Density Perturbations in Heavy-Ion Collisions below the Critical Point
A.~Dumitru
K.~Paech and H.~St\"ocker
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\title{Density Perturbations in Heavy-Ion Collisions below the
Critical Point}
\author{A.~Dumitru, K.~Paech and H.~St\"ocker}
\affil{Institut f\"ur Theoretische Physik, Universit\"at Frankfurt a.M., Germany}
\date{}
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\maketitle
\noindent 

Universality arguments suggest that the chiral phase transition for 
two massless quark flavors is second-order
at baryon-chemical potential $\mu_B=0$~\cite{PW},
which then becomes a crossover for small quark masses.
On the other hand, a first-order phase transition is predicted by a variety
of low-energy effective theories for small
temperature $T$ and large $\mu_B$~\cite{1stO}.
Hence, the first-order phase transition line in the $(\mu_B,T)$ plane must end 
in a second-order critical point~\cite{Stephanov:1999zu}.
For $2+1$ quark flavors the critical point has been located at
$T=160$~MeV and $\mu_B= 725$~MeV~\cite{Fodor:2001pe}.
However, a reliable extrapolation to the continuum limit and to physical
pion mass has not been attempted so far.

There is an ongoing experimental effort to detect that critical point
in heavy-ion collisions at high energies. It is hoped that by varying the
beam energy, for example, one can ``switch'' between the regimes of first-order
transition and cross over, respectively (higher energies correspond to larger
entropy per baryon or $T/\mu_B$).

To investigate collective dynamics in the vicinity of the critical endpoint
we introduce a model for the real-time evolution of a relativistic
fluid of quarks coupled to non-equilibrium dynamics of the long wavelength
(classical) modes of the chiral condensate~\cite{Paech:2003fe}:
\begin{equation}
\partial_\mu\partial^\mu \phi + \partial V_{\rm eff}/\partial\phi 
=0~,~\partial_\mu \left( T_{\rm fl}^{\mu\nu} + T_\phi^{\mu\nu}\right)=0.
\end{equation}
Here, $T_{\rm fl}^{\mu\nu}$ is the energy-momentum tensor of the fluid,
$T_\phi^{\mu\nu}$ that of the classical modes of the chiral condensate, and
$V_{\rm eff}$ is the effective potential obtained by integrating out the
thermalized degrees of freedom.
We focus first on energy-density inhomogeneities for vanishing baryon
density (the nature of the transition is then determined by the effective
quark-field coupling rather than the baryon-chemical
potential~\cite{Paech:2003fe,ove}).
We allow for ``primordial'' Gaussian fluctuations of the condensate $\phi$
on length scales
$\sim 1$~fm on top of a smoothly varying mean field. If propagated
through a first-order chiral phase transition these fluctuations give rise to
a rather inhomogeneous (energy-) density distribution
as seen in Fig.~\ref{edens_chhyd}. Such effects were previously studied
in the context of the QCD transition in the early universe, where
inhomogeneities of the entropy (or baryon to photon ratio) might affect
BBN~\cite{BBN}. However, in the cross-over regime we find much smaller
amplitudes of density perturbations~\cite{Paech:2003fe}.
\begin{figure}[ht]
%\vspace*{-.8cm}
\includegraphics[width=8cm]{g55_ey.eps}
\vspace*{-.3cm}
\caption{Fluid energy density distribution in space-time for a first-order
chiral phase transition~\protect\cite{Paech:2003fe}.}
\label{edens_chhyd}
\vspace*{-.4cm}
\end{figure} 

