Polarized Sum Rule Research Articles

Adler function, Bjorken polarized sum rule: confirmation of elements of the {β}-expansion and the diagrams

Different ways exist to obtain the elements of the {β}-expansion for renormgroup invariant quantities. Here we consider independent confirmation within the standard QCD of a number of our results [1] for the values of elements of this expansion for the nonsinglet Adler DA-function, Bjorken polarized sum rules SBjp up to the order N4LO. We suggest an approach to estimate the results of high order QCD calculations using a smaller number of diagrams of the specific type. This type is based on a proposed generalization of Naive NonAbelianization.

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 {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} in the overline {{text{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 CBK 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 overline {{text{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 {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} for all mentioned renorm-invariant perturbative quantities are discussed.

Adler function, Bjorken Sum Rule and Crewther-Broadhurst-Kataev relation with generic fermion representations at order [formula omitted

We compute the nonsinglet Adler D-function and the coefficient function for Bjorken polarized sum rules SBjp at order O(αs4) in an extended QCD model with arbitrary number of fermion representations. The Crewther-Broadhurst-Kataev (CBK) relation in this order is confirmed.

The {β}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O( {alpha}_s^4 )

We derive explicit expressions for the elements of the {β}-expansion for the nonsinglet Adler DA-function and Bjorken polarized sum rules SBjp in the N4LO 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 DA and SBjp at higher orders which follow from the Crewther [1] and the Broadhurst-Kataev [2] relation.

The overline{mathrm{MS}} -scheme {alpha}_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 αs for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the overline{mathrm{MS}} -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} ( {alpha}_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 ζ4-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.

Bjorken polarized sum rule and infrared-safe QCD couplings

Experimental data obtained for the polarized Bjorken sum rule (BSR) Gamma _1^{p-n}(Q^2) are fitted by using predictions derived within a truncated operator product expansion (OPE) approach to QCD. Four QCD versions are considered: perturbative QCD (pQCD) in the {overline{mathrm{MS}}} scheme, Analytic Perturbation Theory (APT), and 2delta and 3delta analytic QCD versions. In contrast to pQCD, these QCD variants do not have Landau singularities at low positive Q^2, which facilitates the fitting procedure significantly. The fitting procedure is applied first to the experimental data of the inelastic part of BSR, and the known elastic contributions are added after the fitting. In general, when 2delta and 3delta QCD coupling is used the fitted curves give the best results, within the Q^2-range of the fit as well as in extended Q^2-intervals. When the fitting procedure is applied to the total BSR, i.e., to the sum of the experimental data and the elastic contribution, the quality of the results deteriorates significantly.

Computing the dipole polarizability of Ca48 with increased precision

We compute the electric dipole polarizability of 48Ca with an increased precision by including more correlations than in previous studies. Employing the coupled-cluster method we go beyond singles and doubles excitations and include leading-order three-particle-three-hole (3p-3h) excitations for the ground state, excited states, and the similarity transformed operator. We study electromagnetic sum rules, such as the bremsstrahlung sum rule m_0 and the polarizability sum rule alpha_D using interactions from chiral effective field theory. To gauge the quality of our coupled-cluster approximations we perform several benchmarks with the effective interaction hyperspherical harmonics approach in 4He and with self consistent Green's function in 16O. We compute the dipole polarizability of 48Ca employing the chiral interaction N2LOsat [Ekstroem et al., Phys. Rev. C 91, 051301 (2015)] and the 1.8/2.0 (EM) [Hebeler et al., Phys. Rev. C 83, 031301 (2011)]. We find that the effect of 3p-3h excitations in the ground state is small for 1.8/2.0 (EM) but non-negligible for N2LOsat. The addition of these new correlations allows us to improve the precision of our 48Ca calculations and reconcile the recently reported discrepancy between coupled-cluster results based on these interactions and the experimentally determined alpha_D from proton inelastic scattering in 48Ca [Birkhan et al., Phys. Rev. Lett. 118, 252501 (2017)]. For the computation of electromagnetic and polarizability sum rules, the inclusion of leading-order 3p-3h excitations in the ground state is important, while less so for the excited states and the similarity-transformed dipole operator.

Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the \u03b2-factorization property?

