Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies
EEPJ manuscript No. (will be inserted by the editor)
Electromagnetic fields and directed flow in large and smallcolliding systems at ultrarelativistic energies
Lucia Oliva a Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Ger-many Received: date / Revised version: date
Abstract.
The hot and dense QCD matter produced in nuclear collisions at ultrarelativistic energy ischaracterized by very intense electromagnetic fields which attain their maximal strength in the earlypre-equilibrium stage and interplay with the strong vorticity induced in the plasma by the large angularmomentum of the colliding system. A promising observable keeping trace of these phenomena is the directedflow of light hadrons and heavy mesons produced in symmetric and asymmetric heavy-ion collisions as wellas in proton-induced reactions. In particular, the splitting of the directed flow between particles with thesame mass but opposite electric charge as a function of rapidity and transverse momentum gives accessto the electromagnetic response of medium in all collision stages and in the different colliding systems.The highest influence of electromagnetic fields is envisaged in the pre-equilibrium stage of the collision andtherefore a significant imprint is left on the early-produced heavy quarks.The aim of this review is to discuss the current developments towards the understanding of the generationand relaxation time of the electromagnetic fields embedded in both large and small systems and theirimpact on the charge-odd directed flow of light and heavy particles, highlighting the experimental resultsand the different theoretical approaches. Since it is possible to perform realistic simulations of high-energycollisions that incorporate also the generated electromagnetic fields and vorticity, the study of the directedflow can provide unique insight into the early nonequilibrium phase and the ensuing QGP formation andtransport properties.
Nuclear collisions at ultrarelativistic energy represent theonly laboratory on Earth for investigating the deconfinedphase of strongly-interacting matter: the Quark–GluonPlasma (QGP). Its formation and evolution holds some ofthe most extreme properties ever observed in nature: veryhigh temperature T up to several times the pseudocriti-cal value of the transition between hadronic and partonicmatter T c ∼
155 MeV ∼ K [1,2], i.e. five order ofmagnitude higher than the temperature at the centre ofthe Sun; very low value of the viscosity over entropy den-sity ratio η/s close in the vicinity of T c to η/s = 1 / π [3,4],that is more than twenty times lower than that of the wa-ter; huge magnetic field up to about eB ∼ m π ∼ G, i.e. some order of magnitude larger than that expectedon the surface of magnetars [5]; intense vorticity up to ω ∼ . c/ fm ∼ s − [6], that is 14 orders of magni-tude higher than that of any other fluid ever observed.In the last decades the surprising behaviour of theQGP related to its transport properties has been inten-sively studied in Heavy-Ion Collisions (HICs). A new excit-ing era has begun in recent years after the discovery thatsmall-sized and short-lived droplets of QGP are formed a Present address: oliva@fias-uni-frankfurt.de also in small colliding systems; indeed, high-multiplicityevents in proton–proton and proton–nucleus collisions atrelativistic energy present similar collective features asthose found in collisions between two heavy ions.Both large and small colliding systems are character-ized by the presence of extremely intense electromagneticfields (EMF). It was realized more than forty years agothat huge magnetic fields could arise in HICs at high en-ergy [7,8]. The EMF generated since the the early stage ofthe collision are mainly due to the spectator protons and insymmetric nucleus-nucleus collisions, such as Au+Au andPb+Pb, the dominant component is the magnetic fieldorthogonal to the reaction plane B y . The first realisticestimates has been carried out in Refs. [5,9,10]. In asym-metric systems, due to the different number of protons inthe two colliding nuclei, a remarkable electric field alongthe impact parameter axis E x directed from the heaviernucleus towards the lighter nucleus is generated in thecentral collision region. This has been demonstrated andstudied for Cu+Au reactions in Refs. [11,12]. This asym-metry in the EMF profiles is taken to its extreme in thecase of proton-nucleus collisions, where the EMF are al-most completely determined by the heavy nucleus and theproduced electric field E x is comparable to B y , as shownand investigated in Ref. [13] for p+Au collisions. However, a r X i v : . [ nu c l - t h ] J u l Lucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies event-by-event fluctuations of the proton positions in thecolliding ions leads to fluctuations of the EMF and alsothe other components could reach values of the same orderof magnitude of B y and E x [14,15,16].The initial values of the magnetic field attained in periph-eral HICs at top RHIC energy are eB y ≈ m π and aboutone order of magnitude higher at LHC energies. However,the decay rate of this huge fields with time is still un-der debate, being it dependent on the formation timescaleand the electromagnetic response of the medium createdafter the collision. Indeed, semi-analytic calculations ofthe magnetic and electric field evolution in a plasma withconstant electric conductivity [17,18,19] indicate a strongslowdown of the field decay with respect to the evolutionin the vacuum. This description is not reliable in the veryearly stage, before any medium is produced, but suggeststhat, after dropping by some order of magnitude, as soonas the conducting matter is created the EMF freezes outin it and lasts as long as the QGP lifetime. This behaviouris qualitatively in agreement with microscopic simulationswhere the EMF are dynamically generated from spectatorand participant protons as well as newly produced chargedparticles [10,20].The electric and magnetic fields act as accelerators forthe charges present in the expanding fireball and couldmodify the dependence on rapidity y and transverse mo-mentum p T of the final particle distribution with respectto the reaction plane [21]: E d Ndp = d Np T dp T dydφ = d Np T dp T dy × π (cid:34) ∞ (cid:88) n =1 v n ( p t , y ) cos ( n ( φ − Ψ n )) (cid:35) , (1)where φ is the azimuthal angle of the particle momentum p and Ψ n is the n th-order event plane of the collision.Among the Fourier coefficients of the azimuthal particledistribution in Eq. (1) the directed flow v is the mostsensitive observable to the EMF. Indeed the v , given by v = (cid:104) cos ( φ − Ψ ) (cid:105) (2)where the brakets (cid:104)· · · (cid:105) indicate the average over the par-ticles in the collision, is related to a collective sidewardsdeflection of particles on the event plane with respect tothe beam axis ( v ∼ p x /p T ) and the geometry of EMF andfluid velocities requires that the Lorentz force produces anet push of charges along the x axis with opposite di-rections for positively and negatively charges. Hence, thesplitting of the directed flow between particles with thesame mass and different electric charge gives direct accessto the collective electromagnetic response of the medium[19,11,22,20,23,24,25,26,13,27]. Furthermore, the mea-surements of the v produced in different colliding systemsas well as for different energies and centralities [28,29,30,31,32,33] give the opportunity to theoretical models totest their description of the initial state of high-energycollision, especially in view of the combined influence ofEMF and vortical dynamics on the directed flow with a different sensitivity for bulk light particles and early-produced heavy quarks [34,19,11,22,20,35,36,23,24,25,26,13,27]. Besides the EMF and the vorticity, the earlystage is affected by a nontrivial pre-equilibrium dynam-ics [37] which have an impact especially on the heavy-quark propagation [38]. Being possible to embed bothQGP and heavy-flavour dynamics in realistic descriptionsof high-energy collisions that include the generated EMFand vorticity and take into account the pre-equilibriumeffects, further theoretical developments and improved ex-perimental precision could help to solve the current ten-sion between theory and experiment in particular at theLHC energies.The aim of the present article is to discuss the directedflow observable as a probe of the EMF produced in largeand small colliding systems at high energy. The varioustheoretical modelling of the generated EMF are discussed,analysing the spacetime profiles of the fields in the differ-ent systems, from symmetric Au+Au and Pb+Pb colli-sions through the asymmetric Cu+Au to p+Au reactions.The experimental measurements are reviewed along withthe theoretical predictions and investigations for what con-cern the directed flow of light and heavy hadrons, thatconstitutes a gold probe of the collective response of themedium to the EMF as well as of the EMF themselves.Indeed, the charge-dependent v could shed light on thetimescales of the EMF decay and of the formation of elec-tric charges (i.e. qq pairs) in the early pre-equilibriumphase of relativistic collision. Furthermore, a detailed un-derstanding of the (electro-)magnetic fields in HICs is ofgreat importance for the study of other effects driven bythem, such as the Chiral Magnetic Effect (CME) and re-lated transport phenomena [39,40], the splitting in thepolarization of hyperons and anti-hyperons [41,42], theearly-time emission of photons and dileptons [43,44], theSchwinger particle production [17,45]. Moreover, the pres-ence of a magnetic field affects also the QCD phase dia-gram, modifying the behaviour of the pseudocritical tem-perature and the position of the critical endpoint [46,47]. In order to understand the theoretical and experimen-tal results reviewed in the next sections, it is instructiveto discuss the spatial profiles and the temporal evolu-tion of the EMF generated in noncentral HICs as well asin proton-induced reactions, examining the different ap-proaches used for their calculation.
