#  $K_{1}/K^{*}$ enhancement as a signature of chiral symmetry restoration in
heavy ion collisions

Hae-Som Sung<EMAIL_ADDRESS>Department of Physics and Institute of
Physics and Applied Physics, Yonsei University, Seoul 03722, Korea  Sungtae
Cho<EMAIL_ADDRESS>Division of Science Education, Kangwon National
University, Chuncheon 24341, Korea  Juhee Hong<EMAIL_ADDRESS>Department of Physics and Institute of Physics and Applied Physics, Yonsei
University, Seoul 03722, Korea  Su Houng Lee<EMAIL_ADDRESS>Department
of Physics and Institute of Physics and Applied Physics, Yonsei University,
Seoul 03722, Korea  Sanghoon Lim<EMAIL_ADDRESS>Department of Physics,
Pusan National University, Pusan, Korea  Taesoo Song<EMAIL_ADDRESS>GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, 64291
Darmstadt, Germany

###### Abstract

Based on the fact that the mass difference between the chiral partners is an
order parameter of chiral phase transition and that the chiral order parameter
reduces substantially at the chemical freeze-out point in ultra-relativistic
heavy ion collisions, we argue that the production ratio of $K_{1}$ over
$K^{*}$ in such collisions should be substantially larger than that predicted
in the statistical hadronization model. We further show that while the
enhancement effect might be contaminated by the relatively larger decrease of
$K_{1}$ meson than $K^{*}$ meson during the hadronic phase, the signal will be
visible through a systematic study on centrality as the kinetic freeze-out
temperature is higher and the hadronic life time shorter in peripheral
collisions than in central collisions.

##  I Introduction

A recent lattice calculation finds the pseudo-critical temperature for QCD
chiral crossovers at zero baryon chemical potential to be around $156.5\pm
1.5$ MeV Bazavov:2018mes , a value that is consistent with previous results
Borsanyi:2010bp . This temperature is close to the chemical freeze-out
temperature obtained from the statistical hadronization model analysis based
on the yields of hadrons produced in heavy ion collisions at both Relativistic
Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) Andronic:2012dm ;
Stachel:2013zma . Near this temperature, the chiral order parameter associated
with the transition is substantially reduced from its vacuum value
Ding:2013lfa . While the quark condensate is the standard chiral order
parameter, so is the difference between the vector and axial vector
correlation function. That is to say, the mass difference between the $\rho$
and $a_{1}$ is due to the chiral symmetry breaking as is well represented in
the Weinberg relation Weinberg:1967kj . Hence, when hadrons are produced at
the chemical freeze-out point in heavy ion collisions, the chiral partners
will have masses much closer to each other than their vacuum values. This does
not mean that the masses should vanish at the chemical freeze-out point. In
fact, a recent study shows that while chiral symmetry breaking is responsible
for the mass difference between the $\rho$ and $a_{1}$ meson, it only accounts
for a small fraction of the common mass Kim:2020zae .

Statistical hadronization model analysis indicates that the abundances of
hadrons are determined at the chemical freeze-out point Andronic:2005yp . This
suggests that the production of chiral partners at the chemical freeze-out
point will be similar. Unfortunately, $a_{1}$ has a large vacuum decay width
and a large dissociation cross section so that many of the $a_{1}$ produced at
the chemical freeze-out will not survive during the hadronic phase. Indeed,
the measured yields of resonances with large decay widths tend to be smaller
than the statistical hadronization model predictions.

On the other hand, recently it has been emphasized by one of us that the
$K_{1}$ and $K^{*}$ are also chiral partners and that both have vacuum widths
that are smaller than 100 MeV, making them ideal objects to study the effects
of partial chiral symmetry restoration in a nuclear target experiment
Song:2018plu . In this work, we will show that the enhancement of the ratio
$K^{1}/K^{*}$ in heavy ion collisions can be used as a signature of chiral
symmetry restoration in the early stages of heavy ion collisions. Effects of
chiral symmetry restoration in heavy ion collisions were studied in
$K^{+}/\pi^{+}$, $(\Lambda+\Sigma^{0})/\pi^{-}$ ratios Cassing:2015owa . Here,
we will study the $K^{1}/K^{*}$ ratio as their properties are direct order
parameters of spontaneous chiral symmetry breaking.

##  II $K_{1},K^{*}$ couplings in an effective Lagrangian

From a symmetry argument alone, one expects $K_{1}$ to be the chiral partner
of $K^{*}$. In fact, arguments based solely on chiral symmetry predict that
when chiral symmetry is restored, the vector and axial particles become
degenerate Lee:2019tvt . For these effects to be observable in heavy ion
collisions, the vacuum widths as well as the hadronic dissociation of both
vector and axial particles should be small so that the signal will not be
smeared out during the hadronic phase. Since the vacuum widths of both the
$K_{1}$ and $K^{*}$ meson are already smaller than 100 MeV, we focus here on
their hadronic absorptions during the hadronic phase as well as on the
centrality dependence of these effects in a heavy ion collision. Still, in
order to work out the suppression of the initially produced $K_{1}$ mesons
during the hadronic phase of heavy ion collisions, we need to estimate the
hadronic cross sections with other hadrons.

For that purpose, we use the Lagrangian involving spin-1 and spin-0 mesons
following the massive Yang-Mills approach Meissner:1987ge :

$\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{8}F^{2}_{\pi}{\rm{Tr}}\left(D_{\mu}UD^{\mu}U^{\dagger}\right)+\frac{1}{8}F^{2}_{\pi}{\rm{Tr}}\left[M\left(U+U^{\dagger}-2\right)\right]$
(1) $\displaystyle-\dfrac{1}{2}{\rm{Tr}}\left(F_{\mu\nu
L}F^{\mu\nu}_{L}+F_{\mu\nu R}F^{\mu\nu}_{R}\right)$
$\displaystyle+m^{2}_{0}{\rm{Tr}}\left(A_{\mu L}A^{\mu}_{L}+A_{\mu
R}A^{\mu}_{R}\right),$

where $U=\exp\left(2\textit{i}P/\sqrt{2}f_{\pi}\right)$ with the octet of
pseudoscalar mesons $P=P^{a}\lambda^{a}/\sqrt{2}$ and
$F_{\pi}=\sqrt{2}f_{\pi}$ = 132 MeV. The covariant derivative and the field
strength tensor are given by

$\displaystyle D_{\mu}U$ $\displaystyle=$
$\displaystyle\partial_{\mu}U-\textit{i}g\,A_{\mu\,L}U+\textit{i}g\,UA_{\mu\,R},$
$\displaystyle F_{\mu\,\nu}^{L,R}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}^{L,R}-\partial_{\nu}A_{\mu}^{L,R}-ig\left[A_{\mu}^{L,R},A_{\nu}^{L,R}\right],$
(2)

where $V_{\mu}=A_{\mu}^{L}+A_{\mu}^{R}$ and $A_{\mu}=A_{\mu}^{L}-A_{\mu}^{R}$
with the vector and axial-vector mesons
$V_{\mu}=V_{\mu}^{a}\lambda^{a}/\sqrt{2}$ and
$A_{\mu}=A_{\mu}^{a}\lambda^{a}/\sqrt{2}$.

From the first term of Eq. (1), we can read off the $K^{*}K\pi$ and the
$K_{1}K\rho$ interaction terms.

