM particles Jacobian peak (endpoint). Example: W’. EASY –Stable, charged CHAMPs. Examples in J. Feng’s talk. EASY –Stable, neutral Missing energy. Examples: LSP in SUSY, LKP in UED, … DIFFICULT! –Theory: Typically 2 missing particles per event, unknown mass –Experiment: MET is a challenging signature –Sociology: Don’t even try masses/spins at LHC, go to ILC.

5. 5 Why MET signatures are important to study Dark matter? Perhaps, but see J. Feng’s talk for counterexamples. Challenging – need to understand the detector very well. Guaranteed physics in the early LHC data! t t e e W W b b W W e e

6. 6 This talk is being given by a “theorist” The experimentalist asks:The theorist answers: Are there any well motivated such models? You bet. Let me tell you about those. Actually I have a paper… No. Is it possible to have a theory model which gives signature X? Yes. Is there any Monte Carlo which can simulate those models? No. But I’m the wrong person to ask anyway. MC4BSM workshops: http://theory.fnal.gov/mc4bsm/

7. 7 Ask the theorist! Feel free to ask me questions on any topic Some questions that I anticipate: –Suppose we discover SUSY. How would we know it is SUSY and not something else? –Almost all of our SUSY studies are based on LMx study points in MSUGRA. How much model dependence is introduced by the MSUGRA assumptions? Is it possible to design a model-independent SUSY search? –I see you wrote a paper on M T2. I keep hearing about this M T2 and could never understand what it is god for. Can you explain? –What are some safe cuts to use in our skims? Is there any magic (model-independent) cut which would cut the SM background yet preserve all of the (SUSY) signal?

8. 8 MET events: experimentalist’s view What is going on here?

9. 9 Pair production of new particles (conserved R, KK, T parity) Motivated by dark matter + SUSY, UED, LHT –How do you tell the difference? (Cheng, KM, Schmaltz 2002) SM particles x i seen in the detector, originate from two chains –How well can I identify the two chains? Should I even try? What about ISR jets versus jets from particle decays? “WIMPs” X 0 are invisible, momenta unknown, except p T sum –How well can I reconstruct the WIMP momenta? Should I even try? What about SM neutrinos among the x i ’s? MET events: theorist’s view

10. 10 In place of an outline Missing momenta reconstruction? Mass measurementsSpin measurements Inclusive2 symmetric chains NoneInv. mass endpoints and boundary lines Inv. mass shapes M eff, M est,H T Wedgebox ApproximateS min, M Tgen M T2, M 2C, M 3C, M CT, M T2 (n,p,c) As usual (MAOS) Exact?Polynomial method As usual optimism pessimism

11. 11 Today: invariant mass studies Study the invariant mass distributions of the visible particles on one side of the event Does not rely on the MET measurement Can be applied to asymmetric events, e.g. –No visible SM products on the other side –Small leptonic BR on the other side Well tested, will be done anyway. MET Hinchliffe et al. 1997 Allanach et al. 2000 Nojiri et al. 2000 Gjelsten et al. 2004 ATLAS TDR 1999

12. 12 The classic endpoint method Identify a sub-chain as shown. Combinatorics problem? Form all possible invariant mass distributions – M ll, M jll, M jl(lo), M jl(hi) 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 “ATLAS” approach

13. 13 Combinatorics problems Lepton combinatorics Solution: OF subtraction Jet combinatorics Solution: Mixed Event subtraction

14. 14 MAMA MCMC B on-shell B off-shell Example: dilepton invariant mass M LL MBMBMBMB M C -M A

15. 15 Jet-lepton-lepton invariant mass There are 6 different cases to consider: (N jll,-) M JLL

16. 16 Jet-lepton invariant mass M JL But which is near and which is far? Define “low” and “high” pairs as: Allanach et al. 2000

17. 17 “Low” jet-lepton pair invariant mass M JL(lo) 4 additional cases: (-,N jl )

18. 18 “High” jet-lepton pair invariant mass The same 4 cases as “low” jet-lepton pair: (-,N jl ) M JL(hi)

19. 19 Recap So far we measured the upper kinematic endpoints of four invariant mass distributions – M ll, M jll, M jl(lo), M jl(hi) They depend on 4 input masses: M A, M B, M C, M D 4 measurements, 4 unknowns. Should be sufficient. Invert and solve for the masses. However, 2+1 generic problems: –Piecewise defined functions -> multiple solutions? (next) –These four measurements may not all be independent, sometimes – This requires a new measurement. How precise is it?

