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NASA NTRS1994

Proton-induced fragmentation of carbon at energies below 100 MeV

Radiation effects caused by single cosmic ray particles have been studied for many years in radiobiological experiments for different biological objects and biological end-points. Additionally, single event effects in microelectronic devices have gained large interest. There are two fundamental mechanisms by which a single particle can cause radiation effects. On the one hand, a cosmic ray ion with high linear energy transfer can deposit a high dose along its path. On the other hand, in a nuclear collision, a high dose can be deposited by short range particles emitted from the target nucleus. In low earth orbits a large contribution to target fragmentation events originates from trapped protons which are encountered in the South Atlantic Anomaly. These protons have energies up to a few hundred MeV. We study the fragmentation of C, O and Si nuclei - the target nuclei of biological material and microelectronic devices - in nuclear collisions. Our aim is to measure production cross sections, energy spectra, emission directions and charge correlations of the emitted fragments. The present knowledge concerning these data is rather poor. M. Alurralde et al. have calculated cross sections and average energies of fragments produced from Si using the cascade-evaporation model. D.M. Ngo et al. have used the semiempirical cross section formula of Silberberg and Tsao to calculate fragment yields and the statistical model of Goldhaber to describe the reaction kinematics. Cross sections used in these models have uncertainties within a factor of two. Our data will help to test and improve existing models especially for energies below 300 MeV/nucleon. Charge correlations of fragments emitted in the same interaction are of particular importance, since high doses can be deposited if more than one heavy fragment with a short range is produced.

Schmitz, M.↗

NASA NTRS1995

Repair of x-ray-induced DNA double-strand breaks in specific Not I restriction fragments in human fibroblasts: joining of correct and incorrect ends

An assay that allows measurement of absolute induction frequencies for DNA double-strand breaks (dsbs) in defined regions of the genome and that quantitates rejoining of correct DNA ends has been used to study repair of dsbs in normal human fibroblasts after x-irradiation. The approach involves hybridization of single-copy DNA probes to Not I restriction fragments separated according to size by pulsed-field gel electrophoresis. Induction of dsbs is quantitated from the decrease in the intensity of the hybridizing restriction fragment and an accumulation of a smear below the band. Rejoining of dsbs results in reconstitution of the intact restriction fragment only if correct DNA ends are joined. By comparing results from this technique with results from a conventional electrophoresis assay that detects all rejoining events, it is possible to quantitate the misrejoining frequency. Three Not I fragments on the long arm of chromosome 21 were investigated with regard to dsb induction, yielding an identical induction rate of 5.8 X 10(-3) break per megabase pair per Gy. Correct dsb rejoining was measured for two of these Not I fragments after initial doses of 80 and 160 Gy. The misrejoining frequency was about 25% for both fragments and was independent of dose. This result appears to be representative for the whole genome as shown by analysis of the entire Not I fragment distribution. The correct rejoining events primarily occurred within the first 2 h, while the misrejoining kinetics included a much slower component, with about half of the events occurring between 2 and 24 h. These misrejoining kinetics are similar to those previously reported for production of exchange aberrations in interphase chromosomes.

NASA Discipline Radiation Health↗

NASA NTRS2006

The Zero-Degree Detector System for Fragmentation Studies

The measurement of nuclear fragmentation cross sections requires the detection and identification of individual projectile fragments. If light and heavy fragments are recorded in 'ne same detector, it may be impossible distinguish the signal from the light fragment. To overcome this problem, we have developed the Zero-Degree Detector System. The ZDDS enables the measurement of cross sections for light fragment production by using pixelated detectors to separately measure the signals of each fragment. The system has been used to measure the fragmentation of beams as heavy as Fe at the NASA Space Radiation Laboratory at Brookhaven National Laboratory and the Heavy Ion Medical Accelerator in Chiba, Japan.

