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 this chapter
Section titled “Enter this chapter”Begin with From Operator Lists to Independent Bases to distinguish independent interactions from an unreduced operator list.
Topics
Section titled “Topics”- From Operator Lists to Independent Bases — Defines a basis as representatives of an operator quotient.
- Integration by Parts and Equation-of-Motion Redundancy — Removes redundancies with the appropriate boundary and on-shell conditions.
- Local Field Redefinitions and the Equivalence Theorem — Tracks observable invariance under local invertible field changes.
- Representation and Spurion Constraints on Operator Bases — Filters candidate operators using representations and symmetry-breaking spurions.
- Flavor, Hermiticity, and CP Bookkeeping — Controls flavor permutations, conjugation, and CP assignments.
- Fierz Relations and Dimension-Specific Identities — Applies spinor and group identities in a stated dimension.
- Evanescent Operators, Finite Renormalization, and RG Closure — Retains regulator-dependent operators needed for finite loop effects.
- Hilbert-Series Counting and Completeness Diagnostics — Counts independent classes without mistaking a count for explicit representatives.
- On-Shell Amplitude Bases and the Operator Correspondence — Relates local interactions to polynomial contact amplitudes.
- Basis Translation, Scheme Dependence, and Reproducibility — Translates operators, coefficients, schemes, and truncation orders together.
How the ideas fit together
Section titled “How the ideas fit together”An on-shell basis and a subtraction basis need not contain the same operators. Equation-of-motion terms can be removed from scattering interactions by consistent field redefinitions, yet off-shell Green functions change. In dimensional regularization, operators vanishing in four dimensions can multiply ultraviolet poles and leave finite contributions. Thus a correct four-dimensional count does not establish closure of a loop calculation; one must specify the evanescent sector and its finite subtraction. Arzt 1995, §§ 1–2; Dugan and Grinstein 1991, pp. 239–244.
Where to go next
Section titled “Where to go next”Continue to Modes, Factorization, and Multiscale RG to identify which momentum hierarchies require distinct dynamical modes.