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 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.

scholarly journals Relationship between Fujikawa’s Method and the Background Field Method for the Scale Anomaly

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Chris L. Lin ◽  
Carlos R. Ordóñez
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Effective Potential ◽  
Effective Action ◽  
Background Field ◽  
Field Method ◽  
Scale Dependence ◽  
Classical Action ◽  
Loop Expansion ◽  
Background Field Method

We show the equivalence between Fujikawa’s method for calculating the scale anomaly and the diagrammatic approach to calculating the effective potential via the background field method, for anO(N)symmetric scalar field theory. Fujikawa’s method leads to a sum of terms, each one superficially in one-to-one correspondence with a vacuum diagram of the 1-loop expansion. From the viewpoint of the classical action, the anomaly results in a breakdown of the Ward identities due to scale-dependence of the couplings, whereas, in terms of the effective action, the anomaly is the result of the breakdown of Noether’s theorem due to explicit symmetry breaking terms of the effective potential.

scholarly journals FUNCTIONAL RENORMALIZATION OF NONCOMMUTATIVE SCALAR FIELD THEORY

2011 ◽  
Vol 26 (23) ◽  
pp. 4009-4051 ◽  
Author(s):  
ALESSANDRO SFONDRINI ◽  
TIM A. KOSLOWSKI
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
High Energy ◽  
Flow Equation ◽  
Renormalization Group Equation ◽  
Dual Model ◽  
Group Equation ◽  
Scalar Field Theory ◽  
Covariant Model ◽  
The Matrix

In this paper we apply the Functional Renormalization Group Equation (FRGE) to the noncommutative scalar field theory proposed by Grosse and Wulkenhaar. We derive the flow equation in the matrix representation and discuss the theory space for the self-dual model. The features introduced by the external dimensionful scale provided by the noncommutativity parameter, originally pointed out in R. Gurau and O. J. Rosten, J. High Energy Phys.0907, 064 (2009), are discussed in the FRGE context. Using a technical assumption, but without resorting to any truncation, it is then shown that the theory is asymptotically safe for suitably small values of the ϕ4coupling, recovering the result of M. Disertori et al., Phys. Lett. B649, 95 (2007). Finally, we show how the FRGE can be easily used to compute the one-loop beta-functions of the duality covariant model.

scholarly journals CONVERGENCE OF DERIVATIVE EXPANSIONS IN SCALAR FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 2095-2100 ◽  
Author(s):  
TIM R. MORRIS ◽  
JOHN F. TIGHE
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Flow Equation ◽  
Renormalisation Group ◽  
Massless Scalar ◽  
Derivative Expansion ◽  
Scalar Field Theory ◽  
Β Function

The convergence of the derivative expansion of the exact renormalisation group is investigated via the computation of the β function of massless scalar λφ4 theory. The derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. Convergence of the derivative expansion of the Legendre flow equation is trivial at one loop, but also can occur at two loops and in particular converges for an exponential cutoff.

scholarly journals PHASE TRANSITION OF N-COMPONENT SUPERCONDUCTORS

1996 ◽  
Vol 11 (23) ◽  
pp. 4273-4306 ◽  
Author(s):  
B. BERGERHOFF ◽  
D.F. LITIM ◽  
S. LOLA ◽  
C. WETTERICH
Keyword(s):  
Phase Transition ◽  
Three Dimensional ◽  
Gauge Coupling ◽  
Scalar Fields ◽  
Flow Equation ◽  
Second Order ◽  
Complex Scalar ◽  
Gauge Invariant ◽  
Second Order Transition ◽  
Average Action

We investigate the phase transition in the three-dimensional Abelian Higgs model for N complex scalar fields, using the gauge-invariant average action Γk. The dependence of Γk. on the effective infrared cutoff k is described by a nonperturbative flow equation. The transition turns out to be first or second order, depending on the ratio between the scalar and gauge coupling. We look at the fixed points of the theory for various N and compute the critical exponents of the model. Our results suggest the existence of a parameter range with a second order transition for all N, including the case of the superconductor phase transition for N=1.

Phase structure of the Euclidean three-dimensional O(1) ghost model

2019 ◽  
Vol 34 (02) ◽  
pp. 1950021
Author(s):  
Z. Péli ◽  
S. Nagy ◽  
K. Sailer
Keyword(s):  
Wave Function ◽  
Renormalization Group ◽  
Numerical Analysis ◽  
Three Dimensional ◽  
Leading Order ◽  
Wave Function Renormalization ◽  
Flow Equations ◽  
Multicritical Points ◽  
Field Independent ◽  
Average Action

We have treated the Euclidean three-dimensional O(1) ghost model with a modified version of the effective average action (EAA) renormalization group (RG) method, developed by us. We call it Fourier–Wetterich RG approach and it is used to investigate the occurrence of a periodic condensate in terms of the functional RG. The modification involves additional terms in the ansatz of the EAA, corresponding to the Fourier-modes of the periodic condensate. The RG flow equations are derived keeping the terms up to the fourth order of the gradient expansion (GE), however the numerical calculations are conducted in the second order (or next-to-leading order, NLO) of the GE. The expansion of the flow equations around the nontrivial minimum of the local potential takes into account properly the vertices induced by the periodic condensate even if the wave function renormalization is set to be field-independent. The numerical analysis reveals several different phases with three multicritical points.

scholarly journals Near horizon dynamics of three dimensional black holes

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Daniel Grumiller ◽  
Wout Merbis
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Einstein Gravity ◽  
Hamiltonian Reduction ◽  
Total Derivative ◽  
Zero Modes ◽  
Derivative Term ◽  
Total Derivative Term

We perform the Hamiltonian reduction of three dimensional Einstein gravity with negative cosmological constant under constraints imposed by near horizon boundary conditions. The theory reduces to a Floreanini–Jackiw type scalar field theory on the horizon, where the scalar zero modes capture the global black hole charges. The near horizon Hamiltonian is a total derivative term, which explains the softness of all oscillator modes of the scalar field. We find also a (Korteweg–de Vries) hierarchy of modified boundary conditions that we use to lift the degeneracy of the soft hair excitations on the horizon.

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