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 → 3π

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

Advantage of 3-body decays:

• resonance-rich environment
• larger branching fractions

here: B± → K±π∓π± e.g. 3.7σ signal in Kρ

BELLE 2006, BABAR 2008

How to analyse CP violation in Dalitz plots?

• 1. strictly

model-independent extraction from data directly

Gardner et al. 2003, 2004 Bediaga et al. 2009

• 2. theoretical

information

• n

strong amplitudes as input!

)

2

Events / (0.015 GeV/c 20 40 60 80 100 120 )

2

Events / (0.03 GeV/c 20 40 60 80 100

)

2

(GeV/c

π π

m

0.6 0.7 0.8 0.9 1 1.1 )

2

Events / (0.03 GeV/c 20 40 60 80 100 120

)

2

(GeV/c

π π

m

0.7 0.8 0.9 1 1.1 1.2

B− → K−π+π− B+ → K+π−π+ cos(θH) > 0 cos(θH) > 0 cos(θH) < 0 cos(θH) < 0 ρ ρ f0 f0

H C Im

• 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:

DpSCP(i) .

= N(i) − ¯ N(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±ρ0(→ 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

N, ¯ N = α + β Re

• exp(±iδCP )

s − M 2

res + iMresΓres

• 0.5

1 1.5 2 1500 2000 2500 N = 10

6 (500 bins)

Input: δCP = 5◦, Mres = 0.77 GeV, Γres = 0.15 GeV

√s [GeV]

N, ¯ N

• asymmetry:

N − ¯ N = sin δCP × 2βMresΓres (s − M 2

res)2 + (MresΓres)2

• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 5

Illustration: the use of hadronic amplitudes (2)

0,2 0,4 0,6 0,8 1 1,2

• 500

500 1000 1500 N=10

6 (500 bins)

0,2 0,4 0,6 0,8 1 1,2

• 150
• 100
• 50

50 100 150 200 250 N=10

5 (50 bins)

• 4
• 2

2 4 6

DpSCP

0,1 1 10 100

• 4
• 2

2 4 6

DpSCP

0,1 1 10

Input: δCP = 5◦, Mres = 0.77 GeV, Γres = 0.15 GeV

Extracted: δCP = (5.7 ± 0.8)◦ Extracted: δCP = (4 ± 2)◦

√s [GeV] √s [GeV]

N − ¯ N

• no signal in significance

hadronic amplitudes still allow to extract phase δCP

• 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:

T(s, t) = c(t) + 1 π ∞

4M2

π

ds′

• s2

s′2(s′ − s) + u2 s′2(s′ − u)

• ImT(s′, t)
• 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:

T(s, t) = c(t) + 1 π ∞

4M2

π

ds′

• s2

s′2(s′ − s) + u2 s′2(s′ − u)

• ImT(s′, t)
• subtraction function c(t) determined from crossing symmetry
• project onto partial waves tI

J(s) (angular momentum J, isospin I)

⇒ coupled system of partial-wave integral equations tI

J(s) = kI J(s) + 2

• I′=0

∞

• J′=0

∞

4M2

π

ds′KII′

JJ′(s, s′)ImtI′ J′(s′)

Roy 1971

• subtraction polynomial kI

J(s): ππ scattering lengths

can be matched to chiral perturbation theory

Colangelo et al. 2001

• kernel functions KII′

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

400 600 800 1000 1200 1400 s

1/2 (MeV)

50 100 150 200 250 300 CFD Old K decay data Na48/2 K->2 π decay Kaminski et al. Grayer et al. Sol.B Grayer et al. Sol. C Grayer et al. Sol. D Hyams et al. 73

δ0

(0)

1000 1100 1200 1300 1400

s

1/2(MeV)

0.5 1

η0

0(s)

Cohen et al. Etkin et al. Wetzel et al. Hyams et al. 75 Kaminski et al. Hyams et al. 73 Protopopescu et al. CFD .

ππ KK ππ ππ

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 disc FI(s)

= FI(s) × θ(s − 4 M 2

π) × sin δI(s) eiδI(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 disc FI(s)

= FI(s) × θ(s − 4 M 2

π) × sin δI(s) eiδI(s)

⇒ Watson’s final-state theorem

Watson 1954

• solution to this homogeneous integral equation known:

FI(s) = PI(s)ΩI(s) , ΩI(s) = exp s π

∞

• 4M2

π

ds′ δI(s′) s′(s′ − s)

• PI(s) polynomial, ΩI(s) Omnès function

Omnès 1958

completely given in terms of phase shift δI(s)

• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 9

Pion vector form factor and ahvp

µ

• more refined representation:

taken from talk by G. Colangelo 2008

F π

V (s) = Ω1(s)Ωinel(s)Gω(s)

Ωinel(s): inelastic for √s (Mπ + Mω), parametrized using conformal mapping techniques

