

\documentstyle[12pt]{article}
\begin{document}
\title{ Bose-Einstein condensation and independent production of pions}
\author{A.Bialas and K.Zalewski
\thanks{Also at the Institute of Nuclear Physics, Cracow}
\\ M.Smoluchowski Institute of Physics
\\ Jagellonian University, Cracow\thanks{Address: Reymonta 4, 30-059 Krakow,
Poland; e-mail: bialas@thp1.if.uj.edu.pl, zalewski@chall.ifj.edu.pl } }
\maketitle
\begin{abstract}
The influence of the HBT effect on the momentum spectra
 of independently produced pions is studied using the method developed
earlier for discussion of multiplicity distributions.
 It is  shown that in this case all the  spectra and
multiparticle correlation functions are expressible in terms of one function of
two momenta.
 It is also shown that at the  critical point
 all pions are attracted into one quantum state and thus form a Bose-Einstein
condensate.
\end{abstract}

\vspace{0.3cm}
{\bf 1. Introduction}
\vspace{0.3cm}


Several years ago, Pratt \cite{pr1} realized that the well-established
phenomenon of HBT correlations \cite{ha1} can, under certain conditions, lead
to Bose-Einstein condensation in multipion systems, which he called "pion
laser". Since then the effect was investigated by several authors, including
Pratt \cite{pr2,pr3} (see \cite{wi1} for an exhaustive list of references).
Recently \cite{biz}, employing the density matrix formalism, we discussed
multiplicity distributions of independently emitted identical particles for
arbitrary shapes of particle spectra in momentum and in configuration space.
The approach to the condensation point and the conditions for reaching it were
investigated. In the present paper we extend the method of \cite{biz} to
momentum spectra of independently produced identical pions. The explicit
formula for the generating functional is written down and the resulting
momentum spectra are discussed.

Although the independent production mechanism is unlikely to be a realistic
model of pion production, we feel that it is  interesting to investigate
its consequences.

{\bf 2.} The  basic ideas of the HBT effect, as applied to
processes
of particle production, were explained in \cite{biz}, using the approach
developed in \cite{bi1}. The net result is the formula for the
momentum distribution of $n$ identical bosons
\begin{equation}
\Omega(q)=  =\frac1{n!} \sum_{P,P'} \rho^{(0)}(q_P,q_{P'}).  \label{4}
\end{equation}
where $\rho^{(0)}(q,q')$ is the n-particle density matrix {\it calculated
ignoring the identity of particles}\footnote{All  quantities calculated with
the identity of particles ignored will be called {\it uncorrected} and denoted
by a superscript
$^{(0)}$.} and normalized by the condition
\begin{equation}
 Tr[\rho^{(0)}]= \int dq  \rho^{(0)}(q,q) = 1 .  \label{2a}
\end{equation}
 The sum extends over
all permutations  $P$ and $P'$ of particle momenta $[q_1,...,q_n] \equiv q$.


Consider now a system of $n$ particles emitted independently.
 If we ignore the identity of particles, independent emission implies that
 the density matrix factorizes
\begin{equation}
\rho^{(0)}(q,q')= \prod_{i=1}^n \rho^{(0)}(q_i,q_i').   \label{7}
\end{equation}
Introducing this into (\ref{4}) we have
\begin{equation}
\Omega(q)
 =\frac1{n!} \sum_{P,P'}\prod_{i=1}^n \rho^{(0)}((q_P)_i,(q_{P'})_i).
\label{8}
\end{equation}
The information contained in (\ref{8}) is conveniently summarized in the form
of the generating functional $\Phi[u]$ defined as
\begin{equation}
\Phi[u] =\frac{ \sum_{n=0}^{\infty}P^{(0)}(n) W_n[u]}
{ \sum_{n=0}^{\infty}P^{(0)}(n) W_n[1]}
     \label{k2}
\end{equation}
with
\begin{equation}
W_n[u]= \int dq_1...dq_n
\Omega(q_1,...,q_n) u(q_1)...u(q_n).         \label{k3}
\end{equation}
Here $P^{(0)}(n)$ is the {\it uncorrected} multiplicity distribution and
$u(q)$
is an arbitrary real nonnegative  function of $q$. For $u(q)=const\equiv z$ the
generating functional reduces to the generating function of the
multiplicity distribution, $\Phi(z)$ discussed in \cite{biz}.

