Hadronic quark distribution amplitudes from QCD sum-rule moments

We discuss the reliability of hadronic wave functions (quark distribution amplitudes) determined by a finite number of QCD sum-rule moments. Although the expansion coefficients for polynomial models of the wave function are uniquely determined by the moments, the inherent uncertainty in such moments leads to a considerable indeterminacy in the wave functions because minimal changes of the moments can lead to large oscillations of the model function. In particular, the freedom in the moments left by QCD sum rules leads to a nonconverging polynomial expansion. This remains true even if additional constraints on the wave functions are used. As a consequence of this, the widely used procedure of constructing polynomial models of hadronic wave function from QCD sum rule moments does not guarantee even a reasonable approximation to the true wave function. The differences among the model wave functions persist also in the calculations of physical observables like hadronic form factors. This implies that physical observables calculated by means of such model wave functions are in general very unreliable. As specific examples, we examine the pion and nucleon wave functions and show that Gegenbauer as well as Appell polynomial expansions constructed from QCD sum rule moments are ruled out. The implications for the wave functions which are generally used in the literature are discussed.     

Bound-state perturbation method via SVZ sum rules in quantum mechanics

The perturbation method for bound states within the framework of the Shifman-Vainshtein-Zakharov sum rule method is studied on simple systems (linear harmonic oscillator, hydrogen atom) in external electric fields. It is pointed out that for stronger fields reasonable results for the ground-state energy can only be achieved when sum rules are written for the correction to the Euclidean Green function caused by the external field. Moreover, if the system is bound by a singular (Coulomb) potential, one needs to sum higher perturbative corrections to the Green function and to find a realistic approximation of the continuum contribution to the sum rules. The results are of relevance e.g. for calculations of nucleon magnetic moments and toponium properties via SVZ sum rules in QCD.     

On QCD sum rules of the Laplace transform type and light quark masses

We discuss the relation between the usual dispersion relation sum rules and the Laplace transform type sum rules in QCD. Two specific examples corresponding to the ?-coupling constant sum rule and the light quark masses sum rules are considered. An interpretation, within QCD, of Leutwyler's formula for the current algebra quark masses is also given.     

Matrix elements of four-quark operators relevant to life time difference \Delta \Gamma _{B_s} from QCD sum rules

We extract the matrix elements of four-quark operators relevant to the and life time difference from QCD sum rules. We find that the vacuum saturation approximation works reasonably well, i.e., within 10%. We discuss the implications of our results and compare them with a recent lattice QCD determination. Received: 23 April 2001 / Revised version: 27 May 2001 / Published online: 19 July 2001     

QCD sum rules and electromagnetic polarizabilities of a nucleon

We investigate the electromagnetic polarizabilities of a nucleon using the method of QCD sum rules. The diagrams in the operator product expansion are taken into account up to dimension 6. We find that the polarizabilities of a nucleon can be expressed by means of three new kinds of susceptibilities of quark condensates which represent the response under weak external electromagnetic fields. Estimating the susceptibilities with the QCD sum rules itself based on the formalism by Balitsky et al., we see that good agreement is obtained for the neutron electric polarizability.     

Broad sub-continuum resonances and the case for finite-energy sum-rules

There is a need to go beyond the narrow resonance approximation for QCD sum-rule channels which are likely to exhibit sensitivity to broad resonance structures. We discuss how the first two Laplace sum rules are altered when one goes beyond the narrow resonance approximation to include possible subcontinuum resonances with nonzero widths. We show that the corresponding first two finite energy sum rules are insensitive to the widths of such resonances, provided their peaks are symmetric and entirely below the continuum threshold. We also discuss the reduced sensitivity of the first two finite energy sum rules to higher dimensional condensates, and show these sum rules to be insensitive to dimension condensates containing at least one pair. We extract the direct single-instanton contribution to the sum rule for the longitudinal component of the axial-vector correlation function from the known single-instanton contribution to the lowest Laplace sum rule for the pseudoscalar channel. Finally, we demonstrate how inclusion of this instanton contribution to the finite-energy sum rule leads to both a lighter quark mass and to more phenomenologically reasonable higher-mass-resonance contributions within the pseudoscalar channel. Received: 15 September 1997 / Revised version: 15 November 1997 / Published online: 26 February 1998     

