# Chiral symmetry breaking and the masses of hadrons: a review

Su Houng Lee<EMAIL_ADDRESS>Department of Physics and Institute of
Physics and Applied Physics, Yonsei University, Seoul 03722, Korea

###### Abstract

The masses of hadrons in the vacuum, where the chiral symmetry is restored,
and in the medium are in general different even when the changes in the order
parameters of chiral symmetry are the same. Here, we first discuss the
relation between the hadron masses and the chiral symmetry breaking in
approaches based on operator product expansion (OPE). We then discuss what
additional changes occur to the hadron masses when going from the chiral
symmetry restored vacuum to nuclear medium and/or finite temperature. The work
will highlight how we can identify the effects of chiral symmetry restoration
from experimental observations.

##  I Introduction

We typically express the masses of composite particles as the sum of their
constituents and small binding energies. Such decomposition is not possible in
quantum chromodynamics (QCD) as confinement and chiral symmetry breaking are
intricately related to in generating the hadron mass. Yet, it is known that
chiral symmetry breaking is an important ingredient, so isolating its effects
would provide an important cornerstone in understanding the origin of the
hadron massHatsuda (1985); Brown (1991); Hatsuda (1991); Leupold (2009).

In the original application of the QCD sum rules approaches to the vector
meson masses in the nuclear medoumHatsuda (1991), the meson masses were found
to change due to the changes of the light 4-quark condensates for the $\rho$,
$\omega$ mesons and the strange quark condensates for the $\phi$ meson. In the
case of $J/\psi$ in the medium, the changes occur due to the change of gluon
condensate, which is related to the scale-breaking effects through trace
anomaly. Since then, experiments have been performed worldwide to observe mass
shifts of hadrons from heavy ion collisions and nuclear target experiments,
which are expected to reflect finite temperature or density configurations,
respectively Hayano (2008); Ichikawa (2018).

Experimental efforts were concentrated on measuring the electromagnetic
signals from heavy ion collisions and nuclear target experiments. While
experiments trying to measure dilepton spectra from heavy ion collisions are
still going on and planned in the future, these signals have many sources
throughout the evolution of the system. Furthermore, information about chiral
partners is hidden in the continuum. On the other hand, nuclear target
experiments have advantages: the target nuclear density remains almost
constant, and even at these densities, the order parameter is known to
decrease substantially around 30$\%$Lacour:2010ci ; Meissner:2001gz . Also,
signals from hadronic decays are viable as long as one, focuses on small-width
mesons. Furthermore, one can combine excitation function measurements and the
transparency ratios Metag (2017) to estimate the mass and width changes
simultaneously. But in order to identify the effect of chiral symmetry
restoration, it seems essential to first look at the chiral partners and
measure their mass differences. Such measurement will establish the
cornerstone for the relation between mass shift and chiral symmetry
restoration. One should then measure the individual masses.

In this review, we will summarize what happens to hadrons in medium from the
perspective of operator product expansion (OPE). We will first show how to
isolate the chiral symmetry-breaking effects in the hadron masses in the OPE
perspective. The method will be applied to the vector mesons and then to other
hadrons. Through this method, we will be able to isolate the effects of chiral
symmetry breaking and thus calculate the mass of hadrons in the chiral-
symmetry restored vacuum. Here, the chiral-symmetry restored vacuum means the
QCD configuration where the chiral-symmetry breaking effects are taken away
while other non-perturbative effects are left intact. Such a configuration can
be probed in the OPE formalism by taking all the quark operators proportional
to the chiral order parameter to be zero while keeping all other quark or
gluon operators intact. From a lattice point of view, one can also probe such
a configuration by taking away all the quark zero modes when calculating
hadron properties. We will then discuss what other effects will emerge when we
place hadrons in a mediumkim (2020); Kim (2021): these effects do not
contribute when one looks at the differences between the chiral partners.
Finally, we introduce the mass of $K_{1}$ and $K^{*}$ as a suitable example
for chiral partners that can be realistically measured in a nuclear target
experimentSong (2018); Lee (2019).

##  II Chiral order parameter

In this section, we discuss how to isolate the chiral symmetry-breaking part
of any quark operatorkim (2020); Kim (2021): the breaking part is the chiral
order parameter. In any expectation value, the quark part will involve the
quark propagator $S_{q}(x,0)\equiv S_{q}(x,y)|_{y=0}$, which can be written
into a part that is symmetric and anti-symmetric under chiral rotation: this
can be accomplished by adding and subtracting its chiral partner. The
respective parts are also denoted as chiral even $(S)$ and odd $(B)$
components.

$\displaystyle S_{q}(x,0)$ $\displaystyle=$
$\displaystyle\bigg{(}S_{q}^{B}(x,0)+S_{q}^{S}(x,0)\bigg{)},$ (1)
$\displaystyle S_{q}^{B}(x,0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}S_{q}(x,0)-i\gamma^{5}S_{q}(x,0)i\gamma^{5}\bigg{)},$
(2) $\displaystyle S_{q}^{S}(x,0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}S_{q}(x,0)+i\gamma^{5}S_{q}(x,0)i\gamma^{5}\bigg{)}.$
(3)

The separation can also be understood as making a specific chiral
transformation to the quark operator $q\rightarrow\gamma^{5}q$ so that
$S_{q}(x,y)\rightarrow i\gamma^{5}S_{q}(x,y)i\gamma^{5}$ leading to the
symmetric and breaking part as given in Eq.(3) and Eq.(2), respectively.

