Kinematic Space Research Articles

Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles — the Tr(ϕ3) theory — based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the “stringy” Tr(ϕ3) amplitudes secretly contains the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr(ϕ3) field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. These zeros are not manifest in Feynman diagrams but are made obvious by the connection of these amplitudes to the new understanding of associahedra arising from “causal diamonds” in kinematic space. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties for Tr(ϕ3) amplitudes hold and further generalize to the “stringy” Tr(ϕ3) amplitudes. The “kinematic causal diamond” picture suggests a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings [1]: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the n-point amplitude factorizes as the product of two lower-point currents with n+3 external legs in total. We refer to any such subspace of the kinematic space of n massless momenta as “2-split kinematics”, where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint ϕ3 (with string extension known as Z and J integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings [2] and factorizations near zeros [3], to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

Research on Safety Control Technology for Dual Robots Based on Kinematic Models

The industrial robots play an important role in modern industrial manufacturing processes, and their applications are becoming increasingly widespread, greatly improving manufacturing efficiency and significantly enhancing the digitalization level of the manufacturing industry. Therefore, achieving safe control of multi robot collaborative work is of great significance for the stable and efficient operation of robot systems. This article proposes a dual robot safety control technology based on the kinematic model of industrial robots. Firstly, a mathematical model of the kinematic space of the dual robots is constructed using D-H parameters; Then, use recursive traversal algorithm to calculate the minimum distance between each joint arm of the two robots in real time; Finally, the minimum distance between each joint arm will be calculated and compared with the set safety threshold to achieve real-time assessment and control of collision risk during the movement of the dual robots. A dual robot safety control testing platform was built using two KUKA industrial robots, integrated control software was developed, and relevant algorithms were validated. Through experiments, it was verified that this method can evaluate the collision risk of dual robots in real time during motion and effectively control the collision risk during the motion of dual robots.

Perturbative unitarity and the wavefunction of the Universe

Unitarity of time evolution is one of the basic principles constraining physical processes. Its consequences in the perturbative Bunch-Davies wavefunction in cosmology have been formulated in terms of the cosmological optical theorem. In this paper, we re-analyse perturbative unitarity for the Bunch-Davies wavefunction, focusing on: i)i) the role of the i\epsiloniϵ-prescription and its compatibility with the requirement of unitarity; ii)ii) the origin of the different ``cutting rules’’; iii)iii) the emergence of the flat-space optical theorem from the cosmological one. We take the combinatorial point of view of the cosmological polytopes, which provide a first-principle description for a large class of scalar graphs contributing to the wavefunctional. The requirement of the positivity of the geometry together with the preservation of its orientation determine the i\epsiloniϵ-prescription. In kinematic space it translates into giving a small negative imaginary part to all the energies, making the wavefunction coefficients well-defined for any value of their real part along the real axis. Unitarity is instead encoded into a non-convex part of the cosmological polytope, which we name . The cosmological optical theorem emerges as the equivalence between a specific polytope subdivision of the optical polytope and its triangulations, each of which provides different cutting rules. The flat-space optical theorem instead emerges from the non-convexity of the optical polytope. On the more mathematical side, we provide two definitions of this non-convex geometry, none of them based on the idea of the non-convex geometry as a union of convex ones.

Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the singularities of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions ei − ej.Finally we interpret Mizera’s formula for the biadjoint scalar amplitude m(\U0001d540n, \U0001d540n), restricted to a certain dimension n − 2 subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots e1 − e2, … , en−2 − en−1.

Principal Landau determinants

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/. Program summaryProgram title:PLD.jlCPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1Developer's repository link:https://mathrepo.mis.mpg.de/PLD/Licensing provisions: Creative Commons by 4.0Programming language:JuliaSupplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.

