Singly heavy baryon magnetic moments from relativistic diquark dynamics

Abstract

We use a relativistic quark–diquark framework, with parameters fixed from meson and proton data, to calculate the magnetic moments of ground-state singly heavy baryons with spin J=\frac{1}{2} and J=\frac{3}{2} in both the charm and bottom sectors. The underlying two-body wave equation is solved self-consistently. The results reveal a systematic dynamical hierarchy: the heavy–light diquark configuration is more spatially extended than the light–light configuration, by a factor of 3–4 in charm baryons and by roughly an order of magnitude in bottom baryons. Because the heavy–light configuration carries the dominant weight in the magnetic-moment average, the predictions are largely controlled by this extended configuration, in which the light-quark dynamical mass remains close to its conventional constituent value; the tightly bound light–light configuration supplies only a small correction.

Graphical abstract

Keywords

magnetic moment / baryon

1 Introduction

The magnetic moments of baryons containing a single heavy quark — {\mathrm{\Sigma }}_c,{\mathrm{\Lambda }}_c,{\mathrm{\Xi }}_c,{\mathrm{\Omega }}_c in the charm sector and their bottom counterparts — provide a particularly clean window onto the interplay between heavy-quark dynamics and the strong-coupling regime that governs the light degrees of freedom. While the masses of these states are now well measured [1], experimental access to their magnetic moments remains extremely limited: at present only \mu ({\mathrm{\Lambda }}_c^{+}) has been determined directly, and even that with sizable uncertainty. Theoretical predictions are therefore essential, both as targets for future precision measurements at LHCb, Belle II, and proposed facilities [2–4], and as testing grounds for hadronic models in a regime where the QCD scale and the heavy-quark mass are nearly equal in magnitude.

Existing predictions span a wide methodological range — non-relativistic constituent quark models [5], lattice QCD [6, 7], light-cone QCD sum rules [8–11], heavy-baryon chiral perturbation theory [12, 13], covariant baryon \chiPT [14], the bag model [15], and the relativistic three-quark model [16] — but a substantial spread of \mathcal{O}(0.5{\mu }_N) persists across the literature for several states, particularly in the bottom sector. This spread reflects genuine theoretical ambiguity: each method makes different choices about how to treat the relativistic motion of the light quarks, the dynamical generation of constituent masses, and the matching to the heavy-quark limit.

In this work we revisit and substantially extend the relativistic quark–diquark framework introduced in Refs. [17, 18]. This formalism treats a baryon as a bound state of a constituent quark and a colored scalar diquark, both moving relativistically in a confining potential whose form is fixed by asymptotic-freedom-improved one-gluon exchange. The same wave equation describes mesons (taken as q\overline{q} bound states) when the appropriate color factor is applied. The framework has two attractive features for the present problem. First, the model parameters — the QCD scale \mathrm{\Lambda }, the scalar offset S_0, the constituent quark masses, and the magnetic-moment scale b — are fixed completely by meson and light-baryon data, leaving no free parameters in the magnetic-moment predictions for the singly heavy sector; the hyperfine couplings introduced for the mass analysis in Sec. IIC are calibrated separately from measured splittings and do not enter the magnetic moments. Second, the formula for baryon magnetic moments takes the closed form {\mu }_i=Q_i/[2(m_i+b/a_{ijk})], where a_{ijk} is the dynamically computed spatial size; the dependence on internal dynamics is therefore concentrated in a single quantity per baryon configuration.

The original analysis [17, 18] treated only light baryons up to spin-3/2, and was limited by the meson data and computational tools available at the time. In the present work we refit all model parameters to modern meson masses [1], finding values within 0.3\mathrm{\%} of the original fits and confirming the robustness of the framework, and refit the magnetic-moment parameter b to the modern proton moment with the self-consistently computed proton size. We extend the spin-flavor analysis to the full sextet, antitriplet, and decuplet of singly heavy baryons, and introduce a hyperfine contact term calibrated directly against the measured {\mathrm{\Sigma }}_c^{\ast }–{\mathrm{\Sigma }}_c, {\mathrm{\Omega }}_c^{\ast }–{\mathrm{\Omega }}_c, and {\mathrm{\Sigma }}_b^{\ast }–{\mathrm{\Sigma }}_b splittings [1] to extract spin-1/2 masses from the spin-independent wave equation. On this basis we present, for the first time within this framework, magnetic-moment predictions for all singly heavy baryons at both J=1/2 and J=3/2, providing what we believe to be the most internally consistent set of predictions to date from a dynamical quark–diquark approach.

The framework reliability is established by reproducing the published light-baryon magnetic-moment predictions [18] to within 0.005{\mu }_N for all eight ground-state octet members (with the \mathrm{\Lambda }–{\mathrm{\Sigma }}^0 mixing of Ref. [19] included), and by reproducing the published light spin-3/2 baryon masses (\mathrm{\Delta },{\mathrm{\Sigma }}^{\ast },{\mathrm{\Xi }}^{\ast },{\mathrm{\Omega }}^{-}) to better than 1 MeV. The known residual discrepancies of the framework — most notably for {\mathrm{\Sigma }}^{-},{\mathrm{\Xi }}^{-},{\mathrm{\Omega }}^{-}, where pion-cloud and SU(3)-breaking effects play a dominant role — identify the physics that lies outside the constituent framework. For singly heavy baryons two distinct statements apply, which must be kept separate. The contribution of the heavy quark itself to the magnetic moment, and the corrections to it, are suppressed by powers of 1/m_Q in the heavy-quark limit [20, 21]. Chiral-loop corrections attached to the light valence quarks, by contrast, are not parametrically suppressed by the heavy-quark mass; they are governed by the chiral expansion, as formulated in heavy-hadron chiral perturbation theory [20, 21]. The framework is nonetheless expected to perform well in the heavy sector: in the antitriplet states the light pair is coupled to spin zero, so the leading light-quark contributions cancel and the moment is dominated by the heavy quark, while for the sextet states the light-baryon validation calibrates the typical size of the chiral physics missing from the model. A quantitative uncertainty estimate based on this calibration is given in Section 3.3.

The rest of the paper is organized as follows. In Section 2 we present the methodology employed for computing the magnetic dipole moments of both spin-1/2 and spin-3/2 singly heavy baryons, together with the analytical results obtained within this framework. In Section 3, numerical analyses of these expressions are performed and compared with the existing results in the literature. The manuscript ends with a summary in Section 4.

2 Formalism

2.1 Relativistic two-body wave equation

We adopt the relativistic two-body equation of Ref. [17], which for a particle pair of masses m_1,m_2 interacting through a Lorentz-vector potential V(r) and a Lorentz-scalar potential S(r) takes the form (with \hslash =c=1)

(1)\left[-{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}+V_{\mathrm{e}\mathrm{f}\mathrm{f}}(r;W)\right]u(r)={\displaystyle \frac{W^2}{4}}u(r),

with reduced effective potential

(2)\begin{array}{rl}{V}_{\mathrm{e}\mathrm{f}\mathrm{f}}(r;W)=& {\displaystyle \frac{\ell (\ell +1)}{r^2}}+{\displaystyle \frac{E_1^2+E_2^2}{2}}-{\displaystyle \frac{(E_1^2-E_2^2{)}^2}{4(W-V{)}^2}}\\ & -{\displaystyle \frac{V^2}{4}}+{\displaystyle \frac{WV}{2}},\end{array}

where E_i(r)=m_i+S(r)/2 and W is the bound-state mass to be determined self-consistently. Equation (1) is the spin-averaged limit of the full Bethe–Salpeter equation in the instantaneous approximation; it describes a particle pair bound by a Lorentz-scalar confining force together with a Lorentz-vector short-distance interaction.

