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Loyal Durand

Publications and source records attributed to Loyal Durand.

At least 19 recordsLinked to original sources
arXiv2026

Direct determination of the structure functions $F_L$, $F_S$ and $G$ from $F_2$ and $dF_2/dQ^2$ to $O(\alpha_s^2)$

We extend the results of Lappi {\em et al.}, Eur.~Phys.~J.~C {\bf 84}, 84 (2024), to show that it is possible to obtain expressions for the longitudinal, singlet and gluon structure functions $F_L$, $F_S$ and $G$ in deep inelastic scattering directly in terms of the measured functions $F_2$ and $dF_2/\ln(Q^2)$ {\em modulo} non-singlet corrections expected to be small at very small $x$. The latter can be treated at low $x$ using existing quark distributions. Our results are presented consistently to $O(\alpha_s^2)$, correcting and extending the mixed-order results of Lappi {\em et al.}.

hep-ph↗
arXiv2025

Integral representation for a product of two Jacobi functions of the second kind

By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation $\expe^{ix}\expe^{-ix}=1$ to those functions.

math.CA↗
arXiv2024

Some remarks on Coulombic effects in $pp$ and $\bar pp$ scattering and the determination of $\rho$

We point out a very simple method for calculating the mixed Coulomb-nuclear corrections to the $pp$ and $\bar pp$ scattering amplitudes that has been missed in the extensive past work on this problem. The method expresses the correction in terms of a rapidly convergent integral involving the inverse Fourier-Bessel transform of the nuclear amplitude and a known factor containing the Coulomb phase shift with form-factor corrections. The transform can be calculated analytically for the exponential-type model nuclear amplitudes commonly used in fits to the high-energy data at small momentum transfers, and gives very accurate results for the corrections. We examine the possible effects of the Martin zero in the real part of the nuclear amplitude, and the accuracy of the Bethe-West-Yennie phase approximation for the Coulomb-nuclear corrections. We then apply the method to a redetermination of the ratio $\rho$ of the real to the imaginary parts of the forward scattering amplitude in fits to high-energy ISR data previously analyzed using an approximate version of the correction. The only significant changes relative the accuracy of those fits are at 52.8 GeV. Our method is applicable more generally, and can be used also at lower energies and for proton-nucleus scattering.

hep-ph↗
arXiv2024

Simple calculation of the Coulomb-nuclear corrections in $pp$ and $\bar{p} p$ scattering

We present a very simple method for calculating the mixed Coulomb-nuclear effects in the $pp$ and $\bar{p}p$ scattering amplitudes, and illustrate the method using simple models frequently used to describe their differential cross sections at small momentum transfers. Combined with the pure Coulomb and form-factor contributions to the scattering amplitude which are known analytically from prior work, and the unmixed nuclear or strong-interaction scattering amplitude, the results give a much simpler approach to fitting the measured $pp$ and $\bar{p} p$ cross sections and extracting information on the real part of the forward scattering amplitudes than methods now in use.

hep-ph↗
arXiv2023

Double summation addition theorems for Jacobi functions of the first and second kind

In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation between symmetric, antisymmetric and odd-half-integer parameter values for the Jacobi functions with certain Gauss hypergeometric functions which satisfy a quadratic transformation, including associated Legendre, Gegenbauer and Ferrers functions of the first and second kind. We also introduce Olver normalizations of the Jacobi functions which are particularly useful in the derivation of expansion formulas when the parameters are integers. We introduce an application of the addition theorems for the Jacobi functions of the second kind to separated eigenfunction expansions of a fundamental solution of the Laplace-Beltrami operator on the compact and noncompact rank one symmetric spaces.

math.CA↗
arXiv2023

Mehler-Fock transforms and retarded radiative Green functions on hyperbolic and spherical spaces

We develop the theory of causal radiation Green functions on hyperbolic and hyperspherical spaces using a constructive approach based on generalized Mehler-Fock transforms. This approach focuses for $H^d$ on the kernel of the transformation expressed in terms of hyperbolic angles $\theta$ with $0\leq\theta<\infty$. The kernel provides an explicit representation for the generalized delta distribution which acts as the source term for the radiation, and allows easy implementation of the causality or retardation condition and determination of the Green function. We obtain the corresponding kernel distribution on $S^d$ by analytic continuation of the kernel distribution of the Helmholtz equation on $H^d$, then show that this construction leads to the proper retarded Green function for the wave equation. That result is then used to establish the validity of a new generalized Mehler-Fock transformation for $0\leq\theta<\pi$. The present results clarify and extend those obtained recently by Cohl, Dang, and Dunster.