Adams, J. H., Jr.↗

NASA NTRS1995

Evidence on Sizes and Fragmentation of the Nuclei of Comet Shoemaker-Levy 9 from Hubble Space Telescope Images

Central regions on the digital maps of 13 nuclear condensations of Comet Shoemaker-Levy 9, obtained with the Planetary Camera of the Hubble Space Telescope on January 24-25, March 28-30, and July 4, 1994, have been analyzed with the aim to identify the presence of distinct, major fragments in each condensation, to deconvolve their contributions to the signal that also includes the contribution from a surrounding cloud of dust (modeled as an extended source, using two different laws), to estimate the dimensions of the fragments and to study their temporal variations, and to determine the spatial distributions of the fragments as projected onto the plane of the sky. The deconvolution method applied is described and the results of the analysis are summarized, including the finding that sizable fragments die survive until the time of atmospheric entry. This result does not contradict evidence of the comet's continuing, apparently spontaneous fragmentation, which still went on long after the extremely close approach to Jupiter in July 1992 and which, because of the jovian tidal effects, may even have intensified in the final days before the crash on Jupiter. On plausible assumptions, the largest fragments are found to have had effect diameters of 4 km as late as March and event early July 1994. In most condensations, several sizable companions ( 1 km or more across) have been detected within 1000 km of the projected location of the brightest fragment and the surrounding dust cloud has been found to be centered on a point that is shifted in the general direction of the tail, probably due to effects of solar radiation pressure. Since the developed approach is based on certain premises and involves approximations, the results should be viewed as preliminary and the problem should be a subject of further investigation.

Comet Shoemaker-Levy 9↗

NASA NTRS2008

Measurement Techniques for Hypervelocity Impact Test Fragments

The ability to classify the size and shape of individual orbital debris fragments provides a better understanding of the orbital debris environment as a whole. The characterization of breakup fragmentation debris has gradually evolved from a simplistic, spherical assumption towards that of describing debris in terms of size, material, and shape parameters. One of the goals of the NASA Orbital Debris Program Office is to develop high-accuracy techniques to measure these parameters and apply them to orbital debris observations. Measurement of the physical characteristics of debris resulting from groundbased, hypervelocity impact testing provides insight into the shapes and sizes of debris produced from potential impacts in orbit. Current techniques for measuring these ground-test fragments require determination of dimensions based upon visual judgment. This leads to reduced accuracy and provides little or no repeatability for the measurements. With the common goal of mitigating these error sources, allaying any misunderstandings, and moving forward in fragment shape determination, the NASA Orbital Debris Program Office recently began using a computerized measurement system. The goal of using these new techniques is to improve knowledge of the relation between commonly used dimensions and overall shape. The immediate objective is to scan a single fragment, measure its size and shape properties, and import the fragment into a program that renders a 3D model that adequately demonstrates how the object could appear in orbit. This information would then be used to aid optical methods in orbital debris shape determination. This paper provides a description of the measurement techniques used in this initiative and shows results of this work. The tradeoffs of the computerized methods are discussed, as well as the means of repeatability in the measurements of these fragments. This paper serves as a general description of methods for the measurement and shape analysis of orbital debris.

Hill, Nicole E.↗

NASA NTRS2010

Deliberate Satellite Fragmentations and their Effects on the Long-Term Space Environment

Since 1964 at least 56 spacecraft and two launch vehicle upper stages have been deliberately fragmented while in Earth orbit. Many of these events have had no long-lasting effects on the near-Earth space environment, but one represents the most devastating satellite breakup in history that will pose hazards to operational spacecraft in low Earth orbit for decades to come. International space debris mitigation guidelines now call for avoiding the creation of long-lived debris from intentional satellite fragmentations. This paper summarizes the reasons for and environmental consequences of deliberate satellite fragmentations. Contrary to popular belief, only one in five deliberate fragmentations have been related to the testing of anti-satellite weapon systems, for which only one such test has occurred during the past 25 years. Other reasons for deliberate satellite fragmentations range from engineering tests to protecting national security information. Whereas the majority of deliberate satellite fragmentations have occurred in low Earth orbits, some have involved spacecraft in highly elliptical orbits. The former Soviet Union and the current Russian Federation have been responsible for 90% of all identified deliberate on-orbit satellite fragmentations.

