Collider Phenomenology From basic knowledge to new physics searches

Transcription

1 Collider Phenomenology From basic knowledge to new physics searches Tao Han University of Wisconsin Madison BUSSTEPP 2010 Univ. of Swansea, Aug. 23 Sept. 3, 2010

2 Outline: Lecture I: Colliders and Detectors

3 Outline: Lecture I: Colliders and Detectors Lecture II: Basics Techniques and Tools

4 Outline: Lecture I: Colliders and Detectors Lecture II: Basics Techniques and Tools Lecture III: (a). An e + e Linear Collider (b). Perturbative QCD at Hadron Colliders (c). Hadron Colliders Physics

5 Outline: Lecture I: Colliders and Detectors Lecture II: Basics Techniques and Tools Lecture III: (a). An e + e Linear Collider (b). Perturbative QCD at Hadron Colliders (c). Hadron Colliders Physics Lecture IV: From Kinematics to Dynamics

6 Outline: Lecture I: Colliders and Detectors Lecture II: Basics Techniques and Tools Lecture III: (a). An e + e Linear Collider (b). Perturbative QCD at Hadron Colliders (c). Hadron Colliders Physics Lecture IV: From Kinematics to Dynamics Lecture V: Search for New Physics at Hadron Colliders

7 Outline: Lecture I: Colliders and Detectors Lecture II: Basics Techniques and Tools Lecture III: (a). An e + e Linear Collider (b). Perturbative QCD at Hadron Colliders (c). Hadron Colliders Physics Lecture IV: From Kinematics to Dynamics Lecture V: Search for New Physics at Hadron Colliders Main reference: TASI 04 Lecture notes hep-ph/ , plus the other related lectures in this school.

8 III(c). Hadron Collider Physics (A). New HEP frontier: the LHC Major discoveries and excitement ahead... ATLAS (90m underground) CMS (New mission started in March 2010.)

9 LHC Event rates for various SM processes:

10 LHC Event rates for various SM processes: /cm 2 /s 100 fb 1 /yr. Annual yield # of events = σ L int : 10B W ± ; 100M t t; 10M W + W ; 1M H 0... Great potential to open a new chapter of HEP!

11 Theoretical challenges: Unprecedented energy frontier

12 Theoretical challenges: Unprecedented energy frontier (a) Total hadronic cross section: Non-perturbative. The order of magnitude estimate: σ pp = πr 2 eff π/m2 π 120 mb.

13 Theoretical challenges: Unprecedented energy frontier (a) Total hadronic cross section: Non-perturbative. The order of magnitude estimate: σ pp = πr 2 eff π/m2 π 120 mb. Energy-dependence? σ(pp) 21.7 ( < π m 2 π s GeV 2) ln 2 s s 0 Empirical relation Froissart bound.

14 Theoretical challenges: Unprecedented energy frontier (a) Total hadronic cross section: Non-perturbative. The order of magnitude estimate: σ pp = πr 2 eff π/m2 π 120 mb. Energy-dependence? σ(pp) 21.7 ( < π m 2 π s GeV 2) ln 2 s s 0 Empirical relation Froissart bound. σ pp (S) = (b) Perturbative hadronic cross section: dx 1 dx 2 P 1 (x 1, Q 2 )P 2 (x 2, Q 2 ) ˆσ parton (s).

15 Theoretical challenges: Unprecedented energy frontier (a) Total hadronic cross section: Non-perturbative. The order of magnitude estimate: σ pp = πr 2 eff π/m2 π 120 mb. Energy-dependence? σ(pp) 21.7 ( < π m 2 π s GeV 2) ln 2 s s 0 Empirical relation Froissart bound. σ pp (S) = (b) Perturbative hadronic cross section: dx 1 dx 2 P 1 (x 1, Q 2 )P 2 (x 2, Q 2 ) ˆσ parton (s). Accurate (higher orders) partonic cross sections ˆσ parton (s). Parton distributions functions to the extreme (density): Q 2 (a few TeV ) 2, x

16 Experimental challenges: The large rate turns to a hostile environment: 1 billion event/sec: impossible read-off! 1 interesting event per 1,000,000: selection (triggering).

