Electro-magnetic field fluctuation and its correlation with the participant plane in Au+Au and isobaric collisions at $\sqrt{s_{NN}}=200$ GeV

The initial state fluctuations in high-energy heavy-ion collisions play an essential role in understanding several bulk observables. We can attribute the two primary sources of these initial state fluctuations to the event-by-event (e-by-e) geometry fluctuations of the nucleon’s position inside the nuclei due to the nuclear wave function and the fluctuation in impact strong fluctuating transient electro-magentic (EM) fields in the overlap zone of the colliding nucleus. The EM field generated in high-energy heavy-ion collision experiments such as Relativistic Heavy Ion Collider (RHIC) and the Lager Hadron Collider (LHC) is known to be the strongest magnetic field in the universe (e.g., B $\sim 10^{18}-10^{19}$ Gauss for $\sqrt{s_{NN}}=200$ GeV) Ref1 ; Ref2 ; Ref3 ; Ref4 ; Ref5 ; Ref6 . The magnetic field in heavy-ion collisions, while averaged over many events, mostly obey a linear scaling with the centre of mass energy ($\sqrt{s}$) and the impact parameter (b) of collisions, Ref7 i.e., $\big{\langle}eB_{y}\big{\rangle}\sim Zb\sqrt{s}$ for $b\leq 2R_{A}$ where Z is the charge number of the ions, $R_{A}$ is the radius of the nucleus. We take the y axis perpendicular to the reaction plane as per the convention, defined by the impact parameter(chosen as the x-axis) and the beam direction(z-axis). Furthermore, the event-averaged electric fields are also found to be of the same order of magnitude as the magnetic fields (e.g., $eB\approx eE\sim 10m_{\pi}^{2}$ at the topmost RHIC energy Au+Au collisions $\sqrt{s_{NN}}=200$ GeV where $m_{\pi}$ is the pion mass).

It has been conjectured that in addition to the standard ohmic current driven by the electric field, there might appear other new types of current in parity (P) and charge conjugation  (C) odd regions in QGP as responses to the electromagnetic fields. One of this new type of currents is generated along the background magnetic field, a.k.a. the Chiral Magnetic Effect (CME)Ref1 ; Ref8 ; Ref9 ; Ref10 . In other words, in high-energy heavy-ion collisions, special gluonic configurations (sphalerons and instantons) break the P and the CP in the presence of a strong magnetic field. It results in a global electric charge separation with respect to the reaction plane Ref11 ; Ref11a . This charge separation occurs through the transition of the right-handed quarks to the left-handed quarks and vice versa depending on the sign of topological charges Ref1 . Because of their close association with axial anomaly and the topologically nontrivial vacuum structure of QCD, the CME and other associated phenomena such as chiral separation effect (CSE), the chiral electric separation effect (CESE) is known as anomalous effects Huang:2015oca .

It is known that only the lowest Landau level contributes to the CME. In the QED sector, combined electric (E) and magnetic fields (B) are responsible for the transition of chiral fermions from the left-handed chirality branch to the right-handed chirality branch at a rate $\sim e^{2}/(2\pi^{2}){\bf E}\cdot{\bf B}$ Ref10 ; Huang:2015oca .

Similarly, in CSE, the axial current is known to be not conserved due to a source term proportional to $\sim e^{3}/(2\pi^{2}){\bf E}\cdot{\bf B}$. The same term also appears in the Chiral magnetic wave equation if ${\bf E}\cdot{\bf B}$ is non-zero. In other words, ${\bf E}\cdot{\bf B}$ pumps chirality into the system. In ref. Son:2009tf it was shown that the current conservation equation in a relativistic fluid with one conserved charge, with a $U(1)$ anomaly, contains a source term proportional to $E^{\mu}B_{\mu}$. The scalar product of the four vectors $E^{\mu}$ and $B^{\mu}$ in the fluid rest frame is ${\bf E}\cdot{\bf B}$. Hence it is interesting to study ${\bf E}\cdot{\bf B}$ for different collision geometry and its possible correlation with the symmetry (participant) plane, with respect to which we search for the CME signal. It is worthwhile to mention that although the event averaged magnetic field shows a linear behavior with collision centrality, the electric field, on the other hand, shows an opposite trend, i.e., maximum for the central collisions and gradually decreases for higher centralities. In this paper, we focus on the spatial distribution of ${\bf E}\cdot{\bf B}$ for various centrality Au+Au, Ru+Ru, and Zr+Zr collisions at $\sqrt{s_{NN}}=200$ GeV to investigate their angular correlation with the geometry of the fireball. To this end, we introduce the participant plane $\psi_{EB}$ defined with the weight of ${\bf E}\cdot{\bf B}$, and we show the correlation of it with the participant plane $\psi_{pp}$.

