Properties of Full Jet in High-Energy Heavy-Ion Collisions from Parton Scatterings

The properties of fully reconstructed jet are investigated in p + p and Pb + Pb collisions at √s NN = 2.76 TeV within a multiphase transport (AMPT)model with both partonic scatterings and hadronic rescatterings. A large transversemomentum (p T ) asymmetry of dijet or photon-jet arises from the strong interactions between jet and partonic matter. The ξ-dependent jet fragmentation function in Pb + Pb collisions is decomposed into two contributions from different jet hadronization mechanisms, that is, fragmentation versus coalescence. The medium modification of differential jet shape displays that the jet energy is redistributed towards a larger radius owing to jet-medium interactions in heavy-ion collisions. Jet triangular azimuthal anisotropy coefficient, V 3 , which shows a smaller magnitude than the elliptic coefficient V 2 , decreases more quickly with increasing jet p T , which can be attributed to a path-length effect of jet energy loss. All of these properties of full jet are consistent with the jet energy lossmechanism in a stronglyinteracting partonic matter in high-energy heavy-ion collisions.


Introduction
Plenty of experimental data from the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC) have confirmed that a nearly perfect fluid of strongly interacting quark-gluon plasma (QGP) could be created in the early state of high-energy heavy-ion collisions [1][2][3][4].The origin and properties of this almost perfect-fluid QGP are the main subjects of heavy-ion collision studies.Jet, produced by initial hard processes, serves as an important probe to understand the properties of the QGP, since it loses its energy when it passes through the hot partonic medium, the effect which is called jet quenching [5].The nuclear modification factor   , which is the yield ratio of nucleus + nucleus collisions to inelastic p + p collisions normalized by the number of binary inelastic nucleon + nucleon collisions, shows a strong suppression at high transverse momentum   in central nucleus + nucleus collisions at the RHIC [6] and LHC [7] energies.The disappearance of away-side peak in dihadron azimuthal correlation presents the picture that the away-side jet is strongly suppressed by the QGP in central Au + Au collisions at the top RHIC energy [8].On the other hand, the recent LHC measurements on fully reconstructed jets provide a comprehensive characterization of jet quenching.For instance, a larger dijet   asymmetry has been observed in central Pb + Pb collisions than in p + p collisions at the LHC energy [9,10].Since photon does not strongly interact with the QGP, the photon + jet measurements from CMS and ATLAS provide direct and less biased quantitative measures of jet energy loss in the medium, which give a deceasing jet-to-photon momentum imbalance ratio (  ) from peripheral to central centrality bin in Pb + Pb collisions [11,12].The LHC measurements of the modification ratio of jet fragmentation function in Pb + Pb collisions to that in p + p collisions show the interesting features of no modification at low  = ln(1/), a suppression at intermediate , and an enhancement at high  for associated charged hadrons inside the jet cone [13][14][15].The experimental results about differential jet shape show no modification at a small radius but a large enhancement at a large radius in 2 Advances in High Energy Physics central Pb + Pb collisions, relative to that in p + p collisions [14,16].The data on the elliptic anisotropy of reconstructed jets from the ATLAS Collaboration show nonzero V 2 values for the jet   range from 45 to 160 GeV/c for all centrality bins in Pb + Pb collisions [17].
In this review, the properties of reconstructed jets are presented for p + p and Pb + Pb collisions at √  = 2.76 TeV by using a multiphase transport (AMPT) model, which includes both dynamical evolutions of partonic scatterings and hadronic rescatterings.For partonic scatterings, the AMPT model uses an elastic cross section calculated from the pQCD.Several theoretical results are achieved in our numerical simulations.(i) A large   asymmetry of dijet or photon-jet is produced by strong interactions between jets and partonic matter [22,23].(ii) The measured jet fragmentation function ratio of Pb + Pb collisions to p + p collisions is decomposed into two parts, corresponding to the two contributions of jet hadronization from fragmentation and coalescence [24].(iii) The differential jet shapes are significantly modified by the strong interactions between jets and the partonic medium in Pb + Pb collisions relative to that in p + p collisions [25].(iv) Jet V 2 be in a good agreement with the recent ATLAS data and let jet V 3 , which has a smaller magnitude than jet V 2 , decrease quickly with the increasing of jet transverse momentum [26].All of these features support a picture of jet energy loss in a strongly interacting partonic matter.
The review is organized as follows.In Section 2, we give a brief description of AMPT model.The basic mechanisms related to jet in the AMPT model are stated in Section 3. The analysis method for jet reconstruction is explained in Section 4. Results and discussions are presented for dijet, photon-jet, jet fragmentation function, jet shape, and jet azimuthal anisotropy in Section 5. Finally a summary is given in Section 6.

