Dimension-eight Operators Research Articles

Dimension-eight operator basis for universal standard model effective field theory

We present the basis of dimension-eight operators associated with universal theories. We first derive a complete list of independent dimension-eight operators formed with the Standard Model bosonic fields characteristic of such universal new physics scenarios. Without imposing C or P symmetries the basis contains 175 operators—that is, the assumption of universality reduces the number of independent Standard Model effective field theory (SMEFT) coefficients at dimension eight from 44807 to 175. 89 of the 175 universal operators are included in the general dimension-eight operator basis in the literature. The 86 additional operators involve higher derivatives of the Standard Model bosonic fields and can be rotated in favor of operators involving fermions using the Standard Model equations of motion for the bosonic fields. By doing so we obtain the allowed fermionic operators generated in this class of models which we map into the corresponding 86 independent combinations of operators in the dimension-eight basis of [C. W. Murphy, Dimension-8 operators in the standard model effective field theory, .]. Published by the American Physical Society 2024

Top-down and bottom-up: Studying the SMEFT beyond leading order in $1/\Lambda^2$

In order to assess the relevance of higher order terms in the Standard Model effective field theory (SMEFT) expansion we consider four new physics models and their impact on the Drell Yan cross section. Of these four, one scalar model has no effect on Drell Yan, a model of fermions while appearing to generate a momentum expansion actually belongs to the vacuum expectation value expansion and so has a nominal effect on the process. The remaining two, a leptoquark and a Z'Z′ model exhibit a momentum expansion. After matching these models to dimension-ten we study the how the inclusion of dimension-eight and dimension-ten operators in hypothetical effective field theory fits to the full ultraviolet models impacts fits. We do this both in the top-down approach, and in a very limited approximation to the bottom up approach of the SMEFT to infer the impact of a fully general fit to the SMEFT. We find that for the more weakly coupled models a strictly dimension-six fit is sufficient. In contrast when stronger interactions or lighter masses are considered the inclusion of dimension-eight operators becomes necessary. However, their Wilson coefficients perform the role of nuisance parameters with best fit values which can differ statistically from the theory prediction. In the most strongly coupled theories considered (which are already ruled out by data) the inclusion of dimension-ten operators allows for the measurement of dimension-eight operator coefficients consistent with theory predictions and the dimension-ten operator coefficients then behave as nuisance parameters. We also study the impact of the inclusion of partial next order results, such as dimension-six squared contributions, and find that in some cases they improve the convergence of the series while in others they hinder it.

Differential cross-section measurements of the production of four charged leptons in association with two jets using the ATLAS detector

Differential cross-sections are measured for the production of four charged leptons in association with two jets. These measurements are sensitive to final states in which the jets are produced via the strong interaction as well as to the purely-electroweak vector boson scattering process. The analysis is performed using proton-proton collision data collected by ATLAS at sqrt{s} = 13 TeV and with an integrated luminosity of 140 fb−1. The data are corrected for the effects of detector inefficiency and resolution and are compared to state-of-the-art Monte Carlo event generator predictions. The differential cross-sections are used to search for anomalous weak-boson self-interactions that are induced by dimension-six and dimension-eight operators in Standard Model effective field theory.

Impact of dimension-eight SMEFT operators in the electroweak precision observables and triple gauge couplings analysis in universal SMEFT

We perform a complete study of the electroweak precision observables and electroweak gauge boson pair production in terms of the SMEFT up to ${\cal O}(1/\Lambda^4)$ under the assumption of universal, C and P conserving new physics. We show that the analysis of data from those two sectors allows us to obtain closed constraints in the relevant parameter space in this scenario. In particular we find that the Large Hadron Collider data can independently constrain the Wilson coefficients of the dimension-six and -eight operators directly contributing to the triple gauge boson vertices. Our results show that the impact of dimension-eight operators in the study of triple gauge couplings is small.