17 Experimental challenges: The large rate turns to a hostile environment: 1 billion event/sec: impossible read-off! 1 interesting event per 1,000,000: selection (triggering). 25 overlapping events/bunch crossing: Colliding beam n 1 n t = 1/f Severe backgrounds!

18 Triggering thresholds: ATLAS Objects η p T (GeV) µ inclusive (20) e/photon inclusive (26) Two e s or two photons (15) 1-jet inclusive (290) 3 jets (130) 4 jets (90) τ/hadrons (65) /E T Jets+/E T 3.2, ,50 (100,100) (η = ; η = )

19 Triggering thresholds: ATLAS Objects η p T (GeV) µ inclusive (20) e/photon inclusive (26) Two e s or two photons (15) 1-jet inclusive (290) 3 jets (130) 4 jets (90) τ/hadrons (65) /E T Jets+/E T 3.2, ,50 (100,100) (η = ; η = ) With optimal triggering and kinematical selections: p T GeV, η 3 5; /E T 100 GeV.

20 (B). Special kinematics for hadron colliders Hadron momenta: P A = (E A,0,0, p A ), P B = (E A,0,0, p A ), The parton momenta: p 1 = x 1 P A, p 2 = x 2 P B. Then the parton c.m. frame moves randomly, even by event: β cm = x 1 x 2 x 1 + x 2, or : y cm = 1 2 ln 1 + β cm 1 β cm = 1 2 ln x 1 x 2, ( < y cm < ).

21 (B). Special kinematics for hadron colliders Hadron momenta: P A = (E A,0,0, p A ), P B = (E A,0,0, p A ), The parton momenta: p 1 = x 1 P A, p 2 = x 2 P B. Then the parton c.m. frame moves randomly, even by event: β cm = x 1 x 2 x 1 + x 2, or : y cm = 1 2 ln 1 + β cm 1 β cm = 1 2 ln x 1 x 2, ( < y cm < ). The four-momentum vector transforms as ( ) ( E γ γ βcm p = z γ β cm γ ( cosh ycm sinh y = cm sinh y cm cosh y cm This is often called the boost. ) ( ) E p z ) ( E p z ).

22 One wishes to design final-state kinematics invariant under the boost: For a four-momentum p p µ = (E, p), E T = p 2 T + m2, y = 1 2 ln E + p z, E p z p µ = (E T cosh y, p T sin φ, p T cos φ, E T sinh y), d 3 p E = p Tdp T dφ dy = E T de T dφ dy.

23 One wishes to design final-state kinematics invariant under the boost: For a four-momentum p p µ = (E, p), E T = p 2 T + m2, y = 1 2 ln E + p z, E p z p µ = (E T cosh y, p T sin φ, p T cos φ, E T sinh y), d 3 p E = p Tdp T dφ dy = E T de T dφ dy. Due to random boost between Lab-frame/c.m. frame event-by-event, y = 1 2 ln E + p z E p z = 1 2 ln (1 β cm)(e + p z ) (1 + β cm )(E p z ) = y y cm.

24 One wishes to design final-state kinematics invariant under the boost: For a four-momentum p p µ = (E, p), E T = p 2 T + m2, y = 1 2 ln E + p z, E p z p µ = (E T cosh y, p T sin φ, p T cos φ, E T sinh y), d 3 p E = p Tdp T dφ dy = E T de T dφ dy. Due to random boost between Lab-frame/c.m. frame event-by-event, y = 1 2 ln E + p z E p z = 1 2 ln (1 β cm)(e + p z ) (1 + β cm )(E p z ) = y y cm. In the massless limit, rapidity pseudo-rapidity: y η = 1 2 ln 1 + cos θ 1 cos θ = lncot θ 2. Exercise 4.1: Verify all the above equations.

25 The Lego plot: A CDF di-jet event on a lego plot in the η φ plane.

26 The Lego plot: A CDF di-jet event on a lego plot in the η φ plane. φ, y = y 2 y 1 is boost-invariant. Thus the separation between two particles in an event R = φ 2 + y 2 is boost-invariant, and lead to the cone definition of a jet.

