Electromagnetic polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons in heavy baryon chiral perturbation theory

Liang-Zhen Wen; Lu Meng; Shi-Lin Zhu; Yan-Ke Chen

REVIEW 2 major objections 5 minor 100 references

Electromagnetic polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons in heavy baryon chiral perturbation theory

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The spin-$\frac{1}{2}$ singly heavy baryon polarizabilities at $\mathcal{O}(p^3)$: sextet baryons respond through pion clouds and a near-degenerate spin-$\frac{3}{2}$ partner, while the antitriplet turns out point-like.

desk verdict First HBChPT polarizability estimates for singly heavy baryons, but the zero-polarizability claim for the antitriplet misses O(p^3) B3-B6 M1 transitions already present in the paper's own Lagrangian. read the letter →

arxiv 2412.02297 v2 pith:6JQWVVQG submitted 2024-12-03 hep-ph hep-exhep-lat

Yan-Ke Chen , Liang-Zhen Wen , Lu Meng , Shi-Lin Zhu This is my paper · ORCID

classification hep-phhep-exhep-lat

keywords electromagneticpolarizabilitiesheavybaryonchiralperturbationtheorysinglybaryonsComptonscatteringcharmedbottomquarkspinsymmetrylow-energyconstants

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that the electric and magnetic polarizabilities of spin-$\frac{1}{2}$ singly heavy baryons — how much a baryon with one charm or bottom quark deforms under quasistatic electric and magnetic fields — follow systematically from heavy baryon chiral perturbation theory, the low-energy effective theory of QCD, at $\mathcal{O}(p^3)$. It finds a sharp dichotomy: the antitriplet family $\Lambda_c^+$, $\Xi_c^+$, $\Xi_c^0$, where the two light quarks form a spin-zero antisymmetric pair, has exactly vanishing polarizability at this order, because parity and angular momentum conservation forbid its coupling to the pion cloud and it behaves like a charged point particle. The sextet family $\Sigma_c$, $\Xi_c'$, $\Omega_c$, whose light diquark carries spin one, receives its polarizability from two sources: long-range pion and kaon cloud loops, and the magnetic dipole transition into the nearly degenerate spin-$\frac{3}{2}$ partner $B_6^*$, a term proportional to $C_\xi^2/\delta_1$ that dominates the magnetic polarizability because the mass splitting $\delta_1$ is small. That pole term is the same mechanism by which the $\Delta$ resonance shapes nucleon polarizabilities, but it is stronger here because the $B_6$–$B_6^*$ splitting (67 MeV for charmed, 20 MeV for bottom baryons) is smaller than $M_\Delta - M_N$. If the calculation holds up, its tables are the first quantitative chiral perturbation theory predictions for heavy baryon polarizabilities, giving lattice QCD and proposed bent-crystal spin-precession experiments concrete numbers to test.

What carries the argument

The machinery is the spin-averaged forward Compton tensor $\Theta_{\mu\nu} = U(\omega) g_{\mu\nu} + V(\omega) k_\mu k_\nu + \cdots$, whose low-energy coefficients are the polarizabilities themselves: $\alpha_E + \beta_M = -\frac{1}{8\pi} U''(0)$ and $\beta_M = -\frac{1}{4\pi} V(0)$. Heavy baryon chiral perturbation theory removes the baryon mass from the propagators by splitting the field into heavy and light components at a fixed velocity, restoring a power counting in which the $\mathcal{O}(p^3)$ Compton amplitude contains only the tree and one-loop diagrams of Fig. 1. The load-bearing small parameter is the mass splitting $\delta_1 = M_{B_6^*} - M_{B_6}$ that sits in the denominator of the $B_6^*$-exchange diagram: the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ sextet baryons become degenerate in the heavy quark limit, so $\delta_1$ is small (67 MeV for charm, 20 MeV for bottom) and the magnetic polarizability acquires a $\delta_1^{-1}$ pole, the same structure as the $\Delta(1232)$ contribution to nucleon polarizabilities but with a smaller denominator. The loop integrals are evaluated in dimensional regularization through the $J$-functions of Appendix A, and the four low-energy constants are fixed by heavy quark spin symmetry ($g_1^2 = \frac{4}{3} g_3^2$), the quark model ($g_1^2 = \frac{8}{3} g_2^2$), the measured $\Sigma_c$ and $\Sigma_c^*$ decay widths, and a fit of quark magnetic moments to seven lattice QCD baryon magnetic moments.

