1. Mass and Spin from a Sequential Decay with a Jet and Two Leptons Michael Burns University of Florida Advisor: Konstantin T. Matchev Collaborators: Kyoungchul (KC) Kong, Myeonghun Park Burns, Matchev, Park, JHEP 189P 0309 (submitted) [arXiv:0903.4371 [hep-ph]] Burns, Kong, Matchev, Park, JHEP10(2008)081 (published) [arXiv:0808.2472 [hep-ph]] Burns, Kong, Matchev, Park, JHEP 019P 1108 (accepted) [arXiv:0810.5576 [hep-ph]]

2. Contents New Physics and Sequential Decays Mass Determination – Kinematical Endpoint Method – Kinematical Boundaryline Method Spin Determination – Chiral Projections – Basis Functions – Reparametrization

3. New Physics and Sequential Decays At LHC: colored particle production (j), unknown energy and longitudinal momentum (D) Assume OSSF leptons (l n, l f ), missing transverse momentum (A) What is the new physics (assuming the chain “DCBA”)? – What are the masses of A,B,C,D? – What are their spins? “old” physics: “new” physics, “DCBA”:

4. Particle Combinations m 2 ll m 2 jln m 2 jlf m 2 jll For mass determination: For spin determination:

5. MASS

6. Method of Kinematical Endpoints [Bachacou, Hinchliffe, Paige (1999), Figs. 1 and 4] Use extreme kinematical values of invariant mass. (“model-independent”) … however, the dependence is piecewise-defined. [Allanach, Lester, Parker, Webber (2000), Tab. 4, Gjelsten, Miller, Osland (2004), Eqs. 4.3-9, etc.] These values depend on spectrum of A,B,C,D … Offshell B: N jl = 4

7. Inversion and Duplication Experimental ambiguity – Finite statistics, resolution -> “border effect” – Background -> “dangerous feet/drops” Piecewise defintions: hmm… largely ignored – Inversion formulas depend on unknown spectrum – Ambiguity DOES occur! These inversion formulas use the jll threshold! [Burns, Matchev, Park (2009), Eqs. 2.23-6]

8. Example Duplication How to resolve? We have a technique: boundarylines [Burns, Matchev, Park (2009)] [Burns, Matchev, Park (2009), Tab. 2]

9. Two Variable Distribution: -------- We know the expression for the hyperbola. easy to see restricted ( ) distribution [Burns, Matchev, Park (2009), Fig. 10]

10. Two Variable Distribution: ---------- MAIN POINT: shapes of kinematical boundaries reveal Region => no more piecewise ambiguity (from perfect experiment). N jl = 3 N jl = 2N jl = 1 N jl = 4 [Burns, Matchev, Park (2009), Figs. 7,8]

11. (m jl(hi),m jl(lo) ) Resolves Ambiguity [Burns, Matchev, Park (2009), Fig. 9] = 122 GeV = 149 GeV = 200 GeV = 212 GeV

12. SPIN

13. Spin Assignments S = scalar F = spinor V = vector Assume q/qbar jet for spin analysis final-state SM fermions => spin change (+/-)1/2 at each vertex

14. Spins and Chiral Projections Four helicity groupings, depending on RELATIVE (physical) helicities of the jet and two leptons. -> four “basis functions” I J=21 I J=11 I J=12 I J=22 I : relative helicity b/w j and l n J: relative helicity b/w l n and l f spin of antifermion is “opposite of the spinor”?

15. “Near-type” Distributions (“near-type” applies in SM: top decay) The arrow subscripts indicate the relative helicities of the final- state SM fermions. BOTH HELICITY COMBINATIONS CONTRIBUTE! (Notice from the table what happens for equal helicity contributions.) [Burns, Kong, Matchev, Park (2008), Tab. 7]

16. Observable Spin Distributions “cleverly” redefine spin basis functions (like change of basis) Relevant coefficients are the following combinations of couplings: Distribution decomposed into model-dependent ( , ,  ) and model-independent (  ) contribution [Burns, Kong, Matchev, Park (2008), various eqs.]