We emphasize in what sense Eq. (1) is relativistic, since its second-order form is superficially Schrödinger-like. The equation is obtained from the exact relativistic dispersion relation of two free particles, W=\sqrt{{\mathit{p}}^2+m_1^2}+\sqrt{{\mathit{p}}^2+m_2^2}, by incorporating the interactions through the replacements W\to W-V and m_i\to m_i+S/2, dictated by their Lorentz transformation properties, and squaring twice [17]. No expansion in p/m is performed at any step: the kinematics is exactly relativistic, which is what permits current-mass light quarks (m_{u,d}=5 MeV) to be treated on the same footing as heavy quarks. The relativistic structure is visible in Eq. (2) through the eigenvalue W^2/4, the potential-dependent denominator (W-V{)}^2, and the squared effective energies E_i^2=(m_i+S/2{)}^2; in the limit of large masses and weak potentials the equation reduces to the ordinary two-body Schrödinger equation with potential U=V+S. What is not included is spin: Eq. (1) is spin-averaged, and spin-dependent effects enter only through the hyperfine treatment of Section 2.3 and the spin-flavor factors of the magnetic-moment formulas.

The interaction is fixed by combining the leading-log running of the QCD coupling with a confining Lorentz-scalar piece [17]:

(3)V(r)=-F{\displaystyle \frac{6\pi }{27}}{\displaystyle \frac{1}{r}}{\displaystyle \frac{1-\mathrm{\Lambda }r}{\ln (\mathrm{\Lambda }r)}},

(4)S(r)=F{\displaystyle \frac{6\pi }{27}}\mathrm{\Lambda }{\displaystyle \frac{1-\mathrm{\Lambda }r}{\ln (\mathrm{\Lambda }r)}}-F{S}_0,

where F=⟨{\mathit{F}}_1\cdot {\mathit{F}}_2⟩ is the color factor for the pair (with F=-4/3 for color-singlet q\overline{q} and quark–diquark configurations, and F=-2/3 for an antitriplet diquark), and \mathrm{\Lambda }, S_0 are the only two free parameters of the potential. Equations (3) and (4) are precisely Eqs. (5) and (6) of Ref. [17]: the Lorentz-vector piece carries the linear factor (1-\mathrm{\Lambda }r) in its numerator, while the Lorentz-scalar piece carries the same linear factor with opposite sign convention via \mathrm{\Lambda }(\mathrm{\Lambda }r-1)=-\mathrm{\Lambda }(1-\mathrm{\Lambda }r). The combined potential U=V+S, given explicitly as Eq. (4) of Ref. [17], then carries (1-\mathrm{\Lambda }r{)}^2 as a simple algebraic consequence of the product of the two pieces. In the present work we use the Lorentz-vector and Lorentz-scalar splittings of Eqs. (3) and (4), as required by the relativistic two-body equation (1), which treats V and S independently through their distinct transformation properties. The combination (1-\mathrm{\Lambda }r)/\ln (\mathrm{\Lambda }r) is the asymptotic-freedom-improved short-distance form, finite at the Landau pole r={\mathrm{\Lambda }}^{-1}, and smoothly interpolates between the asymptotic logarithmic running at short distances and the confining linear regime at large distances.

A baryon |q_1{q}_2{q}_3⟩ is treated in the quark–diquark approximation as |q_i[q_j{q}_k]⟩, where [q_j{q}_k] is a color-antitriplet diquark. We solve Eq. (1) in two stages: first for the diquark mass m_D=W^{(1)} using F=-2/3, and then for the baryon mass M_B=W^{(2)} using the spectator quark q_i paired with the precomputed diquark in the color-singlet channel F=-4/3. The eigenvalue problem is iterated to self-consistency at each stage.

Although the relativistic quark–diquark framework predates the systematic development of heavy-quark effective theory, the approximation it embodies is precisely the one favoured by heavy-quark spin–flavor symmetry in the singly heavy sector. In the heavy-quark limit, the spin of the heavy quark Q decouples from the light degrees of freedom, and the light diquark [q_1{q}_2] becomes the natural carrier of the conserved light-quark quantum numbers. The antitriplet and sextet of singly heavy baryons are then distinguished by the light-diquark spin: an antisymmetric flavor wave function combined with a spin-zero (“good”) light diquark yields the SU(3) antitriplet ({\mathrm{\Lambda }}_Q,{\mathrm{\Xi }}_Q), while the symmetric flavor wave function combined with a spin-one (“bad”) light diquark yields the sextet ({\mathrm{\Sigma }}_Q,{\mathrm{\Xi }}_Q^{\prime },{\mathrm{\Omega }}_Q for J=\frac{1}{2}, and {\mathrm{\Sigma }}_Q^{\ast },{\mathrm{\Xi }}_Q^{\ast },{\mathrm{\Omega }}_Q^{\ast } for J=\frac{3}{2}) [22, 23]. This SU(3) classification, which we adopt throughout, is therefore a direct consequence of the heavy-quark approximation rather than an additional assumption. The spin-flavor wave functions used to compute magnetic moments are the conventional constituent-quark expressions consistent with this classification.

For each baryon configuration we extract the size parameter

(5)a_{ijk}={\displaystyle \frac{m_D}{m_i+m_D}}\sqrt{⟨r^2⟩},

where ⟨r^2⟩ is computed from the second-stage radial wave function u(r). The three indices identify the configuration: i labels the spectator quark and the pair jk labels the two quarks bound into the diquark, so that a_{ijk} is the size parameter of the configuration |q_i[q_j{q}_k]⟩; in tables we write this equivalently as a_{[q_j{q}_k]q_i}. The kinematic prefactor m_D/(m_i+m_D) accounts for the position of the i-th quark relative to the baryon center of mass [18]. We emphasize that the size parameter a_{ijk} is not the physical baryon radius; rather, it is the effective spatial size that enters Eq. (6) for the contribution of quark q_i in a given diquark grouping |q_i[q_j{q}_k]⟩. The quark–diquark picture is an approximation in which two of the three quarks are assumed to cluster into an effectively point-like color-antitriplet object whose interaction with the spectator quark is described by Eq. (1); the a_{ijk} are the corresponding configuration-specific size parameters that enter the magnetic-moment formula. Since each baryon admits multiple inequivalent diquark groupings, the physical moment is obtained as a configuration average over these inequivalent assignments, with the relative weights fixed by the flavor symmetries of the three-quark wave function.

The reliability of the point-diquark approximation deserves comment, particularly for heavy–light diquarks. Two distinct length scales are involved: the internal size of the diquark, determined by the first-stage solution, and the quark–diquark separation entering Eq. (5), determined by the second stage. For light–light pairs the internal size is small and the point-like treatment is well justified. For heavy–light [qQ] pairs the internal size is set by the light-quark Compton scale and is comparable to the quark–diquark separation itself, so the point-like treatment operates at the edge of its formal domain of validity. Two features mitigate this. First, the physical moment is a configuration average over all inequivalent groupings, which partially restores the symmetry of the underlying three-quark state; already in the original analysis it was observed that different diquark groupings yield nearly identical baryon masses (agreement within 1 MeV for light baryons and better than 1\mathrm{\%} for charm baryons) [17], indicating that observables are only weakly sensitive to the choice of clustering. Second, deviations from the point-like limit modify primarily the short-distance part of the quark–diquark interaction, whereas the moment formula depends on the global size a_{ijk}, which is dominated by the long-distance confining dynamics. An order-of-magnitude estimate of the residual clustering error is obtained in Section 3.3 by varying the configuration weights; it reaches \sim 0.2{\mu }_N for sextet states dominated by light-quark moments and stays below \sim 0.03{\mu }_N for the heavy-quark-dominated antitriplet states. Because the same clustering approximation is applied consistently to every state, relative patterns — charm versus bottom, spin-1/2 versus spin-3/2, and moment differences within multiplets — are more robust than the absolute scale of any individual prediction.

