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Subatomic Physics: Fermionic Bound States in Minkowski Space

This paper explores a fully covariant description for a two-fermion system in Minkowski space using the Bethe-Salpeter equation. It discusses the Nakanishi integral representation and presents numerical results. Conclusions and perspectives, as well as motivations and future directions, are also provided.

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Subatomic Physics: Fermionic Bound States in Minkowski Space

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  1. 1ST Meeting of LIA- Subatomic physics: from theory to applications Fermionicboundstates in Minkowski-space. Wayne de Paula InstitutoTecnológico de Aeronáutica - Brasil R Pimentel, T Frederico – ITA G Salme – INFN/Roma I, M Viviani – INFN/Pisa EPJC (2017) PRD94 (2016) 071901 Work in progress: E Ydrefors, A Castro, G Yabuzaki, J Nogueira – ITA JPBC de Melo – UNICSUL, L Tomio – IFT, O Oliveira - Coimbra, P Maris – Iowa, J Carbonel - CNRS/Orsay, V Karmanov- Lebedev wayne@ita.br

  2. Outline Bethe-SalpeterEquation Nakanishi Integral Representation(NIR) BSE with NIR – TwoFermionsboundstate NumericalResults Conclusionsandperspectives

  3. Motivations To obtain a fully covariant description for a two-fermion system, in Minkowsky space, through the non perturbative framework yield by the Bethe-Salpeter equation. To determine from the BS amplitude, in Minkowsky space, the relevant momentum distributions.

  4. Bethe-SalpeterEquation We start from the four-point Green function which is a solution of the integral equation

  5. Bethe-SalpeterEquation Inserting a complete Fock basis in the Fourier space, the bound state contribution appears as a pole. BSA in configurationspace: Close tothe pole weobtainthe BSE The sameKernelofthe four-point Green function

  6. Nakanishi Integral Representation Let’s take a connected Feynman diagram (G) with N external momenta pi, n internal propagators with momenta lj and masses mj and k loops. The transition amplitude is given by (scalar theory) Feynman parametrization Using properties of Dirac delta and change of variables, we have s = scalar prod. external momenta The dependence upon the details of the diagram moves from the denominator to the numerator. Therefore we have the same formal expression for the denominator of any diagram.

  7. Nakanishi Integral Representation To represent the BSA, we consider the constituent particles with momentum p1, p2 and the bound-state with momentum p. Using the identities we obtain the NIR where

  8. Nakanishi Integral Representation • Nakanishirepresentation: “Parametricrepresentation for anyFeynmanndiagram for interactingbosons, with a denominator carrying the overall analyticalbehavior in Minkowskispace” (1962) Bethe-Salpeter amplitude BSE in Minkowskispacewith NIR for bosons: • Kusakaand Williams, PRD 51 7026 (1995); KarmanovandCarbonell, EPJA 27 1 (2006), EPJA 27 11 (2006), EPJA27 11 (2010); Frederico et al PRD 85 036009(2012), PRD 89, 016010 (2014), EPJC(2015).

  9. NIR for two-fermions CarbonellandKarmanov (EPJA 46, 387 2010) • Bethe-Salpeterequation Ladderapproximation: onebosonexchange • Scalar VertexForm-Factor • Pseudo-scalar • Vector

  10. NIR for two-fermions Bethe-Salpeter amplitude: fermion-fermion 0+state Usingthe NIR System ofcoupled integral equations

  11. Light-Front projection Light-Front variables Weprojectontothenull plane LF amplitudes BistheBindingenergy

  12. NIR for two-fermions The coupledequation system is The Kernel contains singular contributions

  13. NIR for two-fermions Wecan single out the singular contributions For two-fermion BSE withj=1,2,3 and in theworst case Thenonecannot close thearcattheinfinity . The severityofthesingularities (powerj), does notdependontheKernel Weobtainthe singular contributionusing Yan PRD 7 (1973) 1780 Differently, CarbonellandKarmanovintroduced a smoothingfunctiontoperformtheintegration (EPJA 46, 387 (2010)).

  14. BindingEnergiesvsCouplingConstants Vector FulldotsfromCarbonel andKarmanov EPJA (2010). Scalar pseudoscalar The pseudoscalar case has a weakerattraction, whichrequireslargercouplings.

