Off-center collisions in AdS_5 with applications to multiplicity estimates in heavy-ion collisions

TL;DR

The authors analyze off-center collisions of light-like shock waves in AdS$_5$ to bound entropy production via the area of a marginally trapped surface. They derive an analytic expression for the trapped-surface shape in the high-energy, moderate-impact-parameter regime and obtain a leading entropy bound $S_{ m trapped} \sim \pi\left(\frac{4E_+E_- z_+ z_- L^3}{G_5}\right)^{1/3}\frac{\sinh^{-1}\beta}{\beta\sqrt{1+\beta^2}}$, linking bulk geometry to boundary entropy and, hence, to charged-particle multiplicities through $N_{\rm charged} \approx S/7.5$. The paper then confronts data via Glauber-model interpretations, showing that identifying shock energy with participating energy improves agreement with RHIC data, and explores non-conformal effects by slicing away UV and IR regions of AdS$_5$, which alters the energy scaling and highlights limitations of the conformal setup. Overall, the work demonstrates both the promise and the challenges of relating holographic entropy production to heavy-ion phenomenology, and it motivates incorporating confinement into holographic duals for more realistic QCD comparisons.

Abstract

We study the trapped surface produced by an off-center collision of light-like, point-sourced shock waves in anti-de Sitter space. We find an analytic expression for the shape of the trapped surface in the limit where the energy of the shock waves is large and the impact parameter is not too large. We use the area of the trapped surface to estimate a lower bound on the entropy produced in the collision. We compare our results to particle multiplicity measurements in heavy-ion collisions as interpreted through the Glauber model. In an attempt to roughly simulate the effects of asymptotic freedom and confinement in quantum chromodynamics, we also consider the effects of slicing off parts of anti-de Sitter space.

Figures (12)

• Figure 1: (Color online.) Comparisons between the numerics of Lin:2009pn and the analytic formula \ref{['StrappedBetter']}. The black dashed curve represents the leading term in \ref{['StrappedBetter']}; the solid red curve corresponds to the first two terms in \ref{['StrappedBetter']}; the dotted blue curve represents the expression \ref{['StrappedBetter']}, which is correct up to a term of order ${\cal O}(1/\zeta^2)$; the green dots represent the numerical evaluations used in figure 3 of Lin:2009pn; lastly, the vertical green line marks the place where, according to Lin:2009pn, the maximum impact parameter $b_{\rm max} / L$ occurs. We thank S. Lin and E. Shuryak for providing us with the results of their numerical evaluations.
• Figure 2: The energy density of a gold nucleus according to the Woods-Saxon profile, $\epsilon = {\epsilon_0 \over 1 + \exp[(r-R)/a]}$, as a function of the radial distance $r$ to the center of the nucleus. The energy distribution was normalized so that its value at $r = 0$ is one. For a gold nucleus, the parameters $R$ and $a$ take the values $\epsilon_0 = 0.159\, {\rm GeV} / {\rm fm}^3$, $R = 6.38\, {\rm fm}$, and $a = 0.535\, {\rm fm}$Adams:2004rz.
• Figure 3: (Color online.) The impact parameter $b$ as a function of the number of participating nucleons $N_{\rm part}$ in a gold-gold collision, as obtained through optical Glauber calculations at $\sqrt{s_{NN}}=200\,{\rm GeV}$, where $\sigma_{NN}=42\,{\rm mb}$. The blue curve is based on the standard Woods-Saxon distribution, whereas the red curve is based on the conformal distribution, proportional to (\ref{['Tmmidentical']}). Note that in going from $130\,{\rm GeV}$ to $200\,{\rm GeV}$ the scattering cross section decreases to $41\,{\rm mb}$.
• Figure 4: (Color online.) Total number of charged particles $N_{\rm charged}$ as a function of impact parameter $b$. The data was taken from the PHOBOS experiment Back:2005hs. The red curve corresponds to the lower bound on the number of charged particles which is based on the gauge-gravity duality \ref{['GotStrapped']}.
• Figure 5: (Color online.) Plots exhibiting the linear dependence of the total number of charged particles $N_{\rm charged}$ on the number of participating nucleons $N_{\rm part}$. The data was taken from Back:2005hs. The shaded red region shows the allowed values of $N_{\rm charged}/N_{\rm part}$, based on \ref{['GotStrapped']}.