Effects of magnetic field on plasma evolution in relativistic heavy-ion collisions

Very strong magnetic fields can arise in noncentral heavy-ion collisions at ultrarelativistic energies, and may not decay quickly in a conducting plasma. We carry out relativistic magnetohydrodynamics (RMHD) simulations to study the effects of this magnetic field on the evolution of the plasma and on resulting flow fluctuations in the ideal RMHD limit. Our results show that the magnetic field leads to enhancement in elliptic flow for small impact parameters while it suppresses it for large impact parameters (which may provide a signal for the initial stage magnetic field). Interestingly, we find that magnetic field in localized regions can temporarily increase in time as evolving plasma energy density fluctuations lead to reorganization of magnetic flux. This can have important effects on the chiral magnetic effect. The magnetic field has nontrivial effects on the power spectrum of flow fluctuations. For the very strong magnetic field case, one sees a pattern of even-odd difference in the power spectrum of flow coefficients arising from reflection symmetry about the magnetic field direction if initial state fluctuations are not dominant. We discuss the situation of nontrivial magnetic field configurations arising from collision of deformed nuclei and show that it can lead to anomalous elliptic flow. Special (crossed body-body) configurations of deformed nuclei collisions can lead to the presence of a quadrupolar magnetic field, which can have very important effects on the rapidity dependence of transverse expansion (similar to beam focusing from quadrupole fields in accelerators).

Figure 1

A typical situation expected in relativistic heavy-ion collisions with the magnetic field pointing in the $y$ direction. The direction of the group velocity ${\mathbf{v}}_{gr}$ is obtained from $\mathbf{n}$ and $\mathbf{t}$ via Eq. (18). (Figure taken from Ref. [3].)

Figure 2

Effect of magnetic field on buildup of momentum anisotropy ${\epsilon }_p$, showing clear enhancement of ${\epsilon }_p$ with magnetic field, for this set of parameters, in particular for an impact parameter of 4 fm ($B_{\mathrm{time}}=0.4$ fm here).

Figure 3

The top panel shows the plot of $v_2\left(B\right)/v_2\left(0\right)$ vs the impact parameter. The ratio peaks at an impact parameter of about 3 fm, decreasing afterwards, and actually becomes less than 1 (meaning suppression of elliptic flow due to magnetic field) for large impact parameters. The bottom panel shows the plots of $v_2$ for the cases without magnetic field (solid red curve) and with magnetic field (dashed and dotted curves) separately, clearly showing that, for large impact parameters, magnetic field strongly suppresses the elliptic flow.

Figure 4

Central value of magnetic field as a function of the impact parameter. Note that the magnetic field almost monotonically increases with the impact parameter.

Figure 5

The top panel shows the initial magnetic field profile for an impact parameter of 1 fm. The bottom panel shows the initial plasma energy density profile for the same case. Note that, for this small value of impact parameter, the plasma extends well beyond the region along $x$ axis where the magnetic field is significant. Here and in Fig. 6 we show the $y$ component of the magnetic field (in units of $m_{\pi }^2$; the energy density is in units of $\mathrm{MeV}/{\text{fm}}^3$).

Figure 6

The top panel shows the initial magnetic field profile for an impact parameter of 7 fm. The bottom panel shows the initial plasma energy density profile for the same case. Note that, for this large value of impact parameter, the two profiles show opposite behavior compared to Fig. 5. Here we see that the magnetic field profile extends beyond the region along the $x$ axis compared to the plasma energy density profile.

Figure 7

Plot of the central magnetic field in the presence of fluctuations and for relatively smoother plasma background. We see that, for the smoother case, the magnetic field monotonically decreases as expected. However, for the case of stronger fluctuations, the magnetic field initially increases and eventually decreases.

Figure 8

Plot of $v_n^{rms}$ with and without magnetic field. Even though the magnetic field affects the power spectrum, its effects are very tiny here for the magnetic field considered here ($0.1m_{\pi }^2$ and $0.4m_{\pi }^2$).

Figure 9

Plot of $v_n^{rms}$ for a magnetic field with strength $5m_{\pi }^2$. The even-odd power difference is seen in the first few flow coefficients, as fluctuations wash out the effect for large $v_n\mathrm{s}$.

Figure 10

Plot of $v_n^{rms}$ for a very strong magnetic field with strength $15m_{\pi }^2$. Strong differences in the powers of even and odd values of $v_n^{rms}$ are seen. Though such large magnetic fields are completely unexpected here, such effects may provide an unambiguous signal of the presence of any unexpected strong initial magnetic field.

Figure 11

Plot of $v_n^{rms}$ for a magnetic field with strength $m_{\pi }^2$. Here we consider the isotropic region with smooth plasma profile without any fluctuations. Strong differences in the powers of even and odd values of $v_n^{rms}$ are present, arising from the effect of the magnetic field.

Figure 12

Panel (a) shows the situation of the case when the QGP region is elliptical in shape but the magnetic field points along the semi-minor axis. Panel (b) shows the case when the QGP region is roughly spherical, but still a strong magnetic field is present due to strong components coming from spectators.

Figure 13

$v_2$ for the case when the QGP region is elliptical in shape but the magnetic field points along the semi-minor axis, the $x$ axis in this case. Strong suppression of elliptic flow arises with larger momentum flowing in the direction of the semi-major axis of the elliptical QGP shape due to magnetic-field-induced anisotropy.

Figure 14

$v_2$ for the case when the QGP region is roughly isotropic in shape but still nonzero magnetic field is present leading to nonzero $v_2$, monotonically increasing with the strength of the magnetic field, even though no elliptic flow is expected in this case.

Figure 15

Crossed configuration of collision of deformed nuclei. Note that the overlap region will be reasonably isotropic, with possibly a strong $v_4$ component. Importantly now there are four spectator parts whose motion should lead to a quadrupolar magnetic field configuration.

Figure 16

Magnetic field configuration arising from collision of crossed deformed nuclei (uranium) as in Fig. 15. The quadrupolar nature of the field is clear.