Influence of electromagnetic fields in proton-nucleus collisions at relativistic energy
Lucia Oliva, Pierre Moreau, Vadim Voronyuk, Elena Bratkovskaya
aa r X i v : . [ nu c l - t h ] J a n Influence of electromagnetic fields in proton-nucleus collisions at relativistic energy
Lucia Oliva,
1, 2
Pierre Moreau,
2, 3
Vadim Voronyuk,
4, 5 and Elena Bratkovskaya GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany Department of Physics, Duke University, Durham, North Carolina 27708, USA Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Bogolyubov Institute for Theoretical Physics, Metrolohichna str. 14-b, 03143 Kiev, Ukraine
We study proton-gold collisions at RHIC energy √ s NN = 200 GeV within the Parton-Hadron-String Dynamics (PHSD) off-shell transport approach, investigating the influence of the intenseelectromagnetic fields generated in these small systems. We show the space-time evolution of themagnetic and electric components, emphasizing the huge values of the latter one, in particularthe electric field E x along the impact parameter direction whose magnitude is comparable to themagnetic field B y perpendicular to the reaction plane. We find a fair agreement of the chargedparticle pseudorapidity density of the high-multiplicity events with respect to the experimental resultof the PHENIX Collaboration. Focusing on the most central collision, we show rapidity distributionsand spectra as well as the flow coefficients v and v and we discuss the impact of the electromagneticfields on identified particle observables. We compute the directed flow v of π + , π − , K + , K − forcollisions at fixed impact parameter and predict that an electromagnetically-induced splitting in the v of positively and negatively charged particles is generated in the Au-going side of p+Au reactionmainly driven by the huge E x component. We find that this effect is stronger for the strange mesonsand increases for increasing impact parameter. Furthermore, we highlight the amount of directedflow generated in the deconfined phase, finding that it constitutes the main contribution in thecentral rapidity region, especially for kaons. Thus, we support the idea that the directed flow is apromising probe for the electromagnetic fields generated in relativistic nuclear collisions and showthat in proton-induced reactions the electric component along the impact parameter axis is theprimary origin of a charge-odd v of pions and kaons. I. INTRODUCTION
One of the most surprising discoveries of heavy ionexperiments at high energy was that the Quark-GluonPlasma (QGP) created after the collision is a strongly-coupled system that exhibits a fluid behaviour with thedevelopment of anisotropic collective flows. Indeed, ex-periments have observed a large value of the elliptic flow v , which is a measure of the asymmetry in transversemomentum space and is characterized by the second-order harmonic of the Fourier expansion of the particleazimuthal distribution with respect to the reaction plane[1]. The generation of an elliptic flow is connected tothe transport properties of the matter created after thecollision, such as the shear viscosity over entropy densityratio η/s ; the estimated value is very close to the lowerbound η/s = 1 / π conjectured for a strongly interactingsystem [2] establishing the nature of QGP as a nearlyperfect fluid.The QGP was initially expected to be formed only inrelativistic collisions of two heavy ions and small collid-ing systems, such as proton-proton and proton-nucleuscollisions, were only regarded as control measurements.However, in recent years, examining high-multiplicityproton-nucleus and deuteron-nucleus collisions at LHCand RHIC energies respectively it turned out that also inthese small systems particles had a clear preference to beemitted along common transverse directions, though ona smaller scale respect to nucleus-nucleus collisions [3– 6]. Thus a scientific debate started on the nature of thisazimuthal asymmetry, whether it is related to the for-mation of QGP droplets or due to early-time momentumcorrelations. The recent experimental and theoretical re-sults add to a growing body of evidence that even in smallsystems the QGP is created and during its expansion ittranslates efficiently the initial-state geometric eccentric-ity into a final-state momentum anisotropy [7]; see e.g.Ref. [8] for a review.In relativistic collisions of two heavy ion the v shows astrong dependence with the impact parameter since it ismainly related to the global almond shape of the overlapregion. However, it acquires a contribution also from theinitial fluctuations of nucleon positions in the overlap re-gion. These geometrical fluctuations give also rise to theodd harmonics v , v , v , etc... of the Fourier expansionof the particle azimuthal distribution. Nevertheless, thedirected flow v , which refers to a collective sidewardsdeflection of particles, is also strongly connected to thegeneration of vortical patterns in the QGP due to theangular momentum of the system and to the electromag-netic fields produced in the initial stage of the collision.Indeed, in the last decade it has been established thatextremely intense electromagnetic fields are produced innon-central relativistic heavy-ion collisions mainly dueto the motion of spectator charges [9, 10]. In particu-lar, the magnetic field in the very early stage of colli-sions at top RHIC and LHC energies can reach values of | eB y | ∼ − m π , which correspond to about 10 − Gauss, i.e. some order of magnitude higher than thatexpected to be produced in magnetars. The directedflow of both light [11–14] and heavy mesons [15–17] hasbeen considered as a promising probe to characterize thegenerated electric and magnetic fields and the recent ex-perimental results from STAR [18, 19] and ALICE [20]Collaborations challenge the theoretical understanding ofthe v and its connection to the electromagnetic fields indifferent collision systems and energies.The electromagnetic fields have never been studied inthe context of small colliding systems on a basis of micro-scopic transport approaches. With the present work weaim at filling this gap, focusing on proton-gold collisionsat RHIC energy of √ s NN = 200 GeV. Our main goal isthe study of the fields produced in this strongly asym-metric system, investigating their effect on the collectivebehaviour of the created matter, quantifying in particularthe electromagnetically-induced splitting in the charge-dependent directed flow of the most abundant hadronspecies at top RHIC energy.To this end we perform simulations with the Parton-Hadron-String Dynamics (PHSD), which is a microscopicoff-shell transport approach that describes the full space-time evolution of a relativistic nuclear collision fromthe initial hard scatterings and string formation throughthe dynamical onset of the deconfined QGP phase tothe hadronization and subsequent interactions in thehadronic phase [21–23]. PHSD includes also the dynami-cal formation and evolution of the electromagnetic fieldsduring the collision and their influence on quasiparticlepropagation as well as the back-reaction of particle dy-namics on the fields [10–12]. Besides an analysis of theproperties of p+Au collisions and its final particle distri-butions, we present our predictions for the v of chargedand identified hadrons and discuss the influence of elec-tromagnetic fields, which lead to a separation of posi-tively and negatively charged mesons along the impactparameter axis that is far more pronounced in the Au-going side.In an early work, PHSD has been applied for study-ing the dynamics of p+A reaction at ultra-relativisticenergies [24]. The collective anisotropies in these smallsystems have been investigated in many other theoreti-cal works [25–33]. However, the influence of the electro-magnetic fields on particle dynamics and the first flowharmonic v have not been explored in those studies.The plan of the article is as follows. In Sec. II we re-view the PHSD approach reminding also the implemen-tation of the retarded electromagnetic fields. In Sec. III,we discuss how the transverse components of those fieldsare distributed in space and time in proton-gold colli-sions at top RHIC energy. Our results on particle distri-butions and flow harmonics are presented respectively inSec. IV and V, highlighting in particular the role of theelectromagnetic fields. Finally, in Sec. VI we draw ourconclusions. II. PARTICLE AND ELECTROMAGNETICFIELD EVOLUTION IN THE PHSD APPROACH
