QCD Green Functions Research Articles

OPE and quark-hadron duality for two-point functions of tetraquark currents in the 1/Nc expansion

We discuss the Operator Product Expansion (OPE) and quark-hadron duality for two-point Green functions of tetraquark currents. We emphasize that the factorizable part of the OPE series for such Green functions, including nonperturbative contributions described by QCD condensates, is saturated by the full system of ordinary hadrons and therefore cannot have any relationship to the possible tetraquark bound states. Possible tetraquark bound states may be contained in nonfactorizable parts of these Green functions. In the framework of the $1/N_c$ expansion in QCD$(N_c)$, nonfactorizable parts of the two-point Green functions of tetraquark currents provide $N_c$-suppressed contributions compared to the $N_c$-leading factorizable parts. A possible exotic tetraquark state may appear only in $N_c$-subleading contributions to the QCD Green functions, in full accord with the well-known rigorous properties of large-$N_c$ QCD.

The quark spectral functions and the Hadron Vacuum Polarization from application of DSEs in Minkowski space

The hadronic vacuum polarization function $\Pi_h$ for two light flavors is computed on the entire domain of spacelike and timelike momenta using a framework of Dyson-Schwinger equations. The analytical continuation of the function $\Pi_h$ is based on the utilization of the Gauge Technique with the entry of QCD Green's functions determined from generalized quark spectral functions. For the first time, the light quark spectral functions are extracted from the solution of the gap equation for the quark propagator. The scale is set up by the phenomena of dynamical chiral symmetry breaking, which is a striking feature of low energy QCD.

Hadronic vacuum polarization in QCD and its evaluation in Euclidean spacetime

We discuss a new technique to evaluate integrals of QCD Green's functions in the Euclidean based on their Mellin-Barnes representation. We present as a first application the evaluation of the lowest order Hadronic Vacuum Polarization (HVP) contribution to the anomalous magnetic moment of the muon $\frac{1}{2}(g_{\mu}-2)_{\rm\tiny HVP}\equiv a_{\mu}^{\rm HVP}$. It is shown that with a precise determination of the slope and curvature of the HVP function at the origin from lattice QCD (LQCD), one can already obtain a result for $a_{\mu}^{\rm HVP}$ which may serve as a test of the determinations based on experimental measurements of the $e^+ e^-$ annihilation cross-section into hadrons.

Towards tests of quark-hadron duality with functional analysis and spectral function data

The presence of terms that violate quark-hadron duality in the expansion of QCD Green's functions is a generally accepted fact. Recently, a new approach was proposed for the study of duality violations (DVs), which exploits the existence of a rigorous lower bound on the functional distance, measured in a certain norm, between a "true" correlator and its approximant calculated theoretically along a contour in the complex energy plane. In the present paper we pursue the investigation of functional-analysis based tests towards their application to real spectral function data. We derive a closed analytic expression for the minimal functional distance based on the general weighted $L^2$ norm and discuss its relation with the distance measured in $L^\infty$ norm. Using fake data sets obtained from a realistic toy model in which we allow for covariances inspired from the publicly available ALEPH spectral functions, we obtain by Monte Carlo simulations the statistical distribution of the strength parameter that measures the magnitude of the DV term added to the usual operator product expansion (OPE). The results show that, if the region with large errors near the end-point of the spectrum in $\tau$ decays is excluded, the functional-analysis based tests using either $L^2$ or $L^\infty$ norms are able to detect, in a statistically significant way, the presence of DVs in realistic spectral function pseudodata.

Light mesons in QCD and unquenching effects from the 3PI effective action

We investigate the impact of unquenching effects on QCD Green's functions, in the form of quark-loop contributions to both the gluon propagator and three-gluon vertex, in a three-loop inspired truncation of the three-particle irreducible (3PI) effective action. The fully coupled system of Dyson-Schwinger equations for the quark-gluon, ghost-gluon and three-gluon vertices, together with the quark propagator, are solved self-consistently; our only input are the ghost and gluon propagators themselves that are constrained by calculations within lattice QCD. We find that the two different unquenching effects have roughly equal, but opposite, impact on the quark-gluon vertex and quark propagator, with an overall negative impact on the latter. By taking further derivatives of the 3PI effective action, we construct the corresponding quark-antiquark kernel of the Bethe-Salpeter equation for mesons. The leading component is gluon exchange between two fully dressed quark-gluon vertices, thus introducing for the first time an obvious scalar-scalar component to the binding. We gain access to time-like properties of bound states by analytically continuing the coupled system of Dyson-Schwinger equations to the complex plane. We observe that the vector axial-vector splitting is in accord with experiment and that the lightest quark-antiquark scalar meson is above 1 GeV in mass.

