Potential NRQCD for unequal masses and the $B_c$ spectrum at NNNLO

We determine the $1\mathrm{/}m$ and $1\mathrm{/}{m}^2$ spin-independent heavy quarkonium potentials in the unequal mass case with $\mathcal{O}({\alpha }^3)$ and $\mathcal O(\alpha^2)$ accuracy, respectively. We discuss in detail different methods to calculate the potentials, and show the equivalence among them. In particular we obtain, for the first time, the manifestly gauge invariant $1\mathrm{/}m$ and $1\mathrm{/}{m}^2$ potentials in terms of Wilson loops with next-to-leading order (NLO) precision. As an application of our results we derive the theoretical expression for the $B_c$ spectrum in the weak-coupling limit through next-to-next-to-next-to-leading order (N${}^3$LO).