We investigate the non-perturbative degrees of freedom in the class of non-local Higgs theories that have been proposed as an ultraviolet completion 4-D Quantum Field Theory (QFT) generalizing the kinetic energy operators to an infinite series of higher derivatives inspired by string field theory. At the perturbative level, the degrees of freedom of non-local Higgs are the same of the local theory. We prove that, at the non-perturbative level, the physical spectrum of the Higgs mass is actually corrected from the "infinite number of derivatives" present in the action. The technique we use permits to derive the set of Dyson-Schwinger equations in differential form. This proves essentially useful when exact solutions to the local equations are known. We show that all the formalism of the local theory involving the Dyson-Schwinger approach extends quite naturally to the non-local case. Using these methods, the spectrum of non-local theories become accessible and predictable in the non-perturbative regimes. We calculate the N-point correlation functions and predict the mass-gap in the spectrum arising purely from the self-interaction and the non-local scale M. The mass gap generated gets damped in the UV and it reaches conformal limit. We discuss some implications of our result in particle physics and cosmology.
We discuss Liouville field theory in the framework of Schwinger-Dyson approach and derive a functional equation for the three-point structure constant. We argue the existence of a second Schwinger-Dyson equation on the basis of the duality between the screening charge operators and obtain a second functional equation for the structure constant. We discuss the utility of the two functional equations to fix the structure constant uniquely.
We investigate the integrability of the Schwinger-Dyson equations in $c = 1 - \frac{6}{m(m+1)}$ string field theory which were proposed by Ikehara et al as the continuum limit of the Schwinger-Dyson equations of the matrix chain model. We show the continuum Schwinger-Dyson equations generate a closed algebra. This algebra contains Virasoro algebra but does not coincide with $W_{\infty}$ algebra. We include in the Schwinger-Dyson equations a new process of removing from the loop boundaries the operator ${\cal H}(σ)$ which locally changes the spin configuration. We also derive the string field H…
In these lectures I discuss Hopf algebras and Dyson-Schwinger equations. The lectures start with an introduction to Hopf algebras, followed by a review where Hopf algebras occur in particles physics. The final part of these lectures is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
The formalism for Schwinger-Dyson equations of Wilson loops, as developed by Eguchi, Weingarten, and Foerster, is extended to include connected correlators. These should allow one to study propagators and interactions of glueball fields without having to handle gauge dependent fields such as gluons an ghosts. Some steps towards a solution are made.
The Schwinger-Dyson equations connecting free and full Green functions and vertex parts widely were used in QED for finding full Green functions under different conditions. Undoubtedly, the same approach should leads to derivation of many useful information about other models of QFT. In this work we present some technique based on variational equations for effective action to derive many different Schwinger-Dyson type equations in QFT models such as nonlinear sigma model and scalar field theory.
We review a new prescription for calculating the Lagrangian path integral measure directly from the Hamiltonian Schwinger-Dyson equations. The method agrees with the usual way of deriving the measure in which one has to perform the path integration over momenta.
The Schwinger-Dyson equation of fermion self-energy in the linearization approximation is solved exactly in a theory with gauge and effective four-fermion interactions. Different expressions for the indepedent solutions which respectively submit to irregular and regular ultraviolet boundary condition are derived and expounded.
We derive a condition for a nontrivial solution of the Schwinger-Dyson equation to be accompanied by a Goldstone bound state. It implies that, for quenched planar QED, although chiral symmetry breaking occurs when there is a cutoff, the continuum limit fails to exist.
We consider the perturbative renormalization of the Schwinger-Dyson functional, which is the generating functional of the expectation values of the products of the composite operator given by the field derivative of the action. It is argued that this functional plays an important role in the topological Chern-Simons and BF quantum field theories. It is shown that, by means of the renormalized perturbation theory, a canonical renormalization procedure for the Schwinger-Dyson functional is obtained. The combinatoric structure of the Feynman diagrams is illustrated in the case of scalar models. F…
We study the Schwinger-Dyson equations of a matrix model for an open-closed string theory. The free energy with source terms for scaling operators satisfies the same Virasoro conditions as those of the pure closed string and is obtained from that of the pure closed string by giving appropriate nonvanishing background values to all of the sources.
