arXiv:1609.02778v1 [hep-ph] 9 Sep 2016 § ‡ † ∗ eatn ytmo asespros(atngas), (parton partons massless of system teracting collision. the the after of fm/c impact few initial a Europ´een at pour for Conseil Nucl´eaire (CERN) and the Recherche at la (BNL) Collider Laboratory Hadron Large National the (RHIC) Brookhaven Collider the Heavy-Ion at Relativistic heavy-ion pro- the ultra-relativistic been in at has volume collisions QGP sizeable the a decade in en- in last duced relativistic realized the at in be Indeed, collisions can heavy-ion cre- ergies. nowadays by the and laboratory Universe after the the shortly de- of present - ation Such were matter reached. conditions of is stage extreme (QGP)- the new Quark-Gluon-Plasma that a as implies (i.e. which and noted distances hadron), deconfined a larger are of over partons size freely the than move larger can temperatures partons and colour- densities the a since high to at matter confined whereas are hot state, quarks neutral and the phase dense hadronic of the in properties the informa- about obtain to tion theoretically and experimentally both lcrncades [email protected] address: Electronic [email protected] address: Electronic cabrera@fias.uni-frankfurt.de address: Electronic lcrncades ilner@fias.uni-frankfurt.de address: Electronic nieerypeitosta sue ekyin- weakly a assumed that predictions early Unlike studied are energies high at (HIC) collisions Heavy-ion 5 S emotznrmfu cwroefrcugGb Plan GmbH f¨ur Schwerionenforschung Helmholtzzentrum GSI nrjIlner, Andrej HDcluain iheprmna aafrpp uC and Cu+Cu p+p, ap for data G-matrix experimental coupled-channel with √ calculations self-consistent PHSD a from tained ela nlhdoi neatos eivsiaeterole functio the spectral investigate Breit-Wigner We hadronizat employing dynamical by interactions. dynamics a hadronic meson and final - as tr (QGP) well phase Quark-Gluon-Plasma a t - degrees-of-freedom, (PHSD) ter hadronic Dynamics and partonic Parton-Hadron-String rates microscopic the on xeietly r rdcddrn h aehdoi phas hadronic late the during produced are experimentally) hne,sc httefato fthe of fraction the that such channel, rmtefia bevbe.Teiflec ftei-eimeffe in-medium the of influence The observables. final the from pcr.Ti moe eeecntanso h interpreta numbers: PACS the on constraints severe imposes This spectra. u nteivratms eino the of region mass invariant ene the beam a on decreasing production cut with dominant increases their it to however, due densities), modest rather is energies s esuytesrnevco eo ( meson vector strange the study We NN .INTRODUCTION I. 3 2 K nttt eFıiaCruclr(FC,Cnr it Univ Mixto Centro (IFIC), F´ısica Corpuscular de Instituto ntttsd netgc´nd aen,A.Cres22085 Correos Ap. Paterna, Investigaci´on de de Institutos rnfr nttt o dacdSuis(IS,648Fra 60438 (FIAS), Studies Advanced for Institute Frankfurt 0 e.Oraayi hw hta eaiitceege m energies relativistic at that shows analysis Our GeV. 200 = ∗ 4 etrmsnrsnne yaisi ev-o collisions heavy-ion in dynamics resonances meson vector h nvriyo ea tAsi,PyisDprmn,Aust Department, Physics Austin, at Texas of University The 1 ,2, 1, ntttfu hoeicePyi,Jhn ofagGoethe Wolfgang Johann Physik, f¨ur Theoretische Institut ∗ r nfr mMi,648Fakuta an Germany Main, am Frankfurt 60438 Main, am Frankfurt ailCabrera, Daniel ,2, 3, K K K ∗ † rmteQPi ml n a adyb reconstructed be hardly can and small is QGP the from s ∗ ∗ , hitn Markert, Christina K ute nune h hp n h egto h final the of height the and shape the influences further ¯ ∗ yaisi eaiitchayincliin based collisions heavy-ion relativistic in dynamics ) oacs experimentally. easier other com- access and the abundant to quite more On other are many is observables [2]. hadronic origin among it hand, hadronic signal and of QGP channels rare the production very identify heavy- to are of stages plicated they all collisions, to directly ion electromagnetic penetrate that leptons to spite allow hadrons, partonic In probes final measured. initial the are only photons the experiment and probes the about for In looks study information stage. one to matter provide order ap- QGP In which behaves such in- of liquid. and strongly properties ’ideal’ phase the an more hadronic to is close the that proximately in clear medium than shown new have (perturbative teracting [1] a RHIC of pQCD at signatures collisions observa- by Au+Au of described tions experimental be Quantum-Chromo-Dynamics), might which etrmsnresonances vector-meson ohdoiain codnl,dtie tde fthe of studies detailed close Accordingly, stage K QGP the hadronization. and, about [6]) to information earlier provide even to (or freeze-out thus, rela- partonic At the 5]. at [4, duced medium dense the and energies hot tivistic the for probes tive aoaina HCi h atdcd 71] ic the Since [7–10]. decade last vector-mesons the strange in RHIC at laboration neatoswt te eosadbrosi h ex- the in baryons and state mesons final other from suffer with kaons interactions and pions the since mentally n eoacs oiatydcyn oposadkaons and pions to ( decaying dominantly resonances, ing K ∗ pr rmhayhdoswt cam 3 lostrange also [3] ’charm’ with hadrons heavy from Apart ∗ eoacshv enpromdb h TRCol- STAR the by performed been have resonances → g.Mroe,w n htteadditional the that find we Moreover, rgy. ino h xeietlresults. experimental the of tion niinfo arnct atncmat- partonic to hadronic from ansition rah utemr,w ofotthe confront we Furthermore, proach. K o ayndniis(u ihmeson high (but densities baryon low t ktas ,621Drsat Germany Darmstadt, 64291 1, ckstrasse sfrthe for ns ,dmnnl ythe by dominantly e, fi-eimeet nthe on effects in-medium of + 4, asotapoc hc incorpo- which approach ransport uA olsosa nrisu to up energies at collisions Au+Au t nthe on cts π ria eVlni CSIC, - Valencia de ersidad ‡ -67 aeca Spain. Valencia, E-46071 , o fqak n niursas antiquarks and quarks of ion ,i sdffiutt eosrc hmexperi- them reconstruct to difficult is it ), kuta an Germany Main, am nkfurt n ln Bratkovskaya Elena and s ftefinal the of ost K K n ea,USA Texas, in, -Universit¨at ∗ ∗ swt efeege ob- self-energies with ’s K r eeal xetdt epro- be to expected generally are s K ∗ K ∗ yaisa RHIC at dynamics , ∗ K , ¯ K ¯ ∗ K K r eaieysotliv- short relatively are s ∗ r osdrda sensi- as considered are s ∗ + (observed s π K → ∗ 1,5, , K K ¯ ∗ ∗ § 2 panding hadronic phase, i.e. the latter can rescatter or be off-shell behaviour of the kaons K and antikaons K¯ at fi- absorbed. Moreover, the recreation of the K∗ by fusion nite nuclear baryon density in Refs. [22–27]. Previously, of a kaon and pion (K + π K∗) might ’over-shine’ the we have evaluated within this framework the properties of K∗ signal from the QGP and→ lead to additional compli- the strange vector resonances as a function of the nuclear cations. As known from chiral models or coupled-channel baryon density [28] and now incorporate the off-shell be- G-matrix approaches [11–16], the K∗ resonances change haviour of the K∗s in form of a relativistic Breit-Wigner their properties in a hot and dense hadronic medium and spectral function into PHSD. We recall that the PHSD also the properties of the ’daughter’ kaons/antikaons are transport approach incorporates the hadronic and par- affected by hadronic in-medium modifications [17]. All tonic degrees-of-freedom and their interactions, dynam- these effects should be accounted for when addressing or ical hadronization and further off-shell dynamics in the modelling the K∗ production in relativistic HICs. hadronic stage which allows to include in a consistent In order to interpret the experimental results [7–10] it way in-medium effects for hadrons as well as partons. is mandatory to study the dynamics of the K∗s within Throughout this paper we will follow the following con- a proper theoretical framework. Statistical or hydrody- vention: when addressing strange vector mesons consist- namic models can be used to obtain information on the ing of an anti-strange quark, i.e. K∗+ = (us¯) and K∗0 = bulk properties of the medium, however, to properly in- (ds¯) we will refer them as K∗ = (K∗+,K∗0); while for vestigate the interactions of the K∗ with the hadronic mesons with an anti-strange quark, i.e. K∗− = (¯us) and medium one needs a non-equilibrium transport approach K¯ ∗0 = (ds¯ ) we will use notation as K¯ ∗ = (K∗−, K¯ ∗0). to study the influence of the medium on the K∗ and vice This paper is organized as follows: In section II we versa. Furthermore, one needs a consistent theory for the shortly recall the PHSD transport approach and its im- in-medium effects of the K∗ as a function of the nuclear plementations beyond the cascade level. In section III baryon density in order to trace the dynamics of the K∗ we discuss the K∗ vector-meson resonance in-medium ef- during the later stages of the collision in the expansion of fects; the first part describes the calculation of the in- the hadronic fireball in close analogy to the kaon and an- medium effects using the G-matrix model while the sec- tikaon dynamics in heavy-ion reactions at lower energies ond part contains the implementation of these effects into [18]. the PHSD. In section IV we then investigate the proper- ∗ Previously there have been several studies on the ties and the dynamics of the K vector-meson resonances strange vector-meson resonances K∗ within different in the PHSD for Au+Au collisions at the top RHIC en- transport models at relativistic energies. In Refs. [19, 20] ergy, analyze the different production channels, the ac- detailed calculations for Pb+Pb at s = 17.3 GeV tual baryon densities explored as well as the in-medium √ NN ∗ have been performed using the cascade model UrQMD effects of the K spectral functions. In section V we and it was found that the observability of strange parti- present our results in comparison to experimental data; cle resonances is distorted due to the rescattering of the the first part shows a comparison for p+p collisions while strongly interacting decay products. Furthermore, the the second part contains a detailed confrontation with ex- rescattering rate changes with rapidity and transverse perimental data from the STAR Collaboration for A+A collisions including in particular the effect of acceptance momentum pT of the reconstructed resonances, which leads to higher apparent temperatures for these reso- cuts. Finally, we give a summary of our findings in sec- tion VI. nances due to a sizeable depletion at low pT which is most pronounced at midrapidity. Since the rescattering in- creases with baryon density, most resonances that can be reconstructed come from the low nuclear baryon density II. REMINDER OF PHSD region. The authors of Ref. [21] used EPOS3 (option- ally with UrQMD as an afterburner) to investigate the The Parton-Hadron-String Dynamics (PHSD) is a non- hadronic resonance production and interaction in Pb+Pb equilibrium microscopic transport approach [29, 30] that collisions at √sNN = 2.76 TeV and compared their re- incorporates hadronic as well as partonic degrees-of- sults to data from the ALICE Collaboration. Their cal- freedom. It solves generalised (off-shell) transport equa- culations reproduce the data very well for the different tions on the basis of the off-shell Kadanoff-Baym equa- centralities. The simulation also showed that a large part tions [31–33] in first-order gradient expansion. Further- of the resonances at low pT cannot be reconstructed due more, a covariant dynamical transition between the par- to the final state interactions of their decay daughters. tonic and hadronic degrees-of-freddom is employed that Our goal is to investigate the dynamics of K∗ vector- increases the entropy in consistency with the