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DOE OSTI2021

Recent Trends and Innovation in Additive Manufacturing of Soft Functional Materials

The growing demand for wearable devices, soft robotics, and tissue engineering in recent years has led to an increased effort in the field of soft materials. With the advent of personalized devices, the one-shape-fits-all manufacturing methods may soon no longer be the standard for the rapidly increasing market of soft devices. Recent findings have pushed technology and materials in the area of additive manufacturing (AM) as an alternative fabrication method for soft functional devices, taking geometrical designs and functionality to greater heights. For this reason, this review aims to highlights recent development and advances in AM processable soft materials with self-healing, shape memory, electronic, chromic or any combination of these functional properties. Furthermore, the influence of AM on the mechanical and physical properties on the functionality of these materials is expanded upon. Additionally, advances in soft devices in the fields of soft robotics, biomaterials, sensors, energy harvesters, and optoelectronics are discussed. Lastly, current challenges in AM for soft functional materials and future trends are discussed.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Low‐Volume Cores for Fabrication of Compact, Versatile, and Intelligent Soft Systems

Abstract This study introduces the low‐volume core (LVC) fabrication method, which enables the monolithic molding of compact, complex, versatile, and intelligent soft robotic systems. This method uses thin and flexible thermoplastic sheets to mold internal chambers in soft fluidic actuators, valves, and circuits. The LVC fabrication method creates low‐volume networks in soft actuators (LV‐net actuators) that can be made with compact and complex geometries, enabling both low actuation volume input and multi‐degree‐of‐freedom actuators. LVC fabrication can also be used for compact, completely soft, and monolithic logic components (valves with low‐volume core, also called as LV valves) to provide directional resistance as well as a switching mechanism that enables fluidic logic in soft systems. The compatibility of the fabrication methods for both soft actuators and valves facilitates the creation of compact, integrated, and versatile soft robotic systems with embodied intelligence. This study introduces two examples of such intelligent soft robotic systems that integrate both LV‐net actuators and LV valves to demonstrate capability for complex system fabrication.

Yu, Qifan↗

DOE OSTI2019

Soft bootstrap and supersymmetry

The soft bootstrap is an on-shell method to constrain the landscape of effective field theories (EFTs) of massless particles via the consistency of the low-energy S-matrix. Given assumptions on the on-shell data (particle spectra, linear symmetries, and low-energy theorems), the soft bootstrap is an efficient algorithm for determining the possible consistency of an EFT with those properties. The implementation of the soft bootstrap uses the recently discovered method of soft subtracted recursion. We derive a precise criterion for the validity of these recursion relations and show that they fail exactly when the assumed symmetries can be trivially realized by independent operators in the effective action. We use this to show that the possible pure (real and complex) scalar, fermion, and vector exceptional EFTs are highly constrained. Next, we prove how the soft behavior of states in a supermultiplet must be related and illustrate the results in extended supergravity. We demonstrate the power of the soft bootstrap in two applications. First, for the $ \mathcal{N} $ = 1 and $ \mathcal{N} $ = 2 C P 1 nonlinear sigma models, we show that on-shell constructibility establishes the emergence of accidental IR symmetries. This includes a new on-shell perspective on the interplay between $ \mathcal{N} $ = 2 supersymmetry, low-energy theorems, and electromagnetic duality. We also show that $ \mathcal{N} $ = 2 supersymmetry requires 3-point interactions with the photon that make the soft behavior of the scalar O (1) instead of vanishing, despite the underlying symmetric coset. Second, we study Galileon theories, including aspects of supersymmetrization, the possibility of a vector-scalar Galileon EFT, and the existence of higher-derivative corrections preserving the enhanced special Galileon symmetry. The latter is addressed both by soft bootstrap and by application of double-copy/KLT relations applied to higher-derivative corrections of chiral perturbation theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Universality (beyond leading log) of soft radiative corrections to $\hat q$ in $p_\perp$ broadening and energy loss

