For a set of N identical massive boson wavepackets with optimal initial quantum mechanical localization, we derive the Hanbury-Brown/Twiss (HBT) two-particle correlation function. Our result provides finite multiplicity corrections to the coherent state formalism and allows to trace back an error in the so-called cos-prescription. It suggests that what the HBT radius parameters in very small boson emitting systems (e.g. Z_0-decays, p-pbar annihilation) measure is essentially the initial spatial wavepacket width. Both one- and two-particle spectra depend explicitly on this width. Our derivation gives an algorithm for calculating one-particle spectra and two-particle correlations from an arbitrary phase space occupation (q_i,p_i,t_i) as e.g. returned by event generators of heavy ion collisions

Foka, P Y

Kalechofsky, H

Martin, M

Slotta, C

Wiedemann, Urs Achim

Zhang, Q H

arXiv

CERN Document Server

Quantum Mechanical Localization Eects for Bose-Einstein CorrelationsU.A. Wiedemanna;b, P. Fokab, H. Kalechofskyb, M. Martinb, C. Slottaa, Q.H. ZhangaaInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, GermanybDepartment de physique nucleaire et corpusculaire, Universite de Geneve, Switzerland(March 22, 1997)For a set of N identical massive boson wavepackets withoptimal initial quantum mechanical localization, we derive theHanbury-Brown/Twiss (HBT) two-particle correlation func-tion. Our result provides nite multiplicity corrections to thecoherent state formalism and allows to trace back an error inthe so-called cos-prescription. It suggests that what the HBTradius parameters in very small boson emitting systems (e.g.Z0-decays, p-p annihilation) measure is essentially the ini-tial spatial wavepacket width . Both one- and two-particlespectra depend explicitly on this width . Our derivationgives an algorithm for calculating one-particle spectra andtwo-particle correlations from an arbitrary phase space occu-pation (qi;pi; ti)i=1;N as e.g. returned by event generators ofheavy ion collisions.PACS numbers: 25.75.+r, 07.60.ly, 52.60.+hTwo-particle correlations C(Q;K) of identical parti-cles are the only known observables giving access to thespace-time structure of the particle emitting source inheavy ion collisions. Their interpretation is based on theresult of the coherent state formalism [1,2] which readsin the plane wave approximation for a large number ofsourcesC(Q;K) = 1 +R d4xS(x;K) eixQ2Rd4xS(x; P1)Rd4y S(y; P2); (1a)Q = P1 − P2 ; K =12 (P1 + P2) : (1b)In this setting, an Hanbury-Brown/Twiss (HBT) inter-ferometric analysis aims at extracting from the correla-tor C(Q;K) as much information as possible about thespace-time emission function S(x;K). Since this emis-sion function cannot be reconstructed unambiguouslyfromC(Q;K) [3], (1) is mainly used in the study of modelemission functions S(x;K). These studies have clariedto a considerable extent the question which geometri-cal and dynamical source characteristics are reflected inwhich particular momentum dependencies of the corre-lator (cf. [3] and refs. therein). A comparison with mea-sured correlations then allows to constrain the class ofsource models consistent with data.Microscopic event generators are one important tool togenerate model emission functions. Here, we do not dis-cuss in how far existing event generators (e.g. RQMD [4],VENUS [5]) provide an internally consistent calculationof the phase space distribution. None of them propagates(anti)-symmetrized N-particle states from rst principles,and the resulting diculties in calculating 2-particle cor-relations have been discussed recently in great detail [6].The typical event generator output is a set  of phasespace points at given times zi = (qi;pi; ti) which oneassociates with the \points of last interactions". How-ever, the Heisenberg uncertainty principle allows to in-terpret the zi only as mean positions of boson wave