Standard Model Effective Field Theory

Here we present a brief description of the theoretical formalism underlying the SMEFT framework [Wei79b] [BW86] and its application to the analysis of particle physics data, see also [BT19] for a review.

The SMEFT framework

The effects of new heavy BSM particles with typical mass scale \(M\simeq \Lambda\) can under general conditions be parametrized at lower energies \(E\ll \Lambda\) in a model-independent way in terms of a basis of higher-dimensional operators constructed from the SM fields and their symmetries. The resulting effective Lagrangian then admits the following power expansion

\[\mathcal{L}_{\rm SMEFT}=\mathcal{L}_{\rm SM} + \sum_i^{N_{d6}} \frac{c_i}{\Lambda^2}\mathcal{O}_i^{(6)} + \sum_j^{N_{d8}} \frac{b_j}{\Lambda^4}\mathcal{O}_j^{(8)} + \ldots \, ,\]

where \(\mathcal{L}_{\rm SM}\) is the SM Lagrangian, and \(\mathcal{O}_i^{(6)}\) and \(\mathcal{O}_j^{(8)}\) stand for the elements of the operator basis of mass-dimension d=6 and d=8, respectively. Operators with d=5 and d=7, which violate lepton and/or baryon number conservation [DGK+13], are not considered here. Whilst the choice of operator basis used in this expression is not unique, it is possible to relate the results obtained in different bases [FFM+15]. In our approach we adopt the Warsaw basis for \(\mathcal{O}_i^{(6)}\) [GIMR10], and neglect effects arising from operators with mass dimension \(d\ge 8\) .

For specific UV completions, the Wilson coefficients \(_i\) in can be evaluated in terms of the parameters of the BSM theory, such as its coupling constants and masses. However, in a bottom-up approach, they are a priori free parameters and they need to be constrained from experimental data. In general, the effects of the dimension-6 SMEFT operators in a given observable, such as cross-sections at the LHC, differential distributions, or other pseudo-observables, can be written as follows:

\[\sigma=\sigma_{\rm SM} + \sum_i^{N_{d6}}\kappa_i \frac{c_i}{\Lambda^2} + \sum_{i,j}^{N_{d6}} \widetilde{\kappa}_{ij} \frac{c_ic_j}{\Lambda^4} \, ,\]

where \(\sigma_{\rm SM}\) indicates the SM prediction and the Wilson coefficients \(c_i\) are considered to be real for simplicity.

In this equation, the second term arises from operators interfering with the SM amplitude. The resulting \(\mathcal{O}\left(\Lambda^{-2}\right)\) corrections to the SM cross-sections represent formally the dominant correction, though in many cases they can be subleading for different reasons. The third term in representing \(\mathcal{O}\left(\Lambda^{-4}\right)\) effects, arises from the squared amplitudes of the SMEFT operators, irrespectively of whether or not the dimension-6 operators interfere with the SM diagrams. In principle, this second term may not need to be included, depending on if the truncation at \(\mathcal{O}\left(\Lambda^{-2}\right)\) order is done at the Lagrangian or the cross section level, but in practice there are often valid and important reasons to include them in the calculation.

An important aspect of any SMEFT analysis is the need to include all relevant operators that contribute to the processes whose data is used as input to the fit. Only in this way can the SMEFT retain its model and basis independence. However, unless specific scenarios are adopted, the number of non-redundant operators \(N_{d6}\) becomes unfeasibly large: 59 for one generation of fermions [GIMR10] and 2499 for three [AJMT14]. This implies that a global SMEFT fit, even if restricted to dimension-6 operators, will have to explore a huge parameter space with potentially a large number of flat (degenerate) directions.