Spectator induced electromagnetic effect on directed flow
in heavy ion collisions
Abstract
We estimate the electromagnetic effect of the spectator charge on azimuthal anisotropies observed in heavy ion collisions. For peripheral Pb+Pb reactions at the top energy of the CERN Super Proton Synchrotron, GeV, we predict this effect to bring very large distortions to the observed directed flow, , of positive and negative pions emitted close to beam rapidity. The overall magnitude of this effect is comparable to values of reported by the WA98 experiment. We argue that also at lower rapidities, the spectator induced electromagnetic effect may result in the splitting of values of observed for positive and negative pions. Such a splitting is visible in the data reported by the STAR Collaboration from the RHIC Beam Energy Scan.
Both effects are sensitive to the spacetime scenario assumed for pion emission. Therefore, they bring new information on the collision dynamics.
pacs:
25.75.q,25.75Dw,25.75.LdI Introduction
Azimuthal correlations between particles and the reaction plane, observed in noncentral collisions, constitute one of the main subjects of heavy ion collision studies. The primary nucleonnucleon collisions being believed to be insensitive to the reaction plane, the latter phenomena are considered to be the source of information on collective effects present in the heavy ion reaction. A rich phenomenology Snellings1999 (); Csernai2012 (); Luzum2010 (); Retinskaya2012 (); florkowski () has therefore been developed. Among others, the success of the hydrodynamical approach bozek () in describing these phenomena is to be quoted here.
The azimuthal correlations are usually quantified in terms of the Fourier coefficients of the azimuthal distribution of the outgoing particles w.r.t. the reaction plane. These are defined by poskanzerprc ():
(1) 
where denotes the azimuthal angle of the emitted particle, while is the orientation of the reaction plane defined by the impact parameter vector . Specifically, the firstorder coefficient
(2) 
reflects the sidewards collective motion of the particles and is known as directed flow. A very sizeable amount of experimental data on directed flow is at present available from the low energy regime up to the LHC Andronic2001 (); Barrette2000 (); Barrette1998 (); na49prc (); Adams2005 (); Back2005 (); Abelev2008 (); Adamczyk2011 (); Selyuzhenkov2011 (). It is known that depends on collision centrality, particle type, transverse momentum and rapidity. For instance, the rapidity dependence of pion directed flow at SPS energies is known to consist of a smooth passage through zero at midrapidity na49prc (), while large values of , of the order of 0.20.25, are reported in the vicinity of target rapidity, in semicentral collisions wa98 ().
In the present paper we point at the possible electromagnetic component of directed flow. We estimate the size of the spectatorinduced electromagnetic interaction on the collective motion of positive and negative pions. The spectatorinduced electromagnetic effect on charged pion ratios was discussed in our earlier paper twospec (). This effect is in fact observed in experimental data rybickizako (), and results in very large distortions in ratios of charged pions observed at high rapidities and low transverse momenta. Measurements of directed flow offer a new way of spotting the influence of the spectator charge on the emission of final state particles. They also provide a new source of information on the spacetime evolution of the nonperturbative process of pion production^{1}^{1}1The cited paper twospec () also contains a description of various works made in the past on electromagnetic effects in low and high energy nuclear collisions. For more details on this subject, see Bartke09 (); Yagoda52 (); Friedlander62 (); Beneson79 (); Sullivan82 (); Karnaukhov06 (); Ayala97 (); Ayala99 (); Ahle98 (); Lacasse96 (); Xu96 (); Libbrecht79 (); Cugnon81 (); Gyulassy81 (); Bonasera87 (); Li95 (); Barz98 (); Osada96 ()..
Our present analysis will be focussed on peripheral Pb+Pb collisions occurring at the top energy available to SPS heavy ion experiments ( GeV). We choose peripheral collisions because they are characterized by the largest spectator charge and also large values of directed flow. We focus on the SPS energy range by virtue of the abundance of experimental data on the rapidity dependence of , which also gives the possibility of comparison with our results as explained in Section III. We note that the energy range of the SPS partially overlaps with that of the RHIC Beam Energy Scan, addressed in short in Section V.
This paper is organized as follows. Section II contains the description of our Monte Carlo model. In Section III we present the results of our simulation, together with the comparison to experimental data. The discussion of our results is made in Section IV. In Section V, we comment on the experimental data from the RHIC Beam Energy Scan. Our conclusions are presented in Section VI.
Ii The model
A detailed description of our model can be found in twospec (). Only the aspects that are relevant for the present analysis will be listed below. Our aim is to obtain a realistic estimate of the influence of the spectatorinduced electromagnetic interaction on the directed flow of pions. On the other hand, we wish to avoid the detailed discussion of the complex and poorly known mechanism of soft particle production. Therefore we decide on a maximally simplified approach:

