# The masses of hadrons in the chiral symmetry restored vacuum

Jisu Kim<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

###### Abstract

We calculate the masses of the vector and axial vector mesons as well as the
nucleon and the delta resonance in the chiral symmetry restored vacuum. This
is accomplished by separating the quark operators appearing in the QCD sum
rules for these hadrons into the chiral symmetric and symmetry breaking parts
depending on the contributions of the fermion zero modes. We then extract the
vacuum expectation values of all the separated parts of the quark operators
using the QCD sum rule relations for these hadrons with their vacuum masses
and widths. By taking the chiral symmetry breaking parts to be zero while
keeping the symmetric operators to their vacuum values, we obtain the chiral
symmetric part of the hadron masses. We find that the masses of chiral
partners, such as the $(\rho,a_{1})$ and $(K^{*},K_{1})$, become degenerate to
values between 500 and 600 MeV in the chiral symmetry restored vacuum, while
parity partners $(\omega,f_{1})$ that are chiral partners only in the limit
where the disconnected diagrams are neglected remain non-degenerate with
masses $(655,1060)$ MeV, respectively. The masses of the nucleon and the Delta
are also found to reduce to about 500 and 600 MeV, respectively, in the chiral
symmetric vacuum. This shows that while chiral symmetry breaking is
responsible for the mass difference between chiral partner, both the meson and
baryon retain non-trivial fraction of their masses in the chiral symmetry
restored vacuum.

##  I Introduction

Understanding the generation of the masses of hadrons that are larger than
several hundred MeV starting from the current quark masses of less than 10 MeV
is one of the fundamental problems in QCDWilczek:1999be ; Wilczek:2012sb . It
is believed that spontaneous chiral symmetry breakingNambu:1961tp ;
Nambu:1961fr  is partly responsible for the generation of the masses
Hatsuda:1985eb ; Brown:1991kk ; Hatsuda:1991ez ; Leupold:2009kz .

Experiments have been performed worldwide to observe mass shift of hadrons at
finite temperature or density Hayano:2008vn ; Metag:2017yuh ; Ichikawa:2018woh
; Ohnishi:2019cif ; Salabura:2020tou , as chiral symmetry is expected to be
fully and partially restored in the initial states of the relativistic heavy
ion collision and in the nuclear matter probed by nuclear target experiment,
respectively.

It is clear that chiral symmetry breaking is responsible for the mass
difference between chiral partnersWeinberg:1967kj , so that if chiral symmetry
is restored, the mass difference between chiral partners should
vanishSong:2018plu ; Lee:2019tvt . However, how much of the total hadron mass
comes from chiral symmetry breaking is still to be understood.

Recently, we have shown that the masses of the $\rho$ and $a_{1}$ mesons,
which form chiral partners, become degenerate to about 550 to 600 MeV in the
chiral symmetry restored vacuumKim:2020zae . One can can isolate chiral
symmetry breaking effect by separating the quark operators, appearing in the
operator product expansion (OPE) of their correlation functions, into the
chiral symmetric and breaking parts through their dependencies on the fermion
zero mode, which can be directly related to the chiral order parameterBC . As
the four and six-quark operators are separated according to the contribution
of the zero modes, one notes that the chiral symmetry breaking parts are order
parameters of chiral symmetry breaking while the symmetric parts obtain non-
vanishing values from chiral symmetric non-perturbative contributions in
QCDKim:2020zae . Originally, in the in-medium QCD sum rules for the light
vector mesonsHatsuda:1991ez , the four-quark condensates were assumed to be
proportional to the quark condensate square through the vacuum saturation
hypothesisShifman:1978bx  so that they vanished automatically when chiral
symmetry is restored. However, it was found that the chiral symmetric part of
the four-quark condensate is as large as the chiral symmetry breaking part and
that the magnitude of the chiral symmetry breaking part follows that obtained
from using the vacuum saturation hypothesisKim:2020zae . This also explains
why in the previous QCD sum rules, the four-quark operators were multiplied by
a $\kappa$ factor larger than 1 after vacuum saturation hypothesis to
correctly obtain the hadron mass from the sum rule analysisLeupold:1998bt ;
Leupold:2001hj .

In this work, we generalize the the work on ($\rho,a_{1}$)Kim:2020zae  to
study the masses of other chiral partners ($K^{*}(892),K_{1}(1270)$) and
parity partners ($\omega(782)$, $f_{1}(1285)$) as well as the nucleon and the
Delta resonance in the chiral symmetry restored vacuum. In section II, we lay
out the formalism and the analysis method. In section III, we apply the
formalism to the meson sector. The baryons are analyzed in section IV. Section
V summarizes our work. Finally, in the appendix, we provide detailed
explanation of how to separate the quark operators into the chiral symmetric
and breaking parts. We further provide explicit forms for the chiral symmetric
and breaking quark operators appearing in the meson and baryon sum rules and
show relevant relations based on Fiertz transformations.

##  II Sum rules analysis

In this section, we provide an overview of the analysis method used in this
work. We start with the relevant correlation function between currents with
the quantum numbers of the hadrons to study.

$\displaystyle\Pi(q)$ $\displaystyle=$ $\displaystyle i\int
d^{4}x\;e^{iqx}\langle T\\{j^{\Gamma}(x)J^{\Gamma}(0)\\}\rangle,$ (1)

where the subscript $\Gamma$ specifies the current of interest. We then study
the Borel transformed dispersion relation for the invariant part of the
correlation function.

$\displaystyle\widehat{\Pi}(M^{2})=\int_{0}^{\infty}ds\;e^{-s/M^{2}}\rho(s),$
(2)

where $\widehat{\Pi}(M^{2})$ represents the Borel transformed OPE of the
correlator $\Pi$, and $M$ stands for the Borel mass. $\rho(s)$ is the spectral
function for the hadron of interest.

###  II.1 Phenomenological Side

The phenomenological side is constructed using the following form of the
spectral density function $\rho(s)$.

$\begin{split}\rho(s)&=\rho^{\text{pole}}(s)+\rho^{\text{cont}}(s),\\\
\rho^{\mathrm{pole}}(s)&=\frac{1}{\pi}\frac{f\Gamma\sqrt{s}}{(s-m^{2})^{2}+s\Gamma^{2}},\\\
\rho^{\mathrm{cont}}(s)&=\frac{1}{\pi}\theta(s-s_{0})\mathrm{Im}\tilde{\Pi}^{\mathrm{pert}}(s),\end{split}$
(3)

where $m$($\Gamma$) is the Breit-Wigner mass(width) of a hadron and $s_{0}$ is
the threshold parameter. The threshold parameter is chosen to make the Borel
curve flattest. In table 1, we summarize the masses and widths of hadrons we
study and the corresponding interpolating currents used. These parameters with
the spectral density will be used in Eq. (2) to obtain the values of the four
quark operators appearing in the operator product expansion (OPE) up to
dimension-6 operators. In a previous work, using such a method, we were able
to obtain the chiral symmetry breaking and symmetric operators appearing in
the $\rho-a_{1}$ sum rulesKim:2020zae . Here, we will generalize the method to
other hadrons.

Particle | Mass(MeV) | Width(MeV) | Interpolating Current
---|---|---|---
$\rho$ | 775.26 | 149.1 | $\bar{q}\tau^{3}\gamma_{\mu}q$
$a_{1}$ | 1230 | 425 | $\eta_{\mu\nu}\bar{q}\tau^{3}\gamma_{\nu}\gamma_{5}q$
$\omega$ | 782.65 | 8.49 | $\bar{q}\gamma_{\mu}q$
$f_{1}$ | 1281.9 | 22.7 | $\eta_{\mu\nu}\bar{q}\gamma_{\nu}\gamma_{5}q$
$K^{*}$ | 895.81 | 47.4 | $\bar{u}\gamma_{\mu}s$
$K_{1}$ | 1272 | 90 | $\bar{u}\gamma_{\mu}\gamma_{5}s$
$N$ | 938 | 0 | $\epsilon_{abc}(u_{a}C\gamma_{\mu}u_{b})\gamma_{5}\gamma_{\mu}d_{c}$
$\Delta$ | 1232 | 117 | $\epsilon_{abc}(u_{a}C\gamma_{\mu}u_{b})u_{c}$
Table 1: The mass and width for particles used in Eq. (3). The last column
shows the interpolating current used for each particle. $q$ stands for the
isospin doublet and $\eta_{\mu\nu}=q_{\mu}q_{\nu}/q^{2}-g_{\mu\nu}$.

###  II.2 OPE Side

In the original QCD sum rules for light hadrons, the OPE were calculated up to
dimension-6 four quark operators. Then the vacuum saturation approximations
were used to estimate the vacuum expectation values for these four-quark
operators. The total OPE were then used in the Borel transformed sum rules to
estimate the mass of the particle. Unfortunately, factorization assumption is
valid only in the large $N_{c}$ limit and in fact a multiplicative $\kappa$
were often introduce to better reproduce the hadron mass. On the other hand,
if the phenomenological side is modelled with physical observables, one can
use the sum rule to estimate the values of the four-quark operators. The four-
quark operators can then be separated into chiral symmetry breaking and
symmetric parts depending on the contribution of the zero modes.

According to the Banks-Casher formulaBC ; Cohen:1996ng , the chiral order
parameter is proportional to the density of zero eigenvalues in the Euclidean
formalism.

