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At least 865 records · Page 48
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Amplitude/operator basis in chiral perturbation theory

We establish a systematic construction of the on-shell amplitude/operator basis for Chiral Perturbation Theory (ChPT) in D = 4 spacetime dimensions and with an arbitrary number of flavors N f . For kinematic factors, we employ spinor-helicity variables to construct the soft blocks, which are local amplitudes satisfying the Adler’s zero condition, as well as to take into account the reduction in the kinematic basis due to the Gram determinant, which arises at O(p 10 ) when the number of multiplicity N in an amplitude becomes large: N > D. For flavor factors, we include group-theoretic relations at small N f , N f ≤ N, which decreases the flavor basis. The result is obtained by adapting the Young tensor method of constructing the operator basis for generic effective field theories to the case of non-linearly realized symmetries. Working in the massless quark limit, we present purely mesonic operators for both even- and odd-parity at O(p 6 ) and O(p 8 ) for N = 6 and arbitrary N f , and establish a direct correspondence between the amplitude basis and the operator basis. Furthermore, the redundancy due to the Gram determinant is studied at O(p 10 ) for N = 6, 8 and 10.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Improved Cross Section Generation Capability of Griffin in FY22

The Griffin code is a Multiphysics Object-Oriented Simulation Environment (MOOSE) based reactor multiphysics analysis application jointly developed by Idaho National Laboratory and Argonne National Laboratory. The code includes a variety of deterministic steady-state transport solvers for fixed source, k-eigenvalue, adjoint, and subcritical multiplication as well as transient solvers for spatial dynamics with the improved quasi-static method. Griffin uses cross section data in the ISOXML format generated from external deterministic or Monte Carlo cross section generation codes. In recent years, the MC 2 -3 modules have been added to Griffin for fast reactor cross section generation, and the self-shielding application programming interface (SSAPI) was implemented in the ISOXML module for thermal reactor cross section generation. The on-the-fly slowing down method and double-heterogeneity treatment have been implemented to SSAPI and verified against particulate fuel-bearing graphite-moderated thermal reactor problems with high accuracy. This year, work has been focused on improving the cross section generation capability of ISOXML and streamlining the cross section generation procedures. In addition, the form function data were added to ISOXML in order to support the pin power reconstruction capability that was newly implemented in Griffin in this fiscal year. To facilitate the cross section generation using MC 2 -3 and SSAPI in Griffin, the cross section generation workflows have been set up for both fast and thermal spectrum reactors. The MOOSE action system tool was devised for fast spectrum problems, and the MOOSE stochastic tool was adopted to the branch calculation procedure for thermal spectrum problems. Meanwhile, to ensure the accuracy of group-constants, the thermal up-scattering kernel calculator accounting for resonance scattering was implemented in ISOXML, demonstrating the accurate computation of a Doppler-broadened scattering kernel of any Legendre order within a reasonable timescale. Other aspects of ISOXML, such as deletion solver and data, documentation, ISOXML file management, and the interface for Mixture, were improved as well.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

Imaging grain microstructure in a model ceramic energy material with optically generated coherent acoustic phonons

Characterization of microstructure, chemistry and function of advanced energy materials remains a challenge for instrumentation science with tremendous opportunity for new discoveries. This active area of research is starting to make considerable strides with examples ranging from methodologies that employ bright X-rays, electron microscopy and optical spectroscopy. However, further development of instruments, capable of multimodal measurements, is necessary to fully reveal complex microstructure evolution in realistic environments and dynamical systems that are far from equilibrium. In this regard, laser-based instruments have a unique advantage as multiple measurement methodologies are easily combined into a single instrument using standard optics. A new pump-probe method that uses optically generated acoustic phonons is expanding standard optical characterization by providing depth resolved information. Here we report on a noteworthy extension of this pump-probe method to image grain microstructure in ceria. Rich information regarding the orientation of individual crystallites is obtained by monitoring acoustic phonon velocities and noting how the polarization of the probe beam influences the detected signal amplitude. When paired with other optical microscopies, this methodology will provide new perspectives for optical characterization of ceramic materials.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Intercomparison of airborne and surface-based measurements during the CLARIFY, ORACLES and LASIC field experiments