It has been known for many years that soft radiation can give potentially large double-logarithm corrections to $p_\perp$ broadening of a high-energy particle traveling through QCD matter, but that this soft radiation correction can be absorbed into an effective value $\hat q_{eff}$ for the medium $p_\perp$-broadening parameter $\hat q$. Here ``soft'' means high energy compared to medium scales but soft compared to the original high-energy particle traveling through the medium. A similar situation arises in the case of soft corrections to hard splitting of a high-energy particle, such as hard g → gg, where double logarithms can also be absorbed using the same effective $\hat q_{eff}$. In this paper, I study whether the same holds true for potentially large, subleading, {\it single}-logarthim corrections. The correspondence is more indirect for single logarithms, but I show (in the large-$N_c$ limit) that single logarithms from soft radiation in the case of $p_\perp$ broadening also determine the single logarithms from soft radiative corrections to hard g → gg splitting. Along the way, there is an interesting variation of the original BDMPS-Z calculation of splitting rates in the q(hat) approximation. I also discuss how, for soft-radiative corrections to hard splitting processes, there are two different types of ''$\hat q_{eff}$'' that come into play, which differ by ''$i\pi$'' terms that multiply single logarithms.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2018

Recursive Soft Drop

We introduce a new jet substructure technique called Recursive Soft Drop, which generalizes the Soft Drop algorithm to have multiple grooming layers. Like the original Soft Drop method, this new recursive variant traverses a jet clustering tree to remove soft wide-angle contamination. By enforcing the Soft Drop condition N times, Recursive Soft Drop improves the jet mass resolution for boosted hadronic objects like W bosons, top quarks, and Higgs bosons. We further show that this improvement in mass resolution persists when including the effects of pileup, up to large pileup multiplicities. In the limit that N goes to infinity, the resulting groomed jets formally have zero catchment area. As an alternative approach, we present a bottom-up version of Recursive Soft Drop which, in its local form, is similar to Recursive Soft Drop and which, in its global form, can be used to perform event-wide grooming.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Soft Materials Constructed Using Calix[4]pyrrole- and “Texas-Sized” Box-Based Anion Receptors

We present that soft materials have received considerable attention from supramolecular chemists and material scientists alike. This interest reflects the advantages provided by their soft, flexible nature and the convenience of the molecular self-assembly that underlies their preparation. Common soft supramolecular materials include polymeric gels, supramolecular polymers, nanoaggregates, and membranes. Polymeric gels are solidlike networks of cross-linked polymer chains. Supramolecular polymers contain repeat units connected through reversible non-covalent bonds. Nanoaggregates are formed as a result of hydrophobic interactions involving amphiphilic building blocks. Because of the presence of non-covalent interactions, supramolecular soft materials typically display stimuli-responsive or adaptive features. Various macrocyclic hosts, such as cyclodextrins, crown ethers, calixarenes, cucurbiturils, and pillararenes, and many classic non-covalent interactions have been harnessed to construct supramolecular soft materials. Only recently has anion binding been used as the underlying recognition motif. Anions are ubiquitous in the natural world. Their importance has inspired efforts to achieve good anion binding and to exploit anion recognition in a number of fields, including extraction, transport, sensing, and catalysis. Most of this effort has involved the use of stand-alone anion receptors. On the other hand, soft materials with anion recognition features could lead to new macromolecular systems of interest in the context of many application areas. In this Account, we summarize the latest efforts from our laboratory to prepare supramolecular soft materials, including polymeric gels, supramolecular polymers, and nanoaggregates, with bona fide anion recognition features. Two anion receptor systems, namely, calix[4]pyrroles (C4Ps) and a tetraimidazolium macrocycle known as the “Texas-sized” molecular box (TxSB), have been used for this purpose. To date, TxSB-based hydrogels have been utilized to capture anions from water and for coded information applications; C4P-based organic polymeric gels have been used to extract dianions from aqueous source phases and for the on-site detection of chloride anions. Polymers containing C4P and TxSB anion recognition subunits typically display responsive features and can be modified through application of appropriately chosen external stimuli. For instance, nanoaggregates may be formed as a result of the hydrophobic interactions of C4P- and TxSB-based amphiphiles. The resulting aggregates were found to mimic the structural evolution of organelles and could be used as effective anion and ion pair extractants. This Account summarizes progress to date while underscoring potential opportunities associated with combining anion recognition and soft materials chemistry. Finally, the hope is to stimulate further advances in broad areas, including polymer science, supramolecular chemistry, biology, materials research, and information storage.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2015

