-
The correlation coefficient
CBS=−3⟨BS⟩−⟨B⟩⟨S⟩⟨S2⟩−⟨S⟩2 between the baryon number B and the strangeness number S in a strongly interacting matter was first proposed in Refs. [1-3] for probing the properties of the matter produced in relativistic heavy-ion collisions. A later study suggested, however, that the strangeness population factorS3= N3ΛH/NΛN3He/Np measured in these collisions could serve as a better probe of the baryon number and strangeness correlation in the produced matter, owing to its different behaviors in QGP and the hadronic matter [4, 5]. Experimentally,S3 increases from heavy-ion collisions at AGS [6] to RHIC energy [7], and then decreases to a small value in collisions at the LHC energy [8]. Compared with the predictions of the statistical model [5, 9, 10], the values ofS3 extracted from the RHIC data, which has large statistical uncertainties, are larger, which led to questioning the data and its interpretation. As to the different values ofS3 measured at RHIC and LHC, a possible explanation was provided in Ref. [11] by assuming an early freeze-out ofΛ compared with nucleons from hadronic matter, and a longer freeze-out time difference at RHIC than at LHC. This idea was further explored in Ref. [12] to study the production of light nuclei in relativistic heavy-ion collisions by considering their finite sizes compared with the size of the produced hadronic matter at the kinetic freeze out.Since the first theoretical estimate (in the 1970s) of the abundance of hypernuclei that could be produced in heavy-ion collisions [13], many studies addressed this very interesting problem, providing increasingly better estimations [6-8, 14-17] (more details can be found in recent topical reviews [18-20]). For the lightest hypernucleus
3ΛH , the separation energy of itsΛ was very small in early measurements, with a typical value of130±50 keV [21], but a larger value of410±120±110 keV has been suggested based on some recent measurements and using a more precise method [22]. Owing to this significantly smallerΛ separation energy than the nucleon separation energy in normal nuclei with a similar mass number [23],3ΛH can be considered as a loosely boundd−Λ 2-body system. In relativistic heavy-ion collisions, the hypertriton is expected to be in chemical equilibrium withΛ and deuteron as well as with proton and neutron; then, its yield is the same whether it is calculated from the coalescence ofΛ with deuteron or with proton and neutron. As shown in Ref. [17] based on the coalescence model, the yields of hypertritons from these two processes are approximately the same, and the difference is owing to the simplification of taking deuteron as a point particle in this calculation. Therefore, the same3ΛH yield appears in bothS2 andS3 , and the ratioS2= N3ΛHNΛNd can thus also be used as an observable for probing the correlation between baryon and strangeness in relativistic heavy-ion collisions.In this paper, we study the yield ratios
S3= N3ΛH/NΛN3He/Np andS2= N3ΛHNΛNd in the framework of the coalescence model. We first revisit the study ofS3 and find that the discrepancy between the ALICE and STAR measurements may be partially owing to the difference in the primordial proton yield used in the two analyses. We then show that the ratioS2 , particularly its ratioS2/B2 with respect toB2 , which is the coalescence parameter for the production of a deuteron from a proton and neutron pair, is a cleaner probe of the baryon-strangeness correlation in the produced hadronic matter from relativistic heavy-ion collisions. -
In this Section, we first review the experimental data on
S3 from relativistic heavy-ion collisions and then use the coalescence model to extract from these data the correlation betweenΛ and nucleon density fluctuations in the produced matter. -
The value of
