Squeezed K^+ K^- correlations in high energy heavy ion collisions
aa r X i v : . [ nu c l - t h ] J a n Squeezed K + K − correlations in high energy heavy ion collisions Danuce M. Dudek and Sandra S. Padula ∗ Instituto de F´ısica Te´orica–UNESP, C. P. 70532-2, 01156-970 S˜ao Paulo, SP, Brazil (Dated: June 8, 2018)The hot and dense medium formed in high energy heavy ion collisions may modify some hadronicproperties. In particular, if hadron masses are shifted in-medium, it was demonstrated that thiscould lead to back-to-back squeezed correlations (BBC) of particle-antiparticle pairs. Although well-established theoretically, the squeezed correlations have not yet been discovered experimentally. Amethod has been suggested for the empirical search of this effect, which was previously illustrated for φφ pairs. We apply here the formalism and the suggested method to the case of K + K − pairs, sincethey may be easier to identify experimentally. The time distribution of the emission process playsa crucial role in the survival of the BBC’s. We analyze the cases where the emission is supposed tooccur suddenly or via a Lorentzian distribution, and compare with the case of a L´evy distributionin time. Effects of squeezing on the correlation function of identical particles are also analyzed. DOI: 10.1103/PhysRevC.82.034905
INTRODUCTION
Since the beginning of the 1990’s, some people startedcalling attention to the possible existence of a differenttype of correlation, occurring between particles and theirantiparticles. Initially, in 1991, Weiner et al.[1] pointedout to the surprise existence of a new quantum statis-tical correlation between π + π − , which would be simi-lar to the π π case (since π is its own antiparticle),but entirely different from the Bose-Einstein correlations(between π ± π ± ) leading to the Hanbury-Brown & Twiss(HBT) effect. They related those correlations to the ex-pectation values of the annihilation (creator) operators, < ˆ a ( † ) ( k )ˆ a ( † ) ( k ) > = 0, which was then estimated byusing a density matrix containing squeezed states, analo-gous to two-particle squeezing in optics. They predictedthat such squeezed correlations would have intensitiesabove unity, either for charged or neutral pions, i.e., C s ( π + π − ) > C s ( π π ) >
1. Later, Sinyukov[2],discussed a similar effect for π + π − and π π pairs, claim-ing that they would be due to inhomogeneities in thesystem, in opposition to homogeneity regions in HBT,coming from a hydrodynamical description of the systemevolution.Other tentative models tried to formulate the problemmore accurately, and it finally happened at the end ofthat decade, in a proposition made by M. Asakawa et al.[3]. In their approach, these squeezed back-to-back cor-relations (BBC) of boson-antiboson pairs resulted from aquantum mechanical unitary transformation relating in-medium quasi-particles to two-mode squeezed states oftheir free counterparts. We discuss it in some more de-tail below. Shortly after that, P. K. Panda et al.[4] pre-dicted that a similar BBC between fermion-antifermionpairs should exist, if the masses of these particles weremodified in-medium. Both the fermionic (fBBC) and thebosonic (bBBC) back-to-back squeezed correlations are described by analogous formalisms, being both positivecorrelations with unlimited intensity. This last featurecontrasts with the observed quantum statistical correla-tions of identical bosons and identical fermions, whoseintensities are limited to vary between 1 and 2, or 0 and1, respectively. In the remainder of this paper, we focusour discussion on the bosonic case only.The correlation reflecting the squeezing is quantifiedin terms of the ratio of the two-particle distribution bythe product of the single-inclusive distributions, i.e., thespectra of the particle and of the antiparticle. For thesake of comprehension, we first briefly discuss the for-malism for bosons that are their own antiparticles, suchas φφ or π π pairs. In this case, the full correlationfunction, after applying a generalization of Wick’s the-orem for locally equilibrated systems [5, 6] consist of apart reflecting the identity of the particles (HBT), andanother one, reflecting the particle-antiparticle squeezedcorrelation (BBC). This can be written as C ( k , k ) = N ( k , k ) N ( k ) N ( k )= 1 + | G c (1 , | G c (1 , G c (2 ,
