arXiv subjects

Peter Arnold

Publications and source records attributed to Peter Arnold.

At least 19 recordsLinked to original sources

arXiv2026

Calculating extremely high energy bremsstrahlung in matter

Ultra-relativistic electrons initiate electromagnetic showers in ordinary matter that evolve through bremsstrahlung and pair production. At very high energy, the quantum mechanical duration of bremsstrahlung becomes longer than the mean free time to elastically scatter from the medium, leading to a significant suppression known at the Landau-Pomeranchuk-Migdal (LPM) effect. For some ranges of bremsstrahlung photon and initial electron energies $(k_\gamma,E)$, the duration becomes so long that it will also overlap with subsequent pair production by the bremsstrahlung photon, disrupting LPM suppression and drastically changing LPM predictions. We have previously calculated this change for extremely high energies ($k_\gamma \gg 2$ TeV or more, depending on the medium), for which the electron mass and medium-induced photon mass could be ignored. In this paper, we extend that analysis to lower (but still ultra-relativistic) energy by accounting for those masses, leading to a rich map of behavior in different regions of $(k_\gamma,E)$.

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arXiv2026

Extremely high-energy bremsstrahlung in matter

The theory of bremsstrahlung $e \to e\gamma$ by extremely high energy electrons passing through ordinary matter has been qualitatively incomplete. We revisit the suppression of bremsstrahlung by the Landau-Pomeranchuk-Migdal (LPM) effect, here accounting for quantum disruption of that effect from pair production. Our analysis covers the full range of ultra-relativistic electron and photon energies (subject to a few simplifying approximations).

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arXiv2025

Revisiting extremely high energy QED bremsstrahlung in matter: large modifications to the LPM effect

Very high energy electrons initiate electromagnetic showers in ordinary matter that branch and multiply through bremsstrahlung and pair production. At extremely high energies, the quantum mechanical duration of these processes becomes longer than the mean free time to elastically scatter from the medium, which leads to a very significant suppression of bremsstrahlung (and pair production) known as the Landau-Pomeranchuk-Migdal (LPM) effect. We revisit the LPM effect for bremsstrahlung of energy $k_\gamma$ from an electron of energy $E$. We find that there are very large corrections to the LPM bremsstrahlung rate for certain regions of $(k_\gamma,E)$ due to quantum overlap of bremsstrahlung and subsequent pair production. This possibility was first raised in the 1960s, when it was argued qualitatively that pair production would significantly decrease the bremsstrahlung rate in those regions of $(k_\gamma,E)$ compared to the already-suppressed LPM bremsstrahlung rate. We find the opposite -- quantum overlap of bremsstrahlung with pair production significantly *increases* the bremsstrahlung rate compared to the LPM calculation -- and we verify our qualitative arguments with an analytic calculation of the effect.

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arXiv2025

Are Parton Showers in a Quark-Gluon Plasma Strongly Coupled? A Theorist's Test

We study whether in-medium showers of high-energy quarks and gluons can be treated as a sequence of individual splitting processes or whether there is significant quantum overlap between where one splitting ends and the next begins. Accounting for the Landau-Pomeranchuk-Migdal (LPM) effect, we calculate such overlap effects to leading order in high-energy $\alpha_s(\mu)$ for the simplest theoretical situation. We investigate a measure of overlap effects that is independent of physics that can be absorbed into an effective value $\hat{q}_{eff}$ of the jet-quenching parameter $\hat{q}$.

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arXiv2024

Are in-medium quark-gluon showers strongly coupled? Results in the large-$N_f$ limit

Inside a medium, showers originating from a very high energy particle develop via medium-induced splitting processes such as bremsstrahlung and pair production. During shower development, two consecutive splittings sometimes overlap quantum mechanically, so that they cannot be treated independently. Some of these effects can be absorbed into an effective value of a medium parameter known as $\hat q$. Previous calculations (with certain simplifying assumptions) have found that, after adjusting the value of $\hat q$, the leftover effect of overlapping splittings is quite small for purely gluonic large-$N_c$ showers but is very much larger for large-$N_f$ QED showers, at comparable values of $N\alpha$. Here, by investigating the same problem for QCD with *quarks* in the large-$N_f$ limit of many quark flavors, we make a first study of whether the small effect in purely gluonic showers (i) was merely an accident or (ii) is more broadly characteristic of such overlap effects in QCD. We also offer a qualitative explanation for the large size of the effect in QED vs. QCD.