What would settle it

A lattice QCD computation of the electric and magnetic polarizabilities of the singly charmed baryons at or near the physical pion mass would settle the central claim: the framework predicts $\beta_M / \alpha_E = 0.1$ for the chiral-loop part of the sextet, a magnetic polarizability hierarchy ordered by the transition magnetic moments (largest for $\Sigma_c^{++}$, smallest for $\Sigma_c^+$ and $\Xi_c^{\prime +}$), and exactly vanishing polarizabilities for $\Lambda_c^+$ and $\Xi_c$. Measuring the $B_6^* \to B_6 \gamma$ radiative width would independently fix the $C_\xi$ coefficients that control the dominant magnetic contribution. Conversely, a single nonzero polarizability for $\Lambda_c^+$ — for instance from the spin precession of channeled charmed baryons in bent crystals proposed at hadron colliders — would show that the $\mathcal{O}(p^4)$ effects are not negligible.

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Extended reading notes

Core claim

On the paper's own terms, the central result is the complete $\mathcal{O}(p^3)$ prediction for the polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons. For the antitriplet, the spin-averaged forward Compton amplitude reduces to the Thomson term $U_\xi(\omega) = Q_\xi^2 e^2 / M_{\bar{3}}$, so $\alpha_E = \beta_M = 0$ up to this order (Eq. (31)): $\bar{B}_3$ behaves like a charged point particle with no electromagnetic polarizability. For the sextet, the total polarizability is the sum (Eq. (77)) of three pieces: the $B_6\phi$-loop diagrams, the $B_6^*\phi$-loop diagrams, and the tree diagram with a $B_6^*$ intermediate state, whose magnetic polarizability $\beta_M^{(b')}(\xi) = \alpha_{\rm em} C_\xi^2 / (12 M_N^2 \delta_1)$ carries a pole in the small sextet mass splitting. The chiral-loop pieces obey the simple ratio $\beta_M^{(c-g)} = \alpha_E^{(c-g)} / 10$, and in the heavy quark limit the $B_6$ and $B_6^*$ loops satisfy $\alpha_E^{(c-g)} = 2\alpha_E^{(c'-g')}$ and $\beta_M^{(c-g)} = 2\beta_M^{(c'-g')}$, with the finite splitting slightly suppressing the $B_6^*$ loops. Numerically, the charmed sextet baryons come out with electric polarizabilities between $0.90 \times 10^{-4}$ and $9.42 \times 10^{-4}\,\mathrm{fm^3}$ and magnetic polarizabilities between $0.47 \times 10^{-4}$ and $3.78 \times 10^{-4}\,\mathrm{fm^3}$, while the bottom baryons reach much larger magnetic values, up to $18.6 \times 10^{-4}\,\mathrm{fm^3}$ for $\Sigma_b^+$, driven by the 20 MeV splitting.

Load-bearing premise

The load-bearing premise, which the paper itself concedes in its summary, is that the strength constants can be pinned down by heavy quark symmetry, the quark model, and a fit to lattice QCD magnetic moments, and that the third-order chiral expansion remains reliable even when the two sextet states are only 20 MeV apart for bottom baryons.

