Dalitz analysesDalitz analyses IntroductionBelle Analysis School

October 1-2 2009

Hi, could you talk b t D lit

Sure...A couple of months ago, somwehere on

about Dalitzat the BAS?

g ,the net....

A couple of weeks ago, somwehere on

Yes...?

WHO did you saythe net.... WHO did you say I should talk about...?

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 1 BAS, KEK, October 2009

Dalitz analysesDalitz analyses IntroductionBelle Analysis School

Boštjan GolobBelle & Belle II

University of Ljubljana Jožef Stefan Institute

October 1-2 2009

University of Ljubljana, Jožef Stefan Institute

1. Introduction2. Kinematics3. Physics4. Parametrization4. Parametrization5. Experimental issues6. Specifics, Outlook

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 2 BAS, KEK, October 2009

IntroductionHistoryHistory

Richard Henry Dalitz (28 February 1925 – 13 January 2006);Australian physicist;Australian physicist;

@ Cornell introduced “phase space plotsphase space plots”, i.e. Dalitz technique (as called today), e a t tec que (as ca ed today),to study 3π “tau” (kaon) decays;

On the analysis of τ meson data and the nature of the τ mesonOn the analysis of τ-meson data and the nature of the τ-meson Author: R. H. Dalitz a

Affiliation: a Department of Mathematical Physics, University of Birmingham, y gDOI: 10.1080/14786441008520365 Published in: Philosophical Magazine Series 7, Volume 44, Issue 357 October 1953 , pages 1068 - 1080

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 3 BAS, KEK, October 2009

Kinematicsn-body decayJ.D. Jackson, D.R. Tovey, Kinematics, in RPP

( )Φ=Γ nn

d

ppPdM

d

3

12

4

),;(22π

KM

∏∑==

−=Φni i

i

niinn E

pdpPppPd,1

3

3

,1

41 2)2(

)(),;(π

δK LISP:Lorent InvariantPhase Sace

independent variables: 4-vectors: 4 n +

ti l 4 3 7n=3 2n=3 2

conservation laws: 4 = 3 n - 7final state masses: n -arbitrary rotations: 3

n=4 5n=5 8

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 4 BAS, KEK, October 2009

Kinematics3 body decay

J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

take two inv. masses asindependent variables decaying particle:independent variables

213

212

233 32

1)2(

1 dmdmM

d M=Γ

decaying particle:scalar or averagingover spin states33 32)2( Mπ p

mij: inv. mass of part. i,j.2

322

21

2223

213

212 constmmmMmmm =+++=++

211M

dΓ332

13212 32)2(

MMdmdm π

= standard form of Dalitz plot

if |M|2 const ⇒ dΓ/dm 2dm 2 const

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 5 BAS, KEK, October 2009

if |M|2 const. ⇒ dΓ/dm122dm13

2 const.

Kinematics Belle, PRL 99, 131803 (2007)

)

3-body decay

example: D0 → Ksπ -π+ m2 (

KSπ

+ )

cos2

θ

non-uniformity of Dalitz plot ⇒contribution of intermediate states

Ki ti li itKinematic limitsD0 → π+K*- m2(KSπ-)J.D. Jackson, D.R. Tovey, Kinematics, in RPP

θ23*=π

( ) ( ) 222 2mmm +θ23*=0

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 6 BAS, KEK, October 2009

( ) ( )( ) ( )min

223max

223

23max23min23*23

2cos

mmmmm

−−+

=θ 34

KinematicsM1

3-body decay

various intermediate states Mk

contributing to same final state ⇒ interfereinterferenncece

Mn

|M|2 is not incoherent sum, (|M|2 ≠ |M1|2 + |M2|2 + ... ) ,

n

but a coherent sum, ( |M|2 = |M1+ M2 + ...|2 )

example: pp → 3π0

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 7 BAS, KEK, October 2009

PhysicsCleo-c, arXiv:0903.1301

New states and propertiesof known states

,

Dalitz analysis usually not needed for narrow, non-overlapping

K*0

resonances(negligible interference)example: Ds → K+K-π+

(but interf. φπ/KKπ, φπ/f0πimportant in precise Br(Ds → φπ)determination)

φ

determination)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 8 BAS, KEK, October 2009

another example: 37

PhysicsB → KZ+(4430)

Belle, PRD 80, 031104(R) (2009)

New states and propertiesof known states

KZ (4430)