S3 was measured to be0.36±0.26 in central 11.5A GeV/c Au + Pt collisions [6] and increased to1.08±0.22±0.16 for Au+Au collisions at√sNN = 200 GeV with a mixed event sample of a central trigger and a minimal bias trigger [7]. The measured value ofS3 decreases, however, to0.60±0.13±0.21 in central Pb-Pb collisions at√sNN = 2.76 TeV [8]. Preliminary data with an improved precision from Au+Au collisions at√sNN = 200 GeV revealed a 20% reduction in the value ofS3 [24], which makes the results from STAR and ALICE comparable within their experimental uncertainties. The proton yield used in the analysis of theS3 ratio by the STAR Collaboration is based on the subtraction of protons from the decay of hyperons in its measurements. Replacing the proton yield in the STAR analysis with that from the PHENIX data, which is obtained from a theoretical model for the same collision system and energy [25], can also reduce the value ofS3 at√sNN = 200 GeV to a similar value as measured by ALICE [7, 8, 24]. The difference between the results from STAR and ALICE is thus partially owing to the different treatments in the subtraction of the weak decay contribution to the primordial proton yield. Table 1 summarizes the publishedS3 results, together with the value using the proton yield from the PHENIX data in Au+Au collisions at√sNN = 200 GeV. It is seen that the values ofS3 from STAR and ALICE are now comparable within their large uncertainties.experiment S3 AGS 0.36±0.26 STAR 1.08±0.22±0.16 STAR + PH 0.90±0.22±0.15 ALICE 0.60±0.13±0.21 Table 1. Values of
S3 from AGS, STAR and ALICE, with PH indicating that the proton yield is taken from PHENIX [25]. See text for details.Experimentally, there also exists a puzzle related to the
3ΛH/3He ratio [10, 11]. Its value is0.82±0.16±0.12 in the 0-80% centrality of Au+Au collisions at√sNN = 200 GeV at RHIC [7], which is considerably larger than the value of0.47±0.10±0.13 in Pb-Pb collisions at√sNN = 2.76 TeV and the 0-10% centrality at the LHC [8]. Although a preliminary measurement with improved precision by STAR in Au+Au collisions at√sNN = 200 GeV revealed a reduction in the3ΛH/3He ratio to a value comparable with that assessed by the ALICE study, its large uncertainty [26] indicates that more precise measurements of3ΛH in high energy heavy-ion collisions are needed. -
To demonstrate the physics that can be extracted from the ratio
S3 , we adopted the coalescence model for the present study. According to the coalescence formula denoted as COAL-SH in Ref. [27], the yield of a certain nucleus consisting ofNi constituent species i (proton, neutron, andΛ ) of massmi from the kinetically frozen-out hadronic matter of local temperatureTK and volumeVK in a heavy-ion collision can be written asNA=grelgsizegA(A∑imi)3/2[A∏i=1Nim3/2i]×A−1∏i=1(4π/ω)3/2VKx(1+x2)(x21+x2)liG(li,x).
(1) In the above,
gA=(2S+1)/(∏Ai=1(2si+1)) is the statistical factor for A nucleons and/orΛ of spinsi=1/2 to form a nucleus of spin S;grel is the relativistic correction to the effective volume in the momentum space and is set to 1 in the present study, owing to the much larger nucleon mass than the effective temperature of the hadronic matter at kinetic freeze-out; andgsize is the correction owing to the finite size of the produced nucleus and is also taken to be 1 in our study because of the much larger size of the hadronic matter than the sizes of produced light nuclei. The symbolsli andω denote, respectively, the orbital angular momentum of the nucleon orΛ in the nucleus and the oscillator constant used in its wave function. The value ofx=(2TK/ω)1/2 is significantly larger than one because of the much larger size of the nucleus than the thermal wavelength of its constituents in the hadronic matter. Since the light nuclei considered in the present study all involve only thel=0 s-wave, the suppression factorG(li,x) owing to the orbital angular momentum in the above equation is simply one.For the production of
3ΛH and3He , their yields according to Eq. (1), after taking into account the approximations mentioned above, are given byN3ΛH=g3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K,N3He=g3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K.
(2) According to Refs. [28, 29], including possible nucleon and
Λ density fluctuations in heavy-ion collisions at lower energies owing to the spinodal instability during the QGP to hadronic matter phase transition [30-32], modifies the above equations toN3ΛH≈g3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K(1+αΛp+αΛn+αnp),N3He≈g3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K(1+Δp+2αnp).