The rest of this paper is organized as follows: In Sec. II, we describe the detail of calculating electromagnetic fields from the Glauber model on an event-by-event basis. We also discuss their impact parameter dependence and event averaged value. In Sec. III. we discuss the main results, which consists of the impact parameter, space-time, and system size dependence of ${\bf E}\cdot{\bf B}$ and its correlation with the participant plane. Finally, we summarise this study in Sec. IV.

Customarily the electromagnetic field generated by a relativistic charged particle is calculated from the well known Liénard–Wiechert potentials, however, we will calculate it from the second-rank antisymmetric electromagnetic field tensor $F^{\alpha\beta}=\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}$ using the Lorentz transformation. Here $A^{\mu}$ is the four-potential due to an electric charge, in the following calculations we assume the charged protons inside the colliding nuclei move in a straight line trajectory and there are neglegible change in momentum after the collision. The calculation goes as follow Ref12 : first we calculate the component of electromagnetic fields and corresponding $F^{\prime\gamma\delta}$ in the rest frame $S^{\prime}$ of the charge particle. The fields in the laboratory frame is calculated from $F^{\alpha\beta}$ which is obtained from $F^{\prime\gamma\delta}$ through the Lorentz transformation:

$$F^{\alpha\beta}=\frac{\partial x^{\alpha}}{\partial x^{\prime\gamma}}\frac{\partial x^{\beta}}{\partial x^{\prime\delta}}F^{\prime\gamma\delta},$$

(1)

or in matrix notation $F=\Lambda F^{\prime}\tilde{\Lambda}$, where $\Lambda$ is the matrix representation of the Lorentz transformation and $\tilde{x^{\mu}}$ corresponds transpose of $x^{\mu}$. We choose a boost $\beta=v_{z}$ along the $z$ axis. In this case it can be easily shown that the electric fields transform as

	$\displaystyle E_{x}$	$\displaystyle=$	$\displaystyle\gamma E_{x}^{\prime}+\gamma\beta B_{y}^{\prime},$		(2)
	$\displaystyle E_{y}$	$\displaystyle=$	$\displaystyle\gamma E_{y}^{\prime}-\gamma\beta B_{x}^{\prime},$		(3)
	$\displaystyle E_{z}$	$\displaystyle=$	$\displaystyle E_{z}^{\prime},$		(4)

and the magnetic fields transform as,

	$\displaystyle B_{x}$	$\displaystyle=$	$\displaystyle\gamma B_{x}^{\prime}-\gamma\beta E_{y}^{\prime},$		(5)
	$\displaystyle B_{y}$	$\displaystyle=$	$\displaystyle\gamma B_{y}^{\prime}+\gamma\beta E_{x}^{\prime},$		(6)
	$\displaystyle B_{z}$	$\displaystyle=$	$\displaystyle B_{z}^{\prime}.$		(7)

Since the charge is at rest in the $S^{\prime}$ frame $B_{x}^{\prime}=B_{y}^{\prime}=B_{z}^{\prime}=0$, furthermore, it is easy to verify ${\bf B}={\bf\beta}\times{\bf E}$. Next, we calculate ${\bf E}^{\prime}$ at a point P $(x,y,z)$ at time $t$ for a charge at ($x^{\prime}_{c},y^{\prime}_{c},z^{\prime}_{c}$) at time $t^{\prime}$ by noting that $z^{\prime}_{c}\approx\beta t^{\prime}$ (we assume that the origin of the lab frame $S$ and the moving frame $S^{\prime}$ coincide at $t=t^{\prime}=0$). The subscript $c$ corresponds to the charge. For convenience, we denote the transverse distance $\zeta=\sqrt{(x^{\prime}_{c}-x)^{2}+(y^{\prime}_{c}-y)^{2}}$, the distance from the charge to $P$ is $r^{\prime}=\sqrt{\zeta^{2}+\beta^{2}t^{\prime 2}}$ (here we have taken the center of the nucleus to be at the origin of $S^{\prime}$). Also, the positions of the charged particles in the transverse plane are assumed to be frozen due to large Lorentz $\gamma=(1-\beta^{2})^{-1/2}$. A straightforward calculation gives the following values of the electric fields in $S$ frame

