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2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT

Brian Henning, Xiaochuan Lu, Tom Melia, Hitoshi Murayama

TL;DR

The paper develops a conformal-algebra–based Hilbert-series method to enumerate independent SMEFT operator bases while systematically handling equations of motion and integration by parts. By combining the plethystic exponential with group-integral projections and a differential-form perspective, it delivers dimension-by-dimension SMEFT counts for arbitrary numbers of fermion generations, including corrected dim-7 and dim-8 results and complete results up to dim $12$ (with dim $15$ counts discussed). It reveals new operators at dim $7$ for $N_f>1$ and 62 additional dim-$8$ operators for $N_f=1$ beyond prior analyses, and provides a fully automated framework extendable to other four-dimensional gauge theories. The work offers a scalable, general approach to EFT operator enumeration with direct implications for SMEFT phenomenology and model-building.

Abstract

In a companion paper, we show that operator bases for general effective field theories are controlled by the conformal algebra. Equations of motion and integration by parts identities can be systematically treated by organizing operators into irreducible representations of the conformal group. In the present work, we use this result to study the standard model effective field theory (SM EFT), determining the content and number of higher dimension operators up to dimension 12, for an arbitrary number of fermion generations. We find additional operators to those that have appeared in the literature at dimension 7 (specifically in the case of more than one fermion generation) and at dimension 8. (The title sequence is the total number of independent operators in the SM EFT with one fermion generation, including hermitian conjugates, ordered in mass dimension, starting at dimension 5.)

2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT

TL;DR

The paper develops a conformal-algebra–based Hilbert-series method to enumerate independent SMEFT operator bases while systematically handling equations of motion and integration by parts. By combining the plethystic exponential with group-integral projections and a differential-form perspective, it delivers dimension-by-dimension SMEFT counts for arbitrary numbers of fermion generations, including corrected dim-7 and dim-8 results and complete results up to dim (with dim counts discussed). It reveals new operators at dim for and 62 additional dim- operators for beyond prior analyses, and provides a fully automated framework extendable to other four-dimensional gauge theories. The work offers a scalable, general approach to EFT operator enumeration with direct implications for SMEFT phenomenology and model-building.

Abstract

In a companion paper, we show that operator bases for general effective field theories are controlled by the conformal algebra. Equations of motion and integration by parts identities can be systematically treated by organizing operators into irreducible representations of the conformal group. In the present work, we use this result to study the standard model effective field theory (SM EFT), determining the content and number of higher dimension operators up to dimension 12, for an arbitrary number of fermion generations. We find additional operators to those that have appeared in the literature at dimension 7 (specifically in the case of more than one fermion generation) and at dimension 8. (The title sequence is the total number of independent operators in the SM EFT with one fermion generation, including hermitian conjugates, ordered in mass dimension, starting at dimension 5.)

Paper Structure

This paper contains 13 sections, 54 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Computing the Hilbert series
  4. A viewpoint of IBP redundancy using differential forms
  5. Hilbert series for the SM EFT
  6. Selected results
  7. Operator bases at dimension 7 and 8
  8. Counting at higher dimensions with arbitrary $N_{f}$
  9. Discussion
  10. Explicit comparison with Lehman:2015coa at dim-8
  11. A quick example: operator class $H^4 X\mathcal{D}^2$
  12. LM over counts IBP relations
  13. The list of dim-8 operators with two and three derivatives

Figures (1)

  • Figure 1: Growth of the number of independent operators in the SM EFT up to mass dimension 15. Points joined by the lower solid line are for one fermion generation; those joined by the upper solid line are for three generations. Dashed lines are to guide the eye to the growth of the even and odd mass dimension operators in both cases.