dalitz plot analysis techniques klaus peters ruhr universität bochum cornell, may 6, 2004

Dalitz Plot Analysis Techniques

Klaus PetersRuhr Universität Bochum

Cornell, May 6, 2004

2 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques

Overview

Introduction and concepts

Dynamical aspects

Limitations of the models

Technical issues


What is the mission ?

Particle physics at small distances is well understood One Boson Exchange, Heavy Quark Limits

This is not true at large distances Hadronization, Light mesons are barely understood compared to their abundance

Understanding interaction/dynamics of light hadrons will improve our knowledge about non-perturbative QCD parameterizations will give a toolkit to analyze heavy quark processes thus an important tools also for precise standard model tests

We need Appropriate parameterizations for the multi-particle phase space A translation from the parameterizations to effective degrees of

freedom for a deeper understanding of QCD


Goal

For whatever you need the parameterization of the n-Particle phase space

It contains the static properties of the unstable (resonant) particles within the decay chain like

masswidthspin and parities

as well as properties of the initial state and some constraints from the experimental setup/measurement

The main problem is, you don‘t need just a good description,you need the right one

Many solutions may look alike but only one is right


Initial State Mixing - pp Annihilation in Flight

0.5 1.0 1.5 2.0 2.5 3.0

10

1

P [GeV/c]lab

l=4

l=3

l=2l=1

ang. mom. ~ /0.2 GeV/

ll p ccms

ann= l l

l(p)=(2l+1) [1-exp(- (p))] / p l2

l(p)=N(p) exp(-3l(l+1)/4p R ) 2 2100

l [

mb]

pp Annihilation in Flight scattering process:

no well defined initial state maximum angular momentum

rises with energy

Heavy Quark DecaysWeak Decays

B03π D0KSππ


Intermediate State Mixing

Many states may contribute to a final state

not only ones with well defined

(already measured) properties

not only expected ones

Many mixing parameters are poorly known

K-phases SU(3) phases

In addition also D/S mixing

(b1, a1 decays)


Introduction& Concepts


Dynamical aspects


Technical issues


n-Particle Phase space, n=3

2 Observables From four vectors12 Conservation laws -4 Meson masses -3 Free rotation -3 Σ 2

Usual choice Invariant mass m12

Invariant mass m13

π3

π2pp

π1

Dalitz plot


Phase Space Plot - Dalitz Plot

dN ~ (E1dE1) (E2dE2) (E3dE3)/(E1E2E3)

Energy conservation E3 = Etot-E1-E2

Phase space density ρ = dN/dEtot ~ dE1 dE2

Kinetic energies Q=T1+T2+T3

Plot x=(T2-T1)/√3

y=T3-Q/3

Flat, if no dynamics is involved

Q smallQ large


The first plots τ/θ-Puzzle

Dalitz applied it first to KL-decays The former τ/θ puzzle with only a few events goal was to determine spin and parity

And he never called them Dalitz plots


Zemach Formalism

Refs Phys Rev 133, B1201 (1964), Phys

Rev 140, B97 (1965), Phys Rev 140, B109 (1965)

Amplitude M = Σi MI,i MF,i MJP,i

MI,i = isospin dependence

MF,i = form factors

MJP,i = spin-parity factors

Tensors (MJP spin-parity factors) 0T = 1 1Ti = ti

2Tij = (3/2)-1/2 [ti tj - (1/3) t2 δij]

Formalism Multiply tensors for each angular

momentum involved and contract over unobservable indices


Interference problem

PWA The phase space diagram in

hadron physics shows a patterndue to interference and spin effects

This is the unbiased measurement What has to be determined ?