In most cases intermediate states strongly interfere

Z+(4430)example: B → Kπ+ψ’

by fitting fitting DalitzDalitz distributiondistributionobtain evidence of newobtain evidence of new states, measure properties(mass width spin)

B → K*(892)ψ’

B → K0*(1430)ψ’

(mass, width, spin)

projection of Dalitz disribution and fit to

24

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 9 BAS, KEK, October 2009

disribution and fit to M(π+ψ’)

Physics20 )( tAyixqAefDd t +

+=→Γ −

D0 Mixing and CPV

x, y: mixing parameters;

2tA

pAe

dt ff +=

due to mixing, D0 → D0 → fx, y << 1

ΓΓ−Γ

=Γ−

==2

;;' 2121 ymmxttτ

t-dependent pDalitz analyses

different types of interm states;different types of interm. states;example: D0 → π+π- KS

CF: D0 → K*-π+

DCS: D0 → K*+π-

CP: D0 → ρ0 KS

if f = f ⇒ populate same Dalitz plot; l ti h d t i dl ti h d t i d

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 10 BAS, KEK, October 2009

relative phases determined relative phases determined 28

Physics Belle, PRL 99, 131803 (2007)

t dependentt

t-dependent Dalitz analyses

t-dependence:t-dependence:regions of Dalitz plane →specific t dependence F(x, y);

time evolution of Dalitz distribution⇒⇒ x, y x, y

[ ]titi

S

eemm

tDKtmm

21)(1

)(),,(

22

022

λλ

ππ

−−

−++−

++=

=≡

A

MD0→f

28

[ ]

[ ]titi eemmpq

eemm

21),(21

),(2

22 λλ −−+−

+−

−+

++=

A

AD0→f

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 11 BAS, KEK, October 2009

p2λ1,2=F(x,y); m±

2 = m2(KSπ±),

PhysicsBelle, arXiv:0803.3375

φ measurementφ3 measurement

B- →K- D0(→ f)B- →K- D0(→ f)B →K D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS

),(),())(( 22220)(

3±

+±±

−±± +=→→ mmAremmAfDKB ii

mmδϕM(

A: D0 decays; r: ratio of two B amplitudes; δ: strong phase diff. of two B amplitudes

A from fit to Dalitz fit to Dalitz distribution of D0 decays;

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 12 BAS, KEK, October 2009

r, φ3, δ (in principle) from fit to Dalitz fit to Dalitz distribution of B± decays30

sin 2φ eff in b → sqq

Physics SM:

sin 2φ1eff in b → sqq

NP contrib. ⇒ sin 2φ1eff ≠ sin 2φ1

effVtbVts* : no weak phase

quasi twoquasi two--bodybody, B → ρ0(770)KS, f0(980)KSBelle, PRD76, 091103(R) (2007)BaBar, PRL99, 161802 (2007)B B PRL98 051803 (2007)

B→BaBar, PRL98, 051803 (2007)

Vtd* 2: (mixing): sin2φ1

NP:

[ ]tmqStmqAetP CPCP

t

∆∆−∆∆+=∆∆−

sincos14

)(/||

τ

τ

interf. between various states and non-resonant contrib. ⇒ Dalitz analysis

sη’

B0 g

g~b s

+(δ 23

dRR)b

~R

s~

NP:

d d

s

s Ks

B0 gs~R

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 13 BAS, KEK, October 2009

∆S=sin2φ1eff -sin2φ1

Physicssin 2φ eff in b → sqqsin 2φ1

eff in b → sqq

[ ]tmqStmqAetP CPCP

t

∆∆−∆∆+=∆∆−

sincos14

)(/||

τ

τ

4τ

),( 22mmmAA ±=

each point in Dalitz space has a specific time evolution depending onevolution, depending on |A|2-|A|2 (direct CPV)(direct CPV)and ℑ(AA*) (indirect CPVindirect CPV corresponding to

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 14 BAS, KEK, October 2009

ℑ(AA ) (indirect CPVindirect CPV, corresponding to specific two-body contribution sin 2sin 2φφ11

effeff (i)(i))

