K ∗ vector meson resonances dynamics in heavy-ion collisions
Andrej Ilner, Daniel Cabrera, Christina Markert, Elena Bratkovskaya
aa r X i v : . [ h e p - ph ] S e p K ∗ vector meson resonances dynamics in heavy-ion collisions Andrej Ilner,
1, 2, ∗ Daniel Cabrera,
3, 2, † Christina Markert, ‡ and Elena Bratkovskaya § Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨atFrankfurt am Main, 60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies (FIAS), 60438 Frankfurt am Main, Germany Instituto de F´ısica Corpuscular (IFIC), Centro Mixto Universidad de Valencia - CSIC,Institutos de Investigaci´on de Paterna, Ap. Correos 22085, E-46071 Valencia, Spain. The University of Texas at Austin, Physics Department, Austin, Texas, USA GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstrasse 1, 64291 Darmstadt, Germany
We study the strange vector meson ( K ∗ , ¯ K ∗ ) dynamics in relativistic heavy-ion collisions basedon the microscopic Parton-Hadron-String Dynamics (PHSD) transport approach which incorpo-rates partonic and hadronic degrees-of-freedom, a phase transition from hadronic to partonic mat-ter - Quark-Gluon-Plasma (QGP) - and a dynamical hadronization of quarks and antiquarks aswell as final hadronic interactions. We investigate the role of in-medium effects on the K ∗ , ¯ K ∗ meson dynamics by employing Breit-Wigner spectral functions for the K ∗ ’s with self-energies ob-tained from a self-consistent coupled-channel G-matrix approach. Furthermore, we confront thePHSD calculations with experimental data for p+p, Cu+Cu and Au+Au collisions at energies up to √ s NN = 200 GeV. Our analysis shows that at relativistic energies most of the final K ∗ s (observedexperimentally) are produced during the late hadronic phase, dominantly by the K + π → K ∗ channel, such that the fraction of the K ∗ s from the QGP is small and can hardly be reconstructedfrom the final observables. The influence of the in-medium effects on the K ∗ dynamics at RHICenergies is rather modest due to their dominant production at low baryon densities (but high mesondensities), however, it increases with decreasing beam energy. Moreover, we find that the additionalcut on the invariant mass region of the K ∗ further influences the shape and the height of the finalspectra. This imposes severe constraints on the interpretation of the experimental results. PACS numbers:
I. INTRODUCTION
Heavy-ion collisions (HIC) at high energies are studiedboth experimentally and theoretically to obtain informa-tion about the properties of dense and hot matter sincein the hadronic phase the quarks are confined to a colour-neutral state, whereas at high densities and temperaturesthe partons can move freely over larger distances (i.e.larger than the size of a hadron), which implies that thepartons are deconfined and a new stage of matter - de-noted as Quark-Gluon-Plasma (QGP)- is reached. Suchextreme conditions were present shortly after the cre-ation of the Universe and nowadays can be realized inthe laboratory by heavy-ion collisions at relativistic en-ergies. Indeed, in the last decade the QGP has been pro-duced in a sizeable volume in ultra-relativistic heavy-ioncollisions at the Relativistic Heavy-Ion Collider (RHIC)at the Brookhaven National Laboratory (BNL) and atthe Large Hadron Collider at the Conseil Europ´een pourla Recherche Nucl´eaire (CERN) for a few fm/c after theinitial impact of the collision.Unlike early predictions that assumed a weakly in-teracting system of massless partons (parton gas), ∗ Electronic address: ilner@fias.uni-frankfurt.de † Electronic address: cabrera@fias.uni-frankfurt.de ‡ Electronic address: [email protected] § Electronic address: [email protected] which might be described by pQCD (perturbativeQuantum-Chromo-Dynamics), experimental observa-tions of Au+Au collisions at RHIC [1] have shown clearsignatures of a new medium that is more strongly in-teracting than in the hadronic phase and behaves ap-proximately close to an ’ideal’ liquid. In order to studythe properties of such QGP matter one looks for probeswhich provide information about the initial partonicstage. In the experiment only the final hadrons, leptonsand photons are measured. In spite that electromagneticprobes allow to penetrate directly to all stages of heavy-ion collisions, they are very rare and it is quite com-plicated to identify the QGP signal among many otherproduction channels of hadronic origin [2]. On the otherhand, hadronic observables are more abundant and easierto access experimentally.Apart from heavy hadrons with ’charm’ [3] also strangevector-meson resonances K ∗ , ¯ K ∗ s are considered as sensi-tive probes for the hot and dense medium [4, 5]. At rela-tivistic energies the K ∗ s are generally expected to be pro-duced at the partonic freeze-out (or even earlier [6]) and,thus, to provide information about the QGP stage closeto hadronization. Accordingly, detailed studies of the K ∗ resonances have been performed by the STAR Col-laboration at RHIC in the last decade [7–10]. Since thestrange vector-mesons K ∗ , ¯ K ∗ s are relatively short liv-ing resonances, dominantly decaying to pions and kaons( K ∗ → K + π ), it is difficult to reconstruct them experi-mentally since the pions and kaons suffer from final stateinteractions with other mesons and baryons in the ex-panding hadronic phase, i.e. the latter can rescatter or beabsorbed. Moreover, the recreation of the K ∗ by fusionof a kaon and pion ( K + π → K ∗ ) might ’over-shine’ the K ∗ signal from the QGP and lead to additional compli-cations. As known from chiral models or coupled-channelG-matrix approaches [11–16], the K ∗ resonances changetheir properties in a hot and dense hadronic medium andalso the properties of the ’daughter’ kaons/antikaons areaffected by hadronic in-medium modifications [17]. Allthese effects should be accounted for when addressing ormodelling the K ∗ production in relativistic HICs.In order to interpret the experimental results [7–10] itis mandatory to study the dynamics of the K ∗ s withina proper theoretical framework. Statistical or hydrody-namic models can be used to obtain information on thebulk properties of the medium, however, to properly in-vestigate the interactions of the K ∗ with the hadronicmedium one needs a non-equilibrium transport approachto study the influence of the medium on the K ∗ and viceversa. Furthermore, one needs a consistent theory for thein-medium effects of the K ∗ as a function of the nuclearbaryon density in order to trace the dynamics of the K ∗ during the later stages of the collision in the expansion ofthe hadronic fireball in close analogy to the kaon and an-tikaon dynamics in heavy-ion reactions at lower energies[18].Previously there have been several studies on thestrange vector-meson resonances K ∗ within differenttransport models at relativistic energies. In Refs. [19, 20]detailed calculations for Pb+Pb at √ s NN = 17 . p T of the reconstructed resonances, whichleads to higher apparent temperatures for these reso-nances due to a sizeable depletion at low p T which is mostpronounced at midrapidity. Since the rescattering in-creases with baryon density, most resonances that can bereconstructed come from the low nuclear baryon densityregion. The authors of Ref. [21] used EPOS3 (option-ally with UrQMD as an afterburner) to investigate thehadronic resonance production and interaction in Pb+Pbcollisions at √ s NN = 2 .
