General Dyson–Schwinger Equations and Systems

We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faà di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.     

Systems of Linear Dyson–Schwinger Equations

Mathematical Physics, Analysis and Geometry - In this study, we consider a quantum waveguide with random boundary conditions . Precisely we consider Laplace operator restricted to a two dimensional...     

An Efficient Method for the Solution of Schwinger–Dyson Equations for Propagators

Efficient computation methods are devised for the perturbative solution of Schwinger–Dyson equations for propagators. I show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess–Zumino model and the f₆³{\phi_6^3} complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.     

Delta Properties in the Rainbow-Ladder Truncation of Dyson–Schwinger Equations

We present a calculation of the three-quark core contribution to nucleon and Δ-baryon masses and Δ electromagnetic form factors in a Poincaré-covariant Faddeev approach. A consistent setup for the dressed-quark propagator, the quark–quark, quark–’diquark’ and quark–photon interactions is employed, where all ingredients are solutions of their respective Dyson–Schwinger or Bethe–Salpeter equations in a rainbow-ladder truncation. The resulting Δ electromagnetic form factors concur with present experimental and lattice data.     

Higher Loop Corrections to a Schwinger–Dyson Equation

We consider the effects of higher loop corrections to a Schwinger–Dyson equation for propagators. This is made possible by the efficiency of the methods we developed in preceding works, still using the supersymmetric Wess–Zumino model as a laboratory. We obtain the dominant contributions of the three and four-loop primitive divergences at high order in perturbation theory, without the need for their full evaluations. Our main conclusion is that the asymptotic behavior of the perturbative series of the renormalization function remains unchanged, and we conjecture that this will remain the case for all finite order corrections.     

Ghost and Gluon Propagators at Finite Temperatures within a Rainbow Truncation of Dyson–Schwinger Equations

JETP Letters - The finite-temperature behavior of ghost and gluon propagators is investigated within an approach based on the rainbow truncated Dyson–Schwinger equations in Landau gauge. In...     

Complex Langevin equations and Schwinger–Dyson equations

Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger–Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. The relevance to the study of quantum field theory solution space is discussed.     

Resurgent transseries & Dyson–Schwinger equations

We employ resurgent transseries as algebraic tools to investigate two self-consistent Dyson–Schwinger equations, one in Yukawa theory and one in quantum electrodynamics. After a brief but pedagogical review, we derive fixed point equations for the associated anomalous dimensions and insert a moderately generic log-free transseries ansatz to study the possible strictures imposed. While proceeding in various stages, we develop an algebraic method to keep track of the transseries’ coefficients. We explore what conditions must be violated in order to stay clear of fixed point theorems to eschew a unique solution, if so desired, as we explain. An interesting finding is that the flow of data between the different sectors of the transseries shows a pattern typical of resurgence, i.e. the phenomenon that the perturbative sector of the transseries talks to the nonperturbative ones in a one-way fashion. However, our ansatz is not exotic enough as it leads to trivial solutions with vanishing nonperturbative sectors, even when logarithmic monomials are included. We see our result as a harbinger of what future work might reveal about the transseries representations of observables in fully renormalised four-dimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why one should not expect otherwise.     

Fermion family recurrences in the Dyson–Schwinger formalism

We study the multiple solutions of the truncated propagator Dyson–Schwinger equation for a simple fermion theory with Yukawa coupling to a scalar field. Upon increasing the coupling constant g, other parameters being fixed, more than one non-perturbative solution breaking chiral symmetry becomes possible and we find these numerically. These “recurrences” appear as a mechanism to generate different fermion generations as quanta of the same fundamental field in an interacting field theory, without assuming any composite structure. The number of recurrences or flavors is reduced to the question of the value of the Yukawa coupling, and it has no special profound significance in the standard model. The resulting mass function can have one or more nodes and the measurement that potentially detects them can be thought of as a collider-based test of the virtual dispersion relation for the charged lepton member of each family. This requires the three independent measurements of the charged lepton’s energy, three-momentum and off-shellness. We illustrate how this can be achieved for the (more difficult) case of the tau lepton. PACS 12.15.Ff; 11.30.Rd     

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ³ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson–Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator–coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.     

Higher Order Corrections to the Asymptotic Perturbative Solution of a Schwinger–Dyson Equation

Building on our previous works on perturbative solutions to a Schwinger–Dyson for the massless Wess–Zumino model, we show how to compute 1/n corrections to its asymptotic behavior. The coefficients are analytically determined through a sum on all the poles of the Mellin transform of the one-loop diagram. We present results up to the fourth order in 1/n as well as a comparison with numerical results. Unexpected cancellations of zetas are observed in the solution, so that no even zetas appear and the weight of the coefficients is lower than expected, which suggests the existence of more structure in the theory.     

Collective Perspective on Advances in Dyson–Schwinger Equation QCD

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-Q2; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.     

Analytic Results in the Position-Dependent Mass Schrdinger Problem

We investigate the Schro¨dinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the(nontrivial) position-dependent mass V(x) = 0 case whose solutions are hypergeometric functions in tanh2x. Then, we consider an external hyperbolic-tangent potential. We show that the efective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form V(x) = V0 sinh2x.     

Solving the Dyson–Schwinger Equation at Zero and Finite Temperatures

The properties of the solutions of the truncated Dyson–Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail.     

Schwinger–Dyson equation boundary conditions induced by ETC radiative corrections

The technicolor (TC) Schwinger–Dyson equations (SDE) should include radiative corrections induced by extended technicolor (ETC) interactions when TC is embedded into a larger theory including also QCD. These radiative corrections couple the different strongly interacting Dyson equations. We discuss how the boundary conditions of the coupled SDE system are modified by these corrections, and verify that the ultraviolet behavior of the self-energies are described by a function that decreases logarithmically with momentum.     

Determination of Phase Space Spectra of the Generalized Chiral Schwinger model with the Lorentz invariant Mass like Term for the Choice a − r2 = 0

We have considered the generalized version of chiral schwinger model with the Lorentz covariant masslike term for gauge field with the choice a ? r² =?0. We carry out the quantization by the canonical Dirac method of both the gauge-invariant and non-invariant version of this model to determine the phase space structure. Therefore we have shown that the gauge invariant theory has the same physical spectrum as that of the original gauge noninvariant formulation.

     

Potentials with Convergent Schwinger–DeWitt Expansion

Convergence of the Schwinger–DeWittexpansion for the evolution operator kernel for specialclass of potentials is studied. It is shown that thisexpansion, which is in the general case asymptotic,converges for the potentials considered (widely used, inparticular, in one-dimensional many-body problems), andthat convergence takes place only for definite discretevalues of the coupling constant. For other values of the charge, a divergent expansiondetermines the kernels having essential singularity atthe origin (beyond the usual -function). If oneconsiders only this class of potentials, then one can avoid many problems connected withasymptotic expansions, and one gets a theory withdiscrete values of the coupling constant that is incorrespondence with the discreteness of charge innature. This approach can be applied to quantum fieldtheory.