november 9 2009int-jlab workshop amplitude analysis for three- hadron states: historical perspective...
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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