November 9 2009INT-JLab Workshop Amplitude analysis for three- hadron states: Historical perspective...

November 9 2009 INT-JLab Workshop

Amplitude analysis for three-hadron states:

Historical perspectiveIan Aitchison

INT-JLab Workshop, UW Nov 9 2009

November 9 2009 INT-JLab Workshop

Outline

• The isobar model and 3-h analyses in the 1970s….and more recently

• But the isobar model doesn’t satisfy unitarity• Simplest implementation (“K-matrix –like”) of

unitarity relation is wrong• Need also analyticity 3-h dynamics• Qualitative features of corrections to IM• Conclusions (as of mid-1980s)

November 9 2009 INT-JLab Workshop

Illinois group(production density matrix)

• “Partial-wave analysis of the 3 decay of the A2”, G.Ascoli et al., Phys. Rev.Lett. 25 (1970) 962-5

• “Spin-parity analysis of the A3”, G.Ascoli et al., Phys. Rev. D7 (1973) 669-686

• “The reaction at 25 and 40 GeV/c”, Yu. M. Antipov et al., Nucl. Phys. B63 (1973) 141-52, and 153-74 [A1, A2 and A3]

P(event)

e A2

pp

lJPMLl

JPLla

baab

tkinematicanglesEulerXfittedCA

MPJbMPJaAAfitted

),()(

,*,)( 222111

November 9 2009 INT-JLab Workshop

L

l

isobar model amplitude

)(stl

factorization

)(WC PMJlL

November 9 2009 INT-JLab Workshop

SLAC-Berkeley(fit coherent amplitudes)

• “Generalized isobar model formalism” D.J.Herndon et al., Phys.Rev.D11(1975)3165

• “Partial wave analysis of the reaction in the c.m. energy range 1300-2000MeV” D.J. Herndon et al., Phys. Rev. D11(1975) 3183-3213

• “Amplitude analysis of production at 7 GeV/c” M.Tabak et al., Fourth Int. Conf. Exp. Meson Spectroscopy, 1974 AIP Conf Proc 21 46-58

P(event)

NN

)3(

2|),()(| lJPMlL

JPMlL tkinematicanglesXfittedC

November 9 2009 INT-JLab Workshop

And more recently……“Improved measurement of the CKM angle in decays with a Dalitz plot analysis of decays to and ” B.Aubert et al. (BaBar)Phys. Rev. D78 (2008) 034023

The weak phase leads to different and decay rates (direct CPV) and is observable when

and decay to common final states.

About 0.5M events in the Dalitz plot.

(*)(*)(*) KKDB

D 0SK

KKKS0

BB

0(*)D 0(*)D

K

November 9 2009 INT-JLab Workshop

Unitarity (1)

Two-body elastic unitarity

Partial wave amplitude

where power series in

For example

)(st

)/(),/( 222 ifssftssfK rr

KiKtKit

ittttitt

ssststst threshss thresh

11

***

2/1*

)1(/1

2/1/12

)(,)()())((Im

cot1K2

November 9 2009 INT-JLab Workshop

Unitarity(2)Two two-body f.s.i’s

Unitarity: (U) Isobar modelwhere are independent of But this does not satisfy (U)Instead, set

Then

(U)

iC is

)(/),()(/),( 22221111 sDWsCsDWsCF

1211

1

11* ),,()(*

2

1)(2)(

1dxWssFstsiFF

threshss

is a linear function of 2s 1x

)(/)()(/)( 222111 sDWCsDWCF

Not factorized

12222

1

11*

11 )(/),(2

1)(2)(

1dxsDWsCsiCC

threshss

),,(:321 21 WssFX

November 9 2009 INT-JLab Workshop

2s

1s

11 x

11 x1 11 cosx

Integration in unitarity relation

November 9 2009 INT-JLab Workshop

So develops an imaginary part for due to rescattering from the other channel(s)

Implementation

(1) “ -matrix” (eg Ascoli and Wyld, PR D12(1975) 43-

58) Set

Spurious singularities

(IJRA&Golding, Phys.Lett. 59B(1975)288)

iC

threshii ss

122110111 /2

1)()(),( dxDCsiWCWsC

K

s s

2s

1s

November 9 2009 INT-JLab Workshop

Implementation(2) add analyticity dispersion relation

uncorrected isobar model

known function two-body data

Integral equations for

IJRA P. R. 137(1965)B1970, R.Pasquier and J.Y.Pasquier,P.R. 170(1968)1294, IJRA and J J Brehm, P. R. D17(1978)3072

21, cc

adds up all rescatterings

),'(Im'

')(),( 11

11

110111

1

WsCiss

dsWCWsC

threshs

222

22

2)(

101 )(/),(),,()(2

dDWCWsWCmW

November 9 2009 INT-JLab Workshop

Integration for integral equation

2)( mW

1s

2

November 9 2009 INT-JLab Workshop

is essentially the partial wave projection of the OPE 3 3 process

2

1s

and has logarithmic singularities on the boundaryof the Dalitz plot, when all particles in the OPE graph are on-shell. Inside the Dalitz plot, develops an imaginary part.

November 9 2009 INT-JLab Workshop

The rescattering corrections to the IM

)],(1[)(

),(),,()(),(

10

222101

WsWC

dWCWsWCWsC

dzWzWzsdWsWs ),(),,(),,(),( 122

11

uncorrected IM rescattering corrections

where

depends on final state interactions

independent of production parameters

Symbolically, 1)1( And so

Amplitude )(/)(])1(1[ 0

1sDWC

first rescattering correction provides reasonable approx. for dependence of full solution

1s

significant dependence can be generated in full solutionW

November 9 2009 INT-JLab Workshop

Qualitative features of calculations• S-waves

0J0l

0l

is complex “scattering length” Effects of lL,

“Triangle” singularities

1s 1s

2s

All depend on and 1s W

)log( 11 ss

)(),)(1/( 1 WaqWaiconst 2/12 ])(1/[ LQRconst

IJRA & JJBrehm, PR D20(1979)1131; JJB, PR D21(1980)718, D23(1981) 1194, D25(1982)149

November 9 2009 INT-JLab Workshop

Conclusions (as of mid-1980s) The corrections to the IM are unlikely to be larger than of order 20% in magnitude Subenergy corrections can broadly be absorbed into either the two-body parametrizations or the barrier factors, at fixed But (a) there is a - dependence J.J.Brehm, P.R. D25(1982) [ -dependent modulation of ]Study of the heavy-lepton decay

“We can summarize by asserting that rescattering can be a 20%effect relative to the resonance and should be included if the data are refined to that level of accuracy.”

(b) the corrections are final-state dependent (eg versus ) Corrections might be needed to reconcile two-body amplitudes derived from different final states if data good enough

.

1a

W

3

WW

KN