The role of final-state interactions in Dalitz plot studies Bastian - PowerPoint PPT Presentation

The role of final-state interactions in Dalitz plot studies Bastian Kubis Helmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theorie) Bethe Center for Theoretical Physics Universit¨ at Bonn, Germany Hadron 2011 — Munich, June 13th 2011 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 1

The role of final-state interactions in Dalitz plot studies Introduction • Dalitz plots and CP violation • the usefulness of hadronic input What do (low-energy) hadron physicists have on offer? • scattering consistent with analyticity and unitarity: Roy equations • decays linked to scattering: form factors and Omnès solution • low-energy constraints: amplitudes consistent with chiral symmetry (only mentioned in passing) • many-particle dynamics for the example of η → 3 π Les Nabis group input from C. Hanhart gratefully acknowledged B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 2

CP violation in three-body decays B − → K − π + π − B + → K + π − π + H Advantage of 3-body decays: ) 2 Events / (0.015 GeV/c 120 100 • resonance-rich environment 80 • larger branching fractions 60 40 here: B ± → K ± π ∓ π ± 20 ) e.g. 3 . 7 σ signal in Kρ 2 Events / (0.03 GeV/c cos( θ H ) > 0 cos( θ H ) > 0 100 BELLE 2006, BABAR 2008 80 60 C How to analyse CP violation in 40 Im 20 Dalitz plots? ) 2 Events / (0.03 GeV/c cos( θ H ) < 0 cos( θ H ) < 0 120 1. strictly model-independent 100 extraction from data directly 80 60 Gardner et al. 2003, 2004 40 Bediaga et al. 2009 20 0.6 0.7 0.8 0.9 1 1.1 0.7 0.8 0.9 1 1.1 1.2 2. theoretical information on 2 2 m (GeV/c ) m (GeV/c ) π π π π strong amplitudes as input! ρ f 0 ρ f 0 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 3

Direct data analysis: significance Bediaga et al. 2009 Significance in Dalitz plot distributions: N ( i ) − ¯ N ( i ) Dp S CP ( i ) . = N ( i ) + ¯ � N ( i ) N , ¯ N : CP-conjugate decays; i : label of a specific Dalitz plot bin • allows to study local asymmetries • no theoretical input required at all — strictly model-independent • B decays: clear evidence, in particular in B ± → K ± ρ 0 ( → π ± π ∓ ) consistent with Standard Model BELLE 2006, BABAR 2008 • D decays: only upper limits (at few-percent level) Standard Model prediction tiny BABAR 2008 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 4

Illustration: the use of hadronic amplitudes (1) • model: resonance plus CP-violating phase provided by C. Hanhart � exp( ± iδ CP ) � N, ¯ N = α + β Re s − M 2 res + iM res Γ res 6 (500 bins) N = 10 2500 Input: δ CP = 5 ◦ , M res = 0 . 77 GeV, Γ res = 0 . 15 GeV N , ¯ N 2000 1500 0.5 1 1.5 2 √ s [GeV] 2 βM res Γ res N − ¯ • asymmetry: N = sin δ CP × res ) 2 + ( M res Γ res ) 2 ( s − M 2 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 5

Illustration: the use of hadronic amplitudes (2) Input: δ CP = 5 ◦ , M res = 0 . 77 GeV, Γ res = 0 . 15 GeV 6 (500 bins) 5 (50 bins) N=10 N=10 250 Extracted: δ CP = (5 . 7 ± 0 . 8) ◦ 1500 Extracted: δ CP = (4 ± 2) ◦ 200 150 1000 100 N − ¯ N 500 50 0 0 -50 -100 -500 -150 0,2 0,4 0,6 0,8 1 1,2 0,2 0,4 0,6 0,8 1 1,2 √ s [GeV] √ s [GeV] 100 10 10 1 1 0,1 0,1 -4 -2 0 2 4 6 -4 -2 0 2 4 6 Dp S CP Dp S CP • no signal in significance hadronic amplitudes still allow to extract phase δ CP B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 6

ππ scattering constrained by analyticity and unitarity compare also talk by M. Hoferichter on Tuesday Roy equations = coupled system of partial-wave dispersion relations + crossing symmetry + unitarity • twice-subtracted fixed- t dispersion relation: � ∞ s 2 u 2 � � T ( s, t ) = c ( t ) + 1 ds ′ Im T ( s ′ , t ) s ′ 2 ( s ′ − s ) + s ′ 2 ( s ′ − u ) π 4 M 2 π • subtraction function c ( t ) determined from crossing symmetry B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 7