76 TeV and compared their re-sults to data from the ALICE Collaboration. Their cal-culations reproduce the data very well for the differentcentralities. The simulation also showed that a large partof the resonances at low p T cannot be reconstructed dueto the final state interactions of their decay daughters.Our goal is to investigate the dynamics of K ∗ vector-meson resonances using the Parton-Hadron-String Dy-namics transport approach [2] employing the in-mediumeffects of the K ∗ s from the self-consistent coupled-channel unitary G-Matrix approach for the production of K ∗ s from the hadronic channels as well as from a QGPby microscopic dynamical hadronization. The in-mediumeffects have been already successfully used to model the off-shell behaviour of the kaons K and antikaons ¯ K at fi-nite nuclear baryon density in Refs. [22–27]. Previously,we have evaluated within this framework the properties ofthe strange vector resonances as a function of the nuclearbaryon density [28] and now incorporate the off-shell be-haviour of the K ∗ s in form of a relativistic Breit-Wignerspectral function into PHSD. We recall that the PHSDtransport approach incorporates the hadronic and par-tonic degrees-of-freedom and their interactions, dynam-ical hadronization and further off-shell dynamics in thehadronic stage which allows to include in a consistentway in-medium effects for hadrons as well as partons.Throughout this paper we will follow the following con-vention: when addressing strange vector mesons consist-ing of an anti-strange quark, i.e. K ∗ + = ( u ¯ s ) and K ∗ =( d ¯ s ) we will refer them as K ∗ = ( K ∗ + , K ∗ ); while formesons with an anti-strange quark, i.e. K ∗− = (¯ us ) and¯ K ∗ = ( ¯ ds ) we will use notation as ¯ K ∗ = ( K ∗− , ¯ K ∗ ).This paper is organized as follows: In section II weshortly recall the PHSD transport approach and its im-plementations beyond the cascade level. In section IIIwe discuss the K ∗ vector-meson resonance in-medium ef-fects; the first part describes the calculation of the in-medium effects using the G-matrix model while the sec-ond part contains the implementation of these effects intothe PHSD. In section IV we then investigate the proper-ties and the dynamics of the K ∗ vector-meson resonancesin the PHSD for Au+Au collisions at the top RHIC en-ergy, analyze the different production channels, the ac-tual baryon densities explored as well as the in-mediumeffects of the K ∗ spectral functions. In section V wepresent our results in comparison to experimental data;the first part shows a comparison for p+p collisions whilethe second part contains a detailed confrontation with ex-perimental data from the STAR Collaboration for A+Acollisions including in particular the effect of acceptancecuts. Finally, we give a summary of our findings in sec-tion VI. II. REMINDER OF PHSD
The Parton-Hadron-String Dynamics (PHSD) is a non-equilibrium microscopic transport approach [29, 30] thatincorporates hadronic as well as partonic degrees-of-freedom. It solves generalised (off-shell) transport equa-tions on the basis of the off-shell Kadanoff-Baym equa-tions [31–33] in first-order gradient expansion. Further-more, a covariant dynamical transition between the par-tonic and hadronic degrees-of-freddom is employed thatincreases the entropy in consistency with the second lawof thermodynamics. The hadronic part part is equivalentto the HSD transport approach [34, 35] which includesthe baryon octet and decouplet, the 0 − and 1 − mesonnonets and higher resonances. When the mass of thehadrons exceeds a certain value (1 . . ∼ E = 0 . GeV /f m − in line with lQCD [42]. If the en-ergy density is below critical the strings decay to pre-hadrons (as in case of p+p reactions or in the hadroniccorona). The QGP phase is then evolved by the off-shelltransport equations with self-energies and cross sectionsfrom the DQPM. When the fireball expands the proba-bility of the partons for hadronization increases close tothe phase boundary (crossover at all RHIC energies), thehadronisation takes place using covariant transition ratesand the resulting hadronic system is further on governedby the off-shell HSD dynamics incorporating (optionally)self-energies for the hadronic degrees-of-freedom [18].Thus in the PHSD approach the full evolution of arelativistic heavy-ion collision, from the initial hard NNcollisions out of equilibrium up to the hadronisation andfinal interactions of the resulting hadronic particles, isdescribed on the same footing. We recall that this ap-proach has been sucessfully employed for p+p, p+A andA+A reactions from about √ s NN = 8 GeV to 2.76 TeV(see the review [2]).In this study we will concentrate on the strange res-onance dynamics mainly at RHIC energies where thePHSD describes well the bulk observables, i.e. rapidity, p T -spectra, v n coefficients etc. [2]. III. K ∗ IN MEDIUM
Before coming to actual results from PHSD for p+pand A+A collisions at relativistic energies we describe insome detail the evaluation of the kaon, antikaon and K ∗ selfenergies as a function of baryon density and their im-plementation in the PHSD. The medium properties andthe off-shell propagation of the strange vector-meson res-onances K ∗ and ¯ K ∗ are based on the “G-Matrix” ap-proach from Refs. [28, 43]. We use this approach to calculate the self-energy of the K ∗ and ¯ K ∗ which wethen implement in the form of relativistic Breit-Wignerspectral functions into PHSD to model the in-medium ef-fects and off-shell propagation of the K ∗ and ¯ K ∗ withina transport approach. A. G-Matrix approach
In recent years some efforts have been undertaken toassess the interaction of vector mesons with baryons incoupled-channel effective field approaches. In Refs. [11,12] vector mesons are introduced within the Hidden Lo-cal Symmetry approach [13–16] and a tree-level vector-meson baryon s -wave scattering amplitude is derived asthe low-energy limit of a vector-meson exchange mecha-nism. In Ref. [44] the interaction of vector mesons withbaryons is obtained on the basis of a SU (6) spin-flavorsymmetry extension of standard SU (3) meson-baryonchiral perturbation theory, leading to a generalized ( s -wave) Weinberg-Tomozawa interaction between the pseu-doscalar/vector meson octets and the octet and decupletof baryons. These two models share a crucial feature,namely, several N ∗ and hyperon resonances are dynami-cally generated in a broad range of energies upon unita-rization of the leading order (LO), tree-level amplitudes.Such states are indicative of non-trivial meson-baryon dy-namics and one may expect an impact on the in-mediumproperties of strange vector mesons. Incidentally, theproperties of vector mesons within these approaches wereinitially investigated for the case of the ¯ K ∗ (892) [43].The self-energy of the ω meson was also updated recentlyalong the same ideas in [45].The absence of baryonic resonances with S = +1 closeto threshold induces milder nuclear medium effects in theproperties of the K meson [23, 27] as compared to the¯ K meson, whose behaviour is largely dominated by theΛ(1405) resonance appearing in s -wave ¯ KN scattering.A similar situation takes place for the vector partner ofthe K , the K ∗ meson. Note that the K ∗ decays predom-inantly into Kπ , and therefore not only collisional effectsbut its in-medium decay width has to be taken into ac-count (we come back to this point below). However, sincethe K meson itself is barely influenced by nuclear matterone can anticipate small changes in Γ K ∗ , dec as comparedto the vacuum case.An effective Lagrangian approach, as developed in [11],was used in our previous study [28] to calculate the K ∗ meson self-energy at threshold energy. In this approach,the interaction between the octet of light vector mesonsand the octet of J P = 1 / + baryons is built within thehidden local gauge symmetry (HLS) formalism, which al-lows to incorporate vector meson interactions with pseu-doscalar mesons respecting the chiral dynamics of thepseudoscalar meson-meson sector. The interactions ofvector mesons with baryons are assumed to be domi-nated by vector-meson exchange diagrams, which inci-dentally allows for an interpretation of chiral Lagrangiansin the meson-baryon sector as the low-energy limit ofvector-exchange mechanisms naturally occurring in thetheory. Vector-meson baryon amplitudes emerge in thisscheme at leading order, in complete analogy to thepseudoscalar-meson baryon case, via the self-interactionsof vector-meson fields in the HLS approach. At smallmomentum transfer (i.e., neglecting corrections of order p/M , with M the baryon mass), the LO s -wave scatter-ing amplitudes have the same analytical structure as theones in the pseudoscalar-meson baryon sector (Weinberg-Tomozawa interaction), namely V ij = − C ij f (2 √ s − M B i − M B j ) ×× (cid:18) M B i + E i M B i (cid:19) / (cid:18) M B j + E j M B j (cid:19) / ~ε · ~ε ′ ≃ − C ij f ( q + q ′ ) ~ε · ~ε ′ , (1)where q ( q ′ ) stands for the energy of the incom-ing (outgoing) vector meson with polarisation ~ε ( ~ε ′ ), C ij stand for channel-dependent symmetry coefficients [11],and the latin indices label a specific vector-meson baryon( V B ) channel, e.g., K ∗ + p .The K ∗ collisional self-energy follows from summingthe forward K ∗ N scattering amplitude over the allowednucleon states in the medium, schematically Π coll K ∗ = P ~p n ( ~p ) T K ∗ N . Due to the absence of resonant statesnearby, a tρ approximation is well justified at energiessufficiently close to threshold, which leads to the practi-cal result (take T = V here)Π coll K ∗ = 12 (cid:0) V K ∗ + p + V K ∗ + n (cid:1) ρ (cid:18) ρρ (cid:19) ≃ α M K M K ∗ M K ∗ (cid:18) ρρ (cid:19) , (2)with α = 0 .
