An extended $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$ correlator for detecting and characterizing
the Chiral Magnetic Wave

Heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) can lead to a magnetized chiral relativistic quark-gluon plasma (QGP) Kharzeev (2006); Liao (2015); Miransky and Shovkovy (2015); Huang (2016); Kharzeev et al. (2016), in which the mass of fermions are negligible compared to the temperature and/or chemical potential. Such a plasma, which is akin to the primordial plasma in the early Universe Rogachevskii et al. (2017); Rubakov and Gorbunov (2017) and several types of degenerate forms of matter in compact stars Weber (2005), have pseudo-relativistic analogs in Dirac and Weyl materials Vafek and Vishwanath (2014); Burkov (2015); Gorbar et al. (2018). It is further characterized not only by an exactly conserved electric charge but also by an approximately conserved chiral charge, violated only by the quantum chiral anomaly Adler (1969); Bell and Jackiw (1969).

The study of anomalous transport in magnetized chiral plasmas can give fundamental insight not only on the complex interplay of chiral symmetry restoration, axial anomaly and gluon topology in the QGP Moore and Tassler (2011); Mace et al. (2016); Liao et al. (2010); Kharzeev et al. (2016); Koch et al. (2017), but also on the evolution of magnetic fields in the early Universe Joyce and Shaposhnikov (1997); Tashiro et al. (2012). Two of the principal anomalous processes in these plasmas [for electric and chiral charge chemical potential $\mu_{V,A}\neq 0$] are the chiral separation effect (CSE) Vilenkin (1980); Metlitski and Zhitnitsky (2005); Son and Surowka (2009) and the chiral magnetic effect (CME) Fukushima et al. (2008). The CSE is derived from the induction of a non-dissipative chiral axial current:

where $\mu_{V}$ is the vector (electric) chemical potential and $\vec{B}$ is the magnetic field. The CME is similarly characterized by the vector current:

where $\mu_{A}$ is the axial chemical potential that quantifies the axial charge asymmetry or imbalance between right- and left-handed quarks in the plasma Fukushima et al. (2008); Son and Surowka (2009); Zakharov (2012); Fukushima (2013).

The interplay between the CSE and CME in the QGP produced in heavy ion collisions, can lead to the production of a gapless collective mode – termed the chiral magnetic wave (CMW) Kharzeev and Yee (2011), stemming from the coupling between the density waves of the electric and chiral charges. The propagation of the CMW is sustained by alternating oscillations of the local electric and chiral charge densities that feed into each other to ultimately transport positive (negative) charges out-of-plane and negative (positive) charges in-plane to form an electric quadrupole. Here, the reaction plane $\Psi_{\rm RP}$, is defined by the impact vector $\vec{b}$ and the beam direction, so the poles of the quadrupole lie along the direction of the $\vec{B}$-field (out-of-plane) which is essentially perpendicular to $\Psi_{\rm RP}$.

The electric charge quadrupole can induce charge-dependent quadrupole correlations between the positively- and negatively-charged particles produced in the collisions Kharzeev and Yee (2011); Liao (2015); Huang (2016); Kharzeev et al. (2016); Stephanov and Yee (2013); Han and Xu (2019); Zhao et al. (2019). Such correlations can be measured with suitable correlators to aid full characterization of the CMW.

A pervasive approach employed in prior, as well as ongoing experimental studies of the CMW, is to measure the elliptic- or quadrupole flow difference between negatively- and positively charged particles Burnier et al. (2011, 2012):

as a function of charge asymmetry $A_{\rm ch}$. Here, $N^{\pm}$ denotes the number of positively- (negatively-) charged hadrons measured in a given event; the slope parameter $r$, which is experimentally determined from the measurements, is purported to give an estimate of the strength of the CMW signal Kharzeev and Yee (2011); Liao (2015); Voloshin and Belmont (2014); Adam et al. (2016); Huang (2016); Kharzeev et al. (2016); Zhao et al. (2019).