Introduction of Soft X-Ray Spectromicroscopy as an Advanced Technique for Plant Biopolymers Research

Soft X-ray absorption spectroscopy coupled with nano-scale microscopy has been widely used in material science, environmental science, and physical sciences. In this work, the advantages of soft X-ray absorption spectromicroscopy for plant biopolymer research were demonstrated by determining the chemical sensitivity of the technique to identify common plant biopolymers and to map the distributions of biopolymers in plant samples. The chemical sensitivity of soft X-ray spectroscopy to study biopolymers was determined by recording the spectra of common plant biopolymers using soft X-ray and Fourier Transform mid Infrared (FT-IR) spectroscopy techniques. The soft X-ray spectra of lignin, cellulose, and polygalacturonic acid have distinct spectral features. However, there were no distinct differences between cellulose and hemicellulose spectra. Mid infrared spectra of all biopolymers were unique and there were differences between the spectra of water soluble and insoluble xylans. The advantage of nano-scale spatial resolution exploited using soft X-ray spectromicroscopy for plant biopolymer research was demonstrated by mapping plant cell wall biopolymers in a lentil stem section and compared with the FT-IR spectromicroscopy data from the same sample. The soft X-ray spectromicroscopy enables mapping of biopolymers at the sub-cellular (~30 nm) resolution whereas, the limited spatial resolution in the micron scale range in the FT-IR spectromicroscopy made it difficult to identify the localized distribution of biopolymers. The advantages and limitations of soft X-ray and FT-IR spectromicroscopy techniques for biopolymer research are also discussed.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2014

An Overview of the Anomalous Soft Photons in Hadron Production

Measurements of soft photon with low transverse momenta in high-energy hadron-hadron collisions and e+-e- annihilations indicate that they are consistently produced in excess of what are predicted by electromagnetic bremsstrahlung when hadrons (mostly mesons) are produced, but they agree with electromagnetic bremsstrahlung predictions in the absence of hadron production. These excess soft photons are called anomalous soft photons. The occurrence of anomalous soft photons in association with hadron production reveals the presence of additional QED soft photon sources in QCD hadron production. Many different models have been proposed to explain the anomalous soft production. We shall examine specifically a quantum field theory of simultaneous meson and soft photon production in QCD×QED in which the meson production arises from the oscillation of color charge densities of the quarks of the underlying vacuum in a flux tube. As a quark carries both a color charge and an electric charge, the oscillation of the color charge densities will be accompanied by the oscillation of electric charge densities, which will in turn lead to the simultaneous production of anomalous soft photons during the meson production process.

Wong, Cheuk-Yin↗

DOE OSTI2021

Advanced Electron Microscopy of Nanophased Synthetic Polymers and Soft Complexes for Energy and Medicine Applications

After decades of developments, electron microscopy has become a powerful and irreplaceable tool in understanding the ionic, electrical, mechanical, chemical, and other functional performances of next-generation polymers and soft complexes. The recent progress in electron microscopy of nanostructured polymers and soft assemblies is important for applications in many different fields, including, but not limited to, mesoporous and nanoporous materials, absorbents, membranes, solid electrolytes, battery electrodes, ion- and electron-transporting materials, organic semiconductors, soft robotics, optoelectronic devices, biomass, soft magnetic materials, and pharmaceutical drug design. For synthetic polymers and soft complexes, there are four main characteristics that differentiate them from their inorganic or biomacromolecular counterparts in electron microscopy studies: (1) lower contrast, (2) abundance of light elements, (3) polydispersity or nanomorphological variations, and (4) large changes induced by electron beams. Since 2011, the Center for Nanophase Materials Sciences (CNMS) at Oak Ridge National Laboratory has been working with numerous facility users on nanostructured polymer composites, block copolymers, polymer brushes, conjugated molecules, organic–inorganic hybrid nanomaterials, organic–inorganic interfaces, organic crystals, and other soft complexes. This review crystalizes some of the essential challenges, successes, failures, and techniques during the process in the past ten years. It also presents some outlooks and future expectations on the basis of these works at the intersection of electron microscopy, soft matter, and artificial intelligence. Machine learning is expected to automate and facilitate image processing and information extraction of polymer and soft hybrid nanostructures in aspects such as dose-controlled imaging and structure analysis.

