Quantum field theories with many fields

Abstract: The large-$N$ quantum field theories provide a window into the regime of strongly-coupled physics. Our principal object of study in this thesis is the large-$N$ family of melonic QFTs, which contain the Sachdev-Ye-Kitaev-like models, tensor models, and vector models. We begin with a review of this limit of a large number of degrees of freedom (large-$N$) as an approach to the solution of QFTs. Two toy models are used to clarify this approach: a zero-dimensional field theory and the flow of a generalized free field theory. Both models are solvable, and so we can explicitly demonstrate: using the former, the simplifications at large $N$; using the latter, the tools used to study scale-dependence of physics -- the renormalization group. We develop $\tilde{F}$-extremization, a simple method of solution for an arbitrary large-$N$ melonic QFT in its strongly-coupled limit. The infrared conformal field theories show remarkable simplicity, in that they are entirely solved by the requirement that they have as many degrees of freedom as possible, up to a simple constraint arising from the interaction between the fields. We measure the number of degrees of freedom of the conformal infrared theory via $\tilde{F}$, the universal part of the free energy. We then present the example of the quartic Yukawa model in continuous dimension. This model is considered as a tensor field theory, and solved for its conformal limit; we then illustrate its multiplicity of fixed points and their stability, as well as its operator spectrum, matching the data between the large-$N$ and dimensional expansions. These features reflect general characteristics of melonic conformal field theories: their existence, stability, and spectral characteristics. We conclude with future directions of exploration for the melonic theories.

• The paper introduces an F‐extremization method for accurately determining conformal data in large‐N melonic QFTs.
• It demonstrates the resummability of melonic diagrams in tensor, vector, and SYK-like models through rigorous renormalization group and Schwinger-Dyson approaches.
• The study validates the method by matching large‐N and ε‐expansion techniques, revealing detailed fixed point structures and operator spectra.

Quantum Field Theories with Many Fields: An Expert Essay

This work provides an extensive and rigorous analysis of large-$N$ quantum field theories (QFTs), particularly those dominated by melonic diagrams, which include the Sachdev-Ye-Kitaev (SYK)-like models, tensor models, and vector models (2603.04481). The thesis systematically develops theoretical and technical tools to understand strongly-coupled regimes in QFT through the lens of large-$N$ expansions, $F$-extremization, and the renormalization group (RG) flow.

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Foundations: Large-$N$ QFT and Statistical Mechanics

The study begins by reframing QFT as a generalization of statistical mechanics, where the path integral over configurations is weighted by the exponential of the action. The partition function $Z$, observables as expectation values, and the (sphere) free energy $F$ are identified as essential quantities for characterizing QFTs. The renormalization group is emphasized as the fundamental principle that selects relevant operators at different scales, thereby ensuring predictivity despite the a priori infinite space of possible interactions. The interplay between locality, Lorentz invariance, and the continuum limit is elucidated in the construction of a physically meaningful QFT.

Melonic QFTs are introduced as a class of models that become tractable in the large-$N$ limit, which is shown to yield mean-field (factorized) dynamics to leading order. This property underpins the solvability of these theories and motivates the thesis’ focus.

(Figure 1)

Figure 1: Cactus (tadpole) diagrams in the vector model: representative leading large-$N$ contributions to the two-point function, which exhibit efficient resummability.

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Toy Models and the Structure of Scaling Fixed Points

Two toy models are analyzed to clarify renormalization and scaling at large $N$:

1. Zero-dimensional $\phi^4$ theory: Demonstrates that renormalization requires parametrizing the theory by observable correlators rather than bare parameters. The large-$N$ limit makes the theory analytically tractable, with disconnected ("mean-field") contributions dominating all correlators.
2. Generalized free field (GFF) flow: Examines a solvable QFT interpolating between two GFFs with different scaling dimensions. This model serves as a prototype for RG flows in large-$N$ melonic theories and exhibits a beta function structurally similar to Wilson-Fisher fixed points.

The conceptual framework for conformal field theories (CFTs) as RG fixed points is established. Conformal invariance, Weyl invariance, operator spectrum, and crossing symmetry are reviewed, with emphasis placed on the analytic continuation of $d$ and $N$, and the consequences for unitarity in fractional dimensions.

Figure 2: The 4-$\epsilon$ expansion for $N=1$, illustrating the appearance of the Wilson-Fisher fixed point and its relation to the free UV theory.

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The Melonic Theories and Diagrammatic Resummation

The thesis systematically classifies melonic theories and illustrates the resummability of their diagrammatics. Melonic dominance is a structural property: in tensor models (rank $r \geq 3$) with appropriately scaled interactions, the only diagrams surviving at leading order in $1/N$ are iterated melons, which admit closed-form resummation. This is shown with explicit representations, for example:

Figure 3: All graphs in the melonic limit are constructed from the iterated melon; here a high-order vacuum diagram is depicted, contributing to the free energy in a generic quartic melonic theory.

The thesis carefully distinguishes the tensor models, vector models, and SYK-type models within the melonic class. The role of Gurau degree in identifying melonic diagrams is explained, establishing the basis for the universality of this solvability mechanism.

