Parallel scattering, saturation, and generalized Abramovskii-Gribov-Kancheli (AGK) theorem in the EPOS4 framework, with applications for heavy-ion collisions at $\sqrt{s_{NN}}$ of 5.02 TeV and 200 GeV

Ultrarelativistic heavy-ion collisions will first realize many nucleon-nucleon scatterings, happening instantaneously and therefore necessarily in parallel, due to the short collision time. An appropriate quantum mechanical tool to treat that problem is S-matrix theory, and it has been known for a long time how to derive a simple geometric probabilistic picture, still widely used, and here the Abramovskii-Gribov-Kancheli (AGK) theorem plays a crucial role. All this is done in a scenario where energy conservation is not taken care of, but this is needed, in particular for Monte Carlo simulations. When introducing energy-momentum sharing properly, the AGK theorem does not apply anymore, nor do simple geometric concepts such as binary scaling. I will discuss this (very serious) problem, and how it can be solved, in the EPOS4 framework. When connecting the multiple-Pomeron approach (for parallel scatterings) and perturbative QCD, one is actually forced to implement in a very particular way saturation scales in order to get an approach free of contradictions. One recovers a generalized AGK theorem (gAGK), valid at large $p_t$ (larger than the relevant saturation scales). I discuss how gAGK is related to factorization (in proton-proton scatterings) and binary scaling (in heavy-ion collisions). I will show some applications, using this new approach as an initial condition for hydrodynamical evolutions, for heavy-ion collisions at $\sqrt{s_{NN}}$ of 5.02 TeV and 200 GeV, to get some idea about the energy dependence.

Figure 1

Factorization in $pp$ scattering.

Figure 2

Multiple scattering.

Figure 3

Modular structure of multiple scattering.

Figure 4

Nonlinear effects: ladders, which evolve first independently and in parallel, finally fuse.

Figure 5

Nonlinear effects (inside the red ellipses) are “summarized” in the form of saturation scales.

Figure 6

Modular structure of multiple scattering, including saturation scales, indicated by the big red dots.

Figure 7

Exponential function (red squares) and “almost exponential” function (blue circles); see text.

Figure 8

Sketch of a nucleus-nucleus collision in space-time. Here, $t$ is the time and $z$ the longitudinal coordinate, and the blue lines represent the nucleons.

Figure 9

Sketch of a nucleus-nucleus collision at very low energies, with sequential scatterings.

Figure 10

Sketch of a nucleus-nucleus collision at very high energies, with all nucleon-nucleon scatterings happening in parallel, before particle production starts.

Figure 11

Sketch of a nucleus-nucleus collision at intermediate energies, with only some (but not all) $NN$ scatterings being realized, before the particle production starts.

Figure 12

Sketch of a nucleon-nucleon scattering, with successive parton emissions before the hard process in the middle.

Figure 13

Sketch of a parallel scattering in nucleon-nucleon scattering.

Figure 14

Validity of the parallel scattering scenario. The “hybrid” area refers to a partially parallel scenario.

Figure 15

(a) Elastic scattering diagram and (b) the corresponding cut diagram.

Figure 16

Five-Pomeron graph, with two Pomerons being cut.

Figure 17

The only diagram contributing to inclusive cross sections: a single cut Pomeron.

Figure 18

The only diagram contributing to inclusive cross sections in $A+B$ scattering: a single cut Pomeron (multiplied by $AB$).

Figure 19

(a) A single cut Pomeron based on pQCD and (b) the corresponding inelastic process.

Figure 20

(a) Multiple scattering with energy sharing, for two colliding nuclei $A$ and $B$, with Pomerons (cyan vertical lines), remnants (pink horizontal lines), vertices (magenta dots), and cuts (black, vertical). (b) Detailed view of a vertex connected to projectile $i$, associated to pair $k$, with remnants light-cone momentum fraction $x_{\mathrm{remn},i}^{+}$ and Pomeron leg momenta $x_{k\mu }^{+}$.

Figure 21

The only diagram contributing to inclusive cross sections in $A+B$ scattering: a single cut Pomeron (multiplied by $AB$).

Figure 22

The multidimensional variable $\{b,\left\{b_i^A\right\},\left\{b_j^B\right\}\}$, with $b$ being the impact parameter, and with $b_i^A$ and $b_j^B$ being the transverse coordinates of the projectile and target nucleons, respectively (they are meant to be two-dimensional vectors).

Figure 23

Comparing $g\left({\sum }_{\lambda }{\tilde{\beta }}_{\lambda }\right)$ and $c_1{\prod }_{\lambda }{c}_{2}g\left(c_3{\tilde{\beta }}_{\lambda }\right)$ for randomly created sequences ${\tilde{\beta }}_{\lambda }$, for the parameter choice (A) $c_1=2, c_2=0.65$, and $c_3=1$.

Figure 24

Same as Fig. 23, but for the parameter choice (B) $c_1=2, c_2=1.5$, and $c_3=2.8$.