The usual chiral condensate is the dimension 3 two-quark operator. Here,
because of the trace or due to parity, only the chiral symmetry-breaking part
contributes.

$\begin{split}\langle\bar{q}q\rangle&=-\lim_{x\rightarrow 0}\langle{\rm
Tr}[S_{q}^{B}(x,0)]\rangle\\\ &=-\pi\langle\rho(0)\rangle,\end{split}$ (4)

where $\rho(0)$ is the zero mode density, a formula derived by Casher and
Banks Banks (1980).

Let us now move on to a typical four-quark condensate. Such operators have two
distinct forms when using quark connectivity: these are the quark-disconnected
$(dis)$ and quark-connected $(con)$ pieces denoted below by the respective
subscripts,

$\begin{split}\langle(\bar{q}\Gamma q)(\bar{q}\Gamma q)\rangle&=\langle{\rm
Tr}[S_{q}^{i}\Gamma]{\rm Tr}[S_{q}^{i}\Gamma]\rangle_{dis}-\langle{\rm
Tr}[\Gamma S_{q}^{i}\Gamma S_{q}^{i}]\rangle_{con},\end{split}$ (5)

where $\Gamma$ is any combination of Dirac, color and/or flavor matrix. The
cross terms in the summation in $i=B,S$ do not contribute for the $\Gamma$
matrices that we discuss. $S_{q}^{B}$ is the difference between the original
quark operator and its chiral rotated form: This part does not vanish if the
chiral symmetry remains broken. Also, as given in Eq. (4), it is proportional
to the zero mode density.

Let us look at a few examples. For now, let us introduce the flavor matrix
normalized as ${\rm
Tr}\frac{\tau^{a}}{2}\frac{\tau^{b}}{2}=\frac{\delta^{ab}}{2}$,

  * •

When $\Gamma=\gamma^{\mu}$, using the trace property one finds

$\begin{split}\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)\rangle&=-\langle{\rm
Tr}[\gamma^{\mu}S_{q}^{B}\gamma^{\mu}S_{q}^{B}]\rangle_{con}-\langle{\rm
Tr}[\gamma^{\mu}S_{q}^{S}\gamma^{\mu}S_{q}^{S}]\rangle_{con},\end{split}$ (6)

with no cross-term between the symmetric and breaking part of the propagator.
Hence, it contains both the breaking and symmetric operators.

On the other hand, one notes

$\displaystyle\gamma^{5}S_{q}^{S},\gamma^{5}=-S_{q}^{S},~{}~{}~{}~{}~{}\gamma^{5}S_{q}^{B}\gamma^{5}=S_{q}^{B}$
(7)

Hence, to

$\begin{split}\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)\rangle&=-\langle{\rm
Tr}[\gamma^{\mu}S_{q}^{B}\gamma^{\mu}S_{q}^{B}]\rangle_{con}+\langle{\rm
Tr}[\gamma^{\mu}S_{q}^{S}\gamma^{\mu}S_{q}^{S}]\rangle_{con}.\end{split}$ (8)

Therefore, one can isolate the breaking and symmetric parts as follows

$\begin{split}\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)\rangle_{B}&=\frac{1}{2}\bigg{(}\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)\rangle+\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)\rangle\bigg{)},\\\
\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)\rangle_{S}&=\frac{1}{2}\bigg{(}\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}\gamma^{\mu}q)\rangle-\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}i\gamma^{5}\gamma^{\mu}q)\rangle\bigg{)},\end{split}$
(9)

where the subscript $B,S$ respectively indicate the breaking and symmetric
part of the four-quark operator, respectively.

  * •

When $\Gamma=1$, we also have the quark disconnect contribution denoted by the
subscript $dis$ below.

$\begin{split}\langle(\bar{q}q)(\bar{q}q)\rangle&=\langle{\rm
Tr}[S_{q}^{B}]{\rm Tr}[S_{q}^{B}]\rangle_{dis}-\langle{\rm
Tr}[S_{q}^{B}S_{q}^{B}]\rangle_{con}-\langle{\rm
Tr}[S_{q}^{S}S_{q}^{S}]\rangle_{con}.\end{split}$ (10)

To isolate the connected piece above, we introduce the following two quark
operators.

