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DOE OSTI2020

Uncertainties in turbulent statistics and fluxes of CO2 associated with density effect corrections

Density effect corrections (DEC) are applied to adjust raw CO2 fluxes measured by eddy covariance (EC) systems with open-path gas analyzers. DEC is also required for adjusting the measured CO2 concentration fluctuations to obtain the adjusted CO2 for analyzing turbulent statistics or quantifying fluxes. However, our data show that the power spectra of the DEC-adjusted CO2 are distorted in the high frequency range, as compared with the corresponding spectra of temperature and water vapor density. This contradicts the similarity behavior of scalars, as suggested by Monin-Obukhov similarity theory. It is demonstrated that such a distortion is caused by the DEC-induced spikes in the DEC-adjusted CO2, altering turbulent statistics of CO2 and scalar similarity between CO2 and other scalars. Our results suggest that CO2 fluxes are overestimated by applying DEC especially under high Bowen ratio conditions, potentially leading to substantial uncertainties in long-term ecosystem carbon exchange in dry regions.

Gao, Zhongming↗

DOE OSTI2021

A classical, non-singular, bouncing universe

We present a model for a classical, non-singular bouncing cosmology without violation of the null energy condition (NEC). The field content is General Relativity plus a real scalar field with a canonical kinetic term and only renormalizable, polynomial-type self-interactions for the scalar field in the Jordan frame. The universe begins vacuum-energy dominated and is contracting at t=-∞. We consider a closed universe with a positive spatial curvature, which is responsible for the universe bouncing without any NEC violation. An Rφ 2 coupling between the Ricci scalar and the scalar field drives the scalar field from the initial false vacuum to the true vacuum during the bounce. The model is sub-Planckian throughout its evolution and every dimensionful parameter is below the effective-field-theory scale M P , so we expect no ghost-type or tachyonic instabilities. This model solves the horizon problem and extends co-moving particle geodesics to past infinity, resulting in a geodesically complete universe without singularities. We solve the Friedman equations and the scalar-field equation of motion numerically, and analytically under certain approximations.

Astronomy & Astrophysics↗

DOE OSTI2024

${\tt Warm}$SPy: a numerical study of cosmological perturbations in warm inflation

We present ${\tt Warm}$SPy, a numerical code in Python designed to solve for the perturbations' equations in warm inflation models and compute the corresponding scalar power spectrum at CMB horizon crossing. In models of warm inflation, a radiation bath of temperature T during inflation induces a dissipation (friction) rate of strength Q ∝ T c /Φ m in the equation of motion for the inflaton field Φ. While for a temperature-independent dissipation rate (c = 0) an analytic expression for the scalar power spectrum exists, in the case of a non-zero value for c the set of equations can only be solved numerically. For c > 0 (c < 0), the coupling between the perturbations in the inflaton field and radiation induces a growing (decaying) mode in the scalar perturbations, generally parameterized by a multiplicative function G(Q) which we refer to as the scalar dissipation function. Using ${\tt Warm}$SPy, we provide an analytic fit for G(Q) for the cases of c = {3,1,-1}, corresponding to three cases that have been realized in physical models. Compared to previous literature results, our fits are more robust and valid over a broader range of dissipation strengths Q ϵ [10 -7 ,10 4 ]. Additionally, for the first time, we numerically assess the stability of the scalar dissipation function against various model parameters, inflationary histories as well as the effects of metric perturbations. As a whole, the results do not depend appreciably on most of the parameters in the analysis, except for the dissipation index c, providing evidence for the universal behaviour of the scalar dissipation function G(Q).

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Method for designing a combustion system with reduced environmentally-harmful emissions

A method for designing a combustion system which emits less of at least one environmentally-harmful emission is presented. In a describing step, an injector which introduces a fuel into a combustion chamber is described via a CFD code. In a modeling step, combustion kinetics of the fuel are modeled via a pre-processing code as the fuel mixes and reacts with an oxidizer. In a first selecting step, at least one primary scalar is derived during the modeling of the combustion kinetics. In a performing step, a table look-up is performed to obtain at least one data from a look-up database based on the primary scalar. In a second selecting step, at least one secondary scalar is selected in addition to the primary scalar(s). In a specifying step, at least one chemical pathway of formation or destruction for the secondary scalar is specified via a chemistry manager wherein the secondary scalar is representative of the environmentally-harmful emission(s) of the chemical pathway(s). In a utilizing step, the data is utilized to evaluate the chemical pathway(s) to quantify the environmentally-harmful emission(s). In an identifying step, an improvement to the combustion system is identified which reduces the environmentally-harmful emission(s).