$\displaystyle\mathcal{L}^{(3)}$ $\displaystyle=$
$\displaystyle\,\frac{\textit{i}g}{2}{\rm{Tr}}\left(V_{\mu}P\overleftrightarrow{\partial}^{\mu}P\right)$
(3)
$\displaystyle+\frac{\textit{i}g^{2}F_{\pi}}{4}{\rm{Tr}}\left(V_{\mu}PA^{\mu}-V_{\mu}A^{\mu}P\right)$
$\displaystyle+\frac{\textit{i}g}{2}{\rm{Tr}}\left[\left(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}\right)V^{\mu}V^{\nu}\right]$
$\displaystyle+\textit{i}g{\rm{Tr}}\left[\left(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}\right)A^{\mu}A^{\nu}-\partial_{\nu}A_{\mu}A^{\mu}V^{\nu}\right].$

Specifically,

$\displaystyle\mathcal{L}_{VPP}$ $\displaystyle=$
$\displaystyle\frac{\textit{i}g}{2}\bar{K}^{*\,\mu}\left[\left(\vec{\tau}\cdot\partial_{\mu}\vec{\pi}\right)K-\left(\vec{\tau}\cdot\vec{\pi}\right)\partial_{\mu}K\right]$
(4)
$\displaystyle+\frac{\textit{i}g}{2}\bar{K}\left[\left(\vec{\tau}\cdot\vec{\rho}^{\mu}\right)\right]\partial_{\mu}K+H.C.,$
$\displaystyle\mathcal{L}_{AVP}$ $\displaystyle=$
$\displaystyle\textit{i}\frac{g^{2}F_{\pi}}{4}\bar{K_{1}}^{\mu}\left[\left(\vec{\tau}\cdot\vec{\rho_{\mu}}\right)K-\left(\vec{\tau}\cdot\vec{\pi}\right)K^{*}_{\mu}\right]+H.C.,$

with $K\equiv(K^{+},K^{0})^{T}$ , $K^{*}\equiv(K^{+},K^{*0})^{T}$ and
$K_{1}\equiv(K^{+}_{1},K^{0}_{1})^{T}$ being the strange pseudoscalar, vector
and axial-vector meson isospin doublets, respectively. $\vec{\pi}$ and
$\vec{\rho}$, with Pauli matrices $\vec{\tau}$, are the pion and $\rho$ meson
isospin triplets.

As it stands, the interaction terms for $K_{1}K^{*}\pi$ and $K_{1}K\rho$ have
a common coupling as given from Eq. (3). This leads to a larger calculated
decay width of $K_{1}K^{*}\pi$ compared with that of $K_{1}K\rho$, because the
phase space for the former two-body decay is larger than that for the latter.
On the other hand, experimentally one finds $\Gamma_{K_{1}\rightarrow
K^{*}\pi}=25$ MeV and $\Gamma_{K_{1}\rightarrow K\rho}=38$ MeV. The resolution
of this puzzles comes from mixing effects. In the vacuum, the low-lying modes
that couple to the vector current are $K^{*}(892)$ and $K^{*}(1410)$ while
those for the axial vector current are $K_{1}(1270)$ and $K_{1}(1400)$. There
is a subtlety in the nature of the two $K_{1}$ states: they are assumed to be
a mixture of the ${}^{3}P_{1}$ and ${}^{1}P_{1}$ quark-antiquark pair in the
quark model Suzuki:1993yc . We therefore introduce two mixing angles
$\theta_{A}$ and $\theta_{V}$ so that the chiral Lagrangian in Eq. (4) and
(LABEL:eq:LagAVP) are constructed for both $(K^{*}_{A},K_{1A})$ and
$(K^{*}_{B},K_{1B})$ that are related to the physical mesons as follows:

$\displaystyle K_{1A}$ $\displaystyle=$
$\displaystyle\cos\theta_{A}K_{1}(1270)+\sin\theta_{A}K_{1}(1400),$
$\displaystyle K_{1B}$ $\displaystyle=$
$\displaystyle-\sin\theta_{A}K_{1}(1270)+\cos\theta_{A}K_{1}(1400),$
$\displaystyle K^{*}_{A}$ $\displaystyle=$
$\displaystyle\cos\theta_{V}K^{*}(892)+\sin\theta_{V}K^{*}(1410),$
$\displaystyle K^{*}_{B}$ $\displaystyle=$
$\displaystyle-\sin\theta_{V}K^{*}(892)+\cos\theta_{V}K^{*}(1410).$ (6)

Now, we re-express the sum of the two chiral Lagrangians in terms of $K_{1}$
and $K^{*}$ mesons and the two independent couplings $g_{A},g_{B}$ so that the
SU(2) chiral symmetry is still preserved.

$\displaystyle\mathcal{L}^{{}^{\prime}}_{VPP}$ $\displaystyle=$
$\displaystyle\textit{i}g_{K^{*}K\pi}\,\bar{K}^{*\,\mu}\left[\left(\sqrt{2}\vec{\tau}\cdot\partial_{\mu}\vec{\pi}\right)K-\left(\sqrt{2}\vec{\tau}\cdot\vec{\pi}\right)\partial_{\mu}K\right]$
(7) $\displaystyle+H.C.,$ $\displaystyle\mathcal{L}^{{}^{\prime}}_{AVP}$
$\displaystyle=$
$\displaystyle\textit{i}g_{K_{1}K\rho}\bar{K_{1}}^{\mu}\left(\sqrt{2}\vec{\tau}\cdot\vec{\rho}_{\mu}\right)K$
(8)
$\displaystyle-\textit{i}g_{K_{1}K^{*}_{\mu}\pi}\bar{K_{1}}^{\mu}\left(\sqrt{2}\vec{\tau}\cdot\vec{\pi}\right)K^{*}_{\mu}+H.C.,$

where

$\displaystyle g_{K^{*}K\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}}\left(g_{A}\cos\theta_{V}-g_{B}\sin\theta_{V}\right),$
$\displaystyle g_{K^{*}(1410)K\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}}\left(g_{A}\sin\theta_{V}+g_{B}\cos\theta_{V}\right),$
$\displaystyle g_{K_{1}K\rho}$ $\displaystyle=$
$\displaystyle\frac{F_{\pi}}{8\sqrt{2}}\left(g_{A}^{2}\cos\theta_{A}-g_{B}^{2}\sin\theta_{A}\right),$
$\displaystyle g_{K_{1}(1400)K\rho}$ $\displaystyle=$
$\displaystyle\frac{F_{\pi}}{8\sqrt{2}}\left(g_{A}^{2}\sin\theta_{A}+g_{B}^{2}\cos\theta_{A}\right),$
$\displaystyle g_{K_{1}K^{*}\pi}$ $\displaystyle=$
$\displaystyle\frac{F_{\pi}}{8\sqrt{2}}\left(g_{A}^{2}\cos\theta_{A}\cos\theta_{V}+g_{B}^{2}\sin\theta_{A}\sin\theta_{V}\right),$
$\displaystyle g_{K_{1}(1400)K^{*}\pi}$ $\displaystyle=$
$\displaystyle\frac{F_{\pi}}{8\sqrt{2}}\left(g_{A}^{2}\sin\theta_{A}\cos\theta_{V}-g_{B}^{2}\cos\theta_{A}\sin\theta_{V}\right).$

Using the above interaction Lagrangians, we determine the mixing angle
$\theta_{A}$($\theta_{B}$) and coupling constant $g_{A}$($g_{B}$) to best fit
the experimental data for the six decay processes given in Table 1. The table
also summarizes the best fit couplings and the experimental data as well as
the calculated decay widths of the six processes.

ABC | $K^{*}(892)K\pi$ | $K^{*}(1410)K\pi$ | $K_{1}(1270)K\rho$ | $K_{1}(1400)K\rho$ | $K_{1}(1270)K^{*}(892)\pi$ | $K_{1}(1400)K^{*}(892)\pi$
---|---|---|---|---|---|---
$g_{ABC}$ | 3.17 | 0.9 | 2.9 (GeV) |  $-1.02$ (GeV) | 0.649 (GeV) |  $-1.95$ (GeV)
$\Gamma_{A\rightarrow BC}$(experiment) | 50.0 (MeV) | 15.6 (MeV) | 37.8 (MeV) | 5.2 (MeV) | 14.4 (MeV) | 163 (MeV)
$\Gamma_{A\rightarrow BC}$(calculated) | 48.4 (MeV) | 14.8 (MeV) | 39.7 (MeV) | 14.7 (MeV) | 9.42 (MeV) | 98.6 (MeV)
Table 1: The coupling constants are calculated with $g_{A}$=15.5, $g_{B}$=
-10.4, $\cos\theta_{V}$=0.6469, $\sin\theta_{V}$=0.7626,
$\cos\theta_{A}$=0.724 and $\sin\theta_{A}$= -0.6898. With these parameters we
also find four point couplings $g_{K_{1}(1270)K\pi\pi}=$7.67(GeV-1) and
$g_{K_{1}(1400)K\pi\pi}=-34.5$(GeV-1).