Johnson, N. L.↗

NASA NTRS2013

K-Ca Dating of Alkali-Rich Fragments in the Y-74442 and Bhola LL-Chondritic Breccias

Alkali-rich igneous fragments in the brecciated LL-chondrites, Krahenberg (LL5) [1], Bhola (LL3-6) [2], Siena (LL5) [3] and Yamato (Y)-74442 (LL4) [4-6], show characteristic fractionation patterns of alkali and alkaline elements [7]. The alkali-rich fragments in Krahenberg, Bhola and Y-74442 are very similar in mineralogy and petrography, suggesting that they could have come from related precursor materials [6]. Recently we reported Rb-Sr isotopic systematics of alkali-rich igneous rock fragments in Y-74442: nine fragments from Y-74442 yield the Rb-Sr age of 4429 plus or minus 54 Ma (2 sigma) for lambda(Rb-87) = 0.01402 Ga(exp -1) [8] with the initial ratio of Sr-87/Sr-86 = 0.7144 plus or minus 0.0094 (2 sigma) [9]. The Rb-Sr age of the alkali-rich fragments of Y-74442 is younger than the primary Rb-Sr age of 4541 plus or minus 14 Ma for LL-chondrite whole-rock samples [10], implying that they formed after accumulation of LL-chondrite parental bodies, although enrichment may have happened earlier. Marshall and DePaolo [11,12] demonstrated that the K-40 - Ca-40 decay system could be an important chronometer as well as a useful radiogenic tracer for studies of terrestrial rocks. Shih et al. [13,14] and more recently Simon et al. [15] determined K-Ca ages of lunar granitic rocks, and showed the application of the K-Ca chronometer for K-rich planetary materials. Since alkali-rich fragments in the LL-chondritic breccias are highly enriched in K, we can expect enhancements of radiogenic Ca-40. Here, we report preliminary results of K-Ca isotopic systematics of alkali-rich fragments in the LL-chondritic breccias, Y-74442 and Bhola.

Yokoyama, T↗

NASA NTRS2016

DebriSat Fragment Characterization System and Processing Status

The DebriSat project is a continuing effort sponsored by NASA and DoD to update existing break-up models using data obtained from hypervelocity impact tests performed to simulate on-orbit collisions. After the impact tests, a team at the University of Florida has been working to characterize the fragments in terms of their mass, size, shape, color and material content. The focus of the post-impact effort has been the collection of 2 mm and larger fragments resulting from the hypervelocity impact test. To date, in excess of 125K fragments have been recovered which is approximately 40K more than the 85K fragments predicted by the existing models. While the fragment collection activities continue, there has been a transition to the characterization of the recovered fragments. Since the start of the characterization effort, the focus has been on the use of automation to (i) expedite the fragment characterization process and (ii) minimize the effects of human subjectivity on the results; e.g., automated data entry processes were developed and implemented to minimize errors during transcription of the measurement data. At all steps of the process, however, there is human oversight to ensure the integrity of the data. Additionally, repeatability and reproducibility tests have been developed and implemented to ensure that the instrumentations used in the characterization process are accurate and properly calibrated.

Rivero, M.↗

NASA NTRS2017

Fragmentation Increases Impact of Wind Disturbance on Forest Structure and Carbon Stocks in a Western Amazonian Landscape

Tropical second-growth forests could help mitigate climate change, but the degree to which their carbon potential is achieved will depend on exposure to disturbance. Wind disturbance is common in tropical forests, shaping structure, composition, and function, and influencing successional trajectories. However, little is known about the impacts of extreme winds in fragmented landscapes, though second-growth forests are often located in mosaics of forest, pasture, cropland, and other land cover types. Though indirect evidence suggests that fragmentation increases risk of wind damage, few studies have found such impacts following severe storms. In this study, we ask whether fragmentation and forest type (old vs. second growth) were associated with variation in wind damage after a severe convective storm in a fragmented production landscape in western Amazonia. We applied linear spectral unmixing to Landsat 8 imagery from before and after the storm, and combined it with field observations of damage to map wind effects on forest structure and biomass (Figure 4, 5). We also used Landsat 8 imagery to map land cover with the goals of identifying old- and second-growth forest and characterizing fragmentation. We used these data to assess variation in wind disturbance across 95,596 hectares of forest, distributed over 6,110 patches. We find that fragmentation is significantly associated with wind damage, with damage severity higher at forest edges and in edgier, more isolated patches (Figure 7). Damage was more severe in old-growth than in second-growth forests, but this effect was weaker than that of fragmentation (Figure 8). These results illustrate the importance of considering spatial configuration and landscape context in planning tropical forest restoration and predicting carbon sequestration in second-growth forests. Future research should address the mechanisms behind these results, to minimize wind damage risk in second-growth forests so their carbon potential can be maximally achieved.