(3) In the above, the proton relative density fluctuation is denoted by
Δp=⟨(δp)2⟩/⟨p⟩2 , where⟨p⟩=1VK∫np(→r)d→r,⟨(δp)2⟩=1VK∫[np(→r)−⟨np⟩]2d→r,
(4) with
np(→r) being the proton density distribution. The quantitiesαΛp ,αΛn , andαnp are, respectively, theΛ -proton,Λ -neutron, and proton-neutron density fluctuation correlation coefficientsαn1n2=⟨δn1δn2⟩/(⟨n1⟩⟨n2⟩) withn1 andn2 denotingΛ or a nucleon.Taking the same masses for proton and neutron, i.e.,
mp=mn=m , the yield ratioS3= N3ΛH/NΛN3He/Np is thenS3=g1+αΛp+αΛn+αnp1+Δp+2αnp,
(5) with
g=(mΛ+2m3mΛ)3/2≈0.845 . From the above equation, the sum of the correlation coefficientsαΛp +αΛn between theΛ density and the proton or neutron density fluctuations can be determined fromS3 ,Δp , andαnp according toαΛp+αΛn=S3g×(1+Δp+2αnp)−αnp−1.
(6) For the value of
αnp , we follow the method in Ref. [29] by using the deuteron yield after including in the coalescence formula COAL-SH the proton and neutron density fluctuations, i.e.,Nd=23/2gd(2πmTK)3/2NpNnVK(1+αnp).
(7) In terms of
gd−p=123/2gd(2π)3=21/23(2π)3≈0.0019 ,Od−p=Nd/N2p ,Rnp=Np/Nn , andVph=(2πmTK)3/2VK , the value ofαnp can then be calculated fromαnp=gd−pRnpVphOd−p−1.
(8) For the proton density fluctuation
Δp , we consider the ratioN3HeN3p×NpNn=33/2g3He(2πmTK)31V2K(1+Δp+2αnp),
(9) and determine it from the relation
Δp=g3He−pV2phRnpO3He−p−2αnp−1,
(10) where
g3He−p=133/2g3He(2π)6=433/2(2π)6≈1.25×10−5 andO3He−p=N3He/N3p , and the factorsVph ,Rnp andαnp are the same as above.The effective phase-space volume
Vph occupied by nucleons in the hadronic matter at kinetic freeze-out can be evaluated from its value at chemical freeze-out, using the relationT3/2KVK=λT3/2chVch , whereTch andVch are, respectively, the temperature and volume of the system at the chemical freeze-out, andλ is a parameter. For collisions at RHIC energies, we take the value ofTch from the grand canonical ensemble fits to the particle yields in Ref. [33] and that ofVch to beVch=4πR3/3 as in Ref. [33] for collisions at various centralities, except for collisions at the 0-80% centrality, where it is taken to be proportional to the charged particle multiplicity obtained from Ref. [34]. Using the results from Ref. [33] based on the strangeness suppressed canonical ensemble fits gives almost the same results. The different values ofVch extracted from collisions at√sNN = 11.5 and 19.6 GeV are owing to the neglect of experimental uncertainties in our analysis [33]. The values ofTch andVch used in the present study for collisions at the AGS energy are taken from Ref. [35], and for collisions at the LHC energy, they are taken from the COAL-SH model used in Ref. [27]. In our calculations, all hadrons are taken as point particles. Including an exclusive volume for each hadron increasesVch [35]. The value ofλ is determined by assuming that the entropy associated with a single nucleon is the same at the chemical and the kinetic freeze-outs. Explicitly, we use the well-known expression for the entropy associated with a single nucleon in a system in thermal equilibrium, i.e.,S/N=5/2+ln(Vph/N) , whereVph=(2πmT)3/2V is the phase-space volume of N nucleons of mass m in the system. The constancy ofS/N then requiresT3/2chVch/Nch=T3/2KVK/NK , withNch andNK being the numbers of nucleons at the chemical and kinetic freeze-outs, respectively, which then leads toλ=NK/Nch . The values given in Table 2 forλ are obtained from the valueNK measured in experiments and the valueNch given by the statistical hadronization model that includes all the resonances in PDG [36]. Our approach differs from the naive approach of a hadronic matter of constant number of nucleons expanding with a constant total nucleon entropy after a chemical freeze-out, which would giveλ the value of unity, and also that in Ref. [29] based on a multiphase transport model that only includes a small number of resonances and thus gives a smaller value ofλ≈1.6 . We note that the below results of our study are not qualitatively affected by these variations inλ . As to the value ofRnp=Np/Nn , it can be determined from the measured ratio of charged pions according to the relationNp/Nn=(Nπ+/Nπ−)1/2 from the statistical model. In Table 2, we summarize the values of the above parameters for central heavy-ion collisions. For reference, we also provide in Table 3 their values for collisions at the centralities of 0-80% or 0-60%.√sNN/GeV Tch/GeV Vch/fm3 Rnp αnp λ 4.9 0.132 640 0.925 −0.781±0.026 2.23 7.7 0.144 806 0.966 −0.744±0.024 2.60 11.5 0.151 875 0.977 −0.763±0.019 2.76 19.6 0.158 843 0.987 −0.830±0.014 2.92 27 0.160 846 0.988 −0.848±0.012 2.97 39 0.160 951 0.990 −0.834±0.013 3.00 62.4 0.164 1215 0.992 −0.792±0.037 3.16 200 0.168 1334 0.992 −0.726±0.038 3.30 2760 0.156 4320 1.00 −0.717±0.023 2.94 Table 2. Values of parameters used for 0-10% central collisions.
