Quantum mechanical localization effects for Bose-Einstein correlations

Article

Reference

Quantum mechanical localization effects for Bose-Einstein correlations

WIEDEMANN, Urs Achim, et al.

Abstract

For a set of N identical massive boson wave packets with optimal initial quantum mechanical localization, we calculate the Hanbury-Brown–Twiss (HBT) two-particle correlation function.

Our result provides an algorithm for calculating one-particle spectra and two-particle correlations from an arbitrary phase space occupation (qi,pi,ti)i=1,N as, e.g., returned by event generators. It is a microscopic derivation of the result of the coherent state formalism, providing explicit finite multiplicity corrections. Both the one- and two-particle spectra depend explicitly on the initial wave packet width σ which parametrizes the quantum mechanical wave packet localization. They provide upper and lower bounds which suggest that a realistic value for σ has the order of the Compton wavelength.

WIEDEMANN, Urs Achim, et al . Quantum mechanical localization effects for Bose-Einstein correlations. Physical Review. C , 1997, vol. 56, no. 2, p. R614-R618

DOI : 10.1103/PhysRevC.56.R614

Available at:

http://archive-ouverte.unige.ch/unige:112301

Disclaimer: layout of this document may differ from the published version.

1 / 1

Quantum mechanical localization effects for Bose-Einstein correlations

U. A. Wiedemann,^1,2P. Foka,² H. Kalechofsky,² M. Martin,² C. Slotta,¹ and Q. H. Zhang¹

1Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany

2De´partment de Physique Nucle´aire et Corpusculaire, Universite´ de Gene`ve, Gene`ve, Switzerland

~Received 22 April 1997!

For a set of N identical massive boson wave packets with optimal initial quantum mechanical localization, we calculate the Hanbury-Brown–Twiss~HBT!two-particle correlation function. Our result provides an algo- rithm for calculating one-particle spectra and two-particle correlations from an arbitrary phase space occupa- tion (q_i,p_i,t_i)_i51,N as, e.g., returned by event generators. It is a microscopic derivation of the result of the coherent state formalism, providing explicit finite multiplicity corrections. Both the one- and two-particle spectra depend explicitly on the initial wave packet widthswhich parametrizes the quantum mechanical wave packet localization. They provide upper and lower bounds which suggest that a realistic value fors has the order of the Compton wavelength.@S0556-2813~97!50708-4#

PACS number~s!: 25.75.Gz, 02.70.Lq, 24.10.Lx, 52.60.1h

Two-particle correlations C(Q,K) of identical particles are the only known observables giving access to the space- time structure of the particle emitting source in heavy ion collisions. Their interpretation is based on the result of the coherent state formalism@1,2#which reads in the plane wave approximation for a large number of sources

C~Q,K!511 u*d⁴xS~x,K!e^ix•Qu²

*d⁴xS~x, P₁!*d⁴y S~y , P₂!, ~1a! Q5P₁2P₂, K5¹2~P₁1P₂!. ~1b! In this setting, an Hanbury-Brown–Twiss ~HBT! interfero- metric analysis aims at extracting from the correlator C(Q,K) as much information as possible about the space- time emission function S(x,K). Since this emission function cannot be reconstructed unambiguously from C(Q,K) @3#, Eq. ~1!is mainly used in the study of model emission func- tions S(x,K). These studies have clarified to a considerable extent the question which geometrical and dynamical source characteristics are reflected in which particular momentum dependencies of the correlator ~cf. @3# and references therein!. A comparison with measured correlations then al- lows to constrain the class of source models consistent with data.

Microscopic event generators are one important tool to generate model emission functions. Here, we do not discuss in how far existing event generators ~e.g., RQMD @4#, VE- NUS@5#, ARC@6#!provide an internally consistent calcula- tion of the phase space distribution. None of them propagates

~anti!-symmetrized N-particle states from first principles, and the resulting difficulties in calculating two-particle correla- tions have been discussed recently in great detail @7#. The typical event generator output is a set S of phase space points at given times z_i5(q_i,p_i,t_i) which one associates with the ‘‘points of last interactions.’’ However, the Heisen- berg uncertainty principle allows one to interpret the z_i only as mean positions of boson wave packets. To specify the localization of these wave packets in phase space, at least one additional parameter is needed, e.g., the initial spatial wave packet width s. Irrespective of how the phase space

occupation has been obtained, we shall take the setSand the width s as initial condition for the present investigation:S ands define the boson emitting source. For notational sim- plicity, we restrict our discussion to one particle species, negative pions, say.

