[table]capposition=bottom capbtabboxtable[][]

# Revisiting $K_{1}(1270)-K_{1}(1400)$ mixing in QCD sum rules

Yu-Ji Shi<EMAIL_ADDRESS>School of Physics, East China University of
Science and Technology, Shanghai 200237, China  INPAC, Key Laboratory for
Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for
Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China  Jun Zeng<EMAIL_ADDRESS>(Corresponding author) INPAC, Key Laboratory for Particle Astrophysics and
Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Zhi-Fu Deng  INPAC, Key Laboratory for Particle Astrophysics
and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and
Cosmology, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China

###### Abstract

We investigate the $K_{1}(1270)-K_{1}(1400)$ mixing caused by the flavor
$SU(3)$ symmetry breaking. The mixing angle is expressed by a $K_{1A}\to
K_{1B}$ matrix element induced by the operators that breaks flavor $SU(3)$
symmetry. The QCD contribution to this matrix element is assumed to be
dominated and calculated by QCD sum rules. A three-point correlation function
is defined and handled both at the hadron and quark-gluon levels. The quark-
gluon level calculation is based on operator product expansion up to
dimension-5 condensates. A detailed numerical analysis is performed to
determine the Borel parameters, and the obtained mixing angle is
$\theta_{K_{1}}=22^{\circ}\pm 7^{\circ}$ or $\theta_{K_{1}}=68^{\circ}\pm
7^{\circ}$.

##  I Introduction

The flavor $SU(3)$ symmetry plays an important role in the conventional quark
model, which classifies hadrons into various of irreducible representations.
Although the assumption of perfect flavor $SU(3)$ symmetry succeeds in most of
phenomenological analysis on hadron decays and spectrums Zeppenfeld:1980ex ;
Savage:1989ub ; Deshpande:1994ii ; Chau:1986du ; Chau:1990ay ; Lu:2016ogy ;
Wang:2017azm ; He:2018php , its breaking effect still cannot be neglected. One
of the physical effect due to the flavor $SU(3)$ breaking is the hadron
mixing.

According to the quark model, the two axial-vector nonets with $J^{P}=1^{+}$
are expected as the orbital excitation of the $\bar{q}q$ system. There are two
types of P-wave axial-vector mesons: the ${}^{3}P_{1}$ and ${}^{1}P_{1}$ with
the notation ${}^{2S+1}L_{J}$. Generally, in the flavor $SU(3)$ limit these
two nonets cannot be mixed since they have distinct $C$ parities, explicitly
$J^{PC}=1^{++},\ 1^{+-}$ for ${}^{3}P_{1},\ ^{1}P_{1}$ respectively. However,
because of the mass difference between the strange and light quarks, the kaon
nonets $K_{1A}(^{3}P_{1})$ and $K_{1B}(^{1}P_{1})$ are distinguished from the
mass eigenstates $K_{1}(1270)$ and $K_{1}(1400)$. As a result, there emerges a
mixing between these two sets of axial-vector kaons:

$\displaystyle\begin{pmatrix}|K_{1}(1270)\rangle\\\
|K_{1}(1400)\rangle\end{pmatrix}=\begin{pmatrix}\cos\theta_{K_{1}}&\sin\theta_{K_{1}}\\\
-\sin\theta_{K_{1}}&\cos\theta_{K_{1}}\end{pmatrix}\begin{pmatrix}|K_{1B}\rangle\\\
|K_{1A}\rangle\end{pmatrix},$ (1)

where $\theta_{K_{1}}$ is the mixing angle.

Up to now, there are quite a number of studies on $\theta_{K_{1}}$ in the
literature, with most of them based on phenomenological analysis. In
Ref.Suzuki:1993yc , with the use of early experimental information on masses
and the partial rates of $K_{1}(1270)$ and $K_{1}(1400)$, the authors obtained
$\theta_{K_{1}}=33^{\circ}$ or $57^{\circ}$; The Refs. Cheng:2003bn ;
Li:2009tx  phenomenologically analyzed the $\tau\to K_{1}(1270)\nu_{\tau}$ and
$\tau\to K_{1}(1400)\nu_{\tau}$ decays and obtained
$\theta_{K_{1}}=37^{\circ}$ or $58^{\circ}$; The Ref.Close:1997nm  studied the
correlation of the $f_{1}(1285)-f_{1}(1420)$ mixing angle
$\theta_{{}^{3}P_{1}}$ with $\theta_{K_{1}}$, and obtained
$\theta_{K_{1}}=31.7^{\circ}$ or $56.3^{\circ}$; In Ref.Cheng:2013cwa , the
authors used the correspondence between $\theta_{K_{1}}$ and the
$f_{1}(1285)-f_{1}(1420)$, $h_{1}(1170)-h_{1}(1380)$ mixing angles to rule out
unreasonable $\theta_{K_{1}}$ values, and announced a reasonable range as
$28^{\circ}<\theta_{K_{1}}<30^{\circ}$. Besides the phenomenological analysis,
in Ref.Burakovsky:1997dd  the authors obtained
$34^{\circ}<\theta_{K_{1}}<55^{\circ}$ by non-relativistic constituent quark
model with the inputs of the mass difference between the $a_{1}(1260)$ and
$b_{1}(1235)$ mesons, as well as the ratio of the constituent quark masses.
The Ref.Dag:2012zz  related $\theta_{K_{1}}$ with a two-point correlation
function. By calculating this correlation function in QCD sum rules (QCDSR)
the authors obtained $\theta_{K_{1}}=39^{\circ}\pm 4^{\circ}$.

It can be found that the phenomenological predictions on $\theta_{K_{1}}$
mentioned above are not unified. In this work, we desire to give a theoretical
and model independent calculation on $\theta_{K_{1}}$, with the use of QCDSR.
It should be mentioned that the QCDSR calculation in Ref.Dag:2012zz  has a
defect. The authors missed the pseudoscalar kaon contribution when operating
the $K_{1A}$ current on the correlation function. In this work, we will
calculate $\theta_{K_{1}}$ in a different framework of QCDSR from that in
Ref.Dag:2012zz . The $\theta_{K_{1}}$ will be expressed by a $K_{1A}\to
K_{1B}$ matrix element induced by $SU(3)$ breaking operators, with zero
transferring momentum. This method has been successfully applied to the
$\Xi_{c}-\Xi_{c}^{\prime}$ mixing Liu:2023feb ; Deng:2023qaf . The $K_{1A}\to
K_{1B}$ matrix element will be calculated in QCDSR by a three-point
correlation function. Some introductions on QCDSR and its applications can be
found in Refs. Onishchenko:2000wf ; Onishchenko:2000yp ; Kiselev:2001fw ;
Zhang:2008rt ; Wang:2010hs ; Wang:2010vn ; Wang:2010it ; Hu:2017dzi ;
Hu:2017dzi ; Shi:2019hbf .

This paper is arranged as follows: In section II, we introduce the method to
extract $\theta_{K_{1}}$; Section III gives the hadron level calculation;
Section IV gives the quark-gluon level calculation; Section V presents
numerical results; Section VI is a summary of this work.

##  II $K_{1}(1270)-K_{1}(1400)$ mixing

There are two sources of the flavor $SU(3)$ breaking. The first one comes from
the mass difference between $s$ and $u,d$ (nearly massless) quarks, which only
provides QCD contribution to $K_{1}(1270)-K_{1}(1400)$ mixing. Another one
comes from the electric charge difference among the $u,d,s$ quarks, which
involves the QED effect. In this work, we will focus on the QCD contribution
since the QED effect is expected to be tiny as shown by our previous work on
the $\Xi_{c}-\Xi_{c}^{\prime}$ mixing Liu:2023feb ; Deng:2023qaf . The full
QCD Lagrangian of the quark sector contains both the terms conserving and
breaking the flavor $SU(3)$ symmetry: $\mathcal{L}_{\rm
QCD}=\mathcal{L}_{0}+\Delta\mathcal{L}$, where $\mathcal{L}_{0}$ reads as

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$
$\displaystyle\sum_{q}\bar{q}(i\not{D}-m_{u})q,$ (2)

with $D$ being the QCD covariant derivative, $q=u,d,s$, and $m_{u}=m_{d}=0$ is
approximately assumed. The $SU(3)$ symmetry breaking term $\Delta\mathcal{L}$,
which arises from the quark mass difference reads as

$\displaystyle\Delta\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{s}(m_{u}-m_{s})s.$ (3)

Accordingly, the Hamiltonian is decomposed as $H=H_{0}+\Delta H$, with

$\displaystyle\Delta H=\int d^{3}x\Delta\mathcal{H}(x)=-\int
d^{3}x\Delta\mathcal{L}(x).$ (4)

The lowest axial vector kaons $K_{1}(1270)$ and $K_{1}(1400)$ are the mass
eigenstates of the full Hamiltonian $H$:

$\displaystyle H|K_{1}(1270)\rangle=m_{1270}|K_{1}(1270)\rangle,$
$\displaystyle H|K_{1}(1400)\rangle=m_{1400}|K_{1}(1270)\rangle.$ (5)

On the other hand, in the $SU(3)$ symmetry limit, the lowest axial vector
kaons are classified into $K_{1B}(^{1}P_{1})$ and $K_{1A}(^{3}P_{1})$ states,
which are eigenstates of the $SU(3)$ conserved Hamiltonian $H_{0}$:

$\displaystyle H_{0}|K_{1B}\rangle=m_{1B}|K_{1B}\rangle,$ $\displaystyle
H_{0}|K_{1A}\rangle=m_{1A}|K_{1A}\rangle\rangle.$ (6)