math-ph↗
arXiv2022

Fractional operators and multi-integral representations for associated Legendre functions

In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps. These are of potential use in a number of physical problems. We show here how their results can be derived simply from more general relations involving non-integer changes in the order obtained using the fractional group operator methods developed earlier for SO(2,1), E(2,1) and its conformal extension, and SO(3). We also present general integral relations for fractional changes of the degrees of the functions, and related multi-derivative and multi-integral representations.

math-ph↗
arXiv2020

Coulomb-nuclear interference effects in proton-proton scattering: A simple new eikonal approach

We present a simple new approach to the treatment of Coulomb-nuclear interference and form-factor effects in high-energy proton-proton scattering in the context of eikonal models for the scattering amplitude. We show that the corrections to the nuclear and Coulomb amplitudes do not depend sensitively on the details of the eikonal amplitude and can be taken as universal, and present parametrizations for the necessary corrections. We also present a simple model for the nuclear scattering amplitude useful for data analysis at small momentum transfer which builds in the proper nuclear phase and the diffraction zeros in the real and imaginary parts of the amplitude.

hep-ph↗
arXiv2018

Eikonal and asymptotic fits to high energy data for $σ$, $ρ$, and $B$: An update with curvature corrections

We update our eikonal fit and comprehensive asymptotic fits to high energy data on proton--proton and antiproton--proton scattering for $σ_{\rm tot}$, $σ_{\rm elas}$, $σ_{\rm inel}$, $ρ$, and $B$. The fits include the new TOTEM values of total proton-proton cross section, $ρ$, and $B$ at $W=\sqrt{s}$ = 13 TeV and the Telescope Array value of the total proton-proton cross section at $W=\sqrt{s}$ = 95 TeV, data from the latest measurements of the inelastic cross sections at $W$= 8 TeV (by TOTEM and ATLAS) and 13 TeV (by CMS, ATLAS, and TOTEM). An important new feature of this work is the correction of the data to include the effects of curvature in $\ln{(dσ/dt)}$ on the values of $B$, $dσ/dt$ at $t=0$, and $σ_{\rm tot}$ obtained by extrapolation from the larger values of $t$ where the differential cross section is measured, The effects are significant. The stability of the fits is excellent and the new results agree well with the predictions of earlier fits. This work again confirms the evidence for the proton asymptotically becoming a black disk of gluons.

hep-ph↗
arXiv2018

Asymptotic Bessel-function expansions for Legendre and Jacobi functions

We present new asymptotic series for the Legendre and Jacobi functions of the first and second kinds in terms of Bessel functions with appropriate arguments. The results are useful in the context of scattering problems, improve on known limiting results, and allow the calculation of corrections to the leading Bessel-function approximations for these functions. Our derivations of these series are based on Barnes-type representations of the Legendre, Jacobi, and Bessel functions; our method appears to be new. We use the results, finally, to obtain asymptotic Bessel function expansions for the rotation functions needed to describe the scattering of particles with spin.

math-ph↗
arXiv2017

Evidence for a break in the spectrum of astrophysical neutrinos

The announcement by the IceCube Collaboration of the observation of 53 astrophysical neutrino candidates in the energy range 0.03 \alt E_ν/PeV \alt 2 has been greeted with a great deal of justified excitement. Herein we provide fits of single and a broken power-law energy-spectra to these high-energy starting events (HESEs). By comparing our statistical results from fits to (background-free) shower HESE data with the spectral shape of muon neutrinos recently reported by the IceCube Collaboration, we show that there is (3 σ) evidence for a break in the spectrum of astrophysical neutrinos. After that we use the fitted result to predict the rate of Glashow events (in the ~ 6.3 PeV region) and double-bang tau neutrino events (in the PeV region) just at the threshold of IceCube detection.

astro-ph.HE↗
arXiv2016

The slope, curvature, and higher parameters in $pp$ and $\bar{p}p$ scattering, and the extrapolation of measurements of $dσ(s,t)/dt$ to $t=0$

We study the effects of curvature in the expansion of the logarithm of the differential elastic scattering cross section near $t=0$ as $dσ(s,t)/dt=dσ(s,0)/dt\,\times\exp(Bt+Ct^2+Dt^3\cdots)$ in an eikonal model for $pp$ and $\bar{p}p$ scattering, and use the results to discuss the extrapolation of measured differential cross sections and the slope parameters $B$ to $t=-q^2=0$. We find that the curvature effects represented by the parameters $C$ and $D$, while small, lead to significant changes in the forward slope parameter relative to that determined in a purely exponential fit, and to smaller but still significant changes in the forward elastic scattering and total cross sections. Curvature effects should therefore be considered in future analyses or reanalyses of the elastic scattering data.

