arXiv subjects

V. S. Molokoedov

Publications and source records attributed to V. S. Molokoedov.

15 recordsLinked to original sources

arXiv2023

The decomposed photon anomalous dimension in QCD and the $\{β\}$-expanded representations for the Adler function

This work is devoted to the study of the $\{β\}$-expansion of the perturbative expressions for the $e^+e^-$ annihilation Adler function $D(Q^2)$ and for the related renormalization group functions, namely for the photon vacuum polarization function and its anomalous dimension $γ(α_s)$ in QCD at the $\mathcal{O}(α^4_s)$ order. We emphasize that $γ(α_s)$ is not a conformal-invariant contribution to $D(Q^2)$ and, therefore, for a consistent analysis it is necessary to decompose its higher-order PT coefficients in powers of the $β$-function coefficients in the same way as for the Adler function. The arguments in favor of this statement are given. The comparison of the $\overline{MS}$ and PMC/BLM approximants are demonstrated.Theoretical and phenomenologically related consequences of this comparison are briefly commented.

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arXiv2023

The generalized Crewther relation and V-scheme: analytic $O(α^4_s)$ results in QCD and QED

Using the analytical $\rm{\overline{MS}}$-scheme three-loop contribution to the perturbative Coulomb-like part of the static color potential of heavy quark-antiquark system, we obtain the analytical expression for the fourth-order $β$-function in the gauge-invariant effective V-scheme in the case of the generic simple gauge group. Also we present the Adler function of electron-positron annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule in the V-scheme up to $α^4_s$ terms. We demonstrate that at this level of PT in this effective scheme the $β$-function is factorized in the conformal symmetry breaking term of the generalized Crewther relation, which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions. We prove why this relation will be true in other gauge-invariant renormalization schemes as well. The obtained results enable to reveal the difference between the V-scheme $β$-function in QED and the Gell-Man--Low $Ψ$-function. This distinction arises due to the presence of the light-by-light type scattering corrections first appearing in the static potential at the three-loop level.

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arXiv2022

Representation of the RG-invariant quantities in perturbative QCD through powers of the conformal anomaly

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly $β(α_s)/α_s$ in the ${\rm{\overline{MS}}}$-scheme. We remind that such expansion is possible for the Adler function of the process of $e^+e^-$ annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the Crewther-Broadhurst-Kataev relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and ${\rm{\overline{MS}}}$-scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of $β(α_s)/α_s$ for all mentioned perturbative quantities are discussed.

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arXiv2022

Notes on interplay of the QCD and EW perturbative corrections to the pole-running top-quark mass ratio

A specific representation of the known one-loop EW correction to the relation between the pole and running $\msbar$-scheme masses of the top-quark through particle masses of the Standard Model is given within the Fleischer-Jegerlehner tadpole scheme, where the vacuum expectation value of the Higgs field is renormalized. The importance of taking into account both the EW and QCD effects in this relation in the considered case is emphasized. It is noted that the discard of the EW corrections leads to over $10\;{\rm{GeV}}$ shift in the difference between the pole and running $t$-quark masses. This magnitude exceeds essentially the modern uncertainties of the considered relation, following from the treatment of the Tevatron and LHC data where both pole and running $t$-quark masses are defined in the widespread approach when only the QCD corrections are kept in mind between them.

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arXiv2022

The ${\rm{\bar{MS}}}$-scheme $α_s^5$ QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold $\{β\}$-expanded representation

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling $α_s$ for the non-singlet contributions to Adler $D$-function and Bjorken polarized sum rule calculated previously in the $\MSbar$-scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the $\{β\}$-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some $\mathcal{O}(α_s^5)$ corrections to the $D$-function, to the $R$-ratio of $e^+e^-$-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the $SU(3)$ color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the $\{β\}$-expansion procedure the analytical Riemann $ζ_4$-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.

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arXiv2020

Multiloop contributions to the on-shell-$ \overline{\rm{MS}}$ heavy quark mass relation in QCD and the asymptotic structure of the corresponding series: the updated consideration

The asymptotic structure of the QCD perturbative relation between the on-shell and $\overline{\rm{MS}}$ heavy quark masses is studied. We estimate the five and six-loop contributions to this relation by three different techniques. First, the effective charges motivated approach in two variants is used. Second, the results following from the large-$β_0$ approximation are analyzed. Finally, the consequences of applying the asymptotic renormalon-based formula are investigated. We show that all approaches lead to corrections which are qualitatively consistent in order of magnitude. Their sign-alternating character in powers of the number of massless quarks is demonstrated. We emphasize that there is no contradiction in the behavior of the fine structure of the renormalon-based estimates with other approaches if one use the detailed information about the normalization factor included in the renormalon asymptotic formula. The obtained five- and six-loop estimates indicate that in the case of the $b$-quark the asymptotic character of the studied relation manifests itself above the fourth order of PT, whereas for the $t$-quark it starts to reveal itself after the seventh order. This allows to conclude that like the running masses, the pole masses of the $b$ and especially $t$-quark in principle may be used in the phenomenologically-oriented studies.

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arXiv2018

The least squares method: application to analysis of the flavor dependence of the QCD relation between pole and $\rm{\overline{MS}}$-scheme running heavy quark masses

The features of the ordinary least squares method, which gives a possible way to a solution of the overdetermined systems of algebraic equations and allows to estimate the uncertainties of the obtained solutions, are considered. As the important physical example we define four-loop QCD coefficients in the dependence of the relation between pole and running heavy quarks masses on the number of light flavors, using the existing results of numerical supercomputer based calculations of the corresponding four-loop contributions at different fixed numbers of light flavors. Stability of the found solutions to the number of the considered equations and unknowns is demonstrated and supported by the Pearsons's $χ$-squared test.

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arXiv2018

Dependence of five and six-loop estimated QCD corrections to the relation between pole and running masses of heavy quarks on the number of light flavours

In this paper various theoretical approaches are used to define the dependence of the estimated $\mathcal{O}(α^5_s)$ and $\mathcal{O}(α^6_s)$-corrections to the QCD relation between pole and $\rm{\overline{MS}}$ running masses of heavy quarks on the number of light flavours. It is found that recently studied asymptotic formula for the coefficients of this relation, based on the infared-renormalon method, does not reproduce sign-alternating structure in the flavour-dependence of the five and six-loop corrections, which holds in three other used by us approaches.

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arXiv2018

On the relation between pole and running heavy quark masses beyond the four-loop approximation

The effective charges motivated method is applied to the relation between pole and $\rm{\overline{MS}}$-scheme heavy quark masses to study high order perturbative QCD corrections in the observable quantities proportional to the running quark masses. The non-calculated five- and six-loop perturbative QCD coefficients are estimated. This approach predicts for these terms the sign-alternating expansion in powers of number of lighter flavors $n_l$, while the analyzed recently infrared renormalon asymptotic expressions do not reproduce the same behavior. We emphasize that coefficients of the quark mass relation contain proportional to $π^2$ effects, which result from analytical continuation from the Euclidean region, where the scales of the running masses and QCD coupling constant are initially fixed, to the Minkowskian region, where the pole masses and the running QCD parameters are determined. For the $t$-quark the asymptotic nature of the non-resummed PT mass relation does not manifest itself at six-loops, while for the $b$-quark the minimal PT term appears at the probed by direct calculations four-loop level. The recent infrared renormalon based studies support these conclusions.

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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.

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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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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

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.

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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.

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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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