The mixing between the physical doublet
$|K_{P}\rangle=(|K_{1}(1270)\rangle,|K_{1}(1400)\rangle)^{T}$ and the $SU(3)$
doublet $|K_{F}\rangle=(|K_{1B}\rangle,|K_{1A}\rangle)^{T}$ is described by a
unitary transforming matrix $U$ with a mixing angle $\theta_{K_{1}}$:

$\displaystyle|K_{P}\rangle=\begin{pmatrix}\cos\theta_{K_{1}}&\sin\theta_{K_{1}}\\\
-\sin\theta_{K_{1}}&\cos\theta_{K_{1}}\end{pmatrix}|K_{F}\rangle=U|K_{F}\rangle.$
(7)

Here we consider the matrix element for the $SU(3)$ doublet $|K_{F}\rangle$:
$\langle K_{F}(p^{\prime})|H|K_{F}(p)\rangle$. Both the initial and final
states are set to be static ${\vec{p}}={\vec{p}}^{\prime}=0$ and on-shell
$p_{1B}^{0}=m_{1B},\ p_{1A}^{0}=m_{1A}$. With the use of the unitary
transformation $U$ defined in Eq. (7), and the physical masses defined in Eq.
(5), we obtain

$\displaystyle\begin{pmatrix}\langle
K_{1B}(\lambda^{\prime})|H|K_{1B}(\lambda)\rangle&\langle
K_{1B}(\lambda^{\prime})|H|K_{1A}(\lambda)\rangle\\\ \langle
K_{1A}(\lambda^{\prime})|H|K_{1B}(\lambda)\rangle&\langle
K_{1A}(\lambda^{\prime})|H|K_{1A}(\lambda)\rangle\end{pmatrix}$
$\displaystyle=$ $\displaystyle\
2(2\pi)^{3}\delta^{(3)}(\vec{0})\delta_{\lambda\lambda^{\prime}}$
$\displaystyle\times$ $\displaystyle\begin{pmatrix}m_{1270}^{2}\
c_{k}^{2}+m_{1400}^{2}\ s_{k}^{2}&(m_{1270}^{2}-m_{1400}^{2})\ c_{k}s_{k}\\\
(m_{1270}^{2}-m_{1400}^{2})\ c_{k}s_{k}&m_{1270}^{2}\ s_{k}^{2}+m_{1400}^{2}\
c_{k}^{2}\end{pmatrix},$

where $s_{k}=\sin\theta_{K_{1}},c_{k}=\cos\theta_{K_{1}}$ for simplicity the
momentum dependence of the $K_{1B,1A}$ states are not shown in the matrix
element.

It can be found that the upper-right off-diagonal component in the Eq. (II)
leads to the equation

$\displaystyle\langle K_{1B}(\lambda^{\prime})|H|K_{1A}(\lambda)\rangle$ (9)
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta^{(3)}(\vec{0})\delta_{\lambda\lambda^{\prime}}(m_{1270}^{2}-m_{1400}^{2})\sin
2\theta_{K_{1}}.$

Therefore, the mixing angle $\theta_{K_{1}}$ can be extracted as soon as one
calculate the matrix element on the left hand side above, which can be further
expressed as

$\displaystyle\langle K_{1B}(\lambda^{\prime})|H|K_{1A}(\lambda)\rangle$ (10)
$\displaystyle=$ $\displaystyle(2\pi)^{3}\delta^{(3)}({\vec{0}})\langle
K_{1A}(\lambda^{\prime})|\Delta\mathcal{H}(0)|K_{1B}(\lambda)\rangle,$

by integrating out the coordinate. Equating the Eq. (9) and Eq. (10), and
setting $\lambda=\lambda^{\prime}$ we have

$\displaystyle\sin
2\theta_{K_{1}}=\frac{m_{s}-m_{u}}{m_{1270}^{2}-m_{1400}^{2}}\langle
K_{1B}|\bar{s}s(0)|K_{1A}\rangle.$ (11)

Generally, the matrix element $\langle
K_{1A}(p_{2})|\Delta\mathcal{H}(0)|K_{1B}(p_{1})\rangle$ with nonzero initial
and final momentums can be parameterized as

$\displaystyle\langle K_{1B}(p_{2})|\bar{s}s(0)|K_{1A}(p_{1})\rangle$
$\displaystyle=$ $\displaystyle\
\epsilon^{*}_{\mu}(p_{2})\bigg{[}F_{1}g^{\mu\nu}+\frac{F_{2}}{M^{2}}\epsilon^{\mu\nu\alpha\beta}p_{1\alpha}p_{2\beta}+\frac{F_{3}}{M^{2}}p_{1}^{\mu}p_{2}^{\nu}\bigg{]}\epsilon_{\nu}(p_{1}),$
(12)

where $F_{1,2,3}$ are three transition functions of $q^{2}=(p_{1}-p_{2})^{2}$,
and $M^{2}$ can be the mass of $K_{1A}$ or $K_{1B}$. Note that the matrix
element we actually need in Eq. (11) requires $p_{1}=p_{2}$. Thus the
$F_{2},F_{3}$ terms in Eq. (12) vanishes and only $F_{1}(q^{2}=0)$ is
relevant. Accordingly the mixing angle can be obtained as

$\displaystyle\sin
2\theta_{K_{1}}=\frac{m_{s}-m_{u}}{m_{1400}^{2}-m_{1270}^{2}}F_{1}(0).$ (13)

##  III Hadron Level calculation

In this and the next section, we will introduce a QCDSR calculation for the
matrix element in Eq. (12). We firstly define a three-point correlation
function:

$\displaystyle\Pi_{\mu\nu\rho}(p_{1},p_{2})=$ $\displaystyle\ i^{2}\int
d^{4}xd^{4}y\ e^{ip_{2}\cdot x}e^{-ip_{1}\cdot y}$ $\displaystyle\times\langle
0|T\\{J_{\mu\nu}^{1B}(x)\bar{s}s(0)J_{\rho}^{1A\dagger}(y)\\}|0\rangle,$ (14)

where $J_{\mu\nu}^{1A}$ and $J_{\mu\nu}^{1B}$ are the currents of
$K_{1A}(^{3}P_{1})$ and $K_{1B}(^{1}P_{1})$, respectively:

$\displaystyle
J_{\mu\nu}^{1B}=\bar{q}\sigma_{\mu\nu}s,~{}~{}J_{\rho}^{1A}=\bar{q}\gamma_{\rho}\gamma_{5}s.$
(15)

The correlation function defined in Eq. (14) should be calculated both at the
hadron level and the quark-gluon level. At the hadron level, inserting the
complete sets with the same quantum number of $K_{1B}$ and $K_{1A}$ into the
correlation function, and using the following definition of kaon decay
constants:

$\displaystyle\langle 0|J_{\mu\nu}^{1B}(0)|K_{1B}(p,\lambda)\rangle$
$\displaystyle=$ $\displaystyle
if_{K_{1B}}^{\perp}\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha}(p,\lambda)p^{\beta},$
$\displaystyle\langle 0|J_{\rho}^{1A}(0)|K_{1A}(p,\lambda)\rangle$
$\displaystyle=$ $\displaystyle-
if_{K_{1A}}m_{K_{1}A}\epsilon_{\rho}(p,\lambda),$ $\displaystyle\langle
0|J_{\rho}^{1A}(0)|K(p)\rangle$ $\displaystyle=$ $\displaystyle
if_{K}p_{\rho},$ (16)

one can express the correlation function as

$\displaystyle\Pi^{H}_{\mu\nu\rho}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle-
m_{A}f_{A}f_{B}^{\perp}\epsilon_{\mu\nu\alpha\beta}p_{2}^{\beta}\bigg{(}-g^{\alpha\kappa}+\frac{p_{2}^{\alpha}p_{2}^{\kappa}}{m_{B}^{2}}\bigg{)}\frac{1}{p_{2}^{2}-m_{1B}^{2}}\bigg{[}F_{1}g_{\kappa\tau}+\frac{F_{2}}{M^{2}}\epsilon_{\kappa\tau\rho\sigma}p_{1}^{\rho}p_{2}^{\sigma}+\frac{F_{3}}{M^{2}}p_{1\kappa}p_{2\tau}\bigg{]}\bigg{(}-g^{\tau}_{\rho}+\frac{p_{1}^{\tau}p_{1\rho}}{m_{1A}^{2}}\bigg{)}\frac{1}{p_{1}^{2}-m_{1A}^{2}}$
$\displaystyle+f_{k}f_{1B}^{\perp}\epsilon_{\mu\nu\alpha\beta}p_{2}^{\beta}p_{1\rho}\frac{1}{p_{2}^{2}-m_{1B}^{2}}\bigg{(}-g^{\alpha\kappa}+\frac{p_{2}^{\alpha}p_{2}^{\kappa}}{m_{1B}^{2}}\bigg{)}p_{1\kappa}\frac{G(q^{2})}{M}\frac{1}{p_{1}^{2}-m_{K}^{2}}$
$\displaystyle+\int_{s_{1}^{\rm th}}^{\infty}ds_{1}\int_{s_{2}^{\rm
th}}^{\infty}ds_{2}\frac{\rho^{\rm
conti}_{\mu\nu\rho}(s_{1},s_{2},q^{2})}{(s_{1}-p_{1}^{2})(s_{2}-p_{2}^{2})}+\int_{s_{1}^{\rm
th}}^{\infty}ds_{1}\frac{\rho^{\rm
conti}_{1,\mu\nu\rho}(s_{1},p_{2},q^{2})}{s_{1}-p_{1}^{2}}+\int_{s_{2}^{\rm
th}}^{\infty}ds_{2}\frac{\rho^{\rm
conti}_{2,\mu\nu\rho}(p_{1},s_{2},q^{2})}{s_{2}-p_{2}^{2}}.$ (17)

The last three terms above denote the contribution from the excited and
continuous spectrum, which begin at the thresholds $s_{1}^{\rm th}$ and
$s_{2}^{\rm th}$. It should be noted that the axial vector current
$J_{\rho}^{1A}$ can create both an axial vector and a pseudoscalar kaon from
the vaccum. Therefore to obtain Eq. (17) both the $K_{1A}$ and $K$ have been
inserted between $\bar{s}s(0)$ and $J_{\rho}^{1A}(y)$, and we have used the
parameterization for the $K\to K_{1B}$ matrix element:

$\displaystyle\langle
K_{1B}(p_{2})|\bar{s}s(0)|K(p_{1})\rangle=\epsilon^{*}_{\mu}(p_{2})p_{1}^{\mu}\frac{G(q^{2})}{M},$
(18)

where $G(q^{2})$ is the corresponding form factor.

