PhenomenologyTheory of amplitude parameterization

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Partial Wave Analysis and Amplitude Analysis
Phenomenology/Theory of amplitude
parameterization and analysis (2 meson and 3
mesons production)
How to factorize amplitudes
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A Physics Goal three step process
Identify old (a2) and new (p1) states
Resonances appear as a result of amplitude
analysis and are identified as poles on the
un-physical sheet
Use data (physical sheet) as input to
constrain theoretical amplitudes
3. Interpretation composite or fundamental,
structure, relation to
QCD, lattice, etc
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Example of factorization
s/t gtgt 1
T(s,t)
several partial wave in s- channel
s mx2
single partial wave in s- channel
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Two meson production
Single Regge limit
Double Regge limit

• Derive corresponding amplitudes and establish
• singularity structure in sub-channel variables
• Add normal singularities (resonances)
• Use FESR to constrain parameters

Resonance production
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Resonance shape modification
Szczurek et. al
ZEUS W70GeV
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Establishing a P-wave in the hp, hp
Mhp
sgtgtt,Mhp
Regge
Low energy EFT
t
Single resonance pole
Complicated analytical structure
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O(p2p/f2p)
p
p

Low order c expansion
p
p

Higher order c expansion
unitarization
Interplay between elementary (CDD) and
dynamical resonances
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Lesniak et al. (98)
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No FSI
With FSI
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Amplitudes in three particle (meson) production
(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) N1(M)/D1(s1) N2(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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Some comments on the isobar model
7 kinematical variables g polarization
p(1)
p-(3)
p(2)
isobar
s13gtgts23 otherwise channels overlap need
disp.relations (FMSR)
isobar model violates unitarity K-matrix
improvements violate analyticity
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Ambiguities in the 3p system
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Methods for constructing amplitudes (amplitude
analysis)
Analyticity
Data (in principle) allows to determine full
(including unphysical parts) Amplitudes. Bad
news need data for the entire universe
Approximations
Crossing relates unphysical regions of a
channel with a physical region of another
another Unitarity relates cuts to physical data
Other symmetries (kinematical,
dynamicalchiral, U(1), ) constrain low-energy
parts of amplitudes (partial wave expansion, fix
subtraction constant)
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It is important to have a statement from a
significant and respected representation of the
subatomic community to the effect that the
scientific goals of the CLEO-c and GlueX programs
w.r.t. gluonic excitations are fundamental and
important and that there is general agreement
among the theoretical and experimental components
of the community as to what defines clear
results.
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• A plan for theory (v.r.t amplitude analysis)
• Develop amplitude parameterization for selected
• 2 and 3 meson production amplitudes
• Interplay between s and t channel physics
  constrained by
• analyticity and duality
• Prepare a review (manual ?) describing amplitude
  analysis

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Finite energy sum rules
is an analytical function
Continue out of physical region to look for
resonances
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