From OPE to chiral perturbation theory in holographic QCD

The soft wall model in holographic QCD has Regge trajectories but wrong operator product expansion (OPE) for the two-point vectorial QCD Green function. We modify the dilaton potential to comply with the OPE. We study also the axial two-point function using the same modified dilaton field and an additional scalar field to address chiral symmetry breaking. OPE is recovered adding a boundary term and low energy chiral parameters, F π and L 10 , are well described analytically by the model in terms of Regge spacing and QCD condensates. The model nicely supports and extends previous theoretical analyses advocating Digamma function to study QCD two-point functions in different momentum regions.

Assuming Regge trajectories in holographic QCD: from OPE to chiral perturbation theory

The soft wall model in holographic QCD has Regge trajectories but wrong oper- ator product expansion (OPE) for the two-point vectorial QCD Green function. We modify the dilaton potential to comply with the OPE. We study also the axial two-point function us- ing the same modified dilaton field and an additional scalar fie ld to address chiral symmetry breaking. OPE is recovered adding a boundary term and low energy chiral parameters, Fπ and L10, are well described analytically by the model in terms of Regge spacing and QCD condensates. The model nicely supports and extends previous theoretical analyses advocat- ing Digamma function to study QCD two-point functions in different momentum regions.

Unraveling the organization of the QCD tapestry

I review some key aspects of the ongoing progress in our understanding of the infrared dynamics of the QCD Green's functions, derived from the close synergy between Schwinger-Dyson equations and lattice simulations. Particular attention is dedicated to the elaborate nonperturbative mechanisms that endow the fundamental degrees of freedom (quarks and gluons) with dynamical masses. In addition, the recently established connection between the effective interaction obtained from the gauge sector of the theory and that needed for the veracious description of the ground-state properties of hadrons is briefly presented.

Spurious divergences in Dyson-Schwinger equations

We revisit the treatment of spurious ultraviolet divergences in the equation of motion of the gluon propagator caused by a momentum cutoff and the resulting violation of gauge invariance. With present continuum studies of the gluon propagator from its Dyson-Schwinger equation reaching the level of quantitatively accurate descriptions, it becomes increasingly important to understand how to subtract these spurious divergences in an unambiguous way. Here we propose such a method. It is based entirely on the asymptotic perturbative behavior of the QCD Green's functions without affecting non-perturbative aspects such as mass terms or the asymptotic infrared behavior. As a particular example, this allows us to assess the possible influence of the tadpole diagram beyond perturbation theory. Finally, we test this method numerically by solving the system of Dyson-Schwinger equations of the gluon and ghost propagators.

Effects of divergent ghost loops on the Green’s functions of QCD

In the present work we discuss certain characteristic features encoded in some of the fundamental QCD Green's functions, whose origin can be traced back to the nonperturbative masslessness of the ghost field, in the Landau gauge. Specifically, the ghost loops that contribute to these Green's functions display infrared divergences, akin to those encountered in the perturbative treatment, in contradistinction to the gluonic loops, whose perturbative divergences are tamed by the dynamical generation of an effective gluon mass. In d=4, the aforementioned divergences are logarithmic, thus causing a relatively mild impact, whereas in d=3 they are linear, giving rise to enhanced effects. In the case of the gluon propagator, these effects do not interfere with its finiteness, but make its first derivative diverge at the origin, and introduce a maximum in the region of infrared momenta. The three-gluon vertex is also affected, and the induced divergent behavior is clearly exposed in certain special kinematic configurations, usually considered in lattice simulations; the sign of the corresponding divergence is unambiguously determined. The main underlying concepts are developed in the context of a simple toy model, which demonstrates clearly the interconnected nature of the various effects. The picture that emerges is subsequently corroborated by a detailed nonperturbative analysis, combining lattice results with the dynamical integral equations governing the relevant ingredients, such as the nonperturbative ghost loop and the momentum-dependent gluon mass.