Data are presented from intercomparisons between two research aircraft, the FAAM BAe-146 and the NASA Lockheed P3, and between the BAe-146 and the surface-based DOE (Department of Energy) ARM (Atmospheric Radiation Monitoring) Mobile Facility at Ascension Island (8 S, 14.5W, a remote island in the mid-Atlantic). These took place from 17 August to 5 September 2017, during the African biomass burning season. The primary motivation was to give confidence in the use of data from multiple platforms with which to evaluate numerical climate models. The three platforms were involved in the CLouds-Aerosol-Radiation Interaction and Forcing for Year 2017 (CLARIFY-2017), ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES), and Layered Atlantic Smoke and Interactions with Clouds (LASIC) field experiments. Comparisons from flight segments on six days where the BAe-146 flew alongside the ARM facility on Ascension Island are presented, along with comparisons from wing-tip to wing-tip flight of the P3 and BAe-146 on 18 th August 2017. The intercomparison flight sampled a relatively clean atmosphere overlying a moderately polluted boundary layer, while the 6 fly-bys of the ARM site sampled both clean and polluted conditions 2–4 km upwind. We compare and validate characterisations of aerosol physical, chemical, and optical properties, atmospheric radiation, and cloud microphysics between platforms. We assess the performance of measurement instrumentation in the field, under conditions where sampling conditions are not tightly controlled as in laboratory measurements where calibrations are performed. Solar radiation measurements compared well between airborne platforms. Optical absorption coefficient measurements compared well across all three platforms, even though absolute magnitudes were often low (< 10 Mm -1 ) and close to the sensitivity limits of measurement instrumentation thereby confounding assessments of the comparability of absorption Ångström exponent characterisations. Aerosol absorption measurements from airborne platforms were more comparable than aircraft-to-ground observations. Scattering coefficient observations compared well between airborne platforms, but agreement with ground-based measurements was worse, potentially caused by small differences in sampling conditions or actual aerosol population differences. Chemical composition measurements followed a similar pattern, with better comparisons between the airborne platforms. Thermodynamics, aerosol, and cloud microphysical properties generally compared well.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Multipartite entanglement in the one-dimensional spin- 1 2 Heisenberg antiferromagnet

Multipartite entanglement refers to the simultaneous entanglement between multiple subsystems of a many-body quantum system. While multipartite entanglement can be difficult to quantify analytically, it is known that it can be witnessed through the quantum Fisher information (QFI), a quantity that can also be related to dynamical Kubo response functions. In this work, we first show that the finite temperature QFI can generally be expressed in terms of a static structure factor of the system, plus a correction that vanishes as T→0. This implies that the static structure factor witnesses multipartite entanglement near quantum critical points at temperatures below a characteristic energy scale of the system. Therefore, in systems with a known static structure factor, we can deduce finite temperature scaling of multipartite entanglement and low temperature entanglement depth without knowledge of the full dynamical response function of the system. This is particularly useful to study 1D quantum critical systems in which sub-power-law divergences can dominate entanglement growth, where the conventional scaling theory of the QFI breaks down. Furthermore, the 1D spin- 1 2 antiferromagnetic Heisenberg model is an important example of such a system, and we show that multipartite entanglement in the Heisenberg chain diverges nontrivially as ~ln(1/T) 3/2 . We verify these predictions with calculations of the QFI using conformal field theory and matrix product state simulations. Finally, we discuss the implications of our results for experiments to probe entanglement in quantum materials, comparing to neutron scattering data in KCuF 3 , a material well described by the Heisenberg chain.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Effect of dust on rainfall over the Red Sea coast based on WRF-Chem model simulations

