F-Theorem without Supersymmetry

Abstract: The conjectured F-theorem for three-dimensional field theories states that the finite part of the free energy on S^3 decreases along RG trajectories and is stationary at the fixed points. In previous work various successful tests of this proposal were carried out for theories with {\cal N}=2 supersymmetry. In this paper we perform more general tests that do not rely on supersymmetry. We study perturbatively the RG flows produced by weakly relevant operators and show that the free energy decreases monotonically. We also consider large N field theories perturbed by relevant double trace operators, free massive field theories, and some Chern-Simons gauge theories. In all cases the free energy in the IR is smaller than in the UV, consistent with the F-theorem. We discuss other odd-dimensional Euclidean theories on S^d and provide evidence that (-1)^{(d-1)/2} \log |Z| decreases along RG flow; in the particular case d=1 this is the well-known g-theorem.

• The paper demonstrates that free energy on the three-sphere decreases from the UV to the IR in non-supersymmetric settings.
• The analysis employs both perturbative methods and large‑N computations to assess RG flows in various operator deformations.
• Key examples include RG flows driven by weakly relevant operators and double-trace deformations, supporting the F-theorem's validity.

Analysis of the $F$-Theorem in Non-Supersymmetric Field Theories

This paper presents a detailed examination of the so-called $F$-theorem, a conjecture applicable to three-dimensional field theories. The $F$-theorem posits that the finite part of the free energy on a three-sphere, $S^3$, behaves similarly to Zamolodchikov's $c$-theorem in two dimensions, by decreasing along renormalization group (RG) flows and remaining stationary at conformal fixed points. While prior investigations have primarily verified this conjecture in the context of theories possessing $\mathcal{N}=2$ supersymmetry, this study extends the investigation to non-supersymmetric frameworks. The analysis encompasses a wide variety of examples, including weakly relevant operators and relevant double trace operators in large $N$ field theories, ultimately affirming that, in all cases surveyed, the free energy decreases from the ultraviolet (UV) to the infrared (IR), as conjectured by the $F$-theorem.

Methodology and Key Findings

The researchers employ perturbative assessments and various large $N$ computations to scrutinize the conjecture within different settings, ranging from free massless and massive theories, to Chern-Simons gauge theories, and additional odd-dimensional Euclidean models on $S^d$. A noteworthy example discussed is the transformation of CFTs perturbed by scalar operators of dimension $\Delta = d - \epsilon$, illustrating a robust IR fixed point when the coefficient $C$ is positive.

Two particular instances merit attention due to their stronger empirical support for the $F$-theorem: firstly, RG flows instigated by weakly relevant operators in odd-dimensional Euclidean theories, where perturbation theory suggests that the derivative of the free energy with respect to the coupling constant aligns with the beta function, indicating a monotonic decrease; and second, large $N$ gauge theories where relevant double-trace deformations predict an exact decrease in free energy when engaging IR fixed points for a variety of values of $\Delta$.

Implications and Theoretical Progress

The numerical results obtained in numerous examples consistently adhere to the $F$-theorem, suggesting a broader validity of the theorem beyond supersymmetric constraints. In addition to bolstering theoretical field studies, this work provides foundational backing for the conceptual framework of RG flow's constraints in odd dimensions, expanding the understanding gained from anomalies in even dimensions. Moreover, it offers insights aligning with the conjectured dualities between three-dimensional CFTs and various higher-spin gauge theories in AdS spaces, resonating with ideas proposed in the context of the AdS/CFT correspondence.

Future prospects stemming from this work pertain to exploring more intricate configurations, such as supersymmetry breaking scenarios or coupling with real-world systems stirring phase transitions. Additionally, there lies potential in deepening algorithmic approaches to frame systematic proofs of such theorems, potentially through a non-perturbative lens that could ally with existing $a$-maximization paradigms in four dimensions. Altogether, these pathways could unveil richer tapestries interconnecting three-dimensional field theory, holographic principles, and collaborative insights from various quantum systems' analysis.

In conclusion, this investigation substantially enriches the understanding of the $F$-theorem in broader contexts, and its disclosure supports shifting paradigms in how theoretical physics approaches dimensional analysis and field theory continuity across varied parameters.