Elliptic flow of φ-meson at intermediate p_{T}: Influence of mass versus quark number
aa r X i v : . [ nu c l - t h ] F e b Elliptic flow of φ -meson at intermediate p T : Influence of mass versus quark number Subikash Choudhury, ∗ Debojit Sarkar, and Subhasis Chattopadhyay † Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhan Nagar, Kolkata 700064, India (Dated: February 21, 2017)We have studied elliptic flow ( v ) of φ -mesons in the framework of a multi phase transport(AMPT) model at LHC energy. In the realms of AMPT model we observe φ -mesons at intermediatetransverse momentum ( p T ) deviate from the previously observed (at RHIC) particle type groupingof v according to the number of quark content i.e, baryons and mesons. Recent results from theALICE Collaboration have shown that φ -meson and proton v has a similar trend, possibly indicatingthat particle type grouping might be due to the mass of the particles and not the quark content.A stronger radial boost at LHC compared to RHIC seems to offer a consistent explanation to suchobservation. However, recalling that φ -mesons decouple from the hadronic medium before additionalradial flow is build-up in the hadronic phase, similar pattern in φ -meson and proton v may notbe due to radial flow alone. Our study reveals that models incorporating φ -meson production from K ¯ K fusion in the hadronic rescattering phase also predict a comparable magnitude of φ -meson andproton v particularly in the intermediate region of p T . Whereas, v of φ -mesons created in thepartonic phase is in agreement with quark-coalescence motivated baryon-meson grouping of hadron v . This observation seems to provide a plausible alternative interpretation for the apparent mass-like behaviour of φ -meson v . We have also observed a violation of hydrodynamical mass orderingbetween proton and φ -meson v further supporting that φ -mesons are negligibly affected by thecollective radial flow in the hadronic phase due to the small in-medium hadronic interaction crosssections. PACS numbers:Keywords: Quark Gluon Plasma; Elliptic flow; Coalescence; AMPT
I. INTRODUCTION
The primary objective of heavy ion collisions at ultrarelativistic energy is to create and characterize a novelform of QCD matter consisting of strongly interactingand de-confined state of quarks and gluons, the QuarkGluon Plasma (QGP) [1, 2]. Dedicated experiments weredesigned at RHIC and LHC to search for evidences thatensure formation of such new state of matter and studyits properties. One of the key observables, particularlysensitive to the early stage dynamics of the collision andhence to the formation of QGP is the elliptic flow coef-ficient v = h cos[2( ϕ − Ψ RP )] i [3–5]. It quantifies eventand particle averaged anisotropy in the azimuthal ( φ ) dis-tribution of the particles relative to reaction plane angle(Ψ RP ) [6].It is generally perceived that in non-central collisions,the anisotropic emission of final state particles resultsfrom the difference in the pressure gradient in a spatiallyanisotropic but locally thermalized system of quarks andgluons. Below p T < η/s extracted is close to AdS/CFTlower bound of 1/4 π ).Results from RHIC and LHC have revealed that v ∗ Electronic address: [email protected] † Electronic address: [email protected] measured for different particles as a function of p T ex-hibits a characteristic mass ordering up to p T ∼ v and vice-versa atfixed p T . Whereas at intermediate p T , 3 ≤ p T ≤ v ( p T ) exhibits a flavor ordering i.e, baryon andmeson v bifurcates [10, 11, 13]. The observed baryon-meson splitting of identified particles v was found to becompatible with the models invoking hadronization of acollectively expanding partonic medium via a mechanismof quark recombination or coalescence [14–17]. This wasfurther supported by the observation of constituent quarknumber scaling (NCQ) of hadron v , providing a strongindication towards the onset of the partonic collectivityand the dominance of quark degrees