second law meson resonances using the Parton-Hadron-String Dy- of thermodynamics. The hadronic part part is equivalent namics transport approach [2] employing the in-medium to the HSD transport approach [34, 35] which includes effects of the K∗s from the self-consistent coupled- the baryon octet and decouplet, the 0− and 1− meson channel unitary G-Matrix approach for the production of nonets and higher resonances. When the mass of the K∗s from the hadronic channels as well as from a QGP hadrons exceeds a certain value (1.5 GeV for baryons and by microscopic dynamical hadronization. The in-medium 1.3 GeV for mesons) the hadrons are treated as strings effects have been already successfully used to model the (or continuum excitations) that decay to hadrons within 3 a formation time of 0.8 fm/c using the LUND string calculate the self-energy of the K∗ and K¯ ∗ which we decay [36]. In PHSD∼ the partonic, or the QGP phase, is then implement in the form of relativistic Breit-Wigner based on the Dynamical Quasi-Particle Model (DQPM) spectral functions into PHSD to model the in-medium ef- [33, 37] which describes the properties of QCD (in equilib- fects and off-shell propagation of the K∗ and K¯ ∗ within rium) in terms of resummed single-particle Green’s func- a transport approach. tions. Instead of massless partons the gluons and quarks in PHSD are massive strongly-interacting quasi-particles whose masses are distributed according to spectral func- A. G-Matrix approach tions (imaginary parts of the complex propagators). The widths and pole positions of the spectral functions are In recent years some efforts have been undertaken to defined by the real and imaginary parts of the parton assess the interaction of vector mesons with baryons in self-energies and the effective coupling strength in the coupled-channel effective field approaches. In Refs. [11, DQPM is fixed by fitting respective lQCD results from 12] vector mesons are introduced within the Hidden Lo- Refs. [38, 39] (using in total three parameters). cal Symmetry approach [13–16] and a tree-level vector- In the beginning of the nucleus-nucleus collision the meson baryon s-wave scattering amplitude is derived as LUND string model [40] is used to create colour neu- the low-energy limit of a vector-meson exchange mecha- tral strings from the initial hard nucleon scatterings, i.e. nism. In Ref. [44] the interaction of vector mesons with the formation of two strings takes place through pri- baryons is obtained on the basis of a SU(6) spin-flavor mary NN collisions. These early strings dissolve into symmetry extension of standard SU(3) meson-baryon massive coloured quarks and anti-quarks in their self- chiral perturbation theory, leading to a generalized (s- generated mean-field as described by the DQPM [41] if wave) Weinberg-Tomozawa interaction between the pseu- the energy density is above the critical energy density doscalar/vector meson octets and the octet and decuplet = 0.5 GeV/fm−3 in line with lQCD [42]. If the en- E of baryons. These two models share a crucial feature, ergy density is below critical the strings decay to pre- namely, several N ∗ and hyperon resonances are dynami- hadrons (as in case of p+p reactions or in the hadronic cally generated in a broad range of energies upon unita- corona). The QGP phase is then evolved by the off-shell rization of the leading order (LO), tree-level amplitudes. transport equations with self-energies and cross sections Such states are indicative of non-trivial meson-baryon dy- from the DQPM. When the fireball expands the proba- namics and one may expect an impact on the in-medium bility of the partons for hadronization increases close to properties of strange vector mesons. Incidentally, the the phase boundary (crossover at all RHIC energies), the properties of vector mesons within these approaches were hadronisation takes place using covariant transition rates initially investigated for the case of the K¯ ∗(892) [43]. and the resulting hadronic system is further on governed The self-energy of the ω meson was also updated recently by the off-shell HSD dynamics incorporating (optionally) along the same ideas in [45]. self-energies for the hadronic degrees-of-freedom [18]. The absence of baryonic resonances with S = +1 close Thus in the PHSD approach the full evolution of a to threshold induces milder nuclear medium effects in the relativistic heavy-ion collision, from the initial hard NN properties of the K meson [23, 27] as compared to the collisions out of equilibrium up to the hadronisation and K¯ meson, whose behaviour is largely dominated by the final interactions of the resulting hadronic particles, is Λ(1405) resonance appearing in s-wave KN¯ scattering. described on the same footing. We recall that this ap- A similar situation takes place for the vector partner of proach has been sucessfully employed for p+p, p+A and the K, the K∗ meson. Note that the K∗ decays predom- A+A reactions from about √sNN = 8 GeV to 2.76 TeV inantly into Kπ, and therefore not only collisional effects (see the review [2]). but its in-medium decay width has to be taken into ac- In this study we will concentrate on the strange res- count (we come back to this point below). However, since onance dynamics mainly at RHIC energies where the the K meson itself is barely influenced by nuclear matter PHSD describes well the bulk observables, i.e. rapidity, one can anticipate small changes in ΓK∗, dec as compared pT -spectra, vn coefficients etc. [2]. to the vacuum case. An effective Lagrangian approach, as developed in [11], was used in our previous study [28] to calculate the K∗ ∗ III. K IN MEDIUM meson self-energy at threshold energy. In this approach, the interaction between the octet of light vector mesons Before coming to actual results from PHSD for p+p and the octet of J P = 1/2+ baryons is built within the and A+A collisions at relativistic energies we describe in hidden local gauge symmetry (HLS) formalism, which al- some detail the evaluation of the kaon, antikaon and K∗ lows to incorporate vector meson interactions with pseu- selfenergies as a function of baryon density and their im- doscalar mesons respecting the chiral dynamics of the plementation in the PHSD. The medium properties and pseudoscalar meson-meson sector. The interactions of the off-shell propagation of the strange vector-meson res- vector mesons with baryons are assumed to be domi- onances K∗ and K¯ ∗ are based on the “G-Matrix” ap- nated by vector-meson exchange diagrams, which inci- proach from Refs. [28, 43]. We use this approach to dentally allows for an interpretation of chiral Lagrangians 4 in the meson-baryon sector as the low-energy limit of In the case of the K¯ ∗, the collisional part of the selfen- vector-exchange mechanisms naturally occurring in the ergy, related to the quasi-elastic reaction K¯ ∗N K¯ ∗N theory. Vector-meson baryon amplitudes emerge in this and accounting for absorption channels, induces→ a strong scheme at leading order, in complete analogy to the broadening of the K¯ ∗ spectral function as a result of the pseudoscalar-meson baryon case, via the self-interactions mixing with two J P = 1/2− states, the Λ(1783) and of vector-meson fields in the HLS approach. At small Σ(1830), which are dynamically generated in the hidden momentum transfer (i.e., neglecting corrections of order local symmetry approach in a parallel way to the KN¯ p/M, with M the baryon mass), the LO s-wave scatter- interaction and the Λ(1405). Such complicated many- ing amplitudes have the same analytical structure as the body structure of the K¯ ∗N interaction requires a detailed ones in the pseudoscalar-meson baryon sector (Weinberg- analysis of medium corrections such as Pauli blocking on Tomozawa interaction), namely baryons and a self-consistent evaluation of the in-medium K¯ ∗N scattering amplitude (G-matrix) and the K¯ ∗ self- 1 energy, as was done in [43]. For the present study we Vij = Cij (2√s MB MB ) recourse to a suitable parameterization of the resulting − 4f 2 − i − j × K¯ ∗ self-energy and spectral function, which we discuss in 1/2 1/2 M + E MBj + Ej more detail below. Bi i ~ε ~ε ′ As mentioned before, the decay with of the K∗/K¯ ∗ me- × 2MBi 2MBj ·     son (with suitable medium corrections) has to be taken 1 0 ′0 ′ Cij (q + q ) ~ε ~ε , (1) into account to realistically assess production and anihi- ≃− 4f 2 · lation rates. Such effects are readily incorporated for the ∗ where q0 (q′0) stands for the energy of the incom- K¯ within the G-matrix approach which we parametrize ′ ∗ ing (outgoing) vector meson with polarisation ~ε (~ε ), Cij from Ref. [43]. For the K , instead, we evaluate explicitly its medium-modified K∗ Kπ width as follows [28], stand for channel-dependent symmetry coefficients [11], → and the latin indices label a specific vector-meson baryon ∗+ 2 µ−mπ 3 (VB) channel, e.g., K p. µ q (µ,M)Aj (M,ρ) dM ∗ 0 0 0 ΓV, dec(µ,ρ)=Γ , The K collisional self-energy follows from summing V µ0−mπ 3 ∗ µ q (µ0,M)Aj (M, 0) dM the forward K N scattering amplitude over the allowed   RMmin coll (3) nucleon states in the medium, schematically ΠK∗ = R ∗ ∗ ~p n(~p )TK N . Due to the absence of resonant states where j = K and V = K for the present nearby, a tρ approximation is well justified at energies case, q(µ,M) = λ(µ,M,M )/2µ and λ(x,y,z) = P π sufficiently close to threshold, which leads to the practi- x2 (y + z)2 x2 (y z)2 . Γ0 stands for the vec- p V cal result (take T = V here) tor− meson (vacuum)− partial− decay width in the consid-    ered channel and µ0 is the nominal resonance mass, par- 0 coll 1 ρ ticularly ΓK∗,K¯ ∗ = 42 MeV and µ0 = 892 MeV [47]. Π ∗ = V ∗+ + V ∗+ ρ K 2 K p K n 0 ρ Eq. (3) accounts for the in-medium modification of the  0 
resonance width by its decay products. In particular, we MK 2 ρ α MK∗ , (2) consider the fact that kaons and anti-kaons may acquire ≃ MK∗ ρ0   a broad spectral function in the medium, Aj (M,ρ). As with α =0.22, leading to a positive mass shift (equiv- discussed in [28], AK in Eq. (3) is a delta function in alent to a repulsive optical potential) of about δMK∗ vacuum since the kaon is stable in vacuum with respect −≃3 50 MeV at normal matter density ρ = ρ0 = 0.17 fm to the strong interaction, and to a good approximation (recall δMK∗ ReΠK∗ /2MK∗ ). Replacing the lowest the same can be kept at finite nuclear density by us- ≃ ∗ 2 2 order tree level amplitudes V in the former result by ing an effective kaon mass MK (ρ)= MK + ΠK (ρ) with Π (ρ) 0.13M 2 (ρ/ρ ) [27, 48, 49]. In general this may unitarized amplitudes in coupled channels (solving the K ≃ K 0 Bethe-Salpeter equation, T = V + V GT ) one finds a re- not be the case for other vector mesons even in the vac- uum case, e.g., a ρπ, where the ρ meson has a large duction by roughly one third over the previous result, 1 → namely δMK∗ (ρ0) 30 MeV [i.e. α 0.13 in Eq. (2)]. width into two pions (Mmin then stands for the corre- Further medium corrections≃ on the≃ vacuum scattering sponding threshold energy). amplitudes T ∗ , leading to a full G-matrix calcula- Once both collisional and decay self-energies are ob- K N ∗ ¯ ∗ tion, are of marginal relevance in the present case due to tained, the K and K spectral function is readily given the mild corrections introduced by the smoothly energy- as the imaginary part of the vector-meson in-medium dependent K∗N interaction1. propagator, namely, 1 S (ω, ~q; ρ)= ImD (ω, ~q; ρ) V −π V 1 Im ΠV (ω, ~q; ρ) 1 G To be precise, we use the term -matrix for the in-medium effec- = 2 2 . T −π [ω2 ~q 2 M 2 Re Π (ω, ~q; ρ)] + [ImΠ (ω, ~q; ρ)] tive meson-baryon -matrix obtained in DiracBrueckner theory. − − V − V V [46] (4) 5