The problem in associating an emission function S(x,K) to the distribution S is thatS is a discrete phase space dis- tribution of on-shell particles. In contrast, the emission func- tion S(x,K) of the coherent state formalism is a continuous distribution which allows for off-shell momenta K. Often, one circumvents this problem by approximating the Pratt al- gorithm@8#by an ad hoc prescription: each particle pair (i, j ) is weighted with a probability ri j, q_i being 4-vectors q_i5(t_i,q_i),

C~DQ,DK!5 1

N~DQ,DK!_~

(

ri j511cos@~p_i2p_j!~q_i2q_j!#. ~2b! Here, C(DQ,DK) denotes the two-particle correlator for pairs whose relative and average pair momenta p_i2p_j,

1

2(p_i1p_j) lie in the binDQ, DK. N(DQ,DK) is the corre- sponding number of particle pairs. A tentative argument to justify the prescription~2!is thatri j coincides with the for- mal Born probability density C*_C of the Bose-Einstein symmetrized two-particle plane wave

ri j5C*_~q_i,q_j, p_i, p_j!C~q_i,q_j, p_i, p_j!, ~3a!

C~q_i,q_j, p_i, p_j!5 1

A

However, the prescription ~2! based on the ansatz~3! is in- consistent @9# with the result Eq. ~1! of the coherent state formalism: The correlator in ~1!is always larger than unity

@3#. In contrast, the expression ~2! can drop below unity in the region of sufficiently large relative momenta @9#. Also, the prescription ~2! is difficult to reconcile with quantum 56

0556-2813/97/56~2!/614~5!/$10.00 R614 © 1997 The American Physical Society

mechanical localization requirements since the plane wave

~3b! cannot be an eigenstate for both the position and mo- mentum operator.

In what follows, we take the quantum mechanical local- ization of bosons into account by associating to the phase space emission points z_i free Gaussian wave packets of ini- tial spatial width s @10,11#

f_z

i

~s!~X,t!5~ps²!^23/4

S

D

3exp

S

D

q_i~t!5q_i1p_i

m~t2t_i!, Ei5p_i²

2m, ~4b!

si

2~t!5s²1i~t2t_i!

m . ~4c!

This one boson state ~4a! is optimally localized around (q_i,p_i) in the sense that it saturates the Heisenberg uncer- tainty relation Dx•Dp_x51, with Dx_i5s ^{at time t}5t_i. The time evolution of Eq. ~4! is the free unperturbed evolution determined by the Hamiltonian H₀5D/2m, D being the La- placian. Since the ith and j th boson are identical, we associ- ate to the two emission points z_iand z_jthe symmetrized two boson wave functionFi j(X,Y,t) ~the normalization factor is omitted and plays no role in what follows!

Fi j~X,Y,t!5f_z

i

~s!~X,t!f_z

j

~s!~Y,t!1f_z

i

~s!~Y,t!f_z

j

~s!~X,t!.

~5! We now derive an algorithm for calculating one-particle spectran(P) and two-particle correlations C(P₁,P₂) from an arbitrary initial phase space distributionS of best localized boson wave packets f_z

i (s)

. Our first step is to calculate for two identical bosons the detection probability at time t at the positions X and Y with momenta P₁, P₂, respectively. This is given by the two-particle Wigner phase space density@12#

W_{i j}~X,Y,P₁,P₂,t!5Fi j~X,Y,t!~2p!⁶d^~3!~P₁2Pˆ

1! 3d^~3!~P₂2Pˆ₂!Fi j*_~X,Y,t!

5

E

S

D

3e^iP1X1e^iP2Y1F*

S

D

~6! The corresponding probability to detect these bosons with momenta P₁ and P₂ irrespective of their position is

P_{i j}~P₁,P₂!5

E

5w_i~P₁,P₁!w_j~P₂,P₂! 1w_i~P₂,P₂!w_j~P₁,P₁! 12 w_i~P₁,P₂!w_j~P₁,P₂!

3cos@~q_i2q_j!~P₁2P₂!#, ~7a! w_i~P₁,P₂!5e^2~s2/4!~P₁2P₂!²s_i~K!, ~7b! s_i~K!52³~ps²!^3/2e^2s2~pi2K!2. ~7c! Here, P_i denotes 4-vectors P_i5(1/2mP_i²,P_i). We note that P_{i j} is independent of the detection time t, i.e., only the cor- relations which exist already at emission are measured at time t in the detector. This t independence is a consequence of the free time evolution; it is lost if final state interactions are included in the evolution of the wave packets ~4!. Ne- glecting higher-order symmetrizations, we define the~unnor- malized!two pion correlation R(P₁,P₂) for a set of N phase space points z_i by summing the probabilities P_{i j} over all

1

2N(N21) pairs (i, j )

R~P₁,P₂!5_~

(

5n~P₁!n~P₂!22 T_c~P₁,P₂!

