Parametric instability of classical Yang-Mills fields in a color magnetic background

We investigate instabilities of classical Yang-Mills fields in a time-dependent spatially homogeneous color magnetic background field in a nonexpanding geometry for elucidating the earliest stage dynamics of ultrarelativistic heavy-ion collisions. The background field configuration considered in this article is spatially homogeneous and temporally periodic and is introduced by Berges-Scheffler-Schlichting-Sexty (BSSS). We discuss the whole structure of instability bands of fluctuations around the BSSS background gauge field on the basis of Floquet theory, which enables us to discuss the stability in a systematic way. We find various instability bands on the $(p_z,p_T)$ plane. These instability bands are caused by parametric resonance despite the fact that the momentum dependence of the growth rate for $|\mathbf{p}|\le \sqrt{B}$ is similar to a Nielsen-Olesen instability. Moreover, some of the instability bands are found to emerge not only in the low-momentum but also in the high-momentum region, typically of the order of the saturation momentum as $|\mathbf{p}|\sim \sqrt{B}\sim Q_{\mathrm{s}}$.

Figure 1

Floquet analysis for Eqs. (24), (25) and (26). The shaded area denotes the instability bands specified by the $p_z$ or $p_T$ region with $|\mathrm{tr}M|\ge 2$. These are the consequence of parametric resonance. The instability bands of $a_{-}$ (Lamé’s equation with $\epsilon =-1$) are $0\le p_z^2/B_0\le 0.41$ and $0.91\le p_z^2/B_0\le 1.42$. The instability bands of $a_{+}$ (Lamé’s equation with $\epsilon =3$) is $3/2\le p_z^2/B_0\le \sqrt{3}$. The instability bands of $a_{2z}$ (Lamé’s equation with $\epsilon =1$) is $0\le p_T^2/B_0\le 1/2$. Note that exact instability boundaries are known for $\epsilon =1,3$ [28]. Our numerical calculation agrees with that.

Figure 2

The contour map of the instability bands of the classical Yang-Mills equation under oscillating color magnetic fields. The contour lines (white lines) stand for $|\mu {|}_{\max }=100,50,20,10,5,2,1.02$ from top to bottom. The maximum characteristic multiplier is calculated as a function of both $p_z,p_T$, where $p_T=\sqrt{p_x^2+p_y^2}$ is a transverse momentum. Both $p_z$ and $p_T$ are rescaled by the initial strength of the background color magnetic field $B_0$. The dominant instability band in the low-momentum region has an anisotropy. Transverse momentum direction of the dominant band is broader than that of the $p_z$ direction which extends up to $p_T^2/B_0=1.75$, while the $p_z$ direction extends up to $p_z^2/B_0=0.81$.

Figure 3

(top) Instability bands for $p_T=0$. The lowest instability band of ${\mathrm{\Omega }}_A^2$ system (red solid line) and ${\mathrm{\Omega }}_B^2$ system (blue dashed line) located at $0\le p_z^2/B_0\le 0.41$ have the largest growth rate. This dominant instability band is consistent with that found in [25]. (bottom) Instability bands for $p_z=0$. The lowest instability band of ${\mathrm{\Omega }}_D^2$ system (red solid line) and ${\mathrm{\Omega }}_E^2$ system (blue dashed line) have the largest growth rate. They have broader band and reach $p_T^2/B_0=1.75$ and $p_T^2/B_0=0.88$, respectively. The magenta dotted line and the green dot-dashed line stand for the bands of ${\mathrm{\Omega }}_F^2$ and ${\mathrm{\Omega }}_G^2$.

Figure 4

The comparison of instability bands of original EOMs (multicomponent Hill’s equation with ${\mathrm{\Omega }}_I^2$) and single component Hill’s equations (denoted by ${\omega }_{Im}^2$). (top) ${\mathrm{\Omega }}_A^2$ system is effectively reduced to three single component Hill’s equations and one of them is Lamé’s equation with $\epsilon =-1$. The instability bands of the Lamé’s equation are denoted by a blue dashed line. This result shows that most bands of the original EOM, including the first band, are well reproduced by Lamé’s equation. The ${\mathrm{\Omega }}_B^2$ system is exactly reduced to two types of Lamé’s equations. They also lead to quite unstable behavior in the low-momentum region. For the ${\mathrm{\Omega }}_C^2$ system, two Hill’s equations fail to reproduce the original bands. (bottom) The change of instability bands is drastic in ${\mathrm{\Omega }}_D^2,{\mathrm{\Omega }}_E^2$ systems. The time dependence of $U_I$ makes instability regions considerably narrower for these equations.

Figure 5

Comparison of instability bands obtained with a physical initial condition and a unit matrix initial condition. The red solid line denotes the instability bands of ${\mathrm{\Omega }}_A^2$ system calculated by physical initial condition. The blue dashed line results from the unit matrix initial condition.

Figure 6

(In)stability boundaries of Lamé’s equation. The modulus is $k=1/\sqrt{2}$. Solutions are unstable if $|\mathrm{tr}M|\ge 2$. The red dotted lines are determined instability boundaries in a basis of Floquet theory which are characterized by $|\mathrm{tr}M|=2$. The gray solid lines are given by perturbative calculation (multiple-scale analysis) which is valid when $0\le \epsilon <1$.