Parametrization of Bose-Einstein correlations and reconstruction of the source function in hadronic Z-boson decays using the L3 detector

Abstract

Bose-Einstein correlations of pairs of identical charged pions produced in hadronic Z decays are analyzed in terms of various parametrizations. A good description is achieved using a Lévy stable distribution in conjunction with a hadronization model having highly correlated configuration and momentum space, the tau-model. Using these results, the source function is reconstructed.

Bose-Einstein Correlations

Parametrization of Bose-Einstein correlations and reconstruction of the source function in hadronic Z-boson decays using the L3 detector

W.J. Metzger; T. Novák; T. Csörgo^*; W. Kittel for the L3 Collaboration

Radboud University, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands

ABSTRACT

Bose-Einstein correlations of pairs of identical charged pions produced in hadronic Z decays are analyzed in terms of various parametrizations. A good description is achieved using a Lévy stable distribution in conjunction with a hadronization model having highly correlated configuration and momentum space, the t-model. Using these results, the source function is reconstructed.

Key words: Bose-Einstein Correlations

I. INTRODUCTION

In particle and nuclear physics, intensity interferometry provides a direct experimental method for the determination of sizes, shapes and lifetimes of particle-emitting sources (for reviews see [1-5]). In particular, boson interferometry provides a powerful tool for the investigation of the space-time structure of particle production processes, since Bose-Einstein correlations (BEC) of two identical bosons reflect both geometrical and dynamical properties of the particle radiating source.

Here we study BEC in hadronic Z decay. We investigate various static parametrizations in terms of the four-momentum difference, Q = , and find that none give an adequate description of the Bose-Einstein correlation function. However, within the framework of models assuming strongly correlated coordinate and momentum space, a good description is achieved. We then reconstruct the complete space-time picture of the particle emitting source in hadronic Z decay.

The data used in the analysis were collected by the L3 detector [6-10] at an center-of-mass energy of

~ 91.2 . Approximately 36 million like-sign pairs of well-measured charged tracks of about 0.8 million hadronic Z decays are used [11].

We perform analyses on the complete sample as well as on two- and three-jet samples. The latter are found using calorimeter clusters with the Durham jet algorithm [12-14] with a jet resolution parameter y_cut = 0.006. To determine the thrust axis of the event we also use calorimeter clusters.

II. BOSE-EINSTEIN CORRELATION FUNCTION

The two-particle correlation function of two particles with four-momenta p₁ and p₂ is given by the ratio of the two-particle number density, r₂(p₁,p₂), to the product of the two single-particle number densities, r₁ (p₁)r₁(p₂). Since we are here interested only in the correlation R₂ due to Bose-Einstein interference, the product of single-particle densities is replaced by r₀(p₁,p₂), the two-particle density that would occur in the absence of Bose-Einstein correlations:

This r₂ is corrected for detector acceptance and efficiency using Monte Carlo events, to which a full detector simulation has been applied, on a bin-by-bin basis. An event mixing technique is used to construct r₀. This technique removes all correlations, e.g., resonances and energy-momentum conservation, not just Bose-Einstein. Hence, r₀ is corrected for this [11,15] using the JETSET Monte Carlo generator [16].

Since the mass of the two identical particles of the pair is fixed to the pion mass, the correlation function is defined in six-dimensional momentum space. Since Bose-Einstein correlations can be large only at small four-momentum difference Q, they are often parametrized in this one-dimensional distance measure. There is no reason, however, to expect the hadron source to be spherically symmetric in jet fragmentation. Recent investigations have, in fact, found an elongation of the source along the jet axis [15,17-19]. While this effect is well established, the elongation is actually only about 20%, which suggests that a parametrization in terms of the single variable Q, may be a good approximation.

This is not the case in heavy-ion and hadron-hadron interactions, where BEC are found not to depend simply on Q, but on components of the momentum difference separately [5,20-24]. However, in e⁺e^- annihilation at lower energy [25] it has been observed that Q is the appropriate variable. We checked this and confirm that this is indeed the case: We observe [11] that R₂ does not decrease when both q² = (₁ – ₂)² and = (E₁ - E₂)² are large while Q² = q² - is small, but is maximal for Q² = q²- = 0, independent of the individual values of q and q₀. The same is observed in a different decomposition: Q² = , where is the component transverse to the thrust axis and – (E₁ - E₂)² combines the longitudinal momentum and energy differences. Again, R₂ is maximal along the line Q = 0, as is shown in Fig. 1. This is observed both for two-jet and three-jet events. We conclude that a parametrization in terms of Q can be considered a good approximation for the purposes of this article.