The Maxwell equations for the electric field E and themagnetic field B in presence of a charge density ρ and acurrent density J reads: ∇ · E = ρ, ∇ · B = 0 , ∇ × E = − ∂ t B , ∇ × B = ∂ t E + J , (3) ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 3 where we imposed (cid:15) = µ = 1, i.e., the polarization andmagnetization response of matter is disregarded. In thestatic case one obtains from Eqs. (3) the Coulomb andBiot-Savart laws, but the general solution determines thetime-varying fields E ( r , t ) and B ( r , t ) for nonstatic sources ρ ( r , t ) and J ( r , t ).Eqs. (3) can be solved by expressing the electric and mag-netic fields in terms of the scalar potential Φ and the vectorpotential AE = −∇ Φ − ∂ t A , B = ∇ × A , (4)then obtaining from Eqs. (3) inhomogeneous wave equa-tions for the electromagnetic potentials with source terms.These equations can be solved for specified sources, suchas continuous distributions of charge and current densi-ties or a single arbitrarily moving point-like charge. In thelatter case one arrives to the famous Li´enard-Wiechert po-tentials, that can be inserted in Eqs. (4) in order to findthe retarded electric and magnetic fields produced by apoint-like source with charge e at position r ( t ) with ve-locity v ( t ): E ( r , t ) = e π ˆ R − β κ γ R + ˆ R × (cid:104)(cid:16) ˆ R − β (cid:17) × ˙ β (cid:105) κ cR ret B ( r , t ) = (cid:110) ˆ R × E ( r , t ) (cid:111) ret (5)where R = r − r (cid:48) with r (cid:48) ≡ r ( t (cid:48) ) is the relative position, β = v /c and ˙ β = d β / d t are related respectively to veloc-ity and acceleration of the particle, γ = (1 − β ) − / isthe Lorentz factor and κ = 1 − ˆ R · β ; all quantities insidethe braces labelled with “ret” are evaluated at the times t (cid:48) that solves the retardation equation t (cid:48) − t + R ( t (cid:48) ) /c = 0.We see that for a point-like charge the magnetic field isalways perpendicular to the electric field and to the direc-tion ˆ R from the retarded point.The EMF act on the propagation of a particle withcharge q through the Lorentz force : F em = (cid:18) d p d t (cid:19) em = q ( E + β × B ) . (6)In principle, thanks to the superposition principle, theretarded fields (5) and the Lorentz force (6) permit aconsistent computation of the EMF produced in HICsfrom a microscopic point of view, by considering the time-dependent fields induces by all present charges (specta-tors, participants, newly produced particles), taking intoaccount the propagation of the charges themselves in theEMF and the back-reaction of particles on the fields.The temporal evolution of the fields computed in this waywould account naturally for the electric conductivity σ el of We note that throughout the paper we indicate as “Lorentzforce” both terms depending on the electric and the magneticfield; in the literature this nomenclature is also used for refer-ring to only the magnetic part. the system, intended as the proportionality constant be-tween the electric current J induced in the system by theLorentz force and the Lorentz force itself per unit charge: J = σ el ( E + β × B ) . (7)Since in many physical cases the velocity of the chargesis small, the second term can be neglected and Eq. (7) isreduced to the well-know Ohm law J = σ el E . (8)However, in relativistic HICs the particle velocity is highenough (at least along the beam direction) to produce asignificant contribution of the magnetic term in Eq. (7).Moreover, the dominant directions of the magnetic andelectric fields in HICs are such that the two terms inEq. (7) are opposite and there is a partial cancellation.This means that it is important to consider in the theoreti-cal modelling a scenario as close as possible to the physicalsituation, depending on the observables under study. Forthe topic discussed in this paper, i.e. the effect of EMFon the directed flow in HICs, it is not realistic to consideronly the magnetic field B y , since it produces by Faradayinduction a significant E x (to be added to the Coulombcontribution in the asymmetric collision case); moreover,especially at LHC energy, it is plausible that the elec-tric conductivity of the QGP is lower than that given byEq. (8) and usually studied in theoretical models [48,49,50] and lattice QCD calculations [51,52,53,54], because ofthe magnetoresistance, i.e. the contribution of the secondterm in (7). This would in turn have an influence on therelaxation time of the magnetic field.We note that the electric conductivity is not the full storybut other contributions to the charged currents producedin HICs could come from other phenomena, such as theCME, and the corresponding conductivities should be con-sidered. The effect of a chiral conductivity σ χ on the EMFhas been recently studied in Ref. [55] through analytic cal-culations and in Ref. [56] by means of lattice simulations.We see from Eq. (5) that retarded EMF from mov-ing charges are constituted by two contributions: the firstterm is independent from the acceleration and correspondsbasically to elastic Coulomb fields decaying for large dis-tance as R − (“velocity fields”); the second term dependslinearly on the acceleration and represents radiation fieldsvarying for large distances as R − (“acceleration fields”)[57]. Since full computations in the time-dependent caseare very complicated, the acceleration fields in Eqs. (5)are often neglected for practical calculations; the remain-ing term corresponds to the field produced by a chargein uniform motion. Then, the total electric and magneticfields generated in nuclear collisions are a superposition ofthe fields generated from all moving charges q i : e E ( r , t ) = (cid:88) i sgn( q i ) α em R i ( t )(1 − β i ) (cid:110) [ R i ( t ) · β i ] + R i ( t ) (1 − β i ) (cid:111) / ,e B ( r , t ) = (cid:88) i sgn( q i ) α em β i × R i ( t )(1 − β i ) (cid:110) [ R i ( t ) · β i ] + R i ( t ) (1 − β i ) (cid:111) / , (9) Lucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies
Even though with some caution due to the omission ofthe acceleration terms, Eqs. (9) allow a consistent descrip-tion of the medium+fields evolution if charge and currentdistributions are computed dynamically so that the back-reaction of particles on fields is taken into account.This approach is adopted in the Parton-Hadron-StringDynamics (PHSD) framework. PHSD is a covariant dy-namical approach for strongly interacting many-body sys-tems formulated on the basis of off-shell transport equa-tions which determine the temporal evolution of the sys-tem both in the partonic and in the hadronic phase [58].It allows to describe the dynamical evolution of large andsmall colliding systems at relativistic energies in terms ofthe microscopic degrees-of-freedom, explicitly accountingfor the transition between QGP and hadronic phase.The EMF are included in the PHSD model in a dynam-ical manner [10,16]: the formation and evolution of theretarded electric and the magnetic fields are determinedby Eqs. (9) with the sum running over all charged par-ticles, considering spectators and participants protons aswell as newly produced charged hadrons, quarks and an-tiquarks (in PHSD the distinction between participantsand spectators is dynamical, being the latter those nucle-ons which did not undergo initial hard scatterings). Thequasiparticle propagation in the EMF is determined by theLorentz force (6). The fields are calculated time-by-time(including the contributions from all charges) so that theback-reaction of particle dynamics on the EMF is takeninto account. PHSD has been used for studying the influ-ence