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$F$-Extremization: Determining CFT Data in Melonic Theories

A central technical advance is the establishment of $F$-extremization as a method for fully determining the leading-order conformal data in large-$N$ melonic CFTs. For a theory specified by field content and the combinatorial structure of its interactions, the universal part of the sphere free energy $F$ is:

$$F \equiv \sum_{\text{fields}\, \phi} F_{\phi}(\Delta_{\phi})$$

subject to explicit linear constraints from requiring the marginality of each melonic interaction,

$$\forall\, m: \quad -d + \sum_{\phi} q^m_{\phi} \Delta_\phi = 0$$

here $q^m_{\phi}$ denotes the power with which each field appears in interaction $m$.

The $F$-function, up to normalization, counts degrees of freedom (DOF) via the universal part of the sphere partition function. Melonic CFT scaling dimensions are then found by extremizing $F$ with respect to the trial dimensions $\{\Delta_\phi\}$, imposing the interaction constraints.

Figure 4: Specific free energy $F(\lambda)/N$ for the 0-dimensional renormalized $\phi^4$ model, illustrating the reduction in configurational entropy as interaction strength increases.

This principle is proven both using the two-particle-irreducible (2PI) effective action and by mapping the solution of Schwinger-Dyson equations (SDEs) to this variational problem.

Figure 5: The scaling dimension of $\phi$ in the melonic quartic Yukawa theory as a function of dimension $d$; delineation of real and complex solutions associated with different CFT vacua.

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Detailed Study of the Quartic Yukawa Tensor Model

As a concrete, nontrivial example, the tensorial quartic Yukawa model in continuous dimension is solved both via large-$N$ SDEs and $\epsilon$-expansion in $d=3-\epsilon$. The analysis reveals:

• Multiplicity of fixed points: Both bosonic and fermionic sectors display several distinct IR fixed points, some of which correspond to known bosonic (prismatic/sextic) theories, others to new fermionic generalizations.
• Operator spectrum: The SDE approach enables non-perturbative access to the full spectrum of bilinear operators, up to leading order in $1/N$.
• Collision and complexification: Varying $d$, families of fixed points can merge or move off into the complex plane, reflecting transitions between stable/unstable regimes (loss of unitarity or instability of vacua).
• Matching between large-$N$ and $\epsilon$ expansions: Spectral data and scaling dimensions computed in both frameworks agree, confirming the analytical continuation in $d$ and the robustness of the large-$N$ approach.

These findings showcase the power and generality of $F$-extremization in classifying the IR phases and spectra of melonic QFTs.

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General Implications, Limitations, and Future Directions

The results have several important theoretical and practical implications:

• Universality of $F$-extremization: This principle systematically unifies the treatment of large-$N$ melonic CFTs, generalizing established $a$- and $F$-maximization methods from supersymmetric contexts to a broad class of non-supersymmetric models.
• Classification of fixed points and vacua: The approach naturally explains the proliferation and dynamics of fixed points as $d$ and $N$ are varied, and rationalizes phenomena such as complex CFTs and accidental symmetry mixing.
• Computational applications: The explicit form of the $F$-functional and constraints enables efficient (even automated) exploration of the phase space of tensor models, unlocking systematic identification of solvable regimes and stable theories.

However, important limitations are noted:

• Physical realization of CFTs: Not every solution to the $F$-extremization problem necessarily corresponds to a physical, stable, or unitary theory. Symmetry breaking, complex scaling dimensions, and the possibility of non-conformal IR phases mean care is needed in interpreting formal solutions.
• Higher-order $1/N$ effects and non-melonic corrections: The solvability and mean-field picture hold strictly at leading order in $1/N$; subleading effects may qualitatively alter the dynamics or operator spectrum, especially away from the melonic point.
• Relation to gravity and holography: Large-$N$ melonic CFTs have been conjectured to be holographically dual to higher spin or stringy gravity theories. The detailed consequences of $F$-extremization for the landscape of dual gravitational theories deserve further exploration.

(Figure 6)

Figure 6: Schematic of the $(d,N)$ surface of the critical $\mathrm{O}(N)$ model in terms of anomalous dimensions, indicating the rich structure of RG flows and critical points as both parameters are varied.

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Conclusion

This thesis delivers a comprehensive, technical, and conceptually coherent treatment of quantum field theories with many fields, centered on large-$N$ melonic models. The $F$-extremization framework developed provides a universal variational principle that characterizes the IR conformal data, subject to explicit constraints determined by the interaction structure.

The generality of the results, their numerical robustness, and the explicit mapping between distinct perturbative and non-perturbative regimes demonstrate that large-$N$ melonic QFTs serve as a fertile testing ground for both conceptual advances and concrete computations in strongly-coupled physics.

This work opens avenues for further studies, including the classification of exotic CFTs via $F$-landscapes, exploration of non-integer dimensions and non-unitary regimes, and the search for holographic duals of melonic fixed points. The intersection of combinatorial, analytical, and variational methods outlined here stands as a model for future developments in modern quantum field theory.

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Figure 7: The recursive, iterated insertions on a melonic contribution to the two-point function in a multi-field theory, emphasizing the resummable structure foundational to the solvability of melonic QFTs.