scholarly journals Can we make sense out of "Tensor Field Theory"?

2021 ◽  
Vol 4 (4) ◽  
Author(s):  
Vincent Rivasseau ◽  
Fabien Vignes-Tourneret
Keyword(s):  
Field Theory ◽  
Strong Evidence ◽  
Tensor Field ◽  
Slight Modification ◽  
Power Counting ◽  
Careful Study ◽  
Tensor Theory

We continue the constructive program about tensor field theory through the next natural model, namely the rank five tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on \mathbf{U(1)^5}𝐔(1)5. We make a first step towards its construction by establishing its power counting, identifying the divergent graphs and performing a careful study of (a slight modification of) its RG flow. Thus we give strong evidence that this just renormalizable tensor field theory is non perturbatively asymptotically free.

scholarly journals Erratum to: Hints of unitarity at large N in the O(N)3 tensor field theory

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Dario Benedetti ◽  
Razvan Gurau ◽  
Sabine Harribey ◽  
Kenta Suzuki
Keyword(s):  
Field Theory ◽  
Tensor Field ◽  
Normalization Factor ◽  
Large N

The measure in equation (2.11) contains a wrong normalization factor, and it should be multiplied by 21−dΓ(d − 1)/Γ(d/2)2.

scholarly journals The one-loop tadpole in the geoSMEFT

2021 ◽  
Vol 11 (5) ◽  
Author(s):  
Tyler Corbett
Keyword(s):  
Field Theory ◽  
Standard Model ◽  
Effective Field Theory ◽  
Effective Field ◽  
Power Counting ◽  
Leading Order ◽  
The Standard Model ◽  
The One ◽  
Near Future ◽  
Geometric Formulation

Making use of the geometric formulation of the Standard Model Effective Field Theory we calculate the one-loop tadpole diagrams to all orders in the Standard Model Effective Field Theory power counting. This work represents the first calculation of a one-loop amplitude beyond leading order in the Standard Model Effective Field Theory, and discusses the potential to extend this methodology to perform similar calculations of observables in the near future.

scholarly journals Addendum to: A Renormalizable 4-Dimensional Tensor Field Theory

2013 ◽  
Vol 322 (3) ◽  
pp. 957-965 ◽  
Author(s):  
Joseph Ben Geloun ◽  
Vincent Rivasseau
Keyword(s):  
Field Theory ◽  
Tensor Field

Introduction to renormalization theory and renormalization group (RG)

2021 ◽  
pp. 185-219
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Short Distance ◽  
Large Scale ◽  
Relativistic Quantum ◽  
Power Counting ◽  
Fine Tuning ◽  
Quantum Field ◽  
Renormalizable Theory ◽  
Renormalization Theory

A straightforward construction of a local, relativistic quantum field theory (QFT) leads to ultraviolet (UV) divergences and a QFT has to be regularized by modifying its short-distance or large energy momentum structure (momentum regularization is often used in this work). Since such a modification is somewhat arbitrary, it is necessary to verify that the resulting large-scale predictions are, at least to a large extent, short-distance insensitive. Such a verification relies on the renormalization theory and the corresponding renormalization group (RG). In this chapter, the essential steps of a proof of the perturbative renormalizability of the scalar φ4 QFT in dimension 4 are described. All the basic difficulties of renormalization theory, based on power counting, are already present in this simple example. The elegant presentation of Callan is followed, which makes it possible to prove renormalizability and RG equations (in Callan–Symanzik's (CS) form) simultaneously. The background of the discussion is effective QFT and emergent renormalizable theory. The concept of fine tuning and the issue of triviality are emphasized.

The Principles of Renormalisation

2021 ◽  
pp. 304-328
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Power Counting ◽  
Field Theories ◽  
Asymptotic Freedom ◽  
Green Functions ◽  
Quantum Field Theories ◽  
Quantum Field

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

scholarly journals Lorentz-violating gaugeon formalism for rank-2 tensor theory

2019 ◽  
Vol 34 (30) ◽  
pp. 1950245
Author(s):  
Sudhaker Upadhyay ◽  
Mushtaq B. Shah ◽  
Prince A. Ganai
Keyword(s):  
Tensor Field ◽  
Vector Fields ◽  
Fock Space ◽  
Antisymmetric Tensor ◽  
Type I ◽  
Classical Action ◽  
Tensor Theory ◽  
Antisymmetric Tensor Field ◽  
Gauge Fixing ◽  
Rank 2

We develop a BRST symmetric gaugeon formalism for the Abelian rank-2 antisymmetric tensor field in the Lorentz-breaking framework. The Lorentz-breaking is achieved here by considering a proper subgroup of Lorentz group together with translation. In this scenario, the gaugeon fields together with the standard fields of the Abelian rank-2 antisymmetric tensor theory get mass. In order to develop the gaugeon formulation for this theory in very special relativity (VSR), we first introduce a set of dipole vector fields as a quantum gauge freedom to the action. In order to quantize the dipole vector fields, the VSR-modified gauge-fixing and corresponding ghost action are constructed as the classical action is invariant under a VSR-modified gauge transformation. Further, we present a Type I gaugeon formalism for the Abelian rank-2 antisymmetric tensor field theory in VSR. The gauge structures of Fock space constructed with the help of BRST charges are also discussed.

Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie / Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory

1971 ◽  
Vol 26 (4) ◽  
pp. 599-622
Author(s):  
H. von Grünberg
Keyword(s):  
Gauge Group ◽  
Tensor Field ◽  
Mathematical Proof ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Flat Space ◽  
Momentum Tensor ◽  
Field Equations ◽  
Tensor Theory ◽  
Generally Covariant

Abstract In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated.It is shown that a slight restriction of the gauge group of Einstein's linear tensor theory leads to the linearized Jordan-Brans-Dicke-theory. The problem of the inconsistency of the field equations and the equations of motion is solved by introducing the Landau-Lifschitz energy momentum tensor of the gravitational field as an additional source term into the field equations. The second order of the theory together with the corresponding gauge group are calculated explicitly. By means of the structure of the gauge group of the tensor field it is possible to identify the successive orders of the scalar-tensor theory as an expansion of the Jordan-Brans-Dicke-theory in flat space-time. The question of the uniqueness of the procedure is answered by showing that the structure of the gauge group of the tensor field is predetermined by the linear equations of motion. The mathematical proof of this fact confirms formally the meaning of the equations of motion for the geometry of space.

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