III. PARAMETRIZATIONS OF BEC

With a few assumptions [2,5,26], the two-particle correlation function, Eq. (1), is related to the Fourier transformed source distribution:

where f(x) is the (configuration space) density distribution of the source, and (Q) is the Fourier transform (characteristic function) of f(x). The parameter l is introduced to account for several factors, such as the possible lack of complete incoherence of particle production and the presence of long-lived resonance decays if the particle emission consists of a small, resolvable core and a halo with experimentally unresolvable large length scales [27,28]. The parameter g and the (1 + dQ) term parametrize possible long-range correlations not adequately accounted for in the reference sample. While there is no guarantee that (1 + dQ) is the correct form, we will see that it does provide a good description of R₂ in the region Q > 1.5\GeV.

A. Gaussian distributed source

The simplest assumption is that the source has a symmetric Gaussian distribution, in which case (Q) = exp(iµQ - ) and

A fit of Eq. (3) to the data results in an unacceptably low confidence level. The fit is particularly bad at low Q values, as is shown in Fig. 2a for two-jet events and in Fig. 3a for three-jet events, from which we conclude that the shape of the source deviates from a Gaussian.

A model-independent way to study deviations from the Gaussian parametrization is to use [5,29,30] the Edgeworth expansion [31] about a Gaussian. Keeping only the first non-Gaussian term, we have

where k is the third-order cumulant moment and H₃(RQ) º (RQ)³ – 3RQ is the third-order Hermite polynomial. Note that the second-order cumulant corresponds to the radius R.

A fit of Eq. (4) to the two-jet data, shown in Fig. 2b, is indeed much better than the purely Gaussian fit. However, the confidence level is still marginal, and close inspection of the figure shows that the fit curve is systematically above the data in the region 0.6-1.2 and that the data for Q > 1.5 appear flatter than the curve, as is also the case for the purely Gaussian fit. Similar behavior is observed for three-jet events (Fig. 3b) and for all events.

B. Lévy distributed source

The symmetric Lévy stable distribution is characterized by three parameters: x₀, R, and a. Its Fourier transform, (Q), has the following form:

The index of stability, a, satisfies the inequality 0 < a < 2. The case a = 2 corresponds to a Gaussian source distribution with mean x₀ and standard deviation R. For more details, see, e.g., [32].

Then R₂ has the following, relatively simple, form [33]:

from the fit of Eq. (6) to the two-jet data, shown in Fig. 4, it is clear that the correlation function is far from Gaussian: a = 1.34±0.04. The confidence level, although improved compared to the fit of Eq. (3), is still unacceptably low, in fact worse than that for the Edgeworth parametrization. The same is true for three-jet events (Fig. 5) and for all events. The values of a are 1.39±0.04 for three-jet and 1.43±0.03 for all events, respectively.

Both the symmetric Lévy parametrization and the Edgeworth parametrizations do a fair job of describing the region Q < 0.6 GeV, but fail at higher Q. R₂ in the region Q > 1.5 GeV is nearly constant (» 1). However, in the region 0.6-1.5 GeV R₂ has a smaller value, dipping below unity [59], which is indicative of an anti-correlation. This is clearly seen in Figs. 4 and 5 by comparing the data in this region to an extrapolation of a linear fit, Eq. (6) with l = 0, in the region Q > 1.5 GeV. The inability to describe this dip in R₂ is the primary reason for the failure of both the Edgeworth and symmetric Lévy parametrizations.

C. Time dependence of the source

The parametrizations discussed so far, which have proved insufficient to describe the BEC, all assume a static source. The parameter R, representing the size of the source as seen in the rest frame of the pion pair, is a constant. It has, however, been observed that R depends on the transverse mass, m_t = , of the pions [34,35]. It has been shown [36,37] that this dependence can be understood if the produced pions satisfy, approximately, the (generalized) Bjorken-Gottfried condition [38-43], whereby the four-momentum of a produced particle and the space-time position at which it is produced are linearly related:

Such a correlation between space-time and momentum-energy is also a feature of the Lund string model as incorporated in JETSET, which is very successful in describing detailed features of the hadronic final states of e⁺e^- annihilation.