of freedom at thetime of hadronization [18].At RHIC energies, baryon-meson difference in v andNCQ-scaling was taken as a confirmation of quark coales-cence being a plausible mechanism of hadronization at in-termediate values of p T . But at LHC, scaling violation ata level of ± v of φ -mesons and protons in central collisions tend to disfavourcoalescence as a relevant particle production mechanismat this range [11]. In hybrid model calculations wherepartonic and hadronic evolution is modelled by hydrody-namics and hadronic cascade respectively [19, 20], baryonand meson grouping of v , i.e, v baryons2 > v mesons2 at inter-mediate p T may be understood as a manifestation of in-crease in the mean transverse momentum h p T i and hencethe p T -integrated v values of particles as a function ofhadron mass. Some of these hybrid models also predictupto 30% increase in the p T -averaged v due to expectedrise in the radial boost at LHC when compared to Au-Au collisions at top-RHIC energy [4, 19–22, 24, 25]. Thisincrease in total transverse boost could be due to thebuild-up of additional radial flow in the hadronic phasethat boosts massive hadrons to higher p T . As the effect ismore pronounced for high mass particles, observed simi-larity in φ -meson and proton v appears to be consistentwith the increased radial flow in central A-A collisions atLHC relative to RHIC. Further studies on the spectralshapes of proton and φ -meson has revealed that in cen-tral collisions ( p + ¯ p ) /φ ratio is independent of p T upto3-4 GeV/c. The flat p T -dependence of p/φ ratio is seen tobe in agreement with hydrodynamical calculations, sug-gesting the significance of mass over quark number indetermining the shape of p T distributions upto interme-diate values of p T [12]. Thus, the baryon-meson group-ing seems to be congruous with the mass of the particlesrather than the number of quark content [11].Generally, those particles which suffer less interactionsin the hadronic phase are often termed as better probesof partonic phase of heavy ion collisions and may alsobe sensitive to the particle production mechanism. Thehadronic interaction cross section of φ -mesons with non-strange hadrons because of the OZI-suppression rule israther small [26, 27]. Consequently, φ -mesons are notexpected to undergo substantial rescattering in the latehadronic phase and decouple from the medium earlierthat their non-strange counter parts [28, 29]. Fact thatthe φ -mesons are weakly coupled to the medium, ra-dial boost developed during hadronic evolution has less-significant effect on φ -mesons compared to other hadronsof similar masses. Thus, the elliptic flow of φ -mesonsare expected to be more sensitive to the partonic stagesof collision and shown to have negligibly affected byhadronic interactions [30–32].In contrast, recent measurements by the ALICE col-laboration have shown a progressive shift in φ -meson v from meson to baryon band with increasing centralityand interpreted it as a consequence of pick-up of someadditional radial flow in the post-hadronization phase[11]. However, considering that φ -mesons decouple priorto the build-up of radial flow in the hadronic phase, itseems unlikely to be an effect of radial flow only. Itwas shown in [33] that the models incorporating φ -mesonproduction in the hadronic rescattering stage via K ¯ K fusion predict a higher value of φ -meson v relative toother mesons. It would be therefore interesting to testthe effect of hadronic interactions on the elliptic flow of φ -mesons which in-turn may be useful in resolving theambiguity over the origin baryon-meson grouping of v at LHC.Here, using the string melting (SM) version of a multiphase transport model [34] we have calculated v of someselected species of hadrons including φ -mesons for Pb-Pbcollisions at 2.76 TeV. To demonstrate the effect hadronicrescatterings on v , model simulation has been performedby varying the time of hadronic cascade. While dis-cussing our results, emphasis