of the EMF on the directed flow in asymmetric HICsand proton-induced collisions.According to Faraday’s law a strongly decreasing mag-netic field induces an electric field circulating around thedirection of the magnetic field. After the early-times evo-lution, when the system is in the QGP stage, the electricconductivity of the medium is not negligible and the in-duced electric field generates an electric current that cre-ates a magnetic field which according to the Lenz rule op-poses to the decrease of the magnetic field itself, i.e., pointstowards the y direction of the initial B . This collective re-sponse of the produced matter at later stages is takeninto account naturally in the PHSD approach by solv-ing consistently the Maxwell equations and the general-ized transport equations with the inclusion of the Lorentzforce. However, since Eqs. (9) are obtained neglecting theacceleration term in Eqs. (5), the medium electromagneticresponse that can lead to a slowdown of the decrease ofthe magnetic field may be underestimated.The QGP electric conductivity σ el as response of thesystem to an external electric field has been widely studiedin the stationary limit, e.g., in Refs. [48,49,50,51,52,53,54]. In Fig. 1 we show the result of Ref. [50] for the ratio σ el /T as a function of the scaled temperature from the Dy-namical QuasiParticle Model (DQPM) included in PHSDapproach for defining the QGP properties and interactionson the basis of partonic propagators with sizeable imagi-nary parts of the incorporated self-energies. The solid redline is the DQPM result within the relaxation time approx-imation using the on-shell interaction rate Γ on . The ratio -3 -2 -1 DQPM : ON lattice QCD : N f =2+1 N f =2 e l / T T / T c Fig. 1. (Color online) Ratio of electric conductivity over tem-perature σ el /T as a function of the scaled temperature T /T c .The solid red line is the DQPM results within the relax-ation time approximation using the parton interaction rate Γ i ( p , T, µ ) for the inverse relaxation time. The symbols arelattice QCD data for N f = 2 taken from Refs. [51,52] (red andyellow circles) and for N f = 2 + 1 taken from Refs. [53,54](black spheres). Figure adapted from Ref. [50]. σ el /T rises quadratically with temperature above the crit-ical value T c which can be related to the increasing numberof quarks at higher temperatures. We note a good agree-ment of the DQPM result with the available lattice calcu-lations [51,52,53,54]. The electric conductivity can be alsocomputed by solving the relativistic transport equationsfor the partonic matter in a box with periodic boundaryconditions in the presence of an external electric field, asdone in Refs. [48,49].The electric conductivity is a the main ingredient ofanother method used for computing the EMF produced inHICs. It consists in solving analytically the Maxwell equa-tions for a point-like charge e located at the position x (cid:48)⊥ in the transverse plane and travelling along the positive z direction with velocity β in a medium with electric con-ductivity σ el . This is done substituting in Eqs. (3) ρ = ρ ext and J = J ext + J ind , where ρ ext = eδ ( z − βt ) δ ( x ⊥ − x (cid:48)⊥ )and J ext = ˆ zβeδ ( z − βt ) δ ( x ⊥ − x (cid:48)⊥ ) are the externalcharge and current due to the longitudinally-moving pro-tons and J ind is the current induced in the medium andgiven by the Ohm’s law (8). Then, one gets the followingwave equations for the electric and magnetic fields: (cid:0) ∇ − ∂ t − σ el ∂ t (cid:1) B = −∇ × J ext , (cid:0) ∇ − ∂ t − σ el ∂ t (cid:1) E = −∇ ρ ext + ∂ t J ext , (10)which can be solved through the method of Green func-tions in order to find the EMF at an arbitrary spacetimepoint ( t, z, x ⊥ ). The analytic solution is achievable as theelectric conductivity is treated as a constant in all spaceand time. The total EMF in the collision are evaluatedconsidering the fields generated by all spectator and par-ticipant protons in the two colliding nuclei, while σ el ac- ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 5 counts for the conducting QGP created after the collision.This approach has been proposed, with some differences,in Refs. [17,18,19] and is adopted in further studies fordescribing the effect of the EMF on final observables; inparticular Refs. [19,22,24,23] have addressed the topic ofthe EMF influence on particle directed flow in HICs. How-ever, the assumption of a constant conductivity during allthe evolution of the heavy-ion collision may lead to overes-timate the slowdown of the magnetic field decay, especiallyin the early stage when the medium is not yet produced.An improvement would consist in considering the temper-ature dependence of σ el depicted in Fig. 1. In Fig. 2 we represent the time evolution of the magneticfield in the center of the overlap area ( x, y, z ) = (0 , , .In panel (a) we show the results obtained in Ref. [10] withthe Hadron-String Dynamics (HSD) model for Au+Aucollisions at √ s NN = 200 GeV with impact parameter b = 10 fm. The solid red curve is the full computation ofthe magnetic field produced by all charges in the collisions,i.e. spectators, participants and newly produced particles;the dashed turquoise lines indicates the field generatedby only spectator protons. We see that in the first fm /c after the collision of the two nuclei the magnetic field ismainly due to the spectator protons, drops down by threeorders of magnitude and then become comparable withthat from participant protons and newly-formed particles,which represent the dominant contribution at subsequenttimes. The early-stage evolution is close to the field decayin the vacuum, as predicted firstly in Ref. [5], with somedifferences: in Ref. [5] the colliding nuclei are treated asinfinitely thin charged layers, neglecting their finite size inorder to have a semi-analytical form of B , and the rapid-ity degradation of this pancake-shaped nuclei is simulatedby means of a heuristic function, hence taking into ac-count participant baryons but disregarding newly-createdparticles that are instead considered in the full computa-tion (solid red line) with the HSD model. Similar resultsfor the early-stage evolution have been obtained in Ref. [9]through calculations within another microscopic transportmodel, i.e. the Ultrarelativistic Quantum Molecular Dy-namics (UrQMD) approach, even though the latter in-cludes only the spectator contribution and neglects theback reaction on the field of particle propagation, whichare instead accounted for in HSD. The HSD model is theprecursor of PHSD [58] introduced in the previous section,therefore the main results about the generation and evo-lution of the EMF continue to be valid within PHSD with In the original publications different conventions for unitsand colliding frames are used. Here the lines are uniformedto dimensionless unit and the reference frame convention usedthroughout this paper: the nucleus with centre at x > x < y axis. some difference due to the fact that the partonic stage andthe QCD phase transition between hadronic and decon-fined matter are explicitely taken into account in PHSD,so that the contribution of the QGP is included in thecomputation of the EMF.In Fig. 2(b) we plot the temporal evolution of B y in Pb+Pbcollisions at √ s NN = 2 .
76 TeV with b = 7 fm obtained inRef. [19] through numerical calculations based on analyti-cal solutions of the Maxwell equations for different valuesof the electric conductivity, i.e. solving Eq. (10). The solidorange curve representing the B y evolution in a plasmawith electric conductivity σ el = 0 .