##  III The cross section of the $K_{1}$ meson

We assume that $K_{1}$ mesons are in thermal equilibrium when they are
produced at the chemical freeze-out. Then $K_{1}$ mesons interact with other
particles until the kinetic freeze-out point. We consider the hadronic effect
on the $K_{1}$ meson due to interactions with light mesons such as pions and
rho mesons; $K_{1}+\pi\rightarrow K+\pi$, $K_{1}+\pi\rightarrow K^{*}+\rho$,
$K_{1}+\rho\rightarrow K^{*}+\pi$, and $K_{1}+\rho\rightarrow K+\rho$. In
addition to the interaction terms given in Eq. (7) and (8), the Lagrangian
describing the interaction between the $K_{1}$ meson and pions and rho mesons
is given as follows:

$\displaystyle\mathcal{L}^{{}^{\prime}}_{APPP}$ $\displaystyle=$
$\displaystyle
2g_{K_{1}K\pi\pi}\bar{K_{1}}^{\mu}\left[\left(\vec{\tau}\cdot\partial_{\mu}\vec{\pi}\right)\left(\vec{\tau}\cdot\vec{\pi}\right)K+\left(\vec{\tau}\cdot\vec{\pi}\right)^{2}\partial_{\mu}K\right],$
$\displaystyle\mathcal{L}^{{}^{\prime}}_{VPP}$ $\displaystyle=$
$\displaystyle\textit{i}g_{\rho\pi\pi}[\rho^{0\,\mu}(\partial_{\mu}\pi^{+}\,\pi^{-}-\pi^{+}\partial_{\mu}\pi^{-})$
$\displaystyle+$
$\displaystyle\rho^{+\,\mu}(\partial_{\mu}\pi^{-}\,\pi^{0}-\pi^{-}\partial_{\mu}\pi^{0})$
$\displaystyle+$
$\displaystyle\rho^{-\,\mu}(\partial_{\mu}\pi^{0}\,\pi^{+}-\pi^{0}\partial_{\mu}\pi^{+})]+H.C.$
$\displaystyle+$
$\displaystyle\textit{i}g_{KK\rho}\left[\bar{K}\left(\sqrt{2}\vec{\tau}\cdot\vec{\rho}^{\mu}\right)\partial_{\mu}K-\partial_{\mu}\bar{K}\left(\sqrt{2}\vec{\tau}\cdot\vec{\rho}^{\mu}\right)K\right],$
$\displaystyle\mathcal{L}^{{}^{\prime}}_{AVP}$ $\displaystyle=$
$\displaystyle\textit{i}g_{a_{1}\rho\pi}\\{\rho^{+\,\mu}(\pi^{-}a_{1\,\mu}^{0}-\pi^{0}a_{1\,\mu}^{-})$
$\displaystyle+$
$\displaystyle\rho^{-\,\mu}(\pi^{0}a_{1\,\mu}^{+}-\pi^{+}a_{1\,\mu}^{0})+\rho^{0\,\mu}(\pi^{+}a_{1\,\mu}^{-}-\pi^{-}a_{1\,\mu}^{+})\\}$
$\displaystyle\mathcal{L}^{{}^{\prime}}_{VVV}$ $\displaystyle=$
$\displaystyle\sqrt{2}\textit{i}g_{K^{*}K^{*}\rho}[\bar{K^{*}}^{\nu}\vec{\tau}K^{*\,\mu}(\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\nu}\vec{\rho}_{\mu})$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\bar{K^{*}}_{\nu}-\partial_{\nu}\bar{K^{*}}_{\mu})\vec{\tau}\cdot\vec{\rho}^{\mu}K^{*\,\nu}$
$\displaystyle+$
$\displaystyle\bar{K^{*}}^{\mu}\vec{\tau}\cdot\vec{\rho}^{\nu}(\partial_{\mu}K^{*}_{\nu}-\partial_{\nu}K^{*}_{\mu})]$
$\displaystyle+$
$\displaystyle\frac{\textit{i}g}{2}[(\partial_{\mu}(\vec{\tau}\cdot\vec{\rho}_{\nu})-\partial_{\nu}(\vec{\tau}\cdot\vec{\rho}_{\mu}))(\vec{\tau}\cdot\vec{\rho}^{\mu})(\vec{\tau}\cdot\vec{\rho}^{\nu})],$
$\displaystyle\mathcal{L}^{{}^{\prime}}_{AAV}$ $\displaystyle=$
$\displaystyle\textit{i}\sqrt{2}g_{K_{1}a_{1}K^{*}}[\partial_{\nu}\bar{K}_{1\,\mu}\\{(\vec{a}^{\nu}_{1}\cdot\vec{\tau})K^{*\,\mu}$
(10) $\displaystyle-$
$\displaystyle(\vec{a}^{\mu}_{1}\cdot\vec{\tau})K^{*\,\nu}\\}-\bar{K}_{1}^{\mu}\partial_{\mu}(\vec{a}_{1\,\nu}\cdot\vec{\tau})K^{*\,\nu}]$
$\displaystyle+$
$\displaystyle\textit{i}\sqrt{2}g_{K_{1}K_{1}\rho}[\\{\bar{K}_{1}^{\mu}(\vec{\rho}^{\nu}\cdot\vec{\tau})-\bar{K}_{1}^{\nu}(\vec{\rho}^{\mu}\cdot\vec{\tau})\\}\partial_{\nu}K_{1\,\mu}$
$\displaystyle-$
$\displaystyle\textit{i}g\partial_{\mu}\bar{K}_{1}^{\nu}(\vec{\rho}_{\nu}\cdot\vec{\tau})K_{1}^{\mu}].$

The coupling constant $g=8.49$ is determined by fitting the experimental decay
width of $\rho\rightarrow\pi\pi$, from which we find
$g_{\rho\pi\pi}=g_{\rho\rho\rho}=g/\sqrt{2}=6.006$ and
$g_{KK\rho}=g/2\sqrt{2}=3.003$. In the strange vector and axial-vector mesons,
we apply the mixing angle and coupling constants $g_{A}$ and $g_{B}$, which
were determined to give the coupling constants as given in Table 1 and
additionally $g_{K^{*}K^{*}\rho}=0.078$, $g_{K_{1}K_{1}\rho}=1.12$,
$g_{K_{1}a_{1}K^{*}}=4.5$ and $g_{a_{1}\rho\pi}=3.36$ (GeV).

(1-a)

(1-b)

(1-c)

(1-d)

(2-a)

(2-b)

(2-c)

(2-d)

(2-e)

(3-a)

(3-b)

(3-c)

(3-d)

(4-a)

(4-b)

(4-c)

(4-d)

(4-e)

Figure 1: Feynman diagrams for the $K_{1}$ meson absorption by $\pi,\,\rho$
mesons. (1) $K_{1}\pi\rightarrow K\pi$, (2) $K_{1}\pi\rightarrow K^{*}\rho$,
(3) $K_{1}\rho\rightarrow K^{*}\pi$ and (4) $K_{1}\rho\rightarrow K\rho$.