Zambon, Andrea C.↗

DOE OSTI2025

Efficient Decision Trees for Tensor Regressions

Here, we proposed the tensor-input tree (TT) method for scalar-on-tensor and tensor-on-tensor regression problems. We first address scalar-on-tensor problem by proposing scalar-output regression tree models whose input variables are tensors (i.e., multi-way arrays). We devised and implemented fast randomized and deterministic algorithms for efficient fitting of scalar-on-tensor trees, making TT competitive against tensor-input GP models (Yu, Li, and Liu; Sun et al.). Based on scalar-on-tensor tree models, we extend our method to tensor-on-tensor problems using additive tree ensemble approaches. Theoretical justification and extensive experiments, including testing robustness to entrywise input tensor noise, are provided on real and synthetic datasets to illustrate the performance of TT. Our implementation is provided at https://github.com/hrluo/TensorDecisionTreeRegressor. Supplementary materials for this article are available online.

Decision tree regressions↗

DOE OSTI2020

Properties of the new $ \mathcal{N} $ = 1 AdS 4 vacuum of maximal supergravity

The recent comprehensive numerical study of critical points of the scalar potential of four-dimensional N = 8, SO(8) gauged supergravity using Machine Learning software in [1] has led to a discovery of a new N = 1 vacuum with a triality-invariant SO(3) symmetry. Guided by the numerical data for that point, we obtain a consistent SO(3) × Z 2 -invariant truncation of the N = 8 theory to an N = 1 supergravity with three chiral multiplets. Critical points of the truncated scalar potential include both the N = 1 point as well as two new non-supersymmetric and perturbatively unstable points not found by previous searches. Studying the structure of the submanifold of SO(3) × Z 2 -invariant supergravity scalars, we find that it has a simple interpretation as a submanifold of the 14-dimensional Z $3\atop{2}$ -invariant scalar manifold (SU(1, 1)/U(1)) 7 , for which we find a rather remarkable superpotential whose structure matches the single bit error correcting (7, 4) Hamming code. This 14-dimensional scalar manifold contains approximately one quarter of the known critical points. We also show that there exists a smooth supersymmetric domain wall which interpolates between the new N = 1 AdS4 solution and the maximally supersymmetric AdS4 vacuum. Using holography, this result indicates the existence of an N = 1 RG flow from the ABJM SCFT to a new strongly interacting conformal fixed point in the IR.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

The deconstruction of flavor in the privately democratic Higgs sector

The Standard Model (SM) of particle physics fails to explain the observed hierarchy in fermion masses or the origin of fermion-flavor structure. We construct a model to explain these observations in the quark sector. We introduce a spectrum of new particles consisting of six of each — massive singlet vector-like quarks (VLQs), singlet scalars, and SU(2)-doublet scalars. SM quark masses are generated when the neutral components of the SU(2)-doublet scalars acquire non-zero vacuum expectation values (VEVs). We impose global symmetries to ensure that Yukawa couplings stay roughly flavor diagonal and democratic (of the same order), as well as to suppress tree-level flavor-changing neutral currents. Quark-mass hierarchy then follows from a hierarchy in scalar VEVs. The singlet scalars also acquire weak-scale VEVs. Together with the VLQs, they act as messengers between different generations of quarks in the SM. These messenger particles are responsible for generating the elements of the Cabibbo-Kobayashi-Masakawa (CKM) matrix which depend on the ratios of the singlet VEVs and VLQ masses. Constructed this way, the CKM matrix is found to be independent of the SM fermion masses. Using the measured values of the CKM matrix elements and assuming order-one couplings, we derive constraints on the masses of the VLQs and discuss prospects for probing our model in the near future.

CKM Parameters↗

DOE OSTI2020

The cosmological bootstrap: inflationary correlators from symmetries and singularities

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and the particle content of the theory. For example, four-particle amplitudes are analytic for contact interactions and have simple poles with appropriately positive residues for tree-level exchange. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime. After systematically classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities (and the correct normalization of physical ones) completely fixes this correlator. Moreover, a “spin-raising” operator relates it to the correlators associated with the exchange of particles with spin, while “weight-shifting” operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. Using our formalism, we reproduce many classic results in the literature, such as the three-point function of slow-roll inflation, and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. Remarkably, the inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta, which can also be directly connected with the signals for particle production seen in the squeezed limit.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

Quantum dynamics of cosmological particle production: interacting quantum field theories with matrix product states