∗ ∗ with V = K , K¯ , where ΠV contains the sum of the collisional and decay self-energies. 15

+

K* B. Implementation in PHSD

10 ∗ ∗ ¯ / =0.0 In order to implement K and K in-medium proper- 0

ties in PHSD we adopt a relativistic Breit-Wigner pre- / =0.5

0 scription for the strange vector meson spectral functions ) [1/GeV]

/ =1.0 ∗ ¯ ∗ 0 (V = K , K ) [28, 50], 5

2 ∗ A(M, 2 M ΓV (M,ρ) AV (M,ρ)= C1 , π 2 ∗ 2 2 ∗ 2 M MV (ρ) + (MΓV (M,ρ)) − (5) 0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 where M is the invariant mass and C1 is a normalisation constant ensuring that the sum rule M [GeV]

∞ A (M,ρ) dM = 1 (6) FIG. 1: The relativistic Breit-Wigner spectral function V ∗ Z0 A(M,ρ) of the K is shown as a function of the invariant mass M for different nuclear densities. The solid red line cor- is fulfilled. The connection with the spectral function in responds to the vacuum spectral function, whereas the dashed Eq. (4) can be done by setting the vector-meson momen- green and the dash-dotted blue lines correspond to densities tum at zero, ρ/ρ0 = 0.5, 1.0, respectively.

