Functions In QCD Research Articles

Large-Nc QCD meets Regge theory: the example of spin-one two-point functions

We discuss the phenomenological implications of assuming a Veneziano-type spectrum for the vector and axial-vector two-point functions in QCD at large Nc. We also compare the phenomenological results with those of Lowest-Meson Dominance, and find that they agree.

Baryon wave functions in QCD and integrable models

A new theoretical framework is proposed for the description of baryon light-cone distribution amplitudes, based on the observation that their scale dependence to leading logarithmic accuracy is described by a completely integrable model. The physical interpretation is that one is able to find a new quantum number that distinguishes partonic components in the nucleon with different scale dependence.

Improved approximations for the three-loop splitting functions in QCD

We update our approximate parametrizations of the three-loop splitting functions for the evolution of unpolarized parton densities in perturbative QCD. The new information taken into account is given by the additional Mellin moments recently calculated by Retey and Vermaseren. The inclusion of these constraints reduces the uncertainties of our approximations considerably and extends their region of applicability by about one order of magnitude to lower momentum fractions x.

Higher order corrections in perturbative quantum chromodynamics

We present some techniques which have been developed recently or in the recent past to compute Feynman graphs beyond one-loop order. These techniques are useful to compute the three-loop splitting functions in QCD and to obtain the complete second order QED corrections to Bhabha scattering.

Derivation of the effective chiral Lagrangian for pseudoscalar mesons from QCD

We formally derive the chiral Lagrangian for low lying pseudoscalar mesons from the first principles of QCD considering the contributions from the normal part of the theory without taking approximations. The derivation is based on the standard generating functional of QCD in the path integral formalism. The gluon-field integration is formally carried out by expressing the result in terms of physical Green's functions of the gluon. To integrate over the quark-field, we introduce a bilocal auxiliary field Phi(x,y) representing the mesons. We then develop a consistent way of extracting the local pseudoscalar degree of freedom U(x) in Phi(x,y) and integrating out the rest degrees of freedom such that the complete pseudoscalar degree of freedom resides in U(x). With certain techniques, we work out the explicit U(x)-dependence of the effective action up to the p^4-terms in the momentum expansion, which leads to the desired chiral Lagrangian in which all the coefficients contributed from the normal part of the theory are expressed in terms of certain Green's functions in QCD. Together with the existing QCD formulae for the anomaly contributions, the present results leads to the complete QCD definition of the coefficients in the chiral Lagrangian. The relation between the present QCD definition of the p^2-order coefficient F_0^2 and the well-known approximate result given by Pagels and Stokar is discussed.

Off-mass-shell Sudakov-like suppression factor for the fermionic four-point function in QCD

We consider a four-point process, associated with a wide-angle elastic scattering of two off-mass-shell spin-1/2 matter particles, in a non-abelian gauge theory. On the basis of a worldline approach, which reverts the functional to a path-integral description of the system, we factorize an eikonal (“soft”) subsector of the full theory and calculate the Sudakov-like suppression factor for the four-point function as a whole, once we have extracted the associated anomalous dimensions and taken into account the renormalization-group controlled evolution.

Renormalization in lattice QCD: Some three-loop results

We calculate the third coefficient of the lattice beta function in QCD with Wilson fermions, extending the pure gauge results of Lüscher and Weisz; we show how this coefficient modifies the scaling function on the lattice. We also calculate the three-loop average plaquette in the presence of Wilson fermions. This allows us to compute the lattice scaling function both in the standard and energy schemes. Finally, we show how to perform a resummation, to all orders in perturbation theory, of a certain class of gauge invariant diagrams in Lattice QCD. These diagrams are often largely responsible for lattice artifacts. Our resummation leads to an improved perturbative expansion. Applied to a number of cases of interest, this expansion yields results remarkably close to corresponding nonperturbative estimates.

Polarized Structure Functions in QCD

Hadron spin physics is now one of the most active fields of physics. Especially in the last ten years, great progress has been made both theoretically and experimentally that has considerably improved our knowledge of the spin structure of nucleons. We review the nucleon's polarized structure functions from the viewpoint of factorization theorems and the gauge invariant, nonlocal light-cone operators in QCD. We discuss a systematic treatment of the polarized structure functions and the corresponding parton distribution functions, which are relevant to inclusive lepto- and hadro-production. We give a detailed analysis of these spin-dependent distribution functions at the twist-2 and twist-3 level, and present various properties and relations satisfied by the parton distributions, which can be derived directly from QCD. We emphasize unique features of higher twist distributions, and the role of the QCD equations of motion to derive their sensitivity to the quark-gluon correlation and their anomalous dimensions for Q^2-evolution.

