% % 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 7 / Chapter 7 % % 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. References and Solutions of the assignments are % also not included! % \documentstyle[12pt]{book} % Transparencies in frame keretben \textheight=21.5cm \textwidth=19cm \oddsidemargin=10mm \evensidemargin=10mm %\voffset=-0.50cm \hoffset=-0.15cm % \newcommand{\beq}{\begin{equation}} \newcommand{\beqar}{\begin{eqnarray}} \newcommand{\eeq}[1]{\label{#1} \end{equation}} \newcommand{\eeqar}[1]{\label{#1} \end{eqnarray}} %\newcommand{\beq}{$$} %\newcommand{\eeq}{$$} \newcommand{\B}{$\bullet$ } \newcommand{\LT}{$\leadsto$ } %\renewcommand{\vec}[1]{{\bf #1}} \newcommand{\D}{\bigtriangledown} \newcommand{\etal}{{\it et al.,\/}} \newcommand{\lton}{$\raisebox{-.54ex}{$\stackrel{<}{\sim}$}$} \newcommand{\gton}{$\raisebox{-.54ex}{$\stackrel{>}{\sim}$}$} \newcommand{\gfm}{{\sf GeV/fm}^3} \newcommand{\ipint}{\int \frac{d^3 p}{p^0}} \newcommand{\dpi}{ \frac{d^3 p_i}{p_i^0}} \newcommand{\dpj}{ \frac{d^3 p_j}{p_j^0}} \newcommand{\dpk}{ \frac{d^3 p_k}{p_k^0}} \newcommand{\dpl}{ \frac{d^3 p_l}{p_l^0}} \newcommand{\dpkp}{ \frac{d^3 p'_k}{p^{'0}_k}} \newcommand{\dplp}{ \frac{d^3 p'_l}{p^{'0}_l}} \newcommand{\ip}{\int^{\infty}_{-\infty}dp} \newcommand{\pint}{\int d^3 p} \newcommand{\juttner}{ {1\over{(2 \pi \hbar)^3}} \exp \left( {{\mu - p^\mu u_\mu}\over{T}} \right) } \newcommand{\jut}{ e^{ \left( - {{ p^\mu u_\mu}\over{T}} \right) }} \newcommand{\pmum}{p^\mu u_\mu} \newcommand{\amev}{A$\cdot$MeV} \newcommand{\agev}{A$\cdot$GeV} \newcommand{\transparencyframe}[1]{ \phantom{a} \setlength{\unitlength}{1mm} \begin{picture}(10,10)(30,200) \thicklines \put(95,240){\oval(188,18)[t]} \put(95,20){\oval(188,18)[b]} \put(1,20){\line(0,1){220}} \put(189,20){\line(0,1){220}} \thinlines \put(95,240){\oval(186,16)[t]} \put(95,20){\oval(186,16)[b]} \put(2,20){\line(0,1){220}} \put(188,20){\line(0,1){220}} \put(95,8){\scriptsize\sf L.P. 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} %\setlength{\unitlength}{1mm} %}%end tr-page %\newpage % transparency ================================================ p.1 \transparencyframe{ \setcounter{chapter}{6} \vspace*{-1.99cm} \chapter{Measurables} \section{The freeze out process} \B Microscopic models (simulate collisions like event generator) \LT\\ - the evaluation of the measurables: same as in experiments. \B Continuum problems: there are principal difficulties!\\ - when the particles reach the detectors they do not interact\\ - $\exists$ a process: strongly interacting continuum of the matter\\ - \ \ \ \ \ $\longrightarrow$ dilute matter of independent particles\\ - This gradual process handled in FD via source and drain terms [1]. \B Freeze out process - most usually treated as sudden freeze out:\\ - At a given instant in the space-time\\ - \ \ \ \ the constituents of the continuum will become independent,\\ - \ \ \ \ and final interactions and collisions are then neglected. }%end tr-page \newpage % transparency ================================================ p. 2 \transparencyframe{ \vspace*{-0.6cm} \B $\exists$ different levels of possible sophistication:\\ - The sudden freeze out is a hypersurface in the space-time.\\ - The matter flows according FD until this surface is reached.\\ - This surface can be defined by using FD quantities, e.g. $n,\ e,\ T$ 1) \B Simplest approximation: particles follow the flow velocities\\ \LT measurables (disregarding the random thermal motion) (usd. in 70's).\\ - Justified only if the break up happens late\\ - \ \ \ \ \ when the thermal velocity and energy, \& pressure are small.\\ - The break up, particularly in a small system can happen earlier. 2) \B Frequently used method: consider thermal velocities at the break up\\ - random thermal velocities are added to the collective flow velocities\\ \LT measurables. \setlength{\baselineskip}{8pt} {\small\sf This procedure is based on the J\"uttner distribution and in the recent times it was used by Milekhin [2] to calculate measurables in Landau's fluid dynamical model for hadron-hadron collisions. In this chapter we will introduce this approximation. Although this procedure is already taking into account a major part of the neglected thermal energy it is still not exact.} }%end tr-page \newpage % transparency ============================================ p. 3 \transparencyframe{ \vspace*{-1.2cm} \subsection{Formal treatment of the freeze out} \B In fact the freeze out across a hypersurface is a discontinuity,\\ - where the EOS of the matter changes to an ideal gas EOS.\\ \LT The energy - momentum tensor changes discontinuously across this surface. \B If the flow is not orthogonal to this surface\\ \LT \ \ \ \ a CHANGE OF THE FLOW. The importance of this difference between the flow direction and the normal to the freeze out surface was first realized by Cooper and Fry [3]. Many most recent treoretical introductions and works are not aware of this problem and ignoring it. [4,5,etc.] The freeze out discontinuity is a time-like surface in most cases, and so the methods introduced in Chapter 5 for the description of time-like detonations and deflagrations should be used. \B Hence the flow velocity may also be discontinuous\\ - when the matter flows across this surface.\\ (If the pressure at the breakup is small or if the flow is orthogonal to the freeze out surface \LT the effect is negligible.)\\ - A qualitative estimate for this effect is given in ref. [7]. \B The freeze out surface in the space-time is a 3-dim. hyper-surface, $S$,\\ - \ \ \ \ \ \ \ \ with normal vector, $d\sigma^\mu$.\\ - The surface is not a closed surface,\\ - - but it crosses the world-lines of all particles.\\ \B The particles after crossing this surface are frozen out,\\ - i.e. their energies and momenta will not change\\ - \ \ \ (except due to final decays or due to final Coulomb interaction.) }%end tr-page \newpage % transparency ============================================= p. 4 \transparencyframe{ \vspace*{-0.8cm} \B The particles always propagate along time-like paths\\ \LT most of the time (but not always) the freeze out surface \\ . \ \ \ is a time-like surface with a time-like normal vector, $d\sigma^\mu$. \B The invariant number of particles crossing this surface:\\ - \ \ \ \ \ \ \ \ \ \ $dN = N^\mu \ d\sigma_\mu$ \ \ \ \ \ and\\ - - the total number of all the particles crossing this surface is \beq N = \int_S N^\mu \ d\sigma_\mu \ . \eeq{efo1} This total number, $N$, and the total energy and momentum\\ - are of course the same at both sides of the freeze out surface.\\ \B However, the four-current, $N_\mu$, and energy momentum tensor, $T^{\mu\nu}$,\\ - \ \ \ \ \ \ \ \ \ are generally DISCONTINUOUS! This is easy to see, since\\ 1) after the freeze out we consider an ideal gas\\ - \ \ \ interactions are neglected, with an ideal gas EOS. \\ 2) the matter in the flow is described by an EOS where\\ - \ \ \ interactions are included, i.e., by another EOS. \\ 3) Unless the EOS is an ideal gas EOS on both sides of the surface\\ \B\B\B \LT \ \ \ \ \ \ \ \ \ \ \ $N^\mu$ and $T^{\mu\nu}$ are discontinuous. \B Therefore to evaluate the MEASURABLES \\ - \ \ \ we have to use the parameters of the matter after the disconlinuity! The proper calculation of $N^\mu$ and $T^{\mu\nu}$ after such a discontinuity can be performed (see sect. 5.5) if:\\ - the surface, $S$, is known and\\ - the pre freeze out quantities $N^\mu_0$ and $T^{\mu\nu}_0$ are known. }%end tr-page \newpage % transparency ================================================ p. 5 \transparencyframe{ \vspace*{-1.2cm} In fact the correct {\bf determination of the freeze out surface} is an involved problem. In most cases this surface is not obtained as the sulution of the dynamical problem but it is prescribed directly, or defined by requiring some condition(s) to be satisfied. If the normal of the freeze out surface is not identical to the flow velocity\\ \LT the momentum of the matter changes du to the freeze out [7].\\ Even if the two vectors are identical and just the EOS changes,\\ the parameters of the post freeze out matter ($T,\ e$) will be different,\\ - - due to the fact that the energy of the interactions before $t_{BU}$\\ - - should be added tot the kinetic energy of the particles after.