	$\displaystyle E_{x}$	$\displaystyle=$	$\displaystyle\frac{\gamma qx}{\left(\zeta^{2}+\beta^{2}\gamma^{2}\left(t-\beta z\right)^{2}\right)^{3/2}},$		(8)
	$\displaystyle E_{y}$	$\displaystyle=$	$\displaystyle\frac{\gamma qy}{\left(\zeta^{2}+\beta^{2}\gamma^{2}\left(t-\beta z\right)^{2}\right)^{3/2}},$		(9)
	$\displaystyle E_{z}$	$\displaystyle=$	$\displaystyle\frac{q\beta\gamma(t-\beta z)}{\left(\zeta^{2}+\beta^{2}\gamma^{2}\left(t-\beta z\right)^{2}\right)^{3/2}},$		(10)

and the magnetic fields are given by

	$\displaystyle B_{x}$	$\displaystyle=$	$\displaystyle\frac{-\gamma\beta qy}{\left(\zeta^{2}+\beta^{2}\gamma^{2}\left(t-\beta z\right)^{2}\right)^{3/2}},$		(11)
	$\displaystyle B_{y}$	$\displaystyle=$	$\displaystyle\frac{\gamma\beta qx}{\left(\zeta^{2}+\beta^{2}\gamma^{2}\left(t-\beta z\right)^{2}\right)^{3/2}},$		(12)
	$\displaystyle B_{z}$	$\displaystyle=$	$\displaystyle 0.$		(13)

The total electromagnetic field at any point is evaluated using the principle of superposition i.e., calculating fields using Eq.(8) - (13) for all the protons inside the nucleus. We use a cutoff value of $\zeta$ = 0.3 fm while calculating the electric and magnetic field usig Eq.(8) - (13) We note that this cutoff value was chosen as an average effective distance between the quarks inside the nucleons, and it was also reported Ref6 that there is a weak dependence of the field values on $\zeta$ in the range 0.3 to 0.6 fm. Since the colliding nucleus at $\sqrt{s_{NN}}=200$ GeV has Lorentz $\gamma\sim 100$, we can safely assume the nucleus as a flat disk that has a vanishing thickness along the $z$ axis. Also, due to the time-dilation, the nucleons will appear as frozen inside the nucleus, and all nucleons effectively move along $z$ with constant $v_{z}$ i.e ${\bf v_{n}}\equiv$ (0,0,$v_{z}$). As per the convention, we take the velocity of the target nucleus as $+v_{z}$, and the velocity of the projectile nucleus is $-v_{z}$. $v_{z}$ is calculated from the ratio of the relativistic momentum and the energy of a proton

$$v_{z}=\sqrt{1-\Big{(}\frac{2m_{p}}{\sqrt{s_{NN}}}\Big{)}^{2}}.$$

(14)

To obtain the nucleon positions we use the MC-Glauber model Ref13 . We also calculate the initial spatial eccetricity ($\epsilon$, defined later) and the number of participating nucleons $(N_{\rm{part}})$ for a given impact parameter from the MC-Glauber model. In the MC-Glauber model, the positions of the nucleons inside the nucleus are determined by the nuclear density function measured in low-energy electron scattering experimentsRef14 . The functional form of this distribution is:

$$\rho(r,\theta)=\frac{\rho_{0}}{1+exp\big{[}\frac{r-R\big{(}1+\beta_{2}Y_{2}^{0}(\theta)+\beta_{4}Y_{4}^{0}(\theta)\big{)}}{a}\big{]}},$$

(15)

where $\rho_{0}$ corresponds to the nuclear density at the center, $R$ is the radius of the nucleus, $a$ is the skin depth (it controls how quickly the nuclear density falls off near the edge of the nucleus). The spherical harmonics $Y_{l}^{m}(\theta)$ and parameters $\beta_{2}$ and $\beta_{4}$ are used to measure the deformation from spherical shape. For our study, we take $R=6.38$ fm, $a=0.535$ fm, and $\beta_{2}=\beta_{4}=0$ for the $Au_{79}^{197}$ nucleus. We use parameter $\beta_{2}^{Ru}$ =0.158 , $\beta_{2}^{Zr}$=0.08, $R^{Ru}=5.085$ fm, $R^{Zr}=5.02$fm and a = 0.46 fm for both Ru and Zr Ref14a ; Ref14b ; Ref14c . $\beta_{4}$ is taken 0 for both nuclei.

