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A. L. Kataev

Publications and source records attributed to A. L. Kataev.

At least 37 records · Page 2Linked to original sources
arXiv2018

A class of the NSVZ renormalization schemes for ${\cal N}=1$ SQED

For the ${\cal N}=1$ supersymmetric electrodynamics we investigate renormalization schemes in which the NSVZ equation relating the $β$-function to the anomalous dimension of the matter superfields is valid in all loops. We demonstrate that there is an infinite set of such schemes. They are related by finite renormalizations which form a group and are parameterized by one finite function and one arbitrary constant. This implies that the NSVZ $β$-function remains unbroken if the finite renormalization of the coupling constant is related to the finite renormalization of the matter superfields by a special equation derived in this paper. The arbitrary constant corresponds to the arbitrariness of choosing the renormalization point. The results are illustrated by explicit calculations in the three-loop approximation.

hep-th↗
arXiv2018

Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the $β$-factorization property?

The scheme and gauge dependence of the factorization property of the RG $β$-function in the $SU(N_c)$ QCD generalized Crewther relation (GCR), which connects the non-singlet contributions to the Adler and Bjorken polarized sum rule functions, is investigated at the $\mathcal{O}(a^4_s)$ level. In the gauge-invariant $\rm{\overline{MS}}$-scheme this property holds at least at this order. To study whether this property is true in all gauge-invariant schemes, we consider the $\rm{MS}$-like schemes in QCD and the QED-limit of the GCR in the $\rm{\overline{MS}}$-scheme and in the $\rm{MOM}$ and the $\rm{OS}$ schemes. In these schemes we confirm the existence of the $β$-function factorization in the QCD and QED variants of the GCR. The problem of the possible $β$-factorization in the gauge-dependent renormalization schemes in QCD is studied. We consider the gauge non-invariant $\rm{mMOM}$ and $\rm{MOMgggg}$-schemes and demonstrate that in the $\rm{mMOM}$ scheme at the $\mathcal{O}(a^3_s)$ level the $β$-factorization is valid for three values of the gauge parameter $ξ$ only, namely for $ξ=-3, -1$ and $ξ=0$. In the $\mathcal{O}(a^4_s)$ order of PT it remains valid only for case of the Landau gauge $ξ=0$. The consideration of these two schemes for the QCD GCR allows us to conclude that the factorization of RG $β$-function will always be implemented in any $\rm{MOM}$-like schemes with linear covariant gauge at $ξ=0$ and $ξ=-3$ at the $\mathcal{O}(a^3_s)$ level. It is demonstrated that if factorization property for the $\rm{MS}$-like schemes is true in all orders of PT, as theoretically indicated, then the factorization will also occur in the arbitrary $\rm{MOM}$-like scheme in the Landau gauge in all orders of PT as well.

hep-ph↗
arXiv2017

The analytical $\mathcal{O}(a^4_s)$ expression for the polarized Bjorken sum rule in the miniMOM scheme and the consequences for the generalized Crewther relation

The analytical $\mathcal{O}(a^4_s)$ perturbative QCD expression for the flavour non-singlet contribution to the Bjorken polarized sum rule in the rather applicable at present gauge--dependent $\rm{miniMOM}$ scheme is obtained. For the considered three values of the gauge parameter, namely $ξ=0$ (Landau gauge), $ξ=-1$ (anti--Feynman gauge) and $ξ=-3$ (Stefanis--Mikhailov gauge), the scheme-dependent coefficients are considerably smaller than the gauge-independent $\rm{\overline{MS}}$ results. It is found that the fundamental property of the factorization of the QCD renormalization group $β$-function in the generalized Crewther relation, which is valid in the gauge-invariant $\rm{\overline{MS}}$ scheme up to $\mathcal{O}(a^4_s)$-level at least, is unexpectedly valid at the same level in the $\rm{miniMOM}$-scheme for $ξ=0$, and for $ξ=-1$ and $ξ=-3$ in part.

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arXiv2017

The Adler $D$-function for ${\cal N}=1$ SQCD regularized by higher covariant derivatives in the three-loop approximation

We calculate the Adler $D$-function for ${\cal N}=1$ SQCD in the three-loop approximation using the higher covariant derivative regularization and the NSVZ-like subtraction scheme. The recently formulated all-order relation between the Adler function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant is first considered and generalized to the case of an arbitrary representation for the chiral matter superfields. The correctness of this all-order relation is explicitly verified at the three-loop level. The special renormalization scheme in which this all-order relation remains valid for the $D$-function and the anomalous dimension defined in terms of the renormalized coupling constant is constructed in the case of using the higher derivative regularization. The analytic expression for the Adler function for ${\cal N}=1$ SQCD is found in this scheme to the order $O(α_s^2)$. The problem of scheme-dependence of the $D$-function and the NSVZ-like equation is briefly discussed.

