Vladimir Komarov, Bota Baimurzinova, Ainur Kunsafina, Dmitry Tsirkov
A process of inelastic nucleon-nucleon collisions with the impact parameter less than a radius of the nucleon quark core is considered. The criteria for selection of such collisions from a full array of the nucleon-nucleon interactions are suggested.
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The effects of nucleon-nucleon (NN) short range correlations (SRC) and Gribov inelastic shadowing (IS) in high energy collisions are illustrated.
Slow nucleons emitted during a hadron-nucleus interaction can give information on the centrality, impact parameter of the collision. The aim of this note is to provide the reader with the important characteristics of the slow nucleons, focusing on their spectra, correlations. This study tries to build on evidences, hence more weight is put on experimental results than on models.
Different types of high-energy hadron-nucleus cross sections are discussed emphasizing the role played by Nucleon-Nucleon (NN) Short-Range Correlations (SRC) and Gribov Inelastic Shadowing (IS)
The importance of the effects of nucleon-nucleon (NN) short-range correlations (SRC) and Gribov inelastic shadowing (IS) on various high energy scattering processes involving nuclear targets is demonstrated within an improved Glauber-Gribov approach.
The possibility of a reliable extraction of the neutron deep inelastic structure function, $F_2^n(x)$, for $ x < 0.85$ from joint measurements of deep inelastic structure functions of deuteron, $^{3}He$ and $^{3}H$ is investigated. The model dependence in this extraction, linked to the possible different interactions between nucleons in nuclei, is shown to be weak, if the nuclear structure effects are properly taken into account. A combined analysis of the deep inelastic structure functions of these nuclei is proposed to study effects beyond the impulse approximation.
Inclusive $A(e,e')X$ and semi-inclusive $A(e,e'N)X$ deep inelastic electron scattering processes off few-nucleon systems are investigated at $x > 1$, showing some of the relevant features of the cross section which are sensitive to the effects arising from nucleon-nucleon correlations and possible exotic multiquark cluster configurations at short internucleon separations.
One of the main challenges hampering an accurate measurement of the double parton scattering (DPS) cross sections is the difficulty in separating the DPS from the leading twist (LT) contributions. We argue that such a separation can be achieved, and cross section of DPS measured, by exploiting the different centrality dependence of DPS and LT processes in proton-nucleus scattering. We developed a Monte Carlo implementation of the DPS processes which includes realistic nucleon-nucleon (NN) correlations in nuclei, an accurate description of transverse geometry of both hard and soft NN collisions…
Semi-inclusive deep inelastic lepton scattering off few-nucleon systems is investigated assuming that virtual boson absorption occurs on a hadronic cluster which can be either a two-nucleon correlated pair or a six-quark bag. In both cases the relevance of nuclear effects on forward and backward nucleon emissions is illustrated and the differences expected in the energy distribution of the emitted nucleons are analyzed both at x>1 and x<1.
In this paper, the implication of the extreme large $N_c$ limit for nucleon-nucleon cross section is examined. Starting from the $N_c$ scaling of the S-matrix elements, a relation between the total inelastic and total cross sections is derived for the regime in which the momenta are much larger than the QCD scale. A conceptual complication arise from the fact that there is a tower of baryons, such as the $Δ$, with mass splittings from the nucleon which go to zero at large $N_c$. Since these baryons are stable at large $N_c$, the meaning of elasticity must be modified. Processes which only tran…
Geometric Bias and Centrality Dependence of Jet Quenching in High-Energy Nuclear Collisions
通过改进的HIJING初始条件模型考虑几何偏差效应，结合线性玻尔兹曼输运模型，解释了Pb+Pb碰撞中带电强子抑制的中心度依赖性。
Predictions in modified Glauber model of total charged-particle yields centrality dependence in O+O and Ne+Ne collisions at LHC
本研究利用含动量损失参数k的蒙特卡洛修正Glauber模型，预测LHC能区O+O和Ne+Ne碰撞总带电粒子产额的中心度依赖，模型基于Pb+Pb碰撞数据校准，结果可为QGP性质研究提供参考。
Experimental results of inclusive hard-process cross sections in heavy-ion collisions conventionally lean on a normalization computed from Glauber models where the inelastic nucleon-nucleon cross section $σ_{\rm nn}^{\rm inel}$ -- a crucial input parameter -- is simply taken from proton-proton measurements. In this letter, using the computed electro-weak boson production cross sections in lead-lead collisions as a benchmark, we determine $σ_{\rm nn}^{\rm inel}$ from the recent ATLAS data. We find a significantly suppressed $σ_{\rm nn}^{\rm inel}$ relative to what is usually assumed, show the c…
We propose a means to detect the nucleon strange quark-antiquark asymmetry, which is predicted as a non-perturbative effect, but still unchecked directly by available experiments. The difference for the $D(c\bar{q})$ and $\bar{D}(\bar{c}q)$ meson production cross sections in neutrino and antineutrino induced charged current deep inelastic scattering is illustrated to be sensitive to the nucleon strange asymmetry. Prospect is given and the effect due to the light quark fragmentation is also discussed for the extraction of the strange asymmetry in future experiments.
