Dyonic hairy black holes in $U(1)$ gauge-invariant scalar-vector-tensor theories: Cubic and quartic interactions

Abstract

We construct and classify asymptotically flat, static, spherically symmetric hairy black hole solutions in $U(1)$ gauge-invariant scalar-vector-tensor (SVT) theories carrying both electric and magnetic charges. Extending previous studies beyond the quadratic sector, we systematically incorporate cubic and quartic interaction terms in the presence of the magnetic charge. We derive a consistency condition for the quartic interaction that eliminates higher-order derivative terms induced by the magnetic charge, ensuring the theory remains second-order. We classify the obtained solutions based on their symmetry properties: shift-symmetric couplings yield secondary hair governed by the Noether current, whereas $φ$-dependent interactions generate primary hair. Crucially, our analysis reveals that the magnetic charge plays a key role in activating specific interaction sectors such as the cubic coupling $\tilde{f}_3$, which does not appear in the field equations in purely electric configurations. We identify solution branches that are intrinsic to the magnetic charge, as they cease to exist in the vanishing monopole limit ($P\to 0$). Furthermore, we demonstrate that the scalar hair exhibits distinct asymptotic decay rates depending on the interaction type, suggesting possible variations in observational signatures. Finally, we verify the global regularity of these solutions by connecting analytic expansions with numerical integration.

Figures (5)

• Figure 1: Radial profiles of $f$, $h$, $V$, $V'$, $\phi$, and $\phi'$ for the f3-Ia solution. The fields $V$ and $\phi$ are normalized by $M_{\rm pl}$, while their derivatives are normalized by $M_{\rm pl}/r_h$. Boundary conditions are imposed at $r=1.001\,r_h$ based on the near-horizon expansion Eqs. \ref{['f3-sol1-shift-f1h1V1phi1']}-\ref{['f3-sol1-shift-secondorder']}, with parameters $\mu=0.5$, $\phi^{(0)}=M_{\rm pl}$, $P=0.5\,M_{\rm pl}\,r_h$, and $c_3=1.0$.
• Figure 2: Scalar field derivative profiles. We show the radial profile of $\phi'$ normalized by $M_{\rm pl}/r_h$ for three representative magnetic charges, $P=0$ (purely electric), $P=0.5\,M_{\rm pl}\,r_h$, and $P=M_{\rm pl}\,r_h$. Note that the sign of $\phi'$ is reversed (negative) only for the case $P=M_{\rm pl}\,r_h$. Other parameters are fixed as $\mu=0.5$, $\phi^{(0)}=M_{\rm pl}$, and $c_3=1.0$.
• Figure 3: Radial profiles of the metric functions ($f, h$), vector field ($V$), and scalar field ($\phi$), along with their derivatives ($V', \phi'$), in the exterior region. Normalization is the same as in Fig. \ref{['Fig-f3-1a-1']}. Boundary conditions are set at $r=1.001\,r_h$ consistent with Eqs. \ref{['g3-sol1-shift-first-order']}-\ref{['g3-sol1-shift-second-order']}, with parameters $\mu=0.5$, $\phi^{(0)}=M_{\rm pl}$, $P=0.5\,M_{\rm pl}\,r_h$, and $\tilde{c}_3=1.0$. Note the slower decay of $\phi'$ compared to the $f_3$-Ia case (Fig. \ref{['Fig-f3-1a-1']}).
• Figure 4: Radial profiles for the $f_3$-IIa solution. We numerically solve for the radial dependence of the metric functions $f$ and $h$, the vector field $V$, and the scalar field $\phi$, along with their derivatives $V'$ and $\phi'$, in the exterior region of the event horizon. The fields $V$ and $\phi$ are normalized by $M_{\rm pl}$, while their radial derivatives $V'$ and $\phi'$ are normalized by $M_{\rm pl}/r_h$. Boundary conditions are imposed at $r=1.001\,r_h$ consistent with the near-horizon expansions in Eqs. \ref{['f3-sol1-linear']}-\ref{['f3-sol1-phi2']}, with the parameters set to $\mu=0.7$, $\phi^{(0)}=M_{\rm pl}$, $P=0.1\,M_{\rm pl}\,r_h$, and $C_3=-3.0$.
• Figure 5: Radial profiles for the $g_3$-IIb solution. We numerically solve for the radial dependence of the fields from the horizon to the exterior region. Normalization is the same as in Fig. \ref{['Fig-f3-2a']}. Boundary conditions are imposed at $r=1.001\,r_h$ consistent with Eqs. \ref{['g3-sol2-shift-f1h1']} and \ref{['g3-sol2-shift-phi1']} and the Supplemental Material, with parameters $\mu=0.5$, $\phi^{(0)}=M_{\mathrm{pl}}$, $P=0.5\,M_{\mathrm{pl}}\,r_h$, $\hat{C}_3=-10$, and $\hat{D}_3=3000$.