SMEFTs living on the edge: determining the UV theories from positivity and extremality

Abstract

We study the "inverse problem" in the context of the Standard Model Effective Field Theory (SMEFT): how and to what extend can one reconstruct the UV theory, given the measured values of the operator coefficients in the IR? The main obstacle of this problem is the degeneracies in the space of coefficients: a given SMEFT truncated at a finite dimension can be mapped to infinitely many UV theories. We discuss these degeneracies at the dimension-8 level, and show that positivity bounds play a crucial role in the inverse problem. In particular, the degeneracies either vanish or become significantly limited for SMEFTs that live on or close to the positivity bounds. The UV particles of these SMEFTs, and their properties such as spin, charge, other quantum numbers, and interactions with the SM particles, can often be uniquely determined, assuming dimension-8 coefficients are measured. The allowed region for SMEFTs, which forms a convex cone, can be systematically constructed by enumerating its generators. We show that a geometric notion, extremality, conveniently connects the positivity problem with the inverse problem. We discuss the implications of a SMEFT living on an extremal ray, on a k-face, and on the vertex of the positive cone. We also show that the information of the dimension-8 coefficients can be used to set exclusion limits on all individual UV states that interact with the SM, independent of specific model assumptions. Our results indicate that the dimension-8 operators encode much more information about the UV than one would naively expect, which can be used to reverse engineer the UV physics from the SMEFT.

Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics

“…The former method involves constructing the group projectors of irreducible representations from the Clebsch-Gordon coefficients and is very effective for a problem with sufficient symmetries and a manageable number of degrees of freedom, while the latter is useful for generic problems that lack sufficient symmetries and consequently, the bounds are only extracted numerically in general. It is found that only a small fraction of the parameter space of the SMEFT is consistent with the positivity bounds, as discussed in [31][32][33][34][35][36] for the example of constraining the anomalous quartic gauge couplings from vector boson scatterings. In [3], it was found that future electron-positron colliders can be used to probe violations of positivity bounds or the fundamental principles of quantum field theory up to the multi-TeV scale, regardless of the presence of dim-6 operators.…”

Snowmass White Paper: UV Constraints on IR Physics

“…In tree-level UV completions, an extremal ray corresponds to a UV particle with fixed SMEFT quantum numbers, while a ray inside the cone, on the other hand, would indicate presence of multiple UV states. Therefore, positivity bounds can provide important information to reverse-engineer the UV model from the SMEFT data [142,144,[166][167][168]. On the other hand, confronting the experimental data with positivity bounds on dimension-8 operators can also be used to put exclusion limits on relevant BSM physics unambiguously, while excluding dimension-6 operators to a certain energy scale would be ambiguous.…”

“…On the other hand, confronting the experimental data with positivity bounds on dimension-8 operators can also be used to put exclusion limits on relevant BSM physics unambiguously, while excluding dimension-6 operators to a certain energy scale would be ambiguous. This is because a dimension-6 coefficient can vanish due to cancellations between multiple UV states, but that is not possible for a dimension-8 coefficient [144,168]. An intriguing observation is that among the seesaw mechanisms only the heavy right-handed neutrino in the type-I model reaches the extremity of the positivity cone [169], which means that the SMEFT can be used to pin down the seesaw mechanism if it is indeed type-I.…”

Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics

“…In the case of a treelevel UV completion, each of these ERs represents a UV particle with specific quantum numbers or symmetry structure that is a projection of the symmetry of UV theory to that of the SM [8]. (For a loop-level UV completion, an ER may not directly correspond to the UV particle but becomes a component of the UV loop [34].) This implies that the convex cone bounds, or the dim-8 operators, are valuable and pertinent to the inverse-problem of reverse-engineering the UV model from a set of measured Wilson coefficients.…”

“…The fact that the extremal rays of the positivity cone are the group projectors of irreducible representations in the product decomposition of the symmetries of an EFT implies that positivity bounds can be used to infer the UV states [8]. This is most significant if experiments were to find some new physics located close to the boundary of the positivity cone, or even better, close to one of its extremal rays [3,34]. In these favorable circumstances, com-paring the positivity cone with the experimental data would enable the model-independent inference of UV particles' quantum numbers.…”