The amplitude for each scattering of different charge states can be written as
two isospin channels because of isospin conservation. For example, the
scattering of $\pi K_{1}\rightarrow K\pi$ consists of the following charge
states : $K^{-}\pi^{-}\rightarrow K_{1}^{-}\pi^{-}$, $K^{-}\pi^{0}\rightarrow
K_{1}^{-}\pi^{0}$, $\bar{K}^{0}\pi^{-}\rightarrow K_{1}^{-}\pi^{0}$,
$K^{-}\pi^{+}\rightarrow K_{1}^{-}\pi^{+}$, and $\bar{K}^{0}\pi^{0}\rightarrow
K_{1}^{-}\pi^{+}$. These five processes are described by two independent
scattering amplitudes $a_{3/2}$ and $a_{1/2}$ for isospin $3/2$ and $1/2$
channels, respectively, so that

$\displaystyle\begin{cases}\langle
K^{-}\pi^{-}|S|K_{1}^{-}\pi^{-}\rangle\,=\,a_{3/2}\cr\langle
K^{-}\pi^{0}|S|K_{1}^{-}\pi^{0}\rangle\,=\,\frac{2}{3}a_{3/2}+\frac{1}{3}a_{1/2}\cr\langle\bar{K}^{0}\pi^{-}|S|K_{1}^{-}\pi^{0}\rangle\,=\,-\frac{\sqrt{2}}{3}\left(a_{3/2}-a_{1/2}\right)\cr\langle
K^{-}\pi^{+}|S|K_{1}^{-}\pi^{+}\rangle\,=\,\frac{1}{3}a_{3/2}+\frac{2}{3}a_{1/2}\cr\langle\bar{K}^{0}\pi^{0}|S|K_{1}^{-}\pi^{+}\rangle\,=\,\frac{\sqrt{2}}{3}\left(a_{3/2}-a_{1/2}\right)\end{cases}.$
(11)

It should be noted that squaring and adding all five amplitudes, one finds
that the sum is proportional to $2a^{2}_{3/2}+a^{2}_{1/2}$ reflecting the
degeneracy of the isospin.

Using the interaction terms in the Lagrangian discussed above, we evaluated
the amplitudes for all diagrams shown in Fig. 1. These represent the
absorption amplitudes of the $K_{1}$ meson by $\pi$ and $\rho$ mesons. First,
the amplitudes for $K_{1}$ absorption by $\pi$ mesons are represented by the
processes $\pi K_{1}\rightarrow\pi K$ and $\pi K_{1}\rightarrow\rho K^{*}$.
The amplitude for definite isospin channels of the
$\pi(p_{1})K_{1}(p_{2})\rightarrow\pi(p_{3})K(p_{4})$ are given as
$a_{1/2}=g_{K_{1}K\pi\pi}\epsilon_{2}^{\mu}(p_{1}+3p_{3})_{\mu}+3\mathcal{M}^{s}_{1}+2\mathcal{M}^{t}_{1}+\mathcal{M}^{u}_{1}$
and
$a_{3/2}=-2g_{K_{1}K\pi\pi}\epsilon_{2}^{\mu}p_{1\,\mu}-\mathcal{M}^{t}_{1}-2\mathcal{M}^{u}_{1}$
for isospin 1/2 and 3/2, respectively. Here,
$\mathcal{M}^{s}_{1}=\mathcal{M}_{(1-b)}$,
$\mathcal{M}^{t}_{1}=\mathcal{M}_{(1-c)}$, and
$\mathcal{M}^{u}_{1}=\mathcal{M}_{(1-d)}$, where the subscript in the right
hand side represents the diagrams in Fig. 1. The matrix elements are given as

$\displaystyle\mathcal{M}^{s}_{1}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K^{*}\pi}g_{K^{*}K\pi}\,\epsilon_{2}^{\mu}\left(p_{3}-p_{4}\right)^{\nu}$
$\displaystyle\times\frac{-g_{\mu\nu}+(p_{1}+p_{2})_{\mu}(p_{1}+p_{2})_{\nu}/s}{s-m_{K^{*}}^{2}},$
$\displaystyle\mathcal{M}^{t}_{1}$ $\displaystyle=$ $\displaystyle
g_{\rho\pi\pi}g_{K_{1}K\rho}\,\epsilon_{2}^{\mu}\frac{-\left(p_{1}+p_{3}\right)_{\mu}}{t-m_{\rho}^{2}},$
$\displaystyle\mathcal{M}^{u}_{1}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K^{*}\pi}g_{K^{*}K\pi}\,\epsilon_{2}^{\mu}\left(p_{1}+p_{4}\right)^{\nu}$
$\displaystyle\times\frac{-g_{\mu\nu}+(p_{1}-p_{4})_{\mu}(p_{1}-p_{4})_{\nu}/u}{u-m_{K^{*}}^{2}}.$

The two independent amplitudes for the
$\pi(p_{1})K_{1}(p_{2})\rightarrow\rho(p_{3})K^{*}(p_{4})$ are respectively
given as
$a_{1/2}=3\mathcal{M}^{s}_{2}+2\mathcal{M}^{t}_{2}+\mathcal{M}^{u}_{2}$ and
$a_{3/2}=-\mathcal{M}^{t}_{2}-2\mathcal{M}^{u}_{2}$. Here,
$\mathcal{M}^{t}_{2}=\mathcal{M}_{(2-b)}+\mathcal{M}_{(2-d)}$ and
$\mathcal{M}^{u}_{2}=\mathcal{M}_{(2-c)}+\mathcal{M}_{(2-e)}$, with

$\displaystyle\mathcal{M}^{s}_{2}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K^{*}\pi}g_{K^{*}K^{*}\rho}\epsilon_{2\,\alpha}\epsilon^{*\gamma}_{3}\epsilon^{*\delta}_{4}\frac{-g^{\alpha\beta}+(p_{1}+p_{2})^{\alpha}(p_{1}+p_{2})^{\beta}/s}{s-m_{K^{*}}^{2}}[g_{\gamma\beta}(2p_{3}+p_{4})_{\delta}-g_{\delta\beta}(p_{3}+2p_{4})_{\gamma}-g_{\gamma\delta}(p_{3}-p_{4})_{\beta}],$
$\displaystyle\mathcal{M}^{t}_{2}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K^{*}\pi}g_{\rho\pi\pi}\epsilon^{\mu}_{2}\epsilon^{*}_{4\,\mu}\epsilon^{*\gamma}_{3}\frac{1}{t-m_{\pi}^{2}}(2p_{1}-p_{3})_{\gamma}$
$\displaystyle+g_{a_{1}\rho\pi}g_{K_{1}a_{1}K^{*}}\epsilon^{\mu}_{2}\epsilon^{*}_{3\,\alpha}\epsilon^{*\,\delta}_{4}\frac{-g^{\alpha\beta}+(p_{1}-p_{3})^{\alpha}(p_{1}-p_{3})^{\beta}/t}{t-m_{a_{1}}^{2}}\left[g_{\mu\beta}p_{2\,\delta}-g_{\mu\delta}p_{2\,\beta}+g_{\beta\delta}(p_{1}-p_{3})_{\mu}\right],$
$\displaystyle\mathcal{M}^{u}_{2}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K\rho}g_{K^{*}K\pi}\epsilon^{\mu}_{2}\epsilon^{*}_{3\,\mu}\epsilon^{*\delta}_{4}\frac{1}{u-m_{\pi}^{2}}(2p_{1}-p_{4})_{\delta}$
$\displaystyle+g_{K_{1}K^{*}\pi}g_{K_{1}K_{1}\rho}\epsilon^{\mu}_{2}\epsilon^{*\,\gamma}_{3}\epsilon^{*}_{4\,\alpha}\frac{-g^{\alpha\beta}+(p_{1}-p_{4})^{\alpha}(p_{1}-p_{4})^{\beta}/u}{u-m_{K_{1}}^{2}}\left[g_{\mu\beta}(p_{1}-p_{4})_{\gamma}+g_{\mu\gamma}p_{2\,\beta}-g_{\gamma\beta}(p_{1}-p_{4})_{\mu}\right].$