023 fm − is comparedwith the magnetic field decay in the vacuum (i.e., σ el = 0fm − ) which is indicated by the dashed blue line and cor-respond to the calculations of Ref. [5]. We see that thepresence of a conducting plasma with nonzero conductiv-ity strongly delays the decay of the magnetic field, witha more flat time evolution which resembles the behaviourfor t (cid:38) /c of B produced by all charged particles inthe HSD model (solid red curve in Fig. 2(a)). However, inRef. [19] B ( t ) has a mild decay since the very early times.This is due to the simplification used in this approach,as well as in previous similar computations obtained inRefs. [17,18], to treat the electrical conductivity as a con-stant in order to perform an analytic calculation. Thisconsists in dealing with EMF that evolves all the timein a conducting medium. In a more realistic picture, theelectric conductivity is temperature dependent, as shownin Fig. 1, therefore it depends on spacetime points in theplasma and decreases during the plasma expansion andcooling. Moreover, it is not trivial its behaviour in theearly stage before the onset of a mostly thermalized QGP,where σ can not be described by its equilibrium value de-termined by lattice QCD calculations. Hence, in the pre-equilibrium era the blue curve better approximates the B evolution in the collision before the plasma formation.Up to now we have considered the time evolution ofthe event-averaged EMF in symmetric collisions, whichis dominated by the magnetic field B y and the Faraday-induced electric field E x , whereas the others componentsare nearly vanishing. In asymmetric systems the differ-ence in the proton number of the initial nuclei generatesa substantial Coulomb electric field E x directed from theheavier nucleus towards the lighter one. Proton-inducedcollisions represent the most extreme case in which theEMF are basically produced by the heavy nucleus, lead-ing to very similar values for the magnetic field B y andthe electric field E x .Among the various theoretical models addressing thetopic of the EMF generated in non-central collisions, thePHSD approach described in Sec. 2.1 has been used toperform detailed calculations of the EMF and their ob-servable effects in all three cases of symmetric [10,59,16],asymmetric [11,20] and small systems [13]. In Fig. 3 weshow the PHSD calculations of the B y (upper panels) and E x (lower panels) components of the EMF produced attop RHIC energy in Au+Au collisions at b = 7 fm (a–b),Cu+Au at b = 7 fm (c–d) and p+Au collisions at b = 4fm (e–f). The field strength are computed at z = 0 and Lucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies t [fm] -6 -5 -4 -3 -2 -1 - e B / m π all charged particlesonly spectators b = 10 fmAu+Au @RHIC 200 GeV HSD [Voronyuk et al. (2011)] (a) t [fm] -6 -5 -4 -3 -2 -1 - e B / m π σ el = 0.023 fm -1 vacuum b = 7 fmPb+Pb @LHC 2.76 TeV [Gürsoy, Kharzeev and Rajagopal (2014)] (b) Fig. 2. (Color online) Temporal evolution of the magnetic field in the center of the overlapping region for symmetric heavy-ion collisions. Panel (a): results from Ref. [10] obtained with HSD simulations considering the fields produced by all chargedparticles (solid red line) or only spectator protons (dashed turquoise lines). Panel (b): results from Ref. [19] obtained by means ofsemi-analytical computations of the field in the vacuum (dashed blue line) or in a conducting medium with electric conductivity σ el = 0 .
023 fm − (solid orange curve). Au+Au b = 7 fm-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 π (a) Cu+Au b = 7 fm-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 π (c) p+Au b = 4 fm-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 π (e) Au+Au b = 7 fm-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 π (b) Cu+Au b = 7 fm-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 π (d) p+Au b = 4 fm-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 π (f) Fig. 3. (Color online) Transverse profile of the electromagnetic field components B y and E x in z = 0 at the maximum overlaptime of the two nuclei for Au+Au collisions with impact parameter b = 7 fm (a–b), Cu+Au collisions with b = 7 fm (c–d) andp+Au collisions with b = 4 fm (e–f) at √ s NN = 200 GeV. The calculations are performed by means of the PHSD model withthe inclusion of the electromagnetic fields [10,11,13]. Circles representing the position of the colliding nuclei are drawn to guidethe eye. at the maximum overlapping time of the collision, namelywhen the centres of the two colliding nuclei lie in the sametransverse plane. The circles drawn in all panels help toroughly identify the size and position of the nuclei in thethree collision systems and to highlight the interactionarea. We see that in Au+Au collisions the dominant B y component (a) reaches value | eB y | (cid:39) m π whereas E x (b) is nearly vanishing in the intersection area, as well as allother EMF components not shown in the figure.In the case of Cu+Au collisions, besides a huge magneticfield B y (c), also the electric field E x (d) reaches compa-rable values of the order of m π . It is directed from theheavier gold nucleus towards the lighter copper nucleus,hence the overlapping area is interested by an intense and ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 7 asymmetrically distributed electric field.The initial EMF produced in p+Au collisions correspondbasically to that produced by the gold nucleus movingat ultrarelativistic velocity and, consequently, it does notshow a significant dependence on the impact parameter ofthe collision. However, the impinging proton and the par-ticles created after the collision feel a different strength ofthe EMF depending on the collision point. The E x field(f) is strongly asymmetric inside the overlap area of non-central p+Au collisions and has a magnitude similar tothat of the B y component (e), both reaching values ofabout (cid:39) m π in collisions at b = 4 fm; the other EMFcomponents are almost vanishing.We have discussed the spacetime profiles of the EMFaveraged over many events (Fig. 2(a), Fig. 3) or obtainedwith event-averaged charge density (Fig. 2(b)).However, event-by-event fluctuations in proton posi-tions in the colliding nuclei lead to event-by-event fluctu-ations of the generated EMF [14,15,16]. This means that,even though on average the only nonvanishing componentof the field are B y and E x , in a single event other com-ponents of the EMF can reach very high and comparablevalues, (cid:104)| B x |(cid:105) ≈ (cid:104)| B y |(cid:105) ≈ (cid:104)| E x |(cid:105) ≈ (cid:104)| E y |(cid:105) . (11)This happens even in the central point of the overlap areain symmetric collisions, where from symmetry considera-tions one expect that on average the only nonvanishingcomponent of the field is B y , and also for very central col-lisions, where even B y is expected to disappear. Moreover, (cid:104)| B x |(cid:105) , (cid:104)| E x |(cid:105) and (cid:104)| E y |(cid:105) show a very mild dependence onthe impact parameter up to b ∼ R A , being R A the radiusof the nucleus.The estimates of the EMF strengths provide limitedinformation on the influence of the fields on particle prop-agation and final observables. A more clear picture arisesby looking at the total momentum increment ∆ p obtainedby summing over subsequent time steps the mean increaseof charged-particle momenta due to the action of the elec-tric and magnetic forces during the short time interval [59,16]. For all components of ∆ p the contributions from theelectric and magnetic fields in the Lorentz force (6) largelycompensate each other, with a tiny unbalance of the twoterms especially in the momentum increment along the x direction, as clearly shown for Au+Au collision at topRHIC energy in Refs. [59,16] by means of PHSD micro-scopic calculations. This compensation effect can be eas-ily illustrated in the simplified one-dimensional case for aparticle with charge e at position x = x ( t ): eE ∼ − e ∂A∂t = − e ∂A∂x ∂x∂t ∼ − eBv, namely the forces due to the electric and magnetic trans-verse components are roughly equal and inversely directed[59,16]. Thus, the influence of the EMF in relativistic col-lisions can not be reliably studied neglecting the electricfield; moreover, the impact on final observables is expectedto be small not only for the shortness of the interactiontime due to a fast decay of the fields but also for the par- tial cancellation of the forces determined by the transverseelectric and magnetic fields. The rapidity-odd v is sensitive to the vortical structuregenerated in the fireball because of the intense angularmomentum of the two colliding nuclei. In order to extractthe imprint of the EMF one has to study the charge-odddirected flow, by looking at particle and antiparticle ofa specific species separately or considering positively andnegatively charged hadrons. The intense EMF producedby the moving charges of spectators