The effect of hadronic scatterings on the measurement of vector meson spin alignments in heavy-ion collisions
aa r X i v : . [ nu c l - e x ] F e b The effect of hadronic scatterings on the measurement of vector meson spinalignments in heavy-ion collisions
Diyu Shen a,b , Jinhui Chen c , and Zi-Wei Lin d,e a Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China b University of Chinese Academy of Sciences, Beijing 100049, China c Institute of Modern Physics and Key Laboratory of Nuclear Physics andIon-beam Application (MOE), Fudan University, Shanghai 200433, China d Key Laboratory of Quarks and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China and e Department of Physics, East Carolina University, Greenville, North Carolina 27858, USA
Spin alignments of vector mesons and hyperons in relativistic heavy-ion collisions have been pro-posed as signals of the global polarization. The STAR experiment first observed the Λ polarization.Recently, the ALICE collaboration measured the transverse momentum ( p T ) and the collision cen-trality dependence of K ∗ and φ spin alignments in Pb-Pb collisions at √ s NN = 2.76 TeV. A largesignal is observed in the low p T region of mid-central collisions for K ∗ while the signal is muchsmaller for φ , and these have not been understood yet. Since vector mesons have different lifetimesand their decay products have different scattering cross sections, they suffer from different hadroniceffects. In this paper, we study the effect of hadronic interactions on the spin alignment of K ∗ , φ and ρ mesons in relativistic heavy-ion collisions with a multi-phase transport model. We find thathadronic scatterings lead to a deviation of the observed spin alignment matrix element ρ awayfrom the true value for ρ and K ∗ mesons (with a bigger effect on ρ ) while the effect is negligible forthe φ meson. The effect depends on the kinematic acceptance: the observed ρ value is lower thanthe true value when the pseudorapidity ( η ) coverage is small while there is little effect when the η coverage is big. Our study thus provides valuable information to understand the vector meson spinalignment signals observed in the experiments. I. INTRODUCTION
It was predicted that a hot and dense matter, known asa quark-gluon plasma (QGP), will be formed in relativis-tic heavy-ion collisions [1]. This new state of matter [2–4]could have a large angular momentum with the direc-tion perpendicular to the reaction plane in non-centralcollisions [5–7]. The angular momentum is conservedduring the evolution of the system and may result inspin-orbit coupling among the quarks, which will gener-ate a net polarization of hyperon or vector mesons suchas Λ, φ , K ∗ and ρ due to the hadronization process [8].On the other hand, quarks and antiquarks may or maynot be produced with an initial global polarization [9].As the QGP approaches the transition to hadrons, thematter becomes strongly interacting due to nonpertur-bative effects and constitute quarks or antiquarks mayapproach spin and helicity equilibration with the vortic-ity [9]. Therefore, studying the spin polarization can pro-vide additional dynamical information about the hot anddense matter. The topic is gaining increasing interestsboth in theory [10–15] and experiments [16–21]. Moredetails can be found in some of the recent reviews [22–24].The spin alignment of a vector meson is described by a3 × ρ ) as f ( θ ∗ ) ≡ dNd ( cosθ ∗ ) = N × [(1 − ρ ) + (3 ρ − cos θ ∗ ] . (1)In the above, N is a normalization factor, and θ ∗ is theangle between the decayed daughter and the system’sorbit angular momentum ˆ L in the vector meson’s restframe. Experimentally, ρ can be determined by mea-suring the angular distribution of Eq.