The dynamical evolution of heavy-ion collisions as wellas small colliding systems at relativistic energy is de-scribed by means of the Parton-Hadron-String Dynamics(PHSD) approach, which is a covariant dynamical modelfor strongly interacting many-body systems formulatedon the basis of generalized transport equations, which arederived from the off-shell Kadanoff-Baym equations fornon-equilibrium Green functions in phase-space represen-tation [21, 22, 34–37]. The Kadanoff-Baym theory treatsthe field quanta in terms of dressed propagators withcomplex self-energies, whose real and imaginary partscan be related respectively to mean-field potentials andparticle widths [37]. The off-shell transport equationsfully governs the time evolution of the system both inthe partonic and in the hadronic phase, once the propercomplex self-energies of the degrees of freedom are known[34, 35, 37]. See Ref. [37] for a review on off-shell trans-port theory.In the beginning of the nuclear collision, primarynucleon-nucleon hard inelastic scatterings between thetwo impinging nuclei lead to the formation of color-neutral strings described by the FRITIOF model [38, 39]based on the Lund string fragmentation picture. Thesestrings fragment into “pre-hadrons”, which are baryonsand mesons within their formation time τ f (taken to be0.8 fm/ c in their rest frame) and do not interact with thesurrounding medium, and into “leading hadrons”, whichare the fastest residues of the string ends and can rein-teract with other hadrons with reduced cross sectionsin line with quark-counting rules. The fate of the pre-hadrons is determined by the local energy density. If itis above the critical energy density of the deconfinementtransition, which is taken to be ǫ c = 0 . , pre-hadrons dissolve in massive quarks, antiquarks and glu-ons plus a mean-field potential. The properties of thesecolored quasi-particles are determined by the Dynami-cal Quasi-Particle Model (DQPM) [22], which defines theparton spectral functions, i.e. masses M q,g ( ǫ ) and widthsΓ q,g ( ǫ ), and self-generated repulsive mean-field potentials U q,g ( ǫ ). Within the DQPM the local energy density ǫ is related through the lQCD equation of state to thetemperature T in the local cell. In the DQPM model,the temperature-dependent effective masses and widthsof quarks, antiquarks and gluons are fitted to the lQCDthermodynamic quantities, such as energy density, pres-sure and entropy density. Moreover, the partonic inter-action rates derived from the DQPM give rise to trans-port properties of the hot QGP that are in line withthe lQCD results; indeed, the shear and bulk viscositiesas well as electric conductivity agree to a good extentto the corresponding transport coefficients computed onthe lattice [40, 41]. The transition from the partonic tohadronic degrees of freedom is described by dynamicalhadronization, which is modeled by means of covarianttransition rates for the fusion of quark-antiquark pairsto mesonic resonances and three quarks or antiquarksto baryonic states [22, 23]. Thanks to the off-shell na-ture of both partons and hadrons, the hadronization pro-cess fulfil flavor-current conservation, color neutrality aswell as energy-momentum conservation and obey the sec-ond law of thermodynamics of total entropy increase. Inthe hadronic phase, i.e., for energies densities below thecritical energy density ǫ c , the PHSD approach is equiva-lent to the Hadron-Strings Dynamics (HSD) model [42].PHSD has demonstrated its capability to provide a gooddescription of nucleus-nucleus collisions from the lowersuperproton-synchrotron (SPS) to the top LHC energiesfor bulk observables and collective flows [22, 23, 43–45]as well as for electromagnetic probes [46–49].The PHSD approach has been extended by includingfor all binary partonic channels differential off-shell scat-tering cross sections as a function of temperature T andbaryon chemical potential µ B , on the basis of the effec-tive propagators and couplings from the DQPM that ismatched to reproduce the QGP equation of state com-puted on the lattice (PHSD5.0) [50]. Nevertheless, in thiswork we have used the default version (PHSD4.0).PHSD includes the dynamical formation and evolutionof the retarded electromagnetic field (EMF) and its influ-ence on quasi-particle dynamics [10]. In order to obtaina consistent solution of particle and field evolution equa-tions, the off-shell transport equation are supplementedby the Maxwell equations for the electric field E and themagnetic field B . Expressing the fields in terms of thescalar potential Φ and the vector potential A : E = −∇ Φ − ∂ A ∂t , B = ∇ × A , (1)one obtains a wave equation for the potentials whose so-lution for an arbitrarily moving point-like charge is givenby the Li´enard-Wiechert potentials. Inserting them intoEq. (1), the electric and magnetic fields generated by apoint-like source with charge e at position r ( t ) travellingat velocity v ( t ) are given by E ( r , t ) = e π n − β κ γ R + n × h ( n − β ) × ˙ β i κ cR ret (2) B ( r , t ) = { n × E ( r , t ) } ret (3)where R = r − r ′ with r ′ ≡ r ( t ′ ), n = R /R , β = v /c ,˙ β = d β / d t and κ = 1 − n · β ; the subscript “ret” meansthat the quantities inside the braces have to be evalu-ated at the times t ′ that are solutions of the retardationequation t ′ − t + R ( t ′ ) /c = 0. We see from the previousequation that retarded electromagnetic fields from mov-ing charges divide themselves naturally into two contri-butions: the first term represents “velocity fields”, whichare independent of acceleration and are essentially elasticCoulomb fields varying for large R as R − ; the secondterm describes “acceleration fields”, which depend lin-early on the acceleration and are intepreted as radiation fields falling off for large distances as R − [51]. Neglect-ing the acceleration ˙ β in Eqs. (2)-(3) and considering thatin a nuclear collision the total electric and magnetic fieldare a superposition of the fields generated from all mov-ing charges, one obtains the final formulas implementedin the PHSD code: e E ( r , t ) = X i sgn( q i ) α em R i ( t )(1 − β i ) n [ R i ( t ) · β i ] + R i ( t ) (1 − β i ) o / , (4) e B ( r , t ) = X i sgn( q i ) α em β i × R i ( t )(1 − β i ) n [ R i ( t ) · β i ] + R i ( t ) (1 − β i ) o / , (5)where the sum over i runs over all particles with charge q i and α em = e / π ≃ /
137 is the electromagnetic fine-structure constant. The quasiparticle propagation in theelectromagnetic field is determined by the Lorentz force: (cid:18) d p i d t (cid:19) em = q i ( E + β i × B ) (6)It is not clear a priori which is the response to theelectromagnetic field of a particle under formation (aspreviously mentioned a formation time τ f for newly pro-duced particles is considered in PHSD). The suppressionof the electromagnetic coupling to the charge of unformedhadrons and partons can be called as the “inverse LPM”(iLPM) effect [12]. The iLPM effect is incorporated inPHSD as a time delay τ emf in the interaction of theelectromagnetic field with the charged degrees of free-dom. Since the charge is a conserved quantity, preformedcharged particles should sense the electromagnetic fieldlong before being completely formed; hence in PHSD τ emf = τ f /
10 is assumed. See Ref. [12] for a detailedexplanation and discussion of the iLPM effect.Solving the generalized transport equations with theinclusion of the Lorentz force term accounts to havea consistent description of the dynamical evolution ofthe strongly-interacting many-body system with the self-generated electromagnetic fields.
