Charm IX Review of recent results on Amplitude Analyses Novosibirsk T. Evans On behalf of the LHCb collaboration
Why study amplitudes? Window into CP violation, charm mixing .. Measurements of CP violating phase of the CKM matrix, γ . Learn about hadron physics along the way. Presenting results from the LHCb collaboration. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 1/22
Studies of the resonance structure in D 0 → K ∓ π ± π ± π ∓ decays [3] D 0 → K − π + π + π − , largest D 0 → K + π − π − π + , largest contribution from: a 1 (1260) + d V ud contribution from: K 1 (1270 / 1400) + W + V us u u W + c s s V cs c d D 0 V cd D 0 u K − Diagram is O (1) in terms of CKM u π − Diagram has two off-diagonal matrix element → Cabibbo CKM elements → doubly-Cabibbo favoured (CF). suppressed (DCS). BR. ∼ 8% BR. ∼ 2 × 10 − 4 Studied by Mark III [1] and BES III [2]. “Golden modes” for studies of γ and charm mixing. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 2/22
Data samples ν µ X µ − π + slow B D 0 ≈ 0 . 5cm c m 1 ≈ π − π + PV π + K − Reconstruct Uses 2011 + 2012 sample (3 fb − 1 @ 7 B → D ∗ + � D 0 π + � µ − X as a clean source of D 0 and 8 TeV). decays. Charge of ‘slow’ m D ∗ − m D 0 peaks for ‘Right Sign’ (RS) pion and muon relative and ‘Wrong Sign’ (WS). to kaon is used to infer RS sample has ∼ 900 , 000 candidates @ D 0 flavour at > 99 . 9% purity, WS has ∼ 3000@80% production. purity. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 3/22
Phase space acceptance × 10 3 × 10 3 × 10 3 35 60 Entries / (0 . 02GeV 2 /c 4 ) 70 Entries / (0 . 03GeV 2 /c 4 ) Entries / (0 . 02GeV 2 /c 4 ) LHCb LHCb LHCb 30 60 50 Fully Simulated Fully Simulated 25 50 Generator Level Generator Level 40 20 40 30 30 15 20 20 10 10 10 5 1.4 0 1.4 0 1.4 0 ε ( s ) 0.5 1 1.5 2 2.5 ε ( s ) 1 2 3 ε ( s ) 0.5 1 1.5 2 2.5 1.2 1.2 1.2 � GeV 2 /c 4 � � GeV 2 /c 4 � � GeV 2 /c 4 � s K − π + s K − π + π − s K + π − 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.5 1 1.5 2 2.5 1 2 3 0.5 1 1.5 2 2.5 � GeV 2 /c 4 � � GeV 2 /c 4 � � GeV 2 /c 4 � s K − π + s K − π + π − s K + π − Acceptance corrected using simulated events. Corrections are very small due to use of B sample / muonic trigger. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 4/22
Quark level diagrams a 1 (1260) + K ∗ 0 , [ K − π + ] L =0 , ... d π + d V ud c s V ud V cs W + W + d u > > u D 0 W + c s c s u V ud V cd V cd D 0 D 0 u ρ 0 , [ π + π − ] L =0 , ... u K 1 , K ∗ , K ∗ 2 ... u K − ↑ ↑ ↑ s ⇌ d s ⇌ d s ⇌ d ↓ ↓ ↓ K ∗ 0 , [ K + π − ] L =0 , ... K + K 1 (1270 / 1400) + V us V us c V cd d s s W + W + s u > > u W + D 0 c c d d V us u V cd V cd D 0 D 0 u ρ 0 , [ π + π − ] L =0 , ... u a 1 , π, a 2 , ... π − u IX International Workshop on Charm Physics, Novosibirsk, T. Evans 5/22
The Isobar model A ( x ) ∝ F ( q 2 ) S ( x ) T R ( s R ) where: F ( q 2 ) is a form-factor (Blatt-Weisskopf, exponential ...) S ( x ) accounts for the spin/angular momentum configuration T R ( s R ) is a dynamical function that parametrises the isobar (Breit-Wigner, K Matrix ...) IX International Workshop on Charm Physics, Novosibirsk, T. Evans 6/22
Extending to more bodies Quasi two-body topology Cascade topology Turn one of the final state particles into a second isobar → leads to two different decay topologies. Very broadly: A ( x ) ∝ F ( q 2 ) S ( x ) T R ( s R ) T R ′ ( s R ′ ) IX International Workshop on Charm Physics, Novosibirsk, T. Evans 7/22
More about the cascade topology 1.2 GeV 2 /c 4 � π + 1 a 1 (1260) + π − � 0.8 s π + π − ρ (770) 0 0.6 π + 0.4 Three-body decays proceed a pair of quasi two-body decays in isobar 0.2 model, for example: a 1 (1260) + → ρ (770) 0 π + . 0.5 1 Decay amplitude would be: � GeV 2 /c 4 � s π + π − A decay = λ µ ( a 1 ) ∗ q µ T RBW ( s π + π − ) Spin-averaged decay rate for a 1 (1260) → ρπ But what about the dynamical function for the a 1 ? IX International Workshop on Charm Physics, Novosibirsk, T. Evans 8/22
