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Effective description of general extensions of the Standard Model: the complete tree-level dictionary

J. de Blas, J. C. Criado, M. Perez-Victoria, J. Santiago

TL;DR

This work delivers a complete tree-level UV/IR dictionary for the SM EFT (SMEFT) up to dimension $d=6$ by classifying all heavy fields with spins $\le 1$ that can couple linearly to SM fields, including non-renormalizable UV interactions. It provides explicit Wilson coefficients for SMEFT operators in the Warsaw basis, organized top-down by heavy multiplets and bottom-up by operators, and accounts for mixed-spin contributions and Higgs‑sector renormalizations. The dictionary enables model‑independent constraints and precise mappings from UV parameters (masses and couplings) to low-energy observables, illustrated through an LHCb LFU anomaly example. It thereby bridges general UV completions with SMEFT phenomenology, while noting that loop-level matching and strongly coupled UV completions require additional considerations.

Abstract

We compute all the tree-level contributions to the Wilson coefficients of the dimension-six Standard-Model effective theory in ultraviolet completions with general scalar, spinor and vector field content and arbitrary interactions. No assumption about the renormalizability of the high-energy theory is made. This provides a complete ultraviolet/infrared dictionary at the classical level, which can be used to study the low-energy implications of any model of interest, and also to look for explicit completions consistent with low-energy data.

Effective description of general extensions of the Standard Model: the complete tree-level dictionary

TL;DR

This work delivers a complete tree-level UV/IR dictionary for the SM EFT (SMEFT) up to dimension by classifying all heavy fields with spins that can couple linearly to SM fields, including non-renormalizable UV interactions. It provides explicit Wilson coefficients for SMEFT operators in the Warsaw basis, organized top-down by heavy multiplets and bottom-up by operators, and accounts for mixed-spin contributions and Higgs‑sector renormalizations. The dictionary enables model‑independent constraints and precise mappings from UV parameters (masses and couplings) to low-energy observables, illustrated through an LHCb LFU anomaly example. It thereby bridges general UV completions with SMEFT phenomenology, while noting that loop-level matching and strongly coupled UV completions require additional considerations.

Abstract

We compute all the tree-level contributions to the Wilson coefficients of the dimension-six Standard-Model effective theory in ultraviolet completions with general scalar, spinor and vector field content and arbitrary interactions. No assumption about the renormalizability of the high-energy theory is made. This provides a complete ultraviolet/infrared dictionary at the classical level, which can be used to study the low-energy implications of any model of interest, and also to look for explicit completions consistent with low-energy data.

Paper Structure

This paper contains 30 sections, 90 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. General Extensions of the Standard Model
  3. Effective Lagrangian and Tree-Level Matching
  4. Results of the Matching: User Guide
  5. Example: Interpretation of LHCb Anomalies
  6. Conclusions
  7. Explicit BSM Effective Lagrangian
  8. Standard Model Lagrangian
  9. New Scalars
  10. New Fermions
  11. New Vectors
  12. Mixed Terms
  13. Dimension-Six Basis
  14. Operators Generated by Each Field Multiplet
  15. Complete Contributions to Wilson Coefficients
  16. ...and 15 more sections

Figures (1)

  • Figure 1: Feynman diagrams contributing to $\mathcal{L}_{\mathrm{eff}}$ to dimension $n=6$. Non-equivalent permutations of the arrow directions shown here should be considered as well.