III. ELECTROMAGNETIC FIELDDISTRIBUTIONS IN P+AU COLLISIONS
In this section we discuss and compare the distributionin strength and direction of the electromagnetic fieldsfor off-central Au+Au and p+Au collisions. The fielddistributions changes remarkable going from symmetriccollisions (e.g. Au+Au [10]), where the two nuclei havethe same size and number of proton, through asymmetriccollisions (e.g. Cu+Au [11]), where the two nuclei havedifferent size and atomic number, to proton-induced re-actions (e.g. p+Au), when the field distributions is basi-cally the one produced by the heavy nucleus.In Fig. 1 we show the transverse components of theelectromagnetic field produced at top RHIC energy in −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −4−3−2−1 0 1 2 3 4 e E x / m p (a) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −2.5−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 e E x / m p (e) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −4−3−2−1 0 1 2 3 4 e E y / m p (b) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −2.5−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 e E y / m p (f) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −4−3−2−1 0 1 2 3 4 e B x / m p (c) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −2.5−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 e B x / m p (g) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −4−3−2−1 0 1 2 3 4 e B y / m p (d) −15 −10 −5 0 5 10 15x [fm]−15−10−5 0 5 10 15 y [f m ] −2.5−2−1.5−1−0.5 0 0.5 1 1.5 2 2.5 e B y / m p (h) (a)–(d): Au+Au, b = 7 fm (e)–(h): p+Au, b = 4 fm FIG. 1. (Color online) Distribution of the electromagnetic fields E x , E y , B x , B y in the transverse plane at z = 0 at themaximum overlap time of Au+Au collisions with impact parameter b = 7 fm (panels (a)–(d)) and p+Au collisions with b = 4fm (panels (e)–(h)) at √ s NN = 200 GeV. Circles roughly corresponding to the position of the proton and gold nuclei are drawnto guide the eye. Au+Au collisions at b = 7 fm (panels (a)–(d)) and p+Aucollisions at b = 4 fm (panels (e)–(h)). The field strengthare computed at the time when both nuclear centers arein the same transverse plane, namely at the maximumoverlap time of the collision. In all panels circles roughlycorresponding to the size of proton and gold nuclei aredrawn in order to guide the eye highlighting the interac-tion area. It has been pointed out [9, 10] that in sym-metric nucleus-nucleus collisions the electromagnetic fieldproduced in the early stage of the collision is dominatedby the magnetic field along the y direction, i.e. orthog-onal to the reaction plane. We see from Fig. 1 (d) thatthis component reaches value | eB y | ≃ m π whereas the x -component of the magnetic field as well as the elec-tric fields components are nearly vanishing or very low.In Ref. [11], within the PHSD framework, it has beenshown that in Cu+Au collisions a significant electric field E x directed from the heavier gold nucleus towards thelighter copper nucleus is generated in the central regionof the overlap area of the collision. This is due to thedifferent number of protons in the two colliding nuclei.In p+Au collisions the initial electromagnetic field dis-tributions correspond basically to the one produced bythe gold nucleus moving at velocity close to the speed oflight and do not depend significantly on the impact pa-rameter of the collision. Nevertheless, the point in whichthe proton hits the gold nuclei has a strong impact onthe field magnitude filled by it. From Fig. 1 (e) and (f)we see that for non-central p+Au collisions the electricfield produced is strongly asymmetric inside the overlaparea; in this region, for collision at b = 4 fm, both | eB y | (h) and | eE x | (e) reaches values ≃ m π , while the otherelectromagnetic field components are close to zero.From Fig. 2 one can better see how the maximum val-ues of | E x | (panels (a) and (b)) and | B y | (panels (c)and (d)) changes for p+Au collisions at b = 4 fm mov-ing in the transverse plane respect to the centre of theoverlap area { x, y } = { , } . Moreover, thisplot gives information also on the temporal evolutionof the two electromagnetic field components, showingthat both | B y | and | E x | decrease very fast, becomingclose to zero after ∼ .