Forward rapidity $J/ψ$ meson production in proton-nucleus collisions can be an important constraint of descriptions of the small-$x$ nuclear wavefunction. In an earlier work we studied this process using a dipole cross section satisfying the Balitsky-Kovchegov equation, fit to HERA inclusive data and consistently extrapolated to the nuclear case using a standard Woods-Saxon distribution. In this paper we present further calculations of these cross sections, studying the mean transverse momentum of the meson and the dependence on collision centrality. We also extend the calculation to backward …
We investigate the possibility of identifying an explicit pionic component of the nucleon through measurements of polarized $Δ^{++}$ baryon fragments produced in deep-inelastic leptoproduction off polarized protons, which may help to identify the physical mechanism responsible for the breaking of the Gottfried sum rule. The pion-exchange model predicts highly correlated polarizations of the $Δ^{++}$ and target proton, in marked contrast with the competing diquark fragmentation process. Measurement of asymmetries in polarized $Λ$ production may also reveal the presence of a kaon cloud in the nu…
Recent data on the nuclear modification of W and Z boson production measured by the ATLAS collaboration in PbPb collisions at $\sqrt{s_{\rm nn}}=5.02$ TeV show an enhancement in peripheral collisions, seemingly contradicting predictions of the Glauber model. The data were previously explained by arguing that the nucleon-nucleon cross section may be shadowed in nucleus-nucleus collisions, and hence suppressed compared to the proton-proton cross section at the same collision energy. This interpretation has quite significant consequences for the understanding of heavy-ion data, in particular in t…
The centrality dependence of pseudorapidity density of charged particles and transverse energy is studied for a wide range of collision energies for heavy-ion collisions at midrapidity from 7.7 GeV to 5.02 TeV. A two-component model approach has been adopted to quantify the soft and hard components of particle production, coming from nucleon participants and binary nucleon-nucleon collisions, respectively. Within experimental uncertainties, the hard component contributing to the particle production has been found not to show any clear collision energy dependence from RHIC to LHC. The effect of…
The proton structure function is re-deduced from the data of deep inelastic electron-proton scattering after enhanced correction that is made due to the multiple scattering effect. The Glauber approach is used to account for the multiple scattering of electron with the quark constituents of the proton. The results are compared with the prediction of the parton model. Appreciable deviation is observed only at low values of the Bjorken scaling, which correspond to the extremely deep inelastic scattering.
