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

Abstract

We determine the $1/m$ and $1/m^2$ 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 $1/m$ and $1/m^2$ 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).

Comments: 64 pages, 5 figures; v2: Minor changes. Some material moved to appendices. Journal version; v3: Typos corrected. New (but equivalent) eq.(E.1)