We assume, as an example, a peripheral Pb+Pb collision involving participating nucleons at GeV. This is shown in Fig. 1. The two spectator systems are modelled as two homogeneous, Lorentzcontracted spheres. The reaction plane is defined by the collision axis (dashed line in the figure) and the impact parameter vector .

Charged pions are assumed to be emitted from a single point in position space, that is, the original interaction point. The time of pion emission is a free parameter of our model; this parameter sets the initial conditions for the electromagnetic interaction. For peripheral Pb+Pb collisions studied here, the initial spectra of charged pions are assumed to be similar to those in nucleonnucleon collisions.

Charged pions are then numerically traced in the electromagnetic field of the spectator charges until they reach the distance of 10,000 fm away from the interaction point and from each of the two spectator systems. The fragmentation of the spectator systems is neglected. We do not consider the effects of participant charge, strong final state interactions, etc.
Several clarifications should be added to the above.

The simplified geometry of the peripheral collision presented in Fig. 1 was determined on the basis of a dedicated study, discussed in detail in twospec (). A geometrical Monte Carlo simulation served this purpose. This used proper nuclear density profiles mizutori () and assumed the elementary nucleonnucleon crosssection equal to 31.4 mb, in good agreement with experimental p+p data at this collision energy pp (). The geometrical impact parameter corresponding to 60 participating nucleons was found to be fm. Additionally, the center of gravity of each of the two spectator systems was found to be displaced by fm relative to that of the original nucleus. The average spectator charge was elementary units.
Considering the exact shape of the spectator system as unimportant for our subsequent studies, and for the sake of clarity, we modelled the two spectator systems as homogeneously charged spheres. In the rest frame of each sphere, its density was the standard nuclear density /fm. The center of each sphere was additionally displaced by 0.76 fm in order to match the center of gravity of the spectator system. Thus our effective impact parameter (distance of closest approach between the two spheres’ centers) was fm. 
The simplification of initial conditions for pion emission (reduction of the emission zone to a single point in space and time, as discussed above) gives a convenient way to estimate the sensitivity of the electromagnetic effect to the basic characteristics of pion production (the pion formation time, the distance between the pion formation zone and the two spectator systems, etc). The initial momentum spectra of pions (before the action of the electromagnetic field) are assumed to obey wounded nucleon scaling wnm () and to be similar to underlying nucleonnucleon events. As such, they are described by an analytical parametrization of average pion distributions in p+p collisions, recorded by the NA49 experiment at the SPS pp (). More details on this parametrization are given in twospec (). For simplicity, distributions of positive and negative pions are assumed to be identical. The experimental data cited above are expressed in terms of the Feynman variable^{2}^{2}2All the kinematical variables addressed in this paper will be defined in the c.m. system of the collision. and of transverse momentum , and cover the region from to and from to GeV/c. Only a small extrapolation towards higher needs to be applied in the present analysis. The uncertainty of this extrapolation has little or no effect on the results presented in Section III. It should be underlined that full azimuthal symmetry is assumed for the initial emission of pions.