Master constraint approach to quantum-reduced loop gravity

We introduce a master constraint operator on the kinematical Hilbert space of loop quantum gravity representing a set of gauge conditions which classically fix the densitized triad to be diagonal. We argue that the master constraint approach provides a natural and systematic way of carrying out the quantum gauge-fixing procedure which underlies the model known as quantum-reduced loop gravity. The Hilbert space of quantum-reduced loop gravity is obtained as a particular space of solutions of the gauge-fixing master constraint operator. We give a concise summary of the fundamental structure of the quantum-reduced framework, and consider several possible extensions thereof, for which the master constraint formulation provides a convenient starting point. In particular, we propose a generalization of the standard Hilbert space of quantum-reduced loop gravity, which may be relevant in the application of the quantum-reduced model to physical situations in which the Ashtekar connection is not diagonal.

The soaring kite: a tale of two punctured tori

We consider the 5-mass kite family of self-energy Feynman integrals and present a systematic approach for constructing an ε-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the locations of relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be extracted from integrals over maximal cuts. A boundary value is provided such that the differential equation is systematically solved in terms of iterated integrals over g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.

On-shell functions on the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM

We study on-shell functions in the kinematic space for the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM. We construct BCFW bridges that help us build bigger on-shell functions. As a consequence, we provide on-shell diagram formulations for BCFW shifts that correspond to various mass configurations. We will use this to calculate the quadruple cut for the one-loop amplitude on the Coulomb branch and maximal cuts for higher-loops. We make preliminary comments on finding the inequivalent set of on-shell functions for the Coulomb branch.

Planar matrices and arrays of Feynman diagrams: poles for higher k

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes mn(k) for k > 2. In this follow-up work, we investigate the poles of mn(k) from the perspective of such arrays. For general k, we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles, whose number is drastically less than the number of the full arrays. As an example, we first provide all the poles for the cases (k, n) = (3, 7), (3, 8), (3, 9), (3, 10), (4, 8) and (4, 9) in terms of their planar arrays of degenerate Feynman diagrams. We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases. Along the way, we implement hard and soft kinematical limits, which provide a map between the poles in kinematic space and their combinatoric arrays. We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in (k, n) and (n − k, n). We also outline the relation to boundary maps of the hypersimplex Δ k,n and rays in the tropical Grassmannian Tr(k,n) .

First observation and study of the K±→ π0π0μ±ν decay

The NA48/2 experiment at CERN reports the first observation of the K± → π0π0μ±ν decay based on a sample of 2437 candidates with 15% background contamination collected in 2003–2004. The decay branching ratio in the kinematic region of the squared dilepton mass above 0.03 GeV2/c4 is measured to be (0.65 ± 0.03) × 10−6. The extrapolation to the full kinematic space, using a specific model, is found to be (3.45 ± 0.16) × 10−6, in agreement with chiral perturbation theory predictions.

Prescriptive unitarity from positive geometries

In this paper, we define the momentum amplituhedron in the four-dimensional split-signature space of dual momenta. It encodes scattering amplitudes at tree level and loop integrands for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills in the planar sector. In this description, every point in the tree-level geometry is specified by a null polygon. Using the null structure of this kinematic space, we find a geometry whose canonical differential form produces loop-amplitude integrands. Remarkably, at one loop it is a curvy version of a simple polytope, whose vertices are specified by maximal cuts of the amplitude. This construction allows us to find novel formulae for the one-loop integrands for amplitudes with any multiplicity and helicity. The formulae obtained in this way agree with the ones derived via prescriptive unitarity. It makes prescriptive unitarity naturally emerge from this geometric description.

Kinematics and dynamics of non-developable origami

Non-developable origami is a unique type of origami structures that cannot be unfolded into a flat sheet without stretching. In this work, we study the kinematics and dynamics of such origami, theoretically, numerically and experimentally, considering rigid panels and stretchable creases. Unlike developable origami, we find that non-developable origami exhibits distinct folding angle relationships, leading to several separate branches in its kinematic configuration space with snap-through and locking behaviours. By modelling the creases as stretchable torsional springs, we derive a dynamic model to analyse the deployment of non-developable origami structures, from single-vertex origami to origami chains. Our dynamic model unveils the snap-through behaviour between the two kinematics branches of the single-vertex non-developable origami structure, which is further validated by experiments with excellent agreement. We believe that our kinematic and dynamic framework of non-developable origami will greatly expand the current design space of origami structures and guide the design of novel origami actuators.

