Probing the Internal Structure of $X(3872)$ via Magnetic Moment: Distinguishing Color-Singlet and Compact Configurations

Abstract

The nature of the $X(3872)$ exotic hadron remains one of the most debated questions in hadron spectroscopy. We calculate its magnetic moment within a non-relativistic quark model that includes the three-body force arising from the cubic Casimir operator of $SU(3)_C$ with consistent dimensional analysis. Unlike previous works that used an uncontrolled large coupling, we fix the three-body strength using the realistic value of $\sim 10$--$20$ MeV extracted from the recent analysis of Noh et al.~(2024) based on lattice QCD and baryon spectroscopy. We find that the magnetic moment predictions fall into two distinct regions: the pure color-singlet configuration yields $\mu_X = -0.99 \pm 0.02\,\mu_N$, while the compact configurations yield $\mu_X = -1.17$ to $-1.23\,\mu_N$. The difference between the two compact scenarios ($0.06\,\mu_N$) is comparable to the systematic uncertainty of the model ($\sim 0.15\,\mu_N$) and should be interpreted with caution. However, the distinction between the pure color-singlet and the compact scenarios ($\sim 0.18$--$0.24\,\mu_N$) is larger than the estimated model systematic uncertainty and may provide a qualitative structural indicator. We emphasize that the pure color-singlet configuration is a simplified proxy for a molecule and does not represent a physical $D^0\bar{D}^{*0}$ molecule with large spatial extent. A full molecular treatment would require coupled-channel dynamics and long-range pion-exchange potentials, which are beyond the scope of this work. We also resolve a long-standing dimensional inconsistency in the cubic Casimir three-body force formulation and demonstrate through a detailed sensitivity and uncertainty analysis that our conclusions are robust for the physically relevant range of the three-body coupling.