(1). One sees thata deviation of the ρ value from 1/3 means a spin align-ment of the vector meson.Measurements of vector meson spin alignments inheavy-ion collisions have been preformed. Results inAu+Au collisions at √ s NN =200 GeV were initially con-sistent with ρ = 1/3 within uncertainties [20] and thenshow a possible signal with improved event statistics [21].Data in Pb-Pb collisions at √ s NN =2.76 TeV show thatthe ρ of K ∗ mesons could be significantly smaller than1/3 at p T < . ρ of φ mesons ismuch closer to 1/3 [16]. It is proposed that a strangenesscurrent may exist in heavy-ion collisions and gives riseto a non-vanishing mean ϕ field, which explains in partthe φ meson spin alignment but does not apply to the K ∗ meson [26]. The ρ value could also depend on thequark hadronization process and the possible p T depen-dence of vorticity. So far the observed ρ values of vec-tor mesons have not been fully understood, especially inconjunction with the magnitude of the observed Λ globalpolarization [17–19, 27].Another physics process that may affect the observed ρ values is hadronic scatterings of the decay products ofvector mesons. On the one hand, the decay daughters of φ , K ∗ and ρ mesons are different and thus have differenthadronic interactions. On the other hand, the lifetimeof each vector meson species is different. One would ex-pect that the vector meson with a longer lifetime willsuffer less from hadronic interactions since their daugh-ters are produced later at lower densities. Therefore weinvestigate quantitatively the effect of hadronic scatter-ings on vector meson spin alignments in Pb-Pb collisionat √ s NN = 2.76 TeV with the string melting version of amulti-phase transport (AMPT) model [28] (unless spec-ified otherwise). The decay channels used in the currentstudy are φ → K + + K − , K ∗ → K + π and ρ → π + π for the three vector meson species, respectively.The paper is organized as follows. The model andmethodology are introduced in section II. The resultson the spin density matrix element ρ of φ , K ∗ and ρ before and after hadronic scatterings are presented insection III. A summary is then given in Section IV. II. MODEL AND METHODOLOGY d N / d t ( a . u . ) =2.76 TeV NN sPb-Pb, >0 GeV/c T <0.5, p η -0.5< ρ K* φ FIG. 1: The normalized decay time distribution of φ , K ∗ and ρ mesons in the hadronic phase of Pb-Pbcollisions at √ s NN =2.76 TeV from the AMPT model. ρ mesons decay quickly while φ mesons have a relativeflat distribution after 10 fm/c.The AMPT model is a multi-phase transportmodel [28] for studying heavy-ion collisions. In thismodel, the initial conditions are taken from the spatialand momentum distributions of minijet partons and softstring excitations from the HIJING event generator [29],which is followed by two-body elastic parton scatteringsusing the parton cascade model ZPC [30]; the conver-sion from partons to hadrons via either the string frag-mentation [31] for the default version or a quark coa-lescence model for the string melting version [32, 33],and hadronic scatterings based on an extended relativis- d N / d M ( a . u . ) (e) Before selection before hadronic scatterings ) (GeV/c φ → - + K + Invariant mass, K (f) After selection after hadronic scatterings d N / d M ( a . u . ) (c) Before selection before hadronic scatterings ) K* (GeV/c → π Invariant mass, K + (d) After selection after hadronic scatterings d N / d M ( a . u . ) (a) Before selection before hadronic scatterings ) (GeV/c ρ → - π + + π Invariant mass, (b) After selection after hadronic scatterings
FIG. 2: The normalized invariant mass distribution of ρ , K ∗ , φ mesons from the AMPT model. In each panelthe invariant mass is reconstructed by the decaychannel as shown before (filled symbols) or after (opensymbols) hadron scatterings. Open symbols in panels(b), (d) and (f) represent the distributions afterremoving those resonances that have a decay daughterwith momentum change more than 0 .
