QCD Observables Research Articles

Infrared freezing of Euclidean QCD observables

We consider the leading one-chain term in a skeleton expansion for QCD observables and show that for energies ${Q}^{2}>{\ensuremath{\Lambda}}^{2}$, where ${Q}^{2}={\ensuremath{\Lambda}}^{2}$ is the Landau pole of the coupling, the skeleton expansion result is equivalent to the standard Borel integral representation, with ambiguities related to infrared (IR) renormalons. For ${Q}^{2}<{\ensuremath{\Lambda}}^{2}$ the skeleton expansion result is equivalent to a previously proposed modified Borel representation where the ambiguities are connected with ultraviolet (UV) renormalons. We investigate the ${Q}^{2}$-dependence of the perturbative corrections to the Adler-$D$ function, the GLS sum rule and the polarized and unpolarized Bjorken sum rules. In all these cases the one-chain result changes sign in the vicinity of ${Q}^{2}={\ensuremath{\Lambda}}^{2}$, and then exhibits freezing behavior, vanishing at ${Q}^{2}=0$. Finiteness at ${Q}^{2}={\ensuremath{\Lambda}}^{2}$ implies specific relations between the residues of IR and UV renormalon singularities in the Borel plane. These relations, only one of which has previously been noted (though it remained unexplained), are shown to follow from the continuity of the characteristic function in the skeleton expansion. By considering the compensation of nonperturbative and perturbative ambiguities we are led to a result for the ${Q}^{2}$-dependence of these observables at all ${Q}^{2}$, in which there is a single undetermined nonperturbative parameter, and which involves the skeleton expansion characteristic function. The observables freeze to zero in the infrared. We briefly consider the freezing behavior of the Minkowskian ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio.

Super-leading logarithms in non-global observables in QCD?

We reconsider the calculation of a non-global QCD observable and find the possible breakdown of QCD coherence. This breakdown arises as a result of wide angle soft gluon emission developing a sensitivity to emission at small angles and it leads to the appearance of super-leading logarithms. We use the `gaps between jets' cross-section as a concrete example and illustrate that the new logarithms are intimately connected with the presence of Coulomb gluon contributions. We present some rough estimates of their potential phenomenological significance.

Nonpower expansions for QCD observables at low energies

A comprehensive review is presented of the progress made in further developing the ghost-free Analytic Approach to low-energy QCD since “QCD-97” meeting. It is now formulated as a logically closed “Analytic Perturbation Theory” algorithm. Its most essential feature is nonpower functional expansions for QCD observables. Nonpower expansion functions are oscillating in the low energy domain, where the QCD coupling is not weak. This effect suppresses the influence of higher-loop contributions that, in turn, diminishes the scheme dependence and improves the convergence property of perturbative expansion for observables.

Soft gluons at large angles in hadron collisions

A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons. We treat it as the ``fifth form factor'', accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing $t$- and $u$-channels. As an application we revisit the problem of large angle radiation in $gg\to gg$ and give a relatively simple solution and interpretation of the results. We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation.

All-orders infrared freezing of observables in perturbative QCD

We consider a Borel sum definition of all-orders perturbation theory for Minkowskian QCD observables such as the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, and show that both this perturbative component and the additional nonperturbative all-orders operator product expansion (OPE) component can remain separately well-defined for all values of energy $\sqrt{s},$ with the perturbative component dominating as $\stackrel{\ensuremath{\rightarrow}}{s}\ensuremath{\infty},$ and with both components contributing as $\stackrel{\ensuremath{\rightarrow}}{s}0.$ In the infrared $\stackrel{\ensuremath{\rightarrow}}{s}0$ limit the perturbative correction to the parton model result for ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ has an all-orders perturbation theory component which smoothly freezes to the value $\mathcal{R}(0)=2/b,$ where $b=(33\ensuremath{-}{2N}_{f})/6$ is the first QCD beta-function coefficient, with ${N}_{f}$ flavors of massless quark. For freezing one requires ${N}_{f}<9.$ The freezing behavior is manifested by the ``contour-improved'' or ``analytic perturbation theory'' (APT), in which an infinite subset of analytical continuation terms are resummed to all-orders. We show that for the Euclidean Adler-$D$ function, ${D(Q}^{2}),$ the perturbative component remains defined into the infrared if all the renormalon singularities are taken into account, but no analogue of the APT reorganization of perturbation theory is possible. We perform phenomenological comparisons of suitably smeared low-energy data for the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, with the perturbative freezing predictions, and find good agreement.

