Transcript "Amplitude analysis for three-hadron states: Historical perspective"

Amplitude analysis for threehadron states:
Historical perspective
Ian 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)
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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  p     p 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)    ab ( fitted )Aa Ab *, a  J1P1M 1 , b  J 2 P2 M 2
Aa   CLlJP ( fitted ) X LlJPM ( Euler angles , kinematic)tl
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isobar model amplitude

L
CLJ PM
l (W )
t l (s)
factorization
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l
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 N  N in the c.m.
energy range 1300-2000MeV” D.J. Herndon et al., Phys.
Rev. D11(1975) 3183-3213
• “Amplitude analysis of (3 )  production at 7 GeV/c” M.Tabak
et al., Fourth Int. Conf. Exp. Meson Spectroscopy, 1974 AIP
Conf Proc 21 46-58
JPM
JPM
2

|
C
(
fitted
)
X
(
angles
,
kinematic
)
t
|
P(event)  L l
Ll
l
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And more recently……
“Improved measurement of the CKM angle  in
B   D (*) K (*) K (*)decays with a Dalitz plot
analysis of D decays to K S0   and
B.Aubert et al. (BaBar)
K S0 K  K  ”
Phys. Rev. D78 (2008) 034023
The weak phase  leads to different
B  and
B  decay rates (direct CPV) and is observable when
D (*)0and
D
(*) 0
decay to common final states.
About 0.5M events in the
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K Dalitz plot.
INT-JLab Workshop
Unitarity (1)
Two-body elastic unitarity
Partial wave amplitude t (s )
(Im t ( s)) s  sthresh   t * ( s)t ( s) ,   ( s  sthresh )1 / 2
t  t  2it t  1 / t  1 / t  2i
*
*
*
 1 / t  i  K 1  t  (1  iK ) 1 K
where
K
1
2

  cot  power series in
For example
K  f /( sr  s), t  f /( sr  s  if  )
2
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2
Unitarity(2)
Two two-body f.s.i’s X 1  2  3 : F (s1 , s2 ,W )
Unitarity: ( F  F
*
) s1  sth resh
1 1
 2i ( s1 )  t *( s1 ) F ( s1 , s2 , W ) dx1
2 1
(U)
s2 is a linear function of x1
Isobar model F  C1 (W ) / D1 (s1 )  C2 (W ) / D2 (s2 )
where C i are independent of s i
But this does not satisfy (U)
F  C1 (s1,W ) / D1 (s1 )  C2 (s2 ,W ) / D2 (s2 )
Instead, set
Not factorized
Then
(U)  (C1  C1 ) s s
*
1
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thresh
1 1
 2i ( s1 )  C 2 ( s 2 ,W ) / D2 ( s 2 ) dx1
2 1
INT-JLab Workshop
Integration in unitarity relation
s2
x1  1
x1  cos 1
1
x1  1
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s1
So C i develops an imaginary part
for si  si thresh due to rescattering from the other
channel(s)
Implementation
(1) “ K -matrix”
58) Set
(eg Ascoli and Wyld,
PR D12(1975) 43s
C1 ( s1 , W )  C1 0 (W )  i1 ( s1 )
1
C 2 / D2 dx1

2
2
Spurious singularities
s
(IJRA&Golding, Phys.Lett. 59B(1975)288)
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s
s1
Implementation
(2) add analyticity  dispersion relation
C1 (s1 ,W )  C1 0 (W )  
1
 C1 0 (W )  

ds1 '
s1 thresh s1 's1  i Im C1 (s1 ' ,W )
(W  m ) 2

uncorrected
isobar model
(s1 , 2 ,W ) C 2 (2 ,W ) / D2 (2 ) d2
known function
two-body data
adds up all rescatterings
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
Integral equations for c1 , c2
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Integration for integral equation
2
(W  m) 2
s1
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 is essentially the partial wave projection of the
OPE 3
3 process
s1

2
and has logarithmic singularities on the boundary
of the Dalitz plot, when all particles in the OPE
graph are on-shell. Inside the Dalitz plot, 
develops an imaginary part.
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The rescattering corrections to the IM
C ( s1 , W )  C 0 (W )    ( s1 , 2 , W ) C (2 , W ) d2
 C 0 (W ) [1  ( s1 , W )]
uncorrected IM
where
rescattering corrections
( s1 ,W )    ( s1 , 2 ,W ) d2    ( s1 , z,W ) ( z,W ) dz
depends on final state interactions
independent of production parameters
Symbolically,
   (1   ) 1
And so
1

[
1


(
1


)
] C 0 (W ) / D( s )
Amplitude

first rescattering correction    provides reasonable approx.
for s1 dependence of full solution
significant W dependence can be generated in full solution


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Qualitative features of calculations
• S-waves
l 0
J 0
l 0
  const /(1  i a(W ) q1 ), a(W ) is complex “scattering length”

Effects of

“Triangle” singularities
L, l
log( s1  s1 )
  const /[1  (QR ) 2 L ]1 / 2
IJRA & JJBrehm, PR D
20(1979)1131; JJB, PR
D21(1980)718, D23(1981)
1194, D25(1982)149
s2
All depend on s1 and W
s1 
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INT-JLab Workshop
s1
 The
Conclusions (as of mid-1980s)
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 W
 But
(a) there is a W - dependence
J.J.Brehm, P.R. D25(1982) [ W -dependent modulation of a1 ]
Study of the heavy-lepton decay 
 3
“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 K versus
N )
 Corrections might be needed to reconcile two-body
amplitudes derived from different final states if data
good enough
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