The inclusive and exclusive distributions of $n$ identical particles can be
obtained by n-fold functional differentiation of  the generating functional
(\ref{k2}) with respect to $u(q)$ at $u(q)=1$ and $u(q)=0$, respectively.
Similarly $n-$fold differentiation of the logarithm of  $\Phi[u]$ at $u(q)=1$
gives the inclusive correlation function of the n-th order.

To find the explicit expression for $\Phi[u]$
we observe that, given the formula (\ref{8}) for $\Omega(q)$, for each
permutation
$P$ of the momenta
$q_1,...,q_n$, the integral on the
right hand side of (\ref{k3}) factorizes into a product of contributions from
all the cycles of P (as is well known, each permutation can be decomposed
into cycles). Let us denote the contribution from a cycle of length $k$ by
$C_k[u]$. We have
\begin{equation}
C_k[u]= \int d^3q_1 ... d^3q_k u(q_1)\rho^{(0)}(q_1,q_2)u(q_2) \rho^{(0)}(q_2,q_3)
...u(q_k)\rho^{(0)}(q_k,q_1)   \label{10}
\end{equation}



 The rest of the calculation is just combinatorics.

We observe first that any two permutations which have identical partitions into
cycles give equal contributions. Let us consider the set of all
permutations with a given partition into cycles.  Denoting by
$n_k$ the number of
occurrences of a cycle of length $k$ in the  set of permutations considered, the
contribution from all of them can be written as
\begin{equation}
W'_n[u]= \prod_{k=1}^n (C_k[u])^{n_k}
\frac{n!}{(k!)^{n_k}}[(k-1)!]^{n_k}\frac1{n_k!}
= n!\prod_{k=1}^n\frac{\left(\frac{C_k[u]}{k}\right)^{n_k}}{n_k!}.  \label{11}
\end{equation}
In the first equality the first factor is the integral, the second is the
number of partitions of the $n$ particles among the cycles, the third is
the number of ways a cycle can be constructed from $k$ particles and the last
one corrects for the permutations of whole cycles.

$W_n[u]$ is obtained by summing $W_n'[u]$  over
partitions into cycles different from each other.

Until now we have considered a fixed multiplicity. As noted already in
\cite{biz}, the sum over multiplicities can be explicitly performed if the
uncorrected multiplicity distribution $P^{(0)}(n)$ is poissonian (as required
for independent emission)
\begin{equation}
P^{(0)}(n) = e^{-\nu} \frac{\nu^n}{n!}.    \label{13}
\end{equation}
The result is
 an elegant formula for the generating functional (\ref{k2}):
\begin{equation}
\Phi[u]  = \exp\left(\sum_{k=1}^{\infty} \nu^k\frac
{C_k[u] -C_k[1]}{k}\right).   \label{14}
\end{equation}

\vspace{0.3cm}
{\bf 3.} Let us now discuss the general properties of the particle
distributions obtained from  Eq.(\ref{14}).

The main result is that all inclusive distributions can be expressed in terms
of a single function $L(q,q')$, defined as
\begin{eqnarray}
L(q,q') = \sum_{k=1}^{\infty} \nu^k [\rho^{(0)}]^k(q,q')   \label{15}
\end{eqnarray}
where
\begin{equation}
[\rho^{(0)}]^k(q,q')         \equiv
 \int d^3q_2 ... d^3q_k \rho^{(0)}(q,q_2) \rho^{(0)}(q_2,q_3)
...\rho^{(0)}(q_k,q')   \label{k4}
\end{equation}
and the $\nu$-dependence of $L$ is not written explicitly.
The single particle
distribution is given by
\begin{eqnarray}
\omega(q)= L(q,q),    \label{15c}
\end{eqnarray}
and the two-particle correlation function is
\begin{equation}
K_2(q_1,q_2) = L(q_1,q_2) L(q_2,q_1). \label{k12}
\end{equation}
The general formula for the correlation functions reads
\begin{eqnarray}
K_p(q_1,...q_p) = L(q_1,q_2) L(q_2,q_3)....L(q_p,q_1) \nonumber \\  +
permutations\;\;of\;\; (q_2,.....,q_p).             \label{k14}
\end{eqnarray}
It is not difficult to verify that by integrating (\ref{k14}) over all momenta
one recovers the formula for cumulants derived in \cite{biz}.

