[Paper Review] ..., 83106786, 114382724, 1509048322, 2343463290, 27410087742, ... Efficient Hilbert Series for Effective Theories

Coenraad Marinissen, Rahn, Rudi+1 more|Utrecht University Repository (Utrecht University)|Apr 20, 2020

Matrix Theory and Algorithms4 citations

TL;DR

This paper presents Eco, a highly efficient Form-based implementation of the Hilbert series method for counting independent operators in effective field theories, particularly the Standard Model Effective Theory (SMEFT) and its extensions. By optimizing the algorithmic structure and leveraging Form's computational capabilities, the method computes the number of dimension-20 operators in SMEFT in under an hour on a single CPU core, enabling rapid model-building and phenomenological analysis for high-dimensional operators with symmetries and gravity included.

ABSTRACT

We present an efficient algorithm for determining the Hilbert series of an effective theory and provide a companion code called ECO (Efficient Counting of Operators) in FORM. For example, the Hilbert series for the dimension 15 operators in the Standard Model Effective Theory (SMEFT) can be obtained in a minute on a single CPU core. While our implementation focusses on SMEFT, we allow for a flexible user input of the light degrees of freedom. Furthermore, gravity, as well as additional U(1) global or gauge symmetries can be included.

Motivation & Objective

• To develop a fast, scalable algorithm for computing Hilbert series in effective field theories, especially for high-dimensional operator bases.
• To address the computational bottleneck in counting independent operators in SMEFT and its extensions, such as GRSMEFT and models with additional U(1) symmetries.
• To provide a user-friendly, extensible codebase (Eco) that supports flexible field content, including gravity and global symmetries, for use in model-building and phenomenology.

Proposed method

• The method employs the Hilbert series formalism to count gauge- and Lorentz-invariant operators in EFTs, using characters and group representation theory to project onto singlet states.
• It incorporates derivatives via plethystic exponentials and handles integration-by-parts and equations-of-motion redundancies using symmetry and representation-theoretic projections.
• The implementation uses Form, a high-performance symbolic manipulation system, to accelerate tensor and group-theoretic contractions, enabling orders-of-magnitude speedups over prior approaches.
• The code allows users to define fields with their quantum numbers, including spin, gauge charges, and additional U(1) symmetries (e.g., baryon number), via a modular input syntax.
• The algorithm supports inclusion of gravity via the Rarita-Schwinger field and can be extended to include additional light scalars, vectors, and fermions.
• The Hilbert series is computed as a generating function in terms of field and derivative counts, with results organized by operator dimension and field content.

Experimental results

Research questions

• RQ1 How can the Hilbert series computation for effective field theories be accelerated to handle high-dimensional operator bases efficiently?
• RQ2 What is the number of independent operators in SMEFT up to dimension 20, including multiple generations and symmetries?
• RQ3 How does the inclusion of gravity (GRSMEFT) or additional U(1) global symmetries (e.g., B-L) affect the operator count and computational complexity?
• RQ4 Can the Hilbert series method be efficiently implemented in a high-performance symbolic system like Form to enable practical use in model-building?
• RQ5 What is the impact of fermion generation count and field statistics (boson/fermion) on the operator spectrum in SMEFT?

Key findings

• The Hilbert series for dimension-15 SMEFT operators with one generation can be computed in under a minute on a single CPU core, with the full basis up to dimension 20 computed in under an hour.
• For three generations, the number of dimension-20 operators in SMEFT is 27,410,087,742, computed in approximately 3,410 seconds (57 minutes) on a single core.
• The method reproduces known results for SMEFT up to dimension 15 and extends them, including the 22,800 operators in the Two Higgs Doublet Model at dimension 6.
• The inclusion of gravity (GRSMEFT) increases computation time by a factor of about two compared to SMEFT, but the method remains practical for high-dimensional analyses.
• The code correctly identifies that all even-dimension operators in SMEFT up to dimension 10 conserve B-L for one generation, while odd-dimension operators do not, with explicit examples like h⁴ℓ₁²ℓ₂² violating B-L for multiple generations.
• The Eco code enables efficient counting of B-L conserving operators, with a 10% speedup over standard SMEFT computation due to symmetry filtering.