However, a wealth of measurements reported by the ALICE Voloshin and Belmont (2014); Adam et al. (2016), CMS Park (2017); Sirunyan et al. (2019) and STAR Adamczyk et al. (2015); Shou (2019) collaborations, highlight a significant influence from the effects of background, suggesting a need for supplemental measurements with improved correlators that not only suppress background, but are also sensitive to small CMW signals in the presence of these backgrounds.

In prior work, we have proposed Magdy et al. (2018a) and validated the utility Magdy et al. (2018b); Huang et al. (2019); Magdy et al. (2020) of the $R_{\Psi_{m}}(\Delta S)$ correlator for robust detection and characterization of the CME-driven dipole charge separation relative to the $\Psi_{2,3}$ planes. Here, we follow the lead of Ref. Shen et al. (2019) by first, extending the correlator for study of the CMW-driven quadrupole charge separation, followed by detailed sensitivity tests of the correlator with the aid of AMPT model simulations.

The extended correlators, $R^{(d)}_{\Psi_{m}}(\Delta S_{d})$, are constructed for each event plane $\Psi_{m}$, as the ratio:

where $d=$ 1 and 2 denote dipole and quadrupole charge separation respectively, and $C_{\Psi_{m}}(\Delta S_{d})$ and $C_{\Psi_{m}}^{\perp}(\Delta S_{d})$ are correlation functions designed to quantify the dipole and quadrupole charge separation $\Delta S_{d}$, parallel and perpendicular (respectively) to the $\vec{B}$-field, i.e., perpendicular and parallel (respectively) to $\mathrm{\Psi_{RP}}$.

The correlation functions used to quantify the dipole and quadrupole charge separation parallel to the $\vec{B}$-field, are constructed from the ratio of two distributions:

where $N_{\text{real}}(\Delta S_{d})$ is the distribution over events, of charge separation relative to the $\Psi_{m}$ planes in each event:

where $n$ and $p$ are the numbers of negatively- and positively charged hadrons in an event, $\Delta{\varphi_{m}}=\phi-\Psi_{m}$ and $\phi$ is the azimuthal emission angle of the charged hadrons. The $N_{\text{Shuffled}}(\Delta S_{d})$ distribution is similarly obtained from the same events, following random reassignment (shuffling) of the charge of each particle in an event. This procedure ensures identical properties for the numerator and the denominator in Eq. 5, except for the charge-dependent correlations which are of interest.

The correlation functions $C_{\Psi_{m}}^{\perp}(\Delta S_{d})$, used to quantify the dipole and quadrupole charge separation perpendicular to the $\vec{B}$-field, are constructed with the same procedure outlined for $C_{\Psi_{m}}(\Delta S_{d})$, but with $\Psi_{m}$ replaced by $\Psi_{m}+\pi/m^{d}$. Note that this rotation of $\Psi_{m}$ maps the sine terms in Eq. 6 into cosine terms.

The correlator $R^{(d)}_{\Psi_{2}}(\Delta S_{d})=C_{\Psi_{2}}(\Delta S_{d})/C_{\Psi_{2}}^{\perp}(\Delta S_{d})$, gives a measure of the magnitude of the charge separation (dipole and quadrupole) parallel to the $\vec{B}$-field (perpendicular to $\Psi_{2}$), relative to that for charge separation perpendicular to the $\vec{B}$-field (parallel to $\Psi_{2}$). Since the CME- and CMW-driven charge separations are strongly correlated with the $\vec{B}$-field direction, the correlators $R^{(d)}_{\Psi_{3}}(\Delta S_{d})=C_{\Psi_{3}}(\Delta S_{d})/C_{\Psi_{3}}^{\perp}(\Delta S_{d})$ are insensitive to them, due to the absence of a strong correlation between the $\vec{B}$-field and the orientation of the $\Psi_{3}$ plane. For small systems such as $p$/$d$/³He+Au and $p$+Pb, a similar insensitivity is to be expected for $R^{(d)}_{\Psi_{2}}(\Delta S_{d})$, due to the weak correlation between the $\vec{B}$-field and the orientation of the $\Psi_{2}$ plane. For background-driven charge separation however, similar patterns are to be expected for both the $R^{(d)}_{\Psi_{2}}(\Delta S_{d})$ and $R^{(d)}_{\Psi_{3}}(\Delta S_{d})$ distributions.