We sample the nucleon positions assuming that they are randomly distributed with the given distribution $4\pi r^{2}\rho(r)$ (integrating on $\theta$ and $\phi$ ) for Au nucleus and $r^{2}\sin(\theta)\rho(r,\theta)$ for Ru and Zr nuclei respectively.
The impact parameters $b$ of the collisions are randomly selected from the distribution $\frac{dN}{db}\sim b$ upto a maximum value of $\simeq 20$ fm $>2R$. The center of the target and projectile nuclei are shifted to $(-\frac{b}{2},0,0)$ and $(\frac{b}{2},0,0)$ respectively. We use the inelastic nucleon-nucleon cross section $\sigma_{NN}=42$ mb for the top RHIC energy $\sqrt{s_{NN}}=200$ GeV for calculating the probability of an interaction between the target and the projectile nucleonsRef15 ; Ref16 . To show the centrality dependence, we calculate the centrality of the collisions using $N_{\rm{part}}$ and the number of binary collisions $(N_{\rm{coll}})$ obtained from the MC Glauber model . The multiplicity for a given $N_{\rm{part}}$ and $(N_{\rm{coll}})$ is calculated using the two component model as

$$\frac{dN_{ch}}{d\eta}=n_{\rm{pp}}\left[(1-x)\frac{N_{\rm{part}}}{2}+xN_{\rm{coll}}\right],$$

(16)

where $x$ is the fraction of hard scattering, $n_{\rm{pp}}$ is the average multiplicity per unit pseudo-rapidity in pp collisions. The above two-component model of the particle production is based on the assumption that the average particles produced through the soft interactions are proportional to the $N_{\rm{part}}$, and the probability of hard interactions is proportional to $N_{\rm{coll}}$. We calculate the centrality of a given collision in the following way: the number of independent particle emitting sources for a given impact parameter are $(1-x)\frac{N_{\rm{part}}}{2}+xN_{\rm{coll}}$. Each of these sources produces particles following a negative binomial distribution (NBD) with a mean $\mu$ and the width $\sim$ 1/k,

$$\textit{P}_{\mu,k}(n)=\frac{\Gamma(n+k)}{\Gamma(n+1)\Gamma(k)}\Big{(}\frac{\mu}{\mu+k}\Big{)}^{n}\Big{(}\frac{k}{\mu+k}\Big{)}^{k}.$$

(17)

$\textit{P}_{\mu,k}(n)$ is the probability of measuring n hits per independent sources. The mean of this NBD distribution is calculated from the pseudo-rapidity density of the charged multiplicity for the non-single diffractive $\bar{p}p$ collisions at a given $\sqrt{s_{NN}}$ energy Ref17 :

$$\mu=\mathcal{A}\ln^{2}{(s_{NN})}-\mathcal{B}\ln{(s_{NN})}+\mathcal{C},$$

(18)

where $\mathcal{A}=0.023\pm 0.008$, $\mathcal{B}=0.25\pm 0.19$, and $\mathcal{C}=2.5\pm 1.0$. The charged particle multiplicity data for Au+Au 200 GeV collisions measured by the STAR collaboration are explained for $x=0.13$ and $k=1.7$, and $\mu=2.08$ for the pseudo-rapidity range i.e$\mid\eta\mid<0.5$ Ref18 . For example, we show the charged particle multiplicity distribution for Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV for $|\eta|<0.5$ in Fig. 1.

To calculate the centrality of collisions, we subdivide the total area in Fig. 1 into different bins with the condition that the fractional area corresponds to a particular centrality. For example, the bin boundaries $n40$ and $n50$ for the 40$\%$-50$\%$ centrality are defined in such a way that the following relation holds: $\frac{\int_{\infty}^{n40}P(N_{ch})dN_{ch}}{\int_{\infty}^{0}P(N_{ch})dN_{ch}}=0.4$, and $\frac{\int_{\infty}^{n50}P(N_{ch})dN_{ch}}{\int_{\infty}^{0}P(N_{ch})dN_{ch}}=0.5$.

We use three centrality bins 0$\%$-5$\%$, 40$\%$-50$\%$ and 70$\%$-80$\%$ for our calculation of the electric and magnetic field; corresponding impact parameter ranges are $0-3.2$, $9.3-10.6$, and $12.2-13.5$ fm, which are very similar to the values given in Ref18 .