\\ (This temperature change can be neglected only if the freeze out happens\\ when all interaction energies are negligibly small already.) Let us assume that we know the post freeze out quantities, $N^\mu$ and $T^{\mu\nu}$,\\ from the relations (see sect. 5.5): \beq [N^\mu\ d\sigma_\mu] = 0 \ \ \ \ {\sf and}\ \ [T^{\mu\nu}\ d\sigma_\mu] = 0 . \eeq{efo2} Then for an ideal gas we know that \beq N^\mu = \ipint p^\mu \ f_0(x,p) \ , \eeq{efo3} where $f_0$ is the phase space distribution of the ideal gas including the collective flow (see e.g. sect. 2.4). The the total number of particles frozen out can be calculated via eq. (\ref{efo1}). }%end tr-page \newpage % transparency ================================================ p. 6 \transparencyframe{ \vspace*{-0.1cm} \B The experimental measurables are usually differential quantities\\ - (i.e differential cross sections or double differential cross sections)\\ - normalized per event (not per unix incoming flux). E.g.: $$ \frac{dN}{d^3p} $$ satisfying the normalization \beq N = \int_S \frac{dN}{d^3p} d^3p = \int_S \left( \ipint p^\mu \ f_0(x,p)\right) \ d\sigma_\mu = \int_S n(x) \ u^\mu d\sigma_\mu \ . \eeq{efo4} \B Similarly, some component of overall momentum of emitted particles\\ is frequently measured, e.g.: transverse momentum in reaction plane $p^x$ \beq p^x_{tot} = \int_S \frac{dN}{d^3p}p^x\ d^3p = \int_S \left( \ipint p^\mu p^x \ f_0(x,p)\right) \ d\sigma_\mu = \int_S \ T^{\mu x} d\sigma_\mu \ . \eeq{efo5} \B If we are interested in differential quantities\\ - we use relations based on eqs. (\ref{efo4},\ref{efo5}),\\ - where not all the integrals, $dp$, are performed. (We will see examples.) }%end tr-page \newpage % transparency ================================================= p. 7 \transparencyframe{ \vspace*{-1.2cm} \B Frequent simplifying assumption: \\ - the normal of the freeze out surface is pallel to the flow, $u^\mu=d\sigma^\mu$.\\ \LT Simplifies eqs. (\ref{efo4},\ref{efo5}), and\\ - - reduces complications in evaluating $N^\mu$ and $T^{\mu\nu}$ after BU.\\ - - only the energy conservation across the break up surface \\ - - \LT modified freeze out temperature. \B In numerical models with a calculational grid of a finite resolution\\ - - this approximation cannot be done exactly: \\ - - - cells are freezing out at discrete times and they have finite sizes.\\ \B \LT a ragged surface in a way that in each freezing out cell\\ - - the normal of the freeze out surface is parallel to the flow.\\ \smallskip * The surfaces of the neighbouring freeze out cells are connected\\ - - by connecting surfaces which are parallel to the flow\\ - - (\LT there is no matter flux across these connecting surfaces).\\ a) - In this situation the integral over the cell's freeze out hypersufrface\\ - - yields the proper volume of the cell, $\gamma V_{cell}$.\\ b) - The connecting surfaces parallel to the flow do not contribute. \B In analytic- or spherically- or cylindrically symmetric models\\ - the assumed freeze out surface frequently satisfies\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ $u^\mu = d\sigma^\mu$,\\ \LT extra complications in evaluating final quantities do not appear.\\ - We will study examples where the $u^\mu = d\sigma^\mu$ approximation is used. \setlength{\baselineskip}{8pt} {\small\sf In sections 6.2 and 6.3 we evaluate some of the most common measurable quantities used in the relativistic and ultra-relativistic heavy ion collisions. In sections 6.4 and 6.5 quantities frequently used in intermediate energy reactions are discussed, while measurables directly connected with the collective motion are presented in sections 6.6 and 6.7. } }%end tr-page \newpage % transparency ============================================ p. 8 \transparencyframe{ \vspace*{-1.2cm} \section{Baryon measurables} \B Assume that the local baryon momentum distribution at point $x$\\ - is a J\"uttner distribution $f(x,p)$ (after BU).\\ .\ \ (This is a reasonable assumption if the flow velocities are\\ .\ \ relativistic and $T$ is high (e.g., a few hundred MeV), but $T \ll m$).\\ \LT the approximation holds for nucleons and heavy baryons. \B Introduce a parametrization of the flow velocity $u^\mu$: $$ u^\mu = \gamma (1, v_\|, \vec{v}_\perp) = \gamma_\perp (\cosh(y_0), \sinh(y_0), \vec{V}_\perp) , $$ where $y_0$ is the rapidity of the fluid cell. \ \ \ \ \LT\\ $v_\|=\tanh(y_0)$,\ \ \ $\vec{V}_\perp= \vec{v}_\perp / \sqrt{1-v_\|^2}$, \&\\ $\gamma_\perp^2 = (1-v_\|^2)/(1-v_\|^2 - v_\perp^2)$ $=$ $1/(V_\perp^2)$. \B Using the form of $u^\mu$ above, the $\pmum$ product takes a form: $$ E=p^0_{(LR)}= \pmum = \gamma_\perp \left(m_\perp \cosh(y-y_0) - \vec{p}_\perp \vec{V}_\perp \right) , $$ where $y$, $\vec{p}_\perp$ and $m_\perp$ are the rapidity, transverse momentum and transverse mass of a particle of 4-momentum $p^\mu$. Based on ref. [8] we review briefly the different projections. }%end tr-page \newpage % transparency ================================================ p. 9 \transparencyframe{ \vspace*{-1.4cm} \subsection{Rapidity distribution} We can split up the phase space integrals by observing that\\ $ d^3p = dp_{||} d^2p_\perp = p^0 dy \ d^2p_\perp$.\ \ \ \ \ \ \LT $$ {{dN_{cell}}\over{dy}} = \gamma V_{cell} \ \ipint \ {d \over{dy}} \ \left[ \pmum \ f(x,p) \right]. $$ Performing the integrals yields \beq \begin{array}{rcl} {{ dN_{cell}}\over{dy}} & = & {{ \gamma V_{cell} \ g_N}\over{(2\pi \hbar)^3}} \exp({{\mu}\over T}) \ 2 \pi T m^2 \ \sqrt{2\over \pi} \ \sqrt{h m} \times \\ & \sum_{k=0}^{\infty} & \left( {{g^2 m}\over{2h}} \right)^k \ {1 \over{k!}} \left[ \left(1-({ g\over h})^2 \right) K_{k+{5 \over 2}}(hm) \ - \ {1 \over {hm}} K_{k+{3 \over 2}}(hm) \right] , \end{array} \eeq{rap.1} where $V_{cell}$ is the volume of the fluid cell, $g_N$ is the degeneracy of nucleons ($g_N = 4$), $h = \gamma [\cosh(y)-v_\| \sinh(y)]/T$, and $g = \gamma v_\perp/T$. \B If $m g^2 \ll h$ and $ g \ll h$ this expression reduces to $$ {{ dN_{cell}}\over{dy}} = {{ \gamma V_{cell} \ g_N}\over{(2\pi \hbar)^3}} \exp({{\mu}\over T}) \ 2 \pi T m^2 \ \left[1 + {2\over{h m}} + {2\over{(h m)^2}} \right] \ exp (-h m). $$ Integrating this over $y$ yields the baryon number in the fluid cell. \B The final baryon rapidity distribution from all cells is then $$ {{dN}\over{dy}} = \sum_{cell} {{dN_{cell}}\over{dy}}. $$ }%end tr-page \newpage % transparency ============================================ p. 10 \transparencyframe{ \vspace*{-1.2cm} \subsection{Transverse Momentum Spectra} \B Usual quantity measured experimentally. The contribution of a fluid cell to the final baryon transverse momentum distribution is: $$ {{dN_{cell}}\over{p_\perp dp_\perp}} = \gamma V_{cell} \ \ipint \ {d \over{p_\perp dp_\perp}} \ \left[ \pmum \ f(x,p) \right]. $$ Performing the integrals yields \beq {{ dN_{cell}}\over{p_\perp dp_\perp}} = {{ \gamma V_{cell} \ g_N}\over{(2\pi \hbar)^3}} \exp({\mu \over T}) \ T \left( a K_1(a) I_0(b) \ - \ b K_0(a) I_1(b) \right) , \eeq{rap.9} where $a = \gamma_\perp m_\perp/T$, and $b = \gamma_\perp (\vec{V}_\perp \vec{p}_\perp ) /T$. }%end tr-page \newpage % transparency ============================================== p. 11 \transparencyframe{ \vspace*{-1.0cm} \subsection{Collective Sidewards Flow} \B A sensitive method to detect the collective sidewards flow is the\\ - \ \ \ \ \ \ \ \ \ {\bf Transverse Momentum Analysis} \ \ [9]. We can evaluate the transverse momentum flow, $$, in relativistic FD also. Since we know the reaction plane exactly \LT\\ the majority of the complications are nonexistent in a theoretical work. Under the same assumptions that were used above,\\ - the contribution of a fluid cell \\ - - to the transverse momentum projected to the reaction plane $(x,z)$\\ in the C.M. system is: \ \ \ \ \ \ \ \ \ \ \ $ p^x_{tot \ cell} = V_{cell} \int d^3p \ p^x f(x,p) $. The yield falling into a unit rapidity interval around y is $$ {{dp^x_{cell}}\over{dy}} = V_{cell} \int d^2p_\perp \ p^0 p^x f(x,p) . $$ This quantity is not an invariant scalar therefore we are not allowed to evaluate it in an arbitrary frame. }%end tr-page \newpage % transparency =============================================== p. 12 \transparencyframe{ \vspace*{-0.6cm} After a straightforward calculation we can arrive at the result $$ {{dp^x_{cell}}\over{dy}} = 2 \pi \ \cosh(y) m^2\ C\ Q\ V_{cell} $$ \beq \sqrt{2\over\pi} \sum_{k=0}^\infty {{g^{2k+1}}\over{2^k k!