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YITP-14-67

Lambda-Lambda interaction from relativistic heavy-ion collisions

Kenji Morita,1, 2, 3, ∗ Takenori Furumoto,4 and Akira Ohnishi1, †

1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan2Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany

3Institute of Theoretical Physics, University of Wroclaw, PL-50204 Wroc law, Poland4National Institute of Technology, Ichinoseki College, Ichinoseki, Iwate 021-8511, Japan

(Dated: February 5, 2015)

We investigate the two-particle intensity correlation function of Λ in relativistic heavy-ion colli-sions. We find that the behavior of the ΛΛ correlation function at small relative momenta is fairlysensitive to the interaction potential and collective flows. By comparing the results of differentsource functions and potentials, we explore the effect of intrinsic collective motions on the correla-tion function. We find that the recent STAR data gives a strong constraint on the scattering lengthand effective range of ΛΛ interaction as, −1.8 fm−1 < 1/a0 < −0.8 fm−1 and 3.5 fm < reff < 7 fm,respectively, if Λ samples do not include feed-down contribution from long-lived particles. We findthat feed-down correction for Σ0 decay reduces the sensitivity of the correlation function to the de-tail of the ΛΛ interaction. As a result, we obtain a weaker constraint 1/a0 < −0.8 fm−1. Implicationfor the signal of existence of H-dibaryon is discussed. Comparison with the scattering parametersobtained from the double Λ hypernucleus may reveal in-medium effects in the ΛΛ interaction.

PACS numbers: 25.75.Gz, 21.30.Fe

I. INTRODUCTION

Hyperon-hyperon interaction plays an important rolein various aspects of modern nuclear physics such ashypernuclear, exotic particle, neutron star, and strangematter physics. Current most precise information on ΛΛinteraction is obtained from the double Λ hypernuclearmass [1–3], and it is closely related to the existence of theS = −2 dibaryon (the H particle). In neutron star core,hyperons have been believed to emerge and to soften theequation of state [4]. While recent observation of massiveneutron stars disfavors admixture of strange hadrons [5],hypernuclear physics data suggest hyperon admixture atρB = (2 − 4)ρ0 [4]. Deeper understanding of hyperon-hyperon interaction may help solving this massive neu-tron star puzzle. At very high densities, hyperon super-fluid could be continuously connected to the color super-conductor, where the three flavors (uds) and colors (rgb)are entangled.

The existence of the H particle has been one of thelong-standing problems in hadron physics. In 1977, Jaffepointed out that double strange dibaryon made of 6quarks (uuddss) may be deeply bound below the ΛΛthreshold due to the strong attraction from color mag-netic interaction based on the bag model calculation [6].Deeply bound H was ruled out by the existence of dou-ble Λ hypernuclei. A double Λ hypernucleus 6

ΛΛHe wasfound to decay weakly (Nagara event), and the observedenergy of 6

ΛΛHe is 7.25 MeV(= BΛΛ) below the 4He+ΛΛthreshold [1]. Since 6

ΛΛHe should decay to 4He+H if themass of H is MH < 2MΛ − BΛΛ, the deeply bound H

∗Electronic address: kmorita@yukawa.kyoto-u.ac.jp†Electronic address: ohnishi@yukawa.kyoto-u.ac.jp

was ruled out.

The H particle is again attracting much attention dueto recent theoretical and experimental efforts. Recentlattice QCD calculations have demonstrated that H ap-pears as the bound state around the flavor SU(3) limit,and they also suggest the possibility for H to appear as abound state or resonance pole [7]. Experimentally, KEK-E224 and KEK-E522 experiments [8, 9] demonstratedthat ΛΛ invariant mass spectrum is enhanced in the lowenergy region between ΛΛ and ΞN thresholds, comparedwith the phase space estimate and the classical transportmodel calculations [10]. Enhancement of the invariantmass spectrum just above the threshold implies the fi-nal state interaction effects of ΛΛ attraction. KEK-E522data also show the bump structure around 10 MeV abovethe ΛΛ threshold. This bump cannot be explained solelyby the final state interaction effects, but it is not signifi-cant enough (∼ 2σ) to claim the existence of a resonancepole [9].

It is evident that we need higher statistcs data to ob-tain more precise information on ΛΛ interaction, andeventually to conclude the existence/non-existence of theH pole. Higher statistics data will be available from thefuture J-PARC experiments on double Λ hypernuclearobservation and ΛΛ invariant mass measurement, as pro-posed in J-PARC E42 experiment. Recently, an alter-native possibility to access the information on hadron-hadron interactions has been explored in heavy-ion col-lisions at Relativistic Heavy Ion Collider (RHIC) andLarge Hadron Collider (LHC). The large hadron mul-tiplicity, which is achived by hadronization from thequark-gluon plasma (QGP), makes it possible to lookat correlations between hadrons with good experimen-tal statistics. In particular, the intensity correlation ofidentical particles in relative momentum space is knownas the Hanbury-Brown and Twiss (HBT) or Goldhaber-

2

Goldhaber-Lee-Pais (GGLP) effect to give informationon the size of the emission source through the (anti-)symmetrization of the two-boson (fermion) wave func-tion.

The HBT effect of stable hadrons, particularly pions,has been used to estimate the source sizes created in rel-ativistic nucleus-nucleus collisions. On the one hand, ef-fects of interaction between two of those emitted parti-cles on extracting source size can be absorbed into thechaoticity parameter, when the system has a large sizecompared with the interaction range. On the other hand,one expects substantial effects of the interaction on thecorrelation function of identical particles of which inter-action is sufficiently strong in the range comparable tothe effective source size [11, 12]. This implies that onemay be able to use the correlation function to obtain in-formation on the interaction between two identical parti-cles, even if those particles are unstable. For ΛΛ pair, thisidea is not new. It was proposed in ’80s that we can fixresonance parameters, when the source size is small [13].The correlation at low relative momenta was proposed tobe useful to discriminate the sign of the scattering lengtha0, provided that the source size is large [14]; When ΛΛhas a bound state (a0 > 0), the scattering wave functionmust have a node at r ≃ a0 in order to be orthogonalto the bound state wave function, then we may find thesuppression of the correlation. The vertex detectors atRHIC have enabled us to really obtain the ΛΛ corre-lation data in heavy-ion collisions with a good signal-to-noise ratio by choosing weakly decaying Λ off the reactionpoint [15, 16].

In this paper, along with the above expectation, we in-vestigate the interaction between two Λ baryons. Severalhyperon-nucleon (Y N) and hyperon-hyperon (Y Y ) inter-action models have been proposed so far by constrainingparameters from limited number of Y N scattering exper-iments, flavor symmetries, and hypernuclear data. Wecalculate the ΛΛ correlation functions with those inter-action potentials. Given a model source function relevantfor Au+Au collisions at

√sNN = 200 GeV, we discuss the

modification of the correlation function due to the inter-action and collective flow effects, then show that how thebehavior the correlation function constraints the natureof the interaction. A preliminary report of the presentwork can be found in Ref. [17]. In this paper, we presenta detailed systematic analysis with the updated experi-mental data of ΛΛ correlation.

In Ref. [15], the STAR Collaboration has reportedand analyzed their experimental data of ΛΛ correlationwith the Lednicky and Lyuboshitz analytical model [11]which incorporates the effect of the ΛΛ interaction interms of the effective range and the scattering length to-gether with the intercept (chaoticity) parameter λ andnormalization as fitting parameters. Moreover, it hasbeen shown that the inclusion of a residual correlationas an additional gaussian term responsible to the high-momentum tail gives a better description of the data.In this paper, we focus on effects of the ΛΛ interaction

through the modification of the wave function and thedeformation of the emission source function owing to thecollective flow which takes place in relativistic heavy-ioncollisions. We will show that the ΛΛ correlation datameasured by the STAR Collaboration including the inter-cept and the residual correlation can be explained withsome of the recent ΛΛ potentials and flow parametersconstrained by the single particle spectrum of Λ, if we as-sume feed-down correction is negligible. We will then ex-amine effects of Σ0 feed down correction on the obtainedconstraints and discuss their interplay with a residualcorrelation.This paper is organized as follows. In Sec. II, we briefly

summarize models of the ΛΛ interaction. In Sec. III,we introduce the two-particle correlation function withthe final state interaction and show the general propertybased on a simple source model. Intrinsic effects in rela-tivistic heavy-ion collisions are discussed in Sec. IV. Wediscuss feed-down correction and residual correlations inSec. V. We also discuss possible implications for the Hparticle in Sec. VI. Section VII is devoted to concludingremarks.

II. ΛΛ INTERACTION POTENTIAL

We examine several models of ΛΛ interaction pro-posed so far by using ΛΛ correlation in heavy-ion col-lisions. Since experimental information on ΛΛ interac-tion is limited, ΛΛ correlation data are useful to con-strain ΛΛ interaction. ΛΛ interaction is known to beweakly attractive from the ΛΛ bond energy in 6

ΛΛHe,∆BΛΛ = BΛΛ(

6ΛΛHe) − 2BΛ(

5ΛHe) ≃ 1.01 MeV [1].

