# Azimuthally fluctuating magnetic field and its impacts on observables in
heavy-ion collisions

John Bloczynski  Physics Department and Center for Exploration of Energy and
Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN
47408, USA.  Xu-Guang Huang  Physics Department and Center for Exploration of
Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane,
Bloomington, IN 47408, USA.  Xilin Zhang  Physics Department and Center for
Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson
Lane, Bloomington, IN 47408, USA.  Jinfeng Liao  Physics Department and Center
for Exploration of Energy and Matter, Indiana University, 2401 N Milo B.
Sampson Lane, Bloomington, IN 47408, USA.  RIKEN BNL Research Center, Bldg.
510A, Brookhaven National Laboratory, Upton, NY 11973, USA.

###### Abstract

The heavy-ion collisions can produce extremely strong transient magnetic and
electric fields. We study the azimuthal fluctuation of these fields and their
correlations with the also fluctuating matter geometry (characterized by the
participant plane harmonics) using event-by-event simulations. A sizable
suppression of the angular correlations between the magnetic field and the
$2$nd and $4$th harmonic participant planes is found in very central and very
peripheral collisions, while the magnitudes of these correlations peak around
impact parameter $b\sim 8-10\rm fm$ for RHIC collisions. This can lead to
notable impacts on a number of observables related to various magnetic field
induced effects, and our finding suggests that the optimal event class for
measuring them should be that corresponding to $b\sim 8-10$ fm.

###### pacs:

12.38.Mh, 25.75.-q, 25.75.Ag, 24.10.Nz

##  I Introduction

Ultra-relativistic heavy-ion collisions create not only a domain of extremely
high energy density where a new state of matter — the deconfined quark-gluon
plasma (QGP) may form, but also extremely strong (electro)magnetic fields due
to the relativistic motion of the colliding heavy ions carrying large positive
electric charge (Rafelski:1975rf, ). Previous computations showed that the
magnetic fields generated in Au + Au collision at RHIC ($\sqrt{s}=200\rm GeV$)
can reach about $eB\sim m_{\pi}^{2}\sim 10^{18}$ G (arXiv:0711.0950, ;
arXiv:0907.1396, ; arXiv:1003.2436, ; arXiv:1103.4239, ; arXiv:1107.3192, ;
arXiv:1111.1949, ; arXiv:1201.5108, ), which is $10^{13}$ times larger than
the strongest man-made steady magnetic field in the laboratory. The magnetic
field generated at LHC energy can be an order of magnitude larger than that at
RHIC (arXiv:0907.1396, ; arXiv:1201.5108, ), according to a simple scaling law
recently found in (arXiv:1201.5108, ) for the event-averaged magnetic field:
$\langle eB_{y}\rangle\propto Zb\sqrt{s}$ for $b\lesssim 2R_{A}$ , where $Z$
is the charge number of the ions, $b$ is the impact parameter, $R_{A}$ is the
radius of nucleus, and the $y$-axis is perpendicular to the reaction plane.
Thus, heavy-ion collisions provide a unique terrestrial environment with
ultra-strong magnetic fields.

There have been very strong interests and intensive efforts recently in
studying various possible physical effects induced by the presence of strong
magnetic field. Receiving particular enthusiasm is the set of ideas to look
for experimental manifestation of QCD effects stemming from topology and
anomaly and aided by the external magnetic field. These include, e.g., the so-
called Chiral Magnetic Effect (CME) (Kharzeev:2004ey, ; Kharzeev:2007tn, ;
arXiv:0711.0950, ; Fukushima:2008xe, ) in which a nonzero axial charge density
in the matter (presumably from topological transitions via sphalerons)
together with the magnetic field ${\bf B}$ will induce a dipole charge
separation along the ${\bf B}$ direction. A lot of works have been done in the
past few years to experimentally measure this effect by analyzing the charged
particle correlations (Voloshin:2004vk, ; Abelev:2009ac, ;
Selyuzhenkov:2011xq, ) and to understand the interpretation and background
effects related to these measurements (Schlichting:2010qia, ; Pratt:2010zn, ;
Bzdak:2009fc, ; Bzdak:2010fd, ; Liao:2010nv, ; Wang:2009kd, ), as recently
reviewed in Ref. (Bzdak:2012ia, ). There is also the so-called Chiral
Separation Effect (CSE) (Son:2004tq, ), from its interplay with the CME there
arises a gapless collective excitation called the Chiral Magnetic Wave (CMW)
(Kharzeev:2010gd, ). An observable effect of CMW was proposed in
(Burnier:2011bf, ): the CMW with the presence of nonzero vector charge density
and the magnetic field transports the charges in QGP towards an electric
quadrupole distribution with more positive charges near the poles of the
produced fireball (pointing outside of the reaction plane) while more negative
charges near the equator (in the reaction plane), and this leads to a
measurable splitting of negative and positive pions’ elliptic flow. Recently
STAR collaboration reported the first measurement of the charged pion flow
spitting versus the charge asymmetry which appears in agreement with the
predictions from CMW (Burnier:2011bf, ; Burnier:2012ae, ). Yet one more
interesting effect is possible soft photon production through the QCD
conformal anomaly in the external magnetic field as suggested in
(Basar:2012bp, ). (The CME could also possibly cause anisotropic soft photon
production, see Fukushima:2012fg .) In this mechanism the photons are emitted
perpendicular to the ${\bf B}$ direction and there is an appreciable azimuthal
anisotropy of the emitted photons which might partially account for the
unusually large $v_{2}$ of direct photons reported by PHENIX collaboration
(Adare:2011zr, ). In addition to the above anomaly phenomena in magnetic
field, there are also discussions of other novel effects in strong magnetic
fields (arXiv:1008.1055, ; arXiv:1102.3819, ; arXiv:1108.4394, ;
arXiv:1108.0602, ), e.g., the spontaneous electromagnetic superconductivity of
QCD vacuum, the possible enhancement of elliptic flow of charged particles,
possible anisotropic electromagnetic radiation from the QGP, the energy loss
due to the synchrotron radiation of quarks, and the emergence of anisotropic
viscosities.