√sNN/GeV Tch/GeV Vch/fm3 Rnp αnp λ 7.7 0.144 268 0.966 −0.775±0.020 2.60 11.5 0.151 292 0.977 −0.793±0.018 2.76 19.6 0.158 281 0.987 −0.851±0.014 2.92 27 0.160 282 0.988 −0.867±0.012 2.97 39 0.160 317 0.990 −0.859±0.013 3.00 200 0.168 445 0.992 −0.747±0.036 3.30 2760 0.156 1800 1.00 −0.710±0.024 2.94 Table 3. Same as Table 2 for values of parameters used in the calculations for 0-80% centrality in collisions at RHIC energies and for 0-60% centrality at the LHC energy.
We note that if we have assumed instead that the total entropy is the same at the chemical and kinetic freeze-outs, i.e.,
T3KVK=T3chVch [37], a somewhat smallerVK would have been obtained. Since it has been shown in Ref. [38] that the total entropy increases from the chemical to the kinetic freeze-out, and the entropy associated with a baryon is essentially constant, we adopt in the present study the conditionT3/2KVK=λT3/2chVch of constant entropy associated with a nucleon. Although the values ofTK are not used in our calculations, it is useful to show them as references. According to Ref. [33] based on a blast-wave model fit to measured proton, pion, and kaon transverse momentum spectra, the values ofTK are 126, 117, 119, 114, 116, 118, 88, 88 MeV for central heavy-ion collisions at√sNN = 4.9, 7.7, 11.5, 19.6, 27, 39, 200, and 2760 GeV, respectively.As shown in Tables 2 and 3, the extracted values for the neutron and proton density fluctuation correlation
αnp are negative with appreciable magnitude, indicating that neutrons and protons are anti-correlated in the matter produced in relativistic heavy-ion collisions, similar to the findings in Ref. [29]. For the proton density fluctuationΔp , the extracted values are 0.656± 0.049, 0.536± 0.083, and 0.727± 0.085 for collisions at√sNN = 4.9, 200, and 2760 GeV, respectively [6, 7, 39]. It shows a non-monotonic behavior as a function of the collision energy, from√sNN = 7.7 GeV to 200 GeV, using the preliminary data of3He yield from Ref. [24], similar to that of the neutron density fluctuation extracted from the yield ratioNtNpN2d [40]. From the measured values ofS3 and extracted values forαnp andΔp , one can then determine the values ofαΛn+αΛp from heavy-ion collisions at various energies, according to Eq. (6). These results will be shown in the next Section, to compare with the correlation coefficientαΛd of theΛ and deuteron density fluctuations that is extracted from measuredS2= N3ΛHNΛNd ratio. -
In this Section, we first consider the
S2= N3ΛHNΛNd ratio in the framework of the coalescence model. Its ratioS2/B2 with respect to the coalescence parameterB2 for the production of a deuteron from the coalescence of a proton and a neutron is then studied. Based on the experimental data from the RHIC and LHC, we further extract the correlation coefficientαΛd between theΛ and deuteron density fluctuations. -
Approximating
3ΛH as a bound system ofΛ and a deuteron, the yield of3ΛH in heavy-ion collisions can be calculated from the coalescence ofΛ and a deuteron using Eq. (1). Including the effect of the deuteron andΛ density fluctuations, it is given byN3ΛH=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/2NΛNdVK(1+αΛd),
(11) with
αΛd=⟨δnΛδnd⟩/(⟨nΛ⟩⟨nd⟩) being the correlation coefficient between the deuteron andΛ density fluctuations. TheS2 ratio is thenS2=N3ΛHNΛNd=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/21VK(1+αΛd),
(12) with
gS2=(13(mΛ+md)3/2m3/2Λm3/2d(2π)3/2)−1≈0.12 , from which we can express the density fluctuation correlation coefficientαΛd in terms ofS2 asαΛd=gS2S2T3/2KVK−1.