From ∆BΛΛ(6

ΛΛHe), the scattering length and the ef-fective range in the ΛΛ 1S0 channel are suggestedas (a0, reff) = (−0.77 fm, 6.59 fm) [2] or (a0, reff) =(−0.575 fm, 6.45 fm) [3], but in principle one cannot de-termine two low energy scattering parameters from a sin-gle observed number of ∆BΛΛ. For example, while reffvalues are very similar, a0 are different by 25-30 % inthe above two estimates. Low energy ΛΛ scattering pa-rameters are useful to distinguish the models of baryon-baryon (BB) interaction. The long-range part of theNN interaction is dominated by the one pion exchangepotential, which roughly determines the low energy be-havior of NN scattering. By comparison, Λ particle isisoscalar and there is no one-pion exchange in ΛΛ inter-action. Thus the low energy ΛΛ scattering parameters,such as the scattering length a0 and the effective rangereff , are more sensitive to the BB interaction models.There are several types of ΛΛ interactions proposed so

far. Meson exchange model ΛΛ interactions [18–23] havea long history of studies. Nijmegen group have providedseveral versions of NN , Y N and Y Y interactions, modelD (ND) [18], model F (NF) [19], soft core (NSC89 andNSC97) [20, 21], and extended soft core (ESC08) [22].These interactions have been widely used in hypernu-clear structure calculations [2, 3]. Compared with NN

3

TABLE I: ΛΛ potentials. The scattering length (a0) and effective range (reff) are fitted using a two-range gaussian potential,VΛΛ(r) = V1 exp(−r2/µ2

1) + V2 exp(−r2/µ22).

Model a0 (fm) reff (fm) µ1 (fm) V1 (MeV) µ2 (fm) V2 (MeV) Ref.

ND46 4.621 1.300 1.0 −144.89 0.45 127.87 [18] rc = 0.46 fm

ND48 14.394 1.633 1.0 −150.83 0.45 355.09 [18] rc = 0.48 fm

ND50 −10.629 2.042 1.0 −151.54 0.45 587.21 [18] rc = 0.50 fm

ND52 −3.483 2.592 1.0 −150.29 0.45 840.55 [18] rc = 0.52 fm

ND54 −1.893 3.389 1.0 −147.65 0.45 1114.72 [18] rc = 0.54 fm

ND56 −1.179 4.656 1.0 −144.26 0.45 1413.75 [18] rc = 0.56 fm

ND58 −0.764 6.863 1.0 −137.74 0.45 1666.78 [18] rc = 0.58 fm

NF42 3.659 0.975 0.6 −878.97 0.45 1048.58 [19] rc = 0.42 fm

NF44 23.956 1.258 0.6 −1066.98 0.45 1646.65 [19] rc = 0.44 fm

NF46 −3.960 1.721 0.6 −1327.26 0.45 2561.56 [19] rc = 0.46 fm

NF48 −1.511 2.549 0.6 −1647.40 0.45 3888.96 [19] rc = 0.48 fm

NF50 −0.772 4.271 0.6 −2007.35 0.45 5678.97 [19] rc = 0.50 fm

NF52 −0.406 8.828 0.6 −2276.73 0.45 7415.56 [19] rc = 0.52 fm

NSC89-1020 −0.250 7.200 1.0 −22.89 0.45 67.45 [20] mcut = 1020 MeV

NSC89-920 −2.100 1.900 0.6 −1080.35 0.45 2039.54 [20] mcut = 920 MeV

NSC89-820 −1.110 3.200 0.6 −1904.41 0.45 4996.93 [20] mcut = 820 MeV

NSC97a −0.329 12.370 1.0 −69.45 0.45 653.86 [21]

NSC97b −0.397 10.360 1.0 −78.42 0.45 741.76 [21]

NSC97c −0.476 9.130 1.0 −91.80 0.45 914.67 [21]

NSC97d −0.401 1.150 0.4 −445.77 0.30 373.64 [21]

NSC97e −0.501 9.840 1.0 −110.45 0.45 1309.55 [21]

NSC97f −0.350 16.330 1.0 −106.53 0.45 1469.33 [21]

Ehime −4.21 2.41 1.0 −146.6 0.45 720.9 [23]

fss2 −0.81 3.99 0.92 −103.9 0.41 658.2 [25]

ESC08 −0.97 3.86 0.80 −293.66 0.45 1429.27 [22]

and Y N interactions, we have larger uncertainties in Y Yinteraction and there are some rooms to vary model pa-rameters. We regard the hard core radius rc in hardcore models (ND and NF) and the cutoff mass mcut inNSC89 as free parameters. In the case of NSC97, thereare several versions (NSC97a-f) having different spin de-pendence in ΛN interaction, and ΛΛ scattering parame-ters would help discriminating these versions.

Ehime potential is a boson exchange ΛΛ potential [23],whose strength is fitted to the old double Λ hypernuclearbond energy, ∆BΛΛ = 4 MeV [24]. Since this value isproven to be too large, Ehime potential is now known tobe too attractive. Even though, it would be valuable toexamine the ΛΛ correlation with more attractive poten-tial than usually considered.

The quark model BB interactions have a merit thatthe Pauli principle between quarks and the one-gluonexchange give rise to the short range repulsion, whichseems to be consistent with other NN interaction mod-els. At the same time, in order to describe the medium-and long-range part of the BB interaction, we need totake account of the meson exchange between quarks orbaryons. There are several quark model BB interactionswhich include the meson exchange effects. We adopt here

the fss2 model [25], as a typical quark model interaction.This interaction is constructed for the octet-octet BBinteraction and describes the NN scattering data at acomparable precision to meson exchange potential mod-els. For fss2, we use a phase-shift equivalent local poten-tial in the two range gaussian form [25], derived by usingthe inversion method based on supersymmetric quantummechanics [26].

Low energy scattering parameters of the ΛΛ interac-tions considered here are summarized in Table I. In Fig. 1,we show the scattering parameters (1/a0 and reff) of theΛΛ interactions under consideration. These scatteringparameters characterize the low-energy scattering phaseshift in the so-called shape independent form as

k cot δ = − 1

a0+

1

2reffk

2 +O(k4) . (1)

For negatively large 1/a0, the attraction is weak and thephase shift rises slowly at low energy. When we go fromleft to right in the figure, the interaction becomes moreattractive and a bound state appears when a0 becomespositive. We have parameterized the boson exchange ΛΛinteractions, described above in two-range gaussian po-

4

ΛΛ scattering parameters

02468

10121416

-7 -6 -5 -4 -3 -2 -1 0 1 2

r eff [

fm]

1/a0 [fm-1]

ΛΛ scattering parameters

NDNF

NSC89NSC97Ehime

fss2ESC08

FGHKMYY

STAR

FIG. 1: ΛΛ interactions and scattering parameters in the(1/a0, reff) plane. The ΛΛ interations favored by the ΛΛ cor-relation data without feed-down correction are marked withbig circles. The thin big and thick small shaded area corre-spond to the favored regions of scattering parameters withand without feed-down correction, respectively, which showstable and small χ2 minimum. (See text.) The results ofthe analysis by the STAR collaboration is shown by the filledcircle [15], together with systematic error represented by thesurrounding shaded region.

tentials

VΛΛ(r) = V1 exp(−r2/µ21) + V2 exp(−r2/µ2

2) , (2)

then fit the low energy scattering parameters, a0 and reff .

In addition to the ΛΛ potentials listed in Table I, wealso examine the potentials used in Refs. [2] (by Filikhinand Gal; FG) and [3] (by Hiyama, Kamimura, Motoba,Yamada and Yamamoto; HKMYY) with the three-rangeGaussian fit given in those references. The parametersare summarized in Table II.

Before closing the section, we note that the couplingeffects with ΞN and ΣΣ channels are effectively incorpo-rated in the present treatment, since the coupling mod-ifies the low energy scattering parameters of ΛΛ and weuse the low energy phase shift equivalent potential. Alsoin Refs.[2, 3], the coupling effects with ΞN is included inthe ΛΛ potential. The explicit coupling effect on the ΛΛcorrelation would be an interesting subject, but is out ofthe scope of this paper.

III. ΛΛ CORRELATION FUNCTION WITH INTERACTION EFFECTS

A. Formalism

A relevant formulation of the two-proton correlation function is given in [27] which solidates the formula in [28].Here we apply the formula to ΛΛ correlation. Then, for a given relative wave function Ψ12, the correlation functiondefined as two-particle distribution W2(k1,k2) normalized by the one-particle distributions W1(ki)(i = 1, 2) can beexpressed in terms of one-particle phase space density S(x,k) as

C2(Q,K) =W2(k1,k2)

W1(k1)W1(k2)(3)

=

∫

d4x1d4x2S(x1,K)S(x2,K)|Ψ12(Q,x1 − x2 − (t2 − t1)K/m)|2

∫

d4x1d4x2S(x1,k1)S(x1,k2)

(4)

where K = (k1 + k2)/2 and Q = k1 − k2 are the average and the relative momentum of the two identical particles,respectively. Since the expression for the two-particle distribution is derived for smallQ, one can also put k1 ≃ k2 ≃ K

in S(x,ki) in the denominator.