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arXiv2024

Strongly vs. weakly coupled in-medium showers: energy stopping in large-$N_f$ QED

Inside a medium, showers originating from a very high-energy particle may develop via medium-induced splitting processes such as hard bremsstrahlung or pair production. During shower development, two consecutive splittings sometimes overlap quantum mechanically, so that they cannot be treated independently. Some of these effects can be absorbed into an effective value of a medium parameter known as $\hat q$. Previous calculations (with certain simplifying assumptions) have found that, after adjusting the value of $\hat q$, the leftover effect of overlapping splittings is quite small for purely gluonic large-$N_c$ showers but is very much larger for large-$N_f$ QED showers, at comparable values of $N\alpha$. Those works did not quite make for apples-to-apples comparisons: the gluon shower work investigated energy deposition from a gluon-initiated shower, whereas the QED work investigated charge-deposition from an electron-initiated shower. As a first step to tighten up the comparison, this paper investigates energy deposition in the QED case. Along the way, we develop a framework that should be useful in the future to explore whether the very small effect of overlapping splitting in purely gluonic showers is an artifact of having ignored quarks.

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arXiv2023

The LPM effect in sequential bremsstrahlung: gluon shower development

We give details of our study of whether high-energy gluon showers inside a QCD medium can be treated as a sequence of individual splitting processes $g \to gg$, or whether there is significant quantum overlap between where one splitting ends and the next begins (neglecting effects that can be absorbed into an effective value of the jet quenching parameter $\hat q$ that characterizes the medium). The study is carried out by imagining in-medium gluon shower development in the simplest theoretical situation, which includes imagining a very large, static, homogeneous medium and taking the large $N_{\rm c}$ limit. Along the way, we also show how in-medium shower evolution can be written in terms of a "net" splitting rate $[d\Gamma/dx]_{\rm net}$, and we provide a moderately simple analytic fit to our numerical results for the overlap effects included in that rate, which we hope may be of use to others wishing to study possible consequences of overlapping splittings.

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arXiv2022

Are gluon showers inside a quark-gluon plasma strongly coupled? a theorist's test

We study whether in-medium showers of high-energy gluons can be treated as a sequence of individual splitting processes $g{\to}gg$, or whether there is significant quantum overlap between where one splitting ends and the next begins. Accounting for the Landau-Pomeranchuk-Migdal (LPM) effect, we calculate such overlap effects to leading order in high-energy $\alpha_{\rm s}(\mu)$ for the simplest theoretical situation. We investigate a measure of overlap effects that is independent of physics that can be absorbed into an effective value $\hat q_{\rm eff}$ of the jet-quenching parameter $\hat q$.

hep-ph↗

arXiv2022

The LPM effect in sequential bremsstrahlung: incorporation of "instantaneous'' interactions for QCD

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths (formation lengths) of two consecutive splitting processes overlap, avoiding soft-emission approximations. Previous work made a ``nearly-complete'' calculation of the effect of overlapping formation times on gluonic splittings such as $g \to gg \to ggg$ (with simplifying assumptions such as an infinite QCD medium and the large-$N_c$ limit). In this paper, we extend those previous rate calculations from nearly-complete to complete by including processes involving the exchange of longitudinally-polarized gluons. In the context of Lightcone Pertubation Theory, used earlier for the ``nearly-complete'' calculation, such exchanges are instantaneous in lightcone time and have their own diagrammatic representation.

hep-ph↗

arXiv2022

The LPM effect in sequential bremsstrahlung: analytic results for sub-leading (single) logarithms

