---
abstract: 'A simple new algorithm for the calculation of two-particle Bose-Einstein correlations from classical event generators is derived and discussed.'
address: |
     Institut für Theoretische Physik, Universität Regensburg,\
    D-93040 Regensburg, Germany 
author:
- 'Q.H. Zhang[@home], U.A. Wiedemann, C. Slotta, and U. Heinz'
title: 'Bose-Einstein Weights for Event Generators'
---

-1cm

For ultrarelativistic heavy-ion collisions, two-particle Bose-Einstein correlations between identical pions or kaons provide a unique possibility to reconstruct the geometry (size, temporal extension) ${\em and}$ dynamics (collective expansion flow) of the source at the point of hadron freeze-out. (For a recent theoretical review see [@H96].) This reconstruction is, however, not completely model independent; it requires the use of “reasonable” source parametrizations (e.g. [@H96; @CL96; @CN96]) whose parameters are then fixed by a simultaneous analysis of single-particle momentum spectra and two-particle momentum correlations [@CN96; @S96].

Invaluable help for the selection of “reasonable” source parametrization comes from microscopic event generators (e.g. VENUS [@W93], RQMD [@SSG89] or ARC [@PSK92]) which generate the phase-space distribution of hadrons at freeze-out from a dynamical Monte Carlo simulation of the (classical) kinetic phase-space evolution of the collision zone. Unfortunately, it was recently pointed out [@A97; @MKF96] that the direct computation of two-particle correlation functions from classical kinetic codes [@W93; @SSG89; @PSK92] is fraught with severe conceptual and practical problems. These can be simply understood by starting from the general relation [@S73] between the 2-particle correlation function $C(\bbox{q}, \bbox{K})$ and the (real) “emission function” (one-particle Wigner density at freeze-out) $S(x,K)$,

\[1\] $$\begin{aligned}
 \label{1a} 
   \FL C(\bbox{q},\bbox{K}) =
   1 + \frac{{\left\vert\int d^{4}x\, S(x,K)\, e^{iq\cdot x}
              \right\vert}^{2}} 
            {\int d^{4}x\, S(x,p_{a}) \,\int d^{4}y\, S(y,p_{b})} \, ,
 \\
 \label{1b}
   \FL \bbox{q}{=}\bbox{p}_a{-}\bbox{p}_b, \ 
       q^{0}{=}E_{a}{-}E_{b},\ 
       \bbox{K}{=}{\bbox{p}_a{+}\bbox{p}_b \over 2},\
       K^{0}{=}{E_{a}{+}E_{b} \over 2}.
 \end{aligned}$$

$(\bbox{p}_a, E_{a})$ and $(\bbox{p}_b, E_{b})$ are the 4-momenta of the two observed particles. Eq. (\[1a\]) neglects final state interactions which we will leave out in this Letter in order to concentrate on the principal issues.

The numerator in the second term of Eq. (\[1a\]) can be rewritten as $$\begin{aligned}
 \label{2}
   {\rm Num} (q,K) =&& 
   \int d^4x\, d^4y \,S(x,K)\, S(y,K) 
 \nonumber\\
   && \ \times\, \cos(q{\cdot}(x-y)) \, .
 \end{aligned}$$ The problem is the construction of the Wigner density $S(x,K)$ from the output of the event generators. The latter consists of a set of phase-space points $(x_{i},p_{i})$ denoting the (on-shell) momenta $p_{i}$ and points of last interaction $x_{i}$ of the produced particles. According to (\[1b\]) $K$ is the average of two on-shell momenta, but not itself on-shell, $K^{0} \ne \sqrt{m^2+{\bbox{K}^2}}$. Therefore $S(x,K)$ cannot be directly related to the phase-space density generated by the distribution of points $(x_{i},p_{i})$. To overcome this difficulty one ususally imposes [@YK78; @P94] the “smoothness assumption” $S(x,\frac{p_a+p_b}{2}) \approx S(x,p_a) \approx S(x,p_b)$ and rewrites the expression (\[2\]) as $$\begin{aligned}
 \label{3}
   {\rm Num}(q,K) =&& 
   \int d^{4}x\, d^{4}y \, S(x,p_a) \, S(y,p_b) 
 \nonumber\\   
    && \times \cos\bigl((p_a-p_b){\cdot}(x-y)\bigr)\, .
 \end{aligned}$$ One now identifies $S(x,p)$ with the classical output distribution from the event generator, $$\label{4}
   S_{\rm class}(x,p) = \sum_{i=1}^{N} \delta^{(4)}(x-x_{i})\, 
   \delta^{(4)}(p-p_{i})\, ,$$ where $N$ is the total number of pions of a given charge in the event. In the widely used Pratt algorithm [@P94] the expression which results after inserting (\[4\]) into (\[3\]) is simulated by the ad hoc prescription $$\label{5}
   {\rm Num} (q,K) \longmapsto \sum_{i,j \in {\rm bin}} 
   \cos\bigl((p_i-p_j){\cdot}(x_i-x_j)\bigr) \, .$$ Here “bin” denotes a small bin in $(\bbox{q},\bbox{K})$ with $\bbox{p}_i-\bbox{p}_j \approx \bbox{q}$ and $(\bbox{p}_i+\bbox{p}_j)/2
\approx \bbox{K}$. (In practice the bin size depends on event statistics.)