has been given to v of φ -mesons as they are equally massive as protons and Λs but of different quark content. We have also investigatedwhether the v of φ mesons developed at the partonicphase is modified by additional contributions from thehadronic interactions like, K ¯ K → φ -meson production.The presentation of this paper is as follows. In sec-tion II, we briefly discuss about the AMPT model andprocesses of φ -meson production at the partonic andhadronic stage. Results from the model calculation il-lustrating v ( p T ) of φ -mesons and other hadrons for dif-ferent hadronic evolution time are shown in Section IIIand finally we summarize our work in section IV. II. THE AMPT MODELA. Brief description of the model
AMPT is a hybrid transport model that describes dif-ferent stages of a heavy ion collision at relativistic ener-gies. This model has four major steps: the initial con-ditions, the partonic evolution, the hadronization andfinally the hadronic interactions. As initial conditions,AMPT uses spatial and momentum distributions of mini-jet partons and excited soft strings as implemented in theHIJING event generator [35]. Then Zhang’s parton cas-cade (ZPC) [36] is used to model the partonic evolutioncharecterized by two-body parton-parton elastic scatter-ing with parton interaction cross section obtained frompQCD calculations as σ p ≃ πα s / µ . Where α s is theQCD coupling constant for strong interactions and µ isthe Debye screening mass of gluons in the QGP medium.At the end of the partonic evolution, a spatial quark coa-lescence method is implemented to achieve quark-hadronphase transition in the SM version of AMPT. In thismethod, spatially closed quark-antiquark pairs or tripletsare recombined to form mesons and baryons, respectively.Finally, the hadronic interactions are modelled by A Rel-ativistic Transport calculations (ART) [37].In this study, SM version of AMPT has been used tosimulate Pb-Pb collisions with parton scattering crosssections of 1.5 mb and 3 mb by keeping the strong cou-pling constant, α s , fixed at 0.33 and tuning the Debyescreening mass ( µ ) to 3.22 fm − and 2.265 fm − , respec-tively. The parameters for the Lund string fragmentationfunction, i.e, f ( z ) ∝ (1 − z ) a exp ( − bm T /z ) (1)where z denotes the light cone momentum fraction,are kept same as that of the default HIJING values cor-responding to smaller string tension i,e, a=0.5 and b=0.9GeV − . B. Production and interactions of φ mesons In SM version of AMPT φ -mesons are dominantly pro-duced in the partonic stage by coalescence of a strange( s )and an anti-strange(¯ s ) quark. During the hadronic evo-lutions, φ -mesons are also generated from baryon-baryoninteraction channels BB → φN N and baryon-meson in-teraction channels ( π, ρ )B ↔ φ B, where B= N, ∆ , N ∗ [34]. Hadronically, φ -mesons are also produced by kaon-antikaon fusion, K ¯ K → φ , and the production cross sec-tion is obtained from the standard Breit-Wigner form[38].In hadronic rescatterings, φ -mesons also scatter elasti-cally with nucleons and other mesons ( π, K, ρ ). In thismodel, elastic scattering cross section for φ -mesons withnucleons and other mesons are set to 8 mb and 5 mb,respectively [34]. III. RESULTS
Model calculations based on the SM version of AMPThave shown that at top-RHIC energy the elliptic flow of φ mesons are negligibly affected by the hadronic interac-tions. While proton v was found to decrease with theincrease in hadronic rescattering time, v of φ -mesonsremain almost un-altered [19, 23, 30, 32, 39]. Thus at p T < v proton2 < v φ although m φ > m proton ,implying a violation in the hydrodynamically expectedmass ordering. The predicted breaking of the hydro-inspired mass-ordering was corroborated by the recenthigh-statistics measurements of identified particle v atRHIC [13].But a striking difference was noticed at LHC where v of φ -mesons at intermediate p T differs from the well-known baryon-meson hierarchy as mentioned in the ear-lier section. The