1

U ⁽

U

~8a! n~P!5

(

N

s_i~P!. ~8b! Here, s_i(P) is the one-particle probability~7c!that a boson in the state f_z

i (s)

is detected with momentum P. Accordingly, n(P) is the one-particle spectrum of the distributionS with spatial localizations. The contribution T_c to R(P₁,P₂) cor- rects for the fact that the sums in the other two terms of Eq.

~8a!include the N identical pairs (i,i) which are not present in R(P₁,P₂),

T_c~P₁,P₂!5

(

To obtain a normalized two-particle correlation C(P₁,P₂), we choose the normalization

N~P₁,P₂!5n~P₁!n~P₂!2T_c~P₁,P₂!, ~10a! C~P₁,P₂!5R~P₁,P₂!

N~P₁,P₂!. ~10b! This choice is motivated by the experimental praxis of ‘‘nor- malization by mixed pairs:’’ An uncorrelated~mixed!pair is described by an unsymmetrized product state

56 QUANTUM MECHANICAL LOCALIZATION EFFECTS FOR . . . R615

Fi j uncorr

~X,Y,t!5f_z

i

~s!~X,t!f_z

j

~s!~Y,t!, ~11! for which the two-particle Wigner phase space density and the corresponding detection probability P_{i j}^uncorr(P₁,P₂) can be calculated according to Eq.~6!. Taking both distinguish- able states Fi j

uncorr

and Fji uncorr

into account, and summing over all pairs (i, j ), we find

N~P₁,P₂!5_~

(

Hence, the normalization Eq.~10a!is the 2-particle detection probability for uncorrelated pairs. The correlator reads

C~P₁,P₂!511u(i51

N w_i~P₁,P₂!e^iqiQu²2T_c

n~P₁!n~P₂!2T_c . ~13! In contrast to Eq. ~2!, this is a continuous function of the measured momenta P₁, P₂, i.e., no binning of the correlator is required. Note that the normalization~10a!ensures that the correlator ~13! is always smaller than 2 and equals 2 for P₁2P₂50. ~This follows from the fact that the sum of the first two terms in Eq. ~7a! is always larger than the third one.! For any boson source, defined by an arbitrary phase space distributionS and a spatial wave packet widths^{, Eq.}

~13! provides an algorithm of how to calculate the two- particle correlator, using Eqs. ~7b!,~7c!,~8b!, and~9!.

To understand how the correlator~13!relates to the result of the coherent state formalism~1!, the limit of a large num- ber N of emission points is relevant. T_c(P₁,P₂) in Eq.~13!is a sum over N terms while the other terms in the nominator and denominator are sums of N² terms. In this sense, the T_c-dependence of Eq.~13!provides a finite multiplicity cor- rection and can be neglected as a subleading 1/N contribu- tion in the large N limit of Eq. ~13!,

Nlim→`C~P<sub>1</sub>,P<sub>2</sub>!511e<sup>2~s2/2!Q2</sup>u(i`51s_i~K!e^iqiQu²

@(i`51s<sub>i</sub>~P<sub>1</sub>!#~(`j51s_j~P₂!! . ~14! The large N approximation ~14!is clearly justified for pion interferometry in ultrarelativistic ~Pb-Pb! heavy ion colli- sions where typical pion multiplicities are in the hundreds.

For smaller systems@15#, however, one should start from the expression for finite multiplicity Eqs.~13!. The quantitative differences between ~13! and ~14! are discussed following Fig. 2 below. Expression~14!can be obtained from the co- herent state result~1!by inserting

S~x,K!5

(

S_i~x,K!5Nd~t2t_i!e^{2~1/s2!~x2qi!2}

3e^2s2~K2pi!2, ~15b! where N is an arbitrary normalization factor. In this sense, Eq.~15a!is the emission function for a sourceS with initial spatial localizations. It contains the information about how the initial phase space emission points z_i and the measured momenta K are correlated. Spatial and temporal components are not treated equally in Eq. ~15b!, since our derivation is

not Lorentz covariant. The Lorentz covariant setting used in

~1! allows for an additional dependence of S(x,K) on the temporal component of K which does not exist in our deri- vation. In practical applications, however, the emission func- tion ~1! is used in the so-called on-shell approximation, where this additional K₀ dependence is not employed@3#.

