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arXiv · 2608.28253

Revisiting the $U(1)$-$CP$ tension in QCD from a 1PI-action perspective

Paolo Di Vecchia·Francesco Sannino·Graham M. Shore·Gabriele Veneziano·Shimon Yankielowicz
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Abstract

We discuss the tension one encounters in trying to solve, simultaneously, the $U(1)$ and strong-$CP$ problems in QCD with light quarks and a small $\bar{\theta}$ angle within QCD itself. We do so by considering the low-energy expansion of a $1$PI effective action $\Gamma$, a functional of a properly chosen set of gauge-invariant (composite) operators. After enforcing Ward-identity constraints on $\Gamma$, we prove, under minor assumptions, that $CP$ violation at small non-vanishing $\bar{\theta}$ can be avoided only if the anomaly-induced contribution to the pseudoscalar mass matrix is completely negligible when compared to its non-anomalous counterpart - that is, only if the $U(1)$ problem is not solved. The topological susceptibility in QCD and a precise generalization of the one in pure Yang-Mills theory emerge as the most important players in establishing the above tension and in determining the corrections to various known results at leading order in $\bar{\theta}$.

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Paolo Di Vecchia, Francesco Sannino, Graham M. Shore, Gabriele Veneziano, Shimon Yankielowicz. 2026-08-28. Revisiting the $U(1)$-$CP$ tension in QCD from a 1PI-action perspective. https://arxiv.org/abs/2608.28253

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