These formulae represent a formidable constraint on the observed particle
distributions: They basically say that all higher order correlations can be
derived from the two particle correlation function. As they are valid for any
model which assumes independent production, they were found by many authors in
particular cases \cite {an1,by1,csz}, for a full list, see \cite{wi1}. It is
fair to
say, however, that - since the independent production model is not expected to
be a precise description of high energy interactions one expects violations of
these relations, at least to some extent. Recent work by Eggers et al.
\cite{bu1}, points perhaps in this direction. The measurements of deviation of
data from (\ref{15c})-(\ref{k14}) is of great interest, as it may indicate what
is the dominant intraparticle correlation.



Further discussion is greatly simplified if the matrix $\rho^{(0)}(q,q)$ is
expressed in terms of its eigenvalues $\lambda_m$ and
its eigenfunctions $\psi_m(q)$. We have\footnote{
We discuss here only the case of a discrete eigenvalue  spectrum. The
case of a continuous spectrum can be treated along the same lines \cite{biz}.}
\begin{equation}
\rho^{(0)}(q,q')= \sum_m \psi_m(q) \lambda_m \psi_m^*(q')  \label{k9}
\end{equation}
and
\begin{equation}
[\rho^{(0)}]^k(q,q')= \sum_m \psi_m(q) (\lambda_m)^k \psi_m^*(q').  \label{k9c}
\end{equation}
Substituting this into (\ref{15}) and  performing the sum over $k$
we have
\begin{equation}
L(q_1,q_2) = \sum_m \psi_m(q_1) \psi^*_m(q_2)
\frac{\nu\lambda_m}{1-\nu\lambda_m}.       \label{k13}
\end{equation}
It is clear from (\ref{k13}) that $L(q_1,q_2)$ and thus also all inclusive
distributions become singular when
$\nu\lambda_0 \rightarrow 1$ ($\lambda_0$ is the largest eigenvalue). In this
limit, corresponding to  Bose-Einstein condensation,  $L(q,q')$ is dominated
by the first term in the sum and we have
\begin{equation}
L(q_1,q_2) = \frac{ \psi_0(q_1) \psi^*_0(q_2)}
{1-\nu\lambda_0} + \tilde{L}(q_1,q_2),       \label{k15}
\end{equation}
where $\tilde{L}(q_1,q_2)$ remains bounded for $\nu\lambda_0 \rightarrow 1$.
Thus at the condensation point all the particles, except for a negligible
fraction, are in the same state described by the eigenfunction $\psi_0(q)$.
Away from the condensation point, of course, the whole sum in (\ref{k13}) must
be carried out.

Let us now consider  the important example,
when  particles are emitted in a pure state, i.e.
\begin{equation}
\rho(q,q')= \psi(q) \psi^*(q').  \label{17}
\end{equation}
It follows from (\ref{10}) that then
\begin{equation}
C_k[u] =\left( C_1[u]\right)^k   \label{k7}
\end{equation}
and thus the generating functional becomes
\begin{equation}
\Phi[u]  = \exp\left(\sum_{k=1}^{\infty} \frac
{\nu^k((C_1[u])^k-1)}{k}\right) = \frac{1-\nu}{1-\nu C_1[u]}.   \label{18}
\end{equation}
One easily sees that this gives the momentum spectrum $|\psi(q)|^2$ independent
of particle multiplicity and
 the geometric distribution of multiplicities \cite{biz}.


\vspace{0.3cm}
{\bf 4.}
Let us now  consider the uncorrected single particle density matrix of
gaussian form, discussed already in several papers by Pratt
\cite{pr1,pr2,pr3} and recently by other authors \cite{wi1,csz}. To simplify
the notation, we shall restrict ourselves
to the one-dimensional problem. Generalization to the three-dimensional case is
straightforward \cite{biz}. We thus have
\begin{equation}
 \rho^{(0)}(q,q') = \left(\frac1{2\pi \Delta^2}\right)^{\frac12}
e^{-\frac{(q^+)^2}{2\Delta^2} -\frac12R^2(q^-)^2},   \label{21}
\end{equation}
where
\begin{equation}
q^+ \equiv \frac12(q+q'); \;\; q^- \equiv q-q'.
\label{21a}
\end{equation}
As easily seen, $\Delta^2$ is the average
value of the square  of the particle momentum, and $R^2$
is
the average value of the square of the space coordinate of the particle
emission point. As is clear from the context, both $\Delta$ and $R$ refer to
the {\it uncorrected} distributions. The uncertainty principle implies that
\begin{equation}
R \Delta \geq \frac12.   \label{21*}
\end{equation}