(13) In the left plot of Fig. 1, we show by solid symbols the extracted ratio
S2= N3ΛHNΛNd using the experimental data from AGS [6, 41, 42], RHIC [7, 43, 44], and LHC [8, 39, 45]. Open symbols represent the results for the3ΛH yield in the collision at the 0-10% centrality, where the RHIC data are obtained from multiplying the measured3ΛH data at the 0-80% centrality by a factor of 3. We have checked, using data available from the RHIC BES program, that the deuteron toΛ yield ratio is larger by a factor of 3 for collisions at the 0-10% centrality than at the 0-80% centrality, independent of the collision energy [43, 46]. Solid and dashed lines in the left window of Fig. 1 are results from calculations based on the statistical model of Refs. [47-49] as in Ref. [50] using the parameters in Tables 2 and 3 and after taking into account the feed-down correction to theΛ yield. Clearly, although there are missing data points in the large collision energy range, the ratioS2 in collisions at the 0-10% centrality seems to be independent of the collision energy√sNN considering the large uncertainty of the AGS data. In addition, the model calculation describes reasonably well the data at the LHC energy, but over-predicts the results for collisions at the AGS and RHIC energies.Figure 1. (color online) Collision energy dependence of the
S2=N3ΛHNΛNd ratio (left) and the density fluctuation correlation coefficientsαΛp+αΛn andαΛd (right) extracted from the experimental data (symbols) and calculated from the statistical model (horizontal bars) using parameters in Tables 2 and 3. See text for details.The right plot in Fig. 1 shows the value of the correlation coefficients
αΛd as a function of the collision energy for collisions at the 0-10% centrality (solid symbols) and at the 0-80% centrality (open symbols). The large uncertainty at√sNN = 2.76 TeV is owing to the propagation of the standard error from the large volumeVch used in collisions at the LHC energy. Within current experimental uncertainties, the value ofαΛd , which is negative and thus indicates an anti-correlation between theΛ and deuteron density fluctuations, becomes slightly less negative as the collision energy increases and approaches zero at√sNN = 2.76 TeV. The negativeαΛd and the negativeαnp , shown in Tables 2 and 3, respectively, could be owing to the underestimation of the value of theλ parameter or the kinetic freeze-out volume used in our study. Full understanding of these results requires detailed studies based on the microscopic models of light cluster production in high-energy heavy-ion collisions [51, 52], which is, however, beyond the scope of the present study. Compared with the correlation coefficientαΛp+αΛn extracted fromS3 , which seems to vary very little over a broad range of collision energies, the value ofαΛd shows a more visible√sNN dependence. Also, the deviation ofαΛp+αΛn from zero is larger than that ofαΛp+αΛn . This may suggest thatαΛd is a cleaner observable thanαΛp+αΛn for studying the√sNN dependence of baryon density fluctuations and their correlations, as seen from the comparison of Eq. (12) to Eq. (5). Future experimental measurements in a broad range of collision energies from AGS to RHIC will be very useful for shedding light on the underlying physics. -
In the coalescence model, the yield ratio
S2=N3ΛH/(NΛNd) is the coalescence parameter for the production of3ΛH if it is considered as a bound system ofΛ and a deuteron. Because of the strangeness carried byΛ , theS2 may be different from the coalescence parameterB2 for the deuteron production following the coalescence of a proton and a neutron [19, 53-57].From Eq. (12) for
S2 and a similar equation forB2 , given byB2=NdNpNn=gd1m3/2p(2πTK)3/21VK(1+αnp),
(14) taking
Nn=Np then leads toS2B2=N3ΛHNΛNd/NdNpNn=g1+αΛd1+αnp,
(15) where
g=g3ΛHgdm3/2p(mΛ+md)3/2m3/2Λm3/2d≈0.23 . The ratioS2/B2 thus carries information about the difference betweenαnp andαΛd and thus about the difference between the baryon-baryon correlation and the baryon-strangeness correlation. We note that theB2 coalescence parameter here refers to the ratio of integrated yields, whereas in the literature it is determined differentially in momentum [55, 58-60].Similarly, we can introduce the coalescence parameter
