Riemann $ζ(4)$ function contributions to $O\left({α_s}^5\right)$ terms of Adler D-function and Bjorken polarized sum rule in $SU\left(N_c\right)$ QCD: results and consequences
Two renormalization group invariant quantities in quantum chromodinamics (QCD), defined in Euclidean space,namely, Adler D-function of electron-positron annihilation to hadrons and Bjorken polarized deep-inelastic scattering sum rule, are considered. It is shown, that the 5-th order corrections to them in $\overline{MS}$-like renormalization prescriptions, proportional to Riemann $ζ$-function $ζ\left(4\right)$, can be restored by the transition to the C-scheme, with the $β$-function, analogous to Novikov, Shifman, Vainshtein and Zakharov exact $β$-function in $\mathcal{N}=1$ supersymmetric gauge theories. The general analytical expression for these corrections in $SU\left(N_c\right)$ QCD is deduced and their scale invariance is shown. The $β$-expansion procedure for these contributions is performed and mutual cancellation of them in the 5-th order of the generalized Crewther identity are discussed.
We compute the nonsinglet Adler $D$-function and the coefficient function for Bjorken polarized sum rules $S^{Bjp}$ at order $O(α_s^4)$ in an extended QCD model with arbitrary number of fermion representations. The Crewther-Broadhurst-Kataev (CBK) relation in this order is confirmed.
We derive explicit expressions for the elements of the $\{ β\}$-expansion for the nonsinglet Adler $D_A$-function and Bjorken polarized sum rules $S^{Bjp}$ in the N$^4$LO using recent results by Chetyrkin for these quantities computed within extended QCD including any number of fermion representations. We discuss the properties of the $\{ β\}$-expansion for $D_A$ and $S^{Bjp}$ at higher orders which follow from the Crewther [1] and the Broadhurst-Kataev [2] relation.
We compute, for the first time, the order alpha_s^4 contributions to the Bjorken sum rule for polarized electron-nucleon scattering and to the (non-singlet) Adler function for the case of a generic colour gauge group. We confirm at the same order a (generalized) Crewther relation which provides a strong test of the correctness of our previously obtained results: the QCD Adler function and the five-loop beta-function in quenched QED. In particular, the appearance of an irrational contribution proportional to zeta_3 in the latter quantity is confirmed. We obtain the commensurate scale equation r…
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 …
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 consi…
Considering the results of recent distinguished analytical calculations of the 5-loop single-fermion loop corrections to the QED $β$-function we emphasize that to our point of view it is important to perform their independent cross-checks. We propose one of the ways of these cross-check. It is based on the application of the original Crewther relation. We derive the new analytical expressions for the $C_F^4α_s^4$-contributions to the Bjorken polarized sum rule. If results of possible direct calculations will agree with the presented expression, then the appearance of $ζ_3$-term in the 5-loop c…
Considering massless axial-vector-vector triangle diagram in the conformal invariant limit and the the results of recent distinguished analytical calculations of the 5-loop single-fermion loop corrections to the QED $β$-function, we derive the analytical expressions for the $C_F^4α_s^4$-contributions to the Bjorken polarized and Gross-Llewellyn Smith sum rule. This visible in future evaluation can shed extra light on the reliability of the appearance of $ζ_3$-term in the explicitly known part of 5-loop corrections to the QED $β$-function which are proportional to the conformal invariant set of…
The current status of the Adler function and two closely related Deep Inelastic Scattering (DIS) sum rules, namely, the Bjorken sum rule for polarized DIS and the Gross-Llewellyn Smith sum rule are briefly reviewed. A new result is presented: an analytical calculation of the coefficient function of the latter sum rule in a generic gauge theory in order O(alpha_s^4). It is demonstrated that the corresponding Crewther relation allows to fix two of three colour structures in the O(alpha_s^4) contribution to the singlet part of the Adler function.