36 MATERIALS SCIENCE↗

NASA NTRS1989

Solar flare hard and soft x ray relationship determined from SMM HXRBS and BCS data

The exact nature of the solar flare process is still somewhat a mystery. A key element to understanding flares if the relationship between the hard x rays emitted by the most energetic portions of the flare and the soft x rays from other areas and times. This relationship was studied by comparing hard x ray light curved from the Hard X-Ray Burst Spectrometer (HXRBS) with the soft x ray light curve and its derivation from the Bent Crystal Spectrometer (BCS) which is part of the X-Ray Polychrometer (XRP), these instruments being on the Solar Maximum Mission spacecraft (SMM). Data sample was taken from flares observed with the above instruments during 1980, the peak of the previous maximum of solar activity. Flares were chosen based on complete coverage of the event by several instruments. The HXRBS data covers the x ray spectrum from about 25 keV to about 440 keV in 15 spectral channels, while the BCS data used covers a region of the Spectrum around 3 angstroms including emission from the Ca XIX ion. Both sets of data were summed over their spectral ranges and plotted against time at a maximum time resolution of around 3 seconds. The most popular theory of flares holds that a beam of electrons produces the hard x rays by bremsstrahlung while the soft x rays are the thermal response to this energy deposition. The question is whether the rate of change of soft x ray emission might reflect the variability of the electron beam and hence the variability of the hard x rays. To address this, we took the time derivative of the soft x ray light curve and compared it to the hard flares, 12 of them showed very closed agreement between the soft x ray derivative and the hard x ray light curve. The other five did not show this behavior but were similar to each other in general soft x ray behavior. Efforts to determine basic differences between the two kinds of flares continue. In addition the behavior of soft x ray temperature of flares was examined.

Toot, G. David↗

NASA NTRS1994

Impulsive behavior in solar soft X-radiation

The Yohkoh soft X-ray telescope has observed impulsive, thermal, soft X-ray emission at the footpoints of magnetic loops during solar flares. The soft X-ray (thermal) time profiles at the footpoints closely match the hard X-ray (nonthermal) time profiles, directly demonstrating the heating of the lower solar atmosphere on short timescales during the interval of nonthermal energy release. This phenomenon is the rule, rather than the exception, occurring in the majority of flares that we have examined with the Yohkoh data. We illustrate the impulsive behavior with data from the major flare of 1992 January 26. For this flare, the soft X-ray peak times matched the hard X-ray peak times within the time resolution of the soft X-ray measurements (about 10 s), and the soft and hard X-ray locations match within the resolution of the hard X-ray imager. The impulsive soft X-ray emission clearly has a thermal spectral signature, but not at the high temperature of a 'superhot' source. We conclude that the impulsive soft X-ray emission comes from material heated by precipitating electrons at loop footpoints and evaporating from the deeper atmosphere into the flaring flux tube.

Hudson, H. S.↗

NASA NTRS1994

The very soft X-ray emission of X-ray-faint early-type galaxies

A recent reanaylsis of Einstein data, and new ROSAT observations, have revealed the presence of at least two components in the X-ray spectra of X-ray faint early-type galaxies: a relatively hard component (kT greater than 1.5 keV), and a very soft component (kT approximately 0.2-0.3 keV). In this paper we address the problem of the nature of the very soft component and whether it can be due to a hot interstellar medium (ISM), or is most likely originated by the collective emission of very soft stellar sources. To this purpose, hydrodynamical evolutionary sequences for the secular behavior of gas flows in ellipticals have been performed, varying the Type Ia supernovae rate of explosion, and the dark matter amount and distribution. The results are compared with the observational X-ray data: the average Einstein spectrum for six X-ray faint early-type galaxies (among which are NGC 4365 and NGC 4697), and the spectrum obtained by the ROSAT pointed observation of NGC 4365. The very soft component could be entirely explained with a hot ISM only in galaxies such as NGC 4697, i.e., when the depth of the potential well-on which the average ISM temperature strongly depends-is quite shallow; in NGC 4365 a diffuse hot ISM would have a temperature larger than that of the very soft component, because of the deeper potential well. So, in NGC 4365 the softest contribution to the X-ray emission comes certainly from stellar sources. As stellar soft X-ray emitters, we consider late-type stellar coronae, supersoft sources such as those discovered by ROSAT in the Magellanic Clouds and M31, and RS CVn systems. All these candidates can be substantial contributors to the very soft emission, though none of them, taken separately, plausibly accounts entirely for its properties. We finally present a model for the X-ray emission of NGC 4365, to reproduce in detail the results of the ROSAT pointed observation, including the Position Sensitive Proportional Counter (PSPC) spectrum and radial surface brightness distribution. The present data may suggest that the X-ray surface brightness is more extended than the optical profile. In this case, a straightforward explanation in terms of stellar sources could not be satisfactory. The available data can be better explained with three different contributions: a very soft component of stellar origin, a hard component from X-ray binaries, and an approximately 0.6 keV hot ISM. The latter can explain the extended X-ray surface brightness profile, if the galaxy has a dark-to-luminous mass ratio of 9, with the dark matter very broadly distributed, and a SN Ia explosive rate of approximately 0.6 the Tammann rate.