}} ({m\over h})^{k +{3\over2}} \left( K_{k+{3\over2}}(hm) + {{2(k+2)}\over{hm}} K_{k+{5\over2}}(hm) \right), \eeq{rap.2} i) where\ \ \ \ \ \ $Q=\cos(\phi_R)$\\ - \ \ \ is the cosine of the azimuth angle of the fluid cell\\ - \ \ \ measured from the reaction plane,\\ ii) the constant \ \ \ $C$\\ - \ \ \ is given in terms of the baryon density and cell temperature\\ - \ \ \ as $C = n g_N / [ 4 \pi m^2 T K_2(m/T)]$ $=$ $g_N \exp(\mu /T) / (2 \pi \hbar)^3$. The average transverse momentum per nucleon at rapidity $y$ is then $$ = {{ \sum_{cell} dp^x_{cell}/dy} \over {\sum_{cell} dN_{cell}/dy}}. $$ \vfill }%end tr-page \newpage % transparency =============================================== p. 13 \transparencyframe{ \vspace*{-1.2cm} \subsection{Average Transverse Momentum} \B We can also calculate the average transverse momentum, $$,\\ - which is the magnitude of transverse projection of momentum. Similarly to $$, we define \\ - the rapidity distribution of the transverse momentum in the C.M. frame $$ {{ dp_{\perp cell}}\over{dy}} = V_{cell} \int d^2p_\perp \ p^0 p_\perp f(x,p) . $$ This yields after integrations $$ {{ dp_{\perp cell}}\over{dy}} = 2 \pi \ \cosh(y) C V_{cell} $$ \beq \sum_{k=0}^\infty {{g^{2k}}\over{(2k)!!^2}} (2k+3)!! ({m\over h})^{k+2} \left(K_{k+2}(hm) +{{hm}\over{2k+3}} K_{k+1}(hm) \right). \eeq{rap.4} Thus the average $p_\perp$ per baryon is $$ = {{\sum_{cell} dp_{\perp cell}/dy} \over {\sum_{cell} dN_{cell}/dy}} . $$ }%end tr-page \newpage % transparency ================================================ p. 14 \transparencyframe{ \vspace*{-1.2cm} \section{Pion Measurables} \B Light ptcls. \LT J\"uttner distribution: not applicable\\ -\ \ \ \ \ \ e.g. pions\\ -\ \ \ \ \ \ (very high multiplicity at ultra-rel. energies)\\ \B \LT pion distribution at point $x$: rel. Bose distribution $$ f_\pi(x,p) = {{g_\pi}\over{(2\pi\hbar)^3}} {{1}\over{exp\left({{p^{\mu}U_\mu}\over{T}}\right) - 1}}, $$ where $g_\pi$ is the degeneracy factor of pions ($g_\pi=3$).\\ The total pion number can be obtained from the normalization: $$ n_\pi = u_{\mu} N_\pi^{\mu} = u_\mu \ipint p^\mu f_\pi(x,p) = $$ $$ {{g_\pi}\over{(2\pi\hbar)^3}} \ipint {{\pmum}\over{exp\left({{\pmum}\over{T}}\right) - 1}} . $$ - Using \ \ the power series expansion $$ \frac{1}{\exp(\pmum /T) - 1} = \sum_{k=1}^{\infty} \exp(-k \ \pmum /T) $$ we arrive at $$ n_{\pi} = {{4 \pi g_\pi m_\pi^2 T}\over{(2 \pi \hbar)^3}} \ \sum_{k=1}^{\infty} {{1}\over{k}} \ K_2\left({{k m_\pi}\over{T}}\right). $$ }%end tr-page \newpage % transparency ============================================= p. 15 \transparencyframe{ \vspace*{-0.8cm} In the limit of $m_\pi \rightarrow 0$ the modified Bessel function $$ K_2({{k m_\pi}\over{T}}) \rightarrow 2({{T}\over{k m_\pi}})^2 \ . $$ Using $\sum_{k=1}^{\infty} k^{-3} = \zeta(3)$, we end up with the well known expression for the {\bf Stefan-Boltzmann\/} gas: $$ n_\pi = {{g_\pi \zeta(3) T^3}\over{\pi^2 \hbar^3}}. $$ Here $zeta(x)$ is the Riemann $\zeta$-function, $\zeta(3) = 1.202$. \B \LT local ideal gas momentum distribution for pions.\\ \ \ \ Pion number $\propto$ (LR) cell-volume, \& temperature. RAPIDITY DISTRIBUTION: Using power series expansion: \beq \begin{array}{rcl} {{ dN_{\pi,cell}}\over{dy}} & = {{ \gamma V_{cell} \ g_\pi}\over{(2\pi \hbar)^3}} \ 2 \pi T m_\pi^2 \ \sqrt{2\over \pi} \sum_{j=1}^{\infty} \ \sqrt{{h m_\pi}\over j} \sum_{k=0}^{\infty} \left( {{j \ g^2 m_\pi}\over{2h}} \right)^k \ {1 \over{k!}} \times \\ & \times \left[ \left(1-({ g\over h})^2 \right) K_{k+{5 \over 2}}(jhm_\pi) \ - \ {1 \over {jhm_\pi}} K_{k+{3 \over 2}}(jhm_\pi) \right] . \end{array} \eeq{rap.11} TRANSVERSE MOMENTUM SPECTRUM \beq {{ dN_{\pi,cell}}\over{p_\perp dp_\perp}} = {{ \gamma V_{cell} \ T g_\pi}\over{(2\pi \hbar)^3}} \sum_{j=1}^{\infty} \left( a K_1(ja) I_0(jb) \ - \ b K_0(ja) I_1(jb) \right) . \eeq{rap.19} }%end tr-page \newpage % transparency =============================================== p. 16 \transparencyframe{ \vspace*{-1.2cm} COLLECTIVE SIDEWARDS FLOW \beq \begin{array}{rcl} {{ dp^x_{cell, \pi}}\over{dy}} & = 2 \pi \ \cosh(y) m^2_\pi \sqrt{2\over\pi} \ {{g_\pi V_{cell} \ \cos(\phi_R)} \over {(2 \pi \hbar)^3}} \sum_{j=1}^{\infty} \sum_{k=0}^\infty {{(jg)^{2k+1}}\over{2^k k!}} \times \\ & \times ({{m_\pi}\over {jh}})^{k +{3\over2}} \left( K_{k+{3\over2}}(jhm_\pi) + {{2(k+2)}\over{jhm_\pi}} K_{k+{5\over2}}(jhm_\pi) \right). \end{array} \eeq{rap.12} AVERAGE TRANSVERSE MOMENTUM \beq \begin{array}{rcl} {{ dp_{\perp, cell, \pi}}\over{dy}} = & {{2 \pi \ \cosh(y) V_{cell} g_\pi} \over {(2 \pi \hbar)^3}} \sum_{j=1}^{\infty} \sum_{k=0}^\infty {{(jg)^{2k}}\over{(2k)!!^2}} (2k+3)!! \times \\ & \times ({{m_\pi}\over {jh}})^{k+2} \left(K_{k+2}(jhm) +{{jhm}\over{2k+3}} K_{k+1}(jhm) \right). \end{array} \eeq{rap.14} }%end tr-page \newpage % transparency ============================================== p. 17 \transparencyframe{ \vspace*{-1.2cm} \section{Calculation of cross sections} \subsection{Inclusive and exclusive cross sections} \B 1) Inclusive reactions: all impact parameters\\ \B 2) Exclusive collisions: selected subset; criteria. Triggering for inclusive coll.: might miss few events\\ Triggering for exclusive coll.: usually for impact parameter \bigskip \B Assumed: central collisions \LT highest multiplicities\\ -\ \ \ No. of coll. with $b ???????? [b,b+db]$ $ \propto$ $2\pi b\ db$.\\ \B \LT a given, $Q$\%, of highest multiplicity coll.'s includes:\\ -\ \ \ \ \ impact parameter range, $b ??????? [0, \sqrt{{Q\over{100}} }b_{max}]$. \bigskip \B In reality the cut is not sharp\\ - larger impact parameters might lead to smaller multiplicities\\ - and vice versa, due to random fluctuations. \B Impact parameter distribution vs. multiplicity\\ - can be calculated in INC or MD models,\\ - - because these account for random fluctuations [10] }%end tr-page \newpage % transparency =========================================== p. 18 \transparencyframe{ \vspace*{-1.2cm} \subsection{Double and triple differential cross sections} Frequently measured: energy spectra, differential, double differential and in high multiplicity heavy ion reactions triple differential cross sections:\\ $d\sigma/\ dE$,\phantom{mmmmmmmmmmmmm} $d\sigma/\ d\Theta$,\\ $d^2\sigma/\ dE\ d\Theta$ \ \ \ \ \ \ and\ \ \ \ $d^3\sigma/\ dE\ d\Theta\ d\phi$\ \ \ \ \ respectively. \bigskip \B In low energy nuclear physics: - multiplicity is small,\\ \B \LT No reaction plane identified,\\ \LT Triple dimensional cross section not measured.\\ \LT The cross sections are always averaged over azimuth. \bigskip \B Since 1984 in relativistic HI collisions: reaction plane ID [11]\\ \LT triple differential cross sections, etc. \\ --- Reaction plane ID: based on the sidewards flow. \B How do we calculate cross sections in the FD model ?\\ --- based on transformation properties of momentum distributions. \bigskip (An alternative way would be to generate particles randomly according to these distributions and analyse the obtained set of particles, exactly like it is done in experimens [13].) }%end tr-page \newpage % transparency =========================================== p. 19 \transparencyframe{ \vspace*{-1.2cm} \subsection{Boosting thermal distributions} \B Fluid element at break-up: nucleons explode into all directons\\ ---\ \ \ due to their thermal velocities. Thermal distribution of nucleons inside fluid-cell $i$:\\ rel. Fermi-distribution (for nucleons $g_N=4$): \beq f_i(\vec{p}) d^3p = \frac{4}{(2\pi \hbar)^3} \frac{d^3p}{ \exp\left[ \frac{\sqrt{m^2+p^2} - \mu_{ch,i}}{T_i} \right] + 1 }\ , \eeq{xs4} where $\mu_{ch,i}$ is the chemical potential, from the normalization \beq n_i = \int d^3p \ f_i(\vec{p})\ , \eeq{xs5} and $\sqrt{m^2+p^2} = p^0_i$ is the LR-energy of the emitted ptcle.\\ (\B Here we will use the $p^\mu = p^\mu_{LR}$ notation.)\\ (\B Lab. or c.m. momenta denoted by $P^\mu$ or $q^\mu_{cm}$.)\\ The distribution depends on the energy $p^0_{LR}$ only \LT\\ (seen at J\"uttner distribution, i.e. $p^\mu u_\mu \equiv p^0_{LR} = p^0$.)\\ The number of ptcls, $N_i$, in cell of rest vol., $V_i^{(LR)}$,\ \ is $ N_i = V_i^{(LR)} \ n_i$. * For $T=0$ the distribution is sharply cut off at $p=p_F$,\\ \LT $\mu_{ch,i}^0 = \sqrt{m^2+p_F^2}$ $=$ $ \sqrt{ m^2 + \hbar^2 (\frac{3}{2} \pi^2 n_i)^{2/3} }$.\\ For finite temperature the normalization is more complicated. }%end tr-page \newpage % transparency ============================================ p. 20 \transparencyframe{ \vspace*{-1.2cm} GOAL: obtain momentum distributioni, $F(P)$, in Lab. or c.m.\\ --- $f_i(\vec{p})$ should be Lorentz-transformed from LR to the given frame\\ -\ \ \ \ (with boost velocity, $\beta_i^{flow}=\beta_i$, of the cell $i$.) \B Introduce distribution, $F(P)$, which is not an invariant scalar (!)\\ - but normalized to $n=n_i$ in the Lab.: \beqar n = N^\mu u_\mu &=& \ipint \ \underbrace{\pmum}_{= p^0_{LR}} \ f( \underbrace{\pmum}_{= p^0_{LR}} ) = \nonumber \\ \int d^3P \underbrace{\frac{p^0_{LR}}{P^0} f(p^0_{LR}) }_{\equiv F(P)} &=& \ \ \ \int d^3P \ F(P) \ , \eeqar{xs5a} where $F(P)$ is function of $P$ only\\ --- since the LR momentum, $p^0_{LR}$, in terms of $P$ and the flow velocity is $$ p^0 = p^0_{LR} = \pmum = P^\mu U_\mu^{(i)}\ . $$ Then the distribution function in the Lab.: \beq \frac{dN}{d^3P} = \frac{dN}{P^2\ dP\ d\Omega} = F^{Lab.