The prescription (\[5\]) has two severe problems: first, the positivity of Num$(q,K)$ got lost between Eqs. (\[2\]) and (\[3\]) when making the “smoothness assumption”. It was pointed out in Refs. [@CH94; @PRW94] and practically demonstrated in Ref. [@MKF96] that for sources with strong $x$-$p$-correlations (e.g. rapidly expanding sources) this can lead to unphysical oscillations of the simulated correlation function around unity. Second, the intuitive substitution law (\[5\]) is formally incorrect and results in a wrong selection of contributing pairs as well as an incorrect weighting factor for each pair.

To prove this last point let us write in (\[2\]) $$\begin{aligned}
 \label{6}
   S(x,K)\!\!&&\!\!\! S(y,K) = \int d^4P_1\, d^4P_2 \, S(x,P_1)\, S(y,P_2) 
 \nonumber\\
   &\times&\, \delta^{(4)}(P_1-P_2) \,
              \delta^{(4)}\left(K-\frac{P_1+P_2}{2}\right)\, .
 \end{aligned}$$ Inserting the classical expression (\[4\]) one finds instead of (\[5\])

\[7\] $$\begin{aligned}
 \label{7a}
   {\rm Num}(q,K) =&& \sum_{i,j=1}^{N} \delta^{(4)}(p_i-p_j) \,
   \delta^{(4)}\left(K-\frac{p_i+p_j}{2}\right) 
 \nonumber\\
   && \qquad \times \cos\bigl(q{\cdot}(x_i-x_j)\bigr) 
 \\
 \label{7b}
   \longmapsto && 
   \sum_{i,j \in {\rm bin}(K,\epsilon)} \cos\bigl(q{\cdot}(x_i-x_j)\bigr)\, .
 \end{aligned}$$

Here “${\rm bin}(K,\epsilon)$” denotes a small bin around $K$ with width $\epsilon$ in each of the four directions. The prescription (\[7b\]) can be derived rigorously and directly by first generating from Eq. (\[4\]) a piecewise constant function (“histogram”) through “binning”, $$\begin{aligned}
 \label{binning}
   \bar S (x,K) &=& \int_{{\rm bin}(K,\epsilon)} d^4p\, S_{\rm class}(x,p)
 \nonumber\\
   &=& \int_ {K-\epsilon/2}^{K+\epsilon/2} d^4p\, S_{\rm class}(x,p)\, ,
 \end{aligned}$$ and then inserting $\bar S(x,K)$ into Eq. (\[6\]). Eq. (\[binning\]) is a technical step required by finite event statistics; in practice, $\epsilon$ should be chosen as small as technically possible. Note that the selection of pairs in (\[7b\]) differs from the one in (\[5\]): for given $K$ the algorithm (\[7\]) selects pairs with $p_i \approx 
p_j \approx K$, independent of the value $\bbox{q}$ at which the correlation is to be evaluated. For different values of $\bbox{q}$ at fixed $\bbox{K}$, the correlator is obtained by weighting the [*same*]{} set of pairs with different weight factors $\cos\bigl(q{\cdot}(x_i-x_j)\bigr)$ which depend only on the spatial coordinates, but not on the momenta of the particles in the pair. This is consistent with the expectation from Eqs. (\[1a\],\[2\]) that the measured $\bbox{q}$-dependence of the correlator gives access to the distribution of relative distances $x_i-x_j$ in the source (at fixed $K$). Since the steps from Eq. (\[2\]) to Eq. (\[7\]) involve only identical transformations (the difference between (\[7a\]) and (\[7b\]) arising only from a different choice of emission functions (\[4\]) resp. (\[binning\])), they preserve the positivity of the second term in (\[1a\]). Please note that in contrast to (\[5\]), (\[7\]) is a continuous function of $\bbox{q}$, i.e. no binning in $\bbox{q}$ is required.