Water is the single most important element of life. Rainfall plays an important role in the spatial and temporal distribution of this precious natural resource, and it has a direct impact on agricultural production, daily life activities, and human health. One of the important elements that govern rainfall formation and distribution is atmospheric aerosol, which also affects the Earth's radiation balance and climate. Therefore, understanding how dust compositions and distributions affect the regional rainfall pattern is crucial, particularly in regions with high atmospheric dust loads such as the Middle East. Although aerosol and rainfall research has garnered increasing attention as both an independent and interdisciplinary topic in the last few decades, the details of various direct and indirect pathways by which dust affects rainfall are not yet fully understood. Here, we explored the effects of dust on rainfall formation and distribution as well as the physical mechanisms that govern these phenomena, using high-resolution WRF-Chem simulations (~1.5 km × 1.5 km) configured with an advanced double-moment cloud microphysics scheme coupled with a sectional eight-bin aerosol scheme. Our model-simulated results were realistic, as evaluated from multiple perspectives including vertical profiles of aerosol concentrations, aerosol size distributions, vertical profiles of air temperature, diurnal wind cycles, and spatio-temporal rainfall patterns. Rainfall over the Red Sea coast is mainly caused by warm rain processes, which are typically confined within a height of ~6 km over the Sarawat mountains and exhibit a strong diurnal cycle that peaks in the evening at approximately 18:00 local time under the influence of sea breezes. Numerical experiments indicated that dust could both suppress or enhance rainfall. The effect of dust on rainfall was calculated as total, indirect, and direct effects, based on 10-year August-average daily-accumulated rainfall over the study domain covering the eastern Red Sea coast. For extreme rainfall events (domain-average daily-accumulated rainfall of ≥ 1.33 mm), the net effect of dust on rainfall was positive or enhancement (6.05 %), with the indirect effect (4.54 %) and direct effect (1.51 %) both causing rainfall increase. At a 5 % significance level, the total and indirect effects were statistically significant whereas the direct effect was not. For normal rainfall events (domain-average daily-accumulated rainfall < 1.33 mm), the indirect effect enhanced rainfall (4.76 %) whereas the direct effect suppressed rainfall (-5.78 %), resulting in a negative net suppressing effect (-1.02 %), all of which were statistically significant. We investigated the possible physical mechanisms of the effects and found that the rainfall suppression by dust direct effects was mainly caused by the scattering of solar radiation by dust. The surface cooling induced by dust weakens the sea breeze circulation, which decreases the associated landward moisture transport, ultimately suppressing rainfall. For extreme rainfall events, dust causes net rainfall enhancement through indirect effects as the high dust concentration facilitates raindrops to grow when the water vapor is sufficiently available. Our results have broader scientific and environmental implications. Specifically, although dust is considered a problem from an air quality perspective, our results highlight the important role of dust on sea breeze circulation and associated rainfall over the Red Sea coastal regions. Our results also have implications for cloud seeding and water resource management.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Comparison of Orientation Mapping in SEM and TEM

In this work, multiple experimental configurations for performing nanoscale orientation mapping are compared to determine their fidelity to the true 3D microstructure of a sample. Transmission Kikuchi Diffraction (TKD) experiments in an SEM and nanobeam diffraction (NBD) experiments in a TEM were performed on thin electrodeposited hard Au films with two different microstructures. The Au samples either have a grain size that is >50 nm or <20 nm. The same regions of the samples were measured with TKD apparatuses at 30 kV in an SEM with detectors in the horizontal and vertical configurations and in the TEM at 300 kV. Under the proper conditions, we demonstrate all three configurations can produce data of equivalent quality.Each method has strengths and challenges associated with its application and representation of the true microstructure. The conditions needed to obtain high-quality data for each acquisition method and the challenges associated with each will be discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Experimental demonstration of coupled multi-peak Bragg coherent diffraction imaging with genetic algorithms

Bragg coherent diffraction imaging has the potential to provide significant insight into the structure-properties relationship for crystalline materials by imaging, with nanoscale resolution, three-dimensional strain fields within individual grains and nanoparticles. The capability of present-day synchrotrons to locate and measure a multiplicity of Bragg reflections from a single grain makes it possible to recover the full strain tensor with nanometer resolution. Recent methods for coupling reconstructions from several peaks to determine the strain tensor have been developed and applied to synthetic data, but have not been applied to experimental data. Here, using a coupled genetic reconstruction algorithm, we reconstruct an experimental data set and demonstrate improvements in the ability to resolve vector-valued displacement fields internal to the particle as compared to what is achieved with a noncoupled approach. The coupled approach developed in this work was also validated on simulated data sets. In both simulated and experimental data, reconstructions from our coupled Bragg peak algorithm show improvements over the noncoupled independent reconstruction method of 5% in terms of accuracy and 53% in terms of consistency.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Resolving Configurational Disorder for Impurities in a Low-Entropy Phase