2) + | G s (1 , | G c (1 , G c (2 , . (1)The invariant single-particle and two-particle momentumdistributions are given by G c ( i, i ) = ω k i h ˆ a † k i ˆ a k i i = ω k i d Nd k i G c (1 ,
2) = √ ω k ω k h ˆ a † k ˆ a k i ,G s (1 ,
2) = √ ω k ω k h ˆ a k ˆ a k i . (2)In the above equations, h ... i represents thermal averages.The first term in Eq. (2) corresponds to the spectrumof each particle, the second is due to the indinstinguibil-ity of identical particles, reflecting their quantum statis-tics. The third term, in the absence of in-medium massshift is in general identically zero. However, if the parti-cle’s mass is modified in-medium, it can contribute signif-icantly, triggering this novel type of particle-antiparticlecorrelation, yet to be discovered experimentally. This isachieved by means of a Bogoliubov-Valatin (BV) trans-formation, which relates the asymptotic creation (anni-hilation) operators, ˆ a † k (ˆ a k ), of the observed bosons withmomentum k µ = ( ω k , k ), to the in-medium operators, ˆ b † k (ˆ b k ), corresponding to thermalized quasi-particles. TheBV transformation is given byˆ a k = c k ˆ b k + s ∗− k ˆ b †− k ; ˆ a † k = c ∗ k ˆ b † k + s − k ˆ b − k , (3)being c k = cosh( f k ) and s k = sinh( f k ); ( − k ) denotes anopposite sign in the spacial components of the momenta.For conciseness, we keep here the short-hand notationintroduced in Ref.[3] . The coefficient f i,j ( x ) = 12 log " K µi,j ( x ) u µ ( x ) K ∗ νi,j ( x ) u ν ( x ) , (4)is the squeezing parameter, where K µi,j ( x ) = ( k µi + k µj )is the average of the momenta of each particle, and u µ isthe flow velocity of the system. The BV transformationbetween the operators is equivalent to a squeezing oper-ation, from which the name of the resulting correlationis derived.In case of charged mesons, such as π ± or K ± , the termsin Eq. (1) would act independently, i.e., either the firstand the second terms together would lead to the HBTeffect (for π ± π ± or K ± K ± pairs), and the first and thelast terms, to the BBC effect (for π + π − or K + K − pairs).The in-medium modified mass, m ∗ , was originally[3]related quadratically to the asymptotic mass, m , i.e., m ∗ ( | k | ) = m − δM ( | k | ), where the shifting in the mass, δM ( | k | ), could depend on the momenta of the particles.Nevertheless, adopting the same simplified assumption asin a few previous studies [7]-[16], we also consider here aconstant mass-shift, homogeneously distributed all overthe system, and related linearly to the asymptotic massby m ∗ = m ± δM . RESULTS FOR K + K − PAIRS
Initial studies of the problem were performed for astatic, infinite medium [3, 4], later extended to a finite-size system, radially expanding with moderate flow[7].For simplicity, a non-relativistic approach was consid-ered, assuming flow-independent squeezing parameter.The expansion of the system was described by the emis-sion function from the non-relativistic hydrodynamicalparameterization of Ref.[9], later shown to be a non-relativistic hydrodynamical solution. In Fig. 1 we illus-trate these assumptions with a simple sketch. The flowvelocity during the system expansion was considered as v = h u i r /R . The values h u i = 0 and h u i = 0 . FIG. 1: Sketch illustrating the cross-sectional area of theGaussian profile ( ∼ e − r / (2 R ) ) of the system expanding withradial flow. correlation function[7] of K + K − pairs (first and thirdterms in Eq. (1)), as C s ( k , k ) = 1+ ( E | + E ) | c | | s | E E (cid:12)(cid:12)(cid:12) R e − R k1 + k2 )22 +2 n ∗ R ∗ e − ( k1 − k2 )28 m ∗ T exp h(cid:16) − im h u i R m ∗ T ∗ − m ∗ T ∗ − R ∗ (cid:17) ( k + k ) i(cid:12)(cid:12)(cid:12) × nh | s | R + n ∗ R ∗ ( | c | + | s ) | ) exp (cid:16) − k m ∗ T ∗ (cid:17)ih | s | R + n ∗ R ∗ ( | c | + | s ) | ) exp (cid:16) − k m ∗ T ∗ (cid:17)io − . (5)The medium-modified radius and temperature in Eq. (5)are written, respectively, as R ∗ = R p T /T ∗ and T ∗ = T + m h u i m ∗ , as introduced in Ref. [7].As done in the case of φφ correlations[15], it is instruc-tive to analyze the