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Functional renormalization of N scalars with O(N) invariance
High Energy Physics - Theory · Submitted 17 Jun 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential.
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The abstract, sentence by sentence
We discuss general theories of N scalar fields with O(N) symmetry. In addition to the standard case of linearly realized symmetry there are also examples that carry nonlinear realizations, with the topology of a cylinder $R\times S^{N-1}$ or a sphere $S^N$. We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We discuss general theories of N scalar fields with O(N) symmetry. | background 0.74 | 0.70 | 0.04 | asserted 0.87 |
| s1 | In addition to the standard case of linearly realized symmetry there are also examples that carry nonlinear realizations, with the topology of a cylinder $R\times S^{N-1}$ or a sphere $S^N$. | background 0.83 | 0.18 | 0.07 | asserted 1.00 |
| s2 | We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential. | contribution 0.70 | 0.94 | 0.20 | asserted 0.96 |
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The paper
Original abstract
We discuss general theories of N scalar fields with O(N) symmetry. In addition to the standard case of linearly realized symmetry there are also examples that carry nonlinear realizations, with the topology of a cylinder $R\times S^{N-1}$ or a sphere $S^N$. We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential.
Human review
References
How the literature uses it
8 arXiv papers cite this in 13 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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Inclusion of the $O(4)$-violating couplings $P(h)$ also generalizes, at $N=4$, the $O(N)$ model studied in [ref] .
One-loop functional RG flow of scalar theory with electroweak symmetry · Introduction · 1.00
Uses its method or tool 1 paper
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This allows for a straightforward application of the methods developed in [ref] , already employed for the renormalization group study of nonlinear sigma models in [ref] .
One-loop functional RG flow of scalar theory with electroweak symmetry · Flow equations · 0.97
Cites it as background 3 papers
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In particular, in the renormalization group approach of [ref] , the covariant formulation reviewed in section [ref] has been previously applied to nonlinear sigma models [ref] and has more recently received attention in applications to gravity [ref] (see also [ref] ).
Splitting Ward identity · Conclusions · 0.80
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The issue of coordinate transformations (also referred to as field redefinition or reparametrisation invariance) in quantum field theory has been in the past [ref] , and still is [ref] a subject of renewed interest.
Field redefinition and renormalisability in scalar field theories · Introduction · 1.00
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The fixed-point structure and the geometry of the NLSM were thoroughly worked out by Refs. [ref] , also underlining its relevance for gravitational physics.
S-wave pion-pion scattering lengths from nucleon-meson fluctuations · Introduction · 0.97
What it cites · 17 sentences without a confident label
Builds on 1 paper
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In this paper we will lay the foundations for a study of these theories in the second order of the derivative expansion (the case $N=1$ has already been discussed to next order of the derivative expansion [ref] ).
Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order $\partial^4$ · General scalar theories with $O(N)$ symmetry · 0.94
Uses the method or tool of 2 papers
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For convenience we choose the optimized cutoff $R_k(z)=(k^2-z)θ(k^2-z)$ [ref] , for which one can easily calculate Q_n (Ṙ_k+ηR_k/(P_k +q)^l) = k^2(n-l+1)/Γ(n+1)2+η/n+1/(1 +q̃)^l
Critical exponents from optimised renormalisation group flows · Calculation of beta functions · 0.99
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However it is worth mentioning that we have verified numerically, following the same method used for example in [ref] , that in $d=4$ this is the only scaling solution while in $d=3$ there also exists a fixed point with nontrivial potential.
Scaling Solutions in Continuous Dimension · The cylindrical fixed point in the one–loop approximation · 0.95
Compares with 3 papers
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η= -2c_d (d+2) (N-1)/2c_d(N-1)-(d+2)f̃^2 This reproduces the result of [ref] .
Fixed Points of Nonlinear Sigma Models in d>2 · The spherical fixed point in the · 0.99
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In fact our result ([ref]) agrees with [ref] noticing that L^N/2-1_n(z)=(N/2)_n/n! _1F_1 (-n,N/2,z), (α)_n ≡α(α+1)⋯(α+n-1), (α)_0 ≡1,
Critical exponents from optimised renormalisation group flows · Linearised equations at one loop · 1.00
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We will not repeat here the analysis of these solutions, but it will be useful at some later stage to check that the geometrical methods used here agree with the standard ones used in [ref] .
Derivative expansion of the renormalization group in O(N) scalar field theory · General scalar theories with $O(N)$ symmetry · 0.80
Cites as background 6 papers
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For many purposes it has proven more convenient to study instead the $k$–dependence of a one–particle irreducible functional $Γ_k$, called the “Effective Average Action” (EAA), which is defined exactly like the effective action, but with a smooth cutoff in the functional integral (see [ref] for introductory reviews).
An Introduction to the Nonperturbative Renormalization Group · General scalar theories with $O(N)$ symmetry · 0.90
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For many purposes it has proven more convenient to study instead the $k$–dependence of a one–particle irreducible functional $Γ_k$, called the “Effective Average Action” (EAA), which is defined exactly like the effective action, but with a smooth cutoff in the functional integral (see [ref] for introductory reviews).
Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics · General scalar theories with $O(N)$ symmetry · 0.90
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For many purposes it has proven more convenient to study instead the $k$–dependence of a one–particle irreducible functional $Γ_k$, called the “Effective Average Action” (EAA), which is defined exactly like the effective action, but with a smooth cutoff in the functional integral (see [ref] for introductory reviews).
Elements of the Continuous Renormalization Group · General scalar theories with $O(N)$ symmetry · 0.90
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