27 (C). Hadron collider status: The Tevatron rocks, and the LHC delivers!

28 (C). Hadron collider status: The Tevatron rocks, and the LHC delivers! At the Tevatron Run II: Peak luminosity record high cm 2 s 1 ; Integrated luminosity 5 fb 1 /expt, still with potential for discovery.

29 (C). Hadron collider status: The Tevatron rocks, and the LHC delivers! At the Tevatron Run II: Peak luminosity record high cm 2 s 1 ; Integrated luminosity 5 fb 1 /expt, still with potential for discovery. At the LHC: E cm = 7 TeV, integrated luminosity 1.5 pb 1, leading the HEP frontier.

30 ATLAS Z re-discovery:

31 ATLAS Z re-discovery: number of events m(ee) (GeV) D0 Z events.

32 ATLAS W re-discovery:

33 ATLAS W re-discovery: events / 0.5 GeV CDF RUN II PRELIMINARY χ /dof = 64 / m T (µν) (GeV) CDF W events.

34 CMS 1-jet in different rapidities: D0 1-jet in rapidity ranges:

35 CMS 1-jet in different rapidities: D0 1-jet in rapidity ranges: LHC QCD results went BEYOND the Tevatron!

36 CMS W+jets and top events CDF W+jets and top Number of tagged events Background Background errors Background+tt Background+tt errors -1 Data (194 ± 11 pb ) tt scaled to σ =6.7 pb tt Require H T >200 GeV for 3 jets Number of jets in W+jets

37 CMS W+jets and top events CDF W+jets and top Number of tagged events Background Background errors Background+tt Background+tt errors -1 Data (194 ± 11 pb ) tt scaled to σ =6.7 pb tt Require H T >200 GeV for 3 jets Number of jets in W+jets LHC top studies catching up!

38 ATLAS E/ T distribution: CDF E/ T at high end: dn/de/ T CDF preliminary 10 2 data (L=87 pb -1 ) Z+jets W+jets tt+ww+wz+zz E/ T (GeV)

39 ATLAS E/ T distribution: CDF E/ T at high end: dn/de/ T CDF preliminary 10 2 data (L=87 pb -1 ) Z+jets W+jets tt+ww+wz+zz E/ T (GeV) LHC E/ T results rapidly improving!

40 ATLAS E/ T distribution: CDF E/ T at high end: dn/de/ T CDF preliminary 10 2 data (L=87 pb -1 ) Z+jets W+jets tt+ww+wz+zz E/ T (GeV) LHC E/ T results rapidly improving! LHC achieved the first crucial step: The Standard Model rediscovered!

41 ... And have gone on to the physics BSM : 400 GeV < M q (jj) < 1.26 TeV excluded. First BSM physics search, beyond the Tevatron reach!

42 ... And have gone on to the physics BSM : 400 GeV < M q (jj) < 1.26 TeV excluded. First BSM physics search, beyond the Tevatron reach! Anxiously waiting for the new excitement...

43 IV. From Kinematics to Dynamics (A). Characteristic observables: Crucial for uncovering new dynamics.

44 IV. From Kinematics to Dynamics (A). Characteristic observables: Crucial for uncovering new dynamics. Selective experimental events = Characteristic kinematical observables (spatial, time, momentaum phase space) = Dynamical parameters (masses, couplings)

45 IV. From Kinematics to Dynamics (A). Characteristic observables: Crucial for uncovering new dynamics. Selective experimental events = Characteristic kinematical observables (spatial, time, momentaum phase space) = Dynamical parameters (masses, couplings) Energy momentum observables = mass parameters Angular observables = nature of couplings; Production rates, decay branchings/lifetimes = interaction strengths.