On de-globalization in quantum chromodynamics

The recent discovery and resummation of a class of single logarithmic effects (non-global logs), has a significant impact on several QCD observables ranging from the classic Sterman-Weinberg jet definition to currently studied event shapes and rapidity gap observables. The discovery of the above effects overturns, for example, the common wisdom that hadronic energy flow in limited inter-jet regions is dictatedprimarily by the colour flow of the underlying hard partonic subprocess. We discuss some features of non-global logs and the rapid progress being made in estimating and controlling such corrections.

Pion mass dependence of nucleon magnetic moments

The relevance of the pion mass, provenient from a term which explicitely breaks chiral symmetry in the Lagrangian, for nucleon magnetic moment in the frame work of the Skyrme model in two different versions: the usual one model and a modified one which includes a coupling to a light scalar meson field, the sigma s(<img id="_x0000_i1026" src="../../img/revistas/bjp/v34n1a/a08img11.gif" align=absmiddle> 500 - 600 MeV). Results are compared to other calculations. Our main motivation comes from usual extrapolations for values of low energy QCD observables obtained in lattices with large values of pion/quark masses toward realistic value of m p which do not allow it. We do a comparison with results from the Cloudy Bag Model and a chiral hadronic model from chiral perturbation theory. There are several resulting extrapolations from the region of large pion mass to the realistic value depending on the considered model for low energy QCD.

A transverse lattice QCD model for mesons

QCD is analysed with two light-front continuum dimensions and two transverse lattice dimensions. In the limit of large number of colours and strong transverse gauge coupling, the contributions of light-front and transverse directions factorise in the dynamics, and the theory can be analytically solved in a closed form. An integral equation is obtained, describing the properties of mesons, which generalises the 't Hooft equation by including spin degrees of freedom. The meson spectrum, light-front wavefunctions and form factors can be obtained by solving this equation numerically. These results would be a good starting point to model QCD observables which only weakly depend on transverse directions, e.g. deep inelastic scattering structure functions.

Renormalization group summation, spectrality constraints, and coupling constant analyticity for phenomenological applications of two-point correlators in QCD

Analytic structure in the strong coupling constant that emerges for some observables in QCD after duality averaging of renormalization group improved amplitudes is discussed. It is shown that perturbation theory calculations are justified for the proper observables related to the two-point correlators of hadronic currents the analytic properties of which are well-established. A particular case of gluonic current correlators is discussed in detail.

Initial-state interactions and single-spin asymmetries in Drell–Yan processes

We show that the initial-state interactions from gluon exchange between the incoming quark and the target spectator system lead to leading-twist single-spin asymmetries in the Drell–Yan process H 1 H 2 ↑→ℓ +ℓ − X. The QCD initial-state interactions produce a T-odd spin-correlation S → H 2 · P → H 1 × Q → between the target spin and the virtual photon production plane which is not power-law suppressed in the Drell–Yan scaling limit at large photon virtuality Q 2 at fixed x F . The single-spin asymmetry which arises from the initial-state interactions is not related to the target or projectile transversity distribution δq H ( x, Q). The origin of the single-spin asymmetry in πp ↑→ℓ +ℓ − X is a phase difference between two amplitudes coupling the proton target with J z p=± 1 2 to the same final-state, the same amplitudes which are necessary to produce a nonzero proton anomalous magnetic moment. The calculation requires the overlap of target light-front wavefunctions with different orbital angular momentum: ΔL z =1; thus the SSA in the Drell–Yan reaction provides a direct measure of orbital angular momentum in the QCD bound state. The single-spin asymmetry predicted for the Drell–Yan process πp ↑→ℓ +ℓ − X is similar to the single-spin asymmetries in deep inelastic semi-inclusive leptoproduction ℓ p ↑→ℓ′ πX which arises from the final-state rescattering of the outgoing quark. The Bjorken-scaling single-spin asymmetries predicted for the Drell–Yan and leptoproduction processes highlight the importance of initial- and final-state interactions for QCD observables.