}_i(\vec{P}) = \frac{p^0(\vec{P})}{P^0} f_i\left(\vec{p}(\vec{P})\right) \ , \eeq{xs6} where $(p^0,\ \vec{p})$ and $(P^0,\ \vec{P})$ are 4-momenta of ptcles in the cell and Lab. frames, connected by the Lorentz-transformaton: $$ p^0 = p^0(\vec{P}) = \gamma_i P^0 - \gamma_i (\vec{\beta}_i \vec{P}) \ , $$ \beq \vec{p} = \vec{p}(\vec{P}) = \vec{P} - \frac{\gamma_i (p^0+P^0)}{1+\gamma_i} \vec{\beta}_i \ , \eeq{xs7} where $\gamma_i = 1/\sqrt{1-\beta_i^2}$.\\ In the 1-dim. longitudinal FD: $(\vec{\beta_i} \vec{P}) $ $=$ $\beta_i \cos(\Theta) \sqrt{(P^0)^2-m^2}$, }%end tr-page \newpage % transparency =========================================== p. 21 \transparencyframe{ \vspace*{-0.8cm} The the differntial cross section $\propto\ F(P)$,\\ --- but quantities frequently measured are\\ -\ \ \ \ \ energies and angles of the emitted particles.\\ \LT coordinate transformations in the momentum space.\\ The volume element is\ \ \ \ $d^3P = P^2 dP \ d\Omega$,\\ -\ \ and since\ \ \ \ \ \ \ $P^0 = \sqrt{P^2+m^2}$\\ -\ \ the differential\ \ $dP^0 = \frac{P}{\sqrt{P^2+m^2}} dP$ $=$ $\frac{P}{P^0} dP$ \LT \beq \left. \frac{d\sigma}{dP^0\ d\Omega} \right|_{cell\ i} \hspace*{-1pt} = P^0 \sqrt{(P^0)^2 - m^2} \ \frac{d\sigma}{d^3P}\ = P^0 \sqrt{(P^0)^2 - m^2} \ \sigma_0 F(\vec{P}) \ , \eeq{xs7a} where $\sigma_0$ is a constant of dimension fm$^2$ (see later).\\ \B\LT From (\ref{xs6}) and (\ref{xs7}) the double differential cross section\\ -\ \ of primary charged particles, $(p + 2d + 2\alpha + ...)$: \beq \frac{d^2 \sigma_{prim}}{dP^0\ d\Omega} = \frac{Z_T + Z_P}{A_T+A_P} \sum_i \frac{4 V_i^{(LR)} \sigma_0}{(2\pi\hbar)^3} \ \frac{p^0_i(\vec{P}) \sqrt{(P^0)^2-m^2}}{e^{ [p^0_i(\vec{P})-\mu_{ch,i}]/T_i} +1} \ , \eeq{xs8} where\\ \B --- $V_i^{(LR)}$ is the volume of the $i$-th cell in LR \\ \B --- $\sigma_0$ is the target area for selected collisions, $\sigma_0 = 4\pi b_{select}^2$, and\\ \B --- $p^0_i(\vec{P})$ is the particle energy in the $i$th cell's frame, \\ -\ \ expressed in terms of the Lab. frame quantities:\\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $p^0_i$ $=$ $\gamma_i \left[ P^0 - \beta_i \cos(\Theta) \sqrt{(P^0)^2-m^2} \right]$.\\ -\ \ For the general case in 3-dim. FD the distribution (\ref{xs8}) is using\\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $p^0_i$ $=$ $\gamma_i \left[ P^0 - (\vec{\beta_i} \vec{P}) \right] $. }%end tr-page \newpage % transparency ============================================== p. 22 \transparencyframe{ \vspace*{-1.2cm} \subsection{Spherical expansion} The fluid is divided into spherical layers\\ each layer $i$ has a radial velocity $\beta_i^R$. \B Contribution of one layer of thickness $\Delta l$:\\ - Summation over fluid cells in the layer: \beq \sum_{i - cells} V_i^{(LR)} \ ... \ \rightarrow R^2 \Delta l \int d\Omega \ ... \ . \eeq{xs7b} Spherical symmetry \LT cross section: spherically symmetric - \LT \\ - We can choose one direction, the z-axis, to calculate $F(P)$\\ - - at fixed polar angle, $\Theta$, \\ - - the contribution of all azimuthal angles is the same $$ R^2 \Delta l \int d\Omega \ ... \ = R^2 \Delta l \int d\phi \ d\cos\Theta \ ... \ = R^2 \Delta l\ 2\pi \ \int d\cos\Theta \ ... \ $$ \beq = 4\pi R^2 \ \Delta l\ \frac{1}{2} \int d\cos\Theta \ ... \ = V_i^{(LR)} \frac{1}{2} \int d\cos\Theta \ ... \ \eeq{xs7c} Transforming $f(p)$ from a cell to the c.m., and\\ integrating over the layer of cells \LT c.m. distribution \beqar F_i^{c.m.}(\vec{q_{cm}}) = \hspace*{10cm} \nonumber \\ \frac{4}{(2\pi\hbar)^3} \frac{2\pi}{q^0_{cm} \beta_i^r \gamma_i^r \sqrt{(q_{cm}^0)^2-m^2}} \int_{p^0_1}^{p^0_2} \frac{ p^0 \ dp^0 } { \exp\left[\frac{p^0-\mu_{ch\ i}}{T_i}\right] + 1 } \ , \eeqar{xs9} $$ {\sf where}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p^0_{1,2} = \gamma_i^r \ \left[ q^0_{cm} \mp \beta_i^r \sqrt{(q^0_{cm})^2-m^2} \right] \ . $$ }%end tr-page \newpage % transparency ============================================= p. 23 \transparencyframe{ \vspace*{-0.8cm} In the relativistic Boltzmann limit \LT the integral is analytic \\ \B \/LT formula of the Blast Wave model [14]. \B Layers may have different $\beta_i^r, \ \mu_{ch\ i}$ and $T_i$ at $t_{BU}$\\ - Summing for layers, transforming now to the Lab.: \beqar \frac{d^2 \sigma^{BW}_{prim}}{dP^0\ d\Omega} = \frac{Z_T + Z_P}{A_T+A_P} \sum_i \hspace*{6cm} \nonumber \\ \frac{2 V_i^{(LR)}\sigma_0}{(2\pi\hbar)^3} \frac{1}{\beta_i^r \gamma_i^r} \sqrt{ \frac{(P^0)^2-m^2}{q^0_{cm}(\vec{P})^2-m^2} } \int_{p^0_1}^{p^0_2} \frac{ p^0 \ dp^0 }{ \exp\left[\frac{p^0-\mu_{ch\ i}}{T_i}\right] + 1 } \ , \eeqar{xs10} where\\ - $q^0_{cm} = \gamma_L \left[ P^0 - \beta_L \cos(\Theta_L) \sqrt{(P^0)^2-m^2} \right]$,\\ - $\Theta_L$ is the emission angle,\\ - $\beta_L$ the velocity of the c.m. system in the Lab. \setlength{\baselineskip}{9pt}{\small\sf To have the inclusive proton cross section the primary one should be multiplied by the ratio $R_p$ of the emitted protons to the total charge $Z_P+Z_T$.} \bigskip \setlength{\baselineskip}{20pt} Result shown in Fig. 7.1 \\ The so called {\bf invariant} cross section is plotted due to its preferred relativistic transformation propoeties: $ \frac{1}{P} \frac{d\sigma}{dP^0 \ d\Omega}$. Eq. (\ref{xs8}) is the most general form which can be used for any number of fluid cells and for any flow pattern. }%end tr-page \newpage % transparency ============================================ p. 24 \transparencyframe{ \vspace*{-1.2cm} \vspace*{16.5cm} Figure 7.1 {\it Invariant proton cross sections of central Ar + KCl collisions at 800 MeV/nucleon projectile energy $(\Theta_{cm}=90^0)$. Open circles are experimental points. Full (dashed-dotted) curves represent the cross sections obtained in the Blast-Wave (Fireball) models. The presented viscous fluid dynamical model calculations with two different viscosity values yielded the dashed and dashed double dotted results. From [12] } }%end tr-page \newpage % transparency ============================================ p. 25 \transparencyframe{ \vspace*{-1.2cm} \section{Results of three dimensional calculations} Early fluid dynamical calculations (Los Alamos): No thermal velocities\\ \LT sharp peaks in the sidewards direction: sidewards flow effect\\ - very dominant in the calculation but not so much in experiments. \vspace*{14.8cm} Figure 7.2 {\it The angular dependence of double differential cross sections from experiment and from several theoretical calculations. From [15] } }%end tr-page \newpage % transparency ============================================ p. 26 \transparencyframe{ \vspace*{-0.8cm} Later (Frankfurt) thermal distributions were considered in $\sigma$\\ \B \LT thermal smearing\\ - In Fig. 7.2 [15] the results of the two calculations are compared with experimental data and with the results of some Monte-Carlo cascade simulations. Bounce-off or side splash-effects occur only in fluid dynamical scenario. These lead to a peak in the cross section at finite polar angle, $\Theta \approx 30^0 - 60^0 $. The reason is in the flow pattern of non-central collisions, Fig. 7.3. \vspace*{10cm} Figure 7.3 {\it Density ($\rho=n$), temperature ($T$), and velocity (arrows) distributions in a relativistic heavy ion collision (Ne+U 393 MeV/N) in the laboratory system at the breakup moment ($t$=35 fm/c). The impact parameter of the collision is $b$=6 fm. The crosses indicate that the flow velocity is $v < 0.1c$. The full contour lines belong to temperatures $T$=10 and 20 MeV, the dashed ones to nucleon densities $\rho$ =0.05 and 0.1 (1/fm$^3$ ). From [16] } }%end tr-page \newpage % transparency ============================================= p. 27 \transparencyframe{ \vspace*{-1.2cm} \subsection{Fragment emission at the end of the flow} \B In Chapter 4 --- Law of Mass Action:\\ In a Mixture of different isotopes \\ the concentration of isotopes is determined by\\ the common temperature and baryon density \B Use this at break-up for every fluid element\LT\\ - the density, $n_i$, of an isotope, $i$,\\ in terms of the proton and neutron densities and the temperature \beq n_i(n_p,n_n,T) = a_i\ n_p^{Z_i} n_n^{N_i} \eeq{fx8} where $$ a_i = \lambda_T^{3A_i-3} \ A_i^{3/2} \ 2^{-A} \ (2S_i+1) \ \exp \left[\frac{E_0^{(i)}}{T}\right] , $$ and $$ \lambda_T = \frac{h}{\sqrt{2\pi m_p \ T} } $$ is the thermal de Broglie wave length. \B The density of a fragment $i$, of charge $Z_i$, \& neutron number $N_i$\\ ($A_i =N_i +Z_i$) depends on the temperature $T$ ,\\ on the proton an neutron densities, and on the properties of $i$,\\ namely:\\ -- on the spins, $S_i$, and \\ -- on the ground state energy $E_0^{(i)} \\ \ \ \ \ \ \ \ \ (E_o^{(i)} = 2.23, 8.49, 7.72$, and $28.3$ MeV \\ \ \ \ \ \ \ \ \ \ for $i$=d, t, $^3$He, and $^4$He, respectively). }%end tr-page \newpage % transparency =========================================== p. 28 \transparencyframe{ \vspace*{-1.2cm} \B Conservation of local baryon number, charge, and energy \LT \\ - unknowns $n_p, n_n,$ and $T$: \beqar n &=& A_{cell}/V_{cell} = \sum_i n_i A_i \ , \nonumber \\ n \frac{Z}{A} &=& Z_{cell} / A_{cell} = \sum_i n_i Z_i \ , \nonumber \\ e &=& E_{cell} / V_{cell} = \sum_i n_i (m_i + \frac{3}{2} T )\ , \eeqar{fx9} where\\ - $n$ is the baryon density in the fluid cell at the break-up\\ - $E_{cell}$ is the total internal energy of the fluid cell\\ - - including binding energy and rest masses ($k_{Boltzmann}=1$).