The triangular flow in ultrarelativistic $^3$He-Au collisions at RHIC energies is enhanced due to the triangular arrangement of the nucleon configurations in $^3$He. We study the fireball eccentricities in the Glauber Monte Carlo model and find that since the configurations of the projectile $^3$He are elongated triangles, the created fireball has a large ellipticity and a smaller triangularity. The dependence of the triangularity on centrality is weak, so it cannot be extracted from the centrality dependence of the triangular flow $v_3$, as it is dominated by the centrality dependence of the …
The semi-inclusive deep inelastic scattering of electrons off the deuteron ($^2H \equiv D$) and $^3He$ with detection of slow protons and deuterons, respectively, i.e. the processes $D(e,e'p)X$ and $^3He(e,e'D)X$, are calculated within the spectator mechanism, taking into account the final state interaction of the hadronizing quark with the detected protons and deuterons, respectively. It is shown that by a proper choice of the kinematics the origin of the EMC effect and the details of the interaction between the hadronizing quark and the nuclear medium can be investigated at a level which can…
The off-shell behaviors of bound nucleons in deep inelastic lepton nucleus scattering are discussed in two scenarios with the basic constituents chosen to be baryon-mesons and quark-gluons respectively in light-cone formalism. It is found that when taking into account the effect due to internal quark structure of nucleons, the derived scaling variable for bound nucleons and the calculated nuclear structure functions are different from those in considering the baryon-mesons as the effective elementary constituents. This implies that the pure baryon-meson descriptions of nuclei give the inaccura…
The centrality dependence of pseudorapidity density of charged particles ($ dN_{ch}/dη$) in $Cu+Cu$ ($Au+Au$) collisions at RHIC energy of $ \sqrt{s_{_{\rm NN}}}=22.4$ , $62.4 $ and $200$ ($19.6$, $62.4$ and $200$) GeV, is investigated within an improved HIJING code. The standard HIJING model is enhanced by a prescription for collective nucleon-nucleon ($NN$) interactions and more modern parton distribution functions. The collective $NN$-interactions are used to induce both cascade and nucleon shadowing effects. We find collective cascade broadens the pseudorapidity distributions in the tails …
We derive the general structure of the hadronic tensor required to describe deep-inelastic scattering from an off-shell nucleon within a covariant formalism. Of the large number of possible off-shell structure functions we find that only three contribute in the Bjorken limit. In our approach the usual ambiguities encountered when discussing problems related to off-shellness in deep-inelastic scattering are not present. The formulation therefore provides a clear framework within which one can discuss the various approximations and assumptions which have been used in earlier work. As examples, w…
Estimation of collision centrality in terms of the number of participating nucleons in heavy-ion collisions using deep learning
本文首次用 AMPT 模拟数据训练 DNN 与 CNN，依据带电强子的赝快度和方位角信息估计 $N_{\mathrm{part}}$，CNN 精度更高且对能量和参数变化稳健。
We present in this paper a calculation of the average proton-nucleus inelasticity. Using an Iterative Leading Particle Model and the Glauber model, we relate the leading particle distribution in nucleon-nucleus interactions with the respective one in nucleon- proton collisions. To describe the leading particle distribution in nucleon-proton collisions, we use the Regge Mueller formalism. Contribution to 26th ICRC - Salt Lake City, Utah. August, 1999. HE 1.1.14
The Bloom-Gilman local duality of the inelastic structure function of the proton, the deuteron and light complex nuclei is investigated using available experimental data in the squared four-momentum transfer range from 0.3 to 5 (GeV/c)**2. The results of our analysis suggest that the onset of the Bloom-Gilman local duality is anticipated in complex nuclei with respect to the case of the protonand the deuteron. A possible interpretation of this result in terms of a rescaling effect is discussed with particular emphasis to the possibility of reproducing the damping of the nucleon-resonance trans…
Nucleon-Nucleon elastic and inelastic scattering using quantum field theory: a comparative study
本文基于Yukawa理论推导散射矩阵，计算核子-核子弹性与非弹性散射截面，发现标量流计算结果与实验数据高度吻合，可作为复杂场论的有效简化替代方案。
The off-mass-shell behavior of the nucleon structure function, $F_2^N$, is studied within an approach motivated by the Sullivan model. Deep inelastic scattering on the nucleon is considered in the second order in the pion-nucleon coupling constant, corresponding to the dressing of the bare nucleons by the mesonic cloud. The inclusive and semi-inclusive deep inelastic processes on the deuteron involving off-shell nucleons are considered. A deformation of the mesonic cloud for the off-mass-shell nucleon, compared to the free one, generates observable effects in deep inelastic scattering. In part…
Based on the requirement in the simulation of lepton-nucleus deep inelastic scattering (DIS), we construct a fortran program LDCS 1.0 calculating the differential and total cross sections for the unpolarized charged lepton-unpolarized nucleon and neutrino-unpolarized nucleon neutral current (charged current) DIS at leading order. Any set of the experimentally fitted parton distribution functions could be employed directly. The mass of incident and scattered leptons is taken into account and the boundary conditions calculating the single differential and total cross section are studied. The cal…
We present a connection between the modifications induced by the nuclear medium of the nucleon form factors and of the deep inelastic structure functions, obtained using the concept of generalized parton distributions. Generalized parton distributions allow us to access elements of the partonic structure that are common to both the hard inclusive and exclusive scattering processes in nuclei.
