One-pion exchange potential in a strong magnetic field

TL;DR

This work derives the leading-order one-pion exchange potential in a homogeneous magnetic field within chiral perturbation theory with nonrelativistic nucleons, formulating a gauge-invariant NN potential via a dressed-nucleon Wilson line. It shows that the magnetic field shortens the OPEP range in both longitudinal and transverse directions and induces marked anisotropy across isospin channels, with charged-pion propagation encoded through a Schwinger proper-time Green's function and, in the strong-field limit, a lowest-Landau-level dominance. By projecting onto the isospin-singlet (deuteron-relevant) and isospin-triplet channels, the study quantifies channel-dependent modifications to the potential and computes the deuteron energy shift through first-order perturbation theory, finding shifts of order 0.5–1 MeV near $|eB|\sim m_\pi^2$. The results reveal a nontrivial interplay between field-induced anisotropy and tensor forces, and point to important future directions, including Zeeman effects, heavier-meson contributions, lattice QCD approaches, and astrophysical implications for magnetars.

Abstract

We derive the one-pion exchange potential (OPEP) in the presence of a homogeneous magnetic field using chiral perturbation theory with nonrelativistic nucleons. Our approach is applicable not only to weak magnetic fields but also to strong ones up to around the pion-mass scale. The Green's function of charged pions is modified by the magnetic field, leading to changes in the nuclear force. By numerically evaluating the modified OPEP incorporating its spin and isospin dependencies, we show that the range of the potential decreases in both directions parallel and perpendicular to the magnetic field as the field strength increases. We also compute the resulting energy shift of the deuteron due to the modified OPEP, which can reach the order of 1 MeV around $|eB| = m_π^2$, which is comparable to the deuteron binding energy.

Figures (8)

• Figure 1: Magnetic-field dependence of the OPEP matrix element between the $| {S_z=+1} \rangle$ states in the isospin-singlet ($T=0$) and spin-triplet ($S=1$) channel, as given in Eq. \ref{['eq:potential-matrix-element-11']}. Left panel: potential along the magnetic field (longitudinal) direction; right panel: potential along the perpendicular (transverse) direction. For the plots, we use the vacuum values: $g_A=1.27,\; f_\pi=92\text{ MeV}$, and $m_\pi=138\text{ MeV}$.
• Figure 2: Magnetic-field dependence of the OPEP matrix element between the $| {S_z=0} \rangle$ states in the isospin-singlet ($T=0$) and spin-triplet ($S=1$) channel, as given in Eq. \ref{['eq:potential-matrix-element-00']}. Left panel: potential along the magnetic field (longitudinal) direction; right panel: potential along the perpendicular (transverse) direction. For the plots, we use the vacuum values: $g_A=1.27,\; f_\pi=92\text{ MeV}$, and $m_\pi=138\text{ MeV}$.
• Figure 3: Comparison of the exact result \ref{['eq:potential-matrix-element-11']} with the weak-magnetic-field expansion up to $\mathcal{O}( (eB)^2 )$ derived from Eq. (\ref{['eq:OPEP-pn-weak']}) for the OPEP matrix element between the $| {S_z=+1} \rangle$ states in the isospin-singlet ($T=0$) and spin-triplet ($S=1$) channel at $|eB|=m_\pi^2$.
• Figure 4: Comparison of the exact result \ref{['eq:potential-matrix-element-11']} with the strong-field LLL result based on Eq. (\ref{['eq:OPEP-pn-LLL']}) for the OPEP matrix element between the $| {S_z=+1} \rangle$ states in the isospin-singlet ($T=0$) and spin-triplet ($S=1$) channel at $|eB|=25m_\pi^2$.
• Figure 5: Magnetic-field dependence of the OPEP matrix element between the $| {S_z=0} \rangle$ states in the isospin-triplet ($T=1$, especially $T_z=0$) and spin-singlet ($S=0$) channel, as given in Eq. \ref{['eq:potential-matrix-element-00-isotriplet']}. Left panel: potential along the magnetic field (longitudinal) direction; right panel: potential along the perpendicular (transverse) direction. For the plots, we use the vacuum values: $g_A=1.27, f_\pi=92\ \text{MeV}$, and $m_\pi=138\ \text{MeV}$.