$\begin{split}\langle(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)\rangle&=\langle{\rm
Tr}[S_{q}^{B}S_{q}^{B}]\rangle_{con}-\langle{\rm
Tr}[S_{q}^{S}S_{q}^{S}]\rangle_{con},\\\
\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)\rangle&=-\langle{\rm
Tr}[S_{q}^{B}S_{q}^{B}]\rangle_{con}-\langle{\rm
Tr}[S_{q}^{S}S_{q}^{S}]\rangle_{con}.\end{split}$ (11)

Therefore

$\begin{split}\langle(\bar{q}q)(\bar{q}q)\rangle_{B}&=\langle(\bar{q}q)(\bar{q}q)\rangle-\frac{1}{2}\bigg{(}\langle(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)\rangle+\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)\rangle\bigg{)}\\\
\langle(\bar{q}q)(\bar{q}q)\rangle_{S}&=\frac{1}{2}\bigg{(}\langle(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}i\gamma^{5}\frac{\tau^{3}}{\sqrt{2}}q)\rangle+\langle(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)(\bar{q}\frac{\tau^{3}}{\sqrt{2}}q)\rangle\bigg{)}.\end{split}$
(12)

Throughout this work, we will use this method: although a similar separation
can be made by writing the quark operators in terms of right and left-handed
quarks.

  * •

The method can be generalized for all $\Gamma$ matrices. We refer to
references kim (2020); Kim (2021) for details.

  * •

Because the static heavy quark probes all the gluon configurations, it can be
shown that the gluon condensate can also be expressed in terms of the
eigenvalues of the Dirac modesLee (2013).

$\displaystyle\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle=12\langle\sum_{\lambda}\rho(\lambda)\rangle.$
(13)

As can be seen from the above formula, the chiral-symmetry breaking effect
$\rho(0)$ is multiplied by $d\lambda$ and does not contribute to the gluon
condensate. This shows that the physics of the gluon condensate and that of
the chiral-symmetry breaking have different origins.

##  III Vector and axial-vector meson mass

We will now look at how the method can be applied to calculate the hadron
masses in the chiral symmetry-restored vacuum.

###  III.1 $\rho$ and $a_{1}$

Let us start by looking at the OPE of the following correlation function

$\displaystyle\Pi_{\mu\nu}(q)=i\int d^{4}xe^{iqx}\langle
T[J_{\mu}(x),J_{\nu}(0)],$ (14)

where $J_{\mu}=\bar{q}\gamma_{\mu}\tau^{3}q$ for the $\rho$ meson; for the
axial vector meson, we extract the spin 1 part of
$J^{A}_{\mu}=\bar{q}\gamma^{5}\gamma_{\mu}\tau^{3}q$. We use the OPE appearing
in the polarization function defined as $\Pi=\Pi^{\mu}_{\mu}/(-3q^{2})$. The
most important contribution to the vector mesons comes from the dimension-6
four-quark operators. These operators appear as
$-\pi\alpha_{s}\mathcal{M}/Q^{6}$, where for $\rho$ and $a_{1}$ mesons are
given as follows, respectively.

$\begin{split}\mathcal{M}_{\rho}=&\;2\langle(\bar{q}\gamma_{\mu}\gamma^{5}\lambda^{a}\tau^{3}q)^{2}\rangle+\frac{4}{9}\langle(\bar{q}\gamma_{\mu}\lambda^{a}q)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)\rangle,\\\
\mathcal{M}_{a_{1}}=&\;2\langle(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}\rangle+\frac{4}{9}\langle(\bar{q}\gamma_{\mu}\lambda^{a}q)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)\rangle.\end{split}$
(15)

Here $\lambda^{a}$ and $\tau^{3}$ represent the color and SU(2) flavor
matrices, respectively. Hence, $q$ denotes the contribution from both the $u$
and $d$ quarks. On the other hand, the second set of operators has a summation
included as the contribution from strange quark is added. We can now use the
methods discussed in the previous section and divide the operators into chiral
symmetric and breaking pieces. We further introduce the auxiliary parameters
$\kappa_{\rho}$ and $\kappa_{a_{1}}$: these parameters reduce to 1 when the
vacuum saturation hypothesis is used for the quark-quark operators. The four-
quark operators can now be written as below.

$\begin{split}\mathcal{M}_{\rho}=&~{}\kappa_{\rho}\frac{448}{81}\langle\bar{u}u\rangle^{2}\
=\;\frac{28}{9}\langle B_{uu}\rangle_{B}+\langle S_{\rho-a_{1}}\rangle_{S},\\\
\mathcal{M}_{a_{1}}=&~{}-\kappa_{a_{1}}\frac{704}{81}\langle\bar{u}u\rangle^{2}=\;-\frac{44}{9}\langle
B_{uu}\rangle_{B}+\langle S_{\rho-a_{1}}\rangle_{S},\end{split}$ (16)

where

$\begin{split}\langle
B_{uu}\rangle_{B}=&\frac{1}{2}\bigg{(}\langle(\bar{u}\gamma_{\mu}\gamma^{5}\lambda^{a}d)(\bar{d}\gamma_{\mu}\gamma^{5}\lambda^{a}u)\rangle-\langle(\bar{u}\gamma_{\mu}\lambda^{a}d)(\bar{d}\gamma_{\mu}\lambda^{a}u)\rangle\bigg{)}_{B},\\\
\langle S_{\rho-
a_{1}}\rangle_{S}=&\frac{11}{9}\bigg{(}\langle\bar{q}\gamma_{\mu}\gamma^{5}\lambda^{a}\tau^{3}q)^{2}\rangle+\langle(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}\rangle\bigg{)}_{S}+\frac{4}{9}\bigg{(}\langle(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}\rangle-\langle(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}\rangle\bigg{)}_{S}\\\
&+\frac{4}{9}\langle(\bar{q}\gamma_{\mu}\lambda^{a}q)(\bar{s}\gamma_{\mu}\lambda^{a}s)\rangle_{S}.\end{split}$
(17)