different trend of φ -meson v was arguedto be an effect stronger radial flow in Pb-Pb collisionsat √ s NN = 2.76 TeV. Since the earlier measurements attop-RHIC energy have shown that v of φ -mesons remainalmost un-affected because of lower interaction rate in thehadronic medium, we, therefore re-investigate the effectof hadronic rescatterings on the elliptic flow of φ -mesonsat LHC energy by varying the hadronic evolution (cas-cade) time from 0.6 to 30 fm/c. Higher time for hadroniccascade corresponds to larger hadronic rescattering. Inthe figures, the hadronic cascade time of 30 and 0.6 fm/care referred to as w/ had. rescatt. and w/o had. rescatt.,respectively.In Fig. 1 (a) and (b) we have shown the transverse mo-mentum dependence of elliptic flow coefficient ( v ( p T ) )for pions, kaons, φ -mesons and protons in 20-40% Pb-Pbcollisions at √ s NN = 2.76 TeV from the SM version ofAMPT. The elliptic flow coefficient or v is obtained bycalculating the 2 nd order Fourier coefficient of azimuthal( ϕ ) distributions of final state particles with respect to re-action plane angle (Ψ RP ), i.e, v = < cos2( ϕ − Ψ RP ) > .The angular bracket, < ... > , stands for average overmany particles over many events. For all particles in-cluding φ mesons (decay turned-off), particle identifi-cation is done based on their respective PID or parti-cle identification number in AMPT. At this point it is ( GeV/c ) T p ) T ( p v - π + + π - +K + K φ pp+ ART (a)
AMPT -SM, 2.76 TeV
Pb-Pb, 20-40%
No ART(b)
FIG. 1: [Color online] Elliptic flow parameter v forpions( π + + π − ), kaons ( K + + K − ), phi mesons( φ ) and pro-tons ( p + ¯ p ) as a function of transverse momentum calculatedfrom the SM version of AMPT (a) with hadronic rescattering(b) without hadronic rescattering in 20-40% Pb-Pb collisionsat √ s NN = 2.76 TeV. worth mentioning that in experiments identification of φ mesons and its’ v determination differs from the ap-proach presented here. First φ -mesons are identified fromthe invarient mass distribution of their decay daugh-ters ( φ → K + + K − ) by choosing pairs within the 3 σ of φ mass, followed by v determination using invarientmass method [40],etc. By re-calculating our observable,i.e, v ( p T ), using a different technique (scalar productmethod), we have checked further whether the choice ofa particular method biases the final conclusion. We foundthat results obtained from both these methods are con-sistent within statistical error. Having established thatresults are independent of method followed, we now pro-ceed to discuss their physics implications.Figure. 1(a) represents flow coefficient calculated withhadronic rescatterings and Fig. 1(b) shows the samewithout hadronic rescatterings. These results show thatwithout hadronic rescatterings (Fig. 1(b)) the ellipticflow coefficients ( v ( p T )) exhibit a charecteristic massordering, i.e, v π ( p T ) > v K ( p T ) > v p ( p T ) > v φ ( p T ) for m π < m K < m p < m φ at low p T but the mass splittingis small. On the other hand, as shown in Fig. 1(a) masssplitting increases as hadronic rescatterings are switched-on and a violation of mass ordering between protons and φ -mesons ( v p ( p T ) < v φ ( p T ) albeit, m p < m φ ) below p T φ -mesons. As the interac-tion cross section of φ -mesons are much smaller than pro-tons, they decouple from the medium earlier and hence φ -mesons are negligibly affected by the collective expan-sion in the hadronic phase. In contrary, because of signifi-cant hadronic interactions v for protons becomes smallerthan that of the φ -mesons which eventually leads to thebreaking of hydrodynamical mass ordering. A more clearpicture of this behavior can be obtained by studying theratio of v φ ( p T ) to v p ( p T ) as a function transverse mo-mentum. ( GeV/c) T p p / v φ v ARTNo ART
AMPT -SM, 2.76 TeVPb-Pb, 20-40 %