Second, the amplitudes for absorption by $\rho$ mesons are $\rho
K_{1}\rightarrow\pi K^{*}$ and $\rho K_{1}\rightarrow K\rho$. The isospin
channels of $\rho(p_{1})K_{1}(p_{2})\rightarrow\pi(p_{3})K^{*}(p_{4})$ are
$a_{1/2}=3\mathcal{M}^{s}_{3}-2\mathcal{M}^{t}_{3}$ and
$a_{3/2}=\mathcal{M}^{t}_{3}$. Here,
$\mathcal{M}^{s}_{3}=\mathcal{M}_{(3-a)}+\mathcal{M}_{(3-b)}$, and
$\mathcal{M}^{t}_{3}=\mathcal{M}_{(3-c)}+\mathcal{M}_{(3-f)}$ are given as
follows:

$\displaystyle\mathcal{M}^{s}_{3}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K\rho}g_{K^{*}K\pi}\epsilon_{1}^{\mu}\epsilon_{2\,\mu}\epsilon_{4}^{*\,\lambda}\frac{1}{s-m_{K}^{2}}(2p_{3}+p_{4})_{\lambda}$
$\displaystyle+$ $\displaystyle
g_{K_{1}K_{1}\rho}g_{K_{1}K\pi}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4}^{*\,\lambda}\frac{-g^{\mu^{\prime}}_{\lambda}+(p_{1}+p_{2})^{\mu^{\prime}}(p_{1}+p_{2})_{\lambda}/m_{K_{1}}^{2}}{s-m_{K_{1}}^{2}}[g_{\mu^{\prime}\nu}(p_{1}+p_{2})_{\mu}-g_{\mu\mu^{\prime}}(p_{1}+p_{2})_{\nu}+g_{\mu\nu}p_{2\,\mu^{\prime}}],$
$\displaystyle\mathcal{M}^{t}_{3}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K^{*}\pi}g_{\rho\pi\pi}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4\,\nu}^{*}\frac{1}{t-m_{\pi}^{2}}(p_{1}-2p_{3})_{\mu}$
(14) $\displaystyle-$ $\displaystyle
g_{K_{1}a_{1}K^{*}}g_{a_{1}\rho\pi}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4}^{*\,\lambda}\frac{-g^{\nu^{\prime}}_{\mu}+(p_{1}-p_{3})_{\mu}(p_{1}-p_{3})^{\nu^{\prime}}/m_{a_{1}}^{2}}{t-m_{a_{1}}^{2}}[g_{\lambda\nu}p_{2\,\nu^{\prime}}-g_{\nu\nu^{\prime}}p_{2\,\lambda}-g_{\lambda\nu^{\prime}}(p_{1}-p_{3})_{\nu}].$

The isospin channels of $\rho(p_{1})K_{1}(p_{2})\rightarrow
K(p_{3})\rho(p_{4})$ are
$a_{1/2}=-3\mathcal{M}^{s}_{3}-\mathcal{M}^{t}_{4}+2\mathcal{M}^{u}_{4}$ and
$a_{3/2}=2\mathcal{M}^{t}_{4}-\mathcal{M}^{u}_{4}$. Here,
$\mathcal{M}^{s}_{4}=\mathcal{M}_{(4-a)}+\mathcal{M}_{(4-b)}$,
$\mathcal{M}^{t}_{4}=\mathcal{M}_{(4-c)}+\mathcal{M}_{(4-d)}$, and
$\mathcal{M}^{u}_{4}=\mathcal{M}_{(4-e)}$ are given as follows:

$\displaystyle\mathcal{M}^{s}_{4}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K\rho}g_{\rho
KK}\epsilon_{1}^{\mu}\epsilon_{2\,\mu}\epsilon_{4}^{*\,\lambda}\frac{1}{s-m_{K}^{2}}(2p_{3}+p_{4})_{\lambda}$
$\displaystyle+g_{K_{1}K_{1}\rho}g_{K_{1}K\rho}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4}^{*\,\lambda}\frac{-g^{\mu^{\prime}}_{\lambda}+(p_{1}+p_{2})^{\mu^{\prime}}(p_{1}+p_{2})_{\lambda}/m_{K_{1}}^{2}}{s-m_{K_{1}}^{2}}\left[g_{\nu\mu^{\prime}}(p_{1}+p_{2})_{\mu}-g_{\mu\mu^{\prime}}(p_{1}+p_{2})_{\nu}+g_{\mu\nu}p_{2\,\mu^{\prime}}\right],$
$\displaystyle\mathcal{M}^{t}_{4}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K\rho}g_{\rho
KK}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4\,\nu}^{*}\frac{1}{t-m_{K}^{2}}(p_{1}-2p_{3})_{\mu}$
$\displaystyle+g_{K_{1}K_{1}\rho}g_{K_{1}K\rho}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4}^{*\,\lambda}\frac{-g^{\nu^{\prime}}_{\mu}+(p_{1}-p_{3})_{\mu}(p_{1}-p_{3})^{\nu^{\prime}}/m_{K_{1}}^{2}}{t-m_{K_{1}}^{2}}\left[g_{\nu\nu^{\prime}}(p_{1}-p_{3})_{\lambda}-g_{\nu^{\prime}\lambda}(p_{1}-p_{3})_{\nu}+g_{\lambda\nu}p_{2\,\nu^{\prime}}\right],$
$\displaystyle\mathcal{M}^{u}_{4}$ $\displaystyle=$ $\displaystyle
g_{K_{1}K\rho}g_{\rho\rho\rho}\epsilon_{1}^{\mu}\epsilon_{2}^{\nu}\epsilon_{4}^{*\,\lambda}\frac{-g^{\mu^{\prime}}_{\nu}+(p_{1}-p_{4})^{\mu^{\prime}}(p_{1}-p_{4})_{\nu}/m_{\rho}^{2}}{u-m_{\rho}^{2}}\left[g_{\mu\mu^{\prime}}\left(2p_{1}-p_{4}\right)_{\lambda}-g_{\mu^{\prime}\lambda}(p_{1}-2p_{4})_{\mu}-g_{\mu\lambda}(p_{1}+p_{4})_{\mu^{\prime}}\right].$

We keep the convention where particles 1 and 2 stand for initial-state mesons,
and particles 3 and 4 stand for final-state mesons shown on the left and right
sides of the diagrams, respectively. The Mandelstam variables
$s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{3})^{2}$, and $u=(p_{1}-p_{4})^{2}$ have
also been used.

To take into account the finite size of the hadrons when calculating
amplitudes, we introduce form factors shown below at each interaction vertex
for the $u,t-$channel and the $s-$channel, respectively,

$\displaystyle
F_{u,t}(\vec{q})=\frac{\Lambda^{2}-m_{ex}^{2}}{\Lambda^{2}+\vec{q}^{2}},\qquad
F_{s}(\vec{q})=\frac{\Lambda^{2}+m_{ex}^{2}}{\Lambda^{2}+\omega^{2}},$ (16)

with $m_{ex}$ being the mass of the exchanging particle at each diagram. In
Eq. (16) $\vec{q}^{2}$ is the squared three-momentum transfer at
$t,u-$channels, and $\omega^{2}$ is the total energy of the incoming particles
at the $s-$channel in the center of mass frame. On the other hand, we apply
the following form factor for the four point contact interaction,

$\displaystyle
F_{c}(\vec{k})=\left(\frac{\Lambda^{2}}{\Lambda^{2}+\vec{k}^{2}}\right)^{2},$
(17)

where $\vec{k}$ is the average of the squared three-momenta from the form
factors for the given channels at each process. We set the cut off parameter
$\Lambda=$ 1.8 GeV Brown:1991ig .

The cross section after spin averaging is given by

$\displaystyle\sigma=\frac{1}{64\pi s}\int
dt\frac{\overline{\mathcal{M}}^{2}}{|p_{1cm}|^{2}}F^{4},$ (18)

with $\overline{\mathcal{M}}^{2}$ being the squared amplitude of all processes
evaluated by averaging and summing over the degeneracies of the initial and
final particles, respectively. $p_{1cm}$ in Eq. (18) represents for the three
momenta of the initial particles in the center of mass frame. Fig. 2 shows the
cross section for the absorption of the $K_{1}$ meson by $\pi$ and $\rho$
mesons.

