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.

DIMENSIONAL REGULARIZATION IN THE 1/N EXPANSION

1992 ◽  
Vol 07 (14) ◽  
pp. 3265-3289 ◽  
Author(s):  
MASSIMO CAMPOSTRINI ◽  
PAOLO ROSSI
Keyword(s):  
Renormalization Group ◽  
Dimensional Regularization ◽  
Critical Index ◽  
Complete Agreement ◽  
Coupling Constant ◽  
Minimal Subtraction ◽  
Subtraction Scheme ◽  
Group Invariant ◽  
Minimal Subtraction Scheme ◽  
Supersymmetric Version

Two classes of renormalizable 1/N expandable two-dimensional models are analyzed to O(1/N) and the asymptotic behavior of the renormalized two-point functions is nonperturbatively evaluated. These results are taken as a benchmark to study the applicability of dimensional regularization and perturbative minimal subtraction renormalization to the context of the 1/N expansion. Perturbation theory is applied to O(1/N) diagrams to all orders in the weak coupling constant and, after resummation, the same finite renormalization group invariant asymptotic amplitudes are obtained. As a byproduct, the O(1/N) contributions to renormalization group Z functions in the minimal subtraction scheme are extracted and the critical index η is evaluated and compared to previous nonperturbative results, finding complete agreement. The appendix is devoted to the extension of these results to a supersymmetric version of the models.

scholarly journals SCHEME INDEPENDENCE AS AN INHERENT REDUNDANCY IN QUANTUM FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 2071-2074 ◽  
Author(s):  
JOSÉ I. LATORRE ◽  
TIM R. MORRIS
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
Renormalization Group Equations ◽  
Field Transformation ◽  
Exact Renormalization Group ◽  
Infinite Set

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relation. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions.

scholarly journals RENORMALIZATION GROUP EQUATION OF QUARK–LEPTON MASS MATRICES IN THE SO(10) MODEL WITH TWO-HIGGS SCALARS

2003 ◽  
Vol 18 (10) ◽  
pp. 719-731 ◽  
Author(s):  
TAKESHI FUKUYAMA ◽  
TATSURU KIKUCHI
Keyword(s):  
Renormalization Group ◽  
Symmetry Breaking ◽  
Yukawa Coupling ◽  
Intermediate Energy ◽  
Renormalization Group Equation ◽  
Coupling Constants ◽  
Group Equation ◽  
Mass Matrices ◽  
Renormalization Group Equations ◽  
Charged Leptons

The renormalization group equations (RGEs) of the mass matrices of quarks and leptons in an SO(10) model with two-Higgs scalars in the Yukawa coupling are studied. This model is the minimal model of SUSY and non-SUSY SO(10) GUT which can reproduce all the experimental data. Non-SUSY SO(10) GUT model has the intermediate energy phase, Pati–Salam phase, and passes through the symmetry breaking pattern, SO (10) → SU (2)L × SU (2)R × SU (4)C → SU (2)L × U (1)Y × SU (3)C. Though minimal, it has, after the Pati–Salam phase, four Higgs doublets in Yukawa interactions. We consider the RGEs of the Yukawa coupling constants of quarks and charged leptons and of the coupling constants of the dimension-five operators of neutrinos corresponding to the above symmetry breaking pattern. The scalar quartic interactions are also incorporated.

THE BOLTZMANN EQUATION IS A RENORMALIZATION GROUP EQUATION

2000 ◽  
Vol 14 (06) ◽  
pp. 555-561 ◽  
Author(s):  
O. PASHKO ◽  
Y. OONO
Keyword(s):  
Renormalization Group ◽  
Kinetic Equations ◽  
Slow Motion ◽  
Renormalization Group Equation ◽  
Point Of View ◽  
Group Equation ◽  
Amplitude Equations ◽  
Motion Equations ◽  
Reduced Equations ◽  
Renormalization Group Equations

It is known that renormalization group (RG) approaches to partial differential equations give reduced equations, e.g., amplitude equations, as renormalization group equations. Therefore, equations governing slow or global behaviors ought to be derived RG-theoretically. Kinetic equations are slow motion equations from the microscopic dynamics point of view. We demonstrate that kinetic equations are RG equations as expected.

SYMMETRIES OF THE RENORMALIZATION GROUP EQUATIONS AND THE SCALING LAW REVISITED

1993 ◽  
Vol 08 (32) ◽  
pp. 3017-3023
Author(s):  
P. K. JHA ◽  
K. C. TRIPATHY
Keyword(s):  
Renormalization Group ◽  
Operator Product Expansion ◽  
Scaling Laws ◽  
Scaling Law ◽  
Renormalization Group Equation ◽  
Operator Product ◽  
Electromagnetic Current ◽  
Group Equation ◽  
Infinite Dimensional ◽  
Renormalization Group Equations

The symmetry associated with the renormalization group equation satisfied by the Wilson coefficients in the operator product expansion of the electromagnetic current in deep inelastic scattering is re-examined using Blueman-Cole-Obsiannikov-Olver program. It is shown that the system exhibits infinite-dimensional symmetry. From the characteristics, we derive the detailed solutions of the renormalization group equation and the scaling laws for Wilson moments.

scholarly journals EXACT RENORMALIZATION GROUP EQUATIONS AND THE FIELD THEORETICAL APPROACH TO CRITICAL PHENOMENA

2001 ◽  
Vol 16 (11) ◽  
pp. 1825-1845 ◽  
Author(s):  
C. BAGNULS ◽  
C. BERVILLIER
Keyword(s):  
Renormalization Group ◽  
Critical Phenomena ◽  
Theoretical Approach ◽  
Continuum Limit ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Perturbative Approach ◽  
Renormalization Group Equations ◽  
Exact Renormalization Group ◽  
The Continuum

After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the pressure of singularities in the perturbative series are discussed.

scholarly journals TEMPERATURE RENORMALIZATION GROUP AND RESUMMATION

1993 ◽  
Vol 08 (11) ◽  
pp. 1887-1901 ◽  
Author(s):  
PER ELMFORS
Keyword(s):  
Renormalization Group ◽  
Effective Mass ◽  
Renormalization Group Equation ◽  
Higher Order ◽  
Group Equation ◽  
Coupling Constant ◽  
Loop Contribution ◽  
The One

The temperature renormalization group equation (TRGE) is compared with a diagrammatic expansion for the (ϕ4)4-theory. It is found that the one-loop TRGE resums the leading powers of temperature for the effective mass. A two-loop contribution to TRGE is required to do the leading resummation for the coupling constant. It is also shown that the higher order TRGE resums subleading powers of temperature.

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