The AMPT Model
The AMPT model with a string melting scenario [27], which has well described many experimental observables [27][28][29][30][31][32][33], is implemented in this work.The AMPT model includes four main stages of high-energy heavy-ion collisions: the initial condition, parton cascade, hadronization, and hadronic rescatterings.The initial condition, which includes the spatial and momentum distributions of minijet partons and soft string excitations, is obtained from HIJING model [34,35].Then the strings are melted into quarks, and thus a quark and antiquark plasma are formed and start to evolve.The parton cascade process is simulated by Zhang's parton cascade (ZPC) model [36], where the partonic cross section is an elastic cross section controlled by the value of strong-coupling constant   and the Debye screening mass .The AMPT model recombines partons via a simple coalescence model to produce hadrons when the partons freeze out.The dynamics of the subsequent hadronic rescatterings is then described by a relativistic transport (ART) model [37].
In this work, the AMPT model with the newly fitted parameters (such as   = 0.33 and  = 3.2 fm −1 , which corresponds to a parton interaction cross section  = 1.5 mb, and parameters  = 0.9 and  = 0.5 GeV 2 in the Lund string fragmentation model) is used to simulate Pb + Pb collisions at √  = 2.76 TeV.It has shown good descriptions for many experimental observables at the LHC energy, such as pseudorapidity and   distributions [38], and harmonic flows [39,40].Two sets of Pb + Pb simulations are performed by setting the partonic interaction cross section as 1.5 or 0 mb, which corresponds to two different physical scenarios for partonic + hadronic interactions and hadronic interactions only, respectively.

Jet in the AMPT Model
Because the jet production cross section with large transverse momentum is very small, the dijet or photon-jet (denoted as -jet) production is triggered in the initial condition of the HIJING model, which requires that each event has a dijet or -jet with a specified   , in order to increase the simulation efficiency.Several hard QCD processes are taken into account for the initial dijet production with the jet triggering technique in the HIJING model [34,35], which includes  1 +  2 →  1 +  2 ,  1 +  1 →  2 +  2 ,  +  → +,  +  →  + ,  +  →  + , and  +  →  + , with consideration of initial-and final-state radiation corrections.Three prompt photon production processes are taken into account for the -jet study, including  +  →  + ,  +  →  + , and  +  →  +  [41].The high-  primary partons evolve into jet showers full of lower virtuality partons through initial-and final-state QCD radiations.In the string melting mechanism, all excited strings and jets are fragmented into hadrons according to the Lund string fragmentation.Then these hadrons are converted to quarks according to the flavor and spin structures of their valence quarks.After the melting process, the jet parton showers are converted into clusters of on-shell constituent quarks and antiquarks, and a plasma of on-shell constituent quarks and antiquarks is also formed.Next, the jet transport is simulated by the ZPC model including all possible elastic partonic interactions among the medium quarks and jet shower quarks, but without including inelastic parton interactions or further radiations at present.It means that our simulations currently do not include the mechanism of jet radiation energy loss.However a large partonic interaction cross section can partially play an effective role, and we expect to improve it by including many-body interactions into the process of parton cascade in the future.When the quarks freeze out, they are recombined into medium hadrons or jet shower hadrons via a simple coalescence model which combines two nearest quark and antiquark into a meson and three nearest quarks into a baryon.The final-state hadronic interactions, including elastic and inelastic scatterings between jet and hadrons and resonance decays, can be automatically described by the ART model., where the solid (1.5 mb) and dash (0 mb) histograms represent the AMPT results with partonic + hadronic interactions and hadronic interactions only, respectively, while the solid circles represent the data from the CMS experiment [10].The lower part in each panel depicts the ratios of AMPT results to experimental data.

