Warsaw Basis for SMEFT Operators

• The Warsaw basis is a canonical operator basis that defines independent, gauge-invariant dimension-six SMEFT operators by systematically eliminating redundancies using EoMs, integration by parts, and Fierz identities.
• It organizes operators by field content—including bosonic, dipole, fermion–Higgs, and four-fermion sectors—thereby streamlining global SMEFT fits and phenomenological studies.
• The basis facilitates UV matching, renormalization group evolution, and on-shell amplitude mapping, and is implemented in software tools like Rosetta and CoDEx for practical SMEFT analysis.

The Warsaw basis is a canonical operator basis for organizing all independent, gauge-invariant, baryon- and lepton-number–conserving dimension-six operators within the Standard Model Effective Field Theory (SMEFT). Introduced by Grzadkowski et al., the Warsaw basis is favored in SMEFT analyses due to its systematic elimination of redundancies via equations of motion (EoMs), integration by parts (IBP), Fierz transformations, and Hermitian conjugation. It plays a central role in global SMEFT fits and phenomenological studies, forming the backbone for RG evolution, UV matching, basis translation, and amplitude classification frameworks (Cao et al., 2023, Falkowski et al., 2015, Ma et al., 2019, Bakshi et al., 2018, Aoude et al., 2019).

1. Structure and Construction of the Warsaw Basis

The Warsaw basis is defined by extending the Standard Model Lagrangian,

$$\mathcal{L}_\text{SMEFT} = \mathcal{L}_\text{SM} + \sum_i \frac{C_i}{\Lambda^2} O_i + \mathcal{O}(\Lambda^{-4}),$$

where the $O_i$ are dimension-six, SM gauge-invariant operators and $C_i$ are real Wilson coefficients. Construction is performed by first listing all possible monomials in the SM fields and their covariant derivatives that satisfy SM symmetries and mass dimension constraints. Operator redundancies from integration by parts, EoM relations, Fierz identities, and Hermiticity are systematically eliminated, retaining only independent entries.

The canonical set contains 59 baryon- and lepton-number–conserving operators at dimension six (additional baryon-number–violating operators can be included). Operators are grouped by field content and Lorentz structure: purely bosonic, fermion–Higgs, dipole, derivative–current, and four-fermion sectors (Falkowski et al., 2015, Bakshi et al., 2018, Ma et al., 2019).

2. Explicit Operator Content and Organization

The Warsaw basis operators are grouped by their field content:

• Purely bosonic:
  • Pure gauge: $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$.
  • Higgs-kinetic/Higgs self-couplings: $O_H = (H^\dagger H)^3$, $O_{H\Box} = (H^\dagger H)\Box(H^\dagger H)$, $O_{HD} = (H^\dagger D^\mu H)^*(H^\dagger D_\mu H)$.
  • Higgs–gauge: $O_{HG} = (H^\dagger H) G^A_{\mu\nu} G^{A\,\mu\nu}$, $O_{HW}$, $O_i$0, $O_i$1, and CP-odd partners.
• Fermion–Higgs–Yukawa-like: $O_i$2, $O_i$3, $O_i$4, involving bilinear Yukawa structures with Higgs insertions.
• Dipole-type: Eight operators such as $O_i$5, $O_i$6, $O_i$7, $O_i$8, $O_i$9, $C_i$0, $C_i$1, $C_i$2, containing $C_i$3 field structure (Ma et al., 2019).
• Higgs-current × fermion: $C_i$4 and related quark and lepton currents, and their $C_i$5 analogs (Cao et al., 2023).
• Four-fermion operators: Sector subdivided by Lorentz and $C_i$6 index structures: $C_i$7, $C_i$8, $C_i$9, and scalar–tensor types (Falkowski et al., 2015, Bakshi et al., 2018).

A summary table highlights the principal operator classes:

Sector	Number (B, L conserving)	Representative Operators
Purely bosonic	12	$$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$0, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$1, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$2, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$3, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$4
Fermion–Higgs (Yukawa)	3	$$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$5, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$6, $$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$7
Dipole ($$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$8)	8	$$O_G = f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu$$9, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$0, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$1, etc.
Higgs-current × fermion	8	$$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$2, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$3, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$4, etc.
Four-fermion	28	$$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$5, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$6, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$7, $$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$8
Total	59

For complete definitions, see (Falkowski et al., 2015) and (Bakshi et al., 2018).

