Relativistic heavy-ion collisions are an important experimental way of studying the new state of matter as well as the phase diagram of the quantum chromodynamics (QCD) under extremely high temperatures and high densities. In recent years, the beam-energy scan program has been carried out on the relativistic heavy-ion collider at Brookhaven National Laboratory in USA, and the STAR collaboration at relativistic heavy ion collision (RHIC) has measured the difference in the elliptic flow
$v_2$
between
$\pi^-$
and
$\pi^+$
on an event-by-event basis, and found a linear dependence on the charge asymmetry
A
chof the collision system, which is considered as a possible signal of the chiral magnetic wave. Based on the extended multi-phase transport model (AMPT), this paper uses a 3 flavor Nambu-Jona-Lasinio (NJL) model to study the quark isospin Mean-field potential, which provides a new idea for explaining the experimental phenomenon of the linear relationship between the pion elliptic flow splitting
$\Delta v_2=v_2(\pi^-)- $
$ v_2(\pi^+)$
and charge asymmetry
A
ch. Our results show that isovector interaction can cause a splitting of the isospin asymmetric quark matter mean-field potential, manifesting as
$d(\bar{u})$
quarks experiencing a mean-field potential greater than
$u(\bar{d})$
quarks. Therefore,
$d(\bar{u})$
quarks experience more repulsion than
$u(\bar{d})$
quarks in collision processes, resulting in a small increase in the elliptic flow of the partial subflow
$v_2(d)$
and
$v_2(\bar{u})$
, while
$v_2(u)$
and
$v_2(\bar{d})$
decrease slightly. In the hadronization process, the
$\pi$
elliptic flow splitting occurs due to the split of
$d$
and
$u$
elliptic flows combined with the split of
$\bar{u}$
and
$\bar{d}$
elliptic flows. We also found a linear correlation between charge asymmetry and isospin asymmetry at mid-rapidity region from our transport model, and thus explained the experimental phenomenon of the linear relationship between the pion elliptic flow splitting
$\Delta v_2$
and charge asymmetry
A
chby using the isospin mean-field potential of quark matter. Further, isospin properties of quark matter also provide a theoretical basis for isobar collisions and the equation of state of compact star matter.
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