25 fm/ c from the first nucleon-nucleon collisions corresponding to t = 0 fm/ c . Whilemoving in the x − direction the value of | E x | ( x, ,
0) and | B y | ( x, ,
0) changes of about 20-25% in 1 fm, spanningthe y − direction | E x | (2 , y,
0) and | B y | (2 , y,
0) remain al-most homogeneous in a region of at least 6 fm. Hence,for non-central collisions there is a wide area around thepoint where the proton hits the nucleus in which notonly the magnetic field but also the electric field is in-tense, even though they last only for a small fraction onfm/ c . The electromagnetic field acts as an accelerator oncharges that are present during this time according to theLorentz force Eq. (6). However, considering that in a rel-ativistic nuclear collisions the evolution of the fireball isdominated by the longitudinal expansion (at least in theearly stage) and taking into account strengths and direc-tions of the electromagnetic fields, it turns out that the e E x ( x , , ) / m p x=-3 fmx=-2 fmx=-1 fmx= 0 fmx= 1 fmx= 2 fmx= 3 fm - e B y ( x , , ) / m p x=-3 fmx=-2 fmx=-1 fmx= 0 fmx= 1 fmx= 2 fmx= 3 fm t [fm/c] e E x ( , y , ) / m p y=-3 fmy=-2 fmy=-1 fmy= 0 fmy= 1 fmy= 2 fmy= 3 fm t [fm/c] - e B y ( , y , ) / m p y=-3 fmy=-2 fmy=-1 fmy= 0 fmy= 1 fmy= 2 fmy= 3 fm b = 4 fmRHIC 200 GeV p+Au (a) (c)(d)(b) FIG. 2. (Color online) Time evolution of the event-averagedelectromagnetic field components E x (panels (a) and (b)) and B y (panels (c) and (d)) for p+Au collisions at √ s NN = 200GeV with impact parameter b = 4. The different lines corre-spond to different values of x ∈ [ − ,
3] fm at y = z = 0 fm(panels (a) and (c)) and different values of y ∈ [ − ,
3] fm at x = b/ z = 0 fm (panels (b) and (d)). electric and the magnetic part of the Lorentz force pushthe charges in opposite directions, thus ending up with apartial cancellation of the corresponding momentum in-crements of charged particles in each cell [52]. Realisticsimulations are needed in order to extract the dynamicalinfluence of the electromagnetic field on final observables.The cancellation of the electric and magnetic pushes isvery strong in symmetric heavy ion collisions [10]. Wewill see in Sect. V A that in the strongly asymmetricp+Au systems, due to the high value of E x , the elec-tromagnetic field has a remarkable effect on the directedflow of mesons. IV. PROPERTIES OF P+AU COLLISIONSA. Particle distributions and centrality selection
Experimentally, the collision geometry cannot be con-trolled and initial state quantities, such as the impact pa-rameter b , the number of nucleon-nucleon collisions N coll and the number of participating nucleons N part , cannotbe accessed in a direct way. The (MC-)Glauber modelhas been widely used to describe the collision geometry,estimating the initial spatial distribution of nucleons inthe transverse plane, and to connect experimental ob-servables with the theoretically evaluated b , N coll , N part [53]. In heavy-ion collisions, different geometries – dif-ferent b – correspond to different N coll and N part andthe concept of collision geometry is strictly connectedto the concept of collision centrality, which character-izes the size of the overlap area between the two nuclei.
10 20 30 40 50 60N part d N c h / d h ( | h | < . ) −6 −5 −4 −3 −2 −1 FIG. 3. (Color online) Event probability as a function of thenumber of participants and the number of charged particlesat midrapidity ( | η | < .
5) for minimum bias p+Au collisionsat √ s NN = 200 GeV. Nevertheless, there is not a unique definition of central-ity and many observables can be used to its determina-tion, e.g., the number of charged particle N ch producedat midrapidity. Moreover, due to fluctuations in parti-cle production, there are fluctuations between initial andfinal state quantities: for example events with the same N part may correspond to different amount of N ch andviceversa. These centrality fluctuations lead in turn toan uncertainty in the interpretation of experimental mea-surements and its centrality dependence [54–57].In the case of small collision systems, such us proton-induced collisions, the centrality fluctuations become sohuge that the concept itself of centrality changes. Indeed,it is not possible to correlate clearly the collision geom-etry, the size of the fireball and the amount of producedparticle. The centrality determination in small systemsis based on the measured particle multiplicity and losesits strong link to the collision impact parameter [58–61].This implies new efforts from theoretical simulations inorder to compare their results to experimental measure-ments and to make predictions that can be easily verified.Within the PHSD approach we can choose between twodifferent procedures to simulate proton-nucleus collisions:by fixing the value of the impact parameter we can studyand comprehend in a more clear way the influence of thecollision geometry and phenomena strictly related to it(e.g., the generation of electromagnetic fields), whereasreproducing minimum bias collisions with impact param-eter values randomly distributed according with the cor-rect geometric probability allow us a more direct under-standing of experimental measurements (e.g., the particledistributions for given centrality class).In A-A collisions there is a clear scaling behaviourof particle production with the participant number, dN ch /dη ∝ N αpart , with the parameter α depending onthe collision energy [62, 63]. In small system there is stilla correlation between the charged particle multiplicity atmidrapidity and the participant number, but with largedispersion in both quantities respect to A-A collisions[24]. This is shown for p+Au collisions at √ s NN = 200 GeV in Fig. 3, where the probability distribution in thenumber of participant N part and the number of chargedparticles N ch produced at | η | < . | η | < . b = 0 . b = 8 . √ s NN = 200 GeV (panels (a) and (b)) andfor p+Au collisions at the same energy (panels (c) and(d)). While for Au+Au collisions the probability distri-bution of N part and N ch at midrapidity is close to a gaus-sian around a mean value which increases for decreasing b , providing a good correlation between the three quan-tities, for p+Au systems multiplicity fluctuations mixevents from very different impact parameters and colli-sions with b < ∼ | η | < B. Rapidity distributions and transversemomentum spectra