Gaussian sum rules in quantum chromodynamics and local duality

A new type of hadronic sum rules are studied in QCD. They correspond to the Gauss-Weierstrass transform of hadronic spectral functions. There is an interesting analogy between these sum rules and the theory of the heat equation which provides a useful framework for formulating quantitatively the notion of local duality. In particular, finite energy sum rules are shown to follow within this framework from the principle of conservation of total heat. Explicit calculations of these sum rules in QCD - from both perturbative and non-perturbative contributions à la Shifman, Vainshtein and Zakharov [1] - have been made in a rather general way which should cover all the practical cases involving light quark systems. A particular case of the -spectral function and its duality to QCD is discussed in detail as an illustration of this new approach.     

Infrared renormalons and the relations between the Gross-Llewellyn Smith and the Bjorken polarized and unpolarized sum rules

It is demonstrated that the infrared renormalon calculus indicates that the QCD theoretical expressions for the Gross-Llewellyn Smith sum rule and for the Bjorken polarized and unpolarized ones contain an identical negative twist-4 1/Q² correction. This observation is supported by the consideration of the results of calculations of the corresponding twist-4 matrix elements. Together with the indication of the similarity of the perturbative QCD contributions to these three sum rules, this observation leads to simple new theoretical relations between the Gross-Llewellyn Smith and Bjorken polarized and unpolarized sum rules in the energy region Q² ≥ 1 GeV². The validity of this relation is checked using concrete experimental data for the Gross-Llewellyn Smith and Bjorken polarized sum rules.     

Calculation of the kaon B parameter using strictly local sum rules

The kaon B-parameter is computed in the framework of strictly local QCD sum rules for a threepoint function involving pseudoscalar currents. As an application of these sum rules we derive a low energy formula for the B-parameter. We show that strictly local QCD sum rules yield more reliable results than other QCD sum rules, since they need less phenomenological information and do not suffer from stability problems. Our result for the B-parameter isB=0.74±0.17.     

Functional approach to QCD finite-energy sum rules

A new method for the determination of the QCD condensates from low-energy hadronic data is proposed. It generalizes the usual QCD finite-energy sum rules, taking into account explicitely the truncation error of the high-energy QCD expansion. The method is applied to the e⁺e^– annihilation intoI= 1 hadrons, indicating a rather large domain for the values of the gluon and four quark condensates.     

What can we learn from sum rules for vertex functions in QCD?

We demonstrate that the light-cone sum rules for vertex functions based on the operator product expansion and QCD perturbation theory lead to interesting relationships between various non-perturbative parameters associated with hadronic bound states (e.g. vertex couplings and decay constants). We also show that such sum rules provide a valuable means of estimating the matrix elements of the higher spin operators in the meson wave function.     

Light four-quark states and QCD sum rule

The relations among four-quark states, diquarks and QCD sum rules are discussed. The situation of the existing, but incomplete studies of four-quark states with QCD sum rules is analyzed. Masses of some diquark clusters were attempted to be determined by QCD sum rules, and masses of some light tetraquark states were obtained in terms of the diquarks.     

Light four-quark states and QCD sum rule

The relations among four-quark states, diquarks and QCD sum rules are discussed. The situation of the existing, but incomplete studies of four-quark states with QCD sum rules is analyzed. Masses of some diquark clusters were attempted to be determined by QCD sum rules, and masses of some light tetraquark states were obtained in terms of the diquarks.     

Spectral function sum rules in quantum chromodynamics: (I). Charged currents sector

The Weinberg sum rules of the algebra of currents are reconsidered in the light of quantum chromodynamics (QCD). We derive new finite-energy sum rules which replace the old Weinberg sum rules. The new sum rules are convergent and the rate of convergence is explicitly calculated in perturbative QCD at the first non-trivial order. Phenomenological applications of these sum rules in the charged current sector are also discussed.     

Sum rule estimate of the subleading non-perturbative contributions to Bs–$\bar{ B}_{s}$ mixing

We use QCD sum rules to compute the matrix elements of the ΔB=2 operators appearing in the heavy-quark expansion of the width difference of the B _s mass eigenstates. The main focus of our analysis is on the subleading operators R ₂ and R ₃, which appear at next-to-leading order in the 1/m _b expansion. The matrix elements of these operators are already essential for precise phenomenology, but their calculation in lattice QCD is lacking and the values given here provide a first estimate of their values. We conclude that the violation of the factorization approximation for these matrix elements due to non-perturbative vacuum condensates is as low as 1–2%.     