01 GeV/ c due tohadronic scatterings.tic transport model ART [34]. Since the spin degree offreedom is not considered in the current AMPT model,to simulate a spin alignment signal we redistribute thedecay products according to Eq.(1) when resonances de-cay [35] and then study the hadronic scattering effect onvector meson spin alignment with different input ρ val-ues in this work. For hadronic interactions, the extendedART model includes baryon-baryon, baryon-meson, andmeson-meson elastic and inelastic scatterings [28]. Ingeneral, the cross sections of elastic or inelastic scatter-ings depend on the center of mass energy of the scatteredhadrons. Therefore hadrons at different momentum havedifferent cross sections, which may contribute to the p T dependence of the observed ρ value of vector mesons.As the global vorticity is expected to peak at semi-centralcollisions [5], we choose the impact parameter b =8 fm tomimic such collisions in this study.A decay daughter that has a large transverse momen-tum is more likely to come from a parent hadron witha large transverse momentum. When one measures the p T dependence of vector meson spin alignment, it willbe influenced by the p T dependent hadronic interactions.In addition, the decay times of φ , K ∗ and ρ mesons aredifferent as illustrated in Fig. 1. The ρ meson has theshortest lifetime, therefore we expect it to be affectedmost strongly by the hadronic scatterings. However, the φ meson is expected to be less affected due to its longlifetime. Furthermore, pions as the decay products of ρ meson are expected to be scattered more frequently thankaons as the decay products of φ meson. Since the K ∗ meson has a moderate lifetime with kaon and pion as thedecay products, one would expect it to be moderately af-fected by hadronic scatterings. Therefore it is worthwhileto quantify how much hadron scatterings could affect thefinal spin alignment results of different vector mesons.In experiments one uses the n -th event plane ( n = 1 , L . In this paper, we calculate the directionof ˆ L with participant nucleons in each event. The decaydaughters from the aforementioned channels of interestare labelled so that we know explicitly which two de-cay daughters have come from the same parent hadron.The phase space information of the decay daughters isalso recorded when the decay happens, which enables usto distinguish whether a decay product has experiencedscatterings or not by comparing the momentum infor-mation upon decay with that after the hadron cascade.When a decay daughter is destroyed due to a subsequentinelastic scattering, naturally the parent meson cannotbe reconstructed from the final hadron record. In addi-tion, experiments usually reconstruct non-stable vectormesons via the invariant mass distribution of the candi-date decay daughters, where the invariant mass window isdetermined by the vector meson physical width in convo-lution with the detector momentum resolutions [16, 20].As a result, a parent meson with any decay daughter hav-ing subsequent elastic scattering(s) will likely not be re-constructed with the experimental method of backgroundsubtraction within the invariant mass window. There-fore, in the selection procedure of this study we removea resonance if it has a decay daughter with a momentumchange more than 0 .
01 GeV/c due to elastic scattering(s).Figure 2 shows the normalized invariant mass distribu-tions of the vector mesons reconstructed via their decayproducts before and after the selection procedure as de-scribed above. It shows in panels (a), (c) and (e) thatthe hadronic scatterings change the shape of the recon-structed invariant mass distributions. The degrees ofchange are different for φ , K ∗ and ρ mesons, where the ρ meson distribution changes most significantly as ex-pected while the φ meson distribution has little change.After the selection procedure where vector mesons withscattered decay daughters are removed, the distributionsafter hadronic scatterings, as shown in panels (b), (d)and (f), are similar to those before hadronic scatterings.The decay daughters that have passed the selection pro-cedure are then used in our analysis of the vector mesonspin alignment. Note that the distributions of Fig. 2 withdifferent momentum difference cuts have been checkedand similar results are obtained. III. RESULTS AND DISCUSSION