Editorial extensions

If this is right

The antitriplet baryons $\Lambda_c^+$, $\Xi_c^+$, $\Xi_c^0$ are predicted to have exactly zero electric and magnetic polarizability at $\mathcal{O}(p^3)$, so any measured nonzero value would directly signal contributions beyond this order.
Among the charmed sextet baryons, the magnetic polarizability is dominated by the $B_6^* \to B_6 \gamma$ transition for $\Sigma_c^{++}$, $\Sigma_c^0$, $\Xi_c^{\prime 0}$, and $\Omega_c^0$, whereas $\Sigma_c^+$ and $\Xi_c^{\prime +}$, whose transition magnetic moments nearly vanish, are predicted to be markedly less magnetically polarizable.
The bottom baryon magnetic polarizabilities come out several times larger than the charmed ones, up to $18.6 \times 10^{-4}\,\mathrm{fm^3}$ for $\Sigma_b^+$, because the $B_6$–$B_6^*$ splitting shrinks to 20 MeV and the $\delta_1^{-1}$ pole term grows.
In the heavy quark limit the loop contributions obey exact ratio relations $\alpha_E^{(c-g)} = 2\alpha_E^{(c'-g')}$ and $\beta_M^{(c-g)} = 2\beta_M^{(c'-g')}$, and the finite-splitting numerical results approximately preserve them.
The analytical expressions are directly usable as chiral extrapolation formulas for future lattice QCD simulations of heavy baryon electromagnetic properties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the vanishing antitriplet polarizability follows from selection rules rather than from fitted constants, it is the cleanest falsifiable corner of the calculation: a single nonzero polarizability measurement for $\Lambda_c^+$ would directly expose the size of $\mathcal{O}(p^4)$ counterterm contributions.
The $C_\xi^2/\delta_1$ structure implies a testable scaling law — magnetic polarizability growing roughly as $1/\delta_1$ as the heavy quark mass is varied — which lattice QCD could probe by changing the heavy quark mass at fixed lattice spacing.
The baryons with the smallest tree-level contributions, $\Sigma_c^+$ and $\Xi_c^{\prime +}$, are the most sensitive windows into $\mathcal{O}(p^4)$ truncation error, since their predicted polarizabilities are the most likely to shift once the next order is included.
The same Compton-tensor framework extends naturally to the spin-$\frac{3}{2}$ sextet polarizabilities and to doubly heavy baryons, where the relevant partner splittings are even smaller and the pole enhancement would be larger.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First useful take: this is a genuine first step for heavy-baryon polarizabilities in HBChPT, and the sextet loop expressions will likely be reused. But the central claim—zero polarizabilities for the antitriplet—does not survive contact with the paper's own Lagrangian.

The problem is in Eq. (27). The f2 and f4 terms couple B3 to B6 and B6* through magnetic dipole operators. Two such vertices with a B6 or B6* intermediate produce tree graphs at O(p^3), exactly the same order as the diagram the authors keep for the sextet (Fig. 1(b')). The paper does not set f2=f4=0; "decoupled" is a dynamical approximation, not a symmetry. The corresponding transition magnetic moments are not small (order 1 μN from the quark model and lattice inputs), and the resulting βM for Λc+ and Ξc is around 10^-4 fm^3, comparable to the chiral-loop entries in Table V. So Eq. (31) is not correct as stated, and the completeness of Tables V and VI at O(p^3) is not established. The same omitted operators also contribute to the sextet polarizabilities through B3 intermediate states.

What the paper does well: the B6 chiral-loop calculation is new, the heavy quark limit collation in Eq. (76) checks out, and the numerical pattern α(c'-g') ≈ α(c-g)/3 is sensible. The strategy of fixing LECs from lattice QCD magnetic moments and Σc decay widths is reasonable, and the uncertainties are honestly reported. The citation pattern is fair, though the numerical inputs lean on the authors' own earlier chiral and quark-model analyses; that is model dependence, not circularity.

Other soft spots: the η loop is never mentioned; if its coefficients vanish by symmetry that should be stated. For bottom baryons, δ1 = 20 MeV makes the O(p^3) expansion questionable. The reported uncertainties cover the input errors but not the systematic error from the missing B3-B6 transition diagrams.

Bottom line: the framework and the sextet expressions are worth building on, but the antitriplet zero result is a load-bearing gap, not a minor omission. A referee should require the authors to include the f2/f4 contributions or derive a symmetry that sets them to zero. I would not desk-reject this; it deserves a serious referee, but the present version is not ready.

Referee Report

2 major / 5 minor

Summary. The paper presents a heavy-baryon chiral perturbation theory calculation of the spin-averaged electromagnetic polarizabilities alpha_E and beta_M of the spin-1/2 singly heavy baryons through O(p^3). For the antitriplet B3 it claims, in Eq. (31), that alpha_E = beta_M = 0, so that these baryons behave as charged point particles at this order. For the sextet B6 the polarizabilities receive contributions from pion/kaon/eta loops with B6 and B6* intermediate states and from the B6* -> B6 magnetic dipole transition, with the final sums given in Eq. (77). The low-energy constants g1, g3, f6, and f7 are estimated using the experimental Sigma_c decay widths, heavy-quark-spin and quark-model relations, and quark-model transition moments fitted to lattice QCD magnetic moments. Numerical results for charmed and bottom baryons are listed in Tables V and VI, with the bottom sector characterized by the very small splitting delta_1 = 20 MeV and a correspondingly large 1/delta_1 contribution to beta_M.