The Schwinger-Dyson equations in the ladder approximation for $2D$ induced gravity coupled to fermions on a flat background are obtained in conformal gauge. A numerical study of these equations shows the possiblity of chiral symmetry breaking in this theory.
Implementing the requirement that a field theory be invariant under Schwinger-Dyson BRST symmetry in the Hamiltonian formalism, we show the equivalence between Hamiltonian and Lagrangian BRST-formalism at the path integral level. The Lagrangian quantum master equation is derived as a direct consequence of the Fradkin-Vilkovisky theorem in Hamiltonian BRST quantisation.
Counterparts of the Dyson-Schwinger equations for scalar QED in an external electromagnetic field are derived. Exact structure and diagrammatic interpretation of the corresponding mass and polarization operators are obtained. It is shown that in the presence of an external field the final equations include an exact three-photon interaction vertex.
We study quantum chromodynamics from the viewpoint of untruncated Dyson-Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson-Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.
A method for solving Schwinger-Dyson equations for the Green function generating functional of non-Abelian gauge theory is proposed. The method is based on an approximation of Schwinger-Dyson equations by exactly soluble equations. For the SU(2) model the first step equations of the iteration scheme are solved which define a gauge field propagator. Apart from the usual perturbative solution, a non-perturbative solution is found which corresponds to the spontaneous symmetry breaking and obeys infrared finite behaviour of the propagator.
Based on the path integral formalism, we rederive and extend the transverse Ward-Takahashi identities (which were first derived by Yasushi Takahashi) for the vector and the axial vector currents and simultaneously discuss the possible anomaly for them. Subsequently, we propose a new scheme for writing down and solving the Schwinger-Dyson equation in which the the transverse Ward-Takahashi identity together with the usual (longitudinal) Ward-Takahashi identity are applied to specify the fermion-boson vertex function. Especially, in two dimensional Abelian gauge theory, we show that this scheme …
We argue that the second-order gauge-invariant Schwinger-Dyson operator of a gauge theory is the Wheeler-DeWitt operator in the dual string theory. Using this identification, we construct a set of operators in the gauge theory that correspond to excitations of gravity in the bulk. We show that these gauge theory operators have the expected properties for describing the semiclassical local gravity theory.
The Frobenius-Schwinger-Dyson equations are a rather high-brow abstract nonsense type of equations describing n-point functions of arbitrarily high composite insertions. It is not clear how to solve or even find approximate solutions of these equations in general, but they are worth investigating because (a certain preferred type of) renormalization of composite insertions has been performed in advance: it just remains to find solutions given an action and renormalization conditions. Earlier work in this field involved only Gaussian actions or variable transformations thereof. In this work we …
We derive the Schwinger-Dyson/loop equations for the USp(2k) matrix model which close among the closed and open Wilson loop variables. These loop equations exhibit a complete set of the joining and splitting interactions required for the nonorientable Type I superstrings. The open loops realize the SO(2n_f) Chan-Paton factor and their linearized loop equations derive the mixed Dirichlet/Neumann boundary conditions.
Using functional derivatives with respect to the free correlation function we derive a closed set of Schwinger-Dyson equations in phi^4-theory. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the two- and four-point function together with their weights.
We present an algorithm for the derivation of Dyson-Schwinger equations of general theories that is suitable for an implementation within a symbolic programming language. Moreover, we introduce the Mathematica package DoDSE which provides such an implementation. It derives the Dyson-Schwinger equations graphically once the interactions of the theory are specified. A few examples for the application of both the algorithm and the DoDSE package are provided. The package can be obtained from physik.uni-graz.at/~mah/DoDSE.html.
