scholarly journals Progress in Solving the Nonperturbative Renormalization Group for Tensorial Group Field Theory

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 86 ◽  
Author(s):  
Vincent Lahoche ◽  
Dine Ousmane Samary
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Fixed Points ◽  
Expansion Method ◽  
Flow Equation ◽  
Nucl Phys ◽  
Flow Equations ◽  
First Order Phase Transition ◽  
Group Field Theory ◽  
Non Gaussian

This manuscript aims at giving new advances on the functional renormalization group applied to the tensorial group field theory. It is based on the series of our three papers (Lahoche, et al., Class. Quantum Gravity 2018, 35, 19), (Lahoche, et al., Phys. Rev. D 2018, 98, 126010) and (Lahoche, et al., Nucl. Phys. B, 2019, 940, 190–213). We consider the polynomial Abelian U ( 1 ) d models without the closure constraint. More specifically, we discuss the case of the quartic melonic interaction. We present a new approach, namely the effective vertex expansion method, to solve the exact Wetterich flow equation and investigate the resulting flow equations, especially regarding the existence of non-Gaussian fixed points for their connection with phase transitions. To complete this method, we consider a non-trivial constraint arising from the Ward–Takahashi identities and discuss the disappearance of the global non-trivial fixed points taking into account this constraint. Finally, we argue in favor of an alternative scenario involving a first order phase transition into the reduced phase space given by the Ward constraint.

The renormalization group (RG) approach: The critical theory near four dimensions

2021 ◽  
pp. 357-390
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Fixed Points ◽  
Gaussian Approximation ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Flow Equations ◽  
Four Dimensions ◽  
The Mean

In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.

scholarly journals Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Andreas G. A. Pithis ◽  
Johannes Thürigen
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Fixed Points ◽  
Group Method ◽  
Constant Field ◽  
Theory Approach ◽  
Specific Flow ◽  
Functional Renormalization Group ◽  
Spacetime Geometry ◽  
Group Field Theory

Abstract In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

Stability of renormalization group fixed points and decay of correlations

2019 ◽  
pp. 101-110
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Fixed Points ◽  
Gradient Flow ◽  
Scalar Fields ◽  
Effective Field ◽  
Coupling Constants ◽  
Critical Systems ◽  
Universal Properties ◽  
General Physical

Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.

scholarly journals SOLVING NONPERTURBATIVE FLOW EQUATIONS

1995 ◽  
Vol 10 (31) ◽  
pp. 2367-2379 ◽  
Author(s):  
J. ADAMS ◽  
N. TETRADIS ◽  
J. BERGES ◽  
F. FREIRE ◽  
C. WETTERICH ◽  
...  
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Critical Behavior ◽  
Good Accuracy ◽  
Three Dimensional ◽  
Scale Dependence ◽  
Flow Equation ◽  
Flow Equations ◽  
Approximate Form ◽  
Average Action

Nonperturbative exact flow equations describe the scale dependence of the effective average action. We present a numerical solution for an approximate form of the flow equation for the potential in a three-dimensional N-component scalar field theory. The critical behavior, with associated critical exponents, can be inferred with good accuracy.

scholarly journals TOWARDS NONPERATURBATIVE RENORMALIZABILITY OF QUANTUM EINSTEIN GRAVITY

2002 ◽  
Vol 17 (06n07) ◽  
pp. 993-1002 ◽  
Author(s):  
O. LAUSCHER ◽  
M. REUTER
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Einstein Gravity ◽  
Flow Equation ◽  
Planck Length ◽  
Small Length ◽  
Standard Cosmology ◽  
Average Action ◽  
Non Gaussian ◽  
Flatness Problem

We summarize recent evidence supporting the conjecture that four-dimensional Quantum Einstein Gravity (QEG) is nonperturbatively renormalizable along the lines of Weinberg's asymptotic safety scenario. This would mean that QEG is mathematically consistent and predictive even at arbitrarily small length scales below the Planck length. For a truncated version of the exact flow equation of the effective average action we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cutoff-limit. The cosmological implications of this fixed point are discussed, and it is argued that QEG might solve the horizon and flatness problem of standard cosmology without an inflationary period.

Kenneth Wilson — Renormalization and QCD

2014 ◽  
Vol 29 (18) ◽  
pp. 1430043 ◽  
Author(s):  
Franz J. Wegner
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Critical Phenomena ◽  
Gauge Invariance ◽  
Great Pleasure ◽  
Flow Equations ◽  
Kenneth Wilson

Kenneth Wilson had an enormous impact on field theory, in particular on the renormalization group and critical phenomena, and on QCD. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and QCD, and flow equations and similarity renormalization.

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