Non-perturbative conserving approximations and Luttinger's sum rule

Weak-coupling conserving approximations can be constructed by truncations of the Luttinger-Ward functional and are well known as thermodynamically consistent approaches which respect macroscopic conservation laws as well as certain sum rules at zero temperature. These properties can also be shown for variational approximations that are generated within the framework of the self-energy-functional theory without a truncation of the diagram series. Luttinger's sum rule represents an exception. We analyze the conditions under which the sum rule holds within a non-perturbative conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two-site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences for cluster extensions of the dynamical mean-field theory are discussed.     

Gluon polarization in nucleons

In QCD the gauge-invariant gluon polarization in a nucleon can be defined either in a non-local way as the integral over the Ioffe-time distribution of polarized gluons, or in light-cone gauge as the forward matrix element of the local topological current. We have investigated both possibilities within the framework of QCD sum rules. Although the topological current is built from local fields, we have found that its matrix element retains sensitivity to large longitudinal distances. Because QCD sum rules produce artificial oscillations of the Ioffe-time distribution of polarized glue at moderate and large light-like distances, the calculation of the matrix element of the topological current results in a small value of . In a more consistent approach QCD sum rules are used to describe the polarized gluon distribution only at small light-like distances. Assuming that significant contributions to arise only from longitudinal length scales not larger than the nucleon size leads to . Received: 19 August 1997 / Published online: 20 February 1998     

Target independence of the EMC-SMC effect

An approach to deep inelastic scattering is described in which the matrix elements arising from the operator product expansion are factorised into composite operator propagators and proper vertex functions. In the case of polarised μp scattering, the composite operator propagator is identified with the square root of the QCD topological susceptibility √x′(0) , while the corresponding proper vertex is a renormalisation group invariant. We estimate x′(0) using QCD spectral sum rules and find that it is significantly suppressed relative to the OZI expectation. Assuming OZI is a good approximation for the proper vertex, our predictions, f₀¹ d x g₁^p (x; Q²= 10 GeV²) = 0.143 ± 0.005 and G_A⁰ = ΔΣ = 0.353 ± 0.052, are in excellent agreement with the new SMC data. This result, together with one confirming the validity of the OZI rule in the η′ radiative decay, supports our earlier conjecture that the suppression in the flavour singlet component of the first moment of gp observed by the EMC-SMC Collaboration is a target-independent feature of QCD related to the U(1) anomaly and is not a property of the proton structure. As a corollary, we extract the magnitude of higher twist effects from the neutron and Bjorken sum rules.     

Test on Application Range of QCD Sum Rules from Baryon Spectroscopy

By considering the nonperturbative QCD effect via quark and gluon condensates, the confinement terms in effective q-q potential can automatically be given. A potential in the form of (-2α,/3r) + Σ_n^∞ a_2n+1r²ⁿ⁺¹ + (Yukawa-type terms) where positive and negative signs appear alternatively in a_2n+1's is proposed. To testify the validity range of QCD sum rules, the experimental baryon spectrum is fitted and the values of azn+l's as well as their signs are obtained. It is found that in the baryon spectrum calculation, the terms of r³ snd r⁵ are important and higher terms beyond r⁵ can be neglected. Comparing these values with the result calculated from QCD sum rules, some agreements and disagreements are found.     

Quarkonia at Finite T: An Approach Based On QCD Sum Rules and the Maximum Entropy Method

Quarkonium spectral functions at finite temperature are studied, making use of a recently developed method of analyzing QCD sum rules by the maximum entropy method. This approach enables us to directly obtain the spectral function from the sum rules, without having to introduce any specific assumption about its functional form. QCD sum rules for heavy quarkonia incorporate finite temperature effects in form of changing values of the various gluonic condensates that appear in the operator product expansion. These changes depend on the energy density and pressure at finite temperature, which we extract from quenched lattice QCD calculations. As a result, it is found that the charmonium ground states of both S-wave and P-wave channels dissolve into the continuum already at temperatures around or slightly above the critical temperature T _c , while the bottomonium states are less influenced by temperature effects, surviving up to about 2.5 T _c or higher for S-wave and about 2.0 T _c for P-wave states.