It has been pointed out that a finite acceptance in ex-periments will lead to an increase of the observed ρ as the range of pseudorapidity − η max < η < η max issmall [35], where η max denotes the maximum | η | of de-cay daughters. To focus on the hadronic scattering effect,we first correct for the acceptance effect. When the effectof hadron scatterings is neglected, we can write the de-cay daughter distribution in the vector meson rest framewithin a specific kinematic window asf obs ( θ ∗ , ρ in00 , η max , p T ) = f true ( θ ∗ , ρ in00 ) × g( θ ∗ , η max , p T ) , (2)where g( θ ∗ , η max , p T ) denotes the effect of the kinematicwindow, ρ in is the input ρ value, f obs ( θ ∗ , ρ in00 , η max , p T )is the observed distribution, and f true ( θ ∗ , ρ in00 ) is the truedistribution for a given input ρ . Note that in the abovewe assume that the effect of finite acceptance (i.e., the g function) is independent of the input ρ value, which wehave verified numerically to be true to a good accuracy.Since the f true ( θ ∗ , ρ in00 ) distribution is flat for ρ in = 1 / obs ( θ ∗ , ρ in00 , η max , p T )f obs ( θ ∗ , ρ in00 = 1 / , η max , p T ) = f true ( θ ∗ , ρ in00 )f true ( θ ∗ , ρ in00 = 1 / true ( θ ∗ , ρ in00 ) , (3)where A is a constant. Therefore, for each resonancewe divide the cos( θ ∗ ) distribution of interest by thecos( θ ∗ ) distribution obtained for the ρ =1/3 case with-out hadronic scatterings with the same kinematic win-dow. The spin alignment signal is then calculated froma fit to the corrected distribution.Figure 3 shows an example of the acceptance correc-tion. The filled symbols represent the extracted ρ val-ues of the ρ meson without acceptance correction (andwithout hadronic scatterings), where the magnitude in-creases with the decrease of η max [35]. Our results afterthe acceptance correction, shown as the open symbols,are consistent with the input ρ value (solid line). Thereis no η cut on the vector mesons. We have also checkedthe acceptance correction with the method in Ref. [36],which is usually applied in the experimental analysis, andobtained consistent results. max η ρ before correction ρ after correction ρ ρ input =2.76 TeV NN sPb-Pb, FIG. 3: Correction of the η window dependence of ρ for ρ meson in Pb-Pb collisions at √ s NN = 2.76 TeV(without hadronic scatterings and with full p T range),where η max represents the maximum | η | of decaydaughters. The filled and open symbols represent theresults without and with the acceptance correction,respectively; and the solid line represents the inputvalue ρ = 0 . p T dependence ofthe hadronic scattering effect, the vector mesons φ , K ∗ and ρ have been classified into three p T intervals: thefull p T range, 0.4 < p T < p T > p T binning applied in experiments [16, 21].Meanwhile, the decay daughters of each resonance specieshave been selected within different η ranges ( η max = from0.4 to 2) to study the hadronic scattering effect underdifferent experimental acceptance. Figure 4 presents the ρ values of φ , K ∗ and ρ mesons as functions of η max in Pb-Pb collisions at √ s NN =2.76 TeV with the input ρ values of 0.433, 0.333 and 0.183. The global cut-off time for the hadron cascade in the AMPT model isset to 60 fm/ c . The three sub-panels of (a), (b) or (c)show the extracted ρ in the three p T ranges, respec-tively, where filled symbols represent the results with-out hadronic scatterings and open symbols represent theresults after hadronic scatterings. While the ρ valuewithout hadronic scatterings (after the correction for fi-nite acceptance) is consistent with the input value formost cases, sizable effects from hadronic scatterings canbe seen for the ρ meson. The ρ values of K ∗ and ρ mesons after hadronic scatterings are found to decreaseand the decrease is typically bigger for a smaller η range(except for some cases when η max < . ρ mesonsuffers from the most significant hadronic effect as ex- pected, while the φ meson is basically not affected mainlydue to its long lifetime.One also sees that the change of ρ due to hadronicscatterings is quite similar for the three different input ρ values and three p T intervals. The hadronic scat-tering effect on ρ mesons is stronger than that on K ∗ mesons as expected, while the effect on φ mesons is neg-ligible in all p T intervals. The general decrease of ρ may be understood due to the anisotropy of the scatter-ing probabilities, where the decay daughters at θ ∗ ∼ π (which go along the ± y directions for a parent hadronat rest) are more likely to be scattered than those