There is an important point to note in the second set of terms in Eq. (15).
These terms appear with the same coefficients in both the $\rho$ and $a_{1}$
parts. However, the operator has chiral symmetry-breaking contributions: the
zero modes contribute to the connected quark diagrams of this operator.

One notes that the symmetric parts of the $\rho$ and $a_{1}$ channels are
identical while the breaking operator $B_{uu}$ contributes with different
coefficients. Using the observed masses and widths of the $\rho$ and $a_{1}$
meson in their respective sum rules, we can uniquely determine the vacuum
expectation values of the 4-quark operators and thus their breaking and
symmetric parts. Then, one can uniquely determine the respective $\kappa$
values. Using these results, we can estimate the masses of the hadrons in the
chiral symmetry-restored vacuum. This is accomplished for the $\rho$ and
$a_{1}$ mesons by deleting the breaking part of the operator keeping only the
chiral symmetric operators to their vacuum value and performing the QCD sum
rules. Table 1 shows the results for both the matrix elements and the masses
in the vacuum where chiral symmetry is restored. For $\rho-a_{1}$, the result
in Table 1 is obtained by taking the $a_{1}$ width to be 400 MeV. When
$\Gamma_{a_{1}}=250(600)$ MeV is taken, we find that $m_{sym}^{\rho-a_{1}}$
decreases (increases) by about 27.5 MeV.

Particle | $\kappa(\sqrt{s_{0}}({\rm GeV}))$ | $S/B$ |  $\bar{m}_{sym}$(MeV)
---|---|---|---
$\rho$ | 2.60(1.17) | 0.760 | 572.5 $\pm$ 27.5
$a_{1}$ | 0.76(1.55) | -0.485
$\omega$ | 3.20(1.15) | 1.165 | 655 $\pm$ 15
$f_{1}$ | 1.85(1.58) | 0.253 | 1060 $\pm$ 30
$K^{*}$ | 2.097(1.33) | 2.831 | 545 $\mp$ 5
$K_{1}$ | 0.39(1.56) | -0.227
Table 1: The $\kappa$’s (equivalent to Eq. (16) for the respective particles)
are evaluated using QCD sum rules with the observed mass and width. The value
in the bracket shows the threshold parameter $s_{0}$. The ratio $S/B$ shows
the fraction of the chiral symmetric part $S$ to the chiral symmetry-breaking
part $B$. The mass in the chiral symmetry-restored vacuum is given in the
column $\bar{m}_{sym}$. The uncertainties in $\bar{m}_{sym}$ come from taking
the width of $a_{1}$ meson from 250 MeV to 600 MeV, The central values are
those obtained with the $a_{1}$ width of 400 MeVKim (2021).

###  III.2 Other hadrons

As with ($\rho$, $a_{1}$), ($K_{1}$, $K^{*}$) are also chiral partners. On the
other hand, ($\omega$, $f_{1}$) do not form chiral partners: this is due to
the disconnected diagrams. These features can be understood from the OPE
perspective. Nevertheless, we can start from the symmetry-breaking operators
determined from the $\rho$ and the $a_{1}$ sum rule and use the physical mass
and width of the other hadrons to determine the additional 4-quark operators
that are symmetric and breaking chiral under chiral symmetry that appear in
the isospin singlet sum rules. One can then determine the corresponding mass
in the chiral symmetric limit. The details are given in Ref. Kim (2021) and
summarized in Table 1.

##  IV Other effects in medium

In the previous section, we have shown how to estimate the masses of hadrons
in a hypothetical vacuum where the chiral symmetry is restored. Experiments
probe finite temperature and/or density configuration where chiral symmetry is
restored. However, in these configurations, there are other effects that smear
the signal. We will discuss other important effects not related to chiral
symmetry restoration that affect the mass of a hadron in the medium.

###  IV.1 Dispersion effects

The medium can be characterized by its frame of reference, which is usually
taken to be at rest $n^{\mu}=(1,0,0,0)$. The medium introduces effects that
break Lorentz invariance but are not related to chiral symmetry. In general,
the introduction of $n^{\mu}$ makes the correlation function in the momentum
space a function of both $q^{2}$ and $q\cdot n$. Depending on the current and
the medium, the polarization can be even or odd with respect to $q\cdot n$.

####  IV.1.1 Scalar particle

Let us start with a simple example and consider a charge-neutral scalar
particle in an isospin-symmetric medium. The self-energy
$\Sigma(\omega,\vec{q})$ of the scalar particle can be expanded near the mass
shell as follows.

