% Pages up to page 23 are ready, others not! % % Lecture presentation aid for the textbook: % % Laszlo P. Csernai: " Introduction to Relativistic Heavy Ion Collisions" % (John Wiley and Sons Ltd, Chicester, New York, Brisbane, Toronto, % Singapore, 1994; ISBN - 0-471-93420-8) % % Transparencies for Lecture 10 / Chapter 10 % % The following file is containing transparencies in plain LaTeX format. % Figures are usually not included, they can be pasted from the printed % textbook. Empty space is left for this purpose. Some LaTeX figures are % included in the files. 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Csernai: Inroduction to Relativistic Heavy Ion Collisions (Wiley, 1994)} \put(10,235){\parbox[t]{170mm}{\Large\sf \parskip=0.3cm #1}}\end{picture} }%end transparencyframe % \begin{document} \pagestyle{plain} \transparencyframe{ \setcounter{chapter}{9} \vspace*{-1.8cm} \chapter{Search for Quark Gluon Plasma} \section{Introduction} \B\B IN PROGRESS Field theory: the Quantum Chromo Dynamics (QCD). Links between QCD and HI experiments!! $\longleftarrow$ Theory \& Experiment \bigskip HI collisions at ultra-relativistic energies:\\ \B \LT a space-time region $\gg$ fundamental hadronic scale of 1 fm. \B \LT Opportunity to study hot and dense hadronic matter\\ -- \ \ \ the quark-hadron phase transition, predicted by lattice QCD. \B For QGP: 100+ \agev\ is needed (for fixed target experiments), \B First exploratory studies, 1986-1987: CERN-SPS and BNL-AGS. \B At the CERN-SPS, lead beams will be available in Nov. 1994. \B Future: \\ Relativistic Heavy Ion Collider (RHIC) at BNL\\ The CERN Large Hadron Collider, (LHC). }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Theoretical expectations} At extreme densities we expect a deconfinement transition\\ - many color charges will screen the confining potential. Strong interaction thermodynamics, including the critical behaviour at the transition, is described by QCD. QCD PREDICTIONS: {\em Predictions} for equilibrium thermodynamics from lattice QCD: \begin{itemize} \item $\exists$ abrupt change from hadronic to QCD regime.\\ (1st- or 2nd order or smooth (?) --- current research) \item For $N_f=2-3$ one finds $T_c$ = 150-200 MeV, ($\varepsilon_c$ = 1-3 $\gfm$), is necessary to produce QGP. \item The plasma becomes ideal ($\varepsilon \approx 3P$) only for $T/T_c \approx 1.5-2$.\\ (or not at all??) \end{itemize} $\exists$ Alternative approaches:\\ --\ \ \ effective Lagrangian models,\\ --\ \ \ bag models\\ --\ \ \ chiral perturbation theory. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsubsection{GLOBAL CHARACTERISTICS OF THE COLLISION} \B Highest energy density, temperature, entropy density, etc.\\ - cannot be measured directly. \B From experimental observables:\\ multiplicity density, $dN/dy$ of produced hadrons,\\ the abundances of particle species\\ distribution in phase space ($p_t,y$). \paragraph{\B ENERGY DENSITY:} Large multiplicities\\ - Correlated to the transverse energy, $E_T$.\\ - Shape of $M(E_T)$ \LT geometry of nucleus-nucleus collisionsi \LT \\ - Increasing no. of ``participants'' with decreasing impact parameter, $b$.\\ - Energy density, estimated using the Bjorken model:\\ - - $\varepsilon \approx 2 \gfm$ at 200 \agev\ with O+S beams at SPS. \paragraph{\B BARYON DENSITY:} From p-A,at A$\approx$200, $\delta y \approx 2 - 2.5$\ \LT\\ - maximal stopping: $\Delta y \approx 4 -5$.\\ - $n_B/n_0 \approx 10 - 20$ at max. \paragraph{\B FREEZE OUT VOLUME:} Hanbury-Brown and Twiss effect,\\ - - (originally invented to measure star sizes.)\\ - - Freeze-out: when m.f.p. $\lambda\ = $, Size of the system\\ - - Bjorken model: $R_F \approx 0.7{\sf fm}\ \times\ (dN/dy)^\alpha$, where $\alpha = 1/2 - 1/3$.\\ \LT $R_Fi \approx 2-3\ \ R_{Proj}$ !!!\\ \B Thus for Pb+Pb $R_F = $ 17 - 31 fm for SPS - LHC energies (17 \agev\ to 6300 \agev\ in c.m.). }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Experimental facilities} \B Complete coverage in center-of-mass energies from $\sqrt{s} \approx $ 3 to 20 \agev\\ \hfill (see Table 2.1).\\ \B Maximum baryon density at $y_{cm} = 0$: expected at AGS and SPS. Luminosity: excellent at AGS and mostly adequate at SPS,\\ - - ( but not for $J/\Psi$). \B After RHIC (l997-98),\\ - - and LHC ($\approx 2000$)\\ the full spectrum of ion species will be available for experimentation with essentially complete coverage in energy from $\sqrt{s} = 3$ to 200 \agev. The maximum energy density will increase by a factor of two at RHIC,\\ the maximum baryon density could be within the range of the SPS\\ the luminosity will increase by over one order of magnitude at the SPS.\\ The luminosity at RHIC will be adequate for most signals, however,\\ rare hard processes will be only marginally within reach.\\ Therefore these new facilities constitute a significant improvement in all aspects. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{Quarks and gluons} \B\B Quarks and gluons are the constituents of hadrons: \B Quarks are\\ spin-$1\over 2$ fermions, with fractional electric charge and\\ have one of three colors, $N_c=3$. \B Gluons are\\ massless spin 1 bosons, ($\sim$photons), but interact among themselves because of their color charges, as many as $N_c^2-1\ =\ 8$. Quarks come in different flavors, $u,\ d,\ s,\ c,\ b,\ t.$ \\ Ordinary hadrons, $p,\ n,\ \pi,\ \Delta,\ $etc., contain $u$ and $d$ quarks only, \\ strange hadrons, $\Lambda, \Omega, K, $ etc. contain strange quarks also. Table 10.1 {\it Quark properties} \begin{center} \begin{tabular}{lrrrr} \hline\hline Name &Flavor& Electric & Mass $\approx$ \\ & & charge & [MeV] \\ \hline up & u & $2 / 3$& 1.5-5.1 \\ down & d &-$1 / 3$& 8.1-9.5 \\ strange& s &-$1 / 3$& 175$\pm$25 \\ charm & c & $2 / 3$& $\approx$1270 \\ bottom & b &-$1 / 3$& $\approx$4250 \\ top & t & $2 / 3$& 174$\pm$10$\cdot 10^3$\\ \hline\hline \end{tabular} \end{center} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \centerline{SOME ELEMENTS OF QCD} The color gauge group of QCD is SU(3). The generators of the group are $G^a$, where $a=1, .... ,N_c^2-1$, and they satisfy the commutation relation \beq [G^a,G^b] = i\ f^{abc} \ G^c \ , \eeq{kap81} with the group structure constants $f^{abc}$. The generators are conventionally orthogonalized such that \beq {\sf Tr}[G^a G^b] = {1 \over 2} \delta^{ab} \ . \eeq{cr67} The generators of the group can be represented by the 3$\times$3 Gell-Mann matrices, $\lambda^a$, e.g. $G^a = {1 \over 2} \lambda^a$. The gluons are described as the quanta of the gauge field $A^\mu_a$ with the color index $a$, $a=1,...,N_c^2-1$ and space-time index $\mu$. The field strength is \beq F^{\mu\nu}_a = \partial^\mu A^\nu_a - \partial^\nu A^\mu_a - g\ f_{abc} A^\mu_b A^\nu_c \ , \eeq{kap82} where $g$ is the strong coupling constant. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} The gluon field and the field strength are not invariant under an infinitesimal GAUGE TRANSFORMATION $\alpha_a(\vec{r},t)$. The gluon fields transform as \beq A^\mu_a \rightarrow A^\mu_a + g\ f_{abc} A^\mu_b \alpha_c\ -\ \partial^\mu \alpha_a\ , \eeq{kap83} the field strength as \beq F^{\mu\nu}_a \rightarrow F^{\mu\nu}_a + g\ f_{abc} F^{\mu\nu}_b \alpha_c\ , \eeq{kap84} and the quark fields as \beq \psi_k \rightarrow \exp[igG^a\alpha^a] \psi_k . \eeq{kap86} Here $G^a$ is an $N_c\times N_c$ representation of the color group, and $\psi_k$ is an $N_c$-dimensional vector in the color space representing a quark field of flavor $k$. The field strength square is invariant under gauge transformations, $F^{\mu\nu}_a F_{\mu\nu}^a \rightarrow F^{\mu\nu}_a F_{\mu\nu}^a$. The Lagrangian of QCD is \beq {\cal L} = \sum_{k=1}^{N_f} \bar{\psi_k} ( i \gamma_\mu \partial^\mu - M_k - g \gamma_\mu A^\mu_a G_a) \psi_k - {1\over 4} F^{\mu\nu}_a F_{\mu\nu}^a \ , \eeq{kap85} where $\gamma_\mu$'s are the Dirac gamma matrices, $g$ is the coupling constant of quarks to gluons, $M_k$ is the mass of quarks of flavor $k$, and $N_f$ is the number of flavors. If $g=0$ this Lagrangian describes non-interacting, massive quarks, and $N_c^2-1$ non-interacting, massless, free gluons.[2] }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} In the non-interacting case the EOS of the QGP is:\\ - - (the bag model EOS):\\ - - - - ideal fermion gas (quarks)\\ - - - - ideal boson gas (gluons) with the appropriate degeneracy\\ - - - - bag constant, $B\cdot g_{\mu\nu}$, is added to $T^{\mu\nu}$. The EOS can be directly calculated from the microscopic Lagrangian via the Grand Canonical Partition function, \beq Z \equiv Z_{GC} = Z(V,T, \mu_1,\mu_2, ...) = {\sf Tr} \left[\exp \left( -(\hat{H}-\mu_i \hat{N}_i) /T \right) \right] \ , \eeq{kap11} for $i=1,2,...$ conserved charges. All thermodynamical quantities can be calculated from the thermodynamical potential $\Omega \equiv \ln Z$\\ (Chapter 4). \bigskip The functional integral representation of the partition function for high temperatures is presented comprehensively by Kapusta in ref. [2]. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} Table 10.2 {\it Quark composition of some hadrons.} \begin{center} \begin{tabular}{lr} \hline\hline Hadron & Quark content \\ \hline $\pi^+,\ \rho^+ $ &$ u\bar{d}$\\ $\pi^-,\ \rho^- $ &$ d\bar{u}$\\ & \\ $\pi^0,\ \rho^0,\ \omega^0 $ &$^\dagger\ u\bar{u},\ d\bar{d}$\\ $\eta^0,\ \omega_1^0,\ \omega_8^0$ &$^\dagger\ s\bar{s},\ u\bar{u},\ d\bar{d}$\\ $\phi^0$ & $ s\bar{s}$\\ & \\ $K^0,\ K^+,\ K^{0*},\ K^{+*} $ & $^\dagger\ u\bar{s},\ d\bar{s}$\\ $\bar{K}^0,\ K^-,\ \bar{K}^{0*},\ K^{-*} $ & $^\dagger\ u\bar{s},\ d\bar{s}$\\ \hline $p,\ \Delta^+ $ & $ uud $\\ $n,\ \Delta^0 $ & $ udd $\\ & \\ $\Delta^{++} $ & $ uuu $\\ $\Delta^{-} $ & $ ddd $\\ & \\ $\Sigma^+,\ \Sigma^{+*} $ & $ uus $\\ $\Sigma^-,\ \Sigma^{-*} $ & $ dds $\\ $\Sigma^0,\ \Sigma^{0*},\ \Lambda^0 $ & $ uds $\\ & \\ $\Xi^0,\ \Xi^{0*} $ & $ uss $\\ $\Xi^-,\ \Xi^{-*} $ & $ dss $\\ & \\ $\Omega^- $ & $ sss $\\ \hline \end{tabular} \end{center} ($^\dagger$Linear composition of these) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{Lattice QCD} \B Some basic features and recent results\\ - - - just to give an overview\\ - - - - Introduction to lattice QCD: by Creutz [3]. \B Numerical constraints of present computations:\\ - - For QCD with dynamical quarks\\ - - Performance of at least $10^3$ times the present is needed.\\ - - 3-4 years we will know the QCD predictions much better. \B Starts from first principles, the lagrangian of QCDi, to \LT EOS\\ - - There is no analytical method known, which allows this.\\ \LT - Extensive numerical simulations. \B The physical case, two nearly massless and one massive quark,\\ - - is difficult to simulate. The simulations have mainly been performed with some other number of flavors, $N_f$. The calculations have mainly been done in the following cases: \paragraph{PURE GLUON THEORY:} ($N_f=0$) Here one may introduce quarks as\\ - - external sources,\\ - - then this approach is called the valence quark (or quenched) approximation.\\ Due to the relatively easier computation this approach has been widely used, because, the action is local. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \paragraph{FOUR DENERATE FLAVORS:} When discretizing the action one wants to keep chiral symmetry:\\ - - because (an) important mechanism governing the phase transition is\\ - - the restoration of the spontaneously broken ciral symmetry. It is, however, not known how to keep the full chiral symmetry in the lattice discretization\\ - - - without a severe multiplication of the number of flavors. The use of so called Susskind or staggered fermions (see below) is a compromise.\\ (Part of the chiral symmetry is explicit at the prize of having four degenerate flavors. The flavor symmetry is broken on the lattice, but should be restored in the continuum limit.) \bigskip \paragraph{$N_f$ = 2 or 2 + 1\ :} This case is nearer to the physical situation.\\ We can use staggered fermions, however,\\ - - only by assuming that two flavors correspond to \\ - - - taking the square root of the fermion determinant (see below).\\ \LT There is then no fermionic representation on the lattice, and\\ - - we don't know if $\exists$ symmetry corresponding to the number of flavors }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{The lattice formalism} On the lattice we simulate equilibrium thermodynamics. \B The physical temperature, $T$, and volume, $V$, are given by \beqar T &=& 1/(N_\tau\ a) = \beta^{-1} \nonumber \\ V &=& (N_\sigma\ a)^3 \eeqar{pet4} where $a$ is the lattice spacing. \B The parameters which one can vary are:\\ - the coupling constant $g$,\\ - the bare quark mass, $m_q\ a$,\\ - the lattice size in the temperature- direction, $N_\tau$,\\ - \phantom{the lattice size in the} space- direction, $N_\sigma$, \&\\ - the number of dynamical quark flavors, $N_f$. \B The spatial size of the latice is $L^3$ where $$ L=\left( \frac{N_\sigma}{N_\tau} \right) \frac{ (\hbar c)}{T}, $$ thus the total volume of a large calculational lattice ($48^3\times 16$) is $V_{Lattice}=L^3= 61$fm$^3$ at $T_c=150$MeV. \B \LT Large critical fluctuations, not well separated phases! \bigskip \setlength{\baselineskip}{11pt}{\normalsize\sf Because of asymptotic freedom there should exist a continuum limit when $ g \rightarrow 0$. In this limit the correlation lengths go to infinity in lattice units, i.e. for fixed physical masses, which are the inverses of the correlation lengths, the lattice spacing, $a$, goes to zero. Furthermore we want to take the thermodynamical limit $V \rightarrow \infty$ at fixed $T$. } }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \setlength{\baselineskip}{11pt}{\normalsize\sf The lattice discretizes the space-time, so that $x$ becomes a discrete variable, describing the lattice SITE.