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Magnetic moments of the $\Sigma_c$ and $\Sigma_b$ baryons from external-field QCD sum rules

Using external-field QCD sum rules with full heavy-quark mass dependence and a novel normalization strategy to fix uncertainties, this study computes the magnetic moments of $\Sigma_c$ and $\Sigma_b$ baryons, revealing that these moments are primarily carried by the light diquark while the heavy-quark contribution remains small.

Original author: Halil Mutuk

Published 2026-11-10

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CC BY 4.0

Original author: Halil Mutuk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Matter at its most fundamental level is built from particles called quarks, which are bound together by the strong nuclear force to form protons, neutrons, and a vast family of heavier cousins known as baryons. Among these are the singly heavy baryons, exotic particles that contain one very heavy quark—either a charm or a bottom quark—paired with two lighter quarks. These particles are fleeting, decaying almost instantly after creation, yet they hold a key to understanding how the magnetic properties of matter are shared between heavy and light components. Just as a spinning top has a magnetic field, these particles possess a magnetic moment, a measure of how strongly they interact with magnetic fields. While scientists have long measured these properties for ordinary protons and neutrons, the magnetic moments of these heavy baryons have remained unmeasured, leaving a gap in our understanding of how the heavy quark and the lighter pair of quarks cooperate to create the particle's overall magnetic character.

A team of physicists has now filled this gap by calculating the magnetic moments of the Sigma-c and Sigma-b baryons using a sophisticated theoretical framework known as QCD sum rules. Instead of trying to catch these short-lived particles in a detector, the researchers used the mathematical machinery of quantum chromodynamics, the theory that describes how quarks interact, to predict how these particles would behave in a magnetic field. They constructed a detailed model that accounts for the heavy quark's full mass and its complex interactions with both photons and the gluons that bind the quarks together. A crucial step in their work was to calibrate their model using the well-known magnetic moment of the proton, which allowed them to pin down a major source of uncertainty that had previously clouded such calculations. By normalizing their results against the known mass of the baryons, they were able to extract precise predictions for the magnetic moments of the entire family of Sigma-c and Sigma-b particles.

The results reveal a clear picture of how magnetism is distributed within these heavy particles. The calculations show that the magnetic moment is carried almost entirely by the pair of light quarks, while the heavy quark contributes very little. For the Sigma-c baryons, which contain a charm quark, the heavy quark's contribution is a small negative value, slightly reducing the overall magnetic moment. For the Sigma-b baryons, containing a bottom quark, the heavy quark's contribution is even smaller and slightly positive. The team provided specific values for each member of these families: the positively charged Sigma-c has a magnetic moment of 2.37, the neutral one is -1.39, and the singly charged one is 0.50, all measured in nuclear magnetons. The corresponding values for the Sigma-b family are 2.16, -1.06, and 0.55. These numbers come with a range of uncertainty, reflecting the limits of current theoretical methods, but they represent the most comprehensive calculation of its kind to date.

The researchers found that their predictions align well with other theoretical approaches, such as lattice QCD simulations and chiral perturbation theory, particularly when those methods use specific mathematical techniques to account for the behavior of quarks at different energy scales. However, the results also highlight a significant dependence on the choice of the heavy quark's mass, a detail that remains a point of discussion in the field. The study confirms that in the heavy-quark limit, the light diquark acts as the primary source of magnetism, effectively decoupling from the heavy quark's spin. This finding supports the idea that as the heavy quark becomes infinitely massive, its influence on the magnetic moment diminishes, leaving the light quarks to dictate the particle's magnetic response.

While these calculations are theoretical, they are being prepared for a direct test. Experiments are currently being designed to measure the magnetic moments of charm baryons by observing how they precess, or wobble, as they travel through strong electric fields in bent crystals. Although the Sigma particles decay too quickly to be measured directly in these setups, the planned measurements of other charm baryons will help constrain the heavy-quark contribution, providing a real-world check on the theoretical predictions made in this study. The work stands as a significant step forward in mapping the magnetic landscape of heavy matter, offering a detailed map that future experiments can use to verify how the fundamental forces of nature shape the properties of the universe's building blocks.

Technical Summary: Magnetic Moments of the $\Sigma_c$ and $\Sigma_b$ Baryons from External-Field QCD Sum Rules

Problem and Motivation
The magnetic moments of singly heavy baryons ( $\Sigma_Q$ , where $Q=c,b$ ) serve as probes for understanding how magnetic response is distributed between a heavy quark and a light diquark. In the heavy-quark limit, the light-diquark spin decouples from the heavy-quark spin, implying that the light-diquark contribution should be identical for $\Sigma_c$ and $\Sigma_b$ , while the heavy-quark contribution is suppressed by $e_Q/m_Q$ . Despite upcoming experimental efforts to measure charm-baryon moments (e.g., via spin precession in bent crystals at the LHC), no direct measurements of $\Sigma_Q$ moments currently exist. Existing theoretical predictions from quark models, lattice QCD, chiral perturbation theory, and QCD sum rules show significant variation, particularly for the $\Sigma_c$ triplet, where predictions for $\Sigma_c^{++}$ range from 1.3 to 3.1 $\mu_N$ .

Methodology
This work employs the external-field QCD sum rule method to compute the magnetic moments of the $\Sigma_c$ and $\Sigma_b$ isotriplets. The study introduces several refinements over previous external-field analyses (specifically Ref. [22]):

Full Heavy-Quark Mass Dependence: The heavy-quark propagator is retained with its full mass dependence. This includes couplings to the photon, soft gluons, simultaneous photon-gluon couplings, and the gluon condensate. The propagator is generated recursively in momentum space within the Fock–Schwinger gauge.
Normalization Strategy: The magnetic sum rule is normalized by the mass sum rule derived from the same interpolating current and Operator Product Expansion (OPE). This ratio method eliminates the residue $\tilde{\lambda}^2$ and uses the mass sum rule as an internal consistency check, yielding baryon masses within a few percent of experimental values.
Calibration of Susceptibility $\chi$ : The quark-condensate magnetic susceptibility $\chi$ , which typically dominates the uncertainty, is not treated as a free parameter with a wide prior. Instead, it is fixed in each Monte Carlo sample by requiring the sum rule to reproduce the experimental proton magnetic moment ( $\mu_p = 2.793 \, \mu_N$ ) using the same interpolating field (with $Q \to d$ ). This calibration reduces the uncertainty significantly and ensures consistency between nucleon and heavy-baryon sum rules.
Comprehensive Uncertainty Analysis: A Monte Carlo analysis is performed with 1000 input sets to evaluate uncertainties from QCD inputs (condensates, masses). Additional systematic uncertainties are added in quadrature, covering:
- Interpolating field dependence (variation of the mixing parameter $\beta$ ).
- Borel window selection.
- Heavy-quark mass scale ( $\overline{\text{MS}}$ vs. pole mass schemes).
- Normalization procedure (comparing the ratio method with the conventional residue extraction).
- Calibration of $\chi$ .

The OPE includes terms up to operator dimension eight, incorporating the gluon condensate and mixed condensates. The analysis uses $\overline{\text{MS}}$ heavy-quark masses ( $m_c(m_c)$ and $m_b(m_b)$ ).

Key Contributions

First External-Field Analysis of $\Sigma_b$ : This is the first calculation of $\Sigma_b$ magnetic moments using the external-field method.
Separation of Contributions: The linear dependence of the OPE on electric charges allows for the separation of light-diquark and heavy-quark contributions to the magnetic moment.
Refined Error Budget: By calibrating $\chi$ against the proton moment and performing a full Monte Carlo analysis, the study provides a more rigorous error budget than previous works, explicitly accounting for the dependence on the interpolating field and mass schemes.
Validation: The construction is validated by reproducing known perturbative and condensate contributions for pseudoscalar heavy-light correlators and by recovering proton sum rules in the limit $m_Q \to 0$ .

Results
Using $\overline{\text{MS}}$ heavy-quark masses, the computed magnetic moments (in nuclear magnetons, $\mu_N$ ) are:

$\mu_{\Sigma_c^{++}} = 2.37 \pm 0.65$
$\mu_{\Sigma_c^{+}} = 0.50 \pm 0.15$
$\mu_{\Sigma_c^{0}} = -1.39 \pm 0.37$
$\mu_{\Sigma_b^{+}} = 2.16 \pm 0.65$
$\mu_{\Sigma_b^{0}} = 0.55 \pm 0.16$
$\mu_{\Sigma_b^{-}} = -1.06 \pm 0.32$

The results depend sizably on the heavy-quark mass scheme; using pole masses shifts the central values significantly (e.g., $+0.67 \, \mu_N$ for $\Sigma_c^{++}$ ), though pole masses fail to satisfy the pole dominance criteria in the mass sum rule for the chosen windows.

Light vs. Heavy Quark Contributions:
The magnetic moments are found to be carried primarily by the light diquark. The heavy-quark contribution is:

$\Sigma_c$ : $-0.13 \pm 0.02 \, \mu_N$
$\Sigma_b$ : $+0.013 \pm 0.003 \, \mu_N$
These values depend on the choice of the interpolating field (parameter $\beta$ ).

Significance and Comparison
The results agree with the chiral quark-soliton model for all six states and with the quadratic chiral extrapolation of recent lattice QCD calculations for $\Sigma_c$ . They are also consistent with recent light-cone QCD sum rule results within approximately one standard deviation. The study notes that while the central values for $\Sigma_c$ are slightly larger in magnitude than those for $\Sigma_b$ , the uncertainties allow for compatibility with the heavy-quark limit (equal moments for corresponding states).

The paper concludes that the primary limitations of the current analysis are the absence of $O(\alpha_s)$ and anomalous-dimension corrections, the leading-order treatment of the heavy-quark mass scheme, and the use of an effective $\chi$ calibrated on the proton. The dependence on the heavy-quark mass scheme remains the main open issue at this order. The calibration of $\chi$ via the proton moment is identified as the most effective step in reducing the dominant source of uncertainty.

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