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 …
The Crewther-Broadhurst-Kataev (CBK) relation connects the Bjorken function for deep-inelastic sum rules (or the Gross - Llewellyn Smith function) with the Adler function for electron-positron annihilation in QCD; it has been checked to hold up to four loops in perturbation theory. Here we study non-perturbative terms in the CBK relation using a holographic dual theory that is believed to capture properties of QCD. We show that for the large invariant momenta the perturbative CBK relation is exactly satisfied. For the small momenta non-perturbative corrections enter the relation and we calcula…
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-vect…
We first consider the idea of renormalization group-induced estimates, in the context of optimization procedures, for the Brodsky-Lepage-Mackenzie approach to generate higher-order contributions to QCD perturbative series. Secondly, we develop the deviation pattern approach (DPA) in which through a series of comparisons between lower-order RG-induced estimates and the corresponding analytical calculations, one could modify higher-order RG-induced estimates. Finally, using the normal estimation procedure and DPA, we get estimates of $α_s^4$ corrections for the Bjorken sum rule of polarized deep…
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 decompos…
A systematic method is proposed for analyzing the renormalization scheme uncertainties in the next-next-to-leading order QCD predicitions, based on a condition which eliminates schemes that give rise to large cancellations in the expression for the characteristic scheme invariant combination of the expansion coefficients. Using this method it is shown that the QCD corrections to the tau lepton decay are rather stable with respect to change of the scheme, provided that an improved formula is used, which involves numerical evaluation of the contour integral in the complex energy plane with the A…
We consider a phenomenologycal parametrization of the QCD running coupling which arises from the dispersion relation respecting the holomorphic properties of the physical QCD observables in the complex momentum plane. The parameters are fixed by the following requirements: 1) at enough high energies, it reproduces the underlying perturbative coupling, 2) at intermediate energy momenta, it reproduces the experimental semihadronic tau decay ratio, and 3) in the deep IR regime, it satisfies the qualitative properties coming from recent lattice results. Finally, we apply this new coupling to low-e…
We apply the 't Hooft $\bar{MS}$-based scheme to study the scheme-dependence of the QCD generalization of Crewther relation for the product of the normalised non-singlet perturbative contributions to the $e^+e^-$-annihilation Adler function and to the Bjorken sum rule of the polarized lepton-nucleon deep-inelastic scattering process. We prove that after the transformations from the pure $\bar{MS}$-scheme to the 't Hooft scheme the characteristic $\bar{MS}$-scheme theoretical property of this relation, namely the factorization of the $β$-function in its conformal symmetry breaking part, disappe…
We calculate the massive Wilson coefficients for the heavy flavor contributions to the non-singlet charged current deep-inelastic scattering structure functions $F_L^{W^+}(x,Q^2)-F_L^{W^-}(x,Q^2)$ and $F_2^{W^+}(x,Q^2)-F_2^{W^-}(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in Quantum Chromodynamics (QCD) at general values of the Mellin variable $N$ and the momentum fraction $x$. Besides the heavy quark pair production, also the single heavy flavor excitation $s \rightarrow c$ contributes. Numerical results are presented for the charm quark contributions and consequences on th…
We have discovered 7 intimate connections between the published results for the radiative corrections, $\Ck$, to the Gross--Llewellyn Smith (GLS) sum rule, in deep-inelastic lepton scattering, and the radiative corrections, $\Cr$, to the Adler function of the flavour-singlet vector current, in $\ee$ annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED $β$\/-function and the zero-fermion-loop abelian terms in the 3-loop GLS sum rule. The combined effect of all 7 relations is to give the factorization of the 2-loop …
We propose a hypothesis on the detailed structure for the representation of the conformal symmetry breaking term in the basic Crewther relation generalized in the perturbation theory framework in QCD renormalized in the ${\rm \bar{MS}}$ scheme. We establish the validity of this representation in the $O(α_s^4)$ approximation. Using the variant of the generalized Crewther relation formulated here allows finding relations between specific contributions to the QCD perturbation series coefficients for the flavor nonsinglet part of the Adler function $D^{ns}_A$ for the electron-positron annihilation…
As was discovered recently, 5-loop perturbative quenched QED approximation to the QED $β$-function consist from the rational term and the term proportional to $ζ(3)$-function. It is stressed, that this feature is also manifesting itself in the conformal invariant pqQED series for the 4-loop approximation to the anomalous mass dimension. The 4-loop pqQED expression for the singlet contribution into the Ellis-Jaffe polarized sum rule is obtained. It coincides with the similar approximation for the non-singlet coefficient function of Ellis-Jaffe sum rule and of Bjorken polarized sum rule. It is s…
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 …
We have measured the spin-dependent structure function $g_1^p$ in inclusive deep-inelastic scattering of polarized muons off polarized protons, in the kinematic range $0.003 < x < 0.7$ and $1 GeV^2 < Q^2 < 60 GeV^2$. A next-to-leading order QCD analysis is used to evolve the measured $g_1^p(x,Q^2)$ to a fixed $Q^2_0$. The first moment of $g_1^p$ at $Q^2_0 = 10 GeV^2$ is $Γ^p = 0.136\pm 0.013(stat.) \pm 0.009(syst.)\pm 0.005(evol.)$. This result is below the prediction of the Ellis-Jaffe sum rule by more than two standard deviations. The singlet axial charge $a_0$ is found to be $0.28 \pm 0.16$…
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 …
We calculate analytically the flavor non-singlet $O(α_s^2)$ massive Wilson coefficients for the inclusive neutral current non-singlet structure functions $F_{1,2,L}^{ep}(x,Q^2)$ and $g_{1,2}^{ep}(x,Q^2)$ and charged current non-singlet structure functions $F_{1,2,3}^{ν(\barν)p}(x,Q^2)$, at general virtualities $Q^2$ in the deep-inelastic region. Numerical results are presented. We illustrate the transition from low to large virtualities for these observables, which may be contrasted to basic assumptions made in the so-called variable flavor number scheme. We also derive the corresponding resul…
The Principle of Maximal Conformality (PMC) provides a systematic way to set the renormalization scales order-by-order for any perturbative QCD process. The resulting predictions are independent of the choice of renormalization scheme, a requirement of renormalization group invariance. The Crewther relation, which was originally derived for conformal theory, provides a remarkable connection between two observables when the $β$ function vanishes. The "Generalized Crewther Relation" relates these two observables for physical QCD with nonzero $β$ function; specifically, it connects the non-single…
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 …
