Anisotropic heavy quark potential in strongly-coupled $\mathcal{N}=4$ SYM theory in a magnetic field

In this work we use the gauge/gravity duality to study the anisotropy in the heavy quark potential in strongly coupled $\mathcal{N}=4$ super–Yang Mills (SYM) theory (both at zero and nonzero temperature) induced by a constant and uniform magnetic field $\mathcal{B}$. At zero temperature, the inclusion of the magnetic field decreases the attractive force between heavy quarks with respect to its $\mathcal{B}=0$ value and the force associated with the parallel potential is the least attractive force. We find that the same occurs at nonzero temperature and, thus, at least in the case of strongly coupled $\mathcal{N}=4$ SYM, the presence of a magnetic field generally weakens the interaction between heavy quarks in the plasma.

Figure 1

Numerical solution for the functions $W(r)$ and $P(r)$ that appear in the background metric at zero temperature (3), which interpolates between ${\mathrm{AdS}}_3\times {\mathbb{R}}^2$ in the infrared (small $r$) and ${\mathrm{Ads}}_5$ in the ultraviolet (large $r$).

Figure 2

Anisotropy induced by a magnetic field $\mathcal{B}$ in $\mathcal{N}=4$ SYM at $T=0$ (in this plot ${\alpha }^{\prime }=1$) in the heavy quark potential (left panel) and the corresponding force (right panel). The solid black lines denote the isotropic result $\sim -0.228/L$ [23], the dashed red lines correspond to the perpendicular potential $V_{Q\overline{Q}}^{\perp }$ and force $F_{Q\overline{Q}}^{\perp }=-d{V}_{Q\overline{Q}}^{\perp }/dL$, and the dotted-dashed blue lines correspond to the parallel potential $V_{Q\overline{Q}}^{\parallel }$ and force $F_{Q\overline{Q}}^{\parallel }=-d{V}_{Q\overline{Q}}^{\parallel }/dL$.

Figure 3

Numerical solutions for the functions $\ln h(r)$ (solid black line), $W(r)$ (dashed red line), and $G(r)$ (dotted-dashed blue line) in (42) for $b=2.7$. The numerical background interpolates between $\mathrm{BTZ}\times {\mathbb{R}}^2$ in the infrared (small $r$) and ${\mathrm{Ads}}_5$ in the ultraviolet (large $r$).

Figure 4

Interquark separation $LT$ versus the rescaled horizon $y_H$ (see Appendix pp2). In the left panel $\mathcal{B}/T^2=50$ while for the right panel $\mathcal{B}/T^2=1000$. For both panels the solid black line corresponds to the isotropic SYM case while the dashed red line (dotted-dashed blue line) corresponds to the case of anisotropic SYM with $Q\overline{Q}$ axis perpendicular (parallel) to the magnetic field axis.

Figure 5

Anisotropy induced by a strong magnetic field $\mathcal{B}/T^2=1000$ in the heavy quark potential (left panel) and the corresponding force (right panel) experienced by a $Q\overline{Q}$ pair in a strongly coupled $\mathcal{N}=4$ SYM plasma. The solid black lines correspond to the isotropic result $V_{Q\overline{Q}}^{\mathcal{B}=0}$ and isotropic force $F_{Q\overline{Q}}^{\mathcal{B}=0}=-d{V}_{Q\overline{Q}}^{\mathcal{B}=0}/dL$, the dashed red lines correspond to the perpendicular potential $V_{Q\overline{Q}}^{\perp }$ and perpendicular force $F_{Q\overline{Q}}^{\perp }=-d{V}_{Q\overline{Q}}^{\perp }/dL$, and the dotted-dashed curves correspond to the parallel potential $V_{Q\overline{Q}}^{\parallel }$ and force $F_{Q\overline{Q}}^{\parallel }=-d{V}_{Q\overline{Q}}^{\parallel }/dL$. In this plot ${\alpha }^{\prime }=1$.