In heavy-ion collisions the scale of the density perturbations is too small
for them to be resolved in rapidity space. This would require a resolution
$\Delta y<1$, which is about
the thermal width of the local particle momentum distributions.
However, observable consequences of large density inhomogeneities created in
a first-order transition at beam energies below the critical endpoint
may still exist. 
(Inhomogeneities from fluctuations of particle production in the primary
nucleon-nucleon collisions~\cite{Bleicher:wd}
should be largely washed out until decoupling.)
For example, fluctuations of the energy-momentum tensor of matter in
coordinate space are uncorrelated to the reaction plane and
should therefore reduce out-of-plane collective flow ($v_2/\langle
p_t\rangle$) as compared to equilibrium hydrodynamics~\cite{Paech:2003fe}.
Moreover, by analogy to BBN, perturbations of the
entropy per baryon $s/\rho_B$ should affect abundances of rare hadrons:
$\bar{B}/B$, $\bar\Lambda/\bar{p}$~\cite{Back:2001ai}
and $K^+/\pi^+$~\cite{NA49} are larger than for a
homogeneous system with the same total volume, baryon number and entropy.

\begin{thebibliography}{99}
\bibitem{PW}
R.~D.~Pisarski and F.~Wilczek,
%``Remarks On The Chiral Phase Transition In Chromodynamics,''
Phys.\ Rev.\ D {\bf 29} (1984) 338.
%%CITATION = PHRVA,D29,338;%%
\bibitem{1stO}
M.~A.~Halasz {\it et al.},
Phys.\ Rev.\ D {\bf 58} (1998) 096007;
%[arXiv:hep-ph/9804290];
%%CITATION = HEP-PH 9804290;%%
J.~Berges and K.~Rajagopal,
Nucl.\ Phys.\ B {\bf 538} (1999) 215;
%[arXiv:hep-ph/9804233];
%%CITATION = HEP-PH 9804233;%%
T.~M.~Schwarz, S.~P.~Klevansky and G.~Papp,
Phys.\ Rev.\ C {\bf 60} (1999) 055205;
%[arXiv:nucl-th/9903048].
%%CITATION = NUCL-TH 9903048;%%
O.~Scavenius {\it et al.},
%``Chiral phase transition within effective models with constituent quarks,''
Phys.\ Rev.\ C {\bf 64} (2001) 045202.
%[arXiv:nucl-th/0007030].
%%CITATION = NUCL-TH 0007030;%%
\bibitem{Stephanov:1999zu}
M.~Stephanov, K.~Rajagopal and E.~V.~Shuryak,
%``Signatures of the tricritical point in {QCD},''
Phys.\ Rev.\ Lett.\  {\bf 81} (1998) 4816.
%[arXiv:hep-ph/9806219];
%%CITATION = HEP-PH 9806219;%%
\bibitem{Fodor:2001pe}
Z.~Fodor and S.~D.~Katz,
%``Lattice determination of the critical point of QCD at finite T and mu,''
JHEP {\bf 0203} (2002) 014.
%[arXiv:hep-lat/0106002].
%%CITATION = HEP-LAT 0106002;%%
\bibitem{Paech:2003fe}
K.~Paech, H.~St\"ocker and A.~Dumitru,
%``Hydrodynamics near a chiral critical point,''
Phys.\ Rev.\ C {\bf 68} (2003) 044907.
%[arXiv:nucl-th/0302013].
%%CITATION = NUCL-TH 0302013;%%
\bibitem{ove}
O.~Scavenius {\it et al.},
Phys.\ Rev.\ D {\bf 63} (2001) 116003.
%[arXiv:hep-ph/0009171].
%%CITATION = HEP-PH 0009171;%%
\bibitem{BBN}
see e.g.\
D.~J.~Schwarz,
%``The first second of the universe,''
Annalen Phys.\  {\bf 12} (2003) 220.
%[arXiv:astro-ph/0303574].
%%CITATION = ASTRO-PH 0303574;%%
\bibitem{Bleicher:wd}
M.~Bleicher {\it et al.}, Nucl.\ Phys.\ A {\bf 638} (1998) 391.
%%CITATION = NUPHA,A638,391;%%
\bibitem{Back:2001ai}
B.~B.~Back {\it et al.}  [E917 Collaboration],
%``Antilambda production in Au + Au collisions at 11.7-A-GeV/c,''
Phys.\ Rev.\ Lett.\  {\bf 87} (2001) 242301.
%[arXiv:nucl-ex/0101008].
%%CITATION = NUCL-EX 0101008;%%
\bibitem{NA49}
C.~Alt {\it et al.}  [NA49 Collaboration],
%``Strangeness from 20-A-GeV to 158-A-GeV,''
J.\ Phys.\ G {\bf 30} (2004) S119.
%[arXiv:nucl-ex/0305017].
%%CITATION = NUCL-EX 0305017;%%
\end{thebibliography}