Now the hadron level correlation function in Eq. (17) depends on four form
factors $F_{1},F_{2},F_{3},G$. However, only $F_{1}$ is relevant to the mixing
angle as shown in Eq. (13). To remove the irrelevant form factors we operate
the following projection on the correlation function

$\displaystyle\epsilon^{\mu\rho\alpha\beta}p_{1}^{\nu}\Pi_{\mu\nu\rho}(p_{1},p_{2})={\tilde{\Pi}}(p_{1},p_{2})(p_{1}^{\beta}p_{2}^{\alpha}-p_{1}^{\alpha}p_{2}^{\beta}).$
(19)

${\tilde{\Pi}}(p_{1},p_{2})$ is a newly defined scalar correlation function,
which at hadron level only depends on $F_{1}$:

$\displaystyle{\tilde{\Pi}}^{H}(p_{1},p_{2})=\frac{2m_{1A}f_{1A}f_{1B}^{\perp}F_{1}(q^{2})}{(p_{1}^{2}-m_{1A}^{2})(p_{2}^{2}-m_{1B}^{2})}+\cdots,$
(20)

where the ellipse denotes the last three terms in Eq. (17) with the projection
defined in Eq. (19) being operated. In principle, the
${\tilde{\Pi}}(p_{1},p_{2})$ calculated at the hadron level and the quark-
gluon level should be equivalent:

$\displaystyle{\tilde{\Pi}}^{H}(p_{1},p_{2},q^{2})={\tilde{\Pi}}^{\rm
QCD}(p_{1},p_{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int_{0}^{\infty}ds_{1}\int_{0}^{\infty}ds_{2}\frac{{\rm
Im}^{2}{\tilde{\Pi}}^{QCD}(s_{1},s_{2},q^{2})}{(s_{1}-p_{1}^{2})(s_{2}-p_{2}^{2})}.$
(21)

In the second equation above we have expressed the ${\tilde{\Pi}}^{\rm QCD}$
as its dispersive integration form, with $s_{1}^{\rm min}$ and $s_{2}^{\rm
min}$ being the quark level thresholds. According to the quark-hadron duality,
the continuous spectrum contribution at hadron level is equivalent to that at
QCD level. In other words the ellipse term in Eq. (20) is equal to

$\displaystyle\frac{1}{\pi^{2}}$ $\displaystyle\left[\int_{s_{1}^{\rm
th}}^{\infty}ds_{1}\int_{s_{2}^{\rm th}}^{\infty}ds_{2}+\int_{s_{1}^{\rm
th}}^{\infty}ds_{1}\int_{0}^{s_{2}^{\rm th}}ds_{2}\right.$
$\displaystyle\left.+\int_{0}^{s_{1}^{\rm th}}ds_{1}\int_{s_{2}^{\rm
th}}^{\infty}ds_{2}\right]\frac{{\rm Im}^{2}{\tilde{\Pi}}^{\rm
QCD}(s_{1},s_{2},q^{2})}{(s_{1}-p_{1}^{2})(s_{2}-p_{2}^{2})}.$ (22)

Thus the continuous spectrum contribution can be canceled at both the hadron
and QCD levels. After Borel transformation we arrive at the sum rules
equation:

$\displaystyle{\cal B}_{T_{1},T_{2}}\\{{\tilde{\Pi}}^{H}\\}(q^{2})$
$\displaystyle={\cal B}_{T_{1},T_{2}}\\{{\tilde{\Pi}}^{\rm QCD}\\}(q^{2}),$
$\displaystyle 2m_{1A}f_{1A}f_{1B}^{\perp}\
e^{-\frac{m_{1A}^{2}}{T_{1}^{2}}}e^{-\frac{m_{1B}^{2}}{T_{2}^{2}}}F_{1}(q^{2})$
$\displaystyle=\frac{1}{\pi^{2}}\int_{0}^{s_{1}^{\rm
th}}ds_{1}\int_{0}^{s_{2}^{\rm th}}ds_{2}\
e^{-\frac{s_{1}}{T_{1}^{2}}}e^{-\frac{s_{2}}{T_{2}^{2}}}{\rm
Im}^{2}{\tilde{\Pi}}^{\rm QCD}(s_{1},s_{2},q^{2}),$ (23)

where $T_{1},T_{2}$ are the two Borel parameters corresponding to
$p_{1}^{2},p_{2}^{2}$. Now it is clear that $F_{1}$ can be obtained through
Eq. (23) if the imaginary part of ${\tilde{\Pi}}^{\rm QCD}$ is calculated.

##  IV Quark-gluon Level calculation

###  IV.1 Perturbative diagram

Figure 1: The perturbative diagram contribution to the correlation function,
where the lower two vertexes denote the kaon currents defined in Eq. (15)
(left). A general triangle diagram corresponding to a three-point correlation
function (right).

In this section, we present the QCD level calculation for ${\tilde{\Pi}}^{\rm
QCD}$ and extract its imaginary part. In the deep Euclidean region
$p_{1}^{2},p_{2}^{2},q^{2}\ll 0$, ${\tilde{\Pi}}^{\rm QCD}$ can be
analytically calculated by operator product expansion (OPE).

The leading contribution to OPE is from the perturbation diagram as shown by
the left diagram in Fig. 1, with amplitude

$\displaystyle\Pi_{\mu\nu\rho}^{\rm pert}(p_{1},p_{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{iN_{c}}{(2\pi)^{4}}\int d^{4}k_{1}d^{4}k_{2}d^{4}k\
\delta^{4}(p_{2}-k_{2}-k)\delta^{4}(p_{1}-k_{1}-k)$
$\displaystyle\times\frac{{\rm
tr}[\not{k}\sigma_{\mu\nu}(\not{k}_{2}+m_{s})(\not{k}_{1}+m_{s})\gamma_{\rho}\gamma_{5}]}{k^{2}(k_{2}^{2}-m_{s}^{2})(k_{1}^{2}-m_{s}^{2})}.$
(24)

The double imaginary part of the correlation function is related with its
double discontinuity as

$\displaystyle{\rm Im}^{2}\Pi_{\mu\nu\rho}^{\rm
pert}(p_{1},p_{2},q^{2})=\frac{1}{(2i)^{2}}{\rm Disc}^{2}\Pi_{\mu\nu\rho}^{\rm
pert}(p_{1},p_{2},q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{(2i)^{2}}\frac{iN_{c}}{(2\pi)^{4}}(-2\pi i)^{3}\int
d\Phi_{\Delta}(p_{1},p_{2},m_{s},m_{s},0)$ $\displaystyle\times{\rm
tr}[\not{k}\sigma_{\mu\nu}(\not{k}_{2}+m_{s})(\not{k}_{1}+m_{s})\gamma_{\rho}\gamma_{5}],$
(25)

which is obtained by the cutting rules: ${\rm
Disc}\\{1/(p^{2}-m^{2}+i\epsilon)\\}=(-2\pi i)\delta(p^{2}-m^{2})$. In the
above expression, we have introduced a three body phase space integration
measure $d\Phi_{\Delta}$ for a triangle integration as shown by the right
diagram in Fig. 1:

$\displaystyle\int d\Phi_{\Delta}(p_{1},p_{2},m_{1},m_{2},m)$ $\displaystyle=$
$\displaystyle\int d^{4}k_{1}d^{4}k_{2}d^{4}k\
\delta(k_{1}^{2}-m_{1}^{2})\delta(k_{2}^{2}-m_{2}^{2})\delta(k^{2}-m^{2})$
$\displaystyle\times\delta^{4}(p_{2}-k_{2}-k)\delta^{4}(p_{1}-k_{1}-k),$ (26)

where the three internal lines are set on shell. The scalar triangle
integration with unit integrand reads as

$\displaystyle I_{\Delta}=\int d\Phi_{\Delta}(p_{1},p_{2},m_{1},m_{2},m)\cdot
1=\frac{\pi}{2\sqrt{\lambda}}$
$\displaystyle\times\Theta[s_{1},s_{2},q^{2},m_{1},m_{2},m]\theta[s_{1}-s_{1}^{\rm
min}]\theta[s_{2}-s_{2}^{\rm min}],$ (27)

where $\lambda=(s_{1}+s_{2}-q^{2})^{2}-4s_{1}s_{2}$ with
$p_{1}^{2}=s_{1},p_{2}^{2}=s_{2}$. $\Theta$ is a $\theta$ function
constraining the $s_{1},s_{2},q^{2}$:

$\displaystyle\Theta[s_{1},s_{2},q^{2},m_{1},m_{2},m]$ $\displaystyle=$
$\displaystyle\theta\big{[}-m_{2}^{4}s_{1}-m_{1}^{4}s_{2}+m_{2}^{2}[m^{2}(q^{2}+s_{1}-s_{2})$
$\displaystyle+s_{1}(q^{2}-s_{1}+s_{2})]$
$\displaystyle-q^{2}(m^{4}+s_{1}s_{2}-m^{2}(-q^{2}+s_{1}+s_{2}))$
$\displaystyle+m_{1}^{2}[(q^{2}+s_{1}-s_{2})s_{2}$
$\displaystyle+m^{2}(q^{2}-s_{1}+s_{2})+m_{2}^{2}(-q^{2}+s_{1}+s_{2})]\big{]}.$
(28)

The rest two $\theta$ functions ensure that $s_{1},s_{2}$ are above the
corresponding quark level thresholds , namely $s_{1,2}>s_{1,2}^{\rm
min}=(m_{1,2}+m)^{2}$. In the Eq. (25) one have to set $m_{1}=m_{2}=m_{s}$.
The definitions and expressions of higher rank triangle diagram integrations
are given in the Appendix A. The analytical expression of ${\rm
Im}^{2}{\tilde{\Pi}}^{\rm pert}(p_{1},p_{2},q^{2})$ is given in the Appendix
B.

