scholarly journals Four Loop Scalar ϕ4 Theory Using the Functional Renormalization Group

Universe ◽  
2019 ◽  
Vol 5 (1) ◽  
pp. 9 ◽  
Author(s):  
Margaret Carrington ◽  
Christopher Phillips
Keyword(s):  
Renormalization Group ◽  
Effective Theory ◽  
Group Method ◽  
Coupling Constant ◽  
Loop Order ◽  
Quartic Coupling ◽  
Scalar Theory ◽  
Functional Renormalization Group ◽  
Bare Coupling Constant ◽  
Effective Theories

We work with a symmetric scalar theory with quartic coupling in 4-dimensions. Using a 2PI effective theory and working at 4 loop order, we renormalize with a renormalization group method. All divergences are absorbed by one bare coupling constant and one bare mass which are introduced at the level of the Lagrangian. The method is much simpler than counterterm renormalization, and can be generalized to higher order nPI effective theories.

scholarly journals The renormalization group equations revisited

2018 ◽  
Vol 33 (26) ◽  
pp. 1830024 ◽  
Author(s):  
Jean-François Mathiot
Keyword(s):  
Renormalization Group ◽  
Dimensional Regularization ◽  
Direct Consequence ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Coupling Constant ◽  
Loop Order ◽  
Subtraction Scheme ◽  
Minimal Subtraction Scheme ◽  
Renormalization Group Equations

Starting from a well-defined local Lagrangian, we analyze the renormalization group equations in terms of the two different arbitrary scales associated with the regularization procedure and with the physical renormalization of the bare parameters, respectively. We apply our formalism to the minimal subtraction scheme using dimensional regularization. We first argue that the relevant regularization scale in this case should be dimensionless. By relating bare and renormalized parameters to physical observables, we calculate the coefficients of the renormalization group equation up to two-loop order in the [Formula: see text] theory. We show that the usual assumption, considering the bare parameters to be independent of the regularization scale, is not a direct consequence of any physical argument. The coefficients that we find in our two-loop calculation are identical to the standard practice. We finally comment on the decoupling properties of the renormalized coupling constant.

scholarly journals Isotropic Lifshitz scaling in four dimensions

2020 ◽  
Vol 17 (04) ◽  
pp. 2050053 ◽  
Author(s):  
Dario Zappalà
Keyword(s):  
Renormalization Group ◽  
Fixed Points ◽  
Critical Dimension ◽  
Two Dimensional ◽  
Continuous Line ◽  
Scalar Theory ◽  
Functional Renormalization Group ◽  
Four Dimensions ◽  
Dimension Formula

The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.

scholarly journals Lectures on the functional renormalization group method

Open Physics ◽  
2003 ◽  
Vol 1 (1) ◽  
pp. 1-71 ◽  
Author(s):  
Janos Polonyi
Keyword(s):  
Renormalization Group ◽  
Evolution Equation ◽  
Scaling Laws ◽  
General Purpose ◽  
Perturbation Series ◽  
Renormalization Group Equation ◽  
Group Method ◽  
Group Equation ◽  
Critical Systems ◽  
Functional Renormalization Group

AbstractThese introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

scholarly journals Bound states in functional renormalization group

2019 ◽  
Vol 34 (27) ◽  
pp. 1950154 ◽  
Author(s):  
Antal Jakovác ◽  
András Patkós
Keyword(s):  
Renormalization Group ◽  
Type Theory ◽  
Spectral Decomposition ◽  
Bound States ◽  
Scattering Amplitude ◽  
Microscopic Theory ◽  
Effective Theory ◽  
Functional Renormalization Group ◽  
Renormalization Group Equations ◽  
Oppositely Charged

Equivalence criteria are established for an effective Yukawa-type theory of composite fields representing two-particle fermion bound states with the original “microscopic” theory of interacting fermions based on the spectral decomposition of the 2-to-2 fermion scattering amplitude. Functional renormalization group equations of the effective theory are derived exploiting relations expressing the equivalence. The effect of truncating the spectral decomposition is investigated quantitatively on the example of the nonrelativistic bound states of two oppositely charged fermions.

scholarly journals The RG flow of Nambu–Jona-Lasinio model at finite temperature and density