In the previous section we have seen that BEC depend, at least approximately, only on Q and not on its components separately. This is a non-trivial result. For a hydrodynamical type of source, on the contrary, BEC decrease when any of the relative momentum components is large [5,23]. Further, we have seen that R₂ in the region 0.6-1.5 GeV dips below its values at higher Q.

A model which predicts such a Q-dependence while incorporating the Bjorken-Gottfried condition is the so-called t-model, described below.

1. The t model

A model of strongly correlated phase-space, known as the t-model [44], explains the experimentally found invariant relative momentum dependence of Bose-Einstein correlations in e⁺e^- reactions. This model also predicts a specific transverse mass dependence of R₂, that we subject to an experimental test here.

In this model, it is assumed that the average production point in the overall center-of-mass system, , of particles with a given four-momentum k is given by

In the case of two-jet events,

where m_t is the transverse mass and t = is the longitudinal proper time [60]. For isotropically distributed particle production, the transverse mass is replaced by the mass in Eq. (9), while for the case of three-jet events the relation is more complicated. The second assumption is that the distribution of x^m (k^m) about its average, d_D (x^µ(k^m) - ^µ (k^m)), is narrower than the proper-time distribution. Then the emission function of the t-model is

where H(t) is the longitudinal proper-time distribution, the factor d_D(x - dk) describes the strength of the correlations between coordinate space and momentum space variables and r₁(k) is the experimentally measurable single-particle spectrum.

The two-pion distribution, r₂(k₁,k₂), is related to S(x,k), in the plane-wave approximation, by the Yano-Koonin formula [45]:

Approximating the function d_D by a Dirac delta function, the argument of the cosine becomes

Then the two-particle Bose-Einstein correlation function is approximated by

where (w) = òdtH(t)exp(iwt) is the Fourier transform of H(t). Thus an invariant relative momentum dependent BEC appears. Note that R₂ depends not only on Q but also on the average transverse mass of the two pions, _t.

Since there is no particle production before the onset of the collision, H(t) should be a one-sided distribution. We choose a one-sided Lévy distribution, which has the characteristic function [33] (for a ¹ 1)

where the parameter t₀ is the proper time of the onset of particle production and Dt is a measure of the width of the proper-time distribution. For the special case a = 1, see, e.g., [32]. Using this characteristic function in Eq. (13) yields

2. The t model for average m_t

Before proceeding to fits of Eq. (15), we first consider a simplification of the equation obtained by assuming (a) that particle production starts immediately, , t₀ = 0, and (b) an average m_t-dependence, which is implemented in an approximate way by defining an effective radius, R = . This results in:

where R_a is related to R by

Fits of Eq. (16) are first performed with R_a as a free parameter. The fit results obtained, for two-jet, three-jet, and all events are listed in Table I and shown in Fig. 6 for two-jet events and in Fig. 7 for three-jet events. They have acceptable confidence levels, describing well the dip below unity in the 0.6-1.5 GeV region, as well as the low-Q peak.

The fit parameters for the two-jet events satisfy Eq. (17). However, those for three-jet and all events do not. We note that the values of the parameters a and R do not differ greatly between 2- and 3-jet samples, the most significant difference appearing to be nearly 3s for a. However, these parameters are rather highly correlated (in the fit for all events, the correlation coefficients are r(l,R) = 0.95, r(l,a) = -0.67 and r(R,a) = -0.61, which makes the simple calculation of the statistical significance of differences in the parameters unreliable.

Fit results imposing Eq. (17) are given in Table II. For two-jet events, the values of the parameters are the same as in the fit with R_a free—only the uncertainties have changed. For three-jet and all events, the imposition of Eq. (17) results in values of a and R closer to those for two-jet events, but the confidence levels are very bad, a consequence of incompatibility with Eq. (17), an incompatibility that is not surprising given that Eq. (9) is only valid for two-jet events. Therefore, we only consider two-jet events in the remaining sections of this article.