Hematite (α-Fe 2 O 3 ) exerts a strong control over the transport of minor but critical metals in the environment and is used in multiple industrial applications; the photocatalysis community has explored the properties of hematite nanoparticles over a wide range of transition metal dopants. Nonetheless, simplistic assumptions are used to rationalize the local coordination environment of impurities in hematite. Here, we use ab initio molecular dynamics (AIMD)-guided structural analysis to model the extended X-ray absorption fine structure (EXAFS) of Cu 2+ - and Zn 2+ -doped hematite nanoparticles. Specific defect–impurity associations were identified, and the local coordination environments of Cu and Zn both displayed considerable configurational disorder that, in aggregate, approached Jahn–Teller-like distortion for Cu but, in contrast, maintained hematite-like symmetry for Zn. This study highlights the role of defects in accommodating impurities in a nominally low-entropy phase and the limits to traditional shell-by-shell fitting of EXAFS for dopants/impurities in unprecedented bonding environments.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Overview of Nuclear Data Measurement and Analysis at RPI [Slides]

This presentation discusses the experimental, simulation, and nuclear data methods that were validated for the RPI γ-Multiplicity Detector. When the neutron capture γ-cascade data is well-known, the γ-emission spectra can be accurately calculated using the modified simulation tools. The RPI γ-Multiplicity Detector system is now ready for analysis and recommendations for isotopes with deficiencies in γ-ray data. The presentation also discusses future work which includes developing a method for analyzing and adjusting nuclear data for 59 Co, 55 Mn and other measured isotopes including 181 Ta. Additionally, future work includes comparing experimental γ-emission spectra with MCNP-6.2/DICEBOX simulations for 238 U and 235 U. In summation, new capture and transmission measurements for 54 Fe will help improve resonance parameter evaluation. Neutron capture gamma cascade spectra and yields were measured in the resolved resonance region and compared to evaluations. In addition, the pulsed neutron die-away method was developed as a tool to provide data for validation of TSLs.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Tunable Laser-Plasma Amplifier (Final Report)