\\ There are LINKS in the lattice extending from this site to all neighboring sites.\\ A directed link can be characterized by the site of origin and by the direction of the neighbor: $x,\mu$. Four directed links may form a closed loop, a PLAQUETTE.\\ The continuum gluon and quark fields, $A_\mu(x)$ and $\psi_k(x)$, are represented by bosonic and fermionic quantities, $U_{x,\mu}$ and $\chi_{x,k}$ respectively. We can associate a matrix, $U_{x,\mu}$, with every link of the lattice \beq U_{x,\mu} = e^{i g\ A_\mu^c G^c\ a}\ , \eeq{cr73} where a is the lattice spacing and $\mu$ is the direction of the link.\\ $A_\mu^c$ is evaluated at the coordinate of the middle of the link.\\ In fact Wilson [9] proposed an action in terms of the lattice, which returns the QCD in continuum limit. The action is a sum over all elementary squares of the lattice \beq S = \sum_{\Box} S_{\Box} . \eeq{cr75} The action on each of these squares or plaquettes is the trace of the product of the group elements surrounding the plaquette. It consists of a gluonic part, $S_G$, and a fermionic part, $S_F$. The functional integral representation of the partition function, \\ thus can be discretized on the lattice\\ by introducing the contributions of plaquets to these integrals and\\ reducing the functional integrals to DISCRETE SUMS over all possible discrete configurations. $$ Z_{QCD} = \int \prod _{x,\mu} dU_{x,\mu} \prod _{x,k} d\chi_{x,k} d\bar{\chi}_{x,k} \exp[-S_G(U)-S_F] . $$ } }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.8cm} \setlength{\baselineskip}{11pt}{\normalsize\sf The contribution of gluons to $\sum S_{\Box}$ in terms of the $U_{x,\mu}$ matrices \beq S_G(U) = \frac{6}{g^2}\sum_{\nu >\mu} \left( 1 - {1\over 3} Re\left( \ {\sf Tr}[U_{x,\mu}U_{x+\mu,\nu} U^+_{x+\nu,\mu}U^+_{x,\nu}] \right) \right) , \eeq{pet2} is the so called {\em WILSON ACTION}. The contribution of the fermions is more complicated, see [10] and references therein. The so called {\em STAGGERED FERMION ACTION} is given by $$ S_F = \sum_{k=1}^{K} \bar{\chi}_{x,k} Q^{x,y}_k \chi_{y,k} , $$ where $K=N^s_f$ is the number of staggered fermions (in the continuum limit $N_f = 4 N^s_f$), $\chi$ and $\bar{\chi}$ are so called {\em Grassmann variables} [2], and $ Q^{x,y}_k(U) $ is the {\em fermion matrix} defined as $$ Q^{x,y}_k(U) = \sum_{\mu = 0}^3 D_{x,y;\mu} + M_k a \delta_{x,y} . $$ The {\em hopping matrices}, $D_{x,y;\mu}$, have non-zero elements only for $y=x\pm \hat{\mu}$, and they are given by $D_{x,y;\mu} $ $ = $ $ {1\over 2} \eta_\mu(x) \left[ U_{x,\mu} \delta_{x,y;\hat{\mu}} \ - \ U^+_{y,\mu} \delta_{x,y+\hat{\mu}} \right]$, \ \ where the phase factors are $\eta_\mu(x) = $ $ (-1)^{x_0 + x_1 + ... + x_{\mu-1}}$ for $\mu > 0$ and $\eta_0(x) = 1$ [11]. The other representation of fermions on the lattice is the so called {\em Wilson fermion} representation, which we will not discuss here. The contribution of the fermions can be integrated out [2,10] and so only the quantities $U_{x,\mu}$ remain in the contribution \beq Z_{QCD} = \int \prod_{x,\mu} dU_{x,\mu} e^{-S_G(U)} \left(\prod_{k=1}^{K} {\sf det}\left[ Q_k(U) \right] \right) , \eeq{pet1} where the four dimensional lattice has the size $N_\tau \times N_\sigma^3$. The coefficient $6/g^2 \equiv \beta_{Lattice} = 2N_c/g^2$ is frequently used to study the properties of phase transition. By varying $\beta_{Lattice}$ on a fixed lattice a phase transition may be observed. The observables can be obtained by taking expectation values with respect to the integrand in (\ref{pet1}). }%end-normalsize }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} QUANTITIES TO CHARACTERIZE PAHSE TRANSITIONS To characterize the two phases, hadron matter versus quark-gluon plasma, one usually employs order parameters, namely the {\em Polyakov loop} \beq < L > \equiv < {\sf Tr} \prod_{x_0=1}^{N\tau} U_{(\vec{x},x^0),4} > , \eeq{pet5} or the {\em chiral condensate} \beq <\bar{\psi}\psi> = < {\sf Tr}[D(U) + m_q a]^{-1} > . \eeq{pet6} The Polyakov loop, $L$, is related to the free energy of a static quark, and is an exact deconfinement order parameter for the pure SU(3) gauge theory. The chiral condensate $<\bar{\psi},\psi>$ becomes an exact order parameter for the restoration of chiral symmetry for all $ N_f$ when $m_q a\rightarrow 0.$ Other physical quantities measured are, e.g. the energy density, $e$, the pressure, $P$, the entropy density, $s$, and the Debye screening length, $r_D$. These can be defined in terms of the U-matrices. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{The order of the phase transition} UNIVERSALITY arguments relating the field theory to simpler three dimensional spin models [12]. The universality of a phase transition means that transitions belonging to the same symmetry class have similar behaviour of the critical exponents and the same order. For example an SU(N) theory has a global ($Z_N$) symmetry under which the order parameter, e.g. $L$, transforms non-trivially. A spontaneous breakdown of this symmetry is possible. \setlength{\baselineskip}{11pt}{\normalsize\sf The dynamics near the transition may be governed by a (local) $Z_N$ theory. Then one has the prediction that e.g. SU(3) in $d = 3+1$ dimensions is related to $Z_3$ in $d = 3$. For this latter theory only first order transitions are known. Thus SU(3) should have a first order transition. This is not a complete proof, however, because, e.g. the hypothesis of locality of the interaction may be violated. }\bigskip For $N_f > 0$ and $m_q = 0$, there is an effective model of the chiral symmetry restoration.\\ - - QCD with $N_f$ dynamical flavors in 3+1 dimensions should be related to a $U(N_f) \times SU(N_f)$ $\sigma$-model in 3 dimensions\\ - - As was discussed by Pisarski and Wilczek, the transition in this model is expected to be first order for all $N_f > 2$. [13] \bigskip Recent numerical calculations \LT first order phase transitions [14], but in some cases a continuous transition is observed also. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \paragraph{PURE GLUON MATTER:} \B Pure gauge theory ($N_f=0$) \LT\\ - - most calculations predict a first order phase transition. \\ - - latent heat $\Delta e$ is small compared to the Stefan-Boltzmann value\\ - - - - In fact $\Delta e \approx e_{SB}/4$. \B String tension $\sigma$ is small compared to $T=0$\\ - - $\sigma(T_c) \approx\sigma(0) $, and \B Debye screening length $r_D$ is big compared to its value\\ - - further above the transition $r_D(T_c) \approx 3 r_D(1.2\ T_c)$. \paragraph{CALCULATIONS WITH DYNAMICAL QUARKS:} Computer time limitations, \LT few results. \B $\exists$ first order transition when $m_q \leq 0.1\ T_c$.i - - for all $N_f$. (Chiral symmetry --- small m??) Presently there seems to be no evidence for a first order transition for $N_f= 2$. For $N_f = 3$ the first order nature reappears. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \paragraph{FINITE BARYON CHEMICAL POTENTIAL:} No final results yet from lattice QCD (CERN-SPS and BNL-AGS). In an effective potential model of QCD for large baryon chem. pot.\\ - - - a first order phase transition was found. Decreasing the chemical potentiali \LT change\\ - - - the phase transition changed to be of second order. [18,19] The critical temperature (at $\mu_q=0$) was around $T_c \approx 110 $ MeV - - the tricritical point was around $T_t \approx 81-107$ MeV\\ - - and $\mu_t \approx 73-82$ MeV. \B \LT Nucl. matter: a first order phase transition may be expected. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Critical temperature} $T_c$ is important (precision): \beq e = {\sf const.}\ T^4_c \eeq{pet7} sensitive to $T_c$. In Lattice QCD T=fixed, \LT \\ phase transition at a value of the bare coupling $g = g_c(N_\tau,N_f,m_qa)$. We assume $N_\sigma$ to be so big that there is little dependence on $N_\sigma$. The temperature in physical units:\\ - - obtained by setting the scale by another physical quantity.\\ - - - - There exist ``computer measurements'' of $T_c$. According to present numerical estimates for $N_f = 0$\\ - - asymptotic scaling may have been reached for $N_\tau \geq 8$.\\ - - $T_c \approx 220 \pm 40$ MeV.[20] Another possibility to extract $T_c$ in physical units is to assume scaling but not necessarily asymptotic scaling. Then one ``measures'' e.g. the hadron masses at the critical value of the coupling constant but on a lattice with large time extent, i.e. at temperature zero. \B One has a connection between $T_c$ and hadron masses in lattice units\\ - - - which can be extrapolated to the continuum case and then\\ - - - the physical $T_c$ can be extracted.\\ \LT $T_c$ in QCD with $N_f=4$ is only half as big as in for $N_f$ = 0\\ - - - Same tendency of lower $T_c$ was already seen for $N_f$ = 2.[20] }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{The Equation of State} Bag model EOS (chapter 4): Too simple, but right order of magn. Lattice QCD:\\ \B For $N_f = 4$ there is a clear discontinuity, the upper value is essentially in agreement with the free gas value.\\ \B For $N_f = 2$ the phase transition is presumably continuous. but the entropy density goes fairly quickly to its free gas value.\\ \B For $N_f = 0$ the discontinuity is only about a quarter of the Stefan-Boltzmann value, thus indicating a ``weak first order transition''. \vspace*{11cm} Figure 10.1 {\it The Equation of State as function of the coupling constant $g$. The pressure $p$ (as $p/T^4$) and the quantity $(e-3p)/T^4$ is plotted versus $6/g^2$ for $N_f=0$. From Engels et al., reported in ref. [20]} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} NONPERTURBATIVE EFFECTs: \B The deviation of pressure from the free gas, SB, value is substantial:\\ For $T = 2T_c$ it is only about 50 \% of the free gluon gas value. \B The EOS is a relevant concept for HI collisions\\ - - if local equilibratium and thermalization do exist. If this is the case already when the maximum baryon charge density and energy density are reached,\\ \LT global collision characteristics is defined by conservation laws for baryon charge, energy and momentum. \B\B\B In such situation\\ - - - the properties of the EOS are largely irrelevant\\ - - - - - around the transition (1st, 2nd order or smooth) and only\\ - - the asymptotic properties do matter as it was shown in sect 5.5. The final hadronization, on the other hand is sentive to\\ - - - the order of the phase transition.\\ \B\B If the transition is 1st order and metastable states exist characteristic phase transition phenomena:\\ supercooling, delayed phase transition, extra entropy production and critical fluctuations are possible. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Screening lengths} One may, in analogy with the screening in an electromagnetic plasma, expect screening in a quark-gluon plasma. This would then lead to disappearance even of "small" resonances like $J/\psi$ or $\Upsilon$, as was proposed by Matsui and Satz [26]. By resummation of a class of diagrams in perturbation theory one obtains \beq V(r, T) = {{\sf const.} \over {r^2}} e^{-2r/r_D} \ , \eeq{pet17} where the potential $V$ is defined on the lattice from the correlation between Polyakov loops \beq < L(r) \ L(0) > = ^2 e^{- \frac{V(r,T)}{T}} \ \ \ {\sf and} \ \ \ r_D = \frac{1}{g T \sqrt{1+N_f /6}} . \eeq{pet18} This is in analogy with QED where the corresponding Debye screening length can be obtained from classical considerations. In QCD it is uncertain if the thermodynamic limit can be evaluated in perturbation theory, nevertheless, a screening of an effective form \beq V(r,T) = {a \over {r^\alpha}} e^{-r/r_d} \ , \eeq{pet20} is expected, where $a$, $\alpha$ and $r_D$ are unknown functions of $T$. Fixing $\alpha = 1$, one obtains values for $r_D$ between 0.1 - 1 [20,23]. Below $r_D \approx 0.3$ the $J/\psi$ should be dissolved. Further studies, extending, the measurements to larger volumes and distances are still needed to clarify the situation. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Summary of present results} \B Following ref. [20] we can summarize some of the basic results of the ongoing lattice QCD research. \B Calculations with dynamical quarks give $T_c < 200$ MeV, if normalized to the hadron masses. If $T_c \approx 150$ MeV in the physical case, the energy density of the quark-gluon plasma just above the phase transition would be only around 1 $\gfm$. \B For the pure gluon theory, $N_f = 0$, the phase transition is first order, at least for the lattice sizes investigated. For $N_f = 4$ there is also a first order transition for sufficiently small quark masses, thus related to the restoration of chiral symmetry. For $N_f = 2$, however, no discontinuity is observed. The physical case may be a borderline case. \B Free gas behaviour seems to set in in thermodynamical quantities at least for $T > 2T_c$. For $T < 2T_c$ there may be a modification. \B Although the phase transition for $N_f = 0$ is related to deconfinement of static quarks, and for $N_f = 4$ to restoration of chiral symmetry, the properties of the high temperature phase seem to be very similar, in the sense that the quantities $e/e_{SB}, \ \ s/s_{SB}, \ \ r_D$ and other screening lengths are essentially independent of $N_f$. \B First calculations of the surface tension $\sigma$ at the phase transition have been performed, still near the strong coupling region. They give $\sigma \approx 0.24 \cdot T_c^3$. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{Surface tension, viscosity and nucleation} \B First order PHT \ \LT \ Supercooling !