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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Split-helicity tree amplitudes and flag cluster algebras

Recent work has uncovered a connection between the symbol letters of general massless scattering and (permutations of) cluster variables of partial flag varieties. In this paper we explore the cluster adjacency of tree-level gluon amplitudes, specifically focusing on split-helicity amplitudes which can be written in closed form in terms of zigzag diagrams. We check in several cases, and conjecture in general, that the poles in each term satisfy cluster adjacency under a set of permutations that is built from arc permutations of the corresponding zigzag.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Landau singularities from the amplituhedron

We propose a simple geometric algorithm for determining the complete set of branch points of amplitudes in planar N = 4 super-Yang-Mills theory directly from the amplituhedron, without resorting to any particular representation in terms of local Feynman integrals. This represents a step towards translating integrands directly into integrals. In particular, the algorithm provides information about the symbol alphabets of general amplitudes. We illustrate the algorithm applied to the one- and two-loop MHV amplitudes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

We present an all-loop dispersion integral, well-defined to arbitrary logarithmic accuracy, describing the multi-Regge limit of the 2 → 5 amplitude in planar N = 4 super Yang-Mills theory. It follows from factorization, dual conformal symmetry and consistency with soft limits, and specifically holds in the region where the energies of all produced particles have been analytically continued. After promoting the known symbol of the 2-loop N -particle MHV amplitude in this region to a function, we specialize to N = 7, and extract from it the next-to-leading order (NLO) correction to the BFKL central emission vertex, namely the building block of the dispersion integral that had not yet appeared in the well-studied six-gluon case. As an application of our results, we explicitly compute the seven-gluon amplitude at next-to-leading logarithmic accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the two independent NMHV helicity configurations, respectively.

1/N expansion↗
DOE OSTI2018

The double pentaladder integral to all orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N = 4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. Here in this paper, we describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.

1/N Expansion↗
DOE OSTI2022

Vector boson pair production at one loop: analytic results for the process $q\bar{q}$$ℓ\bar{ℓ}$$ℓ'\bar{ℓ}'g$

We present compact analytic results for the one-loop amplitude for the process 0 → $q\bar{q}$$ℓ\bar{ℓ}$$ℓ'\bar{ℓ}'g$, relevant for both the production of a pair of Z and W-bosons in association with a jet. We focus on the gauge-invariant contribution mediated by a loop of quarks. We explicitly include all effects of the loop-quark mass m, appropriate for the production of a pair of Z-bosons. In the limit m → 0, our results are also applicable to the production of W-boson pairs, mediated by a loop of massless quarks. Implemented in a numerical code, the results are fast. The calculation uses novel advancements in spinor-helicity simplification techniques, for the first time applied beyond five-point massless kinematics. We make use of primary decompositions from algebraic-geometry, which now involve non-radical ideals, and p-adic numbers from number theory. We show how to infer whether numerator polynomials belong to symbolic powers of non-radical ideals through numerical evaluations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Bootstrapping a stress-tensor form factor through eight loops

We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planar $\mathcal{N}$ = 4 supersymmetric Yang-Mills theory at six, seven, and eight loops, using boundary data from the form factor operator product expansion. This may represent the highest perturbative order to which multi-variate quantities in a unitary four-dimensional quantum field theory have been computed. In computing this form factor, we observe and employ new restrictions on pairs and triples of adjacent letters in the symbol. We provide details about the function space required to describe the form factor through eight loops. Plotting the results on various lines provides striking numerical evidence for a finite radius of convergence of perturbation theory. By the principle of maximal transcendentality, our results are expected to give the highest weight part of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD through eight loops. These results were also recently used to discover a new antipodal duality between this form factor and a six-point amplitude in the same theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Approximate CFTs and random tensor models

Abstract A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS 3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Physics↗
DOE OSTI2015

Hedgehog bases for A n cluster polylogarithms and an application to six-point amplitudes

Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster algebra structure. In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions, at any weight, whose symbol alphabet consists of cluster coordinates on the A n cluster algebra. As a result, using such a basis we present a new expression for the 2-loop 6-particle NMHV amplitude which makes some of its cluster structure manifest.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Characterizing Topological Order with Matrix Product Operators