The response and the sensitivity of the $R^{(1)}_{\Psi_{2}}(\Delta S_{1})$ correlator to CME-driven charge separation is detailed in Refs. Magdy et al. (2018a, 2020). For CMW-driven charge separation, $R^{(2)}_{\Psi_{2}}(\Delta S_{2})$ is expected to show an approximately linear dependence on $\Delta S_{2}$ for $|\Delta S_{2}|\lesssim 3$, due to a shift in the distributions for $C_{\Psi_{2}}^{\perp}(\Delta S_{d})$ relative to $C_{\Psi_{2}}(\Delta S_{d})$, induced by the CMW. Thus, the slope of the plot of $R^{(2)}_{\Psi_{2}}(\Delta S_{2})$ vs. $\Delta S_{2}$, encodes the magnitude of the CMW signal. This slope is also influenced by particle number fluctuations and the resolution of the $\Psi_{2}$ plane which fluctuates about $\Psi_{\rm RP}$. The influence of the particle number fluctuations can be minimized by scaling $\Delta S_{2}$ by the width $\mathrm{\sigma_{\Delta_{Sh}}}$ of the distribution for $N_{\text{shuffled}}(\Delta S_{2})$ i.e., $\Delta S_{2}^{{}^{\prime}}=\Delta S_{2}/\mathrm{\sigma_{\Delta_{Sh}}}$. Similarly, the effects of the event plane resolution can be accounted for by scaling $\Delta S_{2}^{{}^{\prime}}$ by the resolution factor $\mathrm{\delta_{Res}}$, i.e., $\Delta S_{2}^{{}^{\prime\prime}}=\Delta S_{2}^{{}^{\prime}}/\mathrm{\delta_{Res}}$, where $\mathrm{\delta_{Res}}$ is the event plane resolution. The efficacy of these scaling factors have been confirmed via detailed simulation studies, as well as with data-driven studies.

Our sensitivity studies for $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$, relative to the $\Psi_{2}$ and $\Psi_{3}$ event planes, are performed with AMPT events in which varying degrees of proxy CMW-driven quadrupole charge separation were introduced Ma and Zhang (2011); Shen et al. (2019). The AMPT model is known to give a good representation of the experimentally measured particle yields, spectra, flow, etc.,Lin et al. (2005); Ma and Lin (2016); Ma (2013, 2014a); Bzdak and Ma (2014); Nie et al. (2018). Therefore, it provides a reasonable estimate of both the magnitude and the properties of the background-driven quadrupole charge separation expected in the data collected at RHIC and the LHC.

We simulated Au+Au collisions at $\mbox{$\sqrt{s_{\mathrm{NN}}}$}~{}=~{}200$ GeV with the same AMPT model version used in our prior studies Huang et al. (2019); Shen et al. (2019); Magdy et al. (2020); this version incorporates both string melting and local charge conservation. In brief, the model follows four primary stages: (i) an initial-state, (ii) a parton cascade phase, (iii) a hadronization phase in which partons are converted to hadrons, and (iv) a hadronic re-scattering phase. The initial-state essentially simulates the spatial and momentum distributions of mini-jet partons from QCD hard processes and soft string excitations as encoded in the HIJING model Wang and Gyulassy (1991); Gyulassy and Wang (1994). The parton cascade considers the strong interactions among partons via elastic partonic collisions Zhang (1998). Hadronization is simulated via a coalescence mechanism. After hadronization, the ART model is invoked to simulate baryon-baryon, baryon-meson and meson-meson interactions Li and Ko (1995).