46 (B). Kinematical features: (a). s-channel singularity: bump search we do best. invariant mass of two-body R ab : m 2 ab = (p a + p b ) 2 = M 2 R. combined with the two-body Jacobian peak in transverse momentum: dˆσ dm 2 ee dp2 et Γ Z M Z (m 2 ee M2 Z )2 + Γ 2 Z M2 Z m 2 ee 1 1 4p 2 et /m2 ee

47 (B). Kinematical features: (a). s-channel singularity: bump search we do best. invariant mass of two-body R ab : m 2 ab = (p a + p b ) 2 = M 2 R. combined with the two-body Jacobian peak in transverse momentum: number of events dˆσ dm 2 ee dp2 et Γ Z M Z (m 2 ee M2 Z )2 + Γ 2 Z M2 Z Events m 2 ee 1 Electron E T - W Candidate p 2 et /m2 ee Data PMCS+QCD QCD bkg D0 Run II Preliminary Z e + e m(ee) (GeV) elec (GeV) 80 W eν E T

48 transverse mass of two-body W e ν e : m 2 eν T = (E et + E νt ) 2 ( p et + p νt ) 2 = 2E et ET miss (1 cos φ) m 2 eν. Events Transverse Mass - W Candidate Data PMCS+QCD QCD bkg D0 Run II Preliminary Events Missing E T - W Candidate Data PMCS+QCD QCD bkg D0 Run II Preliminary Transverse mass(gev) MET(GeV) If p T (W) = 0, then m eν T = 2E et = 2E miss T.

49 Exercise 5.1: For a two-body final state kinematics, show that dˆσ dp et = 4p et dˆσ s 1 4p 2 et /s dcos θ. where p et = p e sin θ is the transverse momentum and θ is the polar angle in the c.m. frame. Comment on the apparent singularity at p 2 et = s/4. Exercise 5.2: Show that for an on-shell decay W e ν e : m 2 eν T (E et + E νt ) 2 ( p et + p νt ) 2 m 2 eν. Exercise 5.3: Show that if W/Z has some transverse motion, δp V, then: p et p et [1 + δp V /M V ], m 2 eν T m2 eν T [1 (δp V /M V ) 2 ], m 2 ee = m 2 ee.

50 H 0 W + W j 1 j 2 e ν e : cluster transverse mass (I): m 2 WW T = (E W 1 T + E W2 T) 2 ( p jjt + p et + p miss = ( p 2 jjt + M2 W + p 2 eνt + M2 W )2 ( p jjt + p et + p T miss ) 2 MH 2. T ) 2 where p miss T p/ T = obs p obs T.

51 `2 2 H 0 W + W j 1 j 2 e ν e : cluster transverse mass (I): m 2 WW T = (E W 1 T + E W2 T) 2 ( p jjt + p et + p miss = ( p 2 jjt + M2 W + p 2 eνt + M2 W )2 ( p jjt + p et + p T miss ) 2 MH 2. T ) 2 where p miss T p/ T = obs p obs T. H W W `1 1 H 0 W + W e + ν e e ν e : effecive transverse mass: m 2 eff T = (E e1t + E e2t + ET miss ) 2 ( p e1t + p e2t + p T miss ) 2 m eff T E e1t + E e2t + E miss T

52 `2 2 H 0 W + W j 1 j 2 e ν e : cluster transverse mass (I): m 2 WW T = (E W 1 T + E W2 T) 2 ( p jjt + p et + p miss = ( p 2 jjt + M2 W + p 2 eνt + M2 W )2 ( p jjt + p et + p T miss ) 2 MH 2. T ) 2 where p miss T p/ T = obs p obs T. H W W `1 1 H 0 W + W e + ν e e ν e : effecive transverse mass: m 2 eff T = (E e1t + E e2t + ET miss ) 2 ( p e1t + p e2t + p T miss ) 2 m eff T E e1t + E e2t + E miss T cluster transverse mass (II): m 2 WW C ( p = 2 T,ll + M2 ll + p/ T p 2 T,ll + M2 ll + p/ T m WW C ) 2 ( p T,ll + p/ T ) 2

53 M WW invariant mass (WW fully reconstructable): M WW, T transverse mass (one missing particle ν): M eff, T effetive trans. mass (two missing particles): M WW, C cluster trans. mass (two missing particles):

54 M WW invariant mass (WW fully reconstructable): M WW, T transverse mass (one missing particle ν): M eff, T effetive trans. mass (two missing particles): M WW, C cluster trans. mass (two missing particles): YOU design an optimal variable/observable for the search.