In Fig. 5 we plot the charged-particle pseudorapiditydensity for the 0-5% (solid red curve) and 5-10% (dashedblue curve) most central collisions along with the mini-mum bias result (dotted brown line) for the p+Au systemat √ s NN = 200 GeV. In agreement with the correspond-ing data from the PHENIX Collaboration [65] (red circlesand blue squares), the charged-particle distributions areasymmetric in pseudorapidity η and present an enhance-ment at backward rapidity, i.e., in the Au-going side.Moreover, they vary strongly with centrality, with an in-creasing asymmetry between the proton-going and Au-going directions as the collisions become more central;indeed, the minimum bias result is much flatter respectto the most central collisions, and the 0-5% bin is moreasymmetric respect to the 5-10% one.We have investigated how the use of different pseudo- N part -4 -3 -2 -1 p r obab ili t y b = 0.5 fmb = 2.0 fmb = 3.5 fmb = 5.0 fmb = 6.5 fmb = 8.0 fm Au + AuRHIC 200 GeV (a) N part -3 -2 -1 p r obab ili t y b = 0.5 fmb = 2.0 fmb = 3.5 fmb = 5.0 fmb = 6.5 fmb = 8.0 fm p + AuRHIC 200 GeV (c) N ch (| h |<0.5) -4 -3 -2 -1 p r obab ili t y b = 0.5 fmb = 2.0 fmb = 3.5 fmb = 5.0 fmb = 6.5 fmb = 8.0 fm Au + AuRHIC 200 GeV (b)
20 40 60 80 N ch (| h |<0.5) -3 -2 -1 p r obab ili t y b = 0.5 fmb = 2.0 fmb = 3.5 fmb = 5.0 fmb = 6.5 fmb = 8.0 fm p + AuRHIC 200 GeV (d) FIG. 4. (Color online) Event probability as a function of the number of participants (panels (a) and (c)) and of the number ofcharged particles at midrapidity ((b), (d)) for Au+Au collisions (panels (a) and (b)) and p+Au collisions (panels (c) and (d))at √ s NN = 200 GeV; in all plots the different curves correspond to different impact parameters from b = 0 . b = 8 fm(lines from right to left). rapidity regions to compute charged particles and thendivide the events in centrality classes affects the final re-sults for pseudorapidity distributions of charged parti-cles. This is shown in Fig. 6 for the 5% most centralcollision, comparing the result in Fig. 5 (solid red curve)with that obtained computing charged particles in | η | < − < η < − dN ch /dη distribution. In particular, the latter shows a more pro-nounced asymmetry if we use the range − < η < − | η | < | η | <
2, meaning that some contribu-tion to the asymmetry in rapidity densities comes fromthe use for centrality determination of η -regions far frommidrapidity, where there is the larger difference in parti- cle production between forward and backward rapidity.Consequently, we expect a similar impact of these choicesalso on rapidity distributions of identified particles.The rapidity distributions of π + , π − , K + , K − for 5%central collisions are displayed in Fig. 7. We notice againthat rapidity distributions are asymmetric in rapidity y and present an enhancement in the Au-going side ( y < | η | < . -4 -3 -2 -1 0 1 2 3 4 h d N c h / d h PHENIX, 0-5%PHENIX, 5-10%PHSD, 0-5%PHSD, 5-10%PHSD, minimum bias p + AuRHIC 200 GeV
FIG. 5. (Color online) Pseudorapidity distribution of chargedparticles for p+Au collisions at √ s NN = 200 GeV com-puted with PHSD simulations for minimum bias events (dot-ted brown curve) and for the 5% (solid red curve) and 5-10% (dashed blue curve) most central collisions; for the lat-ter two results the corresponding experimental data from thePHENIX Collaboration [65] are shown for comparison (redcircles and blue squares respectively). -3 -2 -1 0 1 2 3 h d N c h / d h PHENIXPHSD, | h |<2PHSD, | h |<6PHSD, -4< h <-3 p + AuRHIC 200 GeV
5% central
FIG. 6. (Color online) Impact on charged particle pseudora-pidity distribution of using different η -range for select central-ity bins from minimum bias p+Au collisions at √ s NN = 200GeV; the red, green and violet curves correspond the re-sults for the 5% most central events obtained selecting cen-trality with the pseudorapidity ranges | η | < | η | < − < η < − cally no difference in the PHSD results with and withoutthe electromagnetic field for pions, kaons, protons andantiprotons. Nevertheless, our results on rapidity densi-ties and momentum spectra in this strongly asymmetric d N / d y -6 -4 -2 0 2 4 6 y y PHSDPHSD+EMF p + p - K + K - p p_ RHIC 200 GeVp+Au 0-5% (a)(c)(e) (f)(d)(b)
FIG. 7. (Color online) Rapidity distributions of identifiedparticles for 5% central p+Au collisions at √ s NN = 200 GeVobtained with PHSD simulations with (solid red lines) andwithout (dot-dashed blue lines) the inclusion of electromag-netic fields. -3 -2 -1 -4 -3 -2 -1 ( / p p T ) d N / dp T d y [ G e V - ] p T -4 -3 -2 -1 p T PHSDPHSD+EMF p + p - K + K - p p_ RHIC 200 GeVp+Au 0-5% (a) (b)(d)(f)(c)(e)
FIG. 8. (Color online) Transverse momentum spectra of iden-tified hadrons for 5% central p+Au collisions at √ s NN = 200GeV obtained with PHSD simulations with (solid red lines)and without (dot-dashed blue lines) the inclusion of electro-magnetic fields. system can be considered as predictions for the produc-tion of the most abundant hadron species at top RHICenergy.In Fig. 9 we show the channel decomposition of therapidity distributions of π + (a) and K + (b); the resultsfor π − and K − are similar to those of the correspond-ing antiparticle, hence an explicit representation is dis-carded. Regarding pion production, we see that remark-able contributions come from QGP hadronization (solidthick blue line), decay of resonances (∆, ρ , ω , K ∗ ), ini- -6 -4 -2 0 2 4 6 y d N / d y PHSD, totalQGPinitial BB string w decayK* decay r decay D decayother channelselastic scatter. p + RHIC 200 GeVp+Au 0-5% (a) -6 -4 -2 0 2 4 6 y d N / d y PHSD, totalQGPinitial BB string f decayK* decayother channelselastic scatter. K + RHIC 200 GeVp+Au 0-5% (b)
FIG. 9. (Color online) Channel decomposition of the rapidity distributions of π + (a) and K + (b) for 5% central p+Au collisionsat √ s NN = 200 GeV calculated with PHSD. tial baryon-baryon strings (dotted brown line) and otherchannels (dot-dashed green line) such as meson-mesonand meson-baryon strings. Among the resonances, the ρ vector meson (violet squares) constitutes the main pro-duction channel of pions in the central rapidity region,while pions in the target region comes primarily from de-cay of the ∆ baryon (light-green triangles). The dashedblack line represent pions whose last interaction is anelastic scattering with other hadrons, and thus the in-formation about their production channel is lost. Forwhat concerns kaons, we point out that at midrapidity,besides a big contribution from the decay of K ∗ reso-nance (orange diamonds), there is a large production di-rectly from QGP hadronization (solid thick blue line);this is an interesting difference with respect to A–A col-lisions in which the kaons created by K ∗ decay are abouttwice those generated directly from QGP [66]. The othercurves represent kaons coming from φ decay (light-bluetriangles), initial baryon-baryon strings (dotted brownline) and other production channels (dot-dashed greenline) such as hadronic strings; the dashed black line indi-cates kaons which lastly participate in elastic scatterings.It is interesting to note that for both pions and kaonsthere is a noticeable amount of particle escaping fromthe medium just after production, without undergoingfurther rescattering.It may be interesting to look at the highest multiplicityevents that we have analized in this section because thecharged distribution asymmetry coupled with the asym-metry of the distribution of E x could play a bigger rolein other observables. Hence in the following we will focuson the 5% collision with high multiplicity in order to in-vestigate the emergence of collective patterns and grasppossible effects of the generated electromagnetic fields. V. COLLECTIVITY IN P+AU COLLISIONS