Charged pions, with their initial momentum vector defined above in point (b), and weighted by their initial distribution, are subjected to the electromagnetic field of the two spectator systems. The pion trajectory is given by the classical relativistic equation of motion:
(3) where is the pion mass, and the pion momentum is defined by the Lorentz force acting on the pion:
(4) Here, is the pion charge, while and are standard superpositions of fields from the two spectator systems. Our equations take account of relativistic effects, including retardation. Technically, the propagation of the pion is performed numerically by means of an iterative procedure made in small steps in time, with variable step size. The iteration proceeds until the pion is 10,000 fm away from the interaction point and from each of the two spectator systems in their respective rest frames. Negative pions which do not escape from the potential well induced by the spectator system are rejected and do not enter into our final state distributions.
Iii Results
In this Section we present the results of our analysis. These will be displayed as a function of the scaled rapidity , where and are the rapidity of the pion and of the incoming nucleus in the collision c.m. system. We decide on this particular variable in order to simplify the interpretation of our results, and for an easier possible comparison with other collision energies (see also the discussion made in na49prc (); starPandit ()).
iii.1 Electromagnetic effect from one and two spectators
We start by discussing the basic features of the spectatorinduced electromagnetic effect on the directed flow of pions. In particular, this will include the role of each of the two spectator systems in the overall effect. Fig. 2 shows the directed flow of and , induced by their electromagnetic interaction with the spectator charges. The value of is integrated over the transverse momentum of the pion, from to 1 GeV/c. All the simulations presented in the figure assume the simplest situation where the pion emission time is equal to zero (immediate pion creation).
It should be underlined that as our simulation contains no initial azimuthal anisotropy (Sect. II), any nonzero value of apparent in Fig. 2 is indeed solely due to the electromagnetic interaction with the spectator systems. Therefore, we will refer to it as “electromagneticallyinduced directed flow” in order to differentiate it from the “standard” flow phenomena which are caused by the strong interaction. In this context, several remarks are in order.

As it is clearly apparent from the figure, the electromagnetic field induced by the spectator charges exerts a strong influence on the directed flow of charged pions. It can be held responsible for large values of , exceeding 0.2 (20%) for negative pions close to beam rapidity.

The directed flow induced electromagnetically by the two spectators (red solid curve, which goes through the black dots in the figure) exhibits a characteristic structure as a function of rapidity. Qualitatively, this structure recalls the overall behaviour of pion directed flow as it is reported from experimental data (see e.g. wa98 (); na49prc ()). The directed flow displays a smooth transition through zero at , with large values of flow (antiflow) at large negative (positive) rapidity.

The sign of electromagneticallyinduced directed flow for negative pions is opposite with respect to positive pions. For absolute values of , differences in the shape of the two curves are seen close to beam rapidity.

What specifically follows from the above is that the spectatorinduced electromagnetic interaction may result in the splitting of values of observed for positive and negative pions. This splitting exhibits a strong rapiditydependence, and reach very large values close to beam rapidity.

The directed flow which is induced electromagnetically by each of the two spectators separately is also presented in the figure (the two spectator systems are denoted “left” and “right” as in Fig. 1). The contribution of a single spectator does not remain confined to its hemisphere of the collision. On the contrary, it extends into the opposite hemisphere; thus the “right” spectator moving at positive rapidity exerts its influence on the pion down to and beyond , and viceversa for the “left” spectator.

The value of obtained as the result of the electromagnetic interaction with both spectators (red solid curve, which goes through black dots in the figure) appears equal to the result of direct addition of the two singlespectator curves (displayed by the black dots).
Thus, as it it becomes evident from the above, the spectatorinduced electromagnetic interaction appears strong enough to result in sizeable effects, and to play an important role in flow phenomena involving charged pions in heavy ion collisions. This imposes the necessity of more detailed studies which will be presented below.
iii.2 Dependence on initial conditions
The central issue of this paper is the sensitivity of the spectatorinduced electromagnetic effect to the spacetime evolution of the collision. This will be addressed in the present Section.
Fig. 3 shows the rapiditydependence of the electromagneticallyinduced directed flow of positive and negative pions, obtained assuming two different values of the pion emission time: and fm/c. The values of are obtained by integration over of the pion, with integration limits defined as in Section III.1 above. The curves in panels (a) and (b) of the figure are the same as the respective curves in Fig. 2. The following remarks are to be made.

The electromagneticallyinduced directed flow displays a clear dependence on the pion emission time. In the region of beam rapidity, the position of the minimum (maximum) in the valley (peak) in pion shifts from at fm/c to above at fm/c. At the collision energy discussed here, this shift in corresponds to a shift of about 0.6 units in rapidity. The actual shape of the valley (peak) also exhibits sensitivity to .

Also in the region of lower rapidities, and down to midrapidity, the change in results in changes of . For instance, at the absolute values of go from about 2.4% at fm/c down to about 0.5% at fm/c.