Nonperturbative effects in QCD at finite temperature and density

These lecture notes illustrate the application of Dyson-Schwinger equations in QCD. The extensive body of work at zero temperature and chemical potential is represented by a selection of contemporary studies that focus on solving the Bethe-Salpeter equation, deriving an exact mass formula in QCD that describes light and heavy pseudoscalar mesons simultaneously, and the calculation of the electromagnetic pion form factor and the vector meson electroproduction cross sections. These applications emphasise the qualitative importance of the momentum-dependent dressing of elementary Schwinger functions in QCD, which provides a unifying connection between disparate phenomena. They provide a solid foundation for an extension of the approach to nonzero temperature and chemical potential. The essential, formal elements of this application are described and four contemporary studies employed to exemplify the method and its efficacy. They study the demarcation of the phase boundary for deconfinement and chiral symmetry restoration, the calculation of bulk thermodynamic properties of the quark-gluon plasma and the response of pi- and rho-meson observables to T and mu. Along the way a continuum order parameter for deconfinement is introduced, an anticorrelation between the response of masses and decay constants to T and their response to mu elucidated, and a (T,mu)-mirroring of the slow approach of bulk thermodynamic quantities to their ultrarelativistic limit highlighted. These effects too are tied to the momentum-dependent dressing of the elementary Schwinger functions.

Central functions and their physical implications

I define central functions c(g) and c'(g) in quantum field theory, useful to study the flow of the numbers of vector, spinor and scalar degrees of freedom from the UV limit to the IR limit and basic ingredients for a description of quantum field theory as an interpolating theory between pairs of 4D conformal field theories. The key importance of the correlator of four stress-energy tensors is outlined in this respect. Then I focus the analysis on the behaviours of the central functions in QCD, computing their slopes in the UV critical point. To two-loops, c(g) and c'(g) point towards the expected IR directions. As a possible physical application, I argue that a closer study of the central functions might allow us to lower the upper bound on the number of generations to the observed value. Candidate all-order expressions for the central functions are compared with the predictions of electric-magnetic duality.

Three-loop results in QCD with Wilson fermions

We calculate the third coefficient of the lattice beta function in QCD with Wilson fermions, extending the pure gauge results of Lüscher and Weisz; we show how this coefficient modifies the scaling function on the lattice. We also calculate the three-loop average plaquette in the presence of Wilson fermions. This allows us to compute the lattice scaling function both in the standard and energy schemes.

Evolution of chiral-odd spin-independent fracture functions in quantum chromodynamics

We construct evolution equations for the twist-3 chiral-odd spin-independent fracture functions in QCD. The Gribov-Lipatov reciprocity relation is fulfilled at the one-loop level for the quasipartonic two-particle cut vertices only. It is found that the rank of the anomalous dimension matrix is infinite for any given moment of the three-parton fracture function as distinguished from the case of DIS distributions where the rank of the matrix was finite and increases with the number of the moment. In the multicolour limit N c → ∞ the evolution equation for the quark-gluon-quark correlation function decouples from another equation in the system and becomes homogeneous provided we discard the quark mass effects. This fact provides an opportunity to find its analytic solution explicitly in a non-local form similarly to DIS.

From the Feynman-Schwinger representation to the nonperturbative relativistic bound state interaction

We write the four-point Green function in QCD in the Feynman-Schwinger representation and show that all the dynamical information is contained in the Wilson loop average. We work out the QED case in order to obtain the usual Bethe-Salpeter kernel. Finally we discuss the QCD case in the nonperturbative regime giving some insight into the nature of the interaction kernel.

A prediction for the 4-loop β function in QCD

We predict that the four-loop contribution β 3 to the QCD β function in the MS prescription is given by β 3 ⋍ 23600(900) − 6400(200)N f + 350(70)N f 2 + 1.5N f 3 , where N f is the number of flavours and the coefficient of N f 3 is an exact result from large- N f expansion. In the phenomenologically-interesting case N f = 3, we estimate β 3 = (7.6±0.1) × 10 3. We discuss our estimates of the errors in these QCD predictions, basing them on the demonstrated accuracy of our method in test applications to the O( N) Φ 4 theory, and on variations in the details of our estimation method, which goes beyond conventional Padé approximants by estimating and correcting for subasymptotic deviations from exact results.

