Quantum field theory in the large N limit: a review

By M. Moshe, J. Zinn-Justin. 2003

477 citations recorded by the source corpus.

Abstract We review the solutions of O ( N ) and U ( N ) quantum field theories in the large N limit and as 1/ N expansions, in the case of vector representations. Since invariant composite fields have small fluctuations for large N , the method relies on constructing effective field theories for composite fields after integration over the original degrees of freedom. We first solve a general scalar U ( φ 2 ) field theory for N large and discuss various non-perturbative physical issues such as critical behaviour. We show how large N results can also be obtained from variational calculations. We illustrate these ideas by showing that the large N expansion allows to relate the ( φ 2 ) 2 theory and the non-linear σ -model, models which are renormalizable in different dimensions. Similarly, a relation between CP ( N −1) and abelian Higgs models is exhibited. Large N techniques also allow solving self-interacting fermion models. A relation between the Gross–Neveu, a theory with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in four dimensions then follows. We discuss dissipative dynamics, which is relevant to the approach to equilibrium, and which in some formulation exhibits quantum mechanics supersymmetry. This also serves as an introduction to the study of the 3D supersymmetric quantum field theory. Large N methods are useful in problems that involve a crossover between different dimensions. We thus briefly discuss finite size effects, finite temperature scalar and supersymmetric field theories. We also use large N methods to investigate the weakly interacting Bose gas. The solution of the general scalar U ( φ 2 ) field theory is then applied to other issues like tricritical behaviour and double scaling limit.

Subfields: Condensed Matter Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/aa055eaec031298ca426cadb054a9374c094b46e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. David Simmons-Duffin
2. S. Rychkov
3. David Poland
4. A. Eichhorn
5. A. Vichi
6. Ran Yacoby
7. Filip Kos
8. Guy Gur-Ari

In Nuclear and High Energy Physics

1. David Simmons-Duffin
2. David Poland
3. S. Rychkov
4. S. Giombi
5. Ran Yacoby
6. A. Vichi
7. Silviu S. Pufu
8. Filip Kos

Citations in the corpus

• The nonperturbative functional renormalization group and its applications (2020, background): “Thus even high-order perturbative RG expansions cannot relate the two expansions, except in the large-N limit [29].”
• Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models (2016, background): “A well-known example is the O(N) symmetric theory of N scalar fields φ in d dimensions with interaction g(φφ)(2) (for reviews see [1,2]).”
• d = 3 bosonic vector models coupled to Chern-Simons gauge theories (2011, background): “For the theory with only (φ(2))(3) couplings it is easy to see that the large N beta function vanishes to all orders, since there are no contributing diagrams; it is indeed well-known that this coupling is exactly marginal in the large N limit [24] (see [25] for a review).”
• Bootstrapping 3D fermions (2015, background): “When N is even, this theory can be studied perturbatively in d = 4− dimensions (see, for example [50]).”
• Correlation functions of large N Chern-Simons-Matter theories and bosonization in three dimensions (2012, background): “All the theories discussed above have a good 1/N expansion (see [1] for a review), so it is natural to suggest that they could have classical gravitational duals (by the AdS/CFT correspondence [2, 3, 4]) at large N , living on AdS4.”
• The tachyon at the end of the universe (2005, background): “[41]), exhibiting a mass gap, except now there are two vacua for Λ.”
• Bootstrapping the O(N ) vector models (2013, background): “∆φ is known to order 1/N3 while ∆S has only been computed to order 1/N 2 (see [20] and references therein).”
• Interpolating between a and F (2014, background): “The one-loop β-functions for the renormalized couplings g1, g2 are [74]”
• The conformal bootstrap: Theory, numerical techniques, and applications (2018, background): “…point of interest is strongly coupled, requiring one to deform the theory artificially in order to do perturbation theory in a small parameter, with the large-N expansion (see e.g. Moshe and Zinn-Justin (2003) for a review) and the -expansion (Wilson and Fisher, 1972) being two prime examples.”