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arXiv2016

The $\{β\}$-expansion formalism in perturbative QCD and its extension

We discuss the $\{ β\}$-expansion for renormalization group invariant quantities tracing this expansion to the different contractions of the corresponding incomplete BPHZ $R$-operation. All of the coupling renormalizations, which follow from these contractions, should be taken into account for the $\{ β\}$-expansion. We illustrate this feature considering the nonsinglet Adler function $D^\text{NS}$ in the third order of perturbation. We propose a generalization of the $\{ β\}$-expansion for the renormalization group covariant quantities -- the $\{ β,γ\}$-expansion.

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arXiv2016

The NSVZ scheme for ${\cal N}=1$ SQED with $N_f$ flavors, regularized by the dimensional reduction, in the three-loop approximation

At the three-loop level we analyze, how the NSVZ relation appears for ${\cal N}=1$ SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalization group functions defined in terms of the bare coupling constant. However, for the renormalization group functions defined in terms of the renormalized coupling constant, it is possible to impose boundary conditions to the renormalization constants giving the NSVZ scheme in the three-loop order. They are similar to the all-loop ones defining the NSVZ scheme obtained with the higher derivative regularization, but are more complicated. The NSVZ schemes constructed with the dimensional reduction and with the higher derivative regularization are related by a finite renormalization in the considered approximation.

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arXiv2016

On the flavour dependence of the $\mathcal{O}(α_s^4)$ correction to the relation between running and pole heavy quark masses

Recently the four-loop perturbative QCD contributions to the relations between pole and running masses of charm, bottom and top quarks were evaluated in the $\rm{\overline{MS}}$-scheme with identical numerical error bars. In this work the flavour dependence of the $\mathcal{O}(α_s^4)$ correction to these asymptotic series is obtained in the semi-analytical form with the help of the least squares method. The numerical structure of the corresponding asymptotic perturbative relations between pole and running $c$, $b$ and $t$-quark masses is considered and the theoretical errors of the $\mathcal{O}(α_s^4)$-contributions are discussed. The explicit dependence for these relations on the renormalization scale $μ^2$ and the flavour number $n_l$ is presented.

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arXiv2016

Adler function and Bjorken polarized sum rule: Perturbation expansions in powers of $SU(N_c)$ conformal anomaly and studies of the conformal symmetry limit

We consider a new form of analytical perturbation theory expansion in the massless $SU(N_c)$ theory, for the non-singlet part of the $e^+e^-$-annihilation to hadrons Adler function $D^{ns}$ and of the Bjorken sum rule of the polarized lepton-hadron deep-inelastic scattering $C_{ns}^{Bjp}$, and demonstrate its validity at the $O(α_s^4)$-level at least. It is a two-fold series in terms of powers of the conformal anomaly and of $SU(N_c)$ coupling $α_s$. Explicit expressions are obtained for the $\{β\}$-expanded perturbation coefficients at $O(α_s^4)$ level in $\bar{\rm MS}$ scheme, for both considered physical quqantities. Comparisons of the terms in the $\{β\}$-expanded coefficients are made with the corresponding terms obtained by using extra gluino degrees of freedom, or skeleton-motivated expansion, or $R_δ$-scheme motivated expansion in the Principle of Maximal Conformality. Relations between terms of the $\{β\}$-expansion for the $D^{ns}$- and $C_{ns}^{Bjp}$-functions, which follow from the conformal symmetry limit and its violation, are presnted. The relevance to the possible new analysis of the experimental data for the Adler function and Bjorken sum rule is discussed.

hep-ph↗
arXiv2016

From perturbative calculations of the QCD static potential towards four-loop pole-running heavy quarks masses relation

The summary of the available semi-analytical results for the three-loop corrections to the QCD static potential and for the $\mathcal{O}(α_s^4)$ contributions to the ratio of the running and pole heavy quark masses are presented. The procedure of the determination of the dependence of the four-loop contribution to the pole-running heavy quarks mass ratio on the number of quarks flavours, based on application of the least squares method is described. The necessity of clarifying the reason of discrepancy between the numerical uncertainties of the $α_s^4$ coefficients in the mass ratio, obtained by this mathematical method by the direct numerical calculations is emphasised.