Analogy Optics PWA # lamps # level # slits # resonances positions of slits masses sizes of slits widths

bias due to hypothetical spin-parity assumption

Optics

I(x)=|A1(x)+A2(x)e-iφ|2

Dalitz plot

I(m)=|A1(m)+A2(m)e-iφ|2

but only if spins are properly assigned


It’s All a Question of Statistics ...

pp 30

with100 events


It’s All a Question of Statistics ... ...

pp 30

with100 events1000 events


It’s All a Question of Statistics ... ... ...

pp 30

with100 events1000 events10000 events


It’s All a Question of Statistics ... ... ... ...

pp 30

with100 events1000 events10000 events100000 events


Isobar model

Go back to Zemach‘s approach M = Σi MI,i MF,i MJP,i

MI,i = isospin dependence

MF,i = form factors

MJP,i = spin-parity factors

Generalization construct any many-body system

as a tree of subsequent two-body decays the overall process is dominated

by two-body processes the two-body systems behave

identical in each reaction different initial states may interfere need two-body „spin“-algebra

various formalisms need two-body scattering formalism

final state interaction, e.g. Breit-Wigner

s


Particle Decays - Revisited

Fourier-Transform of a short lived state

wave-function + decay transformed from

time to energy spectrum

( ) ( )

( ) ( ) ( )

( ) ( )( )

( )( )

0

0

0

tiE t 2

tiE t iEt2

0

ii E E t2

0

0

t t 0 e e

ff E ~ t 0 e e dt

t 0 e dt

t 0

E E i / 2

τ

τ

τ

ψ ψ

ω ψ

ψ

ψ

τ

--

¥- -

¥- - -

= =

= =

= =

==

- -

ò

ò

mππ

ρ-ω


J/ψπ+π-π0

cosθ

-1 0 +1


Rotations

Single particle states

Rotation RUnitary operator U

D function represents the rotation in angular momentum space

Valid in an inertial system

Relativistic state

[ ] [ ] [ ]

( )

( ) ( )

( ) ( )

( )

( ) ( ) ( )

( )

yz z

jj ' mm'

j,m

2 1 2 1

i Ji J i J

* Jm',m

m'

im' j

* Jm',m

j

im'm'

z

m

m

1

j 'm' jm

1 jm jm

U R R U R U R

U R , , e e e

U R , , jm jm' D , ,

e d

D ,

e

p, jm U R L R jm

, jm' U R , , jm

p, j D p, jm

βα γ

α γ

λ

δ δ

α β γ

α β γ α β γ

β

α β γ α β

Ω

β Ω

γ

λ

Ω

-- -

- -

-

=

=

=

é ù=ë û

é ù =ë û

é ù= ë û=

é ù= ê úë û=

å

å


From decays to the helicity amplitude

J1+2

Basic idea Two state system constructed from

two single particle states

Amplitude Transition from |JM> to the

observed two body system

Helicity amplitude is F and gives the strength of the initial

system to split up into the helicities λ1 and λ2

( )2

1 2

1

J J

1 2

1 2 1 2 1 2

1 2

J1

*J M

2

A p , p J M

w4 JM J M J M

p

and

wF 4 J M J

A N F D

M

, 0

p

,λ λ λ

λ λ

λ λ

π φθλ λ λ λ λ λ

λ λ λ

λ

φ θ

π λ

= -

=

= -

=

=

M

M

M


Properties of the helicity amplitude

LS Scheme

Parity

( )

( )

( )( )

( )

( ) ( )( )

( )

1 2

1 2 1 2

1 2 2

2

1

1

J s sJ J

l

1 2

1 2l

1

,s

1 1 2 2

J

2

1 2

s s

1 2 1

J

2

J J

P J Mls 1 J Mls

2l 1JM l0 s JM

2J 1

s s s JMls

P J M 1 J

F

Mls

1 F

F 1 F

λ λ λ λ

λ λ λ λ

η η

λ λ λ

λ λ λ

λ λ η

η

η

η

η η

+ +

+ +

- -

º

= -

+=

+

-

= -

= -

= -

å

( )

( )( )

( ) ( )( )1 2

JJ 1 1 2 2 ls

l

1 2 1 2ls

ls

1 1

s

2 2

s

l

N F 2l 1 l0 s J M s s s a

wa 4 JMl

JM J M JM JMls JMls J M

2l 1l0 s JM

2J 1

s s s J Mls J

s JMp

M

λ λ

λ λ λ λ

λ

λ λ λ

λ λ λ λ

π

=

+=

+

-

= + -

=

å

å

å

M

M

M

M


Example: f2ππ (Ansatz)

