D P T II Marco Pappagallo

D ^ALITZ P ^LOT T ^ECHNIQUES II

Marco Pappagallo

University of Glasgow

Niccolo’ Cabeo School 22 May 2012

1

O ^UTLINE

2

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

S ^CALAR M ^{ESONS AND} D ^{MESON DECAYS}

3

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

= 0

•  Final spectrum not constrained by isospin and parity conservation

I = 1 I = 1/2 I = 0

f

(600)

a

(980) κ(800)[?] f

(980)

f

(1370)

a

(1450) K*

(1430) f

(1500)

f

(1710)

D

+ à Π ⁺ Π ^- Π ⁺ D ^ALITZ P ^LOT A ^NALYSIS

4

Phys.Rev.D79:032003,2009 Phys.Rev.Lett.:765,2001 Phys.Lett.B585:200,2004

E791

I = 1 I = 1/2 I = 0 f₀(600) a₀(980) κ(800)[?] f₀(980)

f₀(1370) a₀(1450) K*₀(1430) f₀(1500) f₀(1710)

T ^HE D

+ à ^{Π + Π - Π +} D ^ECAY

5

D

D

D

D

f

, f

π

π

π

π

π

π

ρ

ρ

Ø  f

(980), f

(1370), and f

(1270) expected

D

+ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

6

Two identical particles in the final state

Symmetrized DP Two points for each

event on DP

Likelihood must be

symmetric as well

D

+ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

7

f

(980)

f

(980)

D

+ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

8

f

(1270)

f

(1270)

D

+ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

9

f

(1370)

f

(1370)

D

+ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

ρ ???

T ^{HE F} ₀ (980)

11

The f

(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

PHSP

Flatte’ Formula

E791 R ^ESULTS (900 ^EVTS )

12

Ø  Traditional Dalitz plot analysis Ø  f

(980) reference resonance

Ø  f

(980) and f

(1370) masses and widths floated

K ^{MATRIX FORMALISM IN} Π ⁺ Π ^- S- ^WAVE

13

K results Hermitian and symmetric, i.e. real

and symmetric

Resonances in the K-matrix formalism appear

as a sum of poles in the K-matrix

N R ELATIVISTIC B ^REIT -W ^IGNER ’ ^S

14

Consider two poles in a single channel:

K ^{MATRIX FORMALISM IN} Π ⁺ Π ^- S- ^WAVE

15 D0

π

π

D0

π

π

π

π D0

π

π

K

K

[I-iKρ]

P

+

+

+ …

K-Matrix formalism overcomes the main limitation of the BW model to parameterize large and overlapping S-wave ππ resonances.

         

waves D

P S

K P D waves r r i r

r wave

S

D

s s F a e A s s

−

−

−

∑

+

=

, ,,

13 12 1

13

12

, ) ( , )

(

πππ δ ππ

A

D

→ K

π

π

amplitude

P

a b

c R

[ ]

∑

=

j

-1 1j

1

I - i

F Kρ P

Initial production vector

Provided by scattering experiment

I.J.R. Aitchison, Nucl. Phys. A189, 417 (1972)

K ^{MATRIX FORMALISM IN} Π ⁺ Π ^- S- ^WAVE

16 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

F ^{OCUS RESULTS} (1470 ^EVTS )

17

M ODEL I NDEPENDENT P ARTIAL W AVE A NALYSIS

18

         

waves D

, P

13 12 r

r i r

r wave

S 1 13

12

D

( s , s ) F a e A ( s , s )

− ππ

δ

− ππ

∑

+ A =

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

ππ S-wave interpolation

M ODEL I NDEPENDENT P ARTIAL W AVE A NALYSIS

19

http://www.math.ucla.edu/~baker/java/hoefer/Spline.htm Spline DEMO:

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

B ^A B ^AR R ^ESULTS (13 ^{K EVTS} )

