Introduction to analysis techniques: BW, Dalitz plots

DALITZ PLOT TECHNIQUES II
1
Marco Pappagallo
University of Glasgow
Niccolo’ Cabeo School
22 May 2012

OUTLINE
Ø Part I
Ø A bit of history
Ø Kinematic
Ø Dynamic (Breit-Wigner, Barrier factors, angular
distribution)
Ø How to fit a DP
Ø Part II
Ø Examples of DP analysis
2

SCALAR MESONS AND D MESON DECAYS
Scalar meson candidates are too
numerous to fit in a single qq nonet. Some
of them may be multiquark, glueball,
meson-meson bound state, etc….
Charm meson decays are a powerful tool
for investigating light quark spectroscopy:
• Large coupling to scalar mesons
• An initial state well defined J P = 0 -
• Final spectrum not constrained by isospin and
parity conservation
I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800)[?] f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
f 0 (1710)
3

D +
S àΠ + Π-Π + DALITZ PLOT ANALYSIS
E791
Phys.Rev.Lett.:765,2001
Phys.Lett.B585:200,2004
Phys.Rev.D79:032003,2009
I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800)[?] f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
f 0 (1710)
4

THE D +
S àΠ + Π-Π + DECAY
D s +
D s +
π +
f 0 , f 2
ρ 0
π +
D s +
D s +
Ø f 0 (980), f 0 (1370), and f 2 (1270) expected
π +
π -
π +
π +
ρ 0
5

D +
S àΠ + Π-Π + DALITZ PLOT
Two identical particles
in the final state
Symmetrized DP
Two points for each
event on DP
Likelihood must be
symmetric as well
6

D +
S àΠ + Π-Π + DALITZ PLOT
f 0 (980)
f 0 (980)
7

D +
S àΠ + Π-Π + DALITZ PLOT
f 2 (1270)
f 2 (1270)
8

D +
S àΠ + Π-Π + DALITZ PLOT
f 0 (1370)
f 0 (1370)
9

D +
S àΠ + Π-Π + DALITZ PLOT
ρ ???
10

THE F 0 (980)
The f 0 (980) has still uncertain parameters and interpretations because is just sitting
at the KK threshold and strongly coupled to the KK and ππ final states
Flatte’ Formula
PHSP
11

E791 RESULTS (900 EVTS)
Ø Traditional Dalitz plot analysis
Ø f 0 (980) reference resonance
Ø f 0 (980) and f 0 (1370) masses and widths floated
12

K MATRIX FORMALISM IN Π + Π - S-WAVE
Resonances in the K-matrix formalism appear
as a sum of poles in the K-matrix
K results
Hermitian and
symmetric, i.e. real
and symmetric
13

N RELATIVISTIC BREIT-WIGNER’S
Consider two poles in a single channel:
14

K MATRIX FORMALISM IN Π + Π - S-WAVE
+
+
Pj
0
D
0
D
0
D
π
π
K
K
+ …
+
π
−
π
+
π
−
π
+
π
−
π
[I-iKρ] 1j -1
K-Matrix formalism overcomes the main
limitation of the BW model to parameterize
large and overlapping S-wave ππ resonances.
A
D
( s
12
, s
13
)
=
D 0 →K 0 π + π - amplitude
F1
π S−wave
Provided by scattering experiment
Initial production vector
∑
iδr
+ are
Ar
( s12,
s13)
r
π P,
D−waves
Kπ
S,
P,
D−waves
P
∑
R
c
[ -iKρ]
F1 = I 1j Pj
j
I.J.R. Aitchison, Nucl. Phys. A189, 417 (1972)
15
a
b
-1

K MATRIX FORMALISM IN Π + Π - S-WAVE
5 channels: 1=ππ 2=KK 3=multi-meson 4= ηη 5= ηη´
V.V. Anisovich, A.V Sarantsev Eur. Phys. Jour. A16, 229 (2003)
Global fit to the available data
5 poles
16