55 cluster transverse mass (III): H 0 τ + τ µ + ν τ ν µ, ρ ν τ p A lot more complicated with (many) more ν s? H +

56 cluster transverse mass (III): H 0 τ + τ µ + ν τ ν µ, ρ ν τ p A lot more complicated with (many) more ν s? H Not really! τ + τ ultra-relativistic, the final states from a τ decay highly collimated: θ γ 1 τ = m τ /E τ = 2m τ /m H 1.5 (m H = 120 GeV). We can thus take p τ + = p µ + + p ν s +, p ν s + c + p µ +. p τ = p ρ + p ν s, p ν s c p ρ. where c ± are proportionality constants, to be determined. +

57 cluster transverse mass (III): H 0 τ + τ µ + ν τ ν µ, ρ ν τ p A lot more complicated with (many) more ν s? H Not really! τ + τ ultra-relativistic, the final states from a τ decay highly collimated: θ γ 1 τ = m τ /E τ = 2m τ /m H 1.5 (m H = 120 GeV). We can thus take p τ + = p µ + + p ν s +, p ν s + c + p µ +. p τ = p ρ + p ν s, p ν s c p ρ. where c ± are proportionality constants, to be determined. This is applicable to any decays of fast-moving particles, like T Wb lν, b. +

58 Experimental measurements: p ρ, p µ +, p/ T : Unique solutions for c ± exist if c + (p µ +) x + c (p ρ ) x = (p/ T ) x, c + (p µ +) y + c (p ρ ) y = (p/ T ) y. (p µ +) x /(p µ +) y (p ρ ) x /(p ρ ) y. Physically, the τ + and τ should form a finite angle, or the Higgs should have a non-zero transverse momentum.

59 Experimental measurements: p ρ, p µ +, p/ T : Unique solutions for c ± exist if c + (p µ +) x + c (p ρ ) x = (p/ T ) x, c + (p µ +) y + c (p ρ ) y = (p/ T ) y. (p µ +) x /(p µ +) y (p ρ ) x /(p ρ ) y. Physically, the τ + and τ should form a finite angle, or the Higgs should have a non-zero transverse momentum. 1/σ dσ/dm m ττ [ GeV ] m ττ [ GeV ]

60 (b). Two-body versus three-body kinematics Energy end-point and mass edges: utilizing the two-body kinematics Consider a simple case: e + e µ + R µ R with two body decays : µ + R µ+ χ 0, µ R µ χ 0. In the µ + R -rest frame: E0 µ = M2 µ R m 2 χ 2M µr. In the Lab-frame: (1 β)γe 0 µ Elab µ (1 + β)γe0 µ with β = ( 1 4M 2 µ R /s ) 1/2, γ = (1 β) 1/2. Energy end-point: E lab µ M 2 µ R m 2 χ. Mass edge: m max µ + µ = s 2m χ.

61 (b). Two-body versus three-body kinematics Energy end-point and mass edges: utilizing the two-body kinematics Consider a simple case: e + e µ + R µ R with two body decays : µ + R µ+ χ 0, µ R µ χ 0. In the µ + R -rest frame: E0 µ = M2 µ R m 2 χ 2M µr. In the Lab-frame: (1 β)γe 0 µ Elab µ (1 + β)γe0 µ with β = ( 1 4M 2 µ R /s ) 1/2, γ = (1 β) 1/2. Energy end-point: E lab µ M 2 µ R m 2 χ. Mass edge: m max µ + µ = s 2m χ. Same idea can be applied to hadron colliders...

62 l + ~ 0 1 Consider a squark cascade decay: q Z l ~q ~ st edge : M max (ll) M χ 0 2 M χ 0 1 ; 2 nd edge : M max (llj) M q M χ 0 1.