Trocóniz, Ynduráin 2002

Gω(s): ρ − ω mixing

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

E (GeV)

10 20 30 40 50

|Fπ|

2

CMD2 data KLOE data Fit to both sets

• achieve amazing precision

for hadronic contribution to aµ below 1 GeV: ahvp

µ

(√s ≤ 2MK) = (493.7 ± 1.0) × 10−10

Colangelo et al. (preliminary)

• check of data compatibility

with analyticity / unitarity

• 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 → 3π

• interesting due to relation to light quark mass ratios
• M(s, t, u) ∝ A(η → π+π−π0) can be decomposed according to

M(s, t, u) = M0(s)+(s−t)M1(u)+(s−u)M1(t)+M2(t)+M2(u)−2 3M2(s) MI(s) functions of one variable with only a right-hand cut

Stern, Sazdjian, Fuchs 1993; Anisovich, Leutwyler 1998

• I: isospin, i.e. M0,2 S-waves, M1 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:

disc MI(s) =

• MI(s) + ˆ

MI(s)

• × θ(s − 4 M 2

π) × sin δI(s) eiδI(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:

disc MI(s) =

• MI(s) + ˆ

MI(s)

• × θ(s − 4 M 2

π) × sin δI(s) eiδI(s)

• inhomogeneities ˆ

MI(s): angular averages over the MI(s): e.g. ˆ M0 = 2 3M0 + 20 9 M2 + 2(s − s0)M1 + 2 3κzM1 znf(s) = 1 2 1

−1

dz znf

• t(s, z)
• 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:

disc MI(s) =

• MI(s) + ˆ

MI(s)

• × θ(s − 4 M 2

π) × sin δI(s) eiδI(s)

• inhomogeneities ˆ

MI(s): angular averages over the MI(s): e.g. ˆ M0 = 2 3M0 + 20 9 M2 + 2(s − s0)M1 + 2 3κzM1 znf(s) = 1 2 1

−1

dz znf

• t(s, z)
• allows for cross-channel scattering between s-, t-, and u-channel
• "angular averaging" non-trivial

⇒ generates complex analytic structure (3-particle cuts)

• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 12

From unitarity to integral equations: solution

• integral equations including the inhomogeneities ˆ

MI: M0(s) = Ω0(s)

• α0+β0 s+γ0 s2+s3

π ∞

4M2

π

ds′ s′3 sin δ0(s′) ˆ M0(s′) |Ω0(s′)|(s′ − s − iǫ)

• + 2 similar for M1,2(s); 4 subtraction constants to be fixed

Khuri, Treiman 1960; Aitchison 1977; Anisovich, Leutwyler 1998

• solve these equations iteratively by a numerical procedure
• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 13

From unitarity to integral equations: solution

• integral equations including the inhomogeneities ˆ

MI: M0(s) = Ω0(s)

• α0+β0 s+γ0 s2+s3

π ∞

4M2

π

ds′ s′3 sin δ0(s′) ˆ M0(s′) |Ω0(s′)|(s′ − s − iǫ)

• + 2 similar for M1,2(s); 4 subtraction constants to be fixed

Khuri, Treiman 1960; Aitchison 1977; Anisovich, Leutwyler 1998

• solve these equations iteratively by a numerical procedure

2 4 6 8 10

s [Mπ

2]

• 1

1 2 3 4

Re M(s,t=u) tree level 1 iteration 2 iterations final result

2 4 6 8 10

s [Mπ

2]

1 2

Im M(s,t=u) tree level 1 iteration 2 iterations final result

Schneider, Kubis; compare Colangelo et al. 2010

• fast convergence: close to final result after 2 iterations
• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 13

Extensions to heavier decays

• currently extended to other decays: η′ → ηππ, η′ → 3π, ω → 3π

Schneider, Nicknig, Kubis

• challenges for going to heavier meson decays:

⊲ at higher energies: coupled-channel integral equation ⊲ inelasticities certainly not negligible ⊲ perturbative treatment of crossed-channel effects reliable? ⊲ when are higher partial waves non-negligible?

• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 14

Les Nabis

Paul Serusier, The talisman (1888)

Les Nabis informal network to bring together particle ("heavy-quark") and hadron ("light-quark") physicists from theory

• I. Bigi, S. Gardner, C. Hanhart, B. Ku-

bis, T. Mannel, U.-G. Meißner, J.R. Peláez, M.R. Pennington. . .

and experiment

• I. Bediaga, A.E. Bondar, A. Denig, T.J. Ger-

shon, W. Grandl, B.T. Meadows, K. Peters,

• U. Wiedner, G. Wilkinson. . .

to optimize future Dalitz plot CP- studies along these lines!

• B. Kubis, The role of final-state interactions in Dalitz plot studies – p. 15