A (M,ρ)=2 C M S (M,~0,ρ), (7) V · 1 · · V a practical approximation which does not account for 15 explicit energy and momentum dependence of medium corrections (this limitation is consistently dealt with by - considering vector mesons to be at rest in the nuclear K* matter frame when evaluating their self-energy). The in- 10

∗ ∗ ∗ ∗ / =0.0 ¯ 0 medium mass MV and decay width ΓV of the K /K are derived from the vector meson self-energy, as in the case / =0.5 0

of partons in the DQPM, ) [1/GeV] / =1.0

0

∗ 2 2 ∗ 5 (MV ) = MV + Re ΠV (MV ,ρ) , A(M, Γ∗ (M,ρ) = Im Π (M,ρ)/M , (8) V − V where MV denotes the nominal (pole) mass of the reso- 0

nance in vacuum. 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

We briefly comment in the following the essential fea- M [GeV] tures of the K∗ and K¯ ∗ Breit-Wigner spectral functions when including medium effects along the self-energy cal- ¯ ∗ culation in the previous section. In Fig. 1 we depict FIG. 2: Same as Fig. 1 for the K spectral function. the spectral function for the K∗ meson. The K∗ ex- periences a net repulsive interaction with the medium which leads to a shift of the spectral function’s peak at different nuclear densities. The distribution is consid- to higher invariant masses with increasing nuclear den- erably shifted to lower invariant masses when the density sity. Overall, the width of the K∗ becomes slightly is increased, reflecting the overall attractive interaction smaller with increasing density, due to the kaon becom- of the K¯ ∗ with the baryon-rich medium. Furthermore, ing also heavier (this effect is largely compensated by the the K¯ ∗ width is largely enhanced as a consequence of higher K∗ excitation energy) Furthermore, the thresh- the multiple absorption channels that are accounted for old energy for the creation of a K∗ is shifted up, i.e. in the K¯ ∗ self-energy, involving the mixing of the quasi- −1 −1 Mth = MK + Mπ + ∆M(ρ) 0.633GeV + ∆M(ρ), with particle mode with ΛN and ΣN excitations. Conse- ∆M(ρ) Π (ρ)/2M , which≈ amounts to 0.06 M at quently, the threshold energy for the creation of a K¯ ∗ is ≃ K K K normal matter density. considerably diminished, almost down to Mth 2Mπ, ∗ ∼ Interestingly, in case of the vector antikaon K¯ ∗ the ef- which implies that an off-shell K¯ can be created at fects from the medium are rather different as compared rather low invariant masses. to the K∗. Fig. 2 shows the spectral function for the K¯ ∗ The medium corrections discussed before have an 6 impact on the K∗ and K¯ ∗ production rates in the hadronic phase of a heavy-ion collision. The produc- 200 tion/anihilation cross section in PHSD is consistently

180 modified according to -

K* 160

/ =0.0 140 2 ∗ ∗ ∗ 0 6π AK (K¯ )(M,ρ) ΓK∗(K¯ ∗)(M,ρ)

120 σ ∗ ¯ ∗ (M,ρ)= . / =0.5 K (K ) 2 0 q(M,MK ,Mπ) 100

/ =1.0 (9) 0 [mb]

80

Fig. 3 shows the cross-section for K∗ production from 60 Kπ¯ scattering. The evolution of the cross section with 40 ∗ the nuclear density reflects that of the K spectral func- 20

tion, leading to a shift of the energy distribution to higher 0 ∗ invariant masses. The presence of the Γ factor in Eq. 9, 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 ∗ accounting for the effective K¯ Kπ¯ coupling and multi- M [GeV] plying the spectral function, makes the maximum value of the cross section practically unchanged when varying FIG. 4: Same as in Fig. 3 for the K¯ ∗ production/anihilation the density, reaching a value as large as 160-170 mb. The cross section. observed shift of the cross-section implies that, in order to create K∗ at finite densities, larger energies are needed as compared to the same reaction in vacuum. IV. K∗ DYNAMICS IN PHSD

Before we come to the physical observables that can be compared to experimental data we investigate the 200 on/off-shell dynamics of the K∗ vector mesons within 180 the PHSD transport approach for central Au+Au colli- +

160 K* sions at √sNN = 200 GeV. We will investigate the main ∗ 140 production channels of the K mesons, their production in time and record the baryon density at production. In 120 / =0.0 0 order to better understand the experimental results in

100

/ =0.5 0 the next section we also investigate the change of the [mb]

80 / =1.0 transverse momentum spectrum with respect to the ex- 0

60 perimental cuts imposed for the reconstruction the K∗

40 vector mesons.

20

0 A. Production channels and dependence on baryon 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 density

M [GeV]

∗ We start with the production of the strange vector FIG. 3: The cross-section σ for K production/anihilation is mesons as a function of time for the different production shown as a function of the invariant mass M for different nu- channels in PHSD. clear densities. The solid red line corresponds to the vacuum case, whereas the dashed green and the dash-dotted blue lines As can be seen from Fig. 5 there is no sizeable differ- ence between the number of vector kaons K∗ and and vec- correspond to densities ρ/ρ0 = 0.5, 1.0, respectively. tor antikaons K¯ ∗s from PHSD. The production by strings in the hadronic corona occurs early and gives practically Quite different is the behavior of the K¯ ∗ cross section, no further contribution in the late stages. Furthermore, as can be seen in Fig. 4. The energy dependence reflects one can see that the contribution from the QGP is not the attractive nature of the K¯ ∗ interaction with the nu- very large and starts a few fm/c later in the hadroniza- clear medium, as follows from the K¯ ∗ spectral function, tion, when compared to the late K + π channel (blue the peak of the distribution being shifted to lower in- lines). In fact, the QGP contribution is on the same variant masses in the same magnitude as the density is level as the contribution from strings for this system. increased. The fall in the maximum of the cross section, For times larger than 150 fm/c (not shown here) also the though, seems to saturate due to the large increase of the K + π channel decreases rapidly and all vector mesons K¯ ∗ width. At nuclear matter density, the cross section simply decay. reaches a maximum value around 100 mb at invariant With respect to in-medium modifications of the vector masses around 120 MeV below the vacuum case. mesons the baryon density at the production point is of 7

1/2

1/2

(s ) = 200 GeV, Au+Au, all y (s ) = 200 GeV, Au+Au, all y

NN

NN

60 10

+ -

50

K* K* 200 GeV

40

10 GeV

30 N(t)

1 20 tot

10 )/N /

600

all

0

0

50 K*

BB anti-K*

0.1 dN/d(

40 m B

K+ 30 N(t)

QGP

20

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

0

/

1 10 100 1 10 100

t [fm/c] t [fm/c] FIG. 6: The differential distribution of the total number of K∗s 1 dN versus baryon density ρ for Au+Au collisions Ntot d ρ ρ0 FIG. 5: The number of strange vector mesons N(t) is shown  ρ0  as a function of time t for all four isospin channels of the at different cms energies from the PHSD simulations. The K∗ mesons and for all production channels in central Au+Au solid red line shows results for a collision at √sNN = 200 GeV while the dashed blue line shows results for a collision at collisions at √sNN = 200 GeV (all y) from a PHSD calcula- tion. The upper left panel shows the channel decomposition √sNN = 10 GeV. for the K∗+, the upper right panel shows the channel decom- position for the K∗−, the lower left panel shows the channel decomposition of the K∗0 and the lower right panel shows ∗0 ¯ 1/2 the channel decomposition of the K . The legend for all (s ) = 200 GeV, Au+Au, all y

NN four panels is as follows: the solid black line shows all pro- 4 10 duced K∗s, the dash dotted green line shows K∗s produced from baryon-baryon strings, the solid dash double-dotted line ∗ K+

shows K s produced from meson-baryon strings, the dashed 3 light blue line shows K∗s produced from K + π annihilation 10 Strings ∗ and the short dotted red line shows K produced during the all hadronisation of the QGP. )