A covariant formulation of spin-dependent deep-inelastic scattering from nuclei is presented. In the relativistic impulse approximation the inclusive nuclear cross section is described in terms of the amplitude for forward, virtual-photon scattering from an off-mass-shell nucleon. The general structure of the off-shell nucleon hadronic tensor is derived, and the leading behavior of the off-shell nucleon structure functions computed in the Bjorken limit. The formalism, which is valid for nucleons bound inside nuclei with spin 1/2 or 1, is applied to the case of the deuteron.
We have performed several fits to find the ratio $R=σ_{L}/σ_{T}$ in different phenomenological models. Our fits seem to leave no room for possible admixture of scalar partons inside the nucleon.
We discuss the radiative corrections to charged and neutral current deep-inelastic neutrino--nucleon scattering in the minimal supersymmetric standard model (MSSM). In particular, deviations from the Standard Model prediction for the ratios of neutral- to charged-current cross sections, $R^ν$ and $R^{\barν}$, are studied, and results of a scan over the MSSM parameter space are presented.
Neutral current neutrino and antineutrino scattering off the polarized nucleon
研究中性流中微子与反中微子在极化核子上的弹性及单 π 产生非弹性散射，指出自旋不对称性可区分两类过程、识别靶类型并揭示共振—非共振产生机制。
Centrality and cumulative activities in hadron collisions
本文提出强子碰撞弹性事件的累积活性概念，并建立其与中心性在弹性和非弹性事件中的相互关系，为强子散射分析提供新工具。
We construct a model for inelastic neutrino- and electron-nucleon scattering cross sections using effective leading order parton distribution functions with a new scaling variable $ξ_w$. Non-perturbative effects are well described using the $ξ_w$ scaling variable, in combination with multiplicative $K$ factors at low $Q^2$.Our model describes all inelastic charged lepton-nucleon scattering (including resonance) data (HERA/NMC/BCDMS/SLAC/JLab) ranging from very high $Q^2$ to very low $Q^2$ and down to the photo-production region. The model describes existing inelastic neutrino-nucleon scatterin…
We present in this paper a calculation of the average proton-nucleus ine- lasticity. Using an Iterative Leading Particle Model and the Glauber model, we relate the leading particle distribution in nucleon-nucleus interactions with the respective one in nucleon-proton collisions. To describe the leading particle distribution in nucleon-proton collisions, we use the Regge-Mueller formalism. To appear in Journal of Physics G.
The Glauber modeling plays a key role in centrality-dependent measurements of heavy-ion collisions. A central input parameter in Glauber models is the inelastic nucleon-nucleon cross section $σ_{\text{nn}}^{\text{inel}}$ which is nearly always taken from proton-proton measurements. At the LHC energies $σ_{\text{nn}}^{\text{inel}}$ depends on the QCD dynamics at small $x$ and low interaction scales where the shadowing/saturation phenomena are expected to become relatively more important for larger nuclei than for the proton. Thus, $σ_{\text{nn}}^{\text{inel}}$ e.g. in Pb+Pb collisions may well …
The $O(α)$ leptonic QED corrections to neutral current polarized deep inelastic lepton-nucleon scattering are calculated in leptonic variables both for the case of longitudinal and transverse nucleon polarization. The results of the complete calculation are compared with the corresponding leading log expressions. Numerical results are presented for the corrections in the kinematic range of the HERMES experiment and possible future polarized proton beam experiments at HERA.
Numerical results for the degree of polarization of τ^- produced in (CC) neutrino - nucleon Deep Inelastic Scattering (DIS) are presented. Calculations are done in the threshold region, where the τ^- scattered by the small angles and can be partially polarized. The cross sections and polarization are calculated by using the GRV98 parton distributions functions (PDF's) and the GRV98 with modifications of A. Bodek at.al.
We construct a model for inelastic neutrino- and electron-nucleon scattering cross sections using effective leading order parton distribution functions with a new scaling variable $ξ_w$. Non-perturbative effects are well described using the $ξ_w$ scaling variable, in combination with multiplicative $K$ factors at low $Q^2$. Our model describes all inelastic charged lepton-nucleon scattering (including resonance) data (HERA/NMC/BCDMS/SLAC/JLab) ranging from very high $Q^2$ to very low $Q^2$ and down to the photo-production region. The model describes existing inelastic neutrino-nucleon scatteri…
The one-loop electroweak radiative correction of the lowest order to lepton current for deep inelastic scattering of longitudinally polarized leptons by polarized nucleons is obtained in model independent way. The detailed numerical analysis within kinematical requirements of future polarization collider experiments is performed. The possibility to reduce radiative effects by detection of hard photons is studied.
