Estimating the magnetic field strength in heavy-ion collisions via direct photon elliptic flow

There must be electromagnetic fields created during high-energy heavy-ion collisions. As the quark-gluon plasma (QGP) starts to evolve hydrodynamically, although these fields may become weak compared to the energy scales of the strong interaction, they are potentially important to some electromagnetic probes. In this work, we focus on the dissipative corrections in QGP due to the presence of a weak external magnetic field, and calculate accordingly the induced photon radiation in the framework of viscous hydrodynamics. By event-by-event hydrodynamical simulations, the experimentally measured direct photon elliptic flow can be well reproduced. Correspondingly, the direct photon elliptic flow implies a magnetic field strength around $0.1m_{\pi }^2\approx {10}^{16}$ G. This is indeed a weak field in heavy-ion physics that is compatible to the theoretical predictions, however, it is still an ultrastrong magnetic field in nature.

Figure 1

Diagrammatic illustration of the photon emission without and with weak external magnetic fields from QGP in the small angle approximation. The blobs represent the effect of QGP medium that has been integrated and absorbed into the function $\mathcal{I}$.

Figure 2

Pseudorapidity dependent charged hadron yields (left panel) and charged hadron direct flow (right panel) at the top RHIC energy. Solid lines are corresponding results from event-by-event hydrodynamics simulations carried out in this work.

Figure 3

Direct photon yields with also contributions from the weak magnetic photon emissions from AuAu collisions with $\sqrt{s_{NN}}=200$ GeV at RHIC [45]. Comparing to the background radiations, the weak magnetic fields lead to a small increase of photon yields. Given the values of $\rho$ that are compatible with the experimental data of $v_2^{\gamma }$, the increase ranges from less than 1% to a few percents (0–20% centrality), to about 10% (20–40% centrality) and to about 40% (40–60% centrality), respectively.

Figure 4

Direct photon elliptic flow with also contributions from the weak magnetic photon emissions for three centrality classes at the RHIC with $\sqrt{s_{NN}}=200$ GeV. In order to be compatible with the experimental data from the PHENIX collaboration [22], theoretical results are shown as colored bands with respect to varying input values of parameter $\rho ={\sigma }_{\mathrm{el}}/T\times \overline{e{B}_y}/m_{\pi }^2$.

Figure 5

Direct photon triangular flow with also contributions from the weak magnetic photon emissions for three centrality classes at the RHIC with $\sqrt{s_{NN}}=200$ GeV, comparing to experimental data from the PHENIX collaboration [22]. Theoretical results are shown as colored bands with respect to input values of parameter $\rho ={\sigma }_{\mathrm{el}}/T\times \overline{e{B}_y}/m_{\pi }^2$ determined according to $v_2^{\gamma }$.

Figure 6

Direct photon yields with also contributions from the weak magnetic photon emissions from PbPb collisions with $\sqrt{s_{NN}}=2760$ GeV at the LHC [57]. Given the values of $\rho$ that are compatible with the experimental data of $v_2^{\gamma }$, the increase ranges from less than 1% to a few percent (0–20% centrality), to about 10% (20–40% centrality), and to about 40% (40–60% centrality), respectively.

Figure 7

Direct photon elliptic flow with also contributions from the weak magnetic photon emissions for two centrality classes at the LHC with $\sqrt{s_{NN}}=2760$ GeV. In order to be compatible with the experimental data from the ALICE collaboration [23] considering only the statistical errors, theoretical results are shown as colored bands with respect to varying input values of parameter $\rho ={\sigma }_{\mathrm{el}}/T\times \overline{e{B}_y}/m_{\pi }^2$.

Figure 8

Extracted external magnetic field in RHIC AuAu collisions at $\sqrt{s_{NN}}=200$ GeV, from the direct photon elliptic flow. The black line corresponds to the solution of the magnetic field evolution in vacuum in the center of the fireball at $\tau =0.4\mathrm{fm}/c$.