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.     

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.     

Color Glass Condensate in Schwinger–Keldysh QCD

Within the Schwinger–Keldysh representation of many-body QCD, it is shown that the leading quantum corrections to the strong classical color field are “classical” in the sense that the fluctuation field still obeys the classical Jacobi-field equation, while the quantum effects solely reside in the fluctuations of the initial field configurations. Within this context, a systematic derivation of the JIMWLK renormalization group equation is presented. A clear identification of the correct form of gauge transformation rules and the correct form of the matter-field Lagrangian in the Schwinger–Keldysh QCD is also presented.     

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     

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...     

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.     

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.     

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.     

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.     

Role of bare propagator poles in phenomenological Dyson–Schwinger type models

The generation of dressed meson–nucleon scattering-matrix poles is presented. A possible scenario for the interrelation of bare and dressed baryon poles is shown by using a particular version of coupled-channel Dyson–Schwinger type model. These findings are then applied to the Roper resonance, and the conclusion is drawn that it is dynamic in nature. A possible correlation between bare and dressed propagator poles on one side and the quantities of constituent quark-model calculations on the other side are discussed.     

Solving the Dyson–Schwinger equation around its first singularities in the Borel plane

The Dyson–Schwinger equation of the massless Wess–Zumino model is written as an equation over the anomalous dimension of the theory. Its asymptotic behavior is derived and the procedure to compute the perturbations of this asymptotic behavior is detailed. This procedure uses ill-defined objects. To solve this, the Dyson–Schwinger equation is rewritten for the Borel plane. It is shown that the illdefined procedure in the physical plane can be applied in the Borel plane. Other results obtained in the Borel plane are stated and the proof for one result is described.     

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.     

Langevin Equation of Collective Modes of Bose–Einstein Condensates in Traps

A quantum Langevin equation for the amplitudes of the collective modes in Bose–Einstein condensate is derived. The collective modes are coupled to a thermal reservoir of quasi-particles, whose elimination leads to the quantum Langevin equation. The dissipation rates are determined via the correlation function of the fluctuating force and are evaluated in the local-density approximation for the spectrum of quasi-particles and the Thomas–Fermi approximation for the condensate.I take great pleasure in dedicating this paper to Gregoire Nicolis on the occasion of his sixtieth birthday.     

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.