FOCUS RESULTS (1470 EVTS)
17

MODEL INDEPENDENT PARTIAL WAVE ANALYSIS
A
D
( s
12
, s
13
)
=
F
1
ππ S−wave
∑
iδr
+ are
Ar
( s12,
s13)
r
ππ P,
D−waves
ππ S-wave interpolation
Ø N = 30 endpoints
Ø Relaxed Cubic Spline
• Piecewise cubic curve between two
consecutive points
• First and second derivatives are continuous
• Second derivative is zero at each
endpoint(Relaxed condition)
Ø Adaptive binning: same number of π + π -
combinations between two points
18

MODEL INDEPENDENT PARTIAL WAVE ANALYSIS
Spline DEMO:
http://www.math.ucla.edu/~baker/java/hoefer/Spline.htm
Why not S(m) = Re(c(m))+ i Im(c(m)) ?
Pros
• No periodicity in phase
Cons
• Physics is connected to module and phase
• Non linear transformation (issue for the error)
Used for
guidelines and
systematics
19

BABAR RESULTS (13K EVTS)
Results of an unbinned maximum likelihood fit
Reference Mode
Model Independent
Partial Wave Analysis
Sum of BW’s
K-matrix Formalism
• S-wave is the main contribution
• Large f 2 (1270) contribution
• Small ρ’s fit fractions
20

Π + Π - S-WAVE
• The S wave shows in both amplitude and phase the expected behavior for the f 0 (980)
• Activity in the region f 0 (1370) and f 0 (1500) resonances
• The S wave is small in the f 0 (600) region
21

COMPARISON Π + Π - S-WAVE’S
BABAR BABAR
22

PHASE OF Π + Π - S-WAVE
Watson Theorem
Ø ππ phase should be the
same up to the elastic limit
Ø BUT the interaction
between c and P
introduces overall phase
P
R
c
a
b
23

D +
S àK + K-Π + DALITZ PLOT ANALYSIS
E687
Phys.Rev.D83:052001,2011
Phys.Rev.D79:072008,2009
Phys.Lett.B351:591,1995
I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800)[?] f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
f 0 (1710)
24

THE D + àK + K- S Π + DECAY
D s +
D s +
π +
f 0 , f 2
D s +
K +
K -
π +
K *0
K +
25

THE F 0 (980) AND A 0 (980)
f 0 (980) à KK/ππ
Flatte’ Formula
PHSP
a 0 (980) à KK/ηπ
26

D + àK + K- + Π SAMPLE
S
Inclusive D s sample obtained with a likelihood
selection using vertex separation and p*
4σ
Events used to obtain Bkg shape:
(-10σ, -6σ) and (6σ, 10σ)
101k events
• No D wave in K + K -
φ (1020)
• No K - π + structure but a small K* 0 (1430)(~2%)
Partial Wave Analysis
f 0 (1700)
K * (892) 0
27

MOMENTS
Let N be the number of events for a given mass interval:
Using the orthogonality condition, the coefficients in
the expansion are obtained from
θ n is the value of theta for the n-th event
28

PARTIAL WAVE ANALYSIS
S.U. Chung, Phys. Rev. D56, 7299(1997):
0 0 1 0 3 0 5 0 ⎛⎛3 2 1⎞⎞
I( θ)
= ∑ YLYL= Y0+ Y1cosθ+
Y2 ⎜⎜ cosθ−
π π
⎟⎟
L
4π
4 4 ⎝⎝ 2 2⎠⎠
Hp: Lmax
=
M( θ)
= ∑ ALYL = S + P cos θ
L
π π
0
1 1 3
4 4
2 1 3
I( θ ) = M(
θ)
= S+ Pcos θ
4π 4π
⎧⎧ 0 2 2
4π
Y0
= S + P
⎪⎪
⎪⎪
0
⎨⎨ 4π Y1
= 2 S P cosφ
⎪⎪
0 2 2
⎪⎪⎩⎩
4π Y2
= P
5
2
SP
29

