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Dyson–Schwinger equations furnish a Poincaré covariant framework within which to study hadrons. A particular feature is the existence of a nonperturbative, symmetry preserving truncation that enables the proof of exact results. The gap equation reveals that dynamical chiral symmetry breaking is tied to the long-range behavior of the strong interaction, which is thereby constrained by observables, and the pion is precisely understood, and seen to exist simultaneously as a Goldstone mode and a bound state of strongly dressed quarks. The systematic error associated with the simplest truncation has been quantified, and it underpins a one-parameter model efficacious in describing an extensive body of mesonic phenomena. Incipient applications to baryons have brought successes and encountered challenges familiar from early studies of mesons, and promise a covariant field theory upon which to base an understanding of contemporary large momentum transfer data.
We show that the application of a novel gauge-invariant truncation scheme to the Schwinger-Dyson equations of QCD leads, in the Landau gauge, to an infrared finite gluon propagator and a divergent ghost propagator, in qualitative agreement with recent lattice data.
I review recent results on the infrared properties of QCD from Dyson-Schwinger equations. The topics include infrared exponents of one-particle irreducible Green's functions, the fixed point behaviour of the running coupling at zero momentum, the pattern of dynamical quark mass generation and properties of light mesons.
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We study the continuum Schwinger-Dyson equations for nonperturbative two-dimensional quantum gravity coupled to various matter fields. The continuum Schwinger-Dyson equations for the one-matrix model are explicitly derived and turn out to be a formal Virasoro condition on the square root of the partition function, which is conjectured to be the τ function of the KdV hierarchy. Furthermore, we argue that general multi-matrix models are related to the W algebras and suitable reductions of KP hierarchy and its generalizations.
We survey contemporary studies of hadrons and strongly interacting quarks using QCD's Dyson-Schwinger equations,addressing the following aspects: confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small-to large-Q(2); parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.
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Implementation of Dyson orbitals for coupled-cluster and equation-of-motion coupled-cluster wave functions with single and double substitutions is described and demonstrated by examples. Both ionizations from the ground and electronically excited states are considered. Dyson orbitals are necessary for calculating electronic factors of angular distributions of photoelectrons, Compton profiles, electron momentum spectra, etc, and can be interpreted as states of the leaving electron. Formally, Dyson orbitals represent the overlap between an initial N-electron wave function and the N-1 electron wave function of the corresponding ionized system. For the ground state ionization, Dyson orbitals are often similar to the corresponding Hartree-Fock molecular orbitals (MOs); however, for ionization from electronically excited states Dyson orbitals include contributions from several MOs and their shapes are more complex. The theory is applied to calculating the Dyson orbitals for ionization of formaldehyde from the ground and electronically excited states. Partial-wave analysis is employed to compute the probabilities to find the ejected electron in different angular momentum states using the freestanding and Coulomb wave representations of the ionized electron. Rydberg states are shown to yield higher angular momentum electrons, as compared to valence states of the same symmetry. Likewise, faster photoelectrons are most likely to have higher angular momentum.
One of the principal advantages of electron momentum spectroscopy (EMS) is that peaks in the binding-energy spectrum can be assigned with greater certainty than in photoelectron spectroscopy, through a comparison of the EMS triple-differential cross sections with the theoretically calculated spherically averaged momentum distributions (MD's) of Dyson orbitals. While the target Hartree-Fock approximation is commonly used to calculate the Dyson orbital MD's for this purpose, a computationally less demanding method would allow the advantages of EMS to be extended to larger molecules. This paper considers the use of Kohn-Sham density-functional theory for this purpose. Although Dyson orbitals are not among the quantities that can be calculated exactly (in the limit of the exact exchange-correlation functional) within the framework of Kohn-Sham density-functional theory, the Kohn-Sham equation can be regarded as an approximate form of Dyson's quasiparticle equation, with a local self-energy. The well known shortcomings of this approach for estimating ionization potentials and band gaps do not a priori imply a corresponding problem with the orbitals. After discussing these formal considerations, we introduce the ``target Kohn-Sham approximation'' as a means of approximating Dyson orbitals by Kohn-Sham orbitals. The quality of this approximation for the calculation of MD's is assessed by comparison with high-quality configuration-interaction calculations, the target Hartree-Fock approximation, and experiment, for several small molecules. The quality of the target Kohn-Sham approximation MD's is found to be comparable to that of the MD's from the target Hartree-Fock approximation, with evident practical implications for EMS.