FIG. 2: [Color online] Ratio of v φ ( p T ) /v p ( p T ) as a functionof transverse momentum calculated from the SM version ofAMPT with hadronic rescattering (open star) and withouthadronic rescattering (solid star) in 20-40% Pb-Pb collisionsat √ s NN = 2.76 TeV. Filled boxes represent statistical un-certainties. It is evident from Fig.2 that as the hadronic interac-tion time is increased from 0.6 fm/c to 30 fm/c (allowingmore hadronic rescatterings) the ratio of v φ ( p T ) /v p ( p T )exceeds unity below 1.5 GeV/c implying breakdown ofmass ordering.Having observed that AMPT-SM with hadronic rescat-tering has a qualitative agreement with other model cal-culations [19, 39] that reasonably describes the identifiedparticles v at low p T , we now focus on the description ofelliptic flow coefficients at the intermediate p T region. AtRHIC, it was observed that particle production by quarkrecombination manifest itself in an unique particle typegrouping of v according to the number of quark contentin the intermediate p T region, i.e, baryon and meson v are grouped into two separate branches.However, at LHC, latest ALICE results show v of φ -mesons exhibit a different trend from the particle typegrouping. Instead of following the baryon-meson hierar-chy, v values of φ s seem to shifted towards the baryonband [11]. Our model calculation also reveals that φ -meson v follow similar trend as reported by the ALICECollaboration. As shown in Fig 1, v of φ -mesons appearto follow the proton (baryon) in presence hadronic rescat-terings but falls back on the meson band when hadronicinteractions are turned off. A similar observation wasalso reported in this ALICE publication [11], where itwas interpreted as a consequence of strong radial flowthat boosts massive hadrons to higher p T . As φ -mesonsand protons have similar masses, they are expected to beboosted equally.Such observations tend to indicate that baryon-mesongrouping could be due to the mass of the particles ratherthan the number of constituent quarks. However, recall- ing that φ -mesons are weakly coupled to the hadronicmedium because of small interaction cross sections anddecouples prior to the build-up of additional radial flowin the hadronic phase, it seems unlikely to be an effectof radial flow alone. It was shown in [33] that the mod-els with φ -meson production in the hadronic rescatteringstage via K ¯ K fusion predict a higher value of φ -meson v relative to other mesons. It would be therefore inter-esting to test the effect of such processes on the ellipticflow of φ -mesons. ( GeV/c) T p ) T ( p v φ -> K No K/ARTNo ARTART Pb-Pb, 20-40 %
AMPT -SM, 2.76 TeV
FIG. 3: [Color online] Transverse momentum dependence v of φ -mesons calculated from the SM version of AMPTwith hadronic rescattering (solid star), without hadronicrescattering (solid circle) and with hadronic rescattering but K ¯ K → φ forbidden (solid square) in 20-40% Pb-Pb collisionsat √ s NN = 2.76 TeV. Filled boxes and the bands representstatistical uncertainties In this work we have also analyzed φ -meson v byturning-off K ¯K coalescence in the hadronic phase. InFig. 3 solid star represents v of inclusive φ -mesons (all φ -mesons produced in partonic and hadronic phase) andsolid square represents v of φ -mesons excluding thosefrom the K ¯ K fusion process (here we call it primordial φ s). It is interesting to observe that at the end ofhadronic rescattering for 30 fm/c, v of primordial φ -mesons show no change rather it values at intermedi-ate p T > φ -mesons regener-ated hadronically by K ¯ K fusion in the late hadronic stagemay be responsible for the observed increase in v atmoderate p T . But primordial φ -mesons which are domi-nantly produced in the partonic phase are least affectedby hadronic interactions and follow quark-recombinationexpected baryon-meson grouping. In fact, in peripheralcollisions, as shown in the Fig. 4(b), even with hadronicrescattering turned-on, inclusive φ -meson v is seen tofollow meson v instead of baryon. (GeV/c) T p0.5 1 1.5 2 2.5 ) T ( p v - +K + K φ pp+
20 - 40 % (a) 0.5 1 1.5 2 2.50.050.10.150.2
50 - 80 %
AMPT -SM, ARTPb-Pb 2.76 TeV(b)