In most of the phenomenological studies of these effects, the magnetic field
${\bf B}$ in heavy-ion collisions represents the largest source of uncertainty
in their quantitative calculations, and a precise knowledge of the magnetic
field is urgently needed. This is particularly so in the context of the
realization and intensive investigations in the last two or three years that
there are very strong fluctuations in the initial conditions of heavy-ion
collisions. Such fluctuations have been shown to lead to strong observable
effects both in the bulk collective expansions (as “harmonic flows”)
(harmonic_flow, ) and in the anisotropy of penetrating hard probe (as
“harmonic tomography”) (harmonic_tomography, ). Since the magnetic field
directly relies on the initial distributions of protons in both nuclei, a
realistic computation has to take into account such strong initial
fluctuations. A first step has been made in (arXiv:1111.1949, ;
arXiv:1201.5108, ) to compute the magnitude of ${\bf B}$ with event-by-event
fluctuations, which was indeed found to be remarkably modified from the naive
“optical” estimates.

In this paper, we focus on the strong fluctuations in the azimuthal
orientation of the magnetic field ${\bf B}$, and particularly investigate its
angular correlation with the underlying matter geometry (specified by
participant planes) bearing concurrent fluctuations on an event-by-event
basis. This correlation is a very important link between the experimental
measurements and any of the magnetic field induced effects in heavy-ion
collisions. Most (if not all) previous studies rely on the assumption that the
magnetic field is pointing in the out-of-plane direction which would be true
without fluctuations. In the real world, however, both the ${\bf B}$ and the
“planes” (event planes or participant planes) bear strong fluctuations and as
we will show their orientations are never fully “locked” and only strongly
correlated in some circumstances. We will study these correlations in great
details and examine the consequences for a number of observables related to
some aforementioned effects.

The rest of this paper is organized as follows. In Sec. II we show how the
initial fluctuations modify the magnetic and electric fields generation in
heavy-ion collisions. We study the azimuthal fluctuation of the magnetic field
in Sec. III. We discuss the physical implications of these results in Sec. IV.
For completeness we also show results for the electric field in Sec. V. In
sec. VI a new class of charge dependent measurements partly motivated by our
results will be discussed. Finally we summarize in Sec. VII. The natural unit
$\hbar=c=k_{B}=1$ will be used throughout this article.

##  II Event-by-event calculation of the electromagnetic field

The aim of this section is to give an event-by-event calculation of the
electromagnetic fields in heavy-ion collisions. We focus on the fields at the
initial time $t=0$, that is, the moment when the two colliding nuclei overlap
completely. Our starting point is the Lorentz boosted Coulomb formulas which
are equivalent to the Liénard-Wiechert potentials for constantly moving
charges:

$\displaystyle e{\bf B}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}({\bf
R}_{n})\frac{1-v_{n}^{2}}{[R_{n}^{2}-({\bf R}_{n}\times{\bf
v}_{n})^{2}]^{3/2}}{\bf v}_{n}\times{\bf R}_{n},$ $\displaystyle e{\bf
E}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}({\bf
R}_{n})\frac{1-v_{n}^{2}}{[R_{n}^{2}-({\bf R}_{n}\times{\bf
v}_{n})^{2}]^{3/2}}{\bf R}_{n},$ (1)

where ${\bf R}_{n}={\mathbf{r}}-{\mathbf{r}}_{n}(t)$ is the relative position
of the field point ${\mathbf{r}}$ to the $n$th proton at time $t$,
${\mathbf{r}}_{n}(t)$, and ${\bf v}_{n}$ is the velocity of the $n$th proton.
The summations run over all protons in the projectile and target nuclei.
Equations (1)contain singularities at $R_{n}=0$ if we treat protons as point
charges. In practical calculation, to avoid such singularities we treat
protons as uniformly charged spheres with radius $R_{p}$. The charge number
factor $Z_{n}(\bf{R}_{n})$ in Eqs. (1) is introduced to encode this aspect:
when the field point locates outside the $n$th proton (in the rest frame of
the proton) $Z_{n}=1$, otherwise $Z_{n}<1$ depends on ${\bf R}_{n}$. The in-
medium charge radius $R_{p}$ of proton is unknown (the most recent measurement
of the rms charge radius of proton gives $R_{p}=0.84184(67)$ fm in vacuum
Pohl:2010zza ), we choose $R_{p}=0.7$ fm in our numerical simulations. We
checked that varying $R_{p}$ from $0.6$ fm to 0.9 fm will shift the numerical
results within $15\%$ but no qualitative conclusion is altered. The nucleons
in one nucleus move at constant velocity along the beam direction (we choose
it as $z$-direction), with the nucleons in another nucleus moving oppositely
at the same speed. The energy for each nucleon is set to be $\sqrt{s}/2$ in
the center-of-mass frame, therefore the value of the velocity of each nucleon
is given by $v_{n}^{2}=1-(2m_{N}/\sqrt{s})^{2}$, where $m_{N}$ is the mass of
the nucleon. We set the $x$-axis along the impact parameter vector so that the
reaction plane is the $x$-$z$ plane. Finally, the positions of nucleons in the
rest frame of a nucleus are sampled according to the Woods-Saxon distribution.