20

Results of an unbinned maximum likelihood fit Reference Mode

K-matrix Formalism Sum of BW ’ s

•  S-wave is the main contribution

•  Large f

(1270) contribution

•  Small ρ’s fit fractions

Model Independent Partial Wave Analysis

Π ⁺ Π ^- S- ^WAVE

21

•  The S wave shows in both amplitude and phase the expected behavior for the f

(980)

•  Activity in the region f

(1370) and f

(1500) resonances

•  The S wave is small in the f

(600) region

C ^{OMPARISON Π + Π -} S- ^WAVE ’ ^S

22

BABAR BABAR

P ^{HASE OF Π + Π -} S- ^WAVE

23

Watson Theorem

P

a b

c R

Ø   ππ phase should be the

same up to the elastic limit Ø  BUT the interaction

between c and P

introduces overall phase

D

+ àK ⁺ K ^- Π ⁺ D ^ALITZ P ^LOT A ^NALYSIS

24

I = 1 I = 1/2 I = 0 f₀(600) a₀(980) κ(800)[?] f₀(980)

f₀(1370) a₀(1450) K*₀(1430) f₀(1500) f₀(1710)

Phys.Lett.B351:591,1995

E687

Phys.Rev.D79:072008,2009

Phys.Rev.D83:052001,2011

T ^HE D

+ àK ⁺ K ^{- Π +} D ^ECAY

25

D

D

D

f

, f

π

π

K

K

K

K

T ^{HE F} ₀ (980) ^{AND A} ₀ (980)

26 PHSP

Flatte’ Formula

f

(980) à KK/ππ a

(980) à KK/ηπ

D

+ àK ⁺ K ^{- Π + SAMPLE}

27

Inclusive D

sample obtained with a likelihood

selection using vertex separation and p* ^{φ (1020)}

Events used to obtain Bkg shape:

(-10σ, -6σ) and (6σ, 10σ)

101k events

Partial Wave Analysis

f

(1700)

K

(892)

4σ

•  No D wave in K⁺K^-

•  No K^-π⁺ structure but a small K*₀(1430)(~2%)

M ^OMENTS

28

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

is the value of theta for the n-th event

P ^ARTIAL W ^AVE A ^NALYSIS

29

⎧ π = +

⎪ ⎪

π = φ

⎨ ⎪

π =

⎪⎩

2

0 2

SP 0

0

2 2 0

5 1

2

4

4 2 co

S P S P s Y

Y

4 P

Y

( ) ^{θ =} ( ) ^{θ = + θ}

π π

2

2

1 3

4 S 4 P cos

I M

( ) ^{= = + θ + ⎛} _⎜ ^{θ − ⎞} _⎟

π π

π ⎝

θ ∑

¹ ₄

₄ ³

₄ ⁵

³ ₂

¹ ₂ ⎠

L

Y cos Y co s

I Y Y Y

( )

L

A Y Pcos

M = = ⁼ S + θ θ π

∑

1 1 3

4 4

S.U. Chung, Phys. Rev. D56, 7299(1997):

P ^ARTIAL W ^AVE A ^{NALYIS AT} K ⁺ K ^-

30

Events weighted by the spherical harmonic Y

(cos θ

)(ℓ=0,1,2)

•  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 for the P wave and a “Breit-Wigner like” function for the S wave)

Ds⁺

θKK

π⁺

M ODEL I NDEPENDENT E XTRACTION OF P- WAVE

[ I . E . ^Φ (1020)] AND S- WAVE [ I . E . F ₀ (980)] RATIOS

31 S-P interference

integrates to zero Phase space factor

D

+ àK ⁺ K ^{- Π +} R ^ESULTS

32

Results of an unbinned maximum likelihood fit Reference Mode

•  Decay dominated by vector intermediated resonances

•  Contribution from K*

(1410), K

(1430), κ(800), f

(1500),

f

(1270), f

’ (1525) consistent with zero

BaBar CLEO

Ø  K*(892)⁰ and f₀(1370) masses and widths are floated parameters

Ø  f₀(980) parameterized by the effective parameterization extracted by the PWA

Ø  χ2 computed by an adaptive binning algorithm

Ø  K*(892)⁰ and f₀(1370) masses and widths are floated parameters

Ø  f₀(980) parameterized by a Flatte formula

K*(892)

width 5 MeV lower

than PDG08

K ⁺ K ^- M ^OMENTS

33

Each event was weighted by the spherical harmonic Y

(cos θ

) ( ℓ =1,2,3)