In revisiting the influential Dyson and Moore (1983) hypothesis as to why women in South India enjoy relatively more agency than in the North, we conducted an econometric analysis of the determinants of women's mobility and decisionmaking authority. Data for the study come from a household data survey carried out in the Northern state of Uttar Pradesh and in the Southern state of Karnataka in 1995. We find that cross‐cousin and uncle‐niece marriage is more prevalent in Karnataka as expected. Contrary to Dyson and Moore, however, by 1995 a majority of communities in both North and South practiced village exogamy, and dowries in the two regions were of similar size. Reduced‐form, multivariate regressions show that cultural factors affect women's autonomy in ways not earlier predicted. The impact of village exogamy is mixed rather than negative, while that of consanguinity is strongly negative rather than positive as Dyson and Moore surmised. These authors correctly identified the negative effect purdah has on female mobility. Consistent with economic theory, our data show that higher wages for women consistently improve their mobility and authority, while higher male wages decrease them. Improvements in infrastructure–particularly the presence of street lights and schools in the village–are associated with increased women's agency. We conclude, therefore, that economic factors, state action, and restrictions on mobility seem more powerful than kinship structures as explanations of differences in female autonomy between North and South India.
Nonperturbative studies of field theory require the Schwinger-Dyson equations to be truncated to make them tractable. Thus, when investigating the behavior of the fermion propagator, for example, an Ansatz for the three-point vertex has to be made. While the well-known Ward identity determines the longitudinal part of this vertex in terms of the fermion propagator as shown by Ball and Chiu, it leaves the transverse part unconstrained. However, Brown and Dorey have recently emphasized that the requirement of multiplicative renormalizability is not satisfied by arbitrary Ansa$iuml---tze for the vertex. We show how this requirement restricts the form of the transverse part. By considering the example of QED in the quenched approximation, we present a form for the vertex that not only satisfies the Ward identity but is multiplicatively renormalizable to all orders in leading and next-to-leading logarithms in perturbation theory and so provides a suitable Ansatz for the full three-point vertex.
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A proof of Goldstone's theorem is given that highlights the necessary consistency between the exact Schwinger-Dyson equation for the fermion propagator and the exact Bethe-Salpeter equation for fermion-antifermion bound states. The approach is tailored to the case when a global chiral symmetry is dynamically broken. Criteria are provided for maintaining the consistency when the exact equations are modified by approximations. In particular, for gauge theories in which partial conservation of the axial vector current (PCAC) should hold, a constraint on the approximations to the fermion--gauge-boson vertex function is discussed, and a vertex model is given which satisfies both the PCAC constraint and the vector Ward-Takahashi identity.
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We report total and differential cross sections for photodetachment from negative ions using Dyson orbitals calculated from equation-of-motion coupled-cluster wave functions and free wave description of the detached electron. The energy dependence of the cross sections is reproduced well, however, the accuracy of absolute values varies. For F(-), C(-), NH(2)(-), and H(-), the calculated cross sections are within the error bars from the experimental values, whereas the errors for Li(-) and OH(-) are about 20%. The largest errors are observed for O(-) and O(2)(-) for which the calculated cross sections differ from the experimental ones by factors of 3 and 2, respectively. Calculated anisotropy parameters for atomic anions exhibit too slow decrease, which suggests that the diffuseness of the computed Dyson orbitals is underestimated. Moreover, in the asymptotic region, the orbitals exhibit artifactual oscillations probably due to the limitations of Gaussian basis sets. The analysis of the trends in the experimental anisotropy parameters suggests that the interaction of the detached electron with the core, which is neglected in the present model, is important.