Model-Independent Measurements of New Physics Parameters in Missing Energy Events at Hadron Colliders

1. Model-independent measurements of new physics parametersin missing energy eventsat hadron colliders Konstantin Matchev In collaboration with: M. Burns, P. Konar, K. Kong, M. Park arXiv:0808.2472 [hep-ph], arXiv:0810.5576 [hep-ph], arXiv:0812.1042 [hep-ph], arXiv:0903.4371 [hep-ph] University of Maryland March 31, 2009

2. 67 pp 46 pp 32 pp 47 pp Total No of pages : 192 pp These slides cover: • “A general method formodel-independentmeasurements of particle spins, couplings and mixing angles in cascade decays with missing energy at hadron colliders”, JHEP (2008) • Burns, Kong, KM, Park • “Using subsystem MT2 for complete mass determinations in decay chains with missing energy at hadron colliders”, JHEP (2009) • Burns, Kong, KM, Park • “s1/2min – 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 (????) • Burns, KM, Park

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

4. MET events: theorist’s view • 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 xi seen in the detector, originate from two chains • How do you identify the two chains? • What about ISR jets versus jets from particle decays? • “WIMPs” X0 are invisible, momenta unknown, except pT sum • What about SM neutrinos among the xi’s? • How well can you reconstruct the WIMP momenta?

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6. The classic endpoint method • Identify a sub-chain as shown • Form all possible invariant mass distributions • Mll, Mjll, Mjl(lo), Mjl(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?

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

8. Example: dilepton invariant mass B on-shell MLL B off-shell MC-MA MB MA MC

9. Jet-lepton-lepton invariant mass • There are 6 different cases to consider: (Njll,-) MJLL

10. Jet-lepton invariant mass • But which is near and which is far? • Define “low” and “high” pairs as: MJL mjl(lo) = min {mjln, mjlf} mjl(hi) = max {mjln , mjlf}

11. “Low” jet-lepton pair invariant mass • Leads to 4 additional cases: (-,Njl) MJL(lo)

12. “High” jet-lepton pair invariant mass MJL(hi) • The same 4 cases as “low” jet-lepton pair: (-,Njl)

13. How many solutions? • 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: (Njll,Njl) regions • The endpoints are piecewise functions of the masses • 11 cases altogether: (Njll,Njl)

14. An alternative to MJLL MJLL MJLL(Ѳ>π/2) L L in the rest frame of C The MJLL(Ѳ>π/2) invariant mass “threshold”

15. Posing the inverse problem R3 R4 R2 R1 • Njll not used: we have reduced the number of cases to four: • Njl=1, Region R1 • Njl=2, Region R2 • Njl=3, Region R3 • Njl=4, Region R4 • May cross-check the solution with (Njll,Njl) regions Find the spectrum of A,B,C,D, given the 4 endpoints

16. Solving the inverse problem • We found the inverse analytical relations for all four regions. Now test for uniqueness of the solution.

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

18. Our cases of mass ambiguities • Exactspectrum duplication in (3,1), (3,2) and (2,3) Burns, KM, Park (2009)

19. Mathematics of duplication • Compose the two maps • Apply to each pair of different regions • e.g. R2 and R1 • This pair is safe! • Only “boundary” effect due to the finite experimental precision

20. Bad news! • Examples of “real” duplication • Regions R1 and R3, namely (3,1) and (2,3) • Regions R2 and R3, namely (3,2) and (2,3) • The extra measurement of MJLL does not help • Part of region R3 is safe

21. 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!

22. 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 mass2 instead of mass • The shape of the scatter plot reveals the region Ri • Some special points provide additional measurements R1 R2 R3

23. R3 R4 R1 R2 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 • Notice available measurements: n, f, p, perhaps also q

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

25. The four basic JL shapes

26. Animation: Region R1 • Green dot: Mjln endpoint • Blue dot: Mjlf endpoint • Red dot: point P • Endpoints given by (Low,High)=(Near,Far) Region R1 M2jlf M2jl(hi) M2jl(lo) M2jln

27. Animation: Region R2 Region R2 M2jlf M2jl(hi) M2jl(lo) M2jln • Green dot: Mjln endpoint • Blue dot: Mjlf endpoint • Red dot: point P • Black dot: “Equal” endpoint • Endpoints given by (Low,High)=(Equal,Far)

28. Animation: Region R3 Region R3 M2jlf M2jl(hi) M2jl(lo) M2jln • Green dot: Mjln endpoint • Blue dot: Mjlf endpoint • Red dot: point P • Black dot: “Equal” endpoint • Endpoints given by (Low,High)=(Equal,Near)

29. Animation: Region R4 (off-shell) Region R4 (off-shell) M2jlf M2jl(hi) M2jln M2jl(lo) • The shape is fixed: always a triangle • “Low” and “High” endpoints are related:

30. Scatter plots resolve the ambiguity (2,3) (3,1) (3,2) (2,3) “Drop” • R1versus R3 “Foot” • R2versus R3

31. MJLL versus MLL scatter plot Bounded by a hyperbola OWS and a line UV Lester,Parker,White 06 , , The MJLL(Ѳ>π/2) invariant mass “threshold”

32. Animation: MJLL versus MLL scatter plot Several additional measurements besides the 1D endpoints: M2JLL (1,3) (1,2) (2,3) (4,3) (5,4) (1,1) (4,1) (4,2) (3,1) (6,4) (3,2) M2LL (6, 4 ) (5, 4 ) Region (1, - ) Region (2, - ) , (3, - ) Region (4, - )

33. Invariant mass summary as opposed to 5: • Inverse LHC problem solved analytically • Identified dangerous regions of parameter space with exact spectrum duplication • Advertisement: look at scatter plots (in m2) • The shape of the scatter plots determines the type of region (Njll,Njl), removes the ambiguity • The boundaries of the scatter plots offer additional measurements, 11 altogether:

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35. How do we measure mass? We cannot measure spin the same way! Events/unit of spin spin Why is it difficult to measure the spin?

