1. Dalitz Plot Analysis Techniques
Klaus Peters
Ruhr Universität Bochum
Cornell, May 6, 2004

2. Klaus Peters - Dalitz Plot Analysis Techniques
Overview
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

4. Klaus Peters - Dalitz Plot Analysis Techniques
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 don‘t need just a good description,
you need the right one
Many solutions may look alike but only one is right
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

5. Initial State Mixing - pp Annihilation in Flight
scattering process: no well defined initial state
maximum angular momentum rises with energy
Heavy Quark Decays
Weak Decays
B03π
D0KSππ
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

7. Klaus Peters - Dalitz Plot Analysis Techniques
Introduction
& Concepts
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

8. n-Particle Phase space, n=3
2 Observables
From four vectors 12
Conservation laws -4
Meson masses -3
Free rotation -3
Σ 2
Usual choice
Invariant mass m12
Invariant mass m13
Dalitz plot π1 pp π2 π3
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

9. Phase Space Plot - Dalitz Plot
Q small
Q large
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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

10. 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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

12. Klaus Peters - Dalitz Plot Analysis Techniques
Interference problem
PWA
The phase space diagram in hadron physics shows a pattern due 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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

13. It’s All a Question of Statistics ...
pp ® 3p0
with
100 events
Crystal Barrel 3pi0 Plots
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

14. It’s All a Question of Statistics ... ...
pp ® 3p0
with
100 events
1000 events
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

15. It’s All a Question of Statistics ... ... ...
pp ® 3p0
with
100 events
1000 events
10000 events
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

16. It’s All a Question of Statistics ... ... ... ...
pp ® 3p0
with
100 events
1000 events
10000 events
events
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

18. Particle Decays - Revisited
Fourier-Transform of a short lived state
wave-function + decay
transformed from time to energy spectrum
mππ
ρ-ω
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

19. Klaus Peters - Dalitz Plot Analysis Techniques
J/ψπ+π-π0
cosθ -1 +1
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

21. From decays to the helicity amplitude
J®1+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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

22. Properties of the helicity amplitude
Parity
LS Scheme
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

23. Example: f2®ππ (Ansatz)
Initial: f2(1270) IG(JPC) = 0+(2++)
Final: π0 IG(JPC) = 1-(0-+)
Only even angular momenta since ηf=ηπ2(-1)l
Total spin s=2sπ=0
Ansatz
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

24. Klaus Peters - Dalitz Plot Analysis Techniques
Example: f2®ππ (Rates)
Amplitude has to be symmetrized
because of the final state particles
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

25. Klaus Peters - Dalitz Plot Analysis Techniques
Dynamical
Aspects
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

27. 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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

29. Klaus Peters - Dalitz Plot Analysis Techniques
Some Basics a c
Considering two-body processes
Scattering amplitude ffi
Cross section for a partial wave by integration over Ω
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 s b d
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

30. S-Matrix and Unitarity
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
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

31. Klaus Peters - Dalitz Plot Analysis Techniques
Lorentz Invariance
But Lorentz invariance has to be considered
introduces phase space factor ρ
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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

34. Klaus Peters - Dalitz Plot Analysis Techniques
Single Resonance
In the simple case of only one resonance in a single channel
the classical Breit-Wigner is retained
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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

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 βα for each resonance c ? s d
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

38. Klaus Peters - Dalitz Plot Analysis Techniques
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 X0KKπ, Example K* in K+KLπ-
JC=0+
I=0
JC=0+
I=1
JC=1-
I=0
JC=1-
I=1
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

39. Klaus Peters - Dalitz Plot Analysis Techniques
Limitations
of the models
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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

42. Klaus Peters - Dalitz Plot Analysis Techniques
Rescattering
Most severe Problem
Example JLab‘s θ+ of neutrons
No general solution
Specific models needed d ? + n K - p
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

43. Klaus Peters - Dalitz Plot Analysis Techniques
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 <Fl(q)>=∫Fl(q)dBW
Fl(q)~ql
complex even above threshold
meaning of mass and width are mixed up
Im(q)
threshold
Re(q)
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

44. Klaus Peters - Dalitz Plot Analysis Techniques
Technical
Issues
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

45. Klaus Peters - Dalitz Plot Analysis Techniques
Boundary problem I
Most Dalitz plots are symmetric
Problem: sharing of events
Solution: transform DP
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

46. Klaus Peters - Dalitz Plot Analysis Techniques
Boundary Problem II
Efficiencies often factorize in mass and angular distribution
2nd Approach
Use mass and cosθ
Not always applicable
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

47. Fit methods - χ2 vs. Likelihood
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 potential
e.g. CB found events of the type pp®3π0
-logL
more than 2 independent phase space observables
low statistics (compared to size of phase space)
narrow structures [like Φ(1020)]
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

48. Klaus Peters - Dalitz Plot Analysis Techniques
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
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
cut-off
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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

51. Klaus Peters - Dalitz Plot Analysis Techniques
Strategy ?
Where to start the fit ?
Is one more resonance significant ?
Where to stop the sophistication/fit ?
Indications for a bad solution
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

52. Klaus Peters - Dalitz Plot Analysis Techniques
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KSK+K-
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

53. Is one more resonance significant ?
Base your decision on
objective bin-by-bin χ2 and χ2/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!
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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 = ∫|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 region
If the resonances are almost covered by phase space then the argument holds...
...and large residual interference intensities signal overfitting
Klaus Peters - Dalitz Plot Analysis Techniques
Cornell, May 6, 2004

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

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