Jet Reconstruction
An anti-  algorithm from the standard Fastjet package is used to reconstruct full jets [42].The kinetic cuts of reconstructed jet, such as the jet cone size  and jet   , are always chosen to be consistent with the CMS or ATLAS experiment, which will be used for each analysis in the next section.It is very important to remove the underlying event background from the raw reconstructed jet cleanly, since there are thousands of tracks produced in high-energy heavy-ion collisions.Two methods are applied for removing the background of reconstructed jets in our analyses.(i) A pseudorapidity strip of width Δ = 1.0 centered on the jet position, with two highest-energy jets excluded, is used to estimate the background ("average energy per jet area"), which is subtracted from the reconstructed jet energy.It is used to analyze dijet or -jet asymmetry and jet azimuthal anisotropy.(ii) The -reflection method is used for reconstructing jet fragmentation function and differential jet shape, as the CMS experiment did.We select the particles that lie in a background jet cone obtained by reflecting the original jet cone around  = 0 while keeping the same   coordinate.For each signal jet, the background distribution is subtracted from the raw distribution obtained from the jet cone.Jets in an -strip region (|| < 0.3) are excluded to avoid overlap between the signal jet region and the region used for background estimation.

Dijet.
A dijet with   ∼ 120 GeV/c is triggered in the dijet simulations.For the analysis cuts, the transverse momentum of leading jet is required to be larger than 120 GeV/c ( ,1 > 120 GeV/c), while that of subleading jet is required to be larger than 50 GeV/c ( ,2 > 50GeV/c).The azimuthal angle between leading and subleading jets is larger than 2/3 (Δ 1,2 > 2/3), where the subscripts 1 and 2 refer to the leading jet and subleading jet, respectively [10].Only jets within a midrapidity range of |  [43] and CMS experiment [44].
ranges of impact parameters.The AMPT simulations with partonic + hadronic interactions (1.5 mb) present the spectra a little softer than those with hadronic interactions only (0 mb).Note that the measured leading jet   distributions have not been corrected for some detector effects such as detector resolution, fluctuations in and out of the jet cone, or underlying event fluctuations [10].From a quantitative comparison of the ratios of AMPT results to experimental data (lower part in each panel of Figure 1), the AMPT model can reproduce the data to a good degree.
To characterise the transverse momentum balance (or imbalance) of dijet, an asymmetry ratio is defined as   = ( ,1 −  ,2 )/( ,1 +  ,2 ) as LHC experiments did [9,10].The dijet asymmetry   distributions for different centrality bins are shown in Figures 2(a)-2(e).For more peripheral collisions, both sets of AMPT results give similar descriptions of the data, since the partonic interactions are relatively weak in peripheral collisions.However, it is different for more central collisions where the AMPT results with partonic + hadronic interactions give more asymmetric   distribution than those with hadronic interactions only.For instance, for the most central centrality bin (0-10%) in Figure 2(a), the AMPT results (1.5 mb) give a much better description than AMPT results (0 mb). Figure 2(f) presents the mean values ⟨  ⟩ and variances (  ) of   distributions as functions of a number of participant nucleons,  part .The AMPT results (1.5 mb) can well describe the two characteristic quantities for dijet   distributions simultaneously; however the AMPT results (0 mb) underestimate ⟨  ⟩.This indicates that it is the strong interactions between the jets and the partonic matter that yield the observed large   imbalance between the two backto-back jets.

Photon-Jet. A 𝛾-jet of 𝑝
∼ 60 GeV/c is triggered in the -jet simulations.The kinetic cuts for the analysis on -jet transverse momentum imbalance are chosen to be consistent with the CMS experiment [11].The transverse momentum of prompt photon is required to be larger than 60 GeV/c (   > 60 GeV/c) and its pseudorapidity is within a midrapidity gap of 1.44 (|  | < 1.44).A jet cone size is set to be 0.3 ( = 0.3),   of jet is larger than 30 GeV/c ( jet  > 30 GeV/c), and pseudorapidity of jet is within a midrapidity range of | jet | < 1.6.After removing the underlying event background, the triggered -jet events are sampled, with the measured experimental prompt photon   spectra as the weight.Figure 3 shows the final   spectra of prompt photons in comparison with the ATLAS Pb + Pb [43] and CMS p + p data [44].
Compared with dijet, -jet have its unique advantage.Because the prompt photon has no strong interactions with medium, it can provide a natural calibration of initial jet energy.Therefore, the transverse momentum imbalance is defined as the ratio of   =  jet  /   to study jet energy loss mechanism [11,12].Figures 4(a)-4(d) show the imbalance ratio distributions for four centrality bins of Pb + Pb collisions and p + p collisions at √  = 2.76 TeV.The AMPT results with both partonic and hadronic interactions (i.e., 1.5 mb) give a little smaller   and ⟨  ⟩ than those with hadronic interactions only (i.e., 0 mb) and experimental data.To quantitively learn how much jet loses its energy in partonic or hadronic matter, the averaged energy loss fraction of jet, ⟨Δ  /  ⟩, is checked but not shown here, which shows that jet loses its energy from by ∼15% in central collisions down to by ∼5% in peripheral collisions due to decreasing of partonic interactions, whereas hadronic interactions with vanished partonic interactions only can give much smaller energy loss fraction around 4%-2% [23].It indicates that the strong interactions between jet parton shower and partonic matter can produce a larger momentum asymmetry than the interactions between jet hadron shower and hadronic matter, especially for more central collisions.