B3=N3HeNpNpNn for the production of3He from the three-body coalescence of two protons and a neutron, and the coalescence parameterBs3=N3ΛHNΛNpNn for the production of3ΛH from the three-body coalescence ofΛ , a proton, and a neutron. Their ratio is exactly the value ofS3 , as discussed in Section 2. Since the ratioS2/B2 does not involve the proton density fluctuationΔp and other mixed density fluctuation correlations, it seems a more sensitive observable thanS3 for studying theΛ density fluctuation.The left plot in Fig. 2 shows the results for the yield ratio
S2/B2 =N3ΛH/(NΛNd)/(Nd/N2p) from experimental data (solid triangles) and those predicted by the statistical model (solid horizontal bars) using the proton, deuteron, and3ΛH yields from the fullpT range, and including the feed-down correction for the proton yield. It is seen that the measured yield ratio increases slightly with increasing collision energy, as predicted by the statistical model. We note that the value ofB2 has also been determined in experiments from the proton and deuteron momentum spectra in a smallpT window [43]. TheS2/B2 ratio obtained from collisions at the LHC energy for momentum per constituentpT/A=1.4GeV/c [8, 39, 45] is0.899±0.171 . Within their uncertainties, this value is similar to that obtained using yields from the fullpT range. However, theS2/B2 ratios measured for differentpT bins are unavailable from experiments in the energy range available at AGS and RHIC, and this is owing to the lack of3ΛH pT spectra at these energies. Future measurements of3ΛH spectra over a broad range of energies are needed for extracting theΛ and deuteron density correlation coefficientαΛd discussed below.The
Λ and deuteron correlation coefficientαΛd can also be extracted from the yield ratioS2/B2 given in Eq. (15), that isαΛd=(N3ΛHNΛNd(Nd/N2p)/g)×(1+αnp)−1,
(16) by taking advantage of the empirical fact that the neutron and proton density fluctuation correlation
αnp is less affected byTK andVK . Shown in the right plot of Fig. 2 by triangles are the values ofαΛd extracted from the experimental results using Eq. (16). They are seen to have similar values to those obtained from Eq. (13) usingS2=N3ΛHNΛNd , which are shown by closed circles and also in Fig. 1 where it is compared with the density fluctuation correlation coefficientαΛp+αΛn extracted fromS3=N3ΛH/NΛN3He/Np . -
In summary, we have argued that both the ratio
S2 and the ratioS2/B2 , whereS2 andB2 are, respectively, the coalescence parameter for the production of hypertriton fromΛ and a deuteron, and of a deuteron from a proton and a neutron, are more sensitive observables than the previously proposed ratioS3 =N3ΛH/NΛN3He/Np for studying the local baryon-strangeness correlation in the matter produced in relativistic heavy-ion collisions. We have substantiated this argument in the framework of baryon coalescence by demonstrating that the correlation coefficientαΛd betweenΛ and deuteron density fluctuations extracted from measuredS2/B2 shows a stronger dependence on the energy of heavy-ion collisions than the correlation coefficientsαΛp+αΛn betweenΛ and nucleon density fluctuations extracted from the measuredS3 . Although the results in the present study are obtained without including the feed-down contribution to nucleons fromΔ resonances, they will not be qualitatively affected because of the low kinetic freeze-out temperature of ~100 MeV, which only contribues ~20% to the nucleon yield. Experimental measurements of the ratioS2/B2 are expected to provide a promising way to study the strangeness and baryon correlation in the matter produced from heavy-ion collisions as the collision energy or the baryon chemical potential of produced matter is varied, which in turn can shed light on the properties of the QGP to hadronic matter phase transition during collisions.
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