Physics Letters B 749 (2015) 542–546
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Magnetic moment, vorticity-spin coupling and parity-odd conductivity
of chiral fermions in 4-dimensional Wigner functions
Jian-hua Gao
a
, Qun Wang
b,c,∗
a
Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai,
Shandong 264209, China
b
Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
c
Physics Department, Brookhaven National Laboratory, Upt on, NY 11973-5000, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
10 May 2015
Received
in revised form 21 July 2015
Accepted
26 August 2015
Available
online 29 August 2015
Editor:
J.-P. Blaizot
We demonstrate the emergence of the magnetic moment and spin-vorticity coupling of chiral fermions
in 4-dimensional Wigner functions. In linear response theory with space–time varying electromagnetic
fields, the parity-odd part of the electric conductivity can also be derived which reproduces results of
the one-loop and the hard-thermal or hard-dense loop. All these properties show that the 4-dimensional
Wigner functions capture comprehensive aspects of physics for chiral fermions in electromagnetic fields.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Significant progress has been made in understanding the dy-
namics
of chiral fermions in electromagnetic fields. This is partic-
ularly
interesting in high energy heavy ion collisions where very
strong magnetic fields can be created. The magnetic fields are so
strong that quarks can be polarized and their momentum direc-
tions
are parallel or anti-parallel to the magnetic field depending
on quark chiralities and charges. Quarks with the same charge tend
to move in the same direction. Any imbalance in the number of
right-handed and left-handed quarks as a consequence of topo-
logical
configurations of gauge fields may lead to such a charge
separation effect which can be tested in experiments [1]. This is
termed as the Chiral Magnetic Effect (CME) [2,3]. The Chiral Vorti-
cal
Effect (CVE) is also an accompanying effect due to rotation in a
relativistic and charged fluid [4,5]. The interplay of chiral magnetic
and chiral separation effects may lead to a phenomenon called the
Chiral Magnetic Wave [6], whose vortical counter part is the Chiral
Vortical Wave [7].
Kinetic
theory is an important tool to describe these phenom-
ena
in phase space of chiral fermions. The Abelian Berry potential
takes an important role in 3-dimensions (3D) kinetic approach in
accommodation of axial anomaly [8–10]. It has been shown that
the CME, CVE and Covariant Chiral Kinetic Equation (CCKE) can be
*
Corresponding author at: Interdisciplinary Center for Theoretical Study and De-
partment
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China.
E-mail
address: qunwang@ustc.edu.cn (Q. Wang).
derived in quantum kinetic theory from the Wigner function in
4-dimensions (4D) in external electromagnetic fields [11,12]. The
3D chiral kinetic equation [8–10] can be obtained from the CCKE
by integrating out the zero component of the 4-momentum.
In
the 3D chiral kinetic equation, it has been shown that the
fermion energy is shifted by the interaction energy of magnetic
moment with the magnetic field [13]. The magnetic moment and
spin of fermions have relativistic origin [14–16]. It is a natural
conjecture that the magnetic moment should also emerge in the
covariant quantum kinetic approach in 4D Wigner functions. In
this paper, we will demonstrate the emergence of the magnetic
moment as well as spin-vorticity coupling in the framework of
covariant quantum kinetic theory based on 4D Wigner functions.
We will also show that the parity-odd part of electric conductivity
(chiral magnetic conductivity) can also be derived from 4D Wigner
functions in linear response theory with space–time varying elec-
tromagnetic
fields. The result reproduces the chiral magnetic con-
ductivity
of one loop [17] and hard-thermal or hard-dense loop
(HTL or HDL) [18,19] under proper approximations [13,20].
We
adopt the same sign conventions for the fermion charge Q
and
the axial vector component of the Wigner function as in Refs.
[11,12,21].
2. Wigner functions for chiral fermions
In a background electromagnetic field, the quantum mechan-
ical
anologue of a classical phase-space distribution for fermions
is the gauge invariant Wigner function W
αβ
(x, p) which satisfies
the equation of motion [21,22],
γ
μ
K
μ
−m
W (x, p) = 0, where
http://dx.doi.org/10.1016/j.physletb.2015.08.058
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.