22, leading to a positive mass shift (equiv-alent to a repulsive optical potential) of about δM K ∗ ≃
50 MeV at normal matter density ρ = ρ = 0 .
17 fm − (recall δM K ∗ ≃ ReΠ K ∗ / M K ∗ ). Replacing the lowestorder tree level amplitudes V in the former result byunitarized amplitudes in coupled channels (solving theBethe-Salpeter equation, T = V + V GT ) one finds a re-duction by roughly one third over the previous result,namely δM K ∗ ( ρ ) ≃
30 MeV [i.e. α ≃ .
13 in Eq. (2)].Further medium corrections on the vacuum scatteringamplitudes T K ∗ N , leading to a full G -matrix calcula-tion, are of marginal relevance in the present case due tothe mild corrections introduced by the smoothly energy-dependent K ∗ N interaction . To be precise, we use the term G -matrix for the in-medium effec-tive meson-baryon T -matrix obtained in DiracBrueckner theory.[46] In the case of the ¯ K ∗ , the collisional part of the selfen-ergy, related to the quasi-elastic reaction ¯ K ∗ N → ¯ K ∗ N and accounting for absorption channels, induces a strongbroadening of the ¯ K ∗ spectral function as a result of themixing with two J P = 1 / − states, the Λ(1783) andΣ(1830), which are dynamically generated in the hiddenlocal symmetry approach in a parallel way to the ¯ KN interaction and the Λ(1405). Such complicated many-body structure of the ¯ K ∗ N interaction requires a detailedanalysis of medium corrections such as Pauli blocking onbaryons and a self-consistent evaluation of the in-medium¯ K ∗ N scattering amplitude ( G -matrix) and the ¯ K ∗ self-energy, as was done in [43]. For the present study werecourse to a suitable parameterization of the resulting¯ K ∗ self-energy and spectral function, which we discuss inmore detail below.As mentioned before, the decay with of the K ∗ / ¯ K ∗ me-son (with suitable medium corrections) has to be takeninto account to realistically assess production and anihi-lation rates. Such effects are readily incorporated for the¯ K ∗ within the G -matrix approach which we parametrizefrom Ref. [43]. For the K ∗ , instead, we evaluate explicitlyits medium-modified K ∗ → Kπ width as follows [28],Γ V, dec ( µ, ρ ) = Γ V (cid:18) µ µ (cid:19) R µ − m π q ( µ, M ) A j ( M, ρ ) dM R µ − m π M min q ( µ , M ) A j ( M, dM , (3)where j = K and V = K ∗ for the presentcase, q ( µ, M ) = p λ ( µ, M, M π ) / µ and λ ( x, y, z ) = (cid:2) x − ( y + z ) (cid:3) (cid:2) x − ( y − z ) (cid:3) . Γ V stands for the vec-tor meson (vacuum) partial decay width in the consid-ered channel and µ is the nominal resonance mass, par-ticularly Γ K ∗ , ¯ K ∗ = 42 MeV and µ = 892 MeV [47].Eq. (3) accounts for the in-medium modification of theresonance width by its decay products. In particular, weconsider the fact that kaons and anti-kaons may acquirea broad spectral function in the medium, A j ( M, ρ ). Asdiscussed in [28], A K in Eq. (3) is a delta function invacuum since the kaon is stable in vacuum with respectto the strong interaction, and to a good approximationthe same can be kept at finite nuclear density by us-ing an effective kaon mass M ∗ K ( ρ ) = M K + Π K ( ρ ) withΠ K ( ρ ) ≃ . M K ( ρ/ρ ) [27, 48, 49]. In general this maynot be the case for other vector mesons even in the vac-uum case, e.g., a → ρπ , where the ρ meson has a largewidth into two pions ( M min then stands for the corre-sponding threshold energy).Once both collisional and decay self-energies are ob-tained, the K ∗ and ¯ K ∗ spectral function is readily givenas the imaginary part of the vector-meson in-mediumpropagator, namely, S V ( ω, ~q ; ρ ) = − π Im D V ( ω, ~q ; ρ )= − π Im Π V ( ω, ~q ; ρ )[ ω − ~q − M V − Re Π V ( ω, ~q ; ρ )] + [Im Π V ( ω, ~q ; ρ )] . (4)with V = K ∗ , ¯ K ∗ , where Π V contains the sum of thecollisional and decay self-energies. B. Implementation in PHSD
In order to implement K ∗ and ¯ K ∗ in-medium proper-ties in PHSD we adopt a relativistic Breit-Wigner pre-scription for the strange vector meson spectral functions( V = K ∗ , ¯ K ∗ ) [28, 50], A V ( M, ρ ) = C π M Γ ∗ V ( M, ρ ) (cid:0) M − M ∗ V ( ρ ) (cid:1) + ( M Γ ∗ V ( M, ρ )) , (5)where M is the invariant mass and C is a normalisationconstant ensuring that the sum rule Z ∞ A V ( M, ρ ) dM = 1 (6)is fulfilled. The connection with the spectral function inEq. (4) can be done by setting the vector-meson momen-tum at zero, A V ( M, ρ ) = 2 · C · M · S V ( M,~ , ρ ) , (7)a practical approximation which does not account forexplicit energy and momentum dependence of mediumcorrections (this limitation is consistently dealt with byconsidering vector mesons to be at rest in the nuclearmatter frame when evaluating their self-energy). The in-medium mass M ∗ V and decay width Γ ∗ V of the K ∗ / ¯ K ∗ arederived from the vector meson self-energy, as in the caseof partons in the DQPM,( M ∗ V ) = M V + Re Π V ( M ∗ V , ρ ) , Γ ∗ V ( M, ρ ) = − Im Π V ( M, ρ ) /M , (8)where M V denotes the nominal (pole) mass of the reso-nance in vacuum.We briefly comment in the following the essential fea-tures of the K ∗ and ¯ K ∗ Breit-Wigner spectral functionswhen including medium effects along the self-energy cal-culation in the previous section. In Fig. 1 we depictthe spectral function for the K ∗ meson. The K ∗ ex-periences a net repulsive interaction with the mediumwhich leads to a shift of the spectral function’s peakto higher invariant masses with increasing nuclear den-sity. Overall, the width of the K ∗ becomes slightlysmaller with increasing density, due to the kaon becom-ing also heavier (this effect is largely compensated by thehigher K ∗ excitation energy) Furthermore, the thresh-old energy for the creation of a K ∗ is shifted up, i.e. M th = M K + M π + ∆ M ( ρ ) ≈ . GeV + ∆ M ( ρ ), with∆ M ( ρ ) ≃ Π K ( ρ ) / M K , which amounts to 0 . M K atnormal matter density.Interestingly, in case of the vector antikaon ¯ K ∗ the ef-fects from the medium are rather different as comparedto the K ∗ . Fig. 2 shows the spectral function for the ¯ K ∗ / =0.0 / =0.5 / =1.0 A ( M , ) [ / G e V ] M [GeV] K* + FIG. 1: The relativistic Breit-Wigner spectral function A ( M, ρ ) of the K ∗ is shown as a function of the invariantmass M for different nuclear densities. The solid red line cor-responds to the vacuum spectral function, whereas the dashedgreen and the dash-dotted blue lines correspond to densities ρ/ρ = 0 . , .