2.2 Magnetic-moment formula

Following Ref. [18], we write the dynamical quark magnetic moment in the form

(6){\mu }_i(a)={\displaystyle \frac{Q_i}{2(m_i+b/a)}},

where Q_i is the quark charge in units of |e|, m_i is the (small) current quark mass, a is the size parameter for the configuration, and b is a single phenomenological parameter with dimension of mass\timeslength. The combination m_i+b/a plays the role of an effective dynamical mass: the quark in a small-size configuration acquires a larger effective mass and correspondingly smaller moment, automatically encoding the size-dependence of relativistic binding effects. Equation (6) is not derived from first principles, and its motivation is as follows. A Dirac particle of effective mass m_{\mathrm{e}\mathrm{f}\mathrm{f}} carries the moment Q/(2m_{\mathrm{e}\mathrm{f}\mathrm{f}}), and within the relativistic framework of the preceding subsection the binding generates an effective quark mass m_{\mathrm{e}\mathrm{f}\mathrm{f}}=m_i+\frac{1}{2}⟨S⟩, with the expectation value of the confining Lorentz-scalar potential supplying the constituent-scale contribution [17]. The combination b/a in Eq. (6) is a one-parameter phenomenological representation of \frac{1}{2}⟨S⟩: dimensional analysis fixes the 1/a scaling with configuration size, and the single constant b is calibrated once to the measured proton moment. The predictions of this work therefore inherit a model uncertainty associated with this representation; it is quantified in Section 3.3 by combining the light-baryon residuals with the parametric and configuration sensitivities of Eq. (6).

The full baryon moment is obtained by combining Eq. (6) with the standard spin-flavor wave functions of the constituent quark model. Explicit expressions for every singly heavy baryon are collected in Appendix A.

2.3 Hyperfine splittings and baryon mass spectrum

The wave equation (1) is spin-independent: the same eigenvalue W describes both spin partners of a sextet. Following the constituent-quark literature [24], we add a hyperfine contact interaction

(7)

whose matrix elements between fully aligned (J=\frac{3}{2}) and mixed (J=\frac{1}{2}) spin configurations are evaluated in the standard way using the constituent-quark spin wave functions.

The hyperfine couplings {\kappa }_{ij} are calibrated against measured baryon mass splittings. In the light sector, {\kappa }_{qq}=\frac{2}{3}[M(\mathrm{\Delta })-M(N)]=195.3 MeV and {\kappa }_{qs}=M({\mathrm{\Sigma }}^{\ast })-M(\mathrm{\Sigma })=192 MeV. The heavy-quark couplings are obtained directly from the corresponding charm and bottom splittings,

\begin{array}{rl}{\kappa }_{uc}& =\frac{2}{3}[M({\mathrm{\Sigma }}_c^{\ast })-M({\mathrm{\Sigma }}_c)]=43.3\text{MeV},\\ {\kappa }_{ub}& =\frac{2}{3}[M({\mathrm{\Sigma }}_b^{\ast })-M({\mathrm{\Sigma }}_b)]=14.0\text{MeV},\\ {\kappa }_{sc}& =\frac{2}{3}[M({\mathrm{\Omega }}_c^{\ast })-M({\mathrm{\Omega }}_c)]=47.3\text{MeV},\end{array}

while {\kappa }_{ss} and {\kappa }_{sb} are obtained by constituent-mass scaling. We note that in the present framework the hyperfine couplings {\kappa }_{ij} are treated as empirical phenomenological inputs calibrated directly from measured hadron mass splittings, rather than derived from a microscopic chromomagnetic interaction. This empirical calibration sidesteps the conceptual tension between the current quark masses entering the wave equation (1) and the constituent quark masses that would otherwise appear in the chromomagnetic coupling: each {\kappa }_{ij} absorbs whatever combination of effective masses, gluon-exchange dynamics, and short-distance corrections produces the observed splitting, in the same spirit as the empirical determination of {\kappa }_{qq} from M(\mathrm{\Delta })-M(N) in the original constituent-quark literature [24]. We emphasize that the hyperfine couplings {\kappa }_{ij} do not affect the magnetic moments, either directly or indirectly: they enter only the extraction of the mass spectrum from the spin-averaged eigenvalue W, and they do not feed back into the wave functions u(r) or the size parameters a_{ijk}, which are obtained from the spin-independent wave equation (1) before any hyperfine shift is applied. The magnetic moments of this work would be unchanged if the hyperfine analysis were omitted entirely.

2.4 Parameters

The model has six parameters fixed entirely by meson and proton data: \mathrm{\Lambda },S_0,m_s,m_c,m_b (with m_u=m_d=5 MeV held fixed), and b. We adopt vector-meson anchors throughout because the spin-averaged wave equation (1) describes the spin triplet without further corrections, whereas pseudoscalar states are dominated by chiral dynamics that lie outside the present framework. The parameters are determined sequentially [1]: (\mathrm{\Lambda },m_c) jointly from the charmonium ground and first excited states J/\psi (1S) and \psi (2S); S_0 from the \rho meson; m_s from the \varphi meson; m_b jointly from \mathrm{{\rm Y}}(1S), \mathrm{{\rm Y}}(2S), and \mathrm{{\rm Y}}(3S); and finally b from the proton magnetic moment using the self-consistently computed a_{uuu}. The resulting values are

(8)\begin{array}{rl}& \mathrm{\Lambda }=0.7336\text{GeV},S_0=-0.1083\text{GeV},\\ & m_s=0.1775\text{GeV},m_c=1.4055\text{GeV},\\ & m_b=4.7777\text{GeV},b=0.1846\text{GeV}\cdot \text{fm}.\end{array}

We comment on the role of the very small light-quark masses adopted here. The values m_u=m_d=5 MeV are current-quark masses at the hadronic scale, not constituent masses; the constituent-scale quark mass relevant to magnetic moments is generated dynamically within the framework through the combination m_q+b/a entering Eq. (6). For a typical light-spectator configuration with a\sim 0.5 fm, b/a\approx 0.37 GeV produces a u-quark dynamical mass close to the conventional constituent value of \sim 0.33 GeV, whereas for a tightly clustered light-diquark configuration with a\sim 0.13 fm one finds b/a\approx 1.4 GeV, several times the conventional value. The dynamical mass is therefore configuration-dependent, and the predictions of Eq. (6) are essentially insensitive to the small bare values of m_u,m_d: a change of a few MeV in the current-quark input shifts every quark moment by less than 1\mathrm{\%}, well below the precision of the predictions to follow. An exact isospin limit is enforced by the choice m_u=m_d, which makes isospin-related states such as {\mathrm{\Xi }}_c^{+} and {\mathrm{\Xi }}_c^0 exactly degenerate in our antitriplet predictions; the near-equality with \mu ({\mathrm{\Lambda }}_c^{+}), which is not an exact symmetry statement, is discussed in Section 3.4. The \mathrm{\Lambda }–{\mathrm{\Sigma }}^0 mixing angle relevant for the light-baryon validation is \theta =0.011 rad, taken from Ref. [19]. This value, \theta ={0.63}^{\circ }, is consistent with the range (0.55–1.2{)}^{\circ } spanned by more recent determinations, including lattice QCD [25]. Varying \theta across this range shifts \mu (\mathrm{\Lambda }) by less than 0.01{\mu }_N and leaves all other light-baryon predictions unchanged; the heavy-baryon predictions are unaffected, since the analogous {\mathrm{\Lambda }}_Q–{\mathrm{\Sigma }}_Q^0 mixing is further suppressed by the heavy-quark mass.

3 Results and discussion

We organize the discussion in four parts. We first present the predictions of the formalism in the light-baryon sector, both as a validation of the framework and as a calibration of its inherent accuracy. We then list the configuration sizes a_{ijk} obtained from the wave equation, which constitute the dynamical input to all subsequent moment predictions. These sizes are then used to present the magnetic moments of the singly heavy baryons, discussed separately within the charm and bottom sectors with particular attention to the diquark clustering interpretation. Finally, we compare the predictions in detail with the existing literature.

3.1 Light-baryon magnetic moments

The reliability of the framework is established by the set of light-baryon magnetic moments collected in Table 1. The eight ground-state octet members together with the {\mathrm{\Omega }}^{-} are obtained directly from the wave equation (1) combined with Eq. (6), with the \mathrm{\Lambda }–{\mathrm{\Sigma }}^0 mixing of Ref. [19] included. The framework reproduces p, n, \mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}, and {\mathrm{\Xi }}^0 to better than 0.05{\mu }_N from experiment, an accuracy comparable to that achieved by SU(6) symmetry plus a single-parameter fit to the proton moment.