$\begin{split}\langle\bar{q}q\rangle&=\lim_{x\rightarrow
0}-\frac{1}{2}\langle{\rm Tr}[S(0,x)-i\gamma_{5}S(0,x)i\gamma_{5}]\rangle\\\
&=-\pi\langle\rho(\lambda=0)\rangle.\end{split}$ (4)

In a previous work, we showed that the four-quark operator can also be divided
according to the contribution of these zero modesLee:2019tvt ; Kim:2020zae .
The chiral symmetry breaking four quark operators are proportional to the zero
modes while symmetric operator has no contribution from them. In other words,
while the former can be considered as chiral order parameters, the latter
originates from other vacuum structure associated with non-zero eigenvalues.
Therefore, when the zero mode contributions are taken away, chiral symmetry
breaking is restored and the breaking operators vanish while the symmetric
operators remain the same.

Using our prescription for dividing the four quark operators, one can
determine the chiral symmetry breaking part and chiral symmetric part of the
four quark operators appearing as two independent combinations in the sum
rules of chiral partners, such as the $\rho,a_{1}$, and then use the two sum
rules to determine the two values separately. This allows us to estimate the
mass of of vector meson mass by taking the chiral symmetry breaking operator
to zero while keeping the chiral symmetric four quark operator to its vacuum
value and studying the sum rule for the common $\rho,a_{1}$ mass.

The situation is more complicated for the $K^{*},K_{1}$ or the $\omega,f_{1}$
cases. For $K^{*},K_{1}$, they form SU(2) chiral partners, so that the chiral
symmetry breaking operator proportional to both the strange and light quark
condensate can be identified unambiguously. Still, to obtain their mass in the
chiral symmetry restored vacuum, one should use the information about the
chiral symmetry breaking operator obtained in the $\rho,a_{1}$ sum rules. The
mass at the chiral symmetry restored vacuum can then be estimated by taking
all the chiral symmetry breaking operators to zero. As for the $\omega,f_{1}$,
they form chiral partners only when contributions from disconnected diagrams
are neglected. As we will see, combining the results from the $\rho,a_{1}$ sum
rule, we can estimate the magnitude of these operators and estimate the
contribution of chiral symmetry breaking effects on the $\omega,f_{1}$ masses.
For the nucleon and delta masses, we will use results from the meson sum rules
on the four quark operators and try to extract the values for the dimension-8
operators.

A way to parametrize the the sum of the chiral symmetry breaking and symmetric
four quark operator appearing in each sum rules is by introducing an auxiliary
parameter $\kappa$ multiplying the vacuum saturation value of the chiral
breaking operators so that a $\kappa$ value close to 1 means small
contribution from the chiral symmetric operators.

The input values for the low dimensional OPE terms are given in Table 2. We
use the quark condensate value evaluated from the GMOR relation,

$\begin{split}m_{\pi}^{2}f_{\pi}^{2}=-2m_{q}\expectationvalue{\bar{u}u},\end{split}$
(5)

with the values ($m_{\pi}$ = 137.5 MeV, $f_{\pi}$ = 93 MeV, $\bar{m}_{ud}$ =
3.45 $\times$ 1.35 MeV). $m_{\pi}$ and $f_{\pi}$ are the mass and decay
constant of the pion respectively. $m_{q}$ is the averaged mass of $u$ and $d$
quarks. We take the values of $m_{q}$ and $m_{s}$ scaled from 2 GeV to 1 GeV,
as reported in Particle Data GroupParticleDataGroup:2020ssz , implemented here
through the factor of 1.35 multiplying the quark mass. Also, we take the
values of the other sum rule parameters to be values that are frequently used
in QCD sum rule studies. The other parameters are standard values taken from
Hatsuda:1992bv ; Ioffe:1983ju

$m_{q}$ | 3.45 $\times$ 1.35 MeV
---|---
$m_{s}$ | 93 $\times$ 1.35 MeV
$\expectationvalue{\bar{u}u}$ | (0.260 GeV)3
$\expectationvalue{(\alpha_{s}/\pi)G^{2}}$ | 0.012 GeV4
$\alpha_{s}$ | 0.36
$\expectationvalue{\bar{s}s}/\expectationvalue{\bar{u}u}$ | 0.8
$m_{0}^{2}$ | 0.8 $\mathrm{GeV}^{2}$
$\mu$ | 0.5 GeV
$\Lambda_{\mathrm{QCD}}$ | 0.1 GeV
$\expectationvalue{B_{su}}_{B}$/$\expectationvalue{B_{uu}}_{B}$ | 0.565(*)
$\expectationvalue{B_{uu}}_{B}$ |  $(0.3053\;\mathrm{GeV})^{6}$(*)
$\expectationvalue{B_{su}}_{B}$ |  $(0.2776\;\mathrm{GeV})^{6}$(*)
$\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S}$ |  $(0.2826\;\mathrm{GeV})^{6}$(*)
$\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}}_{dis,S}$ |  $-(0.3369\;\mathrm{GeV})^{6}$(*)
Table 2: The upper part shows the OPE parameters used in the sum rule. Lower
part marked with a star marker(*) shows values calculated from the sum rules.
$m^{2}_{0}$ stands for
$g_{s}\expectationvalue{\bar{u}\sigma_{\mu\nu}G^{\mu\nu}_{a}\lambda^{a}u}/\expectationvalue{\bar{u}u}$.
All the values here are obtained with assuming $a_{1}$ width to be 425 MeV.

###  II.3 Borel Window

We now discuss how the Borel windows are obtained in the sum rules. The
extracted sum rule is obtained by setting the phenomenological side
$\widehat{\Pi}^{\mathrm{phen.}}(M^{2})$ equal to the OPE side
$\widehat{\Pi}^{\mathrm{OPE}}(M^{2})$. Each side can be defined as follows.

$\begin{split}\widehat{\Pi}^{\mathrm{phen.}}(M^{2})=&\int^{\infty}_{0}ds\;e^{-s/M^{2}}(\rho^{\mathrm{pole}}(s)+\rho^{\mathrm{cont}}(s)),\\\
\widehat{\Pi}^{\mathrm{OPE}}(M^{2})=&\sum_{j=d_{l}}^{d_{h}}\frac{C_{j}\expectationvalue{O_{j}}}{(M^{2})^{j}},\end{split}$
(6)

where $O_{j}$ and $C_{j}$ are the operator and the corresponding Wilson
coefficient, respectively, and $d_{l}$($d_{h}$) is the lowest(highest) power
of $(M^{2})^{-1}$ considered. As we truncate the OPE, the asymptotic expansion
will break down at small Borel mass region. On the other hand, the
approximation of the continuum approximation and the pole dominance require
the Borel mass to sufficiently small. The constraints give the acceptable
Borel mass range called Borel window. The details are as follows.

In the phenomenological side, one has to make sure that the contribution from
the pole $\rho^{\mathrm{pole}}(s)$ dominates over that from the continuum
$\rho^{\mathrm{cont}}(s)$. The constraint is given by

$\begin{split}\frac{\int_{s_{0}}^{\infty}ds\;e^{-s/M^{2}}\rho^{\mathrm{cont}}(s)}{\int_{0}^{\infty}ds\;e^{-s/M^{2}}\rho^{\mathrm{pole}}(s)}<x_{\mathrm{max}},\end{split}$
(7)

where $x_{\mathrm{max}}$ is a number smaller than 1 and adjusted for the sum
rule for each hadron. This condition determines the upper boundary
$M_{\mathrm{max}}$ in the Borel window.

The other comes from requiring the contribution of the condensate terms to be
smaller than that of the perturbative term. This constraint is obtained by

$\begin{split}\frac{\widehat{\Pi}^{\mathrm{OPE}}_{\mathrm{cond\;terms}}(M^{2})}{\widehat{\Pi}^{\mathrm{OPE}}_{\mathrm{pert\;term}}(M^{2})}<x_{\mathrm{min}},\end{split}$
(8)

where $\widehat{\Pi}^{\mathrm{OPE}}_{\mathrm{cond\;terms}}(M^{2})$ and
$\widehat{\Pi}^{\mathrm{OPE}}_{\mathrm{pert\;term}}(M^{2})$ are the sum of the
condensate terms considered and the perturbative term in OPE side,
respectively. This condition restricts the Borel mass $M$ to be larger than
$M_{\mathrm{min}}$.

The values of $x_{\mathrm{max}}$ and $x_{\mathrm{min}}$ are chosen to make the
extreme point of the Borel curve stable against changes in the threshold
within the Borel window. The chosen values for different hadrons are given in
Table 3, which will also be used in the sum rule analysis in the chiral
symmetry restored vacuum. With these, the ranges of Borel window for each
hadron can be determined. Once the Borel range is determined for a given
hadron, the same range will be used to determine the mass in the chiral
symmetry restored vacuum. In the following Borel curves presented in this
work, Borel curves within(outside) the Borel window are drawn in a
solid(dotted) line.