0, respectively. A ( M , ) [ / G e V ] M [GeV] K* - / =0.0 / =0.5 / =1.0 FIG. 2: Same as Fig. 1 for the ¯ K ∗ spectral function. at different nuclear densities. The distribution is consid-erably shifted to lower invariant masses when the densityis increased, reflecting the overall attractive interactionof the ¯ K ∗ with the baryon-rich medium. Furthermore,the ¯ K ∗ width is largely enhanced as a consequence ofthe multiple absorption channels that are accounted forin the ¯ K ∗ self-energy, involving the mixing of the quasi-particle mode with Λ N − and Σ N − excitations. Conse-quently, the threshold energy for the creation of a ¯ K ∗ isconsiderably diminished, almost down to M th ∼ M π ,which implies that an off-shell ¯ K ∗ can be created atrather low invariant masses.The medium corrections discussed before have animpact on the K ∗ and ¯ K ∗ production rates in thehadronic phase of a heavy-ion collision. The produc-tion/anihilation cross section in PHSD is consistentlymodified according to σ K ∗ ( ¯ K ∗ ) ( M, ρ ) = 6 π A K ∗ ( ¯ K ∗ ) ( M, ρ ) Γ ∗ K ∗ ( ¯ K ∗ ) ( M, ρ ) q ( M, M K , M π ) . (9)Fig. 3 shows the cross-section for K ∗ production from¯ Kπ scattering. The evolution of the cross section withthe nuclear density reflects that of the K ∗ spectral func-tion, leading to a shift of the energy distribution to higherinvariant masses. The presence of the Γ ∗ factor in Eq. 9,accounting for the effective ¯ K ∗ ¯ Kπ coupling and multi-plying the spectral function, makes the maximum valueof the cross section practically unchanged when varyingthe density, reaching a value as large as 160-170 mb. Theobserved shift of the cross-section implies that, in orderto create K ∗ at finite densities, larger energies are neededas compared to the same reaction in vacuum. [ m b ] M [GeV] K* + / =0.0 / =0.5 / =1.0 FIG. 3: The cross-section σ for K ∗ production/anihilation isshown as a function of the invariant mass M for different nu-clear densities. The solid red line corresponds to the vacuumcase, whereas the dashed green and the dash-dotted blue linescorrespond to densities ρ/ρ = 0 . , .
0, respectively.
Quite different is the behavior of the ¯ K ∗ cross section,as can be seen in Fig. 4. The energy dependence reflectsthe attractive nature of the ¯ K ∗ interaction with the nu-clear medium, as follows from the ¯ K ∗ spectral function,the peak of the distribution being shifted to lower in-variant masses in the same magnitude as the density isincreased. The fall in the maximum of the cross section,though, seems to saturate due to the large increase of the¯ K ∗ width. At nuclear matter density, the cross sectionreaches a maximum value around 100 mb at invariantmasses around 120 MeV below the vacuum case. [ m b ] M [GeV] K* - / =0.0 / =0.5 / =1.0 FIG. 4: Same as in Fig. 3 for the ¯ K ∗ production/anihilationcross section. IV. K ∗ DYNAMICS IN PHSD
Before we come to the physical observables that canbe compared to experimental data we investigate theon/off-shell dynamics of the K ∗ vector mesons withinthe PHSD transport approach for central Au+Au colli-sions at √ s NN = 200 GeV. We will investigate the mainproduction channels of the K ∗ mesons, their productionin time and record the baryon density at production. Inorder to better understand the experimental results inthe next section we also investigate the change of thetransverse momentum spectrum with respect to the ex-perimental cuts imposed for the reconstruction the K ∗ vector mesons. A. Production channels and dependence on baryondensity
We start with the production of the strange vectormesons as a function of time for the different productionchannels in PHSD.As can be seen from Fig. 5 there is no sizeable differ-ence between the number of vector kaons K ∗ and and vec-tor antikaons ¯ K ∗ s from PHSD. The production by stringsin the hadronic corona occurs early and gives practicallyno further contribution in the late stages. Furthermore,one can see that the contribution from the QGP is notvery large and starts a few fm/c later in the hadroniza-tion, when compared to the late K + π channel (bluelines). In fact, the QGP contribution is on the samelevel as the contribution from strings for this system.For times larger than 150 fm/c (not shown here) also the K + π channel decreases rapidly and all vector mesonssimply decay.With respect to in-medium modifications of the vectormesons the baryon density at the production point is of all BB mB K+ QGP N ( t ) K* + K* - K* N ( t ) t [fm/c] anti-K* (s NN ) = 200 GeV, Au+Au, all yt [fm/c] FIG. 5: The number of strange vector mesons N ( t ) is shownas a function of time t for all four isospin channels of the K ∗ mesons and for all production channels in central Au+Aucollisions at √ s NN = 200 GeV (all y ) from a PHSD calcula-tion. The upper left panel shows the channel decompositionfor the K ∗ + , the upper right panel shows the channel decom-position for the K ∗− , the lower left panel shows the channeldecomposition of the K ∗ and the lower right panel showsthe channel decomposition of the ¯ K ∗ . The legend for allfour panels is as follows: the solid black line shows all pro-duced K ∗ s, the dash dotted green line shows K ∗ s producedfrom baryon-baryon strings, the solid dash double-dotted lineshows K ∗ s produced from meson-baryon strings, the dashedlight blue line shows K ∗ s produced from K + π annihilationand the short dotted red line shows K ∗ produced during thehadronisation of the QGP. additional interest. Fig. 6 shows the normalised num-ber of K ∗ s for two collision energies as a function of thebaryon density ρρ for central Au+Au collisions. Fromthe previous figures we know that most of the K ∗ s comefrom the K + π channels in the later stage. Accordingly,most of the K ∗ s are created while the baryon density isfairly low, with only very few mesons created above halfnormal nuclear baryon density ρ during a Au+Au col-lision with a cms energy of 200 GeV at RHIC. Thus thein-medium modifications of the strange vector mesons areexpected to be small at this energy as well as observableconsequences in the final spectra. Note, however, thatfor a much lower collisional energy of 10 GeV the baryondensities are much higher and the perspectives to see in-medium modifications of the vector mesons become bet-ter. The situation is not very different from the dileptonmeasurements in heavy-ion reactions with respect to thecontribution from the ρ -meson decay [2].Furthermore, one can see in Fig. 7 the channel decom-position of the K ∗ as a function of the baryon density (inunits of ρ ). As before the dominant channel is the K + π channel. It is important to note that this is the case for (s NN ) = 200 GeV, Au+Au, all y d N / d ( / ) / N t o t /