The accuracy of measuring the total energy of the spectators in the FHCal calorimeter, depending on the fluctuations in the number of nucleons and the hadron shower is obtained. It is shown that, from the point of view of statistics, the process of registering photoelectrons in a photomultiplier (PM) is similar to the process of registering spectator nucleons in a calorimeter. The measurements of single photoelectrons in the PM are in good agreement with the obtained formula for the relative variance of the charge detected by the PM.
We propose new method which allows determination of the strange-sea quark spin distribution in the nucleon through measurement of various inclusive and semi-inclusive polarized deep inelastic electron- or muon-proton reactions. It is shown, that using combinations of inclusive data and semi-inclusive data containing neutral pions in the final state, it is possible to extract the strange-sea quark spin distribution. Similar result can be obtained for charged pions and some other hadrons also.
The processes of lepton-nucleon scattering, including ones with both polarized beams, at high energy provide relevant information about interaction and particles structure, allowing to analyze nucleon spin structure. As energy and experimental accuracy rise, necessity to improve Born cross sections and polarized asymmetries with higher order radiative corrections becomes substantial. In this report we stress on lowest order bremsstrahlung corrections treatment using helicity amplitudes method as applied to actual nowadays charged current lepton-nucleon deep inelastic processes, that allows to …
Possibilities to extract information on the strange form factors of the nucleon from neutrino (antineutrino) inelastic scattering on nuclei, in an energy range from 200 MeV to 1 GeV and more, are investigated in detail. All calculations are performed within two relativistic independent particle models (Fermi gas and shell model); the final state interactions of the ejected nucleon are taken into account through relativistic optical model potentials. We have shown that the values of the cross sections significantly depend on the nuclear model (especially in the lower energy range). However the …
Inclusive electron scattering off few-nucleon systems is investigated at $x>1$ and high momentum transfer, including the contributions from quasi-elastic and deep inelastic scattering processes. It is shown that at $x>1$ the inclusive cross section is dominated by the process of virtual photon absorption on a pair of correlated nucleons both in case of quasi-elastic and deep inelastic scattering events. The sensitivity of the nuclear response to the effects arising from the possible presence of multiquark cluster configurations at short internucleon separations as well as from possible medium-…
The forward-backward multiplicity correlation strength is calculated for arbitrary nucleus-nucleus collision in the framework of the wounded nucleon model. Discussion of our results in the context of the recent STAR data in Au+Au collisions at sqrt{s}=200 GeV is presented. It is suggested that the observed (i) growth of the correlation coefficient with centrality and (ii) approximately flat pseudorapidity dependence of the correlation strength for central collisions are due to the fluctuations of the number of wounded nucleons at a given centrality bin.
We extend the calculations of the spin alignments of vector mesons in $e^+e^-$ annihilation in a recent Rapid Communication to deeply inelastic lepton-nucleon scatterings. We present the results for different mesons in the current fragmentation regions of $μ^- N$$ \to$$ μ^- VX$ at high energies and $ν_μN $$\to$$ μ^- VX$ at both high and low energies. We also present the predictions for $ν_μN$$ \to$$ μ^- VX$ at NOMAD energies in the target fragmentation region using a valence quark model.
Model dependence of the number of participant nucleons and observable consequences in heavy-ion collisions
本文研究了1.23 AGeV下Au-Au碰撞中参与核子数估计的严重模型依赖性，发现碰撞参数和带电粒子数是更优的中心度估计量，并建议实验中采用总重子数平衡作为替代。
A Monte Carlo model, initially developed for soft pp and AA collisions at high energy, is applied for proton-lead interaction at the LHC energy. Elementary collisions are implemented at the partonic level and do not involve the usual Glauber supposition of independent nucleon-nucleon collisions. The average number of participating nucleons and charged multiplicity in p-Pb collisions were calculated and compared with the predictions of Glauber model and experimental data. It was demonstrated that taking into account the energy conservation results in the number of participating nucleons conside…
The positivity bound for the transverse asymmetry $A_2$ may be improved by making use of the fact, that the state of a photon and a nucleon with total spin 3/2, does not participate to the interference. The bound is therefore useful in the case of a longitudinal asymmetry small (say, at low $x$) or negative (like in the neuteron case).