2 /

10 dN/d(

additional interest. Fig. 6 shows the normalised num- 1 ber of K∗s for two collision energies as a function of the 10 baryon density ρ for central Au+Au collisions. From ρ0 ∗ the previous figures we know that most of the K s come 0 10 from the K + π channels in the later stage. Accordingly, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 most of the K∗s are created while the baryon density is / fairly low, with only very few mesons created above half normal nuclear baryon density ρ0 during a Au+Au col- FIG. 7: The differential distribution dN as a function of lision with a cms energy of 200 GeV at RHIC. Thus the d ρ  ρ0  in-medium modifications of the strange vector mesons are the baryon density ρ for Au+Au collisions at a cms energy of ρ0 ∗ expected to be small at this energy as well as observable √sNN = 200 GeV for different production channels of the K . consequences in the final spectra. Note, however, that The solid black line shows the K∗s coming from all production for a much lower collisional energy of 10 GeV the baryon channels, the dashed orange line shows K∗s coming from the densities are much higher and the perspectives to see in- K + π channel and the dash dotted green line shows K∗s medium modifications of the vector mesons become bet- coming from meson-baryon and baryon-baryon strings. ter. The situation is not very different from the dilepton measurements in heavy-ion reactions with respect to the contribution from the ρ-meson decay [2]. Furthermore, one can see in Fig. 7 the channel decom- all baryon densities, since, as opposed to the other chan- position of the K∗ as a function of the baryon density (in nels, K∗s coming from the K + π channel are created units of ρ0). As before the dominant channel is the K +π throughout the whole collision history and can thus oc- channel. It is important to note that this is the case for cur at low and higher baryon densities, respectively. 8

B. Modifications of the K∗ mass distributions in baryon and baryon-baryon strings is practically negligi- the medium ble, however, there is still a sizeable contribution com- ing from the QGP. The shape of the spectral function for vector kaons suggests that there should be some K∗s

1/2 produced at higher invariant masses due to finite baryon

(s ) = 200 GeV, Au+Au, 5% central, all y

NN

4 densities where the pole mass is shifted up. However,

10 the high baryon density region is not much populated at All

BB-string this cms energy as demonstrated above such that these

mB-string effects will be hard to disentangle in the final spectra in comparison to experiment. K+

3

10

QGP As seen from the lower part (”b”) of Fig. 8 the mass

Reconstructed distribution of the vector antikaons is slightly shifted to the low masses which stems from the shift of the spectral + 0 ¯ ∗ K* + K* function of the K to lower masses at finite baryon densi-

2

dN/dM [1/GeV] 10 ties. This means that in-medium effects are still present since the mesons are created at nonzero densities. The contribution of the strings and the QGP does not play (a) a significant role due to the lack of baryons relative to

0.6 0.8 1.0 1.2 1.4 1.6 mesons at the top RHIC energy.

M [GeV] In both of these figures the distribution for the recon- structed particles is shifted from higher to lower invariant

1/2

(s ) = 200 GeV, Au+Au, 5% central, all y mass regions. Furthermore the total number of particles NN

4 10 is reduced to about 40 to 50 % in the reconstructed distri-

All bution as opposed to the particles that were taken from

BB-string the decay point.

mB-string

K+

3

10 QGP C. Transverse momentum distributions and Reconstructed acceptance cuts

2

- 0

dN/dM [1/GeV] 10

K* + anti-K* 1/2

(s ) = 200 GeV, Au+Au, 5% central, |y|<0.5

NN

(b)

1

0 0 10

K* +anti-K*

0.6 0.8 1.0 1.2 1.4 1.6 ] 2

0 0

M [GeV] Decayed K* +anti-K* /GeV

0 2 dN 10 FIG. 8: The differential mass distribution dM for the vector ∗ ¯ ∗ kaons K (”a”, upper part) and antikaons K (”b”, lower dy) [c part) for different production channels as a function of the T dp T

-1

invariant mass M in a Au+Au collision at √sNN = 200 GeV p 10 from a PHSD calculation. The solid black lines show all pro- Reconstructed ∗ ∗ duced K (K¯ )s, the dash dotted green lines indicate produc- no cuts N/(2 2

K* cuts tion from baryon-baryon strings, the dash double-dotted blue d lines – from meson-baryon strings, the dashed light blue lines -2 ¯ 10 – from K + π (K + π) annihilation and the short dotted red 0.5 1.0 1.5 2.0 2.5 3.0 line correspond to the production during the hadronisation of p [GeV/c] the QGP. The dashed orange line with the circles shows the T distribution for K∗s that have been reconstructed from final kaon and pion pairs. FIG. 9: The transverse momentum spectrum 2 d N/(2πpT dpT dy) versus the transverse momentum pT for vector kaons and antikaons in Au+Au collisions at As can be seen in Fig. 8 again the dominant produc- ∗ √sNN = 200 GeV from the PHSD simulation. The dashed tion channel of the vector kaons K (”a”, upper part) and black line shows the spectrum directly at the point when ∗ vector antikaons K¯ (”b”, lower part) is the annihilation the K¯ ∗ decays in the PHSD simulation. The dotted red and of the K +π (K¯ +π) pairs. Due to its broad structure the solid blue lines show the spectrum after the K¯ ∗ has been spectral function of the K∗ and K¯ ∗ allows for the anni- reconstructed from the final kaons and pions. The dotted hilation of K +π pairs also at lower and higher masses as red line does not include any cuts while the solid blue line compared to the vacuum. The contribution from meson- includes cuts on the invariant mass of the K¯ ∗. 9

Before we compare our PHSD results with experimen- 1/2 (s ) = 200 GeV, Au+Au, 5% central, all y

NN tal data, we have to establish the restrictions that arise 3 10 during the reconstruction of the K∗ mesons from final

+ 0

K* + K* K + π pairs on the observables. In Fig. 9 we show 2

] 10 PHSD results for K∗0 and K¯ ∗0 at midrapidity for cen- 2 tral Au+Au collisions at the top RHIC energy. We re-

1 /GeV

10 call that in PHSD we can study the K∗s at their decay 2 point, i.e. at the point in space-time of their decay into

0 dy) [c ∗ 10 K + π pairs. Furthermore, we can reconstruct the K s T

All dp

from their daughter particles, the K and π pairs that T BB-string p -1

10 are affected by final state interactions. In Fig. 9 one mB-string can see that there is a slight difference between the de- K+ N/(2 2 ∗ -2 QGP

10 cayed K s, shown by the dashed black line, and the re- d constructed K∗s, shown by the red line. This difference (a) is due to rescattering and absorption of the final kaons 0.0 0.5 1.0 1.5 2.0 2.5

and pions in the medium. Particles with low transverse p [GeV/c]

T momentum are more affected by this than particles with

1/2 higher transverse momentum, and these two lines merge (s ) = 200 GeV, Au+Au, 5% central, all y

NN

∗ 3 with increasing pT . This implies that fast K s can be 10 reconstructed much more efficiently since their daughter

- 0

K* + anti-K* 2

particles do not interact with the medium as much as ] 10 2 particles with lower pT which rescatter more often due

to their low velocities in the hadronic medium. 1 /GeV

10 Fig. 9 also shows the K∗s that have been recon- 2

structed from final particles (blue line) employing the 0 + 0 dy) [c 10 K* + K* restrictions that have been imposed on the K∗ in the T

All dp form of a restriction to a specific invariant mass region T BB-string p -1

10 of M = [0.8, 1.0] GeV. This is due to the low signal to mB-string ∗ K+

noise ratio during the reconstruction of the K . While N/(2 2 -2

QGP ∗ 10 the spectral function of the K is relatively broad, with d a width of about 42 MeV in vacuum, the experimental (b) signal is very narrow and it is difficult to distinguish cor- 0.0 0.5 1.0 1.5 2.0 2.5