Significance. If the diagram set is complete, the paper provides the first O(p^3) heavy-baryon chiral perturbation theory predictions for singly heavy baryon polarizabilities, together with analytic expressions that can be used for future lattice chiral extrapolation. The authors are transparent about their low-energy-constant estimation, propagate uncertainties, and provide a useful internal consistency check in Eq. (76) relating the B6 and B6* loop contributions in the heavy quark limit. The central quantitative claims are, however, contingent on the completeness of the O(p^3) diagram set; the missing B3-B6(*) magnetic transition diagrams discussed below affect both the antitriplet zero result and the numerical tables.

major comments (2)

[Sec. III A, Eq. (31) and Eq. (27)] The claimed vanishing of the B3 polarizabilities is not established at O(p^3). The L^(2) Lagrangian in Eq. (27) contains the magnetic dipole transition operators proportional to f2 ($\bar{B}_3[S_\mu,S_\nu]\hat{F}^{+}_{\mu\nu}B_6$) and f4 ($\bar{B}_3\hat{F}^{+}_{\mu\nu}S_\nu B_6^{*\mu}$), and the paper nowhere sets f2 = f4 = 0. A tree diagram with two such vertices and an intermediate B6 or B6* is the direct B3 analog of the B6*->B6 diagram in Fig. 1(b'); by the power counting in Eq. (28), with L=0 and two d=2 meson-baryon vertices, this diagram is O(p^3). It generates a nonvanishing beta_M(B3) of order alpha f^2/(12 M_N^2 delta_2), with delta_2 = 127 MeV, plus a delta_3 term. The statement in Sec. III A that B3 and B6(*) are decoupled is therefore an approximation, not a symmetry, and it is inconsistent with the presence of the f2 and f4 terms in the same Lagrangian. Quark-model/lattice transition moments for B6 -> B3 gamma are not suppressed by heavy-quark spin symmetry. The same omitted diagrams also contribute to the B6 polarizabilities through a B3 intermediate. Consequently Eq. (31) and the completeness of Tables V and VI at O(p^3) are not established.
[Sec. IV, Table VI and Eq. (35)] For the bottom baryons the O(p^3) truncation is much less protected than for the charmed sector. With delta_1 = 20 MeV, the magnetic transition contribution in Eq. (35) gives, for example, beta_M = 18.1 x 10^-4 fm^3 for Sigma_b^+, more than four times the chiral-loop contribution and more than an order of magnitude larger than typical nucleon values quoted in Eq. (90). The paper argues in Sec. V from the nucleon precedent that O(p^4) corrections do not change the qualitative conclusions, but the nucleon system has no nearly degenerate partner at the scale of the pion mass. Since the omitted f2/f4 diagrams discussed above involve the analogous 1/delta_2 enhancement, the numerical hierarchy for bottom baryons could change. Please quantify the sensitivity of the bottom predictions to delta_1 and to the neglected 1/delta_2 diagrams before presenting them as quantitative predictions.

minor comments (5)

[Table VI] The last row of Table VI is labeled "Omega^-_c" but should be "Omega^-_b" for the singly bottom baryon.
[Eq. (74)] The definition of S_chi appears to contain a typo: it uses M_pi and R_pi on the right-hand side, whereas the subsequent equations and Eq. (75) require S_chi = M_chi^2(10R_chi - 9 delta_1) + delta_1^2(9 delta_1 - R_chi).
[Sec. III B, text after Eq. (76)] The statement that the finite delta_1 splitting "slightly suppresses" the B6* loop contribution is inconsistent with Table V, where alpha_E^{(c'-g')} / alpha_E^{(c-g)} is about 1/3 rather than slightly below 1/2; please rephrase or give the explicit delta_1-dependent ratio.
[Eq. (25)] The g6 term is included in the leading-order Lagrangian and then set to zero because parity and angular momentum conservation forbid the vertex; it would be clearer to either omit it from Eq. (25) or state explicitly that it is written for completeness before being discarded.
[Eq. (36) and Table I] The sign convention relating C_xi to the leading-order transition magnetic moment mu^{LO}_{xi*->xi+gamma} is not specified; please state the convention so that the entries in Table I and the numerical values in Eqs. (83)-(88) can be reproduced independently.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the polarizabilities are computed from a chiral Lagrangian with LECs fixed by external lattice QCD and decay-width data, not by the target polarizabilities.