Jet Fragmentation Function.
To acquire jet fragmentation functions, the kinetic cuts for jet reconstruction are chosen to be consistent with the CMS experiment [14].The jet cone size is set to 0.3.The transverse momentum of jet is required to be larger than 100 GeV/c (  > 100 GeV/c) within a pseudorapidity range of 0.3 < || < 2 for this analysis, where jets within || < 0.3 are excluded to avoid the overlap between the signal jet region and the jet background estimation region.The jet fragmentation function is obtained by correlating charged hadrons with   > 1 GeV/c falling within the jet cone, with respect to the axis of reconstructed jet.As the CMS experiment defined, the jet fragmentation function, () = 1/ jet  ch /, can be presented as a function of the variable  = ln(1/), where  =  ch ‖ / jet is the fraction of the jet energy carried by the charged particle,  ch ‖ is the momentum component of charged particle along the jet axis,  jet is the magnitude of reconstructed jet momentum, GeV/c, Δ  > 7/8) after background subtraction for four centrality bins in Pb + Pb and p + p collisions, where the solid (1.5 mb) and dash (0 mb) histograms represent the AMPT results with partonic + hadronic and hadronic interactions only, respectively, while the circles represent the data from the CMS experiment [11].and  jet is the total number of jets.All charged particles in the cone of 0.3 around the jet axis are included in this analysis.It should be noted that lower  actually corresponds to higher   .An -reflection method is used to estimate the background, which is subtracted from the reconstructed jet fragmentation function in Pb + Pb collisions.