\\ - - - For simplicity we use nonrelativistic approximations here. \B Then the thermal momentum distributions, $f_i^{cell}(\vec{p}, \vec{r})$ in a fluid cell \beq f_i^{cell} = \frac{n_i(\vec{r})}{[2\pi m_i T(\vec{r})]^{3/2}} \exp \left[ - \ \frac{p^2}{2 m_i T(\vec{r}) } \right] \ . \eeq{fx12} \B Since the fluid velocity is $\vec{v}(\vec{r})$,\\ these distributions should be transformed to the lab. \\ \ \ \ \ (In non-relativistic approximation this is a shift of the variable.) The diff. cross section for $i$ is then: \beq \frac{d^3 \sigma}{dP^3} = \int d^2b \ d^3r \ F_i^{Lab}(\vec{P}, \vec{r}) . \eeq{fx13} }%end tr-page \newpage % transparency ============================================= p. 29 \transparencyframe{ \vspace*{-1.2cm} \B At breakup assume not only nucleons, but $n,\ p,$\ $d,$\ $t,$\ $... $)\\ \ \ \ \ \ (in themal and chemical equlibrium.) \ \ \ \LT \\ Light and heavy fragments are not formed at the same place: \vspace*{12cm} Figure 7.4 {\it Proton ($p$) and alpha particle ($\alpha$) density contour lines calculated for the breakup configuation shown in the previous Figure. The protons are formed in the middle of hot regions opposite to alphas which are formed at the sides. The contour lines belong to $n_{\alpha} =0.005 /fm^3$ and $n_p =0.003$ and $0.006/fm^3$. From [16] } }%end tr-page \newpage % transparency =========================================== p. 30 \transparencyframe{ \vspace*{-0.7cm} This flow pattern \LT typical triple differential cross section:\\ \ \ \ \ \ \ \ \ \ \ \ \ \ FRAGMENT FLOW\\ The contour lines of the cross section\\ in terms of $y$ and the "transverse rapidity" $y_\perp \equiv p_\perp/m$: \vspace*{12cm} Figure 7.5 {\it Contour plots of invariant triple differential invariant cross sections $(1/p)d^3N/dE \ d\phi \ d\cos\Theta$ for the reaction Ne(393 MeV/N)+ U at the impact parameter $b$=6 fm in the reaction plane ($\phi =0^0 /180^0$) and in the plane orthogonal to it ($\phi =90^0$). The contour lines labeled by the parameter $q$ correspond to a value of $10^q$ /(sr MeV$^2$). Parts (a), (b), (c), (d), (e), and (f) correspond to p, n, d, t, $^3$He, and $^4$He cross sections, respectively. The bounce-off effect is predominantly observable in t, $^3$He, and $^4$He spectra. From [16]} }%end tr-page \newpage % transparency =========================================== p. 31 \transparencyframe{ \vspace*{-0.7cm} \B Asymmetric collision: c.s. peaks at the rapidity of the heavy target\\ - $\exists$ secondary peak at the rapidity of the bounced off projectile.\\ \B Although the projectile is completely destroyed\\ - the peak in the cross section is clearly observable \\ for heavier emitted particles like $t$, $^3$He, or $\alpha$, because:\\ \B\B\B T is lower at the periphery where heavier particles are formed\\ \B\B\B collective flow velocities are larger at the periphery\\ \B\B\B for heavy fragments random thermal velocities are smaller \bigskip Protons are light\LT their cross sections are smeared out\\ - due to large random thermal velocities, and\\ - they are originated from central areas of higher temperature.\\ \B \LT \\ \B Thus the bounce off effect is less apparent for protons.\\ - This isotope effect, or ``Fragment flow'' (as it was called later)\\ - - was predicted already in 1983 [16] and \\ - - it was verified and clearly demonstrated later, \\ - - around 1987-88 by several experiments. }%end tr-page \newpage % transparency ============================================ p. 32 \transparencyframe{ \vspace*{-0.8cm} \B Projectile peak's position \LT collective kinematics\\ - selected, exclusive data of the same impact parameter \LT\\ - - the bounce off, or deflection angle can be seen in the c.s. \B Global momentum conservation \LT the target peak on the other side\\ - - line connecting target and projectile passes the N-N c.m. - - target peak is closer to this nucleus-nucleus c.m. for $A_T>A_P$ \B Deflection angle $\approx\ 0$ for large impact parameters, and \\ - - increases as the collision becomes more and more central.\\ \B Deflection at $b$ depends on the stiffness of the nuclear EOS.\\ - - most sensitive measure of the incompressibility \B If the collision would be elastic \LT\\ - - distance from c.m. would be independent of the deflection angle ($b$) \B More central coll. \LT more thermal excitation, ptcl. creation, etc.\\ - - collisions become more inelastic. \B In Fig. 7.6 the projectile peak position at impact parameter, $b= 9, \ 8,\ 7,\ 6,\ ...\ $fm shows a momentum loss gradually exceeding $30$\%. }%end tr-page \newpage % transparency ======================================= p. 33 \transparencyframe{ \vspace*{-1.2cm} \vspace*{16cm} Figure 7.6 {\it The dependence of the c.m. bounce-off deflection angle and inelasticity on the impact parameter $b$. At impact parameters smaller than 3 fm the second local maximum of the spectrum vanishes, and so, the inelasticity cannot be uniquely determined, but the bounce-off angle is measurable. From [16] } }%end tr-page \newpage % transparency ====================================== p. 34 \transparencyframe{ \vspace*{-1.2cm} \section{GLOBAL FLOW ANALYSIS} In relativistic HI collisions multiparticle correlations carry information\\ \B dominant correlation is the final {\em collective flow} pattern \B Collective flow, and shock waves were predicted:\\ - in 1973-74 by Scheid, M\"uller and Greiner [17], and\\ - by Chapline, Johnson, Teller and Weis [18] independently. \B Existence of coll. flow was debated up to 1982-83\\ - Double differential cross sections did not provide strong evidence New and new more sophisticated detectors were built and the analysis of data developed rapidly at the same time too. \B Breakthrough in 1984 with the Plastic Ball detector in Berkeley.\\ - - a large array of close to 1000 detectors\\ - - could detect the simultaneous emission of several hundred particles\\ - - in one nuclear collision.\\ - - \LT Identification of the reaction plane on an event by event basis. }%end tr-page \newpage % transparency ========================================= p. 35 \transparencyframe{ \vspace*{-1.2cm} \B Methods introduced:\\ - - Sphericity tensor,\\ - - Energy flow tensor,\\ - - Thrust analyses. E.g. the sphericity matrix in the c.m. frame: \beq M_{\alpha\beta} = \sum_i w_i p_{i\alpha} p_{i\beta}\ , \ \ \ \ \ \alpha = \beta = x,y,z\ \ \ \ , \eeq{gx27} were\\ - $i$ runs over all emitted charged particles\\ - - - (up to $^4$He for the plastic ball), and\\ - $w_i$ is a weight which may depend on the type of particle $i$.\\ - In case of the ``energy flow tensor'' analysis $w_i=\frac{1}{2m_i}$. The eigenvalues, $Q_i$ and\\ - - eigenvectors, $\vec{e}_1, \ \vec{e}_2, \ \vec{e}_3$, of the tensor\\ - Normalize the sum of eigenvalues to unity so that $Q_3\ge Q_2\ge Q_1$,\\ - \LT commonly used quantities:\\ - - - sphericity: $S= 1.5 (Q_1 +Q_2)$,\\ - - - flatness $F =\sqrt{3} (Q_2-Q_1)/2$,\\ - - - jet angle $\Theta_{c.m.