As S(x,k) represents the one-particle phase space dis-tribution of Λ, effects of interaction are embedded in therelative wave function Ψ12(Q, r). As we are consider-ing the effect of ΛΛ interaction through the potentialV (r), the relative wave function given by solving theSchrodinger equation is time independent. The factor−(t2 − t1)K/m, with m being the mass of Λ, is addedto the relative coordinate in Ψ12 to take into account the

different emission time of two Λ particles.

The two-particle wave function respects the anti-symmetrization for the two identical fermions. Forthe non-interacting spin-singlet (spin-triplet) case, thespatial part of the wave function is symmetric (anti-symmetric) with respect to the exchange of the two par-

5

TABLE II: ΛΛ potentials from Nagara event. The scattering length (a0) and effective range (reff) are fitted using a three-rangegaussian potential, VΛΛ(r) = V1 exp(−r2/µ2

1) + V2 exp(−r2/µ22) + V3 exp(−r2/µ2

3).

Model a0 (fm) reff (fm) µ1 (fm) V1 (MeV) µ2 (fm) V2 (MeV) µ3 (fm) V3 (MeV) Ref.

HKMYY −0.575 6.45 1.342 −10.96 0.777 −141.75 0.35 2136.6 [3]

FG −0.77 6.59 1.342 −21.49 0.777 −250.13 0.35 9324.0 [2]

ticle position,

Ψs =1√2(eik1·x1+ik2·x2 + eik2·x1+ik1·x2)

=1√2e2iK·X(eiQ·r/2 + e−iQ·r/2) (5)

Ψt =1√2(eik1·x1+ik2·x2 − eik2·x1+ik1·x2)

=1√2e2iK·X(eiQ·r/2 − e−iQ·r/2) (6)

where we have introduced the center-of-mass coordinateX = (x1 + x2)/2 and relative one r = x1 − x2.

With the interaction described by a potential V (r), weassume here that only the s-wave is modified. The two-particle wave function in the spin singlet state is repre-sented as, with the solution of the Schrodinger equationin the s-wave χQ(r),

Ψs =√2 [cos(Q · r/2) + χQ(r) − j0(Qr/2)] , (7)

where j0(Qr/2) is the spherical Bessel function at zerothorder and we omit X dependent part as they give unityin |Ψs,t|2. The spin-averaged total wave function squaredis now given by

|Ψ12|2 =1

4|Ψs|2 +

3

4|Ψt|2 (8)

= 1− 1

2cos(Q · r) +

[

χQ(r) − j0

(

Qr

2

)]

× cos

(

Q · r

2

)

+1

2[χQ(r)− j0(Qr/2)]

2(9)

If we neglect the interaction, χQ(r) = j0(Qr/2), onlythe first and second terms remain to give the free HBTcorrelation which has an intercept 1/2 at Q = 0. Thisreflects symmetrization of the spatial wave function inthe spin-singlet channel and the anti-symmetrization ofthe spatial wave function due to the Pauli principle inthe spin-triplet channel, 2 × 1/4 + 0 × 3/4 = 1/2. Onthe other hand, it has been known that the bosonic cor-relation function gives the intercept value two. Thus, weexpect C(Q = 0) < 1/2 for repulsive interactions andC(Q = 0) > 1/2 for attractive ones. Comparisons of thefull correlation function with the free correlation functiongive direct information on the effect of the interaction, aswe shall see below.

B. Static spherically symmetric source

In relativistic heavy-ion collisions, the source emittinghadrons shows collective behaviors. At the RHIC energyconsidered in this paper, the hot medium produced inthe collisions exhibits strongly correlated property whichis understood as nearly perfect fluid of the deconfinedquarks and gluons. The hadrons are produced at thehadronization from the fluid, followed by rather dissipa-tive transport processes [29]. As a result, the emissionsource S(x,k) in Eq. (4) might not be characterized by asimple parametrization. In fact, the two-pion correlationfunctions have been extensively discussed in heavy-ioncollisions and are found to be sensitive not only to theemission source size but to various aspects of the collisionprocesses [30]. This fact made it difficult to fit the mea-sured data within a simple model calculation even withcollective effects being taken into account.We expect that the Λ source may have a simpler form

than that for pions for the following reasons. First, Λis expected to interact weakly with environments mainlyconsisting of pions. Single particle levels of Λ includingthose of deep s-states are clearly observed [31]. This isin contrast with nucleons and pions, whose single parti-cle states have large widths inside nuclei, and suggestsweaker interaction of Λ with pionic environment, sincenuclei contain many virtual pions. Second, the decayfeed effects are expected to be smaller. It is known thatthe feed down effects of Λ → pπ− and pΛ interactionsare important to understand the pp correlation function.For the ΛΛ pair, there is no Coulomb suppression of lowrelative momentum pairs and contribution from particleswhich decay into Λ is limited. In addition, we can in prin-ciple remove those Λ particles from weak decays such asΞ− → Λπ− by using the vertex detector. In the follow-ing, we will show that data on ΛΛ correlation measuredin heavy-ion collisions at the RHIC energy are useful todiscriminate the ΛΛ interaction.To illustrate the capability, we first examine the cor-

relation function from a simple, static and sphericallysymmetric source

Sstat(x,k) = ∗ exp[

−x2 + y2 + z2

2R2

]

δ(t− t0), (10)

where normalization is omitted since it is canceled in thecorrelation function.Putting Eqs. (9) and (10) into Eq. (4) and projecting

onto the function of Q = |Q| by integrating out the anglevariables, the correlation function for the source function

6

0

10

20

30

40

50

0 1 2 3 4 5

χ2 /Nd

of

R [fm]

Data : STAR 0-80%Model : Nijmegen model D

(a)

ND46ND48ND50ND52ND54ND56ND58

0

10

20

30

40

50

0 1 2 3 4 5

χ2 /Nd

of

R [fm]

Data : STAR 0-80%Model : Nijmegen model F

(b)

NF42NF44NF46NF48NF50NF52

0

10

20

30

40

50

0 1 2 3 4 5

χ2 /Nd

of

R [fm]

Data : STAR 0-80%Model : Nijmegen Soft Core 89

(c) NSC89-1020NSC89-920NSC89-820

0

10

20

30

40

50

0 1 2 3 4 5

χ2 /Nd

of

R [fm]

Data : STAR 0-80%Model : Nijmegen Soft Core 97

(d) NSC97aNSC97bNSC97cNSC97dNSC97eNSC97f

0

10

20

30

40

50

0 1 2 3 4 5

χ2 /Nd

of

R [fm]

Data : STAR 0-80%

(e) Ehimefss2

ESC08FG

HKMYY

FIG. 2: χ2 plotted against R with ΛΛ potentials. Nijmegen model D (ND, top left (a)) Nijmegen model F (NF, top middle (b))Nijmegen soft core (NSC89, top right (c)), Nijmegen soft core (NSC97, bottom left (d)), and the rest of potentials in Tables Iand II (bottom right (e)).

(10) becomes

Cstat(Q) = 1− 1

2e−Q2R2

+1

4√πR3

∫ ∞

0

dr r2e−r2/4R2

×[

[χQ(r)]2 − [j0(Qr/2)]2

]

. (11)

Thus the effect of interaction is incorporated as the dif-ference of the squared relative wave function. The freecase is given by a simple Gaussian, which is often usedto obtain the source size.In the static and spherically symmetric source model

(10), the only parameter is the source size R. Thus, it isconvenient to calculate

χ2 ≡∑

i

[

C(Qi)data − C(Qi)

model

σdatai

]2

(12)

as a function of R then search for minimum in order toaddress which potential is favored in data.In the following, we compare the model correlation

function (11) with the data of ΛΛ correlation for 0.01 ≤Qi < 0.5 GeV, measured by the STAR collaboration inAu+Au collisions at

√sNN = 200 GeV with 0-80% cen-

trality [15]. χ2 is calculated for all the potential tabulatedin Tables I and II with the spherical static source modelSstat.The results are shown in Fig. 2, where Ndof = 24.