What follows from the above is that the electromagneticallyinduced splitting of for and , mentioned in Section III.1, will also exhibit sensitivity to .
These observations demonstrate that the spectatorinduced electromagnetic effect is indeed sensitive to the initial conditions, set for the electromagnetic interaction by the emission time . Changes in the pion emission time, or equivalently in the position of the pion formation zone with respect to the two spectator systems, will result in changes of which should be observable in experiment. In other terms, the electromagnetic effect on directed flow is indeed sensitive to the spacetime evolution of pion production, and provides new information on the dynamics of the collision.
iii.3 Dependence on transverse momentum
Up to now, for simplicity, only the rapiditydependence of integrated directed flow was discussed in this paper. In the present Section we study its dependence on the transverse momentum of the pion. This is shown in Fig. 4. The electromagneticallyinduced directed flow exhibits a strong dependence on pion . In particular, this is valid close to beam rapidity where very large absolute values of (of the order of 0.6 for positive pions) are attained at the lowest considered value of transverse momentum, MeV/c. The absolute value of the electromagneticallyinduced rapidly decreases with increasing . On the other hand, little or no dependence on pion is apparent in the region of lower absolute values of , down to midrapidity. The overall pattern of dependence on initial conditions, discussed in Section III.2 above, repeats itself also at fixed values of .
iii.4 Comparison to experimental data
Taking into account the chargeasymmetric nature of the electromagnetic effect discussed here, any comparison of our work to experimental data must involve measurements of directed flow of pions of a given charge, either or . This constitutes an important practical difficulty as most of results published on the rapiditydependence of involve (if particle identification is available at all) only “charged pions”, i.e., summed and . One data set which fulfills our requirement comes from the WA98 experiment, and contains a measurement of positive pions near target rapidity made by means of the Plastic Ball detector, in Pb+Pb collisions at GeV. This measurement is presented in wa98 (), but a more detailed description of the analysis can be found in schlagheck (). The latter reference specifies the corresponding centrality definition as 4080% of the Pb+Pb crosssection. Following the discussion made therein, we expect that differences w.r.t. our definition of the peripheral collision (Section II) would only have a small effect on the actual values of .
Fig. 5 shows the experimental data from WA98, superimposed with the rapiditydependence of the electromagneticallyinduced directed flow as obtained from our work. For the latter, three values of the pion emission time are assumed: , , and fm/c. Our results are integrated over from 0 to 1 GeV/c. The curves corresponding to and fm/c are the same as in Fig. 3.
As it can be seen in the figure, the values of obtained from our simulation, and resulting exclusively from the spectatorinduced electromagnetic interaction, appear comparable to those measured by experiment.
The above statement should clearly be taken with some caution. Account should be taken of the simplicity of our model, and of various detailed issues related to the measurement wa98 (). These problems are beyond the scope of the present paper.
Nevertheless, it seems clear that a very sizeable part of the measured directed flow of may come from the electromagnetic origin. We note that this observation is consistent with our analysis of the influence of spectator charge on ratios in peripheral Pb+Pb collisions at the same energy. Here, pion emission times assumed in the range  fm/c also result in good agreement between our model and experimental data ismd ().
Iv Summary and Discussion
Summing up the principal results of our work, the following can be said:

We devised a simple model of the Pb+Pb collision (Section II). This model does not contain any initial azimuthal anisotropy in pion emission but includes the electromagnetic interaction between the pion charge and the two spectator systems.

This model predicts sizeable values of (electromagneticallyinduced directed flow) for positive and negative pions in the final state of the collision. For positive pions, the predicted values are comparable to those seen in experimental data. As such, a very large part of positive pion directed flow may come from the electromagnetic interaction.

The sign of this effect is opposite for positively and negatively charged pions. This will result in a splitting of values of observed for and .

The size of electromagneticallyinduced directed flow, and therefore also the size of the electromagnetic splitting addressed above, depends on the pion emission time . The latter parameter defines, in our model, the details of the spacetime evolution of pion production, like e.g. the time of pion formation, or the position of the pion formation zone with respect to the spectator system(s).