Consider the in-medium splitting $g \to gg$ of a very high-energy gluon traversing a QCD medium, accounting for the Landau-Pomeranchuk-Migdal (LPM) effect. It has been known for some time that soft radiative corrections to that splitting generate a double-log correction to the splitting rate, whose effects can be absorbed into running of the medium parameter $\hat q$ describing the rate of transverse momentum kicks to high-energy particles due to small-angle scattering from the medium. Less has been known about sub-leading, single logarithms in this context. In this paper, we find analytic formulas for those single logs (with various caveats and clarifications).

hep-ph↗

arXiv2022

The LPM effect in sequential bremsstrahlung: nearly complete results for QCD

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-emission approximations. Previous work has computed overlap effects for double splitting $g \to gg \to ggg$. To make use of those results, one also needs calculations of related virtual loop corrections to single splitting $g \to gg$ in order to cancel severe (power-law) infrared (IR) divergences. This paper provides calculations of nearly all such processes involving gluons and discusses how to organize the results to demonstrate the cancellation. In the soft emission limit, our results reproduce the known double-log behavior of earlier authors who worked in leading-log approximation. We also present a first (albeit numerical and not yet analytic) investigation of sub-leading, single IR logarithms. Ultraviolet divergences appearing in our calculations correctly renormalize the coupling $α_{\rm s}$ in the usual LPM result for leading-order $g \to gg$.

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arXiv2022

The LPM Effect in sequential bremsstrahlung: $1/N_c^2$ corrections

An important question concerning in-medium high-energy parton showers in a quark-gluon plasma or other QCD medium is whether consecutive splittings of the partons in a given shower can be treated as quantum mechanically independent, or whether the formation times for two consecutive splittings instead have significant overlap. Various previous calculations of the effect of overlapping formation times have either (i) restricted attention to a soft bremsstrahlung limit, or else (ii) used the large-$N_c$ limit (where $N_c{=}3$ is the number of quark colors). In this paper, we make a first study of the accuracy of the large-$N_c$ limit used by those calculations of overlap effects that avoid a soft bremsstrahlung approximation. Specifically, we calculate the $1/N_c^2$ correction to previous $N_c{=}\infty$ results for overlap $g \to gg \to ggg$ of two consecutive gluon splittings $g \to gg$. At order $1/N_c^2$, there is interesting and non-trivial color dynamics that must be accounted for during the overlap of the formation times.

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arXiv2021

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

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

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arXiv2020

Towards a More Complete Object-Orientation in Design Grammars

The ongoing digital transformation in industry applies to all product life cycle's stages. The design decisions and dimensioning carried out in the early conceptual design stages determine a huge part of the product's life cycle costs (LCC). The automation of the conceptual design phase promises therefore huge gains in terms of LCC. Design grammars encode design processes in production systems made up of rule sequences which automatically create an abstract central product model (central data model) from given requirements. Graph-based design languages use the Unified-Modeling-Language (UML) to define the product entities (classes) supporting object-oriented inheritance. Graphical rules instantiate the classes and iteratively assemble the central model. This paper proposes to extend the design languages by introducing methods (operations). This allows the use of object-oriented design patterns and interface mechanisms as object-oriented principles are then fully implemented. A graphical mechanism to model the method calls is presented which integrates seamlessly into the graph-based design language's graphical rule specification. The object oriented design grammar enables modularization and reusability of engineering knowledge. The integration of engineering domains is enhanced and multistakeholder collaboration with access control (information security) becomes feasible.

cs.SE↗

arXiv2019

Strong- vs. weak-coupling pictures of jet quenching: a dry run using QED

High-energy partons ($E \gg T$) traveling through a quark-gluon plasma lose energy by splitting via bremsstrahlung and pair production. Regardless of whether or not the quark-gluon plasma itself is strongly coupled, an important question lying at the heart of philosophically different approaches to energy loss is whether the high-energy partons of an in-medium shower can be thought of as a collection of individual particles, or whether their coupling to each other is also so strong that a description as high-energy `particles' is inappropriate. We discuss some possible theorists' tests of this question for simple situations (e.g. an infinite, non-expanding plasma) using thought experiments and first-principles quantum field theory calculations (with some simplifying approximations). The physics of in-medium showers is substantially affected by the Landau-Pomeranchuk-Midgal (LPM) effect, and our proposed tests require use of what might be called `next-to-leading order' LPM results, which account for quantum interference between consecutive splittings. The complete set of such results is not yet available for QCD but is already available for the theory of large-$N_f$ QED. We therefore use large-$N_f$ QED as an example, presenting numerical results as a function of $N_fα$, where $α$ is the strength of the coupling at the relevant high-energy scale characterizing splittings of the high-energy particles.

