Dalitz Plot Analysis Techniques

1
Dalitz Plot Analysis Techniques

• Klaus Peters
• Ruhr Universität Bochum
• Cornell, May 6, 2004

2
Overview

• Introduction and concepts
• Dynamical aspects
• Limitations of the models
• Technical issues

3
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

4
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
• mass
• width
• spin and parities
• as well as properties of the initial state
• and some constraints from the experimental
  setup/measurement
• The main problem is, you dont need just a good
  description,
• you need the right one
• Many solutions may look alike but only one is
  right

5
Initial State Mixing - pp Annihilation in Flight

• pp Annihilation in Flight
• scattering processno well defined initial state
• maximum angular momentum rises with energy
• Heavy Quark Decays
• Weak Decays
• B0?3p
• D0?KSpp

6
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)

7

• Introduction
• Concepts

Introduction and concepts Dynamical
aspects Limitations of the models Technical
issues
8
n-Particle Phase space, n3

• 2 Observables
• From four vectors 12
• Conservation laws -4
• Meson masses -3
• Free rotation -3
• S 2
• Usual choice
• Invariant mass m12
• Invariant mass m13

Dalitz plot
p1
pp
p2
p3
9
Phase Space Plot - Dalitz Plot
Q small Q large

• dN (E1dE1) (E2dE2) (E3dE3)/(E1E2E3)
• Energy conservation E3 Etot-E1-E2
• Phase space density ? dN/dEtot dE1 dE2
• Kinetic energies QT1T2T3
• Plot x(T2-T1)/v3
• yT3-Q/3

Flat, if no dynamics is involved
10
The first plots ? t/?-Puzzle

• Dalitz applied it first to KL-decays
• The former t/? puzzle with only a few events
• goal was to determine spin and parity
• And he never called them Dalitz plots

11
Zemach Formalism

• Refs
• Phys Rev 133, B1201 (1964), Phys Rev 140, B97
  (1965), Phys Rev 140, B109 (1965)
• Amplitude
• M Si 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 dij
• Formalism
• Multiply tensors for each angular momentum
  involved and contract over unobservable indices

12
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-if2 Dalitz
plot I(m)A1(m)A2(m)e-if2
but only if spins are properly assigned
13
Its All a Question of Statistics ...

• pp 3p0
• with
• 100 events

14
Its All a Question of Statistics ... ...

• pp 3p0
• with
• 100 events
• 1000 events

15
Its All a Question of Statistics ... ... ...

• pp 3p0
• with
• 100 events
• 1000 events
• 10000 events

16
Its All a Question of Statistics ... ... ... ...

• pp 3p0
• with
• 100 events
• 1000 events
• 10000 events
• 100000 events

17
Isobar model

• Go back to Zemachs approach
• M Si 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
18
Particle Decays - Revisited

• Fourier-Transform of a short lived state
• wave-function decay
• transformed from time to energy spectrum

mpp
?-?
19
J/??pp-p0
cos?
-1
0
1
20
Rotations

• Single particle states
• Rotation R
• Unitary operator U
• D function represents the rotation in angular
  momentum space
• Valid in an inertial system
• Relativistic state

21
From decays to the helicity amplitude

• J12
• Basic idea
• Two state system constructed from two single
  particle states
• Amplitude
• Transition from JMgt to the observed two body
  system
• Helicity amplitude is F
• and gives the strength of the initial system to
  split up into the helicities ?1 and ?2

22
Properties of the helicity amplitude

• Parity

• LS Scheme

23
Example f2pp (Ansatz)

• Initial f2(1270) IG(JPC) 0(2)
• Final p0 IG(JPC) 1-(0-)
• Only even angular momentasince ?f?p2(-1)l
• Total spin s2sp0
• Ansatz

24
Example f2pp (Rates)

• Amplitude has to be symmetrized
• because of the final state particles

25

• Dynamical
• Aspects

Introduction and concepts Dynamical
aspects Limitations of the models Technical
issues
26
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

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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

28
Unitarity, contd

• 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

29
Some Basics

• Considering two-body processes
• Scattering amplitude ffi
• Cross section for a partial wave by integration
  over O
• Note that TJ has no unit, the unit is carried by
  qi2
• J (spin), M (z-component of J)
• Conservation of angular momentum preserves J and
  M

a
c
s
b
d
30
S-Matrix and Unitarity

• Sfi is the amplitude for an initial state igt
  found in the final state fgt
• 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

31
Lorentz Invariance

• But Lorentz invariance has to be considered
• introduces phase space factor ?