Dynamics of cascade resonances Form of dynamical functions largely constrained by two-body unitarity: m 2 − s − im Γ( s ) � − 1 � T RBW ( s ) ∝ where Γ( s ) ∝ q ( s ) 2 L × phase-space density. Can generalise to the case of unstable decay products by: � � D x |A decay ( x ) | 2 Γ( s ) ∝ pol where the integral is over the phase space of the three body decay. Integrates out the second isobar in the width. Converges to two-body phase-space in limit of narrow resonances. Significantly simplifed model of complicated system → see talk of Mikhail for more advanced treatment. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 9/22
Model building Model(s) have O (100) possible contributing components (different resonances, different orbital configurations...) If model of “reasonable” complexity include O (20) contributions, number of possible models = 100 C 20 ≈ 10 20 . Select plausible contributions to the amplitude using an additive algorithm, results in “forest” of models of comparable fit quality. Models presented include components preferred by a simple majority in the ensemble. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 10/22
D 0 → K − π + π + π − × 10 3 Entries / (0 . 02 GeV 2 /c 4 ) 22 Largest contributions from: LHCb 20 D 0 → a 1 (1260) + K − ∼ 40% 18 16 14 D 0 → K ∗ (892) 0 ρ (770) 0 ∼ 20% 12 10 D 0 → [ K − π + ] L =0 [ π + π − ] L =0 ∼ 20% 8 6 4 Width of bands indicate total systematic 2 uncertainty on model. 0 0.5 1 1.5 2 � GeV 2 / c 4 � s π + π + π − × 10 3 × 10 3 Entries / (0 . 02 GeV 2 /c 4 ) Entries / (0 . 02 GeV 2 /c 4 ) 45 LHCb LHCb 30 40 25 35 30 20 25 15 20 15 10 10 5 5 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 � GeV 2 / c 4 � � GeV 2 / c 4 � s K − π + s π + π − IX International Workshop on Charm Physics, Novosibirsk, T. Evans 11/22
D 0 → K − π + π + π − (II) Fit Fraction [%] K ∗ (892) 0 ρ (770) 0 � L =0 � 7 . 34 ± 0 . 08 ± 0 . 47 K ∗ (892) 0 ρ (770) 0 � L =1 � 6 . 03 ± 0 . 05 ± 0 . 25 K ∗ (892) 0 ρ (770) 0 � L =2 � 8 . 47 ± 0 . 09 ± 0 . 67 ρ (1450) 0 K ∗ (892) 0 � L =0 � 0 . 61 ± 0 . 04 ± 0 . 17 All two-body scalar ρ (1450) 0 K ∗ (892) 0 � L =1 � 1 . 98 ± 0 . 03 ± 0 . 33 contributions ([ hh ′ ] L =0 ) ρ (1450) 0 K ∗ (892) 0 � L =2 � 0 . 46 ± 0 . 03 ± 0 . 15 parametrised using K matrices ρ (770) 0 [ K − π + ] L =0 0 . 93 ± 0 . 03 ± 0 . 05 → no ad-hoc nonresonant K ∗ (892) 0 [ π + π − ] L =0 2 . 35 ± 0 . 09 ± 0 . 33 terms. a 1 (1260) + K − 38 . 07 ± 0 . 24 ± 1 . 38 K 1 (1270) − π + 4 . 66 ± 0 . 05 ± 0 . 39 Uncertainties dominated by K 1 (1400) − π + 1 . 15 ± 0 . 04 ± 0 . 20 systematics. K ∗ 2 (1430) − π + 0 . 46 ± 0 . 01 ± 0 . 03 K (1460) − π + 3 . 75 ± 0 . 10 ± 0 . 37 [ K − π + ] L =0 [ π + π − ] L =0 22 . 04 ± 0 . 28 ± 2 . 09 Sum of Fit Fractions 98 . 29 ± 0 . 37 ± 0 . 84 χ 2 /ν 40483 / 32701 = 1 . 238 IX International Workshop on Charm Physics, Novosibirsk, T. Evans 12/22
D 0 → K + π − π − π + 220 Entries / (0 . 05 GeV 2 /c 4 ) LHCb Largest contributions from: 200 180 D 0 → K 1 (1270 / 1400) + π − ∼ 40% 160 140 D 0 → K ∗ (892) 0 ρ (770) 0 ∼ 20% 120 100 D 0 → [ K + π − ] L =0 [ π + π − ] L =0 ∼ 20% 80 60 40 Backgrounds indicated by filled area 20 0 (combinatorial + mistagged RS decays) 1 2 3 s K + π − π + [ GeV 2 /c 4 ] 250 Entries / (0 . 05 GeV 2 /c 4 ) Entries / (0 . 03 GeV 2 /c 4 ) LHCb LHCb 500 200 400 150 300 100 200 50 100 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 s K + π − [ GeV 2 /c 4 ] s π − π + [ GeV 2 /c 4 ] IX International Workshop on Charm Physics, Novosibirsk, T. Evans 13/22
(quasi) Model Independent Partial Waves R ( A ) 4 3 RBW 2 Binned Linear interpolation 1 Cubic interpolation How much do we really know 0 -1 about dynamics? -2 � (quasi) Model independent -3 methods (QMIPWA). -4 Real and imaginary parts of 0 0.5 1 1.5 s [ GeV 2 /c 4 ] amplitude r n , i n on a discrete 10 set of points are free I ( A ) 9 parameters. RBW 8 Binned Different interpolation 7 Linear interpolation 6 Cubic interpolation schemes (binned, linear, 5 cubic) evaluate the amplitude 4 everywhere else. 3 2 1 0 0.5 1 1.5 s [ GeV 2 /c 4 ] IX International Workshop on Charm Physics, Novosibirsk, T. Evans 14/22
Example usage in other final states ) − Z + 40 Im A LHCb (3/2 LHCb 0.2 30 R Im 20 0 10 1 0 -0.2 0 10 − 5 20 − -0.4 4 30 2 − 3 40 − -0.6 40 20 0 20 40 -0.6 -0.4 -0.2 0 0.2 − − − Z + Re A Re R (3/2 ) Λ c (2860) + argand diagram [4]. Z (4430) argand diagram [5]. See presentations of Anton and Tomasz. IX International Workshop on Charm Physics, Novosibirsk, T. Evans 15/22
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