behavior of the correlation functionfor exactly back-to-back particle-antiparticle pairs, i.e.,pairs with exactly opposite momenta, as a function of the shifted mass parameter, m ∗ , and of the absolute value oftheir momenta. Therefore, we investigate the behaviorof C s ( k , − k , m ∗ ) as a function of m ∗ and | k | . This is ob-tained by imposing the idealized limit of k = − k = k in Eq. (5). Consequently, a few simplifications occur atonce in that equation, i.e., k − k = 2 k and k + k ≡ FIG. 2: (Color online) C ( k , − k ) × m ∗ × | k | comparing the instantaneous and the Lorentzian distributions for both the staticcase, h u i = 0, and for an expanding system with radial flow parameter h u i = 0 . Another essential assumption is underlying the aboveresult. The solution in Eq. (5) follows when an instan-taneous process is considered for the particles’ emission.We adopt throughout the paper ¯ h = c = 1. In the caseof instantaneous emission, the time factor is given by | e − i ( E + E ) τ | = 1 , (6)which results from the Fourier transform of an emissiondistribution described by a delta function. Nevertheless,it is not expected that it this corresponds to a realisticsituation. An emission lasting for a finite time inter-val seems more appropriate. Naturally, a priori it is notknown which functional form better describes the parti-cle emission process. In what follows, we consider twoother types of distribution. One of them is a Lorentzianform, | F (∆ t ) | = [1 + ( ω + ω ) ∆ t ] − , (7)where ω i = p k i + m . The Lorentzian emission distri-bution in Eq. (7) was suggested in Ref.[3] and adoptedin previous studies [4, 7, 8],[11]-[16]. Either of the fac-tors in Eq. (6) or (7) should multiply the second term in Eq. (5). As discussed in Ref. [3], in the adiabatic limit,∆ t → ∞ , the time factor in Eq. (7) completely sup-presses the back-to-back correlation (BBC). On the con-trary, in the instantaneous approximation, either fromEq. (6) or in the limit ∆ t → α -stable L´evy distribution, i.e., | F (∆ t ) | = exp {− [∆ t ( ω + ω )] α } . (8)This functional form was used in the analyses made bythe PHENIX Collaboration[18] to fit two- and three-particle Bose-Einstein correlation functions. Accordingto that analysis, depending on the region investigated ofthe particles’ transverse momentum or transverse mass,good confidence level was obtained in the fit for differentvalues of α . They found α ∼ . < m T < . α = 1 .
35 for 0 . < p T < . α . The L´evydistribution in Eq. (8), should also multiply the secondterm in Eq. (5). We will see in what follows that thereduction effect of this distribution on the squeezed cor-relation function is even more dramatic than the effectof the Lorentzian in (7).We show in Fig. 2 results comparing the time emis-sion distributions of Eq. (6) and (7). The freeze-outtemperature ( T = 177 MeV) and radial flow ( h u i ≈ . C ( k , − k ) × m ∗ × | k | , decreases al-most three orders of magnitude due the Lorentzian timefactor, as compared to the instantaneous emission. How-ever, the resulting signal is still strong enough to allowfor its experimental search. Another interesting outcomeof the calculation is shown in the left panel in Fig. 2,i.e., parts (a) and (c), as compared to the right panel,i.e., parts (b) and (d). In this case, we see the effectof the expansion of the system on the squeezed correla-tion function. The growth of the squeezed correlationfor increasing values of | k | is faster in the static case ascompared to when < u > = 0 .
5, specially at high valuesof | k | . Nevertheless, the presence of flow seems to en-hance the intensity of Cs ( k , − k , m ∗ ) in the whole regionof the ( m ∗ , | k | )-plane investigated, mainly in the lower | k | -region. Naturally, at m ∗ = m K ± ∼
494 MeV, thesqueezing disappears, i.e., C s ( k , − k , m ∗ = m K ± ) ≡ < u > = 0.Therefore, we show only results for h u i = 0 . C ( k , − k ) × m ∗ × | k | for α = 1 and α = 1 . t = 1 fm/c or ∆ t = 2 fm/c. We see that, evenfor a short-lived system, with ∆ t = 1 fm/c and α = 1,the reduction of the squeezed correlation intensity dueto the L´evy distribution is even more dramatic than thatdue to the Lorentzian time emission. For α = 1 .