hep-ph↗
arXiv2015

Structure of three-loop contributions to the beta-function of N=1 SQED with N_f flavors, regularized by the dimensional reduction

In the case of using the higher derivative regularization for $N=1$ SQED with $N_f$ flavors the loop integrals giving the $β$-function are integrals of double total derivatives in the momentum space. This feature allows to reduce one of the loop integrals to an integral of the $δ$-function and to derive the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. In this paper we consider $N=1$ SQED with $N_f$ flavors regularized by the dimensional reduction in the $\overline{\mbox{DR}}$-scheme. Evaluating the scheme-dependent three-loop contribution to the $β$-function proportional to $(N_f)^2$ we find the structures analogous to integrals of the $δ$-singularities. After adding the scheme-independent terms proportional to $(N_f)^1$ we obtain the known result for the three-loop $β$-function.

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arXiv2015

Fourth-order QCD renormalization group quantities in the {\rm{V}}-scheme and the relation of the $β$-function to the Gell-Mann--Low function in QED

The semi-analytical $O(α_s^4)$ expression for the renormalization group $β$-function in the ${\rm{V}}$-scheme is obtained in the case of the $SU(N_c)$ gauge group. In the process of calculations we use the existing information about the three-loop perturbative approximation for the QCD static potential, evaluated in the $\rm{\overline{MS}}$-scheme. The comparison of the numerical values of the third and fourth coefficients for the QCD RG $β$- functions in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-schemes and in minimal MOM scheme in the the Landau gauge is presented. The phenomenologically-oriented comparisons for the coefficients of $O(α_s^4)$ expression for the $e^+e^-$-annihilation R-ratio in these schemes are presented. It is shown, that taking into account of these QCD contributions are of vital importance and lead to the drastic decrease of the scheme-dependence ambiguities of the fourth-order perturbative QCD approximations for the $e^+e^-$ annihilation R-ratio for the number of active flavours,$n_f=5$ in particular. We demonstrate that in the case of QED with $N$-types of leptons the coefficients of the $β^{\rm{V}}$-function are closely related to the ones of the Gell-Mann--Low $Ψ$-function and emphasise that they start to differ from each other at the fourth order due to the appearance of the extra $N^2$-contribution in the V-scheme. The source of this extra correction is clarified. The general all-order QED relations between the coefficients of the $β^{\rm{V}}$- and $Ψ$-functions are discussed.

hep-ph↗
arXiv2015

The four-loop renormalization group QCD and QED $β$-functions in the $V$-scheme and their analogy with the Gell-Mann--Low function in QED and QCD

The semi-analytical expression for the forth coefficient of the renormalization group $β$-function in the ${\rm{V}}$-scheme is obtained in the case of the $SU(N_c)$ gauge group. In the process of calculations we use the three-loop perturbative approximation for the QCD static potential, evaluated in the $\rm{\overline{MS}}$-scheme. The importance of getting more detailed expressions for the $n_f$-independent three-loop contribution to the static potential,obtained at present by two groups, is emphasised. The comparison of the numerical structure of the four-loop approximations for the RG $β$- function of QCD in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-schemes and in the minimal MOM scheme in the Landau gauge are presented. Considering the limit of QED with $N$-types of leptons we discover that the $β^{\rm{V}}$-function is starting to differ from the Gell-Mann--Low function $Ψ(α_{\rm{MOM}})$ at the level of the forth-order perturbative corrections, receiving the proportional to $N^2$ additional term. Taking this feature into account, we propose to consider the $β^{\rm{V}}$-function as the most theoretically substantiated analog of the Gell-Man--Low function in QCD.

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arXiv2015

The generalized BLM approach to fix scale-dependence in QCD: the current status of investigations

