Introduction to Dalitz-plot Gagan Mohanty Student Seminar @ TIFR January 7, 01
Few Diversions
Hierarchical expansion of CKM (1983) + O( 4) magnitudes d s b phases d s b u c t u c t 3
A triangle at the heart V V V V ud cd td V V V us cs ts V V V ub cb tb V Vcb (0,0) d b* = 0 B Mesons * ub * (,h) f 3 f f 1 V V td * cb (1,0) argv arg ub V td Goal: check consistency of the CKM paradigm by overconstraining the Unitarity Triangle: measure three angles and two side lengths look for possible new physics contributions 4
B Meson: A threefold way CP violation in decay: direct Af 1 can take place both for A f neutral and charged B s can have time-dependent and -independent manifestations Need two competing diagrams of different CP-violating and conserving phases CP violation in mixing: indirect only neutral B s are possibly affected SM predicts very small effects q p 1 violation from mixing/decay interference: only neutral B s possibly affected purely time-dependent effect arises due to interference between decay with and without mixing Im 0, = 1 CP CP 5
Enter the Charmless 6
Two vs. Three -body PRO : - Larger BF than two-body decays - Correct way to study interference - Some modes in a well-defined CP eigenstate Gershon, Hazumi PLB 596, 163 3-body BF[10-6 ] CON : - large phase space with low event density; difficult to identify all phase-space structures - mixture of CP-even and -odd final states - more complicated analysis needed BF[10-6 ] 7
Inclusive Background fighting: Continuum (event topology) Other types of B decays (PID, charm and charmonia veto) Signal extraction (kinematics) Full (3body)/partial (QB) Analysis Strategy Dalitz plot technique (three-body decays having reasonable signal size) Time-dependent DP (3body) Time-dependent analysis in neutral B meson decays to determine CP violation parameters at each point of the phase space uds:cc:bb =.1:1.3:1.1 Complexity 8
Three-Particle Phase Space From four vectors 1 Conservation laws 4 Meson masses 3 Free rotation 3 Independent variable p m ( M, 0 r ) p m 1 ( E, p 1 1 ) p m r ( E, p ) r p m ( E, p 3 ) 3 3 1 B 3 Usual choice Invariant mass m 1 Invariant mass m 3 m m m m m 1 p1 p ) ( p p3 ) M m3 m ( ME m m m m 3 p p3 ) ( p p1 ) M m1 ( ME 3 1 Lorentz invariant phase space: dn dm 1 dm3 Decay rate M dm 1 dm 3 Invariant amplitude
A Dalitz plot is nothing but a scatter plot between any two of the three invariant mass squared variables, Decay rate: M Now Enter the Dalitz-plot dm 1 m ij dm 3 If M is constant, the allowed region in the Dalitz-plot will be uniformly populated with events Any non-uniformity gives information about dynamics R. Dalitz had applied this technique for the first time to K L decays To determine spin and parity of the then known τ/θ particle in its decay to three pion final state Q=T 1 +T +T 3 x=(t -T 1 )/ 3 y=t 3 -Q/3
m 3 ME M m 1 1 Pictorially... 1 3 1 3 1 3 1 3 m 1 M m3 ME3 11
Isobar Model in Dalitz-plot Nonresonant B 1 3 1 + {13} Resonance B K* 0 (89) 3 K + π + π - toy expt. Zemach PR 133, B101 (1964) Resonance mass part 1
Angular Part (Helicity Angle) B 1 3 H 34 4 uniform for s = 0 Angular distribution 1 34 : P s 1 (cos H ) cos 3cos H H 1 for s = 1 for s = Helicity angle of a resonance ( m3) ( m3) m cos H ( m ) ( m ) K K 0 ( f 0 * 0 * 0 ( ( ( max 3 89 ), 1430 ), 770 ), 980 ), max s 1 s s 1 s 0 0 min 3 min 3 1 0 +1 cosθ H B B B B B B K K K 0 (770) K f (980) K 0 *0 *0 K c0 (89) (1430) 13
What do we do in the end? Extract c k,nr and θ k,nr by performing a maximum likelihood fit Fit fraction is the ratio of the integral of a single decay amplitude squared to the coherent sum of all Measure CP violation asymmetries by comparing B and B amplitudes 14
How Interference comes to play? Interference of two states M a M b M M a e i M b M a M b Re( M a M * b )cos Im( M a M * b )sin Decay rate interference terms M dm 1 dm 3 The effect of interference terms is proportional to the area of overlap between resonances in DP * 0 * 0 K ( 1430 ) K ( 89 ) 15
