Algorithmic derivation of Dyson–Schwinger equations☆
Introduction
Correlation functions are the basic quantities in local quantum field theories and encode all physical information about the theory. They fulfill quantum equations of motion, conventionally called Dyson–Schwinger equations (DSEs) [1], [2] which are related among each other and form a set of infinitely many coupled equations. Derived from the translational invariance of the path integral they are genuinely non-perturbative and describe the physics of the system on all scales. This makes them a very useful tool for investigating aspects on which some alternative approaches fail. Perhaps the most prominent example is perturbation theory, which is not valid in the strong coupling regime. Since DSEs are likewise applicable in the weak coupling region they successfully extend the results of perturbation theory into the strong coupling domain. An alternative non-perturbative tool, which can be used complementary to DSEs, are Monte-Carlo lattice simulations. Due to the discretization of space–time they have their limits for very low and very high momenta, the former being restricted by the size of the lattice and the latter by finite lattice spacings. DSEs, in contrast, are formulated in continuous spacetime and allow to study also the analytic structure and the infrared regime which is particularly important in an asymptotically free but confining gauge theory like quantum chromodynamics (QCD).
However, DSEs also have their challenges. They represent strongly non-linear integral equations that are numerically involved. Moreover, as they form an infinite tower of equations, they have to be truncated. Recently it turned out in the context of Landau gauge QCD that the leading order truncation based only on the propagator DSEs can miss important qualitative features that are encoded in the equations for the vertices. In particular, the quark–gluon vertex provides a novel mechanism for confinement and chiral symmetry breaking [3], as well as anomalous mass generation [4] in QCD. Yet, the DSEs for the vertices become increasingly complicated and correspondingly hard to obtain algebraically. Another complication is given by the necessity of gauge fixing and the additional degrees of freedom and interactions arising from the corresponding constraints. In particular, in non-covariant gauges like Coulomb gauge or non-linear gauges like the maximally Abelian gauge, this increases the effort to derive the DSEs already at the propagator level considerably, see for example [5], [6], and calls for an algorithmic way to derive these fundamental equations. This is especially useful when working with actions that contain many different fields and interactions, as arise, e.g., when symmetries are not manifestly realized or in the case of necessary additional terms in the action. Examples for the latter are the four-ghost interaction required to ensure renormalizability in maximally Abelian gauge [7], [8] or generalized constructions of Lagrangians allowing additional terms as in ghost–antighost symmetric gauges [9], [10]. A convenient way to derive DSEs also simplifies the comparison of different gauges required to obtain a more gauge independent picture of the basic underlying mechanisms. Finally, it is particularly useful in the context of an IR analysis where the IR scaling, which is important for long-range properties like confinement, can be abstracted from mere power counting.
The aim of this paper is to present such an algorithmic derivation of Dyson–Schwinger equations. A similar aim has been extensively followed in perturbation theory where it resulted in the basically automatic computation of numerous physical processes to a given order, cf. e.g. [11], [12], [13], [14], [15]. Here we partially extend this idea to the non-perturbative regime were such an automatic solution of the created equations is surely beyond our scope. Instead we present an algorithm for the derivation of the equations that is suited for implementation into a symbolic programming language. This algorithm is presented below in Section 2 and is implemented in the Mathematica package DoDSE, which stands for Derivation of Dyson–Schwinger Equations. An example of how to use the algorithm is provided in Section 3. Details on the DoDSE package are presented in Section 4. Whereas a direct algebraic derivation can be quite a tedious task, with the symbolic and graphical notations employed, one can obtain DSEs for general actions with a relatively high number of interactions. Moreover, the presented algorithm operates directly on the level of the effective action and circumvents the tedious step to decompose connected into proper vertices necessary in a derivation on the level of the generating functional of full Green functions. We implemented this algorithm up to the diagrammatic level into DoDSE. From the interactions of the theory, given in symbolic form, the code derives DSEs to the desired order. The outcome are symbolic representations of the Feynman diagrams encoding their topological structure and their symmetry factors. The last step, which has to be done manually, to get the full algebraic form of the DSEs is the replacement of the symbolic form by the explicit integral expression involving proper and bare correlation functions. For some applications of DSEs it is not even necessary to process the symbolic equations even further since they can be used directly, as is, e.g., the case for scaling analyses. Finally, we note that the presented algorithm is in principle also applicable for the generation of a perturbative expansion by re-inserting the Dyson–Schwinger equations for dressed vertices and truncating at a given loop order.
Section snippets
Deriving Dyson–Schwinger equations in symbolic notation
The algorithm presented here involves several abstractions. The first is the use of a symbolic notation that enables us to keep expressions relatively short. Writing out the equations in full detail, i.e. Lorentz and internal group indices as well as all coordinates in position space or momenta, leads to expressions that can easily hide the underlying basic structure. Secondly, we employ a super-field formalism that includes all irreducible fields of the theory into a single reducible
An example: The ghost-gluon vertex in Landau gauge Yang–Mills theory
For the derivation of the ghost-gluon vertex in Landau gauge [21] we start with Fig. 3. As mentioned above, there are two DSEs for this vertex. These arise since the DSEs are derived from the invariance of the initial path integral under changes of the fields. The invariance for the individual fields in the theory yields different generating equations from Eq. (16) that can by appropriate functional differentiation generate topologically distinct DSEs for the same mixed correlation function.
Derivation of Dyson–Schwinger equations using Mathematica
As should have become clear the manual application of the algorithm becomes rather tedious for more complicated Green functions. The algorithm explained in Section 2 is perfectly suitable for an implementation into a symbolic programming language like Mathematica [22]. The functionality of the package DoDSE is that the user enters the interactions of the theory and then all desired DSEs are automatically generated from it. In special cases it may in addition be necessary to provide further
Summary
In this article we presented an algorithm to derive Dyson–Schwinger equations in a convenient way, in which one does not have to deal with the usual abundance of indices and integrals. This is achieved by graphical rules for performing derivatives that allow a quick and straightforward derivation even of higher vertex functions. We used this algorithm in the Mathematica package DoDSE that can give DSEs once the interactions of the theory are specified. It proves especially useful for theories
Acknowledgements
It is a pleasure to thank Axel Maas and Selym Villalba-Chavez for valuable discussions. M.Q.H. is supported by the Doktoratskolleg “Hadrons in Vacuum, Nuclei and Stars” of the Austrian science fund (FWF) under contract W1203-N08. K.S. acknowledges support from the FWF under contract M979-N16, and R.A. from the German research foundation (DFG) under contract AL 279/5-2.
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This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).