pack-ets. To specify the localization of these wavepackets inphase space, at least one additional parameter is needed,e.g. the initial spatial wavepacket width . Irrespectiveof how the phase space occupation has been obtained, weshall take the set  and the width  as initial conditionfor the present investigation:  and  dene the bosonemitting source. For notational simplicity, we restrict ourdiscussion to one particle species, negative pions, say.The problem in associating an emission functionS(x;K) to the distribution  is that  is a discrete phasespace distribution of on-shell particles. In contrast, theemission function S(x;K) of the coherent state formal-ism is a continuous distribution which allows for o-shellmomenta K. Often, one circumvents this problem byan ad hoc prescription, weighting each particle pair (i; j)with a probability ij , qi being 4-vectors qi = (ti;qi),C(Q;K) = 1N(Q;K)P(i;j) ij ; (2a)ij = 1 + cos((pi − pj)  (qi − qj)) : (2b)Here, C(Q;K) denotes the 2-particle correlator forthose pairs whose relative and average pair momentapi−pj ,12 (pi+pj) lie in the bin Q, K. N(Q;K)is the corresponding number of particle pairs. A tenta-tive argument to justify the prescription (2) is that ijcoincides with the formal Born probability density ΨΨof the Bose-Einstein symmetrized 2-particle plane waveij = Ψ(qi; qj ; pi; pj) Ψ(qi; qj ; pi; pj) ; (3a)Ψ(qi; qj ; pi; pj) =1p2(eipiqi+ipjqj + eipjqi+ipiqj: (3b)However, the prescription (2) based on the ansatz (3) isinconsistent [7] with the result (1) of the coherent stateformalism: The correlator in (1) is always larger thanunity [3]. In contrast, the expression (2) can drop belowunity in the region of suciently large relative momenta[7]. Also, the prescription (2) is dicult to reconcile withquantum mechanical localization requirements since theplane wave (3b) cannot be an eigenstate for both theposition and momentum operator.1In what follows, we take the quantum mechanical lo-calization of bosons into account by associating to thephase space emission points zi free Gaussian wavepack-ets of initial spatial width , [8,9]f ()zi (X; t) = (2)−3422i (t) 32exp (ipiX− iEit) exp− 122i(t)(X− qi(t))2(4a)qi(t) = qi +pim(t− ti) ; Ei =p2i2m ; (4b)2i (t) = 2 + i (t−ti)m: (4c)This one boson state (4a) is optimally localized around(qi;pi) in the sense that it saturates the Heisenberg un-certainty relation x  px =12 , with x =  at timet = ti. The time evolution of (4) is the free unperturbedevolution determined by the Hamiltonian H0 =2m , being the Laplacian. Since the i-th and j-th boson areidentical, we associate to the two emission points zi andzj the symmetrized two boson wave function ij(X;Y; t)(the normalization factor is omitted and plays no role inwhat follows)ij(X;Y; t) = f()zi(X; t)f ()zj (Y; t)+f ()zi (Y; t)f()zj(X; t) : (5)We now derive an algorithm for calculating one-particlespectra (P) and two-particle correlations C(P1;P2)from an arbitrary initial phase space distribution  ofbest localized boson wavepackets f()zi . Our rst step isto calculate for two identical bosons the detection proba-bility at time t at the positions X and Y with momentaP1, P2 respectively. This is given by the two-particleWigner phase space densityWij(X;Y;P1;P2; t) = ij(X;Y; t)(2)6(3)(P1 − P^1)(3)(P2 − P^2)ij(X;Y; t)=Zd3X1 d3Y1(X +X12 ;Y +Y12 ; t) eiP1X1eiP2Y1 (X− X12 ;Y −Y12 ; t) : (6)The corresponding probability to detect these bosonswith momenta P1 and P2 irrespective of their positionisPij(P1;P2) =Zd3X d3YWij(X;Y;P1;P2; t)= wi(P1;P1)wj(P2;P2)+wi(P2;P2)wj(P1;P1)+2wi(P1;P2)wj(P1;P2) cos 12ij ; (7a)12ij = (qi − qj)  (P1 − P2) ; (7b)wi(P1;P2) = e−24 (P1 −P2)2si(K) ; (7c)si(K) = 23 (2)32 e−2(pi−K)2: (7d)Here, Pi denotes 4-vectors Pi = (12mP2i ;Pi). We notethat Pij is independent of the detection time t, i.e., onlythe correlations which exist already at emission are mea-sured at time t in the detector. This t-independence isa consequence of the free time evolution; it is lost if -nal state interactions are included in the evolution of thewavepackets (4). Neglecting higher order symmetriza-tions, we dene the (unnormalized) two pion correlationR(P1;P2) for a set of N phase space points zi by sum-ming the probabilities Pij over all12N(N − 1) pairs (i; j)R(P1;P2) =X(i;j)Pij(P1;P2) := (P1) (P2)− 2Tc(P1;P2)+ NXi=1wi(P1;P2)eiti(E1−E2)−iqi(P1−P2)2 ; (8a)(P1) =NXi=1si(P1) : (8b)Here, si(P1) is the one-particle probability that a bo-son in the state f()zi is detected with momentum P1, cf.