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Phuoc Ha

Publications and source records attributed to Phuoc Ha.

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

Transverse momentum dependent gluon density in a proton at low $x$ in the Laplace transform method

We investigate the gluon distribution in a proton at very low $x$, both integrated and transverse momentum dependent, using the Laplace transform technique. By accounting for leading and main next-to-leading contributions, we derive compact analytical expressions for the gluon densities valid in the asymptotic limit $x \to 0$. Our results closely match those from other analytical and numerical approaches, with the main advantage being the simplicity of the expressions, which capture the essential features of more complex calculations.

hep-ph↗
arXiv2025

Evolution of entropy at small $x$

We explore the evolution of the Deep Inelastic Scattering (DIS) entropy, defined as $ S(x,\mu^2) \simeq \ln[xg(x,\mu^2)]$ at small Bjorken variable $x$, where $\mu$ is the observable scale and the gluon distribution $xg(x,\mu^2)$ is derived from the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. We aim to evolve the DIS entropy, which is not directly observable, using a Laplace transform technique. This approach allows us to obtain an analytical solution for the DIS entropy based on known initial gluon distribution functions. We consider both leading-order (LO) and higher-order approximations for the DIS entropy, incorporating the evolved gluon distribution function at the initial scale. The DIS entropy, influenced by purely gluonic emissions, varies with higher-order corrections to the running coupling. By comparing theoretical predictions with charged hadron multiplicity data, we define the evolution. Additionally, we investigate the derivative of the scaling entropy, modeling it as a function of the running coupling, to determine the parameter $\lambda$, known as the Pomeron intercept. We find that the values of $\lambda(x,\mu^2)$ decrease as the order of evolution increases, which is consistent with the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron in the LO and NLO approximations. This investigation provides insights into the dynamics of Quantum Chromodynamics (QCD) at high energies.

hep-ph↗
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

Reduced cross section and gluon distribution in a momentum-space approach

We present a calculation of the reduced cross section in momentum-space approach utilizing the Block-Durand-Ha (BDH) parameterization of the proton structure function $F_{2}(x,Q^2)$ and the leading-order (LO) longitudinal structure function $F_{L}(x,Q^2)$, proposed by Boroun and Ha [G.R. Boroun and P.Ha, Phys. Rev. D {\bf 109} (2024) 094037] using Laplace transform techniques. Our results are compared with the HERA data and extended to the Large Hadron electron Collider (LHeC) domain. We also examine the ratio $F_{L2}(x, Q^2)=F_{L}(x, Q^2)/F_{2}(x, Q^2)$ obtained from our work, comparing it with both the H1 data and the color dipole (CDP) bounds. We find that our results for the reduced cross section and the ratio $F_{L2}(x, Q^2)$ agree with the H1 data. Finally, our evaluation of the gluon distribution functions $G(x,Q^2)$ in momentum-space approach shows very good concordance with the NNPDF3.0LO gluon structure functions for moderate $Q^2$ in the range $10^{-5}{\leq}x{\leq}1$.

hep-ph↗
arXiv2024

Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function $F_{L}(x,Q^2)$, at the leading-order approximation in momentum space, from the structure function $F_{2}(x,Q^2)$ and its derivative with respect to ${\ln}Q^2$ in a kinematical region of low values of the Bjorken variable $x$. Since the $x$ dependence of $F_2(x,Q^2)$ and its evolution with $Q^2$ are determined much better by the data than $F_L(x,Q^2)$, this method provides both a direct check on $F_L(x,Q^2)$ where measured, and a way of extending $F_L(x,Q^2)$ into regions of $x$ and $Q^2$ where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function $F_{2}(x,Q^2)$ [M. M. Block, L. Durand and P. Ha, Phys.Rev.D {\bf89}, 094027 (2014)]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function $F_{L}(x,Q^2)$ and that our obtained results are in line with data from the H1 Collaboration and other results for $F_{L}(x,Q^2)$ obtained using Mellin transform method.