Initial: f2(1270) IG(JPC) = 0+(2++)

Final: π0 IG(JPC) = 1-(0-+)Only even angular momentasince ηf=ηπ

2(-1)l

Total spin s=2sπ=0

Ansatz

( )

( )

( )( )

( )

1 2 1 2

J M J J *J M

1 2

2M 2 2*00 2 00 M0

22 00 20 20

1 1

2M 2*00 20 M0

A N F D ,

0

J 2

A N F D ,

N F 5 20 00 20 00 00 00 a 5a

A 5a D ,

λ λ λ λ λ φ θ

λ λ λ

φ θ

φ θ

=

= - =

=

=

= =

=

E555555F E555555F

( ) ( )( ) ( )

( )( )( )

( )

( ) ( )

( ) ( )

( )

2 2i2 0

2 i1 0

1M 200 20 00

2 i102 2i20

1M 1M'*00 MM' 00

M,M'

2 22 0

21 0

2 200

d ed e

A 5a dd ed e

I A A

112J 1 1

6d sin

4

3d sin cos

23 1

d cos2 2

φ

φ

φ

φ

θθθ

θθ

θ ρ

ρ

θ θ

θ θ θ

θ θ

--

--

±

±

é ùê úê úê úê ú=ê úê úê úê úë û

=

æ ö÷ç ÷ç= ÷ç ÷ç+ ÷çè ø

=

= -

æ ö÷ç= - ÷ç ÷çè ø

å

O


Example: f2ππ (Rates)

Amplitude has to be symmetrized because of the final state particles

( )

( )

4 2 2 4 2

22 4 2 2 2

20

1 3 1 115 sin sin cos cos cos

4 2 1

2

20

4 2

15 3 1I a sin 15s

I a con

in cos 5 cos4 2 2

st

θ θ θ θ θ

θ θ θ

θ

θ θ

æ ö÷ç + + - + ÷ç ÷÷çè ø

æ öæ ö ÷ç ÷ç ÷ç= + + - ÷÷çç ÷÷

= =

çç è ø ÷çè øE555555555555555555555555555555F


DynamicalAspects


Dynamical aspects


Technical issues


Introduction

Search for resonance enhancements is a major tool in meson spectroscopy

The Breit-Wigner Formula was derived for a single resonance appearing in a single channel

But: Nature is more complicated Resonances decay into several channels Several resonances appear within the same channel Thresholds distorts line-shapes due to available phase space

A more general approach is needed for a detailed understanding


Outline of the Unitarity Approach

The only granted feature of an amplitude is UNITARITY Everything which comes in has to get out again no source and no drain of probability

Idea: Model a unitary amplitude Realization: n-Rank Matrix of analytic functions, Tij

one row (column) for each decay channel

What is a resonance? A pole in the complex energy plane

Tij(m) with m being complex Parameterizations: e.g. sum of poles


Unitarity, cont‘d

Goal: Find a reasonable parameterization The parameters are used to model the analytic function to follow the data Only a tool to identify the resonances in the complex energy plane Poles and couplings have not always a direct physical meaning

Problem: Freedom and unitarity Find an approach where unitarity is preserved by construction Leave a lot of freedom for further extension


Some Basics

( )

( ) ( ) ( )

( ) ( )

2 2ffi fi2

i

J J *fi fi

Ji

2J Jf

a b c d

a b

c d

i fi2i

fi

qd 1M f

d

,

;

q(8 )

1f 2J 1 T s D , ,0

q

42J 1

i ab, JM,

f cd, JM,

fi

T sq

λμ

Ω φ θ

λ λ λ μ λ λ

λ λ

λ λ

σΩ π

φ θ

σ

δ

π

=

= -

æ ö÷ç ÷= =ç ÷ç ÷çè ø

= +

æ ö÷ç ÷= +ç ÷ç

= -

=

çè ø

=

=

÷

å

Considering two-body processesScattering amplitude ffi

Cross section for a partial wave by integration over Ω

Note that TJ has no unit, the unit is carried by qi

2

J (spin), M (z-component of J)