(7d). Accordingly, (P1) is the one-particle spectrum ofthe distribution  with spatial localization . The con-tribution Tc to R(P1;P2) corrects for the fact that thesums in the other two terms of (8a) include the N iden-tical pairs (i; i) which are not present in R(P1;P2),Tc(P1;P2) =NXi=1si(P1) si(P2) : (9)To obtain a normalized 2-particle correlation C(P1;P2),we choose the normalizationN(P1;P2) = (P1) (P2)− Tc(P1;P2) ; (10a)C(P1;P2) =R(P1;P2)N(P1;P2): (10b)This choice is motivated by the experimental praxisof \normalization by mixed pairs": An uncorrelated(mixed) pair is described by an unsymmetrized productstateuncorrij (X;Y; t) = f()zi(X; t)f ()zj (Y; t) ; (11)for which the two particle Wigner phase spacedensity and the corresponding detection probabilityP uncorrij (P1;P2) can be calculated according to (6). Tak-ing both distinguishable states uncorrij and uncorrji intoaccount, and summing over all pairs (i; j), we ndN(P1;P2) =X(i;j)P uncorrij (P1;P2) : (12)Hence, the normalization (10a) is the 2-particle detectionprobability for uncorrelated pairs. The correlator reads2C(P1;P2) = 1 +PNi=1 wi(P1;P2)eiqiQ2 − Tc(P1) (P2)− Tc: (13)In contrast to (2), this is a continuous function of themeasured momenta P1, P2, i.e., no binning of the cor-relator is required. Note that the normalization (10a)ensures that the correlator (13) is always smaller than 2and equals 2 for P1 − P2 = 0. (This follows from thefact that the sum of the rst two terms in (7a) is alwayslarger than the third one.) For any boson source, denedby an arbitrary phase space distribution  and a spa-tial wavepacket width , Eq. (13) provides an algorithmof how to calculate the 2-particle correlator, using (7c),(7d), (8b) and (9).To understand how the correlator (13) relates to theresult of the coherent state formalism (1), the limit of alarge number N of emission points is relevant. Tc(P1;P2)in (13) is a sum over N terms while the other terms inthe nominator and denominator are sums of N2 terms.In this sense, the Tc-dependence of (13) provides a nitemultiplicity correction and can be neglected as a sublead-ing 1N-contribution in the large N limit of (13),limN!1C(P1;P2) = 1 +e−22 Q2 P1i=1 si(K)eiqiQ2(P1i=1 si(P1))P1j=1 sj(P2) :(14)We note that in the derivation of (1), subleading 1N -contributions are dropped [2]. The large N approxima-tions (1) and (14) are clearly justied for pion interfer-ometry in ultrarelativistic (Pb-Pb) heavy ion collisionswhere typical pion multiplicities are in the hundreds. Forsmaller systems, however, and especially in studies of themultiplicity dependence of HBT correlations [11], onemight wish to start from the expression for nite mul-tiplicity (13). Expression (14) can be obtained from thecoherent state result (1) by insertingS(x;K) =1Xi=1Si(x;K) ; (15a)Si(x;K) = N (t− ti) e−12(x− qi)2e−2(K− pi)2 ; (15b)where N is an arbitrary normalization factor. In thissense, (15a) is the emission function for a source  withinitial spatial localization . It contains the informa-tion about how the initial phase space emission pointszi and the measured momenta K are correlated. Spa-tial and temporal components are not treated equally in(15b), since our derivation is not Lorentz covariant. TheLorentz covariant setting used in (1) allows for an addi-tional dependence of S(x;K) on the temporal componentof