Figure 1: (Color online) The event-averaged $(eB)^{2}$ and $(eE)^{2}$ (in unit
of $m^{4}_{\pi}$) at $t=0$ and four different points on the transverse plane
as functions of the impact parameter $b$.

Figure 2: (Color online) The event-by-event histograms of $\Psi_{\bf
B}-\Psi_{2}$ at impact parameters $b=0,5,10,12$ fm for Au + Au collision at
RHIC energy. Here $\Psi_{\bf B}$ is the azimuthal direction of ${\bf B}$ field
(at $t=0$ and ${\mathbf{r}}=(0,0,0)$) and $\Psi_{2}$ is the second harmonic
participant plane.

Figure 3: The scatter plots on $\Psi_{B}$-$\Psi_{2}$ plane at impact
parameters $b=0,5,10,12$ fm for Au + Au collision at RHIC energy. Here
$\Psi_{\bf B}$ is the azimuthal direction of ${\bf B}$ field (at $t=0$ and
${\mathbf{r}}=(0,0,0)$) and $\Psi_{2}$ is the second harmonic participant
plane.

In this computation we have obtained in each event all the components of both
the ${\bf B}$ and ${\bf E}$ fields at four different points,
${\mathbf{r}}=(0,0,0),{\mathbf{r}}=(3\,{\rm fm},0,0),{\mathbf{r}}=(0,3\,{\rm
fm},0)$, and ${\mathbf{r}}=(3\,{\rm fm},3\,{\rm fm},0)$, on the transverse
plane for a wide range of impact parameter $b$ in Au + Au collision at RHIC
energy $\sqrt{s}=200$ GeV. We have found that all the $z$ components are
negligibly small, while the transverse components are strong, with their
centrality trends in agreement with previous results in (arXiv:1111.1949, ;
arXiv:1201.5108, ). The absolute magnitudes of these fields are somewhat
smaller than those reported in (arXiv:1111.1949, ; arXiv:1201.5108, ), due to
the important difference that we treat each proton as a charged sphere with
radius $R_{p}$ rather than a point charge, which is both more physical and
mathematically less of singularity problem. As also noted in (arXiv:1111.1949,
; arXiv:1201.5108, ) and confirmed in our computation, although the
$x$-component of the magnetic field as well as the $x$\- and $y$-components of
the electric field vanish after averaging over many events, their magnitudes
in each event can be huge and comparable to the $y$-component of the magnetic
field due to the fluctuations. This already implies that both the magnitude
and the direction of the electromagnetic field (albeit in the unmeasurable
$x$-$y$ plane) fluctuate strongly. In Fig. 1 we show $\langle(eB)^{2}\rangle$
and $\langle(eE)^{2}\rangle$ as functions of $b$ at different points: clearly
the electromagnetic fields bear considerable inhomogeneity on the transverse
plane. We also notice that the electric field can be very strong particularly
for more central collisions. The $\langle(eB)^{2}\rangle$ is particularly
interesting because the signal strengths of several magnetic-field-induced
effects are proportional to $(eB)^{2}$, as we will discuss more in Sec. IV.

##  III Azimuthal correlations between magnetic field and matter geometry

As already mentioned before, on the event-by-event basis the electromagnetic
field fluctuates strongly both in magnitude and in azimuthal direction. The
direction of the ${\bf B}$-field is very important as the ${\bf B}$-field
induced effects inherit this information and occur either along (e.g. CME,
CMW) or perpendicular (e.g. photon emission) to ${\bf B}$.

Even more important from the measurement perspective is the question of the
${\bf B}$ orientation with respect to a frame that could be identified
experimentally rather than to the ideal world reaction plane. In reality what
could be determined is the final-state hadrons’ distribution geometry in
momentum space, in particular the second harmonic $v_{2}$ event plane (EP)
which is relatively more tightly correlated with the initial matter
distribution geometry that can be specified primarily by the second harmonic
$\epsilon_{2}$ participant plane (PP). Therefore the really useful information
is the ${\bf B}$ orientation with respect to the matter geometry per event.
Here we make a first detailed examination of such a kind, studying the
azimuthal correlation between the magnetic field and the participant planes.

