Modified Boltzmann approach for modeling the splitting vertices induced by the hot QCD medium in the deep Landau-Pomeranchuk-Migdal region

Hard probes produced in perturbative processes are excellent probes for the study of the hot and dense QCD matter created in relativistic heavy-ion collisions. Transport theory, allowing for coupling to an evolving medium with fluctuating initial conditions, has become a powerful tool in this endeavor. However, the implementation of the Landau-Pomeranchuk-Migdal (LPM) effect for medium-induced parton bremsstrahlung and pair production poses a challenge to semiclassical transport models based on Boltzmann-type transport equations. In this work, we investigate a possible solution to approximate the LPM effect in a “modified Boltzmann transport” approach, including a prescription for the running coupling constant. By fixing a numerical parameter, this approach quantitatively reproduces the rates of medium-induced parton splitting predicted by the next-to-leading-log solution of the Arnold-Moore-Yaffe equation, which is valid in the deep-LPM regime of an infinite medium. We also find qualitative agreement of our implementation with calculations in a finite and expanding medium, but future improvements are necessary for added precision at small path length. This work benefits transport model-based studies and the usage of these models in the phenomenological extraction of the jet transport coefficient.

Figure 1

The $q\to q+g$ splitting rate in an infinite medium from a quark with $E=1$ TeV and a coupling constant ${\alpha }_s=0.1$. The top plot shows the simulated spectrum $dR/d\omega$ (red dashed line) and power-law fit (green dotted and blue dash-dotted lines) in different gluon energy regions, separated by energy scales ${\omega }_{BH}\approx 2\pi T$. The middle plot compares the simulation to the NLL solution of the AMY equation, and the bottom plot shows the ratio between the simulation and the AMY NLL solution

Figure 2

The splitting rates of $q\to q+g^{*}, g\to g+g^{*}$, and $g\to q^{*}+\overline{q}$ as a function of the parton energy labeled by the star. The mother parton with $E=1$ TeV evolves inside an infinite medium with $T=0.5$ GeV. The simulations (thick dashed lines) are compared to the NLL solutions (thin solid lines).

Figure 3

Top plot: comparison of modified Boltzmann simulation with the NLL solution with running coupling. Three initial guesses of the ratio between simulation and theory are shown in the bottom plot.

Figure 4

Comparison of the path-length-dependent rate $dR/d\omega$ from the simulation using ${\alpha }_s=0.3$ to the theoretical calculation for splitting $q\to q+g$ [32]. The quark energy is 16 GeV.

Figure 5

The ratios of induced splitting rate in an expanding medium to that of a static medium, with expansion parameter $\nu =3/4$, 1, and $3/2$. The analytic results are shown in solid lines and simulations are denoted as symbols. The coupling constant ${\alpha }_s=0.3$; the expansion starts at ${\tau }_0=0.2$ fm/$c$ with an initial temperature $T_0=1$ GeV.

Figure 6

Energy loss per unit path lengh $dE/dL$ as a function of energy $E$, temperature $T$, and coupling constant ${\alpha }_s$. Each column corresponds to a value of the coupling constant: ${\alpha }_s=0.075$, 0.15, 0.3, and 0.6 (from left to right). Each row corresponds to a temperature: $T=0.2$, 0.4, and 0.6 GeV (from top to bottom). $dE/dx$ is divided by the expected scaling ${\alpha }_s^2\sqrt{E{T}^3}$. The MC implementations of the LPM effect referred to as “modified Boltzmann,” “coherence factor,” and “blocking radiation” approaches are shown with red dashed lines, blue dash-dotted lines, and green dotted lines respectively. The AMY NLL results are denoted by black boxes.

Figure 7

Ratios of splitting rate $dR/\omega$ between the modified Boltzmann simulation and the NLL solution for $q\to q+g$ splitting. The quark energies are $E=10$, 100, and 100 GeV from top to bottom plot. Two coupling constants are used: ${\alpha }_s=0.1$ (red solid lines) and ${\alpha }_s=0.3$ (blue dashed lines). $\omega$ stands for the gluon energy. The horizontal dashed lines denote $\pm 10\%$ deviation from unity.

Figure 8

The same as Fig. 8, but for the $g\to g+g$ splitting, and $\omega$ stands for either energy of the final state gluon.

Figure 9

The same as Fig. 8, but for the $g\to q+\overline{q}$ splitting, and $\omega$ stands the energy of the quark.