35, thatstrength is driven to values probably unmeasurable in afirst tentative search. For ∆ t = 2 fm/c, the situation isconsiderably worse, even if α = 1. Finally, combining∆ t = 2 fm/c with α = 1 .
35 reduced the signal basi-cally to unity, the first non-zero decimal digit being toosmall for the precision of the axis scale, if we tried to plot C ( k , − k ) as in the other parts of Fig. 3. That is whyin Fig. 3(d) we plot C ( k , − k ) −
1. We see that the re-sulting squeezed correlation function acquires values toosmall to be measured by this method. Therefore, if Na-ture favors the L´evy distribution and if the emission lastsa short period, i.e., ∆ t = 1 fm/c, the predicted strengthof C ( k , − k ) × m ∗ × | k | from Fig. 3 makes it still possibleto search for the signal, if α = 1. However, if α = 1 . m ∗ , and back-to-back momenta of the pair, k = − k = k . This approach, however, focus the studyon the behavior of the maximum value of C s ( k , − k , m ∗ ).In other words, if we make an analogy to the HBT ef-fect between identical particles, this corresponds to in-vestigate the behavior of the correlation function’s inter-cept. Nevertheless, it is not efficient for the purpose ofsearching for the BBC experimentally, since the modi-fied mass of particles is not an observable quantity, ex-isting only inside the hot and dense medium. Besides,the measurement of particle-antiparticle pairs with ex-actly back-to-back momenta has zero probability to hap-pen in practice. It would be more realistic to look fordistinct values of the momenta of the particles, k and k , and combine in an appropriate manner. Therefore,following previous knowledge of identical particle corre-lations (HBT), the first natural tentative method wouldbe to measure the squeezed correlation function in termsof the momenta of the particles combined as their av-erage, K = ( k + k ), and their relative momenta, q = ( k − k ) [12]-[15]. However, this propositionconsiders non-relativistic momenta and therefore has itsapplication constrained to this limit. For a relativistictreatment, M. Nagy [12] proposed to construct a mo-mentum variable defined as Q µback = ( ω − ω , k + k ) =( q , K ). In fact, it is preferable to redefine this variableas Q bbc = − ( Q µback ) = 4( ω ω − K µ K µ ), whose non-relativistic limit is Q bbc → (2 K ) , returning to the av-erage momentum variable proposed above. Although notinvariant, the advantage of constructing Q bbc as indicatedis that the squeezed correlation function would have itsmaximum around the zero of this variable, keeping a closeanalogy to the HBT procedures and to its non-relativisticcounterpart. In the remainder of this paper, we attainour study to the non-relativistic limit, where the analyt-ical results of the model under discussion, written in Eq.(5) and related ones, are safely applicable.The analogy with the HBT method is not completelytransferred to the study of the BBC effect. In HBT ex-perimental analyses a common practice is to replace theproduct of the two spectra by mixed events, since theseare the reference sample not containing statistically cor-related pairs. However, we see that the second line inEq. (5), representing the product of the particle and theantiparticle spectra in BBC, does contain the squeezingfactor f i,j as well. Therefore, the mixed events tech-nique would not be an appropriate reference sample inconstructing the BBC correlation function. FIG. 3: (Color online) C ( k , − k ) × m ∗ ×| k | for the symmetric, α -stable L´evy distribution with parameters α = 1 . α = 1 . Once defined the choice of plotting variables as K and q , we can proceed to study the squeezed correla-tion function. For emphasizing the characteristics to besearched for, we focus the study to values of the shiftedmass corresponding to the two maxima located moreor less symmetrically below and above the kaon asymp-totic mass, m = m K ± ∼
494 MeV. They correspond to m ∗ = 350 MeV and m ∗ = 650 MeV, respectively. Wethen calculate the squeezed correlation for K + K − pairsusing Eq. (5). From it, is easily envisaged that we shouldreplace k + k = 2 K and k − k = q in the numera-tor, at the same time as replacing k = K + q /