17. Observable Spin Distributions Dilepton: purely “near-type” (nice) Jet-lepton: must include “near-type” and “far-type” together, piecewise defined Only one model-dependent parameter (for each spin case):  !!! Get as much use out of this one as possible (as usual). S fits to same  as L !!! extra constraint D gives charge assymtery; fits to independent model parameters  and  So, in addition to spin, get three measurements of the couplings through , ,  – extra model determination. [Burns, Kong, Matchev, Park (2008), Tab. 4] [Burns, Kong, Matchev, Park (2008), various eqs.]

18. Example: SPS1a LL SS D  We generated “data” from DCBA = SFSF, assuming The fits were determined by minimizing: [Burns, Kong, Matchev, Park (2008), Figs. 4,5,6]

19. Other Spin Assignments D  seems the most promising to discriminate the SPS1a model. However, the most discriminating distribution depends on the masses and spins of the true model. Some models cannot even be discriminated, in principle (using our method). (This does not imply that our method is bad; just general.) [Burns, Kong, Matchev, Park (2008), Tab. 5]

20. Summary Mass determination: – We have inversion formulas using jll threshhold. – We identified the ambiguous endpoint Regions. – We devised the kinematical boundaryline method, which resolves the ambiguity (ideally). Spin determination: – We devised a method that allows the model-dependent parameters to float. – We found a convenient spin basis for these floating parameters. – We identified the problem scenarios (fakers).

21. Appendix: OF Subtraction (leptons) [ATLAS TDR (1999), Figs. 20-9,20-10] desired signal: event selection: 2 OSSF leptons and four “pT-hard” jets chi20 - chi10 = 68 GeV [Hinchliffe, Paige, Shapiro, Soderqvist, Yao (1996), Figs. 15,16] basically same as above

22. Appendix: ME Subtraction (jets) [Ozturk (2007), Fig. 2] jet+lepton distribution desired signal: squark - sneutrino = 284 GeV (different from ours) event selection: one lepton and two “pT-hard” jets

23. Appendix: threshold formulae

24. Appendix: Regions & Configurations in rest-frame of C: (1,.) and (5,.) (2,.) (3,.) (4,.) and (6,.) in (.,1) independently of frame: in rest-frame of B: [Miller, Osland, Raklev (2005), Figs. 2,12] [Burns, Matchev, Park (2009), Fig. 2]

25. Appendix: Dangerous Feet/Drops Background [based on Miller, Osland, Raklev (2005), Figs. 10]

26. Appendix: Inversion Variables [Burns, Matchev, Park (2009), Eqs. 2.28-31]

27. Appendix: Duplication Maps [Burns, Matchev, Park (2009), Fig. 3]

28. Appendix: jll Hyperbola

29. Appendix: (m jll,m ll ) [Burns, Matchev, Park (2009), Fig. 11]

30. Appendix: “Near-type” Basics C’B’A’ = { SFS, SFV, FSF, FVF, VFS, VFV } One of either I or J is irrelevant. C’B’A’f b f a = { CBAl n l f, DCBjl n }. Only the relative helicity between f a and f b is important. Chiral projections allow helicities to be selected by the couplings (because f’s are massless), so that matrix element can be spin-summed. Spin dependence requires either: - chiral imbalance (g L /=g R ) at both vertices, or - B’=V.

31. Appendix: “far-type” log behavior [Miller, Osland, Raklev (2006)] It comes from the Jacobian of the transformation from angles to masses, and the kinematical boundary of the angular variables.

32. Appendix: FSFS vs. FSFV FSFV always fakes FSFS. FSFS can also fake FSFV for some mass spectra. The only condition is: [Burns, Kong, Matchev, Park (2008), Figs. 4,5,6] example of unavoidable false-positive for FSFS, given FSFV

33. Appendix: FVFS vs. FVFV FVFV always fakes FVFS. FVFS can also fake FVFV for some mass spectra. The conditions are: [Burns, Kong, Matchev, Park (2008), Figs. 4,5,6] example of unavoidable false-positive for FVFS, given FVFV