at θ ∗ ∼ ◦ (which go inside the x − z plane for a parenthadron at rest). We also see that for ρ mesons the de-crease of ρ for input ρ =0.433 is slightly stronger thanthat for input ρ =0.183, especially at small η max . Thismay be due the fact that there are more decay daughtersemitted along θ ∗ ∼ π , which further increases thescattering probabilities along these directions.The probability of decay daughters to be scattered, andthus the influence of hadronic scatterings on the vectormeson spin alignment, is not isotropic but depends on theeffective three-dimensional geometry of hadronic matter.Including the hadronic effect, we may write the observedcos( θ ∗ ) distribution approximately asf ′ obs ( θ ∗ , ρ in00 , η max , p T ) = f obs ( θ ∗ , ρ in00 , η max , p T ) × s ( θ ∗ , ρ in , η max , p T ) , (4)where f ′ obs ( θ ∗ , ρ in00 , η max , p T ) is defined as the distributionafter hadronic scattering, f obs ( θ ∗ , ρ in00 , η max , p T ) is the onebefore hadronic scattering, and s( θ ∗ , ρ in00 , η max , p T ) rep-resents the fraction of vector mesons that survive thehadronic scattering. Therefore, the probability of losinga vector meson due to hadronic scattering (i.e., havinga vector meson with scattered daughters) can be writtenas 1 − s( θ ∗ , ρ in00 , η max , p T ) = 1 − f ′ obs ( θ ∗ , ρ in00 , η max , p T )f obs ( θ ∗ , ρ in00 , η max , p T ) . (5)Figure 5 presents the above probability distributions for ρ mesons after the projection to two-dimensional rel-ative momentum planes. Variables p ∗ x and p ∗ z repre-sent the decay daughter’s momentum in the parent’srest frame along the impact parameter and the beamdirection, respectively. The momentum is normalizedwith p ∗ = q p ∗ x + p ∗ y + p ∗ z because the distribution of p ∗ y /p ∗ = cos θ ∗ is directly related to the spin alignmentand the ρ value. We see that the decay daughters of ρ mesons are more likely to be scattered along the p ∗ y axis(also the p ∗ x axis) than along the p ∗ z axis, which leads toa bigger suppression at θ ∗ ∼ π than at θ ∗ ∼ ◦ andthus a decrease of the extracted ρ value. By comparingthe distributions for η max =0.5 and those for η max =2,one observes that the anisotropy of the scattering prob-ability is bigger for the narrower η range, which corre-sponds to a stronger decrease of the extracted ρ value.The observed probability distributions can be under-stood in terms of the effective geometry of the matter. ρ =2.76 TeV NN sPb-Pb, )>0 GeV/c ρ ,K*, φ ( T ) p (a =0.433 ρ input )<1.0 GeV/c ρ ,K*, φ ( T ) 0.4
1.0 GeV/c ρ ,K*, φ ( T ) p (a After hadronic scatterings φ * K ρ max η ρ )>0 GeV/c ρ ,K*, φ ( T ) p (b =0.333 ρ input )<1.0 GeV/c ρ ,K*, φ ( T ) 0.4
1.0 GeV/c ρ ,K*, φ ( T ) p (b max η ρ )>0 GeV/c ρ ,K*, φ ( T ) p (c =0.183 ρ input )<1.0 GeV/c ρ ,K*, φ ( T ) 0.4
1.0 GeV/c ρ ,K*, φ ( T ) p (c max η FIG. 4: The extracted ρ values of φ , K ∗ and ρ mesons as functions of the η range in Pb-Pb collisions at √ s NN =2.76 TeV and b = 8 fm from the AMPT model with three different input values of ρ without and withhadronic scatterings. The x axis center of each point has been slightly shifted to distinguish their error bars. − z p*2 − / p * x p * =2.76 TeV NN sPb-Pb, projection z -p* x ) p* (a − y p*2 − / p * x p * <0.5 η -0.5< projection y -p* x ) p* (a − z p*2 − / p * y p * projection z -p* y ) p* (a − z p*2 − / p * x p * <2 η -2< projection z -p* x ) p* (b − y p*2 − / p * x p * projection y -p* x ) p* (b − z p*2 − / p * y p * projection z -p* y ) p* (b FIG. 5: The probability distribution of ρ mesons lost due to hadronic scatterings in Pb-Pb collisions at √ s NN =2.76TeV with input ρ = 0 . p ∗ represents the decay daughter’s momentum in the ρ meson rest frame. *) θ cos( ) T , p η * , θ s ( / ndf χ N 0.001 ± obs00 ρ ± χ N 0.0013 ± ρ ± χ N 0.0013 ± ρ ± χ N 0.0013 ± ρ ± =2.76 TeV NN sPb-Pb, < 0.5 η =0.333, -0.5< ρ input )>0 GeV/c ρ ( T (a) p *) θ cos( ) T , p η * , θ s ( / ndf χ N 0.001 ± ρ ± χ N 0.001 ± ρ ± χ N 0.001 ± ρ ± χ N 0.001 ± ρ ± )<1 GeV/c ρ ( T (b) 0.4
1 GeV/c ρ ( T (c) p FIG. 6: The effect of hadronic scatterings as a function of cos( θ ∗ ) in Pb-Pb collisions at √ s NN = 2.76 TeV withinput ρ = 0 .