As mentioned earlier, the topology of the electromagnetic fields in heavy-ion collisions has non-trivial dependence on the centrality (possibly also on the $\sqrt{s_{\rm{NN}}}$). Consequently, the source ($\bf{E}\cdot\bf{B}$) of the chiral current in the transverse plane also has a non-trivial centrality dependence. The axial current generated by the magnetic field is supposed to predominantly flow along the direction perpendicular the participant plane. Hence, we investigate here how the sources $\bf{E}\cdot\bf{B}$ of this current is correlated to the participant plane. From the perspective of heavy-ion collisions it is customary to use the Milne co-ordinates i.e., we use the longitudinal proper time $\tau=\sqrt{t^{2}-z^{2}}$ ,$x$,$y$, and the space-time rapidity $\eta=\frac{1}{2}\log(\frac{t+z}{t-z})$ instead of the Cartesian co-ordinate. In Fig.2 we show the distribution of ${\bf E}\cdot{\bf B}$ for 40$\%$-50$\%$ centrality with $\eta$ (left plot) and $\tau$ (right plot). In left plot, we calculate $\bf{E}\cdot\bf{B}$ in the forward light-cone spanned by the region for $\tau$ = 0.4 and $-1.0\leq\eta\leq 1.0$. In right plot, we also calculate $\bf{E}\cdot\bf{B}$ in the forward light-cone. But for this plot, phase space is spanned by the region for $\eta$ = 0.4 and $0.1\leq\tau\leq 0.34$. While calculating the dot product, we transform the field components $B_{x(y)}$ from the Cartesian to the Milne coordinate $\tilde{B}_{x(y)}$ by the following expression $\tilde{B}_{x}=B_{x}/\cosh\eta,\tilde{B}_{y}=B_{y}/\cosh\eta$ and $\tilde{B}_{z}=B_{z}/\tau$. The components of the electric field transform similarly.

To investigate the distribution of $\bf{E}\cdot\bf{B}$ (from now on denoted as $\mathcal{E}$) to the participant plane, we introduce the $\bf{E}\cdot\bf{B}$ symmetry plane $\psi_{\mathcal{E}}$ defined as Ref19 ; Ref20

$$\epsilon_{n}e^{in\psi_{\mathcal{E}}}=-\frac{\int dxdyr^{n}e^{in\varphi}\mathcal{E}(x,y)}{\int dxdyr^{n}\mathcal{E}(x,y)},$$

(19)

where $r^{2}=(x-<x>)^{2}+(y-<y>)^{2}$ and $tan(\varphi)=\frac{y-<y>}{x-<x>}$. Here $(<x>,<y>)$ corresponds to the mean position of the participating nucleons. Using these definitions and Eq.(19) we obtain $\psi_{\mathcal{E}}$ as

$$\psi_{\mathcal{E}}=\frac{1}{n}arctan\frac{\int dxdyr^{n}\sin(n\varphi)\mathcal{E}(x,y)}{\int dxdyr^{n}\cos(n\varphi)\mathcal{E}(x,y)}+\frac{\pi}{n}.$$

(20)

Before going into the main results of this paper, let us very briefly go through the impact parameter dependence of the electric and magnetic field produced in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=200$ GeV. These results are not new and have already been reported in several works Ref6 ; Ref7 ; Huang:2015oca ; Ref20ab , but we include them for the sake of completeness.

In summary, we have studied the event-by-event fluctuations of the electric and the magnetic fields and their possible correlation with the geometry of the high-energy heavy-ion collisions. More particularly we studied the distribution of $\bf{E}\cdot\bf{B}$($=\mathcal{E}$) in the transverse plane for Au+Au, Ru+Ru, and Zr+Zr collisions at $\sqrt{s_{NN}}=200$ GeV. Further, we show the $\tau$ and $\eta$ dependence of $\mathcal{E}$ in Au+Au at 200 GeV per nucleon collisions. As expected, $\mathcal{E}$ is found to be symmetric in $\eta$ (around $\eta=0$), and $\mathcal{E}$ quickly decays as a function of $\tau$ at a given $\eta$. Because $\mathcal{E}$ may contribute to CME as a source of the anomalous current, we investigate the centrality (impact parameter) dependence of the symmetry plane angle $\psi_{\mathcal{E}}$ and its possible correlation with the participant plane. We show that $\psi_{\mathcal{E}}$ is strongly correlated with $\psi_{P}$ for second-fifth order harmonics for Au+Au, Ru+Ru, and Zr+Zr collisions. The second-order planes $\psi_{\mathcal{E}}$ and $\psi_{P}$ are not only correlated but also mostly coincides with each other except for the peripheral collisions, where a rotation by $\pi/2$ is observed for $\psi_{\mathcal{E}}$ irrespective of the collision system size. This phenomenon seems to be happening due to the almost cancellation of electric fields and dominating magnetic field pointing perpendicular to the participant plane in peripheral collisions. To conclude, in this exploratory study, we show that, like the magnetic fields, $\mathcal{E}$ is also correlated to the geometry of the collision even when we consider e-by-e fluctuation of nucleon positions.

We are thankful to the grid computing facility at Variable Energy Cyclotron Centre, Kolkata, for providing us CPU time. VR acknowledges financial support from the DST Inspire faculty research grant (IFA-16-PH-167), India.