Figure 25

$W_{11}(\sqrt{x},\sqrt{x})/x^{{\alpha }_{\mathrm{remn}}}$ as a function of $x=x^{+}{x}^{-}$, for $pp$ scattering at $13\mathrm{TeV}$, for the impact parameters $b=3.5$, 2.5, and 1.5 fm. The solid curves refer to the exact results, and the dotted ones to the calculations based on $g$-factorization [Eq. (90)].

Figure 26

As Fig. 25, but for impact parameters $b=0.5$ and 0 fm.

Figure 27

$1-W_{11}(1,1)$ as a function of $b$, for $pp$ scattering at $13\mathrm{TeV}$. The solid curve refers to the exact result, and the dotted one to the calculations based on $g$-factorization [see Eq. (90)].

Figure 28

$R_{\mathrm{AGK}}$ as a function of the $p_t$ of partons for minimum bias PbPb and $pp$ scatterings at 5.02 TeV.

Figure 29

$R_{\mathrm{PbPb}}$ as a function of the $p_t$ of partons for minimum bias PbPb scatterings at 5.02 TeV.

Figure 30

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for PbPb (central, 0–5%) and $pp$ (event class 16–20 Pomerons) scatterings at 5.02 TeV.

Figure 31

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for different systems, different energies (in TeV), different event classes “M$i$” (M1 = highest and M8 = lowest multiplicity). See text.

Figure 32

A Pomeron connected to projectile nucleon $i$ and target nucleon j, together with other Pomerons connected to one (or both) of these nucleons.

Figure 33

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for PbPb at 5.02 TeV, for the centralities M1 to M8. See text.

Figure 34

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for XeXe at 5.44 TeV, for the centralities M1 to M8. See text.

Figure 35

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for AuAu at 200 GeV, for the centralities M1 to M8. See text.

Figure 36

$R_{\mathrm{deform}}\left(x_{\mathrm{PE}}\right)$ as a function of $x_{\mathrm{PE}}$ for $pp$ at 13 TeV, for the event classes M1 to M8. See text.

Figure 37

A single cut Pomeron.

Figure 38

Cut diagram (a) with nonlinear effects via triple Pomeron vertices and (b) with the nonlinear effects (inside the red circles) “summarized” in the form of saturation scales, which replace these nonlinear parts.

Figure 39

The saturation scale $Q_{\mathrm{sat}}^2$ as a function of $x_{\mathrm{PE}}=x^{+}{x}^{-}$, for several $N_{\mathrm{Pom}}$ event classes (from top to bottom, 10–15, 5–9, and 2–4 Pomerons).

Figure 40

The saturation scale $Q_{\mathrm{sat}}^2$ as a function of $x_{\mathrm{PE}}$, for several event classes (from top to bottom 5–10%, 10–20%, 30–40%, 50–60% centrality).

Figure 41

Sketch of the suppression of low-$p_t$ partons with increasing $Q_{\mathrm{sat}}^2$ for a single Pomeron, where the red line corresponds to some minimum value $Q_{\mathrm{sat}\min }^2$.

Figure 42

The ratio $R_{\mathrm{AGK}}$ (see text) for minimum bias PbPb collisions at 5.02 TeV.

Figure 43

The only diagram contributing to inclusive cross sections in $A+B$ scattering: a single cut Pomeron (multiplied by $AB$).

Figure 44

The contributions (a) $G_{\mathrm{QCD}}^{\mathrm{sea}\text{−}\mathrm{sea}}$ and (b) $G_{\mathrm{QCD}}^{\mathrm{val}\text{−}\mathrm{val}}$.

Figure 45

The two inelastic processes corresponding to the two cut diagrams of Fig. 44.

Figure 46

The two contributions of the parton distribution function (a common remnant vertex factor $V$ is not shown).

Figure 47

Parton yield $dn/d{p}_{t}dy$ for $pp$ at 13 TeV. I show results based on EPOS PDFs (red solid line), CTEQ PDFs (green dashed line), the full EPOS simulation (blue circles), and experimental data from ATLAS (black triangles).

Figure 48

Partonic configuration of two colliding nuclei, each one composed of two nucleons, with three scatterings (from three cut Pomerons). Dark blue lines mark active quarks, red dashed lines active antiquarks, and light blue thick lines projectile and target remnants. One of the target nucleons is just a spectator.

Figure 49

Configuration colorwise equivalent to the one of Fig. 48. The outgoing antiquarks are drawn as incoming quarks (arrows towards vertices).

Figure 50

A possible color flow diagram for the three scatterings of Fig. 49.

Figure 51

A color flow diagram with SL and TL cascades.

Figure 52

The prehadron yield as a function of space-time rapidity, for different centralities in PbPb collisions at 5.02 TeV. The curves refer to all prehadrons (red solid curve), all core prehadrons (red dotted curve), prehadrons from remnant decay (blue solid curve), and core prehadrons from remnant decay (blue dotted curve).

Figure 53

Same as Fig. 52, but for AuAu at 200 GeV.

