The infamous strong $CP$ problem in particle physics can in principle be solved by a massless up quark. In particular, it was hypothesized that topological effects could substantially contribute to the observed nonzero up-quark mass without reintroducing $CP$ violation. Alternatively to previous work using fits to chiral perturbation theory, in this Letter, we bound the strength of the topological mass contribution with direct lattice QCD simulations, by computing the dependence of the pion mass on the dynamical strange-quark mass. We find that the size of the topological mass contribution is inconsistent with the massless up-quark solution to the strong $CP$ problem.

• Received 13 March 2020
• Revised 20 July 2020
• Accepted 29 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.232001

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Published by the American Physical Society