###  II.4 Analysis

To eliminate the $f$-dependence in $\rho^{\mathrm{pole}}(s)$ shown in Eq (3),
we conduct an analysis using the ratio between the extracted sum rule and its
derivative with respect to $-1/M^{2}$. Namely,

$\begin{split}\frac{\frac{d}{d(-1/M^{2})}\widehat{\Pi}^{\mathrm{phen.}}(M^{2})}{\widehat{\Pi}^{\mathrm{phen.}}(M^{2})}=\frac{\int^{\infty}_{0}ds\;se^{-s/M^{2}}\rho(s)}{\int^{\infty}_{0}ds\;e^{-s/M^{2}}\rho(s)}.\end{split}$
(9)

Apart from the Borel mass $M$ dependence, Eq (9) depends on the hadron mass
$m$, the decay width $\Gamma$ and the threshold parameter $s_{0}$. Meanwhile,
its corresponding OPE side

$\begin{split}\frac{\frac{d}{d(-1/M^{2})}\widehat{\Pi}^{\mathrm{OPE}}(M^{2})}{\widehat{\Pi}^{\mathrm{OPE}}(M^{2})}=\frac{-\sum_{j=d_{l}}^{d_{h}}jC_{j}\expectationvalue{O_{j}}/(M^{2})^{j-1}}{\sum_{j=d_{l}}^{d_{h}}C_{j}\expectationvalue{O_{j}}/(M^{2})^{j}}\end{split}$
(10)

depends on the auxiliary parameter $\kappa$ for its dimension-6 operator term.
As explained previously, the mass and decay width values in Table 1 are used.
Equating Eq. (9) to Eq. (10), we obtain the Borel curve for $\kappa$. We then
take its value at the extremum point. Since the $\kappa$ value at the extremum
point varies with $s_{0}$ value, we take the value that reproduce the flattest
and stable mass Borel curve. In summary, the Borel sum rule we will use
throughout the paper is given by

$\begin{split}\frac{\int^{\infty}_{0}ds\;s\;e^{-s/M^{2}}\rho(s)}{\int^{\infty}_{0}ds\;e^{-s/M^{2}}\rho(s)}=\frac{-\sum_{j=d_{l}}^{d_{h}}jC_{j}\expectationvalue{O_{j}}/(M^{2})^{j-1}}{\sum_{j=d_{l}}^{d_{h}}C_{j}\expectationvalue{O_{j}}/(M^{2})^{j}}.\end{split}$
(11)

Once the $\kappa$ value is determined, the chiral symmetry breaking and
symmetric quark operators can be evaluated by combing the sum rules of chiral
partners. By taking chiral symmetric contribution only, one can get the mass
sum rule in chiral symmetry restored vacuum. By taking the $s_{0}$ value that
produce the flattest and stable mass Borel curve, the mass can be extracted.
more details are discussed in the following subsections for different hadrons.

We assume that in the symmetry restored phase, the decay widths for baryon the
Delta, $\omega$ and $f_{1}$ remain the same, while for the chiral partners
$K^{*}$-$K_{1}$ and $\rho$-$a_{1}$, they become that of the corresponding
vector mesons. Our previous workKim:2020zae  shows that changing the width in
the chiral symmetric vacuum, affects the symmetric mass $m_{sym}^{\rho-a_{1}}$
insignificantly, at most 20 MeV. This amount is less than 10% of the total
mass decrease. Furthermore, since the widths of other hadrons we considered
are smaller than that of $\rho$, the effects coming from using different
widths are also expected to be small.

| $\rho$ | $a_{1}$ | $\omega$ | $f_{1}$ | $K^{*}$ | $K_{1}$ | $N$ | $\Delta$
---|---|---|---|---|---|---|---|---
$x_{\mathrm{min}}$ | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.2 | 0.28
$x_{\mathrm{max}}$ | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.75 | 0.96
Table 3:  Values for $x_{\mathrm{max}}$ and $x_{\mathrm{min}}$ defined in Eq
(7) and (8)

##  III Meson

While both ($\rho$, $a_{1}$) and ($K_{1}$, $K^{*}$) are chiral partners,
($\omega$, $f_{1}$) are not because of the contributions from the disconnected
diagrams. These contributions are not zero as $\omega$($f_{1}$) and
$\rho$($a_{1}$) have different widths suggesting that the differences in the
OPE coming from the disconnected four quark contributions, which vanishes in
the vacuum saturation hypothesis, are not zero. Let us first summarize the
four quark condensate appearing in the $\rho,a_{1}$ sum rules.

Particle | $\kappa(\sqrt{s_{0}}(\text{GeV}))$ | $S/B$ |  $\bar{m}_{sym}$(MeV)
---|---|---|---
$\rho$ | 2.60(1.17) | 0.760 | 572.5 $\pm$ 27.5
$a_{1}$ | 0.76(1.58) | -0.485
$\omega$ | 3.20(1.16) | 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 4: The $\kappa$’s are evaluated using sum rules with the physical values
given in Table 1 and the threshold parameter $s_{0}$ in the brackets. $S/B$
indicates the fractional contribution of the chiral symmetric part $S$ to the
contribution of the chiral symmetry breaking part $B$ for each hadron.
$\bar{m}_{sym}$ is the hadronic mass in the chiral symmetry restored vacuum.
The uncertainties of the masses are due to the ranges of the width of $a_{1}$
meson, which we take from 250 MeV to 600 MeV, for the negative and positive
uncertainty values, respectively. Except for the last column, the central
values are evaluated with the $a_{1}$ width of 425 MeV.

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

The sum rules are obtained by using the interpolating currents given in Table
1 and studying the polarization $\Pi=\Pi^{\mu}_{\mu}/(-3q^{2})$. The
dimension-6 four-quark operators from the OPE sides contributing as
$\mathcal{M}/Q^{6}$ are respectively given as followsHatsuda:1992bv .

$\begin{split}\mathcal{M}_{\rho}=&\;-2\pi\alpha_{s}\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}\tau^{3}q)^{2}}\\\
&-\frac{4\pi\alpha_{s}}{9}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)},\\\
\mathcal{M}_{a_{1}}=&\;-2\pi\alpha_{s}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}}\\\
&-\frac{4\pi\alpha_{s}}{9}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)},\end{split}$
(12)

where $q$ denotes the isospin doublet unless stated otherwise. As mentioned in
the previous section, these can be divided into the chiral symmetric and
breaking operators; details are given in Appendix A. By omitting the common
factor ($-\pi\alpha_{s}$) and introducing the auxiliary parameters
$\kappa_{\rho}$ and $\kappa_{a_{1}}$ for each, the results are obtained as the
below.

$\begin{split}\kappa_{\rho}\frac{448}{81}\expectationvalue{\bar{u}u}^{2}=&\;\frac{28}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{\rho-
a_{1}}}_{S},\\\
-\kappa_{a_{1}}\frac{704}{81}\expectationvalue{\bar{u}u}^{2}=&\;-\frac{44}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{\rho-
a_{1}}}_{S},\end{split}$ (13)

where

$\begin{split}\expectationvalue{B_{uu}}_{B}=&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}d)(\bar{d}\gamma_{\mu}\gamma_{5}\lambda^{a}u)}\\\
&\;\;\;\;\;-\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}d)(\bar{d}\gamma_{\mu}\lambda^{a}u)}\bigg{)}_{B},\\\
\expectationvalue{S_{\rho-
a_{1}}}_{S}=&\frac{11}{9}\bigg{(}\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}\tau^{3}q)^{2}}+\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}}\bigg{)}_{S}\\\
&+\frac{4}{9}\bigg{(}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}}-\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}\tau^{3}q)^{2}}\bigg{)}_{S}\\\
&+\frac{4}{9}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)(\bar{s}\gamma_{\mu}\lambda^{a}s)}_{S}.\end{split}$
(14)

Since the symmetric part is identical while the breaking operator $B_{uu}$
contribute with different coefficients, one can identify the breaking and
symmetric part uniquely by optimizing both sum rules with physical quantities.
By taking the breaking part of the operator to be zero while keeping the
symmetric operator to its vacuum value, one can obtain the sum rules in the
chiral symmetry restored vacuum. The results for the matrix elements and the
masses in the chiral symmetry restored vacuum are given in Table 4. The
results for $\rho-a_{1}$ in Table 4 are obtained with the $a_{1}$ width of 425
MeV. Since there are still large uncertainty in the $a_{1}$ width
$\Gamma_{a_{1}}$, we take a larger value. For $\Gamma_{a_{1}}=250(600)$ MeV,
one finds that $m_{sym}^{\rho-a_{1}}$ decreases (increases) by about 27.5 MeV.

###  III.2 $\omega$ and $f_{1}$

The only difference between the OPE sides of $\rho$($a_{1}$) and
$\omega$($f_{1}$) is the disconnected pieceGubler:2016djf .

$\begin{split}\Pi^{\mathrm{OPE}}_{\omega}-\Pi^{\mathrm{OPE}}_{\rho}&=-\frac{2\pi\alpha_{s}}{Q^{6}}\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S},\\\
\Pi^{\mathrm{OPE}}_{f_{1}}-\Pi^{\mathrm{OPE}}_{a_{1}}&=-\frac{2\pi\alpha_{s}}{Q^{6}}\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)}_{dis,S},\end{split}$
(15)

where the two-point correlators $\Pi^{\mathrm{OPE}}_{\omega}$ and
$\Pi^{\mathrm{OPE}}_{f_{1}}$ come from the interpolating currents
$J_{\mu}^{\omega}$ and $J_{\mu}^{f_{1}}$ in Table 1. These disconnected
operators are chiral symmetric and are mainly responsible for the differences
in their masses and widths. As the chiral symmetry operators do not depend on
chiral symmetry breaking effects, the masses of $\omega$ and $f_{1}$ will be
non-degenerate in the chiral symmetry restored vacuum. With the auxiliary
parameter $\kappa$, Eq. (15), apart from (-$\pi\alpha_{s}/Q^{6}$) can be
rewritten as below.

$\begin{split}(\kappa_{\omega}-\kappa_{\rho})\frac{448}{81}\expectationvalue{\bar{u}u}^{2}&=2\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S},\\\
-(\kappa_{f_{1}}-\kappa_{a_{1}})\frac{704}{81}\expectationvalue{\bar{u}u}^{2}&=2\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)}_{dis,S}.\end{split}$
(16)

Figure 1: Borel curves for four-quark operators parametrized by $\kappa$ for
$\omega$(circle) and $f_{1}$(square). The optimal $\kappa$ is taken to be the
value at the extremum.