}=\arccos\left({[\vec{e}_3]_z}/{e_3}\right)$,\\ - - - aspect ratios $R_1 = Q_3/Q_1$ and $ R_2 = Q_2/Q_1$. \B $\vec{e}_3$ and the beam axis define the {bf experimental reaction plane}. \B Important: Distribution of the flow angle $\Theta$ The distribution of the flow angle in this manner is subject to very little experimental and statistical bias, so this was the first generally convincing evidence for the existence of collective flow. }%end tr-page \newpage % transparency ========================================= p. 36 \transparencyframe{ \vspace*{11cm} Figure 7.7 {\it The observed and calculated distribution of the flow angele in collisions of different multplicity. High multiplicity, central, collisions of heavy systems show a clear peak at finite angle, indicating the existence of the sideward flow. The ellipsoid of emitted particles is significantly not aligned with the beam axis. The cascade model does not reproduce this flow effect due to the absence of collective pressure or collective repulsion. From [11] } When the reaction plane is identified the triple differential cross section can be evaluated from the measured data, Fig. 7.8. \setlength{\baselineskip}{11pt}{\normalsize\sf Due to the asymmetry in the detector acceptance the experimental data are not forward-backward symmetric, but the sideward flow is still clearly observable in the data, especially for the heavier systems. The target peak is missing from the observed triple differential cross sections because of the low energy cut of the detector. The random cascade at this energy does not produce a collective azimuthal anicorrelation between the backward and forward directions. } }%end tr-page \newpage % transparency ======================================= p. 37 \transparencyframe{ \vspace*{-1.2cm} \vspace*{12cm} Figure 7.8 {\it Charged particle triple differential cross sections projected to the reaction plane after the reaction plane was identified by the global flow analyses. The experimental plots show the azimuthal anticorrelation while the cascade model does not. Thus a collective flow effect can be suspected in the data. [11]. } \setlength{\baselineskip}{11pt}{\normalsize\sf In CASCADE and MOLECULAR DYNAMICS models the evaluation of global flow parameters is done in the same way as in experiments, since these models create sample events that closely resemble the experimental event sample. These models, however, have direct primary information about the REACTION PLANE. } \LT \setlength{\baselineskip}{19pt}{\Large\sf \B Deviation\\ - - between the principal (theoretical) and experimental reaction plane.\\ \B The agreement between the two reaction planes gets better with increasing multiplicity, and for high multiplicity events the deviation is in the order of 10 degrees only. } }%end tr-page \newpage % transparency =================================== p. 38 \transparencyframe{ \vspace*{-1.2cm} \subsection{Global flow analysis in fluid dynamics} \B In a FD model the summations in eq.(\ref{gx27}) $\rightarrow \ \ \int$ \beq M_{\alpha\beta} = \int d^3r \sum_{j-clust} w_j\ \ \int d^3P^{c.m.}\ P_{j\alpha}^{c.m.}\ P_{j\beta}^{c.m.}\ \ F_i^{c.m.}(\vec{P}^{c.m.}, \vec{r})\ . \eeq{gx28} In a comparison of calculations with the experiments: $\exists$ difficulties:\\ --- limited sensitivity of the detector in momentum space,\\ --- most detectors are not spherically symmetric around c.m.\\ The experimental sphericity matrix [$\neq$ eq. (\ref{gx28})], but rather $$ \left< M_{\alpha\beta} \right>_b = \frac{1}{2\pi b \ \Delta b^2} \int_{b-min}^{b-max} d^2b \int d^3r \sum_{j-clust} w_j\ \hspace*{4cm} $$ \beq \hspace*{4cm} \int_{\mu_{det.}} d^3P^{c.m.} P_{j\alpha}^{c.m.} P_{j\beta}^{c.m.}\ \ F_i^{c.m.}(\vec{P}^{c.m.}, \vec{r})\ . \eeq{gx29} where\\ - $\mu_{det.}$ is the acceptance of the detector in momentum space. }%end tr-page \newpage % transparency ======================================== p. 39 \transparencyframe{ \vspace*{13.5cm} Figure 7.9 {\it Observed and calculated distribution of the flow angele in collisions of different multplicity. Experimental data are compared to calculations in the hydrodynamical model and in the cascade model. From [21] } After the flow tensor: \LT - reaction plane,\ \&\ - ``flow angle'', Fig. 7.9. \B FD reproduces the flow angle distribution as expected.\\ - however, stronger and sharper in the FD model than in the data!\\ - \LT fluctuations are not barely the thermal fluctuations \\ - but fluctuations from {\em finite multiplicity} are also present.\\ - These fluctiations cannot be reproduced by a continuum model (FD) }%end tr-page \newpage % transparency ======================================== p. 40 \transparencyframe{ \vspace*{-1.2cm} \subsection{Decomposition of the global flow tensor} \B Non-relativistic approach:\\ - - $\exists$ the {\em thermal} and {\em collective flow} components\\ \B Calculate the energy flow tensor, $M_{\alpha\beta}$\\ - - in FD approach; Galilei tr. instead of Lorentz tr.\\ \B \LT Very useful result. \B Def.: Collective flow tensor (\ref{gx27}): \beq M_{\alpha\beta} = \sum_i w_i\ p_{i\alpha} p_{i\beta}\ , \ \ \ \ \ \alpha = \beta = x,y,z\ \ \ \ . \eeq{fxx01} - Particle distribution is described by $ f^{cm}_{jH}(\vec{p},\vec{r})$ \LT \beq M_{\alpha\beta} = \int d^3r \sum_{j-clust=p,d,t,..} w_j\ \ \int d^3P^{c.m.} P_{j\alpha}^{c.m.}\ P_{j\beta}^{c.m.}\ \ f_{jH}^{c.m.}(\vec{P}^{c.m.}, \vec{r})\ . \eeq{fxx02} \B $ f_{jH}^{c.m.}(\vec{P}^{c.m.}, \vec{r})$ is known, but (!)\\ --- $\int d^3P^{c.m.} ... $ can only be evaluated numerically.\\ --- (Enormous task for 10,000 fluid cells!) $$ \int d^3r \ \int d^3P^{c.m.} \ \ f_{jH}^{c.m.}(\vec{P}^{c.m.}, \vec{r})\ = $$ \beq \sum_{i-cell} N_{j}^{i-cell} = \sum_{i-cell} \ \int d^3P^{c.m.} \ \ f_{jH}^{c.m.}(\vec{P}^{c.m.}, i-cell)\ , \eeq{fxx03} Now the contribution of one fluid cell.\\ - Assuming Galilei transformation: for momenta: \beq \vec{P} = \vec{P}_{flow} + \vec{P}_{therm} \ . \eeq{fxx04} }%end tr-page \newpage % transparency ====================================== p. 41 \transparencyframe{ \vspace*{-0.5cm} \B The expectation value of $M_{\alpha \beta}$:\\ - - for 1 particle species, $j$,\\ - - from 1 fluid cell, $i$, is $$ \left< M_{\alpha\beta}^{j,i-cell} \right> = N_{j}^{i-cell} \left< P_{\alpha} P_{\beta} \right> = $$ $$ N_{j}^{i-cell} \left\{ \left< P_{\alpha, flow } P_{\beta, flow } \right> + \left< P_{\alpha, therm} P_{\beta, therm} \right> + \right. $$ $$ \hspace*{2cm} \left. \left< P_{\alpha, therm} P_{\beta, flow } \right> + \left< P_{\alpha, flow } P_{\beta, therm} \right> \right\} = $$ \beq N_{j}^{i-cell} \left\{ P_{\alpha, flow } P_{\beta, flow } + \left< P_{\alpha, therm} P_{\beta, therm} \right> + \right. $$ $$ \hspace*{2cm} \left. \left< P_{\alpha, therm} \right> P_{\beta, flow } + P_{\alpha, flow } \left< P_{\beta, therm} \right> \right\} \eeq{fxx07} Since $\vec{P}_{therm}$ follows a symmetric distribution \LT \\ - $\left< P_{\alpha, \ therm} \right> \ = \ 0$, so that \beq \left< M_{\alpha\beta}^{j, \ i-cell} \right> = N_{j}^{i-cell} \left\{ P_{\alpha, \ flow } P_{\beta, \ flow } + \left< P_{\alpha, \ therm} P_{\beta, \ therm} \right> \right\} \eeq{fxx08} }%end tr-page \newpage % transparency ====================================== p. 42 \transparencyframe{ \vspace*{-1.2cm} Since $ f_{jH}^{c.m.}(\vec{P}^{c.m.}, i-cell)$ is a spherically symmetric distribution and\\ $ P_{\alpha, \ therm} $ $ = $ $( \vec{e}_{\alpha} \vec{P}_{therm}) $ where $\vec{e}_{\alpha}$ is a unit vector,\LT $$ \left< P_{\alpha, therm} P_{\beta, therm} \right> = \left( \vec{e}_{\alpha} \vec{P}_{therm}\right) \left( \vec{P}_{therm} \vec{e}_{\beta} \right) = $$ \beq \delta_{\alpha\beta} \left< (\vec{e}_{\alpha} \vec{P}_{therm})^2 \right> = \frac{ \delta_{\alpha\beta}}{3} \left< (P_{therm})^2 \right> . \eeq{fxx09} \B Then \beq \left< M_{\alpha\beta}^{j, \ i-cell} \right> = N_{j}^{i-cell} \left\{ (P_{\alpha, \ flow}^{i-cell} P_{\beta, \ flow }^{i-cell}) + \frac{\delta_{\alpha\beta}}{3} \left< P_{therm}^2 \right> \right\} . \eeq{fxx10} \B Last term: no velocity dependence, \LT\\ - - - spherical contribution (even after sum. over fluid cells). \beq \left< M_{\alpha\beta}^{j} \right> = \sum_{i-cell} N_{j}^{i-cell} ( P_{\alpha, \ flow}^{i-cell} P_{\beta, \ flow }^{i-cell} ) + \frac{\delta_{\alpha\beta}}{3} \sum_{i-cell} N_{j}^{i-cell} \left< P_{therm}^2 \right> . \eeq{fxx11} \B Non-rel. limit: $\varepsilon = \frac{e}{n}=\frac{p^2}{2m}$, \LT\\ - - - $=2 m_j \varepsilon_{therm}$.\\ \B Non-rel. Bolzmann limit: \\ - - - $\left=2 m_j \frac{3}{2} T$ $=$ $3 m_j T$ \\ \B Thus, for the energy flow tensor, $w_j=1/(2m_j)$\\ - - - in the Boltzmann limit: \beq \left< M_{\alpha\beta}^{j} \right> = \frac{1}{2} \left\{ \sum_{i-cell} N_{j}^{i-cell} \left[ m_j ( v_{\alpha, \ flow}^{i-cell} v_{\beta, \ flow }^{i-cell} ) + \delta_{\alpha\beta} T_{i-cell} \right] \right\} . \eeq{fxx12} }%end tr-page \newpage % transparency ====================================== p. 43 \transparencyframe{ \vspace*{-0.5cm} \B Non-spherical structure (non-diagonal part) \\ - - - caused by the collective flow velocities,\\ - - - heavier particles contribute to stronger asymmetry! \B In FD: expectation value of $M_{\alpha\beta}$, can be calculated\\ - - However, $\exists$ FLUCTUATIONS,\\ - - due to the finite MULTIPLICITY\\ \B Finite multiplicity \LT effective asymmetry in thermal distr. also!