Contribution of higher gluon condensates to light-quark vacuum polarization

A method of classifying quark operators in QCD sum rules is suggested. The expansion coefficients of all thed≦8 bilinear quark condensates in gluon condensates are calculated. The coefficient functions at the gluon operators withd≦8 in the polarization operator ∏(q ²) of the light-quark vector current are obtained. A comparison is performed with the calculations in the covariantly constant fields and self-dual fields. The results obtained can be used in the sum rules for the ρ, ω and ? families.     

Theoretical Description and Measurement of the Pion–Photon Transition Form Factor

Detailed predictions for the scaled pion–photon transition form factor are given, derived with the method of light-cone sum rules and using pion distribution amplitudes with two and three Gegenbauer coefficients obtained from QCD sum rules with nonlocal condensates. These predictions agree well with all experimental data that are compatible with QCD scaling (and collinear factorization), but disagree with the high-Q ² data of the BaBar Collaboration that grow with the momentum. A good agreement of our predictions with results obtained from AdS/QCD models and Dyson–Schwinger computations is found.     

OPE analysis of the nucleon scattering tensor: a review including weak interactions and finite mass effects

We perform a systematic operator product expansion of the most general form of the nucleon scattering tensorW _μν including electro-magnetic and weak interaction processes. Finite quark masses are taken into account and a number of higher-twist corrections are included. In this way we derive relations between the lowest moments of all 14 structure functions and matrix elements of local operators. Besides reproducing well-known results, new sum rules for parity-violating polarized structure functions and new mass correction terms are obtained.     

Electroproduction and annihilation scaling laws in a dual model

Scaling laws for large virtual photon mass (q²) in electroproduction and annihilation are studied in the framework of a simple planar dual model. We find, as has recently been conjectured, that the scaling behaviour depends on the number of space-time dimensions spanned by large momenta. In particular, for a certain range of parameters in the model, we find that the annihilation cross section is dominated by the one-dimensional configuration and increases with q² relative to its canonical behaviour while the electroproduction total cross section is dominated by the two-dimensional configuration and has the canonical Bjorken scaling behavior. In general the scaling laws and therefore the structure of events in the model are distinctively different from the conventional parton model. The problem of consistency of planar dual tree diagrams with unitarity sum rules is discussed.     

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.     

Relations between the electroexcitation of nucleon resonances and the deep inelastic continuum for proton and deuteron targets

New data of the electroexcitation cross section on protons and deuterons in the resonance region are compared with the results in the deep inelastic region for four-momentum transfers q² ? 1.5 (GeV/c)². The Bloom-Gilman sum rule is shown to hold for q² ? 1.2 (GeV/c) while the Rittenberg-Rubinstein sum rule is saturated for q² ? 0.4 (GeV/c)². It is shown that the sum rules do not saturate in a local way.     

Collision of Rydberg atom A** with atom B in the ground electronic state. Optical potential

A method for calculating the complex optical potential of slowly colliding Rydberg atom A^** and neutral atom B in the ground electronic state is suggested. The method is based on the asymptotic approach and the theory of multichannel quantum defects, which uses the formalism of renormalized Lippmann-Schwinger equations. The potential is introduced as the 〈q|V _opt|q〉 matrix element of the optical interaction operator, for which the integral equation is derived, and is calculated in the basis set of free particle wave functions |q〉. Fairly simple equations for the shift and broadening of the ionic term are obtained, and the principal characteristics of these equations are analyzed. By way of illustration, the optical potential of the Na^**(nl)+B systems, where B is a rare gas atom, is calculated.     

Operator product expansions in the massless Schwinger model

The known operator solution of the massless Schwinger model is used to calculate exactly, in three operator product expansions, the coefficient functions of the first few operators of low dimension which contribute when vacuum matrix elements are to be taken. A comparison of the results provides a test of the procedure used by M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov [Nucl. Phys. B147 (1979), 385–447] in their study of QCD. It is found that the shift in vacua does not affect the calculation of coefficient functions. The vacuum insertion approximation yields somewhat misleading estimates of vacuum expectation values of some composite operators; however, the standard method used to estimate the errors of vacuum insertion indicates that the approximation is unreliable in this model.     

Effective field theory approach to structure functions at small $x_{\mathrm{Bj}}$

We relate the structure functions of deep inelastic lepton-nucleon scattering to current-current correlation functions in a Euclidean field theory depending on a parameter r. The r-dependent Hamiltonian of the theory is P ⁰ -(1-r)P ³ , with P⁰ the usual Hamiltonian and P³ the third component of the momentum operator. We show that a small in the structure functions corresponds to the small r limit of the effective theory. We argue that for there is a critical regime of the theory where simple scaling relations should hold. We show that in this framework Regge behaviour of the structure functions obtained with the hard pomeron ansatz corresponds to a scaling behaviour of the matrix elements in the effective theory where the intercept of the hard pomeron appears as a critical index. Explicit expressions for various analytic continuations of the structure functions and matrix elements are given as well as path integral representations for the matrix elements in the effective theory. Our aim is to provide a framework for truly non-perturbative calculations of the structure functions at small for arbitrary Q². Received: 16 July 2002 / Published online: 9 December 2002 RID="a" ID="a" e-mail: O.Nachtmann@thphys.uni-heidelberg.de     

Potts q-color field theory and scaling random cluster model

We study structural properties of the q-color Potts field theory which, for real values of q, describes the scaling limit of the random cluster model. We show that the number of independent n-point Potts spin correlators coincides with that of independent n-point cluster connectivities and is given by generalized Bell numbers. Only a subset of these spin correlators enters the determination of the Potts magnetic properties for q integer. The structure of the operator product expansion of the spin fields for generic q is also identified. For the two-dimensional case, we analyze the duality relation between spin and kink field correlators, both for the bulk and boundary cases, obtaining in particular a sum rule for the kink-kink elastic scattering amplitudes.     