The $\kappa$ values can be obtained from the values for the four-quark
operators, which are calculated by substituting the phenomenological side
given in Eq. (3) into Eq. (2) and studying the sum rule for each current.
$\kappa_{\rho}$ and $\kappa_{a_{1}}$ given in Table 4 are obtained in the sum
rule calculation presented in Kim:2020zae . Here the values for
$\kappa_{\rho}$ and $\kappa_{a_{1}}$ are slightly different from those given
in Kim:2020zae  as we use updated values for the quark condensate and mass.
Substituting the spectral density with physical parameters for $\omega$ and
$f_{1}$ into the sum rules, we obtain the corresponding $\kappa$ values as
given in Table 4. Fig.1 shows the $\kappa$ values for $\omega$ and $f_{1}$
meson. We take the $\kappa$ value at the extremum point of the Borel curve.
One can also use the Borel curve for the masses and try to extract the value
for the four-quark operators that best reproduces the $\omega$ and $f_{1}$
mass with their given width. As can be seen in the open circle and square plot
in Fig. 2 the extracted value for the four-quark operators gives the most
stable Borel curve for the mass of $\omega$ and $f_{1}$ mass, respectively.
Using the values of $\kappa$, the values for the disconnected contribution of
the four-quark condensates are given as below.

$\begin{split}\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S}=1.65\expectationvalue{(\bar{u}u)}^{2},\\\
\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}}_{dis,S}=-4.73\expectationvalue{(\bar{u}u)}^{2}.\end{split}$
(17)

The disconnected contributions vanish in the vacuum saturation hypothesis. Our
result in Eq. (17) shows that while such assumptions are approximately valid
in some channels, they are largely violated in some other channels. Assuming
that the disconnected diagrams are mediated by gluon fields, one notes
$\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S}/\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}}_{dis,S}\sim(\alpha_{s})^{2}$
suggesting that estimated values in Eq. (17) are consistent with the
$\alpha_{s}$ counting.

The magnitude of dimension-6 part for $\omega$($f_{1}$) consists of the
disconnected piece and the dimension-6 part of $\rho$($a_{1}$). Now, using the
identification discussed in Appendix A, one notes that the disconnected four-
quark operators appearing in Eq. (15) are chiral symmetric operators.
Therefore, the mass of the $\omega$ and $f_{1}$ mesons in the chiral symmetry
restored vacuum can be obtained by taking the chiral symmetry breaking
operators, which appear in the $\rho$ and $a_{1}$ channels, to zero. Then, as
can be seen in Fig.2 through the Borel curve, one finds that when the symmetry
is restored, the $\omega$ mass becomes 650 MeV, similar to the $\rho-a_{1}$
mass, while the $f_{1}$ mass is 1060 MeV.

Figure 2: The Borel curves for $\omega$(open circle) and $f_{1}$(square)
masses are with each $\kappa$ value in TABLE 4. The Borel curve with
cross(diamond) markers describes the $\omega$($f_{1}$) mass in the chiral
symmetry restored vacuum

###  III.3 $K_{1}$ and $K^{*}$

Unlike the $\omega,f_{1}$ pair, $K^{*}$ and $K_{1}$ form chiral partners, as
is evident when analyzing the operators as given in Appendix A. Hence, when
the chiral symmetry gets restored, the masses become degenerate. As
represented in Song:2018plu , the dimension-6 operators, after omitting the
common factor $-\frac{2\pi\alpha_{s}}{Q^{6}}$ in the OPE, are given as below.

$\begin{split}\mathcal{M}_{K^{*}}=&\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}s)(\bar{s}\gamma_{\mu}\gamma_{5}\lambda^{a}u)}\\\
&+\frac{1}{9}\expectationvalue{(\bar{s}\gamma_{\mu}\lambda^{a}s+\bar{u}\gamma_{\mu}\lambda^{a}u)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)},\\\
\mathcal{M}_{K_{1}}=&\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}s)(\bar{s}\gamma_{\mu}\lambda^{a}u)}\\\
&+\frac{1}{9}\expectationvalue{(\bar{s}\gamma_{\mu}\lambda^{a}s+\bar{u}\gamma_{\mu}\lambda^{a}u)(\sum_{q=u,d,s}\bar{q}\gamma_{\mu}\lambda^{a}q)}.\end{split}$
(18)

One can divide these operators according to chiral symmetric and breaking
parts, shown by the subscripts $S$ and $B$, respectively.

$\begin{split}\mathcal{M}_{K^{*}}=&\expectationvalue{B_{su}}_{B}-\frac{1}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{K}}_{S},\\\
\mathcal{M}_{K_{1}}=&-\expectationvalue{B_{su}}_{B}-\frac{1}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{K}}_{S},\end{split}$
(19)

where

$\begin{split}\expectationvalue{B_{su}}_{B}&=\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}s)(\bar{s}\gamma_{\mu}\gamma_{5}\lambda^{a}u)-(\bar{u}\gamma_{\mu}\lambda^{a}s)(\bar{s}\gamma_{\mu}\lambda^{a}u)}_{B},\\\
\expectationvalue{B_{uu}}_{B}&=\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}d)(\bar{d}\gamma_{\mu}\gamma_{5}\lambda^{a}u)-(\bar{u}\gamma_{\mu}\lambda^{a}d)(\bar{d}\gamma_{\mu}\lambda^{a}u)}_{B},\\\
\expectationvalue{S_{K}}_{S}&=\frac{1}{2}\expectationvalue{(\bar{s}\gamma_{\mu}\gamma_{5}\lambda^{a}u)(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}s)+(\bar{s}\gamma_{\mu}\lambda^{a}u)(\bar{u}\gamma_{\mu}\lambda^{a}s)}_{S}\\\
+&\frac{1}{18}\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}\lambda^{a}d)(\bar{d}\gamma_{\mu}\gamma_{5}\lambda^{a}u)+(\bar{u}\gamma_{\mu}\lambda^{a}d)(\bar{d}\gamma_{\mu}\lambda^{a}u)}_{S}\\\
+&\frac{1}{9}\bigg{(}\langle(\bar{s}\gamma_{\mu}\lambda^{a}s)(\sum_{q=uds}\bar{q}\gamma_{\mu}\lambda^{a}q)\rangle_{S}\\\
&\;\;+\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}u)(\bar{d}\gamma_{\mu}\lambda^{a}d)}_{S}+\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}u)(\bar{s}\gamma_{\mu}\lambda^{a}s)}_{S}\\\
&\;\;+\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}u)^{2}-(\bar{u}\gamma_{\mu}\lambda^{a}d)(\bar{d}\gamma_{\mu}\lambda^{a}u)}_{S}\bigg{)},\end{split}$
(20)

and $S_{K}$ is the symmetric dimension-6 operators for $K^{*}$ and $K_{1}$.
Using the sum rules for the $K^{*}$ and $K_{1}$ mesons one can obtain the
Borel curves and thus determine the magnitudes of the four-quark operators
appearing in these sum rules separately. We parametrize the magnitude of the
obtained matrix elements by identifying them to $\kappa_{K^{*}}$ and
$\kappa_{K_{1}}$ times the vacuum vacuum saturation values, respectively. Eq.
(19) can be rewritten as follows.

$\begin{split}\mathcal{M}_{K^{*}}=&\kappa_{K^{*}}\bigg{(}\frac{16}{9}\expectationvalue{\bar{u}u}\expectationvalue{\bar{s}s}-\frac{16}{81}(\expectationvalue{\bar{u}u}^{2}+\expectationvalue{\bar{s}s}^{2})\bigg{)}\\\
=&\expectationvalue{B_{su}}_{B}-\frac{1}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{K}}_{S},\\\
\mathcal{M}_{K_{1}}=&\kappa_{K_{1}}\bigg{(}-\frac{16}{9}\expectationvalue{\bar{u}u}\expectationvalue{\bar{s}s}-\frac{16}{81}(\expectationvalue{\bar{u}u}^{2}+\expectationvalue{\bar{s}s}^{2})\bigg{)}\\\
=&-\expectationvalue{B_{su}}_{B}-\frac{1}{9}\expectationvalue{B_{uu}}_{B}+\expectationvalue{S_{K}}_{S}.\end{split}$
(21)

Thus, the Borel curves for the four-quark operators become those for the
$\kappa$’s. It should be noted that one can first determine
$\expectationvalue{B_{su}}_{B}$ from the difference between the $K^{*}-K_{1}$
sum rules. As can be seen from Table 5, the value of
$\expectationvalue{B_{su}}_{B}$ thus determined is slightly smaller than
$\expectationvalue{B_{uu}}$ determined from the $\rho$-$a_{1}$ sum rules,
which is what is expected also from the vacuum saturation hypothesis as the
strange quark condensate is expected to be slightly smaller than that of light
quark condensate. Using these values, one can further determine the values for
$\expectationvalue{S_{K}}_{S}$. Fig. 3 shows the Borel curve for the $\kappa$
values for the pair. As in the previous section, the values are taken at the
extremum point and also the threshold parameter ($\sqrt{s})$ values are taken
to make the mass Borel curve flattest and stable. One can use the obtained
values for the four-quark operators and obtain the sum rule for the masses. As
presented in Fig. 4, the Borel curves for the masses of the $K^{*}$(open
circle) and $K_{1}$(square) with the corresponding $\kappa$ value are stable
and reproduce the physical masses.