One sees that the behavior of χ2/Ndof as functions ofR strongly depend on the choice of the potential. Onthe one hand, some potentials exhibit monotonically de-creasing behavior with R then asympotically becomes

flat. These potentials typically have too strong attrac-tion (Ehime, ND50, NF46, ND46, NF44 etc) or too largeeffective range (NSC97, NSC89). On the other hand, astable minimum with small χ2/Ndof is achieved in theregion 1 fm < R < 1.5 fm in potentials such as ND56,NF50, fss2 and ESC08. Looking at the scattering lengthand the effective range of those potentials, we find thatthere is a range of those quantities in which the corre-sponding potentials show the stable and small χ2 mini-mum, as shown in the shaded area in Fig. 1. There arealso marginal potentials (HKMYY, NSC89-820) whichexhibit also a stable minimum but with χ2/Ndof > 5.Since we have compared with the raw data, these po-tentials may yield acceptable fit after appropriate cor-rections, thus should not be ruled out. NSC97d showsa different behavior from any other potentials, due tothe narrow effective range and small scattering length asseen in Fig. 1 which may not be realistic. The small χ2 atR = 0.6fm is achieved by approaching the long tail partof C(Q), rather than the interaction dominated part.Figure 3 displays the correlation functions for poten-

tials considered here compared with the experimentaldata. The size parameter R adopted in the figure cor-responds to the minimum of χ2 in Fig. 2. We also plotthe free correlation function, Eq. (11) without the lastterm, for comparison. While the values of χ2 do notdiffer much, the correlation function at low Q shows asubstantial variation among the potentials. The smalldifference of χ2 is attributed to larger error bars in theexperimental data at low Q region. One sees that re-sults from these potentials have all C(Q) > 1/2 at lowQ, fairly reflecting the attraction between two Λ. Among

7

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

Static source

ND56, R=1.4fmNF50, R=1.2fm

NSC89-820, R=1.4fmfss2, R=1.2fm

ESC08, R=1.3fmFG, R=1.5fm

HKMYY, R=1.5fmFree, R=1.3fm

STAR 0-80%

FIG. 3: ΛΛ correlation function from the static sphericallysymmetric source at the minimum of χ2 together with exper-imental data by STAR.

the potentials with small minimum χ2, on the one hand,NF50 and fss2 show C(Q < 0.1 GeV) ∼ 0.9 and give agood description for the tail part around Q ∼ 0.2 GeV.On the other hand, ND56 and ESC08 exhibit an weakenhancement at Q < 0.1 GeV with a good fit to thedata at 0.1 GeV < Q < 0.3 GeV. Therefore, preci-sion measurement at Q < 0.1 GeV will provide furtherconstraints on the ΛΛ interaction. According to Table1, those potentials have −1.2 fm < a0 < −0.8 fm and3.2 fm < reff < 6.5 fm. However, the effective range isnot contrained well because the model potentials do nothave a combinations of a large scattering length and alarge effective range. Thus, we further construct modelpotentials by varying the effective range with a fixed scat-tering length a0 = −0.8fm in the two-range Gaussian(TRG) form (2). Results for the χ2/Ndof and correspond-ing C(Q) are displayed in Fig. 4. One sees that therealways exists a minimum in the χ2 plot which is partic-ularly sensitive to variation of the effective range aroundreff ≃ 4 fm. The global minimum achived for reff ≃ 4 fmis in accordance with the above model analysis. Since thebehavior of C(Q) with large effective range, reff > 6 fmis similar to FG and HKMYY in Fig. 3, a large effectiverange might be favored if the data receive the correc-tion. Consequently, the present analysis provides ratherlimited constraints on the effective range than the scat-tering length. The favored range of the scattering lengthand the effective range is indicated by the shaded area inFig. 1.

C. Wave functions

In order to characterize the potentials which give rea-sonable description of the measured ΛΛ correlation data,we discuss the corresponding potentials and resultant

TABLE III: ΛΛ potential parameters with various effectiveranges for fixed scattering length, a0 = −0.8fm in the two-range Gaussian (TRG) form (2).

Model reff (fm) µ1 (fm) V1 (MeV) µ2 (fm) V2 (MeV)

TRG02 2.0 0.6 −405.97 0.45 582.29

TRG04 4.0 0.6 −1835.95 0.45 4976.17

TRG06 6.0 0.6 −5569.58 0.45 25435.95

TRG08 8.0 0.8 −889.42 0.45 14595.38

TRG10 10.0 0.8 −1440.02 0.45 43254.42

TRG12 12.0 1.0 −358.38 0.45 20522.96

wave functions.

Since effects of interaction on the correlation functionare incorporated through the difference from the freewave function as seen in Eq. (9), we display the inte-grand of the last term of Eq. (11) rather than the wavefunction itself in the right panel of Fig. 5 as well as the po-tentials in the left panel. The horizontal axis in the rightpanel is normalized by the size parameter R at the mini-mum χ2 to reduce the apparent effect due to the differentsize in the source function. Although the wave functionχQ(r) reflects the remarkable differences of the potentials

at small r, r2e−r2/4R2

acts as a weight factor such thatthe behavior of the correlation function is most sensitiveto the wave function at r ≃ 1 − 3 fm.One sees that thedeviation from the free wave function is fairly reflectedonto C(Q). Namely, ND56 and NSC89-820 which showthe largest attraction in the wave function at r/R ≃ 1.5exhibit the strongest bunching in C(Q), as seen in Fig. 3.ESC08, NF50 and fss2 also follow this trend. HKMYY,one of the two wave functions motivated by the Nagaraevent, has the weakest attraction among the potentialsthus leads to the smallest deviation from C(Q = 0) = 0.5in Fig. 3. The other one, FG, has a somewhat strongerattraction, but the strongest repulsion around the originleads to χ2

Q < χ2free at r < 0.5 fm, which finally gives

C(Q) ≃ 0.8. It is instructive to note that fss2 and NF50give a similar wave function despite the difference be-tween the potentials. As given in Table I, both effectiverange and scattering length have close values in thesepotentials. This indicates that the C(Q) is essentiallydetermined by a0 and reff rather than the detailed formof the potential.

The above consideration demonstrates how sensitivethe correlation function C(Q) is to the relative wave func-tion. From the analysis with the static spherically sym-metric source, strong constraints on the effective rangeand the scattering length of ΛΛ interaction have beenobtained. The next step is to examine whether this capa-bility is affected by the dynamics of relativistic heavy-ioncollisions at the RHIC energy.

8

0

5

10

15

20

25

30

0 1 2 3 4

χ2 /Nd

of

R [fm]

Data : STAR 0-80%

Model : a0 = -0.8fm

(a) TRG02TRG04TRG06TRG08TRG10TRG12

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

Static source(b)

TRG02, R=1.1fmTRG04, R=1.2fmTRG06, R=1.4fmTRG08, R=1.6fmTRG10, R=1.7fmTRG12, R=1.8fm

STAR 0-80%

FIG. 4: Results for TRG potentials. (a) χ2/Ndof against R. (b) C(Q) at the minimum χ2.

-0.1

-0.05

0

0.05

0.1

0.6 0.8 1 1.2 1.4

VΛ

Λ(r

) [G

eV]

r [fm]

ND56NF50

NSC89-820fss2

ESC08FG

HKMYY

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

r2 (χ2 Q

(r)-

χ2 free

(r))

e-r

2 /4R

2 [fm

2 ]

r/R

Q=0.01GeV

ND56NF50

NSC89-820fss2

ESC08FG

HKMYY

FIG. 5: Left: ΛΛ potential for the selected parameter sets which fit the measured data well. Right: Deviation of the relativewave functions from the free one weighted with the source function at Q = 0.01GeV.

IV. EFFECT OF COLLECTIVITY

In the following, we consider source functions incor-porating effects of expansion dynamics of the relativisticheavy-ion collisions. We assume that Λ particles are pro-duced at the chemical freeze-out following the hadroniza-tion from quark-gluon plasma. Although Λ may interactwith other produced hadrons and substantial amount ofΛ multiplicity comes from decay of heavier particles, weassume that the information on ΛΛ interaction encodedin the correlation function is not distorted much by thosehadronic effects. This simplification may be justified inpart by the smaller Λ interaction with the pion dominantenvironment and smaller feed down effects expected inΛΛ correlation. Especially, if most of the weak and elec-tromagnetic decay to Λ is removed by using the vertexdetector, feed down effects on Λ is expected to be small.We examine the feed down effects in the next section and

concentrate on the effects of collectivity in this section.

A. Models

Effects of the collective flow can be studied by mod-ifying the source function S(x,k). We utilize a ther-mal source model to implement the collective effect andchange the source geometry to more relevant one for rel-ativistic heavy-ion collisions. The collective flow velocityat the emission point x is denoted by uµ and the Fermidistribution function nF (E, T ) = (eE/T + 1)−1 for Λ isintroduced to accommodate the thermal distribution inthe rest frame of the emission point.The source function is generally defined through the

invariant spectrum

Ed3N

dk3=

∫

d4xS(x,k). (13)

9

We consider several source functions to discriminate ef-fects on the correlation function. As the simplest exten-sion from the static source model used in the previoussection, we implement the collective flow as

Ssph(x,k) = ∗u · k nF (u · k, T )

× exp

[

−x2 + y2 + z2

2R2

]

δ(t− t0). (14)

The spherically symmetric flow velocity, uµ = γ(1,v)where γ = (1− v2)−1/2 , is assumed to exhibit a Hubble-type expansion v = tanh(ηrr/R). The coefficient ηr con-trols the strength of the expansion. The product u · kbrings the space-momentum correlation into the sourcefunction if v 6= 0. When ηr = 0, the space-momentumcorrelation of the source function is lost, i.e., the sourcefunction is factorized into S(x,k) = A(x)B(k) then theresultant correlation function reduces to that of the staticsource (11) because the momentum dependent part B(k)is canceled by the denominator in Eq. (4).