As explained in \cite{biz}, the eigenfunctions of
 the density matrix (\ref{21}) are of the form
\begin{equation}
\psi_m(q)=a_m e^{-\frac12 \frac{R}{\Delta}q^2}
H_m(\sqrt{\frac{R}{\Delta}}q), \label{32a}
\end{equation}
where $H_m(q)$ is the Hermite polynomial of order $m$ and $a_m$ is the
normalizing factor, given e.g. in \cite{ab1}.

The corresponding eigenvalues are
\begin{equation}
\lambda_m= \lambda_0(1- \lambda_0)^m, \;\;\;\; m=0,1,..., \label{32b}
\end{equation}
where
\begin{equation}
\lambda_0=   \frac2{(1+2\Delta R)} \leq 1 \label{25}
\end{equation}
is the greatest one.

Eqs. (\ref{32a}) and (\ref{32b}) can be now used for the explicit calculation
of $L(q,q')$ [c.f. (\ref{k13})] which, in turn, determines all particle
distributions, as explained in the previous section. This is actually easier
than it looks because the difficult sum over oscillating Hermite polynomials
can be replaced by a sum over Gausssians with positive coefficients. To see
this, we rewrite the formula (\ref{k13}) as a double sum

\begin{equation}
L(q,q')= \sum_m \psi_m(q) \psi^*_m(q')
\sum_{k=1}^{\infty}\left(\nu\lambda_m\right)^k.       \label{k16}
\end{equation}
Introducing (\ref{32b}) and reversing the order of summation we obtain
\begin{equation}
L(q,q')=
\sum_{k=1}^{\infty} \frac {\left(\nu \lambda_0\right)^k}{1-\left(1-
\lambda_0\right)^k} \hat{\rho}_k(q,q'),         \label{k17a}
\end{equation}
where
\begin{equation}
\hat{\rho}_k(q,q') =
 \left(\frac1{2\pi \hat{\Delta}_k^2}\right)^{\frac12}
e^{-\frac{(q^+)^2}{2\hat{\Delta}_k^2} -\frac12\hat{R}_k^2(q^-)^2},
\label{k21}
\end{equation}
and $\hat{\Delta}_k$ and $\hat{R}_k$ are determined from the equations
\begin{equation}
\frac{\hat{R}_k}{\hat{\Delta}_k} =\frac{R}{\Delta};\;\;\;\;
\hat{R}_k\hat{\Delta}_k = \frac12 \frac{1+\omega_k} {1-\omega_k};\;\;\;\;
 \omega_k =
\left(\frac{2R\Delta -1}{2R\Delta +1}\right)^k.   \label{k18}
\end{equation}

The important lesson from this exercise is that even when  the uncorrected
distribution is described by a simple Gaussian, the resulting particle spectra
are fairly complicated superpositions of an infinite number of Gaussians with
varying width.

It follows from (\ref{k18}) that $\hat{R}_k<R$ and $\hat{\Delta}_k < \Delta$
for all $k>1$. Consequently, the observed distributions are always {\it
narrower}  than the assumed
uncorrected ones.

Another interesting quantity is the average value $<(q^-)^2>$ calculated from
the
 two-particle correlation function, i.e.,
\begin{equation}
<(q^-)^2> = \frac1{K_2} \int dq^+ K_2(q,q') (q^-)^2
\label{kk18}
\end{equation}
where the cumulant $K_2$ is the integral of $K_2(q,q')$.
In the standard analysis of the data,  $2<(q^-)^2>$ is usually interpreted as
an inverse of the average squared radius $R_{eff}$ of the particle emission
region.
 The ratio  $R_{eff}^2/R^2$ calculated from (\ref{kk18}) is plotted in Figure
1 for various values of $R\Delta$. One sees that, at a fixed
$R\Delta$, $R_{eff}^2$ decreases from $R^2$ to  $R^2/2R\Delta$  when
$<n>$ varies from
$0$ to $\infty$. Thus we
conclude that, even fairly far from the critical
point, the apparent size of the system  $R_{eff}^2$ (as
determined from the two-particle correlation function) has little to do with
the actual size of the system, given by $R^2$. One sees also that, even at
the fixed particle phase-space density $<n>/R\Delta$, the effect substantially
increases with increasing $R\Delta$.


