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A Euclidean bridge to the relativistic constituent quark model
Nuclear Theory · Submitted 25 Aug 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements.
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The abstract, sentence by sentence
${\bf Background}$ Knowledge of nucleon structure is today ever more of a precision science, with heightened theoretical and experimental activity expected in coming years. At the same time, a persistent gap lingers between theoretical approaches grounded in Euclidean methods (e.g., lattice QCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (such as light-front constituent quark models). ${\bf Purpose}$ Seeking to bridge these complementary worldviews, we explore the potential of a Euclidean constituent quark model (ECQM). This formalism enables us to study the gluonic dressing of the quark-level axial-vector vertex, which we undertake as a test of the framework. ${\bf Method}$ To access its indispensable elements with a minimum of inessential detail, we develop our ECQM using the simplified quark $+$ scalar diquark picture of the nucleon. We construct a hyperspherical formalism involving polynomial expansions of diquark propagators to marry our ECQM with the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters by fitting electromagnetic form factor data. ${\bf Results}$ From this formalism, we define and compute a new quantity --- the Euclidean density function (EDF) --- an object that characterizes the nucleon's various charge distributions as functions of the quark's Euclidean momentum. Applying this technology and incorporating information from BSE analyses, we find the dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero. ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements. The small size we obtain for the impact of the dressed vertex on the axial-singlet charge suggests that models without this effect are on firm ground to neglect it.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | ${\bf Background}$ Knowledge of nucleon structure is today ever more of a precision science, with heightened theoretical and experimental activity expected in coming years. | background 1.00 | 0.03 | 0.01 | asserted 0.91 |
| s1 | At the same time, a persistent gap lingers between theoretical approaches grounded in Euclidean methods (e.g., lattice QCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (such as light-front constituent quark models). | gap 0.72 | 0.04 | 0.02 | asserted 1.00 |
| s2 | ${\bf Purpose}$ Seeking to bridge these complementary worldviews, we explore the potential of a Euclidean constituent quark model (ECQM). | contribution 0.96 | 0.43 | 0.02 | hedged possibility 0.55 |
| s3 | This formalism enables us to study the gluonic dressing of the quark-level axial-vector vertex, which we undertake as a test of the framework. | contribution 0.97 | 0.64 | 0.02 | asserted 0.95 |
| s4 | ${\bf Method}$ To access its indispensable elements with a minimum of inessential detail, we develop our ECQM using the simplified quark $+$ scalar diquark picture of the nucleon. | method 1.00 | 0.93 | 0.02 | asserted 0.99 |
| s5 | We construct a hyperspherical formalism involving polynomial expansions of diquark propagators to marry our ECQM with the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters by fitting electromagnetic form factor data. | method 1.00 | 0.95 | 0.02 | asserted 0.99 |
| s6 | ${\bf Results}$ From this formalism, we define and compute a new quantity --- the Euclidean density function (EDF) --- an object that characterizes the nucleon's various charge distributions as functions of the quark's Euclidean momentum. | contribution 0.75 | 0.96 | 0.05 | asserted 1.00 |
| s7 | Applying this technology and incorporating information from BSE analyses, we find the dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero. | result 1.00 | 0.97 | 0.29 | asserted 0.87 |
| s8 | ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements. | future work 0.46 | 0.46 | 0.49 | asserted 0.82 |
| s9 | The small size we obtain for the impact of the dressed vertex on the axial-singlet charge suggests that models without this effect are on firm ground to neglect it. | interpretation 1.00 | 0.86 | 0.28 | hedged possibility 0.99 |
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The paper
Original abstract
${\bf Background}$ Knowledge of nucleon structure is today ever more of a precision science, with heightened theoretical and experimental activity expected in coming years. At the same time, a persistent gap lingers between theoretical approaches grounded in Euclidean methods (e.g., lattice QCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (such as light-front constituent quark models). ${\bf Purpose}$ Seeking to bridge these complementary worldviews, we explore the potential of a Euclidean constituent quark model (ECQM). This formalism enables us to study the gluonic dressing of the quark-level axial-vector vertex, which we undertake as a test of the framework. ${\bf Method}$ To access its indispensable elements with a minimum of inessential detail, we develop our ECQM using the simplified quark $+$ scalar diquark picture of the nucleon. We construct a hyperspherical formalism involving polynomial expansions of diquark propagators to marry our ECQM with the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters by fitting electromagnetic form factor data. ${\bf Results}$ From this formalism, we define and compute a new quantity --- the Euclidean density function (EDF) --- an object that characterizes the nucleon's various charge distributions as functions of the quark's Euclidean momentum. Applying this technology and incorporating information from BSE analyses, we find the dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero. ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements. The small size we obtain for the impact of the dressed vertex on the axial-singlet charge suggests that models without this effect are on firm ground to neglect it.
Human review
How the literature uses it
6 arXiv papers cite this in 9 sentences
Papers citing it · 5 sentences without a confident label
Uses its method or tool 1 paper
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For the case of bare tensor vertex $Γ^μν(k;0) = σ^μν$ following the calculation method in [ref] (the crucial step is that the authors consider the Gegenbauer polynomial expansion of diquark propagator and then the angular part $k·p$ pieces can be integrated out analytically by the orthogonality.
