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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.20372 |
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| _version_ | 1866915467790123008 |
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| author | Kitano, Ryuichiro Matsudo, Ryutaro Treuer, Lukas |
| author_facet | Kitano, Ryuichiro Matsudo, Ryutaro Treuer, Lukas |
| contents | The large $N$ analysis of QCD states that the potential for the $η'$ meson develops cusps at $η' = π/ N_f$, $3 π/N_f$, $\cdots$, with $N_f$ the number of flavors. Furthermore, the recent discussion of generalized anomalies tells us that even for finite $N$ there should be cusps if $N$ and $N_f$ are not coprime, as one can show that the domain wall configuration of $η'$ should support a Chern-Simons theory on it, i.e., domains are not smoothly connected. On the other hand, there is a supporting argument for instanton-like, smooth potentials of $η'$ from the analyses of softly-broken supersymmetric QCD for $N_f= N-1$, $N$, and $N+1$. We argue that the analysis of the $N_f = N$ case should be subject to the above anomaly argument, and thus there should be a cusp; while the $N_f = N \pm 1$ cases are consistent, as $N_f$ and $N$ are coprime. We discuss how this cuspy/smooth transition can be understood. For $N_f< N$, we find that the number of branches of the $η'$ potential is $\operatorname{gcd}(N,N_f)$, which is the minimum number allowed by the anomaly. We also discuss the condition for s-confinement in QCD-like theories, and find that in general the anomaly matching of the $θ$ periodicity indicates that s-confinement can only be possible when $N_f$ and $N$ are coprime. The s-confinement in supersymmetric QCD at $N_f = N+1$ is a famous example, and the argument generalizes for any number of fermions in the adjoint representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cusps in the $η'$ potential Kitano, Ryuichiro Matsudo, Ryutaro Treuer, Lukas High Energy Physics - Theory High Energy Physics - Phenomenology The large $N$ analysis of QCD states that the potential for the $η'$ meson develops cusps at $η' = π/ N_f$, $3 π/N_f$, $\cdots$, with $N_f$ the number of flavors. Furthermore, the recent discussion of generalized anomalies tells us that even for finite $N$ there should be cusps if $N$ and $N_f$ are not coprime, as one can show that the domain wall configuration of $η'$ should support a Chern-Simons theory on it, i.e., domains are not smoothly connected. On the other hand, there is a supporting argument for instanton-like, smooth potentials of $η'$ from the analyses of softly-broken supersymmetric QCD for $N_f= N-1$, $N$, and $N+1$. We argue that the analysis of the $N_f = N$ case should be subject to the above anomaly argument, and thus there should be a cusp; while the $N_f = N \pm 1$ cases are consistent, as $N_f$ and $N$ are coprime. We discuss how this cuspy/smooth transition can be understood. For $N_f< N$, we find that the number of branches of the $η'$ potential is $\operatorname{gcd}(N,N_f)$, which is the minimum number allowed by the anomaly. We also discuss the condition for s-confinement in QCD-like theories, and find that in general the anomaly matching of the $θ$ periodicity indicates that s-confinement can only be possible when $N_f$ and $N$ are coprime. The s-confinement in supersymmetric QCD at $N_f = N+1$ is a famous example, and the argument generalizes for any number of fermions in the adjoint representation. |
| title | On cusps in the $η'$ potential |
| topic | High Energy Physics - Theory High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2508.20372 |