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i v :n u c l -t h /9907019v 2 31 A u g 1999
Lambda-proton correlations in relativistic heavy ion collisions
Fuqiang Wang 1and Scott Pratt 2
1
Nuclear Science Division,Lawrence Berkeley National Laboratory,Berkeley,CA 94720,U.S.A.
2
Department of Physics and National Superconducting Cyclotron Laboratory,Michigan State University,East Lansing,
MI 48824,U.S.A.
The prospect of using Λp correlations to extract source sizes in relativistic heavy ion collisions is investigated.It is found that the strong interaction induces a large peak in the correlation function that provides more nsitive source size measurements than pp correlations under some circumstances.The prospect of using Λp correlations to measure the time lag between lambda and proton emissions is also studied.
Two-particle correlations have proven to be a powerful tool for determining source sizes and lifetimes in heavy ion collisions.At low energies,correlations of protons,neutrons and intermediate mass fragments have provided information on the space-time extent of the collision sys-tems [1].At relativistic energies,pion,kaon and proton correlations have greatly enhanced our understanding of the dynamics of heavy-ion collisions [2];the correlations provide different but complementary information.For in-stance,heavier particles are more affected by collective flow,thus making the mass-dependence of source sizes a test of our picture of explosive flow in heavy ion col-lisions;freeze-out conditions may be different for pions,kaons and protons,thus comparing parameters inferred from their correlations allows one to test the conjecture of quential freeze-out.
In this letter,we explore lambda-proton (Λp )correla-tions as a candidate of interferometric study.We find that an enhancement to the correlation function at low relative momentum allows one to infer the
size of the emitting source.The inferred lambda source parame-ters may provide valuable information becau lambdas are strangeness carrying baryons.Unlike two-proton (pp )system,the Λp system has no repulsive Coulomb interac-tion.Thus the enhancement from the strong interaction better survives when source sizes become large.We illus-trate the nsitivity of Λp correlations and show that for large sources,they might be more nsitive than pp corre-lations,but not as nsitive as coalescence measurements.We also study the possibility to determine whether lamb-das and protons are emitted simultaneously by compar-ing the correlations for positive and negative values of the projected outward relative momentum.
The correlation of two particles from a chaotic source may be estimated by assuming that they interact only with each other after they are emitted from space time points x a and x b [3],
C (p a ,p b )=
P (p a ,p b )
d 4x a d 4x b S a (p a ,x a )S b (p b ,x b )
(1)
In principle,the correlation depends on the size and shape of the source described by function S (p ,x ),which provides the differential probability of emitting particles of momentum p at a space-time point x .However,for the purpos of our study we will ignore the momentum dependence of the source functions and assume a Gaus-sian form for S .
marry什么意思S (x a )=δ(t )exp −
x 2
+y 2+z 2
4
V σσΛ·σp
T 2π
,(3)
where V C is a Woods-Saxon repulsive core.
孙燕姿英文歌
V C =W C 1+exp
r −R
x
+
3
x
1−e −cr 2
2,
(5)
where x =0.7r and c =2fm −2.The spin-independent part
of the attractive potential is characterized by ¯V
=6.2±0.05MeV,while the spin-dependent part is small,V σ=0.25±0.25MeV,and not well determined.
11.21.41.61.8Λ-p
0.2
number是什么意思0.40.60.811.20
10
20
30
40
reprenting50
p-p
k (MeV/c)
C (k )
FIG.1.Λp (upper panel)and pp (lower panel)correlation functions for R g =4fm (squares),6fm (triangles)and 10fm (circles).For larger sources,Λp correlations provide a more nsitive determination of the size.
The correlation functions for R g =4,6and 10fm sources are illustrated in Fig.1as a function of k ,which is one half of the relative momentum,k =|p Λ−p p |/2,as measured in the pair center-of-mass frame.Also shown are pp correlations for the same source sizes.Clearly,Λp correlations are more nsitive than pp correlations for determining larger source sizes.Despite the fact that the pp scattering length is considerably longer,Coulomb ef-fects obscure the nsitivity of pp correlation for sources larger than approximately 6fm.Thus,even though statistics for Λp correlations are reduced compared to pp statistics,they may provide a more accurate determi-nation of the source size under some circumstances.Cor-relations involving neutrons,such as nn or pn ,are also free of a relative Coulomb interaction,and given the large nn and pn scattering lengths,provide nsitive correla-tions.However,neutrons are notoriously difficult to mea-sure,whereas lambdas can be measured with a charged-particle detector through its π−p decay channel.An-other candidate for source size measurement is deuteron coalescence [5].Statistically,coalescence should provide the most accurat
e determination of source parameters of
cookies是什么意思all baryonic probes as coalescence does not suffer from Coulomb effects,and employs the entire strength of a bound state rather than the fraction reprented by a ri in the scattering pha shift.
For illustration,we have ud the same source size for both lambdas and protons,and assumed thermal mo-mentum distributions.To eliminate the Lorentz factor effect and also for computational reasons,we have ud thermal temperature T =3MeV.However,the Lorentz factor effect is small due to the large lambda and pro-ton mass.For instance,the difference in the correla-tion functions between T =3and 300MeV is less than 5%.For the interest of experimental feasibility of Λp correlation measurements,for T =300MeV,a total of 20million pairs results in,from pure pha space pop-ulation,10pairs in 0<k<5MeV/c (hence 30%statis-tical uncertainty),70pairs in 5<k<10MeV/c (12%),800pairs in 25<k<30MeV/c (4%),and 7000pairs in 0<k<50MeV/c .
An experimental correlation function often resorts to a mixed-event technique for uncorrelated pairs.Since both the height and the width of a Λp correlation function are nsitive to the source size,it is important to normalize the correlation function properly.Since lambda and pro-ton exhibit no correlation at large k ,an experimental Λp correlation function may be normalized to unity at large k .