\\ - - subsequent rapid deflagration to hadronic matter\\ - - \LT large strange anti-baryon abundances. \B Deflagration may be governed by the nucleation rates in the plasma. - - Rate is governed by transport coeff.:\\ - - - surface tension, correlation lengths and the EOS. For an ultra-relativistic gas the coefficients of viscosity \begin{equation} \eta = {4\over {15}} a T^4 \tau \ \ \ {\sf and} \ \ \ \zeta = 4 a T^4 \tau \left[ {1\over 3} - \left( \frac{\partial p}{\partial e} \right) \right]^2 \ , \end{equation} where $\tau$ is the collision time, and $a$ is the Stefan-Boltzmann constant, defined such that the energy density is $e=aT^4$. Since the square of the sound velocity, $(\partial p/\partial e)$, is close to ${1\over 3}$ for ultra-relativistic gases, the bulk viscosity $\zeta$ is usually much smaller than the shear viscosity $\eta$. \B Viscosity is an additive sum of the quark and gluon viscosities, $\eta = \eta_q + \eta_g$. (in terms of viscous relaxation times for quarks and gluons, $\tau_q$ and $\tau_g$.) For zero baryon chemical potential \begin{equation} \tau_g^{-1} = 4.11 \ T \left(1+\frac{N_f}{6}\right) \alpha_s^2 \ |\ln \alpha_s | \ , \end{equation} where $\alpha_s = g^2 /4\pi$ is the QCD fine structure constant. The two terms, 1 and $N_f/6$, come from the contribution of gluon-gluon and gluon-quark scatterings. The viscous relaxation rate for quarks is \begin{equation} \tau_q^{-1} \ = \ 0.39 \ \tau_g^{-1} \ . \end{equation} }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \B At $T_c \approx$ 200 MeV, $\alpha_s\approx 0.23$.\\ - - \LT $\tau \approx $ 1 fm/c.\\ For QCD with 2 flavors the viscosity is \begin{equation} \eta = \frac{1.12\,T^3}{\alpha_s^2\ln(1/\alpha_s)} \ . \end{equation} \B Surface tension, $\sigma$: (for $N_f =0$, and $N_\tau = 2$,)\\ The calculational lattice was divided to two halfs, and $\beta_{Lattice}$ was given different values. \ \LT \\ - - extra energy, $\Delta E = A\cdot\sigma (\Delta\beta_{lattice})$.\\ \B Phase transition value of $\sigma$ is obtained when $\Delta \beta_{Lattice} \rightarrow 0$,\\ so that the dividing surface will be the critical phase transition surface at $T_c$ \ \LT $\sigma = \alpha_0 T_c^3$ where \beq \alpha_0 = 0.24 \pm .06 \ \ \ {\sf or}\ \ \ \alpha_0 = 0.22 \pm .05 \eeq{pet21a} \B $T_c \approx 200$MeV \LT $\sigma \approx 50$MeV/fm$^2$. \B Critical radius, $R^*\approx 0.5 - 3$ fm for $T \approx 0.8-0.97 T_c$ \ \LT\\ - - transition time due to primary nucleation $\approx$ 100 fm.[2] The transition is determined by the interplay between the phase transition dynamics and the global expansion dynamics. \B Completion in 5-100 fm/c. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \section{Nuclear stopping power} \B\B 1 \agev\ energy region: m.f.p. experimentally $\lambda = 2.4 \pm 0.4{\sf fm}$.[33,34]\\ - - Kinetic theory: $2-3\lambda$ for thermalization\\ - - Increased density reduces m.f.p. $\propto n_0/n$, - - - - as well as phase transitions do. \B Viscous or multi-fluid dynamics, kinetic models, like VUU, BUU, etc.,\\ - - molecular dynamics models \B\B Ultra-relativistic energies: stopping is more acute\\ - - - multiparticle formation.[35]\\ - - - forward peaked c.s.\\ - - - non-point like collisions\\ - - - formation time }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Proton nucleus collisions} $p+p$ and the $p+A$ experimental works [36-37] at 100 and 200 A.GeV \B \LT Theoretical works [38-43] on nuclear stopping power\\ - - models are based on 1-dim kinetic equations \\ - - - - for the leading nucleon momentum distribution. \\ The inelastic NN scattering cross section is parametrized as \beq {1 \over {\sigma_{NN}}} {{d\sigma_{NN\rightarrow NX}}\over {dx}} = \xi \eeq{mi521} where $x = p_z/p_i$, \\ - $\xi$ is independent of $p_i$ \& $x$ (According to the Feynman scaling)\\ - - - e.g. for $E_{Lab.} = 19 - 405$ GeV $\xi_p = {1 \over {\sigma_{pp}}} {{d\sigma_{pp\rightarrow pX}}\over {dx}}$\\ - - - - - - varies from 0.43 to 0.55. \B\B The evolution model [35] assumes\\ - - that the avg. longitudinal momentum of the leading baryon\\ - - decreases exponentially with the length of its path, $Z$ \beq = p_i \exp[{{-Z}\over{\Lambda_p}}]\ , \eeq{mi523} where $\Lambda_p$ is the ``momentum degradation length''. \\ It is expressed via $\xi$ as \beq \Lambda_p = 2 \lambda_N/\xi \ , \eeq{mi524} where $\lambda_N$ is the nucleon mean free path.\\ Eqs. (\ref{mi523}) and (\ref{mi524}) are based on the Poisson distribution\\ - - in the number of leading baryon collisions, $\nu$, with $<\nu > = Z/\lambda_N$.\\ (For ultra-relativistic energies eq. Eq. (\ref{mi523}) coincides with the deceleration law used in two fluid dynamics [45].) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \B\B From experimental data $\Lambda_p = 5.7 $fm was extracted[35]. \B\B The evolution or multiple scattering model [35,39]:\\ - - linear deceleration of a proton in nuclear medium in the $z-$direction \\ - - the medium has a thickness of $N$ nucleons in a row.\\ Introduce probability distribution, $Q(x)$,\\ - - - - - - that the incident nucleon has momentum fraction $x$\\ - - - - - - after a collision with one more target nucleon, \LT \\ - - one can calculate the final $x$ distribution of this nucleon\\ - - after colliding with $N$ nucleons in a row, \B --- \LT $H(x,N)$. $x \approx p_z/p_i \approx E/E_i \approx \exp[y-y_0]$,\\ valid for ultra-relativistic incident energies \ \ \ \LT \\ - - - ( outside the narrow region $x\lton m_N/p_i$,)\\ The rapidity distribution of the nucleon after penetrating the target: \beq H'(y,N) = H (e^{y-y_i},N) \ . \eeq{mi009} $H'(y,N)$ is plotted in Fig. 10.2. \B For $N$, $N=5,\ 7$ the distribution is peaked at $\Delta y \approx 1.5,\ 2.5$\\ \B The spread of the distribution is about twice as large as the deceleration.\\ - - $N = 5,\ 7$ represent the diameters of $Cu$ and $Pb$ nuclei, \LT \\ - - - - stopping power is strongly dependent on the size of the nucleus. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{16cm} Figure 10.2 {\it Plots of the distribution of nucleon rapidity deceleration, $H'(y,N)$, after the incident nucleon, with rapidity $y_0=0$, has collided with $N$ target nucleons in a row. From [39]} }%end tr-page \newpage % transparency ============================================= page 30 \transparencyframe{ \vspace*{-1.2cm} \subsection{Heavy ion collisions} \B Based on $p+A$:\\ - Stopping at $\Delta y = 5$ --- $E_{Lab.} = 75$\agev.\\ - - - - For copper nuclei $\Delta y = 3$ --- $E_{Lab.}= 10$\agev\\ - Complete transparency and baryon charge free region for Pb :\\ - - - - above $\Delta y = 10$ or $E_{Lab.} = 11,000$\agev\ - Nuclear effects: density incr., PH.Tr., Critical fluct. etc.