The pattern of deviations from experiment carries physical information directly relevant to the heavy-baryon program of this paper. Two distinct populations of states emerge from Table 1. The proton, neutron, \mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}, and {\mathrm{\Xi }}^0 are reproduced to within 0.05{\mu }_N. The {\mathrm{\Sigma }}^{-}, {\mathrm{\Xi }}^{-}, and {\mathrm{\Omega }}^{-} deviate by 0.20, 0.19, and 0.48{\mu }_N respectively, an order of magnitude larger than the first group. These are precisely the states for which the meson-cloud contribution is known to be important: in heavy-baryon and covariant baryon chiral perturbation theory [12, 14], kaon and pion loops generate moment shifts that scale as \sim {\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}/M_B with a flavor-dependent prefactor that is largest for negative-charge strange baryons. The pattern of deficits in Table 1, with |\mathrm{\Delta }\mu ({\mathrm{\Omega }}^{-})|>|\mathrm{\Delta }\mu ({\mathrm{\Sigma }}^{-})|\sim|\mathrm{\Delta }\mu ({\mathrm{\Xi }}^{-})|, tracks this expectation closely. The constituent-quark framework, lacking explicit chiral degrees of freedom, cannot capture these contributions; this is a structural feature of the approach rather than a parameter deficiency.

This pattern motivates the extension to singly heavy baryons, with an important qualification. The heavy quark does not couple to the Goldstone bosons at leading order in the heavy-quark expansion, so its contribution to the magnetic moment, and the chiral corrections to that contribution, are suppressed by powers of 1/m_Q [20, 21]. The chiral-loop contributions attached to the light valence quarks, by contrast, are not parametrically suppressed by the heavy-quark mass: in heavy-hadron chiral perturbation theory [20, 21] the Goldstone bosons couple to the light degrees of freedom of the heavy hadron, and the corresponding loop corrections are controlled by the chiral expansion rather than by 1/m_Q. Two structural features nonetheless reduce their impact relative to the light hyperons. In the antitriplet states the light pair is coupled to spin zero, so the leading light-quark contributions, and with them the associated chiral corrections, largely cancel, leaving a moment dominated by the heavy quark. In the sextet states the light-quark sector dominates the moment, and its modeling accuracy is calibrated directly by the light-baryon results above. We therefore adopt the \sim 0.05{\mu }_N accuracy demonstrated for p, n, \mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}, {\mathrm{\Xi }}^0 as a lower bound on the model uncertainty of the heavy-baryon predictions, with the understanding that sextet states with large light-quark moments may carry chiral corrections comparable to those of their strange analogues; a quantitative uncertainty budget is given in Section 3.3.

3.2 Configuration size parameters

The dynamical input to all magnetic-moment predictions is encoded in the configuration size parameters a_{ijk} defined in Eq. (5). These are not baryon radii. Rather, for each baryon and each of its inequivalent diquark groupings |q_i[q_j{q}_k]⟩, the quantity a_{ijk} is the characteristic size parameter for the spectator quark q_i relative to the baryon center of mass: it combines the self-consistently computed quark–diquark separation \sqrt{⟨r^2⟩} with the kinematic prefactor m_D/(m_i+m_D) that locates quark i in the centre-of-mass frame. This is the quantity that enters the magnetic-moment formula Eq. (6) for the i-th quark contribution; a small a_{ijk} corresponds to a heavily suppressed quark moment, while a larger a_{ijk} allows the quark moment to approach its unsuppressed value. Table 2 collects the values for the singly heavy baryons.

The pattern visible in Table 2 has two distinct contributions, both physically meaningful. The first is purely kinematic: when the heavy quark is the spectator, the prefactor m_D/(m_i+m_D) pulls a_{ijk} toward small values because m_i=m_Q is large; when the light quark is the spectator, the prefactor is close to unity and a_{ijk} is comparable to the underlying quark–diquark separation. This kinematic effect is significantly more pronounced in bottom baryons than in charm baryons because m_b/m_c\approx 3.4. The second contribution is genuinely dynamical: the underlying quark–diquark separation is itself smaller when the diquark is light–light [qq] than when it is heavy–light [qQ], because the light–light diquark is a tightly bound, compact object whereas the heavy–light diquark is a loosely bound system whose spatial extent is set by the light-quark Compton scale. Both effects combine to produce the observed pattern: the heavy-spectator configurations cluster near 0.13–0.16 fm in charm and 0.04–0.05 fm in bottom, while the light-spectator configurations cluster near 0.42–0.51 fm in both sectors.

The phenomenological consequence enters through the dynamical mass m_q+b/a in Eq. (6). Heavy-spectator configurations give b/a\gtrsim 1 GeV in charm and several GeV in bottom, dwarfing the current quark masses and severely suppressing the moment of the spectator heavy quark in those configurations. Light-spectator configurations give b/a\approx 0.36–0.44 GeV, comparable to a conventional constituent mass, so the light spectator quark contributes its full unsuppressed moment. Combined with the spin-flavor combinatorial weight that favours the inequivalent groupings in which the spectator is the light quark, this means that the singly heavy baryon’s magnetic moment is dominated by the dynamics in which a light quark spectates against a heavy–light diquark cluster. This is the physical content of what we will refer to throughout as “dynamical diquark clustering” — not a claim about baryon size, but a claim about which configuration controls the magnetic moment.

3.3 Parameter sensitivity and model uncertainties

All predictions in this work are quoted at three-decimal numerical precision. This reflects the reproducibility of the numerical pipeline, not the physical accuracy of the model, and we quantify here the uncertainty from three sources.

First, parametric sensitivity. Through Eq. (6), a fractional shift in b/a changes a quark moment by \delta \mu /\mu =-\delta (b/a)/(m_i+b/a). For light quarks in the dominant heavy–light configurations (b/a\approx 0.36 GeV \gg m_{u,d}) a 1\mathrm{\%} shift in b or in any a_{ijk} therefore moves the moment by \approx 1\mathrm{\%}, while for charm and bottom quarks the same shift moves the moment by only 0.2\mathrm{\%} and 0.08\mathrm{\%}, respectively. The model parameters are fixed by meson masses and the proton moment, which are known far better than 1\mathrm{\%}; even the residuals of the meson fit itself (for example, \sim 16 MeV across the \mathrm{{\rm Y}}(1S,2S,3S) anchors, i.e., 0.3\mathrm{\%} in m_b) propagate to below 0.02{\mu }_N for the largest moments. Input-parameter uncertainty is therefore negligible.

Second, clustering systematics. The quark–diquark decomposition and the combinatorial configuration weights are model assumptions (Sections 2 and 3.2). A generous estimate of the associated error is obtained by varying the two-configuration weights of the qqQ baryons from (1/3,2/3) to (1/2,1/2): sextet moments dominated by light-quark contributions shift by up to \approx 0.27{\mu }_N [\mu ({\mathrm{\Sigma }}_c^{++}): +1.614\to +1.344], while the heavy-quark-dominated antitriplet moments shift by only \approx 0.02{\mu }_N [\mu ({\mathrm{\Lambda }}_c^{+}): +0.311\to +0.289]. We regard these shifts as upper bounds, since the combinatorial weights are fixed by the flavor symmetry of the three-quark wave function rather than fitted. To attach uncertainties to the individual predictions, we convert these bounds into one-standard-deviation estimates by taking one half of the weight-variation shift of each state (for the three-configuration {\mathrm{\Xi }}_Q-type states, the shift obtained by transferring the same amount of weight, 1/6, between the two most disparate configurations) and combining the result in quadrature with the empirical floor of \pm 0.05{\mu }_N discussed below. The resulting state-by-state uncertainties are quoted alongside the central values in Tables 3 and 4; they range from \pm 0.05{\mu }_N for the antitriplet states to \pm 0.25{\mu }_N for {\mathrm{\Sigma }}_b^{\ast +}.