The chiral symmetric degenerate mass can be extracted by taking the symmetry
breaking operator to be zero and keeping the magnitude of the symmetric
operator $\expectationvalue{S_{K}}$. Although the $\expectationvalue{B_{uu}}$
value used to solve Eq. (19) depends on the $a_{1}$ width, the extracted mass
$m_{sym}^{K^{*}-K_{1}}$ barely changes as one changes its value. One finds
that $m_{sym}^{K^{*}-K_{1}}$ is 545 MeV, slightly smaller than that of $\rho-
a_{1}$. For $\Gamma_{a_{1}}=250(600)$ MeV, one finds that
$m_{sym}^{K^{*}-K_{1}}$ increases (decreases) by about 5 MeV.

Figure 3: Borel curves for four-quark operators parametrized by $\kappa$ in
$K^{*}$(circle) and $K_{1}$(square) sum rules. The optimal $\kappa$ values are
taken from the value at the extremum point.

Figure 4: Borel curves for $K_{1}$ and $K^{*}$ mass in chiral symmetry
breaking(square and circle) and restored(cross) vacuum.

##  IV Baryon

We use the currents given in Table 1. Since the baryon currents couple to both
parity states, the two-point correlator between the baryonic interpolating
current $\eta(x)$ can be written as followsGubler2013 .

$\begin{split}\Pi(q)=&\;i\int
d^{4}x\;e^{iqx}\expectationvalue{T\\{\eta(x)\bar{\eta}(0)\\}}{0}\\\
=&\;\sum_{n}\bigg{\\{}-\absolutevalue{\lambda^{n}_{+}}^{2}\frac{\not{q}+m^{n}_{+}}{q^{2}-(m_{+}^{n})^{2}+i\epsilon}\\\
&\qquad\;\;\;-\absolutevalue{\lambda^{n}_{-}}^{2}\frac{\not{q}-m^{n}_{-}}{q^{2}-(m^{n}_{-})^{2}+i\epsilon}\bigg{\\}}\\\
\equiv&\;\not{q}\,\Pi_{v}(q^{2})+\mathds{1}\,\Pi_{s}(q^{2}),\end{split}$ (22)

where $m_{+}^{n}$ and $m_{-}^{n}$ represent the mass of the $n$-th excited
positive and negative parity states, respectively, and $\lambda_{\pm}^{n}$
their corresponding overlaps with the currents. As two invariant functions
appear, one can obtain two independent sum rules. However, if chiral symmetry
is restored, $\Pi_{s}(q^{2})$ goes to zero as all the operators in the OPE
side are chiral order parameters and the phenomenological side will be
proportional to the mass difference between chiral partners. Therefore we will
use the sum rule from $\Pi_{v}(q^{2})$, from which we can extract the mass
even in the chiral symmetry restored phase as the two chiral partners will
become degenerate and strengthen the lowest pole contribution.

Since we do not have independent sum rules for chiral partners, we can not
determine the values for the chiral symmetric and breaking quark operators
separately as we did in the meson case. On the other hand, as can be seen in
Table 5, the values of the chiral symmetry breaking four-quark operator
extracted from the meson sum rule are very close to that estimated from the
vacuum saturation hypothesis. Therefore, we will assume that the vacuum
saturation hypothesis is accurate for evaluating the chiral breaking
operators, as was the case for four-quark operators appearing in the meson sum
rules.

Operator | SR | VS | Error
---|---|---|---
$\expectationvalue{B_{uu}}_{B}$ | (0.3056 GeV)6 | (0.2862 GeV)6 | 0.482
$\expectationvalue{B_{su}}_{B}$ | (0.2779 GeV)6 | (0.2758 GeV)6 | 0.038
$\expectationvalue{(\bar{q}\gamma_{\mu}\gamma_{5}\lambda^{a}q)^{2}}_{dis,S}$ | (0.2826 GeV)6 | 0 GeV6 |
$\expectationvalue{(\bar{q}\gamma_{\mu}\lambda^{a}q)^{2}}_{dis,S}$ |  $-$(0.3369 GeV)6 | 0 GeV6 |
Table 5:  Values of the four four-quark operators defined in Eq. (20)
evaluated from our sum rule approach (SR) and from vacuum saturation
hypothesis (VS). The error shown in the last column are the difference between
the two divided by VS value.

Furthermore, while the OPE was calculated up to dimension-6 four quark
operators for the meson case, one needs to consider the higher dimensional
four quark gluon mixed operator for the baryons in order to obtain stability
of the sum rules. We extract and parametrize the magnitude of the dimension-6
and dimension-8 operators by $\kappa_{6}$ and $\kappa_{8}$, respectively, as
was done for the dimension-6 operators in the meson sum rules.

In this case, there is one more independent parameter $\kappa_{8}$. In
principle, since the sum rule is a function of Borel mass, we can determine
both parameters by requiring the phenomenological side to be closest to the
OPE side. Technically, we realize the procedure as follows. To determine its
value, we first select a $\kappa_{6}$ value that ranges from -1 to 4, which is
slightly wider than the range of the $\kappa$ values determined in the meson
cases. Once $\kappa_{6}$ value is chosen, the value of the other parameters
$s_{0}$ and $\kappa_{8}$ can be determined by requiring the Borel curve for
the mass to be flattest and stable. Then we change the $\kappa_{6}$ value and
repeat the procedure and compare the Borel mass curves. The final values of
$\kappa_{6}$ and $\kappa_{8}$ are determined from choosing the most stable
Borel mass curve among the final Borel mass curves. It should be noted that
although we set the $\kappa_{6}$ range, the stable baryonic sum rule for the
mass occurs only for a limited value of $\kappa$’s such that as we approach
the lower and upper limit of the range that we chose for $\kappa_{6}$, the sum
rule become far from being stable. In Table 6, we summarize the determined
$\kappa$ values and masses of nucleon and $\Delta$ in the chiral symmetry
restored vacuum.

Particle | $\kappa_{6}(\sqrt{s_{0}}(\text{GeV}))$ | $\kappa_{8}$ | $S_{6}/B_{6}$ | $S_{6}/B_{8}$ |  $\bar{m}_{sym}$(MeV)
---|---|---|---|---|---
$N$ | 0.05(1.24) | 0.25 | -0.95 | -0.75 | 525
$\Delta$ | 0.26(1.56) | 0.4 | -0.74 | -0.6 | 600
Table 6: The $\kappa$s are evaluated using sum rules using the physical values
given in Table 1 and the threshold parameter $s_{0}$ in the brackets. $S/B$
indicates the fractional contribution of the chiral symmetric part to the
contribution of the chiral symmetry breaking part for each hadron.
$\bar{m}_{sym}$ is the hadronic mass in the chiral symmetry restored vacuum.

###  IV.1 Nucleon

We use the sum rule for nucleon constructed from the Ioffe current, known as
the optimal choiceIoffe:1983ju . Parametrizing the dimension-6 and dimension-8
operators, respectively, with $\kappa_{6}^{N}$ and $\kappa_{8}^{N}$,

$\begin{split}\mathcal{M}^{N}_{6}=&\kappa_{6}^{N}\frac{2}{3}\expectationvalue{\bar{u}u}^{2}=\expectationvalue{B_{6}^{N}}_{B}+\expectationvalue{S_{6}^{N}}_{S},\\\
\mathcal{M}^{N}_{8}=&-\kappa_{8}^{N}\frac{m_{0}^{2}}{6M^{2}}\expectationvalue{\bar{u}u}^{2}=\expectationvalue{B_{8}^{N}}_{B}+\expectationvalue{S_{8}^{N}}_{S},\end{split}$
(23)

where the $\mathcal{M}$’s are defined in the Appendix, and
$B^{N}_{n}$($S^{N}_{n}$) is the dimension-$n$ chiral symmetry
breaking(symmetric) operator and given explicitly in Eq. 43 and Eq. 48. Then,
one can write down the Borel-transformed sum rule as follows.

$\begin{split}\lambda^{2}_{N}\exp(-\frac{m_{N}^{2}}{M^{2}})=&\frac{1}{32\pi^{4}}\bigg{(}1+\frac{\alpha_{s}}{\pi}\bigg{[}\frac{53}{12}+\gamma_{E}\bigg{]}\bigg{)}M^{6}E_{2}L^{-4/9}\\\
&+\frac{1}{32\pi^{2}}\expectationvalue{\frac{\alpha_{s}}{\pi}G^{2}}M^{2}E_{0}L^{-4/9}\\\
&+\kappa_{6}^{N}\frac{2}{3}\expectationvalue{\bar{u}u}^{2}L^{4/9}-\kappa_{8}^{N}\frac{1}{6}\expectationvalue{\bar{u}u}^{2}\frac{m^{2}_{0}}{M^{2}},\\\
\end{split}$ (24)

where

$\begin{split}L=&\frac{\ln(M/\Lambda_{\mathrm{qcd}})}{\ln(\mu/\Lambda_{\mathrm{qcd}})},\\\
m_{0}^{2}=&g_{s}\expectationvalue{\bar{u}\sigma_{\mu\nu}G^{\mu\nu}_{a}\lambda^{a}u}/\expectationvalue{\bar{u}u},\\\
E_{n}=&1-e^{s_{0}/M^{2}}\sum_{j=0}^{n}\frac{1}{j!}\bigg{(}\frac{s_{0}}{M^{2}}\bigg{)}^{j},\end{split}$
(25)

and $\gamma_{E}$ is the Euler-Mascheroni constant. We now use Eq. (11) using
the OPE for the nucleon. The $\kappa$ values are determined using the
procedure discussed in the previous subsection. To asses the validity of the
$\kappa$ values independently, we plot the Borel cures for $\kappa_{8}$ using
Eq. (11) with determined $\kappa_{6}^{N}=0.0425$ value. The resulting Borel
curve is given in Fig 5. The figure also shows the Borel curve for
$\kappa_{6}$ with the best fit $\kappa_{8}^{N}=0.1546$ value. Finally, as
shown in Fig. 6, with the $\kappa$ values, the Borel curve for nucleon
mass(square) is stable and well reproduces the physical mass.

