Group field theory renormalization in the 3D case: Power counting of divergences

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.

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Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions

Communications in Mathematical Physics ◽

10.1007/s00220-014-1928-x ◽

2014 ◽

Vol 330 (2) ◽

pp. 581-637 ◽

Cited By ~ 75

Author(s):

Sylvain Carrozza ◽

Daniele Oriti ◽

Vincent Rivasseau

Keyword(s):

Field Theory ◽

Three Dimensions ◽

Group Field Theory

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Introduction to renormalization theory and renormalization group (RG)

Quantum Field Theory and Critical Phenomena ◽

10.1093/oso/9780198834625.003.0009 ◽

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.

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The Principles of Renormalisation

10.1093/oso/9780192844200.003.0015 ◽

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.

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Quantum gravity, group field theory (GFT), and combinatorics

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0010 ◽

2021 ◽

pp. 121-165

Author(s):

Adrian Tanasa

Keyword(s):

Field Theory ◽

Quantum Theory ◽

Quantum Gravity ◽

Specific Model ◽

Algebraic Combinatorics ◽

Saddle Point Method ◽

Feynman Integrals ◽

Causal Dynamical Triangulations ◽

Group Field Theory ◽

Definition Of

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

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