2, and k = K − q / q region, where C s ( K , q , m ∗ ) reaches much higherintensities for m ∗ = 650 MeV than for m ∗ = 350, in-cluding for its intercept at K = 0. In both cases wesee that the presence of flow enhances the strength of C s ( K , q , m ∗ ), potentially facilitating its detection inan experimental search of the effect. In all the investigation and results discussed above, themass-shifting was considered homogeneously distributedover the entire squeezing region, whose size was fixed to R = 7 fm, the radius of the cross-sectional area depictedin Fig. 1. The squeezed correlation function is actuallysensitive to that size. In fact, this is reflected in its inversewidth of the squeezed correlation functions plotted interms of the average momentum, 2 K . In Ref. [15] weillustrate this sensitivity by considering two values for theradii, R = 7 fm and R = 3 fm. The resulting squeezedcorrelation function is shown to be broader for smallersystems than for larger ones. RESULTS FOR K ± K ± PAIRS
Next, we discuss our findings about the effects of in-medium mass-shift and resulting squeezing on the HBTcorrelation function of K ± K ± pairs. Usual expectationswere that thermalization would wash out any trace ofmass-shift in these type of correlations. However, as itwas demonstrated analytically in Ref. [3, 4], the HBT FIG. 4: (Color online) Behavior of the squeezed correlation function in the ( K , q )-plane, fixing the modified mass to m ∗ =350 MeV. correlation function also depends on the squeezing pa-rameter, f i,j ( m, m ∗ ).In fact, this identical particle correlation is obtainedby inputting in Eq. (1) the chaotic amplitude, G c ( k , k ) = E + E π ) n | s | R e − R q + n ∗ R ∗ ( | c | + | s | ) × exp (cid:16) − K m ∗ T ∗ (cid:17) exp h − (cid:0) R ∗ m ∗ T (cid:17) q i × exp h − im h u i Rm ∗ T ∗ K . q io , (9) as well as the expression for the spectrum of eachparticle, G c ( k i , k i ) = E i (2 π ) n | s | R + n ∗ R ∗ ( | c | + | s | ) exp (cid:16) − k i m ∗ T ∗ (cid:17)o . Since it involves the identicalkaons in this case, the third term in Eq. (1) gives nocontribution. The plots corresponding to such results areshown in Fig. 6, for two values of the average momentum, | K | = 0 . | K | = 2 . t = 2 fm/c). Finite emission intervals are alsodescribed by a Lorentzian distribution similar to that inEq. (7), obtained as the Fourier transform of an expo-nential distribution in time, but in this case, obtained interms of the relative energy, q = ω − ω , i.e., | F (∆ t ) | = [1 + ( ω − ω ) ∆ t ] − , (10)where ω i = p k i + m . The factor in Eq. (10) multipliesthe second term in Eq. (1).In Fig. 6(a), with ∆ t = 0, no sensitivity to the two val-ues of | K | is seen, only the effect of flow is evident. Inthe absence of mass-shift and squeezing, the flow broad-ens the curves, as expected, since it is well-known thatthe expansion reduces the size of the region accessible tointerferometry. In part (b), we see that a finite dura-tion of the emission separates the curves for each valueof | K | , both in presence and in absence of flow. Thiseffect is also well-known, and comes from the coupling ofthe average momentum of the pair to the emission dura-tion, ∆ t . Therefore, when there is no mass-shift and nosqueezing, the relations describe correctly the expansioneffects on the identical particle correlation function. q = k - k K = k + k R = 7 fm T = 177 MeV m * = 650 MeV t = 0 fm/c = 0.5 C s ( K , q ) (b) FIG. 5: (Color online) Behavior of the squeezed correlation function in the ( K , q )-plane, fixing the modified mass to m ∗ =650 MeV. When squeezing is present, the flow broadening is seenin Fig. 6(c) for | K | = 0 . | K | = 2 . | K | . Part (d) essentially repeatswhat is seen in (c), except for devising a modest effect re-lated to the finite duration of the emission, which slightlyseparates the curves corresponding to the two values of | K | , when h u i = 0.We remark that we did not include the Coulomb finalstate interactions in the above analysis. In the case of K + K − pairs, even the Gamow factor which over-predictsthe strength of the effect for finite distances would bevery small. In general, the effect of the Coulomb interac-tions is more pronounced for small values of | q | , whichcorresponds to the region where the hadron-antihadronsqueezing correlation is less favored, therefore being lesssignificant to this analysis. Also in the case of K ± K ± pairs, the squeezing affects the width of the HBT corre-lation function and, since the Coulomb effect is mostlyconcentrated in the region where | q | is small [17], it is not expected to be relevant in this context. SUMMARY AND CONCLUSIONS
