1. Mass Measurement Techniques (for the Large Hadron Collider) Based on “A review of Mass Measurement Techniques proposed for the Large Hadron Collider”, Barr and Lester, arXiv:1004.2732 CERN, February 2011arXiv:1004.2732 Christopher Lester University of Cambridge

2. Extracts from ……… arXiv:1004.2732

3. What mass reconstruction techniques am I going to talk about? –am not interested in fully visible final states as standard mass reconstruction techniques apply –will only consider new particles of unknown mass decaying to invisible particles of unknown mass (and other visible particles) –mainly talk about thinks I’ve worked with Transverse masses, MT2, kinks, kinematic methods. (Not Matrix Element / likelihood methods) –not shameless promotion – focus on faults!

4. Types of Technique Missing transverse momentum M_eff, H_T s Hat Min M_T M_TGEN M_T2 / M_CT M_T2 (with “kinks”) M_T2 / M_CT ( parallel / perp ) M_T2 / M_CT ( “sub-system” ) “Polynomial” constraints Multi-event polynomial constraints Whole dataset variables Cross section Max Likelihood / Matrix Element Few assumptions Many assumptions

5. Types of Technique Missing transverse momentum M_eff, H_T s Hat Min M_T M_TGEN M_T2 / M_CT M_T2 (with “kinks”) M_T2 / M_CT ( parallel / perp ) M_T2 / M_CT ( “sub-system” ) “Polynomial” constraints Multi-event polynomial constraints Whole dataset variables Cross section Max Likelihood / Matrix Element Vague conclusions Specific conclusions

6. Types of Technique Missing transverse momentum M_eff, H_T s Hat Min M_T M_TGEN M_T2 / M_CT M_T2 (with “kinks”) M_T2 / M_CT ( parallel / perp ) M_T2 / M_CT ( “sub-system” ) “Polynomial” constraints Multi-event polynomial constraints Whole dataset variables Cross section Max Likelihood / Matrix Element Robust Fragile

7. The balance of benefits Few assumptions Many assumptions Vague conclusions Specific conclusions Robust Fragile

8. (more details in arXiv:1004.2732 ) The review orders from simple to complicated ….

9. … and from few events required, to many events required …. = M

10. Idealised Hadron Collider Proton 1 Proton 2 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A Remnant 1 Remnant 2 Visible Invisible

11. More Realistic Hadron Collider TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A Proton 1 Proton 2 Remnant 1 Remnant 2 Visible Invisible ISR UE / MPI

12. Few assumptions Vague Conclusions

13. transverse variables without baggage (also known as ETmiss, PTmiss, missing energy, missing momentum etc) (also known as Meff, or the effective mass) (There are no standard definitions of M est and H T authors differ in how many jets are used etc. ) All have some sensitivity to the overall mass scales involved, but interpretation requires a model and more assumptions.

14. M est / M eff example D. R. Tovey, Measuring the SUSY mass scale at the LHC, Phys. Lett. B498 (2001) [hep-ph/0006276] Observable M est sometimes correlates with property of model M eff defined by but correlation is model dependent M est M eff

15. More assumptions Less Vague Conclusions

16. ISR UE / MPI seeks to bound the invariant mass of the interesting part of the collision P. Konar, K. Kong, and K. T. Matchev, rootsmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders, JHEP 03 (2009) 085, [arXiv:0812.1042].

17. Without ISR / MPI HTHT E T miss From arXiv:0812.1042

18. With ISR & MPI etc From arXiv:0812.1042

19. E T miss HTHT From arXiv:0903.2013 Though dependence on ISR Is large, it is calculable and may offer a good test of our understanding. See arXiv:0903.2013 and 1006.0653 Transverse variables are less sensitive to ISR (this is both good & bad)

20. What about (transverse) variables designed to measure the masses of individual particles?

21. A popular new-physics scenario Proton 1 Proton 2 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A Remnant 1 Remnant 2 Invisible Visible Upstream Momentum

22. Example:

23. We have two copies of this: Unknown mass (Visible) (Invisible) A B TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A One copy could be just as relevant!