and newly producedparticles generate a sidewards deflection of quark and an-tiquarks in opposite direction according to their electriccharge, ending up with a different v of the final particlesin which they hadronize.The directed flow of light hadrons in ultrarelativisticHICs has been studied and discussed in many works; inparticular, the effect of the EMF has been highlighted inRefs. [19,24] within the hydrodynamic framework, mainlyfocusing on the rapidity dependence of the v of chargedpions, protons and antiprotons.In Ref. [24] the QGP evolution is described by means ofthe iEBE-VISHNU approach [60] which evolves the equa-tions of relativistic viscous hydrodynamic assuming lon-gitudinal boost-invariance. On top of this hydrodynami-cal background the Maxwell equations are solved throughEq. (10) for describing the evolution of the EMF generatedby electric charges and currents due to spectator protonsand evolving in a plasma with constant electrical conduc-tivity, as explained in Sec. 2.1.From the momentum distribution of particles with differ-ent charge coming out from the calculation, the splittingbetween the directed flow for positively and negativelycharged particles is evaluated by the difference ∆v ≡ v (+) − v ( − ) , (12)in order to isolate the v induced by the EMF from themuch larger contribution due to the background hydrody-namic flow.Some of the results of the calculations of Ref. [24] for the v splitting of π + and π − are shown in Fig. 4. In all pan-els the solid green line correspond to ∆v of pions in thetransverse momentum range 1 < p T < − √ s NN = 200 GeV. Panel (a) rep-resents the comparison with different transverse momen-tum integration range: the ∆v clearly increases as the p T values increase. The authors found the the same trendholds for the ∆v of protons–antiprotons. In panel (b) thecentrality dependence is depicted, showing that the ∆v of pions increases going from central towards peripheralcollisions. This is in line with the fact highlighted alsoin other works [10,19] that the effect of EMF on finalobservables, in particular on the directed flow, is mainlydriven by the fields produced by spectator protons withrespect to the smaller contribution generated by the par-ticipants; indeed, more peripheral collisions correspond to Lucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies -2 -1 0 1 2 y -6-4-20246 ∆ v ( × ) p T ∈ [0, 1] GeVp T ∈ [1, 2] GeVp T ∈ [2, 3] GeVAu+Au @ 200 GeV20 −
30 % (a) -2 -1 0 1 2 y -8-6-4-202468 ∆ v ( × ) − −
20 %20 −
30 %30 −
40 %40 −
50 %Au+Au @ 200 GeVp T ∈ [1, 2] GeV (b) -2 -1 0 1 2 y -3-2-10123 ∆ v ( × ) Au+Au @ 200 GeVPb+Pb @ 2.76 TeVPb+Pb @ 5.02 TeVp T ∈ [1, 2] GeV20 −
30 % (c)
Fig. 4. (Color online) Rapidity dependence of the electromagnetically-induced splitting of the directed flow of pions ∆v π = v ( π + ) − v ( π − ) in the transverse momentum range 1 < p T < −
30% Au+Au collisions at √ s NN = 200 GeV (solidgreen line) in comparison with the v in different transverse momentum ranges (a), centrality classes (b) and collision energies(c). The results are from Ref. [24] and are obtained with hydrodynamical calculations within the iEBE-VISHNU frameworktaking into account the electromagnetic fields. a higher number of spectators. The electromagnetically-induced v splitting of pions decreases as the collisionenergy increases, as shown in the right panel of Fig. 4,where the result at top RHIC energy is compared withPb+Pb collisions at √ s NN = 2 .
76 TeV (dot-dashed blueline) and √ s NN = 5 .
02 TeV (dashed red line). The rea-son is that in collisions at higher energy the spectatorsmove away more quickly from the centre of the collision,hence the EMF generated by them (which is the maincontribution) experience a faster decrease with time; as aconsequence their effect is milder than at lower energy col-lisions. The same qualitative behaviour according to thecharge of the particle has been shown in Ref. [24] for thedifference in the directed flow of protons and antiprotons ∆v p ≡ v ( p ) − v ( p ): at forward rapidity v ( p ) < v ( p )and at backward rapidity v ( p ) > v ( p ).Hence, at y > p , π + ) arepushed downward and negatively charged particles ( p , π − )are pushed upward along the impact parameter directionby the Lorentz force due to the EMF; this small splittingcomes out from the almost complete local cancellation ofthe two opposite forces due to the electric part and themagnetic part of the Lorentz force and indicates that thetotal effect is mildly dominated by the electric field.The results of Ref. [24] shown in Fig. 4 give an overviewof the theoretical expectations for the rapidity dependenceof the electromagnetic splitting of the v of charged pionsas a function of the transverse momentum interval, thecentrality class and the beam energy of the collision. Pi-ons are somehow a more clean probe of the EMF withrespects to other light hadrons in which there are contri-butions of ∆v coming from other sources. However, thefinal effect is tiny and requires very precise experimentalmeasurements.Besides light hadrons, very promising and interestingprobes of the initial EMF are the heavy mesons, in partic-ular the neutral D and D , as predicted in Ref. [22] bymeans of the Langevin dynamics for charm and anticharmquarks in an expanding QGP background. The initial con-ditions for solving the relativistic Langevin equation forthe heavy quarks and the relativistic transport code for the bulk medium are constrained by the experimental dataon the nuclear modification factor R AA ( p T ) and the ellip-tic flow v of D mesons [61] and the transverse momen-tum spectra and the v of the bulk [62,63]. The authorsof Ref. [22] show that the effect of the EMF on the v of charm quarks is significantly larger than that of lightquarks because the heavy quarks, being their formationtime scale of the order of 0.1 fm /c (much smaller than thatof light quarks), are already present when the EMF reachtheir maximal magnitude. Moreover, the relaxation timeof charmed particles is comparable to the QGP lifetime,thus helping them to retain the initial acceleration in thetransverse direction caused by the electromagnetic field.Hence, D and D , in spite of being neutral, are splittedby the EMF due to the electric charges of their heavy con-stituents. This can also be considered a further evidenceof the presence of a deconfined phase in relativistic HICs.In recent years the experimental results for the di-rected flow of charged hadrons as well as of identified lightand heavy mesons has been reported by the STAR [64]and ALICE Collaborations. The recent efforts have beenfocused to measure not only the combined v of particle–antiparticle but also its difference, in order to extract theinfluence of the EMF through the charge-dependence ef-fect that involves the final hadrons and their productionchannels. In particular, we will discuss the experimentaldata for the directed flow of neutral D mesons in compar-ison to the theoretical expectations and to the measure-ments for light hadrons.Fig. 5 shows the STAR data in 10–80% Au+Au colli-sions at RHIC energy of √ s NN = 200 GeV [64] along withpredictions from theoretical models [22,35,23,36]. In panel(a) the rapidity dependence of the averaged directed flow (cid:104) v (cid:105) of neutral D mesons for p T > . p T > . v is often identifiedby its slope dv /dy at midrapidity. By fitting the v ( y ) ofthe averaged D and D with a linear function constrainedto pass through the origin, the STAR Collaboration found ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 9 − − 〉 v 〈 − STAR )cc +uu ( D + D )ss + uu ( + + K − K ) D + Model:(DHydro+EM (Chatterjee et.al.) AMPT =200 GeV, 10 80% NN sAu+Au a) y − − / K ) R a t i o ( D Rapidity (y) − − v ∆ − D D )ss uu ( + K − K) D Model:(DEM (Das et. al.)Hydro+EM (Chatterjee et.al) b) Fig. 5. (Color online) Directed flow average (cid:104) v (cid:105) (a) and dif-ference ∆v (b) of heavy and light mesons as a function ofrapidity for 10–80% Au+Au collisions at √ s NN = 200 GeVmeasured by the STAR Collaboration [64] in comparison tothe theoretical calculations. Filled red symbols represent theexperimental data for D and D mesons in the transversemomentum range p T > . v of K + and K − integrated for p T > . v of neutral D mesons and that of charged kaons. The errorbars and caps represent, respectively, statistical and system-atic uncertainties. In panel (a) the solid red line correspondto hydrodynamic simulations with the inclusion of the elec-tromagnetic fields (“Hydro+EM”) from Refs. [35,23] and thedashed brown line is the results of transport calculations withthe AMPT model [36]. In panel (b) the dotted blue line denotesthe prediction reported in Ref. [22] with a Langevin approachcoupled to the electromagnetic fields (“EM”) and the solid redline is the calculation within the “Hydro+EM” framework [35,23]. The figure is taken from Ref. [64]. the fit d (cid:104) v (cid:105) D /dy = − . ± .