Schwartz, Naomi B.↗

NASA NTRSDate not supplied

Updates on the Debrisat Hypervelocity Experiment and Characterization of Fragments in Support of Environmental Models

To develop, maintain, and update orbital debris environmental and break-up models, such as the NASA Orbital Debris Engineering Model (ORDEM) and the NASA Standard Satellite Breakup Model (SSBM), the NASA Orbital Debris Program Office (ODPO) relies on the most recent validated datasets from various measurement sources, both laboratory and environmental. One key project that will provide insight for break-up events using modern-day spacecraft materials and construction techniques is the DebriSat laboratory hypervelocity-impact test. Based on the mass of the target, the projectile, and the impact velocity, the expected number of fragments greater than 2 mm, generated using the NASA SSBM, was estimated to be close to 85,000. To date, the DebriSat fragment database continues to grow, with over 200,000 fragments collected that will help inform updates to the SSBM. Additionally, the growing fragment ensemble will support key parameters for the next release of NASA’s environmental models, employing fragment shapes, densities, and size distributions. To further support these environmental models, specifically the size parameter used in ground-based optical measurements, optical characterization on a subset of DebriSat fragments is being conducted in ODPO’s Optical Measurement Center (OMC). Broadband bidirectional reflectance distribution (BRDF) measurements will provide insight into the optical-based NASA Size Estimation Model (OSEM). The OSEM equates an object’s brightness to size (e.g., diameter of a disk or sphere) given several assumed parameters, including a defined phase function, albedo, and range. To address the first defined parameter, the ODPO has been using ray-tracing software to simulate light conditions in the OMC and to generate phase functions (i.e., specular, Lambertian, and experimentally-derived) of known shapes, materials, and sizes. This simulated data, in addition to the experimentally derived measurements collected in the OMC, will aid in determining whether a new phase function would be suitable for an updated OSEM. The OSEM also assumes a single-value albedo, thus pre-impact spectral measurements on a subset of DebriSat materials were acquired for baseline material characterization and to provide insight into spacecraft material taxonomies. This DebriSat spectral data, along with spectral measurements from other known spacecraft material samples, will allow for better analyses of albedo variations and the effect on size calculations of defined laboratory targets, thus further supporting OSEM updates. This paper provides an overview of DebriSat, the status of the project, updates on the parameter distributions, an overview of the NASA SSBM, and ongoing fragment characterization efforts within the OMC.

Heather Cowardin↗

NASA NTRSDate not supplied

Updates on the DebriSat Hypervelocity Experiment and Characterization of Fragments in Support of Environmental Models

To develop, maintain, and update orbital debris environmental and break-up models, such as the NASA Orbital Debris Engineering Model (ORDEM) and the NASA Standard Satellite Breakup Model (SSBM), the NASA Orbital Debris Program Office (ODPO) relies on the most recent validated datasets from various measurement sources, both laboratory and environmental. One key project that will provide insight for break-up events using modern-day spacecraft materials and construction techniques is the DebriSat laboratory hypervelocity-impact test. Based on the mass of the target, the projectile, and the impact velocity, the expected number of fragments greater than 2 mm, generated using the NASA SSBM, was estimated to be close to 85,000. To date, the DebriSat fragment database continues to grow, with over 200,000 fragments collected that will help inform updates to the SSBM. Additionally, the growing fragment ensemble will support key parameters for the next release of NASA’s environmental models, employing fragment shapes, densities, and size distributions. To further support these environmental models, specifically the size parameter used in ground-based optical measurements, optical characterization on a subset of DebriSat fragments is being conducted in ODPO’s Optical Measurement Center (OMC). Broadband bidirectional reflectance distribution (BRDF) measurements will provide insight into the optical-based NASA Size Estimation Model (OSEM). The OSEM equates an object’s brightness to size (e.g., diameter of a disk or sphere) given several assumed parameters, including a defined phase function, albedo, and range. To address the first defined parameter, the ODPO has been using ray-tracing software to simulate light conditions in the OMC and to generate phase functions (i.e., specular, Lambertian, and experimentally-derived) of known shapes, materials, and sizes. This simulated data, in addition to the experimentally derived measurements collected in the OMC, will aid in determining whether a new phase function would be suitable for an updated OSEM. The OSEM also assumes a single-value albedo, thus preimpact spectral measurements on a subset of DebriSat materials were acquired for baseline material characterization and to provide insight into spacecraft material taxonomies. This DebriSat spectral data, along with spectral measurements from other known spacecraft material samples, will allow for better analyses of albedo variations and the effect on size calculations of defined laboratory targets, thus further supporting OSEM updates. This paper provides an overview of DebriSat, the status of the project, updates on the parameter distributions, an overview of the NASA SSBM, and ongoing fragment characterization efforts within the OMC.