###  IV.2 $\bar{q}q$ condensate diagrams

Figure 2: The $\bar{q}q$ condensate diagram contribution to the correlation
function, where one of the quark lines is disconnected. The diagrams (b) and
(c) will vanish under the double Borel transformation for $p_{1}^{2}$ and
$p_{2}^{2}$ simultaneously.

The dimension-3 operator contribution to OPE comes from the quark condensing
diagrams as shown in Fig. 2. It can be found that the amplitudes of the $s$
quark condensing diagrams only contain one fraction form, either
$1/(p_{1}^{2}-m_{s}^{2})$ or $1/(p_{2}^{2}-m_{s}^{2})$. Therefore, the
diagrams (b) and (c) will vanish under the double Borel transformation for
$p_{1}^{2}$ and $p_{2}^{2}$ simultaneously. The amplitude of diagram (a) reads
as

$\displaystyle\Pi_{\mu\nu\rho}^{\bar{q}q}(p_{1},p_{2},q^{2})=i^{2}\int
d^{4}xd^{4}y\ e^{ip_{2}\cdot x}e^{-ip_{1}\cdot y}$
$\displaystyle\times[\sigma_{\mu\nu}D^{(0)}_{s}(x,0)D^{(0)}_{s}(0,y)\gamma_{\rho}\gamma_{5}]\langle
0|{\bar{q}}_{a}^{i}(x)q_{b}^{i}(y)|0\rangle,$ (29)

where $D^{(0)}_{s}$ denotes free $s$ quark propagator. The nonlocal $\bar{q}q$
condensing matrix element can be expanded up to dimension-5 local operators as

$\displaystyle\langle 0|{\bar{q}}_{a}^{i}(x)q_{b}^{i}(y)|0\rangle$
$\displaystyle=$ $\displaystyle
N_{c}\left[\langle\bar{q}q\rangle\frac{1}{12}\delta_{ba}+\langle\bar{q}Gq\rangle\frac{1}{192}(x-y)^{2}\delta_{ba}\right].$
(30)

The contribution of dimension-3 operator, namely the $\bar{q}q$ condensate,
only comes from the first term given above. The corresponding amplitude reads
as

$\displaystyle{\rm Im}^{2}{\tilde{\Pi}}^{\bar{q}q}(p_{1},p_{2},q^{2})$
$\displaystyle=$
$\displaystyle\frac{2\pi^{2}}{3}N_{c}m_{s}\langle\bar{q}q\rangle\delta(s_{1}-m_{s}^{2})\delta(s_{2}-m_{s}^{2}).$
(31)

The second term in Eq. (30) provides a contribution from the dimension-5
operator $\bar{q}g_{s}G_{\alpha\beta}q$. The corresponding amplitude reads as

$\displaystyle\Pi_{\mu\nu\rho}^{\bar{q}Gq(1)}(p_{1},p_{2},q^{2})$
$\displaystyle=$
$\displaystyle\frac{N_{c}}{192}\langle\bar{q}Gq\rangle(-1)\left(\frac{\partial^{2}}{\partial
p_{2}^{\alpha}\partial p_{2\alpha}}+\frac{\partial^{2}}{\partial
p_{1}^{\alpha}\partial p_{1\alpha}}+2\frac{\partial^{2}}{\partial
p_{1}^{\alpha}\partial p_{2\alpha}}\right)$ $\displaystyle\times{\rm
tr}[\sigma_{\mu\nu}(\not{p}_{2}+m_{s})(\not{p}_{1}+m_{s})\gamma_{\rho}\gamma_{5}]$
$\displaystyle\times\frac{1}{p_{1}^{2}-m_{s}^{2}}\frac{1}{p_{2}^{2}-m_{s}^{2}},$
(32)

where we have transformed $x,y$ to $-i\partial/\partial
p_{2},i\partial/\partial p_{1}$ through the exponential terms in Eq. (29). To
simplify the calculation of Eq. (32) we can omit the terms suppressed by
$m_{s}^{2}$ and obtain

$\displaystyle\Pi_{\mu\nu\rho}^{\bar{q}Gq(1)}(p_{1},p_{2},q^{2})$
$\displaystyle=$ $\displaystyle-\frac{N_{c}}{12}\langle\bar{q}Gq\rangle
m_{s}\epsilon_{\mu\nu\rho\alpha}(p_{1}^{\alpha}-p_{2}^{\alpha})\frac{\partial}{\partial
M^{2}}$
$\displaystyle\times\left[\frac{1}{(p_{1}^{2}-M^{2})(p_{2}^{2}-m_{s}^{2})}-\frac{1}{(p_{1}^{2}-m_{s}^{2})(p_{2}^{2}-M^{2})}\right]|_{M^{2}=m_{s}^{2}}$
$\displaystyle-$ $\displaystyle\frac{N_{c}}{12}\langle\bar{q}Gq\rangle
m_{s}\epsilon_{\mu\nu\rho\alpha}(p_{1}^{\alpha}+p_{2}^{\alpha})q^{2}$
$\displaystyle\times\frac{\partial^{2}}{\partial M_{1}^{2}\partial
M_{2}^{2}}\left[\frac{1}{(p_{1}^{2}-M_{1}^{2})(p_{2}^{2}-M_{2}^{2})}\right]|_{M_{1}^{2}=M_{2}^{2}=m_{s}^{2}}.$
(33)

To obtain the above expression, we have introduced derivatives on auxiliary
masses $M_{1},M_{2}$ to lower the power of the denominator, which means

$\displaystyle\frac{1}{(p_{1,2}^{2}-m_{1,2}^{2})^{2}}=\frac{\partial}{\partial
M^{2}}\frac{1}{p_{1,2}^{2}-M^{2}}|_{M^{2}\to m_{1,2}^{2}}.$ (34)

Taking the imaginary part, using dispersive integration and conducting Borel
transformation, we arrives at

$\displaystyle{\cal
B}_{T_{1},T_{2}}\\{{\tilde{\Pi}}^{\bar{q}Gq(1)}\\}(q^{2})=\frac{1}{6}N_{c}m_{s}\langle\bar{q}Gq\rangle$
$\displaystyle\times
e^{-m_{s}^{2}/T_{1}^{2}-m_{s}^{2}/T_{2}^{2}}\left[\frac{1}{T_{1}^{2}}-\frac{1}{T_{2}^{2}}+\frac{q^{2}}{T_{1}^{2}T_{2}^{2}}\right].$
(35)

###  IV.3 $\bar{q}Gq$ condensate diagrams

Figure 3: The $\bar{q}Gq$ condensate diagram contribution to the correlation
function, where one of the quark line emits a solft gluon which condensing
with other two disconnected quark fields. The diagrams (b), (c), (e) and (f)
vanishes under double Borel transformation. The diagram (d) vanishes under the
projection introduced in Eq. (19).

The quark-gluon condensing diagrams as shown in Fig. 3, where a quark
interacts with a background gluon field which condensates with the other two
disconnected light quark fields. These diagrams provide the dimension-5
operator contribution in the OPE. The massive and massless quark propagators
in the background gluon field read as

$\displaystyle D_{s}(x,0)=$ $\displaystyle\
i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\left[\frac{\delta_{ij}}{\not{k}-m_{s}}-\frac{g_{s}G_{\alpha\beta}^{A}t_{ij}^{A}}{4}\frac{\sigma^{\alpha\beta}(\not{k}+m_{s})+(\not{k}+m_{s})\sigma^{\alpha\beta}}{(k^{2}-m_{s}^{2})^{2}}\right.$
$\displaystyle\left.-\frac{g_{s}^{2}(t^{A}t^{B})_{ij}G_{\alpha\beta}^{A}G_{\mu\nu}^{B}[f^{\alpha\beta\mu\nu}(k)+f^{\alpha\mu\beta\nu}(k)+f^{\alpha\mu\nu\beta}(k)]}{4(k^{2}-m_{s}^{2})^{2}}+\cdots\right],$
$\displaystyle D_{q}(x,0)=$ $\displaystyle\
\frac{i\delta_{ij}\not{x}}{2\pi^{2}x^{4}}-\frac{ig_{s}G_{\alpha\beta}^{A}t_{ij}^{A}(\not{x}\sigma^{\alpha\beta}+\sigma^{\alpha\beta}\not{x})}{32\pi^{2}x^{2}}+\cdots,$
(36)

where

$\displaystyle f^{\alpha\beta\mu\nu}(k)=\
(\not{k}+m_{s})\gamma^{\alpha}(\not{k}+m_{s})\gamma^{\beta}(\not{k}+m_{s})\gamma^{\mu}(\not{k}+m_{s})\gamma^{\nu}(\not{k}+m_{s}),$
(37)

and we have only present the terms relevant to the OPE up to dimension-5
operators in Eq.(36).

