Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion

Abstract

The tensorial equations for non trivial fully interacting fixed points at lowest order in the $\epsilon$ expansion in $4-\epsilon$ and $3-\epsilon$ dimensions are analysed for $N$-component fields and corresponding multi-index couplings $\lambda$ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For $N=5,6,7$ in the four-index case large numbers of irrational fixed points are found numerically where $||\lambda |{|}^2$ is close to the bound found by Rychkov and Stergiou in arXiv:1810.10541. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For $N⩾6$ the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for $N=5$.

arXiv comment50 pages, 6 figures. v2: 53 pages, 6 figures; Expanded discussion of N=4 case including splitting of fixed point at order ${\epsilon }^2$; v4: Minor corrections and additions

References in corpus (26)

• Critical Phenomena and Renormalization-Group Theory
• Precision Islands in the Ising and $O(N)$ Models
• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• Multicritical phenomena in O(n_1)+O(n_2)-symmetric theories
• Numbers and Functions in Quantum Field Theory
• Bosonic Tensor Models at Large $N$ and Small $\epsilon$
• Large-n Critical Behavior of O(n)xO(m) Spin Models
• Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• Thermal Order in Conformal Theories
• Six-loop $\epsilon$ expansion study of three-dimensional $n$-vector model with cubic anisotropy
• General Properties of Multiscalar RG Flows in $d=4-\epsilon$
• Fixed point stability and decay of correlations
• Analytic and Numerical Bootstrap of CFTs with $O(m)\times O(n)$ Global Symmetry in 3D
• Field theory of bi- and tetracritical points: Statics
• Bootstrapping MN and Tetragonal CFTs in Three Dimensions
• Six-loop $\epsilon$ expansion study of three-dimensional $O(n)\times O(m)$ spin models
• On critical models with $N\le 4$ scalars in $d=4-\epsilon$
• Spectrum of anomalous dimensions in hypercubic theories
• Three-loop renormalization group analysis of a complex model with stable fixed point: Critical exponents up to ${\epsilon }^3$ and ${\epsilon }^4$
• On stability of the three-dimensional fixed point in a model with three coupling constants from the $\epsilon$ expansion: Three-loop results
• The tri-fundamental quartic model
• Platonic Field Theories
• Field theory of bi- and tetracritical points: Relaxational dynamics
• Discovering and quantifying nontrivial fixed points in multi-field models
• Asymptotic freedom from the two loop term of the $\beta$-function in a cubic theory
• Properties of the $\epsilon$-Expansion, Lagrange Inversion and Associahedra and the $O(1)$ Model

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• The bi-conical vector model at $1/N$
• Bootstrapping Mixed MN Correlators in 3D
• The search for the universality class of metric quantum gravity
• Coupled minimal models revisited II: Constraints from permutation symmetry
• Anomalous Dimensions in Hypercubic Theories
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