Exploration at the laser intensity frontier has always offered new avenues for physics and reaching beyond this frontier is a grand challenge. Present-day petawatt-class lasers provide focused intensities on target of 10 22 W/cm 2 , corresponding to electric fields of 200 TV/m, while laser-plasma amplification opens a route for focused intensities well above 10 23 W/cm 2 . Intensities in this range provide the ability to test quantum electrodynamics in the unexplored low-energy, strong-field regime where signatures for new physics may arise. The behavior of matter under such extraordinary conditions is a rich and fascinating subject, not only in its own right in fundamental plasma physics, but for the many potential applications that promise to enrich the natural sciences in the future, including compact electron-beam, ion-beam particle accelerators, and ultra-bright X-ray sources. Although laser systems are now under construction internationally to access intensities of 10 23 W/cm 2 , the current technologies used appear to be fundamentally limited to these intensities. The realization of intensities beyond 10 23 W/cm 2 using parametric amplification in plasmas promises a breakthrough in high-energy density physics. Parametric amplification using Raman scattering in a plasma could provide the enabling technology for the generation of ultra-high-power laser pulses, but a more complete understanding of the nonlinear optics of plasmas is required. There is a significant gap in well-diagnosed laser-plasma instability studies of nonlinear plasma-wave phenomena, which are critical to understand for future laser-plasma devices. To achieve an efficient laser-plasma amplifier, plasma waves must be driven to large amplitude where significant energy can be rapidly transferred from the pump to the seed over the pulse duration of the seed. Simulations suggest that this nonlinear pump depletion regime can be achieved in the “pi-pulse” amplification regime. While simulations show this optimal regime with efficient amplification, it has remained elusive in experiments and there is a growing consensus within the community that thermal effects and pump beam limitations prevent laser-plasma amplifiers from progressing through the linear regime into the nonlinear pump depletion regime. Previous experiments have been significantly limited by the laser power available at the necessary wavelengths for the seed laser; therefore, the amplification is required to start in the linear regime where it is sensitive to many deleterious effects. The enabling technology (currently unique to plasma-wave amplification in the world) at the University of Rochester is the ability to provide a seed pulse with sufficient power (4 mJ/100 fs seed) to immediately drive nonlinear plasma waves into the pi-pulse regime and to tune its wavelength to optimize the efficiency of energy transfer. This in combination with the state-of-the-art OMEGA heater beams providing multiple kilojoules in a nanosecond to sufficiently heat the plasma make this system distinct from previous studies. These heater beams will provide, for the first time in Raman amplification studies, a homogeneous electron temperature high enough to prevent pump beam propagation issues that have plagued previous experiments. These systems will provide a platform for driving electron-plasma waves into the nonlinear regime where pump depletion and pulse shortening are predicted to lead to high amplification efficiencies (>30%). The Team has made significant progress through prior support from DOE Fusion Energy Sciences [DOE Office of Science Award Number DE-SC0016253 (2016-2022)]. This includes twenty-two peer-reviewed manuscripts, one patent, ten contributed talks presented at international conferences, and research that was highlighted as invited talks at fifteen international conferences. The broader impacts of this research are evident in the support of early career scientists, two Ph.D. theses, four current graduate students, a Masters Project, two undergraduate researchers, and an underrepresented minority student hired through the California Alliance for Minority Participation who now works as a Research Engineer in the group. This research met all of the funded research objectives and the highlights from primary Raman amplification thrust of this work are discussed below and form the foundation for the proposed research.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

All-multiplicity one-loop amplitudes in Born-Infeld electrodynamics from generalized unitarity

We initiate a study of non-supersymmetric Born-Infeld electrodynamics in 4d at the quantum level. Explicit all-multiplicity expressions are calculated for the purely rational one-loop amplitudes in the self-dual (+ + … +) and next-to-self-dual (- + … +) helicity sectors. Using a supersymmetric decomposition, d-dimensional unitarity cuts of the integrand factorize into tree-amplitudes in a 4d model of Born-Infeld photons coupled to a massive complex scalar. The two-scalar tree-amplitudes needed to construct the Born-Infeld integrand are computed using two complimentary approaches: (1) as a double-copy of Yang-Mills coupled to a massive adjoint scalar with a dimensionally reduced form of Chiral Perturbation Theory, and (2) by imposing consistency with low-energy theorems under a reduction from 4d to 3d and T-duality. The Born-Infeld integrand is integrated in d = 4 - 2ϵ dimensions at order Ο(ϵ 0 ) using the dimension-shifting formalism. We comment on the implications for electromagnetic duality in quantum Born-Infeld theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Dynamical moment fragmentation in the all-in all-out pyrochlore Nd 2 ⁢Sn 2⁢ O 7

We report single crystal neutron spectroscopy and bulk characterization on hydrothermally grown Nd 2 Sn 2 O 7 , revealing magnetic moment fragmentation embedded within the all-in–all-out ordered state. The spectra reveal a nearly flat band with pinch-point momentum dependence accompanied by dispersive branches that produce half-moon features across multiple Brillouin zones. These defining signatures are captured quantitatively by a minimal dipolar-octupolar spin Hamiltonian, demonstrating excellent agreement between experiment and theory. The higher flat-mode energy helps account for the absence of dynamical interference in prior μSR studies, while the lack of any photon-like excitation imposes strict constraints on the proposed Coulombic antiferromagnet scenario. Furthermore, our results extend moment fragmentation to Nd 2 Sn 2 O 7 and identify it as a clean, tractable platform for quantitative exploration of emergent gauge field physics in frustrated magnets.

