Linear Boltzmann transport for jet propagation in the quark-gluon plasma: Inelastic processes and jet modification

A linear Boltzmann transport (LBT) Monte Carlo model has been developed to describe jet propagation and interaction with the quark-gluon plasma (QGP) in relativistic heavy-ion collisions. A complete set of elastic-scattering processes and medium-induced gluon emissions based on the higher-twist formalism are incorporated for both jet shower and medium recoil partons. It has been employed to describe experimental data on large transverse momentum hadron and jet spectra, correlation and jet substructures in high-energy heavy-ion collisions. We document in detail the structure of the model and validation of the Monte Carlo implementations of the physics processes in LBT, in particular, the inelastic process of medium-induced gluon radiation. We carry out a comprehensive examination of the jet-medium interaction as implemented in LBT through energy loss and momentum broadening of a single hard parton, the energy and transverse momentum transfer from leading partons to medium-induced gluons and jet-induced medium excitation, and medium modification of reconstructed jets in a static and uniform medium. With realistic and event-by-event hydrodynamic medium in heavy-ion collisions, we compute and compare with experimental data on the jet cone-size dependence of the single inclusive jet suppression at both the BNL Relativistic Heavy-Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC), the dijet asymmetry at the LHC and $\gamma$-jet correlation at RHIC. Effects of medium-induced gluon emissions and jet-induced medium excitation on jet observables are systematically examined. Rescatterings of the radiated gluons and recoil partons with the QGP are found essential to account for the enhancement of soft particle yield toward the edge of the jet cone.

Figure 1

Flow chart of the LBT Monte Carlo model. The green shaded boxes represent modules that can be replaced by users with alternatives.

Figure 2

$h\left(v\right)$ defined in Eq. (21) as a function of fractional momentum $z$ and $(t-t_i)E$.

Figure 3

$h\left(v\right)$ defined in Eq. (21) as a function of fractional momentum $z$ for four different values of $(t-t_i)E$.

Figure 4

The effective medium-induced parton splitting function $P_a\left(z\right)h\left(v\right)/\left[z(1-z)\right]$ for a quark (upper panel) and a gluon (lower panel) for four different values of $(t-t_i)E=1$, 4, 20, 60 (from bottom to top) as compared with the asymptotic limit (dashed line) when $(t-t_i)E\to \infty$.

Figure 5

The rate of gluon emission from a gluon jet parton at $t-t_i=1$ fm/$c$ as a function of temperature $T$ and initial parton energy $E$. The strong coupling constant ${\alpha }_{\mathrm{s}}$ is set to 0.3.

Figure 6

Fractional energy distributions of the radiated gluons in $q\to q+g$ (upper) and $g\to g+g$ (lower) processes with an initial parton energy $E_0=20$ GeV in a static medium at temperature $T=400$ MeV from semi-analytical calculations (black open circles), direct MC sampling (red dashed lines), and LBT simulations for $2\to 3$ (green solid lines) and $2\to 2+n(n\ge 1)$ (blue solid lines).

Figure 7

The angular distribution of medium-induced radiated gluon for a typical parton energy and length $\sqrt{\mathrm{\Delta }tE}$ and Debye screening mass ${\mu }_D/E$.

Figure 8

Polar angle distributions of the radiated gluons in $q\to q+g$ process with an initial parton energy $E_0=20$ GeV in a static medium at temperature $T=400$ MeV from semi-analytical calculations (black open circles), direct MC sampling (red dashed line), LBT simulations for $2\to 3$ (green solid line), and $2\to 2+n(n\ge 1)$ (blue solid line) processes.

Figure 9

The time dependence of the fractional energy loss $\mathrm{\Delta }E/E_0$ (upper panel) and energy loss per unit length $dE/dz$ (lower panel) of a quark (up and down triangle) or a gluon (circle and square) with initial energy $E_0=100$ GeV through a static medium with temperature $T=300$ MeV, with only elastic (black and blue) and elastic $+$ gluon radiation processes (red and magenta).

Figure 10

Quark energy loss per unit length as a function of time in a static medium with temperatures $T=200$ MeV (upper panel) and $T=400$ MeV (lower panel), for different initial energies $E_0$. Both elastic and gluon radiation processes are included.

Figure 11

The same as Fig. 10 except for a gluon.

Figure 12

Transverse momentum distribution of a quark with an initial energy $E_0=100$ GeV at different evolution times through a static medium at temperature $T=300$ MeV. The upper panel includes only elastic scatterings, while the lower panel includes both elastic scatterings and gluon emissions.

Figure 13

The same as Fig. 12 except for a gluon.

Figure 14

Energy density distribution of a jet shower originating from a 100 GeV quark at $x=y=z=0$ and propagating along the $z$ direction in a static medium at $T=300$ MeV. Results are shown at $t=6$ fm/$c$ with contributions from only elastic scatterings with rescatterings (upper panel), elastic $+$ inelastic scatterings without rescatterings (middle panel), and elastic $+$ inelastic scatterings with rescatterings (lower panel).