K which does not exist in our derivation. In practicalapplications however, the emission function (1) is usedin the so-called on-shell approximation, where this addi-tional K0-dependence is not employed, [3].In the derivation of (13), no averaging was involved.However, Eq. (15) is easily generalized to continuousphase space distributions (q;p; t) which can encode sta-tistical assumptions. In this case, one extends the sum(15a) to a phase space integral weighted by the distribu-tion ,S(x;K) =Zd3qi d3pi dti (qi;pi; ti)Si(x;K) : (16)Both the 2-particle correlator (13) and the 1-particlespectrum (P1) in (8b) depend on the initial spatial lo-calization  which is an additional free parameter. Wenow discuss this -dependence. We rst consider thelimit  ! 0, in which the Gaussian wavepacket (4a) de-scribes at freeze out (t = t(i)) a state with position uncer-tainty x = 0, i.e., the source is sharply (\classically")localized in conguration space. The prize for this opti-mal spatial information is that nothing can be said aboutthe initial momenta pi at emission. The measured mo-mentum correlations contain spatial information aboutthe source, namelylim!0C(P1;P2) = 1 +1N(N − 1)X(i;j)cos 12ij : (17)Due to the cos 12ij -term, cf. (7b), the dependence of the2-particle correlator (17) on the measured relative mo-mentum P1 −P2 gives information on the initial relativedistances qi − qj in the source. This is the HBT eect.Eq. (17) diers signicantly from the cos-prescription(2): here, P1 − P2 is the measured relative pair momen-tum, while pi − pj in (2) denotes the initial momentumdierence. As a consequence, the sumP(i;j) in (17) goesover all pairs irrespective of the momenta pi, pj sincein the limit  ! 0, all information about these initialmomenta is lost, while the sum in (2) goes only overthose pairs for which the initial relative pair momentumpi − pj lies in the same bin as the measured P1 − P2.Since the correlator (17) is a limiting case of (14), it isalways larger than unity. In contrast, due to the wrongpair selection criterion, the correlator (2) can drop below1. This insuciency of (2) becomes more signicant forsources with strong q-p position-momentum correlation,as was noticed in [7].The other limiting case of (13) is the plane wave limitlim!1C(P1;P2) = 1 + P1;P2 : (18)In this limit, nothing can be said about the spatio-temporal extension of the source since the 2-particle sym-metrized wave functions (5) contain no space-time infor-mation.3The dierence between (17) and (18) shows that the-dependence of the 2-particle correlator cannot be ne-glected. As pointed out in [8,9], however, none of the twolimits is realistic. For  ! 0, one has sharp informationin conguration space but the momentum space infor-mation is lost and hence, the set  of phase space emis-sion points (qi;pi; ti) contains no information about theone-particle momentum spectrum (P) - the one-particlespectrum is flat. In the limit  !1, on the other hand,no space-time information is contained in . A realisticwidth  hence lies in between these two extremes.Also, a narrow spatial width  leads to a signif-icant broadening of the one-particle spectrum. Fora given statistical phase space distribution (q;p; t),the one-particle spectrum is obtained from (16) viaRd4xS(x;K). Especially, for a Boltzmann distributionof temperature T , (q;p; t) / exp(− p22mT ), one obtainsfrom (16) a one-particle spectrum of eective tempera-ture [9]Te = T +12m2 : (19) is a free parameter which has to be determined froma comparison to data. How can this be done? One ideais to look at systems which can be expected to providevery small, almost pointlike boson emission regions. Can-didates are e.g. the p-p annihilation process [10], Z0-decays [11], or elementary p-p collisions. The width ofthe HBT-correlator determined for these systems shouldbe dominated by the width . To obtain an argumentsupporting this idea, we