The amplitude for the neutrinoless double $beta$ ($0 ubetabeta$) decay of the two-neutron system, $nnto ppe^-e^-$, constitutes a key building block for nuclear-structure calculations of heavy nuclei employed in large-scale $0 ubetabeta$ searches. Assuming that the $0 ubetabeta$ process is mediated by a light-Majorana-neutrino exchange, a systematic analysis in chiral effective field theory shows that already at leading order a contact operator is required to ensure renormalizability. In this work, we develop a method to estimate the numerical value of its coefficient in analogy to the Cottingham formula and validate the result by reproducing the charge-independence-breaking contribution to the nucleon-nucleon scattering lengths. Our central result, while derived in the $overline{text{MS}}$ scheme, is given in terms of the renormalized amplitude $mathcal{A}_ u(|mathbf{p}|,|mathbf{p}^prime|)$, matching to which will allow one to determine the contact-term contribution in regularization schemes employed in nuclear-structure calculations. Our results thus greatly reduce a crucial uncertainty in the interpretation of searches for $0 ubetabeta$ decay.
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