Reexamining the gluon spectrum in the boost-invariant glasma from a semianalytic approach

In high energy heavy-ion collisions, the degrees of freedom at the very early stage can be effectively represented by strong classical gluonic fields within the color glass condensate framework. As the system expands, the strong gluonic fields eventually become weak such that an equivalent description using the gluonic particle degrees of freedom starts to become valid. In this paper, the spectrum of these gluonic particles is reexamined by solving the classical Yang-Mills equations semianalytically, with the solutions having the form of power series expansions in the proper time. A different formula for the gluon spectrum, which is consistent with energy density during the whole time evolution, is proposed. One finds that the chromoelectric fields have larger contributions to the gluon spectrum than the chromomagnetic fields do. Furthermore, the large momentum modes take less time to reach the weak-field regime while smaller momentum modes take more time. The resulting functional form of the gluon spectrum is exponential in nature and the spectrum is close to a thermal distribution, with effective temperatures around $0.6Q_s$ to $0.9Q_s$ late in the glasma evolution. The sensitiveness of the gluon spectrum to the infrared and the ultraviolet cutoffs is discussed.

Figure 1

The coefficient functions $\sqrt[2n+1]{{\mathcal{C}}_1(n,k_{\perp })},\sqrt[2n+1]{{\mathcal{C}}_2(n,k_{\perp })},\sqrt[2n+1]{{\tilde{\mathcal{C}}}_2(n,k_{\perp })},\sqrt[2n-1]{{\mathcal{D}}_1(n,k_{\perp })},\sqrt[2n-1]{{\mathcal{D}}_2(n,k_{\perp })}$, and $\sqrt[2n-1]{{\tilde{\mathcal{D}}}_2(n,k_{\perp })}$ at different orders $n$ for $k_{\perp }=Q_s$. The input parameters are $Q=4.0\mathrm{GeV},m=0.2\mathrm{GeV}$, and $Q_s=1.2\mathrm{GeV}$.

Figure 2

Time evolution of the four terms ${\mathfrak{E}}^i{\mathfrak{E}}^i,{\mathfrak{B}}^i{\mathfrak{B}}^i,{\mathfrak{E}}^z{\mathfrak{E}}^z$, and ${\mathfrak{B}}^z{\mathfrak{B}}^z$ for the momentum mode $k_{\perp }=Q_s$.

Figure 3

Time evolution of chromoelectric part ${\mathfrak{E}}^i{\mathfrak{E}}^i+{\mathfrak{E}}^z{\mathfrak{E}}^z$ and the chromomagnetic part ${\mathfrak{B}}^i{\mathfrak{B}}^i+{\mathfrak{B}}^z{\mathfrak{B}}^z$ for the momentum mode $k_{\perp }=Q_s$.

Figure 4

Four different momentum modes of the gluon spectrum evolve with time.

Figure 5

The gluon spectrum and the energy density spectrum at $\tau =0.6\mathrm{fm}/c$. The blue dots are the numerical results while the red curves are from the thermal fitting functions (22). (a) The gluon spectrum fitted with a thermal function and (b) The energy density spectrum fitted with a thermal function.

Figure 6

The gluon spectrum and the energy density spectrum at $\tau =0.6\mathrm{fm}/c$. The blue dots are the numerical results while the red curves are from the nonthermal function (23). (a) The gluon spectrum fitted with a nonthermal function and (b) The energy density spectrum fitted with a nonthermal function.

Figure 7

The gluon spectrum and the energy density spectrum at $\tau =0.6\mathrm{fm}/c$. The blue dots are the numerical results while the red curves are from the nonthermal function (25). (a) The gluon spectrum fitted with a nonthermal function and (b) The energy density spectrum fitted with a nonthermal function.

Figure 8

The normalized gluon spectrums for three different values of $Q_s$. The area under each curve is normalized to be 1.

Figure 9

The normalized gluon spectrums for three different values of $Q$. The area under each curve is normalized to be one.

Figure 10

The normalized gluon spectrums for three different values of $m$. The area under each curve is normalized to be 1.