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.

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.

scholarly journals CONSISTENCY OF THE HYBRID REGULARIZATION WITH HIGHER COVARIANT DERIVATIVE AND INFINITELY MANY PAULI–VILLARS

2001 ◽  
Vol 16 (22) ◽  
pp. 3755-3783
Author(s):  
KOH-ICHI NITTOH
Keyword(s):  
Covariant Derivative ◽  
Vacuum Polarization ◽  
Gauge Theories ◽  
Polarization Tensor ◽  
Logarithmic Divergence ◽  
Yang Mills ◽  
Higher Covariant Derivative ◽  
Group Functions ◽  
Supersymmetric Gauge ◽  
Vacuum Polarization Tensor

We study the regularization and renormalization of the Yang–Mills theory in the framework of the manifestly invariant formalism, which consists of a higher covariant derivative with an infinitely many Pauli–Villars fields. Unphysical logarithmic divergence, which is the problematic point on the Slavnov method, does not appear in our scheme, and the well-known value of the renormalization group functions are derived. The cancellation mechanism of the quadratic divergence is also demonstrated by calculating the vacuum polarization tensor of the order of Λ0 and Λ-4. These results are the evidence that our method is valid for intrinsically divergent theories and is expected to be available for the theory which contains the quantity depending on the space–time dimensions, like supersymmetric gauge theories.

scholarly journals Supersymmetry, quantum corrections, and the higher derivative regularization

2018 ◽  
Vol 191 ◽  
pp. 06002
Author(s):  
Konstantin Stepanyantz
Keyword(s):  
Anomalous Dimension ◽  
Quantum Corrections ◽  
Explicit Calculation ◽  
Yukawa Couplings ◽  
Higher Derivative ◽  
Anomalous Dimensions ◽  
Higher Covariant Derivative ◽  
Derivative Method ◽  
Abelian Case ◽  
Supersymmetric Theories

We investigate the structure of quantum corrections in N = 1 supersymmetric theories using the higher covariant derivative method for regularization. In particular, we discuss the non-renormalization theorem for the triple gauge-ghost vertices and its connection with the exact NSVZ β-function. Namely, using the finiteness of the triple gauge-ghost vertices we rewrite the NSVZ equation in a form of a relation between the β-function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev-Popov ghosts, and of the matter superfields. We argue that it is this form that follows from the perturbative calculations, and give a simple prescription how to construct the NSVZ scheme in the non-Abelian case. These statements are confirmed by an explicit calculation of the three-loop contributions to the β-function containing Yukawa couplings. Moreover, we calculate the two-loop anomalous dimension of the ghost superfields and demonstrate that for doing this calculation it is very important that the quantum gauge superfield is renormalized non-linearly.

scholarly journals The NSVZ relations for $$ \mathcal{N} $$ = 1 supersymmetric theories with multiple gauge couplings

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
D. S. Korneev ◽  
D. V. Plotnikov ◽  
K. V. Stepanyantz ◽  
N. A. Tereshina
Keyword(s):  
Gauge Coupling ◽  
Coupling Constants ◽  
Abelian Gauge ◽  
Covariant Derivatives ◽  
Higher Derivative ◽  
Anomalous Dimensions ◽  
New Form ◽  
Gauge Couplings ◽  
Supersymmetric Gauge ◽  
Supersymmetric Theories

Abstract We investigate the NSVZ relations for $$ \mathcal{N} $$ N = 1 supersymmetric gauge theories with multiple gauge couplings. As examples, we consider MSSM and the flipped SU(5) model, for which they easily reproduce the results for the two-loop β-functions. For $$ \mathcal{N} $$ N = 1 SQCD interacting with the Abelian gauge superfield we demonstrate that the NSVZ-like equation for the Adler D-function follows from the NSVZ relations. Also we derive all-loop equations describing how the NSVZ equations for theories with multiple gauge couplings change under finite renormalizations. They allow describing a continuous set of NSVZ schemes in which the exact NSVZ β-functions are valid for all gauge coupling constants. Very likely, this class includes the HD+MSL scheme, which is obtained if a theory is regularized by Higher covariant Derivatives and divergences are removed by Minimal Subtractions of Logarithms. That is why we also discuss how one can construct the higher derivative regularization for theories with multiple gauge couplings. Presumably, this regularization allows to derive the NSVZ equations for such theories in all loops. In this paper we make the first step of this derivation, namely, the NSVZ equations for theories with multiple gauge couplings are rewritten in a new form which relates the β-functions to the anomalous dimensions of the quantum gauge superfields, of the Faddeev-Popov ghosts, and of the matter superfields. The equivalence of this new form to the original NSVZ relations follows from the extension of the non-renormalization theorem for the triple gauge-ghost vertices, which is also derived in this paper.