$\displaystyle\Sigma(\omega,\vec{q})=a(q^{2}-m^{2})+b\vec{q}^{2}+S.$ (18)

Substituting this into the dispersion relation,
$q^{2}-m^{2}=\Sigma(\omega,\vec{q})$, will lead to pole mass shift that
depends on the momentum of the particle

$\Delta m^{2}=\frac{b\vec{q}^{2}+S}{1-a}.$ (19)

The question relating the mass shift in medium and chiral symmetry breaking is
where the effects of the latter are hidden: $a,b$ or $S$. It is in $S$.

####  IV.1.2 Nucleon in medium

For spin 1/2 particle, there is the additional dependence coming from the
gamma matrix. To understand the complexity, we just assume a self-energy
typically parameterized in Walecka type model in nuclear matter at rest.

$\displaystyle/q-m=S+\gamma^{0}V.$ (20)

This leads to the following pole mass shift at $\vec{q}=0$.

$\displaystyle\Delta m=S+V.$ (21)

It is known phenomenologically that there are a large scalar attraction
($\sim-400$ MeV) and a vector repulsion of similar magnitude due to the
nuclear density; both have small energy dependencies.

When analyzing the OPE for the nucleon, one introduces a nucleon interpolating
field $\eta$ composed of three quarks and studies the two-point correlation
function.

$\displaystyle\Pi_{N}(\omega,\vec{q})=i\int d^{4}xe^{iqx}\langle
T[\eta(x)\bar{\eta}(0)]\rangle.$ (22)

The phenomenological side of Eq. (22) can be parametrized using Eq. (20) as

$\displaystyle\Pi_{N}(\omega,\vec{q})=-\lambda_{N}^{2}\frac{(\omega-V)\gamma^{0}+(m+S)}{(\omega-V)^{2}-(m+S)^{2}}+,$
(23)

where $\lambda_{N}$ is the coupling between the nucleon interpolating field
and the nucleon state. Performing the OPE for Eq. (22), it was shown that the
leading operators that contribute to the scalar and vector self-energy are
respectively given as

$\displaystyle\begin{split}m+S&\propto-\langle\bar{q}q\rangle,\\\
V&\propto\langle\bar{q}\gamma^{0}q\rangle.\end{split}$ (24)

Therefore, while the scalar attraction is partly related to the decrease of
the quark condensate, the vector repulsion is related to the quark density
effect. Hence, while the scalar attraction is related to the condensate, its
quenching can not be directly seen in the pole mass due to the presence of
baryon (quark) density. Furthermore, as the nucleon interpolating field is not
an eigenstate of the parity, there will be contributions from the negative
parity nucleon state that contributes with an opposite sign in the scalar
mass. Hence, quenching of the chiral condensate in the OPE will induce the
vanishing of the mass difference between the parity partner nucleon states but
will not tell us anything about the nucleon mass change just by looking at the
scalar part of the correlation function.

####  IV.1.3 Vector particles

There are even more complications for the vector particles due to the
different responses of the moving vector meson with respect to the medium. The
vector meson self-energy in the medium can in general be written as followsLee
(1997).

$\displaystyle P_{\mu\nu}(q^{2}-m^{2})$
$\displaystyle=P^{T}_{\mu\nu}\Pi^{T}(\omega,\vec{q})+P^{L}_{\mu\nu}\Pi^{L}(\omega,\vec{q}),$
(25)

where
$P_{\mu\nu}=(q_{\mu}q_{\nu}/q^{2}-g_{\mu\nu}),~{}P^{T}_{ij}=(\delta_{ij}-\vec{q}_{i}\vec{q}_{j}/\vec{q}^{2})$,
where $i,i$ are spatial indices and the remaining polarization parts begin
zero, and $P^{L}_{\mu\nu}=(q_{\mu}q_{\nu}/q^{2}-g_{\mu\nu}-P_{\mu\nu}^{T})$.

  * •

When $\vec{q}=0$ then $\Pi^{T}=\Pi^{L}=\Pi(\omega)$ and the mass in medium can
be obtained from Eq. (25), which reduces to below.

$(\omega^{2}-m^{2})=\Pi(\omega).$ (26)

  * •

When $\vec{q}\neq 0$ then $\Pi^{T}\neq\Pi^{L}$ and the mass in the medium can
be obtained from Eq. (25) separately for the transverse and longitudinal
modes, which after the projection reduces to below.

$\displaystyle\begin{cases}{\rm
Transverse~{}~{}mode:}&(q^{2}-m^{2})=\Pi^{T}(\omega,\vec{q})\cr{\rm
Longitudinal~{}~{}mode:}&(q^{2}-m^{2})=\Pi^{L}(\omega,\vec{q})\end{cases}$
(27)

By analyzing the OPE for light vector particles, one notes that the chiral
symmetry breaking operators constitute an important contribution to $\Pi$ but
do not contribute significantly in $\Pi^{T}-\Pi^{L}$. Hence when measuring
chiral symmetry-breaking effects from the medium, it is important to measure
vector particles with low momenta with respect to the medium.