Figure 54

Energy density at the initial proper time ${\tau }_0$ as a function of the transverse coordinate $r$, for an azimuthal angle $\varphi =0$ (red solid curve) and $\varphi =\pi /2$ (red dotted curve). The blue dashed lines represent the freeze-out energy density. I show results for different centralities (defined via impact parameter) in PbPb collisions at 5.02 TeV (upper plot) and in AuAu collisions at 200 GeV (lower plot).

Figure 55

The X/“core+corona” ratio as a function of $p_t$ (for $\left|\eta \right|<1$), with X being the “corona” contribution (blue), the “core” (green), and the “full” contribution (red), for four centrality classes and four different particle species, for PbPb at 5.02 TeV (upper plot) and AuAu at 200 GeV (lower plot).

Figure 56

Pseudorapidity distributions of charged particles in AuAu collisions at 200 GeV, comparing EPOS4 (lines) to BRAHMS data (points).

Figure 57

Transverse mass distributions of ${\pi }^{+}, {\pi }^{-}$ in AuAu collisions at 200 GeV for different centrality classes. EPOS4 simulation (lines) are compared to data from STAR. From top to bottom, one multiplies the curves by $3^{-i}$ ($i=0,1,2,3,...$).

Figure 58

Same as Fig. 57, but for $K^{+}, K^{-}, p$, and $\overline{p}$.

Figure 59

Transverse momentum distributions of ${\pi }^{+}, {\pi }^{-}, K^{+}$, and $K^{-}$ in central (0–5%) AuAu collisions at 200 GeV for different rapidities. EPOS4 simulations (lines) are compared to data from BRAHMS. From top to bottom, the curves are multiplied by $3^{-i}$ ($i=0,1,2,3,...$).

Figure 60

Transverse momentum distributions of ${\pi }^{+}, {\pi }^{-}$ in AuAu collisions at 200 GeV at central rapidity for different centralities. EPOS4 simulations (lines) are compared to data from PHENIX [42]. From top to bottom, the curves are multiplied by $3^{-i}$ ($i=0,1,2,3,4$).

Figure 61

Same as Fig. 60, but for $K^{+}, K^{-}, p$, and $\overline{p}$.

Figure 62

Transverse momentum distributions of $K_0, \mathrm{\Lambda }, \overline{\mathrm{\Lambda }}{\mathrm{\Xi }}^{-}, {\overline{\mathrm{\Xi }}}^{+}$, and $\mathrm{\Omega }$ in AuAu collisions at 200 GeV at central rapidity for different centralities. EPOS4 simulations (lines) are compared to data from STAR. From top to bottom, one multiplies the curves by $3^{-i}$ ($i=0,1,2,3,4$).

Figure 63

Pseudorapidity dependence of $v_2$ of charged particles for different multiplicity classes in AuAu collisions at 200 GeV. I compare the simulations (red lines) with data from PHOBOS (black points).

Figure 64

Centrality dependence (using $N_{\mathrm{part}}$) of $v_2$ of charged particles in AuAu collisions at 200 GeV. I compare the simulations (red lines) with data from PHOBOS (black points).

Figure 65

Transverse momentum dependence of $v_2$ for pions (left column), kaons (middle column), and protons (right column) in AuAu collisions at 200 GeV, for different centralities (from top to bottom): 0–10%, 10–20%, 20–30%, 30–40%, 40–50%, and 50–60%. I compare the full simulations (red lines) and those without hadronic cascade (yellow lines) with data from PHENIX (black points).

Figure 66

Transverse momentum dependence of $v_3$ (upper plot) and $v_4$ (lower plot) for pions (left column), kaons (middle column), and protons (right column) in AuAu collisions at 200 GeV, for different centralities. I compare the full simulations (red lines) with data from PHENIX (black points).

Figure 67

Transverse momentum distributions of pions in PbPb collisions at 5.02 TeV for different centrality classes. EPOS4 simulation data (lines) are compared to data from ALICE. From top to bottom, one multiplies the curves by $2^{-i}$ ($i=0,1,2,3,4,...$).

Figure 68

Same as Fig. 67, but for kaons (upper plot) and protons (lower plot).

Figure 69

Pseudorapidity dependence of $v_2\left\{2\right\}$ and $v_2\left\{4\right\}$ for different multiplicity classes (using percentiles) in PbPb collisions at 2.76 GeV. I compare the EPOS4 simulations (red lines) with data from ALICE (black points).

Figure 70

$p_t$ dependence of $v_2$ for different multiplicity classes (using percentiles) in PbPb collisions at 5.02 TeV, for (from left to right) pions, protons, kaons ($K_s), \varphi$ mesons, and lambdas. I compare the simulations (red lines) with data from ALICE (black points).

Figure 71

$p_t$ dependence of $v_3$ for different multiplicity classes in PbPb collisions at 5.02 TeV. I compare the simulations (red lines) with data from ALICE (black points).

Figure 72

Contour integrations in $j$ plane.

Figure 73

Contour integration in $z$ plane.