In quantum field theory with three-point and four-point couplings the Feynman diagrams of perturbation theory contain momentum independent subdiagrams, the ``tadpoles'' and ``snails''. With the help of Dyson-Schwinger equations we show how these can be summed up completely by a suitable modification of the mass and coupling parameters. This reduces the number of diagrams significantly. The method is useful for the organisation of perturbative calculations in higher orders.
In three-dimensional QED, which is analyzed in the 1/$N$ expansion, we obtain a sufficient and necessary condition for a nontrivial solution of the Dyson-Schwinger equation to be chiral symmetry breaking solution. In the derivation, a normalization condition of the Goldstone bound state is used. It is showed that the existent analytical solutions satisfy this condition.
We derive an infinite sequence of Schwinger-Dyson equations for $N=1$ supersymmetric Yang-Mills theory. The fundamental and the only variable employed is the Wilson-loop geometrically represented in $N=1$ superspace: it organizes an infinite number of supersymmetrizing insertions into the ordinary Wilson-loop as a single entity. In the large $N_{c}$ limit, our equation becomes a closed loop equation for the one-point function of the Wilson-loop average.
In this short note we extend the results of Alfaro and Damgaard on the origin of antifields to theories with a gauge algebra that is open or reducible.
We formulate the Schwinger-Dyson equations in the ladder approximation for 2D induced quantum gravity with fermions using covariant gauges of harmonic type. It is shown that these equations can be formulated consistently in a gauge of Landau type (for negative cosmological constant). A numerical analysis of the equations hints towards the possibility of chiral symmetry breaking, depending on the value of the coupling constant.
Supersymmetric models, in most cases, suffer from the lack of non-perturbative techniques. Recently, an approach based on Dyson-Schwinger equations has been proposed for the massless Wess-Zumino model. In this case, the equations for the self-energies of the fields were solved with the strong ansatz to take them all equal. We show, by numerically solving the equations, that this is a too strong choice as also solutions with all different self-energies are acceptable and more generic. This could have interesting implications for supersymmetry breaking.
In Dyson-Schwinger studies of the Yang-Mills propagators the four-gluon interaction has been usually neglected due to the related technical difficulties with the associated two-loop terms and especially their renormalization. A possible scenario to renormalize these fully-dressed two-loop terms is presented. Preliminary results for the Landau gauge gluon propagator are shown. Implications for the gluon propagators in maximally Abelian gauge are discussed.
The dymamical chiral symmetry breaking in higher- derivative quantum gravity has been investigated on the flat background. The Schwinger- Dyson equations numerical solutions have been found in the ladder approximation. Both two- and four- dimensional cases have been considered. The dymamical fermion mass generation accompanied by the second- order phase transition has been shown to take place in a different gauges.
The two- and three-point functions and the four-gluon vertex of three-dimensional Yang-Mills theory are calculated from their Dyson-Schwinger equations and the 3PI effective action. Within a self-contained truncation various effects of truncating Dyson-Schwinger equations are studied. Estimates for the errors induced by truncations are derived from comparisons between results from different equations, comparisons with lattice results, and varying higher Green functions. The results indicate that the two-loop diagrams are important in the gluon propagator, where they are explicitly calculated, …
The general method for treating non-Gaussian wave functionals in the Hamiltonian formulation of a quantum field theory, which was previously developed and applied to Yang--Mills theory in Coulomb gauge, is generalized to full QCD. The Hamiltonian Dyson-Schwinger equations as well as the quark and gluon gap equations are derived and analysed in the IR and UV momentum regime. The back-reaction of the quarks on the gluon sector is investigated.
We present a way for calculating the Lagrangian path integral measure directly from the Hamiltonian Schwinger--Dyson equations. The method agrees with the usual way of deriving the measure, however it may be applied to all theories, even when the corresponding momentum integration is not Gaussian. Of particular interest is the connection that is made between the path integral measure and the measure in the corresponding 0-dimensional model. This allows us to uniquely define the path integral even for the case of Euclidean theories whose action is not bounded from below.