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution

a matter of statistics102 pp → 3π0 events10 pp → 3π events

Adopted from K. Peters, talk at Charm 2006, Beijing

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 15 BAS, KEK, October 2009

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution

a matter of statistics103 pp → 3π0 events10 pp → 3π events

Adopted from K. Peters, talk at Charm 2006, Beijing

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 16 BAS, KEK, October 2009

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution

a matter of statistics104 pp → 3π0 events104 pp → 3π0 events

Adopted from K. Peters, talk at Charm 2006, Beijing

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 17 BAS, KEK, October 2009

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution

a matter of statistics105 pp → 3π0 events

larger stat. ⇒ larger sensitivity to model details

Breit-Wigner resonances

10 pp → 3π events

Breit Wigner resonancessimple consideration of spin 0 elastic scatteringleads to the Breit-Wigner amplitude for a b → r → a b

Adopted from K. Peters, talk at Charm 2006, Beijing

// 2

4/)(4/

2/2/

22

22

Γ+−Γ

=Γ−−

Γ=

EmT

iEmT

rr

|T|2 |T|2

mr=1.0Γ=0.3

mr=0.5Γ=0.2

| |

Γ

mR

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 18 BAS, KEK, October 2009

E E

ParametrizationBreit Wigner resonances

|T1+ T2 |2

Breit-Wigner resonances

several intermediate statesb ba b → r1 → a b

r2model amplitude as sum of BW amplitudes;

E

sum of BW’s amplitudes; - the approach violates unitarity for wide overlapping resonances;

- the BW shape is distorted close

(“isobar model”)most commonly used most commonly used to model Dalitz distributionsthe BW shape is distorted close

to the thresholds;

parametrization of Dalitz distribution is

distributions

22

D0, B0 → ABCpa model, phenomenological object;

it should provide adequate description adequate description ∑ Φ

=

rACAB

BWr

ir

ACAB

mmAea

mmr ),(

),(22

22A

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 19 BAS, KEK, October 2009

of dataof data; caution needed in interpretation

r

ParametrizationBW resonancesBW resonances

parametrization usually follows Cleo PRD63 092001 (2001) J 0Cleo, PRD63, 092001 (2001) JA,B,C=0

resonance spin

J=0

J=1

J=2

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 20 BAS, KEK, October 2009

ParametrizationBW resonancesBW resonances

parametrization usually follows Cleo PRD63 092001 (2001)Cleo, PRD63, 092001 (2001)

JA,B,C=0

mass dependent width;pAB: momentum of A/B in AB rest frame;pr : momentum of A/B in rest frame of r;

Fi

Ji=0

pr ;(note: MAB may not equal Mr)

i

Ji=1 form factors (Blatt-Weisskopf)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 21 BAS, KEK, October 2009

Ji=2 R: phenomenological factor(0 GeV-1 – 10 GeV-1 for D0, 0 GeV-1 – 3 GeV-1 for r)

ParametrizationBW resonancesBW resonances

Refinements - FlattFlattéé parametrizationFlattFlattéé parametrization

if 2nd channel for r decays opens close to mr;example f0(980) → ππ, +−−

=0

220

0 )(1

KKKKfff ggimmm

Aρρ ππππππ

S.M. Flatté, PLB63, 224 (1976)

p 0( ) ,KK threshold close to mf0

- parametrization of nonnon resonantresonant contribution

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +−= 2

2

2

2 )(1

)(1

ij

ji

ij

jiij

ff

mmm

mmm

ρ

nonnon--resonantresonant contribution in D0 decays usually a constantterm; in B0 charmless decays phase

gij: couplings r/ij(note: mismatch in several papers on Flatté parameters of f0(980) and a (980); contact A Zupanc Zhao Li orin B charmless decays phase

space larger, some variations of ANR; example B0 →KSπ+π- ANR ∝ exp(-αmij

2) Belle, PRD79, 072004 (2009)

a0(980); contact A. Zupanc, Zhao Li orB.G.)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 22 BAS, KEK, October 2009

p S NR p( ij )B → Dππ ANR = const.