Three-dimensional kinematics of leaf-cutter ant mandibles: not all dicondylic joints are simple hinges.

Insects use their mandibles for a variety of tasks, including food processing, material transport, nest building, brood care, and fighting. Despite this functional diversity, mandible motion is typically thought to be constrained to rotation about a single fixed axis. Here, we conduct a direct quantitative test of this 'hinge joint hypothesis' in a species that uses its mandibles for a wide range of tasks: Atta vollenweideri leaf-cutter ants. Mandible movements from live restrained ants were reconstructed in three dimensions using a multi-camera rig. Rigid body kinematic analyses revealed strong evidence that mandible movement occupies a kinematic space that requires more than one rotational degree of freedom: at large opening angles, mandible motion is dominated by yaw. But at small opening angles, mandibles both yaw and pitch. The combination of yaw and pitch allows mandibles to 'criss-cross': either mandible can be on top when mandibles are closed. We observed criss-crossing in freely cutting ants, suggesting that it is functionally important. Combined with recent reports on the diversity of joint articulations in other insects, our results show that insect mandible kinematics are more diverse than traditionally assumed, and thus worthy of further detailed investigation. This article is part of the theme issue 'Food processing and nutritional assimilation in animals'.

Resonance control to eliminate reactive power of an improved dual-port direct-drive wave energy converter

The single-port direct-drive wave energy converter (SPDDWEC) is widely utilized for harvesting ocean power among the existing wave energy converters (WECs) designs. Due to its working principle of generating reactive power inevitably and intermittent nature of the waves, the SPDDWEC is unable to provide continuous and stabilized energy to the grid. This paper proposed a dual-port resonance (DPR) control, which can achieve an improved dual-port direct-drive WEC (DPDDWEC) resonance by adjusting the dual electromagnetic force and electromagnetic spring force. The proposed DPR control can effectively avoid reactive power flow of the DPDDWEC, which can generate more energy than the SPDDWEC. Based on the established DPDDWEC kinematic state space model, the proposed DPR control is simulated and verified. The study results confirm the effectiveness of the proposed control method and correctness of theoretical analysis.

Articulatory working space in first and second languages

Articulatory working space (acoustic and kinematic) is often studied to understand the overall size (limits) of a speaker’s articulatory behaviors. For example, prior research has shown that the magnitude of articulators’ movement (e.g., maximum tongue advancement, lip aperture) changes as a function of speech effort (loud, clear, and slow speech). To better understand second language acquisition in adults (i.e., articulatory working space determined by the language or anatomical differences), we compare both acoustic and kinematic working space of adult learners of English between their first language (L1) and second language (L2), which is also compared with that of native speakers of English. Specifically, the articulatory convex hull is measured during passage reading for both acoustic (F1 and F2 trajectories) and kinematic (tongue trajectories on x- and y-dimensions) data. Participants include 11 adult learners of English (four men and seven women) with a Korean language background and 10 adult L1 speakers of English (six men and four women). In the presentation, the findings will be discussed to address whether articulatory space is (1) different between native and nonnative languages (determined by linguistic needs) or (2) rather a constant articulatory characteristic of speakers between the languages, regardless of the speaker’s English proficiency.