CMS data
Figure 5(a) shows the jet fragmentation function () in p + p collisions at √  = 2.76 TeV.From a quantitative comparison of the ratio of AMPT result to experimental data shown in Figure 5(b), the result obtained from AMPT simulations with hadronic interactions can only basically describe the jet fragmentation function in p + p collisions, which provides a reliable baseline for the following studies in Pb + Pb collisions with the AMPT model with partonic + hadronic interactions.
Because heavy-ion collisions involve many important stages, the evolution of the jet fragmentation function during different stages can provide the important information about the mechanism of medium modifications of jet fragmentation functions in Pb + Pb collisions.Figure 6  fragmentation function ratios of the most central events (0-10%) in Pb + Pb collisions to p + p collisions; that is, () =  Pb+Pb ()/ p+p (), at different evolution stages from AMPT simulations with partonic + hadronic interactions (1.5 mb).Some points are slightly shifted along the -axis for better representation.The initial jet fragmentation function ratio is around unity which indicates no modification in the initial state of Pb + Pb collisions.Two basic features of modification, an enhancement at low  and a suppression at intermediate , appear in the jet fragmentation function ratio after the process of parton cascade.The enhancement at low  area is due to the fact that the energy loss of the jet is more significant than that of leading-like partons, which relatively decrease their .On the other hand, the suppression is the result of the decrease of associated particles with intermediate   owing to the jet energy loss in the partonic medium, which are probably shifted to lower   or even thermalized.However, the expected high- enhancement owing to the shift or thermalization is hard to be seen for the current statistics.A significant enhancement around intermediate and high  and small suppression at low  are observed after coalescence.It is because the coalescence mechanism in the AMPT slightly increases the total momentum of jet, owing to the involution of medium partons, and also increases the momenta of shower hadrons in comparison with the previous stage.And there is little effect from hadronic rescatterings.The solid squares represent the data from the CMS experiment [14].Some points are slightly shifted along the -axis for better representation.
However, neither of the final jet fragmentation function ratios in the simulations can fit the experimental data for the whole  range.The reason why the AMPT results can not match the measured jet fragmentation function ratio for the whole  range is that the AMPT model with a string-melting scenario only uses a coalescence model for hadronization but misses the other important one, that is, fragmentation.Actually, the interplay of fragmentation and coalescence indeed can give very good descriptions such as   spectra and elliptic flow in a wide   range [45].To well describe the experimental data of the jet fragmentation function ratio in the whole  range, it is proposed to decompose the measured jet fragmentation function ratio to where     () and     () are fragmentation and coalescence parts, respectively, which are assumed to coexist in the measured jet fragmentation function ratio ().  and   are the contribution factors for fragmentation and coalescence parts, respectively.The functional form of   () is assumed to be the same as that of jet fragmentation function ratio after parton cascade, based on the parton-hadron duality or the subleading correction effect of fragmentation on the nuclear modification factor [46].The functional form of   () is assumed to be equal to the jet fragmentation function ratio after hadronic rescatterings, which includes both effects of coalescence and hadronic rescatterings.Thus, the two contribution parts can be obtained by fitting the experimental data of () with (1).It should be noted that   and   are assumed to be independent of  for simplicity in this work, which also can be understood as averaged values.a dominant contribution from fragmentation in the low- range.For the most peripheral collisions in Figure 7(d), it is hard to conclude due to the large uncertainties of two contributions.In general, the effect of coalescence tends to be more dominant for the high- range in more central collisions, while the contribution from fragmentation becomes more important for the low- range in more peripheral collisions.

Jet Shape.
To reconstruct jet shape, the kinematic cuts are chosen to be consistent with the CMS experiment [14,16].The transverse momentum of a jet is required to be larger than 100 GeV/c within a pseudorapidity  range of 0.3 < || < 2 for this analysis.Jets within || < 0.3 are excluded in order to avoid the overlap between the signal jet region and the jet background estimation region.The jet cone sizes  are set to be 0.3.The differential jet shape () is defined as the fraction of the transverse momentum carried by particles (  > 1 GeV/c) associated with the jet, which are contained inside an (,  (azimuthal angle)) annulus of inner and outer radii of  ± /2 around the jet axis, where  is chosen to be 0.05 and 0 ≤  ≤ .() satisfies the normalization condition ∫  0 () = 1.The -reflection method is used to estimate the background, which is subtracted from the reconstructed differential jet shape.
Figure 8 shows the comparison of the differential jet shapes () between the AMPT result with hadronic interactions only (0 mb) and the experimental data for p + p collisions at √  = 2.76 TeV.The AMPT result basically can describe p + p data, which provides a qualified baseline for the following calculations in Pb + Pb collisions.
Figures 9(a)-9(d) present the differential jet shape ratios of different centrality bins in Pb + Pb collisions to p + p collisions from AMPT simulations with hadronic interactions only (0 mb) and with both partonic and hadronic interactions (1.5 mb), in comparisons with the experimental data.Some points are slightly shifted along the -axis for better representation.It is found that including only hadronic interactions between jets and hadronic medium can hardly modify the energy distributions of reconstructed jets for all centrality bins in Pb + Pb collisions.On the other hand, the AMPT results with both partonic and hadronic interactions give relatively larger modifications, which indicates that the observed medium modifications of differential jet shapes mainly result from the interactions between jets and partonic medium.The emergent modifications in central Pb + Pb collisions show a suppression at a small radius and an enhancement at a large radius, which implies that the jet energy is redistributed towards a larger radius via the strong interactions between jet and the partonic medium.