Probe of the anomalous neutral triple gauge couplings in photon-induced collision at future muon colliders

The anomalous ZZγ and Zγγ neutral triple gauge couplings occurred by dimension-eight operators are investigated through the process μ+μ−→μ+γ⁎μ−→μ+Z(νν¯)μ− at the muon collider with s=3, 6, 10 and 14 TeV. The charged lepton pseudo-rapidity, the charged lepton transverse momentum and the transverse missing energy distributions are taken in consideration for the final state of the process in the analysis. The sensitivities of the anomalous couplings are obtained at 95% Confidence Level with integrated luminosity of Lint=1, 4, 10 and 20 ab−1, respectively, according to center-of-mass energies of muon collider taking into account the effects of systematic uncertainties 0%, 3% and 5%. The best limits of anomalous CBB/Λ4, CBW/Λ4, CB˜W/Λ4 and CWW/Λ4 couplings without systematic uncertainty at center-of-mass energy of 14 TeV and integrated luminosity of 20 ab−1 are found to be [-0.01026; 0.00636] TeV−4, [-0.02482; 0.03053] TeV−4, [-0.01830; 0.02510] TeV−4, [-0.06981; 0.07387] TeV−4, respectively.

Integrating out heavy scalars with modified equations of motion: Matching computation of dimension-eight SMEFT coefficients

The shift in focus towards searches for physics beyond the Standard Model (SM) employing model-independent Effective Field Theory (EFT) methods necessitates a rigorous approach to matching to guarantee the validity of the obtained results and constraints. The limits on the leading dimension-six EFT effects can be rather inaccurate for LHC searches that suffer from large uncertainties while exploring an extensive energy range. Similarly, precise measurements can, in principle, test the subleading effects of the operator expansion. In this work, we present an algorithmic approach to automatise matching computations for dimension-eight operators for generic scalar extensions with proper implementation of equations of motion. We devise a step-by-step procedure to obtain the dimension-eight Wilson coefficients (WCs) in a non-redundant basis to arrive at complete matching results. We apply this formalism to a range of scalar extensions of the SM and provide tree-level and loop-suppressed results. Finally, we discuss the relevance of the dimension-eight operators for a range of phenomenological analyses, particularly focusing on Higgs and electroweak physics.

Origin of neutrino masses on the convex cone of positivity bounds

We exhibit the geometric structure of the convex cone in the linear space of the Wilson coefficients for the dimension-eight operators involving the left-handed lepton doublet $L$ and the Higgs doublet $H$ in the Standard Model effective field theory (SMEFT). The boundary of the convex cone gives rise to the positivity bounds on the Wilson coefficients, while the extremal ray corresponds to the unique particle state in the theory of ultraviolet completion. Among three types of canonical seesaw models for neutrino masses, we discover that only right-handed neutrinos in the type-I seesaw model show up as one of the extremal rays, whereas the heavy particles in the type-II and type-III seesaw models live inside the cone. The experimental determination of the relevant Wilson coefficients close to the extremal ray of the type-I seesaw model will unambiguously pin down or rule out the latter as the origin of neutrino masses. This discovery offers a novel way to distinguish the most popular seesaw model from others, and it also strengthens the SMEFT as an especially powerful tool to probe new physics beyond the Standard Model.

Role of dimension-eight operators in an EFT for the 2HDM

The Standard Model effective field theory (SMEFT) is the tool of choice for studying deviations of Higgs couplings from the Standard Model predictions. The SMEFT is an expansion in an infinite tower of higher dimension operators, which is typically truncated at dimension-six. We consider the effective theory including dimension-eight operators and examine the matching to the 2 Higgs doublet model (2HDM) that is assumed to be valid at some high scale. Both the limits from the direct production of single Higgs bosons and the indirect limits on the Higgs trilinear coupling are considered in the context of the SMEFT matched to the 2HDM, and the importance of the assumptions about the expansions in powers of the high scale are examined numerically.

Unambiguously Testing Positivity at Lepton Colliders.

The diphoton channel at lepton colliders, e^{+}e^{-}(μ^{+}μ^{-})→γγ, 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 Zγ/ZZ processes which are related to the γγ ones, but are more complicated due to the massive Z boson.

Positivity bounds in the standard model effective field theory beyond tree level

Focusing on four-Higgs interactions, we analyse the robustness of tree-level-derived positivity bounds on Standard Model effective field theory (SMEFT) operators under quantum corrections. Among other results, we demonstrate that: (i) Even in the simplest extensions of the Standard Model, e.g. with one new scalar singlet or with a neutral triplet, some positivity bounds are strictly violated; (ii) the mixing of the dimension-eight operators under renormalisation, which we compute here for the first time, can drive them out of their positivity region; (iii) the running of the dimension-eight interactions triggered by solely dimension-six terms respects the positivity bounds. Our results suggest, on one hand, that departures from positivity within the SMEFT, if ever found in the data, do not necessarily imply the breaking of unitarity or causality, nor the presence of new light degrees of freedom. On the other hand, they lead to strong constraints on the form of certain anomalous dimensions.