Figure 2: The cross sections for the dissociation of the $K_{1}$ meson by
$\pi$ and $\rho$ mesons via processes $K_{1}\pi\rightarrow K\pi$,
$K_{1}\pi\rightarrow K^{*}\rho$, $K_{1}\rho\rightarrow K\rho$ and
$K_{1}\rho\rightarrow K^{*}\pi$.

For the hadronic production cross section of the $K_{1}$ meson during the
hadronic stage in heavy-ion collisions, we will use detailed balance relation
in the rate equation.

##  IV Time evolution of the $K_{1}$ meson abundance

We now investigate the time evolution of the $K_{1}$ meson abundance using the
cross sections evaluated in the previous section. We construct an evolution
equation, similar to that in Ref. Cho:2015qca , composed of densities and
abundances for mesons participating in all the processes shown in Fig. 2: the
dissociation processes of $K_{1}$ are due to $\pi$ and $\rho$ mesons.

$\displaystyle\frac{dN_{K_{1}}(\tau)}{d\tau}$ $\displaystyle=$
$\displaystyle\left<\sigma_{K_{1}\pi\rightarrow
K\pi}\,v_{K_{1}\pi}\right>n_{\pi}\bigg{[}-N_{K_{1}}+\frac{N_{K_{1}}^{T}}{N_{K}^{T}}N_{K}\bigg{]}$
(19) $\displaystyle+$ $\displaystyle\left<\sigma_{K_{1}\pi\rightarrow
K^{*}\rho}\,v_{K_{1}\pi}\right>\bigg{[}-n_{\pi}N_{K_{1}}+\frac{n_{\pi}^{T}N_{K_{1}}^{T}}{n_{\rho}^{T}N_{K^{*}}^{T}}n_{\rho}N_{K^{*}}\bigg{]}$
$\displaystyle+$ $\displaystyle\left<\sigma_{K_{1}\rho\rightarrow
K^{*}\pi}\,v_{K_{1}\rho}\right>\bigg{[}-n_{\rho}N_{K_{1}}+\frac{n_{\rho}^{T}N_{K_{1}}^{T}}{n_{\pi}^{T}N_{K^{*}}^{T}}n_{\pi}N_{K^{*}}\bigg{]}$
$\displaystyle+$ $\displaystyle\left<\sigma_{K_{1}\rho\rightarrow
K\rho}\,v_{K_{1}\rho}\right>\bigg{[}-n_{\rho}N_{K_{1}}+\frac{n_{\rho}^{T}N_{K_{1}}^{T}}{n_{\rho}^{T}N_{K}^{T}}n_{\rho}N_{K}\bigg{]}$
$\displaystyle+$ $\displaystyle\left<\Gamma_{K_{1}\rightarrow
K\rho}\right>\bigg{[}-N_{K_{1}}+\frac{N_{K_{1}}^{T}}{n_{\rho}^{T}N_{K}^{T}}n_{\rho}N_{K}\bigg{]}$
$\displaystyle+$ $\displaystyle\left<\Gamma_{K_{1}\rightarrow
K^{*}\pi}\right>\bigg{[}-N_{K_{1}}+\frac{N_{K_{1}}^{T}}{n_{\pi}^{T}N_{K^{*}}^{T}}n_{\pi}N_{K^{*}}\bigg{]},$

with $n_{i}(\tau)$ being the density of the light meson $i$ in the hadronic
phase at proper time $\tau$. $N_{j}(\tau)=n_{j}(\tau)V(\tau)$, where $V(\tau)$
is the volume of the hadronic matter given in Eq. (V). The ones with the
superscript $T$ are the thermal abundance at temperature $T$, which depends on
the proper time $\tau$ as given in Eq. (V).

$\displaystyle n_{i}^{T}(\tau)$ $\displaystyle=$
$\displaystyle\frac{g_{i}}{2\pi^{2}}\int^{\infty}_{0}\frac{p^{2}dp}{\exp\left(\sqrt{p^{2}+m_{i}^{2}}/T(\tau)\right)-1}$
(20) $\displaystyle\approx$
$\displaystyle\frac{g_{i}}{2\pi^{2}}m_{i}^{2}T(\tau)K_{2}\left[\frac{m_{i}}{T(\tau)}\right],$

where $K_{2}$ is the modified Bessel function of the second kind. We take
$n_{i}(\tau)=n_{i}^{T}(\tau)$ for pions and $\rho$ mesons.

We use the following two constraint equations involving the $\tau$ dependence
for the numbers of $K$ and $K^{*}$.

$\displaystyle N_{K_{1}}+N_{N_{K^{*}}}+N_{K}$ $\displaystyle=$ $\displaystyle
N_{0},$ (21) $\displaystyle\frac{N_{K^{*}}+N_{K_{1}}\times{\rm BR}}{N_{0}}$
$\displaystyle=$ $\displaystyle aT+b,$ (22)

where $N_{0}$ is a constant determined at the chemical freeze-out point. Eq.
(21) follows from neglecting strangeness annihilation during the hadronic
phase. BR=0.16 is the branching ratio of $K_{1}$ decaying into $K^{*}$ so that
the ratio in Eq. (22) is the ratio between the final $K^{*}$ and $K$ numbers
when these mesons freeze-out at temperature $T$. This equation is motivated by
the experimental observation that show the approximately linear decrease in
the observed ratio between $K^{*}$ and $K$ numbers as a function of the cube
root of multiplicity in heavy ion collision Acharya:2019qge , which can be
understood in a hadronic model calculation Cho:2015qca  that finds the ratio
to decreases linearly with the temperature of the hadronic phase. The
coefficients $a$ and b in Eq. (22) were determined to reproduce the
statistical model prediction for the ratio at $T=T_{c}$ and its linear
reduction to 0.158 at $T=90$ MeV, which are the observed values for the ratio
and the extracted freeze-out temperature Acharya:2019yoi  for the lowest and
highest centrality Pb-Pb collision Acharya:2019qge , respectively. $a=$ 1.816
GeV-1 and $b=-$0.005444 are obtained for the first scenario, where we assume
that the number of $K_{1}$ is equal to that of $K^{*}$ at the chemical freeze-
out point, whereas $a=$ 2.272 GeV-1 and $b=-$0.04643 in the second scenario,
where we assume that both yields of $K_{1}$ and $K^{*}$ follow statistical
model values at the chemical freeze-out point. Solving Eq. (19) with the two
constraints given in Eq. (21) and Eq. (22), we can determine the $\tau$
dependence and the final freeze-out numbers of $N_{K_{1}},N_{K^{*}},N_{K}$.

When substituting in the rate equation, we take the degeneracy factor
$g_{i}=(2S+1)$, where $S$ is the spin, as the isospin effect is taken into
account in the cross section which includes the contribution from all charge
states. The production contribution has been taken into account by a detailed
balance condition, which requires each square bracket in Eq. (19) to be zero
in thermal equilibrium.

