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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
3 Cornell, May 6, 2004Klaus 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
4 Cornell, May 6, 2004Klaus 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
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
5 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
6 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Introduction& Concepts
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
8 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
9 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
10 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
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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
12 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
13 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
It’s All a Question of Statistics ...
pp 30
with100 events
14 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
It’s All a Question of Statistics ... ...
pp 30
with100 events1000 events
15 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
It’s All a Question of Statistics ... ... ...
pp 30
with100 events1000 events10000 events
16 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
It’s All a Question of Statistics ... ... ... ...
pp 30
with100 events1000 events10000 events100000 events
17 Cornell, May 6, 2004Klaus 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
18 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
19 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
J/ψπ+π-π0
cosθ
-1 0 +1
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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
21 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
22 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
23 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
24 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
25 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
DynamicalAspects
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
26 Cornell, May 6, 2004Klaus 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
27 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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 Cornell, May 6, 2004Klaus 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
29 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
30 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
31 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
32 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
33 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
34 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
35 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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= +
- -
36 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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=
- - +
=
+ =
37 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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?
38 Cornell, May 6, 2004Klaus 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 X0KKπ, Example K* in K+KLπ-
JC=0+
I=0JC=0+
I=1JC=1-
I=0JC=1-
I=1
39 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Limitations of the models
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
40 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
42 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Rescattering
Most severe Problem Example JLab‘s θ+ of neutrons
No general solution Specific models needed
d
? ? +n
K+
K -p
K -p
n
43 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
44 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Technical Issues
Introduction and concepts
Dynamical aspects
Limitations of the models
Technical issues
45 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Boundary problem I
Most Dalitz plots are symmetricProblem: sharing of events
Solution: transform DP
r
f(r)
46 Cornell, May 6, 2004Klaus 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
47 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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)]
48 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
49 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
50 Cornell, May 6, 2004Klaus 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
51 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
Strategy
Where to start the fit
Is one more resonance significant
Indications for a bad solution
Where to stop the sophistication/fit
?
??
?
52 Cornell, May 6, 2004Klaus 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
( ) ( )
( ) ( )
LM0
LM0
t LM D φ,θ,0
I Ω D φ,θ,0 dΩ
=
= òD0KSK+K-
53 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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!
54 Cornell, May 6, 2004Klaus 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
55 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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
56 Cornell, May 6, 2004Klaus Peters - Dalitz Plot Analysis Techniques
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 Cornell, May 6, 2004Klaus 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