Responses of quark-antiquark interactions and heavy quark dynamics to magnetic fields

We investigate the impact of the magnetic field generated by colliding nuclei on heavy quark-antiquark interactions and heavy quark dynamics in the quark-gluon plasma (QGP). By means of the hard-thermal-loop resummation technique combined with dimension-two gluon condensates, the static heavy quark potential and heavy quark momentum diffusion coefficient, which incorporate both perturbative and nonperturbative interactions between heavy quarks and the QGP medium, are computed beyond the lowest Landau level approximation. We find that the imaginary part of the heavy quark potential in the magnetic field exhibits significant anisotropy. Specifically, the absolute value of the imaginary part is larger when the quark-antiquark separation is aligned perpendicular to the magnetic field direction, compared to when it is aligned parallel to the magnetic field direction. The heavy quark momentum diffusion coefficient in the magnetized QGP medium also becomes anisotropic. As the temperature rises, the influence of higher Landau levels becomes increasingly significant, resulting in a decrease in the anisotropy ratio of the heavy quark momentum diffusion coefficient to values even below 1. At sufficiently high temperatures, this ratio ultimately approaches 1. The nonperturbative interactions are indispensable for understanding heavy quark dynamics in the low-temperature region. We also study the response of viscous quark matter to the magnetic field and explore its implications for heavy quark potential, thermal decay widths of quarkonium states, as well as heavy quark momentum diffusion coefficient.

Figure 1

The one-loop diagram for quark contribution to the retarded gluon self-energy ${\mathrm{\Pi }}_{R,\mathrm{quark}}^{\mu \nu }$. The solid (curly) line represents a quark (gluon) propagator, where the quark can occupy different Landau levels. $k$ and $p=k+Q$ are the loop momenta. $Q=(q_0,\mathit{q})$ is the external momentum and denotes the four-momentum transfer. The bottom row corresponds to the retarded gluon self-energy, with indices “1” and “2” being two types of vertices. $\mu ,\nu$ are Lorentz indices, $a$, $b$ are color indices.

Figure 2

Upper panel: the $t$-channel scatterings between heavy quarks (blue lines) and thermal partons (light quarks and gluons) from QGP medium. Lower panel: the corresponding squared scattering amplitude is directly related to the imaginary part of the HQ self-energy with the effective propagator of the exchanged gluon between a heavy quark and a thermal parton from the QGP medium. Both quark-loop and gluon-loop contributions to the gluon self-energy are considered in the computation of the effective gluon propagator. The blue blob denotes the HTL resummation.

Figure 3

Left panel: temperature ($T$) dependence of squared Debye mass $m_D^2$ for different numbers of maximum Landau level ($n_{\max }$) at $eB=15m_{\pi }^2$. Middle panel: temperature dependence of $m_D^2$ for different values of scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$) at $eB=15m_{\pi }^2$ and $n_{\max }=3$. Right panel: the variation of $m_D^2$ with magnetic field ($eB$) for different values of ${\zeta }_{\parallel }/s$ at fixed $T=0.2\mathrm{GeV}$ and $n_{\max }=3$.

Figure 4

Left panel: the variation of the real part of the HQ potential $\mathrm{Re}V$ with quark-antiquark separation distance ($r$) for different numbers of maximum Landau level ($n_{\max }$) and different temperatures ($T$) at $eB=15m_{\pi }^2$. Middle panel: the variation of the $\mathrm{Re}V$ with $r$ for different scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$)and different magnetic fields ($eB$) at fixed $T=0.2\mathrm{GeV}$ and $n_{\max }=3$. Right panel: the variation of the $\mathrm{Re}V$ with $r$ for several values of the angle ($\theta$) between the quark-antiquark separation and the magnetic field direction, where the parametrized forms of the angular-dependent strong coupling constant and string tension are quoted from Ref. [34].

Figure 5

Left panel: the variation of the imaginary part of the HQ potential related to gluon-loop contribution of the gluon self-energy $\mathrm{Im}{V}^{\mathrm{gluon}}$ with quark-antiquark separation distance ($r$) for different numbers of maximum Landau level ($n_{\max }$) at fixed $T=0.2\mathrm{GeV}$ and $eB=15m_{\pi }^2$. Right panel: the variation of the $\mathrm{Im}{V}^{\mathrm{gluon}}$ with $r$ for different scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$) and different magnetic fields at fixed $T=0.2\mathrm{GeV}$.

