Partial Wave Analysis

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Partial Wave Analysis
Adam Szczepaniak Indiana University

• What is it
• Relation to physical properties
• Properties of the S-matrix
• Limitation and perspectives
• Examples peripheral production of 2- and
  3-particle final states

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depending on how much we know about the
amplitude
Know everything !

Y(xi) includes it all kinematics and
dynamics, There is nothing to fit !
This is best case scenario
Know nothing

Worst case scenario
Usually somewhere in between

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Example
p- p ! hp0 n

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the less you know the more ambiguous the
answer
0 physics input maximal ambiguity
some physics input moderate ambiguities
know everything no ambiguities
You do it in all possible way to study
systematics
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p- p ! p0p0 n
(J. Gunter et al.) 2001
f2(1270)
s(400-1200)
p0p0 spectrum
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p- p ! hp0 n
(A.Dzierba et al.) 2003
Assume a0 and a2 resonances
( i.e. a dynamical assumption)
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.. so, how much we know about the S-matrix

• kinematical constraints

• dynamics is much harder
• Heisenberg-Mandelstam program (ca. 1960-1970)
• QCD (ca. 1970)
• Jlab upgrade (ca. Now !)

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HM reconstruct S given Mandelstam representation
and the unitarity
condition)

• S matrix has specific analytic properties
  (causality)

G(E) is analytic for ImEgt0
ImE
ReE
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Mandelstam hypothesis
1. There is an analytical representation for S
(which includes poles and cuts of physical
origin)
(so far unknown)
2. Given a representation a unique solution can
be found
(not true)
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Complete representation complete dynamics
Non-relativistic example need the potential
Relativistic example
p
p
N
K
p
p
p
K
p
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a given representation can have multiple
solutions ! (incomplete
knowledge of dynamics)
Non-relativistic example Blaschke product
k
2a
k
Relativistc example (Castillejo,Daliz,Dyson
(CDD) poles)
fl
Have the same representation
fl
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Good news

• Low energy
• Effective range expansion (low energy)
• Two body unitarity
• Small number of (renormalized) parameters
• QCD input

• High energy
• Regge behavior
• Asymptotic freedom

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Illustration pp (SI0)
pp
2 Resonaces _at_ 1.3, 1.5 GeV
ppKK
pp only (no KK, no resonances)
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Regge poles
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combine low (chiral) and high energy
information
sgtgtt,M
Chiral
Regge
t
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p- (18GeV) p ? X p ? h p- p
? h p- p
30 000 events
Nevents N(s, t, Mhp , W)
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p- p ! hp0 p
DATA (from E852)
PWA determined by maximizing likelihood
function over an even sample
adam_at_mantrid00 data ls -l total
835636 -rw-r--r-- 1 adam adam 87351564
May 10 1040 ACC -rw-rw-r-- 1 adam adam
4882894 May 10 1039 DAT -rw-r--r-- 1 adam
adam 762596488 May 10 1042
RAW adam_at_mantrid00 data
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Assume BW resonance in all, M1,0, P-waves
p1(900 5GeV) emerges
Intensity in the weak P-waves is strongly
affected by the a2(1320), strong wave due to
acceptance corrections
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E852 hp- analysis
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p- p ! hp- p
Results of coupled channel analysis of
p- p ! hp- p
D
D
P
S
P
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f(P)-f(D)
P2
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Peripheral production of hybrid mesons
Excited Flux Tube
Quarks
Hybrid Meson
like
like
So only parallel quark spins lead to exotic JPC
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Photo production enhances exotic mesons
OPE
Afanasev, AS, (00)
Agrees with Condo93
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Problems with the isobar (sequential decay)
model
(1)
p-
a2-(1320) (J)
L
CJLS
S
M
p-
(2)
r0
Breit-Wigner
s1
(3)
p
A(s1,s2,M) CJLS (M) / D1(s1)
F(s1,s2,M) f1(M)/D1(s1) f2(M)/D2(s2)
1 ?? 2
Independent on 2-particle sub-channel energy
violates unitarity !
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Has to depend on 2-particle energy
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F1(s1,M)i H(s1,M)ij Cj(M)
i,js,r
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3p sample
Available !
Analyzed
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Summary

• Need theory input to minimize (mathematical)
  ambiguities
• and to understand systematic errors
• Need more theory input to determine physical
  states
• (coherent background vs resonances)

• There is lots of data to work with and there
  will be more
• especially needed for establishing gluonic
  excitations
• p1(1400) and p1(1600) in hp are most likely due
  to
• Residual interactions much like the s meson in
  the pp S-wave