, , ( )

Belle, PRD76, 012006 (2007)

ParametrizationK matrix

E.P. Wigner, Phys. Rev. 70, 15 (1946);S.U. Chung et al., Ann. Phys. 4, 404 (1995)

K-matrix

- sum of BW’s violates unitaritybroad S wave ππ Kπ states are

ρρ TiIS

iSfS fi

+=

=2/12/12

- broad S-wave ππ, Kπ states are not simple BW’s

scattering operator Sρ

ρ

iTKKiKIT

+=

−=−−

−

11

1)(

scattering operator Stransition operator TK matrix

++ =⇔= KKISS

iijij = ρδρoriginally developed for description of scattering

can be adopted to production processproduction process, smm ii

i

221 )(1 +

−=ρ

e.g. D, B decays, using scattering data scattering data

kfkf KiKIT 1)( −−= ρ D

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 23 BAS, KEK, October 2009

I.J.R. Aitchison, Nucl. Phys. A189, 514 (1972); kfkf PiKIF 1)( −−= ρ P (I-iKρ)-1

ParametrizationK matrix PiKIF 1)( −−= ρK-matrix

assume we are parametrizing a 2 body

Drescattering of k→ ππ

production of k K

kk PiKIF ,)(= ππππ ρ

parametrizing a 2-body intermediate state of a 3-body final state(example: ππ in KSπ+π-):

P (I-iKρ)-1of k→ ππ of k KS

k=ππ, KK, ηη, ...

(example: ππ in KSπ π ):Belle, PRL99, 131803 (2007)BaBar, hep-ex/0507101 scatt

scattprod

kr r

rkr

k sssf

smgP

0

0,2

1−−

+−

= ∑ ππβ

s=m2ππ

scattrr 1 ⎪⎫⎪⎧

resonances - poles;(may not coincide withphysical states)

slow-varying part;

factorkinemsssf

smgg

K scatt

scattscatt

ijr r

rj

ri

ij .1

0

02 ⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−

+−

= ∑

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 24 BAS, KEK, October 2009

gir, mr, fijscatt, s0

scatt : from compilation of ππ scattering datascattering data

ParametrizationK matrix gg ρρ D. Asner, Dalitz plot analysis K-matrix

for single channel (e g ππ)

)()1(2

1

2

gKiKT

smggK

ρππ

ρ

ππππ

==

−=

−

formalism, in RPP

(e.g. ππ) and single pole(e.g. ρ0): )(

)()1( 22

smg

gismKiKT

ρρρππ

ρππρ

Γ=

−−=−=

KK--matrix = single BWmatrix = single BW

for multiple channels )(

)(2 simsm

smT

ρρρ

ρρ

Γ−−Γ

=

pand/or multiple poles:

KK--matrix matrix ≠≠ multiple BW multiple BW

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 25 BAS, KEK, October 2009

S. Malvezzi, talk at Charm07, Cornell

Experimental issues

Inv. mass resolutiontypically σ << Γ;e g B → Dππ: σ ~3 MeV;e.g. B → Dππ: σππ 3 MeV;non-negligible for φ(K+K-), ω(π+π-)→ ABW⊗G(σ) numerically (or included in syst.)

also non-negligible at Dalitz boundaries→ mass constraint fit mass constraint fit of ABC to D0, B0;also same Dalitz region signal/sideband (background description)

EfficiencyABW(mAB

2,mAC2) → ABW (mAB

2,MAC2) • ε (mAB

2,mAC2);

ε (mAB2,mAC

2) from MC generated uniformly over phase space;parametrization (factorize mAB

2,mAC2 dependence)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 26 BAS, KEK, October 2009

Experimental issues

BackgroundD0, B0 mass sidebands 2222

,

22,

22 //

ACAB

side

ACABside

MCbkg

ACABsig

MCbkg

ACAB

bkg

dmdmdN

dmdmdNdmdmdN

dmdmdN

⋅=

Fitmany issues (talk by E. White)(talk by E. White);- unbinned sometimes binned;

Dalitz model testunbinned, sometimes binned;

- fit quality bin Dalitz plot, χ2;model appropriateness

test

pp p(hypothesis testing)→ compare lnL for various models;significances of ind. contributions

Dalitz + decay tfit test(talk by B. (talk by B. YabsleyYabsley););

- biastest using MC with generated Dalitz model,

l D0 K +

fit test

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 27 BAS, KEK, October 2009

example D0 →KSπ+π- Belle, PRL99, 131803 (2007)Belle Note 9799

Experimental issuesFit

Belle, PRD79, 072004 (2009)

Fit- multiple solutions,

example B0 →KSπ+π-

test for multiple solutions by performing numerous fits with varying starting values of solution 1 solution 2y g gparameters;

toy MC to checks if some solutions due to stat fluctuations;

K0*(1430)+π- fit fraction:~ 65% ~ 17%

due to stat. fluctuations;