The most direct experimental evidence of the gener-ation of collective flow comes from the observation ofanisotropic radial flow in the x − y plane perpendicu-lar to the beam z − axis. It is often characterized by theFourier expansion in momentum space of the azimuthalparticle distribution, whose first two coefficients are thedirected flow v and the elliptic flow v respectively givenby v = h cos φ i ≡ h p x /p T i , (7) v = h cos 2 φ i ≡ (cid:10) ( p x − p y ) /p T (cid:11) , (8)where φ is the azimuthal angle of the particle (in mo-mentum space) and the brakets indicate average over allevents.In order to take into account the event-by-event flowfluctuations, we compute in each event the n -th or-der event-plane angles Ψ n from the final-state momen-tum distribution and the azimuthal anisotropy harmon-ics with respect to the correspondent event-plane angle;then, the final result is obtained averaging over all eventsin the centrality class of interest. The n -th order flowharmonics are given by v { Ψ n } n = h cos [ n ( φ − Ψ n )] i Res(Ψ n ) , (9)with the n-th order event-plane angle Ψ n computed asΨ n = 1 n atan2( Q yn , Q xn ) , (10)being Q xn = P i cos [ nφ i ] and Q yn = P i sin [ nφ i ] respec-tively the x and y projection of the flow vector Q n , wherethe sum runs over all particles in the chosen pseudorapid-ity range. Since the finite number of particles produces0 p T [GeV] c ha r ged pa r t i c l e s v { Y } PHENIX, | h |<0.35PHSD, w/ resolutionPHSD, w/o resolution RHIC 200 GeVp+Au 0-5%
FIG. 10. (Color online) Comparison of the elliptic flow vstransverse momentum of charged particles at midrapidity for5% central p+Au collisions at √ s NN = 200 GeV measured byPHENIX Collaboration [7, 60] (black dots) with that obtainedwith PHSD simulations (red curves): the solid thick line is the v { Ψ } computed with Eq. (9), whereas the dashed thin curveis the observed elliptic flow h cos [2 ( φ − Ψ )] i . limited resolution in the determination of Ψ n – and thisis important especially for small colliding system, such asp-A – the v n must be corrected up to what they wouldbe relative to the real reaction plane [1]; this is done bydividing the observed v n by the event-plane angle reso-lution Res(Ψ n ), which can be computed by means of thethree-subevent method that correlate independent deter-minations of Ψ n in different pseudorapidity regions . Forthe determination of Ψ n and Res(Ψ n ) we have used thefollowing three pseudorapidity ranges: − < η < − − < η < − − . < η < +0 .
5. The values are cho-sen in order to be similar to those of the detectors usedby the PHENIX Collaboration for the determination ofthe 2nd order event plane [7, 60, 65].In Fig. 10 we show with a solid red line the PHSDresults for the elliptic flow of charged particles respectto the 2nd order event plane v { Ψ } as a function of thetransverse momentum for the most central p+Au colli-sions at √ s NN = 200 GeV. For comparison we plot theexperimental data from PHENIX Collaboration [7, 60](black dots). The resolution of the 2nd order event-plane angle in 5% central p+Au collisions determined byPHENIX Collaboration is Res(Ψ F V T X − S ) = 0 .
171 [60],where the FVTX-S detector covers the pseudorapidity Denoting with A, B and C three different pseudorapidity regions(subevents), the resolution of the nth-order event-plane angle inthe subevent A is given by [67]Res(Ψ An ) = s (cid:10) cos (cid:2) n (cid:0) Ψ An − Ψ Bn (cid:1)(cid:3)(cid:11) (cid:10) cos (cid:2) n (cid:0) Ψ An − Ψ Cn (cid:1)(cid:3)(cid:11) h cos [ n (Ψ Bn − Ψ Cn )] i (11) -2 -1 0 1 2 h c ha r ged pa r t i c l e s v { Y } PHSD, w/ resolutionPHSD, w/o resolution
RHIC 200 GeVp+Au 0-5%
FIG. 11. (Color online) Pseudorapidity dependence of thedirected flow of charged particles for 5% central p+Au colli-sions at √ s NN = 200 GeV obtained with PHSD simulations(red curves); the solid thick line is the v { Ψ } computed withEq. (9), whereas the dashed thin curve is the observed di-rected flow h cos [ φ − Ψ ] i . range − < η < −
1. Within PHSD simulations we havefound a very close value, i.e., Res(Ψ − <η< − ) ≃ . h cos [2 ( φ − Ψ )] i in PHSD simulations, i.e., without di-vision for the event-plane angle resolution, showing howimportant is the method used to compute the elliptic flowfor a meaningful comparison with experimental data.The final result is comparable in magnitude with the el-liptic flow found in collisions between heavy nuclei, giv-ing indications of the fact that even in proton-nucleuscollisions, despite the volume smallness and the lifetimeshortness of the fireball, the formation of QGP dropletsallow the generation of collective patterns visible in apreferential direction of particle emission. A. Prediction for the directed flow
In this section we present our prediction of the directedflow of charged and identified particles and we discuss theeffect of electromagnetic fields. Indeed, the directed flowis very promising observable to investigate the influenceof electromagnetic fields, which could lead to a separationof positively and negatively charged particles along theimpact parameter axis.In Fig. 11 we plot with a solid line the v { Ψ } ofcharged particles versus pseudorapidity for 5% centralp+Au collisions at √ s NN = 200 GeV, with the event-plane angle Ψ computed in the pseudorapidity range − < η < −
3, where the resolution is found to beRes(Ψ − <η< − ) ≃ . v { Y } ( w / r e s o l u t i on ) -2 -1 0 1 2 y y PHSDPHSD+EMFRHIC 200 GeVp+Au 0-5% p + p - K - K + p_p (a)(c)(e) (b)(d)(f) FIG. 12. (Color online) Directed flow of identified particles, π + , π − , K + , K − , p and p , as a function of rapidity for 5%central p+Au collisions at √ s NN = 200 GeV obtained withPHSD simulations with (solid red lines) and without (dot-dashed blue lines) electromagnetic fields. directed flow h cos [ φ − Ψ ] i corresponding to the dashedline. We found a big directed flow, with a value of about0.18 at midrapidity; however, as for the elliptic flow, themagnitude of the directed flow as well depends on themethod used for the determination of the event plane.Indeed, the η -ranges used for computing Ψ and Res(Ψ )are regions of particle production and this ends up in cor-relations between v and Ψ .One main focus of this work is the investigations ofthe impact of electromagnetic fields and a possible effectmay be a splitting of the directed flow versus rapidity ofhadrons with the same mass but opposite electric charges.In Fig. 12 we plot the rapidity dependence of v { Ψ } of π + , π − , K + , K − , p and p for 5% central p+Au collisions at √ s NN = 200 GeV calculated with PHSD with and with-out the electromagnetic fields, represented respectivelyby solid red lines and dot-dashed blue lines. Within thepresent statistics, there is no visible difference in this ob-servable between simulations of 5% most central p+Aucollisions with and without the electromagnetic field.This could be explained in the following way. First ofall we have seen in Sec. IV A that multiplicity fluctua-tions in the final state mixes events from different impactparameters and in the 0-5% centrality bin there are con-tributions from peripheral to frontal collisions and eventswith b ≈ E x ≈ E y ≈ B x ≈ By ≈ respectto which the directed flow is computed is very weakly correlated with the reaction plane defined by the beamaxis and the impact parameter direction; this lead to acancellation, at least partial, of opposite contributions inthe directed flow.In order to further explore this point we have per-formed simulations of √ s NN = 200 GeV p+Au collisionsat fixed impact parameter and computed the directedflow simply by means of Eq. (7) (corresponding to com-pute v in each event respect to the true reaction plane).Even though the investigation of such selection would bevery challenging from the experimental side, this case hasthe advantage to draw very clean predictions from thetheoretical point of view, far from possible repercussionsof the particular choices adopted for centrality selectionand event-plane determination.In Fig. 13 we plot the PHSD results for the rapiditydependence of v (in percentage) of pions (panels (a) and(c)) and kaons (panels (b) and (d)) for p+Au collisions at √ s NN = 200 GeV with impact parameter b = 2 fm (pan-els (a) and (b)) and b = 6 fm (panels (c) and (d)). Simu-lations with and without the inclusion of electromagneticfields are labelled by solid and dashed curves respectively.Each plot shows v in a wide rapidity window with azoom at more central rapidities in the inset panel. Firstly,we focus on the upper panels where the directed flow of π + (blue lines with squares) and π − (red lines with cir-cles) is plotted. We notice that within simulations with-out electromagnetic fields the two oppositely-charged pi-ons show basically the same v , with some difference onlyat backward rapidity y < ∼ − π + and π − in the target region. Switching on theelectromagnetic fields in the PHSD simulations, we canclearly see that π + are pushed by the electromagneticfields in the positive x − direction and conversely π − geta kick along the negative x , hence leading to a splittingin the v curves of the two particles. Moreover, the pushof the electromagnetic field results stronger for higherimpact parameters, how can be deduced by comparingthe left and right panels corresponding respectively to b = 2 fm and b = 6 fm. The effect on the kaons shownin the lower panels is even more interesting because theelectromagnetic fields generate a flip in the directed flowof K + (green lines with squares) and K − (orange lineswith circles). Indeed, the two mesons present a differentdirected flow even in simulations that do not account forthe electromagnetic fields. This is due to the fact that K + ( su system) receive more contributions from trans-ported u and d quarks from the initial colliding nucleiwith respect to K − ( su system) [18, 68]; hence, at back-ward rapidity K + displays a smaller v with respect to K − . The electromagnetically-induced splitting in the di-rected flow of charged kaons is turned over respect tothat discussed above and dominates over it.For both pions and kaons, the direction of the splittingin v results from the contrast between sideways kicks oncharged particles by electric and magnetic forces: withinour convention for the reference frame, the electric part ofthe Lorentz force (6) pushes positively charged particles2 -5 -4 -3 -2 -1 0 1 2 3 4 5 y v ( % ) p + p _ p + , EMF p _ , EMF -2 -1 0 1 200.20.40.60.8 b = 2 fm p + AuRHIC 200 GeV (a) -5 -4 -3 -2 -1 0 1 2 3 4 5 y v ( % ) p + p _ p + , EMF p _ , EMF -2 -1 0 1 20123 b = 6 fm RHIC 200 GeVp + Au (c) -5 -4 -3 -2 -1 0 1 2 3 4 5 y v ( % ) K + K _ K + , EMFK _ , EMF -2 -1 0 1 2-0.4-0.200.20.40.60.81 b = 2 fm RHIC 200 GeVp + Au (b) -5 -4 -3 -2 -1 0 1 2 3 4 5 y v ( % ) K + K _ K + , EMFK _ , EMF -2 -1 0 1 20123 b = 6 fm RHIC 200 GeVp + Au (d)
FIG. 13. (Color online) Directed flow of pions (panels (a) and (c)) and kaons (panels (b) and (d)) as a function of rapidity for b = 2 fm (panels (a) and (b)) and b = 6 fm (panels (c) and (d)) p+Au collisions at √ s NN = 200 GeV obtained with PHSDsimulations with (solid curves) and without (dashed curves) electromagnetic fields. The inset panels are zooms of the rapiditywindow | y | <
2, with arrows highlighting in which direction the presence of the electromagnetic fields affect the v observable. along the positive x − direction and negatively chargedparticles along the negative x − direction while the mag-netic Lorentz force does the opposite [11–17]. As high-lighted by the arrows in the insets of Fig. 13, the winnerof this force balance in proton-nucleus reactions is theelectric field, whose effects on directed flow could be dis-tinguished as due to Faraday induction and Coulomb in-teraction [14]; while the Faraday effect and the Coulombcontribution within the plasma are the main origin of the v splitting in the central rapidity window, the Coulombforce exerted by proton spectators affect mainly particle v close to the target region, leading to an attraction of π − and K − and a repulsion of π + and K + .In order to pinpoint the magnitude of the directed flowof π + , π − , K + , K − induced by the electromagnetic fieldin p+Au collisions at √ s NN = 200, we show in Fig. 14(in percentage) for each particle species the quantity v emf ≡ v ( P HSD + EMF )1 − v ( P HSD )1 (12)i.e., the difference of v in PHSD simulations with andwithout the electromagnetic field, therefore removing the -2 -1 0 1 2 y -1.2-0.9-0.6-0.300.30.60.91.2 v m f ( % ) p + p _ K + K _ -2 -1 0 1 2 y b = 2 fm b = 6 fm RHIC 200 GeVp + Au (a) (b)
FIG. 14. (Color online) Directed flow induced by the elec-tromagnetic fields for pions and kaons in p+Au collisions at √ s NN = 200 with impact parameter b = 2 fm (a) and b = 6fm (b). -2 -1 0 1 2 y D v m f ( % ) totalprod. at hadronizat.prod. in HG phase -2 -1 0 1 2 y b = 6 fmb = 2 fm RHIC 200 GeV
PIONS p + Au (a) (b) -2 -1 0 1 2 y D v m f ( % ) totalprod. at hadronizat.prod. in HG phase -2 -1 0 1 2 y b = 6 fmb = 2 fm RHIC 200 GeV
KAONS p + Au (c) (d)