\vspace{0.3cm}
{\bf 5.}
Our conclusions can be summarized as follows.

(i) It has been shown that the formulation of the problem of HBT interference
in terms of the density matrix provides an effective tool for discussion  not
only of particle multiplicities \cite{biz} but also of the momentum spectra.

(ii) For independent particle production, the effects of HBT symmetrization
are expressed in terms of one function of two momenta. Consequently, all
multiparticle correlations are expressible in terms of the single particle
distribution and the two-particle correlation function. It would be very
interesting to verify if this general relation holds -- at least approximately
-- in the data.

(iii) The symmetrized distributions reveal the existence of a critical point
corresponding to the Bose-Einstein condensation, when almost all the particles
fall into one quantum state, corresponding to the lowest eigenvalue of the
(uncorrected) density matrix.

(iv) Due to symmetrization, the width of the momentum spectrum decreases and
the
width of the two-particle correlation function increases. This effect becomes
stronger, when the system approaches criticality. Close to the critical point
the determination of the size of the emission region from the width of the
correlation function is no longer possible.

(v) For a Gaussian uncorrected density matrix, the corrected momentum
distribution is Gaussian both in the low density and in the high density limit,
but non-Gaussian in the intermediate region. At low density the parameters
$\Delta$ and $R$
  are determined by the widths of the momentum distribution and of the
correlation function, respectively. At high density both these widths depend on
the ratio $\frac{R}{\Delta}$ only.

\vspace{0.3cm}
{\bf Acknowledgements}
\vspace{0.3cm}

We would like to thank P.Bialas, K. Fia?lkowski, J.Pisut and A.Staruszkiewicz
for very useful
comments and discussions. This work was supported in part by the KBN Grant 2
P03B 086 14.

\vspace{0.5cm}

\begin{thebibliography} {99}
\bibitem{pr1}
S.Pratt, Phys. Letters B301 (1993) 159.
\bibitem{ha1}
R. Hanbury-Brown and R.Q.Twiss, Nature 177 (1957) 27.
\bibitem{pr2}
S.Pratt and V.Zelevinsky, Phys. Rev. Letters 72 (1994) 816.
\bibitem{pr3}
S.Pratt, Phys. Rev. C50 (1994) 469.
\bibitem{wi1}
U.A.Wiedemann, .
\bibitem{biz}
A.Bialas and K.Zalewski,   E.J.Phys. C, in print.
\bibitem{bi1}
A.Bialas and A.Krzywicki, Phys.Letters B354 (1995) 134.
\bibitem{we1}
For a review, see, e.g., D.H. Boal, C.-K. Gelbke and B.K. Jennings, Rev.
Mod.Phys. 62 (1990) 553. G.Baym, Acta Phys. Pol B29 (1998) 1839.
\bibitem{an1}
I.V.Andreev, M.Pluemer and R.M.Weiner, Int. J. of Mod. Phys. A8 (1993) 4577.
\bibitem{by1}
M.Biyajima et al., Prog. of Theor. Phys. 84 (1990) 931.
\bibitem{csz}
T. Cso\"rg\"o and J. Zim\'anyi, Phys. Rev. Letters 80 (1998) 916.
\bibitem{bu1}
H.C.Eggers, P.Lipa nad B.Buschbeck, Phys. Rev. Lett. 79 (1997) 197.
\bibitem{ab1}
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, N.Y.
\end{thebibliography}

\vspace{0.3cm}
{\bf Figure captions}
\vspace{0.3cm}


Figure 1. $R_{eff}/R^2$ plotted versus $<n>$ for different
values of $R\Delta$. The marked points correspond to a fixed value of the
particle phase-space density $<n>/R\Delta = 8$.

\end{document}