The Λp correlation function is one fourth spin singlet (S =0)and three fourths spin triplet (S =1).To illustrate the spin dependence,correlation functions are prented for a R g =4fm source in the upper panel of Fig.2,p-arately for S =0and S =1pairs.If V σin Eq.(3)were zero,the two contributions would be identical.As V σis not well understood,0<V σ<0.5MeV [4],measuring the spin dependence of the correlation function could in principle determine V σ.
The correlation function is largely determined by the scattering length and effective range of the potential [6].The scattering length and effective range corresponding
to the potential (3)(¯V
建设项目管理
=6.2MeV,V σ=0.25MeV)are −2.88fm and 2.92fm for the spin singlet,and −1.66fm and 3.78fm for the spin triplet,respectively.They are in reasonable agreement with tho from Refs.[4,7].Using the values and an analytical approximation similar for neutron-proton correlations [6],we obtain Λp correlation functions that are consistent with the results in the upper panel of Fig.2.
6350The nsitivity of the spin-averaged correlation func-tion to the parameters ¯V
and V σis illustrated in the lower panel of Fig.2.Using the stated uncertainties [4],¯V
=6.2±0.05MeV and V σ=0.25±0.25MeV,the correlation function for a R g =4fm source is shown for a range of parameters.The results suggest that the un-certainties in the potential parameters translate into an approximately ±0.5fm uncertainty in the source size ex-tracted from the correlation function at k<25MeV/c ,
while the large k tail has better constrain on the source size.
建筑安全技术与管理1.2
1.41.61.8
22.2
高一物理教材1
1.21.41.61.8k (MeV/c)
C (k )
FIG.2.Upper panel:the spin decomposition of the Λp cor-relation function for a R g =4fm source,with the spin-averaged result reprented by the thick solid line.Lower panel:the nsitivity of the correlation function to the uncertainties
scroll lock是什么意思
in the potential parameters.Default values (¯V
=6.2MeV,V σ=0.25MeV)are reprented by the thick solid line.Chang-ing ¯V
yields the results reprented by the squares;changing V σyields the results reprented by the circl
es.The uncer-tainties in the parameters translate to a ±0.5fm uncertainty in the extracted source size.
Recently,Lednicky et al.[8]have shown that correla-tions of non-identical particles can provide information revealing whether the particles are emitted simultane-ously.If the lambdas are emitted before the protons in such a way that the probability cloud describing the pro-tons lags that for the lambdas of the same velocity,the correlation function then depends on the sign of the rela-tive momentum in the direction defined by that of the dis-placement of the lambda and proton clouds.An example is illustrated in Fig.3,where both sources are assumed to be characterized by a size of 4fm,but are parated by ∆z =1.5fm or 3.0fm.In Fig.3the difference of the correlation functions,C +(k )−C −(k ),is plotted against k where the transver component of k is required to be less than 10MeV/c .Here,C +refers to the correlation function constructed with the requirement that k z >0,while C −is constructed with the opposite constraint.A displacement ∆z can result from a displacement in time,∆τ=∆z/v ,where v is the velocity of the Λp pair.
The displacement direction,ˆz ,is then defined by the di-rection of v ,or that of the pair momentum relative to the source.Thus,if Λp correlations can be measured with an accuracy of a few percent at k ∼30MeV/c ,the conjecture that strange and non-strange baryons are emitted simul-taneously can be
addresd quantitatively.
00.05
0.1
0.15
0.2
10
20
30
40
50
k (MeV/c)
C +(k ) − C −(k )
FIG.3.The difference in the Λp correlation functions for positive and negative values of k z is shown assuming that both lambdas and protons are characterized by 4fm sources,but are parated by 1.5fm (squares)or 3fm (circles)in z .Such a paration might result if the lambdas would escape earlier than the protons.Small-scale fluctuations are due to fluctuations in the Monte Carlo procedure.
One should remember that the residual interaction of the proton with the Coulomb field of the nuclear sources might distort the result.Unlike pp correlations,where both particles feel the identical force,only the proton experiences the Coulomb field.This issue has been pre-viously considered in the context of pn correlations where the distortion was shown to be small for fast-moving pairs [9].This effect should be smaller at RHIC where the excess charge at midrapidity is expected to be smaller than obrved at SPS and AGS energies.
Finally,we make a brief note regarding ΛΛand ¯Λ
p cor-relations.In the view of our Λp results,we expect that the ΛΛstrong interaction would also give sizeable corre-lations.The correlations are of great interest becau of the predicted existence of
a ΛΛ-like particle [10].In
conjunction with Λp ,¯Λ
p correlations may reveal valuable information on low energy ¯Λ
p annihilation cross ctions which are prently unknown but indispensable in mod-eling certain aspects of heavy-ion collisions [11].
In summary,Λp correlations may provide a uful char-acterization of the space-time structure of relativistic heavy ion collisions should one be able to gather suffi-cient statistics.The lack of a relative Coulomb inter-action allows the strong interaction to produce a large peak in the correlation function even for large sources,to which pp correlation los its nsitivity.Furthermore,by binning according to the sign of the projected relative
momentum,one might address the question of whether lambdas and protons are emitted simultaneously.How-ever,the interpretation of the correlation function could benefit from a more preci parameterization of theΛp interaction.
ACKNOWLEDGMENTS
F.W.acknowledges Drs.V.Koch,R.Lednicky,J.C. Peng,A.M.Poskanzer and N.Xu for uful discussions. This work was supported by the U.S.Department of En-ergy under contract DE-AC03-76SF00098and the Na-tional Science Foundation under grant PHY-96-05207.