200 GeV 10 GeV
FIG. 6: The differential distribution of the total number of K ∗ s N tot dNd (cid:16) ρρ (cid:17) versus baryon density ρρ for Au+Au collisionsat different cms energies from the PHSD simulations. Thesolid red line shows results for a collision at √ s NN = 200 GeVwhile the dashed blue line shows results for a collision at √ s NN = 10 GeV. (s NN ) = 200 GeV, Au+Au, all y d N / d ( / ) / K+ Strings all
FIG. 7: The differential distribution dNd (cid:16) ρρ (cid:17) as a function ofthe baryon density ρρ for Au+Au collisions at a cms energy of √ s NN = 200 GeV for different production channels of the K ∗ .The solid black line shows the K ∗ s coming from all productionchannels, the dashed orange line shows K ∗ s coming from the K + π channel and the dash dotted green line shows K ∗ scoming from meson-baryon and baryon-baryon strings. all baryon densities, since, as opposed to the other chan-nels, K ∗ s coming from the K + π channel are createdthroughout the whole collision history and can thus oc-cur at low and higher baryon densities, respectively. B. Modifications of the K ∗ mass distributions inthe medium (a) All BB-string mB-string K+ QGP Reconstructed(s NN ) = 200 GeV, Au+Au, 5% central, all y d N / d M [ / G e V ] M [GeV] K* + + K* All BB-string mB-string K+ QGP Reconstructed(s NN ) = 200 GeV, Au+Au, 5% central, all y d N / d M [ / G e V ] M [GeV] K* - + anti-K* (b) FIG. 8: The differential mass distribution dNdM for the vectorkaons K ∗ (”a”, upper part) and antikaons ¯ K ∗ (”b”, lowerpart) for different production channels as a function of theinvariant mass M in a Au+Au collision at √ s NN = 200 GeVfrom a PHSD calculation. The solid black lines show all pro-duced K ∗ ( ¯ K ∗ )s, the dash dotted green lines indicate produc-tion from baryon-baryon strings, the dash double-dotted bluelines – from meson-baryon strings, the dashed light blue lines– from K + π ( ¯ K + π ) annihilation and the short dotted redline correspond to the production during the hadronisation ofthe QGP. The dashed orange line with the circles shows thedistribution for K ∗ s that have been reconstructed from finalkaon and pion pairs. As can be seen in Fig. 8 again the dominant produc-tion channel of the vector kaons K ∗ (”a”, upper part) andvector antikaons ¯ K ∗ (”b”, lower part) is the annihilationof the K + π ( ¯ K + π ) pairs. Due to its broad structure thespectral function of the K ∗ and ¯ K ∗ allows for the anni-hilation of K + π pairs also at lower and higher masses ascompared to the vacuum. The contribution from meson- baryon and baryon-baryon strings is practically negligi-ble, however, there is still a sizeable contribution com-ing from the QGP. The shape of the spectral functionfor vector kaons suggests that there should be some K ∗ sproduced at higher invariant masses due to finite baryondensities where the pole mass is shifted up. However,the high baryon density region is not much populated atthis cms energy as demonstrated above such that theseeffects will be hard to disentangle in the final spectra incomparison to experiment.As seen from the lower part (”b”) of Fig. 8 the massdistribution of the vector antikaons is slightly shifted tothe low masses which stems from the shift of the spectralfunction of the ¯ K ∗ to lower masses at finite baryon densi-ties. This means that in-medium effects are still presentsince the mesons are created at nonzero densities. Thecontribution of the strings and the QGP does not playa significant role due to the lack of baryons relative tomesons at the top RHIC energy.In both of these figures the distribution for the recon-structed particles is shifted from higher to lower invariantmass regions. Furthermore the total number of particlesis reduced to about 40 to 50 % in the reconstructed distri-bution as opposed to the particles that were taken fromthe decay point. C. Transverse momentum distributions andacceptance cuts -2 -1 K* +anti-K* Decayed K* +anti-K* d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c](s NN ) = 200 GeV, Au+Au, 5% central, |y|<0.5Reconstructed no cuts K* cuts FIG. 9: The transverse momentum spectrum d N / (2 πp T dp T dy ) versus the transverse momentum p T for vector kaons and antikaons in Au+Au collisions at √ s NN = 200 GeV from the PHSD simulation. The dashedblack line shows the spectrum directly at the point whenthe ¯ K ∗ decays in the PHSD simulation. The dotted red andsolid blue lines show the spectrum after the ¯ K ∗ has beenreconstructed from the final kaons and pions. The dottedred line does not include any cuts while the solid blue lineincludes cuts on the invariant mass of the ¯ K ∗ . Before we compare our PHSD results with experimen-tal data, we have to establish the restrictions that ariseduring the reconstruction of the K ∗ mesons from final K + π pairs on the observables. In Fig. 9 we showPHSD results for K ∗ and ¯ K ∗ at midrapidity for cen-tral Au+Au collisions at the top RHIC energy. We re-call that in PHSD we can study the K ∗ s at their decaypoint, i.e. at the point in space-time of their decay into K + π pairs. Furthermore, we can reconstruct the K ∗ sfrom their daughter particles, the K and π pairs thatare affected by final state interactions. In Fig. 9 onecan see that there is a slight difference between the de-cayed K ∗ s, shown by the dashed black line, and the re-constructed K ∗ s, shown by the red line. This differenceis due to rescattering and absorption of the final kaonsand pions in the medium. Particles with low transversemomentum are more affected by this than particles withhigher transverse momentum, and these two lines mergewith increasing p T . This implies that fast K ∗ s can bereconstructed much more efficiently since their daughterparticles do not interact with the medium as much asparticles with lower p T which rescatter more often dueto their low velocities in the hadronic medium.Fig. 9 also shows the K ∗ s that have been recon-structed from final particles (blue line) employing therestrictions that have been imposed on the K ∗ in theform of a restriction to a specific invariant mass regionof M = [0 . , .
