Electromagnetic fields from the extended Kharzeev-McLerran-Warringa model in relativistic heavy-ion collisions
Introduction
The study of strongly interacting matter and its properties in the presence of strong electromagnetic (EM) fields has been a hot topic for more than a decade [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. Ultra-relativistic heavy-ion collisions provide a unique way for creating and exploring the strongly interacting matter at extremely high temperature and high energy density, where the matter is expected to be deconfined and reach a new state of matter, which is so-called the “quark-gluon plasma” (QGP) [36], [37], [38], [39]. Properties of strongly interacting matter are governed by quantum chromodynamics (QCD), which have been widely and experimentally studied both on the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and on the Large Hadron Collider (LHC) at CERN. In heavy-ion collisions, e.g., at top RHIC energy at or LHC energy at , the two oppositely fast moving (almost at the speed of light) colliding nuclei in non-central nucleus-nucleus (A-A) collisions can generate hitherto the strongest EM fields [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], which are usually estimated to be at the order of magnitude of at top RHIC energy, or at LHC energies, where is the pion mass. Here we should note that event-by-event fluctuations of generated EM fields have been widely studied in recent years in Refs. [9], [10], [11], [15], [21], [22], [23], [29], [30], [31], [32], [33], [34], which can give rise to non-vanishing components of EM fields such as and due to the fluctuations of proton positions in the two colliding nuclei.
In the QCD vacuum, topologically non-trivial gluon field configurations with non-zero winding number [40], [41], [42] of deconfined QGP in the presence of such a strong magnetic field B can induce a non-conserved axial current and also a non-conserved vector current along the direction of magnetic field, which are respectively called the “chiral separation effect” (CSE) and “chiral magnetic effect” (CME) [2], [6], [43], [44], [45], [46], [47], [48], [49], [50], [51]. Since the axial current requires a charge-conjugate -odd environment while vector current requires a chirality imbalanced parity -odd environment, an asymmetry between the amount of positive and negative charges along the direction of magnetic field B in heavy-ion collisions is expected. Experimental observations of the CME can be regarded as direct evidences of topologically non-trivial gluon field configurations, and furthermore can be interpreted as indications of event-by-event local and violations of QCD at quantum level [2], [43].
Besides the CME and CSE, it is well known that strong magnetic fields can also influence many QCD processes [49], e.g. the induction of chiral symmetry breaking [52], influences on chiral condensation [53], and modifications of in-medium particle's mass [54], [55], [56], [57], [58], [59], [60]. As an important consequence, the QCD phase diagram may be dynamically modified by such a strong magnetic field [61], [62], [63], [64], [65], e.g. color-superconducting phases at very high baryon densities will be strongly affected by the magnetic field [61]. When anomaly processes are coupled with strong magnetic fields, many interesting effects [49], e.g. the formation of -domain walls [66], will also be induced and generated.
Since the generated magnetic field B in heavy-ion collisions can not be directly measured event-by-event, it is thus of enormous challenge to measure the magnetic field induced chiral anomalous effects in experiments. In heavy-ion collisions, the magnetic field is generated along the direction roughly but preferentially perpendicular to the reaction plane (RP), experimental measurements are therefore usually conducted using the two-particle correlator firstly proposed by Voloshin [67], where α and β denote the electric charge sign of particles α and β, and are respectively their azimuthal angles, is the azimuthal angle of the constructed reaction plane for a given event, and denotes the average over all particle pairs and then over all events. Therefore, the same-sign (SS) and opposite-sign (OS) correlators can be respectively defined as and . Based on Refs. [5], [67], the magnetic field driven CME is expected to contribute to a negative but a positive . The STAR Collaboration [68], [69], [70], [71], [72], [73] and ALICE Collaboration [74] have independently measured the and correlators, which indeed show the expected features of the CME. However, there exit some ambiguities [75], [76], [77], [78], [79], [80], [81], [82], [83] in the interpretation of experimental data due to large background contaminations, potentially arising from the elliptic-flow () driven background contributions, e.g., the transverse momentum conservation (TMC) [78], [82], [83] and local charge conservation (LCC) [79], [80], [81]. Hence, a dedicated run of and isobar colliding systems at RHIC has been proposed [5], which is expected to yield unambiguous evidence for the CME signal by varying the signal but with the -driven backgrounds roughly fixed [5], [84], [85], [86], [87], [88], [89].