24. Can get a long way just using the (full) transverse mass!

25. Recall the (full!) W transverse mass W e ν TT !! NOT THIS !! !! This is NOT the transverse mass !!

26. W transverse mass : why used? In every event m T < m W if the W is on shell In every event m T is a lower bound on m W There are events in which m T can saturate the bound on m W. The above properties motivate m T in W mass measurements.

27. I.W.Kim: “Algebraic singularity method of mass measurement with missing energy” arXiv:0910.1149v2 [hep-ph] arXiv:0910.1149v2 Observable Momentum Phase Space Formalising an old idea … kinematic boundaries, creases, edges, cusps etc

28. But outside standard model Don’t usually know mass of invisible final state particle! (neutralino?) Chi parameter “χ” to represent the hypothesized mass of invisible particle So for new physics need:

29. (most commonly on x-axis of many 2D plots which occur later) Chi parameter “χ” (mass of “invisible” final state particle) is EVERYWHERE!

30. Reminder: We define the “full” transverse mass in terms of “χ”, a hypothesis for the mass of the invisible particle, since it is unknown. where and A B

31. A B Schematically, all we have guaranteed so far is the picture below: mT()mT()  mBmB mAmA Since “χ” can now be “wrong”, some of the properties of the transverse mass can “break”: m T (  ) max is no longer invariant under transverse boosts! (except when  =m B ) m T (  )<m A may no longer hold! (however we always retain: m T (m B ) < m A )

32. It turns out that one actually gets things more like this: mT()mT()  mBmB mAmA A B mBmB

33. In fact, we get this very nice result: mBmB mAmA A B mT(  ) Minimal Kinematic Constraints and m(T2), Hsin-Chia Cheng and Zhenyu Han (UCD) e-Print: arXiv:0810.5178 [hep-ph] andarXiv:0810.5178 [hep-ph] “Transverse masses and kinematic constraints, from the Boundary to the Crease” arXiv:0908.3779arXiv:0908.3779 The “full” transverse mass curve is the boundary of the region of (mother,daughter) masses consistent with the observed event!

34. Digression (Salutary Tale)

35. Cricket

36. The Stumps

37. The Ashes

38. Kinematic Boundary

39. Transverse masses and kinematic constraints: from the boundary to the crease arXiv:0908.3779v2arXiv:0908.3779v2 [hep-ph]

40. Back to the nice result mBmB mAmA A B mT(  ) Minimal Kinematic Constraints and m(T2), Hsin-Chia Cheng and Zhenyu Han (UCD) e-Print: arXiv:0810.5178 [hep-ph] andarXiv:0810.5178 [hep-ph] “Transverse masses and kinematic constraints, from the Boundary to the Crease” arXiv:0908.3779arXiv:0908.3779 The “full” transverse mass curve is the boundary of the region of (mother,daughter) masses consistent with the observed event!

41. Event 1 of 8 mT()mT()  mBmB mAmA A B mBmB

42. Event 2 of 8 mT()mT()  mBmB mAmA A B mBmB

43. Event 3 of 8 mT()mT()  mBmB mAmA A B mBmB

44. Event 4 of 8 mT()mT()  mBmB mAmA A B mBmB

45. Event 5 of 8 mT()mT()  mBmB mAmA A B mBmB

46. Event 6 of 8 mT()mT()  mBmB mAmA A B mBmB

47. Event 7 of 8 mT()mT()  mBmB mAmA A B mBmB

48. Event 8 of 8 mT()mT()  mBmB mAmA A B mBmB

49. Overlay all 8 events mT()mT()  mBmB mAmA A B mBmB

50. Overlay many events CASE 2  arXiv: 0711.4008 mBmB mAmA mT()mT()

51. Here is a transverse mass “KINK” CASE 2  mT()mT() arXiv: 0711.4008 mBmB mAmA

52. Alternatively, look at M T distributions for a variety of values of chi. mTmT arXiv: 0711.4008 Each curve has a different value of chi Where is the kink now?