Both the two-particle correlator ~13!and the one-particle spectrumn^{(P) in Eq.}~8b!depend on the initial spatial local- ization s which is an additional free parameter. We now discuss this s dependence. We first consider the limit s→0, in which the Gaussian wave packet~4a!describes at freeze out (t5t⁽ⁱ⁾) a state with position uncertaintyDx50, i.e., the source is sharply ~‘‘classically’’! localized in con- figuration space. The price for this optimal spatial informa- tion is that nothing can be said about the initial momenta p_i at emission, the one-particle spectrum n(P) is flat. The measured momentum correlations contain spatial informa- tion about the source, namely

s→lim0C~P₁,P₂!511(_~i, j!cos@~q_i2q_j!~P₁2P₂!#

N~N21! .

~16! Due to the cos term, the dependence of the two-particle cor- relator ~16! on the measured relative momentum P₁2P₂ gives information on the initial relative distances q_i2q_j in the source. This is the HBT effect. Equation ~16! differs significantly from the cos prescription ~2!: here, P₁2P₂ is the measured relative pair momentum, while p_i2p_j in Eq.

~2! denotes the initial momentum difference. As a conse- quence, the sum ((i, j ) in Eq. ~16! goes over all pairs irre- spective of the momenta p_i, p_j since in the limit s→0, all information about these initial momenta is lost, while the sum in Eq.~2!goes only over those pairs for which the initial relative pair momentum p_i2p_j lies in the same bin as the measured P₁2P₂. Since the correlator~16!is a limiting case of Eq.~14!, it is always larger than unity. In contrast, due to the wrong pair selection criterion, the correlator~2!can drop below 1. This insufficiency of Eq.~2!becomes more signifi- cant for sources with strong q-p position-momentum corre- lation, as was noticed in@9#.

The other limiting case of Eq.~13!is the plane wave limit

slim→`C~P₁,P₂!511dP₁,P₂. ~17! In this limit, nothing can be said about the spatiotemporal extension of the source since the two-particle symmetrized wave functions ~5!contain no space-time information.

The difference between Eqs.~16!and~17!shows that the s dependence of the two-particle correlator cannot be ne- glected. As pointed out already in @10,11#, none of the two limits is realistic. For s→0, one has sharp information in configuration space, but the momentum space information is lost and hence, the set S of phase space emission points (q_i,p_i,t_i) contains no information about the one-particle momentum spectrum n(P). In the limits→`, on the other hand, no space-time information is contained inS. A realis- tic widths hence lies in between these two extremes.

For illustration, let us consider a simple model for the phase space distribution r(q,p,t) of the emission points z_i, r~q,p,t!5e^2~q2/2R2!e^2~p2/2mT!d~t2t₀!. ~18!

This model assumes that the wave packets are emitted in- stantaneously at time t₀ from a spatial region of Gaussian width R with Boltzmann distributed momenta. To discuss in the above formalism such continuous phase space distribu- tions which encode statistical assumptions, we extend the sum~15a!to a phase space integral weighted by the~classi- cal!distributionr^,@2,13#

S~x,K!5

E

For the model~18!, the calculation can be done analytically.

The unnormalized one-particle spectrum *d⁴xS(x,K) is found to be a Boltzmann distribution with effective tempera- ture @11#

T_eff5T1 1

2ms^{2. ~20!}

For the two-particle correlator of the model~18!, we find C~P₁,P₂!511e^{2Reff2 Q2}, ~21a! R_eff² 5R²1s²

2

2mTs²

112mTs^{2. ~21b!} Figure 1 shows thes dependence of the correlator for input parameters R55 fm, T5150 MeV. As long as the source size is significantly larger than the wave packet width, R@s, the correlator is essentially determined by R while in the opposite case, i.e., for very small source sizes, s ^deter- mines the width of C. Also, the one-particle spectrum shows a clear s dependence. According to Eq. ~20!, it broadens significantly for narrow spatial widths s^.s is a free param- eter which has to be determined from a comparison to data.

How can this be done? One idea is to look at systems which can be expected to provide very small, almost pointlike bo- son emission regions. Candidates are, e.g., the p- p¯ annihila-

tion process@14# or Z₀ decays@15#. The width of the HBT correlator determined for these systems should be dominated by the widths. In the extreme case of a ‘‘pointlike source’’

r^(qi,p_i,t_i)5d^(3)(qi2q˜)d^{( t˜}2t_i) with no momentum de- pendence, for which all particles are emitted from the same space-time position q˜, t˜, the HBT radius parameter reads, e.g.,

R_point^HBT5s^/

A

The difference between Eqs.~21b!and~22!indicates already the model dependence of this discussion. Both the geometry and dynamics of the very small sources are presumably more complicated. Also, the widthscould in principle depend on the emission points z_i, the localized wave packets could have a different, non-Gaussian shape, etc. Still, Eq.~22!sug- gests that the size of the HBT radius parameters measured for very small boson emitting systems is essentially given by s^.