A formal mechanism for generation of the CMW is not implemented in the AMPT model. However, a proxy CMW-induced quadrupole charge separation can be implemented Ma and Zhang (2011); Ma (2014b) by interchanging the the position coordinates ($x$, $y$, $z$) for a fraction ($f_{q}$) of the in-plane light quarks ($u$, $d$ and $s$) carrying positive (negative) charges with out-of-plane quarks carrying negative (positive) charges, at the start of the partonic stage. This procedure lends itself to two quadrupole charge configurations, relative to the in-plane and out-of-plane orientations. The first or Type (I), is for events with negative net charge ($A_{\rm ch}<-0.01$) in which the $u$ and $\bar{d}$ are set to be concentrated on the equator of the quadrupole (in-plane), while $\bar{d}$ and $u$ quarks are set to be concentrated at the poles of the quadrupole (out-of-plane). The second or Type (II), is for events with positive net charge ($A_{\rm ch}>-0.01$) in which the in-plane and out-of-plane quark configurations are swapped. The latter configuration was employed for the bulk of the AMPT events generated with proxy input signals. The magnitude of the proxy CMW signal is set by the fraction $f_{q}$, which serves to characterize the strength of the quadrupole charge separation.

The AMPT events with varying degrees of proxy CMW signals were analyzed with the $R^{(2)}_{\Psi_{2,3}}(\Delta S_{2})$ correlators to identify and quantify their response to the respective input signals, following the requisite corrections for particle number fluctuations ($\Delta S_{2}^{{}^{\prime}}=\Delta S_{2}/\mathrm{\sigma_{\Delta_{Sh}}}$) and event-plane resolution ($\Delta S_{2}^{{}^{\prime\prime}}=\Delta S_{2}^{{}^{\prime}}/\mathrm{\delta_{Res}}$), as described earlier.

The top panels of Fig. 1 confirm the expected Gaussian distributions for $N(\Delta S^{{}^{\prime\prime}}_{2})^{\bot}$, as well as the shift in its mean value as $f_{q}$ increases; the mean value is zero for $f_{q}=0$ (a) and progressively shifts to $\Delta S^{{}^{\prime\prime}}_{2}<0$ for $f_{q}>0$ (b and c). These CMW-induced shifts for $f_{q}>0$, are made more transparent in Figs. 1 (d)-(f) where the shift of $C_{\Psi_{2}}^{\perp}(\Delta S^{{}^{\prime\prime}}_{2})$ relative to the $C_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ correlation function is apparent c.f. Fig. 1 (f).

The $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ and $R^{(2)}_{\Psi_{3}}(\Delta S^{{}^{\prime\prime}}_{2})$ correlators, obtained for several input values of $f_{q}$, are shown in Fig. 2. They indicate an essentially flat distribution for $R^{(2)}_{\Psi_{3}}(\Delta S^{{}^{\prime\prime}}_{2})$ irrespective of the value of $f_{q}$. These patterns are consistent with the expected insensitivity of $R^{(2)}_{\Psi_{3}}(\Delta S^{{}^{\prime\prime}}_{2})$ to CMW-driven charge separation due to the absence of a strong correlation between the $\vec{B}$-field and the orientation of the $\Psi_{3}$ plane. Figs. 2 (a)-(f) show that the $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ correlator evolves from a flat distribution for $f_{q}=0$, to an approximately linear dependence on $\Delta S^{{}^{\prime\prime}}_{2}$ (for $|\Delta S^{{}^{\prime\prime}}_{2}|\lesssim 3$) with slopes that reflect the increase in the magnitude of the input CMW-driven charge separation with $f_{q}$. These patterns not only confirm the input quadrupole charge separation signal in each case; they suggest that the $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$ correlator is relatively insensitive to a possible $v_{2,3}$-driven background [and their associated fluctuations] as well as the local charge conservation effects implemented in the AMPT model. Note the essentially flat distributions for $R^{(2)}_{\Psi_{3}}(\Delta S^{{}^{\prime\prime}}_{2})$ and for $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ when the input signal is set to zero.