53. What causes the kink? Two entirely independent things can cause the kink: –(1) Variability in the “visible mass” –(2) Recoil of the “interesting things” against Upstream Transverse Momentum Which is the dominant cause depends on the particular situation … let us look at each separately:

54. Kink cause 1: Variability in visible mass m Vis can change from event to event Gradient of m T (  ) curve depends on m Vis Curves with low m Vis tend to be “flatter” mT()mT()  mBmB mAmA A B mBmB

55. Kink cause 1: Variability in visible mass m Vis can change from event to event Gradient of m T (  ) curve depends on m Vis Curves with high m Vis tend to be “steeper” mT()mT()  mBmB mAmA A B mBmB

56. Kink cause 2 : Recoil against Upstream Momentum TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A Invisible Visible Upstream Momentum

57. Kink cause 2: Recoil against UTM UTM can change from event to event Gradient of m T (  ) curve depends on UTM Curves with UTM opposite to visible. momenta tend to be “flatter” mT()mT()  mBmB mAmA A B mBmB Upstream Momentum

58. Kink cause 2: Recoil against UTM UTM can change from event to event Gradient of m T (  ) curve depends on UTM Curves with UTM parallel to visible momenta tend to be “steeper” mT()mT()  mBmB mAmA A B mBmB Upstream Momentum

59. What do we do in events with a pair of decays? Invisible Visible Upstream Momentum Invisible Visible Upstream Momentum MT works for :

60. A A B B MT2 : the stransverse mass one can generalize m T to m T2 (“Transverse” mass to “Stransverse” mass) arXiv: hep-ph/9906349 For a pair of decays

61. MT2 distribution over many events: m T2 (m B ) mBmB mAmA MT2 endpoint structure is weaker than MT (due to more missing information in the event)

62. MT2 (like MT) is also a mass-space boundary mBmB mAmA mT2(  ) Minimal Kinematic Constraints and m(T2), Hsin-Chia Cheng and Zhenyu Han (UCD) e-Print: arXiv:0810.5178 [hep-ph]arXiv:0810.5178 [hep-ph] The MT2(chi) curve is the boundary of the region of (mother, daughter) mass-space consistent with the observed event!

63. MT2 and MT behave in exactly the same way as each other, and consequently they share the same kink structure. Somewhat surprisingly, MT and MT2 kink-based methods are the only(*) methods that have been found which can in principle determine the mass of the invisible particles in short chains! (see arXiv: 0810.5576) (*) There is evidence (Alwall) that Matrix Element methods can do so too, though at the cost of model dependence and very large amounts of CPU.

64. This should worry you … CASE 2

65. Are kinks observable ? Expect KINK only from UTM Recoil (perhaps only from ISR!) Expect stronger KINK due to both UTM recoil, AND variability in the visible masses. arXiv: 0711.4008

66. More hopeful news ….. “Top Quark Mass Measurement using mT2 in the Dilepton Channel at CDF” (arXiv:0911.2956 and PRD) reports that the mT2 measurement of the top-mass has the “smallest systematic error” in that channel. Top-quark physics is an important testing ground for mT2 methods, both at the LHC and at the Tevatron.

67. MT2 as search variable (utilising shape of differential distribution) m T2 (m B ) mAmA More likely use at the LHC for these variables. SELECT

68. (more details in arXiv:1004.2732 ) Not all proposed new-physics chains are short!

69. If chains a longer use “edges” or “Kinematic endpoints” Plot distributions of the invariant masses of what you can see

70. What is a kinematic endpoint? Consider M LL

71. Dilepton invariant mass distribution Di-Lepton Invariant Mass (GeV) Relative Frequency Straight line This is the Endpoint!