θKK

D_s⁺

f₀(980)/φ (1020) interference

π⁺

K ^{- Π +} M ^OMENTS

34

Each event was weighted by the spherical

harmonic and Y

(cos θ

) ( ℓ =1,2,3)

K⁺

Small S-P interference ⇒ No κ(800) ?

θ_Kπ

Π ⁺ Π ^- AND K ⁺ K ^- S- ^WAVE C ^OMPARISON

35 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 D⁰ and D_s decays. Are f₀(980) and a₀(980) 4-quark states?*

K

K

and π

π

S waves extracted by a Model Independent way. Opportunity to obtain new information about f

(980)!

*L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett.B651, 129 (2007)

D ⁺ àK ^{- Π + Π +} D ^ALITZ P ^LOT A ^NALYSIS

36

Phys.Rev.D78:052001,2008

Phys.Rev.D73:032004,2005 E791

I = 1 I = 1/2 I = 0 f₀(600) a₀(980) κ(800)[?] f₀(980) f₀(1370) a₀(1450) K*₀(1430) f₀(1500) f₀(1710)

D ⁺ àK ^{- Π + Π +} D ^ALITZ P ^LOT

37

K

(892)

K

(892)

S wave interfering with

K^*(892)⁰

140k events

E791

S ^CALAR M ^{ESON K} (800) ^AND D ⁺ àK ^{- Π + Π +}

38

E791 CLEO-c

No. Events 15090(Purity=94%) 140793(Purity=99%)

Isobar MIPWA Isobar Isobar MIPWA

Fit Fraction(%) Fit Fraction(%)

K*(892)⁰π⁺ 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)⁰π⁺ 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)⁰π⁺ 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)⁰π⁺ 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

χ²/ν 1.08 1.00 1.36 1.10 1.03

CLEO-c: “The addition of the I=2 ππ S wave to either the isobar model or the model-independent partial wave approach is the key piece that gives good agreement with data in both cases”

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”

Reference Mode

Breit-Wigner

K ^{- Π +} S- ^WAVE

39

Low Kπ enhancement removed by including I=2 ππ S-wave

E791

K*₀(1430) phase motion

Flat distribution if K*₀(1430) removed Dashed curves = ±1σ of S wave

parameters in the isobar model

T ^{EST OF} W ^ATSON ’ ^S T ^HEOREM

40

E791 S wave

P wave

D wave

S, P, D Kπ wave (I=1/2) from LASS

(K

p → K

π

n)

Solid curves = ±1σ of S wave parameters in MIPWA

I = 1 I = 1/2 I = 0 f₀(600) a₀(980) κ(800)[?] f₀(980)

f₀(1370) a₀(1450) K*₀(1430) f₀(1500) f₀(1710)

D ⁰ àK ⁺ K ^- Π ⁰ D ^ALITZ P ^LOT A ^NALYSIS

41

Phys.Rev.D76:011102,2007

Phys.Rev.D74:031108(R),2006

D ⁰ àK ⁺ K ^{- Π 0} D ^ALITZ P ^LOT

42

Motivation

•  Nature of Kπ S wave below 1.4 GeV

•  Is there a charged κ(800)

state?

•  Nature of f

(980)/a

(980): KK wave

K*(982)

11.2k events

K*(982)

φ (1020)

F ^{IT RESULTS OF} D ⁰ àK ⁺ K ^{- Π 0}

43

CLEO-c BaBar

No. Events 735 11200

Decay Mode Fit Fraction(%) Fit Fraction(%)

f₀(980)π⁰ --- --- 6.7±1.4±1.2 10.5±1.1±1.2 [a₀(980)π⁰] --- --- [6.0±1.8±1.2] [11.0±1.5±1.2]

φ(1020)π⁰ 14.9±1.6 16.1±1.9 19.3±0.6±0.4 19.4±0.6±0.5 f'₂(1525)π⁰ --- --- 0.08±0.04±0.05 --- κ(800)⁺ K^- --- 12.6±5.