Pellegrini, S.↗

DOE OSTI2025

Soft and Stretchable Thienopyrroledione–Based Polymers via Direct Arylation

π-conjugated polymers (CPs) that are concurrently soft and stretchable are needed for deformable electronics. Molecular-level modification of indacenodithiophene (IDT) copolymers, a class of CPs that exhibit high hole mobilities (μ hole ), is an approach that can help realize intrinsically soft and stretchable CPs. Numerous examples of design strategies to adjust the stretchability of CPs exist, but imparting softness is comparatively less studied. In this study, a systematic molecular weight (MW) series is constructed on a promising candidate for soft CPs, poly(indacenodithiophene-co-thienopyrroledione) (p(IDT C16- TPD C8 )), by optimizing direct arylation polymerization conditions in hopes of improving stretchability and μhole without significantly impacting softness. We found p(IDT C16- TPD C8 ) at a degree of polymerization of 32 shows high stretchability (crack onset strain, CoS > 100%) without significantly impacting softness (elastic modulus, E = 32 MPa), which to the best of our knowledge outperforms previously reported stretchable and soft CPs. To further study how molecular-level modifications impact polymer properties, a MW series of a new extended donor unit polymer, poly(indacenodithienothiophene-co-thienopyrroledione) (p(IDTT C16- TPD C8 )), was synthesized. The IDTT C16 copolymers did not result in a greater average μ hole when comparing between p(IDTT C16- TPD C8 ) and p(IDT C16- TPD C8 ) despite their higher crystallinity observed by GIWAXS. While these findings warrant further investigation, this study points toward unique charge transport properties of IDT-based polymers.

direct arylation polymerization↗

DOE OSTI2014

In Situ SIMS and IR Spectroscopy of Well-Defined Surfaces Prepared by Soft Landing of Mass-Selected Ions

Soft landing of mass-selected ions onto surfaces is a powerful approach for the highly-controlled preparation of materials that are inaccessible using conventional synthesis techniques. Coupling soft landing with in situ characterization using secondary ion mass spectrometry (SIMS) and infrared reflection absorption spectroscopy (IRRAS) enables analysis of well-defined surfaces under clean vacuum conditions. The capabilities of three soft-landing instruments constructed in our laboratory are illustrated for the representative system of surface-bound organometallics prepared by soft landing of mass-selected ruthenium tris(bipyridine) dications, [Ru(bpy)3]2+, onto carboxylic acid terminated self-assembled monolayer surfaces on gold (COOH-SAMs). In situ time-of-flight (TOF)-SIMS provides insight into the reactivity of the soft-landed ions. In addition, the kinetics of charge reduction, neutralization and desorption occurring on the COOH-SAM both during and after ion soft landing are studied using in situ Fourier transform ion cyclotron resonance (FT-ICR)-SIMS measurements. In situ IRRAS experiments provide insight into how the structure of organic ligands surrounding metal centers is perturbed through immobilization of organometallic ions on COOH-SAM surfaces by soft landing. Collectively, the three instruments provide complementary information about the chemical composition, reactivity and structure of well-defined species supported on surfaces.