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↗
arXiv2022

Some applications of the Eikonal model with Coulomb and curvature corrections in $pp$ and $\bar{p}p$ scattering

Using a simple eikonal approach to the treatment of Coulomb-nuclear interference and form-factors effects and taking into account the curvature effects in high-energy $pp$ and $\bar{p}p$ scattering, we determine the basic parameters $B$, $\rho$ and $\sigma_{\rm tot}$ from fits to experiment at $W=\sqrt s=$ 53 GeV, 62.3 GeV, 8 TeV, and 13 TeV. We then investigate the differential cross sections in the dip region for $pp$ and $\bar{p}p$ elastic scattering at $W=$ 53 GeV and 1.96 TeV. We find that the results of the basic parameters calculated using the simple eikonal approach agree well with the values determined in other analyses. We find that Coulomb effects are significant in the dip region at 53 GeV and 1.96 TeV, and must be taken into account in searches for odderon effects through cross section differences in that energy region.

hep-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↗
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↗
arXiv2014

Connection of the virtual $γ^*p$ cross section of $ep$ deep inelastic scattering to real $γp$ scattering, and the implications for $νN$ and $ep$ total cross sections

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{γp}$ at small values of Bjorken $x$, including the data at {\em very low} $Q^2$, using a new model for $F_2^{γp}$ which both includes an asymptotic (high energy) part that satisfies a saturated Froissart bound behavior, with a vector-dominance like mass factor in the parameterization, and extends smoothly to $Q^2=0$. We require that the corresponding part of the virtual $γ^* p$ cross section match the known asymptotic part of the real $γp$ cross section at $Q^2=0$, a cross section which is determined by strong interactions and asymptotically satisfies a saturated Froissart bound of the form $α+β\ln s+γ\ln^2s$. Using this model for the asymptotic part of $F_2^{γp}$ plus a known valence contribution, we fit the asymptotic high energy part of the HERA data with $x\le 0.1$ and $W\ge 25$ GeV; the fit is excellent. We find that the mass parameter in the fit lies in the region of the light vector mesons, somewhat above the $ρ$ meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high energy $ep$ and isoscalar $νN$ total cross sections. Both cross sections obey an analytic expression of the type $a +b \ln E +c \ln^2 E +d \ln^3 E$ at large energies $E$ of the incident particle, reflecting the fact that the underlying strong interaction parts of the $γ^*p$, $Z^*N$ and $W^*N$ cross sections satisfy the saturated Froissart bound. Since approximately 50% of the $νN$ center of mass (cms) energy is found in $W$---the cms energy of the strongly interacting intermediate vector boson-nucleon system---a study of ultra-high-energy neutrino-nucleon cross sections would allow us, for the first time, to explore {\em strong interactions at incredibly high energies}.

hep-ph↗
arXiv2013

Implications of a Froissart bound saturation of $γ^*$-$p$ deep inelastic scattering. Part II. Ultra-high energy neutrino interactions

In Part I (in this journal) we argued that the structure function $F_2^{γp}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $γ^*p$ scattering, has a saturated Froissart-bounded form behaving as $\ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $\ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_ν\sim 10^{16}$-$10^{17}$ GeV where there are now limits on neutrino fluxes. We estimate that these cross sections are accurate to $\sim$2% at the highest energies considered, with the major uncertainty coming from the errors in the parameters that were needed to fit $F_2^{γp}(x,Q^2)$. We compare our results to recently published neutrino cross sections derived from NLO parton distribution functions, which become much larger at high energies because of the use of power-law extrapolations of quark distributions to small $x$. We argue that our calculation of the UHE $νN$ cross sections is the best one can make based the existing experimental deep inelastic scattering data. Further, we show that the strong interaction Froissart bound of $\ln^2 (1/x)$ on $F_2^{γp}$ translates to an exact bound of $\ln^3E_ν$ for leading-order-weak $νN$ scattering. The energy dependence of $νN$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.

hep-ph↗