0] GeV. This is due to the low signal tonoise ratio during the reconstruction of the K ∗ . Whilethe spectral function of the K ∗ is relatively broad, witha width of about 42 MeV in vacuum, the experimentalsignal is very narrow and it is difficult to distinguish cor-related from uncorrelated K + π pairs below and abovethe selected invariant mass region.It is also important to note the different productionchannels of the K ∗ in actual observables, e.g. thetransverse momentum spectrum where different channelsmight reflect different spectral slopes. This is shown inFigs. 10 for vector kaons K ∗ (upper part)and vector an-tikaons ¯ K ∗ (lower part), respectively. Similarly to thefigures in Fig. 8 one can see that K ∗ s coming frommeson-baryon and baryon-baryon strings contribute onlya small part to the overall spectrum. The far dominantchannel contribution comes from K + π annihilation inthe final hadronic stage. The second-largest contributioncomes from the QGP, however, it is smaller by about anorder of magnitude for all p T which makes the K ∗ lesssuitable as a probe for the QGP. Furthermore, the spec-tral slopes from all channels are very similar in the high p T region. This is true for both K ∗ s and ¯ K ∗ s . D. Production and decay time of the K ∗ In order to understand better which K ∗ s can be seen orreconstructed in the detector at an experiment like STARone can look at when the K ∗ ’s are produced and whenthey decay during a heavy-ion collision. As can be seen -2 -1 (a) K* + + K* (s NN ) = 200 GeV, Au+Au, 5% central, all y d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c] All BB-string mB-string K+ QGP0.0 0.5 1.0 1.5 2.0 2.510 -2 -1 (b) (s NN ) = 200 GeV, Au+Au, 5% central, all y K* - + anti-K* d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c] K* + + K* All BB-string mB-string K+ QGP
FIG. 10: The channel decomposition of the transverse mo-mentum spectrum d N / πp T dp T dy for K ∗ (upper part) andfor the ¯ K ∗ (lower part) for different productions channelsin Au+Au collisions at a cms energy of √ s NN = 200 GeVfrom a PHSD calculation: the solid black lines show all pro-duced K ∗ ( ¯ K ∗ )s, the dash dotted green lines indicate produc-tion from baryon-baryon strings, the dash double-dotted bluelines – from meson-baryon strings, the dashed light blue lines– from K + π ( ¯ K + π ) annihilation and the short dotted redlines correspond to the production during the hadronisationof the QGP. in fig. 5 different channels dominate the K ∗ productionduring certain times of a heavy-ion collision. Thus onecould deduce from what source the reconstructed Kπ pairwould originate.Fig. 11 shows the production and the decay rates of all K ∗ s and ¯ K ∗ which existed in a heavy-ion collisions (solidblack and dashed red lines) and which of these K ∗ s couldbe reconstructed in the detector from the final pions andkaons ( π + K ) – short-dashed green and dash-dotted bluelines.As can be seen many of the K ∗ s that decay during theearly stages of the collision, up to a time of t = 20 fm/c,0 (s NN ) = 200 GeV, Au+Au, 5% central, all y d N / d t [f m - ] t [fm/c]K* (all) Creation DecayReconstructed K* (all) Creation Decay FIG. 11: The creation and decay rates versus time for centralAu+Au collision at a cms energy of √ s NN = 200 GeV. Boththe K ∗ s and ¯ K ∗ s are included. The solid black and dashedred lines show the creation and decay rates for all K ∗ s in asystem. The short-dashed green and dash-dotted blue linesshow the creation and decay rates respectively of K ∗ s thatcould be reconstructed from the final pions and kaons ( π + K ). decay into Kπ pairs that do not reach the detector andthus cannot be reconstructed due to absorption by themedium or rescattering of the final pions and kaons. Dur-ing later stages of the collision, however, the K ∗ s thatare created (most probably from Kπ collisions, as canbe seen in fig. 5) can also be seen in the detector, sincethe medium is already dilute and absorption of the decayparticles by the medium is rare. V. RESULTS FROM PHSD IN COMPARISONTO DATA FROM STAR
In this section we present our results from the PHSDtransport approach in comparison to the data from theSTAR Collaboration at RHIC. [7–10]The STAR collaboration at RHIC has investigated thefollowing hadronic decay channels: K (892) ∗ → K + π − ,¯ K (892) ∗ → K − π + and K (892) ∗± → K S π ± → π + π − π ± [7]. As has been mentioned above the K ∗ reconstruc-tion relies on the final particles observed in the detector,i.e. the kaons and the pions. Since both decay productssuffer from rescattering and absorption the reconstruc-tion becomes difficult. Furthermore, the time resolutionand accuracy in momentum and invariant mass of thedetector itself adds another hurdle to a reconstruction ofthe K ∗ signal due to a possible misidentification of theparticles.The procedure for obtaining the K ∗ signal is based onthe combination of all kaons and pions with respect tothe hadronic decay channels mentioned above, i.e. onlychannels that are physically possible are taken. To get rid of the background there are several techniques that canbe employed: i) One can take all unphysical channels andcombine the kaons and pions to get the background spec-trum. Another method to get the background is ii) to flipthe x and the y component of the momentum of eitherthe kaons and pions and combine all physical channels.A third method iii) consists of pairing all kaons and pionsfrom two different events. Unlike the first two methodsthis method ensures that there can be no possible corre-lation between the kaons and the pions and thus is mostsuited to construct the background spectrum [51] since itis not possible that kaons and pions from different eventscan be correlated.For the reconstruction of the K ∗± vertex cuts also needto be taken into account due to the second decay vertexwhich stems from the decay of the K S to a π + π − pair.However, in PHSD this is not accounted for because the K ∗± directly decay to a K instead of a K S [7].In principle, the experimental reconstruction proce-dure could exactly be repeated with the PHSD final par-ticle spectra on an event-by-event basis. However, thiswould imply to generate a huge amount of events which isvery costly due to limited computing power. On the otherhand the transport approach offers information that isnot available in the experiment. We can precisely iden-tify in PHSD the kaons and pions that correspond to acertain K ∗ decay and we can reconstruct the final K ∗ spectrum much more efficiently and also with high pre-cision.These different methods will not lead to a sizeable dif-ferences for p+p collisions, since the number of createdparticles is relatively small and the risk of mismatchingparticles is comparably low. It is possible (and likely)that there is a difference in A+A collisions. Our stud-ies, however, have not shown a significant difference ascompared to the correlation method. A. p+p collisions