36. Why is it difficult to measure the spin? • Missing energy signatures arise from something like: • Several alternative explanations:

37. Spin measurements – Iproduction cross-section Kane,Petrov,Shao,Wang 2008 • G. Kane et al: ”Basically, if the mass and the production cross-section are measured, the spin is then determined” • Are we really measuring the production cross-section? • How can we be sure that • There is no contribution from indirect production of particle Y? • Think of W pair production from top quarks • The branching fraction B(X->SM) is 100 %?

38. Spin measurements – Iproduction cross-section • Even if we were able to measure • It depends on the mass of X (OK, it will be measured) • It depends on the mass of heavy t-channel paticles (?) • It depends on the coupling of X to gluons (?) • It depends on the coupling of X to quarks (?) • It depends on the representation (Number of colors) of X (?) • It depends on the spin of X (Yes! That’s what we want) • Conclusion: it is very unlikely that by measuring a single number (such as a cross-section) we will be able to determine the spin. We have to look at distributions. What about those question marks?

39. What is a good distribution to look at? Athanasiou et al 06, Kilic,Wang,Yavin 07, Csaki,Heinonen,Perelstein 07, S. Thomas (KITP) • Invariant mass distributions! • Advantages: • well studied • know about spin • Disadvantage: know about many other things, not all of which are measured! • Masses MA, MB, MC, MD (x,y,z) • Couplings and mixing angles (gL and gR) • Particle-antiparticle (D/D*) fraction (f/f*) (f+f*=1) • Previous spin studies compare two sets of spin assignments, but fix everything else • This is not a true measurement of the spin • It’s the wrong chronological order

40. Most likely chronology • Measure the SM backgrounds (rediscover SM) • Find a missing energy signal • Convince yourself it is a real signal • Measure the cross-section (times BR) • Measure the overall mass scale • Observe structures, measure individual masses • Measure spins? • Then perhaps also measure • Chirality of the couplings (L versus R) • Mixing angles • Particle-antiparticle fraction

41. What is the relevant question? • We ask: Given the data, which spin configuration gives a good fit for arbitrary values of the yet unknown parameters? • We fix: mass spectrum • We let spins, couplings, mixing angles, particle/antiparticle fraction f, etc. to float • Previously people had asked: Given the data, which spin configuration gives a good fit for fixed values (the true ones) of the yet unknown parameters? • They fix: everything but the spins • Then let spins to float

42. Does this really make any difference? Yes! Dilepton invariant mass distribution. Data from SPS1a. Spins vary Everything else fixed to SPS1a values Easy to distinguish! Mass spectrum fixed to SPS1a values Everything else varies Difficult to distinguish! Athanasiou, Lester, Smillie, Webber 06 Burns, Kong, KM, Park 08

43. Does this really make any difference? Yes! Lepton charge (Barr) asymmetry. Data: “UED” with SPS1a mass spectrum. Spins vary Everything else fixed to SPS1a values Easy to distinguish! Mass spectrum fixed to SPS1a values Everything else varies Difficult to distinguish! Athanasiou, Lester, Smillie, Webber 06 Burns, Kong, KM, Park 08

44. How to let the couplings float? • Scan by brute force. Mobilize an army of graduate students, consider all (105?) possible values of MSSM parameters, fit each set of (105?) parameter values to the data. Decide if it fits or not. Continue until all possible values (or graduate students) are exhausted. Obviously impractical. • Mobilize some theorists to figure out which of the 105 parameters are really relevant for the invariant mass distributions. Consider a “constrained” MSSM where only the relevant parameters are varied. Still too many parameters (15?). • Ask the theorists to figure out which combinations of parameters are really relevant for the invariant mass distributions. Our claim: • For a three-step (two-step) decay chain there are only three (one) parameters which encode all of the model-dependence

45. Model-independent approach • Most general parameterization of the couplings • Each vertex has 4 real parameters, |aL|, |aR|, 2 phases • The phases are unobservable • The product |aL|.|aR| absorbed in the normalization • The ratios |aR|/|aL| defined as new parameters

46. Helicity combinations • We cannot measure the fermion helicity • Previous work considered only certain helicity combinations • We also consider all remaining possibilities • Each block gives rise to a specific function FS;IJ • The functions FS;IJ provide a model independent basis for the spin analysis

47. Final parameter count • How many parameters altogether? • No. Only three! • The quark vertex is parameterized by a combination of f and • At the Tevatron f=0.5 and there are only 2 parameters A completely model-independent approach requires only up to 3 continuous parameters!

48. ~ Choice of parameters • What is the best parameterization? • No. Even better: change basis • Write any invariant mass distribution as

49. The advantage of the new basis • Nice (intermediate) result • The jlf distribution is also predicted! • Note the opposite signs for beta and gamma: those terms will cancel in the sum!

50. What are all these functions?