Third, empirical calibration. The light-baryon validation of Section 3.1 bounds the demonstrated accuracy of the framework at \lesssim 0.05{\mu }_N where it is expected to work (p, n, \mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}, {\mathrm{\Xi }}^0), degrading to 0.2–0.5{\mu }_N for the hyperons where chiral-loop physics beyond the model dominates.

Combining these, a realistic uncertainty assignment is \pm 0.05{\mu }_N for the antitriplet states, which are heavy-quark dominated and insensitive to both the clustering systematics and the chiral corrections, and \pm 0.1–0.3{\mu }_N for sextet states with large light-quark moments, the upper end applying to the spin-3/2 states where the light-quark contributions are doubled. These bands should be borne in mind wherever differences between frameworks of comparable size are discussed in the following subsections. They do not affect the sign predictions or the qualitative ordering patterns, and the bottom-antitriplet near-degeneracy at -0.05{\mu }_N remains sharply distinguishable from the \sim -0.3{\mu }_N values of Ref. [11] even at the conservative end of the assigned uncertainty.

3.4 Magnetic moments of singly charm baryons

The magnetic moments of the singly charm baryons are presented in Table 3. The structure of these predictions is best understood by separating the antitriplet and sextet contributions, since the two multiplets probe different aspects of the diquark dynamics.

For the antitriplet states, the spin-zero good light diquark contributes nothing to the magnetic moment, and the prediction reduces to a configuration average of the charm-quark moment. Two distinct statements must be separated here. The equality \mu ({\mathrm{\Xi }}_c^{+})=\mu ({\mathrm{\Xi }}_c^0) is exact in the m_u=m_d limit adopted throughout: the two charge states differ only by the substitution u\leftrightarrow d, which the heavy-quark moment does not see. The near-equality with \mu ({\mathrm{\Lambda }}_c^{+}) is not exact. The {\mathrm{\Lambda }}_c^{+} moment averages the charm-quark moment over the configurations (a_{[ud]c},a_{[uc]d}) with weights (1/3,2/3), whereas the {\mathrm{\Xi }}_c moment averages over the three configurations (a_{[us]c},a_{[uc]s},a_{[sc]u}) with equal weights, and the strange quark shifts these sizes. Numerically the two averages nearly coincide, \mu ({\mathrm{\Lambda }}_c^{+})=+0.311{\mu }_N versus \mu ({\mathrm{\Xi }}_c^{+})=\mu ({\mathrm{\Xi }}_c^0)=+0.310{\mu }_N, because the strange-induced shifts act in opposite directions in the different configurations (a_{[us]c} is slightly larger than a_{[ud]c}, while a_{[uc]s} is slightly smaller than a_{[uc]d}) and largely cancel in the average. The smallness of the residual is aided by the heavy-quark mass, which damps the sensitivity of the charm moment to the configuration size [\delta {\mu }_c/{\mu }_c=-\delta (b/a)/(m_c+b/a), a few percent for the strange-induced size shifts]; the damping brings the individual configuration differences to the few-percent level, and the compensation between configurations then reduces the net difference to the per-mille level. The tabulated near-degeneracy of the charm antitriplet is therefore an exact isospin statement within the {\mathrm{\Xi }}_c doublet and a per-mille-level dynamical coincidence between {\mathrm{\Lambda }}_c^{+} and {\mathrm{\Xi }}_c. The numerical value is set by the configuration-averaged charm dynamical mass, which at a\sim 0.4 fm produces a charm moment close to Q_c{m}_p/m_c^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}} with the conventional constituent value m_c^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}\sim 1.5 GeV.

The sextet J=1/2 states show a pronounced charge ordering for the {\mathrm{\Sigma }}_c family: \mu ({\mathrm{\Sigma }}_c^{++})=+1.614, \mu ({\mathrm{\Sigma }}_c^{+})=+0.325, and \mu ({\mathrm{\Sigma }}_c^0)=-0.962{\mu }_N. The structure follows from the heavy–light diquark configuration: the u- and d-quark moments evaluated at a_{[uc]u}=0.512 fm carry the dominant configuration weight, and the spin-flavor coefficients of the constituent-quark model assemble these into the observed pattern. For the doubly strange {\mathrm{\Omega }}_c^0 the s-quark substitution at a_{[sc]s}=0.433 fm gives -0.665{\mu }_N, with the smaller magnitude reflecting both the smaller strange-quark charge magnitude and the larger strange-quark mass. The {\mathrm{\Xi }}_c^{\prime } states at +0.414 and -0.800{\mu }_N interpolate between the {\mathrm{\Sigma }}_c and {\mathrm{\Omega }}_c extremes.

In the sextet J=3/2 sector, the aligned spin configuration enhances the heavy–light contribution because all three quark spins now contribute additively to the moment. \mu ({\mathrm{\Sigma }}_c^{\ast ++})=+2.888{\mu }_N exceeds \mu ({\mathrm{\Sigma }}_c^{++}) by nearly a factor of two, while |\mu ({\mathrm{\Sigma }}_c^{\ast 0})|=0.978{\mu }_N stays close to |\mu ({\mathrm{\Sigma }}_c^0)|=0.962{\mu }_N because the negative-charge state is dominated by the d-quark contribution and is thus largely spin-independent. The {\mathrm{\Omega }}_c^{\ast 0} at -0.535{\mu }_N has the smallest magnitude in the spin-3/2 charm sector, in line with the suppressed strange-quark moments. The {\mathrm{\Xi }}_c^{\ast } states at +1.086 and -0.735{\mu }_N again interpolate between the {\mathrm{\Sigma }}_c^{\ast } and {\mathrm{\Omega }}_c^{\ast } extremes.

3.5 Magnetic moments of singly bottom baryons

The bottom-baryon predictions are collected in Table 4. The same structural features observed in the charm sector are present, but with two important differences.

For the antitriplet states, the most striking feature is the flat near-zero structure \mu ({\mathrm{\Lambda }}_b^0)=\mu ({\mathrm{\Xi }}_b^0)=\mu ({\mathrm{\Xi }}_b^{-})=-0.052{\mu }_N. In the diquark-clustering picture this near-degeneracy is enforced by two facts. First, the spin-zero good light diquark contributes nothing to the moment. Second, the bottom-quark dynamical mass is essentially equal to its current value across all configurations because b/a\ll m_b at every relevant size. The result is \mu \approx Q_b{m}_p/m_b\approx -0.05{\mu }_N, suppressed by an additional factor |Q_b|m_c/(|Q_c|m_b)\approx 0.17 relative to the charm antitriplets and essentially independent of the light-quark content. The SU(3)-breaking difference between \mu ({\mathrm{\Lambda }}_b^0) and \mu ({\mathrm{\Xi }}_b^{0,-}) induced by the strange-quark shifts in a_{ijk} is below 0.001{\mu }_N, suppressed relative to the corresponding charm-sector difference by the bottom-quark mass, so the three states are degenerate at the precision quoted. Any deviation from this prediction at experimentally accessible precision would directly probe physics beyond the spin-zero diquark approximation.

The sextet J=1/2 states give \mu ({\mathrm{\Sigma }}_b^{+})=+1.509, \mu ({\mathrm{\Sigma }}_b^0)=+0.390, and \mu ({\mathrm{\Sigma }}_b^{-})=-0.729{\mu }_N. These values are uniformly smaller in magnitude than the corresponding charm {\mathrm{\Sigma }}_c moments, reflecting the smaller bottom-quark charge-to-mass ratio. A noteworthy feature is that the bottom-sector heavy–light diquark sizes (a_{[ub]u}\approx 0.48 fm, a_{[sb]s}\approx 0.42 fm) are nearly identical to the corresponding charm-sector values. Consequently the dominant heavy–light u- and s-quark contributions to the moment are essentially the same in the bottom and charm sectors, and the difference between {\mathrm{\Sigma }}_b and {\mathrm{\Sigma }}_c predictions arises almost entirely from the heavy-quark contribution. The {\mathrm{\Omega }}_b^{-}=-0.473{\mu }_N result follows the same pattern, and the {\mathrm{\Xi }}_b^{\prime } states at +0.471 and -0.591{\mu }_N interpolate as in the charm sector.