In this work we discuss an effective way to search for K + K − squeezed correlations in heavy ion collisions, cur-rently at RHIC, and soon at the LHC. We use suit-able variables introduced previously [8],[10]-[15] to in-vestigate the expected behavior of the squeezed corre-lation function in an experimental search of the effect.This is studied by plotting C s ( K , q , m ∗ ) in terms ofthe average momentum of the pair, 2 | K | , and its rel-ative momentum, | q | . These variables are combina-tions of the momenta of the particle and the antiparticleof each pair, and 2 | K | , is the non-relativistic limit of Q bbc = 4( ω ω − K µ K µ ), as discussed previously[12].We started by investigating the general behavior of C s ( k , − k , m ∗ ) for exactly back-to-back K + K − pairs, asa function of both | k | and the in-medium shifted mass, m ∗ . This was showing in Fig. 2 comparing the casesof sudden particle emission and a finite emission intervaldescribed by a Lorentzian distribution. A L´evy distribu- C ( k , k ) q (GeV/c) K = 0.5 GeV/c K = 2.0 GeV/cSqueezing OFFt = 0 fm/cR = 7 fm= 0 = 0.5 = 0 (a) C ( k , k ) q (GeV/c) K = 0.5 GeV/c K = 2.0 GeV/cSqueezing OFF = 0.5 = 0 t = 2 fm/cR = 7 fm = 0 (b) (c) = 0.5 = 0K = 0.5 GeV/cK = 2.0 GeV/cm * = 350 MeV Squeezing ON C ( k , k ) q (GeV/c) K = 0.5 GeV/c K = 2.0 GeV/ct = 0 fm/c R = 7 fm = 0 m * = 350 MeV (d) Squeezing ON = 0.5 = 0
R = 7 fm = 0 C ( k , k ) q (GeV/c) K = 0.5 GeV/c K = 2.0 GeV/c t = 2 fm/c FIG. 6: Identical particle correlation functions for two values of | K | , both for sudden emission (∆ t = 0) and for a Lorentziandistribution in time, from Eq. (10), with ∆ t = 2 fm/c. Parts (a) and (b) show results in the absence of in-medium massmodification. Parts (c) and (d) consider a shifted mass of m ∗ = 350 MeV. tion was also studied, with results shown in Fig. 3. Wecould see the striking reduction effect of finite emissionintervals, even for the Lorentzian distribution. The L´evytype causes an even more dramatic suppression of theeffect. If this distribution is the one favored by Nature,the hadronic squeezed correlation function could still besearched for, if the duration of the emission process isshort, not longer than ∆ t ≃ C s ( K , q , m ∗ ), in the ( K , q )-plane.We find that, in the presence of flow, the signal is ex-pected to be stronger over the momentum regions shownin the plots, i.e., roughly for 0 ≤ | K | ≤ − ≤ | q | ≤ | K | .Finally, it is worth emphasizing that the results shownhere correspond to the signals of the squeezing expectedif the particles have their mass shifted in the hot anddense medium formed in high energy collisions. If theparticle’s properties, such as its mass, are not modified inthe medium, the squeezed correlation functions would beunity for all values of 2 | K | , and therefore, no signal wouldbe observed. It that is the case, then the HBT correlationfunctions would behave as usual, both in the presence orabsence of flow. However, if the particles’ masses areindeed shifted in-medium, the experimental discovery ofsqueezed particle-antiparticle correlation (and the distor-tions pointed out in the HBT correlations) would be anunequivocal signature of these modifications, by meansof hadronic probes. The values of the modified mass, m ∗ , adopted here for illustrating the squeezing effects for K + K − pairs, correspond approximately to the maximumvalues shown in Fig. 2. However, if the modified massturns to be shifted away from the maximum values con-sidered in the above calculations, C s (2 K , q ) wouldattain smaller intensities than the ones shown, but thesignal could still be high enough to be observed experi-mentally. The squeezed correlations are very sensitive tothe form of the emission distribution in time, as shownabove. Instant emissions would fully preserve the signal.Lorentzian time distributions would drastically reduce itand L´evy-type distributions would attenuate it more dra-matically or even make the searched signal unmeasurable.Another important point that needs emphasis is that thesqueezed correlation function should be plotted in the( K , q )-plane. If plotted as function of K only, thismeans that all the variations in each bin of q are av-eraged out, as they are projected in the K -axis. Thiscould enlarge the error bars and decrease the signal sub-stantially, depending on the region of q selected for theplot. Therefore, the experimental search for the squeezedhadronic correlations should aim at good statistics of theevents for enhancing the chances of its discovery. Acknowledgments
We are grateful to Tam´as Cs¨org˝o and Mart´on Nagy formotivating us to investigate the K + K − squeezed corre-lations in the case of a L´evy distribution of the particle’semission as well. DMD is also thankful to CAPES andFAPESP for their financial support during the develop-ment of this work. ∗ Electronic address: [email protected], [email protected][1] I. V. Andreev, M. Pl¨umer and R. M Weiner, Phys. Rev.Lett. 67 (1991) 3475.[2] Yu. Sinyukov, Nucl. Phys.