One can estimate the importance of the momentum dependence of the mass shift
by looking at the OPE term that contributes to this effect. To leading order
in $\alpha_{s}$, nuclear density $\rho_{n}$ and $\vec{q}$ the lowest order
operator that contributes to the difference isFriman:1999wu

$\displaystyle\Pi^{T}(\omega,\vec{q})-\Pi^{L}(\omega,\vec{q})=\frac{mA_{2}^{u+d}\vec{q}^{2}}{\omega^{6}}\rho_{n}.$
(28)

Here $m$ is the nucleon mass and $A_{2}^{q}=2\int dxx[q(x)+\bar{q}(x)]$, where
$q(x)$ is the quark distribution function inside the proton at scale $\mu$
taken to be 1 GeV for this work. It is found that this contributes to about 2%
correction to the mass at $\vec{q}\sim 0.5$ GeV at nuclear matter densityLee
(1997). Hence, it is important to study the mass shift for vector meson
traveling with small momentum.

##  V Chiral partners: $K_{1}$ and $K^{*}$ mesons

As discussed above, isolating the effects of chiral symmetry restoration from
experimental measurements of hadron masses is problematic as there are many
different medium effects. On the other hand, it can be shown from OPE and/or
general arguments that the mass difference between chiral partners depends
only on chiral symmetry breaking. Therefore, it is important to first measure
the mass differences between chiral partners. There are several candidates.
The obvious choice is the pion and the sigma meson. These particles have been
analyzed frequently in a medium, where these particles are expected to become
degenerate when chiral symmetry is restored. On the other hand, the sigma
meson has a large width decaying into two pions and hence is very difficult to
measure. While the decay width into two pions is expected to decrease as the
phase space becomes smaller when the sigma meson mass decreases in a dense
medium, the decaying pions will interact with the medium smearing any
significant observable signals. Hence, it is important to identify chiral
partners with reasonably small vacuum widths.

The masses and widths of the vector mesons are given in table 2. The $\rho$
and $a_{1}$ are chiral partners: on the other hand, because they both have
large widths, previous attempts failed to measure their mass shift. The small
width $\omega$ and $f_{1}(1285)$ seem to be more accessible experimentally:
but they are isospin singlets and thus are not chiral partners. Gubler (2016).
Nevertheless, $\omega$ and $f_{1}(1285)$ are chiral partners when we neglect
the flavor-changing disconnected contributions. Hence, when experiments can
measure their masses, it would still be useful as we can use this information
to extract the density dependence of flavor-changing disconnected four-quark
condensates. Finally, the best candidates seem to be the $K^{*}$ and $K_{1}$.
They are chiral partners and both have reasonably small widths. Furthermore,
the chiral symmetry-breaking effects are in the ground states as the spectra
of excited states are similar. Below, we will now discuss $K^{*}$ and $K_{1}$
states.

$J^{PC}=1^{--}$ | Mass | Width | $J^{PC}=1^{++}$ | Mass | Width
---|---|---|---|---|---
$\rho$ | 770 | 150 | $a_{1}$ | 1260 | 250 -600
$\omega$ | 782 | 8.49 | $f_{1}$ | 1285 | 24.2
$\phi$ | 1020 | 4.266 | $f_{1}$ | 1420 | 54.9
$K^{*}(1^{-})$ | 892 | 50.3 | $K_{1}(1^{+})$ | 1270 | 90
$K^{*}(1^{-})$ | 1410 | 236 | $K_{1}(1^{+})$ | 1400 | 174
Table 2: Physical parameters of the vector and axial vector mesons. Units are
in MeV.

###  V.1 $K_{1}$ and $K^{*}$ correlation functions

We will now study the correlation functions for the $K^{*}$ and $K_{1}$
mesonsSong (2018). The time-ordered current correlation function is given by

$\displaystyle\Pi_{\mu\nu}(\omega,{\bf q})=i\int d^{4}xe^{iq\cdot
x}\langle|T[j_{\mu},\bar{j}_{\nu}]|\rangle,$ (29)

where $q^{\mu}=(\omega,{\bf q})$ and

$\displaystyle j_{\mu}^{K_{1}^{+}}=\bar{s}\gamma_{\mu}\gamma_{5}u$ ,
$\displaystyle~{}~{}~{}j_{\mu}^{K_{1}^{-}}=\bar{u}\gamma_{\mu}\gamma_{5}s$
$\displaystyle j_{\mu}^{K^{*+}}=\bar{s}\gamma_{\mu}u$ ,
$\displaystyle~{}~{}~{}j_{\mu}^{K^{*-}}=\bar{u}\gamma_{\mu}s.$ (30)

We will only consider isospin symmetric nuclear matter. Therefore, the result
obtained by interchanging the $u$ quark with the $d$ quark will be the same.

The polarization functions for both the vector and axial vector are not
conserved. Therefore they will have contributions from the scalar and pseudo-
scalar mesons respectively. To extract the spin-1 part, we use the following
projection.

