On the dynamical origin of the η potential and the axion mass

Csaba Csáki, Raffaele Tito D’Agnolo, Rick S. Gupta, Eric Kuflik, Tuhin S. Roy, Maximilian Ruhdorfer*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 languageAmerican English
Article number139
JournalJournal of High Energy Physics
Issue number10
StatePublished - Oct 2023

Bibliographical note

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


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


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