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\journal{Nuclear Physics A}

\begin{document}

\begin{frontmatter}

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\title{Shocks and sounds in Quark-Gluon Plasma}

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\author{Edward Shuryak }

\address{Department of Physics and Astronomy, Stony Brook University \\
Stony Brook NY 11794 USA}

\begin{abstract}
%% Text of abstract
Large energy deposition from LHC quenching jets restarted interest
to shock formation. Shocks also have theoretical significance as
the simplest out-of-equilibrium setting without time dependence.
While weak shocks have small gradients and can be
treated hydrodynamically in the Navier-Stokes (NS) approximation, 
the ones without a small parameter (strong shocks) needs other methods.
Two of those will be applied: (i) the ``resummed hydrodynamics" proposed
earlier by Lublinsky and myself; and (ii) AdS/CFT correspondence, which
uses the gravitational setting. In the latter case we apply 
novel variational approach and find approximate solution. %good to within
%fraction of a percent.
 The conclusion from both treatments is that the
strong shocks deviate from NS only be few percent. We then propose a novel mechanism of shock production
at hadronization, from Raileigh collapse of the QGP bubbles.
Further discussion  of the ``fireball sonograms" 
deals with shocks/sounds produced by the quenching jets.
\end{abstract}

\begin{keyword}
%% keywords here, in the form: keyword \sep keyword
QGP, hydrodynamics, equilibration, AdS/CFT
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%% or \MSC[2008] code \sep code (2000 is the default)

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\section{Strong shocks in AdS/CFT \cite{Shuryak:2012sf}}
\label{sec_ads}
The shocks are jumps from one state of matter to the other,  and thus they are important methodically, for 
the understanding of the main unsolved problem of the field, a very rapid equilibration of sQGP.
{\em Weak shocks} are small jumps  across the wide fronts, 
those are basically sounds which are parametrically well described by hydrodynamics. {\em Strong shocks} are jumps $O(1)$ and
their shape/width is the issue we study. In gases,
amenable to kinetic treatment, this width is defined by the particles' mean free paths. In liquids, of which sQGP is one, the width is defined by gradient expansions and viscosities. 

In the frame comoving with a shock there is no time dependence. The fluxes of momentum and energy $T_{11}, T_{01}$  are simply constant, so 
in the Navier-Stokes (NS) approximation for relativistic fluid one has
  \be  
   (\epsilon+p)u_1^2+p +\eta {4\over 3} \partial_x u_1 +O(\partial^2)=C_{11};  \,\,\,\, 
 (\epsilon+p)  u_0 u_1 +\eta  \partial_x u_0 +O(\partial^2)=C_{10}
  \ee
where $u_0=cosh(y(x)),u_1=sinh(y(x))$ are expressed via the rapidity of the flow, $\eta$ is the shear viscosity and two  constants in the r.h.s.  can be inferred e.g. from flows (fluxes) far before/after
the shock, at $x\rightarrow \pm  \infty$, where the matter is homogeneous and the gradient terms vanish. It is not
hard to solve  these equations for $p(x),y(x)$ and obtain the shock profile in the NS approximation.
The question is how to include higher gradient terms.  While not being small for strong shocks, their contribution
is controlled by  the  coefficients -- ``higher viscosities"  -- and may or may be small. 
 
   One approach to related problems is  the  so called ``improved hydrodynamics" suggested by Lublinsky and myself
\cite{Lublinsky:2009kv} 
(LS) which proposed a particular approximate resummation of all high gradient terms deduced from the AdS/CFT stress tensor correlators.
Based on sign-alternating nature of higher viscosities, the resumption put the gradients in the denominator, like
in charge screening problem.
  Its simplest form (called model 2 in \cite{Lublinsky:2009kv} ) is an
operator $ [1-  \partial_x^2/(8\pi^2 T^2)]^{-1} $. Applying LS hydrodynamics to shock problem one comfortably finds corrections  
to NS profile of only  a percent
magnitude, even for strong shocks.

