Deep Inelastic Scattering Sum Rules Research Articles

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 βi of the QCD β-function, which are evaluated in the MS-like schemes. As a next step all βi-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 CBjpNS (as) the βi-dependent structure of the NNLO approximation for CBjpNS (as) is predicted in QCD with ngl-multiplet of gluino degrees of freedom, which appear in SUSY extension of QCD. The importance of performing the analytical calculation of the N3LO additional contributions of ngl gluino multiplet to CBjpNS (as) 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(a4s)-level is emphasised.

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\left( {\alpha_s^2} \right) $ 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.

Adler Function, DIS sum rules and Crewther Relations

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.

Gravitational coupling to two-particle bound states and momentum conservation in deep inelastic scattering

The momentum conservation sum rule for deep inelastic scattering (DIS) from composite particles is investigated using the general theory of relativity. For two $(1+1)\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$ examples, it is shown that covariant theories automatically satisy the DIS momentum conservation sum rule provided the bound state is covariantly normalized. Therefore, in these cases the two DIS sum rules for baryon conservation and momentum conservation are equivalent.

Heavy flavour contributions to the deep inelastic scattering sum rules

We have calculated the first and second order corrections to several deep inelastic sum rules which are due to heavy flavour contributions. A comparison is made with the existing perturbation series which has been computed up to third order for massless quarks only. In general it turns out that the effects of heavy quarks are very small except when Q∼ m or Q≫ m. Here Q and m denote the virtual mass of the vector boson and the mass of the heavy quark, respectively. For Q≫ m the radiative corrections reveal large logarithms of the type ln Q 2/ m 2 which have to be absorbed in the running coupling constant so that the number of light flavours n f is enhanced by one unit. However this has to happen at much larger values of Q i.e. Q∼6.5 m than one usually assumes for the flavour thresholds which appear in the running coupling constant. An alternative description for the heavy flavour dependence of the running coupling constant in the context of the MOM-scheme is discussed.

RS-invariant all-orders renormalon resummations for some QCD observables

We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS)-invariant effective charge beta-function coefficients by the portion containing the highest power of b= 1 6 (11N−2N f) for SU( N) QCD with N f quark flavours. This can be accomplished using exact large- N f all-orders results. The resulting resummation is RS invariant and the exact next-to-leading order (NLO) and next-to-NLO (NNLO) coefficients in any RS are included. This improves on a previously employed naive resummation of the leading- b piece of the perturbative coefficients which is RS dependent, making its comparison with fixed-order perturbative results ambiguous. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation theory for the e + e − R-ratio, the analogous τ-lepton decay ratio R τ and deep inelastic scattering (DIS) sum rules, by comparing it with the exact NNLO results in the effective charge RS. For the R-ratio and R,, where large-order perturbative behaviour is dominated by a leading ultra-violet renormalon singularity, the comparison indicates fixed-order perturbation theory to be very reliable. For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. In this way we estimate that at LEP/ SLD energies ideal data on the R-ratio could determine α s ( M z ) to three-significant figures, and for the R τ we estimate a theoretical uncertainty δα s ( m τ ) ≅ 0.008 corresponding to δα s ( M z ) ≅ 0.001 This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.

All-orders renormalon resummations for some QCD observablest

Exact large- N f results for the QCD Adler D-function and Deep Inelastic Scattering sum rules are used to resum to all orders the portion of QCD perturbative coefficients containing the highest power of b = 1 6 (11 N-2N f) , for SU ( N) QCD with N f quark flavours. These terms correspond to renormalon singularities in the Borel plane and are expected asymptotically to dominate the coefficients to all orders in the 1/ N f expansion. Remarkably, we note that this is already apparent in comparisons with the exact next-to-leading order (NLO) and next-to-NLO (NNLO) perturbative coefficients. The ultra-violet (UV) and infra-red (IR) renormalon singularities in the Borel transform are isolated and the Borel sum (principal value regulated for IR) performed. Resummed results are also obtained for the Minkowski quantities related to the D-function, the e + e − R-ratio and the analogous τ-lepton decay ratio, R τ . The renormalization scheme dependence of these partial resummations is discussed and they are compared with the results from other groups and with exact fixed-order perturbation theory at NNLO. Prospects for improving the resummation by including more exact details of the Borel transform are considered.

Relativistic corrections to the electromagnetic and axial moments of nuclei and other composite systems

We calculate the electromagnetic and axial nuclear moments of the deuteron and triton as a function of their radius using a relativistic two-nucleon and three-nucleon model formulated on the light-cone. The results also provide an estimate of the nuclear binding corrections to helicity-dependent deep inelastic scattering sum rules. At large nucleon radius, the moments are given by the usual nonrelativistic formulae modified by finite binding effects. At small radius, the moments take the canonical values given by the generalization of the Drell-Hearn-Gerasimov sum rule. In addition, as R → 0, the constituent helicities become completely disoriented, and the Gamow-Teller matrix element vanishes. Thus, in the pointlike limit MR → 0, the moments of spin-one bound states coincide with the canonical couplings of elementary spin-one bosons of the Standard Model, μ = e M , Q = −e M 2 , and g a = 0. Similarly, in the case of spin-half systems, the magnetic moment approaches the Dirac value μ = e 2M , and the axial coupling g a vanishes.

Estimates of the higher-order QCD corrections: theory and applications

We consider the further development of the formalism of the estimates of higher-order perturbative corrections in the Euclidean region, which is based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We present the estimates of the order $O(\alpha^{4}_{s})$ QCD corrections to the Euclidean quantities: the $e^+e^-$-annihilation $D$-function and the deep inelastic scattering sum rules, namely the non-polarized and polarized Bjorken sum rules and to the Gross--Llewellyn Smith sum rule. The results for the $D$-function are further applied to estimate the $O(\alpha_s^4)$ QCD corrections to the Minkowskian quantities $R(s) = \sigma_{tot} (e^{+}e^{-} \to {\rm hadrons}) / \sigma (e^{+}e^{-} \to \mu^{+} \mu^{-})$ and $R_{\tau} = \Gamma (\tau \to \nu_{\tau} + {\rm hadrons}) / \Gamma (\tau \to \nu_{\tau} \overline{\nu}_{e} e)$. The problem of the fixation of the uncertainties due to the $O(\alpha_s^5)$ corrections to the considered quantities is also discussed.

ESTIMATES OF THE HIGHER ORDER QCD CORRECTIONS TO R(s), Rτ AND DEEP INELASTIC SCATTERING SUM RULES

We present the attempt to study the problem of the estimates of higher order perturbative corrections to physical quantities in the Euclidean region. Our considerations are based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We emphasize that in order to obtain the concrete results for the physical quantities in the Minkowskian region the results of application of this formalism should be supplemented by the explicit calculations of the effects of the analytical continuation. We present the estimates of the order [Formula: see text] QCD corrections to the Euclidean quantities: the e+e−-annihilation D-function and the deep inelastic scattering sum rules, namely the nonpolarized and polarized Bjorken sum rules and to the Gross–Llewellyn Smith sum rule. The results for the D-function are further applied to estimate the [Formula: see text] QCD corrections to the Minkowskian quantities R(s) = σ tot (e+e− → hadrons )/σ(e+e− → µ+µ−) and [Formula: see text]. The problem of the fixation of the uncertainties due to the [Formula: see text] corrections to the considered quantities is also discussed.