Figure 6

Right panel: the variation of the imaginary part of the HQ potential related to the LLL quark-loop contribution of the gluon self-energy $\mathrm{Im}{V}^{\mathrm{LLL\; quark}}$ with respect to quark-antiquark separation distances for different numbers of maximum Landau level ($n_{\max }$). Middle panel: the variation of the imaginary part of the HQ potential related to the HLL quark-loop contribution of the gluon self-energy $\mathrm{Im}{V}^{\mathrm{HLL\; quark}}$ with separation distances for different $n_{\max }$, where the solid lines and dotted lines represent the cases for the quark-antiquark separations perpendicular ($\rho \ne 0$, $z=0$) and parallel ($\rho =0$, $z\ne 0$) to the magnetic field direction, respectively. Right panel: the variation of the imaginary part of the HQ potential related to total quark-loop contributions of the gluon self-energy $\mathrm{Im}{V}^{\mathrm{quark}}=\mathrm{Im}{V}^{\mathrm{LLL\; quark}}+\mathrm{Im}{V}^{\mathrm{HLL\; quark}}$ with quark-antiquark separation distances for different $n_{\max }$ in the case of $\rho \ne 0$, $z=0$. All of the computations are performed at fixed $T=0.2\mathrm{GeV}$ and $eB=15m_{\pi }^2$.

Figure 7

Similar to Fig. 6 but for different scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$) at fixed $n_{\max }=3$.

Figure 8

The contour spatial maps for $\mathrm{Im}{V}^{\mathrm{HLL\; quark}}$ (left panel), $\mathrm{Im}{V}^{\mathrm{quark}}=\mathrm{Im}{V}^{\mathrm{LLL\; quark}}+\mathrm{Im}{V}^{\mathrm{HLL\; quark}}$ (middle panel), and $\mathrm{Im}V=\mathrm{Im}{V}^{\mathrm{quark}}+\mathrm{Im}{V}^{\mathrm{gluon}}$ (right panel) at $T=0.2\mathrm{GeV}$ and $eB=15m_{\pi }^2$ for $n_{\max }=50$.

Figure 9

Thermal decay widths for charmonium $J/\mathrm{\Psi }$ (left panel) and bottomonium $\mathrm{\Upsilon }$ (right panel) as a function of the magnetic field ($eB$) at different scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$) and different temperatures ($T$). The magenta lines are the thermal decay width results without considering the imaginary part of the HQ potential from the quark-loop contribution.

Figure 10

The LLL quark contribution to the perturbative term of scaled transverse HQ momentum diffusion coefficient ${\kappa }_{\perp ,\mathrm{HTL}}^{\mathrm{LLL\; quark}}/T^3$ (left panel), the HLL quark contribution to the perturbative term of scaled transverse HQ momentum diffusion coefficient ${\kappa }_{\perp ,\mathrm{HTL}}^{\mathrm{HLL\; quark}}/T^3$ (middle panel), as well as the HLL quark contribution to the perturbative term of scaled longitudinal HQ momentum diffusion coefficient ${\kappa }_{\parallel ,\mathrm{HTL}}^{\mathrm{HLL\; quark}}/T^3$ (right panel) as functions of $T$ for different $n_{\max }$ at $eB=15m_{\pi }^2$.

Figure 11

Similar to Fig. 10 but for different scaled longitudinal bulk viscosities (${\zeta }_{\parallel }/s$) at fixed $n_{\max }=3$.

Figure 12

The comparison between the perturbative terms of ${\kappa }^{\mathrm{quark}}$ and the nonperturbative terms of ${\kappa }^{\mathrm{quark}}$ at $n_{\max }=3$ and $eB=15m_{\pi }^2$. The solid lines and dotted lines are the transverse and longitudinal parts of ${\kappa }^{\mathrm{quark}}$, respectively. “String” and “String2” represent the nonperturbative terms employing $\sigma (T)={\sigma }_0\sqrt{1-\frac{\pi T^2}{3{\sigma }_0}}$ [85, 86] and employing $\sigma (T)=a{T}_1^2/T^2\sigma (T_1)$, respectively.

Figure 13

The gluon contribution to the perturbative term of scaled longitudinal HQ momentum diffusion coefficient ${\kappa }_{\parallel ,\mathrm{HTL}}^{\mathrm{gluon}}/T^3$ as a function of temperature for different numbers of maximum Landau level (left panel) and different scaled longitudinal bulk viscosities (right panel) at $eB=15m_{\pi }^2$. In the right panel, we also display the nonperturbative terms of the gluon contribution to the scaled longitudinal HQ momentum diffusion coefficient ${\kappa }_{\parallel ,\text{String}}^{\mathrm{gluon}}/T^3$, utilizing different $T$-dependent string tensions.

Figure 14

Temperature dependence of the anisotropy ratio of the HQ momentum diffusion coefficient ${\kappa }_{\perp }/{\kappa }_{\parallel }$ for different numbers of maximum Landau level ($n_{\max }$) at $eB=15m_{\pi }^2$. The red dotted line is the result incorporating finite longitudinal bulk viscous effect i.e., ${\zeta }_{\parallel }/s=0.03$ at $n_{\max }=3$.

Figure 15

The ratio of ${\kappa }_{\perp ,\mathrm{HTL}}^{\mathrm{HLL\; quark}}$ to ${\kappa }_{\parallel ,\mathrm{HTL}}^{\mathrm{HLL\; quark}}$ as a function of temperature for different numbers of maximum Landau level ($n_{\max }$) at $eB=15m_{\pi }^2$.