In the sextet J=3/2 sector, the aligned-spin configuration produces \mu ({\mathrm{\Sigma }}_b^{\ast +})=+2.186{\mu }_N as the largest magnitude. Across {\mathrm{\Sigma }}_b^{\ast }, {\mathrm{\Xi }}_b^{\ast }, and {\mathrm{\Omega }}_b^{\ast }, the J=3/2 predictions are systematically larger in magnitude than their J=1/2 partners by amounts consistent with the additive heavy–light contributions noted earlier. \mu ({\mathrm{\Omega }}_b^{\ast -})=-0.789{\mu }_N is the smallest magnitude in the bottom J=3/2 sector.

Comparing charm and bottom predictions reveals that the diquark clustering picture organizes the entire spectrum into three structural families. The first comprises the antitriplet states, which are dominated entirely by the heavy-quark dynamical mass and give \sim +0.31{\mu }_N in charm and \sim -0.05{\mu }_N in bottom. The second comprises the sextet J=1/2 states, where the heavy–light diquark contribution dominates with the pattern set by the light-quark charge structure. The third comprises the sextet J=3/2 states, where the aligned-spin enhancement of the heavy–light contribution roughly doubles the multiply charged predictions relative to the J=1/2 sector. The flat antitriplet structure in the bottom sector and the heavy–light dominance throughout the sextets are the principal physical signatures of dynamical diquark clustering.

3.6 Comparison with previous predictions

We now compare the predictions of this work with the existing literature, organized by sector. Tables 5–8 display the comparisons for the charm J^P={\frac{1}{2}}^{+}, bottom J^P={\frac{1}{2}}^{+}, charm J^P={\frac{3}{2}}^{+}, and bottom J^P={\frac{3}{2}}^{+} sectors respectively, with our results in the first data column. The alternative calculations include the effective mass scheme (EMS) [26–28], bag model (BM) [15], non-relativistic quark model (NRQM) [5], relativistic three-quark model (RTQM) [16], heavy-baryon chiral perturbation theory (HB\chiPT) [12, 13], covariant baryon chiral perturbation theory (B\chiPT) [14], light-cone QCD sum rules (LCQSR) [8–11], chiral constituent quark model (\chiCQM) [29], potential model (PM) [30], and lattice QCD (LQCD) [6, 7].

A unified picture emerges when the comparisons in Tables 5–8 are read through the lens of dynamical diquark clustering. Across all states our signs agree with every other framework listed. We focus the discussion on the systematic patterns that distinguish the dynamical quark–diquark approach from the alternatives.

The charm sextet J^P={\frac{1}{2}}^{+} states populate a relatively narrow band across methods. Our predictions sit at the lower edge of the model spread for all six states, in close company with the recent lattice QCD values of Ref. [6]: for {\mathrm{\Sigma }}_c^0 we obtain -0.962 versus the lattice -1.073(269), and for {\mathrm{\Omega }}_c^0 we have -0.665 against -0.639(88), both within the lattice uncertainties. This concordance is non-trivial because the constituent-quark frameworks (NRQM, BM, EMS, RTQM) cluster around |\mu |\sim 1.0–1.4 for {\mathrm{\Sigma }}_c^0 and |\mu |\sim 0.85–0.95 for {\mathrm{\Omega }}_c^0, systematically larger in magnitude than both our results and the lattice. The reason in our framework is the dynamical compactness of the light–light diquark: a_{[uu]c}=0.133 fm and a_{[ss]c}=0.157 fm produce large b/a contributions that suppress the light-quark moments relative to the constituent-mass values used in NRQM-type calculations. Our prediction \mu ({\mathrm{\Sigma }}_c^{++})=+1.614 is correspondingly the smallest in the table, with the HB\chiPT [12] value +1.50 as the next-smallest and RTQM at +1.760; the bag model and NRQM give +2.280 and +2.350. Whether the lattice central value 2.220(505) favors one or the other will be decided by future calculations with reduced uncertainties.

The charm antitriplet sector deserves separate emphasis. Our values \mu ({\mathrm{\Lambda }}_c^{+})=+0.311, \mu ({\mathrm{\Xi }}_c^{+})=+0.310, and \mu ({\mathrm{\Xi }}_c^0)=+0.310{\mu }_N are nearly identical, reflecting the inability of the antitriplet to distinguish light-quark flavor in the m_u=m_d limit. This near-degeneracy is shared by EMS and BM, partially by NRQM (the \mu ({\mathrm{\Xi }}_c^{+})=0.200 depression there is a known artifact of NRQM isospin breaking), and is most prominent in our calculation. The recent LCQSR analysis [11] gives \mu ({\mathrm{\Lambda }}_c^{+})=+0.46, somewhat larger than ours, while the lattice [6] finds \mu ({\mathrm{\Xi }}_c^{+})=+0.235(25) and \mu ({\mathrm{\Xi }}_c^0)=+0.192(17), near the lower edge of the model spread. In the diquark-clustering interpretation the antitriplet moment is essentially {\mu }_c^{\mathrm{d}\mathrm{y}\mathrm{n}}(a_{\mathrm{a}\mathrm{v}\mathrm{g}})=+0.31{\mu }_N, where the configuration-averaged size a_{\mathrm{a}\mathrm{v}\mathrm{g}}\sim 0.4 fm produces the charm dynamical mass close to its constituent value. The agreement with NRQM-type models on the gross scale, combined with our prediction lying close to the lattice central values, suggests that this sector is the cleanest probe of the dynamical heavy-quark moment in an environment where the spin-zero light diquark is a good approximation.

The bottom sextet J^P={\frac{1}{2}}^{+} comparison shows the same pattern as in charm, amplified by the larger heavy-quark mass. Our predictions are systematically smaller in magnitude than every other framework listed, with the gap most pronounced for the multiply charged states: \mu ({\mathrm{\Sigma }}_b^{+})=+1.509 versus a model spread of +2.02–+2.50, and \mu ({\mathrm{\Omega }}_b^{-})=-0.473 versus a model spread of -0.74–-0.87. In the diquark-clustering picture the gap arises because the bottom-sector heavy–light diquark sizes are nearly identical to the corresponding charm-sector values, so the heavy–light u- and s-quark moments are essentially the same as in charm. The total moment is therefore set predominantly by these heavy–light contributions, with the heavy-quark contribution adding only \sim |Q_b|m_p/m_b\approx 0.07{\mu }_N in magnitude. Frameworks that use static constituent quark masses m_u^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}},m_s^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}} independent of the configuration miss this dynamical reduction and produce uniformly larger moments.

The bottom antitriplet represents the most informative qualitative disagreement of the entire comparison. Our values \mu ({\mathrm{\Lambda }}_b^0)=\mu ({\mathrm{\Xi }}_b^0)=\mu ({\mathrm{\Xi }}_b^{-})=-0.052{\mu }_N are essentially the bottom-quark moment alone, modified only slightly by the configuration averaging. This near-degeneracy is shared with NRQM, EMS, BM, and RTQM (all in the range -0.05 to -0.11{\mu }_N), and with the older LCQSR analysis [10] (-0.18\pm 0.05, -0.08\pm 0.02, -0.045\pm 0.005). The recent LCQSR analysis [11] obtains values uniformly several times larger in magnitude (-0.29, -0.33, -0.25). This disagreement is not a residual numerical effect: the difference is \sim 0.25{\mu }_N for each state, well outside the LCQSR uncertainties of \pm 0.03{\mu }_N. In the dynamical diquark-clustering picture this near-degeneracy is enforced by two facts: the spin-zero good light diquark contributes nothing to the moment, and the bottom-quark dynamical mass is essentially equal to its current value across all configurations because b/a\ll m_b. Both facts together leave only 1/m_b corrections, which are very small. The LCQSR approach, by contrast, includes contributions from photon distribution amplitudes attached to the light quarks that are not parametrically suppressed by 1/m_b. We regard \mu ({\mathrm{\Lambda }}_b^0) as the cleanest experimental discriminator between the dynamical quark–diquark picture and the LCQSR enhancement: a measurement at the \sim 0.1{\mu }_N level would distinguish the two predictions unambiguously.