DC susceptibility measurements↗
DOE OSTI2022

Gold-gold luminosity increase in RHIC for a beam energy scan with colliding beam energies extending below the nominal injection energy

The Beam Energy Scan phase II (BES-II), performed in the Relativistic Heavy Ion Collider (RHIC) from 2019 to 2021, explored the phase transition between quark-gluon plasma and hadronic gas. BES-II exceeded the goal of a fourfold increase in the average luminosity over that achieved during Beam Energy Scan phase I (BES-I), at five gold beam energies: 9.8, 7.3, 5.75, 4.59, and 3.85 GeV / nucleon. This was accomplished by addressing several beam dynamics effects, including intrabeam scattering, beam-beam, space charge, beam instability, and field errors induced by superconducting magnet persistent currents. Some of these effects are especially detrimental at low energies. BES-II achievements are presented, and the measures taken to improve RHIC performance are described. These measures span the whole RHIC complex, including ion beam sources, injectors, beam lifetime improvements in RHIC, and operation with the world’s first bunched beam Low Energy RHIC electron Cooler (LEReC).

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Application of sulfur SAD to small crystals with a large asymmetric unit and anomalous substructure

The application of sulfur single-wavelength anomalous dispersion (S-SAD) to determine the crystal structures of macromolecules can be challenging if the asymmetric unit is large, the crystals are small, the size of the anomalously scattering sulfur structure is large and the resolution at which the anomalous signals can be accurately measured is modest. Here, as a study of such a case, approaches to the SAD phasing of orthorhombic Ric-8A crystals are described. The structure of Ric-8A was published with only a brief description of the phasing process [Zeng et al. (2019), Structure , 27 , 1137–1141]. Here, alternative approaches to determining the 40-atom sulfur substructure of the 103 kDa Ric-8A dimer that composes the asymmetric unit are explored. At the data-collection wavelength of 1.77 Å measured at the Frontier micro-focusing Macromolecular Crystallography (FMX) beamline at National Synchrotron Light Source II, the sulfur anomalous signal strength, |Δ ano |/σΔ ano ( d ′′/sig), approaches 1.4 at 3.4 Å resolution. The highly redundant, 11 000 000-reflection data set measured from 18 crystals was segmented into isomorphous clusters using BLEND in the CCP 4 program suite. Data sets within clusters or sets of clusters were scaled and merged using AIMLESS from CCP 4 or, alternatively, the phenix.scale_and_merge tool from the Phenix suite. The latter proved to be the more effective in extracting anomalous signals. The HySS tool in Phenix , SHELXC / D and PRASA as implemented in the CRANK 2 program suite were each employed to determine the sulfur substructure. All of these approaches were effective, although HySS , as a component of the phenix.autosol tool, required data from all crystals to find the positions of the sulfur atoms. Critical contributors in this case study to successful phase determination by SAD included (i) the high-flux FMX beamline, featuring helical-mode data collection and a helium-filled beam path, (ii) as recognized by many authors, a very highly redundant, multiple-crystal data set and (iii) the inclusion within that data set of data from crystals that were scanned over large ω ranges, yielding highly isomorphous and highly redundant intensity measurements.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

A two-loop four-point form factor at function level

Recently, the maximally-helicity-violating four-point form factor for the chiral stress-energy tensor in planar $\mathcal{N}$ = 4 super Yang-Mills was computed to three loops at the level of the symbol associated with multiple polylogarithms. It exhibits antipodal self-duality, or invariance under the combined action of a kinematic map and reversing the ordering of letters in the symbol. Here we lift the two-loop form factor from symbol level to function level. We provide an iterated representation of the function’s derivatives (coproducts). In order to do so, we find a three-parameter limit of the five-parameter phase space where the symbol’s letters are all rational. We also use function-level information about dihedral symmetries and the soft, collinear, and factorization limits, as well as limits governed by the form-factor operator product expansion (FFOPE). We provide plots of the remainder function on several kinematic slices, and show that the result is compatible with the FFOPE data. We further verify that antipodal self-duality is valid at two loops beyond the level of the symbol.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