  However, LS is an approximation and one still would like to have    
 the first-principle solution of this  {\em out-of-equilibrium} problem, which the AdS/CFT correspondence
 provides. It means solving a very complex gravitational problem of colliding black branes. While those are notoriously difficult,
 the shock setting  has a distinction of having a {\em stationary} (time-independent) setting.
 It is impossible to describe it here --  the
 corresponding Einstein equations take many pages and turned out to be too long even in the full-size PRC article. Let me only state here
 that the method used is an (original) variational approach using elliptic nature of the Einstein equations in this setting. 
 
  The main lesson from that is that while 
 the NS profile needs  to be corrected by higher gradients, those are small  O(few  percents). 
 Small viscosity $\eta/s=1/4\pi$  plus {\em good  convergence of series over higher viscosities/gradients}
 do lead to thin  width of the shock, $O(1/\pi T)$,
confirming  rapid
transition between two very
different states of sQGP. 

\section{Mini-Bangs from Rayleigh collapse of the QGP bubbles?}
\label{sec_mini}
The fluctuations of the initial conditions result in  event-by-event fluctuations
  of the second (elliptic) \cite{Mrowczynski:2002bw} and of  the higher   \cite{Alver:2010gr} flow harmonics.
At the previous QM2011 we had presented  our first predictions for higher moments
 \cite{Staig:2010pn} which at this conference  became a large industry. Their good agreement with RHIC and LHC data 
  confirmed that, in spite of dissipative effects during their propagation $\tau_{freezeout} \sim 10 \, fm/c$, they still have an observable magnitude. 
 One may ask if there can be other sources of the shocks/sounds $during$ the evolution. Indeed, the fluctuation-dussipation
theorem insists that  fluctuations in matter should appear at any time. As a particular
example we now propose a new idea  
 \cite{bubble}, the collapse of the QGP bubbles. As everyone knows, a coffee pot ``sings" just before boiling,
 and we suggest that the sQGP does the same.
 
 There are many examples that dissipation -- and thus fluctuations -- peak at $T_c$.
While the QCD phase transition is a cross-over, it is close to the 1st order one,
and formation of some inhomogeneous intermediate state in the near-$T_c$ domain is probable.
If so, the following sequence of events has to follow: (i) the surface tension should lead   to near-spherical ``QGP drops", (ii) as the temperature gets a bit below $T_c$ they undergo the so called {\em Rayleigh collapse} 
(iii) quite efficiently transferring their energy into the outgoing shocks. 

Historically, lord  Rayleigh discovered collapse of the bubbles investigating demand from British navy during the World War I,
on the origin of mysterious small holes in ship's propellers. About a decade  ago the phenomenon
 has been studied extensively in the lab, in connection with the so called  {\em sonoluminescence}. Indeed,
weak sound can induce bubble collapse,  in volume by a  
factor $\sim 10^6$, producing  temperatures  $O(eV)$ (thus  light) and supersonic shocks with high pressures easily capable to
destroy the best steel of the ship's propellers.

  One can use hydrodynamics to derive the following equation for the bubble radius $R(t)$ and its time derivatives (dots)
\be
\ddot{R} R + (3/2) \dot{R}^2 =\delta p_{volume} +\delta p_{surface} -{4 \eta \dot{R} \over \rho R} + \delta p_{sound \, radiation}
\ee
where the r.h.s. contains two effective pressures, the NS term and the sound radiation reaction term (containing higher time derivatives of $R$). If all these terms
are neglected, the equation leads to the analytic  Rayleigh's solution producing infinite velocity at certain time.  
As our studies found, a sufficient viscosity does stop  the Rayleigh collapse. In any case, rather sudden stop of inward flow leads to the outgoing shocks,   the ``mini-bang" as we call them in this paper. 

Finding observable consequences of those events is not simple. First of all,
being produced at $T\approx T_c$,
 these shocks/sounds have to propagate
 in hadronic phase: we thus think that LHC conditions have better chance to see them than 
 those at RHIC due to later kinetic freezeout. Second, they propagate 
  much shorter time than those from initial perturbations, of few fm/c only, and thus much have size much smaller than
  ``sound circles" from the initial state fluctuations (studied before). Preliminary estimates show
that   it is hard to see them in $\phi$ harmonics as they get hidden in a peak near zero relative $\phi$.

We thus propose a novel approach,  focusing at the correlation functions in the
$rapidity$ direction.
Unlike the  initial state perturbations, which are rapidity independent,  the ``mini-bangs" at $T_c$ 
 are born well localized in particular rapidity of the QGP bubbles. 
 