3. The t model with m_t dependence

Fits of Eq. (15) to the two-jet data are performed in several m_t intervals. The resulting fits are shown for several m_t intervals in Fig. 8, and the values of the parameters obtained in the fits are listed in Fig. 9. The quality of the fits is seen to be statistically acceptable and the fitted values of the model parameters, a, t₀ and Dt, are stable and within errors independent of m_t, confirming the expectation of the t-model. We conclude that the t-model with a one-sided Lévy proper-time distribution describes the data with parameters t₀» 0 fm, a » 0.38±0.05 and Dt » 3.5±0.6 fm. These values are consistent with the fit of Eq. (16) in the previous section, including the value of R, which, combined with the average value of m_t (0.563), corresponds to Dt = 3.5 fm. Just as in the fit of Eq. (16), the parameters of the Lévy distribution are highly correlated. Typical values of the correlation coefficients are r(l,Dt) = 0.95, r(l,a) = -0.67 and r(Dt,a) = -0.9.

IV. THE EMISSION FUNCTION OF TWO-JET EVENTS

Within the framework of the t-model, we now reconstruct the space-time picture of the emitting process for two-jet events. The emission function in configuration space, S(x), is the proper time derivative of the integral over k of S(x,k), which in the t-model is given by Eq. (10). Approximating d_D by a Dirac delta function, we find

To simplify the reconstruction of S(x) we assume that it can be factorized in the following way:

where I(r) is the single-particle transverse distribution, G(h) is the space-time rapidity distribution of particle production, and H(t) is the proper-time distribution. With the strongly correlated phase-space of the t-model, h = y and r = p_tt/m_t. Hence,

where N_y and N_pt are the single-particle inclusive rapidity and p_t distributions, respectively. The factorization of transverse and longitudinal distributions has been checked. The distribution of p_t is, to a good approximation, independent of the rapidity [11].

With these assumptions and using H(t) as obtained from the fit of Eq. (15) (shown in Fig. 10) together with the inclusive rapidity and p_t distributions [11], the full emission function is reconstructed. Its integral over the transverse distribution is plotted in Fig. 11. It exhibits a "boomerang" shape with a maximum at low t and z but with tails reaching out to very large values of t and z, a feature also observed in hadron-hadron [46] and heavy ion collisions [47].

The transverse part of the emission function is obtained by integrating over z and azimuthal angle. Figure 12 shows the transverse part of the emission function for various proper times. Particle production starts immediately, increases rapidly and decreases slowly. A ring-like structure, similar to the expanding, ring-like wave created by a pebble in a pond, is observed. These pictures together form a movie that ends in about 3.5 fm, making it the shortest movie ever made of a process in nature. An animated gif file covering the first 0.3 fm (10^-24 sec) is available [48].

V. DISCUSSION

BEC of all events as well as two- and three-jet events are observed to be well-described by a Lévy parametrization incorporating strong correlations between configuration- and momentum-space. A Lévy distribution arises naturally from a fractal, or from a random walk or anomalous diffusion [49], and the parton shower of the leading log approximation of QCD is a fractal [50-52]. In this case, the Lévy index of stability is related to the strong coupling constant, a_s, by[53,54]

Assuming (generalized) local parton hadron duality [55-57], one can expect that the distribution of hadrons retains the features of the gluon distribution. For the value of a found in fits of Eq. (16) we find a_s = 0.37±0.04 for two-jet events, This is a reasonable value for a scale of 1-2 GeV, which is where the production of hadrons takes place. For comparison, from t decay, a_s(m_t» 1.8 GeV) = 0.35±0.03 [58].

It is of particular interest to point out the m_t dependence of the "width" of the source. In Eq. (15) the parameter associated with the width is Dt. Note that it enters Eq. (15) as DtQ²/_t. In a Gaussian parametrization the radius R enters the parametrization as R²Q². Our observance that Dt is independent of _t thus corresponds to R µ 1/ and can be interpreted as confirmation of the observance [34,35] of such a dependence of the Gaussian radii in 2- and 3-dimensional analyses of Z decays. The lack of dependence of all the parameters of Eq. (15) on the transverse mass is in accordance with the t-model.

Using the BEC fit results and the t-model, the emission function of two-jet events is reconstructed. Particle production begins immediately after the collision, increases rapidly and then decreases slowly, occurring predominantly close to the light cone. In the transverse plane a ring-like structure expands outwards, which is similar to the picture in hadron-hadron interactions but unlike that of heavy ion collisions.

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Received on 1 November, 2006

*

Visitor from Budapest, Hungary, sponsored by the Scientific Exchange between Hungary (OTKA) and The Netherlands (NWO), project B64- 27/N25186.

Publication Dates

• Publication in this collection
  24 Sept 2007
• Date of issue
  Sept 2007

History

• Received
  01 Nov 2006