Structure of heavy quarkonia in a strong magnetic field

TL;DR

The paper addresses how strong magnetic fields modify heavy quarkonia structure by solving a nonrelativistic quark model with cylindrical symmetry using CGEM to obtain rest-frame wave functions and then mapping them to light-front wave functions. Key findings include pronounced transverse momentum broadening of the LFWFs driven by Landau-level dynamics, while the leading-order longitudinal momentum distributions of ground states remain largely unchanged; excited states show significant reshaping and nodal reorganization near avoided crossings due to spin-magnetic mixing. A phenomenological relativistic correction to the Landau term is shown to modify longitudinal PDFs and can reverse trends in $k_z$-squared as a function of the field, highlighting relativistic effects not captured in the base nonrelativistic framework. These results help connect nonrelativistic quark-model insights with relativistic light-front approaches, offering qualitative guidance for lattice QCD studies and providing momentum-space observables relevant to high-energy processes in strong magnetic backgrounds.

Abstract

We investigate the structural modifications of heavy quarkonia in the presence of strong magnetic fields using a constituent quark model. By incorporating the effects of spin mixing and quark Landau levels, we employ a nonrelativistic Hamiltonian that captures the essential features of quark dynamics in a magnetic field. The two-body Schrödinger equation is solved using the cylindrical Gaussian expansion method, which respects the cylindrical symmetry induced by a magnetic field. We extract the corresponding light-front wave function (LFWF) densities and analyze their transverse and longitudinal structures, revealing characteristic features such as transverse momentum broadening. While the longitudinal structure is only slightly modified within the nonrelativistic Hamiltonian, we discuss some corrections that can significantly affect its longitudinal structure. Furthermore, we discuss the structure modifications of excited states and find notable changes in the LFWF densities, and state reshuffling near avoided crossings. These results demonstrate the sensitivity of hadron structure to external magnetic fields and help bridge our understanding to relativistic approaches.

Figures (15)

• Figure 1: Magnetic-field dependence of the mass spectra: (left panel) $J/\psi_T$ and $\psi(2S)_T$ states; (right panel) $\eta_c(1S, 2S)$, $J/\psi_L$, and $\psi(2S)_L$ states. The Landau level primarily affects the masses of the transverse vector charmonia. Level repulsion and avoided crossings appear in the right panel due to state mixing. Note that we plot a dense set of data points so that the curves appear continuous.
• Figure 2: LFWF densities of the transverse vector charmonia at several different magnetic field strengths: (Top panels) $J/\psi$ states and (Bottom panels) $\psi(2S)$ states. The densities become more elongated in the transverse direction as the magnetic field increases.
• Figure 3: LFWF densities of pseudoscalar and longitudinal vector charmonia at several different magnetic field strengths: (Top panels) $2^\text{nd}$ state and (Bottom panels) $1^\text{st}$ state. The shape of the $2^\text{nd}$ state at certain magnetic fields changes significantly due to state mixing and avoided crossing.
• Figure 4: Magnetic-field dependence of $\expval{k^2}$ and the anisotropy parameter $\epsilon_{\mathrm{LF}}$ for $J/\psi_T$ and $\psi(2S)_T$ states. Transverse momentum broadening is one of the dominant effects of the magnetic field. The rapid increase in the longitudinal momentum of $\psi(2S)_T$ at low magnetic field arises from the redistribution of the LFWF density.
• Figure 5: Close-up view of the weak-field region in Fig. \ref{['fig:charm_asym']}, where $\left\langle \tfrac{1}{2}k_\perp^2 \right\rangle$ for $J/\psi_T$ and $\psi(2S)_T$ are plotted, and the typical LFWFs of $\psi(2S)_T$ in three different regimes: (i) momentum increase due to elongation of LFWFs with transverse nodes, (ii) momentum decrease due to the node disappearance, and (iii) momentum increase due to elongation of LFWFs with no transverse node.