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of $\mathsf G$- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation

Motivated by various applications, unbounded Hamiltonian simulation has recently garnered great attention. Quantum Magnus algorithms, designed to achieve commutator scaling for time-dependent Hamiltonian simulation, have been found to be particularly efficient for such applications. When applied to unbounded Hamiltonian simulation in the interaction picture, they exhibit an unexpected superconvergence phenomenon. However, existing proofs are limited to the spatially continuous setting and do not extend to discrete spatial discretizations. Here, in this work, we provide the first superconvergence estimate in the fully discrete setting with a finite number of spatial discretization points N, and show that it holds with an error constant uniform in N. The proof is based on the two-parameter symbol class, which, to our knowledge, is applied for the first time in algorithm analysis. The key idea is to establish a semiclassical framework by identifying two parameters through the discretization number and the time step size rescaled by the operator norm, such that the semiclassical uniformity guarantees the uniformity of both. This approach may have broader applications in numerical analysis beyond the specific context of this work.

Borns-Weil, Yonah [University of California, Berke↗
DOE OSTI2012

New type of possible high-pressure polymorphism in NiAs minerals in planetary cores

The nickel arsenide (B81) and related crystal structures are among the most important crystallographic arrangements assumed by Fe and Ni compounds with light elements such as Si, O, S, and P, expected to be present in planetary cores. Despite the simple structure, some of these materials like troilite (FeS) exhibit complex phase diagrams and rich polymorphism, involving significant changes in interatomic bonding and physical properties. NiP (oP16) represents one of the two principal structure distortions found in the nickel arsenide family and is characterized by P–P bonding interactions that lead to the formation of P2 dimers. In the current study, the single-crystal synchrotron X-ray diffraction technique, aided by first principles density functional theory (DFT) calculations, has been applied to examine the compression behavior of NiP up to 30 GPa. Two new reversible displacive phase transitions leading to orthorhombic high-pressure phases with Pearson symbols oP40 and oC24 were found to occur at approximately 8.5 and 25.0 GPa, respectively. The oP40 phase has the primitive Pnma space group with unit cell a = 4.7729(5) Å, b = 16.6619(12) Å, and c = 5.8071(8) Å at 16.3(1) GPa and is a superstructure of the ambient oP16 phase with multiplicity of 2.5. The oC24 phase has the acentric Cmc21 space group with unit cell a = 9.695(6) Å, b = 5.7101(9) Å, and c = 4.7438(6) Å at 28.5(1) GPa and is a superstructure of the oP16 phase with multiplicity of 1.5. DFT calculations fully support the observed sequence of phase transitions. The two new phases constitute logical next stages of P sublattice polymerization, in which the dilution of the P3 units, introduced in the first high-pressure phase, decreases, leading to compositions of Ni20(P3)4(P2)4 and Ni12(P3)4, and provide important clues to understanding of phase relations and transformation pathways in the NiAs family.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Spatiotemporal pattern detection, generation, and computation with circuits

Abstract Implementations of neurons, delays, and synapse circuits are presented with simulations. These neural elements are used to create two small spiking neural networks, the Rate-Window and Order-Biased clusters, which are capable of detecting simple two-spike spatiotemporal patterns. A simple pattern detecting network (SPDN) is created by combining the Rate-Window and Order-Biased clusters, where clusters are small spiking neural networks, and its simple pattern detection ability is demonstrated in simulation. The SPDN is used to implement a complex pattern detecting network (CPDN) and its complex pattern detection ability is demonstrated in simulation. Methods for generating arbitrary spatiotemporal patterns are presented. The CPDN and spatiotemporal pattern generation methods are then used to implement a novel spatiotemporal computing paradigm based on detecting and responding to spatiotemporal symbols. A simulation of a spatiotemporal half adder is presented to demonstrate the computing paradigm.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Combining static analysis and state transition graphs for verification of event-condition-action systems in the RERS 2012 and 2013 challenges

