Contact interaction treatment of the nucleon Faddeev equation

TL;DR

This work develops a symmetry-preserving, vector⊗vector contact-interaction (SCI) framework to solve the three-body nucleon Faddeev equation at leading rainbow-ladder order, yielding algebraic tractability while preserving essential QCD symmetries. The SCI formulation reveals that scalar diquark (MA) and axialvector diquark (MS) correlations jointly shape the nucleon’s bound-state amplitude, which simplifies to three independent expansion coefficients under S$_3$ symmetry. Electromagnetic form factors are computed from a conserved current, with flavour-separated contributions providing insight into the roles of up and down quarks; comparisons with realistic QCD-connected 3-body results show similar $Q^2\to0$ behavior but considerably stiffer $Q^2$-evolution in SCI, highlighting sensitivity to hadron mass generation mechanisms. The study identifies clear avenues for improvement and extension, including SU(3) octet/baryon analyses and semileptonic transitions, to sharpen understanding of emergent hadron mass and to benchmark more sophisticated continuum approaches.

Abstract

Working with a symmetry-preserving treatment of a vector $\otimes$ vector contact interaction (SCI), a largely algebraic three-body Faddeev equation treatment of the nucleon bound state problem is introduced and used to deliver results for all nucleon charge and magnetisation distributions and their flavour separation. A strength of the SCI treatment is that it provides for a transparent understanding of this three-body approach to developing predictions for baryon observables. Comparisons of SCI results with predictions obtained in realistic-interaction Faddeev equation studies reveal the sensitivities of given observable to phenomena associated with the emergence of hadron mass.

Figures (8)

• Figure 1: Rainbow-ladder truncation Faddeev equation. Filled circle: Faddeev amplitude, $\Psi$, the matrix-valued solution. Spring: dressed-gluon interaction that mediates quark+quark scattering; see Eqs. \ref{['EqRLInteraction']}, \ref{['defcalG']}. Solid line: dressed-quark propagator, $S$, calculated from the rainbow gap equation. Lines not carrying a shaded circle are amputated. Isospin symmetry is assumed. The sum runs over each of the cases involving quark "$i=3,1,2$" as a spectator to the exchange interaction.
• Figure 2: Sketch of the $q(qq)$ Faddeev equation used, e.g., in Refs. Roberts:2011cfXu:2015kta. $\Psi$ is the Poincaré-covariant solution amplitude for a baryon with total momentum $P=p_q+p_d=k_q+k_d$, constituted from three valence quarks, two of which are paired in a fully-interacting nonpointlike diquark correlation. Here, $\Psi$ expresses the relative momentum correlation between the dressed-quarks and -diquarks. Legend. Single line -- dressed-quark propagator; $\Gamma$ -- diquark correlation amplitude; double line -- $(qq)$ (diquark) propagator; and shaded box -- Faddeev kernel, which explicitly shows the quark exchange binding mechanism, which is the horizonal quark line at the top of the diagram.
• Figure 3: The nucleon has three valence quarks; hence, the complete RL nucleon electromagnetic current has three terms: $J_\mu(Q) = \sum_{a=1,2,3} J_\mu^a(Q)$. Symmetries mean that one can readily obtain the $a=1,2$ components once the $a=3$ component is known Eichmann:2011pv. $\delta$, $\delta^\prime$ are spinor indices and $\rho$, $\rho^\prime$ are isospin indices. $\Gamma_\mu$ is the dressed-photon+quark vertex, see, e.g., Ref. Xu:2019ilh.
• Figure 4: Proton electromagnetic form factors. Legend. Solid purple curve -- SCI results obtained herein; dashed blue curve -- realistic-interaction $3$-body prediction, reproduced from Ref. Yao:2024uej; dot-dashed green -- data parametrisation in Ref. Kelly:2004hm. Orange points -- experimental data taken from compilation in Ref. Arrington:2007ux.
• Figure 5: Neutron electromagnetic form factors. Legend. Solid purple curve -- SCI results obtained herein; dashed blue curve -- realistic-interaction $3$-body prediction, reproduced from Ref. Yao:2024uej; dot-dashed green -- data parametrisation in Ref. Kelly:2004hm. $G^n_E$ experimental data: Refs. Passchier:1999cjHerberg:1999udE93026:2001cssBermuth:2003qhWarren:2003maGlazier:2004nyPlaster:2005cxBLAST:2008bubRiordan:2010id. $G^n_M$ data: Refs. Lung:1992buAnklin:1998aeKubon:2001rjJeffersonLabE95-001:2006daxCLAS:2008idi