full rationale

The central derivation is not circular. The electromagnetic polarizabilities are defined through the second-order Compton amplitude (Eqs. (1)-(13)) and then computed from HBχPT tree and loop diagrams. The required LECs are fixed from external inputs: g2 from the measured Σc decay widths via Eq. (78), g1 and g3 from standard heavy-quark-spin and quark-model relations, and f6/f7 (through Cξ) from quark magnetic moments fitted to lattice QCD baryon magnetic moments in Eq. (81). The target observables—the polarizabilities in Tables V and VI—do not enter these fits, so there is no fitted-input-called-prediction reduction. Eq. (77) sums tree and loop contributions whose sizes are controlled by these externally constrained parameters; this is a genuine prediction, not a rearrangement of the input. The vanishing antitriplet polarizabilities in Eq. (31) follow from the stated selection rule g6=0 and the adopted decoupling between B̄3 and B6(∗); whether that decoupling is complete at O(p^3) is a physical/correctness concern, not a circularity, because Eq. (31) is not obtained by defining the polarizability to be zero. The paper does rely on several prior works by the same group (Refs. [37,40-42,52]) for LEC relations and chiral-convergence arguments, but these are used as background approximations and are not an unverified uniqueness theorem that forces the result. The minority self-citation is therefore not load-bearing in the circular sense, and the paper is self-contained against external lattice QCD and experimental inputs. The modest score of 2 reflects only the presence of these minor self-citations, not any circular reduction of the derivation.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper's numerical predictions rest on several parameters that are fitted or model-estimated rather than derived from the EFT: g1 and g3 via quark model and heavy quark spin symmetry relations, and Cξ via quark-model transition moments fitted to lattice QCD. No new entities are introduced. The axioms are mostly standard EFT assumptions; the ad hoc ones are the model relations used to close the system.

free parameters (6)

g1 (axial coupling of B6 to Goldstone bosons) = 0.928 ± 0.093 ± 0.037
Obtained from g2 via quark model symmetry g1^2 = 8/3 g2^2; controls the B6 loop contributions to polarizabilities.
g3 (axial coupling of B6* to Goldstone bosons) = 0.804 ± 0.081 ± 0.032
Obtained from heavy quark spin symmetry g1^2 = 4/3 g3^2; controls the B6* loop contributions.
g2 (B6-B3-pion coupling) = |g2| = 0.568 ± 0.023
Extracted from the measured Σ_c -> Λ_c π decay widths using Eq. (78); used as the experimental anchor for g1 and g3.
Constituent quark magnetic moments μu, μd, μs, μc = μu=1.078(88), μd=-0.539(44), μs=-0.456(23), μc=0.205(15) in μN
Fitted to seven lattice QCD baryon magnetic moments via Eq. (81) with χ^2/d.o.f = 1.71; these determine the quark-model transition magnetic moments.
Bottom quark magnetic moment μb = (-0.05 ± 0.05) μN
Estimated by hand because no lattice or experimental constraints are available; enters the bottom transition magnetic moments.
Transition coefficients Cξ (standing in for LECs f6, f7) = Cξ = -√6 μ_{ξ*→ξ+γ}: 0.82(10), 0.06(3), -0.70(5), 0.10(6), -0.66(3), -0.62(3)
Computed from quark model with fitted quark moments; β_M(b') is proportional to Cξ^2, so these effectively replace the undetermined LECs f6 and f7.

assumptions (6)

domain assumption Heavy baryon chiral perturbation theory with the given power counting (Eq. (28)) describes singly heavy baryon low-energy physics.
The entire calculation is performed in this effective field theory; the framework is standard but not derived here.
domain assumption The φ B̄3 B̄3 vertex is forbidden, so g6 = 0.
Stated in Section II B; follows from the spin-zero light diquark and parity/angular momentum conservation. Standard result.
ad hoc to paper Quark model relation g1^2 = 8/3 g2^2 and heavy quark spin symmetry relation g1^2 = 4/3 g3^2 hold within the quoted 10% uncertainty.
Used in Section IV to fix g1 and g3; these are model assumptions, not derived from the effective field theory.
domain assumption The mass splittings δ1 = 67 MeV (charm) and 20 MeV (bottom) can be treated perturbatively at O(p^3).
The paper uses δ1 as a small expansion parameter; for bottom baryons δ1 << Mπ, and the resulting 1/δ1 enhancement raises convergence questions.
ad hoc to paper O(p^4) corrections do not change the qualitative conclusions.
Stated in Section V by analogy with the nucleon case (Refs. [69,70]); no heavy-baryon O(p^4) estimate is given.
ad hoc to paper Quark model expressions with fitted quark moments provide a valid estimate for the LECs f6 and f7 via Cξ.
Section IV computes Cξ from quark-model transition magnetic moments rather than from an EFT determination of f6 and f7.

