Super-extremal black holes in the quasitopological electromagnetic field theory

It has recently been proved that a simple generalization of electromagnetism, referred to as quasitopological electromagnetic field theory, is characterized by the presence of dyonic black-hole solutions of the Einstein field equations that, in certain parameter regions, are characterized by four horizons. In the present compact paper we reveal the existence, in this non-linear electrodynamic field theory, of super-extremal black-hole spacetimes that are characterized by the four degenerate functional relations $[g_{00}(r){]}_{r=r_{\text{H}}}=[d{g}_{00}(r)\mathrm{/}dr{]}_{r=r_{\text{H}}}=[d^2{g}_{00}(r)\mathrm{/}d{r}^2{]}_{r=r_{\text{H}}}=[d^3{g}_{00}(r)\mathrm{/}d{r}^3{]}_{r=r_{\text{H}}}=0$, where $g_{00}(r)$ is the $tt$-component of the curved line element and $r_{\text{H}}$ is the black-hole horizon radius. In particular, using analytical techniques we prove that the quartically degenerate super-extremal black holes are characterized by the universal (parameter-{\it independent}) dimensionless compactness parameter $M\mathrm{/}{r}_{\text{H}}=\frac{2}{3}(2\gamma +1)$, where $\gamma \equiv {}_2{F}_1(1\mathrm{/}4,1;5\mathrm{/}4;-3)$.