p [GeV/c] related from uncorrelated K + π pairs below and above T the selected invariant mass region. It is also important to note the different production FIG. 10: The channel decomposition of the transverse mo- ∗ 2 ∗ channels of the K in actual observables, e.g. the mentum spectrum d N/2πpT dpT dy for K (upper part) and transverse momentum spectrum where different channels for the K¯ ∗ (lower part) for different productions channels might reflect different spectral slopes. This is shown in in Au+Au collisions at a cms energy of √sNN = 200 GeV Figs. 10 for vector kaons K∗ (upper part)and vector an- from a PHSD calculation: the solid black lines show all pro- ∗ ¯ ∗ tikaons K¯ ∗ (lower part), respectively. Similarly to the duced K (K )s, the dash dotted green lines indicate produc- figures in Fig. 8 one can see that K∗s coming from tion from baryon-baryon strings, the dash double-dotted blue meson-baryon and baryon-baryon strings contribute only lines – from meson-baryon strings, the dashed light blue lines – from K + π (K¯ + π) annihilation and the short dotted red a small part to the overall spectrum. The far dominant lines correspond to the production during the hadronisation channel contribution comes from K + π annihilation in of the QGP. the final hadronic stage. The second-largest contribution comes from the QGP, however, it is smaller by about an ∗ order of magnitude for all pT which makes the K less in fig. 5 different channels dominate the K∗ production suitable as a probe for the QGP. Furthermore, the spec- during certain times of a heavy-ion collision. Thus one tral slopes from all channels are very similar in the high could deduce from what source the reconstructed Kπ pair ∗ ¯ ∗ pT region. This is true for both K s and K s. would originate. Fig. 11 shows the production and the decay rates of all K∗s and K¯ ∗ which existed in a heavy-ion collisions (solid ∗ D. Production and decay time of the K black and dashed red lines) and which of these K∗s could be reconstructed in the detector from the final pions and In order to understand better which K∗s can be seen or kaons (π +K) – short-dashed green and dash-dotted blue reconstructed in the detector at an experiment like STAR lines. one can look at when the K∗’s are produced and when As can be seen many of the K∗s that decay during the they decay during a heavy-ion collision. As can be seen early stages of the collision, up to a time of t = 20 fm/c, 10

1/2 of the background there are several techniques that can

(s ) = 200 GeV, Au+Au, 5% central, all y NN be employed: i) One can take all unphysical channels and 30 combine the kaons and pions to get the background spec-

K* (all) trum. Another method to get the background is ii) to flip 25

Creation the x and the y component of the momentum of either Decay the kaons and pions and combine all physical channels. 20

Reconstructed K* (all) ] A third method iii) consists of pairing all kaons and pions -1 Creation from two different events. Unlike the first two methods 15 Decay this method ensures that there can be no possible corre- lation between the kaons and the pions and thus is most 10 dN/dt [fm suited to construct the background spectrum [51] since it is not possible that kaons and pions from different events 5 can be correlated. ∗±

0 For the reconstruction of the K vertex cuts also need

0 20 40 60 80 100 to be taken into account due to the second decay vertex 0 + − t [fm/c] which stems from the decay of the KS 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 [7]. FIG. 11: The creation and decay rates versus time for central S In principle, the experimental reconstruction proce- Au+Au collision at a cms energy of √sNN = 200 GeV. Both the K∗s and K¯ ∗s are included. The solid black and dashed dure could exactly be repeated with the PHSD final par- red lines show the creation and decay rates for all K∗s in a ticle spectra on an event-by-event basis. However, this system. The short-dashed green and dash-dotted blue lines would imply to generate a huge amount of events which is show the creation and decay rates respectively of K∗s that very costly due to limited computing power. On the other could be reconstructed from the final pions and kaons (π+K). hand the transport approach offers information that is not available in the experiment. We can precisely iden- tify in PHSD the kaons and pions that correspond to a decay into Kπ pairs that do not reach the detector and certain K∗ decay and we can reconstruct the final K∗ thus cannot be reconstructed due to absorption by the spectrum much more efficiently and also with high pre- medium or rescattering of the final pions and kaons. Dur- cision. ing later stages of the collision, however, the K∗s that These different methods will not lead to a sizeable dif- are created (most probably from Kπ collisions, as can ferences for p+p collisions, since the number of created be seen in fig. 5) can also be seen in the detector, since particles is relatively small and the risk of mismatching the medium is already dilute and absorption of the decay particles is comparably low. It is possible (and likely) particles by the medium is rare. that there is a difference in A+A collisions. Our stud- ies, however, have not shown a significant difference as compared to the correlation method. V. RESULTS FROM PHSD IN COMPARISON TO DATA FROM STAR A. p+p collisions In this section we present our results from the PHSD transport approach in comparison to the data from the First of all, the experimental reconstruction of the K∗ STAR Collaboration at RHIC. [7–10] in p+p collisions doesn’t lead to a strong distortion of the The STAR collaboration at RHIC has investigated the K∗ signal since p+p collisions produce only a low amount ∗0 + − following hadronic decay channels: K(892) K π , of particles and the rescattering and absorption of the ¯ ∗0 − + ∗± 0 ± → + − ± K(892) K π and K(892) KSπ π π π final kaons and pions is practically vanishing compared to [7]. As has→ been mentioned above→ the K∗ →reconstruc- A+A collisions. Furthermore, there are no modifications tion relies on the final particles observed in the detector, of the kaons by a medium. i.e. the kaons and the pions. Since both decay products As can be seen from both figures in Fig. 12 the results suffer from rescattering and absorption the reconstruc- from the PHSD calculations reproduces the experimen- tion becomes difficult. Furthermore, the time resolution tal data very well. The K∗ spectrum from the recon- and accuracy in momentum and invariant mass of the structed final kaons and pions matches the K∗ spectrum detector itself adds another hurdle to a reconstruction of from the K∗, which were directly taken from the decay the K∗ signal due to a possible misidentification of the point. This holds true for K∗s with both a zero and a particles. non-zero electric charge. Experimentally all the possible The procedure for obtaining the K∗ signal is based on K∗ reconstruction channels are considered, through ei- the combination of all kaons and pions with respect to ther a direct decay into a K +π pair or indirectly through 0 + − the hadronic decay channels mentioned above, i.e. only the second vertex calculation of the K S π π decay. channels that are physically possible are taken. To get rid We note in passing that that PHSD appears→ to slightly 11

1/2

is much more more distorted. The number of produced (s ) = 200 GeV, |y|<0.5

NN

-1 particles in an Au+Au collisions is much higher than in a 10 p+p collision which leads to a lot more background and PHSD to a higher probability of misidentification of particles.

] Decayed K*s 2 -2 Furthermore, the cuts on the invariant mass have a large 10 Reconstructed K*s effect, as shown in IV. /GeV 2

-3

(a) 10 1/2

dy) [c (s ) = 200 GeV, Au+Au, 10% central, |y|<0.5

T NN

2

10 dp T p

-4

STAR pp minimum bias K*0 + anti-K*0 10

1 ]

10 2

+ - N/(2

2 PHSD (K* + K* )/2 d

Decayed /GeV

-5

2 0

10

10

Reconstructed

0.0 0.5 1.0 1.5 2.0 2.5 3.0

dy) [c p [GeV/c] T T

-1

10 dp T

1/2

p

(s ) = 200 GeV, |y|<0.5

NN

-1

-2

10

10

N/(2 STAR Au+Au 10% central 2

PHSD d

Decayed K*s ] -3 2

10

-2

Reconstructed K*s 10

0.5 1.0 1.5 2.0 2.5 3.0 /GeV

p [GeV/c] 2

T

-3

10

(b) dy) [c T FIG. 13: The transverse momentum spectrum 2 dp

T d N/2πpT dpT dy versus the transverse momentum pT

p ∗0 ∗0 STAR pp minimum bias -4 for K + K¯ mesons in a Au+Au collision at a cms energy 10 of √sNN = 200 GeV. The solid black circles show the data

