We present the Standard Model Effective Field Theories (SMEFT) from purely on-shell arguments. Starting from few basics assumptions such as Poincaré invariance and locality, we classify all the renormalisable and non-renormalisable interactions at lowest order in the couplings. From these building blocks, we review how locality and unitarity enforce Lie algebra structures to appear in the S-matrix elements together with relations among couplings (and hypercharges). Furthermore, we give a fully on-shell algorithm to compute any higher-point tree-level amplitude (or form factor) in generic EFTs, bypassing BCFW-like recursion relations which are known to be problematic when non-renormalisable interactions are involved. Finally, using known amplitudes techniques we compute the mixing matrix of SMEFT marginal interactions up to mass dimension 8, to linear order in the effective interactions.

The eikonal approximation is an ideal tool to extract classical observables in gauge theory and gravity directly from scattering amplitudes. Here we consider effective theories of gravity where in addition to the Einstein-Hilbert term we include nonminimal couplings of the type R 3 , R 4 and FFR. In particular, we study the scattering of gravitons and photons of frequency ω off heavy scalars of mass m in the limit m ≫ ω ≫ j⃗ qj, where ⃗ q is the momentum transfer. The presence of nonminimal couplings induces helicityflip processes which survive the eikonal limit, thereby promoting the eikonal phase to an eikonal phase matrix. We obtain the latter from the relevant two-to-two helicity amplitudes that we compute up to oneloop order, and confirm that the leading-order terms in ω exponentiateà la Amati, Ciafaloni and Veneziano. From the eigenvalues of the eikonal phase matrix we then extract two physical observables, to 2PM order: the classical deflection angle and Shapiro time delay/advance. Whenever the classical expectation of helicity conservation of the massless scattered particle is violated, i.e., the eigenvalues of the eikonal matrix are nondegenerate, causality violation due to time advance is a generic possibility for small impact parameter. We show that for graviton scattering in the R 4 and FFR theories, time advance is circumvented if the couplings of these interactions satisfy certain positivity conditions, while it is unavoidable for graviton scattering in the R 3 theory and photon scattering in the FFR theory. The scattering processes we consider mimic the deflection of photons and gravitons off spinless heavy objects such as black holes.

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We consider Einstein gravity with the addition of R 2 and R μν R μν interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorization and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature.

We present the Standard Model Effective Field Theories (SMEFT) from purely on-shell arguments. Starting from few basics assumptions such as Poincaré invariance and locality, we classify all the renormalisable and non-renormalisable interactions at lowest order in the couplings. From these building blocks, we review how locality and unitarity enforce Lie algebra structures to appear in the S-matrix elements together with relations among couplings (and hypercharges). Furthermore, we give a fully on-shell algorithm to compute any higher-point tree-level amplitude (or form factor) in generic EFTs, bypassing BCFW-like recursion relations which are known to be problematic when non-renormalisable interactions are involved. Finally, using known amplitudes techniques we compute the mixing matrix of SMEFT marginal interactions up to mass dimension 8, to linear order in the effective interactions.