We have checked for the simple analytically solvable model presented in [@MKF96] that the algorithm (\[7\]) indeed allows to reconstruct from a classical Monte Carlo simulation the correct analytical expression for $C(\bbox{q},\bbox{K})$; in particular it removes the unphysical oscillations found in [@MKF96]. In Fig. 1 we show various approximations for the correlation function for a 1-dimensional source in $z$-direction with emission function $$\label{model}
  S(z,t;K) = e^{-z^2/R^2}\, \delta(K-\alpha z)\, 
                                   \delta(t)\, ,$$ with $R{=}10$ fm and $\alpha{=}0.02$ GeV/fm. This classical source features perfect $z$-$K$-correlations, $K{=}\alpha z$, which are, of course, quantum mechanically forbidden (see below). For the source (\[model\]), the exact correlator (\[1\]) can be calculated analytically, yielding $C(q) = 1 + \exp[q^2/(2\alpha^2 R^2)]$ (solid line in Fig. 1). The pathological rise of $C(q)$ [ *above*]{} the value 2 at $q{=}0$ is due to the violation of the uncertainty relation between $z$ and $p_z$ by the model (\[model\]); we selected this model because we believed that such a feature may be particularly difficult to reproduce in an event generator. Indeed, reconstructing the correlator from a Monte Carlo simulation of (\[model\]) via the Pratt prescription (\[5\]) yields the long-dashed line in Fig. 1 [@MKF96]; it can be calculated analytically from Eq. (\[3\]) as $C(q) = 1 + 
\cos(q^2/\alpha)$. This is always less than 2, but oscillates wildly around 1, becoming even 0 at regular $q$-intervals. This contradicts the positivity of the second term in (\[1\]) and is due to the smoothness approximation (\[3\]). – The two remaining lines in Fig. 1 show results from the same Monte Carlo simulation of (\[model\]) but reconstructing the correlator through the new algorithm (\[7b\]) instead of (\[5\]). For the dot-dashed line the bin width $\epsilon$ was chosen as $\epsilon= 10$ MeV, for the dotted line as $\epsilon = 5$ MeV. In both cases the simulated result deviates from the exact one (solid line); this is not a failure of the algorithm, but a result of the binning procedure (\[binning\]) applied to the source (\[model\]) – a purely technical step required by finite event statistics. As seen, the discrepancy decreases with decreasing bin width $\epsilon$.

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-2.5cm

While Eq. (\[7\]) thus solves the technical problems of the prescription (\[5\]), it does not address the principal physical problem that the classical distribution (\[4\]) is not a valid Wigner density since it violates the uncertainty relation by simultaneously fixing the coordinates $x_{i}$ and momenta $p_{i}$ of the emitted particles. It has been repeatedly suggested [@PGG90; @Z96; @MP97] that this can be remedied by replacing the sharply localized $\delta$-functions in (\[4\]) by minimum-uncertainty wave packets. In the remainder of this Letter we will discuss how such a procedure will modify the algorithm for calculating single-particle spectra and two-particle correlations from event generators, thereby rendering it quantum mechanically consistent.

Let us start from the folding relation for the emission function, derived in Ref. [@CH94] within the covariant current formalism [@GKW79]: $$\label{8}
   S(x,K) = \int d^4z \, d^4Q \, \rho(x-z,Q) \, S_{0}(z,K-Q) \, .$$ Here $$\label{9}
   S_{0}(x,p) = \int d^4v\, e^{-ip{\cdot}v}\, 
   j_0^*\left(x+{\textstyle{v\over 2}}\right) \, 
   j_0\left(x-{\textstyle{v\over 2}}\right)$$ is the Wigner density associated with an elementary source current amplitude $j_0(x)$, taken below as a Gaussian wavepacket, and $\rho(x,p)$ is a classical phase-space distribution for the centers $x_i$ and (average) momenta $p_i$ of these wave packets, here taken as $$\label{10}
   \rho(x,p) = \sum_{i=1}^N \delta^{(4)}(x-x_i) \, \delta^{(4)}(p-p_i)\, .$$ For the elementary source amplitude we make the ansatz $$\label{11}
   j_0(x) = {\cal N} \, \exp\left(-{\bbox{x}^2 \over 2\sigma^2}\right)
   \, \delta(x^0) \, ;$$ this source emits a Gaussian wave-packet with width parameter $\sigma$ at “freeze-out time” $x^0$. With this ansatz the elementary Wigner density $S_0$ becomes $$\label{12}
   S_0(x,p) =  8 (\pi\sigma^2)^{\frac{3}{2}} \,\vert{\cal N} \vert^2 
   \delta(x^{0}) \exp\left(-{\bbox{x}^2 \over \sigma^2}
                              -\sigma^2\bbox{p}^2 \right) \, .$$ Inserting this and (\[10\]) into (\[8\]) one finds $$\begin{aligned}
 \label{13}
   S(x,K) = {\cal N'} \sum_{i=1}^N &&\delta(x^0-x_i^0) \,
   \exp\left(-{(\bbox{x}-\bbox{x}_i)^2 \over \sigma^2}\right)
 \nonumber\\
   &\times& 
     \exp\left(-\sigma^2(\bbox{K}-\bbox{p}_i)^2 \right)\, , \end{aligned}$$ with ${\cal N'} = 8(\pi\sigma^2)^{\frac{3}{2}} \vert {\cal N}\vert^2$. This generalizes the classical ansatz (\[4\]) into a quantum mechanically consistent source Wigner density; with the free parameter $\sigma$ one can choose the relative degree of localization in coordinate space $(\sigma \to 0)$ or momentum space $(\sigma \to \infty)$, always preserving $\Delta x \cdot \Delta p = \hbar$.