0 0 N/(2

2 (K* + anti-K* )/2 from the STAR Collaboration for 10% central collisions d while the connected symbols show results from the PHSD -5 10 calculation. The solid green squares show results where the 0.0 0.5 1.0 1.5 2.0 2.5 3.0 K∗s were taken at the point of their decay while the solid

p [GeV/c] ∗ T red stars show results for K s that have been reconstructed from the final K + π pairs. The STAR data are taken from FIG. 12: The transverse momentum spectrum Ref. [7]. 2 d N/2πpT dpT dy versus the transverse momentum pT ∗+ ∗− ∗0 ¯ ∗0 for K + K (”a”, upper part) and K + K (”b”, lower Fig. 13 highlights the difference between the recon- part) in a p+p collision at a cms energy of √sNN = 200 GeV. structed K∗ spectrum and the spectrum of K∗s taken The solid black circles denote minimum bias p+p data from the STAR experiment from Ref. [7]. The connected open directly from their decay point. While the reconstructed symbols represent results from a PHSD simulation. The open spectrum follows the experimental data very well after ∗ green squares show results where the spectrum was obtained applying the acceptance cuts, the K spectrum as taken from the K∗(K¯ ∗)’s coming directly from the decay point in at the decay point is sizeably higher at low transverse PHSD. The open red stars show results from K∗(K¯ +)’s that momentum and has a lower slope. As has been discussed have been reconstructed from the final K(K¯ ) and π pairs. in the previous section only a small part of the change in the spectrum is due to rescattering and absorption by the medium. We have found that the dominant changes ∗ underestimate the pT slope for charged K s whereas it arise from the different cuts imposed in the reconstruc- ∗ slightly overestimates the pT slope for neutral K s. Since tion. This implies that parameters such as an effective in PHSD the slopes for charged and neutral K∗s are the temperature T ∗, which can be extracted from an expo- same within statistical accuracy one might ’see’ a slightly nential fit to the pT spectra, do not necessarily correlate different slope in the STAR data. However, this is within to the value that reflects the actual K∗ decays during the systematic uncertainties. a collision. Both the cuts and the rescattering and ab- sorption by the medium only affect the lower part of the transverse momentum spectrum while both the green and B. A+A collisions the red lines converge towards each other for higher pT . When considering rescattering and absorption effects The experimental reconstruction of the K∗ spectrum one may argue that the faster kaons and pions, i.e. parti- in A+A collisions is a lot more complicated and the signal cles with a higher pT , are able to escape the medium and 12 reach the detector without sizeable interaction with the pairs also lead to a lowering of the spectrum at small medium while particles with lower transverse momentum transverse mass. experience stronger final state interactions which lead to ∗ a distortion of the reconstructed K spectrum. Since 1/2 (s ) = 200 GeV, Au+Au, |y|<0.5 the size of the ’fireball’ changes with centrality the K∗ NN 0.6 spectra for different centrality classes provide further in-

p+p formation. Au+Au

STAR

STAR

PHSD

PHSD

1/2

0.4

(s ) = 200 GeV, Au+Au, |y|<0.5

NN

3

10 -

PHSD /K

2 PHSD

0 0 0

10

(K* +anti-K* )/2

+ -

(K* +anti-K* )/2 ] K*

central*2 2

1 [50,80]% 0.2

10 [0,10]%

[30,50]%

0

[50,80]% 10

-1

10 dy) [1/Gev T

0.0

-2 dm 10 0 100 200 300 400 500 600 700 T STAR

0 0 m

(K* +anti-K* )/2 dN /d -3

ch 10

STAR central*2

+ -

[0,10]% (K* +anti-K* )/2

-4 N/(2

10

2 [30,50]% [50,80]% ∗0 − d

[50,80]% FIG. 15: The K /K ratio versus dNch/dη for Au+Au col-

-5 10 lisions at a cms energy of √sNN = 200 GeV. The solid black 0.0 0.5 1.0 1.5 2.0 2.5 squares show data from the Au+Au collision at the STAR

m -m [GeV]

T 0 experiment while the orange circle shows data from p+p col- lisions at STAR. The solid red stars show results from Au+Au 2 PHSD calculations and the open green star shows results from FIG. 14: The transverse mass spectrum d N/2πmT dmT dy p+p PHSD calculations. The STAR data are taken from as a function of the transverse mass mT m0 for different [7, 9]. centralities for K∗0 + K¯ ∗0 as well as for− peripheral colli- sions for K∗+ + K∗− in Au+Au collisions at a cms energy ∗0 of √sNN = 200 GeV. The symbols show data from the STAR In Figs. 15 and 16 we show the ratios between K and experiment while the solid lines show results from the PHSD K− as a function of the charged particle pseudorapidity ∗0 ¯ ∗0 calculations. For the K + K STAR data the legend is as and the cms energy, respectively, in comparison to the follows: the solid black circles show data from central Au+Au STAR data. The reason to study this particular ratio is collisions, the open stars show data for [0, 10]% central colli- that the quark content of K∗s and K is the same, the dif- sions, the open black squares show data for [30, 50]% central collisions and the open black circles show data for [50, 80]% ference lies in the mass and the relative orientation of the central collisions. For the K∗+ + K¯ ∗− STAR data the up- quark spin. By studying this ratio one hoped to find out ∗ side down solid black triangles show data for [50, 80]% central more about the K production properties and the freeze- collisions. For the K∗0 + K¯ ∗0 PHSD results the legend is as out conditions in relativistic heavy-ion collisions. The follows: the solid red line shows results from central Au+Au PHSD results in Fig. 15 reproduce STAR data very well collisions, the dashed olive line shows results for [0, 10]% cen- within error bars which indicates the production channels tral collisions, the dotted green line shows results for [30, 50]% in PHSD are in line with the experimental observation. central collisions and the dashed blue line shows results for The ratios in Fig. 15 are shown as the real ratios, i.e. ∗+ ¯ ∗− [50, 80]% central collisions. For the K + K PHSD results they have not been normalised to the ratio measured in the solid light blue line shows results for [50, 80]% central col- minimum bias p+p collisions, as was done in [7]. We lisions. The STAR data are taken from [7, 8]. obtain a similar value for the K∗0 to K− ratio in p+p collisions as in the experiment. Furthermore, the PHSD As can be seen in Fig. 14 the experimental data have results match the experimental data from Ref. [7] as also been taken for Au+Au collisions at a cms energy of well. Furthermore, from Fig. 15 one can see that there ∗0 − √sNN = 200 GeV for different centralities. We have used is almost no dependence of the K /K ratio on the the same K∗ reconstruction procedure to calculate the centrality and accordingly the impact parameter. ∗ reconstructed mT spectrum of the K form PHSD simu- Fig. 16 shows the same ratio as a function of different lations for the same centrality classes. The experimental cms energies. One has to note that we show Au+Au data can be reproduced by the results from PHSD very and p+p data in accordance with the original publication well, separately for the different centralities. We note where this data have appeared. As seen the PHSD results that the same restrictions and effects are also present at can reproduce these STAR data within error bars well, higher impact parameters. Rescattering and absorption too. There seems to be no strong dependence of the ratio effects play a role at central as well as at peripheral col- with cms energy. lisions and the cuts imposed on the reconstructed Kπ Figs. 17 and 18, furthermore, show the average trans- 13

1/2

5% central, |y|<0.5 (s ) = 200 GeV, 5% central, |y|<0.5

NN

0.7 1.4

p+p

Au+Au

1.2

0.6

STAR

STAR

NA27

1.0 PHSD

0.5 PHSD -

0

0.8

K* /K 0

0.4 K*

0.6 > > [GeV] T

0.3

0.4 Au+Au

p+p

STAR

STAR

0.2

0.2

PHSD

PHSD

0.1 0.0

1 2 3

0 100 200 300 400 500 600 700 10 10 10

1/2

dN /d (s ) [GeV]

ch

NN

∗0 − FIG. 16: The K /K ratio as a function of the cms energy FIG. 17: The average transverse momentum < pT > as a √sNN . The black squares show data from Au+Au collisions function of dNch/dη. The black and orange symbols show at the STAR experiment. The orange circle shows STAR data data for K∗0 from the STAR experiment. The solid black from p+p collisions. Additionally data from the NA27 exper- squares show data for Au+Au collisions while the solid or- iment is shown for lower cms energies as open blue diamonds. ange circle shows data for p+p collisions. The green and red The red and green symbols show results from a PHSD calcu- symbols show results for K∗0 from a PHSD simulation. The lation. The solid red stars show results for Au+Au collisions solid red stars show results for Au+Au collisions while the while the open green stars show results for p+p collisions. open green star shows results for p+p collisions. The STAR The STAR data are taken from [7, 8]. data are taken from Ref. [7]. verse momentum as a function of the charged particle the azimuthal particle distributions in momentum space pseudorapidity and the average number of participants, ∞ respectively. Again, the average transverse momentum d3N 1 d2N E = 1+2 v cos(n(φ Ψ )) results from PHSD agree with the data from the STAR d3~p 2π p dp dy n − R T T n=1 ! experiment very well, both for p+p and also for Au+Au X (10) collisions. Furthermore, in Fig. 18 one can see that our re- which we have taken as sults agree with the data for peripheral collisions, with only few participating nucleons, and for central collisions px + py v2 = (11) where the number of participants is high. The STAR px py Collaboration has found that the average transverse mo- − mentum of the K∗s is higher than the average transverse since the reaction plane is ficed in PHSD. In this fig- momentum of kaons and pions. This might indicate that ure the STAR data are taken from two different runs at the average transverse momentum is more strongly re- RHIC. When considering the error bars of the experimen- lated to the mass of the particle observed. tal data from the first (RunII) and the second (RUNIV) So far, most of the collisions were performed for both data are in agreement and a significant non-zero v2 ∗0 Au+Au or p+p collisions. However, there are also data for the K can be extracted in minimum bias Au+Au Cu+Cu which have been provided in form of the average collisions at a cms energy of √sNN = 200 GeV. Again transverse momentum in Fig. 19. The data are available within error bars the PHSD results are compatible with the measurements. for a cms energy of √sNN = 200 GeV and for a lower energy of √sNN = 62.4 GeV. In Fig. 19 one can see that the results from PHSD reproduce the STAR data for Cu+Cu collisions also very well. Our average mo- VI. SUMMARY mentum for the reconstructed K∗0s as a function of the average number of participants agrees very well both for In this study we have investigated the strange vector ∗ ∗ the higher energy of √sNN = 200 GeV and also for lower meson (K , K¯ ) dynamics in heavy-ion collisions based energy of √sNN = 62.4 GeV. on the microscopic off-shell PHSD transport approach Finally, Fig. 20 shows experimental data and PHSD which incorporates partonic and hadronic degrees-of- results for the elliptic flow v2, which is defined as the freedom and includes a crossover phase transition from second harmonic coefficient of the Fourier expansion of partons to hadrons and vice versa. We have studied 14