\\ - - \LT separation of thermal and collective flow terms is not trivial. \B This was demonstrated in [24]: \\ - - in a simple model, 3 thermal sources assumed,\\ - - these were moving with velocities like 3 cells\\ - - (see Fig. 7.10). }%end tr-page \newpage % transparency ====================================== p. 44 \transparencyframe{ \vspace*{-0.8cm} \vspace*{13cm} Figure 7.10 {\it Three particle emitting sources simulating the final state of a heavy ion collision, depicted in the configuration and in the momentum space. There is a larger central source at rest in the center of mass frame with higher temperature, representing the hot spectator matter. On the sides there are two colder regions representing the less excited spectator matter with a velocity that has a sidewards pointing component. This sidewards motion (or sidewards flow) is caused by the high central pressure during the intermediate stages of the reaction. Thus, the so called ``flow angle'' is finite due to this collective pressure. In the momentum space it is apparent that the spectators did suffer a highly inelastic collision because the absolute value of their velocities is less than the initial target and projectile velocity in the c.m. frame. } }%end tr-page \newpage % transparency ====================================== p. 45 \transparencyframe{ \vspace*{-1.2cm} \B Impact parameter dependence of \\ - - flow angle, $\Theta_f(b)$, and inelasticity:\\ - - - extracted from 3 dimensional FD\\ \B Simplified model --- there are only 3 fluid cells!! \B \LT Flow tensor, its eigen values and eigen vectors\\ - - in TWO WAYS: - - - 1) before the simulation of experimental events\\ - - - - - (from thermo \& FD parameters, eq. (7.41)\\ - - - 2) same way as in experiments after generating random events\\ - - - - - (using FREESCO \& eq. (7.32)) FREESCO: Fragments are generated randomly according to thermal expectation values. Conservation laws are satisfied. \B 1) and 2) are compared See Fig. 7.11 }%end tr-page \newpage % transparency ======================================= p. 46 \transparencyframe{ \vspace*{-1.2cm} \vspace*{13.5cm} Figure 7.11 {\it Fluid dynamical ecpectation values (full curves with open circles) and statistically generated events (dots and open squares) yielding flow diagrams: Flow angles versis the aspect ratio of the largest to the smallest eigenvalue of the flow tensor, $R_{1/3}$, fot Nb+Nb reaction at 400 MeV/nucleon lab. beam energy. Four different impact parameters, $b=s=0.1, \ 0.3, \ 0.5, \ 0.7 s_{max}$, and 3 different particle species $j = p,\ p-\alpha, \ all$ are shown. At each impact parameter the whole fluid dynamical expectation curve is plotted for all impact parameters, the particular impact parameter of the actual case is indicated by an open circle. This fluid dynamical expectation value depends on the particle species, $j$. From [24] } }%end tr-page \newpage % transparency ======================================= p. 47 \transparencyframe{ \vspace*{-0.8cm} The impact parameter dependence of the random scattering results are shown in Fig. 7.12 \vspace*{11cm} Figure 7.12 {\it Fluid dynamical expectation values (full curves with dots) and statistically generated events (open circles) yielding flow diagrams: Flow angles versis the aspect ratio of the largest to the smallest eigenvalue of the flow tensor, $R_{1/3}$, fot Nb+Nb reaction at 400 MeV/nucleon lab. beam energy. The parameters indicate the impact parameter in units of $10s/s_{max}$. The fluid dynamical expectation value depends on the particle species, $p, \ p-\alpha, \ all$. From [24] } \B Random fluctuations: not only around the FD expectation,\\ - but $\exists$ an unmistakable shift: \\ - - - the aspect ratios are shifted: 0 $\longrightarrow$ Finite values. }%end tr-page \newpage % transparency ======================================= p. 48 \transparencyframe{ \vspace*{-1.2cm} Danielewicz and Gyulassy [25] analysed generally\\ \B - how random fluctuations shift the expectation values of observables.\\ - Generated random events of multiplicity, $M$,\\ - - corresponding to the same FD expectation values:\\ - - $\Theta_{flow} = 0$, and $R_{1/3} \neq 0$.\\ \B \LT finite multiplicity shifts the observable average. \vspace*{11cm} Figure 7.13 {\it Expectation values (full curves) for a given finite multiplicity, $M$, and statistically generated events (dots and triangles) yielding flow diagrams: Flow angles versis the aspect ratio of the largest to the smallest eigenvalue of the flow tensor, $R_{1/3}$. The original fluid dynamical expectation value did not have any transverse flow in, $\Theta_{flow} = 0$. From [25] } }%end tr-page \newpage % transparency ========================================== p. 49 \transparencyframe{ \vspace*{-0.8cm} \B Thus the observed FINITE FLOW ANGLE in itself is NOT A PROOF of the existence of a transverse flow!. \B Danielewicz and Gyulassy [25] analysed the same\\ - - for flow angles with nonzero expectation value, $\Theta_{flow} \neq 0$.\\ - For M=40 the observed flow angle was larger for small original angles\\ - - - and smaller when the original fluid dynamical expectation was close to $\Theta_{flow} = 90^o$.\\ \B The observed flow angle tends to $57^o$ when the aspect ratio tends to one: \vspace*{10.5cm} Figure 7.14 {\it Expectation values of the observed flow angle, $\Theta '$, (full curves) for multiplicity, $M=40$, of flow diagrams when the original fluid dynamical distribution has the theoretical flow angle indicated by the parameters along the curves. From [25] } }%end tr-page \newpage % transparency ========================================== p. 50 \transparencyframe{ \vspace*{-0.82cm} \B Finite multiplicity distortions\LT \\ - - - difficult to identify the transverse flow experimentally.\\ \B Danielewicz and Gyulassy have succeded, however,\\ - to find a quantity which showed qualitatively the transverse flow.\\ \B\B\B --- The distribution $dN/d\Theta_{flow}$\\ of the flow angles in the experimental event sample. While the average arising from this distribution function looses the information about the original flow angle,\\ the distribution shows a peak at a finite $\Theta_{flow}$ in the presence of a transverse flow.\\ If there is no inherent transverse flow present the distribution peakes at $\Theta_{flow} = 0$, but it can be a wide distribution yielding a large mean value: \vspace*{9.2cm} Figure 7.15 {\it Expectation values of the observed flow angle distributions $dN_{event}/d\Theta_{flow}$ (full curves) for multiplicities, $M=20, \ 40, \ 100$, when the original fluid dynamical flow angle is $20^o$. From [25] } }%end tr-page \newpage % transparency ======================================== p. 51 \transparencyframe{ \vspace*{-0.1cm} \B \B Actually these studies led in 1984 finally to a breakthrough:\\ - the experimatal group at the ``Plastic-Ball'' detector system in Berkeley\\ - succeded to identify the collective transverse flow beyond doubt\\ - for the 1st time after 10 years of persistent research \& scientific debate. \B The flow tensor "$dN/d\cos \Theta$" method was used for high multiplicity $^{93}Nb+\ ^{93}Nb$ reactions at 400 MeV projectile energy. \B The setup of the detector system is shown in Fig. 7.16. }%end tr-page \newpage % transparency ======================================= p. 52 \transparencyframe{ \vspace*{-1.2cm} \vspace*{21.8cm} Figure 7.16 {\it The ``Plastic-Ball'' detector. } }%end tr-page \newpage % transparency ======================================== p. 53 \transparencyframe{ \vspace*{-1.2cm} The detector consisted od 655+160 Plastic Scintillator telescopes (The Plastic Ball) and 60 pairs of scintillator counters plus 36 single counters (The Plastic Wall), with the goal to identify as many particles as possible. Of course the Plastic Ball has a certain range of sensitivity.\\ The target is in vacuum chamber \LT low energy particles cannot leave. \LT the target peak cannot be seen on the rapidity distribution measured!!\\ (To avoid the inaccuracies the best is to study symmetric A+A collisions.\\ In this case a symmetrization can be performed around the c.m. and the\\ limited sensitivity range of the detector can be circumvented.) \vspace*{12.7cm} Figure 7.17 {\it The sensitivity range of the ``Plastic-Ball'' detector in rapidity variables ($u_\perp = p_\perp/m$). From [26] } }%end tr-page \newpage % transparency =========================================== p. 54 \transparencyframe{ \vspace*{-1.2cm} \B The near to 4$\pi$ detector sensitivity made it possible to perform\\ - \ \ \ EVENT BY EVENT MEASUREMENTS,\\ which was unheard of in conventional nuclear physics before.