[Paper Review] Field reparametrization in effective field theories

Giampiero Passarino|arXiv (Cornell University)|Oct 30, 2016

Black Holes and Theoretical Physics16 references4 citations

TL;DR

This paper investigates field reparametrizations in Effective Field Theories (EFTs), particularly focusing on how non-linear, non-invariant field transformations affect the equivalence of different operator bases at dimension six. It establishes that while Lagrangians may change under such reparametrizations, the S-matrix remains invariant, ensuring physical observables are preserved. The key contribution is a rigorous framework for understanding basis equivalence in SMEFT, showing that Wilson coefficients mix under renormalization and that only S-matrix elements—not Lagrangians—define physical predictions.

ABSTRACT

Debate topic for Effective Field Theory (EFT) is the choice of a "basis" for $\mrdim = 6$ operators Clearly all bases are equivalent as long as they are a "basis", containing a minimal set of operators after the use of equations of motion and respecting gauge invariance. From a more formal point of view a basis is characterized by its closure with respect to renormalization. Equivalence of bases should always be understood as a statement for the S-matrix and not for the Lagrangian, as dictated by the equivalence theorem. Any phenomenological approach that misses one of these ingredients is still acceptable for a preliminar analysis, as long as it does not pretend to be an EFT. Here we revisit the equivalence theorem and its consequences for EFT when two sets of higher dimensional operators are connected by a set of non-linear, noninvariant, field reparametrizations.

Motivation & Objective

• To clarify the role of field reparametrizations in Effective Field Theories, especially in the context of basis equivalence for dimension-6 operators.
• To resolve confusion about whether different operator bases in SMEFT are physically distinct or equivalent.
• To demonstrate that physical predictions are determined by the S-matrix, not the Lagrangian, even under non-linear field reparametrizations.
• To analyze how Wilson coefficients transform under field reparametrizations and why S-matrix equivalence is the true criterion for physical equivalence.
• To show that certain contact interactions introduced via field reparametrizations do not alter physical amplitudes, as cancellations (e.g., Phoenix diagrams) preserve unitarity and on-shell amplitudes.

Proposed method

• The paper employs the Equivalence Theorem (ET) to analyze how non-linear field reparametrizations affect S-matrix elements in SMEFT.
• It uses off-shell Green's functions, source normalization, and amputation to derive on-shell S-matrix elements, ensuring consistency with LSZ formalism.
• The analysis includes explicit computation of fermionic contact interactions and their cancellation mechanisms via special vertices (e.g., Z-boson self-energy contributions).
• It examines the role of wave-function normalization and field redefinitions (Φ′ = ZΦ) in ensuring canonical normalization of kinetic terms in the Lagrangian.
• The paper studies the behavior of non-local and local field transformations, showing that only local ones preserve the structure of dimensional regularization.
• It proves that field reparametrizations cannot alter the physical content of a theory, as the S-matrix remains invariant under such transformations.

Experimental results

Research questions

• RQ1 How do non-linear, non-invariant field reparametrizations affect the physical predictions of an Effective Field Theory at dimension six?
• RQ2 To what extent are different bases of dimension-6 operators physically equivalent, and what conditions ensure this equivalence?
• RQ3 Why is the S-matrix the correct object for defining physical equivalence, rather than the Lagrangian?
• RQ4 How do special vertices (e.g., from Higgs-Kibble ghosts) ensure cancellation of unphysical contributions in scattering amplitudes?
• RQ5 What is the impact of field reparametrizations on Wilson coefficient mixing under renormalization in SMEFT?

Key findings

• Field reparametrizations do not alter the S-matrix, confirming that physical observables are invariant under such transformations, even when the Lagrangian changes.
• The S-matrix is the only physically meaningful object for comparing bases; Lagrangian-level differences do not imply physical differences.
• Wilson coefficients in one basis can be mapped to linear combinations in another, but such mappings must preserve S-matrix elements to maintain physical consistency.
• Special vertices introduced by field reparametrizations cancel unphysical contributions in amplitudes, ensuring that contact interactions do not alter on-shell scattering amplitudes.
• Phoenix diagrams—resurgent contact interactions from redefined fields—do not change physical amplitudes, as cancellations are enforced by the full set of terms, including ghost and gauge interactions.
• The paper confirms that field redefinitions cannot generate or remove interactions in a way that alters physical predictions, as long as the full S-matrix is preserved.