Instantons and meson correlation functions in QCD

Various QCD correlators are calculated in the instanton liquid model in zeromode approximation and $1/N_c$ expansion. Previous works are extended by including dynamical quark loops. In contrast to the original "perturbative" $1/N_c$ expansion not all quark loops are suppressed. In the flavor singlet meson correlators a chain of quark bubbles survives the $N_c\to\infty$ limit causing a massive $\eta^\prime$ in the pseudoscalar correlator while keeping massless pions in the triplet correlator. The correlators are plotted and meson masses and couplings are obtained from a spectral fit. They are compared to the values obtained from numerical studies of the instanton liquid and to experimental results.

From QCD to Quark Potential Model11The project supported in part by National Natural Science Foundation of China and the Nuclear Science Foundation of China

The connection between phenomenological parameters in the quark potential model and fundamental QCD parameters is established by studying the fermionic three-point Green's functions in QCD incorporating the effects of nonperturbative vacuum. The scale of chiral-symmetry-breaking leading to the constituent quark picture is estimated.

Physically motivated approximation to a parton distribution function in QCD

It has been suggested that parton distributions in coordinate space, so called Ioffe-time distributions, provide a more natural object for non-perturbative methods compared to the usual momentum distributions. In this paper we argue that the shape of experimentally determined Ioffe-time distributions of quarks in a nucleon target clearly indicates separation of longitudinal scales, which is not easily recognizable in terms of conventional longitudinal momentum space considerations. We demonstrate how to use this observation to determine parton distributions, using non-perturbative information about the first few moments and the Regge asymptotics at small x.

Nonperturbative condensate contributions to the effective quark-gluon coupling

We study the effective quark-gluon coupling at low-energy scale, which is defined as the amplitude of a quark emitting or absorbing a gluon with some momentum at low-energy scale. This amplitude is determined from the fermionic three-point Green’s functions of QCD including the leading order contributions of nonperturbative condensates through use of the operator-product expansion. By this approach, we discuss the relationship between the constituent quark and the quark of QCD Lagrangian, and estimate the scale of chiral symmetry breaking and the size of a constituent quark in participating the strong interaction process, such as form factors and radii.

ONCE MORE ON THE RADIATIVE CORRECTIONS TO THE NUCLEON STRUCTURE FUNCTIONS IN QCD

A new representation of the next-to-leading QCD corrections to the nucleon structure functions is given in terms of parton distributions. All O(αs) corrections to the leading logarithmic approximation (LLA) are included. In contrast to the similar representations in the literature terms of order [Formula: see text] do not appear in our expressions for the nucleon structure functions taken in the next-to-leading logarithmic approximation. This result is generalized for any order in αs beyond the LLA. Terms of order [Formula: see text] which belong only to the approximation considered are present in such a representation for the structure functions.

Quark model, large order behavior and nonperturbative wave functions in QCD

We discuss a few, apparently different (but actually, tightly related) problems: 1. 1. the relation between QCD and valence quark model, 2. 2. the evaluation of the nonlocal condensate 〈 q(x)q(0)〉 , its relation to the heavy-light qQ quark system and to constituent quark mass, 3. 3. the asymptotic behavior of the nonperturbative pion wave function ψ( k ⊥ 2, x) at x → 0, 1, k ⊥ 2 → ∞ and 4. 4. the large order behavior of perturbative series. The analysis is based on such general methods as dispersion relations, duality and PCAC. We use the steepest descent method (also known as semiclassical, or instanton calculus), introduced by Lipatov to calculate the n-th moment of the ψ( k ⊥ 2, x) with result 〈 k ⊥ 2 n 〉 ∼ n!.. This information fixes the asymptotic behavior of wf at large k ⊥ 2. This behavior turned out to be Gaussian, as commonly used in the phenomenological analyses. The same method determines the asymptotic behavior of the mixed local vacuum condensates 〈 qG μν nq〉 ∼ n! at large n as well as the nonlocal vacuum condensate 〈 q(x)q(0)〉 which naturally arises in the description of the heavy-light qQ quark system. The relation between the nonlocal condensate and the constituent quark mass is also discussed.