The single particle spectrum, which occurs in the denominator of the second term of (\[1a\]), is now given by

\[14\] $$\begin{aligned}
 \label{14a}
   E_a {dN\over d^3p_a} &=& \int d^4x \, S(x,p_a)  = 
   {\cal N''} \sum_{i=1}^{N} v_i(\bbox{p}_a) \, ,
 \\
 \label{14b}
   {\cal N''} &=& (2\pi\sigma^2)^2 \vert {\cal N} \vert^2 \, ,
 \\
 \label{14c}
   v_ i (\bbox{p}_a) &=& 
   \exp\bigl(-\sigma^2 (\bbox{p}_a-\bbox{p}_i)^2\bigr) \, ,
 \end{aligned}$$

and similarly for $\bbox{p}_b$. It is normalized to the total number $N$ of pions in the event; this fixes the normalization constant ${\cal N}$ above. The exchange term (\[2\]) in the two-particle spectrum (with $q$ and $K$ defined in Eq. (\[1b\])) is similarly derived as

\[15\] $$\begin{aligned}
 \label{15a}
   {\rm Num}(\bbox{q},\bbox{K})\! &=& \!({\cal N''})^2 
   \exp(-{\textstyle{1\over 2}}\sigma^2\bbox{q}^2) 
   \sum_{i,j=1}^N w_{ij}(\bbox{q},\bbox{K})  ,
 \\
 \label{15b}
   w_{ij}(\bbox{q},\bbox{K})\! &=& \! v_i(\bbox{K}) \, v_j(\bbox{K}) \, 
   \cos\bigl(q{\cdot}(x_i-x_j)\bigr)  .
 \end{aligned}$$

The normalization ${\cal N''}$ drops out in the correlator (\[1a\]). Eq. (\[15\]) should be compared with the classical expressions (\[5\]) and (\[7\]). Like (\[7\]) (but contrary to (\[5\])) it is positive definite and thus free of spurious oscillations around 0. The sum in (\[15a\]) is now over [*all*]{} pairs $(i,j)$; the sharp restriction to the bin “${\rm bin}(K,\epsilon)$” in (\[7\]) is replaced by the Gaussian weight factors $v_i(\bbox{K})$ and $\exp(-{1\over 2} 
\sigma^2 \bbox{q}^2)$. Please note, however, that the former also occur in the new definition (\[14a\]) for the single particle spectrum, and must be kept in both places for consistency. Eq. (\[15b\]) shares with Eq. (\[7\]) the property (which was already discussed) that the cosine weight factor for each pair $(i,j)$ depends only on $x_i-x_j$, but not on $p_i$ and $p_j$. By combining Eqs. (\[14\]) and (\[15\]) it can be shown that now the correlator $C(\bbox{q,K})$ is always between 1 and 2, i.e. that the pathological rise above 2 shown in Fig. 1 cannot happen for an emission function which respects the uncertainty relation. In Ref. [@Wetal] the results (\[14\]) and (\[15\]) (but not (\[7\])) were derived with different methods, including finite multiplicity corrections.