Naturalness and the muon magnetic moment

We study a predictive model for explaining the apparent deviation of the muon anomalous magnetic moment from the Standard Model expectation. There are no new scalars and hence no new hierarchy puzzles beyond those associated with the Higgs; the only new particles at the TeV scale are vector-like singlet and doublet leptons. Interestingly, this simple model provides a calculable example violating the Wilsonian notion of naturalness: despite the absence of any symmetries prohibiting its generation, the coefficient of the naively leading dimension-six operator for (g − 2) vanishes at one-loop. While effective field theorists interpret this either as a surprising UV cancellation of power divergences, or as a delicate cancellation between matching UV and calculable IR corrections to (g − 2) from parametrically separated scales, there is a simple explanation in the full theory: the loop integrand is a total derivative of a function vanishing in both the deep UV and IR. The leading contribution to (g − 2) arises from dimension-eight operators, and thus the required masses of new fermions are lower than naively expected, with a sizeable portion of parameter space already covered by direct searches at the LHC. The viable parameter space free of fine-tuning for the muon mass will be fully covered by future direct LHC searches, and all of the parameter space can be probed by precision measurements at planned future lepton colliders.

Complete set of dimension-eight operators in the standard model effective field theory

We present a complete list of the dimension 8 operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group, and identify the Lorentz structures as states in a $SU(N)$ group. In this way, redundancy from equations of motion is absent and that from integration-by-part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the $SU(N)$ group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can be easily applied to higher dimensional standard model effective field theory and other effective field theories, making these studies more approachable.

Effective field theory versus UV-complete model: vector boson scattering as a case study

Effective field theories (EFT) are commonly used to parameterize effects of BSM physics in vector boson scattering (VBS). For Wilson coefficients which are large enough to produce presently observable effects, the validity range of the EFT represents only a fraction of the energy range covered by the LHC, however. In order to shed light on possible extrapolations into the high energy region, a class of UV-complete toy models, with extra SU(2) multiplets of scalars or of fermions with vector-like weak couplings, is considered. By calculating the Wilson coefficients up to energy-dimension eight, and full one-loop contributions to VBS due to the heavy multiplets, the EFT approach, with and without unitarization at high energy, is compared to the perturbative prediction. For high multiplicities, e.g. nonets of fermions, the toy models predict sizable effects in transversely polarized VBS, but only outside the validity range of the EFT. At lower energies, dimension-eight operators are needed for an adequate description of the models, providing another example that dimension-eight can be more important than dimension-six operators. A simplified VBFNLO implementation is used to estimate sensitivity of VBS to such BSM effects at the LHC. Unitarization captures qualitative features of the toy models at high energy but significantly underestimates signal cross sections in the threshold region of the new particles.

Unitarity constraints on anomalous quartic couplings

We obtain the partial-wave unitarity constraints on the lowest-dimension effective operators which generate anomalous quartic gauge couplings but leave the triple gauge couplings unaffected. We consider operator expansions with linear and nonlinear realizations of the electroweak symmetry and explore the multidimensional parameter space of the coefficients of the relevant operators: 20 dimension-eight operators in the linear expansion and 5 ${\cal O}(p^4)$ operators in the derivative expansion. We study two-to-two scattering of electroweak gauge bosons and Higgs bosons taking into account all coupled channels and all possible helicity amplitudes for the $J=0,1$ partial waves. In general, the bounds degrade by factors of a few when several operator coefficients are considered non-vanishing simultaneously. However, this requires to consider constraints from both $J=0$ and $J=1$ partial waves for some sets of operators.

Sum rules for CP-violating operators of Weinberg type

We estimate the size of the hadronic matrix elements of CP-violating three-gluon and four-gluon Weinberg operators using sum-rule techniques. In the three-gluon case, we are able to reproduce the expressions given in earlier works, while the four-gluon results obtained in this article are new. Our paper therefore represents the first systematic study of contributions to the electric dipole moment of the neutron due to CP-violating dimension-six and dimension-eight operators. We provide many details on both the derivation of the sum rules as well as the analysis of the uncertainties that plague our final predictions.

