Resurgent transseries & Dyson–Schwinger equations
Abstract
We employ resurgent transseries as algebraic tools to investigate two selfconsistent 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 logfree 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 oneway 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 fourdimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why onemore »
 Authors:
 Publication Date:
 OSTI Identifier:
 22617383
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics; Journal Volume: 372; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FOURDIMENSIONAL CALCULATIONS; GREEN FUNCTION; MATHEMATICAL SOLUTIONS; QUANTUM ELECTRODYNAMICS; RENORMALIZATION
Citation Formats
Klaczynski, Lutz, Email: klacz@mathematik.huberlin.de. Resurgent transseries & Dyson–Schwinger equations. United States: N. p., 2016.
Web. doi:10.1016/J.AOP.2016.06.003.
Klaczynski, Lutz, Email: klacz@mathematik.huberlin.de. Resurgent transseries & Dyson–Schwinger equations. United States. doi:10.1016/J.AOP.2016.06.003.
Klaczynski, Lutz, Email: klacz@mathematik.huberlin.de. Thu .
"Resurgent transseries & Dyson–Schwinger equations". United States.
doi:10.1016/J.AOP.2016.06.003.
@article{osti_22617383,
title = {Resurgent transseries & Dyson–Schwinger equations},
author = {Klaczynski, Lutz, Email: klacz@mathematik.huberlin.de},
abstractNote = {We employ resurgent transseries as algebraic tools to investigate two selfconsistent 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 logfree 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 oneway 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 fourdimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why one should not expect otherwise. This paper is considerably selfcontained. Readers with little prior knowledge are let in on the basic reasons why perturbative series in quantum field theory eventually require an upgrade to transseries. Furthermore, in order to acquaint the reader with the language utilised extensively in this work, we also provide a concise mathematical introduction to gridbased transseries.},
doi = {10.1016/J.AOP.2016.06.003},
journal = {Annals of Physics},
number = ,
volume = 372,
place = {United States},
year = {Thu Sep 15 00:00:00 EDT 2016},
month = {Thu Sep 15 00:00:00 EDT 2016}
}

Assuming an ansatz for the vertex function and the electron propagator suggested by the Ward identity we solve the DysonSchwinger equations in quantum electrodynamics in the infrared domain. The nonperturbative results obtained in this way are in agreement with perturbation theory. In our approach the loop integrals are finite. Our procedure is selfconsistent, no divergences arise during calculations, and only a finite renormalization has to be carried out.

Selfconsistent solution of the SchwingerDyson equations for the nucleon and meson propagators
The SchwingerDyson equations for the nucleon and meson propagators are solved selfconsistently in an approximation that goes beyond the HartreeFock approximation. The traditional approach consists in solving the nucleon SchwingerDyson equation with bare meson propagators and bare mesonnucleon vertices; the corrections to the meson propagators are calculated using the bare nucleon propagator and bare nucleonmeson vertices. It is known that such an approximation scheme produces the appearance of ghost poles in the propagators. In this paper the coupled system of SchwingerDyson equations for the nucleon and the meson propagators are solved selfconsistently including vertex corrections. The interplay of selfconsistency andmore » 
Dynamical chiral symmetry breaking, Goldstone`s theorem, and the consistency of the SchwingerDyson and BetheSalpeter equations
A proof of Goldstone`s theorem is given that highlights the necessary consistency between the exact SchwingerDyson equation for the fermion propagator and the exact BetheSalpeter equation for fermionantifermion 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 fermiongaugeboson vertex function is discussed, and a vertex model is given which satisfies both the PCACmore » 
DysonSchwinger equations and hadronic observables in QCD
The DysonSchwinger equations (DSEs) are a tower of coupled integral equations that relate the Green functions of QCD to one another and include the equation for the quark selfenergy, the analogue of the gap equation in superconductivity; the BetheSalpeter equation, the solution of which yields meson bound state amplitudes; and the covariant Fadde`ev equation, whose solution provides baryon bound state amplitudes. Solving the tower of DSEs provides the solution of QCD; a field theory being completely defined when all of its npoint functions are known. The nonperturbative DSE approach is being developed as a complement and computationally lessintensive alternative tomore » 
Calculation of Hadron Form Factors from Euclidean DysonSchwinger Equations
We apply Euclidean time methods to phenomenological DysonSchwinger models of hadrons. By performing a Fourier transform of the momentum space correlation function to Euclidean time and by taking the large Euclidean time limit, we project onto the lightest onmassshell hadron for given quantum numbers. The procedure, which actually resembles lattice gauge theory methods, allows the extraction of moments of structure functions, moments of lightcone wave functions, and form factors without {ital ad hoc} extrapolations to the onmassshell points. We demonstrate the practicality of the procedure with the example of the pion form factor. {copyright} {ital 1997} {ital The American Physicalmore »