Understanding hadron structure using lattice QCD

An elementary introduction is presented to the study of hadron structure using lattice QCD. Following a brief review of relevant aspects of path integrals, the discrete lattice path integral is presented for gluon and quark fields and used to calculate physical observables. Essential aspects of instanton physics are reviewed, and it is shown how the instanton content is extracted from lattice gluon configurations. Finally, both comparison of results including all gluons with those including only instantons and the study of quark zero modes associated with instantons and their contributions to hadronic observables are used to show the dominant role of gluons in hadron structure. MIT CTP# 2701 hep-lat/9804017 1. – Introduction and Motivation Although a quarter of a century has passed since the experimental discovery of quarks and the formulation of QCD, we are only now beginning to understand the essential physics of the structure of light hadrons. To truly understand hadron structure, one must solve rather than model QCD, and the only known means to do so is the numerical solution of lattice field theory. But obtaining accurate numerical results for observables from a computer is not enough — we also need to obtain physical insight. Hence, our strategy is to use numerical evaluation of the QCD path integral on a lattice to identify the configurations that dominate the action as well as to calculate observables. In recent years, the algorithms and techniques of lattice QCD and the performance of massively parallel computers have developed to the point that we are now on the threshold of reliable, quantitative calculations of QCD observables. Furthermore, there is strong evidence from lattice calculations that the topological excitations of the gluon field corresponding in the semiclassical limit to instantons play a dominant role in the structure of light hadrons. The purpose of these lectures is to describe at an elementary level the basic elements of lattice QCD and how numerical solution of QCD on a lattice is elucidating the role of instantons in light hadrons. To appreciate the significance of the current lattice results, it is useful to recall the wide range of disparate physical pictures that have arisen from the different QCD inspired models introduced to model hadrons. For example, non-relativistic quark models focus on constituent quarks interacting via an adiabatic potential. Bag models postulate a

Analytic perturbation theory in analyzing some QCD observables

The paper is devoted to application of recently devised ghost-free Analytic Perturbation Theory (APT) for analysis of some QCD observables. We start with the discussion of the main problem of the perturbative QCD -- ghost singularities and with the resume of this trouble solution within the APT. By a few examples in the various energy and momentum transfer regions (with the flavor number f=3, 4 and 5) we demonstrate the effect of improved convergence of the APT modified perturbative QCD expansion. Our first observation is that in the APT analysis the three-loop contribution $\sim \alpha_s^3$) is as a rule numerically inessential. This gives raise a hope for practical solution of the well-known problem of asymptotic nature of common QFT perturbation series. The second result is that a usual perturbative analysis of time-like events with the big $\pi^2$ term in the $\alpha_s^3$ coefficient is not adequate at $s\leq 2 \GeV^2$. In particular, this relates to $\tau$ decay. Then, for the "high" (f=5) region it is shown that the common two-loop (NLO, NLLA) perturbation approximation widely used there (at $10 \GeV\lesssim\sqrt{s}\lesssim 170 \GeV$) for analysis of shape/events data contains a systematic negative error of a 1--2 per cent level for the extracted $\bar{alpha}_s^{(2)} $ values. Our physical conclusion is that the $\bar{alpha}_s(M^2_Z)$ value averaged over the f=5 data appreciably differs $<bar{alpha}_s(M^2_Z)>_{f=5}\simeq 0.124$ from the currently accepted "world average" (=0.118).}

Resummation of non-global QCD observables

We discuss issues related to the resummation of non-global observables in QCD, those that are sensitive to radiation in only a part of phase space. Examples of such observables are certain single-hemisphere event shapes in e+e- and DIS. Compared to global observables (those sensitive to all emissions, e.g. the e+e- thrust) a new class of single-logarithmic terms arises. These have been neglected in recent calculations in the literature. For a whole set of single hemisphere e+e- and DIS event shapes, we analytically evaluate the first such term, at order alpha_s^2, and give numerical results for the resummation of these terms in the large-Nc limit.

Analytic perturbation theory for QCD observables

We investigate the connection between ghost-free formulations of the RG-invariant QCD perturbation theory in the spacelike and timelike regions. Our basic tool is the “double spectral representation,” similar to the representation for the Adler function, which stems from the first principles of local QFT and relates real functions in the Euclidean and Minkowskian (i.e., timelike) regions. On this base, we establish a simple relation between the approach (known from the early 1980s) of resumming the π2 terms for the invariant coupling function \(\widetilde\alpha (s)\) and QCD observables in the timelike region and the invariant analytic approach (devised a few years ago) leading to the “analyticized” coupling function αan(Q2) and nonpower expansion for observables in the spacelike domain. The function αan(Q2) and the expansion are free of unphysical singularities. The formulated self-consistent scheme, analytic perturbation theory (APT), relates renorm-invariant, effective coupling functions αan(Q2) and \(\widetilde\alpha (s)\), as well as nonpower perturbation expansions for observables in the Euclidean and Minkowskian domains, free of extra singularities and with better convergence in the infrared region. We present a global generalization of the new APT scheme in the case of real QCD, including the domain with various numbers of active quarks. Preliminary estimates indicate that calculations in the framework of the global scheme can produce results quite different from the usual ones for \(\overline \alpha _s \), even in the five-quark region. Numerical examples are given.