FIG. 15. (Color online) Splitting of directed flow between positively and negatively charged mesons induced by the electro-magnetic fields estimated by Eq. (13) for pions (panels (a) and (b)) and kaons (panels (c) and (d)) in p+Au collisions at √ s NN = 200 with impact parameter b = 2 (panels (a) and (c)) fm and b = 6 fm (panels (b) and (d)). In each panel the totalsplitting (solid black lines) is shown along with the contributions given by mesons coming directly from hadronization of theQGP (dashed magenta curves) and mesons produced in the hadron gas (HG) phase (dot-dashed blue curves). directed flow due to other causes such as vortical effectsand fluctuations. We notice that the effect increases forincreasing impact parameter, as it is evident comparingthe simulations at b = 2 fm (a) to those at b = 6 fm (b),and it is bigger in magnitude for kaons (green diamondsand orange triangles) respect to pions (blue circles andred lines squares), at least in the rapidity region | y | < v ; moreover, for slowerparticles is bigger the duration of the influence of theelectromagnetic fields.The magnitude of the splitting in the directed flow ofhadrons with opposite charge can be measured by thequantity ∆ v ≡ v +1 − v − , where v +1 and v − are the di-rected flow of the positively and negatively charged par-ticles respectively. Then we consider the quantity∆ v emf ≡ ∆ v ( P HSD + EMF )1 − ∆ v ( P HSD )1 (13)which gives information on the magnitude of the directedflow splitting induced by the electromagnetic fields. Thisquantity (in percentage) is presented in Fig. 15 for p+Aucollisions at √ s NN = 200 with two different impact pa-rameter b = 2 fm (panels (a) and (c)) and b = 6 fm(panels (b) and (d)); the results for pions and kaons areshown respectively in the left and right panels by solidblack curves.In order to understand how much of this electromagnet-ically induced splitting keeps trace of the splitting pro-duced at partonic level, we have distinguished betweenmesons created directly through hadronization of quark-gluon plasma (dashed magenta lines) and mesons pro-duced in the hadronic phase (dot-dashed blue lines); see channel decomposition in Fig. 9. We see that in the ra-pidity window | y | <
2, for both pions and kaons, the split-ting generated by the electromagnetic fields at partoniclevel is higher than that induced in the hadronic phase;this difference is far more pronounced in the strange sec-tor, where the splitting induced in the QGP dominatesover that produced in the confined phase for about a fac-tor of two at backward rapidity.In all cases the electromagnetic signal becomes weakergoing in the forward direction, due to the fact that fromone hand particle production is smaller in this region andfrom the other hand all spectators come from the Aunucleus, hence the fields produced by participants andspectators imprints their influence mainly at backwardrapidity. Furthermore, we notice that the electromagnet-ically induced splitting increases with increasing impactparameter following the increasing trend of E x . VI. CONCLUSIONS
In this work we have studied p+Au collisions at √ s NN = 200 GeV with the PHSD approach, which de-scribes the entire dynamical evolution of the collision andincludes in a consistent way the dynamical generationof retarded electromagnetic fields and their influence onquasi-particle propagation.We have analysed the electromagnetic field generatedby all charged particles, both spectator protons andcharged hadronic and partonic particles produced in thecollision. We have shown the distribution in the trans-verse plane of the transverse components of the field aswell as their time evolution. The x − component of theelectric field is comparable in magnitude to B y , that insymmetric colliding systems is the only dominant com-ponent of the electromagnetic field. Both E x and B y are4strongly asymmetric inside the overlap region and de-crease very fast, approaching zero after about 0.25 fm/ c from the first nucleon-nucleon collision.We have performed simulations by means of PHSDwith and without the inclusion of electromagnetic fieldsand compared the corresponding outcomes in order todisentangle the possible impact of the fields on final ob-servables.The PHSD results for the charged particle rapidity dis-tribution fairly agree with the experimental data from thePHENIX Collaboration: particle production is enhancedin the Au-going directions and the asymmetry betweenforward and backward rapidity increases with the cen-trality of the collision. Moreover, we have shown our pre-dictions for rapidity densities and momentum spectra ofpions, kaons, protons and antiprotons, highlighting alsothat these observables are not modified by the presenceof electromagnetic fields.We have studied the first two flow harmonics of theazimuthal particle distribution of charged and identifiedparticles. We have found a good agreement of the ellipticflow of charged particle in the 5% most central collisionscomputed with PHSD with respect to the PHENIX ex-perimental result. We have shown the sensitivity of theresult on the event-plane reconstruction accounting forthe resolution in line with the experimental procedure.Furthermore, we have presented our prediction of the di-rected flow v of charged and identified particles for col-lisions in the 0-5% centrality bin as well as for collisionsat fixed impact parameter b = 2 fm and b = 6 fm. Dis-tributions of particles with the same mass but oppositeelectric charge could be splitted by the electromagneticfields and we have clearly observed this effect in the di-rected flow of pions and kaons in collisions at fixed impactparameter: the v of π + and K + is pushed upward andthe v of π − and K − is pushed downward with respect tothe case without electromagnetic fields. This trend is vis-ible in a wide rapidity window, but is more pronouncedin the Au-going side and the splitting increases for moreperipheral collisions.Moreover, we have investigated the amount of splittinggenerated in the partonic and hadronic stages, distin-guishing between mesons formed by hadronization of thequark-gluon plasma or produced by hadronic interaction.At rapidities | y | < v generated at partonic level ishigher than that built up in the confined phase; for thestrange mesons the first contribution is rather dominantover the latter at backward rapidity (Au-going side).Thus, we conclude that the study of the directed flowof charged hadrons can shed light on the influence ofelectromagnetic fields on the dynamics of proton-inducedcollisions. ACKNOWLEDGEMENTS
The authors appreciate useful discussions with Wolf-gang Cassing, Darren McGlinchey, Ilya Selyuzhenkov, Olga Soloveva, Taesoo Song and Qiao Xu. L.O. and E.B.acknowledge support by the Deutsche Forschungsgemein-schaft (DFG) through the grant CRC-TR 211 ’Strong-interaction matter under extreme conditions’, from theCOST Action THOR CA15213 and by the DeutscherAkademischer Austauschdienst (DAAD). L.O. has beenin part financially supported by the Alexander von Hum-boldt Foundation. The computational resources havebeen provided by the LOEWE-Center for Scientific Com-puting and the Green IT Cube at GSI.5 [1] A. M. Poskanzer and S. A. Voloshin, Phys. Rev.
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