Using functional derivatives with respect to free propagators and interactions we derive a closed set of Schwinger-Dyson equations in quantum electrodynamics. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the $n$-point functions and the vacuum energy together with their correct weights.
We discuss the relationship between the boundary conditions of the Schwinger-Dyson equations and the phase diagram of a bosonic field theory or matrix model. In the thermodynamic limit, many boundary conditions lead to the same solution, while other boundary conditions have no such limit. The list of boundary conditions for which a thermodynamic limit exists depends on the parameters of the theory. The boundary conditions of a physical solution may be quite exotic, corresponding to path integration over various inequivalent complex contours.
Using the exact path integral solution of the Schwinger model -- a model where instantons are present -- the Dyson-Schwinger equation is shown to hold by explicit computation. It turns out that the Dyson-Schwinger equation separately holds for every instanton sector. This is due to Theta-invariance of the Schwinger model.
By means of a formal expression of the Cornwall-Jackiw-Tomboulis effective potential for quark propagator at finite temperature and finite quark chemical potential, we derive the real-time thermal Schwinger-Dyson equation for quark propagator in Landau gauge. Denote the inverse quark propagator by $A(p^2)\not{p}-B(p^2)$, we argue that, when temperature $T$ is less than the given infrared momentum cutoff $p_c$, $A(p^2)=1$ is a feasible approximation and can be assumed in discussions of chiral symmetry phase transition problem in QCD.
We solve the two-point function of the lowest dimensional scalar operator in the critical $ϕ^4$ theory on $4-ε$ dimensional real projective space in three different methods. The first is to use the conventional perturbation theory, and the second is to impose the crosscap bootstrap equation, and the third is to solve the Schwinger-Dyson equation under the assumption of conformal invariance. We find that the three methods lead to mutually consistent results but each has its own advantage.
In this work we present a number of generalizations of Wick's theorems on integrals with Gaussian weight to a larger class of weights which we call subgaussian. Examples of subgaussian contractions are that of Kac-Moody or Virasoro type, although the concept of a subgaussian weight does not refer a priori to two-dimensional field theory. The generalization was chosen in such a way that the contraction rules become a combinatorical way of solving the Schwinger-Dyson equation. In a still more general setting we prove a relation between solutions of the Schwinger-Dyson equation and a map $N$, whi…
Dyson--Schwinger equations are an established, powerful non-perturbative tool for QCD. In the Hamiltonian formulation of a quantum field theory they can be used to perform variational calculations with non-Gaussian wave functionals. By means of the DSEs the various $n$-point functions, needed in expectation values of observables like the Hamilton operator, can be thus expressed in terms of the variational kernels of our trial ansatz. Equations of motion for these variational kernels are derived by minimizing the energy density and solved numerically.
Coulomb gauge Yang-Mills theory is considered within the first order formalism. It is shown that the action is invariant under both the standard BRS transform and an additional component. The Ward-Takahashi identity arising from this non-standard transform is shown to be automatically satisfied by the equations of motion.
Using simple models in D=0+0 and D=0+1 dimensions we construct partition functions and compute two-point correlations. The exact result is compared with saddle-point approximation and solutions of Schwinger-Dyson equations. When integrals are dominated by more than one saddle-point we find Schwinger-Dyson equations do not reproduce the correct results unless the action is first transformed into dual variables.
We study combinatorial Dyson Schwinger equations, expressed in the Hopf algebra of words with a quasi shuffle product. We map them into an algebra of polynomials in one indeterminate L and show that the leading log expansion one obtains with such a mapping are simple power law like expression
We use the Schwinger-Dyson equations as a starting point to derive renormalization flow equations. We show that Katanin's scheme arises as a simple truncation of these equations. We then give the full renormalization group equations up to third order in the irreducible vertex. Furthermore, we show that to the fifth order, there exists a functional of the self-energy and the irreducible four-point vertex whose saddle point is the solution of Schwinger-Dyson equations.
