Perturbative Renormalization of Neutron-Antineutron Operators

Two-loop anomalous dimensions and one-loop renormalization scheme matching factors are calculated for six-quark operators responsible for neutron-antineutron transitions. When combined with lattice QCD determinations of the matrix elements of these operators, our results can be used to reliably predict the neutron-antineutron vacuum transition time, $\tau_{n\bar{n}}$, in terms of basic parameters of baryon-number violating beyond-the-Standard-Model theories. The operators are classified by their chiral transformation properties, and a basis in which there is no operator mixing due to strong interactions is identified. Operator projectors that are required for non-perturbative renormalization of the corresponding lattice QCD six-quark operator matrix elements are constructed. A complete calculation of $\delta m = 1/\tau_{n\bar{n}}$ in a particular beyond-the-Standard-Model theory is presented as an example to demonstrate how operator renormalization and results from lattice QCD are combined with experimental bounds on $\delta m$ to constrain the scale of new baryon-number violating physics. At the present computationally accessible lattice QCD matching scale of $\sim$ 2 GeV, the next-to-next-to-leading-order effects calculated in this work correct the leading-order plus next-to-leading-order $\delta m$ predictions of beyond-the-Standard-Model theories by $< 26\%$. Next-to-next-to-next-to-leading-order effects provide additional unknown corrections to predictions of $\delta m$ that are estimated to be $< 7\%$.