Expressions (\[14\]) and (\[15\]) depend on one free parameter, the Gaussian width $\sigma$ of the wave-packets. It is instructive to discuss the two obvious limits, $\sigma \to 0$ and $\sigma \to 
\infty$. For $\sigma=0$ the elementary sources are sharply localized in space (cf. Eq. (\[11\])); as a result, the single-particle momentum spectrum (\[14\]) is completely flat. Furthermore, one sees from Eqs. (\[15\]) that the correlation function becomes $K$-independent, even for expanding sources of type (\[10\]) where the $x_i$ and $p_i$ are strongly correlated. Both features are clearly unrealistic. In the opposite limit, $\sigma \to \infty$, the Gaussian smearing factors in the single particle spectrum (\[14\]) disappear, and Eq. (\[14a\]) degenerates to a sum over $\delta$-functions; this is the usually employed algorithm for computing single-particle spectra from event generators with classical particle propagation [@W93; @SSG89; @PSK92]. The two-particle correlation term (\[15\]), on the other hand, is then sharply concentrated at $q=0$, i.e. the correlator $C(\bbox{q},\bbox{K})$ drops from 2 to 1 over a $q$-range of order $1/\sigma$. This translates into a source radius $\sim \sigma$ and reflects the diverging spatial extension of the elementary wavepackets in this limit, irrespective of the (localized) spatial distribution $\rho(x,p)$ of their centers.

It is thus clear that in practice $\sigma$ must be kept finite, but non-zero. As pointed out in [@PGG90; @Z96; @MP97] this implies a broadening of the single-particle momentum distributions relative to the one derived from the classical phase-space distribution $\rho (x,p)$ of freeze-out points. Using the algorithms (\[14\],\[15\]) for a quantum mechanically consistent computation of spectra and Bose-Einstein correlations from classical event generators thus requires a retuning of the codes to elementary $e^+e^-$ and $pp$ collisions, using the same algorithms there.

This last step imposes rather restrictive limits for the value of $\sigma$ [@PGG90]. Since the pions from high energy $pp$ and $e^+e^-$ collisions have an average transverse momentum $\langle 
p_\perp \rangle \simeq 0.35$ GeV, the Gaussian width in momentum space of the elementary wave packets must be below this value. This implies $\sigma \gtrsim 0.5$ fm. On the other hand, the effective source radii for such collisions extracted from 2-particle correlations are of the order of only $0.8 - 2$ fm [@L89]. This implies $\sigma 
\lesssim 1$ fm. It was already pointed out in the pioneering GGLP paper [@GGLP60] that this value roughly agrees with the pion’s Compton wavelength. This suggests the following interpretation of the measured pion spectra and Bose-Einstein correlations from elementary hadron-hadron and $e^+e^-$ collisions: each elementary collision produces a small number of elementary Gaussian wave packets with width $\sigma$, 0.5 fm $< \sigma <$ 1 fm. For $\sigma \sim 0.5$ fm the width of the Gaussian momentum distribution in (\[14c\]) nearly exhausts the measured $\langle p_\perp 
\rangle$; the measured single-pion spectra thus reflect mostly the intrinsic momentum distribution of the elementary pion wave packet. On the other hand, the HBT radius $R_{\perp}^{\rm HBT} = 
\sigma/\sqrt{2}$ corresponding to the $q$-Gaussian in (\[15a\]) nearly exhausts the values extracted from two-particle correlation measurements; the latter thus mirror the intrinsic spatial width of the wave packets. There is very little additional room in the data for random (thermal) motion of the wave packets relative to each other, nor for their spatial distribution over a larger volume. If the total source is much bigger than 1 fm, it must expand very rapidly, with homogeneity regions which are not much larger than the size of an elementary wave packet. In this sense, pion spectra from $e^{+}e^{-}$ and $pp$ collision measure the smallest sources compatible with the uncertainty relation.

In summary, we have derived a new algorithm for the computation of single-particle spectra and two-particle correlations from classical event generators. The classical algorithm (\[7\]) removes the recently discovered deficiencies of the presently employed Pratt algorithm (\[5\]). The quantum mechanical algorithm (\[14\],\[15\]) additionally ensures that the uncertainty relation is not violated. We also showed how the free parameter $\sigma$ in the latter algorithm can be fixed from elementary $e^{+}e^{-}$ and hadron-hadron collisions.

We acknowledge stimulating conversations with P. Foka, H. Kalechofsky, M. Martin, and S. Pratt, as well as many discussions during the HBT96 Workshop at the ECT\* in Trento, Sept. 16-27, 1996, which brought the problems with (\[5\]) into the open. This work was supported by BMBF, GSI and DFG. Q.H.Z. acknowledges support by the Alexander von Humboldt Foundation through a Research Fellowship.

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