\\ - - - - Increased stopping!! \B Experimentally:\\ - rapidity distribution - baryons \& mesons\\ - - - - Peak $\sim$ stopping\\ - - - - Flat $\sim$ transparency. \B Problems:\\ - - Bjorken model $\sim$ transparency\\ - - Landau model $\sim$ stopping\\ - - - - Nevertheless both yield similar y-distr. !!! \B Problems:\\ - - Perfect one fluid hydro $\sim$ stopping\\ - - - - However, $\exists$ Bounce back\\ - - - - \LT central dip in y-distr, imitating transparency !!! }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} \B Experimentally in details:\\ - - S+S $S+S$ collisions at 200 \agev\ \\ - - to quantify the stopped energy $<$ energy loss per nucleon $>$\\ - - - - from the rapidity and $p_t$ \\ - - - - Protons in centr. : $ = 5.8 \pm 0.3$ GeV $\sim$ stopping\\ - - - - Protons in perif. : $ = 4.8 \pm 0.3$ GeV $\sim$ NN data\\ - - - - Total energy deposition of $54 \times 5.8 = 313$ GeV in centrali c.\\ = However, negative multipl. per nucleon pair goes up by 10\% only\\ - - - - when comparing central S+S with N+N data.\\ - - - - mean transverse momentum is unchanged.\\ \B \LT conclusion: only small fraction of energy loss is in the pions,\\ - - - - both with regard to their number and their $$. \vspace*{9cm} Figure 10.3 {\it The ratio of the rapidity densities in central $^{32}S+^{32}S$ and $p+p$ collisions. From ref. [48]. } }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.2cm} In Fig. 10.3 the ratio $ dN/dy (S + S)/dN/dy (p + p)$ is shown for negatives, $K^0_s$, and $\Lambda$ particles as a function of rapidity. \B Difference in width --- inverse parabolic shape of the ratio.\\ \B Ratio($\pi$) is 30.6, 12\% more than ratio of charges (27)\\ \B Enhanced strangeness production: 60 and 55 for $\Lambda$ and $K_s^0$,\\ \B K enhancement at mid-rapidity. \B\B Energy loss of the nucleons \LT strange particles near mid-rapidity. - - - - \& baryon-antibaryon production\\ - - - - (an enhanced production of di-quark pairs) Hence, nuclear transparency turned out to be not so important up to now. Owing to the large stopping power of nuclear matter, large baryon densities are observed even in Sulphur - Sulphur collisions at the SPS. \paragraph{EXPECTATIONS FOR HIGHER ENERGIES} - \\ \B in p-A collisions at A$\approx$200 the proton projectile loses\\ - - - - approximately 2 - 2.5 units in rapidity, $\delta y \approx 2 - 2.5$.\\ \LT heavy ions: maximal stopping for $\Delta y \approx 4 -5$. \B In c.m.: projectile at $Y_0$, target at $-Y_0$.\\ - - - - After collision: baryons at $\pm (Y_0 \mp \delta y)$,\\ - - - - - - $\delta y$ is the rapidity loss. \\ \LT baryon free region: $\Delta Y_{baryon \ free} \approx 2Y_0 - 4 \delta y$.\\ - - - (assumed: spread of decelerated baryons $\sim$ deceleration.) \B \LT baryon free region of y = 1 - 2 at the BNL-RHIC and\\ - - - - - - - - - - - - - 6 - 8 at the CERN-LHC.\\ If $\delta y > 2.5$ \LT baryon free region at LHC only. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Stopping in theoretical models} \B One fluid dynamical models assume complete stopping\\ - - Landau model and the spherical model\\ \B Bjorken model is also a one-fluid modela\\, - - however, the initial condition: from nonequilibrium stage\\ - - - - where the projectile and target baryons penetrated each other.\\ \B Two or more component fluid dynamical models, and explicitly microscopic cascade or molecular dynamics models are able to describe transparency explicitly. \paragraph{STRING MODELS, stopping:} - \\ \B cascade models on the quark parton level\\ - - - - - applicable at CERN-SPS and BNL-AGS energies.\\ - - Soft quark-quark collisions: ``string'' formation and hadronization. \B Problems: String density: 10/ fm$^2$ in CERN-SPS lead beam experiments\\ - - - - - independent strings ????? (string fusion) Experiments do demonstrate rather nice stopping (even at CERN energies) in terms of nucleon energy loss. Pion rapidity distribution is surprisingly close to predictions of Landau hydrodynamics. Does it really mean, that there is rapid thermalization? }%end tr-page \newpage % transparency ============================================ page 34 \transparencyframe{ \vspace*{-1.2cm} \section{Reaction models} \B Rel. HI collisions: borderline of applicability of theoretical approaches\\ - - the smaller than macroscopic size and\\ - - the short reaction time compared to microscopic equilibration. \B Primary goal: conclusions on global features of the hot and dense matter\\ \LT - largest colliding systems are the most favorable\\ \B Expectations:\\ - - - Final particle multiplicity is in the order of several thousand.\\ - - - during the reaction local thermal and mechanical eq. are established\\ - - - - - - (not necessarily phase or chemical equilibrium)\\ - - - for local volume elements of the size of $\approx$ 10-100 fm$^3$\\ \B \LT complicated --- QCD is beyond our possibilities \B\B Reaction models --- two groups: - - - 1) assume the local equilibrium at some stage of the reaction\\ - - - - - - (not necessarily at the initial moments)\\ - - - - - - appropriate boundarty and initial conditions\\ - - - - - - \B Fluid dynamical models and continuum models. - - - 2) No local equilibrium, but microscopic dynamics\\ - - - - - - interactions, transitions, internal dynamics, etc.\\ - - - - - - Transport theoretical behaviour of these systems is studied\\ - - - - - - usually, by Monte-Carlo simulation. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Fluid dynamical results} \B Simple models:\\ - - Bjorken model, Landau model\\ - - - - Initial conditions are different\\ - - - - Bjorken model corresponds to a later physical situation \B Experiments at the CERN-SPS and at BNL-AGS resemble Landau model\\ - - - - - stopping is large\\ - - - - - frequent use of Bjorken's formula for the initial energy density at the present energies is not justified \B Simple models: successfull for transverse momentum spectra\\ - - - assuming a collective fluid dynamical expansion which is spherically and/or cylindrically symmetric }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsubsection{Detailed numerical models} \B 3-Dim detailed fluid dynamical models\\ --- applied to ultra-relativistic HI reactions\\ - - - Problem:\\ - - - proper initial state and the selection of EOS\\ \B Allows the study of collective transverse flow \B Example: $Pb+Pb$ collision at 160 \agev\ \\ - - - EOS including a strong first order phase transition to QGP!\\ - - - Initial state: moment of impact of Lorentz contracted nuclei\\ - - - Assuming immediate local equilibrium\\ \B \LT high temperature ($T>400$MeV) and density 0.5 fm/c\\ - - - By 1.5 fm/c the central region is over the maximum compression,\\ - - - the temperature starts to decrease,\\ - - - but most of the central region is still in the QGP phase ($T>T_{cr}$). }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{14cm} Figure 10.4 {\it Contour lines of the temperature distribution, T [MeV], in the reaction plane of a Pb+Pb reaction of impact parameter b=4fm, at 160 \agev\ beam energy at c.m. times 0.34, 0.68, 1.03, 1.37, 1.72 fm/c. The figures are distorted for better recognizability, the size of the frame in the beam direction is 5 fm, while in the transverse direction is 20 fm. From ref. [58]. } \B i) Development of the collective sidewards flow, starting around 1fm/c\\ \B ii) Final hadronization and freeze-out between 5-50 fm/c (Involved)\\ \B Compared to string models,\\ - - - transverse flow is developing earlier and it is stronger }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Microscopic string models} \B Important basic differences from low energy Monte Carlo INC, VUU, ..