The direct effect of a toroidal magnetic field on stellar oscillations. A revised expression for the general matrix element

Magnetic fields affect stellar oscillations through eigenmode mixing. If the contribution of a magnetic field to the stellar oscillation frequency is weak, this mode coupling can be determined using perturbation theory. The general matrix element between two modes quantifies their coupling strength. This study revises the calculation of the direct effect of a subsurface toroidal magnetic field on solar-like stellar oscillations and provides corrected expressions for the general matrix element and the corresponding angular kernels. We perturbed a non-magnetic, non-rotating equilibrium stellar model using a superposition of toroidal magnetic fields. Applying quasi-degenerate perturbation theory and generalised spherical harmonics, we derived the general matrix element and computed the shifts and splitting of the multiplet frequencies. We identified selection rules for mode coupling by analysing the symmetries of the angular kernels. Previous studies underestimated the direct effect of a toroidal magnetic field by neglecting non-zero contributions in integrals over products of scalar generalised spherical harmonics. We have developed an analytical method to calculate these integrals. For a strong toroidal field located at the base of the solar convection zone, the frequency shift due to the direct effect can reach 14 nHz, which is twice as large as previously reported. Modes couple through the toroidal field only if their azimuthal orders are equal and the sum of the harmonic degrees of the modes and the magnetic field components is even. The toroidal field only partially lifts the degeneracy of multiplet frequencies, leaving modes with the same absolute azimuthal order m within a multiplet degenerate. We confirm that deep toroidal magnetic fields are not responsible for the observed frequency shifts between the maximum and minimum of the solar cycle. However, although their detection is strongly impeded by the much stronger effects of near-surface fields, their existence cannot be ruled out.