 Rapidity correlations in $pp$ collisions are known to reveal ``clusters" of certain size, decaying isotropically in their rest frame.
 RHIC data on AA collisions (especially from PHOBOS \cite{Alver:2008aa}) show  clusters with larger multiplicity.  Even more importantly, they have larger extension in rapidity
 than isotropic decays  would give. Furthermore,
   at Hard Probes 2012 and this meeting ALICE reported appearence of a double-hump structure in rapidity correlation function, provided
    secondaries with few GeV $p_t$ are selected. 
   We suggest that those phenonemna are due to propagation of shocks/sounds  in the rapidity direction.
   We are currently trying to  describe those quantitatively,  by a model 
including the sounds from the ``mini-bangs".  Complexity of the problem is related to the fact that sounds near the fireball
edge are strongly distorted by radial flow, and also are sensitive to the shape of the freeze out surface. 


\section{Shocks and sounds from quenching jets}
\label{sec_mach}
Sound emission from quenching jets \cite{CasalderreySolana:2004qm} has been proposed in 2004
 as an explanation of the peaks in the two-particle correlation functions at $\phi=\pi\pm 1 rad$.
 However it since turned out that the data used, at   $p_t\sim 3 \, GeV$,  
were  unrelated to jets but are instead well explained by the hydrodynamical response to the initial state fluctuations. 
 Of course, setting the trigger momentum higher, e.g. $p_t\sim
10 \, GeV$,  one goes to the domain dominated by jets. And we still maintain that certain energy deposited into matter
by  trigger and associate jets should go to 4 peaks at $\phi=\pm 1, \pi\pm 1 rad$, visible provided the associate particle has 
 $p_t\sim 3 \, GeV$, optimal to see higher hydro harmonics. While at RHIC those are obscured by ``punch- through" associate jets,
at LHC preliminary indications from ALICE (private communications) finds rather empty $\phi\approx \pi$ region and 
even some indication  for the four peaks.

A striking phenomenon discovered at LHC  is existence of 
 very asymmetric events, in which the trigger jet has 
the transverse energy $E_T$ significantly larger than that of the associate jet $E_A$.
Energy of the jet is  deposited into QGP can be as large as 
$ \Delta E\approx E_T-E_A \sim 100 \, GeV $.
(The first equality we write as approximate since the trigger also looses certain energy,
as well as picking up some from fluctuations/trigger bias, both O(10\, GeV) or so.)

My two comments on that   \cite{Shuryak:2011vf} are as follows. First: in this case one has so-to-say two underlying events.
The inside of the Mach cones is the place where  $\Delta E$ is dissipated. It increases $T$ and hydro flow by few percents,
which should be visible even on event-by-event basis in spectra of secondaries with  $p_t\sim 3 \, GeV$.

The second comment is that such energy deposition does not lead to sounds but to shocks. 
As the volume in which this energy is deposited is
%Basic unit of length we use is the holographic  distance to the horizon $1/(\pi T)\approx .15 \, fm$.
%The strong shock example we used above produces a width-at-half maximum of about 4 such units,
%making it about $\delta r \approx 0.6 fm$ for $T=.4 GeV=1/(0.5\, fm)$ . 
%Energy of the jet is  deposited into a cylinder of length $L$ and radius given by the shock width. Its volume is
\be V=L \pi r^2 \sim (10 fm)*3.14*(0.36 \, fm^2 ) \approx 10 fm^3 \ee
and its   deposited energy is as above, extra energy density $\delta \epsilon \approx 10 \,\,GeV/fm^3$ is comparable to 
the QGP stress in the bulk.
% \be p_{QGP}\approx 4 T^4 & =& 4 *(0.4\, GeV) /(0.5\, fm)^3 \nonumber \\
%\sim  \,\, 10 \,\,GeV/fm^3 \ee  
Thus such strong quenching events
should create {\em strong shocks}. Their velocity, known from QGP thermodynamics \cite{Shuryak:2012sf}  exceeds  
the speed of  sound,  which leads to interesting and predictable  dependence
of the size/shape of the ``heated region" on the deposited energy.   