\\ - - - lower energies: point like or compact particles \paragraph{HADRON - HADRON REACTIONS:} --\\ - - low energy picture is not applicable for $p+p$ (100 \agev\ or more)\\ - - collisions are far from being binary and\\ - - large number of reaction products are created\\ - - - - transverse momentum spread of secondaries is limited\\ - - - - beam directed distribution is widely spread \\ - - - - nearly flat distribution in rapidity.\\ \B\B \LT Introduction of composite intermediate objects, so called\\ - - - - - - STRINGS or flux tubes (internal structure and space-time dyn.) \paragraph{SRINGS:} composite objects \\ - - - strings or flux tubes spanned between two (groups of) quarks and\\ - - - {\em chromo-electric field} between them\\ - - - secondary hadrons are formed later by pair creation from the field \B\B Thus, microscopic Monte-Carlo models have new constituents!!\\ - - Quite succesfull for all hadron-hadron collisions.\\ - - (The number of strings in a collision was very low and the system was a dilute system of these constituents.) }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.4cm} \paragraph{HEAVY ION REACTIONS:} problems became obvious in a short time.\\ - - The density of constituents became so large that\\ - - interactions among them became important. \B Part of these string models which allowed for\\ - - - interactions of secondary hadrons, \&secondary string formations\\ - \LT reproduced data on rapidity and $p_\perp$ distributions of $p,\ \pi,\ K, ..$\\ - - - (without secondary scattering and string formation underpredicted the observed stopping power, i.e., they failed to describe the proton rapidity distributions) \B By 1991- 1992:\\ - i) string-string interactions and string fusion are necessary\\ - - - - strange antibaryons, because\\ - ii) the inclusion of hard partons is necessary to describe high $p_\perp$ phenomena, minijets, gluon jets, particularly at highly ultra-relativistic energies. [59-61] }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.4cm} \subsubsection{Monte-Carlo model families:} \paragraph{A) Naive string models:} extrapolations of hadron-hadron models\\ - - - supplemented by the baryon distributions in projectile and target\\ - - - widely used by experimentalists to simulate data, for design etc.\\ - - - e.g. FRITIOF model (based on the Lund model) \paragraph{B) String models with rescattering:} more realistic models\\ - - - string formation in prim. \& scnd. h+h collisions are identical\\ - - - e.g. RQMD, VENUS, QGSM, ARC, DPM, etc.\\ - - - nice: No additional ad hoc assumptions are involved, just $h+h$\\ \LT closely identical results !!!\\ - Problems: massive heavy ion collisions like $Au+Au$ or $Pb+Pb$,\\ - - - at enegies 200 \agev\ or higher\\ - - - string density becomes unrealistically high [72] \paragraph{C) String models with string fusion:} remedy to the problems\\ - - - fusion of strings or the formation of composite objects\\ - - - e.g. RQMD versions with string fusion, VENUS versions with double strings, [75] String Fusion Model (SFM), [76] etc. \paragraph{D) Parton cascade models:} for higher energies, (RHIC, LHC)\\ - - abandon the idea of strings or flux-tubes alltogether\\ - - Parton cascade is happening in the perturbative vacuum. [77]\\ - - P.: transition from the physical vacuum to the perturbative one\\ - - P.: the final hadronization and return to the physical vacuum\\ - - - - test of the models: when RHIC and LHC }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsubsection{Monte-Carlo models and Quark Gluon Plasma} \B The sequence of the previously mentioned models\\ - illustrates a gradual approach to QGP.\\ - The objects (strings) are becoming larger $\sim$ bigger and bigger chunks of non-equilibrated quark matter. \B\B String models should have a stationary equilibrium solution:\\ - - \LT an Equation of State. \B EOS for any of these Monte-Carlo models:\\ - - - model prescriptions should be applied to a large fixed container\\ - - - Unfortunately this is not done in any of the models up to now !!! Nevertheless, it cannot be excluded that these string models have a first order phase transition in their EOS, since the strings have a substantial amount of energy, which is latent, i.e. it is an internal energy and it is subtracted from the kinetic energy of the constituents leading to the pressure of the material. }%end tr-page \newpage % transparency ========================================== page 42 \transparencyframe{ \vspace*{-1.4cm} \section{On some suggested signals} \B A) Primordial remnants in observed hadron features:\\ - - strangeness enhancement, which is significantly larger for QGP\\ - - discontinuities in momentum distribution (excitation f.)\\ - - \LT first order phase transition \B B) Signals produced early, not affected by the subsequent hadronization: - - thermal dileptons and thermal photons,\\ - - - - which are emitted by the plasma and then escape.\\ - - heavy quark bound states, like $J/\Psi$ suppression or hard jets\\ - - - - effect of the produced dense medium on these particles. In the moment it is impossible to give a complete review of quark-gluon plasms signatures here. However, one can group the quark gluon plasma signatures in the following categories: \begin{itemize} \item Transverse flow and thermodynamic variables measuring the EOS. \item Strangeness and Anti-baryon enhancement. \item Strangelets and other "exotic" signatures of the quark-gluon plasma. \item Photons, and lepton pairs. \item $J/\Psi$ suppression. \end{itemize} }%end tr-page \newpage % transparency ========================================= page 43 \transparencyframe{ \vspace*{-1.2cm} \subsubsection{Collective flow, $p_\perp$ spectra, thermodynamic variables} \B The aim of this group: EOS and thermodynamic parameters\\ - These quantities would exhibit a discontinuity for a first-order ph.Tr.\\ - - - (However, we face small, dynamic systems.)\\ - - - In real heavy ion collisions, we may expect a steep, continuous rise even if the phase transition is 1st order because only a part of the matter is converted into plasma, and this part will increase with energy. \paragraph{Transverse flow}: \\ - Is the most obvious to measure the EOS\\ - - - when one approaches the ph.tr. the EOS becomes soft\\ - \LT only small increase of the transverse flow velocity is expected.\\ - - - Only when the energy density significantly exceeds QGP threshold\\ - - - collective flow is expected to increase noticeably again. Calculations of hydrodynamical expansion with bag-model EOS yield:\\ - three stages of rapid, modest and again rapid transverse flow increase\\ - - - with the increase of beam or internal energy.