This insensitivity can be further checked via the event-shape engineering, through fractional cuts on the distribution of the magnitude of the $q_{2}$ flow vector Schukraft et al. (2013). Here, the underlying notion is that elliptic flow ${v_{2}}$, which is a major driver of background correlations, is strongly correlated with $q_{2}$ Acharya et al. (2018); Zhao (2018). Thus, the magnitude of the background correlations can be increased(decreased) by selecting events with larger(smaller) $q_{2}$ values. Such selections were made by splitting each event into three sub-events; $A[\eta<-0.3]$, $B[|\eta|<0.4]$, and $C[\eta>0.3]$, where sub-event $B$ was used to evaluate $q_{2}$, and the other sub-events used to evaluate $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ via the methods described earlier.

Figure 3 shows a comparison of the $q_{2}$-selected $R^{(2)}_{\Psi_{2}}$ distributions (a), $v_{2}$ (b) and the slopes (c) extracted from the distributions shown in panel (a), respectively. These results were obtained for 10-50% central Au+Au collisions with $f_{q}$=5%. They indicate that while $v_{2}$ increases with $q_{2}$, the corresponding slope for the $R^{(2)}_{\Psi_{2}}$ correlators (Fig. 3 (c)) show little, if any, change. This insensitivity to the value of $q_{2}$ is incompatible with a dominating influence of background-driven contributions to $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$. It is noteworthy that a further analysis performed for background-driven charge separation with strong local charge conservation, also indicated that $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ is essentially insensitive to this background.

The $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ distributions shown in Fig. 2, indicate slopes that visibly increase with $f_{q}$. To quantify the measured signal strengths, we extracted the slope $S$, of the respective $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ distributions shown in the figure. Fig. 4 indicates a linear dependence of these slopes on $f_{q}$. It also shows that the magnitude and trends of $S$ are independent of the event plane used in the analysis. These results suggests that the $R^{(2)}_{\Psi_{2}}$ correlator not only suppresses background, but is sensitive to small CMW-driven charge separation in the presence of such backgrounds.

The slopes of the $R^{(2)}_{\Psi_{2}}(\Delta S^{{}^{\prime\prime}}_{2})$ vs. $\Delta S^{{}^{\prime\prime}}_{2}$ distributions can also be explored as a function of the charge asymmetry $A_{\rm ch}$ as shown in Fig. 5. Here, the $A_{\rm ch}$ distribution shown in the inset, hints at the fact that the model parameters used in the AMPT simulations were chosen to give a positive net charge, when averaged over all events. Fig. 5 shows the expected decrease of $S$ with $A_{\rm ch}$ for $A_{\rm ch}<0$. It also shows that the sign of $S$ can even be flipped for sufficiently large negative values of $A_{\rm ch}$, in accord with expectations. Fig. 5 also shows that the slopes for $R^{(2)}_{\Psi_{3}}(\Delta S^{{}^{\prime\prime}}_{2})$ vs. $\Delta S^{{}^{\prime\prime}}_{2}$ are insensitive to $A_{\rm ch}$ as might be expected. These dependencies could serve as further aids to CMW signal detection and characterization in experimental measurements.

In summary, we have extended the $R^{(1)}_{\Psi_{m}}(\Delta S_{1})$ correlator, previously used to measure CME-induced dipole charge separation, to include the $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$ correlator, which can be used to measure CMW-driven quadrupole charge separation. Validation tests involving varing degrees of proxy CMW signals injected into AMPT events, show that the $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$ correlator provides discernible responses for background- and CMW-driven charge separation which could aid robust identification of the CMW. They also indicate a level of sensitivity that would allow for a robust experimental characterization of the purported CMW signals via $R^{(2)}_{\Psi_{m}}(\Delta S_{2})$ measurements in heavy-ion collisions.