32
Resonances in K-Matrix Formalism

• 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

33
Blatt-Weisskopf Barrier Factors

• 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

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Single Resonance

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

35
Overlapping resonances
Sum of Breit-Wigner
Sum of K-matrix poles

• This simple example of resonances
• at 1270 MeV/c2 and 1565 MeV/c2
• illustrate the effect of nearby resonances

36
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

37
Production of Resonances P-Vector

• So far only s-channel resonances
• Generalization 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 ßa
  for each resonance

c
?
s
d
38
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
• pp- and p0p0 (as intermediate states)
• or combining pp, pn and nn
• or X0?KKp, Example K in KKLp-

JC0 I0
JC0 I1
JC1- I0
JC1- I1
39

• Limitations
• of the models

Introduction and concepts Dynamical
aspects Limitations of the models Technical
issues
40
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
41
t-channel Effects (also u-channel)

• They may appear resonant and non-resonant
• Formally 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
42
Rescattering

• Most severe Problem
• Example JLabs ? of neutrons
• No general solution
• Specific models needed

43
Barrier factors
Resonant daughters

• 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 ltFl(q)gt?Fl(q)dBW
• Fl(q)ql
• complex even above threshold
• meaning of mass and width are mixed up

Im(q)
Re(q)
threshold
44

• Technical
• Issues

Introduction and concepts Dynamical
aspects Limitations of the models Technical
issues
45
Boundary problem I

• Most Dalitz plots are symmetric
• Problem sharing of events
• Solution transform DP

46
Boundary Problem II

• Efficiencies often factorize in mass and angular
  distribution
• 2nd Approach
• Use mass and cos?
• Not always applicable

47
Fit methods - ?2 vs. Likelihood

• ?2
• small of independent phase space observables
• usually not more than 2
• High statistics
• gt10k if there are only a few well known
  resonances
• gt50k for complicated final states with discovery
  potential
• e.g. CB found 752.000 events of the type pp3p0
• -logL
• more than 2 independent phase space observables
• low statistics (compared to size of phase space)
• narrow structures like F(1020)

48
Adaptive binning

• 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
• ?2 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-binPoisson-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

49
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 fit
• fit to sidebands and fix for Dalitz plot fit
• Try all to get a feeling on the systematic error

50
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 F or f1(1285) or
  f0(980)
• Wrongly matched tracks
• Cures phase-smearing and non-isotropic
  resolution effects

51
Strategy
?
Where to start the fit
?
Is one more resonance significant
?
Where to stop the sophistication/fit
?
Indications for a bad solution
52
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

D0?KSKK-
53
Is one more resonance significant ?

• Base your decision on
• objective bin-by-bin ?2 and ?2/Ndof
• visual quality
• is the trend right?
• is there an imbalance between different regions
• compatibility with expected DL structure
• Produce Toy MC for Likelihood Evaluation
• many sets with full efficiency and Dalitz plot
  fit
• each set of events with various amplitude
  hypotheses
• calc DL expectation
• DL expectation is usually not just ½/dof
• sometimes adding a wrong (not necessary)
  resonance can lead to values over 100!
• compare this with data
• Result a probability for your hypothesis!

54
Where to stop

• Apart from what was said before
• Additional hypothetical trees (resonances,
  mechanisms) do not improve the description
  considerably
• Dont try to parameterize your data with
  inconsistent techniques
• If the model dont match, the model might be the
  problem
• reiterate with a better model

55
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 ?Aj2dO/?SiAi2
• But due to interferences, something is missing
• Incoherent IA2B2
• Coherent IAeifB2 A2B2
  2Re(AB)sinfIm(AB)cosf
• If SjBFj is much different from 100 there might
  be a problem
• The sum of interference terms must vanish if
  integrated from -8 to 8
• But phase space limits this region
• If the resonances are almost covered by phase
  spacethen the argument holds...
• ...and large residual interference intensities
  signal overfitting

56
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

57
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