333 and η max =0.5.In the case of η max =0.5, the effective geometry of thehadronic matter as seen by the decay daughters in theparent’s rest frame may be considered as a short cylinderwith the axis along the z axis. Therefore scatterings ofdecay daughters in the transverse plane are more likelythan those along the z axis, as illustrated in the Fig. 5.In particular, as shown by the projection in the p ∗ y - p ∗ z plane, scatterings along the p ∗ z axis are less likely, whichleads to relatively more vector mesons to be observedaround θ ∗ ∼ ◦ and a smaller observed ρ value. Onthe other hand, from panels ( a ) and ( b ) of Fig. 5 onesees that the scattering probability is about symmetric along p ∗ x and p ∗ y axes, indicating that the transverse spa-tial anisotropy in the hadronic stage (or its effect on thescattering probability versus cos( θ ∗ )) is small.Figure 6 shows the surviving functions( θ ∗ , ρ in00 , η max , p T ) of ρ mesons in the case of input ρ =0.333 and η max =0.5. We see that a ρ meson athigher p T is more likely to survive the hadronic scatter-ings. A non-uniform distribution of s( θ ∗ , ρ in00 , η max , p T )indicates the hadronic effect, and the decreasing trendversus cos θ ∗ corresponds to a decrease of the extracted ρ value. Also note that the cos θ ∗ distribution afterhadronic scatterings and the finite acceptance correction max η − ρ ∆ K* string melting AMPTK* default AMPT =2.76 TeV NN sPb-Pb, (K*)>0 GeV/c T =0.183, p ρ input FIG. 7: The change of the extracted ρ values of K ∗ mesons due to hadronic scatterings from the stringmelting version (triangles) and the default version(diamonds) of the AMPT model for Pb-Pb collisions at √ s NN =2.76 TeV.may not follow the shape of Eq.(1) well; this can bereflected in the sometimes large χ values in the fit withEq.(1), as shown in Fig. 6(a) as an example.We have also calculated Au+Au collisions at √ s NN =200 GeV and the hadronic effect there is found tobe smaller than that in Pb-Pb collisions at √ s NN =2.76TeV. This could come from the lower hadron multiplicityand consequently the smaller scattering probability of de-cay daughters in the hadron cascade of the lower-energyAu+Au collisions.In principle, the hadronic effect on vector meson spinalignments shall be present in any hadronic transportmodel [37, 38], although the magnitude of the effect couldbe different due to the different treatments of the hadrontransport. For example, the string melting version of theAMPT model starts the quark coalescence process afterthe kinetic freezeout of partons. Therefore, the magni-tude of the parton cross section affects the average den-sity at the start of hadron cascade [39] and may thusaffect the magnitude of the effect of hadronic scatteringson spin alignment observables. Here we check the ef-fect in the AMPT model with a different configuration.Figure 7 presents the results from the default version ofAMPT, where the parton cascade only includes minijetpartons and is thus shorter and the hadronization is mod-eled by the Lund string fragmentation [31]. Since the ρ meson global spin alignment has not been measured yet,we choose the K ∗ for illustration. We see that the generalfeature is the same, while the decrease of the extracted ρ in the default AMPT model is larger, which is a result of the earlier start of the hadron cascade phase in the de-fault AMPT model. Future experimental measurementon the ρ meson spin alignment is called for.Note that this study shows that hadronic scatteringscould lead to a finite deviation of the extracted ρ valuefrom the true value at the time of the vector meson de-cays. We have not addressed what the true value shallbe or how the true value is developed. One may expectthat vector mesons with a longer lifetime would expe-rience more hadronic scatterings and their later freeze-out time could affect their polarization including the ρ value. However, an explicit calculation would require theinclusion of the spin degree of freedom in the partonicand hadronic transport, which is beyond the scope of thecurrent study. IV. SUMMARY
We have studied the effect of hadronic scattering onthe spin alignment of φ , K ∗ and ρ mesons at LHC en-ergies with a multi-phase transport model. We findthat hadronic interactions will lead to a deviation ofthe extracted spin density matrix element ρ from itstrue value, where the deviation depends on the effec-tive three-dimensional geometry of the hadronic matter.With finite acceptance, the observed ρ decreases dueto hadronic scattering because it is less likely for the de-cay daughters to be scattered along the z axis, whichtends to around θ ∗ ∼ ◦ . The hadronic effect on ρ is more significant for ρ mesons because of their shorterlifetime and the larger scattering cross section of the π decay daughters, therefore measurements of the ρ mesonspin alignment will be interesting. The hadronic effect on K ∗ mesons is moderate, while φ mesons are almost notaffected by hadronic scatterings due to their longer life-time and the relatively small scattering cross sections ofkaons. Furthermore, the hadronic effect on ρ is biggerfrom the default version of the AMPT model than thatfrom the string melting version because of the earlier anddenser hadron matter in the default AMPT model. Ourstudy suggests that hadronic scatterings could affect vec-tor meson spin alignment observables in addition to thespin-orbit coupling and vorticity. ACKNOWLEDGEMENTS
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