Study on Anomalous Neutral Triple-gauge Boson Couplings from Dimension-eight Operators at the HL-LHC

The anomalous neutral triple gauge boson couplings (aNTGCs) for the Z gamma gamma and Z gamma Z vertices described by dimension-eight operators are examined through the process pp to l^{+}l^{-} gamma at the High-Luminosity Large Hadron Collider (HL-LHC). We performed an analysis on transverse momentum of photon and angular distribution of charged lepton in the final state including detector effects. Sensitivity limits of the C_{\widetilde{B}W}, C_{BB} couplings are obtained at 95 % C.L. to constrain for the range [-1.88: 1.88] TeV^{-4}, [-1.47: 1.47] TeV^{-4} and [-1.14:1.14] TeV^{-4}, [-0.86: 0.86] TeV^{-4} with an integrated luminosity of 300 fb^{-1} and 3000 fb^{-1}, respectively.

Probing the effects of dimension-eight operators describing anomalous neutral triple gauge boson interactions at FCC-hh

The effects of dimension-eight operators giving rise to anomalous neutral triple gauge boson interactions of Zγγ and ZγZ vertices in pp→l−l+γ and pp→νν¯γ are investigated at 100 TeV center of mass energy of future circular hadron collider (FCC-hh). The transverse momentum of photon, invariant mass of l−l+γ and angular distribution of charged lepton in the rest frame of l−l+ and Missing Energy Transverse (MET) are considered in the analysis. The realistic detector effects are also included with Delphes simulation. Sensitivity limits obtained at 95% C.L. for CB˜W/Λ4 and CBB/Λ4 couplings are [−0.52;0.52]([−0.40;0.40]) TeV−4, [−0.43;0.43]([−0.33;0.33]) TeV−4 in the dilepton+photon channel and [−0.11;0.11]([−0.084;0.084]) TeV−4, [−0.092;0.092]([−0.072;0.072]) TeV−4 in the MET+photon channel with Lint=1 (3) ab−1, respectively.

Scattering of W and Z bosons at high-energy lepton colliders

We present a new study of quasi-elastic W and Z scattering processes in high-energy e^+e^- collisions, based on and extrapolating the low-energy effective theory which extends the standard model with a 125;text {GeV} Higgs boson. We parameterize deviations in the low-energy range in terms of the dimension-eight operators that arise in the effective theory. Smoothly extending this to higher energy, we study a set of simplified models of new physics in W / Z scattering, (1) a structureless extrapolation of the effective theory, and (2) scalar and tensor resonance multiplets. The high-energy asymptotics of all models is regulated by a universal unitarization procedure. This enables us to provide benchmark scenarios which can be meaningfully evaluated off shell and in exclusive event samples, and to determine the sensitivity of an e^+e^- collider to the model parameters. We analyze the longitudinal vector-boson scattering modes, where we optimize the cuts for the fiducial cross section for different collider scenarios. Here, we choose energy stages of 1.0, 1.4 and 3 TeV, as motivated by the extendability of the ILC project and the staging scenario of the CLIC project.

$$\mu \rightarrow e \gamma $$ μ → e γ in the 2HDM: an exercise in EFT

The 2 Higgs Doublet Model of type III has renormalisable Lepton Flavour-Violating couplings, and its one- and two-loop (“Barr–Zee”) contributions to $$\mu \rightarrow e \gamma $$ are known. In the decoupling limit, where the mass scale M of the second doublet is much greater than the electroweak scale, the model can be parametrised with an Effective Field Theory (EFT) containing dimension-six operators. The $$1/M^2$$ terms of the exact calculation are reproduced in the EFT, provided that the four-fermion operator basis below the weak scale is enlarged with respect to the SU(2)-invariant Buchmuller–Wyler list. It is found that the dominant two-loop “Barr–Zee” contributions arise mostly in two-loop matching and running, and that dimension-eight operators might be numerically relevant.

The DBI action, higher-derivative supergravity, and flattening inflaton potentials

In string theory compactifications it is common to find an effective Lagrangian for the scalar fields with a non-canonical kinetic term. We study the effective action of the scalar position moduli of Type II Dp-branes. In many instances the kinetic terms are in fact modified by a term proportional to the scalar potential itself. This can be linked to the appearance of higher-dimensional supersymmetric operators correcting the Kahler potential. We identify the supersymmetric dimension-eight operators describing the α′ corrections captured by the D-brane Dirac-Born-Infeld action. Our analysis then allows an embedding of the D-brane moduli effective action into an $$ \mathcal{N}=1 $$ supergravity formulation. The effects of the potential-dependent kinetic terms may be very important if one of the scalars is the inflaton, since they lead to a flattening of the scalar potential. We analyze this flattening effect in detail and compute its impact on the CMB observables for single-field inflation with monomial potentials.