scholarly journals Approximate differential equations for renormalization group functions in models free of vertex divergencies

2010 ◽  
Vol 826 (3) ◽  
pp. 522-531 ◽  
Author(s):  
Marc P. Bellon
Keyword(s):  
Renormalization Group ◽  
Differential Equations ◽  
Group Functions

scholarly journals Three-loop vertex integrals at symmetric point

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Andrey Pikelner
Keyword(s):  
Renormalization Group ◽  
Differential Equations ◽  
Basis Functions ◽  
Composite Operator ◽  
Operator Matrix ◽  
Matrix Elements ◽  
Root Of Unity ◽  
Loop Level ◽  
Symmetric Point ◽  
Group Functions

Abstract This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the ε-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.

RENORMALIZATION AND ASYMPTOTICS

2000 ◽  
Vol 14 (12n13) ◽  
pp. 1327-1361 ◽  
Author(s):  
Y. OONO
Keyword(s):  
Renormalization Group ◽  
Differential Equations ◽  
Perturbation Methods ◽  
Explicit Calculation ◽  
Group Approach ◽  
New Approach ◽  
Asymptotic Behaviors ◽  
Singular Perturbation Methods ◽  
Renormalization Group Approach ◽  
Long Time

After a gentle introduction to the Stückelberg–Petermann style (i.e. field-theoretical) renormalization group (RG) theory, its application to the study of asymptotic behaviors of differential equations is explained through simple examples. The essence of singular perturbation methods to study asymptotic behaviors of differential equations is to reduce it to equations governing long time scale behaviors (i.e. the so-called reductive perturbation). The RG approach gives the reduced equation as an RG equation (this is called the reductive renormalization group approach). Once the RG equation is written down, the asymptotic behavior can be obtained by solving it. The RG equation also facilitates the error analysis of the asymptotic solutions. A new approach via "proto-RG equation" explained in this article further simplifies the reductive use of RG. For example, to the lowest nontrivial order the approach does not require any explicit calculation of perturbative results.

scholarly journals Six-dimensional ultraviolet completion of the ℂℙ(N) σ model at two loops

2020 ◽  
Vol 35 (22) ◽  
pp. 2050188
Author(s):  
J. A. Gracey
Keyword(s):  
Renormalization Group ◽  
Quantum Electrodynamics ◽  
Matter Field ◽  
Coupling Constants ◽  
Universal Theory ◽  
Gauge Parameter ◽  
Loop Analysis ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Group Functions

We extend the recent one-loop analysis of the ultraviolet completion of the [Formula: see text] nonlinear [Formula: see text] model in six dimensions to two-loop order in the [Formula: see text] scheme for an arbitrary covariant gauge. In particular we compute the anomalous dimensions of the fields and [Formula: see text]-functions of the four coupling constants. We note that like Quantum Electrodynamics (QED) in four dimensions the matter field anomalous dimension only depends on the gauge parameter at one loop. As a nontrivial check we verify that the critical exponents derived from these renormalization group functions at the Wilson–Fisher fixed point are consistent with the [Formula: see text] expansion of the respective large [Formula: see text] exponents of the underlying universal theory. Using the Ward–Takahashi identity we deduce the three-loop [Formula: see text] renormalization group functions for the six-dimensional ultraviolet completeness of scalar QED.

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