## Abstract

We investigate the dynamics responsible for generating the potential of the η ^{ ′}, the (would-be) Goldstone boson associated with the anomalous axial U(1) symmetry of QCD. The standard lore posits that pure QCD dynamics generates a confining potential with a branched structure as a function of the θ angle, and that this same potential largely determines the properties of the η ^{ ′} once fermions are included. Here we test this picture by examining a supersymmetric extension of QCD with a small amount of supersymmetry breaking generated via anomaly mediation. For pure SU(N) QCD without flavors, we verify that there are N branches generated by gaugino condensation. Once quarks are introduced, the flavor effects qualitatively change the strong dynamics of the pure theory. For F flavors we find |N − F| branches, whose dynamical origin is gaugino condensation in the unbroken subgroup for F < N – 1, and in the dual gauge group for F > N + 1. For the special cases of F = N – 1, N, N + 1 we find no branches and the entire potential is consistent with being a one-instanton effect. The number of branches is a simple consequence of the selection rules of an anomalous U(1)_{ R} symmetry. We find that the η ^{ ′} mass does not vanish in the large N limit for fixed F/N, since the anomaly is non-vanishing. The same dynamics that is responsible for the η ^{ ′} potential is also responsible for the axion potential. We present a simple derivation of the axion mass formula for an arbitrary number of flavors.

Original language | American English |
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Article number | 139 |

Journal | Journal of High Energy Physics |

Volume | 2023 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2023 |

### Bibliographical note

Publisher Copyright:© 2023, The Author(s).

## Keywords

- 1/N Expansion
- Axions and ALPs
- Chiral Lagrangian
- Supersymmetric Gauge Theory

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