QCD analysis of twist-4 contributions to the g i structure functions

We analyze the twist-4 contributions to Bjorken and Ellis-Jaffe sum rules for spin-dependent structure function g ₁(x, Q ²). We investigate the anomalous dimensions of the twist-4 operators which determine the logarithmic correction to the 1/Q ² behavior of the twist-4 contribution by evaluating off-shell Green’s functions in both flavor non-singlet and singlet case. It is shown that the operators which are proportional to the equation of motion play an important role to extract the anomalous dimensions of physical operators. The calculations to solve the operator mixing of higher-twist operators are given in detail     

A class of modified sum rules in QCD

We consider a class of modifications to the exponential QCD sum rules, replacing the energy variableq ² by (q ²) ^k withk positive integer. As a result of this modification, (i) the relative role of the condensates on the theoretical side is changed, (ii) the integral on the phenomenological side acquires a rapidly decreasing weight function and (iii) the way of approaching infinity in theq ² complex plane splits intok rays. We explicitly discuss the modified sum rules withk=1,k=2 andk=3 for the ρ meson and conclude that, unlike to thek=1 case, thek=2 andk=3 sum rules do not yield a satisfactory scheme for the prediction of the resonance parameters unless many higher condensates are known. We show that, for practical applications with the present knowledge of condensates, the standard exponential QCD sum rules are the most suitable ones out of the class considered.     

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.     

Current and constituent quarks: Their implications for resonance excitations polarized and unpolarized inelastic structure functions

The transformation between constituent and current quarks is discussed and applied to the calculation of matrix elements for nucleon to resonance transitions induced by arbitrary currents belonging to an octet. In particular, previous results on πN and γN transitions are subsumed while weak interactions (ΔQ = 0 or 1) and longitudinal photon induced transitions are discussed for the first time. The implications of a non-trivial Melosh transformation upon previous calculations in quark parton models of the non-diffractive component of the deep inelastic structure functions are discussed. The magnitudes of the unpolarized structure functions are found to be unchanged but for the polarized structure functions significant corrections to previous calculations are discovered. In particular it is found that it is not necessary that the polarization asymmetry A^γP be positive in the deep inelastic region. Our approach is valid for all Q², even for photoproduction, and is not restricted to deep inelastic scattering in contrast to earlier parton model calculations. The saturation of various current algebra sum rules is discussed.     

Duality in the current propagator

The idea of local duality between generalized vector dominance and light cone algebra or parton model is applied to the propagator of vector currents. A set of sum rules reminiscent of the generalized first Weinberg sum rules but involving certain unknown parameters are derived. These parameters are fixed from the leptonic decay modes of ?, ω, ?. The sum rules are used to calculate the ω-? mixing angle and thereby predict $\text{Γ(?→}\text{K}\text{K}\text{) = 3.24±0.34}\text{MeV}$, in agreement with experiment. Simple mass mixing or current mixing model for the vector mesons seem to be inconsistent with data. The total cross section for e⁺e^?→hadrons is calculated in the scaling region. If scaling is achieved at $\text{s ? 9–16}\text{GeV}{}^2$ then it is argued that about 22% of the events in this energy region will include strange particles.     

Non-commutative Euclidean and Minkowski structures

A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SO _q (3) or SO _q(1, 3) is introduced. The generating elements of this algebra are hermitean and can be identified with coordinates, momenta and angular momenta. In addition a unitary scaling operator is part of the algebra.     

Fixed-mass sum rules in a non-perturbative parton-model framework

The formulation of the parton model due to Landshoff and Polkinghorne is used to discuss the various fixed-mass sum rules that can be derived for the structure functions measurable in inelastic electroproduction, including those dependent on the target spin and also allowing for an extension to the non-Abelian case of isovector currents. The model is re-expressed in terms of light-cone variables which achieves some simplification and gives some new results, in particular concerning the leading terms of the longitudinal structure function in the scaling limit for $\text{spin}\text{-}\text{1}\text{2}$ theories. Careful attention is paid to verifying the consistency of the results with current conservation, which is achieved by embedding the model in simple field theories so as to introduce the necessary dynamics. Those sum rules which appear to diverge are shown to be made finite by becoming statements concerning the residue of fixed J-plane poles, for which an analysis is given.     

Sum rules in QCD for g ηΛΛ and g πΛΣ

New relations between the Borel QCD sum rules for the strong couplings of the Σ⁰ and Λ hyperons are derived. It is shown that, using the sum rules for g _MΣΣ, M = π ⁰, η, as a starting point, one can readily obtain the corresponding sum rules for the coupling constants g _ηΛΛ and g _πΛΣ ⁰. The values of these constants are calculated in a specific parameter region.