In order to do that, we determine the participant planes for various harmonics
from the Monte Carlo Glauber simulations of the initial condition and analyze
the angular correlations between the ${\bf B}$ and the participant plane
orientations from the same event. The $n$th harmonic participant plane angle
$\Psi_{n}$ and eccentricity $\epsilon_{n}$ are calculated from participant
density $\rho({\mathbf{r}})$ as in the literature (e.g. (harmonic_flow, )):
$\epsilon_{1}e^{i\Psi_{1}}=-(\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{3}e^{i\phi})/({\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{3}}),$ and
$\epsilon_{n}e^{in\Psi_{n}}=-({\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{n}e^{in\phi})/(\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{n}})$ for $n>1$.
In this Section we focus on the 2nd harmonic participant plane $\Psi_{2}$ as
it is the most prominent anisotropy from both geometry and fluctuations.
Correlations of ${\bf B}$ with other harmonics will be discussed in Sec. IV.
By determining ${\bf B}$ and $\Psi_{2}$ in each event we can examine the
distribution of their relative angle over many events: in Fig. 2 we plot the
event-by-event histograms of $\Psi_{\bf B}-\Psi_{2}$ at $b=0,5,10$ and $12$ fm
for Au + Au collisions at $\sqrt{s}=200$ GeV where $\Psi_{\bf B}$ is the
azimuthal direction of the magnetic field at $t=0$ and ${\mathbf{r}}=(0,0,0)$.
Strikingly at $b=0$ the histogram of $\Psi_{\bf B}-\Psi_{2}$ is almost uniform
indicating that $\Psi_{\bf B}$ and $\Psi_{2}$ are basically uncorrelated. For
$b=5,10$ and $12$ fm, the histograms have Gaussian shapes peaking at $\pi/2$
with the corresponding widths not small at all. The width first decreases when
$b$ increased from $5$ to $10$ fm and then increases again toward $b=12$ fm.
So although the ${\bf B}$ field indeed points at $\pi/2$ with respect to
$\Psi_{2}$ plane on average, it fluctuates significantly in each event. The
correlation between $\Psi_{\bf B}$ and $\Psi_{2}$ is the strongest in middle-
centrality while weakens much in the most central and most peripheral
collisions.

To further reveal the correlation pattern between $\Psi_{\bf B}$ and
$\Psi_{2}$ and to understand better the non-monotonous centrality trend of the
widths of the histograms in Fig. 2, we show the scatter plots from all events
at given $b$ on the $\Psi_{\bf B}$-$\Psi_{2}$ plane in Fig. 3 which visualize
the 2D probability distribution density. Again for $b=0$, the events are
almost uniformly distributed indicating negligible correlation between
$\Psi_{\bf B}$ and $\Psi_{2}$. For $b=5,10$, and $12$ fm, the event
distributions evidently concentrate around $(\Psi_{\bf B},\Psi_{2})=(\pi/2,0)$
indicating a correlation between the two. Going from $b=5$ to $10$ and to $12$
fm, the spread in $\Psi_{\bf B}$ keeps shrinking while the spread in
$\Psi_{2}$ clear grows with larger $b$. This is because for non-central
collisions with increasing $b$, the ${\bf B}$ is mostly from the spectators
whose number increases and bears less fluctuations while the $\Psi_{2}$ is
determined by participants whose number decreases and fluctuates more. This
explains the non-monotonic trend of the widths in the histograms in Fig. 2. In
short, we have found that the event-by-event fluctuations of the initial
condition bring azimuthal fluctuations in both $\Psi_{\bf B}$ and $\Psi_{2}$,
and the angular correlation between them is smeared out significantly in the
very central and very peripheral collisions while stays strong for middle-
centrality collisions. This observation certainly influences the
interpretation of observables related with ${\bf B}$-induced effects, as will
be discussed in the next section.

##  IV Impact on various observables

We now discuss the impacts of the azimuthal fluctuations of the magnetic field
with respect to matter geometry (the participant planes) on a number of
pertinent observables recently measured in heavy-ion collisions. We recap a
few points already discussed: the ${\bf B}$-induced effects are sensitive to
the azimuthal direction of the ${\bf B}$ field, which however is not
experimentally known; in the past when observables are proposed and
interpreted for measuring certain $\bf B$-related effects, it is often assumed
that the $\bf B$ direction is perpendicular to the reaction plane; as already
shown in last section, this assumption is not true, and the azimuthal
orientation between $\Psi_{\bf B}$ of $\bf B$ and $\Psi_{2}$ fluctuates with
sizable spread in their relative angle $(\Psi_{\bf B}-\Psi_{2})$. In what
follows we evaluate the impacts of such fluctuations on the experimentally
measured quantities.

First, let us consider the pair correlation
$\gamma=\langle\cos(\phi_{1}+\phi_{2}-2\Psi_{2})\rangle$ with $\phi_{1,2}$ the
azimuthal angles of the particle 1 and 2 where the average is taken over
events. The measurements of $\gamma$ are motivated by the search for the
Chiral Magnetic Effect (arXiv:0711.0950, ). (The $\gamma$ is actually
extracted through three particle correlations
$\langle\cos(\phi_{1}+\phi_{2}-2\phi_{3})\rangle$ divided by elliptic flow
$v_{2}$ with the third particle serving as a “projector” toward the Event
Plane Voloshin:2004vk . ) Now let us focus on the same-charge pairs, and
suppose the CME indeed gives rise to the same-side azimuthal correlations
along the $\bf B$-field direction for the same-charge pairs. The two-particle
density then receives the following contribution (with $A_{++}$ the signal
strength)