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Cite this review

Pith. "Pith review of Electromagnetic polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons in heavy baryon chiral perturbation theory." pith.science (2026). https://pith.science/paper/6JQWVVQG

@misc{pith2026241202297,
  author       = {Pith},
  title        = {Pith review of: Electromagnetic polarizabilities of the spin-$\frac12$ singly heavy baryons in heavy baryon chiral perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JQWVVQG}},
  note         = {Machine review of arXiv:2412.02297}
}

abstract

We calculate the electromagnetic polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons in the heavy baryon chiral perturbation theory up to $\mathcal{O}(p^3)$. We estimate the low-energy constants using the magnetic moments of singly charmed baryons from lattice QCD simulations and the experimental decay widths of $\Sigma_c$ and $\Sigma_c^*$. Our results indicate that the long-range chiral corrections make significant contributions to the polarizabilities. Additionally, the magnetic dipole transitions $\mathcal{B}_6^* \to \mathcal{B}_6 +\gamma $ also provide large contribution to the magnetic polarizabilities.

Figures

Figures reproduced from arXiv: 2412.02297 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

100 extracted references · 35 canonical work pages

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Tree diagrams The tree diagrams in Figs. 1(a) and (b) yield the Thomson amplitude U (a+b) ξ (ω) = Q2 ξe2 MB6 , V (a+b) ξ (ω) = 0, (32) which does not contribute to the electromagnetic polarizabilities: α(a+b) E (ξ) = β(a+b) M (ξ) = 0. (33) The tree diagram with B∗ 6 as the intermediate state, shown in Fig. 1(b′), yields: U (b′) ξ (ω) = − e2C 2 ξ ω2 12M 2 ...
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heavy” and “light

We use the notations ψ¯3, ψ6 and ψ∗ 6 to denote the antitriplet, spin- 1 2 sextet, and spin- 3 2 sextet baryons, respectively. These baryon fields are represented as [80]: 3 ψ¯3,c =   0 Λ + c Ξ+ c −Λ+ c 0 Ξ 0 c −Ξ+ c −Ξ0 c 0   , ψ 6,c =   Σ++ c Σ+ c√ 2 Ξ′+ c√ 2 Σ+ c√ 2 Σ0 c Ξ′0 c√ 2 Ξ′+ c√ 2 Ξ′0 c√ 2 Ω0 c   , ψ ∗µ 6,c =  ...
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X χ D(d+e) ξ,χ F 2χ Z 1 0 dx J ′ 2(ωx, 0, M2 χ) # , (38) U (f ) ξ (ω) =A

Loop diagrams Using the J -function defined in Appendix A, we can obtain the form factors of the B6ϕ-loop diagrams in Figs. 1(c)–(g): U (c) ξ (ω) =A X χ D(c) ξ,χ F 2χ J0(ω, 0, M2 χ), (37) U (d+e) ξ (ω) =A "X χ D(d+e) ξ,χ F 2χ Z 1 0 dx J ′ 2(ωx, 0, M2 χ) # , (38) U (f ) ξ (ω) =A "X χ D(f ) ξ,χ F 2χ Z 1 0 dx (1 − x)(d + 1)J ′′ 6 (ωx, 0, M2 χ) − X χ D(f ) ξ,...
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+ αemg2 3RK 288π2F 2 K(M 2 K − δ2
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1(b′) purely represents two consecutive magnetic dipole transitions, so it does not contribute to the electric polarizability

At the same time, Fig. 1(b′) purely represents two consecutive magnetic dipole transitions, so it does not contribute to the electric polarizability. We summarize the (transition) magnetic dipole moments obtained from the quark model [40], leading order HB χPT calcu- lations [41], and lattice QCD simulations [46, 47, 49] in Tables II and III. The coeffici...
[7]