The stellar halo in Local Group Hestia simulations

Recent progress in understanding the assembly history of the Milky Way (MW) is driven by the tremendous amount of high-quality data delivered byGaia(ESA), revealing a number of substructures potentially linked to several ancient accretion events. In this work we aim to explore the phase-space structure of accreted stars by analysing six M31/MW analogues from the HESTIA suite of cosmological hydrodynamics zoom-in simulations of the Local Group. We find that all HESTIA galaxies experience a few dozen mergers but only between one and four of those have stellar mass ratios > 0.2, relative to the host at the time of the merger. Depending on the halo definition, the most massive merger contributes from 20% to 70% of the total stellar halo mass. Individual merger remnants show diverse density distributions atz = 0, significantly overlapping with each other and with the in situ stars in theLz − E, (VR, Vϕ) and (R, vϕ) coordinates. Moreover, merger debris often shifts position in theLz − Espace with cosmic time due to the galactic mass growth and the non-axisymmetry of the potential. In agreement with previous works, we show that even individual merger debris exhibit a number of distinctLz − Efeatures. In the (VR, Vϕ) plane, all HESTIA galaxies reveal radially hot, non-rotating or weakly counter-rotating, Gaia-Sausage-like features, which are the remnants of the most recent significant mergers. We find an age gradient inLz − Espace for individual debris, where the youngest stars, formed in the inner regions of accreting systems, deposit to the innermost regions of the host galaxies. The bulk of these stars formed during the last stages of accretion, making it possible to use the stellar ages of the remnants to date the merger event. In action space (Jr, Jz, Jϕ), merger debris do not appear as isolated substructures, but are instead scattered over a large parameter area and overlap with the in situ stars. We suggest that accreted stars can be best identified usingJr > 0.2−0.3(104 kpc km s−1)0.5. We also introduce a new, purely kinematic space (Jz/Jr-orbital eccentricity), where different merger debris can be disentangled better from each other and from the in situ stars. Accreted stars have a broad distribution of eccentricities, peaking atϵ ≈ 0.6 − 0.9, and their mean eccentricity tends to be smaller for systems accreted more recently.

Research on the Model Predictive Trajectory Tracking Control of Unmanned Ground Tracked Vehicles

This article summarizes the research significance and the development status of the unmanned ground tracked vehicles (UGTVs). According to the speed and steering principle of the UGTVs in plane motion, the kinematic state space equation of the vehicle is established. Based on the model predictive control (MPC), the UGTVs trajectory tracking controller is also established. After introducing the overall control system solution, based on the vehicle model, the track speed on both sides is used as the control input to solve the predicted output of the system. After constraining the control amount, the model prediction controller is established by using S function. Several simulations with different preset speeds are performed under linear conditions, and the numerical simulation with different prediction time domains and control time domains are performed under continuous curves. The test validates the effectiveness of the controller, and the effects of speed and time domain parameters on the deviation are analyzed based on the results. Based on satellite positioning technology, trajectory tracking tests of the UGTVs are carried out. After analyzing the positioning principle and common errors, the real-time kinematic technology is used to improve the positioning accuracy of the vehicle prototype. The hardware of the test platform includes a data acquisition system, a control execution system and a walking execution system, and the software part includes data processing and optimization control. The trajectory tracking experiments under different driving conditions are conducted, and the results show that the navigation tracking system can achieve good tracking performance.

$\nu$-flows: Conditional neutrino regression

We present \nuν-Flows, a novel method for restricting the likelihood space of neutrino kinematics in high-energy collider experiments using conditional normalising flows and deep invertible neural networks. This method allows the recovery of the full neutrino momentum which is usually left as a free parameter and permits one to sample neutrino values under a learned conditional likelihood given event observations. We demonstrate the success of \nuν-Flows in a case study by applying it to simulated semileptonic t\bar{t}tt‾ events and show that it can lead to more accurate momentum reconstruction, particularly of the longitudinal coordinate. We also show that this has direct benefits in a downstream task of jet association, leading to an improvement of up to a factor of 1.41 compared to conventional methods.

Loop quantum cosmology of nondiagonal Bianchi models

The non-diagonal Bianchi models are studied in the loop framework for their classical and quantum formulation. The expressions of the Ashtekar-Barbero-Immirzi variables and their properties are found to provide a loop quantization of these models. In the special case of Bianchi I Universe, it is shown that the geometrical operators result invariant from the diagonal description. Hence, the kinematical Hilbert space of the non-diagonal Bianchi I model has similar features to the diagonal one.