20. 20 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

21. 21 An alternative to M JLL The M JLL( Ѳ >π/2) invariant mass “threshold” M JLL in the rest frame of C L L M JLL(Ѳ>π/2) Nojiri et al, 2000, Allanach et al. 2000

22. 22 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)

23. 23 Posing the LHC 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

24. 24 Solving the LHC inverse problem Find the four masses of A, B, C, D, given the 4 endpoints Solution: Burns, KM, Park (2009)

25. 25 Multiple solutions? Previously multiple solutions arose due to insufficient experimental precision or using an incomplete data set Gjelsten, Miller, Osland (2005); Gjelsten, Miller, Osland, Raklev (2006)

26. 26 Mass ambiguities Exact spectrum duplication in (3,1), (3,2) and (2,3) Burns, KM, Park (2009)

27. 27 What have we learned so far? How the classic (ATLAS) endpoint method works The inverse problem can be solved analytically 5 endpoint measurements may not be enough to uniquely determine 4 masses –Good news: in theory, 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! old new

28. 28 One 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) Costanzo, Tovey (2009)

29. 29 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” Burns, KM, Park (2009)

30. 30 Precision problem Gjelsten, Miller, Osland (2004) Lester (2006) In theory OK, but –scatter plots require more statistics –the M JLL( Ѳ >π/2) “threshold” is hard to read

31. 31 Back to the drawing board Redesign the classic endpoint method –do not use distributions whose endpoints are piecewise-defined functions: M jll, M jl(lo) or M jl(hi) –do not use the poorly measured M JLL( Ѳ >π/2) “threshold” –do not use scatter plots –derive the shapes of all differential distributions Sounds impossible? Must introduce new observable distributions. KM, Moortgat, Pape, Park (2009)

32. 32 New jet-lepton distributions But which is near and which is far? ATLAS: define “low” and “high” as: Allanach et al. 2000 Don’t ask, don’t tell: always use the two jet-lepton entries in a symmetric fashion KM, Moortgat, Pape, Park (2009)

33. 33 The combined jet-lepton distribution Simply plot “near” and “far” together KM, Moortgat, Pape, Park (2009) Read the two endpoints These two are not piecewise defined

34. 34 The generalized sums Plot the combination Alpha is a continuous parameter: infinitely many possibilities! Alpha=1 is not piecewise defined: KM, Moortgat, Pape, Park (2009)

35. 35 The product and the difference Unfortunately, both endpoints piecewise defined

36. 36 The bottom line If we use only the 4 unambiguous endpoints The masses are found from Despite the 2-fold near-far confusion, the answers for A, C and D are unique! Remember that there are (infinitely) many more endpoint measurements –Allow measurement of M B –Improve precision KM, Moortgat, Pape, Park (2009)

37. 37 Summary There now exists a “CMS” version of the invariant mass endpoint method. It uses a different set of (in principle, infinitely many) invariant mass distributions It avoids multiple solution ambiguities (Allegedly) it leads to better precision –more measurements –better measured endpoints

38. 38 BACKUPS

39. 39 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

40. 40 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

41. 41 Understanding shapes Let’s 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

42. 42 From “near-far” to “low-high” This reordering is simply origami: a 45 degree fold

43. 43 The four basic JL shapes

44. 44 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

45. 45 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)

46. 46 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)

47. 47 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:

48. 48 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”

49. 49 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”

50. 50 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:

51. 51 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: