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.

China

Cen Zhang

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

In the standard model effective field theory, operators involving the top quark are generally difficult to probe and can generate sizable loop contributions to electroweak precision observables measured by past and future lepton colliders. Could the high precision of electroweak measurements compensate for loop suppression and provide competitive reaches on these operators? Would the inclusion of these contributions introduce too many additional parameters for a meaningful global electroweak analysis to be performed? In this paper, we perform a detailed phenomenological study to address these two important questions. Focusing on eight dimension-6 operators that generate anomalous couplings between electroweak gauge bosons and third-generation quarks, we calculate their one loop contributions to   $ e^+e^- \to f\bar{f} $   processes, both on and off the Z-pole, and the   $ e^+e^- \to f\bar{f} $   process. A global analysis is performed with these eight operators and those that contribute to the above processes at tree level using measurements at the LEP, SLC, and several low energy experiments. We find that although current electroweak precision measurements are sensitive to the one-loop effects of top-quark operators, it is difficult to separate them from the operators that contribute at tree level, making a global analysis rather challenging. Under further assumptions (for instance, new physics contributes to only third generation quark operators and the S and T parameters), competitive reaches may be obtained in a global fit. Another important finding of our study is that the two operators that generate the dipole interactions of the bottom quark have a significant impact on the Z-pole measurements and should not be omitted. We also discuss the implications of the recently reported W-boson mass measurement at the CDF for our results. Finally, we estimate the reaches of future lepton colliders in probing top-quark operators with precision electroweak measurements.

Yiming Liu

Yuhao Wang

Lei Zhang

Jiayin Gu

Probing top-quark operators with precision electroweak measurements  

Moments of the leptonic angular distribution in the Drell-Yan process have recently been shown to be sensitive probes of a specific class of dimension-8, four-fermion operators in the Standard Model Effective Field Theory, involving a pair of quarks and leptons. The same operators are also subject to positivity bounds, when requiring the associated (unknown) UV completion to obey basic principles of quantum field theory. We perform a phenomenological study to quantify the sensitivity of the high-luminosity LHC to this set of operators and, by extension, the positivity bounds. We further extend the angular basis of moments and consider double differential information to improve the ability to disentangle the different operators, leading to a sensitivity to new physics scales up to 3 TeV. We use this information to explore the violation of positivity at the LHC as a way to test the underlying principles of quantum field theory. Finally, we present a case study which combines our results with information from other (current and prospective) experiments, as well as the positivity cone to infer the properties of possible tree-level UV completions. The data lead to robust, model-independent lower bounds on the   <math> <mi>M</mi> <mo>/</mo> <msqrt> <mi>g</mi> </msqrt> </math>  $$ M/\sqrt{g} $$  combination of the particle mass and coupling, for states that couple to right-handed leptons and/or up quarks.

Xu Li

United Kingdom

Ken Mimasu

South Korea

Kimiko Yamashita

Chengjie Yang

Shuang-Yong Zhou

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

The diphoton channel at lepton colliders, <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>(</mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo><mo>→</mo><mi>γ</mi><mi>γ</mi></mrow></math>, has a remarkable feature that the leading new physics contribution comes only from dimension-eight operators. This contribution is subject to a set of positivity bounds, derived from the fundamental principles of quantum field theory, such as unitarity, locality, analyticity and Lorentz invariance. These positivity bounds are thus applicable to the most direct observable: the diphoton cross section. This unique feature provides a clear, robust, and unambiguous test of these principles. We estimate the capability of various future lepton colliders in probing the dimension-eight operators and testing the positivity bounds in this channel. We show that positivity bounds can lift certain flat directions among the effective operators and significantly change the perspectives of a global analysis. We also discuss the positivity bounds of the <math><mrow><mi>Z</mi><mi>γ</mi><mo>/</mo><mi>Z</mi><mi>Z</mi></mrow></math> processes which are related to the <math><mi>γ</mi><mi>γ</mi></math> ones, but are more complicated due to the massive <math><mi>Z</mi></math> boson.

Germany

United States

Lian-Tao Wang

Unambiguously Testing Positivity at Lepton Colliders

We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of su symmetric dispersion relations supplemented with positivity of the partial waves, st null constraints and the generalized optical theorem. This generalizes the convex cone approach to constrain the s 2 coefficient space to higher orders. Optimal positive bounds can be extracted by semi-definite programs with a continuous decision variable, compared with linear programs for the case of a single field. As an example, we explicitly compute the positivity constraints on bi-scalar theories, and find all the Wilson coefficients can be constrained in a finite region, including the coefficients with odd powers of s, which are absent in the single scalar case.

Zong-Zhe Du

Triple crossing positivity bounds for multi-field theories

We present a global interpretation of Higgs, diboson, and top quark production and decay measurements from the LHC in the framework of the Standard Model Effective Field Theory (SMEFT) at dimension six. We constrain simultaneously 36 independent directions in its parameter space, and compare the outcome of the global analysis with that from individual and two-parameter fits. Our results are obtained by means of state-of-the-art theoretical calculations for the SM and the EFT cross-sections, and account for both linear and quadratic corrections in the 1/Λ2 expansion. We demonstrate how the inclusion of NLO QCD and   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (Λ −4) effects is instrumental to accurately map the posterior distributions associated to the fitted Wilson coefficients. We assess the interplay and complementarity between the top quark, Higgs, and diboson measurements, deploy a variety of statistical estimators to quantify the impact of each dataset in the parameter space, and carry out fits in BSM-inspired scenarios such as the top-philic model. Our results represent a stepping stone in the ongoing program of model-independent searches at the LHC from precision measurements, and pave the way towards yet more global SMEFT interpretations extended to other high-p  T  processes as well as to low-energy observables.

Netherlands

Jacob Ethier

Giacomo Magni

Belgium

Italy

Fabio Maltoni

Luca Mantani

Emanuele Nocera

Juan Rojo

Emma Slade

Eleni Vryonidou

Combined SMEFT interpretation of Higgs, diboson, and top quark data from the LHC

We discuss the general method for obtaining full positivity bounds on multifield effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that, for a generic EFT containing three or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions, and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.

Hao Xu

Positivity in Multifield Effective Field Theories

We present the automation of one-loop computations in the standard-model effective field theory at dimension 6. Our general implementation, dubbed smeft@nlo, covers all types of operators: bosonic and two- and four-fermion ones. Included ultraviolet and rational counterterms presently allow for fully differential predictions, possibly matched to the parton shower, up to the one-loop level in the strong coupling or in four-quark operator coefficients. Exact flavor symmetries are imposed among light quark generations, and an initial focus is set on top-quark interactions in the fermionic sector. We illustrate the potential of this implementation with novel loop-induced and next-to-leading-order computations relevant for top-quark, electroweak, and Higgs-boson phenomenology at the LHC and future colliders.

Céline Degrande

Israel

Gauthier Durieux

CERN

Automated one-loop computations in the standard model effective field theory

We use generalized elastic positivity bounds to constrain the parameter space of multi-field spin-2 effective field theories. These generalized bounds involve inelastic scattering amplitudes between particles with different masses, which contain kinematic singularities even in the t = 0 limit. We apply these bounds to the pseudo-linear spin-2 theory, the cycle spin-2 theory and the line spin-2 theory respectively. For the pseudo-linear theory, we exclude the remaining operators that are unconstrained by the usual elastic positivity bounds, thus excluding all the leading (or highest cutoff) interacting operators in the theory. For the cycle and line theory, our approach also provides new bounds on the Wilson coefficients previously unconstrained, bounding the parameter space in both theories to be a finite region (i.e., every Wilson coefficient being constrained from both sides). To help visualize these finite regions, we sample various cross sections of them and estimate the total volumes.

Zi-Yue Wang

Generalized elastic positivity bounds on interacting massive spin-2 theories

We consider the positivity bounds on dimension-8 four-electron operators and study two related phenomenological aspects at future lepton colliders. First, if positivity is violated, probing such violations will revolutionize our understanding of the fundamental pillars of quantum field theory and the S-matrix theory. We observe that positivity violation at scales of 1-10 TeV can potentially be probed at future lepton colliders even if one assumes that dimension-6 operators are also present. Second, the positive nature of the dimension-8 parameter space often allows us to either directly infer the existence of UV-scale particles together with their quantum numbers or exclude them up to certain scales in a model-independent way. In particular, dimension-8 positivity plays an important role in the test of the Standard Model. If no deviations from the Standard Model are observed, it allows for simultaneous exclusion limits on all kinds of potential UV-complete models. Unlike the dimension-6 case, these limits apply regardless of the UV model setup and cannot be removed by possible cancellations among various UV contributions. This thus consists of a novel and universal test to confirm the Standard Model. We demonstrate with realistic examples how all the previously mentioned possibilities, including the test of positivity violation, can be achieved. Hence, we provide an important motivation for studying dimension-8 operators more comprehensively.

France

Benjamin Fuks

Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states  