Operator Bases and Field Redefinitions

An EFT basis is not merely a list of symmetry-allowed monomials. At a fixed field content and order, it is a normalized set of representatives for local operators modulo relations that do not change the declared physical object. Integration by parts, leading equations of motion, algebraic identities, and perturbative field redefinitions remove redundant directions; dimensional regularization can then require evanescent directions for loop closure. This chapter separates those operations so that a count, an explicit basis, an anomalous-dimension calculation, and a translation file cannot be mistaken for one another.

Enter through the reduction problem

If the immediate question is…	Start with…	Leave with…
What mathematical object is an EFT basis?	From Operator Lists to Independent Bases	An ambient invariant space, a declared quotient, explicit representatives, and independent spanning checks
Why may total derivatives and leading equations of motion be removed?	Integration by Parts and Equation-of-Motion Redundancy	An order-consistent coefficient map with boundary, source, and off-shell qualifications
When does a field redefinition preserve observables?	Local Field Redefinitions and the Equivalence Theorem	A perturbatively invertible change of variables, Jacobian treatment, and S-matrix check
How are Lorentz, gauge, flavor, and discrete symmetries imposed?	Representation and Spurion Constraints on Operator Bases	A singlet-construction procedure with spurion assumptions declared
How are flavor labels, conjugates, Hermiticity, and CP counted?	Flavor, Hermiticity, and CP Bookkeeping	Independent coefficient tensors with no hidden conjugation or flavor double count
Which Fierz identities are legal in the chosen dimension?	Fierz Relations and Dimension-Specific Identities	A dimension-labeled reduction and the evanescent directions it creates away from that dimension
Why retain operators that vanish in four dimensions?	Evanescent Operators, Finite Renormalization, and RG Closure	A physical-plus-evanescent renormalization system and a finite projection back to observables
What does a Hilbert series prove?	Hilbert-Series Counting and Completeness Diagnostics	A graded count with explicit construction, normalization, and independence obligations still visible
How do local contact amplitudes encode operator classes?	On-Shell Amplitude Bases and the Operator Correspondence	A contact-term basis modulo on-shell and momentum-conservation relations, with color and regulator caveats
How are published bases and anomalous dimensions translated?	Basis Translation, Scheme Dependence, and Reproducibility	A rank-checked coefficient map, finite scheme data, checksums, and round-trip tests

The natural linear route is the order shown. Readers who need only a tree-level on-shell basis can stop before evanescent operators, but loop matching in dimensional regularization cannot: a four-dimensional reduction performed too early can discard $O(\epsilon)$ structures whose product with a $1/\epsilon$ pole is finite.

The operator quotient

Fix the spacetime dimension, fields and representations, conserved charges, polynomial or derivative order, and any spurion assignments. Let $\mathcal V_{d,\mathbf q}^{\mathrm{inv}}$ be the finite-dimensional vector space of local invariant monomials in that sector. For an action or on-shell S-matrix problem without physical boundary terms, define the redundancy space

$$\mathcal R = \mathcal R_{\mathrm{IBP}} +\mathcal R_{\mathrm{EOM}} +\mathcal R_{\mathrm{alg}},$$

where the three terms are generated by total derivatives, leading-equation-of-motion insertions, and algebraic identities such as Fierz, Schouten, Bianchi, and group-tensor relations. The physical operator classes are

$$\mathcal B_{d,\mathbf q} \cong \frac{\mathcal V_{d,\mathbf q}^{\mathrm{inv}}} {\mathcal R}.$$

This quotient states what is equivalent; it does not choose representatives or their normalization. An explicit basis is a section of the quotient map: one operator is selected for each class, with coefficients defined by a Lagrangian convention such as $\mathcal L_{\mathrm{EFT}}=C^T O$. Spanning means every candidate reduces to those representatives. Independence means no nonzero linear combination of the representatives lies in $\mathcal R$.

Henning, Lu, Melia, and Murayama formulate integration by parts and equations of motion as equivalence relations and show how polynomial-ring quotients encode both counts and representatives in Henning et al. 2016, §§ 1–2, preprint pp. 1–7, Open PDF. Their construction also illustrates why a Hilbert-series coefficient gives a dimension of a graded quotient, not by itself a normalized list suitable for matching.

The quotient depends on the object being preserved. Total derivatives can contribute on a physical boundary. Equation-of-motion operators alter off-shell Green functions and source contact terms even when they vanish in on-shell amplitudes. A basis for background-field matching can therefore be larger than a reduced on-shell basis, with an explicit projection connecting them.

A scalar reduction in one line

For a real scalar with

$$\mathcal L_0 = \frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4,$$

consider the dimension-six candidates

$$O_1=\phi^6, \qquad O_2=\phi^2(\partial\phi)^2, \qquad O_3=\phi^3\Box\phi.$$

The total derivative

$$\partial_\mu(\phi^3\partial^\mu\phi) =3\phi^2(\partial\phi)^2+\phi^3\Box\phi$$

gives $O_3\simeq-3O_2$ in the action when the boundary term vanishes. The leading equation of motion,

$$\Box\phi=-m^2\phi-\frac{\lambda}{6}\phi^3,$$

then gives

$$O_3\simeq-m^2\phi^4-\frac{\lambda}{6}\phi^6, \qquad O_2\simeq\frac{m^2}{3}\phi^4+\frac{\lambda}{18}\phi^6.$$

Thus this three-element list contains only one new dimension-six on-shell class after lower-dimensional parameter shifts are included. The reduction is order dependent: if these operators enter with $1/\Lambda^2$, the equation of motion used here comes from $\mathcal L_0$. Feeding the dimension-six correction back into the equation of motion would generate dimension-eight changes and must be tracked there.

