Gagan Mohanty

Introduction to Dalitz-plot

Student Seminar @ TIFR January 27, 2012

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2

Few Diversions

3

Hierarchical expansion of CKM

d s b

u

c

t

d s b

u

c

t

magnitudes phases

(1983)

+ O(¸ 4)

4

tbtstd

cbcscd

ubusud

VVV

VVV

VVV

V

d•b* = 0 B Mesons

A triangle at the heart

tdub VV argarg

(1,0) f3

f2

f1

(,h)

(0,0)

*

cb

td

V

V

*

*

cb

ub

V

V

Goal: check consistency of the CKM paradigm by overconstraining the Unitarity Triangle:

measure three angles and two side lengths

look for possible new physics contributions

5

B Meson: A threefold way

• CP violation in decay: direct – can take place both for

neutral and charged B’s – can have time-dependent and -independent

manifestations – Need two competing diagrams of different

CP-violating and –conserving phases

• CP violation in mixing: indirect – only neutral B’s are possibly

affected – SM predicts very small effects

• violation from mixing/decay interference: – only neutral B’s possibly affected – purely time-dependent effect – arises due to interference between decay

with and without mixing

1f

f

A

A

1q

p

Im 0 =, 1PCP C

6

Enter the Charmless

7

PRO : - Larger BF than two-body decays - Correct way to study interference - Some modes in a well-defined CP eigenstate

CON : - large phase space with low event density; difficult to identify all phase-space structures - mixture of CP-even and -odd final states - more complicated analysis needed

Two vs. Three

Gershon, Hazumi PLB 596, 163

3-body

2-body

BF[10-6]

BF[10-6]

8

Analysis Strategy

Signal extraction (kinematics)

Background fighting:

Continuum (event topology)

Other types of B decays (PID,

charm and charmonia veto)

Dalitz plot technique (three-body decays having reasonable

signal size)

Time-dependent analysis in neutral B meson decays to

determine CP violation parameters at each point of the

phase space

Inclusive

Full (3body)/partial (Q2B)

Time-dependent DP (3body)

uds:cc:bb =

2.1:1.3:1.1

Complexity

From four vectors 12

Conservation laws −4

Meson masses −3

Free rotation −3

Independent variable 2

1

2

3

Usual choice

Invariant mass m12

Invariant mass m23

r ) , ( 3 3 3

p E p m

) 0 , ( M p m

r ) , ( 1 1 1

p E p m

) , ( 2 2 2

p E p m r

3

2

3

22

3

2

21

2

12 2)()( MEmMppppm mmmm

1

2

1

22

1

2

32

2

23 2)()( MEmMppppm mmmm

B

Lorentz invariant phase space:

2

23

2

12dmdmdN

Decay rate

2

23

2

12

2dmdmM

Invariant amplitude

Three-Particle Phase Space

R. Dalitz had applied this technique for the first

time to KL decays

To determine spin and parity of the then known

τ/θ particle in its decay to three pion final state

Q=T1+T2+T3

x=(T2-T1)/√3

y=T3-Q/3

A Dalitz plot is nothing but a scatter plot between any two of the three invariant

mass squared variables, 2

ijm

Decay rate:

2

23

2

12

2dmdmM

If |M|2 is constant, the allowed

region in the Dalitz-plot will be

uniformly populated with events

Any non-uniformity gives

information about dynamics

Now Enter the Dalitz-plot

3

2

3

22

12 2MEmMm

1

2

1

2

2

23

2

ME

m

M

m

1 2

3

1

3

2

2

1

3

1

2

3

Pictorially...

11

12

Isobar Model in Dalitz-plot

B

1

2

3

B

2 1

3

+

Resonance

{13}

K+π+π- toy expt.

Zemach PR 133,

B1201 (1964)

K*0(892)

Resonance mass part

Nonresonant

3

2

4

H12B

34

Angular distribution 1 34 : 2

1 )(cos HsP

for s = 0

for s = 1

for s = 2 2

2

2

1cos3

cos

H

H

uniform

min

2

23max

2

23

2

23min

2

23max

2

23

)()(

2)()(cos

mm

mmmH

Helicity angle of a resonance

KB

KfB

KB

KB

KB

KB

c0

0

0

0*

0*

)980(

)770(

)1430(

)892(

cosθ

H

−1

0

+1 0 ), 980 (

1 ), 770 (

0 ), 1430 (

1 ), 892 (

0

0

0 *

0 *

s f

s

s K

s K

Angular Part (Helicity Angle)

13

14

What do we do in the end?

Extract ck,NR and θk,NR by performing a maximum

likelihood fit

Measure CP violation asymmetries by comparing B

and B amplitudes

Fit fraction is the ratio of the integral of a single decay

amplitude squared to the coherent sum of all

Interference of two states Ma Mb

sin)Im(2cos)Re(2 **2222

bababab

i

a MMMMMMMeMM

2

23

2

12

2dmdmM

Decay rate interference terms

The effect of interference

terms is proportional to the

area of overlap between

resonances in DP

) 1430 ( 0 *

K ) 892 ( 0 *

K

How Interference comes to play?

