Feynman polytopes and the tropical geometry of UV and IR divergences
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72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
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Soft factorisation and exponentiation from Schwinger-space geometry
Infrared divergences in Quantum Field Theory govern the low-energy dynamics of many physical theories, and their understanding is a crucial ingredient in predicting the outcomes of collider experiments. We present a novel approach to deriving the structure of these divergences by employing the Schwinger parametrization of Feynman integrals. After using tropical geometry to identify divergent limits, we study the all-orders asymptotic properties of Feynman diagrams via matrix manipulations of graph Laplacians, which allows us to analyse their IR behaviour systematically. We explicitly demonstrate the soft-hard factorization of the integrand for a broad class of diagrams, and reveal that when written in terms of worldline distances, topologically distinct diagrams asymptote to the same integrand at leading order in the soft limit. In particular, for the case of Quantum Electrodynamics (with massive fermions), we use this fact to show how ladder-type diagrams combine in Schwinger-parameter space to yield the correct exponentiated soft anomalous dimension. This framework provides a foundation for extending these methods to more complex theories like Quantum Chromodynamics and offers a pathway towards a systematic understanding of infrared divergences in perturbative amplitudes.
Factorization↗
Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints
The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.
97 MATHEMATICS AND COMPUTING↗
Widespread windthrow in Southeast Asian tropical forests verified by satellite observations
Windthrow, defined as abrupt tree mortality caused by intense winds, is well documented in the Neotropics but remains largely unverified in Asian tropical forests. Using Landsat 8 imagery for the period 2020-2022 and established spectral (shortwave-infrared, near-infrared, and red band composites) and morphological criteria (fan-shaped geometry and sharp boundaries), this study verified the presence of windthrow across Sumatra, Borneo, and New Guinea and produced the first georeferenced inventory of 60 events in the region. Event sizes were right-skewed, with many small gaps and few large (>200 ha) disturbances. Orientations were anisotropic, with concentrations of windthrow events pointing west to northwest-to-north and eastward, both aligning with known pathways of organized convection in the Maritime Continent of Southeast Asia. Detection was conservatively biased low by cloud cover, long revisit intervals, and the exclusion of land-use areas. This study provides a verified baseline that enables reproducible mapping with higher resolution sensors and field observations to quantify windthrow frequency, severity, and ecological significance.
Asia’s tropical forests↗
Stronger Tropical Cyclone–Induced Ocean Cooling in Near-Coastal Regions Compared to the Open Ocean
Abstract Tropical cyclones (TC) often induce strong mixing in the upper ocean that generates a trail of cooler sea surface temperature (Twake) in their wakes. The Twake can affect TC intensity, so its prediction is important, especially in coastal regions where TCs can make landfall. Coastal Twakes are often more complex than those in the open ocean due to the influences of coastline geometry, highly variable water depth, continental runoff, and shelf processes. Using observational data since 2002, here we show a significantly stronger global mean Twake in coastal regions compared to offshore regions. Temperature stratification is the main driver of stronger coastal Twakes in the North Atlantic and east Pacific. In the northwest Pacific and north Indian Ocean, the differences between coastal and offshore Twakes are smaller due to compensation between TC forcings and ocean stratification. The north Indian Ocean is unique in the Northern Hemisphere because salinity stratification plays a major role on the spatial distribution of Twake. In the South Pacific Ocean, TC intensity and translation speed are crucial for explaining coastal–offshore Twake differences, while ocean stratification and mixed layer depth are more important for the coastal–offshore Twake differences in the south Indian Ocean. These findings suggest that coastal–offshore differences in ocean stratification need to be properly represented in models in order to capture changes in TC-induced ocean cooling as storms approach landfall. Significance Statement Landfalling tropical cyclones (TCs) often cause considerable damage in coastal regions with dense human populations. Understanding TC–ocean interaction and how it differs between coastal and offshore regions can help predict TC intensity prior to landfall. Sea surface cooling after TC passage is an important proxy for TC–ocean interaction. A global evaluation of coastal TC-induced cooling has not been conducted. Using data covering two decades, we show significantly stronger TC-induced surface cooling in coastal regions compared to offshore regions at the global scale and in all basins except the northwest Pacific and north Indian Ocean. The difference is driven mainly by upper-ocean conditions in the North Atlantic, east Pacific, and south Indian Ocean, and by TC characteristics in the South Pacific.