2015 ◽  
Vol 30 (27) ◽  
pp. 1550180 ◽  
Author(s):  
Ken-Ichi Aoki ◽  
Masatoshi Yamada
Keyword(s):  
Renormalization Group ◽  
Phase Boundary ◽  
Finite Temperature ◽  
Quantum Corrections ◽  
Coupling Constant ◽  
Density Region ◽  
Chiral Phase ◽  
Functional Renormalization Group ◽  
Njl Model ◽  
High Density Region

We study the Nambu–Jona-Lasinio model at finite temperature and finite density by using the functional renormalization group. The RG flows of the four-Fermi coupling constant in the NJL model are investigated. We obtain the chiral phase boundary in cases of the large-[Formula: see text] leading approximation and an improved approximation. The large-[Formula: see text] nonleading term at the vanishing temperature has a singularity at the Fermi surface. We show that the quantum corrections by the large-[Formula: see text] nonleading term largely influence the phase boundary at low temperature and high density region.

scholarly journals INVARIANT FORMULATION OF THE FUNCTIONAL RENORMALIZATION GROUP METHOD FOR U(n)×U(n) SYMMETRIC MATRIX MODELS

2012 ◽  
Vol 27 (36) ◽  
pp. 1250212 ◽  
Author(s):  
A. PATKÓS
Keyword(s):  
Renormalization Group ◽  
Symmetric Matrix ◽  
The Other ◽  
Group Method ◽  
Renormalization Group Method ◽  
Potential Approximation ◽  
Functional Renormalization Group ◽  
Local Potential Approximation ◽  
Invariant Formulation ◽  
Symmetric Theory

The Local Potential Approximation (LPA) to the Wetterich-equation is formulated explicitly in terms of operators, which are invariant under the U (n)× U (n) symmetry group. Complete formulas are presented for the two-flavor ( U (2)× U (2)) case. The same approach leads to a unique natural truncation of the functional driving the renormalization flow of the potential of the three-flavor case ( U (3)× U (3)). The procedure applied to the SU (3)× SU (3) symmetric theory, results in an equation, which potentially allows an RG-investigation of the effect of the 't Hooft term representing the U A(1) anomaly, disentangled from the other operators.

scholarly journals Optimized regulator for the quantized anharmonic oscillator

2015 ◽  
Vol 30 (12) ◽  
pp. 1550058 ◽  
Author(s):  
J. Kovacs ◽  
S. Nagy ◽  
K. Sailer
Keyword(s):  
Renormalization Group ◽  
Ground State ◽  
Evolution Equations ◽  
Energy Gap ◽  
Anharmonic Oscillator ◽  
Group Method ◽  
Renormalization Group Method ◽  
Functional Renormalization Group ◽  
First Excited State ◽  
Symmetric Phase

The energy gap between the first excited state and the ground state is calculated for the quantized anharmonic oscillator in the framework of the functional renormalization group method. The compactly supported smooth regulator is used which includes various types of regulators as limiting cases. It was found that the value of the energy gap depends on the regulator parameters. We argue that the optimization based on the disappearance of the false, broken symmetric phase of the model leads to the Litim's regulator. The least sensitivity on the regulator parameters leads, however, to an IR regulator being somewhat different of the Litim's one, but it can be described as a perturbatively improved, or generalized Litim's regulator and provides analytic evolution equations, too.

COUNTING STATISTICS OF NONEQUILIBRIUM TUNNELING PROCESS USING FUNCTIONAL RENORMALIZATION GROUP

2010 ◽  
Vol 24 (05) ◽  
pp. 575-585 ◽  
Author(s):  
QING-QIANG XU ◽  
BEN-LING GAO ◽  
SHI-JIE XIONG
Keyword(s):  
Renormalization Group ◽  
Charge Transfer ◽  
Quantum Dot ◽  
Group Method ◽  
Renormalization Group Method ◽  
Functional Renormalization Group ◽  
Nonequilibrium Transport ◽  
Cumulant Generating Function ◽  
Full Counting Statistics ◽  
Counting Statistics

We develop a scheme to investigate the flux of nonequilibrium transport based on the full counting statistics and nonequilibrium functional renormalization group method. As an illustrative example, we study the charge transfer in the system of an interacting quantum dot connected to two noninteracting reservoirs via tunneling. Within the lowest approximation in functional renormalization group, we obtain the cumulant generating function analytically.

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