In Eq. (19), we have considered the thermally averaged cross section,
$\left<\sigma_{ab\rightarrow cd}v_{ab}\right>$ given below for initial two
particles in a two-body process $ab\rightarrow cd$,

$\displaystyle\left<\sigma_{ab\rightarrow cd}v_{ab}\right>$ $\displaystyle=$
$\displaystyle\frac{\int
d^{3}\vec{p}_{a}d^{3}\vec{p}_{b}f_{a}(\vec{p}_{a})f_{b}(\vec{p}_{b})\sigma_{ab\rightarrow
cd}v_{ab}}{\int
d^{3}\vec{p}_{a}d^{3}\vec{p}_{b}f_{a}(\vec{p}_{a})f_{b}(\vec{p}_{b})}$ (23)
$\displaystyle=$
$\displaystyle\left[64\pi\,Tm_{a}^{2}m_{b}^{2}K_{2}(m_{a}/T)K_{2}(m_{b}/T)\right]^{-1}$
$\displaystyle\times\int^{\infty}_{\sqrt{s_{0}}}d\sqrt{s}\,K_{1}(\sqrt{s}/T)\int_{t_{0}}^{t_{1}}dt\,|\mathcal{M}|^{2},$

with $\sqrt{s_{0}}={\rm max}(m_{a}+m_{b},m_{c}+m_{d})$, $K_{1}$ and $K_{2}$
being the modified Bessel function of the second kind. The integration limits
are $t_{0}(t_{1})$ =
$(\frac{m_{1}^{2}+m_{3}^{2}-m_{2}^{2}+m_{4}^{2}}{2\sqrt{s}})^{2}-(p_{1cm}\mp
p_{3cm})^{2}$. Here, we approximate $f_{i}$ to be the Boltzmann momentum
distribution of a particle $i$,
$f_{i}(\vec{p})=e^{-\sqrt{\vec{p}^{2}+m^{2}}/T}$. $v_{ab}$ is the relative
velocity of particles of species $a$ and $b$, $v_{ab}=\sqrt{\left(p_{a}\cdot
p_{b}\right)^{2}-m_{a}^{2}m_{b}^{2}}/(E_{a}E_{b})$.
$\left<\Gamma_{K_{1}}\right>=\Gamma_{K_{1}}(m_{K_{1}})K_{1}(m_{K_{1}}/T)/K_{2}(m_{K_{1}}/T)$
is the thermally averaged decay width of the $K_{1}$ meson Cho:2015qca .

Figure 3: Thermally averaged cross sections for the absorption of a $K_{1}$
meson via processes $K_{1}\pi\rightarrow K\pi$, $K_{1}\pi\rightarrow
K^{*}\rho$, $K_{1}\rho\rightarrow K\rho$, $K_{1}\rho\rightarrow K^{*}\pi$,
$K_{1}\rightarrow K\rho$, and $K_{1}\rightarrow K^{*}\pi$.

##  V initial temperature and time evolution in heavy ion collision

The initial temperature inside the nuclear matter produced from heavy ion
collisions is studied with hydro simulation. We will consider Pb-Pb collision
at $\sqrt{s_{NN}}=5.02$ TeV in different centrality ranges: 0–5 %, 40–50 % and
70–80 %. The initial energy density in the transverse plane is calculated by
using a two-dimensional Gaussian distribution of $\sigma=0.4~{}\mathrm{fm}$
for each participating nucleon obtained from the Monte Carlo Glauber framework
Miller:2007ri . A scale factor as a function of the number of participating
nucleons is multiplied to the initial density for a hydrodynamic evolution
with the sonic model Habich:2014jna . This tune is required to match the
multiplicity with the measured data in various centrality ranges Adam:2016ddh
.

Figure 4 shows the initial temperature distribution at $\tau_{0}=0.5~{}{\rm
fm}/c$ of example events in three centrality ranges from the hydrodynamic
simulation. As can be seen in the figure, the fraction of area where initial
temperature is higher than 156 MeV is 84 %, 75 %, and 67 % in the 0–5 %, 40–50
%, and 70–80 % centrality ranges, respectively. Furthermore, the fraction of
initial energy that comes from regions where the temperature is higher than
156 MeV is larger than 98% in all three centrality ranges. Therefore we can
assume that almost all $K_{1}$ and $K^{*}$ are produced through the QGP phase
even in 70–80 % central collisions.

Figure 4: Initial temperature distribution for example events in 0–5%, 40–50%
and 70–80% centrality ranges of Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV.

For calculation of the number of $K_{1}$ and $K^{*}$ mesons inside the
expanding medium, a more simple approach is used by assuming an isentropic
expansion of uniform matter as follows Song:2010fk :

$\displaystyle\partial_{\tau}(A\tau\langle T^{\tau\tau}\rangle)=-pA,$ (24)
$\displaystyle\partial_{\tau}(A\tau s\langle\gamma_{r}\rangle)=0,$ (25)

where $T^{\tau\tau}=(e+p)u_{\tau}^{2}-p$ is the energy momentum tensor in the
Milne coordinate system with $u_{\tau}$, $e$ and $p$ being the fluid velocity,
energy density and pressure, respectively. $s$ is the entropy density,
$\gamma_{r}=1/\sqrt{1-v_{r}^{2}}$ with $v_{r}$ being the radial velocity of
fluid cell, $A=\pi R^{2}$ with $R$ being the transverse radius of the uniform
matter and $\langle\cdots\rangle$ denotes average over the transverse area.
Eq. (25) clearly shows that the total entropy is conserved during the
expansion with Lorentz contraction of transverse area due to the flow velocity
taken into account.

It should be noted that while the entropy conservation is a reasonable
approximation for the QGP phase, the effect of larger viscosity has to be
taken into account for the hadronic phaseDemir:2008tr . While the simple
hydrodynamic model could be supplemented through hadronic transport models to
take into account the increase of entropy during the hadronic phaseXu:2017akx
, we will leave such improvements for later study and neglect such effects in
this work as the main tendency of the enhancement as a function of centrality
will remain valid.

Assuming that the radial flow velocity is a linear function of the radial
distance from the center, i.e., $\gamma_{r}v_{r}=\gamma_{R}\dot{R}(r/R)$,
where $\dot{R}=\partial R/\partial\tau$ and
$\gamma_{R}=1/\sqrt{1-\dot{R}^{2}}$, we then have Song:2010fk

$\displaystyle\langle
u_{\tau}^{2}\rangle|_{\eta=0}=\langle\gamma_{r}^{2}\rangle=1+\frac{\gamma_{R}^{2}\dot{R}^{2}}{2},$
$\displaystyle\langle\gamma_{r}\rangle=\frac{2}{3\gamma_{R}^{2}\dot{R}^{2}}\left(\gamma_{R}^{3}-1\right).$
(26)

Since the energy density $e$ and pressure $p$ are related by the equation of
state of the matter through its temperature $T$ Borsanyi:2010cj ,
Eqs.(24)-(25) are thus simultaneous equations for $T$, $\dot{R}$.

The results of the numerical calculations can be parametrized in the following
form for the hadronic phase:

$\displaystyle V(\tau)$ $\displaystyle=$
$\displaystyle\pi\left[R+v\left(\tau-\tau_{c}\right)+a/2\left(\tau-\tau_{c}\right)^{2}\right]^{2}\tau
c,$ $\displaystyle T(\tau)$ $\displaystyle=$ $\displaystyle
T_{c}-\left(T_{h}-T_{f}\right)\left(\frac{\tau-\tau_{h}}{\tau_{f}-\tau_{h}}\right)^{\alpha},$
(27)

with $T_{h}$ and $T_{f}$ being the hadronization and kinetic freeze-out
temperature. We take $T_{h}$ = 156 ${\rm MeV}$ to be the same as the cross
over temperature $T_{c}=T_{h}$. Eq. (V) can be thought to describe the system
of the hadronic phase of the nuclear matter expanding with transverse velocity
$v$ and transverse acceleration $a$ starting from its transverse size $R$ at
the chemical freeze-out time $\tau_{c}$. The values of $T_{f}$ according to
centrality are taken from Ref. Acharya:2019yoi . The values used in Eq. (V)
are summarized in Table 2. The schematic hydrodynamics in Eqs. (24) - (26)
assumes the expansion of a uniform matter whose temperature or particle
density is the same anywhere in the local rest frame. In the Lab. frame,
however, the particle density increases with the radial distance due to
Lorentz contraction. We note that Eq. (27) is not the parametrization for the
volume in the Lab frame but for the volume with the Lorentz contraction taken
into account, that is, $A\tau\langle\gamma_{r}\rangle$ in Eq. (25).