It can be found that the diagram (b), (c), (e) and (f) in Fig. 3 vanish after
the Borel transformation due to the same reason as that happens in the
$\bar{q}q$ condensate diagrams. The diagram (d) vanishes after the projection
introduced in Eq. (19). Thus diagram (a) is the only non-vanishing diagram
with amplitude:

$\displaystyle\Pi_{\mu\nu\rho}^{\bar{q}Gq(2)}(p_{1},p_{2},q^{2})=$
$\displaystyle\ i^{2}\int d^{4}xd^{4}y\ e^{ip_{2}\cdot x}e^{-ip_{1}\cdot
y}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\
e^{ik_{1}\cdot y}e^{-ik_{2}\cdot x}$
$\displaystyle\times\left[\sigma_{\mu\nu}\left(-i\over
4\right)\frac{\sigma^{\alpha\beta}(\not{k}_{2}+m_{s})+(\not{k}_{2}+m_{s})\sigma^{\alpha\beta}}{(k_{2}^{2}-m_{s}^{2})^{2}}\frac{i(\not{k}_{1}+m_{s})}{k_{1}^{2}-m_{s}^{2}}\gamma_{\rho}\gamma_{5}\right]_{ab}$
$\displaystyle\times t^{A}_{ij}\langle
0|{\bar{q}}_{a}^{i}(x)g_{s}G_{\alpha\beta}^{A}(0)q_{b}^{i}(y)|0\rangle.$ (38)

Using the the quark-gluon condensate formula, keeping the leading term:

$\displaystyle\langle
0|{\bar{q}}_{a}^{i}(x)g_{s}G_{\alpha\beta}^{A}(0)q_{b}^{i}(y)|0\rangle=\frac{1}{192}\langle\bar{q}Gq\rangle(\sigma_{\alpha\beta})_{ba}t_{ji}^{A}+\cdots,$
(39)

and conducting the projection introduced in Eq. (19), we can obtain

$\displaystyle{\tilde{\Pi}}^{\bar{q}Gq(2)}(p_{1},p_{2},q^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{24}m_{s}\langle\bar{q}Gq\rangle\frac{\partial}{\partial
M^{2}}\frac{1}{(p_{1}^{2}-m_{s}^{2})(p_{2}^{2}-M^{2})}|_{M^{2}=m_{s}^{2}}.$
(40)

The Borel transformed form reads as

$\displaystyle{\cal B}_{T_{1},T_{2}}\\{{\tilde{\Pi}}^{\bar{q}Gq(2)}\\}(q^{2})$
$\displaystyle=$
$\displaystyle-\frac{1}{6}m_{s}\langle\bar{q}Gq\rangle\frac{1}{T_{2}^{2}}e^{-m_{s}^{2}/T_{1}^{2}}e^{-m_{s}^{2}/T_{2}^{2}}.$
(41)

###  IV.4 GG condensate diagrams

Figure 4: The $GG$ condensate diagram contribution to the correlation
function, where two soft gluons are emitted from the internal quark lines and
condensate with each other. The diagrams (d), (e) and (f) can be neglected in
this calculation since the diagrams (d), (e) are suppressed by $m_{s}^{2}$,
while the diagram (f) is suppressed by $m^{2}$.

The gluon-gluon condensate diagrams are shown in Fig. 4, where the internal
quarks interact with two soft background gluon fields which condensate in the
vaccum. These diagrams provide the dimension-4 operator contribution in the
OPE. In the diagrams (a), both the two $s$ quark interact with the background
gluons, and the corresponding amplitude reads as

$\displaystyle\Pi_{\mu\nu\rho}^{GG(a)}(p_{1},p_{2},q^{2})$ $\displaystyle=$
$\displaystyle\int d^{4}xd^{4}y\ e^{ip_{2}\cdot x}e^{-ip_{1}\cdot
y}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{d^{4}k}{(2\pi)^{4}}\
e^{ik_{1}\cdot y}e^{-ik_{2}\cdot
x}e^{-ik\cdot(y-x)}\left(-\frac{i}{4}\right)^{2}$ $\displaystyle\times{\rm
tr}\left[\frac{-i\not{k}}{k^{2}}\sigma_{\mu\nu}\frac{\sigma^{\alpha\beta}(\not{k}_{2}+m_{s})+(\not{k}_{2}+m_{s})\sigma^{\alpha\beta}}{(k_{2}^{2}-m_{s}^{2})^{2}}\frac{\sigma^{\kappa\tau}(\not{k}_{1}+m_{s})+(\not{k}_{1}+m_{s})\sigma^{\kappa\tau}}{(k_{2}^{2}-m_{s}^{2})^{2}}\gamma_{\rho}\gamma_{5}\right]$
$\displaystyle\times{\rm tr}[t^{A}t^{A}]g_{s}^{2}\langle
0|G_{\alpha\beta}^{A}(0)G_{\kappa\tau}^{A}(0)|0\rangle.$ (42)

Using the gluon condensate formula:

$\displaystyle g_{s}^{2}\langle
0|G_{\alpha\beta}^{A}(0)G_{\kappa\tau}^{A}(0)|0\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{96}\langle
GG\rangle\delta_{AB}(g_{\alpha\kappa}g_{\beta\tau}-g_{\alpha\tau}{\beta\kappa}),$
(43)

and extracting the imaginary part by cutting rules, we arrive at

$\displaystyle{\rm Im}^{2}\Pi_{\mu\nu\rho}^{GG(a)}(p_{1},p_{2},q^{2})$
$\displaystyle=$ $\displaystyle\frac{1}{3072\pi}\frac{\partial^{2}}{\partial
M_{1}^{2}\partial M_{2}^{2}}\int d\Phi_{\Delta}[p_{1},p_{2},M_{1},M_{2},0]$
$\displaystyle\times{\rm
tr}\left[\not{k}\sigma_{\mu\nu}(\sigma^{\alpha\beta}(\not{k}_{2}+m_{s})+(\not{k}_{2}+m_{s})\sigma^{\alpha\beta})\right.$
(44)
$\displaystyle\left.\times(\sigma^{\kappa\tau}(\not{k}_{1}+m_{s})+(\not{k}_{1}+m_{s})\sigma^{\kappa\tau})\gamma_{\rho}\gamma_{5}\right]$
$\displaystyle\times(g_{\alpha\kappa}g_{\beta\tau}-g_{\alpha\tau}{\beta\kappa})|_{M_{1}^{2}=m_{s}^{2},M_{2}^{2}=m_{s}^{2}}.$
(45)

Note that before doing the derivative on the auxiliary masses we must
temporary change the invariant mass square of the $k_{1},k_{2}$ lines to be
$M_{1}^{2},M_{2}^{2}$. The analytical result of ${\rm
Im}^{2}\tilde{\Pi}^{GG(k_{1}k_{2})}$ is given in the Appendix B.

In the diagrams (b) and (c), one of the two condensing gluons comes from the
massless quark. The massless quark propagator in the background gluon field
has been given by Eq. (36) with the use of coordinate space. The amplitude of
diagram (b) reads as

$\displaystyle\Pi_{\mu\nu\rho}^{GG(b)}(p_{1},p_{2},q^{2})$ $\displaystyle=$
$\displaystyle\int d^{4}xd^{4}y\ e^{ip_{2}\cdot x}e^{-ip_{1}\cdot
y}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\
e^{ik_{1}\cdot y}e^{-ik_{2}\cdot
x}\left(-\frac{i}{32\pi^{2}}\right)\left(-\frac{i}{4}\right){\rm
tr}[t^{A}t^{B}]$ $\displaystyle\times{\rm
tr}\left[\frac{(\not{y}-\not{x})\sigma^{\kappa\tau}+\sigma^{\kappa\tau}(\not{y}-\not{x})}{(y-x)^{2}}\sigma_{\mu\nu}\frac{i(\not{k}_{2}+m_{s})}{k_{2}^{2}-m_{s}^{2}}\frac{\sigma^{\alpha\beta}(\not{k}_{1}+m_{s})+(\not{k}_{1}+m_{s})\sigma^{\alpha\beta}}{(k_{1}^{2}-m_{s}^{2})^{2}}\gamma_{\rho}\gamma_{5}\right]$
$\displaystyle\times g_{s}^{2}\langle
0|G_{\alpha\beta}^{A}(0)G_{\kappa\tau}^{B}(0)|0\rangle.$ (46)

Redefining the coordinate: $w=y-x$, and using the integration formula in the
coordinate space:

$\displaystyle\int d^{4}w\ e^{-i(p_{1}-k_{1})\cdot
w}\frac{1}{w^{2}}=(-4\pi^{2}i)\frac{1}{(p_{1}-k_{1})^{2}},$ (47)

we have the imaginary part as

$\displaystyle\ {\rm
Im}^{2}{\tilde{\Pi}}^{GG(b)}(p_{1},p_{2},q^{2})(p_{1}^{\beta^{\prime}}p_{2}^{\alpha^{\prime}}-p_{2}^{\alpha^{\prime}}p_{2}^{\beta^{\prime}})$
$\displaystyle=$ $\displaystyle-\frac{1}{6144\pi}\langle
GG\rangle\epsilon^{\mu\rho\alpha^{\prime}\beta^{\prime}}p_{1}^{\nu}\frac{\partial}{\partial
M_{1}^{2}}\left(\frac{\partial}{\partial
p_{1}^{\sigma}}+\frac{\partial}{\partial p_{2}^{\sigma}}\right)$
$\displaystyle\times\int
d\Phi_{\Delta}[p_{1},p_{2},M_{1},m_{s},0][g_{\alpha\kappa}g_{\beta\tau}-g_{\alpha\tau}g_{\beta\kappa}]$
$\displaystyle\times{\rm
tr}[(\gamma^{\sigma}\sigma^{\kappa\tau}+\sigma^{\kappa\tau}\gamma^{\sigma})\sigma_{\mu\nu}(\sigma^{\alpha\beta}(\not{k}_{2}+m_{s})$
$\displaystyle+(\not{k}_{2}+m_{s})\sigma^{\alpha\beta})(k_{1}^{2}-m_{s}^{2})\gamma_{\rho}\gamma_{5}]|_{M_{1}^{2}=m_{s}^{2}},$
(48)

where the linear term of $w$ has been transformed to the derivatives of
$p_{1},p_{2}$. The calculation of diagram (c) is almost the same so it will
not be present here. On the other hand, it can be found that the diagrams (d),
(e) and (f) can be neglected in this calculation, since the diagrams (d), (e)
are suppressed by $m_{s}^{2}$, while the diagram (f) is suppressed by $m^{2}$.
Therefore these three diagrams will not be considered here. The analytical
results for the diagrams (a), (b) and (c) are given in the Appendix B.