$\displaystyle\frac{1}{3}(q^{\mu}q^{\nu}/q^{2}-g^{\mu\nu})\Pi_{\mu\nu}(q)\stackrel{{\scriptstyle{\bf
q}\rightarrow 0}}{{\longrightarrow}}\Pi(\omega,0).$ (31)

Also, because the medium is at rest, the particle dispersion relation will
have both a transverse and longitudinal polarization component. As discussed
in the previous section, we will therefore choose the external three-momentum
to be zero $q^{\mu}=(\omega,\bf{0})$.

####  V.1.1 4-quark operators in the $K^{*}-K_{1}$ sector

When we choose the currents to be $J^{K^{*}}_{\mu}=\bar{u}\gamma_{\mu}s$ and
$J^{K_{1}}_{\mu}=\bar{u}\gamma_{\mu}\gamma^{5}s$, the corresponding
dimension-6 4-quark operators are given below.

$\displaystyle\Pi^{K^{*}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha_{s}}{Q^{6}}\langle(\bar{u}\gamma_{\mu}\gamma^{5}\lambda^{a}s)(\bar{s}\gamma_{\mu}\gamma^{5}\lambda^{a}u)\rangle+\frac{2\pi\alpha_{s}}{9Q^{6}}\langle(\bar{s}\gamma_{\mu}\lambda^{a}s+\bar{u}\gamma_{\mu}\lambda^{a}u)(\bar{q}\gamma_{\mu}\lambda^{a}q)\rangle,$
$\displaystyle\Pi^{K_{1}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha_{s}}{Q^{6}}\langle(\bar{u}\gamma_{\mu}\lambda^{a}s)(\bar{s}\gamma_{\mu}\lambda^{a}u)\rangle+\frac{2\pi\alpha_{s}}{9Q^{6}}\langle(\bar{s}\gamma_{\mu}\lambda^{a}s+\bar{u}\gamma_{\mu}\lambda^{a}u)(\bar{q}\gamma_{\mu}\lambda^{a}q)\rangle.$

These operators can also be decomposed in terms of chiral symmetry breaking
and symmetric pieces.

####  V.1.2 Weinberg relations for $K_{1},K^{*}$

We first study the difference between the correlation functions for the vector
and axial vector currents. Up to dimension 6 operators, it has the following
form.

$\displaystyle-\frac{2}{Q^{2}}m_{s}\langle\bar{u}u\rangle_{0}+\frac{2\pi}{Q^{4}}\alpha_{s}\bigg{(}\langle(\bar{u}\gamma_{\mu}\lambda^{a}s)((\bar{s}\gamma_{\mu}\lambda^{a}u)\rangle-c.p\bigg{)}$
$\displaystyle-\frac{8\pi}{3Q^{6}}q^{\mu}q^{\nu}\alpha_{s}\bigg{(}\langle(\bar{u}\gamma_{\mu}\lambda^{a}s)((\bar{s}\gamma_{\nu}\lambda^{a}u)\rangle_{\cal
ST}-c.p.\bigg{)}.$ (33)

Here $c.p$ is the operator where the $\gamma_{\mu}$ is replaced by
$\gamma_{\mu}\gamma_{5}$ in the 4-quark operator. The operators appearing in
the second line of Eq. (33) are the twist-4 matrix element. Since the
difference between the chiral partners is an order parameter of the chiral
symmetry breaking, Eq. (33) is also a chiral order parameter. In fact, all the
operators to higher dimensions are all order parameters of the chiral
symmetry. When the vacuum saturation hypothesis is applied the operators
appearing in Eq. (33) are $m_{s}\langle\bar{u}u\rangle$ at dimension 4 and
$\langle\bar{s}s\rangle\langle\bar{u}u\rangle$ at dimension 6. As one can see,
in this case, the operators are chiral symmetry-breaking parts in the
strangeness sector multiplied by those in the light quark sector. Therefore,
irrespective of whether the vacuum saturation hypothesis is valid, the fact
that it should be proportional to the chiral order parameter remains valid.
Therefore, to estimate their changes in the nuclear medium, one can
approximate their values in terms of the changes in the light and strange
condensates.

To obtain the Weinberg type sum rule in the current case, we follow the same
set of approximations as in the original worksWeimberg (1967). That is, for
the phenomenological side, we take the vector and axial vector ground state to
be different, but take the form for the excited states including the continuum
to be the same for both cases. We then make an asymptotic expansion in
$1/Q^{2}$ and equate the phenomenological side to the OPE. Since we work up to
dimension 6 operators, we obtain the following two relations.

$\displaystyle
f_{K^{*}}^{2}m_{K^{*}}^{4}-f_{K_{1}}^{2}m_{K_{1}}^{4}=-2m_{s}\langle\bar{u}u\rangle,$
$\displaystyle
f_{K^{*}}^{2}m_{K^{*}}^{6}-f_{K_{1}}^{2}m_{K_{1}}^{6}=-\frac{64}{9}\pi\alpha_{s}\langle\bar{u}u\rangle\langle\bar{s}s\rangle.$
(34)

When the two relations are combined, we obtain the following relation.