PARTIAL WAVE ANALYIS AT K + K -
Events weighted by the spherical harmonic Y 0 ℓ (cos θ KK )(ℓ=0,1,2)
π +
Ds +
θKK
• Background subtracted
• Efficiency corrected
• Phase space corrected
Phenomenological description
of the S wave to be used in the
DP analysis
Solid line is the result of a
binned fit (Breit-Wigner 30 for the
P wave and a “Breit-Wigner
like” function for the S wave)

MODEL INDEPENDENT EXTRACTION OF P-WAVE
[I.E. Φ(1020)] AND S-WAVE[I.E. F 0 (980)] RATIOS
S-P interference
integrates to zero
Phase space factor
31

D + àK + K- S Π + RESULTS
Reference Mode
Results of an unbinned maximum likelihood fit
BaBar CLEO
Ø K*(892) 0 and f 0 (1370) masses and widths are
floated parameters
Ø f 0 (980) parameterized by the effective
parameterization extracted by the PWA
Ø χ2 computed by an adaptive binning algorithm
• Decay dominated by vector
intermediated resonances
Ø K*(892) 0 and f0 (1370) masses and widths are
floated parameters
Ø f 0 (980) parameterized by a Flatte formula
• Contribution from K* 1 (1410),
K 2 (1430), κ(800), f 0 (1500),
f 2 (1270), f 2 ’(1525) consistent
with zero
K*(892) 0 width
5 MeV lower
than PDG08
32

K + K - MOMENTS
Each event was weighted by the spherical
harmonic Y0 ℓ (cos θKK ) (ℓ=1,2,3)
f 0 (980)/φ (1020)
interference
π +
D s +
θKK
33

K - Π + MOMENTS
Each event was weighted by the spherical
harmonic and Y0 ℓ (cos θKπ ) (ℓ=1,2,3) K
D +
s +
Small S-P
interference ⇒
No κ(800) ?
θ Kπ
34

Π + Π - AND K + K - S-WAVE COMPARISON
Remarks:
• Agreement between K + K - S wave and π + π - S waves
despite the interferences with the other scalar mesons
(especially in the π + π - system)
• Agreement between the KK S waves extracted by D 0
and D s decays. Are f 0 (980) and a 0 (980) 4-quark states?*
*L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett.B651, 129 (2007)
K
35
+ K- and π + π- S waves extracted by a Model
Independent way. Opportunity to obtain new
information about f0 (980)!

D + àK - Π + Π + DALITZ PLOT ANALYSIS
Phys.Rev.D73:032004,2005 E791
Phys.Rev.D78:052001,2008
I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800)[?] f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
f 0 (1710)
36

D + àK - Π + Π + DALITZ PLOT
K * (892) 0
140k events
K * (892) 0
E791
S wave
interfering with
K * (892) 0
37

SCALAR MESON K(800) AND D + àK - Π + Π +
Reference Mode
Breit-Wigner
E791 CLEO-c
No. Events 15090(Purity=94%) 140793(Purity=99%)
Isobar MIPWA Isobar Isobar MIPWA
Fit Fraction(%) Fit Fraction(%)
K*(892) 0 π + 12.6±1.6 11.9±0.2±2.6 11.2±0.2±2.0 5.15±0.24 4.94±0.23
K*1(1680) 0 π + 2.1±0.4 1.2±0.6±1.2 1.28±0.04±0.28 0.238±0.024 0.098±0.059
K*2(1430) 0 π + 0.5±0.1 0.2±0.1±0.1 0.38±0.02±0.03 0.124±0.011 0.102±0.020
NR 16.1±5.3 ------- 8.9±0.3±1.4 38.0±1.9 -------
κ(800)π + 45.6±10.7 ------- 33.2±0.4±2.4 8.5±0.5 -------
K* 0 (1430) 0 π + 12.2±1.3 ------- 10.4±0.6±0.5 7.53±0.65 6.65±0.31
K - π + S-wave ------- 78.6±1.4±1.8 ------- ------- 41.9±1.9
I=2 π + π + S wave ------- ------- ------- 13.4±2.3 15.5±2.8
χ 2 /ν 1.08 1.00 1.36 1.10 1.03
E791:“At the statistical level of this experiment,
differences between the MIPWA and the isobar model
result are not found to be significant, and both provide
good descriptions of the data”
CLEO-c: “The addition of the I=2 ππ S wave to
either the isobar model or the model-independent
38
partial wave approach is the key piece that gives
good agreement with data in both cases”