- individual contribution∑ Φ= 2222 ),(),(

rACAB

BWr

irACAB mmAeamm rA

- individual contribution → fit fraction, fi

Σfi ≠ 100%

∫ ∑

∫ Φ

= 2

22222 ),( ACABACABBWi

ii

i

r

dmdmmmAeaf

i

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 28 BAS, KEK, October 2009

∫ ∑ Φ 2222 ),( ACABr

ACABBWr

ir dmdmmmAea r

Experimental issuesFitFit

- systematic uncertainty due to Dalitz model

variation of resonance parameters (width mass );variation of resonance parameters (width, mass, ...);variation of form factors, dependence of width on mass; inclusion/exclusion of intermediate states with low signif.; difference isobar / K-matrix model;;possible biases (MC)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 29 BAS, KEK, October 2009

Specifics, outlookZ+(4430)

Belle, PRD 80, 031104(R) (2009)

Z (4430)New states and propertiesof known states

Larger statistics important for significance;

ibilit t di ti i hpossibility to distinguish JZ=1 and JZ=0

(hypothesis testing)(hypothesis testing)

Outlook:Outlook:

more surprises

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 30 BAS, KEK, October 2009

BaBar, PRD 79, 112001 (2009)

Specifics, outlookD0 Mi i d CPV

Belle, PRL 99, 131803 (2007), 540fb-1

D0 Mixing and CPVK*X(1400)+

)%290800( 130±±x

K*(892)+

)%24.033.0(

)%29.080.0(

14.010.0

16.013.0

±±=

±±=

y

x

(89 )

K*(892)- ρ/ωpq 09.010.0

29.030.086.0/ ±±=

rad)09.030.028.024.0( ±±−=ϕ

relative sign of x, y; K-matrix used for systematics

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 31 BAS, KEK, October 2009

1038

Specifics, outlookBelle PRL 96 151801 (2006)D0 Mixing and CPV

direct determination of x, y0

∑ +−Φ

+− =r

BWr

ir mmAeamm r ),(),( 2222A

Belle, PRL 96, 151801 (2006)

instead of x’, y’ as in D0 →K+π-

(rotated for unknown δ between CF and DCS)

r

a1=1, Φ1=0relative phases determined from the fit to Dalitz distribution

2⎤⎡

Outlook:

fit to Dalitz distribution

[ ] 2

2

2 08030.0⎥⎤

⎢⎡

[ ] 2

0

2 06.0/

30.0 radrad+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

LLϕσ need to reduce Dalitzmodel systematics;

[ ] 2

0

2|/| 08.0

/30.0

+⎥⎥⎦⎢

⎢⎣

=LLpqσ other modes: KSKK, πππ0,...

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 32 BAS, KEK, October 2009

Specifics, outlookφ3 measurement

B- →K- D0(→ f) ))(( 0)(−

±± =→→ fDKBM(

Belle, arXiv:0803.3375

( )B- →K- D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS

),(),(

))((2222 3±

+±± +

→→

mmAremmA

fDKBii

mmδϕ

M(

S

use x±, y± instead of r, φ3, δ;→ more free parameters; )sin(

)cos(

3

3

δφδφ

+±=+±=

±±

±±

ryrx

lower bias, simple distributions

φ3=76o (+12o -13o) ± 4o ± 9o

Outlook:model independent determination;

22222

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 33 BAS, KEK, October 2009from Cleo-c

)),(),((|| 22222±± −= mmmm DD mm δδFM

12

Specifics, outlooksin 2φ eff in b → sqq BaBar PRD 79 112001 (2009)

example B →J/ψKπ

Belle, PRD79, 072004 (2009)

sin 2φ1eff in b → sqq

change of variables →rectangular Dalitz plot

22

BaBar, PRD 79, 112001 (2009)

22

122arccos1'

''det θ

πππ ⎞⎜⎜⎛

−−

=

→±

mmm

ddmJdmdm m

01'

2

θθ

π π

=

⎠⎜⎝ −− mmm KsB

K

π-

θ

K*0

π

B

KSθ0

a oid concentration of e ents

π+

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 34 BAS, KEK, October 2009

avoid concentration of events at Dalitz boundaries 39

Specifics, outlooksin 2φ eff in b → sqq

Belle, PRD79, 072004 (2009)

sin 2φ1eff in b → sqq

φ1eff(f0)=(12.7 ±6.9

6.5±2.8±3.3)o

Outlook:

⎨⎧≤∆ sKfS 03.0)( 0

⎩⎨⎧≤∆

s

sK

fS ϕσ 05.003.0)( 0

@ 50ab-1 for ∆S=0

b D lit ln.b.: Dalitz analyses measuring φ1

eff

⇒σ(S) = 2 cosφ1eff σ(φ1

eff)∆S

SuperBelle, 50 ab-1

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 35 BAS, KEK, October 2009

Summaryi t d f D lit l i flinstead of summary.... Dalitz analysis flow....

improved methodsimproved methodsgive you larger luminosity!

is Belle note on t-dependent Dalitzready?