$\displaystyle f_{++}=A_{++}\cos(\phi_{1}-\Psi_{\bf
B})\,\cos(\phi_{2}-\Psi_{\bf B}).$ (2)

This translates into the following form after re-defining angles
$\bar{\phi}_{i}=\phi_{i}-\Psi_{2}$ and $\bar{\Psi}_{\bf B}=\Psi_{\bf
B}-\Psi_{2}$:

$\displaystyle f_{++}$ $\displaystyle=$
$\displaystyle\frac{A_{++}}{2}\cos(\bar{\phi}_{1}-\bar{\phi}_{2})+\frac{A_{++}}{2}[\cos(2\bar{\Psi}_{\bf
B})]\,\cos(\bar{\phi}_{1}+\bar{\phi}_{2})$ (3)
$\displaystyle+\frac{A_{++}}{2}[\sin(2\bar{\Psi}_{\bf
B})]\,\sin(\bar{\phi}_{1}+\bar{\phi}_{2}).$

We therefore see that the CME’s contribution to the $\gamma_{++}$:

$\displaystyle\gamma_{++}\sim\frac{\langle A_{++}\cos(2\bar{\Psi}_{\bf
B})\rangle}{2}.$ (4)

If the $\bf B$-direction were to be always perfectly aligned with the out-of-
plane, i.e. $\bar{\Psi}_{\bf B}=\pi/2$, then we simply have
$\gamma_{++}\to-\langle A_{++}\rangle/2$. But the fluctuations in magnetic
field as well as in matter geometry will blur the angular relation between the
two and modify the signal by the factor $\sim\langle\cos(2\bar{\Psi}_{\bf
B})\rangle$. Similarly, if one measures the charge separation with respect to
higher harmonic participant plane, e.g. the fourth harmonic plane $\Psi_{4}$,
the corresponding correlation will be
$\langle\cos[2(\phi_{1}+\phi_{2}-2\Psi_{4})]\rangle$ (as recently proposed
with the hope to disentangle the collective-flow and CME contributions to
$\gamma$ Voloshin:2011mx ). The azimuthal fluctuation of ${\bf B}$ field with
respect to $\Psi_{4}$ will again contribute a modification factor
$\sim\langle\cos[4(\Psi_{\bf B}-\Psi_{4})]\rangle$ in the above correlation.

Figure 4: (Color online) The correlations $\langle\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle$ as functions of impact parameter for $n=1,2,3,4$ at four
different positions on the transverse plane: (from left to right)
${\mathbf{r}}=(0,0,0)$ fm; ${\mathbf{r}}=(3,0,0)$ fm; ${\mathbf{r}}=(0,3,0)$
fm; ${\mathbf{r}}=(3,3,0)$ fm.  Figure 5: (Color online) The $(e{\bf
B})^{2}$-weighted correlations $\langle(e{\bf B})^{2}\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle/\langle(e{\bf B})^{2}\rangle$ as functions of impact
parameter for $n=1,2,3,4$ at ${\mathbf{r}}=(0,0,0)$ fm.

Recently it has been suggested that a splitting between the elliptic flow of
the $\pi^{-}$ and $\pi^{+}$ could occur in heavy-ion collisions due to the
Chiral Magnetic Wave (Burnier:2011bf, ; Burnier:2012ae, ), as briefly
discussed in Sec. I. Specifically the CMW will induce an electric quadruple in
the net charge distribution along the $\bf B$ field:

$\displaystyle\rho_{e}(\phi)\sim 2r_{e}\cos[2(\phi-\Psi_{\bf
B})]=2r_{e}\cos[2(\bar{\phi}-\bar{\Psi}_{\bf B})],$ (5)

where $r_{e}\propto\int d\phi\rho_{e}(\phi)\cos[2(\phi-\Psi_{\bf B})]$
quantifies the quadruple from CMW. This modifies the final charged hadron
distribution by an amount proportional to $r_{e}$:

$\displaystyle\frac{dN_{\pm}(\phi)}{d\phi}\sim
N_{\pm}\left\\{1+2v_{2}\cos(2\bar{\phi})\pm\frac{r_{e}}{N}\cos[2(\bar{\phi}-\bar{\Psi}_{\bf
B})]\right\\},\quad$ (6)

where $N=N_{+}+N_{-}$ is the charge multiplicity. Thus one obtains a splitting
of the charged pions $v_{2}$

$\displaystyle
v_{2}^{\pi^{-}}-v_{2}^{\pi^{+}}=-\,\langle\frac{r_{e}}{N}\cos(2\bar{\Psi}_{\bf
B})\rangle.$ (7)

Again, we see the modification factor $\sim\langle\cos(2\Psi_{\bf B})\rangle$
arising from relative orientation between $\bf B$ and $\Psi_{2}$. If
$\bar{\Psi}_{\bf B}$ were to be simply $\pi/2$ (i.e. $\bf B$ always out-of-
plane) then the sign of splitting is always $v_{2}^{\pi^{-}}>v_{2}^{\pi^{+}}$.
However due to the fluctuation, $\bar{\Psi}_{\bf B}$ deviates from $\pi/2$ and
leads to a smaller splitting.