Conservation of angular momentum preserves J and M

sa

b

c

d


S-Matrix and Unitarity

[ ]

† †

† † †

† 1 1

1 †

f

1

i

1 1

†

S I 2iT

SS S S I

T T 2iT T 2iTT

(T ) T 2

S f S

iI

(T iI) T iI

T K iTK K

i

K T iI

K K

K T

KT

, 0

i

- -

- -

- -

=

=

= +

= =

- = =

- =

+ =

+

=

+

= + =

+

=

Sfi is the amplitude for an initial state |i> found in the final state |f>

An operator K can be defined Caley Transform which is hermitian by construction from time reversal invariance it follows

that K is symmetric and commutes with T


Lorentz Invariance

( ) ( ) ( ) ( )

( )

†

J J *fi fi

J

2fi f

fi2i

2 2

a b a bii

f

1

n

2ii i

2q

ˆT T

ˆ ˆT 2J 1 T s D , ,0

d 4T̂

d m

m m m m2q

; 1 as

1 1m m m

m

2

m

q

0

0

Σλμ

Σ

ρ ρ

Ω φ θ

σ ρΩ

Ω ρ

ρ

ρ

ρ

ρ

ρ ρ

ρ

=

= +

æ öæ ö ÷ç÷ç ÷= ç÷ç ÷÷çç ÷çè øè ø

é ùé ùæ ö æ ö+ -ê ú

æ ö÷ç ÷ç ÷ç ÷= ç ÷ç ÷ç

ê ú÷ ÷ç ç

÷ç ÷÷

= = - -÷ ÷ç çê úê ú÷ ÷÷ ÷ç çè ø

çè ø

= ®

è øê úê úë ûë

¥

û

®

=

å

O

2 2

c d c df m m m m1 1

m m m

é ùé ùæ ö æ ö+ -ê úê ú÷ ÷ç ç= - -÷ ÷ç çê úê ú÷ ÷÷ ÷ç çè ø è øê úê úë ûë û

But Lorentz invariance has to be considered

introduces phase space factor ρ


Resonances in K-Matrix Formalism

( ) ( )( )

( ) ( )( ) ( )

( )( )

( )

( ) ( )

i jij 2 2

i

ij ij ij

2i i

ii

0 li

j

22i 2

i i i

0 li i i i

K K c

g m m m

m m

g m g mK

m m

g mm B q,q

g m m B q,q

m

α α

α α

αα

α α α

α α

α α α

α α α α

α

α

α α

Γ

Γ

ρρ

Γ γ

Γ

γ ρ

ρ

Γ

Γ

=-

é ù=

® +

=

=

= ê úë û

=

å

å

Resonances are introduced as sum of poles, one pole per resonance expected

It is possible to parameterize non-resonant backgrounds by additional unit-less real constants or functions cij

Unitarity is still preserved

Partial widths are energy dependent


Blatt-Weisskopf Barrier Factors

( )( )( )

( )

( )

( )

( )

( )

lli

l

0

1

2

2 2

3

3 2

4

4 2 2

2

R

R

F q 1

2zF q

z 1

13zF q

(z 3) 9z

277zF q

z(z 15) 9(2z 5)

12746zF q

(z 45z 105) 2

F qB q,q

F q

qz

q

MeVq 1

5z

97,

)

3

(2z 1

c

2

α αα

=

æ ö÷ç ÷= ç ÷

=

=+

=- +

=- + -

=- + + -

ç ÷çè ø

=

The energy dependence is usually parameterized in terms of Hankel-Functions

Normalization is done that Fl(q) = 1 at the pole position

Main problem is the choice of the scale parameter qR


Single Resonance

In the simple case of only one resonance in a single channelthe classical Breit-Wigner is retained