Resonance saturation in the odd-intrinsic parity sector of low-energy QCD

Using the large ${N}_{C}$ approximation we have constructed the most general chiral resonance Lagrangian in the odd-intrinsic parity sector that can generate low-energy chiral constants up to $\mathcal{O}({p}^{6})$. Integrating out the resonance fields these $\mathcal{O}({p}^{6})$ constants are expressed in terms of resonance couplings and masses. The role of ${\ensuremath{\eta}}^{\ensuremath{'}}$ is discussed and its contribution is explicitly factorized. Using the resonance basis we have also calculated two QCD Green functions of currents, $⟨VVP⟩$ and $⟨VAS⟩$, and found, imposing high-energy constraints, additional relations for resonance couplings. We have studied several phenomenological implications based on these correlators which provided, for example, a prediction for the ${\ensuremath{\pi}}^{0}$-pole contribution to the muon $g\ensuremath{-}2$ factor: ${a}_{\ensuremath{\mu}}^{{\ensuremath{\pi}}^{0}}=65.8(1.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$.

QCD Green's Functions and Phases of Strongly-Interacting Matter

After presenting a brief summary of functional approaches to QCD at vanishing temperatures and densities the application of QCD Green's functions at non-vanishing temperature and vanishing density is discussed. It is pointed out in which way the infrared behavior of the gluon propagator reflects the (de-)confinement transition. Numerical results for the quark propagator are given thereby verifying the relation between (de-)confinement and dynamical chiral symmetry breaking (restoration). Last but not least some results of Dyson-Schwinger equations for the color-superconducting phase at large densities are shown.

Relations between SU(2)- and SU(3)-LECs in chiral perturbation theory

Chiral perturbation theory in the two--flavour sector allows one to analyse Green functions in QCD in the limit where the strange quark mass is considered to be large in comparison to the external momenta and to the light quark masses m_u and m_d. In this framework, the low--energy constants of SU{2}_R \times SU{2}_L depend on the value of the heavy quark masses. For the coupling constants which occur at order p^2 and p^4 in the chiral expansion, we worked out the dependence on the strange quark mass at two--loop accuracy, and provided analogous relations for some of the couplings c_i which are relevant at order p^6. This talk comments on the methods used, and illustrates implications of the results obtained.

Screening masses in quenched [formula omitted] Yang–Mills theory: Universality from dynamics?

We compute the spectrum of gluonic screening-masses in the 0 + + channel of quenched 3 d Yang–Mills theory near the phase-transition. Our finite-temperature lattice simulations are performed at scaling region, using state-of-art techniques for thermalization and spectroscopy, which allows for thorough data extrapolations to thermodynamic limit. Ratios among mass-excitations with the same quantum numbers on the gauge theory, 2 d Ising and λ ϕ 4 models are compared, resulting in a nice agreement with predictions from universality. In addition, a gauge-to-scalar mapping, previously employed to fit QCD Green's functions at deep IR, is verified to dynamically describe these universal spectroscopic patterns.

Integrating out strange quarks in ChPT: Terms at order [formula omitted

Chiral perturbation theory in the two-flavour sector allows one to analyse Green functions in qcd in a limit where the strange quark mass is considered to be large in comparison to the external momenta and to the light quark masses m u and m d . In this framework, the low-energy constants of SU ( 2 ) R × SU ( 2 ) L depend on the value of the heavy quark masses. In a recent article, we have worked out, for the coupling constants l i which occur at order p 4 in the chiral expansion, the dependence on the strange quark mass at two-loop accuracy. Here, we provide analogous relations for some of the couplings c i which are relevant at order p 6 . To keep the calculations somewhat reasonable in size, we consider only those 28 couplings which enter the Green functions built from vector and axial vector quark currents in the chiral limit m u = m d = 0 , m s ≠ 0 . This provides the matching for 27 linear combinations of the 28 couplings.