hep-ph↗
arXiv2016

Comment on "More on Heisenberg's model for high energy nucleon-nucleon scattering"

We comment on the treatment of asymptotic black-disk scattering in a recent paper of Nastase and Sonnenschein, Phys.\ Rev.\ D\ {\bf 92}, 015028 (2015), on scattering in an updated version of the Heisenberg model which gives $pp$ and $\bar{p}p$ cross sections which increase at very high energies as $\ln^2s$. We show that the total cross section they define does not correspond to that measured in experiments, with the result that their limit for the ratio $σ_{\rm elas}/σ_{\rm tot}$ is too small by a factor 2. The correct ratio for black-disk scattering, $σ_{\rm elas}/σ_{\rm tot} \rightarrow 1/2$ for $s\rightarrow\infty$, is strongly supported by experiment.

hep-ph↗
arXiv2015

Comprehensive fits to high energy data for $σ$, $ρ$, and $B$ and the asymptotic black-disk limit

We demonstrate that the entirety of the data on proton--proton and antiproton--proton forward scattering between 6 GeV and 57 TeV center-of-mass energy is sufficient to show that $σ_{\rm elas}/σ_{\rm tot} \rightarrow 1/2$, and that $8πB/σ_{\rm tot}\rightarrow 1$ at very high energies, where $B$ the forward slope parameter for the differential elastic scattering cross sections. The relations demonstrate convincingly that the asymptotic $pp$ and $\bar{p}p$ scattering amplitudes approach those of scattering from a black disk. This result obviously has implications for any new physics that modifies the forward scattering amplitudes.

hep-ph↗
arXiv2015

Eikonal fit to $pp$ and $\bar{p}p$ scattering and the edge in the scattering amplitude

We make a detailed eikonal fit to current data on the total and elastic scattering cross sections, the ratios $ρ$ of the real to the imaginary parts of the forward elastic scattering amplitudes, and the logarithmic slopes $B$ of the differential cross sections $dσ/dt$ at $t=0$, for proton-proton and antiproton-proton scattering at center-of-mass energies $W$ from 5 GeV to 57 TeV. The fit allows us to investigate the structure of the eikonal amplitudes in detail, including the impact-parameter structure of the energy-independent edge in the scattering amplitude shown to exist by Block {\em et al.} \cite{edge}. We show that the edge region has an essentially fixed shape with a peak at approximately the "black disk" radius $R_{\rm tot}=\sqrt{σ_{\rm tot}/2π}$ of the scattering amplitude, a constant width $t_{\rm edge}\approx 1$ fm, and migrates to larger impact parameters with increasing energy proportionally to $R_{\rm tot}$. We comment on possible physical mechanisms which could lead to the edge. We show that the eikonal results for the cross sections and $ρ$ values are described to high accuracy by analytic expressions of the forms used in earlier analyses by Block and Halzen, and extend the result to the elastic-scattering slope parameter $B$. These expressions provide simple extrapolations of the results to much higher energies. Finally, we calculate the survival probabilities for large rapidity gaps in the scattering.

hep-ph↗
arXiv2015

Evidence for a Constant `Edge' in Proton-Proton Scattering at Very High Energies

Accurate fits to $pp$ and $\bar pp$ cross section data up to Tevatron energies, incorporating the constraints imposed by analyticity and unitarity, successfully predict the results of recent LHC and cosmic ray measurements, and suggest that the cross sections approach a black disc limit asymptotically. The approach to the limit is, however, very slow. We present a simple geometric picture which explains these features in a natural way. A black disc of logarithmically growing radius is supplemented by a soft `edge' whose properties are invariant with energy. The constancy of the edge results in the prediction that the quantity $(σ^{TOT}-2σ^{El})/\surdσ^{TOT}$ approaches a constant at high energy. Using the existing fits, this prediction appears to be verified. The value of the limiting constant allows an estimate of the thickness of the edge, which turns out to be on the order of $1\,{\rm fm}$. One thus arrives at a picture where the proton-proton scattering at lower energies is dominated by what becomes the edge, while at higher energies it is dominated by the disc. The crossover between the two regimes is only at $\surd s\geq $ 10 TeV, accounting for the slow approach to asymptotic behavior. Some questions as to the nature of the edge are discussed.

hep-ph↗