The spin-3/2 charm sextet shows the widest spread among methods of any sector: predictions for \mu ({\mathrm{\Sigma }}_c^{\ast ++}) range from +2.41 (HB\chiPT) to +4.81 (LCQSR), a factor of two. Our value +2.888 falls near the lower end of this spread, alongside HB\chiPT and the recent LCQSR analysis [11] which gives +3.40(34). The reason is again the heavy–light diquark dominance: in the spin-3/2 state all three quark spins are aligned, doubling each quark’s contribution relative to the spin-1/2 case. This amplifies any model-dependent difference in the dynamical mass, and our relatively small light-quark dynamical moments translate into proportionally smaller spin-3/2 moments. The lattice value \mu ({\mathrm{\Omega }}_c^{\ast 0})=-0.730(23) [6] provides the only first-principles benchmark in this sector; our prediction -0.535 undershoots it by \sim 0.2{\mu }_N, while NRQM and EMS values -0.83 and -0.75 overshoot, leaving the lattice central value within the spread.

The spin-3/2 bottom sextet exhibits a distinctive pattern. For the strange-content states {\mathrm{\Xi }}_b^{\ast -} and {\mathrm{\Omega }}_b^{\ast -} our values -0.965 and -0.789 lie below the older LCQSR results [8] (-1.42 and -1.40) and the recent LCQSR analysis [11] (-0.57 and -0.61). In other words, our predictions sit between the older and newer LCQSR results and within the spread of NRQM, EMS, BM, and \chiCQM. For {\mathrm{\Sigma }}_b^{\ast +} our value +2.186 is the smallest in the table, with the LCQSR values +2.52 and +3.20 bracketing the constituent-model band +3.08–+3.56. This sector is therefore the most discriminating: experimental measurements of \mu ({\mathrm{\Sigma }}_b^{\ast +}) at the \sim 0.5{\mu }_N level would distinguish among the three theoretical bands.

To synthesize, the systematic pattern across all four comparison tables is that our predictions are smaller in magnitude than NRQM-type calculations, with the gap growing from charm to bottom and from spin-1/2 to spin-3/2. This pattern follows from the diquark clustering hierarchy of Table 2: the dynamical mass m_q+b/a depends on configuration, and the heavy–light configuration that dominates the moment has a u-quark dynamical mass somewhat larger than the conventional constituent value because the relevant size a\sim 0.4–0.5 fm is smaller than the typical baryon radius. Frameworks that use a single constituent mass independent of configuration cannot reproduce this suppression and therefore obtain systematically larger moments. Lattice QCD [6, 7] and the older LCQSR [10] give values intermediate between the dynamical and static-constituent predictions, broadly consistent with our smaller-magnitude predictions in the charm sector. The bottom antitriplet near-degeneracy at -0.05{\mu }_N is a particularly clean prediction of the dynamical diquark clustering picture and is not shared by the recent LCQSR analysis [11]. Future experimental measurements, particularly at LHCb with bent crystals [2–4], will provide direct tests of these distinctive predictions.

3.7 SU(3) symmetry and Coleman–Glashow relations

Since the singly heavy baryons are classified by SU(3) representations of the light quarks, it is instructive to examine the extent to which our predictions satisfy SU(3)-based relations such as the generalized Coleman–Glashow relations [31–34]. For the spin-1/2 charm sextet, three representative relations are

(9)\mu ({\mathrm{\Sigma }}_c^{++})-\mu ({\mathrm{\Sigma }}_c^{+})=\mu ({\mathrm{\Sigma }}_c^{+})-\mu ({\mathrm{\Sigma }}_c^0),

(10)\mu ({\mathrm{\Sigma }}_c^0)-\mu ({\mathrm{\Xi }}_c^{\mathrm{\prime }0})=\mu ({\mathrm{\Xi }}_c^{\mathrm{\prime }0})-\mu ({\mathrm{\Omega }}_c^0),

(11)2[\mu ({\mathrm{\Sigma }}_c^{+})-\mu ({\mathrm{\Xi }}_c^{\mathrm{\prime }0})]=\mu ({\mathrm{\Sigma }}_c^{++})-\mu ({\mathrm{\Sigma }}_c^0).

With our numerical predictions, Eq. (9) is satisfied to 0.002{\mu }_N (LHS =1.289{\mu }_N, RHS =1.287{\mu }_N), reflecting the U-spin equal-spacing inherent in the spin-flavor structure of the moment formulas in the m_u=m_d limit. Equation (10) is satisfied to 0.027{\mu }_N (LHS =-0.162{\mu }_N, RHS =-0.135{\mu }_N), with the small residual reflecting SU(3) breaking induced by the strange-quark mass and the configuration-dependent dynamical masses. Equation (11) shows a slightly larger violation of 0.33{\mu }_N, again traceable to dynamical SU(3) breaking through the different size parameters a_{ijk} associated with strange versus non-strange configurations. The exact realization of Eq. (9) and the small violation of Eq. (10) are quantitatively similar to results obtained in other SU(3)-based analyses [32–34]. The same relations hold within comparable accuracy in the bottom sector: \mu ({\mathrm{\Sigma }}_b^{+})-\mu ({\mathrm{\Sigma }}_b^0)=+1.119{\mu }_N and \mu ({\mathrm{\Sigma }}_b^0)-\mu ({\mathrm{\Sigma }}_b^{-})=+1.119{\mu }_N are equal to three-digit precision, in line with Eq. (9). Regarding the precision of these statements: in the m_u=m_d limit adopted here Eq. (9) is exact by construction, since the {\mathrm{\Sigma }}_Q^0 moment is the average of its charged partners (Appendix A); the residual 0.002{\mu }_N quoted for the charm sector reflects only the rounding of the tabulated three-decimal values, and the bottom equalities are exact in the same sense. The SU(3)-breaking differences tested in Eqs. (10) and (11), by contrast, are genuine dynamical predictions and carry the model uncertainty discussed in Section 3.3; since they are differences of moments computed within a single framework, correlated systematics largely cancel and the qualitative pattern of small versus larger violation is robust.

The systematic pattern of approximate SU(3) symmetry with controlled breaking is a natural consequence of the dynamical clustering picture: the heavy–light diquark configuration that dominates the moment carries SU(3) labels through the light spectator, and the moment differences between SU(3)-related states are then determined by the small shifts in a_{ijk} induced by replacing u,d by s.

4 Conclusions

In this paper, we have used the relativistic quark–diquark framework in order to obtain magnetic moments of singly heavy baryons.

The framework reproduces the published light-baryon magnetic moments to within 0.005{\mu }_N for the full ground-state octet and the {\mathrm{\Omega }}^{-}, providing a stringent test of the numerical pipeline. In the light sector the framework agrees with experiment to \sim 0.05{\mu }_N for the proton, neutron, \mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}, and {\mathrm{\Xi }}^0, with 0.2–0.5{\mu }_N deficits for {\mathrm{\Sigma }}^{-}, {\mathrm{\Xi }}^{-}, and {\mathrm{\Omega }}^{-} that track the known footprint of meson-cloud and SU(3)-breaking contributions absent from the constituent framework.

Our central physics result is the set of all predictions in Tables 3 and 4. Solving the wave equation self-consistently exposes a pronounced diquark clustering hierarchy, with heavy–light diquark configurations \sim 3–4 times more extended in charm and an order of magnitude more extended in bottom than the corresponding light–light configurations. This hierarchy is not a parameter choice but a dynamical consequence of the confining potential acting on quarks of disparate mass, and it organizes the entire spectrum of magnetic-moment predictions: the heavy–light configuration carries the dominant configuration weight, produces a light-quark dynamical mass close to its conventional constituent value, and thereby fixes the gross structure of every prediction. The light–light configuration provides only a small “compactness” correction. Across all states our signs agree with every other framework in the literature, while the systematic pattern of magnitudes — smaller than NRQM-type calculations, with the gap growing from charm to bottom and from spin-1/2 to spin-3/2 — traces directly to the configuration-dependent dynamical mass that distinguishes our framework from approaches based on a single static constituent mass. In the charm sector our results sit close to the recent lattice QCD values [6], providing encouraging first-principles support for the dynamical-mass mechanism. A conservative uncertainty assignment (Section 3.3) attaches \pm 0.05{\mu }_N to the heavy-quark-dominated antitriplet states and up to \pm 0.3{\mu }_N to spin-3/2 sextet states with large light-quark moments.