A566 , 589c (1994). [3] M. Asakawa, T. Cs¨org˝o and M. Gyulassy Phys. Rev. Lett. , 4013 (1999).[4] P. K. Panda, T. Cs¨org˝o, Y. Hama, G. Krein and SandraS. Padula, Phys. Lett. B512 , 49 (2001).[5] M. Gyulassy, S. K. Kaufmann, and L. W. Wilson, Phys.Rev.
C 20 , 2267 (1979).[6] A. Makhlin and Yu Sinyukov, Sov. J. Nucl. Phys. , 354(1987); Yu Sinyukov, Nucl. Phys. A566 , 589c (1994).[7] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda andT. Cs¨org˝o, Phys. Rev.
C73 , 044906 (2006).[8] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda andT. Cs¨org˝o, Proc. Quark Matter 2005, Nucl. Phys.
A774 ,615 (2006).[9] T. Cs¨org˝o, B. L¨orstad, and J. Zim´anyi, Phys. Lett. B338,134 (1994); P. Csizmadia, T. Cs¨org˝o and B. Luk´acs, ibid.B443, 21 (1998).[10] Sandra S. Padula, Y. Hama, G. Krein, P. K. Panda andT. Cs¨org˝o, Proc. Workshop on Particle Correlations andFemtoscopy (WPCF), AIP Conf. Proc. , 645 (2006).[11] T. Cs¨org˝o and Sandra S. Padula, Proc. WPCF 2006,
Braz. J. Phys. (2007) 949.[12] Sandra S. Padula, O. Socolowski Jr., T. Cs¨org˝o and M.Nargy, Proc. Quark Matter 2008, J. Phys. G: Nucl. Part.Phys. , 104141 (2008).[13] Sandra S. Padula, Danuce M. Dudek and O. SocolowskiJr., Proc. WPCF 2008, A. Phys. Pol. , N. 4, 1225(2009).[14] Sandra S. Padula, O. Socolowski Jr. and DanuceM. Dudek, in Proc. of the XXXVIII InternationalSymposium on Multiparticle Dynamics (ISMD 2008),DESYPROC200901, 271 (2009), [arXiv:0812.1784v1(nucl-th)] and [arXiv:0902.0377 (hep-ph), p. 271 (2009)].[15] Sandra S. Padula, O. Socolowski Jr., “Searching forsqueezed particle-antiparticle correlations in high energyheavy ion collisions”, arXiv:1001.0126 [nucl-th] .[16] Danuce M. Dudek, Squeezed Hadronic Correlations of K + K − pairs in Relativistic Heavy Ion collisions , MasterDissertation presented to the Instituto de F´ısica Te´orica- UNESP (March/2009).[17] Miklos Gyulassy and Sandra S. Padula, Phys. Rev. C41,R21 (1990).[18] M. Csan´ad [PHENIX Collaboration], Proc. Quark Mat-ter 2005, Nucl. Phys. A774 , 611 (2006).[19] S. S. Adler et al., PHENIX Collaboration, Phys. Rev.