  15. Scalarexchange: Nakanishiweight-function

  16. ScalarExchange: LF amplitudes WeakBinding Strong Binding

  17. Vector Exchange: LF amplitudes WeakBinding Strong Binding

  18. Valence momentum distribution Usingcreationandannihilationoperatorsontothenull plane, weconstruct a generic LF Fockstate. Fromthis, wecanwritethe Valence probability as We define theValence momentum distributiondensity as The valence longitudinal andtransverse LF momentum distributiondensities are

  19. Vector Exchange: Momentum Distribution WP, E Ydreford, J Alvarenga, T Frederico andGSalme Fig. 5 Valence Longitudinal (leftpanel) andTransverse (rightpanel) LF Momentum distribution.

  20. Conclusionsand Perspectives • A method for solving the fermionic BSE has been given and also a general rule for the expected singularities. • Our numerical results confirm the robustness of the Nakanishi Integral Representation for solving the BSE. • We intend to calculate Hadronic observables: Form-Factors, GPD’s, TMD’s…

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  22. StieltjesTransformationvsUniqueness The LF amplitude isgivenby StieltjesTransformation Nakanishishowed (bosons) thattheweightfunctionisunique. It meansthatifboth LHS and RHS havethesame integral operator, theycanbeextracted

  23. StieltjesTransformation Wecan compare withtheUniquenessmethod The BSE iswritten as StieltjesTransformation: Uniqueness Bosons: Carbonell, Frederico andKarmanov PLB 2017.

  24. Extreme Binding Energy (B=2m) Using Feynman parametrization, Dirac delta propertiesandUniquenesswehave Solving numerically we obtain g2=68, which is consistent with the solution of the BSE for B close to 2m For thefollowingvalues for thevariables The Kernels are Abigail Castro – masterproject

  25. StieltjesTransformationvsUniqueness The LF amplitude isgivenby StieltjesTransformation Nakanishishowed (bosons) thattheweightfunctionisunique. It meansthatifboth LHS and RHS havethesame integral operator, theycanbeextracted

  26. StieltjesTransformation Wecan compare withtheUniquenessmethod The BSE iswritten as StieltjesTransformation: Uniqueness Bosons: Carbonell, Frederico andKarmanov PLB 2017.

  27. Extreme Binding Energy (B=2m) Using Feynman parametrization, Dirac delta propertiesandUniquenesswehave Solving numerically we obtain g2=68, which is consistent with the solution of the BSE for B close to 2m For thefollowingvalues for thevariables The Kernels are Abigail Castro

  28. StieltjesTransformation StieltjesTransformation: Uniqueness For thefollowingvalues for thevariables The Kernels are

  29. WickRotation Wecanperform a wickrotationand solve the BSE in Euclideanspace The bindingenergies are thesameoftheMinkowskispace, butthe BSA isnot. ToobtainFormFactorsandTransverse Momentum Distributionweneedthe BSA. Lattice QCD

  30. ScalarExchange: WeakBinding

  31. Scalar Exchange: Strong Binding

  32. Vector Exchange: WeakBinding

  33. Vector Exchange: Strong Binding

  34. NumericalMethod Basisexpansion for theNakanishiweightfunction Gegenbauerpolynomials Laguerre polynomials Orthonormality

  35. NIR for two-fermions Bethe-Salpeter amplitude: fermion-fermion 0+state Multiplying BSE by Siandtakingthe trace

  36. Vector coupling Fig. 1 Bindindenergyvscouplingconstant for a massless vector exchange. FulldotsfromCarbonelandKarmanov EPJA (2010).

  37. Scalarandpseudoscalarexchange The pseudoscalar case has a weakerattraction, whichrequireslargercouplings.

  38. UniquenessofNakanishRepresentation Nakanishishowedthattheweightfunctionisunique. It meansthatifboth LHS and RHS havethesame integral operator, theycanbeextracted

  39. Vector Exchange: LF amplitudes MockPion – LQCD: Dyn. quark mass = 250 MeV, Dyn. gluonmass= 500 MeV WeakBinding Strong Binding

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