While the above model (14) is useful to understandeffects of the collective flow, the geometric part of thesource function still posses the spherical symmetry whichis not appropriate for heavy-ion collisions at the RHICenergy. Owing to the huge colliding energy, the systemundergoes rapid expansion along the collision axis (wetake it as z axis) followed by slower one in the perpen-dicular direction. Therefore, we consider a cylindricallysymmetric expanding system with the longitudinal boostinvariance. The boost invariance also implies the infiniteextent of the source in the collision axis. In the pres-ence of the strong longitudinal flow, however, this doesnot mean that source function have infinite width in thelongitudinal direction since the thermal factor naturallygives a finite extent. In the Boltzmann approximation,one can derive an approximate but analytic expressionfor the longitudinal source size, RL ≃ τ0

√

T/mT [32],where τ0 denotes the freeze-out proper time and mT is

the transverse mass mT =√

p2x + p2y +m2. The boost

invariant approximation is of course valid only aroundmidrapidity region. At the RHIC energy, effects of pos-sible deviation were found to be small [33].

In reality, collisions have finite impact parameters evenin the highest multiplicity bin and event-by-event fluctua-tions may induce further asymmetry to the source. Thoseeffects may be important for understanding experimen-tal data on the ππ HBT correlation, but we shall ignorethem since we are not aiming at extracting the source ge-ometry but examining the final state interaction betweenΛ. The present statistics does not seem enough to projectΛΛ correlation function onto each directions of Q. Thus,we assume the azimuthally symmetric gaussian sourceprofile. We expect this only leads to a smaller effectivesource size resulting from averaging over the azimuthalangle. In this case, we can put the average momentumK as K = (KT , 0,Kz) without loss of generality. Then aconvenient source function has been used for the analysis

of pion HBT radii in Ref. [34],

Scyl(x,k) =mT cosh(y − YL)

(2π)3√

2π(∆τ)2nf (u · k, T )

× exp

[

− (τ − τ0)

2(∆τ)2− x2 + y2

2R2

]

. (15)

where y = 1/2 log[(Ek + kz)/(Ek − kz)] is the rapidity ofthe emitted particle. The gaussian form of temporal partof the source function in the proper time τ =

√t2 − z2

takes into account possible emission duration ∆τ arounda freeze-out time τ0.

The four flow velocity can be parametrized by the lon-gitudinal and transverse rapidity

ut = coshYT coshYL (16)

uz = coshYT sinhYL (17)

ux = sinhYT cosφ (18)

uy = sinhYT sinφ. (19)

The longitudinal flow rapidity is given by the scaling so-lution [35]

YL = ηs =1

2ln

t+ z

t− z(20)

The transverse flow rapidity is assumed to be

YT = ηfrTR

(21)

where rT =√

x2 + y2 and ηf controls the strength of thetransverse flow.

We fix the transverse flow strength parameter ηf fromthe single Λ spectra in 200GeV Au+Au collisions mea-sured by the STAR collaboration [36]. We use the20−40% data since the detailed spectrum after feed-downsubtraction is given. As we have ignored the baryon andthe strangeness chemical potentials in the thermal de-scription, we average the Λ and Λ data for the fit. Fix-ing T = 160MeV, for the chemical freeze-out, we obtainthe minimum of χ2/Ndof is about 1.9 for ηf = 0.33. Weuse this value in the calculations below.1 Note that thiscan be determined independent of the size parameters,R, τ0, and ∆τ . These parameters are regarded as freeparameters to study the influence of the source geometryon ΛΛ correlation function. Relations to the HBT radiihave been extensively studied in Refs. [34] and [37] in thecontext of the ππ correlation.

1 We checked that the choice of the temperature and the resul-

tant flow strength do not influence our main objective in this

paper, by repeating the same calculations for T = 200MeV and

120MeV.

10

B. Effects of collectivity

Now we calculate the ΛΛ correlation function for theexpanding sources. Owing the space-time correlation in-duced by the collective flow, the correlation function de-pends on the average momentum K in addition to therelative one. We integrate the numerator and the denom-inator of Eq. (4) as

C(Q) =

∫

d3KW2(k1,k2)∫

d3KW1(k1)W1(k2). (22)

For the cylindrical source (15), the integration is carriedout for the rapidity and the transverse momentum withinthe experimental acceptance ranges. For the sphericalsource, integration with respect to |K| is done for thesame range as the transverse momentum in the cylin-drical case, for simplicity. Although the following resultsmay change quantitatively due to the average momentumdependence of the correlation function, these integrationsdo not lead to any change in our discussion below.

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

Free, R=1.2fm

ηr=0.0, Free Sphericalηr=0.33, Free Sphericalηf=0.0, Free Cylindrical

ηf=0.33, Free Cylindrical

FIG. 6: Free correlation functions for the spherically sym-metric source (14) and for the cylindrically symmetric boost-invariant source (15). Closed and open symbols stand forC(Q) with and without flow, respectively.

We begin with examining effects of collective flow inthe spherically symmetric source to make a clear con-nection with the analyses in the previous section. Wechoose R = 1.2 fm in the model source function (14).Although not fitted to the experimental data, we applythe result of the fit to pT spectrum in the cylindricalsource model and put ηr = 0.33. Figure 6 illustrates theeffect of the flow strength on the free correlation func-

tions. Dotted line stands for the static case 1− 12e−Q2R2

.The expanding case is represented by open squares. Onesees that the flow makes the correlation extended in thehigher momentum region. In other words, the effectivesource size becomes smaller, as is well known for pionHBT radii [38, 39]. In Fig. 7, we turn on the interac-tion between Λ by taking the fss2 potential. Since the

difference of the potentials results in the relative wavefunction, our discussion in this section does not dependon the choice of the potential. By comparing with theηr = 0 case, which is the same as fss2 result in Fig. 3,one finds that the correlation function resembles the moreattractive potential. Since the interaction is not altered,one can understand this behavior as the effect of the col-lective flow. As shown in Fig. 6, effective source size isdecreased by the collective flow. This can be understoodas the result of position-dependent suppression of emis-sion probability by the thermal factor [37]. This makes Λpairs with small Q concentrated inside small region thusthey become more sensitive to the short range interac-tion.

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

fss2, R=1.2fm

ηr=0.0, Free Sphericalηr=0.0, Spherical

ηr=0.33, Sphericalηf=0.0, Cylindrical

ηf=0.33, Cylindrical

FIG. 7: Correlation functions with fss2 interaction. Symbolsare similar to the previous figure 6.

Results for the cylindrical source function Scyl (15) arealso displayed in Figs. 6 and 7. Here we put τ0 = 5fm and∆τ = 2fm. The free case with ηf = 0 is shown as closedcircles in Fig. 6. The narrower width of the correlationfunction than that of the spherical source is due to thefact that ηf controls the transverse flow only and thatthe boost-invariant longitudinal expansion takes place.As seen in Eq. (15), the boost-invariant source functionhas infinite extent in the longitudinal direcion if one ig-nores the thermal distribution. Although the longitudi-nal boost invariant expansion makes the effective sourcesize finite, the source function has still a large longitu-dinal extent such that the behavior of C(Q) resembleslarger source size than the spherically symmetric source.Since the longitudinal flow effect is dominant, the effectof the transverse flow seen through ηf is small. Note,however, that the effective source size becomes smallerby increasing ηf , because the same discussion as the caseof the spherical source applies to the present result. Asa result, the whole shape of the C(Q) does not changemuch by the interaction. Nevertheless, the behavior ofC(Q) at low Q remains sensitive to the interaction, as de-picted in Fig. 7. If we introduce a finite longitudinal size

11

into the source function, such as e−z2/2R2

, the behavior ofthe correlation function becomes closer to the case of thespherical source since the effective source size is reduced.We note that increasing τ0 as well as ∆τ also makes theeffective source size larger, since τ0 corresponds to thesource extent in z direction and ∆τ contributes to thecorrelation function through the emission time differencein Eq. (4).

0

5

10

15

20

0 0.5 1 1.5 2 2.5

χ2 /Nd

of

R [fm]

t0=5fm/c, ∆τ = 2fm/c

fss2Static Spherical

ηr=0.33, Sphericalηf=0, Cylindrical

ηf=0.33, Cylindrical

FIG. 8: χ2/Ndof against the size parameter for the fss2 inter-action.