54 ENVIRONMENTAL SCIENCES↗
Stringy canonical forms
Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α'. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α'→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite α', they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the α'→ 0 limit of tree-level string amplitudes, and scattering equations that appear when studying the α'→ ∞ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A corresponding to usual string amplitudes), and other natural integrals over the positive Grassmannian.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
FORTE Measurements of Global Lightning Altitudes
While multiple lightning detection systems provide geographical locations of lightning events across the globe, robust lightning altitude measurements on a global scale have proven elusive. Space-based platforms have an advantageous viewing geometry for making these measurements, but prior studies with the Fast On-orbit Recording of Transient Events (FORTE) satellite were limited to a few thousand events. In this study, we apply the same technique for calculating source altitude from the previous efforts to a large catalog of hundreds of thousands of global FORTE in-cloud lightning events that were coincident with flashes geolocated by its lightning imager between 1997 and 2003. We use this new data set to document global variations in lightning altitude. As in previous studies, we find that FORTE primarily resolves sources from the upper (positive) charge layer at ~11 km altitude in normal thunderstorms. However, sources are also recorded from other charge layers in the storm and from leaders developing between layers. In particular, we note a pronounced increase in source altitude in the first 20 ms of FORTE flashes from the negative leader developing upward into the upper positive charge layer. Regions known for wintertime and/or stratiform lightning have increased contributions from low-altitude sources, while tropical regions particularly around Panama and the Maritime Continent have the greatest concentrations of high-altitude sources.
54 ENVIRONMENTAL SCIENCES↗
Comparison of scattering ratio profiles retrieved from ALADIN/Aeolus and CALIOP/CALIPSO observations and preliminary estimates of cloud fraction profiles
The space-borne active sounders have been contributing invaluable vertically resolved information of atmospheric optical properties since the launch of Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) in 2006. To build long-term records from space-borne lidars useful for climate studies, one has to understand the differences between successive space lidars operating at different wavelengths, flying on different orbits, and using different viewing geometries, receiving paths, and detectors. In this article, we compare the results of Atmospheric Laser Doppler INstrument (ALADIN) and Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) lidars for the period from 28 June to 31 December 2019. First, we build a dataset of ALADIN–CALIOP collocated profiles (Δdist<1°; Δtime<6 h). Then we convert ALADIN's 355 nm particulate backscatter and extinction profiles into the scattering ratio vertical profiles SR(z) at 532 nm using molecular density profiles from Goddard Earth Observing System Data Assimilation System, version 5 (GEOS-5 DAS). And finally, we build the CALIOP and ALADIN globally gridded cloud fraction profiles CF(z) by applying the same cloud detection threshold to the SR(z) profiles of both lidars at the same spatial resolution. Before comparing the SR(z) and CF(z) profiles retrieved from the two analyzed lidar missions, we performed a numerical experiment to estimate the best achievable cloud detection agreement CDA norm (z) considering the differences between the instruments. We define CDA norm (z) in each latitude–altitude bin as the occurrence frequency of cloud layers detected by both lidars, divided by a cloud fraction value for the same latitude–altitude bin. We simulated the SR(z) and CF(z) profiles that would be observed by these two lidars if they were flying over the same atmosphere predicted by a global model. By analyzing these simulations, we show that the theoretical limit for CDA$_{norm}^{theor}$(z) for a combination of ALADIN and CALIOP instruments is equal to 0.81±0.07 at all altitudes. In other words, 19 % of the clouds cannot be detected simultaneously by two instruments due to said differences. The analyses of the actual observed CALIOP–ALADIN collocated dataset containing ~78 000 pairs of nighttime SR(z) profiles revealed the following points: (a) the values of SR(z) agree well up to ~3 km height. (b) The CF(z) profiles show agreement below ~3 km, where ~80 % of the clouds detected by CALIOP are detected by ALADIN as expected from the numerical experiment. (c) Above this height, the CDA$_{norm}^{obs}$(z) reduces to ~50 %. (d) On average, better sensitivity to lower clouds skews ALADIN's cloud peak height in pairs of ALADIN–CALIOP profiles by ~0.5±0.6 km downwards, but this effect does not alter the heights of polar stratospheric clouds and high tropical clouds thanks to their strong backscatter signals. (e) The temporal evolution of the observed CDA$_{norm}^{obs}$(z) does not reveal any statistically significant change during the considered period. This indicates that the instrument-related issues in ALADIN L0/L1 have been mitigated, at least down to the uncertainties of the following CDA$_{norm}^{obs}$(z) values: 68±12 %, 55±14 %, 34±14 %, 39±13 %, and 42±14 % estimated at 0.75, 2.25, 6.75, 8.75, and 10.25 km, respectively.
54 ENVIRONMENTAL SCIENCES↗