Relativisric Description of Two and Threebody Systems'''

1
Bogoliubov Lab. Theor. Phys. JINR, Dubna
Leonid P. Kaptari(ab), Sergey M.
Dorkin(a) a)
Bogoliubov Lab. Theor. Phys., JINR, Dubna
b) University
of Perugia INFN, Sez. Perugia

2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
2
L.P. Kaptari, S.M. Dorkin Two-fermion bound
states...
BSE
Ladder approximation
Wick stability of single particle states
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
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• Wick rotation
• Matrix form
• 3) Complete set of 4x4 matrices

2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
states...
Y-scaling
D(e,e)X reactions
Dirac basis, advantages 1) simple matrix
structure
2) fast convergence of the iteration method
disadvantages a) nontransparent
physical interpretation of the partial
amplitudes,
study of relativistic effects...
b) comparison with
other approaches (LF,Gross...)
and with
nonrelativistic calcualtions...
c) two-dimensional array
(p0 ,p), interpolation and
numerical inverse Wick
rotation
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
Spin-angular harmonics (Cubis, Tjon...)
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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Spin-angular harmonics
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
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BS
Gross
Paris
Bonn
Bonn
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
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2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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Covariant representation (boost effects...)
Example backward DppD elastic scattering
(one-nucleon exchange approximation)
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
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BS positive waves
MAPLE V
Lorentz boost
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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isobar
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Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Two-fermion bound
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Mass-spectrum of the BS equation
F (p) lK(p,p) F(p)
Exhousting (depletion) method
ln
K
F0
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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The algorithm

neither l nor F known !
Consider the ratio
Construct a new kernel
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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BS kernel
Back to BS kernel
Gram-Schmidt biorthogonal basis
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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ground -------- first excited
J2
J1
J0
At g10 Mg (J0) and M1 (J0)
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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One Iteration Approximation (V. Karmanov)
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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One Iteration Approximation
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P .Kaptari, S.M. Dorkin Twofermion bound
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One Iteration Approximation
pp (00 ) n
pD
FSI (BS OIA )
NR
RIA (BS)
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Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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Hyperspherical harmonics
scalar case
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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Hyperspherical harmonics (spinor case)
new basis
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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ga1..4 into the Gegenbauer polynomials Xj
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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parametrization
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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Three commonly accepted expressions
Rainbow approximation ( )
effective model for gluon propagator
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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Angular integration
The main equation
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento
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L.P. Kaptari, S.M. Dorkin Twofermion bound
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• A variety of mathematical methods and
  representations to solve numerically the BSE has
  been considered
• the Dirac matrices representation appears to
  have the simplest matrix structure and to
  provide the fastest convergence of the iteration
  procedure. However, physical interpretation of
  results is hindered, the parametrization of the
  numerical solution is cumbersome and
  inconvenient.
• the spin-angular harmonics, along with the
  covariant representation allow for a more
  transparent physical meaning and for a
  straightforward treatments of relativistic
  effects in the computed observables.
• A mathematical method, based on the
  Gramm-Schmidt orthogonalization procedure, is
  proposed to find the mass-spectrum of the BSE.
  The method assures a reasonable fast convergence
  of the numerical procedure and appears to be
  suitable for describing the energy spectrum of
  heavy mesons
• The hyperspherical harmonics method is used to
  find numerical solutions in form of
  one-dimensional arrays such solutions are rather
  convenient in further calculations involving the
  BS amplitudes
• The possibility to describe the known mesons as
  two-quark bound states within the BS formalism
  is considered. To generate dynamically the quark
  masses, Dayson-Schwinger equation is solved
  within the rainbow approximation with an
  effective model gluon propagator by the
  hyperspherical harmonics method. It is shown that
  with the obtained quark masses the BSE provide a
  reasonable description of the meson ground
  states.
• A detailed investigation of meson properties
  (transition FFs, decay widths etc) within DSE
  BSE is planned in the nearest future (work is
  in progress).

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L.P. Kaptari, S.M. Dorkin Twofermion bound
states...
2009
Relativisric Description of Two and Three-body
Systems...
19-23 October, Trento