Finally, we consider the example of recently suggested soft photon production
from conformal anomaly in the presence of an external ${\bf B}$ field
(Basar:2012bp, ). The photon emitted via this mechanism carries azimuthal
information of the external magnetic field. If one assumes a homogeneous
external $\bf B$ field, then the produced photon spectrum goes like
$q_{0}dN/d^{3}{\bm{q}}\sim(\bm{q}\times\bm{B})^{2}$ leading to the following
azimuthal distribution:

$\displaystyle
dN_{\gamma}/d\phi\propto\left\\{1-\cos[2(\phi-\Psi_{B})]\right\\}=\left\\{1-\cos[2(\bar{\phi}-\bar{\Psi}_{B})]\right\\}\quad$
(8)

which will contribute the following to the elliptic flow of these photons
(with $V_{\gamma}$ being the signal strength)

$\displaystyle v_{2}^{\gamma}=-\langle V_{\gamma}\cos(2\bar{\Psi}_{\bf
B})\rangle.$ (9)

Once again we see the same modification factor
$\sim\langle\cos(2\bar{\Psi}_{\bf B})\rangle$ due to the mismatch between $\bf
B$ field and the out-of-plane direction from event to event. If
$\bar{\Psi}_{\bf B}$ were to be simply $\pi/2$ (i.e. $\bf B$ always out-of-
plane) then indeed these photons will have positive and sizable elliptic flow.
However the fluctuation may bring $\bar{\Psi}_{\bf B}$ to depart from $\pi/2$
considerably.

Figure 6: (Color online) The correlations $\langle\cos[n(\Psi_{\bf
E}-\Psi_{n})]\rangle$ as functions of impact parameter for $n=1,2,3,4$ at four
different positions on the transverse plane: (from left to right)
${\mathbf{r}}=(0,0,0)$ fm; ${\mathbf{r}}=(3,0,0)$ fm; ${\mathbf{r}}=(0,3,0)$
fm; ${\mathbf{r}}=(3,3,0)$ fm.

In all three examples, the fluctuating $\Psi_{\bf B}-\Psi_{2}$ brings in a
reduction to the intrinsic strength of the signal by

$\displaystyle R_{1}=\langle\cos(2\bar{\Psi}_{\bf B})\rangle.$ (10)

It is therefore worth a close examination of this factor: see Fig. 4 for the
computed average values of $\langle\cos[n(\Psi_{\bf B}-\Psi_{n})]\rangle$ for
varied centralities from event-by-event determination of the $\bf B$-field
direction $\Psi_{\bf B}$ (at several different spatial points) and the
participants harmonics, $\Psi_{n}$, $n=1,2,3,4$. The plots suggest that the
correlations between $\Psi_{\bf B}$ and the odd harmonics $\Psi_{1},\Psi_{3}$
are practically zero (in accord with parity invariance), while the the
correlations of $\Psi_{\bf B}$ with even harmonics $\Psi_{2},\Psi_{4}$ are
nonzero but get suppressed as compared with the results in optical geometry
limit (without fluctuations) e.g. $\langle\cos[2(\Psi_{\bf
B}-\Psi_{2})]\rangle_{\rm opt}=-1$. The centrality dependence of
$\langle\cos[2(\Psi_{\bf B}-\Psi_{2})]\rangle$ is in perfect agreement with
the patterns seen in the histograms Fig. 2 and scatter plots Fig. 3: it is
significantly suppressed in the most central and most peripheral cases
(indicating no correlations) while is maximized around $b=8\sim 10$ fm with
peak values $-0.6\sim 0.7$. Similar behavior is observed for
$\langle\cos[4(\Psi_{\bf B}-\Psi_{4})]\rangle$ too albeit with a weaker
correlation strengths, e.g. with the peak values $\sim 0.2$ or so. Although
suppressed, the correlation between $\Psi_{\bf B}$ and $\Psi_{4}$ appears
still sizable for moderate values of $b$. This may imply complications and
caveats that must be seriously addressed for the proposal in Ref.
Voloshin:2011mx  to use the two-particle correlation with respect to the
fourth harmonic event plane to disentangle the CME contribution and the flow
contribution.

The plots in Fig.4 for four different field points demonstrate the spatial
dependence of the correlations $\langle\cos[n(\Psi_{\bf B}-\Psi_{n})]\rangle$
on the transverse plane. We first see minor differences when the field point
deviates from the origin and such differences become sizable for the outer
most point ${\mathbf{r}}=(3,3,0)$ fm. The comparison suggests that further
away from the origin the correlations between $\Psi_{\bf B}$ and $\Psi_{n}$
become even weaker. However, near the central overlapping region, the
correlations are almost homogeneous over, for example, the distance of the
typical size of a sphaleron, $(\alpha_{s}T)^{-1}\lesssim 1$ fm for RHIC and
LHC McLerran:1990de .