Understanding real-time dynamics of interacting quantum fields in curved space-time remains a major theoretical challenge. We employ tensor network methods to study such dynamics using interacting scalar and gauge theories in 1+1 spacetime dimensions, subject to a quench modeling a homogeneously expanding gravitational background. The models considered are the scalar λϕ 4 theory and the Schwinger model, i.e. a Dirac fermion coupled to a U(1) gauge field which is equivalent via bosonization to a scalar field with a cosine self-interaction. In the free scalar limit, both theories reproduce known analytical results, providing a nontrivial numerical validation of bosonization in curved spacetime for the Schwinger model. Our central finding is that self-interactions lead to a suppression of gravitational particle production compared to the free-field case, as evidenced by two-point functions and the spectra of produced particles. We further examine the behavior of entanglement generation and find that interactions suppress entanglement growth in the λϕ 4 theory, while in the Schwinger model, the interplay between suppressed particle production and enhanced inter-particle correlations leads to more complex entanglement behavior. Our results pave the way for further explorations of nonperturbative quantum real-time dynamics of interacting scalar and gauge theories in arbitrary gravitational backgrounds.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Sharpening the Distance Conjecture in diverse dimensions

The Distance Conjecture holds that any infinite-distance limit in the scalar field moduli space of a consistent theory of quantum gravity must be accompanied by a tower of light particles whose masses scale exponentially with proper field distance ||$\phi$|| as m ~ exp(–λ||$\phi$||), where λ is order-one in Planck units. While the evidence for this conjecture is formidable, there is at present no consensus on which values of λ are allowed. In this paper, we propose a sharp lower bound for the lightest tower in a given infinite-distance limit in d dimensions: λ ≥ $1/\sqrt{d-2}$. In support of this proposal, we show that (1) it is exactly preserved under dimensional reduction, (2) it is saturated in many examples of string/M-theory compactifications, including maximal supergravity in d = 4 – 10 dimensions, and (3) it is saturated in many examples of minimal supergravity in d = 4 – 10 dimensions, assuming appropriate versions of the Weak Gravity Conjecture. We argue that towers with λ < $1/\sqrt{d-2}$ discussed previously in the literature are always accompanied by even lighter towers with λ ≥ $1/\sqrt{d-2}$, thereby satisfying our proposed bound. We discuss connections with and implications for the Emergent String Conjecture, the Scalar Weak Gravity Conjecture, the Repulsive Force Conjecture, large-field inflation, and scalar field potentials in quantum gravity. In particular, we argue that if our proposed bound applies beyond massless moduli spaces to scalar fields with potentials, then accelerated cosmological expansion cannot occur in asymptotic regimes of scalar field space in quantum gravity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Generating rotating spacetime in Ricci-based gravity: naked singularity as a black hole mimicker

Motivated by the lack of rotating solutions sourced by matter in General Relativity as well as in modified gravity theories, in this work we extend a recently discovered exact rotating solution of the minimal Einstein-scalar theory to its counterpart in Eddington-inspired Born-Infeld gravity coupled to a Born-Infeld scalar field. This is accomplished with the implementation of a well-developed mapping between solutions of Ricci-Based Palatini theories of gravity and General Relativity. The new solution is parametrized by the scalar charge and the Born-Infeld coupling constant apart from the mass and spin of the compact object. Compared to the spacetime prior to the mapping, we find that the high-energy modifications at the Born-Infeld scale are able to suppress but not remove the curvature divergence of the original naked null singularity. Depending on the sign of the Born-Infeld coupling constant, these modifications may even give rise to an additional timelike singularity exterior to the null one. In spite of that, both of the naked singularities before and after the mapping are capable of casting shadows, and as a consequence of the mapping relation, their shadows turn out to be identical as seen by a distant observer on the equatorial plane. Even though the scalar field induces a peculiar oblateness to the appearance of the shadow with its left and right endpoints held fixed, the closedness condition for the shadow contour sets a small upper bound on the absolute value of the scalar charge, which leads to observational features of the shadow closely resembling those of a Kerr black hole.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

Abstract If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologiesE 1 –E 6 ,E 11 ,E 12 ,E 16 , andE 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Astronomy & Astrophysics↗