##  V Numerical Results

The masses of $K_{1A},K_{1B}$ and their decay constants are taken from
Ref.Yang:2007zt : $m_{K_{1A}}=1.31\pm 0.06\ {\rm GeV}$, $m_{K_{1B}}=1.34\pm
0.08\ {\rm GeV}$, $f_{1A}=0.25\pm 0.013\ {\rm GeV}$ and $f_{1B}=0.19\pm 0.01\
{\rm GeV}$. In this work we set $m_{u}=m_{d}=0$, and $m_{s}=(0.1\pm 0.005)\
{\rm GeV}$ at the energy scale $\mu=m_{K_{1}}=1.3\ {\rm GeV}$
ParticleDataGroup:2020ssz . The condensate parameters are taken as
Ioffe:2005ym ; Colangelo:2000dp : $\langle\bar{q}q\rangle=-(0.24\pm
0.01\mathrm{GeV})^{3},\left\langle\bar{q}Gq\right\rangle=m_{0}^{2}\langle\bar{q}q\rangle\
{\rm with}\ m_{0}^{2}=(0.8\pm 0.2)\mathrm{GeV}^{2},\langle
GG\rangle=(4\pi^{2})(0.012\pm 0.004)\mathrm{GeV}^{4}$.

In terms of the threshold parameters, note that since $J^{1A}_{\rho}$ can
create both pseudoscalar and axial vector kaons, $s_{1}^{\rm th}$ should
correspond to the excited pseudoscalar $K(1460)$. On the other hand,
$J^{1B}_{\mu\nu}$ can only create axial vector kaons, thus $s_{2}^{\rm th}$
should correspond to the excited axial vector $K_{1}(1650)$. Therefore we
choose $s_{1}^{\rm th}=1.46^{2}\ {\rm GeV}^{2}$ and $s_{2}^{\rm th}=1.65^{2}\
{\rm GeV}^{2}$.

The determination of Borel parameters depends on two criteria. Firstly, the
contribution from the continuous spectrum must be suppressed so that be
smaller than the pole contribution. Quantitatively, this criterion can be
expressed by the constraint:

$\displaystyle\xi_{\rm conti}$ $\displaystyle\equiv\frac{\int_{s_{1}^{\rm
th}}^{\infty}ds_{1}\int_{s_{2}^{\rm th}}^{\infty}ds_{2}\
e^{-\frac{s_{1}}{T_{1}^{2}}}e^{-\frac{s_{2}}{T_{2}^{2}}}{\rm
Im}^{2}{\tilde{\Pi}}^{\rm
QCD}(s_{1},s_{2},q^{2})}{\int_{0}^{\infty}ds_{1}\int_{0}^{\infty}ds_{2}\
e^{-\frac{s_{1}}{T_{1}^{2}}}e^{-\frac{s_{2}}{T_{2}^{2}}}{\rm
Im}^{2}{\tilde{\Pi}}^{\rm QCD}(s_{1},s_{2},q^{2})}$ $\displaystyle\lesssim
0.5,$ (49)

where the nominator denotes the contribution from the continuous spectrum,
while the denominator denotes all the spectrum contribution. Note that the
Borel parameter is related with the corresponding hadron mass. Since the mass
difference of the intial and final kaon is little, one can simply set
$T_{1}=T_{2}=T$. Without loss of generality, $q^{2}$ can be chosen by an
arbitrary value in the deep Euclidean region when determining $T^{2}$. Here we
choose $q^{2}=-6\ {\rm GeV}^{2}$ and present $\xi_{\rm conti}$ as a function
of $T^{2}$ in Fig.5, where the blue and red bands denote the errors from the
uncertainties of condensate parameters and $m_{K_{1A}},\ f_{1A},\
f_{1B}^{\perp}$, respectively. It can be found that $\xi_{\rm conti}$
increases with the increasing of $T^{2}$. $\xi_{\rm conti}=50\%$ occurs at
$T^{2}=1.16\thicksim 1.81\ {\rm GeV}^{2}$, which gives the range of the upper
limit for the Borel parameter: $1.16\ {\rm GeV}^{2}<T_{\rm upper}^{2}<1.81\
{\rm GeV}^{2}$.

Figure 5: $\xi_{\rm conti}$ as a function of $T^{2}$ with $q^{2}=-6\ {\rm
GeV}^{2}$. The blue and red bands denote the errors from the uncertainties of
condensate parameters and $m_{K_{1A}},\ f_{1A},\ f_{1B}^{\perp}$,
respectively.

Figure 6: Absolute $F_{1}$ contributed from various of condensate diagrams
(left) and $\eta_{\rm cond}$ as a function of $T^{2}$ (right) with $q^{2}=-6\
{\rm GeV}^{2}$. The blue and red bands denote the errors from the
uncertainties of condensate parameters and $m_{K_{1A}},\ f_{1A},\
f_{1B}^{\perp}$, respectively.

The second criterion demands the convergence of OPE, which means that the
$F_{1}$ contributed from the higher dimension condensates must be smaller than
that from the lower ones, namely:

$\displaystyle|F_{1}^{\rm
pert}|>|F_{1}^{\bar{q}q}|>|F_{1}^{GG}|>|F_{1}^{\bar{q}Gq}|.$ (50)

The left diagram in Fig.6 shows the absolute $F_{1}$ contributed from various
of condensate diagrams with $q^{2}=-6\ {\rm GeV}^{2}$. It is obvious that the
perturbative diagram contribution is larger than all the other condensate
contributions. The right diagram in Fig.6 shows the fraction of the condensate
and the perturbative contribution:

$\displaystyle\eta_{\rm
cond}\equiv\frac{|F_{1}^{\bar{q}q}+F_{1}^{GG}+F_{1}^{\bar{q}Gq}|}{|F_{1}^{\rm
pert}|}.$ (51)

It can be found that $\eta_{\rm cond}<1$ is safely satisfied in a wide $T^{2}$
range. Furthermore, the comparison between the condensate contribution
themselves leads to

$\displaystyle|F_{1}^{\bar{q}q}|>|F_{1}^{GG}+F_{1}^{\bar{q}Gq}|$
$\displaystyle\ \Longrightarrow\ T^{2}\gtrsim 1.58\ {\rm GeV}^{2},$
$\displaystyle|F_{1}^{GG}|>|F_{1}^{\bar{q}Gq}|$ $\displaystyle\
\Longrightarrow\ T^{2}\gtrsim 1.68\ {\rm GeV}^{2}.$ (52)

Note that the critical points obtained above are within the upper limit range
given by the first criterion $1.16\ {\rm GeV}^{2}<T_{\rm upper}^{2}<1.81\ {\rm
GeV}^{2}$, which determines a narrow window for $T^{2}$: $1.6\ {\rm
GeV}^{2}\lesssim T^{2}\lesssim 1.8\ {\rm GeV}^{2}$.

Figure 7: Absolute value of $\theta_{K_{1}}$ as a function of $T^{2}$. The
$\theta_{K_{1}}$ is calculated from Eq.(13), with $F_{1}(0)$ being fitted by
Eq. (53) in the region $-10\ {\rm GeV}^{2}<q^{2}<-3\ {\rm GeV}^{2}$. The blue
and red bands denote the errors from the uncertainties of condensate
parameters and $m_{K_{1A}},\ f_{1A},\ f_{1B}^{\perp}$, respectively.