63 dσ/dm llq (Events/100fb -1 /5GeV) dσ/dm lq (Events/100fb -1 /5GeV) dσ/dm ll (Events/100fb -1 /0.375GeV) (a) mll (GeV) (c1) High m lq (GeV) (d) mllq (GeV) dσ/dm hq (Events/100fb -1 /5GeV) dσ/dm lq (Events/100fb -1 /5GeV) dσ/dm llq (Events/100fb -1 /5GeV) (b) mllq (GeV) (c2) Low m lq (GeV) (e) mhq (GeV)

64 (c). t-channel singularity: splitting. Gauge boson radiation off a fermion: The familiar Weizsäcker-Williams approximation f f a p γ / f σ(fa f X) X dx dp 2 T P γ/f (x, p2 T ) σ(γa X), P γ/e (x, p 2 T ) = α 2π 1 + (1 x) 2 x ( 1 p 2 ) E m e. T

65 (c). t-channel singularity: splitting. Gauge boson radiation off a fermion: The familiar Weizsäcker-Williams approximation f f a p γ / f σ(fa f X) X dx dp 2 T P γ/f (x, p2 T ) σ(γa X), P γ/e (x, p 2 T ) = α 2π 1 + (1 x) 2 x ( 1 p 2 ) E m e. T The kernel is the same as q qg generic for parton splitting; The high energy enhancement dp 2 T /p2 T ln(e/m e) reflects the collinear behavior.

66 Generalize to massive gauge bosons: PV/f T (x, p2 T ) = g2 V + g2 A 1 + (1 x) 2 8π 2 x P L V/f (x, p2 T ) = g2 V + g2 A 4π 2 1 x x p 2 T (p 2 T + (1 x)m2 V )2, (1 x)m 2 V (p 2 T + (1 x)m2 V )2.

67 Generalize to massive gauge bosons: At low-p jt, PV/f T (x, p2 T ) = g2 V + g2 A 1 + (1 x) 2 8π 2 x P L V/f (x, p2 T ) = g2 V + g2 A 4π 2 1 x x p 2 T (p 2 T + (1 x)m2 V )2, (1 x)m 2 V (p 2 T + (1 x)m2 V )2. Special kinematics for massive gauge boson fusion processes: For the accompanying jets, At high-p jt, p 2 jt (1 x)m2 V E j (1 x)e q dσ(v T ) dp 2 jt dσ(v L ) dp 2 jt 1/p 2 jt 1/p 4 jt } f orward jet tagging central jet vetoing has become important tools for Higgs searches, single-top signal etc.

68 (C). Charge forward-backward asymmetry A FB : The coupling vertex of a vector boson V µ to an arbitrary fermion pair f i L,R τ g f τ γ µ P τ crucial to probe chiral structures. The parton-level forward-backward asymmetry is defined as A i,f FB N F N B N F + N B = 3 4 A ia f, A f = (gf L )2 (g f R )2 (g f L )2 + (g f R )2. where N F (N B ) is the number of events in the forward (backward) direction defined in the parton c.m. frame relative to the initial-state fermion p i.

69 At hadronic level: A LHC FB = dx1 q A q,f FB ( Pq (x 1 )P q (x 2 ) P q (x 1 )P q (x 2 ) ) sign(x 1 x 2 ) ( Pq (x 1 )P q (x 2 ) + P q (x 1 )P q (x 2 ) ). dx1 q

70 At hadronic level: A LHC FB = dx1 q A q,f FB ( Pq (x 1 )P q (x 2 ) P q (x 1 )P q (x 2 ) ) sign(x 1 x 2 ) ( Pq (x 1 )P q (x 2 ) + P q (x 1 )P q (x 2 ) ). dx1 q Perfectly fine for Z/Z -type: In p p collisions, p proton is the direction of p quark. In pp collisions, however, what is the direction of p quark?

71 At hadronic level: A LHC FB = dx1 q A q,f FB ( Pq (x 1 )P q (x 2 ) P q (x 1 )P q (x 2 ) ) sign(x 1 x 2 ) ( Pq (x 1 )P q (x 2 ) + P q (x 1 )P q (x 2 ) ). dx1 q Perfectly fine for Z/Z -type: In p p collisions, p proton is the direction of p quark. In pp collisions, however, what is the direction of p quark? It is the boost-direction of l + l.