Heather Cowardin↗

DOE OSTI2021

Habitat fragmentation alters the distance of abiotic seed dispersal through edge effects and direction of dispersal

Habitat loss and fragmentation are leading causes of species declines, driven in part by reduced dispersal. Isolating the effects of fragmentation on dispersal, however, is daunting because the consequences of fragmentation are typically intertwined, such as reduced connectivity and increased prevalence of edge effects. We used a large-scale landscape experiment to separate consequences of fragmentation on seed dispersal, considering both distance and direction of local dispersal. We evaluated seed dispersal for five wind- or gravity-dispersed, herbaceous plant species that were planted at different distances from habitat edges, within fragments that varied in their connectivity and shape (edge-to-area ratio). Dispersal distance was affected by proximity and direction relative to the nearest edge. For four of five species, dispersal distances were greater further from habitat edges and when seeds dispersed in the direction of the nearest edge. Connectivity and patch edge-to-area ratio had minimal effects on local dispersal. Here our findings illustrate how some, but not all, landscape changes associated with fragmentation can affect the key population process of seed dispersal.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Landscape fragmentation overturns classical metapopulation thinking

Habitat loss and isolation caused by landscape fragmentation represent a growing threat to global biodiversity. Existing theory suggests that the process will lead to a decline in metapopulation viability. However, since most metapopulation models are restricted to simple networks of discrete habitat patches, the effects of real landscape fragmentation, particularly in stochastic environments, are not well understood. To close this major gap in ecological theory, we developed a spatially explicit, individual-based model applicable to realistic landscape structures, bridging metapopulation ecology and landscape ecology. This model reproduced classical metapopulation dynamics under conventional model assumptions, but on fragmented landscapes, it uncovered general dynamics that are in stark contradiction to the prevailing views in the ecological and conservation literature. Notably, fragmentation can give rise to a series of dualities: a) positive and negative responses to environmental noise, b) relative slowdown and acceleration in density decline, and c) synchronization and desynchronization of local population dynamics. Furthermore, counter to common intuition, species that interact locally (“residents”) were often more resilient to fragmentation than long-ranging “migrants.” This set of findings signals a need to fundamentally reconsider our approach to ecosystem management in a noisy and fragmented world.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Strong Hilbert space fragmentation and fractons from subsystem and higher-form symmetries

Here, we introduce a route to Hilbert space fragmentation in high dimensions leveraging the group-word formalism. We show that taking strongly fragmented models in one dimension and “lifting” to higher dimensions using subsystem symmetries can yield strongly fragmented dynamics in higher dimensions, with subdimensional (e.g., lineonic) excitations. This provides a route to higher-dimensional strong fragmentation, and also a route to fractonic behavior. Meanwhile, lifting one-dimensional fragmented models to higher dimensions using higher-form symmetries yields models with topologically robust fragmentation. In three or more spatial dimensions, one can also “mix and match” subsystem and higher-form symmetries, leading to canonical fracton models such as X cube. We speculate that this approach could also yield a route to non-Abelian fractons. These constructions unify a number of phenomena that have been discussed in the literature, as well as furnishing models with unique properties.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Measurement of inclusive and leading subjet fragmentation in pp and Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV

This article presents new measurements of the fragmentation properties of jets in both proton–proton (pp) and heavy-ion collisions with the ALICE experiment at the Large Hadron Collider (LHC). We report distributions of the fraction $z_r$ of transverse momentum carried by subjets of radius $r$ within jets of radius $R$. Charged-particle jets are reconstructed at midrapidity using the anti-$k_T$ algorithm with jet radius $R$ = 0.4, and subjets are reconstructed by reclustering the jet constituents using the anti-$k_T$ algorithm with radii $r$ = 0.1 and $r$ = 0.2. In proton–proton collisions, we measure both the inclusive and leading subjet distributions. We compare these measurements to perturbative calculations at next-to-leading logarithmic accuracy, which suggest a large impact of threshold resummation and hadronization effects on the $z_r$ distribution. In heavy-ion collisions, we measure the leading subjet distributions, which allow access to a region of harder jet frag- mentation than has been probed by previous measurements of jet quenching via hadron fragmentation distributions. The $z_r$ distributions enable extraction of the parton-to-subjet fragmentation function and allow for tests of the universality of jet fragmentation functions in the quark–gluon plasma (QGP). We find no significant modification of $z_r$ distributions in Pb–Pb compared to pp collisions. However, the distributions are also consistent with a hardening trend for $z_r$ < 0.95, as predicted by several jet quenching models. As $z_r$ → 1 our results indicate that any such hardening effects cease, exposing qualitatively new possibilities to disentangle competing jet quenching mechanisms. By comparing our results to theoretical calculations based on an independent extraction of the parton-to-jet fragmentation function, we find consistency with the universality of jet fragmentation and no indication of factorization breaking in the QGP.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

The evolution of dam induced river fragmentation in the United States

It is established that dams decrease river connectivity; however, previous global scale studies of river fragmentation focused on a small subset of the largest dams. In the United States, mid-sized dams, which are too small for global databases, account for 96% of major anthropogenic structures and 48% of reservoir storage. We conduct a national evaluation of the evolution of anthropogenic river bifurcation over time that includes more than 50,000 nationally inventoried dams. Mid-sized dams account for 73% of anthropogenically created stream fragments nationally. They also contribute disproportionately to short fragments (less than 10 km), which is particularly troubling for aquatic habitats. Here we show that dam construction has essentially reversed natural fragmentation patterns in the United States. Prior to human development, smaller river fragments and less connected networks occurred in arid basins while today we show that humid basins are the most fragmented due to human structures.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Probing fission fragment angular momenta by photon measurements

We discuss how the measurement of photon angular correlations can reveal information about the orientation of the fission fragment angular momenta. Photons from identified stretched E2 collective transitions in even-even fission product nuclei are particularly suitable because they do not affect the orientation of the nuclear spin. Their angular distribution relative to the direction of a fission fragment may reveal the orientation of the fragment spins relative to the fission axis. A novel means of probing the correlated fission fragment spins is the distribution of the opening angle between E2 photons from even-even partner fragments which reveals the mutual correlation of the fragment spins, if the photon helicities can be determined, demonstrating the potential power of helicity measurements in fission.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Definition of fragmentation functions and the violation of sum rules

We point out a problem with the formulation and derivations of sum rules for quark fragmentation functions that impacts their validity in QCD, but which potentially points toward an improved understanding of final states in inclusive hard processes. Fragmentation functions give the distribution of final-state hadrons arising from a parton exiting a hard scattering, and the sum rules for momentum, electric charge, etc. express conservation of these quantities. The problem arises from a mismatch between the quark quantum numbers of the initial quark and the fact that all observed final-state hadrons are confined bound states with color zero. We point that, in a confining theory like QCD, the Wilson line in the operator definition of a fragmentation function entails that the final state in a fragmentation function includes a bound state in the external field generated by the Wilson line. We justify this with the aid of general features of string hadronization. The anomalous bound states are restricted to fractional momentum z = 0 . They tend to invalidate sum rules like the one for charge conservation when applied to the fragmentation functions inferred from experimental data but not the momentum sum rule. We propose to exploit our ideas in future studies as a way to relate the fragmentation functions extracted from inclusive cross sections to more detailed nonperturbative descriptions of final state hadronization. We also describe scenarios wherein the traditional sum rules might remain approximately valid with a reasonably high degree of accuracy. Published by the American Physical Society 2024

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