017 (stat . ) ± .
016 (syst . );the fit for the slope of charged kaons is d (cid:104) v (cid:105) K /dy = − . ± . . ) ± . . ) . Therefore, the STAR Collaboration observed an absolutevalue of the neutral D mesons dv /dy that is about 25times larger than that of the charged kaons with a 3 . σ significance [64] and is the largest among that of all theeleven particle species measured at top RHIC energy [28,30,32]. This supports the idea that heavy mesons are ex-cellent probes to investigate the early dynamics of nuclear collisions.The “antiflow” behaviour – the negative slope at midra-pidity – of the directed flow of D mesons at top RHICenergy has been predicted fifteen years ago in transportcalculations within the Hadron-String Dynamics (HSD)approach [34]. The authors of Ref. [35,23], by means of aLangevin dynamics for heavy quarks coupled to hydrody-namic equations for the bulk medium, have highlightedfor the first time that the v of D mesons in noncen-tral collisions is several times larger than that of chargedparticles; their prediction (solid red line labelled as “Hy-dro+EM”) is shown in Fig. 5(a) along with the result fromA Multi-Phase Transport (AMPT) model [36] (dashedbrown curve). Even though the qualitative behaviour ofthe D -meson v is captured by the theoretical calculations,the experimental data are quantitatively underestimated.In Fig. 5(b) the experimental data for the directed flowsplitting of neutral D mesons ∆v D = v ( D ) − v ( D )measured by the STAR Collaboration [64] is shown withred squares together with the theoretical predictions fromRef. [22] labelled by the dotted blue line (“EM”) and fromRef. [23] represented by the solid red line (“Hydro+EM”).The authors of Ref. [22], by solving the Langevin equa-tion in an expanding QGP background described with arelativistic Boltzmann approach, have predicted that the v splitting of D and D mesons is orders of magnitudelarger than that of light hadron species, since the heavy c and c quarks are produced in hard scatterings and hencefeel the high early EMF. A similar result has been ob-tained within the Langevin+hydrodynamics approach ofRef. [35] with the inclusion of the EMF [23]. The linearfit of the ∆v slope (constrained to intersect the origin)provided by STAR is given by d∆v D /dy = − . ± .
034 (stat . ) ± .
020 (syst . );within the present uncertainties the data are consistentalso with a zero slope.The theoretical results of the ∆v of D mesons are de-pendent on the assumed electric conductivity of the bulkmedium which affect the time evolution of the EMF, asdiscussed in Sec. 2, and on the description of the dynamicsof charm and anticharm quarks in the QGP. Both mod-els predicted a nonzero slope [22,23] which is in the rightballpark considering the current experimental errors.The ALICE measurements of the directed flow v ofcharged hadrons and heavy mesons as a function of pseu-dorapidity η in Pb+Pb collisions at √ s NN = 5 .
02 TeV [33]are shown in Fig. 6.In panel (a) red circles and blue diamonds represent, re-spectively, the directed flow of positively and negativelycharged hadrons integrated for transverse momenta p T > . √ s NN = 2 .
76 TeV and √ s NN = 5 . ∆v h = v ( h + ) − v ( h − ) isshown in panel (c) with violet squares. The ALICE Collab- -0.500.5 v h + × h - × D D -0.5 0 0.5 η -0.500.5 ∆ v (h + - h - ) × -0.5 0 0.5 η D - D T < 6 GeV Pb+Pb @ LHC 5.02 TeV ALICE T < 6 GeVp T > 0.2 GeVp T > 0.2 GeV 10-40%5-40% 10-40% (a) (b)(c) (d) Fig. 6. (Color online) Pseudorapidity dependence of the di-rected flow and its splitting measured by ALICE Collabora-tion [33] for Pb+Pb collisions at √ s NN = 5 .
02 TeV. Leftpanels: (a) v of positively (red circles) and negatively (bluediamonds) charged hadrons and (c) its difference ∆v ( h ) = v ( h + ) − v ( h − ) (violet squares) in the 5–40% centrality classintegrated for transverse momenta p T > . v of D (green circles) and D (orange diamonds)mesons and (d) its difference ∆v ( D ) = v ( D ) − v ( D )(brown squares) for 10–40% central collisions and 3 < p T < oration extracted a fit of its slope with a linear function: d∆v h /dη = [1 . ± .
49 (stat . ) ± .