, (68) β(c′−g′) M Σ+ c = αemg2 3Rπ 144π2F 2π (M 2π − δ2
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, (69) β(c′−g′) M Σ0 c = αemg2 3Rπ 288π2F 2π (M 2π − δ2

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, (70) β(c′−g′) M Ξ′+ c = αemg2 3Rπ 576π2F 2π (M 2π − δ2
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+ αemg2 3RK 144π2F 2 K(M 2 K − δ2
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, (71) 11 β(c′−g′) M Ξ′0 c = αemg2 3Rπ 576π2F 2π (M 2π − δ2
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+ αemg2 3RK 576π2F 2 K(M 2 K − δ2
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, (72) β(c′−g′) M Ω0 c = αemg2 3RK 288π2F 2 K(M 2 K − δ2
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(74) In the heavy quark limit with g2 1 = 4 3 g2 3 [37, 80, 85, 86] and δ1 = 0 , we have Rχ = πMχ 2 , S χ = 5πM 3 χ

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Tree diagrams The tree diagrams in Figs. 1(a) and (b) yield the Thomson amplitude U (a+b) ξ (ω) = Q2 ξe2 MB6 , V (a+b) ξ (ω) = 0, (32) which does not contribute to the electromagnetic polarizabilities: α(a+b) E (ξ) = β(a+b) M (ξ) = 0. (33) The tree diagram with B∗ 6 as the intermediate state, shown in Fig. 1(b′), yields: U (b′) ξ (ω) = − e2C 2 ξ ω2 12M 2 ...

[2] [2]

heavy” and “light

We use the notations ψ¯3, ψ6 and ψ∗ 6 to denote the antitriplet, spin- 1 2 sextet, and spin- 3 2 sextet baryons, respectively. These baryon fields are represented as [80]: 3 ψ¯3,c =   0 Λ + c Ξ+ c −Λ+ c 0 Ξ 0 c −Ξ+ c −Ξ0 c 0   , ψ 6,c =   Σ++ c Σ+ c√ 2 Ξ′+ c√ 2 Σ+ c√ 2 Σ0 c Ξ′0 c√ 2 Ξ′+ c√ 2 Ξ′0 c√ 2 Ω0 c   , ψ ∗µ 6,c =  ...

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M. A. Ivanov, J. G. Korner, V . E. Lyubovitskij, and A. G. Rusetsky, Strong and radiative decays of heavy flavored baryons, Phys. Rev. D 60, 094002 (1999), arXiv:hep-ph/9904421

arXiv 1999

[4] [4]

X χ D(d+e) ξ,χ F 2χ Z 1 0 dx J ′ 2(ωx, 0, M2 χ) # , (38) U (f ) ξ (ω) =A

Loop diagrams Using the J -function defined in Appendix A, we can obtain the form factors of the B6ϕ-loop diagrams in Figs. 1(c)–(g): U (c) ξ (ω) =A X χ D(c) ξ,χ F 2χ J0(ω, 0, M2 χ), (37) U (d+e) ξ (ω) =A "X χ D(d+e) ξ,χ F 2χ Z 1 0 dx J ′ 2(ωx, 0, M2 χ) # , (38) U (f ) ξ (ω) =A "X χ D(f ) ξ,χ F 2χ Z 1 0 dx (1 − x)(d + 1)J ′′ 6 (ωx, 0, M2 χ) − X χ D(f ) ξ,...

[5] [5]

+ αemg2 3RK 288π2F 2 K(M 2 K − δ2

[6] [6]

1(b′) purely represents two consecutive magnetic dipole transitions, so it does not contribute to the electric polarizability

At the same time, Fig. 1(b′) purely represents two consecutive magnetic dipole transitions, so it does not contribute to the electric polarizability. We summarize the (transition) magnetic dipole moments obtained from the quark model [40], leading order HB χPT calcu- lations [41], and lattice QCD simulations [46, 47, 49] in Tables II and III. The coeffici...

[7] [7]

, (68) β(c′−g′) M Σ+ c = αemg2 3Rπ 144π2F 2π (M 2π − δ2

[8] [9]

, (69) β(c′−g′) M Σ0 c = αemg2 3Rπ 288π2F 2π (M 2π − δ2