I present a brief review of the generalized Brodsky-Lepage-McKenzie (BLM) approaches to fix the scale-dependence of the renormalization group (RG) invariant quantities in QCD. At first, these approaches are based on the expansions of the coefficients of the perturbative series for the RG-invariant quantities in the products of the coefficients $β_i$ of the QCD $β$-function, which are evaluated in the MS-like schemes. As a next step all $β_i$-dependent terms are absorbed into the BLM-type scale(s) of the powers of the QCD couplings. The difference between two existing formulations of the above mentioned generalizations based on the seBLM approach and the Principle of Maximal Conformality (PMC) are clarified in the case of the Bjorken polarized deep-inelastic scattering sum rule. Using the conformal symmetry-based relations for the non-singlet coefficient functions of the Adler D-function and of Bjorken polarized deep-inelastic scattering sum rules $C^{\rm Bjp}_{\rm NS}(a_s)$ the $β_i$-dependent structure of the NNLO approximation for $C^{\rm Bjp}_{\rm NS}(a_s)$ is predicted in QCD with $n_{gl}$-multiplet of gluino degrees of freedom, which appear in SUSY extension of QCD. The importance of performing the analytical calculation of the N$^3$LO additional contributions of $n_{gl}$ gluino multiplet to $C^{\rm Bjp}_{\rm NS}(a_s)$ for checking the presented in the report NNLO prediction and for the studies of the possibility to determine the discussed $\{β\}$-expansion pattern of this sum rule at the $O(a_s^4)$-level is emphasised.

hep-ph↗
arXiv2015

$\{β\}$-expansion in QCD, its conformal symmetry limit: theory + applications

The basis of the $\{β\}$-expansion for the perturbative series evaluated in the $\overline{MS}$ scheme for the renormalization group invariant quantities is summarized.Comparison with a similar representation,used within the BLM-motivated Principle of Maximal Conformality,is discussed.We stress that the original $\{β\}$-expansion contains a completed list of terms rather than its PMC analog. The arguments in favour of the complete $\{β\}$-expansion are presented. They are based on the relations which follow from the power $β$-function generalization of the Crewther relation for the nonsinglet $\overline{MS}$ contributions to the Adler $D^{NS}$-functionand to the Bjorken sum rule $C^{Bjp}_{NS}$ of the polarized lepton-nucleon scattering. The terms of the complete $\{β\}$-expansionat the $O(α_s^3)$ level for $D^{NS}$ and $C^{Bjp}_{NS}$ are presented. These perturbative results are expressed in the PMC-type form. The problem of applications of these expressions for phenomenological applications is summarized.

hep-ph↗
arXiv2015

Generalization of the Brodsky-Lepage-Mackenzie optimization within the $\{β\}$-expansion and the Principle of Maximal Conformality

We discuss generalizations of the BLM optimization procedure for renormalization group invariant quantities. In this respect, we discuss in detail the features and construction of the $\{β\}$--expansion representation instead of the standard perturbative series with regards to the Adler $D$-function and Bjorken polarized sum rules obtained in order of ${\cal O}(α_s^4)$. Based on the $\{β\}$--expansion we analyse different schemes of optimization, including the corrected Principle of Maximal Conformality, numerically illustrating their results. We suggest our scheme for the series optimization and apply it to both the above quantities.

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arXiv2014

The NSVZ beta-function in supersymmetric theories with different regularizations and renormalization prescriptions

We briefly review the calculations of quantum corrections related with the exact NSVZ $β$-function in ${\cal N}=1$ supersymmetric theories, paying especial attention to the scheme dependence of the results. It is explained, how the NSVZ relation is obtained for the renormalization group functions defined in terms of the bare coupling constant if a theory is regularized by higher derivatives. Also we describe, how to construct a special renormalization prescription which gives the NSVZ relation for the renormalization group functions defined in terms of the renormalized coupling constant exactly in all orders for Abelian supersymmetric theories, regularized by higher derivatives. The scheme dependence of the NSVZ $β$-function (for the renormalization group functions defined in terms of the renormalized coupling constant) is discussed in the non-Abelian case. It is shown that in this case the NSVZ $β$-function leads to a certain scheme-independent equality.

hep-th↗
arXiv2014

Conformal symmetry limit of QED and QCD and identities between perturbative contributions to deep-inelastic scattering sum rules

Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken, the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axial-vector and two non-singlet vector currents and the non-singlet axial-vector, vector and singlet vector currents in the limit, when the conformal symmetry of the gauge models with fermions is considered unbroken. We specify the perturbative conditions for this symmetry to be valid in the case of the $U(1)$ and $SU(N_c)$ models. The all-order perturbative identity following from the conformal invariant limit between the concrete contributions to the Bjorken, the Ellis-Jaffe and the Gross-Llewellyn Smith sum rules is proved. The analytical and numerical $O(α^4)$ and $O(α_s^2)$ conformal symmetry based approximations for these sum rules and for the Adler function of the non-singlet vector currents are summarized. Possible theoretical applications of the results presented are discussed.

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