41 (syst . )] × − , indicating a positive value with a significance of 2 . σ .The theoretical prediction for the v splitting between π + and π − at LHC energy obtained in Ref. [24] with hydro-dynamic simulations coupled to the EMF and shown inFig. 4(c) gives a ∆v of the same order of magnitude ofthat between positively and negatively charged hadronsmeasured by ALICE but with the opposite sign. How-ever, those calculations are able to capture the effect ofthe EMF on the v but do not account for another sourceof v that is the vorticity and the angular momentum ofthe colliding system and the baryon stopping mechanism,which act in a different way on K + and p from one handand on K − and p from the other hand [30,32,29]; indeed,the latters are constituted of only newly produced par-tons and follow the “antiflow” behaviour of the directedflow at midrapidity (negative slope), while K + and p getcontributions from the flow of the initial nuclei due tothe up and down quarks from which they hadronize. Thissource of v gives similar contribution to π + and π − sothat the splitting due to it is absent or negligible for pi-ons. This effect is clearly seen in transport simulationswith the PHSD model for the v of pions and kaons inp+Au collisions at top RHIC energy [13] shown and dis-cussed in Sec. 4. This source of ∆v should be added tothat induced by the EMF and the opposite sign betweentheoretical calculation that do not include this effect [24]and the experimental data [33] may indicate that in sym- metric collisions the electromagnetically-induced v split-ting is subdominant with respect to the contribution fromthe flow of the initial nuclei (this is not the case for asym-metric collisions as we will discuss in the next section). Itcan be surprising that this holds even at LHC energies,since the baryon stopping effects are observed to decreasewith increasing collision energy, as indicated by the STARdata at different √ s NN [30,32] and by the measurementof a smaller magnitude of v at LHC energy [29] with re-spect to lower energies. Nevertheless, the contribution tothe flow from the initial nucleons can be significant, espe-cially in the proton and antiproton v splitting as well asin those between charged kaons, therefore influencing thecharge dependence of the inclusive hadron v in oppositeway to the prediction for the EMF effect [24]. Anotherpossible reason of the discrepancy between theory and ex-periment for LHC collisions is that the description of theEMF is not enough realistic to account for the tiny un-balance between the magnetic contribution in the Lorentzforce and the electric one and one needs to perform a fullcalculation which accounts for the back-reaction of theaccelerated particles on the fields themselves and the pre-equilibrium effects of the field evolution in the early stage.The ALICE Collaboration presented also the data forthe v of D and D in the 10–40% centrality class [33].The directed flow of neutral D mesons is shown in Fig. 6(b):the data favour a positive slope for the pseudorapidity de-pendence of the v of D (green circles) and a negativeslope for that of D (orange diamonds) with a significanceof about 2 σ . This is different from the observations at topRHIC energy, where a negative slope is found for bothparticles [64]. Furthermore, the v for D mesons is aboutthree orders of magnitude larger than that measured forcharged particles shown in panel (a) and this should notbe explainable with the different transverse momentumintegration ranges.The theoretical predictions, based on the Langevin ap-proach embedded in a kinetic [22] or hydrodynamic [23]framework, underestimate for about one order of magni-tude the ALICE data for the directed flow splitting be-tween D and its antiparticle depicted in Fig. 6(d); thecorresponding linear fit extracted by ALICE is d∆v /dη = [4 . ± . . ) ± . . )] × − , suggesting a positive slope with a significance of 2 . σ . Thetheoretical calculations [22,23] predicted a negative slopealso at LHC energies, as for the RHIC case. Hence, thesign of the slope of the d∆v D observed by ALICE is indisagreement with both the theoretical expectations andthe STAR measurements at √ s NN = 200 GeV [64] shownin Fig. 5(b). As discussed for the case of charged hadrons,also for the v splitting of D mesons the tension betweentheory and experiment at LHC energy may be related toan underestimated influence of the magnetic field. Nev-ertheless, the much larger effect observed for the heavymesons supports the role attributed to them to be a sensi-tive probe of the initial EMF produced in ultrarelativisticHICs. ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 11 v − − |<1 η CuAu 200GeV | PHSD+EMF iLPM + h PHSD+EMF iLPM h
10 20%
STAR ) PHSD (h ) v + (h v [GeV/c] T p0 0.5 1 1.5 2 2.5 3 v ∆ − −
20 30% [GeV/c] T p0 0.5 1 1.5 2 2.5 3
30 40% [GeV/c] T p0 0.5 1 1.5 2 2.5 3
40 50% [GeV/c] T p0 0.5 1 1.5 2 2.5 3
50 60% [GeV/c] T p0 0.5 1 1.5 2 2.5 3 Fig. 7.
Transverse momentum dependence of the directed flow of positively and negatively charged hadrons (upper panels)and their difference ∆v h = v ( h + ) − v ( h − ) (lower panels) for Cu+Au collisions at √ σ NN = 200 GeV and various centralitiescomputed by the PHSD model in comparison to the experimental data (star markers) from STAR Collaboration [31,65]. Thefigure is taken from Ref. [20]. In the previous section we have discussed the theoreticaland experimental results for the effect on the directed flowof the EMF produced in symmetric Au+Au and Pb+Pbcollisions, where the dominant components are the mag-netic field B y mainly produced by the spectator chargesand the electric field E x generated by Faraday inductiondue the time decrease of the magnetic field. In asymmetricsystems, e.g. Cu+Au, the larger components are still B y and E x , but the latter includes a big contribution from theCoulomb field due to the different number of protons in-side the colliding nuclei, so that a substantial electric fielddirected from the heavier to the lighter nucleus is createdin the interaction area, as shown in Fig. 3(d).The EMF produced in Cu+Au collisions along with theirinfluence on the directed flow of light hadrons as a functionof rapidity and transverse momentum p T has been studiedwith the PHSD approach in Refs. [11,20]. For collisions attop RHIC energy they found that without the EMF the v of charged pions as a function of transverse momen-tum p T varies between 0 . −
1% in absolute value. Theinclusion of the EMF in the simulation splits the distribu-tions of positively and negatively charged pions, pushingthe v ( π + ) upward and v ( π − ) downward with respectto the case without EMF; moreover, the charge splitting ∆v π = v ( π + ) − v ( π − ) increases with increasing p T .In Fig. 7 we show the PHSD results from Ref. [20] forthe transverse momentum dependence of the directed flowof positive ( h + ) and negative ( h − ) charged hadrons inte-grated in the pseudorapidity interval | η | < √ σ NN = 200 GeV in five central-ity intervals. In the upper panels the v of h + and h − isplotted separately with magenta circles and blue squares respectively, whereas the lower panels show the difference ∆v h = v ( h + ) − v ( h − ) in comparison to the experimen-tal data from the STAR Collaboration [31]. The directedflow splitting of charged hadrons is about 10% of the v magnitude [31]. For p T < ∆v h is negative, namely h − particles have a larger v than h + with respect to theevent-plane determined from spectators in the Au-goingdirection . This is in line with the expectation from thedirection of the x component of the electric field whichpoints from the Au nucleus towards the Cu nucleus [11,12]; therefore, in the considered reference frame (Cu at x > ∆v becomesconsistent with zero. It tends to disappear also going to-wards more peripheral collisions, as visible for the 50%–60% centrality class.Since in asymmetric collisions the Coulomb electricfield due to the difference in proton number of the collidingnuclei governs the influence of the EMF effect on particle v , one can expect that the impact increases going to ex-tremely asymmetric systems, such as proton-nucleus colli-sions. The directed flow of light mesons in p+Au collisionsat top RHIC energy has been studied, for the first timewithin the transport framework, by means of the PHSDapproach [13]. The effect of the EMF has been disentan-gled performing simulation with and without the inclusionof those fields and the impact on final observables has been In Refs. [20] and [31] the directions of the initial Au andCu nuclei are inverted and different conventions for the event-plane determination are used.2 Lucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies -5 -4 -3 -2 -1 0 1 2 3 4 5 y v ( % ) π + π _ π + , EMF π _ , 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 ( % ) π + π _ π + , EMF π _ , 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. 8. (Color online) Directed flow in percentage of pions (a-c) and kaons (b-d) as a function of rapidity for b = 2 fm (a-b)and b = 6 fm (c-d) p+Au collisions at √ s NN = 200 GeV obtained with PHSD simulations with (solid curves) and without(dashed curves) electromagnetic fields. The inset panels are zooms of the rapidity window | y | <