PHASE TRANSITION IN CONTINUOUS SYMMETRY MODEL IN GENERAL DIMENSIONS — FIXED DIMENSION RENORMALIZATION GROUP APPROACH

1993 ◽  
Vol 08 (30) ◽  
pp. 5329-5351 ◽  
Author(s):  
YU. HOLOVATCH
Keyword(s):  
Phase Transition ◽  
Renormalization Group ◽  
Symmetric Model ◽  
Continuous Symmetry ◽  
Group Approach ◽  
Fixed Dimension ◽  
Loop Approximation ◽  
Renormalization Group Approach ◽  
Borel Transformation ◽  
Group Functions

Critical exponents of the O(m)-symmetric model are calculated in the case when dimension of space is noninteger. Calculations are performed in the frames of the field-theoretical approach using the three-loop approximation. Renormalization group functions in the Callan-Symanzik scheme are considered directly in noninteger dimensions. Perturbation theory expansions are resummed with the use of Padé-Borel transformation.

scholarly journals APPLICATIONS OF EXACT RENORMALIZATION GROUP TECHNIQUES TO THE NON-PERTURBATIVE STUDY OF SUPERSYMMETRIC GAUGE FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 1811-1824 ◽  
Author(s):  
S. ARNONE ◽  
S. CHIANTESE ◽  
K. YOSHIDA
Keyword(s):  
Renormalization Group ◽  
Beta Function ◽  
Ultra Violet ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Finite Mass ◽  
Yang Mills ◽  
Supersymmetric Gauge ◽  
Exact Renormalization Group ◽  
Massless Fields

Exact Renormalization Group techniques are applied to supersymmetric models in order to get some insights into the low energy effective actions of such theories. Starting from the ultra-violet finite mass deformed N=4 supersymmetric Yang Mills theory, one varies the regularising mass and compensates for it by introducing an effective Wilsonian action. (Polchinski's) renormalization group equation is modified in an essential way by the presence of rescaling (a.k.a. Konishi) anomaly, which is responsible for the beta-function. When supersymmetry is broken up to N=1 the form of effective actions in terms of massless fields is quite reasonable, while in the case of the N=2 model we appear to have problems related to instantons.

scholarly journals One-loop polarization operator of the quantum gauge superfield for 𝒩 = 1 SYM regularized by higher derivatives

2017 ◽  
Vol 32 (36) ◽  
pp. 1750194 ◽  
Author(s):  
A. E. Kazantsev ◽  
M. B. Skoptsov ◽  
K. V. Stepanyantz
Keyword(s):  
Polarization Operator ◽  
Special Choice ◽  
Yang Mills ◽  
Feynman Gauge ◽  
Covariant Derivatives ◽  
Higher Derivative ◽  
Loop Approximation ◽  
Higher Derivatives ◽  
Supersymmetric Gauge ◽  
Brst Invariance

We consider the general [Formula: see text] supersymmetric gauge theory with matter, regularized by higher covariant derivatives without breaking the BRST invariance, in the massless limit. In the [Formula: see text]-gauge we obtain the (unrenormalized) expression for the two-point Green function of the quantum gauge superfield in the one-loop approximation as a sum of integrals over the loop momentum. The result is presented as a sum of three parts: the first one corresponds to the pure supersymmetric Yang–Mills theory in the Feynman gauge, the second one contains all gauge-dependent terms, and the third one is the contribution of diagrams with a matter loop. For the Feynman gauge and a special choice of the higher derivative regulator in the gauge fixing term, we analytically calculate these integrals in the limit [Formula: see text]. In particular, in addition to the leading logarithmically divergent terms, which are determined by integrals of double total derivatives, we also find the finite constants.

scholarly journals GAUGE CONSISTENT WILSON RENORMALIZATION GROUP II: THE NON-ABELIAN CASE

2000 ◽  
Vol 15 (14) ◽  
pp. 2153-2179 ◽  
Author(s):  
M. SIMIONATO
Keyword(s):  
Renormalization Group ◽  
Gauge Theories ◽  
Beta Function ◽  
Ward Identities ◽  
Abelian Gauge ◽  
Axial Gauge ◽  
Abelian Case ◽  
Group Ii ◽  
The One ◽  
Wilson Renormalization Group

We give a Wilsonian formulation of non-Abelian gauge theories explicitly consistent with axial gauge Ward identities. The issues of unitarity and dependence on the quantization direction are carefully investigated. A Wilsonian computation of the one-loop QCD beta function is performed.

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