\\ - The existence of some ``plateau'' in the middle\\ - - - is the consequence of softness of EOS in the ``mixed phase''.\\ Detailed numerical studies in the context of the hydrodynamical model have shown that this characteristic feature is rather weak in realistic models, [49] unless rehadronization occurs like an explosive process. [27,29,78] FERMILAB: The shape of the observed multiplicity dependence was similar to hydrodynamical calculations. L\'evai and M\"uller analysed these data in terms of a simple model, [80] and found stong indication of QGP EOS. }%end tr-page \newpage % transparency ========================================= page 43 \transparencyframe{ \vspace*{-1.0cm} \paragraph{The shape of the pion spectra}: \\ \B at CERN\\ - - - two exponentialss, with slopes ($T_{eff}$) of 50 \& 200 MeV,\\ - - - while $K_s^0, \ \rho$ or $\Lambda$ show the higher slope only. \B \LT suggests a lower break-up temperature, \& hydrodynamical expansion \B Monte Carlo Cascade or Srting model calculations\\ \LT contribution of the particles evaporated in the whole process and\\ - - low $P_\perp$ enhancement is due to resonance decays \B Thus transverse momentum spectra at a single beam energy do not provide sufficient evidence for a plasma- or even a collective flow signal. \B HBT:\\ - space-time dynamics needs independent confirmation\\ - - - e.g. by particle interferometry. Transverse sizes found in heavy ion collisions are larger than the radii of incident nuclei, indicating that produced hadrons rescatter before breakup. Interferometric size determinations will be possible on an, event-by-event basis when Pb or Au beams become available, and can be correlated with global parameters like $p_\perp$ and $dN/dy$. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Strangeness and Anti-baryon enhancement} \B Due to the reduction of the production threshold in GGP\\ - - - for strange hadrons from $\approx$700 to $\approx$300 MeV\\ - - - for baryon-antibaryon pairs from $\approx$2 GeV to almost zero. \\ \B\B \LT Strongest signal for strange antibaryons\\ - - - which combine both effects [82-84]. Enhanced strange quark production in deconfined QGP \LT chemical eq.\\ - - strangeness abundance for hadronic matter in ch.eq. is smaller. \B In a rapid hadronization QGP strangeness abundance is conserved\\ - \LT larger than the hadronic equilibrium abundance at the breakup.\\ - - (Slow hadronization and long expansion in hadronic phase before breakup will reduce the strangeness abundance to the hadronic eq. value.) $\exists$ alternative suggestions: strange particles, and especially antibaryons, would be produced more abundantly, if their masses would be reduced in dense hadronic matter due to medium effects without QGP. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-0.4cm} Strangeness production has been measured:\\ - - Ratios like $K/\pi$ and inclusive $\varphi$, $\Lambda$ and $\bar{\Lambda}$ production turned out to be a factor 2-3 larger than the values in $p+p$ interaction at the same energy. Also a strong increase in multi-strange baryon production WA85 CERN, [85]\\ at mid-rapidity and for momenta $p_T > 1$GeV/c: \beq \frac{ \bar{\Lambda}}{ \Lambda } = 0.13 \pm 0.0 3, \ \ \frac{ \bar{\Xi} }{ \Xi} = 0.39 \pm 0.07, \ \ \frac{ \Xi }{ \bar{\Lambda} } = 0.6 \pm 0.2, \ \ \frac{ \Xi }{ \Lambda } = 0.2 \pm 0.4 . \eeq{wa85} \B can be explained either by QGP or hadronic gas with\\ - - - $T = 220$MeV and $\mu_B = 340$MeV.\\ * It is, however, difficult to imagine chemical eq. during the short life\\ - - - of a hadronic fireball. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Heavy quark bound states} $J/\Psi$ particles decay electromagnetically into observable muon pairs. A suppression of the $J/\Psi$ signal\\ - - - relative to the Drell-Yan di-muon continuumi\\ - - - had been predicted as a signature for deconfinement,\\ - - - based on Debye screening between color charges. The ground state of the ($c\bar{c}$) pair does not exist if the color screening length, $\lambda_D = 1/gT$ is less than the ground state radius. Lattice simulations of SU(3) gauge theory show that this condition should be satisfied slightly above the deconfinement temperature (see sect. 10.3.5). }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.6cm} \subsection{Electromagnetic probes} \paragraph{Photons} : \\ QGP emits thermal photons.\\ - - - the main background at low momenta comes from the decay of hadrons, mainly $\pi^0$ and $\eta$ of high momenta. - - - in addition, from direct photons from Compton scattering. No clear signal has been seen within experimental sensitivities so far. \paragraph{Di-leptons} : \\ - - produced when $\pi^+ \pi^-$ or $q\bar{q}$ pairs annihilate\\ - - - - in a hot pion or quark gas, respectively.\\ - - produced also in decay of low mass vector mesons, $\rho$, $\omega$, $\varphi$\\ - - and in hard interactions between incident partons at early stages\\ - - - - leading to so called Drell-Yan pairs or\\ - - - - to the production of heavy ($c\bar{c}$ or $b\bar{b}$) vector mesons\\ - - - - - - which subsequently decay into lepton pairs. \B $\exists$ competition btwn thermal di-leptons and Drell-Yan production\\ - - - - thermal di-leptons and Drell-Yan pairs have different\\ - - - - - - functional dependencies on di-lepton pair mass, M.\\ - - - - - - \LT distinguishable. \B $\exists$ clear-cut window to see $J/\Psi$ via high mass dileptons, between resonance decays and Drell-Yan production. \B Detection of low mass thermal dileptons, from $\pi^+ \pi^-$ annihilation in a pion gas or from $q\bar{q}$ annihilation in a quark gas, is not so easy,\\ - - large backgrounds from $\pi^0$ and $\eta$ decays and from virtual Brems\-strahl\-ung must be subtracted. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \vspace*{-1.2cm} \subsection{Exotic signals} Most probable exotic objects formed in QGP are strangelets [87]. \B metastable objects with baryon number $A > 2$\\ - - - that contain several strange quarks. \B The simplest such object is the strangeness $S = -2$ dibaryon\\ - - the H-particle,\\ which is predicted to be metastable in the original MIT bag model and might be produced in relativistic nuclear collisions. Experiments searching for strangelets produced in relativistic heavy ion reactions are in progress at BNL, and in preparation at CERN [88]. }%end tr-page \newpage % transparency ===================================================== \transparencyframe{ \section{Assignment 10} \begin{description} \item[10.a] Show that in the Bjorken model the local flow velocity, $u^\mu = {1 \over \tau} (t, 0, 0, z)$, is orthogonal to the $\tau =$const. hyperbola. \end{description} }%end tr-page % transparency ================================================================ \end{document}