DEFT: A program for operators in EFT

TL;DR

DEFT is a Python-based toolkit that automates the construction, validation, and basis-change of effective field theory operator spaces, with a focus on SMEFT and potential generalizations to other gauge groups. By formulating the operator content as a quotient $U \cong V/W$ and implementing systematic redundancy removals from integration by parts, Fierz identities, and EOM, it enables rapid generation of operator bases and explicit basis transformations. Cross-checks against established counts and Hilbert-series results demonstrate reliability, and explicit basis conversions (e.g., SILH↔Warsaw) show practical utility for data interpretation and new-physics searches. The framework provides a scalable, basis-agnostic approach to precision SM tests and EFT-based phenomenology, with clear pathways to extensions to other dimensions or gauge structures.

Abstract

We describe a Python-based computer program, DEFT, for manipulating operators in effective field theories (EFTs). In its current incarnation, DEFT can be applied to 4-dimensional, Poincaré invariant theories with gauge group $SU(3)\times SU(2) \times U(1)$, such as the Standard Model (SM), but a variety of extensions (e.g. to lower dimensions or to an arbitrary product of unitary gauge groups) are conceptually straightforward. Amongst other features, the program is able to: (i) check whether an input list of Lagrangian operators (of a given dimension in the EFT expansion) is a basis for the space of operators contributing to S-matrix elements, once redundancies (such as Fierz-Pauli identities, integration by parts, and equations of motion) are taken into account; (ii) generate such a basis (where possible) from an input algorithm; (iii) carry out a change of basis. We describe applications to the SM (where we carry out a number of non-trivial cross-checks) and extensions thereof, and outline how the program may be of use in precision tests of the SM and in the ongoing search for new physics at the LHC and elsewhere. The code and instructions can be downloaded from http://web.physics.ucsb.edu/~dwsuth/DEFT/.

Figures (5)

• Figure 1: The fields of the one generation Standard Model in component form, along with their mass dimensions, and their representations under the SM symmetries.
• Figure 2: Two schematic amplitudes whose dimension $n$ parts are equal: the square and circle denote higher and lower derivative dimension $n$ operators.
• Figure 3: The number of independent operators at each mass dimension $d$, for various combinations of fields. $\{H,B,W,G,l_L,e_R,q_L,u_R,d_R\}$ are those of the one generation Standard Model (cf. Figure \ref{['fig:smfieldreps']}); $\phi$ is a real scalar singlet under the symmetries of the Standard Model. $G^4$ and $\{q_L^4,u_R^4,d_R^4\}$ are respectively the gauge boson and matter fields of an $SU(4)$ gauge group, with the same electroweak charges as their $SU(3)$ charged counterparts in the Standard Model. In the penultimate line of the Table, we treat the $SU(4)$ as a global symmetry.
• Figure 4: The absolute values of the non-trivial components of $r_{ij}$, defined in (\ref{['eq:rrefexample']}), for the one generation SM. Each marker is positioned in line with the basis operator corresponding to column $j$, and formatted according to the field composition of the redundant operator corresponding to the leading coefficient in the row $i$. We define the marginal couplings in terms of the measured coefficients of the three generation Standard Model: the Higgs quartic and gauge couplings equal those measured at the $Z$ pole, whereas the Yukawa couplings are set by measurements of the heaviest generation: $y_u = \frac{m_t}{v}$, $y_d = \frac{m_b}{v}$, $y_e = \frac{m_\tau}{v}$.
• Figure 5: A schematic 'map' of all dimension 6 operators allowed by Lorentz symmetry, cut in half about its axis of symmetry $\sum h =0$ (reflected in this line are the hermitian conjugates of the operators shown). Arrows show the movement induced by equation of motion relations and commutation of derivative relations in the space of dim 6 operators, colour-coded to show $\phi$ EOMs (blue), $\psi$ EOMs (green, short dashed), $F$ EOMs (red, long dashed), and replacing derivatives with field strengths (grey, dash-dotted).