Octet baryons in large magnetic fields

Magnetic properties of octet baryons are investigated within the framework of chiral perturbation theory. Utilizing a power counting for large magnetic fields, the Landau levels of charged mesons are treated exactly giving rise to baryon energies that depend nonanalytically on the strength of the magnetic field. In the small-field limit, baryon magnetic moments and polarizabilities emerge from the calculated energies. We argue that the magnetic polarizabilities of hyperons provide a testing ground for potentially large contributions from decuplet pole diagrams. In external magnetic fields, such contributions manifest themselves through decuplet-octet mixing, for which possible results are compared in a few scenarios. These scenarios can be tested with lattice QCD calculations of the octet baryon energies in magnetic fields.

Figure 1

Charged meson propagator in a large magnetic field. Free meson propagators are shown as solid lines; external fields are depicted as wiggly lines ending in crosses; charge couplings appearing in the $\mathcal{O}({\epsilon }^2)$ chiral Lagrangian density are depicted by filled circles, and these must be summed to obtain the propagator (dashed line) in the large-field power counting.

Figure 2

Decuplet pole diagram that produces $\mathcal{O}({\epsilon }^3)$ contributions to the octet baryon energies. The octet baryons are shown with solid lines, while decuplet baryons are shown with double lines. External fields are shown with wiggly lines terminating in crosses.

Figure 3

Loop contributions to the octet baryon energies at $\mathcal{O}({\epsilon }^3)$. The single lines represent octet baryons, the double lines represent decuplet baryons, and wiggly lines represent the external magnetic field. The dashed lines represent mesons propagating in the magnetic field; see the diagrammatic depiction in Fig. 1. While generated from couplings in the Lagrangian density, diagrams in the first row identically vanish.

Figure 4

Behavior of the nonanalytic loop functions $F_1(x,y)$ and $F_2(x,y)$ appearing in Eqs. (26) and (29), respectively. The left panel shows the $x$-dependence of $F_1(x,0)$ (solid), along with the small- and large-$x$ asymptotic behavior (dashed and dotted). Asymptotic approximations are labeled by the order to which they are valid. Notice that the small-$x$ expansions have been plotted beyond their range of applicability. The right panel shows the $y$-dependence of each function evaluated at $x=1$, which corresponds to a magnetic field strength satisfying $|eB|=m_{\varphi }^2$. The range of $y$ values plotted encompasses the values required by the intermediate-state baryons in loop diagrams. Notice that the functions become identical for $y=0$.

Figure 5

Assessment of decuplet-octet mixing arising from the Hamiltonian in Eq. (38). The spin-up (black) and spin-down (blue) eigenstate energies are plotted as a function of the magnetic field. These energy shifts, $\mathrm{\Delta }E=E-M_B$, are given in units of the nucleon mass, with the solid curves corresponding to the full solution in Eq. (39), and the dashed curves representing the approximation in Eq. (20), which retains mixing only perturbatively through the decuplet-pole contribution.

Figure 6

Assessment of perturbative versus nonperturbative mixing in the ${\mathrm{\Delta }}^0–n$ system. On the left, the mixing angle in Eq. (42) is plotted for the spin-up (black) and spin-down (blue) states. On the right, the energy shifts are plotted as a function of the magnetic field, with solid curves corresponding to the result in Eq. (39) for the two spin states. Color correlated dashed and dotted curves show the $\mathcal{O}(B^2)$ and $\mathcal{O}(B^4)$ approximations to Eq. (39), respectively. Despite larger mixing, the spin-up energy is well described by perturbation theory in the magnetic field. The expansion for the spin-down energy can be considerably improved by the resummation into an expansion in $\xi$, see Eq. (43), which is shown to $\mathcal{O}({\xi }^2)$ for both spin states as the (red) dotted curves. In the spin-up case, this resummation introduces a pole, which is responsible for the rapid divergence from the solid black curve.

Figure 7

Magnetic-field dependence of the $I_3\ne 0$ baryon energy shifts, $\mathrm{\Delta }E=E-M_B$. The black curves correspond to spin-up states, while the lighter (blue) curves correspond to spin down. The dashed curves only account for effects up to $\mathcal{O}(B^2)$, see Eq. (56), where the magnetic polarizability is taken to be that in Eq. (55). The solid curves additionally include nonperturbative effects beyond $\mathcal{O}(B^2)$, which include loop corrections and mixing with decuplet baryons; see Eq. (57).

Figure 8

Energy eigenvalues in the ${\mathrm{\Sigma }}^0\text{−}\mathrm{\Lambda }$ system as a function of the magnetic field for spin-up (black) and spin-down (blue) states. These energies are shifted relative to the mass of the $\mathrm{\Lambda }$ baryon, $\mathrm{\Delta }E=E-M_{\mathrm{\Lambda }}$, and plotted in units of the nucleon mass. The dashed curves indicate results using the $\mathcal{O}(B^2)$ approximation in Eq. (b5), while the solid curves include contributions from charged-meson loops and mixing with the ${\mathrm{\Sigma }}^{*0}$ baryon, determined from the Hamiltonian in Eq. (b4).

Figure 9

Assessment of the magnetic field dependence of eigenstate energies in the coupled ${\mathrm{\Sigma }}^{*0}–{\mathrm{\Sigma }}^0–\mathrm{\Lambda }$ system. On the left, solid curves correspond to the energy shifts, $\mathrm{\Delta }E=E-M_{\mathrm{\Lambda }}$, of the eigenstates of Eq. (b1), which treats the magnetic moments of the $I_3=0$ baryons to all orders, with black curves for spin-up states and blue curves for spin-down states. The dashed curves correspond to the energies obtained by treating mixing with the ${\mathrm{\Sigma }}^{*0}$ baryon perturbatively, according to Eq. (b2). On the right, the mixing angles $\theta$ and $\varphi$ of Eq. (b3) are plotted as a function of the magnetic field, with black for spin-up states and blue for spin-down states.