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arXiv2019

The LPM effect in sequential bremsstrahlung: from large-N QCD to N=3 via the SU(N) analog of Wigner 6-j symbols

Consider a high-energy parton showering as it traverses a QCD medium such as a quark-gluon plasma. Interference effects between successive splittings in the shower are potentially very important but have so far been calculated (even in idealized theoretical situations) only in soft emission or large-$N_{\rm c}$ limits, where $N_{\rm c}$ is the number of quark colors. In this paper, we show how one may remove the assumption of large $N_{\rm c}$ and so begin investigation of $N_{\rm c}{=}3$ without soft-emission approximations. Treating finite $N_{\rm c}$ requires (i) classifying different ways that four gluons can form a color singlet and (ii) calculating medium-induced transitions between those singlets, for which we find application of results for the generalization of Wigner 6-$j$ symbols from angular momentum to SU($N_{\rm c}$). Throughout, we make use of the multiple scattering ($\hat q$) approximation for high-energy partons crossing quark-gluon plasmas, and we find that this approximation is self-consistent only if the transverse-momentum diffusion parameter $\hat q$ for different color representations satisfies Casimir scaling (even for strongly-coupled, and not just weakly-coupled, quark-gluon plasmas). We also find that results for $N_{\rm c}{=}3$ depend, mathematically, on being able to calculate the propagator for a coupled non-relativistic quantum harmonic oscillator problem in which the spring constants are operators acting on a 5-dimensional Hilbert space of internal color states. Those spring constants are represented by constant $5{\times}5$ matrices, which we explicitly construct. We are unaware of any closed form solution for this type of harmonic oscillator problem, and we discuss prospects for using numerical evaluation.

hep-ph↗

arXiv2019

Multi-particle potentials from light-like Wilson lines in quark-gluon plasmas: a generalized relation of in-medium splitting rates to jet-quenching parameters $\hat q$

A powerful historical insight about the theory of in-medium showering in QCD backgrounds was that splitting rates can be related to a parameter $\hat q$ that characterizes the rate of transverse-momentum kicks to a high-energy particle from the medium. Another powerful insight was that $\hat q$ can be defined (with caveats) even when the medium is strongly coupled, using long, narrow Wilson loops whose two long edges are light-like Wilson lines. The medium effects for the original calculations of in-medium splitting rates can be formulated in terms of 3-body imaginary-valued `potentials' that are defined with three long, light-like Wilson lines. Corrections due to overlap of two consecutive splittings can be calculated using similarly defined 4-body potentials. I give a simple argument for how such $N$-body potentials can be determined in the appropriate limit just from knowledge of the values of $\hat q$ for different color representations. For $N > 3$, the $N$-body potentials have non-trivial color structure, which will complicate calculations of overlap corrections outside of the large-$N_c$ or soft bremsstrahlung limits.

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arXiv2018

In-medium loop corrections and longitudinally polarized gauge bosons in high-energy showers

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-emission approximations. In this particular paper, we show (i) how the "instantaneous" interactions of Light-Cone Perturbation Theory must be included in the calculation to account for effects of longitudinally-polarized gauge bosons in intermediate states, and (ii) how to compute virtual corrections to LPM emission rates, which will be necessary in order to make infrared-safe calculations of the characteristics of in-medium QCD showering of high-energy partons. In order to develop these topics in as simple a context as possible, we will focus in the current paper not on QCD but on large-$N_f$ QED, where $N_f$ is the number of electron flavors.

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