In a previous paper \cite{Hatta:2020ltd}, we have pointed out the connection between the $CP$-odd three-gluon (Weinberg) operator and certain twist-four corrections in polarized deep inelastic scattering. Based on this observation, we give a numerical estimate of the electric dipole moment of the proton and neutron induced by the Weinberg operator. Our result is smaller than the previous estimates based on QCD sum rules by a factor of about 3 or more.
We present a unified description of nuclear deep inelastic scattering (DIS) over the whole region $0<x<1$ of the Bjorken variable. Our approach is based on a relativistically covariant formalism which uses analytical properties of quark correlators. In the laboratory frame it naturally incorporates two mechanisms of DIS: (I) scattering from quarks and antiquarks in the target and (II) production of quark-antiquark pairs followed by interactions with the target. We first calculate structure functions of the free nucleon and develop a model for the quark spectral functions. We show that mechanis…
We study charged lepton flavor violation (CLFV) associated with heavy quark pair production in lepton-nucleon deep-inelastic scattering $\ell_i N \to \ell_j q\bar{q} X$. Here $\ell_i$ and $\ell_j$ denote the initial and final leptons; $N$ and $X$ are respectively the initial nucleon and arbitrary final hadronic system. We employ a model Lagrangian in which a scalar and pseudoscalar mediator generates the CLFV. We derive heavy quark structure functions for scalar and pseudoscalar currents and compute momentum distributions of the final lepton for the process. Our focus is on the heavy quark mas…
We present the results of the covariant spectator quark model applied to the nucleon structure function $f(x)$ measured in unpolarized deep inelastic scattering, and the structure functions $g_1(x)$ and $g_2(x)$ measured in deep inelastic scattering using polarized beams and targets ($x$ is the Bjorken scaling variable). The nucleon is modeled by a valence quark-diquark structure with $S,P$ and $D$ components. The shape of the wave functions and the relative strength of each component are fixed by making fits to the deep inelastic scattering data for the structure functions $f(x)$ and $g_1(x)$…
We present a detailed phenomenological study of charged-current-mediated deep-inelastic scattering off longitudinally polarized nucleons at a future Electron-Ion Collider. A new version of the event generator package DJANGOH, extended by capabilities to handle processes with polarized nucleons, is introduced and used to simulate charged current deep-inelastic scattering including QED, QCD, and electroweak radiative effects. We carefully explore the range of validity and the accuracy of the Jacquet-Blondel method to reconstruct the relevant kinematic variables from the measured hadronic final s…
We derive a general framework for describing semi-inclusive deep-inelastic lepton-nucleon scattering in terms of the unintegrated parton distributions and other higher twist parton correlations. Such a framework provides a consistent approach to the calculation of inclusive and semi-inclusive cross sections including higher twist effects. As an example, we calculate the azimuthal asymmetries to the order of 1/Q in semi-inclusive process with transversely polarized target. A non-vanishing single-spin asymmetry in the ``triggered inclusive process'' is predicted to be 1/Q suppressed with a part …
We address the question whether the nucleon's antiquark sea can be attributed entirely to its virtual meson cloud and, in essence, whether there exists a smooth transition between hadronic and quark-gluon degrees of freedom. We take into account contributions from $π$ and $K$ mesons and compare with the nucleon's antiquark distributions which serve as a non-perturbative input to the QCD evolution equations. We elucidate the different behavior in the flavor singlet and non-singlet channels and study the dependence of our results on the scale $Q^2$. The meson-nucleon cut-offs that we determine g…
Using a recent parametrization of nuclear effects in parton distribution functions we calculate the neutrino-nucleon cross section at energies relevant for ultra high energy neutrino telescopes. The modification of the cross section in comparison with the calculation using parton densities in free nucleons is of the order of few per cent for the parameter range of interest in neutrino telescopes (A=10 and E=10$^6$ GeV) and it reaches 20 % at the highest energies (E=10$^{12}$ GeV) and for the largest nuclear size (A=190) considered.
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