Large $N_f$ quantum field theory

Abstract

We review the development of the large $N$ method, where $N$ indicates the number of flavours, used to study perturbative and nonperturbative properties of quantum field theories. The relevant historical background is summarized as a prelude to the introduction of the large $N$ critical point formalism. This is used to compute large $N$ corrections to $d$-dimensional critical exponents of the universal quantum field theory present at the Wilson-Fisher fixed point. While pedagogical in part the application to gauge theories is also covered and the use of the large $N$ method to complement explicit high order perturbative computations in gauge theories is also highlighted. The usefulness of the technique in relation to other methods currently used to study quantum field theories in $d$-dimensions is also summarized.

62 latex pages, 22 figures; invited review article

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• Anomalous dimensions for ${\varphi }^n$ in scale invariant $d=3$ theory
• Towards the QED beta function and renormalons at $1/N_f^2$ and $1/N_f^3$
• Analytic Coupling Structure of Large $N_f$ (Super) QED and QCD
• Critical properties of the Néel/algebraic-spin-liquid transition
• The bi-conical vector model at $1/N$
• Critical properties of the valence-bond-solid transition in lattice quantum electrodynamics
• Safe Glueballs and Baryons
• A critical look at $\beta$-function singularities at large $N$
• Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\epsilon )$
• The background field method and critical vector models
• CFT data in the Gross-Neveu model
• Twist-tuned quantum criticality in moiré bilayer graphene
• Bubble-resummation and critical-point methods for $\beta$-functions at large $N$
• Dirac quantum criticality in twisted double bilayer transition metal dichalcogenides
• One-loop divergences in the six-dimensional $\mathcal{N}=(1,0)$ hypermultiplet self-coupling model
• Critical properties of three-dimensional many-flavour QEDs
• Two-loop mass anomalous dimension in reduced quantum electrodynamics and application to dynamical fermion mass generation