72 How about W ± /W ± (l ± ν)-type? In p p collisions, p proton is the direction of p quark, AND l + (l ) along the direction with q (q) OK at the Tevatron,

73 How about W ± /W ± (l ± ν)-type? In p p collisions, p proton is the direction of p quark, AND l + (l ) along the direction with q (q) OK at the Tevatron, But: (1). cann t get the boost-direction of l ± ν system; (2). Looking at l ± alone, no insight for W L or W R!

74 How about W ± /W ± (l ± ν)-type? In p p collisions, p proton is the direction of p quark, AND l + (l ) along the direction with q (q) OK at the Tevatron, But: (1). cann t get the boost-direction of l ± ν system; (2). Looking at l ± alone, no insight for W L or W R! In p p collisions: (1). a reconstructable system; (2). with spin correlation: Only tops: W t b l ± ν b: dσ/dcos θ la (pb) W L W R cos θ la 0.5 1

75 (D). CP asymmetries A CP : To non-ambiguously identify CP-violation effects, one must rely on CP-odd variables.

76 (D). CP asymmetries A CP : To non-ambiguously identify CP-violation effects, one must rely on CP-odd variables. Definition: A CP vanishes if CP-violation interactions do not exist (for the relevant particles involved). This is meant to be in contrast to an observable: that d be modified by the presence of CP-violation, but is not zero when CP-violation is absent. e.g. M (χ ± χ 0 ), σ(h0, A 0 ),...

77 (D). CP asymmetries A CP : To non-ambiguously identify CP-violation effects, one must rely on CP-odd variables. Definition: A CP vanishes if CP-violation interactions do not exist (for the relevant particles involved). This is meant to be in contrast to an observable: that d be modified by the presence of CP-violation, but is not zero when CP-violation is absent. e.g. M (χ ± χ 0 ), σ(h0, A 0 ),... Two ways: a). Compare the rates between a process and its CP-conjugate process: R(i f) R(ī f) R(i f) + R(ī f), e.g. Γ(t W + q) Γ( t W q) Γ(t W + q) + Γ( t W q).

78 b). Construct a CP-odd kinematical variable for an initially CP-eigenstate: M M 1 + M 2 sin θ, A CP = σ F σ B = 1 0 dσ dcos θ dcos θ 0 1 dσ dcos θ dcos θ

79 b). Construct a CP-odd kinematical variable for an initially CP-eigenstate: M M 1 + M 2 sin θ, A CP = σ F σ B = 1 0 dσ dcos θ dcos θ 0 E.g. 1: H Z(p 1 )Z (p 2 ) e + (q 1 )e (q 2 ), µ + µ Z µ ( p 1 ) 1 dσ dcos θ dcos θ Z ν ( p 2 ) Γ µν ( p 1, p 2 ) h Γ µν (p 1, p 2 ) = i 2 v h[a M2 Z gµν +b (p µ 1 pν 2 p 1 p 2 g µν )+ b ǫ µνρσ p 1ρ p 2σ ] a = 1, b = b = 0 for SM. In general, a, b, b complex form factors, describing new physics at a higher scale.

80 For H Z(p 1 )Z (p 2 ) e + (q 1 )e (q 2 ), µ + µ, define: O CP ( p 1 p 2 ) ( q 1 q 2 ), or cos θ = ( p 1 p 2 ) ( q 1 q 2 ) p 1 p 2 q 1 q 2 ).

81 For H Z(p 1 )Z (p 2 ) e + (q 1 )e (q 2 ), µ + µ, define: O CP ( p 1 p 2 ) ( q 1 q 2 ), or cos θ = ( p 1 p 2 ) ( q 1 q 2 ) p 1 p 2 q 1 q 2 ). E.g. 2: H t(p t ) t(p t ) e+ (q 1 )ν 1 b 1, e (q 2 )ν 2 b 2. thus define an asymmetry angle. m t v t(a + bγ 5 )t H O CP ( p t p t ) ( p e + p e ).