From the pion interferometric measurements of systems like the p- p¯ annihilation process or Z₀ decays@14,15#, one infers on the basis of Eq. ~22!a pion wave packet width of the orders'1 fm. This is in good agreement with the natu- ral localization scale of the pion, its Compton wavelength

@14#. Remarkably, for such a localization, the additional quantum contribution to the temperature T_effin Eq.~20!is of order 1/2ms²'100 MeV. This indicates that the initial spa- tial localization widths plays an important role in account- ing for the slope of the measured one-particle spectra.

To mimic the role of an event generator, we have used the one-dimensional version of Eq. ~18! for a Monte Carlo model which produces sets ~‘‘events’’! S of phase space points according to the distribution r. This allows for the

FIG. 2. Multiplicity dependence of the approximation ~14!, in which the finite multiplicity corrections T_c are dropped. All lines show the correlator of the model~18!for input parameters R55 fm, m5139 MeV, T5150 MeV, ands50.5. Results are averaged over 1000 events. The thick solid line depicts the result for Eq. ~13! which agrees irrespective of the multiplicity N with the analytical result~21!. All other lines show results of the large N approxima- tion~14!for different multiplicities: N53~dashed line!, N55~dot- ted line!, N510 ~dash-dotted line!, N520 ~thin dashed line!, N540~thin dotted line!.

FIG. 1. The two particle correlator ~21a! of the phase space distribution ~18!for input parameters R55 fm, m5139 MeV and T5150 MeV. The different lines are for wave packet widths of s50.1 fm~thick solid line!, s51.0 fm ~dashed line!, s52.0 fm

~dotted line!, s54.0 fm ~dash-dotted line! and s58.0 fm ~solid line!.

56 QUANTUM MECHANICAL LOCALIZATION EFFECTS FOR . . . R617

study of the multiplicity dependence of the correlator ~13! and its large N approximation ~14! in which the correction terms T_c are neglected. Figure 2 shows the correlators cal- culated from Eqs.~13!and~14!for a sample of 1000 events of multiplicity N. Note that the prescriptions ~13! and ~14! provide continuous functions for the setsS of discrete phase space points and hence, no binning is required in Fig. 2.

Since the initial setsS have been obtained by Monte Carlo techniques, a statistical error is associated with the obtained results. In the present study, however, we have chosen a sufficiently large event sample to neglect statistical fluctua- tions. If included, they would lead to a minor broadening of the lines presented in Fig. 2 without changing our conclu- sions.

The main message of Fig. 2 is that irrespective of the event multiplicity, the correlator can be calculated from the corresponding continuous phase space distributionr^{via Eqs.}

~19!and~1!. The obtained expression always coincides with the correlator calculated from Eq. ~13!. Dropping the finite multiplicity corrections T_c in Eq. ~13! introduces an error which rapidly decreases with increasing event multiplicity.

Figure 2 hence supports the counting argument made above that T_c is a subleading 1/N contribution.

The Monte Carlo study presented here gives only a very simplified toy model in the spirit of @9#. In the present for- malism, the role of an event generator for the boson emitting source is to provide a dynamical calculation of the phase space occupationSfrom some more fundamental initial con-

dition. The current praxis for event generators of heavy ion collisions amounts to determining the one-particle spectrum in the limit s→`. We have shown that this limit is unreal- istic and that a realistic spatial wave packet width leads to a substantial broadening of the one-particle spectrum. Our main result is an algorithm which allows for the calculation of both the one-particle spectra vian^{(P) in Eq.}~8b!, and the two-particle correlations via C(P₁,P₂) in Eq.~13!, starting from an arbitrary initial phase space distribution S of wave packets with arbitrary spatial localization s. The number of numerical operations in Eqs. ~8b! and ~13! increases only linearly with the event multiplicity N. This is in contrast to the quadratic increase with N in previously used Bose- Einstein prescriptions like Eq. ~2! and makes the proposed algorithm very fast. The spatial widthsis an additional free parameter which has to be fixed in comparison with experi- mental data. The slope of the transverse mass spectra ~20! provides a lower bound fors while the two-particle correla- tors provide an upper bound. For pions these bounds are very tight and a realistic widthsis of the order of the pion Comp- ton wavelength.

We thank U. Heinz for helpful remarks and a critical reading of the manuscript. Stimulating discussions with J.-P.

Naef and L. Rosselet are acknowledged. This work was sup- ported by Alexander von Humbold Stiftung, BMBF, DFG, GSI, and Fonds national suisse de la recherche scientifique.

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