The problem of scheme and gauge dependence of the factorization property of the renormalization group β-function in the SU(Nc) QCD generalized Crewther relation (GCR), which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions, is investigated at the mathcal{O}left({a}_s^4right) level of perturbation theory. It is known that in the gauge-invariant renormalization overline{mathrm{MS}} -scheme this property holds in the QCD GCR at least at this order. To study whether this factorization property is true in all gauge-invariant schemes, we consider the MS-like schemes in QCD and the QED-limit of the GCR in the overline{mathrm{MS}} -scheme and in two other gauge-independent subtraction schemes, namely in the momentum MOM and the on-shell 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. To investigate this problem we consider the gauge non-invariant mMOM and MOMgggg-schemes. We demonstrate that in the mMOM scheme at the mathcal{O}left({a}_s^3right) 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}left({a}_s^4right) order of PT it remains valid only for case of the Landau gauge ξ = 0. The consideration of these two gauge-dependent schemes for the QCD GCR allows us to conclude that the factorization of RG β-function will always be implemented in any MOM-like renormalization schemes with linear covariant gauge at ξ = 0 and ξ = −3 at the mathcal{O}left({a}_s^3right) approximation. It is demonstrated that if factorization property for the MS-like schemes is true in all orders of PT, as theoretically indicated in the several works on the subject, then the factorization will also occur in the arbitrary MOM-like scheme in the Landau gauge in all orders of perturbation theory as well.

Bjorken sum rule in QCD with analytic coupling

We present details of study of the Bjorken polarized sum rule carried out recently in [1] within the range of energies where the data were collected by JLAB collaboration, 0.05 GeV2 < Q2 < 3 GeV2. Three approaches to QCD with analytic (holomorphic) coupling are considered: Analytic Perturbation Theory (APT), Two-delta analytic QCD (2δanQCD), and Three-delta lattice-motivated analytic QCD in the three-loop and four-loop MiniMOM schemes (3l3δanQCD, 4l3δanQCD). The new frameworks (2δ and 3δ) with respective couplings give results which agree well with the experimental data for 0.5 GeV2 < Q2 < 3 GeV2 already when only one higher-twist term is taken into account.

The analytical expression for the polarized Bjorken sum rule in the miniMOM scheme and the consequences for the generalized Crewther relation

The analytical perturbative QCD expression for the flavour non-singlet contribution to the Bjorken polarized sum rule in the rather applicable at present gauge–dependent 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 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 scheme up to -level at least, is unexpectedly valid at the same level in the miniMOM-scheme for ξ = 0, and for ξ = −1 and ξ = −3 in part.

On a realization of {\u03b2}-expansion in QCD

We suggest a simple algebraic approach to fix the elements of the {β}-expansion for renormalization group invariant quantities, which uses additional degrees of freedom. The approach is discussed in detail for N2LO calculations in QCD with the MSSM gluino — an additional degree of freedom. We derive the formulae of the {β}-expansion for the nonsinglet Adler D-function and Bjorken polarized sum rules in the actual N3LO within this quantum field theory scheme with the MSSM gluino and the scheme with the second additional degree of freedom. We discuss the properties of the {β}-expansion for higher orders considering the N4LO as an example.

Effect of a cosmological constant on propagation of vacuum polarized photons in stationary spacetimes

Consideration of vacuum polarization in quantum electrodynamics may affect the momentum dispersion relation for photons for a non-trivial background, due to the appearance of curvature dependent terms in the effective action. We investigate the effect of a positive cosmological constant $$\Lambda $$ on this at one-loop order for stationary $$\Lambda $$ -vacuum spacetimes. To the best of our knowledge, so far it only has been shown that $$\Lambda $$ affects the propagation in a time dependent black hole spacetime. Here we consider the static de Sitter cosmic string and the Kerr–de Sitter spacetime to show that there might occur a non-vanishing effect due to $$\Lambda $$ for physical polarizations. The consistency of these results with the polarization sum rule is discussed.

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 $\{\beta\}$--expansion representation instead of the standard perturbative series with regards to the Adler $D$-function and Bjorken polarized sum rules obtained in order of ${\cal O}(\alpha_s^4)$. Based on the $\{\beta\}$--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.

The singlet contribution to the Bjorken sum rule for polarized deep inelastic scattering

It is shown that the existing four-loop result for the Bjorken polarized sum rule for deep inelastic electron–nucleon scattering obtained within perturbative Quantum Chromodynamics should be supplemented by the calculation of the diagrams of the so-called singlet type. We also give an explanation of the interesting coincidence of two different classes of diagrams, one of the non-singlet and one of the singlet type, contributing the αs4-approximation to the total cross-section of electron–positron annihilation into hadrons.

Proton Spin Structure from Measurable Parton Distributions

We present a systematic study of the proton spin structure in terms of measurable parton distributions. For a transversely polarized proton, we derive a polarization sum rule from the leading generalized parton distributions appearing in hard exclusive processes. For a longitudinally polarized proton, we obtain a helicity decomposition from well-known quark and gluon helicity distributions and orbital angular-momentum contributions. The latter are shown to be related to measurable subleading generalized parton distributions and quantum-phase space Wigner distributions.