Jet Azimuthal Anisotropy.
To reconstruct jet azimuthal anisotropy, the kinematic cuts are chosen to be the same as those in the ATLAS experiment [17].The jet cone size  is set to be 0.2.The average energy per jet area, as the underlying event background, is subtracted from the reconstructed jet energy in Pb + Pb collisions.Only jets within a midrapidity range of || < 2 are considered in this analysis.
It is well known that all orders of harmonic flows can arise from the initial geometry fluctuations through finalstate interactions [47,48].Futhermore, the even orders of harmonic flows can be affected by initial fluctuations in the collision geometry [49].To calculate the th Fourier coefficient of jets V jet  , the th event plane Ψ   can be defined as where  and  are the coordinate position and azimuthal angle of each parton in the AMPT initial state and the average ⟨⋅ ⋅ ⋅ ⟩ denotes density weighting.Then V jet  can be obtained from the following equation: Note that although the definition for V jet  is the same as that for a single hadron, it is, however, expected to have a smaller bias because the reconstructed jet has kinematic properties that are more closely related to those of the parent partons [17].
Jets V 2 and V 3 as a function of  part for two typical   bins of 45 <   < 60 GeV/c and 60 <   < 80 GeV/c, calculated by ( 2) and (3) and denoted as V  } (open circles), though it has a little higher magnitudes due to the initial fluctuation contribution [49].For jet V 3 , it is smaller than jet V 2 .By comparing jet V 3 between two different jet   bins, jet V 3 tends to vanish with the increase of jet   .On the other hand, we checked that the jet passes a longer path length through the medium in the direction of Δ ∼ /2 or Δ ∼ /3 for an elliptic or triangle shape profile, which is consistent with the path-length effect of jet energy loss [50].and solid squares represent the data from the CMS experiment [14].Some points are slightly shifted along the -axis for better representation.

Summary
In summary, the properties of fully reconstructed jet are investigated within the AMPT model with both elastic partonic scatterings and hadronic rescatterings.A large   asymmetry of dijet or photon-jet is produced by the strong interactions between jets and partonic matter.These partonic scatterings lead to medium modifications of full jets in Pb + Pb collisions with respect to p + p collisions.The measured jet fragmentation function ratio of Pb + Pb collisions to p + p collisions, which depends on  = ln(1/), can be decomposed into two contributions of jet hadronization from fragmentation and coalescence.The medium modification of differential jet shape indicates that the jet energy is redistributed towards a larger radius through the strong scatterings between jet and the partonic medium.A jet azimuthal anisotropy coefficient V jet 2 is in good agreement with the recent ATLAS data.V jet 3 has a smaller magnitude than V jet 2 and decreases quickly with increasing jet   , owing to the path-length effect of jet energy loss.The properties of full jet in Pb + Pb collisions disclose the fact that jet loses energy

Figure 4 :
Figure 4: The distributions of imbalance ratio   =  jet  /   between the photon (   > 60 GeV/c) and jet ( jet  > 30 GeV/c, Δ  > 7/8)after background subtraction for four centrality bins in Pb + Pb and p + p collisions, where the solid (1.5 mb) and dash (0 mb) histograms represent the AMPT results with partonic + hadronic and hadronic interactions only, respectively, while the circles represent the data from the CMS experiment[11].

Figure 5 :
Figure 5: (a) The jet fragmentation function () in p + p collisions at √  = 2.76 TeV, where the histograms represent the AMPT (0 mb) result with hadronic interactions only and the squares represent the data from the CMS experiment [14].(b) The ratio of AMPT result to experimental data.

Figure 6 :
Figure6: The jet fragmentation function ratios of the most central centrality bin (0-10%) in Pb + Pb collisions to p + p collisions at different evolution stages.The solid squares represent the data from the CMS experiment[14].Some points are slightly shifted along the -axis for better representation.

Figure 7 :Figure 8 :
Figure7: The jet fragmentation function ratios of four centrality bins in Pb + Pb collisions to p + p collisions.The solid curves show twocomponent (fragmentation + coalescence) fitting functions, while the two kinds of hatched areas give the fragmentation and coalescence contribution parts, that is,     () and     (), for the jet fragmentation function ratios measured by the CMS experiment[14].
are shown in Figures 10(a) and 10(b), respectively.V jet 2 {Ψ  2 } (open triangles) is consistent with the jet V 2 calculations of V jet 2 {Ψ

Figure 9 :
Figure9: The differential jet shape ratios of four centrality bins in Pb + Pb collisions to p + p collisions, where solid circles represent the AMPT results with partonic + hadronic interactions (1.5 mb), open circles represent the AMPT results with hadronic interactions only (0 mb), and solid squares represent the data from the CMS experiment[14].Some points are slightly shifted along the -axis for better representation.