opbasis -- a Python package to derive minimal operator bases

TL;DR

This work presents opbasis, a Python package designed to automate the derivation of complete and minimal on-shell operator bases for lattice EFTs with discrete, non-continuum symmetries, addressing the challenge of lattice artifacts described by SymEFT. The approach combines model definitions of symmetries, generation of operator templates at a target mass-dimension, construction of an overcomplete basis, and systematic reduction to a minimal, linearly independent set using EOM and integration-by-parts relations. The authors illustrate the method with concrete examples, including the $O(a)$ axial-vector basis in Wilson QCD, the SymEFT basis for unrooted Staggered quarks, and an explicit $B^*(\mathbf{p})\pi(-\mathbf{p})$ interpolator, while showcasing the flexibility to incorporate custom Blocks and momentum twists. The tool significantly lowers the barrier for systematic lattice-artifact analyses and opens pathways to automate operator mixing studies across different lattice discretisations and extended EFTs.

Abstract

Finding a complete and yet minimal on-shell basis of operators of a given mass-dimension that are compatible with a specific set of transformation properties is the first step in any Effective Field Theory description. This step is the main bottleneck for systematic studies of leading logarithmic corrections to integer-power lattice artifacts in Symanzik Effective Field Theory targeting various local fields and lattice actions. The focus on discrete symmetry transformations in lattice field theory, especially reduced hypercubic spacetime symmetry with Euclidean signature, complicates the use of standard continuum field theory tools. Here, a new Python package is being presented that targets the typical lattice field-theorist's use cases. While the main target lies on continuum EFTs describing 4D non-Abelian lattice gauge theories, the applicability can be extended beyond Effective Field Theories. New discrete symmetries, twisted masses, or the introduction of boosts are just a few examples of possible extensions that can be easily implemented by the user. This should allow for a wider range of theories and applications beyond the initial focus of this package. The general functionality of the package is explained along the lines of three examples: The $\mathrm{O}(a)$ operator basis of the axial-vector in Wilson QCD, operator bases compatible with the symmetries of unrooted Staggered quarks as well as a pedestrian derivation of a $B^*(\mathbf{p})π(-\mathbf{p})$ operator with pseudo-scalar quantum numbers. Each example makes use of an increasing range of features and requires user-defined extensions show-casing the versatility of the package.

Figures (7)

• Figure 1: Vector
• Figure 2: Pseudo-scalar
• Figure 3: Schematic of the internal representation of LinearComb by various commuting terms carrying distinct factors and products of different instances of Trace and / or Bilinear. The latter are sketched in more detail in figures \ref{['fig:Bilin']}--\ref{['fig:otherTrace']}.
• Figure 4: Schematic of the three major parts forming a Bilinear namely the left and right flavour dressed with covariant derivatives (or fermion EOMs D0l and D0) stored in covl and covr respectively as well as everything in between those flavours stored in blocks. The main customisability offered by this package is due to the custom implementation of Block that can be stored in blocks. _AlgebraBlock is a protected implementation taking care of an element of the algebra optionally with covariant derivatives acting on it. The three possible insertions of elements of the algebra are the dummy generator Colour, the field strength F, and the gluon EOM DF.
• Figure 5: Schematic of the algebra Trace which only allows elements of _AlgebraBlock to be present. _AlgebraBlock is a protected implementation taking care of an element of the algebra optionally with covariant derivatives acting on it. The three possible insertions of elements of the algebra are the dummy generator Colour, the field strength F, and the gluon EOM DF.