We review the status of calculations of Yang-Mills Green functions from Dyson-Schwinger equations. The role of truncations is discussed and results for the four-gluon vertex are presented.
We establish a novel method of obtaining the gauge-invariant bilocal condensate of gluon fields from the solutions of the Dyson--Schwinger equations (DSE) which works in the infrared and ultraviolet limits. We present explicit results for SU(3) case in four dimensions, and compare to other related analytical and lattice results.
The variational Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge is investigated within the framework of the canonical recursive Dyson--Schwinger equations. The dressing of the quark propagator arising from the variationally determined non-perturbative kernels is expanded and renormalized at one-loop order, yielding a chiral condensate compatible with the observations.
In perturbative SU(N) Chern-Simons gauge theory, it is shown that the Schwinger-Dyson equations assume a quite simplified form. The generating functional of the correlation functions of the curvature is considered; it is demonstrated that the renormalized Schwinger-Dyson functional is related with the generating functional of the correlation functions of the gauge connections by some kind of duality transformation.
The gluon propagator plays a central role in determining the dynamics of QCD. In this work we demonstrate for BRST quantised QCD that the Dyson-Schwinger equation imposes significant analytic constraints on the structure of this propagator. In particular, we find that these constraints control the appearance of massless components in the gluon spectral density.
Using kinematic renormalization, we derive the Schwinger--Dyson equations for a massive Yukawa model and a Wess--Zumino-like one. Both have linear Schwinger--Dyson equations and a massive renormalized particle. An explicit solution is found in the IR limit of the non-supersymmetric case. Parametric solutions are found in the UV limit of the same model and for the supersymmertic model.
We study the Schwinger-Dyson equation associated with a chirality- changing fermion 4-point function in a strongly-coupled $U(1)$ gauge theory. After making appropriate simplifications, we solve the equation numerically via a relaxation method. Our analysis provides an estimate of the critical coupling and it gives some indication as to the general momentum dependence of the 4-point function.
We provide a rigorous construction of the Schwinger functions for the massive and massless Thirring models. We use the renormalization group approach, controlling its flow through the Ward-Takahashi identities combined with the Schwinger-Dyson equation. The role of the anomalies is discussed.
Probing Probability Geometry with Schwinger--Dyson Identities: Score Mismatch, Fisher Information, and Configurational Temperature
National Institute for Theoretical and Computational Sciences, School of Physics, and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand（威特沃特斯兰德大学理论计算科学国家研究所、物理学院及曼德尔斯坦理论物理研究所）；Brown Center for Theoretical Physics and Innovation, Department of Physics, Brown University（布朗大学物理系理论物理与创新中心）
In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.
An Abelian gauge theory with Chern-Simons term is investigated for a four-component Dirac fermion in 1+2 dimensions. The Ball-Chiu (BC) vertex function is employed to modify the rainbow-ladder approximation for the Schwinger-Dyson (SD) equation. We numerically solve the SD equation and show the gauge dependence for the resulting phase boundary for the parity and the chiral symmetry.
We propose an ansatz which solves the Dyson-Schwinger equation for the real scalar fields in Poincare patch of de Sitter space in the IR limit. The Dyson-Schwinger equation for this ansatz reduces to the kinetic equation, if one considers scalar fields from the principal series. Solving the latter equation we show that under the adiabatic switching on and then off the coupling constant the Bunch-Davies vacuum relaxes in the future infinity to the state with the flat Gibbons-Hawking density of out-Jost harmonics on top of the corresponding de Sitter invariant out-vacuum.
In this work we demonstrate that non-zero neutrino masses can be generated from gravitational interactions. We solve the Schwinger-Dyson equations to find a non-trivial vacuum thereby determining the scale of the neutrino condensate and the number of new particle degrees of freedom required for gravitationally induced dynamical chiral symmetry breaking. We show for minimal beyond the Standard Model particle content, the scale of the condensation occurs close to the Planck scale.