Finally, we have to check the behavior of $F_{1}(0)$ in the Borel parameter
window $1.6\ {\rm GeV}^{2}\lesssim T^{2}\lesssim 1.8\ {\rm GeV}^{2}$. The
physical region result: $F_{1}(q^{2}\geq 0)$ can be obtained by using the deep
Euclidean values $F_{1}(q^{2}\leq 0)$ and analytic continuation. In this work,
a single pole formula is used:

$\displaystyle F_{1}(q^{2})=\frac{F_{1}(0)}{1-q^{2}/m_{\rm pole}^{2}}$ (53)

to fit $F_{1}(q^{2})$ in a region $-10\ {\rm GeV}^{2}<q^{2}<-3\ {\rm
GeV}^{2}$, which is chosen so that the spectral integrals can be calculated
safely by applying cutting rules. In Eq.(53), $F_{1}(0)$ and $m_{\rm pole}$
play the role of fitting parameters, and the mixing angle $\theta_{K_{1}}$ can
be obtained from $F_{1}(0)$ through Eq. (13). It should be mentioned that the
decay constants $f_{1A}$ and $f_{1B}^{\perp}$ in Eq. (23) are calculated by
QCDSR in Ref.Yang:2007zt , which have sign ambiguity. The reason is that when
using a two-point correlation in QCDSR to calculate $f_{1A}$ or
$f_{1B}^{\perp}$, one can only determine their square and thus the exact sign
cannot be determined. Considering the sign ambiguity, we present the absolute
value of $\theta_{K_{1}}$ as a function of $T^{2}$ in Fig.7. Including the
effect of error bands, we obtain the mixing angle as:
$|\theta_{K_{1}}|=22^{\circ}\pm 7^{\circ}$. Note that both $\theta_{K_{1}}$
and $90^{\circ}-\theta_{K_{1}}$ are the solutions to Eq. (13). Therefore
another possible mixing angle value is:
$|\theta_{K_{1}}^{\prime}|=68^{\circ}\pm 7^{\circ}$.

Table 1: Comparing the $|\theta_{K_{1}}|$ obtained in this work with those from literature.  Refs. | $|\theta_{K_{1}}|$
---|---
This Work |  $22^{\circ}\pm 7^{\circ}$ or $68^{\circ}\pm 7^{\circ}$
(1) Suzuki:1993yc  |  $33^{\circ}$ or $57^{\circ}$
(2) Burakovsky:1997dd  | $34^{\circ}<|\theta_{K_{1}}|<55^{\circ}$
(3) Cheng:2003bn ; Li:2009tx  |  $37^{\circ}$ or $58^{\circ}$
(4) Close:1997nm  |  $\left(31.7_{-2.5}^{+2.8}\right)^{\circ}$ or $\left(56.3_{-4.1}^{+3.9}\right)^{\circ}$
(5) Cheng:2013cwa  | $28^{\circ}<|\theta_{K_{1}}|<30^{\circ}$
(6) Dag:2012zz  | $39^{\circ}\pm 4^{\circ}$

In Table 1, we compare our result on $|\theta_{K_{1}}|$ with those obtained in
literature by various of methods:

  1. (1)

Using early experimental information on masses and the partial rates of
$K_{1}(1270)$ and $K_{1}(1400)$ Suzuki:1993yc ;

  2. (2)

Using non-relativistic constituent quark model with the inputs of the mass
difference between the $a_{1}(1260)$ and $b_{1}(1235)$ mesons, as well as the
ratio of the constituent quark masses Burakovsky:1997dd ;

  3. (3)

Phenomenologically analyzing on the $\tau$ weak decays: $\tau\to
K_{1}(1270)\nu_{\tau}$ and $\tau\to K_{1}(1400)\nu_{\tau}$ Cheng:2003bn ;
Li:2009tx ;

  4. (4)

Analyzing the $f_{1}(1285)-f_{1}(1420)$ mixing angle $\theta_{{}^{3}P_{1}}$
and its correlation to $\theta_{K_{1}}$ Close:1997nm ;

  5. (5)

Analyzing both the mixing angle of $f_{1}(1285)-f_{1}(1420)$ and
$h_{1}(1170)-h_{1}(1380)$ Cheng:2013cwa ;

  6. (6)

Relating $\theta_{K_{1}}$ with a two-point correlation function, which is
studied by QCDSR.Dag:2012zz .

It can be found that most of the $|\theta_{K_{1}}|$ value in the literature
are in the vicinity of either $33^{\circ}$ or $57^{\circ}$. Our result:
$|\theta_{K_{1}}|=22^{\circ}\pm 7^{\circ}$ is slightly below this range but
consistent with that given by Ref. Cheng:2013cwa  within the error. In Ref.
Cheng:2013cwa , to determine $\theta_{K_{1}}$ the authors found the
correspondence between $\theta_{K_{1}}$ and the $f_{1}(1285)-f_{1}(1420)$,
$h_{1}(1170)-h_{1}(1380)$ mixing angles, and ruled out unreasonable
$\theta_{K_{1}}$ values in previous literature. In Ref. Dag:2012zz , a
different QCDSR program was performed to extract $\theta_{K_{1}}$, where the
authors related $\theta_{K_{1}}$ with a two-point correlation function (Eq.[2]
in Ref. Dag:2012zz ) and calculated it by OPE. However, when introducing the
interpolation current of $K_{1A}$, the authors missed the contribution from
the pseudoscalar $K$ as illustrated in section III, and wrongly extracted the
longitudinal component of the two-point correlation function.

##  VI summary

In this work we investigate the $K_{1}(1270)-K_{1}(1400)$ mixing caused by the
flavor $SU(3)$ symmetry breaking. The mixing angle is expressed by a
$K_{1A}\to K_{1B}$ matrix element induced by the $s$ quark mass operator that
breaks flavor $SU(3)$ symmetry. We focus on the QCD contribution to this
matrix element and calculate it by QCDSR, where a three-point correlation
function is defined and calculated both at the hadron and quark-gluon levels.
In the calculation at the quark-gluon level, the operator product expansion is
up to dimension-5 condensates. A detailed numerical analysis is performed to
determine the Borel parameters, and the obtained mixing angle is
$\theta_{K_{1}}=22^{\circ}\pm 7^{\circ}$ or $\theta_{K_{1}}=68^{\circ}\pm
7^{\circ}$, which is consistent with the phenomenological analysis on the
relation between $\theta_{K_{1}}$ and the mixing angle of strangeless axial
vector mesons.

## Acknowledgement

We thank Wei Wang and Zhen-Xing Zhao for valuable discussions. This work is
supported in part by Natural Science Foundation of China under Grant
No.12305103, 12205180, and 12147140. The work of Y.J. Shi is also supported by
Opening Foundation of Shanghai Key Laboratory of Particle Physics and
Cosmology under Grant No.22DZ2229013-2. The work of J.Zeng is also partially
supported by the Project funded by China Postdoctoral Science Foundation under
Grant No. 2022M712088.

##  Appendix A Triangle Diagram Integration

Here we present the rank one and two triangle diagram integrations. They are
defined as

$\displaystyle\int d\Phi_{\Delta}(p_{1},p_{2},m_{1},m_{2},m)k^{\mu}$
$\displaystyle=(A_{1}p_{1}^{\mu}+B_{1}p_{2}^{\mu})I_{\Delta},$
$\displaystyle\int d\Phi_{\Delta}(p_{1},p_{2},m_{1},m_{2},m)\
k^{\mu_{1}}k^{\mu_{2}}$
$\displaystyle=[A_{2}p_{1}^{\mu_{1}}p_{1}^{\mu_{2}}+B_{2}p_{2}^{\mu_{1}}p_{2}^{\mu_{2}}+C_{2}(p_{1}^{\mu_{1}}p_{2}^{\mu_{2}}+p_{2}^{\mu_{1}}p_{1}^{\mu_{2}})+D_{2}g^{\mu_{1}\mu_{2}}]I_{\Delta},$
(54)