Figure 15

The same as Fig. 14 except for a gluon.

Figure 16

The same as the contour plot in Fig. 14 except for the energy density carried by recoil partons, negative partons, radiated gluons and their sum. Note the color scales in upper and lower panels are different for clarity.

Figure 17

Energy distribution of partons inside a jet shower at different times developed from a 100 GeV quark in a static medium at $T=300$ MeV, with only elastic scatterings with rescatterings (upper panel), elastic $+$ inelastic scatterings without rescatterings (middle panel), and elastic $+$ inelastic scatterings with rescatterings (lower panel).

Figure 18

The same as Fig. 17 except for an initial gluon.

Figure 19

Projection from azimuthal angle $\phi$ relative to the jet parton direction $y$ to the azimuthal angle $\varphi$ in the transverse plane of the collision frame.

Figure 20

Projected azimuthal angle distribution of partons within different energy intervals inside a jet shower developed from a 100 GeV gluon in a static medium at $T=300$ MeV. The upper panel shows results at time $t=2$ fm/$c$, and the lower panel at $t=6$ fm/$c$. Only elastic scatterings are included, with rescatterings of recoil partons.

Figure 21

The same as Fig. 20 except that both elastic and inelastic scatterings are included without rescatterings of radiated gluons and recoil partons.

Figure 22

The same as Fig. 20 except that both elastic and inelastic scatterings are included with rescatterings of radiated gluons and recoil partons.

Figure 23

Accumulated fractional energy loss $\mathrm{\Delta }E/E_0$ of a jet with cone size $R=0.4$ as a function of the evolution time including only elastic scatterings with rescatterings (solid), elastic $+$ inelastic scatterings without rescatterings (dotted), and elastic $+$ inelastic scatterings with rescatterings (dot-dashed). The jet is initiated with a 100 GeV quark (upper panel) or gluon (lower panel), propagating through a static medium at $T=300$ MeV.

Figure 24

Fractional longitudinal momentum ($z_{\mathrm{jet}}=p_{\mathrm{L}}/E_{\mathrm{jet}}$) distribution of constituent partons within a jet with a cone size $R=0.4$ at different evolution times, with (solid) and without (dashed) rescatterings of radiated gluons and recoil partons. The jet is initiated with a 100 GeV quark (upper panel) or gluon (lower panel), propagating through a static medium at $T=300$ MeV with both elastic and inelastic scatterings switched on.

Figure 25

The same as Fig. 24 except that the fractional momentum $z_{0,\mathrm{jet}}$ is defined with respect to the initial parton energy.

Figure 26

Transverse energy distribution within a jet with a cone size $R=0.4$ at different times during the jet propagation with (solid lines) and without (dashed lines) rescatterings of radiated gluons and recoil partons. The jet is initiated with a 100 GeV quark (upper panel) or gluon (lower panel), propagating through a static medium at $T=300$ MeV with both elastic and inelastic scatterings switched on.

Figure 27

LBT simulations of the nuclear modification factor $R_{\mathrm{AA}}$ for single inclusive jets in 0%–10% $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV with jet cone sizes $R=0.2$ (top panel), 0.3 (middle panel), and 0.4 (bottom panel) as compared with STAR data [83].

Figure 28

The ratio of the inclusive jet spectra with jet cone size $R=0.2$ and 0.4 in 0%–10% $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV from the LBT calculations as compared with the STAR data [83] and the corresponding $p+p$ result from Pythia simulation. The black dashed line, red solid line, and green solid line represent LBT results with minimal $p_{\mathrm{T}}$ trigger bias as 0, 5, and 7 GeV/$c$ for the leading parton respectively. The purple solid line represents the average of LBT results between minimal $p_{\mathrm{T}}$ trigger bias of 5 and 7 GeV/$c$ for the leading parton, while the purple dashed line represents the Pythia result for $p+p$ collisions.

Figure 29

LBT results on the nuclear modification factor $R_{\mathrm{AA}}$ for single inclusive jets in 0%–10% $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, with different jet cone sizes and two different values of ${\alpha }_{\mathrm{s}}=0.13$ (red) and 0.15 (blue) as compared with the CMS [84] and ATLAS [85] data.

Figure 30

The nuclear modification factor $I_{\mathrm{AA}}$ from LBT simulations for $\gamma$-triggered jets in 0%–15% $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV with different jet cone sizes as compared with the STAR data [91].

Figure 31

Dijet asymmetry ($x_{\mathrm{J}}$) distribution in $p+p$ (blue) and $\text{Pb}+\text{Pb}$ collisions (red) at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV from LBT calculations as compared with the ATLAS data [92]. The transverse momentum of the leading jet is selected within $158<p_{\mathrm{T},1}<178$ GeV/$c$ in the upper panel and $398<p_{\mathrm{T},1}<562$ GeV/$c$ in the lower panel.

Figure 32

Illustration of a $2\to 3$ process.

Figure 33

Illustration of multiple gluon emissions.