Dressed Quark Tensor Vertex and Nucleon Tensor Charge · Dress Effects on The Tensor Charge · 1.00
Uses its data or values 1 paper
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The parameters adapted from [ref] are showed in Table.[ref]
Dressed Quark Tensor Vertex and Nucleon Tensor Charge · Dress Effects on The Tensor Charge · 0.99
Cites it as background 2 papers
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Historically, there have been diverse attempts to simulate these form factors in QCD-inspired models [ref] , compute them using lattice QCD or other theoretical methods (see, e.g., [ref] ), or to extract them phenomenologically based on empirical data, analogously to the quark-gluon PDFs themselves.
The photon PDF within the CT18 global analysis · Contributions to the photon PDF and its uncertainty · 1.00
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Other recent approaches to the nucleon emphasize diquark degrees of freedom (see e.g. [ref] ).
Proton quark distributions from a light-front Faddeev-Bethe-Salpeter approach · Introduction · 0.97
What it cites · 16 sentences without a confident label
Uses the method or tool of 4 papers
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We derive the EDFs of Eqs. ([ref]) and ([ref]) by closely following the analogous calculation for the hadronic vacuum polarization effect in the muon's anomalous magnetic moment [ref] ; viz., we now evaluate Eq. ([ref]) for $p ^'= p$ in Sec. [ref] and Eq. ([ref]) in Sec. [ref] using a hyperspherical formalism originally adapted to QED [ref] .
Lattice calculation of the lowest order hadronic contribution to the muon anomalous magnetic moment · Euclidean density function · 0.91
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We proceed using the Minkowski $↔$ Euclidean transcription dictionary as outlined in, Refs. [ref] , wherein $4$-momenta and Dirac matrices transform according to
Dyson-Schwinger Equations and the Application to Hadronic Physics · Euclidean space conventions · 0.92
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We proceed using the Minkowski $↔$ Euclidean transcription dictionary as outlined in, Refs. [ref] , wherein $4$-momenta and Dirac matrices transform according to
Explanation and Prediction of Observables using Continuum Strong QCD · Euclidean space conventions · 0.92
Uses data or values from 4 papers
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We therefore make the identification $f_g(k) ≡F_R(k;0)$ mentioned in Sec. [ref], and directly insert the numerical results reported in Ref. [ref] to smear the bare model axial charge as in Eq. ([ref]).
Dressed-quarks and the nucleon's axial charge · Gluon Dressing Effect · 0.99
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For this purpose, we treat the phenomenological parametrization of Kelly [ref] as a proxy for the world's experimental data and global fits thereof [ref] , rather than preferencing individual sets; we may then minimize the numerical badness-of-fit measure
Nucleon Electromagnetic Form Factors · Electromagnetic form factors · 0.91
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For this purpose, we treat the phenomenological parametrization of Kelly [ref] as a proxy for the world's experimental data and global fits thereof [ref] , rather than preferencing individual sets; we may then minimize the numerical badness-of-fit measure
Nucleon electromagnetic form factors · Electromagnetic form factors · 0.91
Compares with 1 paper
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Incidentally, this figure is in accord with the moment of the scalar diquark contribution to the quark helicity PDF obtained in a typical light-front quark model (see Eqs. (61) and (62) of Ref. [ref] ):
Nucleon form factors and spin content in a quark-diquark model with a pion cloud · Quark helicity · 0.99
Cites as background 31 papers
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In the context of BSE analyses [ref] , the dressed axial-vector vertex is represented by the structure $Γ^fg_5μ(K;P)$, which is understood to connect an incoming quark of flavor $g$ and momentum $K_- = K - (1-η)P$ to an outgoing quark of flavor $f$ and momentum $K_+ = K + ηP$; here $P$ and $K$ represent the total and relative momentum of the quark pair, and $η$ is a dimensionless parameter upon which calculations cannot depend.
Confinement Phenomenology in the Bethe-Salpeter Equation · Gluon Dressing Effect · 0.93
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In the context of BSE analyses [ref] , the dressed axial-vector vertex is represented by the structure $Γ^fg_5μ(K;P)$, which is understood to connect an incoming quark of flavor $g$ and momentum $K_- = K - (1-η)P$ to an outgoing quark of flavor $f$ and momentum $K_+ = K + ηP$; here $P$ and $K$ represent the total and relative momentum of the quark pair, and $η$ is a dimensionless parameter upon which calculations cannot depend.
Dressed-quarks and the nucleon's axial charge · Gluon Dressing Effect · 0.93
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In the context of BSE analyses [ref] , the dressed axial-vector vertex is represented by the structure $Γ^fg_5μ(K;P)$, which is understood to connect an incoming quark of flavor $g$ and momentum $K_- = K - (1-η)P$ to an outgoing quark of flavor $f$ and momentum $K_+ = K + ηP$; here $P$ and $K$ represent the total and relative momentum of the quark pair, and $η$ is a dimensionless parameter upon which calculations cannot depend.
Flavour symmetry breaking and meson masses · Gluon Dressing Effect · 0.93
Discussion
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