Time-dependent CP asymmetry E e+ =3.1 GeV l + K + s - Recoil products are used for tagging B: B 0 or B 0 -bar E e- = 9 GeV B tag B CP l + 056 Dt = Dz/c l - + Time Evolution: Dt / B0 e Im f Dt 1 sin 4 0 B 1 Dm Dt cosdm Dt Tagged as B 0 or B 0 -bar S d K S 1 1 C - d 16
Time-dependent DP Time-dependent decay rate of B 0 (B 0 ) three-body Include detector effects (mistagging and resolution) Determine mixing-induced CP [sine coefficient] and direct CP [cosine coefficient] at each point in the DP direct CP 17
Dalitz plot analyses of B + π + π - π + and K + π + π - 18
B + π + π + π - Dalitz plot 3M B pairs 468±35 PRD 7, 0500 (005) Coupled BW ρ(770) f (170) Fit BB qq Single BW Phase-space 19
PRD 7, 0500 (005) B + π + π + π - : Summary ρ 0 (770) is the dominant component 3σ indication for f (170) and NR Little evidence for σ (seen by BES in the decay J/ψ ωπ + π - ) No contribution from χ c0 not feasible to measure γ with analysed dataset Bediaga et al., PRL 81, 4067 (1998) σ PLB 597, 39 (004) f (170) f 0 (980) 0
Time-dependent Dalitz plot analysis of B 0 K S π + π - 1
Motivations Dominantly b s penguin transition NP effects prone to Provides a test if mixing-induced CP asymmetry equals to that of tree-level transition b ccs Measure β eff in QB modes unambiguously interference term allows determination of cosine term (beauty of DP) We can determine the relative phase between B 0 K *+ (89)π - and B 0 K *- (89)π + access to CKM angle γ Deshpande et al., PRL 90, 06180 (003) Ciuchini et al., PRD 74, 051301 (006) Gronau et al., PRD 75, 01400 (007)
Time-dependent QB Existing Measurements Time-integrated QB Time-integrated DP Both agree reasonably well Discrepancy in the nonresonant contribution Belle also observes structure near 1.3 GeV/c in the π + π - spectrum 3
Signal Yield Simultaneous fit including m ES, ΔE, NN, Δt and tagged (B 0 /B 0 ) DP variables 383M B pairs arxiv:0708.097 Contrib. LP007 BB qq Signal: (17±70) in total candidate sample of 55 4
ρ 0 (770)K S f 0 (980)K S K * (89)π Dalitz plot Content Gounaris-Sakurai Coupled BW Single BW K *0 (1430)π LASS shape f x (1300)K S Single BW f (170)K S,, χ c0 K S,, K * (89) K *0 (1430) arxiv:0708.097 Contrib. LP007 f 0 (980) ρ(770) f x (1300)+f (170) χ c0 BB qq 5
Time-dependent CP violation Time-dependent CP asymmetry measured at each point in the K S π + π - Dalitz plot for the first time arxiv:0708.097 Contrib. LP007 Charmonium Stat only Stat + Syst Charmonium f 0 (980)K S value.1σ above charmonium ρ 0 K S consistent with the world-average 6
CKMfitter Group, J. Charles et al., EPJ C41, 1 (005) Overall Picture Confirmation of the CKM paradigm as the only source of CP violation in the SM insufficient to explain the matter-antimatter asymmetry Need additional source(s) beyond the realm of the SM 7
Summary First measurement of the inclusive mode B + K + K - π + DP measurements in the charged Kππ and πππ modes Evidence of direct CP violation in the ρ 0 (770)K ± decay β eff measured without any sign ambiguity (thanks to the time-dependent DP technique) Measured CP violation parameters agree well with SM predictions (modulo discrepancy in the loop diagrams) Look forward to more data (Super flavour factory) to pin down these holy grails Because we know already the SM is not the ultimate theory 8
Look another way 1. CP violation in the mixing : q p 1 q p M i D i D * * 1 1 Exp: N(B 0 B 0 l + l + X) N(B 0 B 0 l - l - X) The mass eigenstates are not CP eigenstates Very small in B system. CPV due to interference between mixing & decay: q A q i m( ) 0 ~ e p A p 3. CP violation in the decay (aka "direct" CP-violation) A A A A A A 1 1 1 amplitudes, with different weak & strong phases, to the same final state f Experimentally seen both in B 0 and B ± decays e.g. b c (+) b u 9
A 1 = A 1 Direct CP asymmetry A 1 = A 1 i f i f CP A = A e i e if A = A e i e if (CP-conserving) f f (CP-violating) Time-integrated direct CP asymmetry requires at least two amplitudes and 0: A = A 1 + A + f A = A 1 + A A f 30