As a result of geometry and flow effects, optimal sourcesize is modified from that in the spherical static sourcecase. In Fig. 8, we show the size parameter dependenceof χ2/Ndof with fss2 interaction. For the spherical ge-ometry, the optimal source size is shifted to the largerdirection by 30-40 %. Figure 9 shows the correspondingcorrelation functions at the minimum of χ2. Interest-ingly, despite the difference in the source size, the resul-tant correlation functions for ηr = 0 and 0.33 with thespherical source are almost the same. The effect of theflow is absorbed into the larger optimal source size.For the cylindrical and boost invariant source, the op-

timal transverse source size is also shifted upwards, whilethe shift is smaller as discussed above. The behavior ofthe χ2 in small R reflects the effect of the geometry andthe flow. Since the source function is elongated in thelongitudinal direction, the source has a longitudinal ex-tent even when R is so small. Thus Λ particles feel lessattraction than in the case of the spherical source withsmall R. The transverse flow further reduces the vari-ation of C(Q) against R. It is interesting to note thatdespite almost the same value of χ2/Ndof of the cylindri-cal source as that of the spherical source, the optimizedcorrelation function shown in Fig. 9 has a different fea-ture. The result of the cylindrical source indicates thatC(Q) with the small width also describes the data wellif this shape of C(Q) can be obtained, though it was notpossible in the spherical source as shown in Sec.III.Then, we have performed the same analysis for other

ΛΛ interactions. We found that the favored ΛΛ interac-

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

fss2, Optimized

ηr=0.0, Spherical, R=1.2fmηr=0.33, Spherical, R=1.6fmηf=0.0, Cylindrical, R=0.6fm

ηf=0.33, Cylindrical, R=0.7fmSTAR 0-80%

FIG. 9: ΛΛ correlation function with fss2 interaction for thespherical and the cylindrical boost-invariant source optimizedfor minimum χ2.

tions are the same as those in the case of the sphericalstatic source Sstat. However, we also found that interac-tions with larger effective range and smaller scatteringlength are favored. For example, among ND interac-tions, ND58 is now better than ND56, contrary to thecase of Sstat (Fig. 2), while ND56 still gives a good fit,too. NSC97 potentials are also found to be improved,but χ2/Ndof ≃ 8 at the best in NSC97c and NSC97e.NSC97d no longer shows a good fit with the cylindri-cal source model Scyl. This indicates that the good fit inSstat (Fig. 2) was spurious one; not due to the interactionbut to an accidental geometric effect.

χ2 and the optimized C(Q) for all the potential thatgive χ2/Ndof < 8 are summarized in Fig.10. Since ND58and FG have almost same a0 and reff, the result is so.Other ΛΛ interactions give larger χ2. In Fig. 1, we markthe favored ΛΛ interation with big circles. The scat-tering parameters of these interactions are in the range−1.8 fm−1 < 1/a0 < −0.8 fm−1 and 3.5 fm < reff < 7 fm.Potentials studied in this work and outside of this regiongive larger χ2, χ2/Ndof > 5, at any value of the sourcesize and with any geometry and flow values. One seesthat the small size is favored owing to somewhat scat-tered data points and larger errors in small Q region.The difference of the correlation function in those po-tentials are of the same size as experimental errors for0.1 < Q < 0.2 GeV and even larger at Q < 0.1 GeV.Therefore, the data give strong constraint on the inter-action potential of the ΛΛ system. Indeed, the favoredpotentials obtained from this analysis agrees well withthe results obtained in the analysis of 12C(K−,K+)ΛΛXreaction [9]. Enhancement of the ΛΛ invariant mass spec-trum at low energies are well described by the final stateinteraction effects with fss2 [25] and ESC04d [40]. Thescattering length and the effective range in ESC04d in-teraction are a0 = −1.323 fm and reff = 4.401 fm, re-

12

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5

χ2 /Nd

of

R [fm]

ND56ND58NF50

NSC89-820fss2

ESC08FG

HKMYY 0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

ND56,R=1.1fmND58,R=1.1fmNF50,R=0.8fm

NSC89-820,R=1.0fmfss2,R=0.7fm

ESC08,R=0.9fmFG,R=1.1fm

HKMYY,R=1.0fmSTAR 0-80%

FIG. 10: Left: χ2/Ndof for the favored potentials in the cylindrical source model with collective flow (15). Right: correspondingΛΛ correlation functions at the minimum χ2.

spectively, which are inside the region obtained in thepresent analysis. The analyses of ∆BΛΛ in the Nagaraevent [2, 3] show ΛΛ interactions which are inside theallowed region.

V. FEED-DOWN CONTRIBUTION

A. Estimate of feed-down contribution

So far we have assumed that Λ baryons are directlyemitted from the hot matter. This assumption would bevalid if one could remove decay contribution from parentparticles such as Ξ and Σ0 or the correlation functionC(Q) was not affected by such feed-down contribution.In the case of the ππ correlation function, it has beenknown that long-lived parents give a sharp correlationnear Q ≃ 0 which cannot be resolved thus cause an ap-parent reduction of the intercept C(Q = 0) [41–43]. Thesame argument applies to the ΛΛ correlation as well. ForNtot being the total number of measured Λ and Np beingthe long-lived parents decaying into Λ, respectively, theeffective intercept λ is given by

λ =

(

1− Np

Ntot

)2

. (23)

The correlation function after the feed-down correctionto the bare one reads

Ccorr(Q) = 1 + λ(Cbare(Q)− 1), (24)

which should be confronted with the data.The contribution to Ntot mainly consists of Σ(1385),

Σ0, Ξ as well as direct Λ and heavier resonances such asΩ are negligible. Since decay width of Σ(1385) is 36-40MeV, this contribution will give the Λ source functionan effectively long lifetime and might influence low Q

behavior of C(Q), but will not affect λ. Thus, we donot treat daughters from Σ(1385) as a part of the long-lived parents but regards them as a part of direct Λ. Oneneeds to invoke dynamical simulations for more seriousestimates of these short-lived resonance effects.

Then Np consists of contribution from Σ0 and Ξ. Thefraction of Σ0 to Λ in heavy ion collisions is not experi-mentally known, because of the difficulty in reconstruc-tion of Σ0 → Λγ process. Here we utilize an experi-mental result in p+Be collisions at plab = 28.5GeV [44],NΣ0/NΛ = 0.278. Although the production process ofthe hyperons in pA collisions could be different from thatin heavy ion collisions, we note that this ratio is consis-tent with thermal model calculations [45].

Ξ yields in Au+Au collisions at√sNN = 200 GeV

has been shown to 15% of total Λ [36]. Here we assumethat a part of Ξ decay contribution to Λ is excluded bythe candidate selection employed in the STAR measure-ment [15], according to the distance of closest approachless than 0.4cm which is comparable to minimum decaylength of Ξ. Combining these fractions of resonances toΛ and subtracting the Ξ contributions from Ntot, We ob-tain λ = (0.67)2. If we take account of the Ξ contributioninto the total yields, λ = (0.572)2. Since the above se-lection does not exclude all of Ξ, the realistic value couldbe a little smaller than (0.67)2, but larger than (0.572)2.We estimate the higher resonance contribution to Ntot towithin a few percent. Therefore, we reanalyze the databy correcting the correlation function obtained from thecylindrical source function (15), with λ = (0.67)2. Weconfirmed that the lower value λ = (0.572)2 only givesquantitative change in χ2/Ndof values in the followinganalyses.

13

0

5

10

15

20

0 0.5 1 1.5 2 2.5

χ2 /Nd

of

R [fm]

Cylindrical, λ=(0.67)2

Data : STAR 0-80%

ND58NF52

NSC89-1020NSC97a

NSC97bNSC97cNSC97eNSC97f

fss2ESC08

FGHKMYY

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

Cylindrical, λ=(0.67)2

ND58,R=0.2fmNF52,R=0.4fm

NSC89-1020,R=0.4fmNSC97a,R=0.4fmNSC97b,R=0.4fmNSC97c,R=0.4fmNSC97e,R=0.4fmNSC97f,R=0.5fm

fss2,R=0.1fmESC08,R=0.1fm

FG,R=0.5fmHKMYY, R=0.1fm

STAR 0-80%

FIG. 11: Same as Fig. 10, but corrected for Σ0 feed-down contribution by λ = (0.67)2.

B. Effects on Correlation function

Results of reanalysis with the feed-down contributionare shown in Fig. 11. The correction factor λ = (0.67)2

leads to C(Q = 0) = 0.776. This implies the reduc-tion of not only the intercept, but also whole correlationfunction. As a result, C(Q) becomes less sensitive tothe difference among the potentials and we have morepotentials which can fit the data. For instance, NSC97potentials now reproduce the data except for NSC97d.However, one immediately notes that the potentials re-produce the data with unphysically small transverse size,R ≤ 0.5fm. Owing to the multiplicative factor λ, the tailof C(Q) becomes closer to unity. Then smaller size ispreferred to fit the long tail in the STAR data. Thus. ifwe adopt the smaller value of λ, the minimum of χ2 shiftsto smaller R values. As explained in Sec. IV, the smallersize is accompanied by the stronger effect of the attrac-tion. ND56 and NSC89-820, which already overshoot thedata in the uncorrected case due to too strong attractionin Fig. 10, can no longer reproduce the data while someof those with weaker attaction fit the data well. fss2 andESC08 overshoot the data despite the weaker attaction.Non-monotonic behavior in χ2/Ndof indicate that bothmay fit the low Q region with R ≃ 0.4fm, but minimumχ2 cannot be achived due to the tail of C(Q).