In this work, we present a combination of approaches for the verification of event-condition-action (ECA) systems. The analyzed ECA systems range from structurally simple to structurally complex systems. We address the verification of reachability properties and behavioral properties. Reachability properties are represented by assertions in the program and we determine statically whether an assertion holds for all execution paths. Behavioral properties are represented as linear temporal logic formulas specifying the input/output behavior of the program. Our approach assumes a finite state space. We compare a symbolic analysis with an exhaustive state space exploration and discuss the trade-offs between the approaches in terms of the number of computed states and run-time behavior. All variants compute a state transition graph which can also be passed to an LTL verifier. The variants have a different impact on the number of computed states in the state transition graph which in turn impacts the run-time and memory consumption of subsequent phases. Finally, we evaluate the different analysis variants with the RERS benchmarks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Speeding-up fuzzing through directional seeds

Abstract Fuzzing is an automated process for discovering inputs in a program that may trigger unexpected behavior. Today, fuzzing has become a standard practice for the discovery of bugs and security vulnerabilities. However, the main issue with such practices is that the exploration of the input space of programs can often be prohibitively expensive. Therefore, several alternative fuzzing strategies have been introduced during the last few years. Some fuzzing techniques rely on human expertise to provide a plausible set of initial input examples, namely, seeds. However, the process of handcrafting seeds for fuzzing purposes often becomes strenuous for humans as it requires a deeper understanding of the Program-Under-Test (PUT). Also, the use of known inputs to programs often does not trigger vulnerable program behavior or may not reach potentially vulnerable code locations. To address those issues, we propose a seed generation framework that enables Human-In-The-Loop (HITL) directed fuzzing where the human assumes a more active role in the creation of seeds that can penetrate and assess desired locations of the PUT. Our proposed framework uses Symbolic Execution (SE) to generate seeds that exercise paths to target program locations. Moreover, our framework enables the visualization of the explored execution paths in the binary of the PUT for the generated seeds. We evaluated our approach on a set of 12 carefully designed C programs with diverse characteristics that mimic real-world programs. The experimental results show the effectiveness of the proposed approach in improving the performance of standard fuzzing tools such as the American Fuzzy Lop ("Image missing" <#comment/> ). Specifically, our solution can generate seeds that substantially enhance the performance of the fuzzer, achieving speedups ranging from $$1.46\times $$ 1.46 × to $$68.53\times $$ 68.53 × for branch conditions, $$1.39\times $$ 1.39 × to $$254.62\times $$ 254.62 × for branch depths, $$14,879.59\times $$ 14 , 879.59 × to $$30,295.88\times $$ 30 , 295.88 × for branch widths over traditional seeds. Additionally, the speedup increases with the number of target function ranging from $$12,260\times $$ 12 , 260 × to $$22,856.07\times $$ 22 , 856.07 × over traditional seeds while only requiring less than 15 seconds on average for the seed generation step.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Formal Modules for Relative Formal Lubin–Tate Groups

Relative formal Lubin–Tate groups are studied, namely, their structure, the ring of endomorphisms, and the group of points. The primary elements are considered, and an explicit formula for the generalized Hilbert symbol is derived.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Model Elliptic Boundary-Value Problems for Pseudodifferential Operators in Canonical Nonsmooth Domains

We consider a simplest elliptic pseudodifferential equation in a multi-dimensional cone (multi-dimensional angle) and describe all possible structures of its solutions related to the wave factorization of the elliptic symbol. Depending on the index of wave factorization, we consider various statements of well-posed boundary-value problems. The existence of solutions is studied in Sobolev–Slobodetskii spaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

SUSPECT: MINLP special structure detector for Pyomo

We present SUSPECT, an open source toolkit that symbolically analyzes mixed-integer nonlinear optimization problems formulated using the Python algebraic modeling library Pyomo. We present the data structures and algorithms used to implement SUSPECT. SUSPECT works on a directed acyclic graph representation of the optimization problem to perform: bounds tightening, bound propagation, monotonicity detection, and convexity detection. We show how the tree-walking rules in SUSPECT balance the need for lightweight computation with effective special structure detection. SUSPECT can be used as a standalone tool or as a Python library to be integrated in other tools or solvers. Here, we highlight the easy extensibility of SUSPECT with several recent convexity detection tricks from the literature. We also report experimental results on the MINLPLib 2 dataset.

97 MATHEMATICS AND COMPUTING↗