First of all, the experimental reconstruction of the K ∗ in p+p collisions doesn’t lead to a strong distortion of the K ∗ signal since p+p collisions produce only a low amountof particles and the rescattering and absorption of thefinal kaons and pions is practically vanishing compared toA+A collisions. Furthermore, there are no modificationsof the kaons by a medium.As can be seen from both figures in Fig. 12 the resultsfrom the PHSD calculations reproduces the experimen-tal data very well. The K ∗ spectrum from the recon-structed final kaons and pions matches the K ∗ spectrumfrom the K ∗ , which were directly taken from the decaypoint. This holds true for K ∗ s with both a zero and anon-zero electric charge. Experimentally all the possible K ∗ reconstruction channels are considered, through ei-ther a direct decay into a K + π pair or indirectly throughthe second vertex calculation of the K S → π + π − decay.We note in passing that that PHSD appears to slightly1 -5 -4 -3 -2 -1 (a)(K* + + K* - )/2 (s NN ) = 200 GeV, |y|<0.5 STAR pp minimum bias d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c] PHSD Decayed K*s Reconstructed K*s0.0 0.5 1.0 1.5 2.0 2.5 3.010 -5 -4 -3 -2 -1 (K* + anti-K* )/2 STAR pp minimum bias d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c] PHSD Decayed K*s Reconstructed K*s (s NN ) = 200 GeV, |y|<0.5 (b) FIG. 12: The transverse momentum spectrum d N / πp T dp T dy versus the transverse momentum p T for K ∗ + + K ∗− (”a”, upper part) and K ∗ + ¯ K ∗ (”b”, lowerpart) in a p+p collision at a cms energy of √ s NN = 200 GeV.The solid black circles denote minimum bias p+p data fromthe STAR experiment from Ref. [7]. The connected opensymbols represent results from a PHSD simulation. The opengreen squares show results where the spectrum was obtainedfrom the K ∗ ( ¯ K ∗ )’s coming directly from the decay point inPHSD. The open red stars show results from K ∗ ( ¯ K + )’s thathave been reconstructed from the final K ( ¯ K ) and π pairs. underestimate the p T slope for charged K ∗ s whereas itslightly overestimates the p T slope for neutral K ∗ s. Sincein PHSD the slopes for charged and neutral K ∗ s are thesame within statistical accuracy one might ’see’ a slightlydifferent slope in the STAR data. However, this is withinthe systematic uncertainties. B. A+A collisions
The experimental reconstruction of the K ∗ spectrumin A+A collisions is a lot more complicated and the signal is much more more distorted. The number of producedparticles in an Au+Au collisions is much higher than in ap+p collision which leads to a lot more background andto a higher probability of misidentification of particles.Furthermore, the cuts on the invariant mass have a largeeffect, as shown in IV. -3 -2 -1 PHSD Decayed Reconstructed STAR Au+Au 10% central d N / ( T dp T d y ) [ c / G e V ] p T [GeV/c](s NN ) = 200 GeV, Au+Au, 10% central, |y|<0.5 K*0 + anti-K*0
FIG. 13: The transverse momentum spectrum d N / πp T dp T dy versus the transverse momentum p T for K ∗ + ¯ K ∗ mesons in a Au+Au collision at a cms energyof √ s NN = 200 GeV. The solid black circles show the datafrom the STAR Collaboration for 10% central collisionswhile the connected symbols show results from the PHSDcalculation. The solid green squares show results where the K ∗ s were taken at the point of their decay while the solidred stars show results for K ∗ s that have been reconstructedfrom the final K + π pairs. The STAR data are taken fromRef. [7]. Fig. 13 highlights the difference between the recon-structed K ∗ spectrum and the spectrum of K ∗ s takendirectly from their decay point. While the reconstructedspectrum follows the experimental data very well afterapplying the acceptance cuts, the K ∗ spectrum as takenat the decay point is sizeably higher at low transversemomentum and has a lower slope. As has been discussedin the previous section only a small part of the changein the spectrum is due to rescattering and absorption bythe medium. We have found that the dominant changesarise from the different cuts imposed in the reconstruc-tion. This implies that parameters such as an effectivetemperature T ∗ , which can be extracted from an expo-nential fit to the p T spectra, do not necessarily correlateto the value that reflects the actual K ∗ decays duringa collision. Both the cuts and the rescattering and ab-sorption by the medium only affect the lower part of thetransverse momentum spectrum while both the green andthe red lines converge towards each other for higher p T .When considering rescattering and absorption effectsone may argue that the faster kaons and pions, i.e. parti-cles with a higher p T , are able to escape the medium and2reach the detector without sizeable interaction with themedium while particles with lower transverse momentumexperience stronger final state interactions which lead toa distortion of the reconstructed K ∗ spectrum. Sincethe size of the ’fireball’ changes with centrality the K ∗ spectra for different centrality classes provide further in-formation. -5 -4 -3 -2 -1 STAR(K* + +anti-K* - )/2 [50,80]% PHSD(K* + +anti-K* - )/2 [50,80]% (s NN ) = 200 GeV, Au+Au, |y|<0.5 d N / ( m T d m T d y ) [ / G e v ] m T -m [GeV] STAR(K* +anti-K* )/2 central*2 [0,10]% [30,50]% [50,80]% PHSD(K* +anti-K* )/2 central*2 [0,10]% [30,50]% [50,80]% FIG. 14: The transverse mass spectrum d N/ πm T dm T dy as a function of the transverse mass m T − m for differentcentralities for K ∗ + ¯ K ∗ as well as for peripheral colli-sions for K ∗ + + K ∗− in Au+Au collisions at a cms energyof √ s NN = 200 GeV. The symbols show data from the STARexperiment while the solid lines show results from the PHSDcalculations. For the K ∗ + ¯ K ∗ STAR data the legend is asfollows: the solid black circles show data from central Au+Aucollisions, the open stars show data for [0 , , , K ∗ + + ¯ K ∗− STAR data the up-side down solid black triangles show data for [50 , K ∗ + ¯ K ∗ PHSD results the legend is asfollows: the solid red line shows results from central Au+Aucollisions, the dashed olive line shows results for [0 , , , K ∗ + + ¯ K ∗− PHSD resultsthe solid light blue line shows results for [50 , As can be seen in Fig. 14 the experimental data havealso been taken for Au+Au collisions at a cms energy of √ s NN = 200 GeV for different centralities. We have usedthe same K ∗ reconstruction procedure to calculate thereconstructed m T spectrum of the K ∗ form PHSD simu-lations for the same centrality classes. The experimentaldata can be reproduced by the results from PHSD verywell, separately for the different centralities. We notethat the same restrictions and effects are also present athigher impact parameters. Rescattering and absorptioneffects play a role at central as well as at peripheral col-lisions and the cuts imposed on the reconstructed Kπ pairs also lead to a lowering of the spectrum at smalltransverse mass. p+p STAR PHSD K * / K - dN ch /d Au+Au STAR PHSD (s NN ) = 200 GeV, Au+Au, |y|<0.5 FIG. 15: The K ∗ /K − ratio versus dN ch /dη for Au+Au col-lisions at a cms energy of √ s NN = 200 GeV. The solid blacksquares show data from the Au+Au collision at the STARexperiment while the orange circle shows data from p+p col-lisions at STAR. The solid red stars show results from Au+AuPHSD calculations and the open green star shows results fromp+p PHSD calculations. The STAR data are taken from[7, 9]. In Figs. 15 and 16 we show the ratios between K ∗ and K − as a function of the charged particle pseudorapidityand the cms energy, respectively, in comparison to theSTAR data. The reason to study this particular ratio isthat the quark content of K ∗ s and K is the same, the dif-ference lies in the mass and the relative orientation of thequark spin. By studying this ratio one hoped to find outmore about the K ∗ production properties and the freeze-out conditions in relativistic heavy-ion collisions. ThePHSD results in Fig. 15 reproduce STAR data very wellwithin error bars which indicates the production channelsin PHSD are in line with the experimental observation.The ratios in Fig. 15 are shown as the real ratios, i.e.they have not been normalised to the ratio measured inminimum bias p+p collisions, as was done in [7]. Weobtain a similar value for the K ∗ to K − ratio in p+pcollisions as in the experiment. Furthermore, the PHSDresults match the experimental data from Ref. [7] aswell. Furthermore, from Fig. 15 one can see that thereis almost no dependence of the K ∗ /K − ratio on thecentrality and accordingly the impact parameter.Fig. 16 shows the same ratio as a function of differentcms energies. One has to note that we show Au+Auand p+p data in accordance with the original publicationwhere this data have appeared. As seen the PHSD resultscan reproduce these STAR data within error bars well,too. There seems to be no strong dependence of the ratiowith cms energy.Figs. 17 and 18, furthermore, show the average trans-3
