Nexttoleading order lightfront threebody dynamics

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LC2008 Mulhouse July 6-11
Next-to-leading order light-front three-body
dynamics
J. A. O. Marinho, Tobias Frederico ITA, São
José dos Campos PhD Thesis
Collaborators JPBC de Melo, E Pace, S Pisano, G.
Salme, P.U. Sauer
OUTLINE

• Introduction Motivation - three-body valence WF
• Review LF Quasi-Potential Approach 4d B.-S.
  equation
• LF three-boson dynamics ladder 4d B.-S.
  equation
• Bosonic Yukawa model LO and NLO
• Zero-range model LO and NLO
• RemarksConclusions

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• I. Introduction Motivation
• Dynamics of few-constituents valence state on the
  Light-Front (LF)
• ?? 4d Bethe-Salpeter eq.
• Iterated Resolvents- Brodsky, Pauli, Pinsky
  Phys. Rept. 301(98)299
• 
  Frederico et al NPA737(04)260c
• LF two-boson/ two-fermion systems - (Quasi
  Potential Approach)
• 
  Sales et al PRC61(00)044003
  63(01)064003
• LF conserved current operators - Kvinikhidze
  Blankleider PRD68(03)02581
• WTI -QP two-boson/two-fermion - Marinho et al
  PRD76(07)096001PRD77(08)116010
• Electroweak observables, form-factors GPD
  (Nonvalence contributions)
• Tiburzi Miller PRD67(03)054014 054015
  two-body/GPD
• De Melo et al PRD73(06) 074013 pion ff q gt 0
  (SL TL)
• Ji, Mischenko, Radyuskin PRD73(06)114013
  pion/GPD

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II. Review LF Quasi-Potential Approach 4d
B.-S. equation
Woloshyn Jackson NPB64(1973)269
two-boson systems
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Sales et al. PRC61(2000)044003
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valence propagator in global LF time
Valence? Valence scattering amplitude
Effective interaction
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Reconstruction 4-d B.S. amplitude from the
valence wf
(i e ?0)
(projecting back to the LF retrieves the valence
wf.)
Reversing LF time projection operation!
ltBS Ampl. 4d operator BS Amplgt ? ltval.3d
operator val.gt
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ExampleBosonic Yukawa model
Mass2 eigenvalue eq. valence wf
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IV. Three-boson systems and ladder 4d BS equation
Integration over k- for 1 and 2 ? free 3-boson
resolvent
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Faddeev decomposition
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In practice Wi is obtained from a power
expansion in V
LO
NLO
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Bosonic Yukawa model LO
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Bosonic Yukawa model NLO
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V. Zero-range model LO and NLO
4d ladder
Laddercontact?
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Leading order bound-state 3-boson equation
Bound state eq.
Vi Faddeev components
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The divergence is eliminated by fixing lambda
through the 2-boson bound state mass!
Frederico 92 regularization Mjk real ? no
collapse Carbonell Karmanov 03
no-regularizaton collapse for M2Blt critical value
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Next-to-Leading order bound-state 3-boson equation
LO kernel
NLO kernel
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Three-body reducible diagrams
i
i
i
i
j
j
j

...
j
k
k
k
k
Three-body irreducible diagrams
i
i
i
i
j
j

j
j
k
k
k
k

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NLO 3-boson bound-state equation
LO kernel
????
NLO kernel
Effective 3-body forces
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• VI. Remarks Conclusions
• LF Few-body dynamics dynamics of the valence
  w.-f.
• Quasi-potential Approach to LF ? LF dynamics
• 4-d Bethe-Salpeter amplitude ?? valence w.-f.
• 4-d operators ?? 3-d operators acting valence
  w.f.
• Conserved current operator WTI
• Quasi-potential Approach systematic expansion of
  the LF dynamics of three-body bound-states
• NLO effective 3-body interactions with pair
  creation
• Zero-range model LO retrieves previous valence
  bound-state eq. ( also
  covariant LF approach)
• NLO three-body interaction ? 4-particle
  intermediate state
• Chalenges dependence on the regularization in
  NLO? Conserved current operator? Spin?

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