Table 2: Values for the volume and temperature profiles in the phenomenological model Eq.(V). Centrality | $T_{f}$ | $t_{c}$ | $t_{f}$ | R | v | a | $\alpha$
---|---|---|---|---|---|---|---
(%) | (MeV) | (fm/c) | (fm/c) | (fm) | (c) | ($c^{2}/fm$) |
$0-5\%$ | 90 | 8.7 | 28.1 | 14.9 | 0.93 | 0.04 | 0.835
$40-50\%$ | 108 | 4.9 | 13 | 7.8 | 0.78 | 0.052 | 0.9
$70-80\%$ | 147 | 2.2 | 2.9 | 4.43 | 0.481 | 0.161 | 0.847

##  VI The abundance of $K_{1}$ mesons in heavy-ion collisions

Table 3 shows the abundance of the strange mesons at the chemical freeze-out
point for different centrality ranges. The yield ratio between the $K_{1}$ and
$K^{*}$ mesons at the kinetic freeze-out point is also given. $N_{K_{1}}$ is
the solution of the rate equation Eq. (19) with the initial number equal to
$N_{K^{*}}=N_{K*}^{T}$ at $T_{c}=156$ MeV in the presence of chiral symmetry
restoration. On the other hand, $N_{K_{1}}^{T}$ is the expected yield without
chiral symmetry restoration when the initial number is given by its thermal
equilibrium number at $T_{c}$. Hence, $N_{K_{1}}^{T}$ at the kinetic freeze-
out point is the number expected in the statistical model when hadronic
dissociation is taken into account. For a clear signature of chiral symmetry
restoration at $T_{c}$, we want $N_{K_{1}}/N_{K^{*}}$ at the kinetic freeze-
out temperature to be sufficiently larger than $N^{T}_{K_{1}}/N_{K^{*}}$ at
$T_{c}$, which is the standard statistical model prediction for the ratio. One
notes that this is true for 40-50 % and 70-80 % centrality ranges. In fact, as
can be seen in the last column of Table 3, for 70-80 %, one finds that
$N_{K_{1}}/N_{K^{*}}$ at the kinetic freeze-out point is 0.905, which is much
larger than $N^{T}_{K_{1}}/N_{K^{*}}$ at any time for all centralities: the
time dependencies of $N^{T}_{K_{1}}$ and $N_{K^{*}}$ are given in Fig. 5. Fig.
6 summarizes the yield ratios between $K_{1}$ and $K^{*}$ mesons for different
centralities with (red dots) and without (black squares) chiral symmetry
restoration at $T_{c}$. As can be seen in the figure, the ratio increases
sharply as the centrality decreases (because of the higher kinetic freeze-out
temperature and thus the shorter lifetime of the hadronic phase at more
peripheral collisions Cho:2015exb ; Motornenko:2019jha ) when chiral symmetry
restoration at $T_{c}$ is taken into account. It is possible that at the
chemical freeze-out point, the chiral order parameter will partly acquire its
vacuum value. Then, the initial ratio of $N_{K_{1}}/N_{K^{*}}$ at $T_{c}$
would be slightly less than 1. Furthermore, the number of pions and rho mesons
should be larger than estimated in Eq. (20) due to the effective chemical
potential during the hadronic phaseRapp:2000gy , leading to slightly larger
dissociation cross sections than estimated in this work. Nevertheless, the
large increase in the yield ratio towards peripheral collisions would still be
visible so that one can identify chiral symmetry restoration. Hence, a
systematic study of the ratio $N_{K_{1}}/N_{K^{*}}$ for different centralities
in a heavy ion collision will unambiguously show the evidence of chiral
symmetry restoration at $T_{c}$.

Table 3: The kaon, $K^{*}$ and $K_{1}$ meson abundances under hadronic interaction at chemical and kinetic freeze out temperatures. $N_{K_{1}}^{T}$ is the number of $K_{1}$ meson assuming thermal equilibrium while $N_{K_{1}}$ is the abundance assuming chiral symmetry at $T_{c}$. The last column shows our prediction when chiral symmetry is restored.  Centrality | T(MeV) | $N_{K}$ | $N_{K^{*}}$ | $N_{K_{1}}^{T}$ | $N_{K_{1}}$ | $\frac{N_{K_{1}}^{T}}{N_{K^{*}}}$ | $\frac{N_{K_{1}}}{N_{K^{*}}}$
---|---|---|---|---|---|---|---
$0-5\%$ | 156 | 80.5 | 37.0 | 5.62 | 37.0 | 0.152 | 1.00
| 90 |  |  | 0.157 | 1.37 | 0.006 | 0.057
$40-50\%$ | 156 | 12.4 | 5.72 | 0.867 | 5.72 | 0.152 | 1.00
| 108 |  |  | 0.149 | 1.17 | 0.034 | 0.268
$70-80\%$ | 156 | 1.80 | 0.827 | 0.125 | 0.827 | 0.152 | 1.00
| 147 |  |  | 0.108 | 0.714 | 0.137 | 0.905

Figure 5: The time evolution of the $K_{1}$ meson abundance under hadronic
interaction when the initial number is equal to $N_{K*}$(circle) or
$N_{K_{1}}^{T}$(triangle) at $T_{c}$ for 40-50%(upper graph) and 70-80%(lower
graph) centrality ranges.  Figure 6:  The yield ratio of $K_{1}$ over $K^{*}$
with and without chiral symmetry restoration for 70-80%, 40-50%, and 0-5%
centrality ranges. $K_{1}$ feed down to $K^{*}$ should be added when comparing
to experiment.

##  VII Summary

We studied effects of chiral symmetry restoration at the chemical freeze-out
point in a relativistic heavy ion collision. As the mass differences between
the chiral partners are order parameters of the chiral symmetry restoration,
the production of the chiral pairs will be degenerate at the chemical freeze-
out point, where the chiral order parameter is found to be substantially
reduced from the vacuum value in lattice calculations. For the effects to be
observable, the vacuum width as well as the hadronic dissociation of both
particles should be small so that the signal will not be smeared out during
the hadronic phase. The vacuum widths of both the $K_{1}$ and $K^{*}$ meson
are smaller than 100 MeV and thus potential candidates to be studied. We have
performed a systematic study on their hadronic absorption during the hadronic
phase as well as on the centrality dependence of these effects in a heavy ion
collision. Our findings suggest that while the anomalously large initial ratio
between the $K_{1}$ and $K^{*}$ meson compared to that of the statistical
model prediction will most likely be smeared out in a central collision, the
signal will be visible in a peripheral heavy ion collision due to the shorter
life time of the hadronic phase and higher freeze-out temperature.

The chiral partnership between the $K^{*}$ and $K_{1}$ exists between the same
charge states when the baryon chemical potential is non-zero Song:2018plu . In
ultra-relativistic heavy ion collisions, the initial state will have almost
zero baryon chemical potential. Hence, one can compare the production of any
of the charge states through the decay products given below or their charge
conjugation. For the $K_{1}$ meson they are

$\displaystyle
K_{1}^{-}\to\begin{cases}\rho^{0}K^{-}\cr\rho^{-}\bar{K}^{0}\cr\pi^{0}K^{*-}\cr\pi^{-}\bar{K}^{*0}\end{cases},~{}~{}~{}~{}\bar{K}_{1}^{0}\to\begin{cases}\rho^{+}K^{-}\cr\rho^{0}\bar{K}^{0}\cr\pi^{+}K^{*-}\cr\pi^{0}\bar{K}^{*0}\end{cases},$

and for $K^{*}$

$\displaystyle
K^{*-}\to\begin{cases}\pi^{0}K^{-}\cr\pi^{-}\bar{K}^{0}\end{cases},~{}~{}~{}~{}\bar{K}^{*0}\to\begin{cases}\pi^{+}K^{-}\cr\pi^{0}\bar{K}^{0}\end{cases}.$

A systematic study of the production of these states depending on the
centrality will lead us to identify chiral symmetry restoration in heavy ion
collision.

## Acknowledgments

This work was supported by Samsung Science and Technology Foundation under
Project Number SSTF-BA1901-04. S. Lim acknowledge support from the National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
under Contract No. NRF-2008-00458. S. Cho was supported by the National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(No. 2019R1A2C1087107).

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