In this paper, we will review an analytical model for the estimation of strong EM fields generated in relativistic heavy-ion collisions from the original work initiated by Kharzeev, McLerran and Warringa [2], which we refer to as the original KMW model. On the ground of it, we formulate our generalizations of the estimated EM fields. We first start from generalizing the three-dimensional charge distributions, e.g. three-parameter Fermi (3pF) model, by incorporating Lorentz contraction effect, based on which we point out that the formulas of EM fields in the original KMW model can be properly extended by incorporating the longitudinal position dependence through the generalized charge distribution models for both spherically symmetric and axially deformed colliding nuclei used in heavy-ion collisions. Also, we make retardation correction to the estimated EM fields contributed by participants. Thus our formulation of the estimated EM fields can be easily applied to lower energy regions, such as the current beam-energy-scan (BES) program at RHIC, the under planning FAIR, NICA and J-PARC programs. It is due to the fact that the Lorentz contraction is not so large that the “pancake-shaped disk” approximation used in the original KMW model [2] is no longer appropriate in these lower energy regions. Moreover, we further extend the formulas of EM fields by incorporating medium feedback effects according to the Faraday's induction law, thus a constant Ohm conductivity is properly and analytically embedded for the pure QGP medium and also a time-dependent conductivity for the realistic QGP evolution. Finally, we make numerical evaluations of the generalized magnetic field strength in the extended KMW model for detailed comparisons of time evolution, impact parameter b dependence of the estimated magnetic field strength with those from the original KMW model.
This paper is organized as follows. We present detailed formulations of the estimated EM fields in the extended KMW model for heavy-ion collisions in Sec. 2, which consists of four subparts. We first present in Sec. 2.1 a formal generalization of the charge distributions from the widely used 3pF model in which the relativistic Lorentz contraction effects on the geometries of both spherically formed and axially deformed colliding nuclei are taken into account. We then in Sec. 2.2 generalize the EM fields in the vacuum starting from the widely used Liénard-Wiechert equations, and naturally extend the formulas of EM fields with generalized charge distributions and retardation correction. We further extend the EM fields with medium feedback effects by incorporating a constant Ohm electric conductivity in Sec. 2.3, a time-dependent electric conductivity in Sec. 2.4 and also an alternative solution for simulations in Appendix A. Some evaluations and comparisons about the time evolution, centrality (impact parameter b) dependence of the estimated magnetic field from the extended KMW model in comparison with those from the original KMW model are presented in Sec. 3. We finally summarize the main process of such generalization along with the conclusions in Sec. 4. The notation we use in this paper is the rationalized Lorentz-Heaviside units within the natural units, with and .
Section snippets
Electromagnetic fields from the extended KMW model
Before the detailed discussion, let us first emphasize that physical situations of heavy-ion collisions consist of event-by-event local and violation processes due to the effects of topological charge fluctuations with non-trivial QCD gauge field configurations (characterized by the topological invariant, the winding number ) in the vicinity of the deconfined QGP phase [2]. These and violation processes can only locally happen under some special and even extreme conditions in QCD,
Results and discussions
Since we have explicitly formulated the estimations of generated EM fields for heavy-ion collisions in the vacuum in Eqs. (10)-(14), in the pure QGP medium in Eqs. (22)-(26) and also during the realistic QGP evolution in Eqs. (27)-(31), as well as an alternative solution to doing Monte-Carlo simulations during the realistic QGP evolution in Eqs. (A.1)-(A.3), let us first give some pre-analysis before performing the ab integration of charge distribution or event-by-event simulations of
Summary
In summary, based on the Kharzeev-McLerran-Warringa (KMW) model [2] for the estimation of strong EM fields generated in relativistic heavy-ion collisions, we make an attempt to generalize the formulas of estimated EM fields in the original KMW model, which eventually turns out to be above formulations in the extended KMW model consisting of three scenarios: the pure vacuum scenario, the pure conducting medium scenario and the relatively realistic QGP evolution scenario.
We first start from
CRediT authorship contribution statement
Yi Chen: Conceptualization, Formal analysis, Methodology, Software, Writing – original draft. Xin-Li Sheng: Conceptualization, Formal analysis, Methodology, Software, Writing – original draft. Guo-Liang Ma: Conceptualization, Funding acquisition, Project administration, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to gratefully thank Yu-Gang Ma, Jin-Hui Chen, Song Zhang and Qi-Ye Shou for their stimulating and helpful discussions and comments, and Chen Zhong for maintaining high-quality performance of computational facilities. Y.C. thankfully acknowledges the helpful and fruitful discussions with Wei-Tian Deng, Huan-Xiong Yang, Heng-Tong Ding, Xu-Guang Huang, and Qun Wang, as well as the timely discussions and helps from Xin-Li Zhao, Yi-Lin Cheng, Xin Ai, Bang-Xiang Chen, Yu Guo,
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