The same relation follows from the perturbative field redefinition $\phi\mapsto\phi+a\phi^3/\Lambda^2$, which removes one derivative operator and shifts the $\phi^4$ and $\phi^6$ coefficients. Arzt proves that local equation-of-motion reductions can be used consistently in loop calculations and displays this scalar change of variables in Arzt 1995, §§ 1–2, preprint pp. 2–7, Open PDF. The underlying S-matrix equivalence for suitable local changes of variables goes back to Kamefuchi, O’Raifeartaigh, and Salam 1961, pp. 529–549.

Five different deliverables

Deliverable	What it establishes	What it does not establish
Count	$\dim\mathcal B$ in a declared grading and symmetry sector	Explicit representatives, normalization, or loop closure
Construction	A spanning set of representative operators	Independence unless a rank or normal-form check is supplied
Normalization	Field, generator, symmetry-factor, flavor, and coefficient conventions	Equivalence to another basis without an explicit map
Renormalization closure	Counterterms remain in the physical, redundant, and evanescent space retained for the calculation	Scheme-independent individual coefficients
Translation	Operators, coefficients, parameters, and finite renormalizations map between conventions	Correctness unless rank, observable, and round-trip checks pass

A complete tree-level basis needs the first three. A loop-ready basis needs all five. The renormalization space can be larger than the final physical quotient because equation-of-motion, BRST-exact, and evanescent insertions may be needed as intermediate counterterms.

In $D=4-2\epsilon$, an identity that is exact only at $D=4$ defines an evanescent operator $E=O_a-f_{ab}O_b$ rather than the zero vector. An $O(\epsilon)$ tree matrix element of $E$ can multiply a UV pole and generate a finite physical term. Eliminating $E$ before subtraction therefore changes the finite scheme. Dugan and Grinstein explain how evanescent operators can be projected away only with compensating finite renormalizations in Dugan and Grinstein 1991, pp. 239–244.

A reproducible basis construction

1. Declare the spacetime dimension, field content, gauge and global representations, flavor range, spurions, derivative or mass dimension, and whether the target is an action, Green-function basis, or on-shell amplitude.
2. Generate Lorentz and internal singlets before using integration by parts or equations of motion. Record conjugation, Hermiticity, and CP conventions at this stage.
3. Build the relation matrix for total derivatives, leading equations of motion, algebraic identities, and any permitted perturbative field redefinitions.
4. Compute a rank, row-reduced form, Gröbner normal form, or independent on-shell contact-amplitude basis. The quotient dimension and the number of representatives must agree.
5. Fix operator and coefficient normalizations, including factorials, group generators, flavor tensor symmetries, and powers of couplings or heavy scales.
6. For dimensional regularization, append a declared evanescent sector and verify counterterm closure before projecting to a four-dimensional physical basis.
7. Translate to every comparison basis with a matrix of known rank. Transform coefficients contragrediently and test an invariant amplitude or correlator.
8. Perform a round trip, preserve exact rational or symbolic entries where possible, and attach software versions and checksums to generated maps.

This construction exposes where automation helps. Symbolic algebra can generate singlets, reduce relation matrices, and verify round trips; it cannot decide the physical boundary conditions, the matching object, the evanescent prescription, or which finite scheme a published coefficient uses.

Scope and connections

Matching, Decoupling, and Threshold Evolution supplies coefficients in a generating or reduced basis and must state which one. Composite Operators and Mixing develops the renormalization matrices whose closure is tested here. Model-wide inventories such as SMEFT, HEFT, LEFT, heavy-particle theories, and gravity EFT are organized in Effective-Theory Architecture Atlas; this chapter develops the reusable construction and translation methods rather than reproducing a catalog.

The next chapter, Modes, Factorization, and Multiscale RG, introduces homogeneous momentum modes and factorized operator structures. Its operators still require the quotient and normalization discipline developed here, but overlap subtraction, rapidity evolution, and resonant modes are separate problems.

References

• Arzt, Christopher. “Reduced Effective Lagrangians.” Physics Letters B 342, no. 1–4 (1995): 189–195. DOI; arXiv
• Dugan, Michael J., and Benjamin Grinstein. “On the Vanishing of Evanescent Operators.” Physics Letters B 256, no. 2 (1991): 239–244. DOI
• Henning, Brian, Xiaochuan Lu, Tom Melia, and Hitoshi Murayama. “Hilbert Series and Operator Bases with Derivatives in Effective Field Theories.” Communications in Mathematical Physics 347, no. 2 (2016): 363–388. DOI; arXiv
• Kamefuchi, S., L. O’Raifeartaigh, and Abdus Salam. “Change of Variables and Equivalence Theorems in Quantum Field Theories.” Nuclear Physics 28, no. 4 (1961): 529–549. DOI