consider the extreme case of a\pointlike source"  with no momentum dependence, forwhich all particles are emitted from the same space-timeposition ~q, ~t. Calculating the emission function (16) forthe corresponding (qi;pi; ti) = (3)(qi − ~q)(~t− ti), wendC(P1;P2) = 1 + e−22 Q2; (20a)RHBTpoint = =p2 : (20b)Several assumptions enter this result: for pointlikesources with an additional momentum dependence, thecorrelation is in general more complicated. Also, thewidth  could in principle depend on the emission pointszi, the localized wavepackets could have a dierent, non-Gaussian shape, etc. Still, Eq. (20) suggests that the sizeof the HBT radius parameters measured for very smallboson emitting systems is essentially given by .The measured HBT radii provide via (20b) an upperbound on the size of  while the eective slope (19) of theone-particle spectrum provides a lower bound. From thepion interferometric measurements of systems like the p-p annihilation process or Z0-decays [10,11], one infers onthe basis of (20b) a pion wavepacket width of the order  1 fm. This is in good agreement with the naturallocalization scale of the pion, its Compton wavelength.Remarkably, for such a localization, the additional quan-tum contribution to the temperature Te in (19) is oforder 12m2  100 MeV. This indicates that the initialspatial localization width  plays an important role in ac-counting for the slope of the measured one-particle spec-tra. For elementary systems [10,11], the bounds obtainedfrom (19) and (20) seem to leave only little leeway forthermal excitations or spatial extensions of the collisionwhich can not be accounted for by the nite quantumlocalization .In the present formalism, the role of an event generatorfor the boson emitting source is to provide a dynamicalcalculation of the phase space occupation  from somemore fundamental initial condition. The current praxisfor event generators of heavy ion collisions amounts todetermining the 1-particle spectrum in the limit  !1.We have shown that this limit is unrealistic and that arealistic spatial wavepacket width leads to a substantialbroadening of the one-particle spectrum. Our main re-sult is an algorithm which allows for the calculation ofboth the one-particle spectra via (P) in (8b), and thetwo-particle correlations via C(P1;P2) in (13), startingfrom an arbitrary initial phase space distribution  ofwavepackets with arbitrary spatial localization . Here,the spatial width  is an additional free parameter whichcan be determined by tuning the output of a microscopicevent generator to e.g. elementary p-p collisions. Wehave argued that the measured one- and two-particlepion spectra of such systems provide upper and lowerconstraints for the width , a realistic width being of theorder of the pion Compton wavelength.We thank Ulrich Heinz for stimulating discussions anda critical reading of the manuscript. This work was sup-ported by BMBF, DFG, GSI and DPNC.[1] E. Shuryak, Phys. Lett. B44, 387 (1973).[2] S. Chapman and U. Heinz, Phys. Lett. B340, 250 (1994).[3] U. Heinz, Lectures given at the NATO ASI on "Correla-tions and Clustering Phenomena in Subatomic Physics"at Dronten, Netherlands, nucl-th/9609029.[4] H. Sorge, H. Sto¨cker and W. Greiner, Ann. Phys. 192,266 (1989).[5] K. Werner, Phys. Rep. 232, 87 (1993).[6] J. Aichelin, nucl-th/9609006.[7] M. Martin, H. Kalechofsky and P. Foka, nucl-th/9612023.[8] S. Padula, M. Gyulassy and S. Gavin, Nucl. Phys. B329,357 (1990).[9] H. Merlitz and D. Pelte, nucl-th/9702005.[10] G. Goldhaber, S. Goldhaber, W. Lee and A. Pais, Phys.Rev. 120, 300 (1960).[11] OPAL Collab., G. Alexander et al., CERN-PPE/96-90.4

Quantum Mechanical Localization Effects for Bose-Einstein Correlations

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

Quantum Mechanical Localization Effects for Bose-Einstein Correlations

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