FIG. 4: [Color online] Elliptic flow parameter v for kaons( K + + K − ), phi mesons( φ ) and protons ( p + ¯ p ) as a func-tion of transverse momentum calculated from AMPT SM withhadronic rescattering in (a) 20-40% and (b) 50-80% centralityclasses of Pb-Pb collisions at √ s NN = 2.76 TeV. This could be because of relatively lesser number of re-generated φ -mesons in peripheral collisions than in cen-tral or mid-central collisions at same √ s NN . Thus, theapparent mass-like behaviour of φ -meson v may alsobe understood as a consequence of φ -meson regenerationfrom K ¯ K fusion.To further substantiate that φ -meson v in AMPT is con-sistent with quark number and not mass, we compare v of primordial φ -mesons with pions and protons. Resultspresented in Fig. 5 and 6 clearly show, despite mass of φ -meson being comparable to that of proton (baryon), φ -meson v ( p T ) at intermediate p T region exhibit similarflow pattern as that of the lighter mesons irrespective ofparton scattering cross section. Further confirming thatin AMPT particle species dependence of the v ( p T ) is abaryon-meson effect and not because of the mass of theparticle. However, any deviation from the observed pat-tern may be attributed to the modification in the spec-tral shape and/or v itself by hadronic interactions in thelater stages of collision. IV. DISCUSSION
In summary, we have studied elliptic flow of φ -mesonsat low and intermediate ranges of transverse momentumfor 20-40% Pb-Pb collisions at 2.76 TeV using a hybridtransport model AMPT. φ -meson v has generated lotsof interest at LHC since it was observed to deviate fromparticle type dependent flow patteren at intermediate p T .This observation led to interpretation of baryon-mesonordering of v as a mass effect rather the quark number.As separate flow patterns for baryons and mesons arenaturally accounted by the hadronization models wherehadrons are formed by coalescing quark from a collec-tively expanding partonic medium, mass-like flow pat-tern for φ mesons would suggest that baryon-meson or-dering is simply an interplay between particle mass and (GeV/c) T p ) T ( p v - π + + πφ AMPT -SM , 2.76 TeV (a) 3 mb
Pb-Pb, 20-40%
No ART (b) 1.5 mb
FIG. 5: [Color online] Transverse momentum dependence of φ -meson and pion v for 20-40% Pb-Pb collisions at √ s NN =2.76 TeV. Results obtained from SM version of AMPT modelfor parton scattering cross section of (a) 3 mb and (b) 1.5 mbwithout hadronic rescatterings. ( GeV/c ) T p ) T ( p v pp + φ AMPT -SM, 2.76 TeV(a) 3 mb
Pb-Pb, 20-40 %
No ART (b) 1.5 mb
FIG. 6: [Color online] Transverse momentum dependenceof φ -meson and proton v for 20-40% Pb-Pb collisions at √ s NN = 2.76 TeV. Results obtained from SM version ofAMPT model for parton scattering cross section of (a) 3 mband (b) 1.5 mb without hadronic rescatterings. radial flow, which can be explained in the hydrodynam-ical framework without requiring different hadronizationschemes such as recombination.However, our model calculation shows that regenera-tion of φ during the hadronic phase through hadronicinteractions of K/ ¯ K fusion could be responsible for thisapparent mass-like behaviour. Whereas those created inthe partonic phase by s -¯ s coalescence perfectly follow thebaryon-meson grouping. Inspite of having mass compa-rable to that of proton, similarity in the v ( p T ) of φ andother lighter mesons ( π, K ) further supports that ellipticflow developed at the partonic phase is inherited by thehadrons via a mechanism of quark recombination.At low p T , violation in the traditional hydrodynamicmass ordering between proton and φ -meson v is ob-served. This is attributed to small interaction cross sec-tion of φ -mesons compared protons resulting in a de-crease in proton v keeping φ -meson v almost unaffectedduring hadronic rescatterings. This observation is sup-ported by RHIC data for Au-Au collisions at 200 GeVbut could not be verified at LHC due to lack of databelow 0.9 GeV/c in p T . Acknowledgements