DOE OSTI2024

Longer-lived mediators from charged mesons and photons at neutrino experiments

Since many of the dark-sector particles interact with Standard Model (SM) particles in multiple ways, they can appear in experimental facilities where SM particles appear in abundance. In this study, we explore a particular class of longer-lived mediators that are dominantly produced from photons and charged mesons that arise in proton-beam fixed-target-type neutrino experiments. This class of mediators encompasses light scalars that appear in theories like extended Higgs sectors, muon(electro)philic scalars, etc. We evaluate the sensitivities of these mediators at beam-based neutrino experiments such as the finished ArgoNeuT, ongoing MicroBooNE, SBND, ICARUS, and the upcoming DUNE experiment. We find that muonphilic scalars are more enhanced while produced from three-body decay of charged mesons. The above-mentioned experiments can probe unexplored regions of parameter space that can explain the current discrepancy in the anomalous magnetic moment of muons. We further find that Compton-like scattering of photons is the largest source of electrophilic scalars. By utilizing this, the DUNE Near Detector can explore new regions in the sensitivity space of electrophilic scalars. We also show that Bethe-Heitler scattering processes can be used to probe flavor-specific lepton final states even for the mediator masses below twice the lepton mass. Published by the American Physical Society 2024

Astronomy & Astrophysics↗

DOE OSTI2024

Staircase resiliency in a fluctuating cellular array

Inhomogeneous mixing by stationary convective cells set in a fixed array is a particularly simple route to layering. Layered profile structures, or staircases, have been observed in many systems, including drift-wave turbulence in magnetic confinement devices. The simplest type of staircase occurs in passive-scalar advection, due to the existence and interplay of two disparate timescales, the cell turn-over (τ H ), and the cell diffusion (τ D ) time. In this simple system, we study the resiliency of the staircase structure in the presence of global transverse shear and weak vortex scattering. The fixed cellular array is then generalized to a fluctuating vortex array in a series of numerical experiments. The focus is on regimes of low-modest effective Reynolds numbers, as found in magnetic fusion devices. By systematically perturbing the elements of the vortex array, we learn that staircases form and are resilient (although steps become less regular, due to cell mergers) over a broad range of Reynolds numbers. The criteria for resiliency are (a) τ D >>τ H and (b) a sufficiently high profile curvature (κ ≥ 1.5). We learn that scalar concentration travels along regions of shear, thus staircase barriers form first, and scalar concentration "homogenizes" in vortices later. The scattering of vortices induces a lower effective speed of scalar concentration front propagation. The paths are those of the least time. We observe that if background diffusion is kept fixed, the cell geometric properties can be used to derive an approximation for the effective diffusivity of the scalar. Furthermore, the effective diffusivity of the fluctuating vortex array does not deviate significantly from that of the fixed cellular array.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Holography vs. Scale Separation

In this work, we point out a contradiction between holography and scale-separated AdS (i.e. parametrically large mass gap) in string theory, making the standard assumption that the holographic CFT describes the IR degrees of freedom on a brane that decouple from gravity. We show that the CFT can only decouple from gravity if the scalar potential in the dual AdS satisfies a certain criterion. Namely, there must exist a scalar field trajectory that follows the gradient of the scalar potential to the asymptotic region of the scalar field space in which limit $\partial_ϕ\ln(V)\partial_ϕ\ln(Λ_s)\leq2/(d-2)$, where $Λ_s$ is the quantum gravity cut-off. This condition, which generically implies lack of scale-separation, is satisfied in the standard examples of AdS/CFT. However, proposed attempts at achieving scale separation, such as DGKT, employ scalar potentials that violate this condition. We therefore conclude that the CFT duals of DGKT vacua cannot exist in string theory. Barring fine-tuning, our conclusions apply to other Ricci-flat flux compactifications including the KKLT scenario which relies on scale separation to obtain a metastable de Sitter uplift.

FOS: Physical sciences↗

DOE OSTI2024

Resonant multiscalar production in the generic complex singlet model in the multi-TeV region