OPE-RχT matching at order αs: hard gluonic corrections to three-point Green functions

In this work we push the matching between the QCD operator product expansion (OPE) and resonance chiral theory (RχT) to order αs. To this end we compute two- and three-point QCD Green functions (GFs) in both theories and compare the results. GFs which are order parameters of chiral symmetry breaking make this matching more transparent and thus we concentrate on those. On the OPE side one needs to calculate the hard-gluon virtual corrections to the quark condensate, and in particular for three-point GFs this computation was hitherto missing. We also discuss the need for including the infinite tower of hadronic states in the hadronic representation of the GF when non-analytic terms such as logarithms are present in the OPE Wilson coefficients.

QCD Green functions and their application to hadron physics

In a functional approach to QCD the infrared behaviour of Landau gauge Green functions is investigated. It can be proven that the ghost Dyson-Schwinger equation implies the Gribov-Zwanziger horizon condition. Its relation to the Kugo-Ojima confinement scenario is elucidated. Positivity violation for gluons is demonstrated, and the analytic structure of the gluon propagator is studied. Quark confinement is related to an infrared divergence of the quark-gluon vertex. It is shown that in the latter various components are non-vanishing due to the dynamical breaking of chiral symmetry. As a result an infrared finite running coupling in the Yang-Mills sector is derived whereas the running coupling related to the quark-gluon vertex is infrared divergent. In Coulomb gauge QCD already the one-gluon-exchange (over-)confines. This leads to a vanishing quark propagator, and thus quarks are confined. Nevertheless colour singlet quantities derived from the quark propagator are well-defined. Especially the expression for the quark condensate proves that chiral symmetry is dynamically broken. As expected the properties of mesons can be directly calculated whereas the mass of coloured diquarks diverges, and thus diquarks are confined. The latter nevertheless possess a well-defined size. In the third part the results obtained so far will be used to formulate a covariant Faddeev approach to nucleons. The resulting amplitudes describe the quark core of the nucleon. Besides the mass of this state also the electromagnetic form factors are calculated. The results for charge radii and magnetic moments as a function of the quark current mass provide some indication what the missing pion cloud may contribute to the nucleons' properties.

The infrared behavior of lattice QCD Green's functions

We investigate different aspects of lattice QCD in Landau gauge using Monte Carlo simulations. In particular, we focus on the low momentum behavior of gluon and ghost propagators. The gauge group is SU(3). Different systematic effects on the gluon and ghost propagators are studied, e.g. the dependence on the choice of Gribov copies or the influence of dynamical Wilson fermions. We compare our data with results from studies of Dyson-Schwinger equations for the gluon and ghost propagators. We demonstrate that the infrared behavior of both propagators, as found in this thesis, is consistent with different criteria for confinement. However, the running coupling constant, given as a renormalization-group-invariant combination of the gluon and ghost dressing functions, does not expose a finite infrared fixed point. We also report on a first nonperturbative computation of the SU(3) ghost-gluon-vertex renormalization constant and on an investigation of the spectral properties of the Faddeev-Popov operator.

Low-Energy Dynamics in Ultradegenerate QCD Matter

We study the low-energy behavior of QCD Green functions in the limit that the baryon chemical potential is much larger than the QCD scale parameter LambdaQCD. We show that there is a systematic low-energy expansion in powers of (omega/m)(1/3), where omega is the energy and m is the screening scale. This expansion is valid even if the effective quark-gluon coupling g is not small. The expansion is purely perturbative in the magnetic regime |k| >> k0. If the external momenta and energies satisfy |k| approximately k0, planar, Abelian ladder diagrams involving the full quark propagator have to be resummed but the corresponding Dyson-Schwinger equations are closed.

Towards a consistent estimate of the chiral low-energy constants

Guided by the large- N C limit of QCD, we construct the most general chiral resonance Lagrangian that can generate chiral low-energy constants up to O ( p 6 ) . By integrating out the resonance fields, the low-energy constants are parametrized in terms of resonance masses and couplings. Information on those couplings and on the low-energy constants can be extracted by analysing QCD Green functions of currents both for large and small momenta. The chiral resonance theory generates Green functions that interpolate between QCD and chiral perturbation theory. As specific examples we consider the 〈 V A P 〉 and 〈 S P P 〉 Green functions.