\\ The great success of the detector created enormous response in 1984. The achievemens were soon got publicity even in the New York Times. The Plastic Ball detector was used later extensively for many quantitative studies of the nuclear Equation of State. Three years later the Plastic Ball detector was transported to the CERN and it was used in the experimental set up of the WA-80 collaboration. \B Immediately after the Plastic Ball the collective sidewards flow was detected by the other optical 4$\pi$ detectors as well:\\ - by the streamer chamber and\\ - by nuclear emulsions. \B These latter detectors have a much smaller statistics, - fewer collisions can be quantitatively analysed by them\\ \LT mucl more sensitive statistical method,\\ - developed originally for streamer chamber by Danielewicz and Odyniecz [9]. \B This method became widely used and sort of a standard for measuring collective flow. \B It could even be used for nuclear emulsions, for a sample of few hunderd collisions [27]. The method will be introduced in the next section. }%end tr-page \newpage % transparency =========================================== p. 55 \transparencyframe{ \vspace*{-0.1cm} \section{Transverse Flow Analysis} \B The method we present here was introduced by\\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Danielewicz and Odyniecz[9].\\ - became a widely used sensitive method to detect the transverse flow\\ - could be used in experiments with small statistical sample:\\ - \ \ \ \ in streamer chamber \& even in emulsion experiments [27]. \B Relies on the determination of the reaction plane,\\ - then transverse momenta of ptcls. are projected (or rotated) to this. The determination of the reaction plane is crutial in an experiment,\\ - \ \ \ possible only with large observed multiplicity, M. \B Involves two basic ideas:\\ - A) to select the rapidity range and rapidity dependent waiting factors\\ -\ \ \ in the c.m. which provide the reaction plane\\ -\ \ \ closest to the real reaction plane, and\\ - B) to remove trivial and spurious self correlations from the projections. }%end tr-page \newpage % transparency ========================================== p. 56 \transparencyframe{ \vspace*{-1.2cm} \subsection{Determination of the reaction plane (A)} The reaction plane is defined by the transverse vector $\vec{Q}$: \beq \vec{Q} = \sum_{\nu=1}^M w_\nu \ \vec{p}_{\perp \nu} \eeq{tfa1} where the weight factor, $w_\nu$,\\ - depends on the rapidity of the emitted particle, $\nu$,\\ i) - so that the central rapidity region, \\ - \ \ \ (where the particle emission is azimuthally symmetric) is omitted, \&\\ ii) - forward and backward rapidity regions get weights with opposite signs. \B \LT forward and backward moving particles, \\ - which are azimuthally anticorrelated if $\exists$ a collective tr. flow, \\ will contribute equally to $\vec{Q}$: \beq w_\nu = \left\{ \begin{array}{ll} +1 &: if y > y_c \\ 0 &: if -y_c y_c \\ 0 &: if -y_c \left. \right|_{y,y+\Delta y} =\frac{1} { < | \vec{Q} | > } \left[ w_\nu \left<\vec{p}_{\perp \nu}^{\ 2}\right> + \left< \sum_{\mu\neq \nu} w_\mu (\vec{p}_{\perp \nu} \vec{p}_{\perp \mu}) \right> \right] , \eeq{tfa6} the expectation value in the second term would vanish \ $< \vec{p}_{\perp \nu} \vec{p}_{\perp \mu}> = 0$\\ due to symmetry reasons, \B but the first term would not\\ -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ since $\vec{p}_{\perp \nu}^{\ 2} \geq 0$. }%end tr-page \newpage % transparency ======================================= p. 60 \transparencyframe{ \vspace*{-0.8cm} Thus this definition would yield a finite $< p^{^{x'}}_\nu> \left. \right|_{y,y+\Delta y}$\\ -\ \ even if there are no real collective correlations in the sample. -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ An example for this is shown in Fig. 7.19:\\ \B where the real sample shows a {\bf clear anticorrelation}\\ -\ \ \ \ between the forward and backward moving particles (a).\\ \B But, if we take the emitted particles from all events in the sample\\ - \ and mix them randomly to form an artificial sample\\ - \ where the real physical correlations should be lost,\\ -\ \ \ \ the $p^x/a$ plot {\bf still shows} an azimuthal anticorrelation\\ -\ \ \ \ du to the selfcorrelation effect (b). \vspace*{10cm} Figure 7.19 {\it Comparision of $p^{x'}/a$ plots of a real physical sample (a) and on a randomized artificial sample (b). Due to the selfcorrelation azimuthal anticorrelation appears between forward and backward moving particles in both cases. From [9] } }%end tr-page \newpage % transparency ========================================== p. 61 \transparencyframe{ \vspace*{0.8cm} \B We can remove these self correlations\\ -\ \ \ by removing the first term in expression (\ref{tfa6}).\\ -\ \ \ This can be done if we project each particle's transverse momentum\\ -\ \ \ \ \ \ \ \ \ to a reaction plane determined by all other particles: \beq \vec{Q}_\nu = \sum_{\mu \neq \nu} w_\mu \vec{p}_{\perp \mu}. \eeq{tfa7} Now the projection does not contain the selfcorrelation term \beq p^{^{x'}}_\nu =\frac{1} { < | \vec{Q} | > } \sum_{\mu\neq \nu} w_\mu (\vec{p}_{\perp \nu} \vec{p}_{\perp \mu}) . \eeq{tfa8} \B Thus $p^{^{x'}}_\nu$ is nonzero only if real physical correlations exist. Fig. 7.20 shows that after removing the selfcorrelation from $\vec{p}_{\perp \nu} \vec{Q}$\\ -\ \ \ the real sample still shows the azimuthal anticorrealtion (a),\\ -\ \ \ while the arificial sample shows no effect (b). }%end tr-page \newpage % transparency ======================================= p. 62 \transparencyframe{ \vspace*{-0.8cm} \vspace*{10cm} Figure 7.20 {\it Comparision of $p^{x'}/a$ and $dP^x/dy$ plots of a real physical sample (a) and on a randomized artificial sample (b). Due to the removed selfcorrelation azimuthal anticorrelation appears between forward and backward moving particles only for the real physical sample (a). From [9] } \B The quantity $p^{x'}/a$ underestimates the transverse momentum\\ - because the vector $\vec{Q}_\nu$ fluctuates around the real reaction plane\\ - by some azimuthal dispersion angle, $\Delta \varphi$. \B If we would know this real reaction plane\\ - we could estimate the value of $p^x$ projected to the real plane as: $$ < p^{x'} > = < p^x > < \cos \Delta \varphi > . $$ In experiments $ \Delta \varphi $ is approximated by the width of the distribution of the azimuthal angle differnce, in Fig. 7.18. }%end tr-page \newpage % transparency ====================================== p. 63 \transparencyframe{ \vspace*{-1.2cm} A practical definition was introduced by the Plastic Ball team later,\\ \B\B the {\bf ``Flow''}, $F$, to measure the transverse momentum transfere: $$ F = \frac{\partial [ p^x/a ] }{ \partial y } \left. \right|_{y=0} \ . $$ $F$ is subject to less experimental bias than, e.g., the maximum of $p^x/a$,\\ - and it enabled us to compare different reactions and\\ - results of different experimental devices to each other. \vspace*{12cm} Figure 7.21 {\it Mean transverse momentum per nucleon projected into the reaction plane as a function of the normalized center-of-mass rapidity for 400 MeV per nucleon Nb+Nb in the muliplicity bin, between 50\% and 75\% of $N_p^{max}$. The slope of the solid line represents the ``flow'', $F$, obtained from fitting the data. From [19] } }%end tr-page \newpage % transparency ======================================== p. 64 \transparencyframe{ \vspace*{-1.2cm} \section{Assignment 7} \begin{description} \item[7.a] Calculate the double differential cross section by boosting the local J\"uttner (relativistic Boltzmann) distribution spherically by a collective flow velocity. (See the Blast-Wave Model of ref. [14].) \item[7.b] Calculate the $p^x (y)$ distribution of nucleons (i.e., the average transverse momentum of particles at rapidity $y$ projected to the reaction plane) if you have 3 thermal particle emitting sources, described by the non-relativistic Boltzmann distributions. Two of the 3 sources are moving, one forward and one backward in the c.m. frame. These two have opposite transverse flow velocities. The third source is at rest in the c.m. frame. Use Galilei transformation instead of Lorentz transformation. \end{description} }%end tr-page % transparency ================================================================ \end{document}