$\displaystyle
f_{K^{*}}^{2}m_{K^{*}}^{4}(m_{K_{1}}^{2}-m_{K^{*}}^{2})=-2m_{s}\langle\bar{u}u\rangle
m_{K_{1}}^{2}+\frac{64}{9}\pi\alpha_{s}\langle\bar{u}u\rangle\langle\bar{s}s\rangle.$
(35)

Hence, we obtain $m_{K_{1}}^{2}=m_{K^{*}}^{2}$ when chiral symmetry is
restored. We expect that the chiral order parameter changes by $30\%$ in
nuclear matter, which means that there will be a non-trivial change in the
mass difference between $m_{K_{1}}$ and $m_{K^{*}}$. In a detailed QCD sum
rule analysis in Ref. Song (2018), it was found that the mass shift in the
nuclear matter will be maximally -249 (-35) MeV for $K_{1}^{-}$ ($K_{1}^{+}$).

##  VI Phenomenological observations

The decay channel that is dominant for the $K^{*}$ and $K_{1}$ mesons are
given in table 3. Using the coupling, we find that both $K^{*}$ ($K_{1}$) can
be produced by a kaon beam via the $\pi$ ($\rho$) exchange with a nucleon.

$1^{-}$ | Decay Mode | $1^{+}$ | Decay mode
---|---|---|---
$K^{*}(892)$ |  $K\pi$ (100%) | $K_{1}(1270)$ |  $K\rho$ (42%)
|  |  |  $K^{*}\pi$ (16%)
Table 3: Dominant hadronic decay channels of $K^{*}$ and $K_{1}$ meson.

There are different charge states for the $K^{*}$ and $K_{1}$. The chiral
partners are between the same charge states. Therefore when a $K^{-}$ beam is
used on a nuclear target at the JPARC facility, for example, the produced
these particles and their final states for $K_{1}$ will be as follows.

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

and for $K^{*}$ by

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

Similar production of both $K^{*},K_{1}$ can be achieved by a pion beam at
GSIParyev (2020).

The degeneracy between $K^{*}$ and $K_{1}$ mass when chiral symmetry is
restored can also be probed in a relativistic heavy ion collisionSung (2021).
This is so because, at the initial stages of the collision, the hadronic phase
will undergo a phase transition to the quark-gluon plasma state, where chiral
symmetry is expected to be restored. As the system cools down, one will cross
the hadronization point where hadrons are formed: statistical hadronization
model (SHM) predicts the hadronization temperature at LHC energies to be
around 156 MeVStachel:2013zma . At this point, however, the order parameter is
still quite smallDing:2013lfa . As the mass difference depends only on the
chiral order parameter, the masses between chiral partners will be similar. It
is well-known that particle production follows the statistical hadronization
model (SHM). The prediction for the individual particles will depend on the
details of the model that is determined by the system size, strangeness
enhancement, and whether the system should be treated as a canonical or grand
canonical ensemble. However, $K^{*}$ and $K_{1}$ have the same number of
strangeness and their production ratio will be similar when their masses
become degenerate. Hence we expect that the initial number of $K^{*}$ and
$K_{1}$ at the chemical hadronization point will be similar. One still has to
consider the possible changes in their number as the systems go through the
hadronic phaseCho (2015). This effect is called the hadronic rescattering
effect, which typically depends on the vacuum width of the hadron of interest.
Fortunately, here again, the small and similar vacuum width between $K_{1}$
and $K^{*}$ will lead to a similar reduction so that the ratio can be
estimated perturbatively. If the vacuum masses are used in the SHM, the
expected particle ratio between $K_{1}$ and $K^{*}$ will be very small due to
the much larger mass of $K_{1}$ in the vacuum. The anomalously large ratio is
more enhanced in a peripheral collision, as the hadronic lifetime will be
shorter there. Therefore measuring the production of both the $K_{1}$ and
$K^{*}$ from heavy ion collision in both central and peripheral collisions and
then comparing the observed production ratios to those obtained in the SHM
with vacuum masses will provide a signature for chiral symmetry restoration in
heavy ion collisions.

##  VII Conclusions

Understanding the origin of the masses of hadrons is a key topic of interest.
It is believed that chiral symmetry breaking is partly responsible for the
generation of the hadron mass in QCD. Here, using OPE perspective, we
discussed how one can isolate the effects of chiral symmetry breaking in the
hadron mass. While chiral symmetry is expected to be resorted in the medium,
there are difficulties in measuring the mass shift and isolating the effects
of chiral symmetry breaking. We have used OPE perspective to analyze the other
effects. We have further emphasized that to isolate the chiral symmetry-
breaking effects, one has to start by measuring the mass differences between
chiral partners. Finally, we showed that experimentally measuring the mass
shift between $K^{*}$ and $K_{1}$ mass should be the first priority.

###### Acknowledgements.

I would like to thank Jisu Kim for collaborating on many works cited in the
references. This work was supported by Samsung Science and Technology
Foundation under Project Number SSTF-BA1901-04.

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