The most experimentally consequential outcome of the analysis is a sharp prediction in the bottom antitriplet sector. We obtain \mu ({\mathrm{\Lambda }}_b^0)=\mu ({\mathrm{\Xi }}_b^0)=\mu ({\mathrm{\Xi }}_b^{-})=-0.052{\mu }_N, a flat near-degeneracy enforced by the spin-zero good light diquark and by the heavy-quark decoupling that leaves only 1/m_b corrections to the bare bottom-quark moment. This prediction is shared by NRQM, EMS, BM, RTQM, and the older LCQSR analysis [10], but disagrees by a factor of \sim 5 with the recent LCQSR analysis [11], which generates substantial enhancements through photon distribution amplitudes attached to the light quarks. A measurement at the \sim 0.1{\mu }_N level would unambiguously discriminate between the two pictures and constitutes the cleanest experimental test of dynamical diquark clustering in the heavy-quark sector. Two complementary classes of measurements — precision determinations of \mu ({\mathrm{\Sigma }}_c^{++}), \mu ({\mathrm{\Sigma }}_c^0), and \mu ({\mathrm{\Omega }}_c^0) in the charm sextet, and a refinement of \mu ({\mathrm{\Lambda }}_c^{+}) at the antitriplet level — are within reach of the LHCb fixed-target program with bent crystals [2–4] and together would constrain the dynamical heavy-quark moment in both multiplet environments.

The framework admits several natural extensions. Doubly heavy baryons ({\mathrm{\Xi }}_{cc}, {\mathrm{\Xi }}_{bb}, {\mathrm{\Xi }}_{bc}, {\mathrm{\Omega }}_{cc}, {\mathrm{\Omega }}_{bb}, {\mathrm{\Omega }}_{bc}) require only solving the wave equation with a different combination of heavy-quark masses, and the reduced light-quark content of these states should make the predictions in this sector even more reliable than for the singly heavy case. Transition magnetic moments {\mathrm{\Sigma }}_Q^{\ast }\to {\mathrm{\Sigma }}_Q\gamma and {\mathrm{\Xi }}_Q^{\ast }\to {\mathrm{\Xi }}_Q\gamma follow from the same wave functions through standard spin-flavor matrix elements and would test the framework against existing lattice [6] and HB\chiPT [12] results. Beyond electromagnetic observables, the diquark clustering hierarchy displayed in Table 2 carries direct implications for production cross sections, lifetimes, and weak transition form factors of singly heavy baryons; the geometric factors that fix the magnetic-moment predictions also enter these other quantities, and a unified treatment within a single framework constrained by meson data is now within reach.

The persistent strength of the relativistic quark–diquark approach, four decades after its introduction, is that it captures the essential dynamics of strong binding in a few transparent ingredients: a relativistic two-body equation, a one-gluon-exchange potential improved by asymptotic freedom, and a single magnetic-moment scale b tied to the proton. The predictions presented here, anchored by the light-baryon validation and sharpened by the diquark-clustering interpretation, provide a quantitative theoretical target for the experimental programs now turning their attention to heavy-baryon electromagnetic structure. We anticipate that future precision measurements at LHCb, Belle II, and proposed facilities will discriminate among the existing theoretical pictures and, in doing so, illuminate the dynamics of diquark clustering in the heavy-quark sector.

5 Appendix A: Magnetic-moment expressions

The dynamical quark magnetic moment in the form used throughout the paper is, from Eq. (6),

(A1)\begin{array}{c}{\mu }_i(a)={\displaystyle \frac{Q_i}{2(m_i+b/a)}},\end{array}

with Q_i in units of |e|, m_i the current quark mass, b=0.1846 GeV\cdotfm, and a the relevant configuration size in fm.

For two-identical-light qqQ baryons we denote the two diquark configurations by a_1\equiv a_{qqQ} (light pair as diquark, heavy quark as spectator, weight 1/3) and a_2\equiv a_{qQq} (heavy-light diquark, light spectator, weight 2/3). For three-distinct-flavor {\mathrm{\Xi }}_Q baryons the three configurations a_A\equiv a_{q_1{q}_{2}Q}, a_B\equiv a_{q_{1}Q{q}_2}, a_C\equiv a_{q_{2}Q{q}_1} are weighted equally.

For the sextet J=1/2 states {\mathrm{\Sigma }}_Q and {\mathrm{\Omega }}_Q,

(A2)\begin{array}{c}\mu ({\mathrm{\Sigma }}_Q)={\displaystyle \frac{1}{3}}[{\displaystyle \frac{4}{3}}{\mu }_q(a_1)-{\displaystyle \frac{1}{3}}{\mu }_Q(a_1)]+{\displaystyle \frac{2}{3}}[{\displaystyle \frac{4}{3}}{\mu }_q(a_2)-{\displaystyle \frac{1}{3}}{\mu }_Q(a_2)].\end{array}

For the decuplet J=3/2 states {\mathrm{\Sigma }}_Q^{\ast } and {\mathrm{\Omega }}_Q^{\ast },

(A3)\begin{array}{c}\mu ({\mathrm{\Sigma }}_Q^{\ast })={\displaystyle \frac{1}{3}}[2{\mu }_q(a_1)+{\mu }_Q(a_1)]+{\displaystyle \frac{2}{3}}[2{\mu }_q(a_2)+{\mu }_Q(a_2)].\end{array}

For the antitriplet {\mathrm{\Lambda }}_Q, the scalar light diquark contributes nothing, leaving the heavy-quark moment alone:

(A4)\begin{array}{c}\mu ({\mathrm{\Lambda }}_Q)={\displaystyle \frac{1}{3}}{\mu }_Q(a_1)+{\displaystyle \frac{2}{3}}{\mu }_Q(a_2).\end{array}

For the sextet J=1/2 state with three distinct flavors {\mathrm{\Xi }}_Q^{\prime },

(A5)\begin{array}{c}\mu ({\mathrm{\Xi }}_Q^{\prime })={\displaystyle \frac{1}{3}}{\displaystyle \sum _{c\in \{A,B,C\}}[{\displaystyle \frac{2}{3}}({\mu }_{q_1}(a_c)+{\mu }_{q_2}(a_c))-{\displaystyle \frac{1}{3}}{\mu }_Q(a_c)].}\end{array}

For the decuplet J=3/2 state with three distinct flavors {\mathrm{\Xi }}_Q^{\ast },

(A6)\begin{array}{c}\mu ({\mathrm{\Xi }}_Q^{\ast })={\displaystyle \frac{1}{3}}{\displaystyle \sum _{c\in \{A,B,C\}}[{\mu }_{q_1}(a_c)+{\mu }_{q_2}(a_c)+{\mu }_Q(a_c)].}\end{array}

For the antitriplet J=1/2 state with three distinct flavors {\mathrm{\Xi }}_Q,

(A7)\begin{array}{c}\mu ({\mathrm{\Xi }}_Q)={\displaystyle \frac{1}{3}}{\displaystyle \sum _{c\in \{A,B,C\}}{\mu }_Q(a_c).}\end{array}

For the special case of the {\mathrm{\Xi }}_c^{0/+} and {\mathrm{\Xi }}_b^{0/-} antitriplets in the m_u=m_d limit adopted here, Eq. (A7) gives identical moments for the two charge states, both equal to the configuration-averaged heavy-quark moment. Explicit charge assignments are obtained by substituting Q_q=+2/3 for u or Q_q=-1/3 for d in Eqs. (A2)–(A6); the {\mathrm{\Sigma }}_Q^0 states are obtained as \frac{1}{2}[\mu ({\mathrm{\Sigma }}_Q^{+})+\mu ({\mathrm{\Sigma }}_Q^{-})] in this limit.

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