The last issue we want to address is the possible correlation between the
${\bf B}$ orientation and its own strength. It is conceivable that there could
be correlation (on an event-by-event basis) between the signal strength and
the $\bf B$ field direction. Typically the signal strength of these $\bf
B$-field induced effects will scale as $\sim\bf B^{2}$. For example, recalling
that the CME current is proportional to ${\bf B}$ and the signal strength
$A_{++}$ as mentioned earlier would be nearly proportional to ${\bf B}^{2}$.
The quadrupole strength $r_{e}$ through CMW also scales similarly
$r_{e}\propto{\bf B}^{2}$. So does the signal strength $V_{\gamma}$ in the
case of soft photon emissions. If the field strength and the orientation of
${\bf B}$ do correlate with each other, then one may not be able to factorize
the signal strength and the field orientation factor $\cos(2\bar{\Psi}_{\bf
B})$ when taking the event average. We therefore further examine the
correlations between the two by evaluating the factor

$\displaystyle R_{2}=\frac{\langle(e{\bf B})^{2}\cos(2\bar{\Psi}_{\bf
B})\rangle}{\langle(e{\bf B})^{2}\rangle}.$ (11)

In Fig. 5 we show the impact parameter dependence of the $(e{\bf
B})^{2}$-weighted correlations $\langle(e{\bf B})^{2}\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle/\langle(e{\bf B})^{2}\rangle$ for $n=1,2,3,4$. We find
little difference between the two factors $R_{1}$ in Fig.4(the most left
panel) and $R_{2}$ in Fig.5 at all centralities for all $n$ values. While only
results for field point ${\mathbf{r}}=(0,0,0)$ are shown in Fig.5, we have
checked all other three points and the observation is the same. Therefore, we
conclude that the magnitude of the magnetic field has no noticeable
correlation to its azimuthal direction with respect to the matter geometry.

##  V Azimuthal correlation between electric field and matter geometry

In this section, we briefly discuss how the event-by-event fluctuation affects
the correlation between the orientation of ${\bf E}$-field and the
participants harmonic planes, $\Psi_{n}$. The motivation is that from Fig. 1
we notice that the electric field can be as strong as the magnetic one, and
thus they may lead to observable effects too. An obvious example is possible
multiple charge distributions induced by the strong electric field. Similarly
to the magnetic field case, possible ${\bf E}$-induced effects would be
affected by the azimuthal correlations between ${\bf E}$ and matter geometry.
It is therefore interesting to also study these correlations. In Fig. 6, we
show the correlations $\langle\cos[n(\Psi_{\bf E}-\Psi_{n})]\rangle$,
$n=1,2,3,4$, as functions of the impact parameter at the four field points as
before. There are a number of interesting features that differ from the
magnetic field case: 1) there is a clear negative correlation (i.e. back-to-
back) between $\Psi_{\bf E}$ and $\Psi_{1}$ and it is most strong in the more
central collisions and at the field point near the origin — this is
understandable as the pole of $\Psi_{1}$ with more matter will concurrently
have more positive charges from protons generating ${\bf E}$ pointing in the
opposite; 2) at field points away from the origin the $\Psi_{\bf E}$ is
strongly correlated to $\Psi_{2}$ (and also weakly correlated to $\Psi_{4}$) —
the direction of ${\bf E}$ is more in-plane for the field point on $x$-axis
while more out-of-plane for points away from $x$-axis; 3) there is also a weak
correlation between $\Psi_{\bf E}$ and $\Psi_{3}$. Finally we have also
checked the $(eE)^{2}$-weighted correlations $\langle(e{\bf
E})^{2}\cos[n(\Psi_{\bf E}-\Psi_{n})]\rangle/\langle(e{\bf E})^{2}\rangle$ and
find no visible correlation between the ${\bf E}$ magnitude and orientation.

##  VI Discussion on charge-dependent measurements

The studies of correlations between ${\bf B}$ and ${\bf E}$ fields and matter
geometry suggest in general that there may be nontrivial charge distributions,
particularly in azimuthal angles (e.g. charged dipole and quadrupole), induced
by varied ${\bf B}$\- and ${\bf E}-$induced effects. There are also other
effects not related the initial ${\bf E}$ and ${\bf B}$ fields, e.g., the
local charge conservation effect (Schlichting:2010qia, ) that can lead to
nontrivial charge azimuthal correlations when coupled with various harmonic
flows. It is therefore tempting to think about possible measurements that may
fully extract information for the azimuthal charge distributions. In parallel
to the measurements of (charge-inclusive) global azimuthal particle
distributions that can be subsequently Fourier-decomposed into various
harmonic components, here we suggest a class of observables, the charged
multiple vectors $\hat{Q}^{c}_{n}$. Consider the measured charged hadrons in
an event we can construct the charged multiple vector $\hat{Q}_{n}^{c}$ with
magnitude $Q^{c}_{n}$ and azimuthal angle $\Psi^{c}_{n}$:

$\displaystyle Q^{c}_{n}\,e^{in\Psi^{c}_{n}}=\sum_{i}\,q_{i}\,e^{in\phi_{i}}$
(12)

where the summation runs over all particles with $q_{i}$ and $\phi_{i}$ the
electric charge and azimuthal angle of the $i$-th particle. This idea
generalizes the earlier proposal of the charged dipole vector analysis
suggested in (Liao:2010nv, ) in the context of CME observables. One may
introduce properly $p_{t}$-weighed definition. One may also think of sub-event
version of this analysis or possible multi-particle correlation improved
version. We emphasize that these are different and independent from the
existing $\hat{Q}_{n}$ vectors related to flow measurements, i.e.
$Q_{n}\,e^{in\Psi_{n}}=\sum_{i}\,e^{in\phi_{i}}$. The $\hat{Q}_{n}$ is charge
blind and includes all charges similarly while the $\hat{Q}_{n}^{c}$ takes the
difference between positive and negative charges therefore yields information
on the charge distribution. The technical difficulty of $\hat{Q}_{n}^{c}$
measurements should be at the same level as previous $\hat{Q}_{n}$ analysis
and quite feasible, while clearly the $\hat{Q}_{n}^{c}$ analysis provides
orthogonal and unique information on the charge distributions. With a joint
$\hat{Q}_{n}$ and $\hat{Q}_{n}^{c}$ analysis one can study the strength and
azimuthal correlations among all harmonics and charged multipoles. Therefore
it is very valuable to do a systematic charged multiple vector analysis,
leading toward a quantitative “charge landscape survey” in heavy-ion
collisions.