C. Residual correlation

The agreement between the data and models at smallsource size cannot be physically reasonable result. Itrather suggests that there exist additional sources of thecorrelation which give the long tail of C(Q). In Ref. [15],a Gaussian term with two additional fitting parametersares and rres,

Cres(Q) = arese−r2

resQ2

, (25)

is employed in the fitting function Cfit(Q) to account forthe long tail as a residual correlation, presumably causedby parent particles. While this prescription is found toimprove the quality of the fit, the origin is not known.Here we investigate effects of the additional term by eval-uating χ2 as a function of ares and rres for each of sourcesize R.

Figure 12 shows results including the residual cor-relation term for potentials which give reasonable fitsto the data before the feed-down correction. One seesthat χ2/Ndof ≃ 1 independent of the size parameter forR > 0.5fm. The strength ares and the spatial size param-eter rres of the residual correlation exhibit R dependence.In particular, for R < 1fm, rres becomes larger as R isdecreased and eventually rres > R. On the other hand,rres and ares for different potentials approach to commonvalues as R increases. These tendencies might indicate atwo-source structure in the data. Although unrealistic,in case of small R and large rres, the large variation in thestrength ares (see the middle panel of Fig. 12) indicatesthat the low Q behavior is dominated by the residualterm and the long tail of C(Q) is fitted with the interac-tion and the collective effects in the source function bysmall R. In this case, the role of each terms is invertedbut the general structure is kept such that the low Q partis sensitive to the interaction and the high Q tail is at-tributed to a correlation in a small size. The two sourcestructure might be more natural for small rres and largeR, in which rres approaches to 0.6fm and R is comparableto source sizes extracted from proton-proton correlationmeasurements.

We find that the same behavior is also seen in otherpotentials with 1/a0 ≤ −0.8fm. The feed-down correc-tion reduces the sensitivity to ΛΛ interaction in low Qregion. Consequently, the data no longer constraints theeffective range, as demonstrated in Fig. 13, where thecorrelation functions are plotted for R = 2.5 fm, in ac-cordance with the χ2 analysis in Fig. 12. All the shown

14

0

1

2

3

4

0 0.5 1 1.5 2 2.5

χ2 /Nd

of

R [fm]

τ0=5fm/c, ∆τ = 2fm/c

λ=(0.67)2, +Cres(Q)

Data : STAR 0-80%

fss2ESC08

FGHKMYY

-0.1

-0.08

-0.06

-0.04

-0.02

a res

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6r r

es [

fm]

FIG. 12: Residual correlation parameters as functions of Rfor fss2, ESC08, FG and HKMYY potentials. Bottom: mini-mum of χ2. Top and middle panels : rres and ares giving theminimum χ2, respectively.

potentials fit the data well and no difference is seen inQ > 0.15GeV/c. At low Q, albeit small, C(Q) exhibitinteraction dependence fairly reflecting the strength ofattraction, similarly to the right panel of Fig. 10. Note,however, that FG and fss2 have a factor of two differenteffective range while they have almost the same scatter-ing length. We also find that all the NSC97 potentials, ofwhich reff is broadly ranged from 1.15fm to 16.33fm andthe scattering length is 1/a0 < −2fm−1, can reproducethe data with χ2/Ndof ≃ 1.In general, results with the residual correlation term

(25) depends on the feed-down contribution λ. Since thevalue λ = (0.67)2 takes only Σ0 into account, this servesa minimal correction owing to possible Ξ contribution.We confirmed the present result for the constraint on the

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

C(Q

)

Q [GeV/c]

Cylindrical, +Cres(Q)

λ=(0.67)2

fss2, R=2.5fmESC08, R=2.5fm

FG, R=2.5fmHKMYY, R=2.5fm

STAR 0-80%

FIG. 13: Correlation functions combined with the residualterm and the feed-down correction for R = 2.5fm.

scattering length, 1/a0 < −0.8fm, holds for smaller λ byrepeating the same analyses for λ = (0.572)2 in which Ξcontribution is included.

The above discussion applies to all the potentials witha0 < 0 analyzed here. We note that there are two ex-ceptions. We find that ND46 and NF42, which have thepositive largest 1/a0 thus have a bound state, can fit thedata when R < 0.7fm with the residual correlation taking2 < rres < 4fm and −0.2 < ares < −0.08. We considerit to be coincidence, since it is accompanied with rreslarger than the source size and χ2/Ndof ∼ 1 is achivedonly in small R region. As we shall discuss below, theappearance of the bound state should lead to suppressionof C(Q) at low Q when the source size is larger than a0.Therefore, one may be able to confirm or rule out thispossibility by analyzing data of more central collisions,which are expected to have a larger source size.

VI. DISCUSSION

A. Possible signal of H resonance

On the basis of the scattering length and the effectiverange of the ΛΛ interaction obtained in the present anal-yses, the existence ofH particle as a bound state of ΛΛ isnot preferred. This can be understood from the enhancedΛΛ correlation function observed in the data comparedwith the free case. If we had a bound state in ΛΛ, thecorrelation function would be suppressed from the freecase. The scattering wave function has the asymptoticform, χq(r) = e−iδ sin(qr + δ)/qr, where q = Q/2 is therelative momentum of Λ. In the case of small enoughinteraction range compared with the source size, we cansubstitute the asymptotic form for the scattering wavefunction χq(r) in Eq. (11) and obtain the low energy

15

limit of the correlation function,

C(Q) → 1

2− 1√

π

a0R

+1

4

(a0R

)2

(Q → 0) , (26)

where the phase shift is given approximately as δ ≃ −a0q.For ΛΛ interaction with a bound state (a0 > 0), Thescattering wave function has a node at r ≃ a0 at low en-ergies, then the correlation function is suppressed com-pared with the free case in the low energy limit, as long asthe second term dominates in Eq. (26). Thus we wouldsee suppressed Q region if we have a bound state. Inpractice, the interaction range is not small enough com-pared with the source size considered here, thus the aboveestimate might not be precise. It should be noted thatthe above argument is not valid, when ΛΛ is not thedominant component of H .

0.8

0.9

1.0

0 100 200 300 400 500

C(Q

)

Q (MeV/c)

Expected H signal

STAR (0-80 %)fit

(EH, ΓH)=(14 MeV, 4.5 MeV), x 10(1.8 MeV, 1.5 MeV), x 2

FIG. 14: Possible resonance H signal in the ΛΛ correlationfunction. Signal for (EH ,ΓH) = (14 MeV, 4.5 MeV) and(EH ,ΓH) = (1.8 MeV, 1.5 MeV) are multiplied by 10 and2, respectively.

The existence of H as a resonance pole above the ΛΛthreshold is another interesting possibility, as suggestedin KEK experiments [8, 9]. While ΛΛ potentials con-sidered here do not have H as an s−wave resonance,a quark model calculation with instanton induced in-teraction allow the existence of resonance H below theΞN threshold [46]. In order to evaluate the strengthof the resonance H signal in the correlation function,we have invoked the statistical model results. In thestatistical model [47], H (Λ) yield is calculated to beNH ≃ 1.3 × 10−2 (NΛ ≃ 30) per event per unit rapid-ity. We here assume that the resonance H is produced ina different mechanism from the ΛΛ potential scattering.We also assume that the mass of H is distributed accord-ing to the Breit-Wigner function, then the contributionof resonance H in the ΛΛ relative momentum spectrumis given as

dNH

dydQ= NHfBW(EQ)

dEQ

dQ, (27)

where fBW(E) = ΓH/[(E − EH)2 + Γ2H/4]/2π is the

Breit-Wigner function. In Fig. 14, we show the strengthof the resonance H signal. We have fitted the STARdata in a simple smooth function, and added theratio of dNH/dydQ to the thermal ΛΛ distribution,dNΛΛ/dydQ = 4πq2NΛΛ exp(−q2/2µT )/(2πµT )3/2/2,where q = Q/2, NΛΛ = N2

Λ and µ = MΛ/2. The signalis multiplied by a factor. The resonance parameters andthe multiplication factor are chosen so as to fit the bumpstructure in the data as if the bump is the resonance Hsignal.We find that the resonance H signal relative to the

thermal background is small and higher statistics is nec-essary to confirm its existence, especially when the reso-nance energy is large, EH > 10 MeV. In order to demon-strate this point, we show the results with (EH ,ΓH) =(14 MeV, 4.5 MeV) and (EH ,ΓH) = (1.8 MeV, 1.5 MeV),where the signal is multiplied by 10 and 2, respectively.In the case where the resonance energy is large, thesignal is in the same order of the bump height aftermultiplied by 10. The bump structure around 14 MeV(Q ∼ 250 MeV/c) in the correlation function data forthe 0-80 % centrality seems to come from statistical fluc-tuations and is not considered to be the signal of theresonance H , since the position and height depend onthe centralilty [48]. If the resonance H exists at smallenergy above the ΛΛ threshold, it may be possible todetect the signal. The bump in the data at around 1.8MeV (Q ∼ 90 MeV/c) is also considered to come fromthe statistical fluctuation, and its strength is about twiceof the statistical model estimate of the H signal. If wecan reduce the error to a half, it would be possible toconfirm or rule out the existence of resonance H at lowenergies.