( )( )

i

210 002 2

00 0

T e sin

mT B q,q

m m im m

δ δ

Γ ρρΓ

=é ù æ ö÷çê úé ù ÷= çê ú ÷ê úë ûç ÷ç- - è øê úë û


Overlapping resonances

This simple example of resonances at 1270 MeV/c2 and 1565 MeV/c2

illustrate the effect of nearby resonances

Sum of Breit-Wigner

Sum of K-matrix poles

2 2A B

2 2 2 2A B

g gK

m m m m= +

- -


Resonances near threshold

Line-shapes are strongly distorted by thresholds

if strong coupling to the opening channel exists

unitarity implies cusp in one channel to give room for the other channel

( )

21 1 2

0 0 21

22

21

2 21

2 2

2 2 2 20 0 0 1 1 2 2

2

m

Tm m im

r

1

γ γ γΓ

γ γ γ

Γ

γ

ρ ρ

γ

γ γ

γ

γ

æ ö÷ç ÷ç ÷ç ÷çè ø=

- - +

=

+ =


Production of Resonances: P-Vector

( )

( )( )

1 1

0i

i 2 2i

i i i

F I iK P TK P

g mP

m

P P

m

d

α α

α α

ρ

β

ρ

- -=

-

®

= -

+

= å

So far only s-channel resonancesGeneralization for production processes

Aitchison approach T is used to propagate the

production vector P to the observed amplitude F

P contains the same poles as K An arbitrary real function may be

added to accommodate for background amplitudes

The production vector P has complex strengths βα for each resonance

sc

d?


Coupled channels

K-Matrix/P-Vector approach imply coupled channel functionality same intermediate state but different final states

Isospin relations (pure hadronic) combine different channels of the same gender, like π+π- and π0π0 (as intermediate states) or combining pp, pn and nn or X0KKπ, Example K* in K+KLπ-

JC=0+

I=0JC=0+

I=1JC=1-

I=0JC=1-

I=1




Dynamical aspects


Technical issues


Limits of the Isobar approach

The isobar model implies s-channel reactions all two-body combinations undergo FSI FSI dominates all amplitudes can be added coherently

Failures of the model t-channel exchange Rescattering s


t-channel Effects (also u-channel)

They may appear resonant and non-resonantFormally they cannot be used with Isobars

But the interaction is among two particles

To save the Isobar Ansatz (workaround) they may appear as unphysical poles in K-Matrices or as polynomial of s in K-Matrices background terms in unitary form

t


Rescattering

Most severe Problem Example JLab‘s θ+ of neutrons

No general solution Specific models needed

d

? ? +n

K+

K -p

K -p

n


Barrier factors

Scales and Formulae formula was derived from

a cylindrical potential the scale (197.3 MeV/c) may be

different for different processes valid in the vicinity of the pole definition of the breakup-

momentum

Breakup-momentum may become complex

(sub-threshold) set to zero below threshold

need <Fl(q)>=∫Fl(q)dBW Fl(q)~ql

complex even above threshold meaning of mass and width are

mixed up

Resonant daughters

( )( )( )

( )2 2

lli 2 2

l R

F q q 13zB q,q ; z ; F q

F q q (z 3) 9zα αα

æ ö÷ç ÷= = =ç ÷ç ÷ç - +è ø

2 2

2 a b a bii i

m m m m2q1 as m ; 1 1

m m mρ ρ

é ùé ùæ ö æ ö+ -ê úê ú÷ ÷ç ç® ® ¥ = = - -÷ ÷ç çê úê ú÷ ÷÷ ÷ç çè ø è øê úê úë ûë ûRe(q)

Im(q

)

threshold


Technical Issues


Dynamical aspects


Technical issues


Boundary problem I

Most Dalitz plots are symmetricProblem: sharing of events

Solution: transform DP

r

f(r)


Boundary Problem II

Efficiencies often factorize in mass and angular distribution

2nd Approach Use mass and cosθ Not always applicable


Fit methods - χ2 vs. Likelihood

χ2

small # of independent phase space observables usually not more than 2

High statistics >10k if there are only a few well known resonances>50k for complicated final states with discovery potentiale.g. CB found 752.000 events of the type pp3π0