Here, to compensate for the absence of the sum rule for the chiral partner, we
use the vacuum saturation hypothesis to estimate the value of the chiral
symmetry breaking contribution. Hence, the difference between the total quark
operators and the corresponding chiral symmetry breaking contribution can be
identified as the contributions from the chiral symmetric operators. Then the
symmetric dimension-6 and dimension-8 contributions are given, respectively,
as below.

Figure 5: Borel curves for nucleon $\kappa_{6}$(triangle and right y-axis) and
$\kappa_{8}$(circle and left y-axis). The curve for one of $\kappa$’s is drawn
with the other $\kappa$ taken to be the extremum value inside the Borel
window. Unit of $\sqrt{s_{0}}$ is in GeV

Figure 6: Borel curves for nucleon mass in chiral symmetry broken(square) and
symmetry restored(star) vacuum.

$\begin{split}\expectationvalue{S_{6}^{N}}_{S}=&(\kappa_{6}^{N}-1)\frac{2}{3}\expectationvalue{\bar{u}u}^{2}L^{4/9},\\\
\expectationvalue{S_{8}^{N}}_{S}=&-(\kappa_{8}^{N}-1)\frac{1}{6}\expectationvalue{\bar{u}u}^{2}\frac{m_{0}^{2}}{M^{2}},\end{split}$
(26)

where $\expectationvalue{S_{n}^{N}}_{S}$ is the contribution from the chiral
symmetric part of the dimension-$n$ operators with $\kappa=1$ in the vacuum
saturation hypothesis. The matrix form of the dimension-6 and dimension-8
operators are given in Eq. (43).

The nucleon mass $m^{N}_{sym}$ in the chiral symmetric vacuum can now be
estimated by taking the breaking contributions to be zero while keeping the
symmetric operators to their vacuum value. As can be seen in lower Borel curve
in Fig. 6, the mass of the nucleon and $N^{*}(1535)$ $m^{N}_{sym}$ converge to
500-550 MeV when the symmetry is totally restored. The Borel curve with a star
marker in Fig. 6 describes the mass, which is also stable.

###  IV.2 $\Delta$ isobar

As mentioned in Ref. PhysRevC.51.2260 ; Marques:2018mic , the current $\eta$
for $\Delta^{++}$ given in Table 1 is a unique choice at lowest dimension but
couples to both the spin 3/2 and 1/2 states. So to extract the spin $3/2$
state, we concentrate on the $g_{\mu\nu}$ tensor structure, which has no
contribution from the spin $1/2$ state. Introducing $\kappa$s as before,

$\begin{split}\mathcal{M}^{\Delta}_{8}=&\kappa_{6}^{\Delta}\frac{4}{3}\expectationvalue{\bar{u}u}^{2}=\expectationvalue{B_{6}^{\Delta}}_{B}+\expectationvalue{S_{6}^{\Delta}}_{S},\\\
\mathcal{M}^{\Delta}_{8}=&\kappa_{8}^{\Delta}\frac{13}{18M^{2}}m_{0}^{2}\expectationvalue{\bar{u}u}^{2}=\expectationvalue{B_{8}^{\Delta}}_{B}+\expectationvalue{S_{8}^{\Delta}}_{S},\end{split}$
(27)

where the $\mathcal{M}$’s are defined in the Appendix, and
$B^{N}_{n}$($S^{N}_{n}$) is the dimension-$n$ chiral symmetry
breaking(symmetric) operator and given explicitly in Eq. (43) and Eq. (48)
Then, the sum rule can be rewritten as follows.

$\begin{split}F_{\Delta}&I^{(0)}_{\Delta}(m_{\Delta},\Gamma_{\Delta},M^{2})=\\\
&-\frac{1}{80\pi^{4}}M^{6}E_{2}L^{4/27}+\frac{5}{288\pi^{2}}M^{2}E_{0}\expectationvalue{\frac{\alpha_{s}}{\pi}G^{2}}\\\
&-\kappa_{6}^{\Delta}\frac{4}{3}\expectationvalue{\bar{u}u}^{2}L^{28/27}+\kappa_{8}^{\Delta}\frac{13}{18M^{2}}m_{0}^{2}\expectationvalue{\bar{u}u}^{2}L^{16/27},\end{split}$
(28)

where

$\begin{split}I_{\Delta}^{(n)}=\int^{s_{0}}_{4m_{\pi}^{2}}e^{-s/M^{2}}\rho^{\mathrm{pole}}(m_{\Delta},\Gamma_{\Delta},s)s^{n}ds.\end{split}$
(29)

Figure 7: Borel curves for $\Delta$ $\kappa_{6}$(triangle) and
$\kappa_{8}$(circle). The curve for one of $\kappa$s is drawn with the other
$\kappa$ taken to be the extremum value. Unit of $\sqrt{s_{0}}$ is in GeV

Figure 8: Borel curves for $\Delta$ mass in the chiral symmetry broken(square)
and symmetry restored(diamond and star) vacuum. the curves with star markers
is the expected $\Delta$ mass. Unit of $\sqrt{s_{0}}$ is in GeV.

We first construct the sum rule to estimate the values of the dimension-6 and
dimension-8 operators parametrized by $\kappa$’s as before. As shown in Fig 7,
the Borel curves for $\kappa_{6}$(triangle) and $\kappa_{8}$(open circle) has
an extremum value at the same point. We can again show that the determined
values appropriately reproduces the $\Delta$ mass in the Borel sum rule. Using
the $\kappa_{6}^{\Delta}=0.26$ and $\kappa_{8}^{\Delta}=0.4$ one finds a
stable Borel curve for the mass as given in the lines marked by squares in
Fig. 8.

As in the nucleon case, using the extracted values dimension-6 and dimension-8
quark operators, one can estimate the values for the corresponding chiral
symmetric operators appearing in the $\Delta$ sum rule.

$\begin{split}\expectationvalue{S_{6}^{\Delta}}_{S}=&-(\kappa_{6}^{\Delta}-1)\frac{4}{3}\expectationvalue{\bar{u}u}^{2}L^{28/27},\\\
\expectationvalue{S_{8}^{\Delta}}_{S}=&(\kappa_{8}^{\Delta}-1)\frac{13}{18M^{2}}m_{0}^{2}\expectationvalue{\bar{u}u}^{2}L^{16/27}.\end{split}$
(30)

By keeping the value of the chiral symmetric operator to be the same as in the
vacuum and taking the breaking contribution to be zero, one can extract the
$\Delta$ mass in the chiral symmetry restored vacuum $m_{sym}^{\Delta}$ from
the Borel sum rule for the mass. One finds that the mass of $\Delta$ in the
symmetry restored vacuum is about 600 MeV.

##  V Summary

We have studied the masses of hadrons in the chiral symmetry restored vacuum.
This is accomplished through first dividing the quark-operators into chiral
symmetry breaking and symmetric operator depending on the contributions from
the quark zero modes, which according to Casher-Banks formula is responsible
for spontaneous chiral symmetry breaking. We then estimated the values of
these operators separately by studying the sum rules between chiral partners,
which have the same contribution from the chiral symmetric operators but where
the difference is proportional to chiral symmetry breaking operators. We have
further estimated the contributions of disconnected quark operators by
studying the sum rule for $\omega$ and $f_{1}$ mesons. We find that the
magnitude of chiral symmetry operators are as big as the chiral symmetry
breaking operators. We further find that the value of chiral symmetry breaking
operators are close to the values obtained in the vacuum saturation
hypothesis, which automatically relates them to the quark condensate. On the
other hand the value of the chiral symmetric operators could be as large as
the breaking operator, which explains the need of effective $\kappa$
parameters to modify the values of the four-quark operators estimated within
the vacuum saturation hypothesis to that necessary to reproduce the meson
masses in the QCD sum rule approach. We then calculated the masses of the
hadrons in the chiral symmetry restored vacuum using the QCD sum rule approach
by taking the chiral symmetry breaking operators to zero while keeping the
values of the other quark operators to their vacuum values.

We also applied our method to the baryon sector. In this case, since the
chiral partners contribute in the same sum rule, we estimated the values of
the dimension-6 and dimension-8 quark operators which contain both the chiral
symmetric and breaking operators. We then used the vacuum saturation
hypothesis to estimate the values of the chiral symmetry breaking operators
from which we were able to estimate the values for the chiral symmetric
operators for both the dimension-6 and dimension-8 operators. Again, taking
the chiral symmetry breaking operators to zero, we find that the values of the
chiral symmetric nucleon and delta masses to be around 500 MeV and 600 MeV,
respectively.

We therefore conclude while chiral symmetry breaking is responsible for the
mass difference between chiral partners, a large fraction of the common masses
have other non-perturbative origins. We further emphasize that our findings
show that the vacuum saturation hypothesis that relates quark operators to
quark condensate only works for the chiral symmetry breaking operators, which
are chiral order parameters as are the quark condensate.

We have shown that masses of all hadron studied in this work tend to decrease
and the masses of chiral partners become degenerate when the chiral symmetry
is restored. Thus the result explicitly demonstrates that the spontaneous
breaking of chiral symmetry responsible for the mass splitting of chiral
partners, but is also partly responsible for common hadronic mass of the
chiral partners. Despite their strangeness, $K_{1}$ and $K^{*}$ have similar
mass as that of $\rho$ and $a_{1}$ in the chiral symmetry restored vacuum.
Meanwhile, the mass difference between $\omega$ and $f_{1}$ is found to remain
large even when chiral symmetry is restored. Both the nucleon and the $\Delta$
mass will reduce to about 500 and 600 MeV, respectively, in the chiral
symmetry restored vacuum.

