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  • 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

    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 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

    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

    ( )= 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

    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=

    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

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

    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

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

    ( ) ( )( ) ( )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

    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-+

    (but interf. /KK, /f0important in precise Br(Ds )determination)

    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 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

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

    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

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

    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;

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

    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

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

    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

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

    (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

    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 30 events10 pp 3 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 30 events104 pp 30 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 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

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

    // 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

    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 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