A limiting velocity for quarkonium propagation in a strongly coupled plasma via AdS/CFT

TL;DR

The paper analyzes heavy quarkonium–like mesons in a strongly coupled plasma using the AdS/CFT correspondence with D3/D7-branes. It derives the Minkowski-space dispersion relations for mesons, showing that at large momentum they approach a linear form with a universal limiting velocity $v_0$ determined by the temperature-to-quark-mass ratio $T/m_q$ and equal to the local speed of light at the D7-brane tip. Analytically and numerically, the authors compute the subleading parameters $a$ and $b$, establish a precise large-$k$ dispersion expansion, and confirm a velocity-dependent dissociation temperature $T_{\rm diss}(v) \approx (1-v^2)^{1/4} T_{\rm diss}$ that agrees with previous screening-length analyses. The results illuminate how strong coupling modifies quarkonium propagation and dissociation and suggest qualitative lessons applicable to QCD quarkonia in heavy-ion collisions, including potential observable consequences related to slowed meson motion in the plasma. The work also generalizes to other Dp-Dq setups, reinforcing the idea that the limiting velocity is governed by local geometric properties at the brane tip, with broad implications for the dynamics of bound states in strongly coupled plasmas.

Abstract

We study the dispersion relations of mesons in a particular hot strongly coupled supersymmetric gauge theory plasma. We find that at large momentum k the dispersion relations become omega = v_0 k + a + b/k + ..., where the limiting velocity v_0 is the same for mesons with any quantum numbers and depends only on the ratio of the temperature to the quark mass T/m_q. We compute a and b in terms of the meson quantum numbers and T/m_q. The limiting meson velocity v_0 becomes much smaller than the speed of light at temperatures below but close to T_diss, the temperature above which no meson bound states at rest in the plasma are found. From our result for v_0, we find that the temperature above which no meson bound states with velocity v exist is T_diss(v) \simeq (1-v^2)^(1/4) T_diss, up to few percent corrections.We thus confirm by direct calculation of meson dispersion relations a result inferred indirectly in previous work via analysis of the screening length between a static quark and antiquark in a moving plasma. Although we do not do our calculations in QCD, we argue that the qualitative features of the dispersion relation we compute, including in particular the relation between dissociation temperature and meson velocity, may apply to bottomonium and charmonium mesons propagating in the strongly coupled plasma of QCD. We discuss how our results can contribute to understanding quarkonium physics in heavy ion collisions.

Figures (6)

• Figure 1: ${\epsilon}_\infty$ (determined by the embedding $y$ at infinity) versus $\varepsilon$ (determined either by $y(0)$, for Minkowski embeddings with $\varepsilon<1$, or by where the embedding intersects the horizon, for $\varepsilon>1$). The right panel zooms in on the vicinity of the critical embedding at $\varepsilon=1$. The stable embeddings and the first order phase transition are indicated by the thick curve; the metastable embeddings are indicated by the thin curves.
• Figure 2: The squared "masses" of the two orthonormal geometric modes of the D7-brane fluctuations for Minkowski embeddings (left panel) and black hole embeddings (right panel). In each figure, $m_1^2$ ($m_2^2$) is plotted as a solid (dashed) line for three values of ${\epsilon}_\infty$. The Minkowski embeddings have ${\epsilon}_\infty=0.587$, 0.471 and 0.249 (top to bottom) and the black hole embeddings have ${\epsilon}_\infty=1.656$, 0.647 and 0.586 (again top to bottom, this time with temperature increasing from top to bottom.) The Minkowski embedding is plotted as a function of $\rho$ and the black hole embedding as a function of $u$ with the horizon on the left at $u=1$.
• Figure 3: Potentials $V_s(z)$ for Minkowski embeddings at various temperatures, all with $k=\ell=0$. The left (right) panel is for $s=1$ ($s=2$). In each panel, the potentials are drawn for ${\epsilon}_\infty=0.249$, 0.471, 0.586 and 0.5948, with the potential widening as the critical embedding is approached, i.e. as ${\epsilon}_\infty$ is increased. The ${\epsilon}_\infty=0.586$ potential is that for the Minkowski embedding at the first order transition; the widest potential shown describes the fluctuations of a metastable Minkowski embedding very close to the critical embedding. The potential becomes infinitely wide as the critical embedding is approached, but it does so only logarithmically in ${\epsilon}_\infty^c-{\epsilon}_\infty$. Note that the tip of the D7-brane is at $z=0$, on the left side of the figure, whereas $\rho=\infty$ has been mapped to a finite value of the tortoise coordinate $z=z_{\rm max}$, corresponding to the "wall" on the right side of each of the potentials in the figure.
• Figure 4: Potentials $V_s(z_{bh})$ for black hole embeddings at various temperatures, all with $k=\ell=0$. The left (right) panel is for $s=1$ ($s=2$). In each panel, the potentials are drawn for ${\epsilon}_\infty=3584.$, 0.647, 0.586, 0.586, 0.5940 and 0.5948, from narrower to wider, with the potential widening as the critical embedding is approached from the right along the curve in Fig. \ref{['fig:epsepsinf']}. Note that $z_{bh}$ is defined such that the horizon is at $z_{bh}=\infty$, and $\rho=\infty$ is at $z_{bh}=0$. The narrower (wider) of the two potentials with ${\epsilon}_\infty=0.586$ is that for the stable (unstable) black hole embedding: at this ${\epsilon}_\infty$, there is a first order transition (see Fig. \ref{['fig:epsepsinf']}) between the stable Minkowski embedding (whose potential is found in Fig. \ref{['fig3']}) and the stable black hole embedding. The potentials at ${\epsilon}_\infty=0.5940$ and 0.5948 describe fluctuations of metastable black hole embeddings, with the latter being a black hole embedding very close to the critical embedding.
• Figure 5: Potential and ground state wave function for $\psi_1$ (left three panels) and $\psi_2$ (right three panels) for $k$ given by 5, 20 and 100 (top to bottom). All plots have $\varepsilon=0.756$, corresponding to the Minkowski embedding at the dissociation transition. $V(z)$ and the ground state ($n=\ell=0$) solutions to the Schrödinger equation in the potentials $V$ are both shown as solid lines, and the ground state energies are indicated by the horizontal (red) lines. The dashed lines show the approximation (\ref{['eq:eig2']}) to the wave functions.