Achieving renormalization-scale- and scheme-independence in Padé-related resummation in QCD

Previously developed Padé-related method of resummation for QCD observables, which achieves exact renormalization-scale-invariance, is extended so that the scheme-invariance is obtained as well. The dependence on the leading scheme parameter c 2 is eliminated by a variant of the method of the principle of minimal sensitivity. The subleading parameter c 3 in the approximant is then fixed in such a way that the correct known location of the leading infrared renormalon pole is reproduced. Thus, β-functions which go beyond the last perturbatively calculated order in the observable are used. The β-functions in the approximant are quasianalytically continued by Padé approximants. Two aspects of nonperturbative physics are accounted for in the presented resummation: a mechanism of quasianalytic continuation from the weak- into the strong-coupling regime, and the (approximant-specific) contribution of the leading infrared renormalon. The case of the Bjorken polarized sum rule is considered as a specific example of how the method works.

Power corrections in the single dressed gluon approximation - the average thrust as a case study

Infrared power corrections for Minkowskian QCD observables are analyzed in the framework of renormalon resummation, motivated by analogy with the skeleton expansion in QED and the BLM approach. Performing the ``massive gluon'' renormalon integral a renormalization scheme invariant result is obtained. Various regularizations of the integral are studied. In particular, we compare the infrared cutoff regularization with the standard principal value Borel sum and show that they yield equivalent results once power terms are included. As an example the average thrust < T > in e+e- annihilation is analyzed. We find that a major part of the discrepancy between the known next-to-leading order calculation and experiment can be explained by resummation of higher order perturbative terms. This fact does not preclude the infrared finite coupling interpretation with a substantial 1/Q power term. Fitting the regularized perturbative sum plus a 1/Q term to experimental data yields alpha_s^{MSbar}(M_Z) = 0.110 \pm 0.002.

Renormalization and factorization scale dependence of observables in QCD

Scale dependence of physical quantities computed in perturbative QCD is reviewed and discussed.

Renormalon phenomena in jets and hard processes

The ‘renormalon’ or ‘dispersive’ method for estimating non-perturbative corrections to QCD observables is reviewed. The corrections are power-suppressed, i.e. of the form A Q p where Q is the hard process momentum scale. The renormalon method exploits the connections between divergences of the QCD perturbation series and low-momentum dynamics to predict the power, p. The further assumption of an approximately universal low-energy effective strong coupling leads to relationships between the coefficients A for different observables. Results on 1 Q 2 corrections to deep inelastic structure functions and 1 Q corrections to event shapes are presented and compared with experiment. Shape variables that could be free of 1 Q and α s (Q 2) Q corrections are suggested.

Tuning actions and observables in lattice QCD

We propose a strategy for conducting lattice QCD simulations at fixed volume but variable quark mass so as to investigate the physical effects of dynamical fermions. We present details of techniques which enable this to be carried out effectively, namely the tuning in bare parameter space and efficient stochastic estimation of the fermion determinant. Preliminary results and tests of the method are presented. We discuss further possible applications of these techniques.

Optimal renormalization scale and scheme for exclusive processes

We use the Brodsky-Lepage-Mackenzie (BLM) method to fix the renormalization scale of the QCD coupling in exclusive hadronic amplitudes such as the pion form factor and the photon-to-pion transition form factor at large momentum transfer. Renormalization-scheme-independent commensurate scale relations are established which connect the hard scattering subprocess amplitudes that control exclusive processes to other QCD observables such as the heavy quark potential and the electron-positron annihilation cross section. The commensurate scale relation connecting the heavy quark potential, as determined from lattice gauge theory, to the photon-to-pion transition form factor is in excellent agreement with $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{e}{\ensuremath{\pi}}^{0}e$ data assuming that the pion distribution amplitude is close to its asymptotic form $\sqrt{3}{f}_{\ensuremath{\pi}}x(1\ensuremath{-}x)$. We also reproduce the scaling and normalization of the $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ data at large momentum transfer. Because the renormalization scale is small, we argue that the effective coupling is nearly constant, thus accounting for the nominal scaling behavior of the data. However, the normalization of the space-like pion form factor ${F}_{\ensuremath{\pi}}{(Q}^{2})$ obtained from electroproduction experiments is somewhat higher than that predicted by the corresponding commensurate scale relation. This discrepancy may be due to systematic errors introduced by the extrapolation of the ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{p}{\ensuremath{\pi}}^{+}n$ electroproduction data to the pion pole.