NP!NP!

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 36 BAS, KEK, October 2009

Bckups

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 37 BAS, KEK, October 2009

Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

decaying particle:scalar or averaging

i t tover spin states⇒integration over dΩ1*, dΩ3

213

212dmdmd

=Γ

233 32

1)2(

1M

Mπ=

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 38 BAS, KEK, October 2009

6

Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

2/)()( 222*2**2 MEEM + v

2/)(;2/)(),(

1222

21

2*2

*2

*1

*2

*112

1223

212

23

23312

2

mmmMEppEEmmmmMEpEmM

+−=⇒=+=−−=⇒+=

)cos,(cos22

*23

212

223

*23

23

2*3

22

2*2

*3

*2

23

22

223

ϑϑ

mfmmEmEEEmmm

=−−−++=

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 39 BAS, KEK, October 2009

6

Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,

3-body decay

( ) ( )( ) ( )22

223max

223min

223*

232

cosmm

mmm−

−+=ϑ ( ) ( )min23max23 mm −

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 40 BAS, KEK, October 2009

6

Physics moreB → K ψ(2S)

Belle, PRL 91, 262001 (2003), 140fb-1

New states

Dalitz analysis not needed B → K X(3872)

B → K ψ(2S)(→ J/ψ π -π+)

for narrow, non-overlaping resonances(negligible interference)example: B → K J/ψ π -π+

( )(→ J/ψ π -π+)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 41 BAS, KEK, October 2009

8

Physics moret d d t

Belle, PRL 99, 131803 (2007)

t-dependent Dalitz analyses

t d d [ ]titi

S tDKtmm

1

)(),,(

22

022

λλ

ππ −++− =≡M

t-dependence:regions of Dalitz plane →specific t dependence F(x, y);

[ ]

[ ]titi

titi

eemmq

eemm

21

21

),(1

),(21

22

22

λλ

λλ

−−+

−−+−

−+

++=

A

A

time evolution of Dalitz distribution⇒⇒ x, y x, y

[ ]eemmp

),(2 +−+ A

2,12121

Γ−= imλ

22,12,1

)()(

),(),(

2222

2222

Φ

+−Φ

+−

=

=

∑

∑mmAeamm

mmAeamm

BWi

r

BWr

ir

r

r

A

A

,,

),(),( −++−

Φ=Φ=⎯⎯⎯ →⎯

= ∑aa

mmAeamm

rrrrDCPVno

rrrA

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 42 BAS, KEK, October 2009

),(),( 2222−++− = mmmm AA 28

Physics moreBelle, PRD79, 072004 (2009), , ( )

sin 2φ1eff in b → sqq

ci determines amount of DCPVci determines amount of DCPV

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 43 BAS, KEK, October 2009

31

Physics moreBelle, PRD79, 072004 (2009), , ( )

sin 2φ1eff in b → sqq

vtx reconstr (non-scaling)vtx reconstr. (non-scaling)improved with better tracking

bkg model (scaling)CPV in bkg; improved with measurements of contributing modes e g ’Ks

⎩⎨⎧≤

s

sKKfS ϕσ 06.0

04.0)( 0 @ 50 ab-1

of contributing modes – e.g. η Kssig. model (non-scaling)misreconstructed events; comparison with MC with correct reconstr.

D lit d l ( li )

some syst. errors cancel in ∆S = S(sqq)-S(J/ψKs)

⎧ Kf030Dalitz model (scaling)parametr. of NR; inclusion of other resonances;

⎩⎨⎧≤∆

s

sKKfS ϕσ 05.0

03.0)( 0

for ∆S=0

n.b.: Dalitz analyses measuring φ1eff ⇒σ(S) = 2 cosφ1

eff σ(φ1eff)

B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 44 BAS, KEK, October 2009