8 --- ---

K⁺π⁰ 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^-π⁰ 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 --- --- ---

Reference Mode

LASS parameterization

Breit-Wigner BaBar:

•  Lass amplitude is favoured. E791 amplitude used for systematics

•  Ambiguity between a large K⁺π⁰ S wave and K*(1410), f’₂(1525)

•  No higher f₀ states

•  No charged kappa

C L E O : “A m b i g u i t y between a NR and a Breit-Wigner kappa”

P ^ARTIAL W ^AVE A ^{NALYSIS IN} K ⁺ K ^-

44

D ⁰ à Π ⁺ Π ^- Π ⁰ D ^ALITZ P ^LOT A ^NALYSIS

45

Phys.Rev.Lett.99,251801,2007

D ⁰ à ^{Π + Π - Π 0} D ^ALITZ P ^LOT

46

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

Six-fold symmetry

•  No natural explanation in the usual factorization picture

•  Unknown broad state coupling strongly to 3 pions?

45k events

ρ (770)

-

ρ(770)

0

Zemach, Phys.Rev.133(1964), B1201

*M. Gaspero et al., Phys. Rev.D78, 014015 (2008)

*B. Bhattacharya et al., Phys.rev.D81, 096008 (2010)

ρ(770)

+

I = 1 I = 1/2 I = 0 f₀(600) a₀(980) κ(800) f₀(980)

f₀(1370) a₀(1450) K*₀(1430) f₀(1500) f₀(1710)

D ⁺ à Π ⁺ Π ^- Π ⁺ D ^ALITZ P ^LOT A ^NALYSIS

47

Phys.Lett.B585:200,2004

Phys.Rev.Lett.:765,2001 E791

Phys.Rev.D76:012001,2007

D ⁺ à ^{Π + Π - Π +} D ^ALITZ P ^LOT

48 No visible structures:

Ø  Decay dominated by S wave Ø  50% of events are background

K

veto

F ^{IT RESULTS OF} D ⁺ à ^{Π + Π - Π +}

49

E791 FOCUS CLEO-c

No. Events ≈1170 ≈1500 ≈2600(Purity=55%)

Model Isobar K-Matrix Isobar Schechter Achasov

f₀(600)π⁺ 46.3±9.0±2.1 --- 41.8±2.9 --- --- f₀(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 f₀(1370)π⁺ 2.3±1.5±0.8 --- 2.6±1.9 2.6±1.7 3.2±0.7 f₀(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)⁰π⁺ 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)⁰π⁺ 0.7±0.7±0.3 --- --- --- --- f₂(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

Π ⁺ Π ^- S- ^WAVE

50

“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

Dotted curves = ±1σ of S wave parameters in the isobar model

With a 10x statistics, a

MIPWA could distinguish

between different models

S EARCH FOR CP VIOLATION IN D ⁺ àK ⁺ K ^- Π ⁺

51

Phys.Rev.D84:112008,2011

M ^IRANDA P ^ROCEDURE

52

CP violation No CP violation

Number of particles Number of anti-particles Model-independent

search for CP violation

S ^{EARCH FOR} CP ^VIOLATION

53

Luminosity = 35 pb

N. Evts = 350k

No CP violation so far…..but data

collected in 2011 ~ 1fb

and more in 2012

F ^ROM D ^TO B ^MESONS

54

D

àK

π

π

B

àK

π

π

Same model?

Ø  Large NR component

Ø  Large (destructive) interferences as well Ø  Total fit fraction ~ 300%

C ^ONCLUSION

55

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

C ^ONCLUSION (2)

56

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