The work [1] addressed the radiative energy loss of fast quarks in a finite-size QCD matter within the light-cone path integral formalism developed in [2]. There was considered the case of gluon emission from a quark produced inside the matter, which corresponds to the situation of partons with large transverse momenta produced in hard reactions in the initial stage of $AA$-collisions and undergoing final state  interactions in the quark-gluon plasma, and the case of gluon emission from a quark approaching the medium from outside, which corresponds to the situation of hadron-nucleus collisions. In [2] the gluon  spectrum in the Feynman variable $x=\omega/E$ (here $\omega$ and $E$ are the gluon and initial quark energies) has been expressed through the Green function of a two dimensional Schrodinger  equation describing evolution of a fictitious $gq\bar{q}$ system, in which $\bar{q}$ is located at the center of mass of the $gq$ pair, from point-like to point-like state. In this Schrodinger equation the coordinate $z$ plays the role of time, and the Schrodinger mass equals $x(1-x)E$, which is the reduced mass of the $gq$ pair in the impact parameter plane (where the role of masses are played by the particle energies). The imaginary potential is proportional to the product of the matter density and the three-body cross section $\sigma_{gq\bar{q}}$  of interaction of the $gq\bar{q}$  system with a medium constituent. At $\rho\ll r_{D}$ (here $\rho$ is the transverse size of the $gq$ pair, and  $r_{D}$ is color screening radius of the matter) $\sigma_{gq\bar{q}}\propto \rho^{2}$, up to a factor slowly (logarithmically) varying  with $\rho$. In [1] this factor was replaced by its value at the typical size of the $gq$ pair. In this case the Green function is reduced to the oscillator Green function, which has an analytic representation. It simplifies greatly the numerical calculations.

By analytically investigating the limit $L\to 0$ ($L$ is the quark path length in the matter) it was demonstrated that for a quark produced in the matter the radiated energy $\propto L^{2}$. Such an unusual $L$-dependence is a consequence of strong suppression of the emission probability for gluons with the formation length that exceeds the quark path length in the matter. It was demonstrated that the $L^{2}$ behavior occurs both for regimes of strong and weak LPM effect. For a quark approaching the matter outside the effect is absent and the energy loss is $\propto L$. The numerical calculations in [1] were performed accounting for the finite masses of the quasiparticles. It was demonstrated that the energy loss is insensitive to the gluon quasiparticle mass at $E \gtrsim 50$ GeV and depends weakly on the quark mass in the whole energy range.

The topic of the work [1] at the time of its publication was of great interest due to its connection to the future experiments on heavy ion collisions at RHIC and LHC, where there was expected production of the quark-gluon plasma. It was believed that the energy loss of the fast quarks and gluons, produced in hard processes, as they traverse the quark-gluon plasma would suppress the hadron spectra at high transverse momenta (similarly to the reduction of radiation in a concrete nuclear reactor containment), and the magnitude of this suppression would give information about the plasma density. Subsequently it became a reality, and analysis of the nuclear suppression of hadron spectra (this phenomenon is usually called jet quenching) is now one of the major methods for diagnostics of the QCD matter produced in heavy ion collisions at RHIC and LHC.

At present the treatment of the radiative energy loss within the path integral method in the oscillator approximation used in [1] is widely used in studies of jet quenching. This approximation is used in the popular AMW (Armesto, Salgado and Wiedemann [5]) approach to jet quenching, and in investigation of the jet modification in the quark-gluon plasma accounting for multiple gluon emission [6, 7].
However, it worth noting that later on in [8]  it was shown that the oscillator approximation has a shortcoming in the case of partons produced inside the matter. But it does not affect the $L^{2}$ dependence of the energy loss that persists beyond the oscillator approximation as well.

[1] B.G. Zakharov, JETP Lett.  65, 615 (1997).

[2] B.G. Zakharov, JETP Lett.  63, 952 (1996).

[3] L.D. Landau and I.Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92, 535, 735 (1953).

[4] A.B. Migdal,  Phys. Rev. 103, 1811 (1956).

[5] N. Armesto, C.A. Salgado, and U.A. Wiedemann, Phys. Rev. D 69, 114003 (2004).

[6] Y. Mehtar-Tani, C.A. Salgado, and K. Tywoniuk, JHEP 1210, 197  (2012).

[7] J.P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani, JHEP 1301, 143 (2013).

[8] B.G. Zakharov, JETP Lett.  73, 49 (2001).