Euclidean bridge to the relativistic constituent quark model

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).

Purpose: Seeking to bridge these complementary world views, 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.

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.

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 quenched dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero, while use of recent determinations of unquenched BSEs results in a large suppression.

Conclusions: The quark $+$ scalar diquark ECQM is a step toward a realistic quark model in Euclidean space, and needs additional refinements. The substantial effect we obtain for the impact on the axial-singlet charge of the unquenched dressed vertex compared to the quenched demands further investigation.

Figure 1

(a) The triangle diagram responsible for the nucleon's electromagnetic current $J_{\mu }$ and first nontrivial contribution to $F_{1,2}\left(\tilde{q}{}^2\right)$. (b) The main graph for the quark contributions to the nucleon's axial-singlet charge, $a_0$. In both cases, solid internal lines represent the propagation of the interacting quark, while the dashed lines are for the scalar spectator diquark. The ovate blobs symbolize our prescription for the momentum dependence of the nucleon-quark-diquark interaction as given by $\phi \left({\tilde{k}}^2\right)$ in Eq. (3).

Figure 2

(a) A plot of the fitted electromagnetic form factors $G_{E,M}\left(\tilde{q}{}^2\right)$, where we constrain fits with the phenomenological parametrization of Kelly [62] for $\tilde{q}\le 1$ GeV. Here, solid lines give the result of our fitted model for the parameters listed in Table 1, while the dashed lines are the phenomenological fits of Ref. [62], with $G_E$ given in black and $G_M$ in red in both cases. (b) A similar comparison, but in this case for the form factor ratios with respect to the well-known dipole parametrization [62] $G_D\left(\tilde{q}{}^2\right)\equiv {(1+\tilde{q}{}^2/{\mathrm{\Lambda }}_D^2)}^{-2}$, where ${\mathrm{\Lambda }}_D^2=0.71{\mathrm{GeV}}^2$.

Figure 3

A comparison of EDFs for the proton's charge $2\tilde{k}{\overline{f}}_1\left({\tilde{k}}^2\right)$ [Eq. (35), black solid] and axial-singlet charge $2\tilde{k}{\overline{a}}_0\left({\tilde{k}}^2\right)$ [Eq. (42), maroon dashed] carried by the struck quark in the scalar diquark ECQM as functions of its Euclidean momentum $\tilde{k}$; for illustration, we contrast these with the result of using the Gaussian wave function, $2\tilde{k}{\overline{f}}_1^{\mathrm{WF}}\left({\tilde{k}}^2\right)$ from Eq. (23) (red dotted). The thin lines and associated shaded regions at bottom correspond to the integrands of these distributions' first moments in ${\tilde{k}}^2$, i.e., $M^1\sim \langle {\tilde{k}}^2\rangle$ of Eqs. (20) and (21). Note that these latter moments have been rescaled by a factor of 2 for comparison.

Figure 4

(a) The diagram leading to the DSE for a quark of momentum $\tilde{k}$ dressed by a nonperturbative gluon carrying loop momentum $\tilde{L}$. (b) The corresponding diagram for the quark axial-vector vertex BSE, responsible for the $\tilde{k}$-dependent gluonic dressing correction to the axial charge of an individual quark. In the flavor-singlet channel, additional unquenching loop diagrams also contribute, as described in Ref. [55].

Figure 5

(a) The gluon dressing function $f_g\left(\tilde{k}{}^2\right)$ under several different scenarios: the perturbative limit, $f_g\left(\tilde{k}{}^2\right)=1$ (solid black); using an improved dynamical chiral symmetry-breaking kernel in the quenched BSE, $f_g^{\mathrm{DB}}\left(\tilde{k}{}^2\right)$ (red dashed); with the quenched rainbow-ladder truncation method, $f_g^{\mathrm{RL}}\left(\tilde{k}{}^2\right)$ (blue dotted); and using the result of a calculation [55] that included the effects of unquenching which contribute to the axial-singlet charge (green short dashed). (b) A plot of the integrand of Eq. (24) $2\tilde{k}{\overline{a}}_0\left(\tilde{k}{}^2\right)f_g\left(\tilde{k}{}^2\right)$, taking for $f_g\left(\tilde{k}{}^2\right)$ the gluon dressing functions shown in panel (a), and retaining the aforementioned line styles.