72. What about these invariant masses?

73. Might therefore need to define: OR ? Some extra difficulties – may not know order particles were emitted There are many other possibilities for resolving problems due to position ambiguity. For example, compare hep-ph/0007009 with arXiv:0906.2417

74. Kinematic Edges ll lq high llqXq lq low llq Xqllq lq lowlq high llqll

75. Determine how edge positions depend on sparticle masses hep-ph/0007009

76. Different parts of model space behave differently: m QLL max Where are the big mass differences? hep-ph/0007009

77. Solve all edge position for masses!..... mainly constrain mass differences

78. “Just”-constrained events (and over-constrained events) Even if there are invisible decay products, events can often be fully reconstructed if decay chains are long enough. (mass-shell constraints must be >= unknown momenta) Left: case considered in hep-ph/9812233 massless

79. For example (hep-ph/0312317) quintuples of events of the form: are exactly constrained similarly pairs of events of the form: (arXiv:0905.1344) are exactly constrained. Small collections of under-constrained events can be over-constrained!

80. (more details in arXiv:1004.2732 ) Once again – needed an index.

81. Not time to talk about many things Parallel and perpendicular MT2 and MCT Subsystem MT2 and MCT methods Solution counting methods (eg arXiv:0707.0030) Hybrid Variables Phase space boundaries (arXiv:0903.4371)0903.4371 Cusps and Singularity Variables (Ian-Woo Kim) and many more! In 30 minutes I have only scratched the surface of the variables that have been discussed. Even the recent review of mass measurement methods arXiv:1004.2732 makes only a small dent in 70+ pages. However it provides at least an index …

82. Let’s stop here!

83. Take home messages Lots of approaches to kinematic mass measurement –some very general, some very specific. –very little of the “detailed stuff” is tested in anger. Experimentalists not universally convinced of utility! –very often BGs present serious impediment. –theorists and experimenters should pay close attention to zone of applicability BUT –Finding sensible variables buys more than just mass measurements - e.g. signal sensitivity

84. Extras if time …

85. Other MT2 related variables (1/3) MCT (“Contralinear-Transverse Mass”) (arXiv:0802.2879) –Is equivalent to MT2 in the special case that there is no missing momentum (and that the visible particles are massless). –Proposes an interesting multi-stage method for measuring additional masses –Can be calculated fast enough to use in ATLAS trigger.

86. Other MT2 related variables (2/3) MTGEN (“MT for GENeral number of final state particles”) (arXiv:0708.1028) –Used when each “side” of the event decays to MANY visible particles (and one invisible particle) and it is not possible to determine which decay product is from which side … all possibilities are tried Inclusive or Hemispheric MT2 (Nojirir + Shimizu) (arXiv:0802.2412) –Similar to MTGEN but based on an assignment of decay product to sides via hemisphere algorithm. –Guaranteed to be >= MTGEN

87. Other MT2 related variables (3/3) M2C (“MT2 Constrained”) arXiv:0712.0943 (wait for v3... there are some problems with the v1 and v2 drafts) M2CUB (“MT2 Constrained Upper Bound”) arXiv:0806.3224 There is a sense in which these two variables are really two sides of the same coin. –if we could re-write history we might name them more symmetrically –I will call them m Small and m Big in this talk.

88. m Small and m Big Basic idea is to combine: –MT2 with –a di-lepton invariant mass endpoint measurement (or similar) providing:  = M A – M B (or M Y -M N in the notation of their figure above)

89. m T2 (  )  mBmB mAmA  m  LB  m Big m Small m  UB “Best case” (needs SPT, i.e. large recoil PT) Both m Big and m Small are found.

90. m T2 (  )  mBmB mAmA  m  LB  m Small “Typical ZPT case” (no m Big is found)

91. m T2 (  )  mBmB mAmA   “Possible ZPT case” (neither m Big nor m Small is found) * * Except for conventional definition of m Small to be  in this case.

92. m T2 (  )  mBmB mAmA   m Big m Small m  UB “Possible SPT case” (no m Small is found) * * Except for conventional definition of m Small to be  in this case.

93. What m Small and m Big look like, and how they determine the parent mass m Big m Small arXiv:0806.3224 Here is the true value of the parent mass … determined nicely

94. Outcome: m Big provides the first potentially-useful event- by-event upper bound for m A –(and a corresponding event-by-event upper bound for m B called m  UB ) m Small provides a new kind of event-by-event lower bound for m A which incorporates consistency information with the dilepton edge m Big is always reliant on SPT (large recoil of interesting system against “up-stream momentum”) – cannot ignore recoil here!