## Acknowledgements

This work was supported by by Samsung Science and Technology Foundation under
Project Number SSTF-BA1901-04.

## Appendix

###  V.1 Four-quark operators

In general, a four-quark operator can be written as follows.

$\begin{split}\expectationvalue{(\bar{q}\Gamma_{a}q)(\bar{q}\Gamma_{b}q)}=&\expectationvalue{{\rm
Tr}[S\Gamma_{a}]{\rm Tr}[S\Gamma_{b}]}-\expectationvalue{{\rm
Tr}[S\Gamma_{a}S\Gamma_{b}]]},\end{split}$ (31)

where $\Gamma$ are combinations of gamma, color and flavor matrices, and $S$
is the quark propagator. The first term in Eq. (31) is referred to as the
disconnected contribution while the second as the connected.

First, it is easy to check if the disconnected term is chiral symmetric or
breaking operator. This is accomplished by making use of the first line in Eq.
(4) and using the following.

$\begin{split}{\rm Tr}[S\Gamma_{a}]=&\frac{1}{2}\bigg{(}{\rm
Tr}[(S+i\gamma_{5}Si\gamma_{5})\Gamma_{a}]+{\rm
Tr}[(S-i\gamma_{5}Si\gamma_{5})\Gamma_{a}]\bigg{)}.\end{split}$ (32)

The first term is chiral symmetric and the second breaking term. In terms of
the Dirac zero modes, only the second term is proportional to the them.
However, one of them is zero depending on $\Gamma_{a}$. For example, if
$\Gamma_{a}=\gamma_{\mu}$, the breaking term is zero so that the total
disconnect contribution is chiral symmetric. The term is automatically zero if
$\Gamma_{a}$ includes a flavor matrix.

Second, as for the connected contribution, one can decompose the matrix into
chiral symmetric and breaking part. If $\Gamma_{a}$ does not include a flavour
matrix, one can subtract and add a similar matrix with the flavour matrix
inserted so as to isolate the connected contribution.

$\begin{split}\expectationvalue{(\bar{q}\Gamma_{a}q)(\bar{q}\Gamma_{b}q)}=&\bigg{(}\expectationvalue{(\bar{q}\Gamma_{a}q)(\bar{q}\Gamma_{b}q)}-\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}\tau_{i}q)}\bigg{)}\\\
&+\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}\tau_{i}q)},\end{split}$
(33)

where the first line contains only disconnected contribution while the second
line only connected. Therefore, one is left with determining the chiral
property of the connected diagrams. This is accomplished by the following
separation.

$\begin{split}\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}\tau_{i}q)}=&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}q)}-\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}i\gamma_{5}q)(\bar{q}i\gamma_{5}\Gamma_{b}\tau_{i}q)}\bigg{)}+\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}q)}+\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}i\gamma_{5}q)(\bar{q}i\gamma_{5}\Gamma_{b}\tau_{i}q)}\bigg{)}\\\
=&\frac{1}{2}\bigg{(}\expectationvalue{{\rm
Tr}[S\Gamma_{a}\tau_{i}(S-i\gamma_{5}Si\gamma_{5})\Gamma_{b}\tau_{i}]}\bigg{)}+\frac{1}{2}\bigg{(}\expectationvalue{{\rm
Tr}[S\Gamma_{a}\tau_{i}(S+i\gamma_{5}Si\gamma_{5})\Gamma_{b}\tau_{i}]}\bigg{)}\\\
=&\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}\tau_{i}q)}_{B}+\expectationvalue{(\bar{q}\Gamma_{a}\tau_{i}q)(\bar{q}\Gamma_{b}\tau_{i}q)}_{S},\end{split}$
(34)

where the subscript $B$ and $S$ represent the chiral symmetry breaking and
symmetric parts respectively.

###  V.2 Fierz Transformation

Every four-quark operator that appears in baryonic OPE sides has the following
formKoike:1993sq .

$\begin{split}\expectationvalue{\mathcal{O}_{6}}=&\frac{2}{3}\expectationvalue{(\bar{q}_{1}\Gamma_{o}q_{2})(\bar{q}_{3}\Gamma_{p}q_{4})}\\\
&-\frac{1}{2}\expectationvalue{(\bar{q}_{1}\Gamma_{o}\lambda_{a}q_{2})(\bar{q}_{3}\Gamma_{p}\lambda^{a}q_{4})},\end{split}$
(35)

where $\Gamma_{\alpha}$ and $\lambda^{a}$ are a Dirac matrix and a Gell-Mann
matrix, respectively. Using the Fierz completeness relation for the color
$SU(3)$,

$\begin{split}\mathds{1}_{ij}\mathds{1}_{kl}=\frac{1}{3}\mathds{1}_{il}\mathds{1}_{kj}+\sum_{A=1}^{8}\frac{1}{2}\lambda^{A}_{il}\lambda^{A}_{kj},\end{split}$
(36)

the Gell-Mann matrices in the operator can be eliminated.

$\displaystyle\langle(\bar{q}_{1}\Gamma_{o}\lambda_{a}q_{2})$
$\displaystyle(\bar{q}_{3}$ $\displaystyle\Gamma_{p}\lambda^{a}q_{4})\rangle=$
(37) $\displaystyle-\frac{2}{3}$
$\displaystyle\expectationvalue{(\bar{q}_{1}\Gamma_{o}q_{2})(\bar{q}_{3}\Gamma_{p}q)}$
$\displaystyle-\frac{1}{8}$
$\displaystyle\Tr[\Gamma_{o}\Gamma^{r}\Gamma_{p}\Gamma^{s}]\expectationvalue{(\bar{q}_{1}\Gamma_{r}q_{4})(\bar{q}_{3}\Gamma_{s}q_{2})},$

where the Dirac and Gell-Mann matrices are normalized as
$\Tr[\Gamma_{r}\Gamma^{s}]=4\delta_{r}^{s}$ and
$\Tr[\lambda^{a}\lambda_{b}]=2\delta^{a}_{b}$, respectively. Applying the
transformation reduces the number of four-quark operators in the baryonic OPE
sides by half. When it comes to the pure-flavored four-quark operators, the
transformation can be represented linearly.

$\begin{split}\begin{pmatrix}\expectationvalue{(\bar{u}\lambda^{a}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{5}\lambda^{a}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{\mu}\lambda^{a}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{5}\gamma_{\mu}\lambda^{a}u)^{2}}\\\
\expectationvalue{(\bar{u}\sigma_{\mu\nu}\lambda^{a}u)^{2}}\end{pmatrix}=\begin{pmatrix}-7/6&-1/2&-1/2&1/2&-1/4\\\
-1/2&-7/6&1/2&-1/2&-1/4\\\ -2&2&1/3&1&0\\\ 2&-2&1&1/3&0\\\
-6&-6&0&0&1/3\end{pmatrix}\begin{pmatrix}\expectationvalue{(\bar{u}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{5}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{\mu}u)^{2}}\\\
\expectationvalue{(\bar{u}\gamma_{5}\gamma_{\mu}u)^{2}}\\\
\expectationvalue{(\bar{u}\sigma_{\mu\nu}u)^{2}}\end{pmatrix}.\end{split}$
(38)

###  V.3 Separation of breaking and symmetric operators

Using the prescription discussed in Appendix A, one can categorize all the
quark operators appearing in the nucleon sum rule as follows. With the
following identity,

$\begin{split}\langle(\bar{u}\Gamma_{1}d)(\bar{d}\Gamma^{1}u)\rangle=\expectationvalue{(\bar{u}\Gamma_{1}u)(\bar{u}\Gamma^{1}u)}-\expectationvalue{(\bar{u}\Gamma_{1}u)(\bar{d}\Gamma^{1}d)},\end{split}$
(39)

there are 5(Dirac structure) $\times$ 2(pure or mixed flavor) types of four-
quark condensates in the baryonic OPE sides. These four-quark condensates also
can be divided into chiral symmetry breaking and symmetric condensates
according to the contribution of the zero modes. The operators denoted with
subscript $S$ ($\expectationvalue{...}_{S}$) and $B$
($\expectationvalue{...}_{B}$) denote chiral symmetric and breaking operators,
respectively.

$\begin{split}1.\expectationvalue{(\bar{u}u)(\bar{u}u)}=&\expectationvalue{(\bar{u}u)(\bar{u}u)-(\bar{u}d)(\bar{d}u)}_{B}\\\
&+\expectationvalue{(\bar{u}d)(\bar{d}u)}\\\
=&\expectationvalue{(\bar{u}u)(\bar{d}d)}_{B}\\\
&+\frac{1}{2}\expectationvalue{(\bar{u}d)(\bar{d}u)+(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{B}\\\
&+\frac{1}{2}\expectationvalue{(\bar{u}d)(\bar{d}u)-(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{S}.\end{split}$
(40)

Since the pure flavored condensates contain connected and disconnected pieces,
we first divide the condensate into the two. Then, the connected piece is
separated into the breaking and symmetric condensates. In the same way, the
other remaining operators can be rewritten as the following.