K - Π + S-WAVE
Dashed curves = ±1σ of S wave
parameters in the isobar model
E791
K* 0 (1430)
phase motion
Low Kπ enhancement removed
by including I=2 ππ S-wave
Flat distribution if
K* 0 (1430) removed
39

TEST OF WATSON’S THEOREM
S wave
P wave
D wave
E791
Solid curves = ±1σ of S wave
parameters in MIPWA
S, P, D Kπ wave
(I=1/2) from LASS
(K - p → K - π + n)
40

I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800)[?] f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
D 0 àK + K - Π 0 DALITZ PLOT ANALYSIS
Phys.Rev.D76:011102,2007
Phys.Rev.D74:031108(R),2006
f 0 (1710)
41

D 0 àK + K - Π 0 DALITZ PLOT
φ (1020)
K*(982) -
11.2k events
Motivation
• Nature of Kπ S wave below 1.4 GeV
• Is there a charged κ(800) + state?
• Nature of f 0 (980)/a 0 (980): KK wave
K*(982) +
42

FIT RESULTS OF D 0 àK + K - Π 0
Breit-Wigner
Reference Mode
LASS
parameterization
CLEO-c BaBar
No. Events 735 11200
Decay Mode Fit Fraction(%) Fit Fraction(%)
f 0 (980)π 0 ------- ------- 6.7±1.4±1.2 10.5±1.1±1.2
[a 0 (980)π 0 ] ------- ------- [6.0±1.8±1.2] [11.0±1.5±1.2]
φ(1020)π 0 14.9±1.6 16.1±1.9 19.3±0.6±0.4 19.4±0.6±0.5
f' 2 (1525)π 0 ------- ------- 0.08±0.04±0.05 -------
κ(800) + K - ------- 12.6±5.
8
------- -------
K + π 0 S wave ------- ------- 16.3±3.4±2.1 71.1±3.7±1.9
K*(892) + K - 46.1±3.1 48.1±4.
5
45.2±0.8±0.6 44.4±0.8±0.6
K*(1410) + K - ------- ------- 3.7±1.1±1.1 -------
κ(800) - K + ------- 11.1±4.7 ------- -------
K - π 0 S wave ------- ------- 2.7±1.4±0.8 3.9±0.9±1.0
K*(892) - K + 12.3±2.2 12.9±2.
6
16.0±0.8±0.6 15.9±0.7±0.6
K*(1410) - K + ------- ------- 4.8±1.8±1.2 -------
NR 36.0±3.7 ------- ------- -------
CLEO: “ A m b i g u i t y
between a NR and a
Breit-Wigner kappa”
BaBar:
• Lass amplitude is
favoured. E791 amplitude
used for systematics
• Ambiguity between a
large K + π 0 S wave and
K*(1410), f’ 2 (1525)
• No higher f 0 states
• No charged kappa
43

PARTIAL WAVE ANALYSIS IN K + K -
44

D 0 àΠ + Π - Π 0 DALITZ PLOT ANALYSIS
Phys.Rev.Lett.99,251801,2007
45

ρ(770)
+
D 0 àΠ + Π - Π 0 DALITZ PLOT
ρ(770)
-
Six-fold symmetry
45k events
ρ(770)
0
Although states of isospin zero, one, and two are
possible in principle, the Dalitz plot shows strong
depopulation along each of its symmetry axes,
characteristic of a final state with isospin zero
Zemach, Phys.Rev.133(1964), B1201
• No natural explanation in the
usual factorization picture
• Unknown broad state coupling
strongly to 3 pions?
46
*M. Gaspero et al., Phys. Rev.D78, 014015 (2008)
*B. Bhattacharya et al., Phys.rev.D81, 096008 (2010)