in situ↗

NASA NTRS2016

Soft-Robotic Rover with Electrodynamic Power Scavenging: NIAC Final Report - Phase 1

The purpose of studying the capabilities of Electrodynamic Tethers (EDT) and Soft Robotics is to ascertain the feasibility of using cross-cutting EDT and soft robotics technologies to achieve future NASA mission objectives with mass and power budgets orders of magnitude lower than conventional spacecraft. In this context, the Phase I study focuses on three technological elements: the design of a soft-robotic rover that can operate in extraterrestrial oceans, demonstrating feasibility of electrodynamic tethers for power scavenging in the Europa environment, and utilizing electrolysis to power biomimetic propulsion.The Phase I results show that a soft robotic, underwater rover has many advantages over a traditional view of autonomous underwater vehicles. Many of these advantages stem from its ability to collapse or expand the body, which carries two key benefits: (i) cost savings in transport and (ii) buoyancy control. Furthermore, this rover's material offers properties that enable it to survive most oceanic conditions, withstand a likely radiation environment, and retard ice formation. The use of these soft robots under water is very attractive because buoyancy enables very large robots without the need for skeletal structures that limit their shape-changing ability. The prime limitation of soft robots for underwater exploration is their nascent state of development, an issue that this study has begun to address and that we hope to continue in Phase II. The theoretical calculations and experimental investigation on electrodynamic tethers discussed in this report show that their use in saltwater environments is feasible. However, magneto hydrodynamic effects require attention, which will be a priority in Phase II. A possible approach involves magnetic shielding of a portion of the EDT array to generate significant current from imposed alternating magnetic fields. The Phase I experiments show that this approach may enable enough power to be generated for a soft robotic rover of the scale contemplated here. This power is in the range of 1mW to 1W and determines the time required to collect and transmit science data.

Soft Robotics↗

DOE OSTI2018

Soft factorization in QED from 2D Kac-Moody symmetry

The soft factorization theorem for 4D abelian gauge theory states that the S -matrix factorizes into soft and hard parts, with the universal soft part containing all soft and collinear poles. Similarly, correlation functions on the sphere in a 2D CFT with a U(1) Kac-Moody current algebra factorize into current algebra and non-current algebra factors, with the current algebra factor fully determined by its pole structure. In this paper, we show that these 4D and 2D factorizations are mathematically the same phenomena. The soft ‘t Hooft-Wilson lines and soft photons are realized as a complexified 2D current algebra on the celestial sphere at null infinity. The current algebra level is determined by the cusp anomalous dimension. The associated complex U(1) boson lives on a torus whose modular parameter is τ=$\frac{2πi}{e^2}$ + $\frac{θ}{2π}$. The correlators of this 2D current algebra fully reproduce the known soft part of the 4D S -matrix, as well as a conjectured generalization involving magnetic charges.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Soft-collinear gravity beyond the leading power

We construct “soft-collinear gravity”, the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion. Despite the absence of collinear divergences in gravity at leading power, the construction exhibits remarkable similarities with soft-collinear effective theory of QCD (gauge fields). It reveals an emergent soft background gauge symmetry, which allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative, the soft Riemann tensor, and a covariant generalisation of the collinear light-cone gauge metric field. The gauge symmetries control both the unsuppressed collinear field components and the inherent inhomogeneity in λ of the invariant objects to all orders, resulting in a consistent expansion.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Soft gluon emission at two loops in full color

The soft emission factor is a central ingredient in the factorization of generic n-particle gauge theory amplitudes with one soft gluon in the external state. We present the complete two-loop soft factor, capturing the leading power behavior in the soft-gluon momentum. At two loops, the color structure and the kinematic dependence of the soft factor become nontrivial as the soft gluon can couple to three hard partons for the first time (tripole terms). The nontrivial kinematic dependence of the tripole terms is of uniform, maximal transcendental weight, and can be expressed (in a “Euclidean” region) in terms of single-valued harmonic polylogarithms. Our results are consistent with the behavior of the recently computed symbol of the two-loop five-particle amplitude in $Ν$ = 4 super-Yang-Mills theory. In the limit where the outgoing soft gluon is also collinear with an incoming hard parton, potentially dangerous factorization-violating terms can arise.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