15

16

Time-dependent CP asymmetry

Dt = Dz/c

l+

l-

+

-

l+

s-

K+

Ee+=3.1 GeV

Ee-= 9 GeV

KS

Recoil products are used for tagging B: B0 or B0-bar

S C

Time Evolution:

Btag BCP

Tagged as B0 or

B0-bar

DD

DD

D

D

tmtme

tf dd

B

t B

cos1

1sin

1

Im21

42

2

2

/

0

0

056

17

Time-dependent DP

Time-dependent decay rate of B0(B0) → three-body

Determine mixing-induced CP [sine coefficient] and

direct CP [cosine coefficient] at each point in the DP

direct CP

Include detector effects (mistagging and resolution)

18

Dalitz plot analyses of

B+ → π+π-π+ and K+π+π-

19

B+→ π+π+π- Dalitz plot

Phase-space

Coupled BW

ρ(770)

f2(1270)

Single BW

qq BB

PRD 72, 052002 (2005) 232M B pairs

468±35

Fit

20

B+→ π+π+π-: Summary

ρ0(770) is the dominant component

3σ indication for f2(1270) and NR

Little evidence for σ (seen by BES

in the decay J/ψ→ωπ+π -)

No contribution from χc0 not

feasible to measure γ with analysed

dataset

σ

f0(980)

f2(1270)

PLB 597, 39 (2004)

Bediaga et al., PRL 81, 4067 (1998)

PRD 72, 052002 (2005)

21

Time-dependent Dalitz plot

analysis of B0 → KSπ+π-

22

Motivations • Dominantly b → s penguin transition prone to

NP effects

• Provides a test if mixing-induced CP asymmetry

equals to that of tree-level transition b → ccs

• Measure βeff in Q2B modes unambiguously

interference term allows determination of cosine

term (beauty of DP)

• We can determine the relative phase between B0→

K*+(892)π- and B0→ K*-(892)π+ access to CKM

angle γ Deshpande et al., PRL 90, 061802 (2003)

Ciuchini et al., PRD 74, 051301 (2006)

Gronau et al., PRD 75, 014002 (2007)

23

Existing Measurements

Both agree reasonably well

Discrepancy in the nonresonant contribution

Belle also observes structure near 1.3 GeV/c2 in the π+π- spectrum

Time-dependent Q2B

Time-integrated Q2B Time-integrated DP

24

Signal Yield • Simultaneous fit including

– mES, ΔE, NN, Δt and tagged (B0/B0) DP variables

383M B pairs arXiv:0708.2097

Contrib. LP2007

BB

qq

Signal: (2172±70) in total candidate sample of 22525

25

Dalitz plot Content

qq BB

f0(980) ρ0(770)KS Gounaris-Sakurai

f0(980)KS Coupled BW

K*(892)π Single BW

K*0(1430)π LASS shape

fx(1300)KS Single BW

f2(1270)KS ,,

χc0KS ,,

ρ(770)

fx(1300)+f2(1270)

χc0

K*(892)

K*0(1430)

arXiv:0708.2097

Contrib. LP2007

26

Time-dependent CP violation

Charmonium

Charmonium

Time-dependent CP asymmetry measured at each point in

the KSπ+π- Dalitz plot for the first time

f0(980)KS value 2.1σ

above charmonium

ρ0KS consistent with

the world-average

arXiv:0708.2097

Contrib. LP2007

Stat only

Stat + Syst

27

Overall Picture

• Confirmation of the CKM paradigm as the only

source of CP violation in the SM insufficient

to explain the matter-antimatter asymmetry

• Need additional source(s) beyond the realm of the

SM

CKMfitter Group, J. Charles

et al., EPJ C41, 1 (2005)

28

Summary First measurement of the inclusive mode B+ → K+K-π+

DP measurements in the charged Kππ and πππ modes

Evidence of direct CP violation in the ρ0(770)K± decay

βeff measured without any sign ambiguity (thanks to the

time-dependent DP technique)

Measured CP violation parameters agree well with SM

predictions (modulo discrepancy in the loop diagrams)

Look forward to more data (Super flavour factory) to

pin down these holy grails

Because we know already the SM is not the ultimate theory

29

Look another way

Very small in B system

3. CP violation in the decay

(aka "direct" CP-violation)

2 amplitudes, with different weak & strong phases,

to the same final state f

Experimentally seen both in B0 and B± decays

e.g. b c (+) b u

A AA

A A A

1 2

1 2

≠1

2. CPV due to interference between mixing & decay:

( ) q A

mp A

0 iqe

p

2~

1. CP violation in the mixing : * *M iq

p i

D D

12 122

Exp: N(B0B0 ℓ+ℓ+X) ≠ N(B0B0 ℓ-ℓ-X)

The mass eigenstates are not CP eigenstates

≠1 q

p

30

Direct CP asymmetry

i f i f CP

A1 = |A1|

A2 = |A2| ei eif

A1 = |A1|

A2 = |A2| ei eif

(CP-conserving)

f f (CP-violating)

f

+ f

A = A1+ A2 A = A1+ A2 A

Time-integrated direct CP asymmetry requires at least two

amplitudes and 0: