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Flavor decomposition of electromagnetic multipole moments in $[q_1q_2][\bar q_3 \bar c]$ tetraquarks: Light-quark analogs of $T_{cs1}(2900)^0$ with $J^P = 1^+$

This paper employs light-cone QCD sum rules to demonstrate that in $J^P=1^+$ $[q_1q_2][\bar q_3 \bar c]$ tetraquarks with a pseudoscalar antidiquark, the charm quark algebraically decouples from electromagnetic moments, leaving a distinct light-flavor pattern and non-zero quadrupole moments that serve as structural signatures to discriminate between compact and molecular internal architectures.

Original authors: Ulaş Özdem

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ulaş Özdem

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have mapped the subatomic world by sorting particles into neat families based on how they are built from quarks. This standard map works well for most matter, but in the last twenty years, experiments have uncovered a flood of strange new particles that do not fit the old categories. These are exotic states, often called tetraquarks, which appear to be made of four quarks bound together rather than the usual two or three. While scientists have confirmed their existence, a major mystery remains: how are these four pieces actually arranged inside? Are they tightly packed into a single, compact cluster, or are they more like two separate particles loosely orbiting each other, similar to a molecule? Answering this question is crucial because the internal arrangement dictates how the particle behaves, how it decays, and what forces hold it together. To solve this, researchers need to look beyond just the particle's mass and instead examine its electromagnetic personality—specifically, how it responds to magnetic fields and how its electric charge is distributed in space.

In a new study, Ulaş Özdem investigates a specific family of these exotic particles that contain a heavy charm quark and three lighter quarks. These particles are the light-quark cousins of a recently discovered state called Tcs1(2900)0, which has been observed by the LHCb collaboration. The researcher's goal was to calculate two key properties for these particles: their magnetic dipole moment, which measures how strongly they act like tiny magnets, and their electric quadrupole moment, which reveals whether their internal charge shape is perfectly round or stretched out like a rugby ball. To do this, the study used a sophisticated theoretical method called light-cone QCD sum rules. This approach allows physicists to connect the fundamental laws of quarks and gluons to the observable properties of the particle by simulating how the particle interacts with an external electromagnetic field. The study focused on a specific theoretical model where the four quarks are arranged into two pairs: a light diquark and a heavy antidiquark, with the heavy charm quark sitting inside a spin-zero pair.

The calculations produced a striking and unexpected result regarding the role of the heavy charm quark. In most theoretical models, the heavy quark contributes to the particle's magnetic properties, though its influence is often small because of its large mass. However, in this specific arrangement, the researchers found that the charm quark's contribution to the magnetic moment vanishes completely. It is not merely small; it is mathematically zero. This happens because of the specific way the charm quark is paired with another antiquark in a spin-zero state. The mathematical structure of this pairing causes the electric charge of the charm quark to cancel itself out entirely when calculating the magnetic response. This means that for this specific type of compact particle, the magnetic behavior is determined almost entirely by the three light quarks, while the heavy charm quark effectively disappears from the magnetic picture. This finding stands in sharp contrast to other theoretical models where the charm quark sits in a different type of pair, or to models where the particle is a loose molecule; in those cases, the charm quark does contribute, albeit with a different strength.

Beyond the magnetic moment, the study also mapped out the electric shape of these particles. The results showed that the charge distribution is never perfectly spherical; it is always deformed. For particles containing two up quarks, the shape is elongated along the axis of rotation, resembling a prolate spheroid. For those with two down quarks, the shape is flattened, like an oblate spheroid. When the particle contains a mix of up and down quarks, the deformation is smaller due to partial cancellations between the different quark charges. The study also calculated the ratio between the electric shape and the magnetic strength. This ratio serves as a structural fingerprint that does not depend on the overall size of the particle, offering a clean way to compare different theoretical models. The researchers found that for the compact models they studied, this ratio falls within a specific, consistent range, whereas models treating the particle as a loose molecule predict a ratio that is vastly different, with much larger magnetic moments and much smaller electric deformations.

The significance of these findings lies in their ability to distinguish between competing theories of what these particles actually are. Because the charm quark's contribution is exactly zero in this specific compact model but non-zero in others, measuring the magnetic properties of these particles in a future experiment could tell scientists which internal structure is correct. If an experiment finds that the charm quark plays no role in the magnetic moment, it would strongly support the idea that these particles are compact clusters with the heavy quark in a spin-zero pair. If the charm quark does contribute, it would point toward a different internal arrangement or a molecular structure. The study provides a clear set of predictions for the magnetic and electric moments of these particles, offering a concrete target for experimentalists at facilities like the LHCb, Belle II, and BESIII. By comparing these precise theoretical numbers with future measurements, physicists will be able to finally determine the true architecture of these exotic states, moving from simply knowing they exist to understanding exactly how they are built.

Technical Summary: Flavor Decomposition of Electromagnetic Multipole Moments in $[q_1q_2][\bar{q}_3\bar{c}]$ Tetraquarks

Problem and Motivation
Recent hadron spectroscopy has revealed a proliferation of states, particularly in the charm-strange sector (e.g., the $T_{cs}(2900)$ family), that defy conventional quark-model classification. While masses and decay widths provide initial characterization, they are often insufficient to distinguish between competing structural interpretations, such as compact tetraquarks, hadronic molecules, or other configurations. Electromagnetic multipole moments—specifically the magnetic dipole moment ( $\mu$ ) and electric quadrupole moment ( $D$ )—serve as critical diagnostic tools. The magnetic moment probes the internal spin and current structure, while the quadrupole moment measures deviations from spherical symmetry in the charge distribution. This work addresses the need for such observables in the light-quark analogs of the $T_{cs1}(2900)^0$ family, specifically $[q_1q_2][\bar{q}_3\bar{c}]$ tetraquark states with quantum numbers $J^P = 1^+$ .

Methodology
The study employs the Light-Cone QCD Sum Rule (LCSR) framework in the presence of an external electromagnetic field.

Interpolating Current: The author utilizes a diquark–antidiquark interpolating current where the light diquark ( $q_1q_2$ ) is axial-vector and the heavy antidiquark ( $\bar{q}_3\bar{c}$ ) is pseudoscalar. The current is constructed as:
$J_\mu(x) = [q_1^T C \gamma_\mu q_2][\bar{q}_3 \gamma_5 C \bar{c}^T] - [q_1^T C \gamma_\mu q_2][\bar{q}_3 \gamma_5 C \bar{c}^T]$
(with appropriate color indices).
Correlation Function: The calculation is based on the correlation function of two currents in an external electromagnetic field, expanded via the Operator Product Expansion (OPE) near the light cone.
Photon Couplings: The analysis includes both perturbative photon emission (photon coupling directly to quark lines) and non-perturbative effects encoded in photon distribution amplitudes (DAs) up to twist-4. Non-perturbative couplings are restricted to light quarks ( $u, d$ ), as heavy-quark DAs are suppressed by $1/m_c$ .
Flavor Decomposition: A systematic flavor decomposition is performed across all admissible $u/d$ configurations ($uu, dd, ud, du$) for the light diquark and the light antiquark ( $\bar{q}_3$ ).
Numerical Analysis: The sum rules are evaluated using standard input parameters (charm mass, condensates, photon DAs) and auxiliary parameters (Borel mass $M^2$ and continuum threshold $s_0$ ). Stability windows are defined by ensuring pole contribution dominance (>40%) and OPE convergence (<5% for dimension-7 terms).

Key Results
The numerical analysis yields the magnetic dipole and electric quadrupole moments for eight distinct flavor configurations.

Magnetic Moments ( $\mu$ ):
- Flavor Pattern: States with $uu$ light diquarks exhibit large positive moments ( $0.66 - 0.87 \, \mu_N$ ). States with $dd$ diquarks show negative moments ($-0.33$ to $-0.54 \, \mu_N$ ). Mixed-flavor ($ud/du$) configurations yield significantly smaller magnitudes ( $0.06 - 0.27 \, \mu_N$ ) due to partial cancellations.
- Charm Quark Contribution: A critical finding is that the charm-quark contribution to the magnetic moment vanishes algebraically ( $\mu_c = 0$ ) across all configurations. This is not a result of dynamical $1/m_c$ suppression but an operator-level consequence of the specific spin organization where the charm antiquark resides in a pseudoscalar antidiquark. The Dirac trace associated with the $\gamma_5$ coupling eliminates the charm charge $e_c$ from the sum rules.
Electric Quadrupole Moments ( $D$ ):
- All channels yield non-zero quadrupole moments, indicating intrinsically deformed charge distributions.
- $uu$-dominated states are prolate ( $D > 0$ , $\approx 2.0 \times 10^{-2} \, \text{fm}^2$ ), while $dd$-dominated states are oblate ( $D < 0$ , $\approx -1.0 \times 10^{-2} \, \text{fm}^2$ ). Mixed configurations show smaller magnitudes ( $\approx 0.49 \times 10^{-2} \, \text{fm}^2$ ).
Multipole Ratio ( $D/\mu$ ):
- The dimensionless ratio $D/\mu$ is identified as a structural observable independent of the current normalization.
- For non-suppressed configurations, the ratio falls in the range $1.2 - 2.1$.
- Configurations with suppressed magnetic moments (due to cancellation) yield anomalously high ratios ( $\approx 5.6$ ), which are attributed to kinematic divergence rather than structural properties.

Significance and Claims
The paper posits that the algebraic decoupling of the charm contribution is a "methodological signature" that discriminates among different internal organizations of singly heavy tetraquarks.

Operator-Level Discriminator: The result $\mu_c = 0$ is specific to the current architecture where the heavy quark is in a pseudoscalar antidiquark. In contrast, alternative compact constructions (e.g., where the heavy quark is in a vector antidiquark, as in the $X_{AV}$ state) yield non-vanishing charm contributions via standard OPE. Hadronic molecular models yield dynamically suppressed (but non-zero) charm contributions.
Structural Probe: The author argues that while the flavor decomposition itself is not directly observable, the distinct total moments predicted by inequivalent architectures serve as a diagnostic. If a physical candidate is dominated by the specific spin organization analyzed here, its electromagnetic moments should exhibit the predicted algebraic decoupling and flavor patterns.
Phenomenological Utility: The predicted moments and the $D/\mu$ ratio provide targets for future experimental analyses of radiative transitions ( $T \to T\gamma$ ) at facilities like LHCb and Belle II. The paper suggests that combining these low-energy electromagnetic observables with high- $p_T$ fragmentation production data could constrain the compact vs. molecular nature of such states from opposite kinematic ends.

The work concludes that the electromagnetic flavor patterns and the specific algebraic behavior of the charm contribution constitute a coherent structural signature for this specific class of compact tetraquark configurations.

Technical Summary: Flavor Decomposition of Electromagnetic Multipole Moments in [q1q2][qˉ3cˉ][q_1q_2][\bar{q}_3\bar{c}][q1​q2​][qˉ​3​cˉ] Tetraquarks

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Technical Summary: Flavor Decomposition of Electromagnetic Multipole Moments in $[q_1q_2][\bar{q}_3\bar{c}]$ Tetraquarks