$\begin{split}&2.\expectationvalue{(\bar{u}\gamma_{5}u)(\bar{u}\gamma_{5}u)}=\expectationvalue{(\bar{u}\gamma_{5}u)(\bar{d}\gamma_{5}d)}_{B}+\frac{1}{2}\expectationvalue{(\bar{u}d)(\bar{d}u)+(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{B}-\frac{1}{2}\expectationvalue{(\bar{u}d)(\bar{d}u)-\expectationvalue{\bar{u}\gamma_{5}d}(\bar{d}\gamma_{5}u)}_{S},\\\
&3.\expectationvalue{(\bar{u}\gamma_{\mu}u)(\bar{u}\gamma_{\mu}u)}=\expectationvalue{(\bar{u}\gamma_{\mu}u)(\bar{d}\gamma_{\mu}d)}_{S}+\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)+(\bar{u}\gamma_{5}\gamma_{\mu}d)(\bar{d}\gamma_{5}\gamma_{\mu}u)}_{S}\\\
&\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)-(\bar{u}\gamma_{5}\gamma_{\mu}d)(\bar{d}\gamma_{5}\gamma_{\mu}u)}_{B},\\\
&4.\expectationvalue{(\bar{u}\gamma_{5}\gamma_{\mu}u)(\bar{u}\gamma_{5}\gamma_{\mu}u)}=\expectationvalue{(\bar{u}\gamma_{5}\gamma_{\mu}u)(\bar{d}\gamma_{5}\gamma_{\mu}d)}_{S}+\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)+(\bar{u}\gamma_{5}\gamma_{\mu}d)(\bar{d}\gamma_{5}\gamma_{\mu}u)}_{S}\\\
&\qquad\qquad\qquad\qquad\qquad-\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)-(\bar{u}\gamma_{5}\gamma_{\mu}d)(\bar{d}\gamma_{5}\gamma_{\mu}u)}_{B},\\\
&5.\expectationvalue{(\bar{u}\sigma_{\mu\nu}u)(\bar{u}\sigma_{\mu\nu}u)}=\expectationvalue{(\bar{u}\sigma_{\mu\nu}u)(\bar{d}\sigma_{\mu\nu}d)}_{B}+\expectationvalue{(\bar{u}\sigma_{\mu\nu}d)(\bar{d}\sigma_{\mu\nu}u)}_{B}.\end{split}$
(41)

For mixed flavor operators, only disconnected contributions exist so that the
chiral symmetry properties can be determined as discussed in Appendix A. Note
that since the color matrices $\lambda^{a}$ have no impact on the separation
rules, the same formula as given in Eq. (41) applies when the quark anti-quark
pair contains a Gell-Mann matrix with the index contracted with that in the
other pair.

###  V.4 Baryon quark operators

In this subsection, we summarize the quark operators separated into chiral
symmetric and breaking parts in the Baryon sum rules. The dimension-6 four
quark operators appearing in the baryonic OPE sides are given, explicitly.

$\begin{split}\mathcal{M}_{6}^{N}=&\expectationvalue{B_{6}^{N}}_{B}+\expectationvalue{S_{6}^{N}}_{S},\\\
\mathcal{M}_{6}^{\Delta}=&\expectationvalue{B_{6}^{\Delta}}_{B}+\expectationvalue{S_{6}^{\Delta}}_{S},\end{split}$
(42)

where

$\begin{split}\expectationvalue{B^{N}_{6}}_{B}=&-\expectationvalue{(\bar{u}u)(\bar{d}d)}_{B}+\expectationvalue{(\bar{u}\gamma_{5}u)(\bar{d}\gamma_{5}d)}_{B}-\frac{1}{2}\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)-(\bar{u}\gamma_{\mu}\gamma_{5}d)(\bar{d}\gamma_{\mu}\gamma_{5}u)}_{B},\\\
\expectationvalue{S^{N}_{6}}_{S}=&-3\expectationvalue{(\bar{u}\gamma_{\mu}u)(\bar{d}\gamma_{\mu}d)}_{S}-\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}d)(\bar{d}\gamma_{\mu}\gamma_{5}u)}_{S}+2\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)+(\bar{u}\gamma_{\mu}\gamma_{5}d)(\bar{d}\gamma_{\mu}\gamma_{5}u)}_{S},\\\
\expectationvalue{B^{\Delta}_{6}}_{B}=&-2\expectationvalue{(\bar{u}u)(\bar{d}d)}_{B}+2\expectationvalue{(\bar{u}\gamma_{5}u)(\bar{d}\gamma_{5}d)}_{B}-\expectationvalue{(\bar{u}\gamma_{\mu}d)(\bar{d}\gamma_{\mu}u)-(\bar{u}\gamma_{\mu}\gamma_{5}d)(\bar{d}\gamma_{\mu}\gamma_{5}u)}_{B},\\\
\expectationvalue{S^{\Delta}_{6}}_{S}=&-2\expectationvalue{(\bar{u}d)(\bar{d}u)-(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{S}-\expectationvalue{(\bar{u}\gamma_{\mu}u)(\bar{d}\gamma_{\mu}d)}_{S}+\expectationvalue{(\bar{u}\gamma_{\mu}\gamma_{5}u)(\bar{d}\gamma_{\mu}\gamma_{5}d)}_{S},\end{split}$
(43)

$\mathcal{M}_{N}^{6}$ and $\mathcal{M}_{\Delta}^{6}$ are the dimension-6 terms
that appear in OPE side of nucleon and $\Delta$, respectively. For simplicity,
the anomalous dimensional correction factors are omitted.

For notational convenience, the followings are introduced.

$\begin{split}D\\{\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}\\}\equiv&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}_{1}\Gamma_{a}D^{2}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}D^{2}q_{4})}\bigg{)},\\\
G\\{\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}\\}\equiv&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}_{1}\Gamma_{a}\sigma\cdot
Gq_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}\sigma\cdot
Gq_{4})}\\\ &+\expectationvalue{(\bar{q}_{1}\Gamma_{a}\sigma\cdot
G^{A}q_{2})(\bar{q}_{3}\Gamma_{b}\lambda_{A}q_{4})}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}\lambda_{A}q_{2})(\bar{q}_{3}\Gamma_{b}\sigma\cdot
G^{A}q_{4})}\bigg{)},\end{split}$ (44)

where

$\displaystyle D^{2}$ $\displaystyle=$ $\displaystyle D_{\mu}D^{\mu},$
$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}^{A}\lambda_{A},$ $\displaystyle(\sigma\cdot G)^{ab}_{ij}$
$\displaystyle=$
$\displaystyle(\sigma_{\mu\nu})_{ij}G^{\mu\nu}_{A}(\lambda^{A})^{ab},$
$\displaystyle\sigma_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}].$ (45)

Since gluon field strength tensor and gauge covariant derivative operators do
not affect separation rules, the dimension-8 operators in right-hand side of
Eq (44) have the same chiral separation property as the dimension-6 operator
inside the curly bracket. Namely

$\begin{split}D\\{\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}_{B,S}\\}=&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}_{1}\Gamma_{a}D^{2}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}_{B,S}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}D^{2}q_{4})}_{B,S}\bigg{)},\\\
G\\{\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}_{B,S}\\}=&\frac{1}{2}\bigg{(}\expectationvalue{(\bar{q}_{1}\Gamma_{a}\sigma\cdot
Gq_{2})(\bar{q}_{3}\Gamma_{b}q_{4})}_{B,S}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}q_{2})(\bar{q}_{3}\Gamma_{b}\sigma\cdot
Gq_{4})}_{B,S}\\\ &+\expectationvalue{(\bar{q}_{1}\Gamma_{a}\sigma\cdot
G^{A}q_{2})(\bar{q}_{3}\Gamma_{b}\lambda_{A}q_{4})}_{B,S}+\expectationvalue{(\bar{q}_{1}\Gamma_{a}\lambda_{A}q_{2})(\bar{q}_{3}\Gamma_{b}\sigma\cdot
G^{A}q_{4})}_{B,S}\bigg{)}.\end{split}$ (46)

The dimension-8 operators appearing in nucleon and $\Delta$ OPE side are
given, respectively, as

$\begin{split}\mathcal{M}_{8}^{N}=&\expectationvalue{B_{8}^{N}}_{B}+\expectationvalue{S_{8}^{N}}_{S},\\\
\mathcal{M}_{8}^{\Delta}=&\expectationvalue{B_{8}^{\Delta}}_{B}+\expectationvalue{S_{8}^{\Delta}}_{S}.\end{split}$
(47)

where

$\begin{split}\expectationvalue{B_{8}^{N}}_{B}=&-\frac{1}{M^{2}}D\\{B_{6}^{N}\\}+\frac{1}{6M^{2}}G\\{\expectationvalue{(\bar{u}u)(\bar{d}d)}_{B}\\}+\frac{1}{12M^{2}}G\\{\expectationvalue{(\bar{u}d)(\bar{d}u)+(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{B}\\},\\\
\expectationvalue{S_{8}^{N}}_{S}=&-\frac{1}{M^{2}}D\\{S_{6}^{N}\\}+\frac{1}{12M^{2}}G\\{\expectationvalue{(\bar{u}d)(\bar{d}u)-(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{S}\\},\\\
\expectationvalue{B_{8}^{\Delta}}_{B}=&-\frac{1}{M^{2}}D\\{B_{6}^{\Delta}\\}+\frac{1}{18M^{2}}G\\{\expectationvalue{(\bar{u}u)(\bar{d}d)}_{B}\\}+\frac{1}{36M^{2}}G\\{\expectationvalue{(\bar{u}d)(\bar{d}u)+(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{B}\\},\\\
\expectationvalue{S_{8}^{\Delta}}_{S}=&-\frac{1}{M^{2}}D\\{S_{6}^{\Delta}\\}+\frac{1}{36M^{2}}G\\{\expectationvalue{(\bar{u}d)(\bar{d}u)-(\bar{u}\gamma_{5}d)(\bar{d}\gamma_{5}u)}_{S}\\}.\end{split}$
(48)

For simplicity, the anomalous dimensional correction factors are omitted.

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