Initial value problem for magnetic fields in heavy ion collisions

When the quark-gluon plasma emerges in the wake of a heavy-ion collision, a magnetic field created by the valence charges has already permeated the entire interaction region. Evolution of this “initial” field in the plasma is governed by the Maxwell equations in an electrically conducting medium. As the plasma expands, external valence charges induce a magnetic field that also contributes to the total magnetic field in the plasma. I solve the initial value problem describing these processes and argue that the initial magnetic field often dominates over the one induced by the valence charges. In particular, it grows approximately proportional to the collision energy, unlike the induced component, which is energy independent. As a result, the magnetic field has a significant phenomenological influence on the quark-gluon plasma at CERN Large Hadron Collider energies over its entire lifetime.

Figure 1

Two counterpropagating charges $e$. One charge moves along the positive $z$ axis at $z=vt, x=-b/2, y=0$ while another one moves in the opposite direction at $z=-vt, x=b/2, y=0$.

Figure 2

Magnetic field in units of $m_{\pi }^2/e$. $\sigma =5.8$ MeV, $z=0$ fm ($\eta =0$). Left panel: $t_0=0.2$ fm; right panel: $t_0=0.5$ fm. The valence current does not contribute at all ($B_{\text{val}}=0$).

Figure 3

Magnetic field in units of $m_{\pi }^2/e$. $\sigma =5.8$ MeV, $z=0.6$ fm ($\eta =0.086$). Left panel: $t_0=0.2$ fm; right panel: $t_0=0.5$ fm. Solid, dashed, and dotted lines stand for $B, B_{\text{init}}$, and $B_{\text{val}}$.

Figure 4

Magnetic field in units of $m_{\pi }^2/e$. $\sigma =5.8$ MeV, $z=0.2$ fm, $t_0=0.2$ fm. Solid, dashed, and dotted lines stand for $B, B_{\text{init}}$, and $B_{\text{val}}$. Left panel: $\gamma =100$ (RHIC); right panel: $\gamma =2000$ (LHC).

Figure 5

Magnetic field in units of $m_{\pi }^2/e$. $z=0.2$ fm, $t_0=0.2$ fm. Left panel: model A. Right panel: model B. Solid, dashed, and dotted lines stand for $B, B_{\text{init}}$, and $B_{\text{val}}$.