scholarly journals The NSVZ β-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
K. V. Stepanyantz
Keyword(s):  
Covariant Derivatives ◽  
Higher Covariant Derivatives ◽  
Β Function

scholarly journals On higher covariant derivatives of the curvature tensors of Kählerian C-spaces

1983 ◽  
Vol 91 ◽  
pp. 1-18 ◽  
Author(s):  
Ryoichi Takagi
Keyword(s):  
Symmetric Spaces ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Covariant Derivatives ◽  
Higher Covariant Derivatives ◽  
Group Of Isometries ◽  
Complex Homogeneous Manifold ◽  
Curvature Tensors ◽  
Simply Connected ◽  
Derivatives Of

A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

scholarly journals The all-loop perturbative derivation of the NSVZ $$\beta $$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
K. V. Stepanyantz
Keyword(s):  
Renormalization Group ◽  
Beta Function ◽  
Covariant Derivatives ◽  
Anomalous Dimensions ◽  
Higher Covariant Derivative ◽  
Higher Covariant Derivatives ◽  
Loop Approximation ◽  
Abelian Case ◽  
Group Functions ◽  
Supersymmetric Gauge

AbstractThe perturbative all-loop derivation of the NSVZ $$\beta $$ β -function for $${{\mathcal {N}}}=1$$ N = 1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $$\beta $$ β -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $$\beta $$ β -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

scholarly journals Three-loop contribution of the Faddeev–Popov ghosts to the $$\beta $$-function of $$\mathcal{N}=1$$ supersymmetric gauge theories and the NSVZ relation

2019 ◽  
Vol 79 (9) ◽  
Author(s):  
M. D. Kuzmichev ◽  
N. P. Meshcheriakov ◽  
S. V. Novgorodtsev ◽  
I. E. Shirokov ◽  
K. V. Stepanyantz
Keyword(s):  
Gauge Theories ◽  
Anomalous Dimension ◽  
Beta Function ◽  
Supersymmetric Gauge Theories ◽  
Loop Contribution ◽  
Covariant Derivatives ◽  
Anomalous Dimensions ◽  
Higher Covariant Derivatives ◽  
Supersymmetric Gauge ◽  
Multiloop Calculations

Abstract We find the three-loop contribution to the $$\beta $$β-function of $$\mathcal{N}=1$$N=1 supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev–Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $$\beta $$β-function is compared with the two-loop anomalous dimension of the Faddeev–Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $$\beta $$β-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.

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