Dalitz analysesDalitz analyses Introduction - IJS golob/sola/Dalitz_BAS09_ · PDF...

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
Botjan GolobBelle & Belle II
University of Ljubljana Joef Stefan Institute
October 1-2 2009
University of Ljubljana, Joef Stefan Institute
1. Introduction2. Kinematics3. Physics4. Parametrization4. Parametrization5. Experimental issues6. Specifics, Outlook

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 3 tau (kaon) decays;
On the analysis of meson data and the nature of the mesonOn the analysis of -meson data and the nature of the -meson Author: R. H. Dalitz aAffiliation: 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

Kinematicsn-body decayJ.D. Jackson, D.R. Tovey, Kinematics, in RPP
( )= nn
d
ppPdM
d
3
12
4
),;(22
KM
==
=ni i
i
niinn E
pdpPppPd,1
3
3
,1
41 2)2(
)(),;(
K 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

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=
decaying particle:scalar or averagingover spin states33 32)2( M p
mij: inv. mass of part. i,j.23
22
21
2223
213
212 constmmmMmmm =+++=++
211 Md 33213
212 32)2(
MMdmdm
= standard form of Dalitz plot
if |M|2 const d/dm 2dm 2 const
if |M|2 const. d/dm122dm132 const.

Kinematics Belle, PRL 99, 131803 (2007)
)
3-body decay
example: D0 Ks -+ m2 (
KS
+ )
cos2
non-uniformity of Dalitz plot contribution of intermediate states
Ki ti li itKinematic limitsD0 +K*- m2(KS-)J.D. Jackson, D.R. Tovey, Kinematics, in RPP
23*=
( ) ( ) 222 2mmm +23*=0
( ) ( )( ) ( )min223max223
23max23min23*23
2cos
mmmmm
+
= 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 30

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-+
(but interf. /KK, /f0important in precise Br(Ds )determination)
determination)
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 K+
by fitting fitting DalitzDalitz distributiondistributionobtain evidence of newobtain evidence of new states, measure properties(mass width spin)
B K*(892)
B K0*(1430)
(mass, width, spin)
projection of Dalitz disribution and fit to
24
disribution and fit to M(+)

Physics20 )( tAyixqAefDd t ++= D0 Mixing and CPV
x, y: mixing parameters;
2tA
pAe
dt ff+=
due to mixing, D0 D0 fx, y

Physics Belle, PRL 99, 131803 (2007)t dependent
tt-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
[ ]titiS
eemm
tDKtmm
21)(1
)(),,(
22
022
++
++=
=
A
MD0f
28
[ ]
[ ]titi eemmpq
eemm
21),(21
),(2
22 +
+
+
++=
A
AD0f
p21,2=F(x,y); m2 = m2(KS),

PhysicsBelle, arXiv:0803.3375
measurement3 measurement
B- K- D0( f)B- K- D0( f)B K D0( f)interference |M|2 = F(3) f= +- KS
),(),())(( 22220)(
3
+
+= mmAremmAfDKB ii mm
M(A: D0 decays; r: ratio of two B amplitudes; : strong phase diff. of two B amplitudes
A from fit to Dalitz fit to Dalitz distribution of D0 decays;
r, 3, (in principle) from fit to Dalitz fit to Dalitz distribution of B decays 30

sin 2 eff in b sqq
Physics SM:sin 21eff in b sqq
NP contrib. sin 21eff sin 21eff VtbVts* : no weak phase
quasi twoquasi two--bodybody, B 0(770)KS, f0(980)KSBelle, PRD76, 091103(R) (2007)BaBar, PRL99, 161802 (2007)B B PRL98 051803 (2007)
BBaBar, PRL98, 051803 (2007)
Vtd* 2: (mixing): sin21
NP:
[ ]tmqStmqAetP CPCPt
+=
sincos14
)(/||
interf. between various states and non-resonant contrib. Dalitz analysis
s
B0 g
g~b s
+( 23dRR)b
~R
s~
NP:
d d
s
s Ks
B0 gs~R
S=sin21eff -sin21

Physicssin 2 eff in b sqqsin 21eff in b sqq
[ ]tmqStmqAetP CPCPt
+=
sincos14
)(/||
4
),( 22 mmmAA =
each point in Dalitz space has a specific time evolution depending onevolution, depending on |A|2-|A|2 (direct CPV)(direct CPV)and (AA*) (indirect CPVindirect CPV corresponding to
(AA ) (indirect CPVindirect CPV, corresponding to specific two-body contribution sin 2sin 211effeff (i)(i))

Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution
a matter of statistics102 pp 30 events10 pp 3 events
Adopted from K. Peters, talk at Charm 2006, Beijing

a matter of statistics105 pp 30 events
larger stat. larger sensitivity to model details
Breit-Wigner resonances
10 pp 3 events
Breit Wigner resonancessimple consideration of spin 0 elastic scatteringleads to the Breit-Wigner amplitude for a b r a b
// 2
4/)(4/
2/2/
22
22
+
=
=
EmT
iEmT
rr
|T|2 |T|2
mr=1.0=0.3
mr=0.5=0.2
| |
mR
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 BWs 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

Dalitz analysesDalitz analyses Introduction - IJS golob/sola/Dalitz_BAS09_ · PDF...

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