##  VII Summary

In summary, we have performed a detailed study of the event-by-event
fluctuations of both the azimuthal orientation of the magnetic and electric
fields as well as the matter geometry (which is specified by the participant
planes of a series of harmonics) in the initial condition of heavy-ion
collisions. Such fluctuations suppress the azimuthal correlations between the
magnetic field and the second harmonic participant plane particularly in the
very central and very peripheral collisions, while leaving a window around
$b=8\sim 10$ fm with still sizable correlation between the two. We have
further studied similar azimuthal correlations between ${\bf B}$ and other
harmonic participant planes, and found similar yet weaker correlation with
$\Psi_{4}$ while no correlation with $\Psi_{1}$ and $\Psi_{3}$. The
correlation between $\Psi_{\bf B}$ and $\Psi_{4}$ may indicate that the CME
can also contribute to the charge-pair correlation
$\langle\cos[2(\phi_{1}+\phi_{2}-2\Psi_{4})]\rangle$ as opposed to the
assumption in Ref. Voloshin:2011mx . Examination of these correlations at
different field points shows notable dependence on spatial positions. For
completeness we have presented similar studies for electric field ${\bf E}$
azimuthal correlations with matter geometry which show quite different
patterns from that for the ${\bf B}$ field. We have evaluated the impact of
such azimuthal fluctuations and correlations on a number of observables
related to magnetic-field induced effects in heavy-ion collisions: the charged
pair azimuthal correlations, the charge dependent elliptic flow of pions, as
well as the azimuthal anisotropy of soft photons due to conformal anomaly.
Specifically we have quantified the modification factors
$R_{1}=\langle\cos(2\bar{\Psi}_{\bf B})\rangle$ and the $(e{\bf
B})^{2}$-weighted reduction factor $R_{2}=\langle(e{\bf
B})^{2}\cos(2\bar{\Psi}_{\bf B})\rangle/\langle(e{\bf B})^{2}\rangle$ at
different centralities (see Figs. 4 and 5) and found sizable reduction in both
very central and very peripheral collisions. From these results we conclude
that the optimal centrality class for observing the above mentioned ${\bf
B}$-induced effects corresponds to impact parameter range $b\sim 8-10$ fm. The
qualitative conclusion should hold also for Pb + Pb collisions at LHC.

We end with a few pertinent remarks:
(1) In our computations (as well as all the previous computations
arXiv:0711.0950 ; arXiv:0907.1396 ; arXiv:1003.2436 ; arXiv:1103.4239 ;
arXiv:1107.3192 ; arXiv:1111.1949 ; arXiv:1201.5108 ), the classical Liénard-
Wiechert potentials have been used. One may worry about quantum corrections to
the field equations as the magnitude of the electromagnetic field is much
larger than the electron and light quark masses. In principle a calculation
including all relevant QED processes is needed, but a one-loop Euler-
Heisenberg effective lagrangian (see Ref. Dunne:2004nc  for review) may give a
good indication on the size of such corrections. At strong-field limit the
field equation derived from this lagrangian can be regarded as linear Maxwell
equations with the renormalized charge
$e^{2}\rightarrow\tilde{e}^{2}=e^{2}/\left[1-\frac{e^{2}}{24\pi^{2}}\ln\frac{e^{2}|F|^{2}}{m_{e}^{4}}\right]$.
An estimate implies that the computed electromagnetic field may be amended
only by a few percent even for $e|F|\sim 100m_{\pi}^{2}$.
(2) As shown in Refs. arXiv:1103.4239 ; arXiv:1201.5108 , the magnetic field
magnitude bears strong time dependence. Furthermore, the electromagnetic
response of QGP could dramatically modify the time evolution of the ${\bf B}$
and ${\bf E}$ fields. It would be important to incorporate these into
realistic modeling of effects induced by these fields, as recently attempted
in Toneev:2012zx  to incorporate the time evolution into the event-by-event
estimates of the CME by using the parton-hadron-string-dynamics approach.
(3) As suggested in Ref. Voloshin:2010ut , in order to disentangle the effects
of the magnetic field and the collective flow, it is very useful to study the
U + U collisions where the geometry becomes a nontrivial level arm. It will be
interesting to study in the future the various correlations among the field
strengths, field azimuthal orientations with the matter geometric
eccentricities as well as the harmonic participant planes.

Acknowledgments: We are grateful to Y. Burnier, A. Bzdak, U. Heinz, D.
Kharzeev, V. Koch, V. Skokov, H. Yee, and Z. Qiu for helpful communications
and discussions. We also thank the RIKEN BNL Rsearch Center for partial
support.

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