As it follows from the above, the electromagneticallyinduced directed flow, and the resulting splitting of for and , bring new information on the spacetime dynamics of the Pb+Pb reaction.
This has, in our view, important consequences for future studies of directed flow. It is of course clear that our model contains important simplifications. Among others, it intentionally neglects the initial pion flow induced by the strong rather than the electromagnetic force^{3}^{3}3We keep in mind that the inclusion of the latter is necessary in order to describe the summed charged pion data (see, e.g., na49prc ()).. The inclusion of such phenomena is the domain of more detailed models and we leave it for future studies. However, it is clear from our work that the electromagnetic splitting of and addressed above will remain present also in more sophisticated phenomenological descriptions of the heavy ion collision. This inspires a set of proposals and remarks listed below.

The study of rapiditydependence of directed flow made separately for pions of a given charge, i.e., and , will bring additional information on the dynamics of the heavy ion collision. This information will add up to the already rich phenomenological content of summed charged pion flow studies. The electromagnetic splitting of for positive and negative pions will provide information about the position of the pion formation zone with respect to the spectator systems. Specifically, also the lack of such splitting at a given rapidity would imply that the formation of pions occurs relatively far from the nearest spectator.

The situation should be similar for other produced particles, like e.g., charged kaons. The situation for final state protons should be at least partially different as protons at forward rapidity will come also from spectator fragmentation Bartke09 (); na49spec (); na49pc (). However, also in this case the role played by the electromagnetic repulsion of protons from the spectator systems should be investigated in detail.

Last but not least, our analysis brings attention to asymmetric nuclear collisions, be it A+B, d+A, or p+A reactions. In such collisions, the electromagnetic influence of the two spectators would not cancel out at midrapidity, and therefore the role of single spectators could be isolated, yielding further information on the spacetime evolution of the collision (see Fig. 2 for comparison).
All of the above observations are consistent with the results of our study of spectatorinduced electromagnetic forces acting on charged pion ratios twospec (). However, we feel that the precision of the present experimental measurements of the size of directed flow, the key role played by flow measurements in the present phenomenology of heavy ion collisions, and the overall interest of the community in such studies, make the consideration of the role played by electromagnetic interactions important for future analyses.
V Data from the RHIC Beam Energy Scan
In the context of the discussion made above, and based on the comparison between the results of our simulation and existing SPS experimental data, it is important to note that the RHIC Beam Energy Scan (BES) program can also address the search of the spectatorinduced electromagnetic effects discussed here.
Specifically, preliminary data on directed flow of positive and negative pions from the STAR experiment have already been reported in Panditidentifiedcharged (). The measured values of and indeed display a signal of splitting which increases with increasing rapidity. At positive rapidity, remains below , which follows our predictions for the electromagneticallyinduced directed flow, formulated in Sections III.1 and III.2 above.
Presenting final conclusions on the nature of this effect probably requires more precise modelling of the spacetime mechanism of charged pion production, including also its centrality dependence. However, we note that the overall magnitude of the splitting of values seen by experiment looks (roughly) comparable to that predicted in the preceding Sections, for the largest assumed values of the pion emission time and in the relatively limited range of available to the measurement Panditidentifiedcharged (). Thus, we consider that the observed splitting is indeed a signature of electromagneticallyinduced directed flow.
Vi Conclusions
The electromagnetic field induced by the presence of spectator systems may result in sizeable distortions of the directed flow pattern observed for pions of a given charge (that is, and ) produced in heavy ion collisions. This implies the presence of the electromagnetic splitting of for particles of opposite charges. The size of this splitting depends on the spacetime scenario of pion production and therefore brings new information on the collision dynamics. The comparison to experimental data suggests that a very large part of directed flow observed for positive pions may come from the electromagnetic origin.
New experimental and phenomenological studies are needed in order to
elucidate these questions, and possibly bring new insight into the
evolution of the heavy ion reaction in space and time. On the
experimental side, this implies measurements of particles of specific
charges, for symmetric and asymmetric nuclear reactions. On the
theoretical side, the possibility of verification of existing models of
the heavy ion reaction by taking into account spectatorinduced
electromagnetic effects should be investigated. Furthermore, studies of
the role of spectatorinduced electromagnetic interactions for higher
harmonics of the azimuthal distribution would also be highly indicated.
Acknowledgments
This work was supported by the Polish National Science Centre (on the basis of decision no. DEC2011/03/B/ST2/02634).
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