B. Implication to ΛΛN three-body interaction

Another interesting implication of the present analysesis the difference between the vacuum and in-medium ΛΛinteractions. The ΛΛ interactions in Refs. [2, 3] may beless attractive than those expected from the ΛΛ corre-lation data; as seen in Figs. 3 and 10, ΛΛ correlationsfrom these two interactions tend to be smaller than thedata. The strengths of these ΛΛ interaction are fittedto the ΛΛ bond energy in the Nagara event ∆BΛΛ =1.01±0.20+0.18

−0.11 MeV [1]. The ΛΛ bond energy is recentlyupdated to be a smaller value, ∆BΛΛ = 0.67± 0.16 MeV(BΛΛ = 6.91 MeV) [49] following the Ξ− mass updateby the Particle Data Group [50]. The updated ∆BΛΛ

value could imply further weaker ΛΛ interaction. For ex-ample, the HKMYY interaction was updated to fit thenew ∆BΛΛ [51], and the scattering parameters become(a0, reff) = (−0.44 fm, 10.1 fm),which is outside of the fa-vored region based on the analyses without the feed-downcorrection.While it is still premature to draw any conclusion, less

attractive ΛΛ interaction in nuclei than the interaction in

16

vacuum would suggest the density dependence of the ΛΛinteraction. A part of ΛΛ attraction comes from the cou-pling with the ΞN channel ΛΛ ↔ ΞN , since the thresholdenergy difference is small, MΞ +MN − 2MΛ ∼ 28 MeV.The Pauli blocking in the ΞN channel leads to reduc-ing the attraction in the ΛΛ channel. All the nucleon 0sstates are occuplied in 6

ΛΛHe, the nucleon in the inter-mediate state has to be in the p state, and the couplingeffect is suppressed. Myint, Shinmura and Akaishi foundthat the coupled channel Pauli suppression effect resultsin the reduction of the bond energy in 6

ΛΛHe by 0.09,0.43 and 0.88 MeV for the ND, NSC97e and NF couplingstrength, respectively [52]. Their result with NSC97e in-teraction ∆BΛΛ = 0.64 MeV roughly coincides with theupdated value, but their estimate of the scattering pa-rameters (a0, reff) = (−0.5 fm, 8.41 fm) is outside thefavored region by the STAR dat in which the feed downcorrection is not taken into account. It would be an inter-esting issue whether ΛΛ interaction with strong couplingwith ΞN channel can consistently explain the Nagaraevent and the RHIC data.The coupled channel Pauli suppression is found to be

one of the important origins to generate repulsive contri-bution in nuclear matter at high density [53]. Thus fur-ther investigations of ΛΛ correlation and double hypernu-clei would open a way to access the density dependenceof the ΛΛ interaction, or the ΛΛN three-body interac-tion, which would be important to understand the originof the additional repulsion required to support massiveneutron stars with strange hadrons [5].

C. Comparison with the previous work

Before closing the section, we briefly comment on theanalysis by the STAR Collaboration in [15]. Using theLednicky and Lyuboshitz analytical model, they foundthat the same data favor, albeit weak, a repulsive in-teraction a0 > 0 in contrast to our analysis suggestingweakly attractive interaction; the phase shift at low en-ergies, δ ≃ −a0k, increases when a0 < 0 as suggested inthe present work, while it decreases when a0 > 0 as inthe analysis by the STAR collaboration. 2

The main reason for this discrepancy would come fromthe difference in the treatment of the intercept parame-ter λ. In the analysis in [15], it is a fitting parameter toobtain minimum χ2 while we fix it from yields of parentparticles decaying to λ. Their result λ = 0.18± 0.05+0.12

−0.06

is much smaller than ours, and gives C(0) closer tounity. A small value of λ leads to a small bare corre-lation Cbare(Q) at small Q as found from Eq. (24), andthe low energy value would be less than the free value,Cbare(Q → 0) < 0.5, implying a repulsive interaction.

2 Note that sign convention of a0 is different from that in the

present paper.

This indicates the importance to understand the λ valuein the analysis of ΛΛ correlation aiming at extracting theinteractions.

VII. CONCLUDING REMARKS

We have studied the ΛΛ correlation function in rel-ativistic heavy-ion collisions. By using a simple staticsource model, we illustrate how the correlation functionis sensitive to differences in interaction potentials. Fromfits to the measured experimental data, it turned outthat the favored potentials have small negative scatter-ing length and effective range around 4 fm. Then weexamined effects of the collective expansion on the be-havior of the correlation function by making use of athermal source model with the boost-invariant expansionalong the collision axis and transverse expansion fittedto pT spectrum. We point out that the strong expan-sion modifies the behavior of the correlation function atsmall Q, but it remains sensitive to the potential. Inparticular, it turns out that the same potentials as thestatic source case are favored. We have obtained a set ofthe potentials which give reasonable fit to the data witha small transverse size, under an assumption that feeddown correction to the correlation function is negligible.Such potentials are characterized by the scattering length−1.8 fm−1 < 1/a0 < −0.8 fm−1 and the effective range3.5 fm < reff < 7 fm, as represented by a thick shadedarea in Fig. 1.In the above analysis, we treat the size parameter R

as a fitting parameter to the experimental data. The ob-tained source size is found to be somewhat smaller thanthe HBT radii for protons. Discussion on the collision dy-namics based on the size, as done in pion HBT studies, isbeyond our scope in this paper. A possible reason mightbe small scattering cross sections of Λ with other par-ticles, which give the correlation function less sensitiveto the later stage of the collision process. Although theagreement of the favored potentials between the staticsource model and the boost-invariant source model anal-yses seems to suggest irrelevance of the detailed dynamicsof collisions, our findings should be confirmed by studieswith more realistic source models.We have also studied effects of feed down correction

from Σ0 → Λγ decay in analogy with effects of long-livedparticles in the pion HBT. The reduction of the inter-cept parameter λ is found to reduce the sensitivity ofthe correlation function to ΛΛ interaction in low Q andlead to unphysically smaller source size to fit high Q tail.To remedy this, we have considered an additional Gaus-sian term in the correlation function and found that thedata is well described by the two source structure. Weconfirmed that the low Q part remains sensitive to theinteraction with coarse resolution such that the correla-tion function is no longer sensitive to the effective range.The resultant constraint on the scattering length is foundto be 1/a0 < −0.8fm−1. In this case, the source size is

17

roughly consistent with proton HBT radii, but the longtail in Q is attributed to the additional term of whichorigin is not known.Since this term can be attributed to a source with small

size, it might suggest a different source of Λ produc-tion such as a two-step hadron production mechanism,in which inhomogeneous matter is first formed and de-cays into hadrons later. Nevertheless, the agreement ofthe favored potentials in this work with the most recentΛΛ potential models (fss2 and ESC08) indicate relevanceof our study and demonstrates the feasibility of using theΛΛ correlation function to extract the interaction. Fur-thermore, channel coupling needs to be taken care of fora better understanding of data. Preliminary analysis [17]shows that the coupling with ΞN channel is not signifi-cant as long as the coupling potential is not very strong.A more consistent treatment and data with higher statis-tics are desired to pin down the ΛΛ interaction.

Acknowledgments

The authors would like to thank N. Shah and H. Z.Huang for providing them the STAR data. They also

would like to acknowledge T. Rijken, S. Shinmura, Y. Ya-mamoto, E. Hiyama, A. Gal, Y. Akaishi and B. Muller forhelpful discussions. K.M. would like to thank K. Redlichfor discussion on Σ0 decay and hyperon yields in ther-mal models. He also acknowledges other members of In-stitute of Theoretical Physics in University of Wroclawfor discussion. Numerical computations were carried outon SR16000 at YITP in Kyoto university. This work issupported in part by the Grants-in-Aid for Scientific Re-search from JSPS (Nos. (B) 23340067, (B) 24340054,(C) 24540271), by the Grants-in-Aid for Scientific Re-search on Innovative Areas from MEXT (No. 2404:24105001, 24105008), by the Yukawa International Pro-gram for Quark-hadron Sciences, by HIC for FAIR andby the Polish Science Foundation (NCN), under Maestrogrant 2013/10/A/ST2/00106.

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