I = 1 I = 1/2 I = 0
f 0 (600)
a 0 (980) κ(800) f 0 (980)
f 0 (1370)
a 0 (1450) K* 0 (1430) f 0 (1500)
D + àΠ + Π - Π + DALITZ PLOT ANALYSIS
Phys.Rev.D76:012001,2007
Phys.Lett.B585:200,2004
Phys.Rev.Lett.:765,2001 E791
f 0 (1710)
47

D + àΠ + Π - Π + DALITZ PLOT
K 0 s veto
No visible structures:
Ø Decay dominated by S wave
Ø 50% of events are background
48

FIT RESULTS OF D + àΠ + Π - Π +
E791 FOCUS CLEO-c
No. Events ≈1170 ≈1500 ≈2600(Purity=55%)
Model Isobar K-Matrix Isobar Schechter Achasov
f 0 (600)π + 46.3±9.0±2.1 ------- 41.8±2.9 ------- -------
f 0 (980)π + 6.2±1.3±0.4 ------- 4.1±0.9 ------- -------
Low S wave ------- ------- “45.9±3.0” 43.4±11.8 75±7
f 0 (1370)π + 2.3±1.5±0.8 ------- 2.6±1.9 2.6±1.7 3.2±0.7
f 0 (1500)π + ------- ------- 3.4±1.3 4.3±2.4 4.0±0.8
NR 7.8±6.0±2.7 ------- ------- ------- -------
S wave π + ------- 56.0±3.2±2.1 ------- ------- -------
ρ(770) 0 π + 33.6±3.2±2.2 30.8±3.1±2.3 20.0±2.5 19.6±7.4 18.4±4.0
ρ(1450) 0 π + 0.7±0.7±0.3 ------- ------- ------- -------
f 2 (1270)π + 19.4±2.5±0.4 11.7±1.9±0.2 18.2±2.7 18.4±7.4 23.2±5.0
I=2 π + π + S wave ------- ------- ------- ------- 16.6±3.2
49

Π + Π - S-WAVE
Dotted curves = ±1σ of
S wave parameters in
the isobar model
“The S wave shapes are quite
similar up to the interplay
with other resonances, and
with the data set we have in
hand we are not sensitive to
the details of the S wave
parametrization.”
Magnitude Phase
With a 10x statistics, a
MIPWA could distinguish
between different models
50

SEARCH FOR CP VIOLATION IN D + àK + K - Π +
Phys.Rev.D84:112008,2011
51

MIRANDA PROCEDURE
Model-independent
search for CP violation
No CP violation
Number of particles Number of anti-particles
CP violation
52

SEARCH FOR CP VIOLATION
Luminosity = 35 pb -1
N. Evts = 350k
No CP violation so far…..but data
collected in 2011 ~ 1fb -1 and more in 2012
53

FROM D TO B MESONS
D 0 àK - π + π 0 B 0 àK - π + π 0
Same model?
B Dalitz plot features:
54
Ø Large NR component
Ø Large (destructive) interferences as well
Ø Total fit fraction ~ 300%

CONCLUSION
World effort towards understanding the 3-body decay
International cooperation “Les Nabis”
Experimentalists:
Ø Improve selection of signals
Ø Understand efficiency and background
Ø Binning analysis (Miranda procedure and many others)
Ø Make fitter faster (DP with several million of events are
already on tape)
Ø Improve minimization
Ø Parallel computation
Ø Graphics Processing Unit (GPU)
Theorists:
Ø Rescattering effects
Ø 3-body unitary
Ø Model independent analysis
55

CONCLUSION(2)
Dalitz plots play a fundamental role into the front line
of flavour physics.
High statistical data are available and allow to explore:
Ø Angles of The Unitary Triangle
Ø Physics beyond the SM by the search of CP violation
Ø Light meson spectroscopy
Ø Discovery of new states
56