ππ scattering constrained by analyticity and unitarity compare also talk by M. Hoferichter on Tuesday Roy equations = coupled system of partial-wave dispersion relations + crossing symmetry + unitarity • twice-subtracted fixed- t dispersion relation: � ∞ s 2 u 2 � � T ( s, t ) = c ( t ) + 1 ds ′ Im T ( s ′ , t ) s ′ 2 ( s ′ − s ) + s ′ 2 ( s ′ − u ) π 4 M 2 π • subtraction function c ( t ) determined from crossing symmetry • project onto partial waves t I J ( s ) (angular momentum J , isospin I ) ⇒ coupled system of partial-wave integral equations � ∞ 2 ∞ � � ds ′ K II ′ JJ ′ ( s, s ′ )Im t I ′ t I J ( s ) = k I J ′ ( s ′ ) J ( s ) + 4 M 2 I ′ =0 J ′ =0 π Roy 1971 • subtraction polynomial k I J ( s ) : ππ scattering lengths can be matched to chiral perturbation theory Colangelo et al. 2001 • kernel functions K II ′ JJ ′ ( s, s ′ ) known analytically B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 7

ππ scattering constrained by analyticity and unitarity • elastic unitarity ⇒ coupled integral equations for phase shifts • modern precision analyses: ⊲ ππ scattering Ananthanarayan et al. 2001, García-Martín et al. 2011 ⊲ πK scattering Büttiker et al. 2004 • example: ππ I = 0 S-wave phase shift & inelasticity 0 (s) 300 η 0 (0) δ 0 1 250 CFD Old K decay data Na48/2 K->2 π decay 200 Kaminski et al. Grayer et al. Sol.B Grayer et al. Sol. C Grayer et al. Sol. D 150 Hyams et al. 73 0.5 Cohen et al. Etkin et al. ππ KK 100 Wetzel et al. Hyams et al. 75 Kaminski et al. ππ ππ Hyams et al. 73 50 Protopopescu et al. CFD . 0 0 1000 1100 1200 1300 1400 400 600 800 1000 1200 1400 1/2 (MeV) 1/2 (MeV) s s García-Martín et al. 2011 • strong constraints on data from analyticity and unitarity! B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 8

Analyticity and unitarity: form factor • just two particles in final state (form factor); from unitarity: disc = F I ( s ) × θ ( s − 4 M 2 π ) × sin δ I ( s ) e iδ I ( s ) disc F I ( s ) = ⇒ Watson’s final-state theorem Watson 1954 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 9

Analyticity and unitarity: form factor • just two particles in final state (form factor); from unitarity: disc = F I ( s ) × θ ( s − 4 M 2 π ) × sin δ I ( s ) e iδ I ( s ) disc F I ( s ) = ⇒ Watson’s final-state theorem Watson 1954 • solution to this homogeneous integral equation known: � s ∞ δ I ( s ′ ) � � ds ′ F I ( s ) = P I ( s )Ω I ( s ) , Ω I ( s ) = exp s ′ ( s ′ − s ) π 4 M 2 π P I ( s ) polynomial, Ω I ( s ) Omnès function Omnès 1958 completely given in terms of phase shift δ I ( s ) B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 9

Pion vector form factor and a hvp µ • more refined representation: taken from talk by G. Colangelo 2008 F π V ( s ) = Ω 1 ( s )Ω inel ( s ) G ω ( s ) Ω inel ( s ) : inelastic for √ s � ( M π + M ω ) , parametrized using conformal mapping techniques Trocóniz, Ynduráin 2002 G ω ( s ) : ρ − ω mixing 50 • achieve amazing precision CMD2 data KLOE data for hadronic contribution to Fit to both sets 40 a µ below 1 GeV: ( √ s ≤ 2 M K ) 30 a hvp 2 µ |F π | = (493 . 7 ± 1 . 0) × 10 − 10 20 Colangelo et al. (preliminary) 10 • check of data compatibility with analyticity / unitarity 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 E (GeV) B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 10

Dispersion relations for three-body decays compare also following talk by P . Magalhães Example: η → 3 π • interesting due to relation to light quark mass ratios • M ( s, t, u ) ∝ A ( η → π + π − π 0 ) can be decomposed according to M ( s, t, u ) = M 0 ( s )+( s − t ) M 1 ( u )+( s − u ) M 1 ( t )+ M 2 ( t )+ M 2 ( u ) − 2 3 M 2 ( s ) M I ( s ) functions of one variable with only a right-hand cut Stern, Sazdjian, Fuchs 1993; Anisovich, Leutwyler 1998 • I : isospin, i.e. M 0 , 2 S-waves, M 1 P-wave • decomposition exact if discontinuities in D- and higher partial waves neglected B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 11

From unitarity to integral equations: inhomogeneities • more complicated unitarity relation for 4-point functions: M I ( s ) + ˆ × θ ( s − 4 M 2 π ) × sin δ I ( s ) e iδ I ( s ) � � disc M I ( s ) = M I ( s ) B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 12

From unitarity to integral equations: inhomogeneities • more complicated unitarity relation for 4-point functions: M I ( s ) + ˆ × θ ( s − 4 M 2 π ) × sin δ I ( s ) e iδ I ( s ) � � disc M I ( s ) = M I ( s ) • inhomogeneities ˆ M I ( s ) : angular averages over the M I ( s ) : e.g. M 0 = 2 3 �M 0 � + 20 9 �M 2 � + 2( s − s 0 ) �M 1 � + 2 ˆ 3 κ � z M 1 � � 1 � z n f � ( s ) = 1 dz z n f � � t ( s, z ) 2 − 1 B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 12