We are not currently working on modeling shocks, but sounds, as there is well developed analytic solution for those,
as perturbations of the ``Gubser flow" solution. 
Those allow to obtain Mach cone solutions, including the transverse flow and realistic viscosity. 


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%\cite{Shuryak:2012sf}
\bibitem{Shuryak:2012sf} 
  E.~Shuryak,
  %``Shocks in Quark-Gluon Plasmas,''
  Phys.\ Rev.\ C {\bf 86}, 024907 (2012)
  [arXiv:1203.6614 [hep-ph]].
  %%CITATION = ARXIV:1203.6614;%%

%\cite{Lublinsky:2009kv}
\bibitem{Lublinsky:2009kv} 
  M.~Lublinsky and E.~Shuryak,
  %``Improved Hydrodynamics from the AdS/CFT,''
  Phys.\ Rev.\ D {\bf 80}, 065026 (2009)
%  [arXiv:0905.4069 [hep-ph]].
  %%CITATION = ARXIV:0905.4069;%%


%\cite{Mrowczynski:2002bw}
\bibitem{Mrowczynski:2002bw} 
  S.~Mrowczynski and E.~V.~Shuryak,
  %``Elliptic flow fluctuations,''
  Acta Phys.\ Polon.\ B {\bf 34}, 4241 (2003)
  [nucl-th/0208052].
  %%CITATION = NUCL-TH/0208052;%%


  
  %\cite{Alver:2010gr}
\bibitem{Alver:2010gr} 
  B.~Alver and G.~Roland,
  %``Collision geometry fluctuations and triangular flow in heavy-ion collisions,''
  Phys.\ Rev.\ C {\bf 81}, 054905 (2010)
  [Erratum-ibid.\ C {\bf 82}, 039903 (2010)]
  [arXiv:1003.0194 [nucl-th]].
  %%CITATION = ARXIV:1003.0194;%%
  
  %  \cite{CasalderreySolana:2004qm}
\bibitem{CasalderreySolana:2004qm} 
  J.~Casalderrey-Solana, E.~V.~Shuryak and D.~Teaney,
  %``Conical flow induced by quenched QCD jets,''
  J.\ Phys.\ Conf.\ Ser.\  {\bf 27}, 22 (2005)
  [Nucl.\ Phys.\ A {\bf 774}, 577 (2006)]
%  [hep-ph/0411315].
  %%CITATION = HEP-PH/0411315;%%
  
  %\cite{Staig:2010pn}
\bibitem{Staig:2010pn} 
  P.~Staig and E.~Shuryak,
  %``The Fate of the Initial State Fluctuations in Heavy Ion Collisions. II The Fluctuations and Sounds,''
  Phys.\ Rev.\ C {\bf 84}, 034908 (2011)
  [arXiv:1008.3139 [nucl-th]].
  %%CITATION = ARXIV:1008.3139;%%
%
%\cite{Staig:2011wj}
%\bibitem{Staig:2011wj} 
%  P.~Staig and E.~Shuryak,
  %``The Fate of the Initial State Fluctuations in Heavy Ion Collisions. III The Second Act of Hydrodynamics,''
  Phys.\ Rev.\ C {\bf 84}, 044912 (2011)
  [arXiv:1105.0676 [nucl-th]].
  %%CITATION = ARXIV:1105.0676;%%

\bibitem{bubble} 
  P.~Staig and E.~Shuryak, The ``Mini-Bangs" as Signals of the QCD Phase Transition, in progress

%\cite{Alver:2008aa}
\bibitem{Alver:2008aa} 
  B.~Alver {\it et al.}  [PHOBOS Collaboration],
  %``System size dependence of cluster properties from two-particle angular correlations in Cu+Cu and Au+Au collisions at s(NN)**(1/2) = 200-GeV,''
  Phys.\ Rev.\ C {\bf 81}, 024904 (2010)
  [arXiv:0812.1172 [nucl-ex]].
  %%CITATION = ARXIV:0812.1172;%%  


%\cite{Shuryak:2011vf}
\bibitem{Shuryak:2011vf} 
  E.~Shuryak,
  ``Jet/Fireball Edge should be observable!,''
  arXiv:1101.4839 [hep-ph].
  %%CITATION = ARXIV:1101.4839;%%

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