-logL more than 2 independent phase space observables low statistics (compared to size of phase space) narrow structures [like Φ(1020)]


Adaptive binning

cut-off

Finite size effects in a bin of the Dalitz plot limited line shape sensitivity for narrow

resonances

Entry cut-off for bins of a Dalitz plots χ2 makes no sense for small #entries cut-off usually 10 entries

Problems the cut-off method may deplete

important regions of the plot to much circumvent this by using a bin-by-bin

Poisson-test for these areas

alternatively: adaptive Dalitz plots, but one may miss narrow depleted regions, like the f0(980) dip

systematic choice-of-binning-errors


Background subtraction and/or fitting

Experiments at LEAR did a great job, but backgrounds were low and statistics were extremely high

Background was usually not an issue In D(s)-Decays we know this is a severe problem Backgrounds can exceed 50%

Approaches Likelihood compensation

add logLi of all background events (from sidebands) Background parameterization (added incoherently)

combined fitfit to sidebands and fix for Dalitz plot fit

Try all to get a feeling on the systematic error


Finite Resolution

Due to resolution or wrong matching: True phase space coordinates of MC events

are different from the reconstructed coordinates In principle amplitudes of MC-events have to be calculated at the

generated coordinate, not the reconstructed location But they are plotted at the reconstructed location

Applies to: Experiments with “bad” resolution (like Asterix) For narrow resonances [like Φ or f1(1285) or f0(980)] Wrongly matched tracks

Cures phase-smearing and non-isotropic resolution effects


Strategy

Where to start the fit

Is one more resonance significant

Indications for a bad solution

Where to stop the sophistication/fit

?

??

?


Where to start

Problem dependent start with obvious signatures

Sometimes a moment-analysis can help to find important contributions best suited if no crossing bands occur

( ) ( )

( ) ( )

LM0

LM0

t LM D φ,θ,0

I Ω D φ,θ,0 dΩ

=

= òD0KSK+K-


Is one more resonance significant ?

Base your decision on objective bin-by-bin χ2 and χ2/Ndof

visual qualityis the trend right?is there an imbalance between different regions

compatibility with expected L structure

Produce Toy MC for Likelihood Evaluation many sets with full efficiency and Dalitz plot fit

each set of events with various amplitude hypothesescalc L expectation

L expectation is usually not just ½/dofsometimes adding a wrong (not necessary) resonance

can lead to values over 100!compare this with data

Result: a probability for your hypothesis!


Where to stop

Apart from what was said before Additional hypothetical trees (resonances, mechanisms) do not

improve the description considerably Don‘t try to parameterize your data with inconsistent techniques If the model don‘t match, the model might be the problem reiterate with a better model


Indications for a bad solution

Apart from what was said before one indication can be a large branching fraction of interference terms Definition of BF of channel j

BFj = ∫|Aj|2dΩ/∫|ΣiAi|2

But due to interferences, something is missing Incoherent I=|A|2+|B|2

Coherent I=|A+eiφB|2 = |A|2+|B|2 +2[Re(AB*)sinφ+Im(AB*)cosφ If ΣjBFj is much different from 100% there might be a problem

The sum of interference terms must vanish if integrated from -∞ to +∞

But phase space limits this regionIf the resonances are almost covered by phase spacethen the argument holds......and large residual interference intensities signal overfitting


Other important topics

Amplitude calculation Symbolic amplitude manipulations (Mathematica) On-the-fly amplitude construction (Tara)

CPU demand Minimization strategies and derivatives

Coupled channel implementation Variants, Pros and Cons Numerical instabilities Unitarity constraints Constraining ambiguous solutions with external information

Constraining resonance parameters systematic impact if wrong masses are used


Summary & Outlook

Dalitz plot analysis is an important tool for Light and Heavy Hadron spectroscopy CP-Violation studies Multi-body phase space parameterization

Stable solutions need High statistics Good angular coverage Good efficiency knowledge

High Statistics need Precise modeling Huge amount of CPU and Memory Joint Spin Analysis Group

dalitz plot analysis techniques klaus peters ruhr universität bochum cornell, may 6, 2004

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