We develop benchmarks for resonant discalar production in the generic complex singlet scalar extension of the Standard Model (SM) with no additional symmetries, which contains two new scalars. These benchmarks maximize discalar resonant production modes at future 𝑝⁢𝑝 colliders: 𝑝⁢𝑝 → ℎ 2 → ℎ 1⁡ ℎ 1 , 𝑝⁢𝑝 → ℎ 2 → ℎ 1 ⁡ℎ 3 , and 𝑝⁢𝑝 → ℎ 2 → ℎ 3 ⁡ℎ 3 , where ℎ 1 is the observed SM-like Higgs boson and ℎ 2,3 are new scalars. The decays ℎ 2 → ℎ 1 ⁡ℎ 3 and ℎ 2 → ℎ 3 ⁡ℎ 3 may be the only way to discover ℎ 3 , leading to a discovery of two new scalars at once. Current LHC and projected future collider (HL-LHC, FCC-ee+HL-LHC, IL⁢C⁢500+HL-LHC) constraints on this model are used to produce benchmarks at the HL-LHC for ℎ 2 masses between 250 GeV and 1 TeV and a future 𝑝⁢𝑝 collider (FCC-hh) for ℎ 2 masses between 250 GeV and 12 TeV. We update the current LHC bounds on the singlet-Higgs boson mixing angle for these benchmarks. As the mass of ℎ 2 approaches the multi-TeV region, certain limiting behaviors of the maximum rates are uncovered due to theoretical constraints on the parameters. These limits, which can be derived analytically, are BR⁡(ℎ 2 → ℎ 1⁡ ℎ 1 ) → 0.25, BR⁡(ℎ 2 → ℎ 3 ⁡ℎ 3 ) → 0.5, and BR⁡(ℎ 2 → ℎ 1⁡ ℎ 3 ) → 0. It can also be shown that the maximum rates of 𝑝⁢𝑝 → ℎ 2 → ℎ 1 ⁡ℎ 1 and 𝑝⁢𝑝 → ℎ 2 → ℎ 3⁡ ℎ 3 approach the same value. Hence, all three ℎ 2 → ℎ 𝑖 ⁡ℎ 𝑗 decays are promising discovery modes for ℎ 2 masses at and below 𝒪⁡(1 TeV), while above 𝒪⁡(1 TeV) the decays ℎ 2 → ℎ 1 ⁡ℎ 1 and ℎ 2 → ℎ 3 ⁡ℎ 3 are more encouraging. We choose benchmark masses for ℎ 3 to produce a large range of decay signatures including multi-𝑏, multivector boson, and multi-SM-like Higgs production. As we will show, the behavior of the maximum rates leads to the surprising conclusion that in the multi-TeV region this model may be discovered in the Higgs quartet production mode via ℎ 2 → ℎ 3 ⁡ℎ 3 → 4⁢ℎ 1 decays before Higgs triple production is observed. The maximum di- and four Higgs production rates are similar in the multi-TeV range.

Extensions of Higgs sector↗

DOE OSTI2026

Data-based filtered dissipation rate modelling for multi-modal turbulent combustion: evaluating a priori model generalizability

Manifold-based models offer a computationally efficient alternative to directly transporting the thermochemical state in computational simulations of turbulent reacting flows, projecting the high-dimensional thermochemical state-space onto a low-dimensional manifold. Recent efforts have yielded a manifold-based model applicable to multi-modal combustion, enabling reconstruction of the thermochemical state from solutions to two-dimensional manifold equations in mixture fraction and generalized progress variable that are parameterised by three scalar dissipation rates. In coarse-grained simulations such as Large Eddy Simulation (LES), closure of the multi-modal manifold equations and subfilter variances/covariance requires closure of three filtered scalar dissipation rates. Here, the present work adopts a data-based approach, providing closure for the three filtered scalar dissipation rates via deep neural networks (DNNs). High-fidelity datasets corresponding to an autoigniting n-dodecane jet flame and a bluff body swirl-stabilized confined lifted spray flame of two aviation fuels (Jet-A and C1) with different ignition propensities are leveraged to generate training data that spans a diverse range of thermodynamic conditions and combustion modes, including low- and high-temperature ignition regimes in addition to premixed and nonpremixed behaviour. A final DNN model is trained to enforce inherent physical constraints by learning nonlinear functional transformations of the three filtered scalar dissipation rates. The generalizability of this constrained DNN model is demonstrated a priori via conditional statistics evaluated on the lifted spray flame with C1–a configuration that had not been included in the training data. Excellent DNN agreement with conditional DNS statistics is observed, and integrated gradients are computed to identify the most sensitive input variables. The similarity of the marginal PDFs of the most informative input variables and outputs across configurations are quantified via the Wasserstein metric, demonstrating that data-based models may successfully generalize to unseen parametric conditions so long as the most informative input variables share similar distributions across training and testing datasets.

Data-based modelling↗

DOE OSTI2023

Renormalization of the Standard Model Effective Field Theory from geometry

S-matrix elements are invariant under field redefinitions of the Lagrangian. They are determined by geometric quantities such as the curvature of the field-space manifold of scalar and gauge fields. We present a formalism where scalar and gauge fields are treated together, with a metric on the combined space of both types of fields. Scalar and gauge scattering amplitudes are given by the Riemann curvature Rijkl of this combined space, with indices i, j, k, l chosen to be scalar or gauge indices depending on the type of external particle. One-loop divergences can also be computed in terms of geometric invariants of the combined space, which greatly simplifies the computation of renormalization group equations. We apply our formalism to the Standard Model Effective Field Theory (SMEFT), and compute the renormalization group equations for even-parity bosonic operators to mass dimension eight.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