1/2

(s ) = 200 GeV, 5% central, |y|<0.5

Cu+Cu, 5% central, |y|<0.5 NN

1.4 1.1

0 PHSD 0

K* K*

62.4 GeV

1.2

200 GeV 1.0

1.0

0.9 > T > > [GeV] 0.8

Au+Au p+p

0.8 STAR

STAR 0.6 STAR

62.4 GeV

PHSD PHSD 200 GeV

0.4 0.7

0 100 200 300 400 0 25 50 75 100 125 150

part

part

FIG. 18: The average transverse momentum < pT > versus FIG. 19: The average transverse momentum < pT > as a the average number of participants < Npart >. The black function of the average number of participants < Npart > and orange symbols show data for K∗0 from the STAR ex- for Cu+Cu collisions at different energies. The open and periment. The solid black squares show data for Au+Au solid black triangles show K∗0 from the STAR experiment collisions while the solid orange circle shows data for p+p for √sNN = 62.4 GeV and for √sNN = 200 GeV respec- collisions. The green and red symbols show results for K∗0 tively. The open red and solid green stars show results for from a PHSD simulation. The solid red stars show results for K∗0s from a PHSD simulations from reconstructed K and Au+Au collisions while the open green star shows results for π pairs for √sNN = 62.4 GeV and for √sNN = 200 GeV p+p collisions. The STAR data are taken from Ref. [8]. respectively. The STAR data are taken from Ref. [8]. the influence of in-medium effects on the strange reso- and used reconstruction methods in close analogy to the nance dynamics by implementing the in-medium K∗, K¯ ∗ experimental ones - by looking at the decay products of K∗ mesons to pions and kaons K∗ K +π and compare spectral functions for the production of strange vector ∗ → mesons by hadronic channels as well as by hadroniza- to the real spectra of all K formed in A+A collisions. By tion of strange and light quarks. For that we employed that we could pin down the influence of final-state inter- the results of our previous study in Ref. [28] where we actions of the decay products and quantify the distortion obtained the self-energies for the K∗ and K¯ ∗ from a self- of the ’true’ signal in heavy-ion collisions. As a ’reference consistent coupled-channel chiral unitary G-Matrix ap- frame’ for a check of our model we used the experimen- proach. The full G-matrix result we have approximated tal data for pp collisions and found a good agreement in terms of relativistic Breit-Wigner spectral functions between the PHSD calculations and data. which are easy to incorporate in the off-shell PHSD trans- Our findings can be summarized as follows: port approach. The real and imaginary parts of these The fraction of K∗ mesons produced by hadroniza- self-energies are related to a mass shift and collisional • tion from the QGP in the final observables is rela- broadening of the spectral functions. Furthermore, the tively small compared to the hadronic production decay width of the spectral function was calculated using channels. the in-medium spectral functions of the K and K¯ on the K∗ and K¯ ∗ decays, respectively. Most of the final K∗’s stem from the hadronic • We mention that the PHSD approach treats the par- phase, i.e dominantly from pion-kaon annihilation tonic stage on a microscopic basis and accounts for the (K + π K∗). → dynamical hadronization of quarks based on transition The decay products of the K∗ mesons - final pions rates from the DQPM model. This allows to investigate • and kaons - suffer from partly resonant final state the properties of the K∗ mesons produced from the QGP interactions which lead to a significant distortion of by hadronization without assumptions on ”freeze-out” the K∗ spectra. The low transverse momentum p criteria as used in hydrodynamic descriptions of heavy- T part is most sensitive to these ’loses’ of kaon-pion ion collisions. Moreover, even after hadronization in some correlations while the high p K∗ mesons are less area of the expanding fireball the K∗ mesons can enter T affected. That is in line with the conclusions from other ”hot sports” or blobs of QGP and dissolve to quarks the previous studies [19–21]. and antiquarks again and vice versa. We have confronted the results of our microscopic cal- At RHIC energies most of the K∗’s are produced culations with the experimental data at RHIC energies • at relatively low baryon densities due to the rapid 15

1/2 it difficult to distinguish experimentally the ”ori-

(s ) = 200 GeV, Au+Au, |y|<0.5 NN gin” of K∗ mesons by applying different cuts e.g. 30 STAR min. bias AuAu on transverse momentum.

1/2 0

(s ) =200 GeV

NN K*

RunIV

RunII The restriction of the invariant mass region for the 20 • possible reconstructed K∗s leads to a further size- able distortion of the K∗ signal and artificially en- hance the effective slope parameter of the pT or mT

(%) 10 2 spectrum. v

The number of K∗s that can be reconstructed from

PHSD • 0 final pions and kaons is heavily reduced in compar- Decayed ∗

Reconstructed ison to the number of K s that have been produced during a heavy-ion collisions; especially it applies to 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 the K∗’s produced and decayed at the early stages

p [GeV] T of heavy-ion collision, whereas the number of recon- structable K∗s in the later stages is much closer to ∗ FIG. 20: The elliptic flow v2 as a function of the transverse the number of K s that have been produced due momentum pT . The open and solid black circles show data to the fat that the system expands rapidly and the from Run II and Run IV of the STAR experiment respec- final state interaction is less probable. ∗0 tively. The runs were done for K in minimum bias Au+Au Furthermore, we mention that the dominance of final collisions at a cms energy of √sNN = 200 GeV. The solid hadronic channels in the strange vector-meson channel is green stars and solid red squares show results from a PHSD ∗0 ∗0 in line with our findings for the dilepton production from simulation. The solid green stars show the K v2 for K s non-strange ρ and ω decays in these reactions [2]. directly at their decay point while the red solid squares show ∗0 v2 for K s which have been reconstructed from the final Ks and πs. The STAR data are taken from Ref. [7, 9]. Acknowledgements expansion of the fireball, thus in-medium effects in the hadronic phase are small at these energies. The authors acknowledge inspiring discussions with With decreasing beam energy, the fraction of the J¨org Aichelin, Wolfgang Cassing, Taesoo Song, Pierre hadronic phase and its duration increases, which Moreau, Anders Knospe, Laura Tolos and Vadim leads to the fact that the K∗ mesons probe much Voronyuk. A.I. acknowledges support by HIC for FAIR high baryon densities. In this respect the future and HGS-HIRe for FAIR. D.C. acknowledges support by FAIR and NICA facilities are optimally set up to Ministerio de Econom´ıay Competitividad (Spain), Grant study in-medium K∗ resonance dynamics. Nr. FIS2014-51948-C2-1-P. This work was supported by BMBF and HIC for FAIR. The computational resources The differences in the slope of the pT spectra from have been provided by LOEWE-CSC at the Goethe Uni- • various channels is small (cf. Fig. 10) which makes versity Frankfurt.

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