3. Operator Redundancies and Basis Equivalence

All redundancies in the set are eliminated by enforcing:

• EoMs: Use of classical field equations—total derivatives and terms proportional to vanishing EoM operators are excluded.
• Integration by parts: Surface terms and total derivatives are removed, which can relate operators differing by derivative placement.
• Fierz identities: Rewrites involving fermion bilinears are utilized to avoid operator overcounting in four-fermion sectors.
• Hermiticity: Non-Hermitian operators are paired with their Hermitian conjugates where required.

This rigorous procedure ensures the linear independence and completeness of the Warsaw basis for SMEFT applications (Falkowski et al., 2015, Cao et al., 2023, Ma et al., 2019).

4. Basis Translations and Relation to SILH

Transformation between the Warsaw and other bases—such as SILH (Strongly Interacting Light Higgs)—is achieved via explicit linear maps derived from field redefinitions and EoMs. The basis translation rules are realized algorithmically within tools such as Rosetta, which encode both the operator content and the explicit mapping between Wilson coefficients (Falkowski et al., 2015).

For example, the mapping from Warsaw to SILH for the subset relevant to vector boson scattering involves relations such as

$$O_W = \epsilon^{IJK} W^I_\mu{}^\nu W^J_\nu{}^\rho W^K_\rho{}^\mu$$9

as in (Cao et al., 2023). If the mapping involves only operators in mutually orthogonal blocks (under the coupled-channel scattering analysis), bounds on Wilson coefficients can be translated entrywise; otherwise, constrained maximizations must be solved due to partial wave interference (Cao et al., 2023).

5. Applications: Unitarity, RG Evolution, and Matching

Unitarity Analysis

The Warsaw basis is a standard choice for computing unitarity bounds on dimension-six SMEFT operators. Unitarity conditions for $O_H = (H^\dagger H)^3$0 and $O_H = (H^\dagger H)^3$1 scattering are imposed by partial wave analysis: $O_H = (H^\dagger H)^3$2 and the analytic maximal coefficient bounds for each Warsaw operator can be tabulated accordingly (Cao et al., 2023).

Renormalization Group Evolution

The anomalous dimension matrix for the Warsaw basis, as computed by Jenkins–Manohar–Trott, provides the structure for one-loop RGEs: $O_H = (H^\dagger H)^3$3 with $O_H = (H^\dagger H)^3$4 exhibiting block/subsector structure corresponding to the operator classes. Dedicated tools such as CoDEx automate this evolution, enabling consistent running from the UV matching scale down to the weak scale (Bakshi et al., 2018).

Matching to UV Theories

The Covariant Derivative Expansion (CDE) method, encoded in packages like CoDEx, enables both tree and one-loop matching from arbitrary renormalizable or non-renormalizable BSM Lagrangians to the Warsaw basis. The results yield explicit expressions for all $O_H = (H^\dagger H)^3$5 at the high scale in terms of heavy field properties and couplings. RG evolution then flows these to relevant low scales (Bakshi et al., 2018).

6. On-shell Amplitudes and the Operator–Amplitude Map

Recent developments have established a transparent correspondence between the Warsaw operator basis and the space of unfactorizable on-shell amplitudes. Each operator in the Warsaw basis maps uniquely (often via simple linear relations) to contact amplitudes classified by external legs, quantum numbers, and symmetries. The amplitude-based approach elegantly eliminates operator redundancies and provides a powerful computational tool for SMEFT collider phenomenology. In particular, the on-shell basis recovers the canonical Warsaw set, modulo linear combinations in the Higgs-derivative and current sectors (Ma et al., 2019, Aoude et al., 2019).

7. Software Implementations and Translation Tools

The Warsaw basis is natively implemented in several computational tools facilitating SMEFT analyses:

• Rosetta: Implements the full set of Warsaw-basis dim-6 operators as a Python class, with translation dictionaries to other bases (SILH, etc.), and provides decorator-based basis-translation routines (Falkowski et al., 2015).
• CoDEx: Automates UV matching (tree/loop level) and RG running in the Warsaw basis (Bakshi et al., 2018).
• Amplitude Mappings: On-shell basis construction is algorithmically realized, matching amplitude contact terms to Warsaw operators for n-point SMEFT amplitude analysis (Ma et al., 2019).

These implementations ensure both code-level consistency and conceptual clarity for global SMEFT studies, collider data analyses, and BSM matching scenarios.

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References:

• (Cao et al., 2023)
• (Falkowski et al., 2015)
• (Bakshi et al., 2018)
• (Aoude et al., 2019)
• (Ma et al., 2019)