Erratum: Various versions of analytic QCD and skeleton-motivated evaluation of observables [Phys. Rev. D74, 114030 (2006)

We present skeleton-motivated evaluation of QCD observables. The approach can be applied in analytic versions of QCD in certain classes of renormalization schemes. We present two versions of analytic QCD which can be regarded as low-energy modifications of the ``minimal'' analytic QCD and which reproduce the measured value of the semihadronic tau decay ratio r{tau}. Further, we describe an approach of calculating the higher order analytic couplings Ak (k=2,3,...) on the basis of logarithmic derivatives of the analytic coupling A1(Q^2). This approach can be easily applied in any version of analytic QCD. We adjust the free parameters of the afore-mentioned two analytic models in such a way that the skeleton-motivated evaluation reproduces the correct known values of r{tau} and of the Bjorken polarized sum rule (BjPSR) db(Q^2) at a given point (e.g., at Q^2=2 GeV^2). We then evaluate the low-energy behavior of the Adler function dv(Q^2) and the BjPSR db(Q^2) in the afore-mentioned evaluation approach, in the three analytic versions of QCD. We compare with the results obtained in the ``minimal'' analytic QCD and with the evaluation approach of Milton et al. and Shirkov.

Riemann [formula omitted]-terms in perturbative QED series, conformal symmetry and the analogies with structures of multiloop effects in [formula omitted] supersymmetric Yang–Mills theory

As was discovered recently, the 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 functions of the Ellis–Jaffe sum rule and of the Bjorken polarized sum rule. It is stressed that this property is valid in all orders of perturbation theory thanks to the conformal symmetry of pqQED series and to the Crewther relation, which relates non-singlet and singlet coefficient functions of the Ellis–Jaffe sum rule with the coefficient functions of the non-singlet and singlet Adler D-functions. The basic steps of derivation of the Crewther relation in the singlet channel from the AVV triangle diagram are outlined. The similarities between analytical structures of asymptotic series for the coefficient functions in pqQED and for the anomalous dimensions in N=4 conformal invariant supersymmertic Yang–Mills theory are observed. The guess is proposed that the appearance of ζ(3)-terms in the pqQED expressions and the absence of ζ(5)-terms at the same level is the indication of absence of “wrapping” interactions in pqQED.

Is it possible to check urgently the 5-loop analytical results for the [formula omitted]-annihilation Adler function?

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 CF4αs4-contributions to the Bjorken polarized sum rule. If results of possible direct calculations will agree with the presented expression, then the appearance of ζ3-term in the 5-loop correction to the QED β-function and in the CF4αs4 contribution into the e+e− annihilation Adler function will get independent support and may be analysed within the framework of the recently introduced concept of “maximal transcendentality”.

Various versions of analytic QCD and skeleton-motivated evaluation of observables

We present skeleton-motivated evaluation of QCD observables. The approach can be applied in analytic versions of QCD in certain classes of renormalization schemes. We present two versions of analytic QCD which can be regarded as low-energy modifications of the ``minimal'' analytic QCD and which reproduce the measured value of the semihadronic $\ensuremath{\tau}$ decay ratio ${r}_{\ensuremath{\tau}}$. Further, we describe an approach of calculating the higher-order analytic couplings ${\mathcal{A}}_{k}$ $(k=2,3,\dots{})$ on the basis of logarithmic derivatives of the analytic coupling ${\mathcal{A}}_{1}({Q}^{2})$. This approach can be applied in any version of analytic QCD. We adjust the free parameters of the aforementioned two analytic models in such a way that the skeleton-motivated evaluation reproduces the correct known values of ${r}_{\ensuremath{\tau}}$ and of the Bjorken polarized sum rule (BjPSR) ${d}_{b}({Q}^{2})$ at a given point (e.g., at ${Q}^{2}=2\text{ }\text{ }{\mathrm{GeV}}^{2}$). We then evaluate the low-energy behavior of the Adler function ${d}_{v}({Q}^{2})$ and the BjPSR ${d}_{b}({Q}^{2})$ in the aforementioned evaluation approach, in the three analytic versions of QCD. We compare with the results obtained in the minimal analytic QCD and with the evaluation approach of Milton et al. and Shirkov.

Photonuclear sum rules and the tetrahedral configuration ofHe4

Three well-known photonuclear sum rules (SR), i.e., the Thomas-Reiche-Kuhn, the bremsstrahlungs and the polarizability SR are calculated for $^{4}\mathrm{He}$ with the realistic nucleon-nucleon potential Argonne V18 and the three-nucleon force Urbana IX. The relation between these sum rules and the corresponding energy weighted integrals of the cross section is discussed. Two additional equivalences for the bremsstrahlungs SR are given, which connect it to the proton-neutron and neutron-neutron distances. Using them, together with our result for the bremsstrahlungs SR, we find a deviation from the tetrahedral symmetry of the spatial configuration of $^{4}\mathrm{He}$. The possibility to access this deviation experimentally is discussed.