5% central, |y|<0.5 K * / K - (s NN ) [GeV]Au+Au STAR PHSDp+p STAR NA27 PHSD FIG. 16: The K ∗ /K − ratio as a function of the cms energy √ s NN . The black squares show data from Au+Au collisionsat the STAR experiment. The orange circle shows STAR datafrom p+p collisions. Additionally data from the NA27 exper-iment is shown for lower cms energies as open blue diamonds.The red and green symbols show results from a PHSD calcu-lation. The solid red stars show results for Au+Au collisionswhile the open green stars show results for p+p collisions.The STAR data are taken from [7, 8]. verse momentum as a function of the charged particlepseudorapidity and the average number of participants,respectively. Again, the average transverse momentumresults from PHSD agree with the data from the STARexperiment very well, both for p+p and also for Au+Aucollisions.Furthermore, in Fig. 18 one can see that our re-sults agree with the data for peripheral collisions, withonly few participating nucleons, and for central collisionswhere the number of participants is high. The STARCollaboration has found that the average transverse mo-mentum of the K ∗ s is higher than the average transversemomentum of kaons and pions. This might indicate thatthe average transverse momentum is more strongly re-lated to the mass of the particle observed.So far, most of the collisions were performed forAu+Au or p+p collisions. However, there are also dataCu+Cu which have been provided in form of the averagetransverse momentum in Fig. 19. The data are availablefor a cms energy of √ s NN = 200 GeV and for a lowerenergy of √ s NN = 62 . K ∗ s as a function of theaverage number of participants agrees very well both forthe higher energy of √ s NN = 200 GeV and also for lowerenergy of √ s NN = 62 . v , which is defined as thesecond harmonic coefficient of the Fourier expansion of (s NN ) = 200 GeV, 5% central, |y|<0.5 < p T > [ G e V ] dN ch /dp+p STAR PHSD Au+Au STAR PHSD K* FIG. 17: The average transverse momentum < p T > as afunction of dN ch /dη . The black and orange symbols showdata for K ∗ from the STAR experiment. The solid blacksquares show data for Au+Au collisions while the solid or-ange circle shows data for p+p collisions. The green and redsymbols show results for K ∗ from a PHSD simulation. Thesolid red stars show results for Au+Au collisions while theopen green star shows results for p+p collisions. The STARdata are taken from Ref. [7]. the azimuthal particle distributions in momentum space E d Nd ~p = 12 π d Np T dp T dy ∞ X n =1 v n cos ( n ( φ − Ψ R )) ! (10)which we have taken as v = p x + p y p x − p y (11)since the reaction plane is ficed in PHSD. In this fig-ure the STAR data are taken from two different runs atRHIC. When considering the error bars of the experimen-tal data from the first (RunII) and the second (RUNIV)both data are in agreement and a significant non-zero v for the K ∗ can be extracted in minimum bias Au+Aucollisions at a cms energy of √ s NN = 200 GeV. Againwithin error bars the PHSD results are compatible withthe measurements. VI. SUMMARY
In this study we have investigated the strange vectormeson ( K ∗ , ¯ K ∗ ) dynamics in heavy-ion collisions basedon the microscopic off-shell PHSD transport approachwhich incorporates partonic and hadronic degrees-of-freedom and includes a crossover phase transition frompartons to hadrons and vice versa. We have studied4 K* (s NN ) = 200 GeV, 5% central, |y|<0.5 < p T > [ G e V ]
The authors acknowledge inspiring discussions withJ¨org Aichelin, Wolfgang Cassing, Taesoo Song, PierreMoreau, Anders Knospe, Laura Tolos and VadimVoronyuk. A.I. acknowledges support by HIC for FAIRand HGS-HIRe for FAIR. D.C. acknowledges support byMinisterio de Econom´ıa y Competitividad (Spain), GrantNr. FIS2014-51948-C2-1-P. This work was supported byBMBF and HIC for FAIR. The computational resourceshave been provided by LOEWE-CSC at the Goethe Uni-versity Frankfurt. [1] Proceedings of Quark Matter-2014, Nucl. Phys. A ,1 (2014).[2] O. Linnyk, E. L. Bratkovskaya, W. Cassing, Prog. Part.Nucl. Phys. 87 (2016) 50[3] A. Adare et al. (PHENIX Collaboration), Phys. Rev.Lett. , 172301 (2007); B. Abelev et al. [ALICE Col-laboration], JHEP , 112 (2012).[4] J. Rafelski, J. Letessier and G. Torrieri, Phys. Rev. C (2001) 054907 Erratum: [Phys. Rev. C (2002) 069902][5] C. Markert, G. Torrieri and J. Rafelski, AIP Conf. Proc. , 533 (2002)[6] C. Markert, R. Bellwied and I. Vitev, Phys. Lett. B (2008) 92[7] J. Adams et al. [STAR Collaboration], Phys. Rev. C (2005) 064902[8] M. M. Aggarwal et al. [STAR Collaboration], Phys. Rev. C (2011) 034909[9] L. Kumar [STAR Collaboration], EPJ Web Conf. (2015) 00017[10] B. B. Abelev et al. [ALICE Collaboration], Phys. Rev. C (2015) 024609[11] E. Oset and A. Ramos, Eur. Phys. J. A (2010) 445.[12] E. Oset, A. Ramos, E. J. Garzon, R. Molina, L. Tolos,C. W. Xiao, J. J. Wu and B. S. Zou, Int. J. Mod. Phys.E (2012) 1230011.[13] M. Bando, T. Kugo, S. Uehara, K. Yamawaki andT. Yanagida, Phys. Rev. Lett. (1985) 1215.[14] M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. (1988) 217.[15] M. Harada and K. Yamawaki, Phys. Rept. (2003) 1[16] U. G. Meissner, Phys. Rept. (1988) 213.[17] E.L. Bratkovskaya, J. Aichelin, M. Thomere, S. Vogel, M. Bleicher, Phys. Rev. C 87 (2013) 064907[18] W. Cassing, L. Tol´os, E. L. Bratkovskaya, A. Ramos,Nucl. Phys. A727 (2003) 59[19] M. Bleicher and J. Aichelin, Phys. Lett. B (2002) 81[20] S. Vogel, J. Aichelin and M. Bleicher, J. Phys. G (2010) 094046[21] A. G. Knospe, C. Markert, K. Werner, J. Steinheimerand M. Bleicher, Phys. Rev. C (2016) no.1, 014911[22] M. Lutz, Phys. Lett. B (1998) 12[23] A. Ramos and E. Oset, Nucl. Phys. A (2000) 481[24] L. Tolos, A. Ramos, A. Polls and T. T. S. Kuo, Nucl.Phys. A (2001) 547[25] L. Tolos, A. Ramos and E. Oset, Phys. Rev. C (2006)015203[26] M. F. M. Lutz, C. L. Korpa and M. Moller, Nucl. Phys.A (2008) 124[27] L. Tolos, D. Cabrera and A. Ramos, Phys. Rev. C (2008) 045205[28] A. Ilner, D. Cabrera, P. Srisawad and E. Bratkovskaya,Nucl. Phys. A (2014) 249[29] W. Cassing and E. L. Bratkovskaya, Nucl. Phys. A (2009) 215[30] E. L. Bratkovskaya, W. Cassing, V. P. Konchakovski andO. Linnyk, Nucl. Phys. A (2011) 162[31] L. P. Kadanoff and G. Baym, Quantum Statistical Me-chanics , Benjamin, New York, 1962.[32] S. Juchem, W. Cassing and C. Greiner, Nucl. Phys. A (2004) 92[33] W. Cassing, Nucl. Phys. A (2007) 70[34] W. Ehehalt and W. Cassing, Nucl. Phys. A (1996)449. [35] W. Cassing and E. L. Bratkovskaya, Phys. Rept. (1999) 65.[36] B. Nilsson-Almqvist and E. Stenlund, Comp. Phys.Comm. , 387 (1987); B. Andersson, G. Gustafson, andH. Pi, Z. Phys. C , 485 (1993).[37] W. Cassing, Nucl. Phys. A (2007) 365[38] Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz,S. Krieg and K. K. Szabo, JHEP (2009) 088[39] M. Cheng et al. , Phys. Rev. D (2008) 014511[40] B. Andersson, G. Gustafson and H. Pi, Z. Phys. C (1993) 485.[41] W. Cassing, Eur. Phys. J. ST (2009) 3[42] S. Borsanyi et al. , Phys. Rev. D (2015) no.1, 014505[43] L. Tolos, R. Molina, E. Oset and A. Ramos, Phys. Rev.C (2010) 045210[44] D. Gamermann, C. Garcia-Recio, J. Nieves and L. L. Sal-cedo, Phys. Rev. D (2011) 056017[45] A. Ramos, L. Tolos, R. Molina and E. Oset, Eur. Phys.J. A (2013) 148[46] K. A. Brueckner, Phys. Rev. (1955) 1353.[47] J. Beringer et al. [Particle Data Group Collaboration],Phys. Rev. D (2012) 010001.[48] E. Oset and A. Ramos, Nucl. Phys. A (1998) 99[49] N. Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A (1995) 325[50] E. L. Bratkovskaya and W. Cassing, Nucl. Phys. A (2008) 214[51] R. Shahoyan [NA60 Collaboration], Nucl. Phys. A827