\end{document}

nucl-th0402038.extracted_bib
\begin{thebibliography}{99}
\bibitem{PW}
R.~D.~Pisarski and F.~Wilczek,
%``Remarks On The Chiral Phase Transition In Chromodynamics,''
Phys.\ Rev.\ D {\bf 29} (1984) 338.
%%CITATION = PHRVA,D29,338;%%
\bibitem{1stO}
M.~A.~Halasz {\it et al.},
Phys.\ Rev.\ D {\bf 58} (1998) 096007;
%[arXiv:hep-ph/9804290];
%%CITATION = HEP-PH 9804290;%%
J.~Berges and K.~Rajagopal,
Nucl.\ Phys.\ B {\bf 538} (1999) 215;
%[arXiv:hep-ph/9804233];
%%CITATION = HEP-PH 9804233;%%
T.~M.~Schwarz, S.~P.~Klevansky and G.~Papp,
Phys.\ Rev.\ C {\bf 60} (1999) 055205;
%[arXiv:nucl-th/9903048].
%%CITATION = NUCL-TH 9903048;%%
O.~Scavenius {\it et al.},
%``Chiral phase transition within effective models with constituent quarks,''
Phys.\ Rev.\ C {\bf 64} (2001) 045202.
%[arXiv:nucl-th/0007030].
%%CITATION = NUCL-TH 0007030;%%
\bibitem{Stephanov:1999zu}
M.~Stephanov, K.~Rajagopal and E.~V.~Shuryak,
%``Signatures of the tricritical point in {QCD},''
Phys.\ Rev.\ Lett.\  {\bf 81} (1998) 4816.
%[arXiv:hep-ph/9806219];
%%CITATION = HEP-PH 9806219;%%
\bibitem{Fodor:2001pe}
Z.~Fodor and S.~D.~Katz,
%``Lattice determination of the critical point of QCD at finite T and mu,''
JHEP {\bf 0203} (2002) 014.
%[arXiv:hep-lat/0106002].
%%CITATION = HEP-LAT 0106002;%%
\bibitem{Paech:2003fe}
K.~Paech, H.~St\"ocker and A.~Dumitru,
%``Hydrodynamics near a chiral critical point,''
Phys.\ Rev.\ C {\bf 68} (2003) 044907.
%[arXiv:nucl-th/0302013].
%%CITATION = NUCL-TH 0302013;%%
\bibitem{ove}
O.~Scavenius {\it et al.},
Phys.\ Rev.\ D {\bf 63} (2001) 116003.
%[arXiv:hep-ph/0009171].
%%CITATION = HEP-PH 0009171;%%
\bibitem{BBN}
see e.g.\
D.~J.~Schwarz,
%``The first second of the universe,''
Annalen Phys.\  {\bf 12} (2003) 220.
%[arXiv:astro-ph/0303574].
%%CITATION = ASTRO-PH 0303574;%%
\bibitem{Bleicher:wd}
M.~Bleicher {\it et al.}, Nucl.\ Phys.\ A {\bf 638} (1998) 391.
%%CITATION = NUPHA,A638,391;%%
\bibitem{Back:2001ai}
B.~B.~Back {\it et al.}  [E917 Collaboration],
%``Antilambda production in Au + Au collisions at 11.7-A-GeV/c,''
Phys.\ Rev.\ Lett.\  {\bf 87} (2001) 242301.
%[arXiv:nucl-ex/0101008].
%%CITATION = NUCL-EX 0101008;%%
\bibitem{NA49}
C.~Alt {\it et al.}  [NA49 Collaboration],
%``Strangeness from 20-A-GeV to 158-A-GeV,''
J.\ Phys.\ G {\bf 30} (2004) S119.
%[arXiv:nucl-ex/0305017].
%%CITATION = NUCL-EX 0305017;%%
\end{thebibliography}