2, with arrows highlighting inwhich direction the presence of the electromagnetic fields affects the v observable. The figure is taken from Ref. [13]. investigated both in minimum bias collisions and for fixedimpact parameter.In Fig. 8 we show the results of Ref. [13] obtained withthe PHSD calculations for the rapidity dependence of the v of pions (a-c) and kaons (b-d) in p+Au collisions at √ s NN = 200 GeV with impact parameter b = 2 fm (a-b)and b = 6 fm (c-d). Along with the complete simulationscorresponding to the solid curves, we plot by dashed linesthe results without the inclusion of EMF. For each plot azoom of the v at central rapidity is depicted in the insetpanel. In this calculations the incoming proton moves at x > x < π + (dashed blue line withsquares) and π − (dashed red line with circles) have thesame v , with mild difference only at rapidity y (cid:46) − π + (solid blue line with squares) are pushed to-wards the positive x − direction and oppositely π − (solidred line with circles) are kicked along the negative x ; thewhole effect is to split the v of the pions with oppositeelectric charge. Comparing simulation at b = 2 fm (a-b)with those at b = 6 fm (c-d), we see that the push ofthe EMF is stronger for higher impact parameters; this iscomprehensible by looking to the plots of the electric andmagnetic fields in Fig. 3(e-f).We now discuss the directed flow of kaons shown in panels(b) and (d), which is not only influenced by the EMF butkeep trace also of the strong vorticity embedded in thesystem. From the dashed curves, we see that K + (greenline with squares) and K − (orange line with circles) showa splitting in the v even in simulations in which the EMFare switched off. This interesting effect, already discussedin Sec. 3, can be linked to the intense angular momentumowned by the two impinging nuclei, that is strongly keptby the u and d quarks inside the initial nucleons and partlytransferred to the produced hot QGP. The huge angularmomentum of the initial partons is visible in K + ( su )which receive more contributions with respect to K − ( su ) ucia Oliva: Electromagnetic fields and directed flow in large and small colliding systems at ultrarelativistic energies 13 from the sidewards deflection of the heavy nucleus [66,32];as a consequence, K + has a smaller v with respect to K − at backward rapidity, that is the Au-going direction. How-ever, accounting for the EMF (solid lines) the distributionof K + is pushed upward and that of K − is pushed down-ward at y <
0; this electromagnetically-induced splittingdominates over that generated by the initial vorticity ofthe system, so that the final result presents the same trendas seen for the pions.The role of the EMF on the charge-dependent v in proton-nucleus collisions can be understood in a similar way as forthe Cu+Au colliding system. For both pions and kaons,the sign of the splitting in v results from the balancebetween sideways pushes on charged particles by the elec-tric and magnetic part of the Lorentz force (6) [11,20,19,24,22,25,23]. We can distinguish two contributions in theelectric field: the Faraday induction due to the decreaseof the magnetic field and the Coulomb interaction dueto the difference in proton number of the colliding sys-tem. The force exerted by the Faraday-induced E x hasopposite directions at forward and backward rapidity andgoes in both regions against the push of the magnetic field B y . The x -projected Coulomb electric field, instead, hasthe same direction at both positive and negative y . Inthe backward hemisphere, where the main QGP produc-tion occurs, Coulomb and Faraday electric fields sum up:according to our convention for the reference frame, theelectric part of the Lorentz force pushes positively (neg-atively) charged particles along the positive (negative) x direction while the magnetic contribution has the oppositeimpact. As highlighted by the arrows drawn in the insetsof Fig. 8, the winner of this force competition in p+Aureactions is E x , attracting π − and K − towards the beamaxis and pushing far π + and K + . The splitting increasesfor higher (absolute) values of rapidity and for larger im-pact parameters. Furthermore, it depends also on particlespecies, having a stronger impact on kaons with respectto pions. This can be related to the different mass andconsequently different velocity of the two meson species:slower particles (the kaons) feel longer the influence of theEMF and undergo to a smaller force from the magneticfield ( F B ∝ v × B ) that leads to a smaller compensationof the electric field push. In this review the main approaches adopted in theoret-ical calculations for describing the generation and timeevolution of the electromagnetic fields (EMF) in ultrarel-ativistic nuclear collisions has been discussed, highlightingthe significant differences in the fields produced in sym-metric and asymmetric systems. Indeed, in the latter case,the different number of protons of the two colliding nucleigenerates a more inhomogeneous distribution of the EMFand give rise to substantial values of some componentsrelatively smaller in the symmetric case. This asymmetryis taken to its extreme in the case of proton-nucleus colli-sions, where the profiles of the fields are almost completelydetermined by the heavy ion. The EMF drive many interesting effects, some of whichare also affected by the fascinating early-time dynamicsof nuclear collisions when the EMF attain their maximalstrength and the system is in a very anisotropic and out-of-equilibrium state characterized also by the intense vor-ticity induced by the huge angular momentum owned bythe colliding nuclei.The presence of the EMF breaks the spherical symme-try in the plane perpendicular to the beam direction andinduces an azimuthal asymmetry in the final particle dis-tribution. In particular, due to the combined geometry ofEMF and QGP expansion, the directed flow v has beenconsidered as the most promising flow harmonics to seethe impact of the EMF in heavy-ion collisions and smallcolliding systems. Its magnitude is determined also by theinitial-state fluctuations and the QGP vortical patterns,but its charge-dependent behaviour gives direct access tothe EMF strength and decay rate, which in turn give alsoinformation on the presence of electric charges in the early-stage as well as the electromagnetic response of the QGP.The recent theoretical and experimental results on thedirected flow of light hadrons and heavy mesons has beenreviewed. Kinetic calculations of the directed flow of lightmesons in p+Au collisions at top RHIC energy clearlyshow the importance for the numerical codes to be ableto capture the effect of the EMF on the v but also to donot miss other sources of v such as the vorticity of themedium and the baryon stopping mechanism, which actin different way on some particles (e.g., K + and p ) withrespect to their antiparticles ( K − and p ), hence inducinga charge-dependent v which has to be combined to theelectromagnetically-induced splitting. In heavy-ion colli-sions a much larger effect on the neutral D mesons withrespect to light particles has been predicted by the theo-retical models for the v generated by the EMF and theinitial tilt of the fireball; the recent experimental obser-vations support the role attributed to the heavy flavourto be a sensitive probe of the early dynamics of ultrarel-ativistic collisions, due to their strong sensitivity to thevorticity and the EMF there produced. However, the re-cent experimental measurements of the v of D and D at LHC energies cannot be explained by the current theo-retical calculations and show opposite behaviour with re-spect to the data in RHIC collisions, suggesting that atthe highest energies there may be a not trivial and notyet understood interplay between the initial-state effectsand the EMF; this could lead to a dominant impact of themagnetic field over those caused by the electric field andby the fireball vorticity. It would be interesting to see ifthe trend observed at LHC in symmetric Pb+Pb collisionsis maintained in asymmetric systems (e.g., p+Pb) wherethe huge electric field should oppose more to the magneticfield.More efforts are needed both on the theoretical andthe experimental side to disentangle the role of the EMFfrom the other sources on the generation of a charge-odddirected flow. A more detailed scan of various systems andenergies is desirable, from heavy-ion collisions to proton-induced reactions at the different beam energies. The cur- rent tension between theory and experiment makes clearthe importance to perform precise measurements and tofurther develop the numerical codes able to describe allthe phenomena taking place especially in the early-stageof the collisions. Acknowledgements
The author appreciates very useful discussions with ElenaBratkovskaya, Wolfgang Cassing, Salvatore Plumari, MarcoRuggieri, Olga Soloveva, Vadim Voronyuk. The authoris financially funded by the Alexander von Humboldt-Stiftung and acknowledges support from the COST ActionTHOR CA15213 and from the Deutsche Forschungsge-meinschaft (DFG) through the grant CRC-TR 211 ’Strong-interaction matter under extreme conditions’.
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