where

$\displaystyle A_{1}=$
$\displaystyle\frac{-\left(m^{2}(q^{2}-s_{1}+s_{2})\right)+s_{2}\left(2m_{1}^{2}-q^{2}-s_{1}+s_{2}\right)+m_{2}^{2}(q^{2}-s_{1}-s_{2})}{q^{4}-2q^{2}(s_{1}+s_{2})+(s_{1}-s_{2})^{2}},$
$\displaystyle B_{1}=$
$\displaystyle\frac{-\left(m^{2}(q^{2}+s_{1}-s_{2})\right)+m_{1}^{2}(q^{2}-s_{1}-s_{2})+s_{1}\left(2m_{2}^{2}-q^{2}+s_{1}-s_{2}\right)}{q^{4}-2q^{2}(s_{1}+s_{2})+(s_{1}-s_{2})^{2}},$
$\displaystyle A_{2}=$
$\displaystyle\frac{1}{(q^{4}+(s_{1}-s_{2})^{2}-2q^{2}(s_{1}+s_{2}))^{2}}\big{[}m^{4}(q^{4}-2q^{2}(s_{1}-2s_{2})+(s_{1}-s_{2})^{2})$
$\displaystyle+(6m_{1}^{4}+q^{4}+q^{2}(4s_{1}-2s_{2})+(s_{1}-s_{2})^{2}-6m_{1}^{2}(q^{2}+s_{1}-s_{2}))s_{2}^{2}$
$\displaystyle+m_{2}^{4}(q^{4}+s_{1}^{2}+4s_{1}s_{2}+s_{2}^{2}-2q^{2}(s_{1}+s_{2}))$
$\displaystyle-2m_{2}^{2}s_{2}(q^{4}-2s_{1}^{2}+q^{2}(s_{1}-2s_{2})+s_{1}s_{2}+s_{2}^{2}+3m_{1}^{2}(-q^{2}+s_{1}+s_{2}))$
$\displaystyle-2m^{2}(m_{2}^{2}(q^{4}+s_{1}^{2}+s_{1}s_{2}-2s_{2}^{2}+q^{2}(-2s_{1}+s_{2}))$
$\displaystyle+s_{2}(-2q^{4}+(s_{1}-s_{2})^{2}+3m_{1}^{2}(q^{2}-s_{1}+s_{2})+q^{2}(s_{1}+s_{2})))\big{]},$
$\displaystyle B_{2}=$
$\displaystyle\frac{1}{(q^{4}+(s_{1}-s_{2})^{2}-2q^{2}(s_{1}+s_{2}))^{2}}\big{[}m^{4}(q^{4}+q^{2}(4s_{1}-2s_{2})+(s_{1}-s_{2})^{2})$
$\displaystyle+s_{1}^{2}(6m_{2}^{4}+q^{4}-2q^{2}(s_{1}-2s_{2})+(s_{1}-s_{2})^{2}-6m_{2}^{2}(q^{2}-s_{1}+s_{2}))$
$\displaystyle+m_{1}^{4}(q^{4}+s_{1}^{2}+4s_{1}s_{2}+s_{2}^{2}-2q^{2}(s_{1}+s_{2}))$
$\displaystyle-2m_{1}^{2}s_{1}(q^{4}+s_{1}^{2}+s_{1}s_{2}-2s_{2}^{2}+q^{2}(-2s_{1}+s_{2})+3m_{2}^{2}(-q^{2}+s_{1}+s_{2}))$
$\displaystyle-2m^{2}(m_{1}^{2}(q^{4}-2s_{1}^{2}+q^{2}(s_{1}-2s_{2})+s_{1}s_{2}+s_{2}^{2})$
$\displaystyle+s_{1}(-2q^{4}+(s_{1}-s_{2})^{2}+3m_{2}^{2}(q^{2}+s_{1}-s_{2})+q^{2}(s_{1}+s_{2})))\big{]},$
$\displaystyle C_{2}=$
$\displaystyle\frac{1}{(q^{4}+(s_{1}-s_{2})^{2}-2q^{2}(s_{1}+s_{2}))^{2}}\big{[}3m_{1}^{4}(q^{2}-s_{1}-s_{2})s_{2}+m^{4}(2q^{4}-(s_{1}-s_{2})^{2}-q^{2}(s_{1}+s_{2}))$
$\displaystyle-m^{2}(-q^{6}+q^{4}s_{1}+q^{2}s_{1}^{2}-s_{1}^{3}+q^{4}s_{2}-6q^{2}s_{1}s_{2}+s_{1}^{2}s_{2}+q^{2}s_{2}^{2}+s_{1}s_{2}^{2}-s_{2}^{3}$
$\displaystyle+2m_{2}^{2}(q^{4}+q^{2}s_{1}-2s_{1}^{2}-2q^{2}s_{2}+s_{1}s_{2}+s_{2}^{2})+2m_{1}^{2}(q^{4}+s_{1}^{2}+s_{1}s_{2}-2s_{2}^{2}+q^{2}(-2s_{1}+s_{2})))$
$\displaystyle+2m_{1}^{2}(-s_{2}(q^{4}-2s_{1}^{2}+q^{2}(s_{1}-2s_{2})+s_{1}s_{2}+s_{2}^{2})+m_{2}^{2}(q^{4}+s_{1}^{2}+4s_{1}s_{2}+s_{2}^{2}-2q^{2}(s_{1}+s_{2})))$
$\displaystyle-
s_{1}(3m_{2}^{4}(-q^{2}+s_{1}+s_{2})+2m_{2}^{2}(q^{4}+s_{1}^{2}+s_{1}s_{2}-2s_{2}^{2}+q^{2}(-2s_{1}+s_{2}))$
$\displaystyle+s_{2}(-2q^{4}+(s_{1}-s_{2})^{2}+q^{2}(s_{1}+s_{2})))\big{]},$
$\displaystyle D_{2}=$
$\displaystyle\frac{1}{2(q^{4}+(s_{1}-s_{2})^{2}-2q^{2}(s_{1}+s_{2}))}\big{[}m^{4}q^{2}+m_{1}^{4}s_{2}+m_{1}^{2}(m_{2}^{2}(q^{2}-s_{1}-s_{2})$
$\displaystyle+s_{2}(-q^{2}-s_{1}+s_{2}))+s_{1}(m_{2}^{4}+q^{2}s_{2}-m_{2}^{2}(q^{2}-s_{1}+s_{2}))$
$\displaystyle-m^{2}(m_{2}^{2}(q^{2}+s_{1}-s_{2})+m_{1}^{2}(q^{2}-s_{1}+s_{2})+q^{2}(-q^{2}+s_{1}+s_{2}))\big{]}.$
(55)

##  Appendix B Analytical Results

Here we present the analytical results for the calculation of perturbative
diagram and $GG$ condensate diagrams. The imaginary part of the correlation
function contributed by perturbative diagram reads as

$\displaystyle{\rm Im}^{2}{\tilde{\Pi}}^{\rm pert}(p_{1},p_{2},q^{2})$
$\displaystyle=$
$\displaystyle\frac{N_{c}}{4\pi}I_{\Delta}[m_{s}^{2}(2A_{1}+10B_{1}-1)+B_{1}(-3q^{2}-s_{1}+s_{2})+s_{1}].$
(56)

The imaginary parts of the correlation function contributed by the $GG$
condensate diagrams read as

$\displaystyle{\rm Im}^{2}{\tilde{\Pi}}^{GG(a)}(p_{1},p_{2},q^{2})$
$\displaystyle=$ $\displaystyle\frac{\langle
GG\rangle}{48\pi}\frac{\partial^{2}}{\partial M_{1}^{2}\partial M_{2}^{2}}$
$\displaystyle\times
I_{\Delta}[M_{1}^{2}(2A_{1}+8B_{1}-1)+B_{1}(6M_{2}^{2}+24m_{s}^{2}-7q^{2}-s_{1}+s_{2})+s_{1}]|_{M_{1}^{2}=M_{2}^{2}=m_{s}^{2}}.$
(57) $\displaystyle{\rm Im}^{2}{\tilde{\Pi}}^{GG(b)}(p_{1},p_{2},q^{2})$
$\displaystyle=$ $\displaystyle\frac{\langle
GG\rangle}{4\pi}\frac{\partial}{\partial
M_{1}^{2}}I_{\Delta}(A_{1}+B_{1}-1)|_{M_{1}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{96\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{1}^{2}}$
$\displaystyle\times[M_{1}^{2}(q^{2}-5s_{1}+5s_{2})+m_{s}^{2}(q^{2}+s_{1}-s_{2})+q^{2}(-3q^{2}+7s_{1}+3s_{2})]$
$\displaystyle\times\theta[(M_{1}^{2})^{2}(-s_{2})+M_{1}^{2}(m_{s}^{2}(-q^{2}+s_{1}+s_{2})+s_{2}(q^{2}+s_{1}-s_{2}))-s_{1}\left(m_{s}^{4}-m_{s}^{2}(q^{2}-s_{1}+s_{2})+q^{2}s_{2}\right)]$
$\displaystyle\times\theta(s_{1}-M_{1}^{2})\theta(s_{2}-m_{s}^{2})|_{M_{1}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{48\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{1}^{2}}$
$\displaystyle\times[(M_{1}^{2})^{2}s_{2}+M_{1}^{2}(q^{2}-s_{1}-s_{2})\left(m_{s}^{2}-q^{2}+s_{1}-s_{2}\right)+m_{s}^{4}s_{1}-2m_{s}^{2}\left(q^{4}-q^{2}(s_{1}+2s_{2})+s_{2}(s_{2}-s_{1})\right)$
$\displaystyle+q^{2}\left(q^{4}-2q^{2}(s_{1}+s_{2})+s_{1}^{2}+\text{s1}s_{2}+s_{2}^{2}\right)]$
$\displaystyle\times\frac{s_{2}^{(2)}-s_{2}^{(1)}}{|s_{2}^{(2)}-s_{2}^{(1)}|}[\delta(s_{2}-s_{2}^{(1)})-\delta(s_{2}-s_{2}^{(2)})]|_{M_{1}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{48\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{1}^{2}}$
$\displaystyle\times[s_{1}(M_{1}^{2}(q^{2}-s_{1}+s_{2})+m_{s}^{2}(q^{2}+s_{1}-s_{2})+q^{2}(-q^{2}+s_{1}+s_{2}))]$
$\displaystyle\times\frac{s_{1}^{(2)}-s_{1}^{(1)}}{|s_{1}^{(2)}-s_{1}^{(1)}|}[\delta(s_{1}-s_{1}^{(1)})-\delta(s_{1}-s_{1}^{(2)})]|_{M_{1}^{2}=m_{s}^{2}}.$
(58) $\displaystyle{\rm Im}^{2}{\tilde{\Pi}}^{GG(c)}(p_{1},p_{2},q^{2})$
$\displaystyle=$ $\displaystyle\frac{\langle
GG\rangle}{8\pi}\frac{\partial}{\partial
M_{2}^{2}}I_{\Delta}(A_{1}+B_{1}-1)|_{M_{2}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{48\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{2}^{2}}$
$\displaystyle\times[3M_{2}^{2}(q^{2}+s_{1}-s_{2})-m_{s}^{2}(q^{2}+7s_{1}-7s_{2})+q^{2}(-3q^{2}+7s_{1}+3s_{2})]\theta[-(M_{2}^{2})^{2}s_{1}+m_{s}^{2}(M_{2}^{2}(-q^{2}+s_{1}+s_{2})$
$\displaystyle+s_{2}(q^{2}+s_{1}-s_{2}))+M_{2}^{2}s_{1}(q^{2}-s_{1}+s_{2})+m_{s}^{4}(-s_{2})-q^{2}s_{1}s_{2}]\theta(s_{1}-m_{s}^{2})\theta(s_{2}-M_{2}^{2})|_{M_{2}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{24\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{1}^{2}}$
$\displaystyle\times[(M_{2}^{2})^{2}s_{1}+M_{2}^{2}(q^{2}-s_{1}-s_{2})\left(m_{s}^{2}-q^{2}-s_{1}+s_{2}\right)+m_{s}^{4}s_{2}-2m_{s}^{2}\left(q^{4}-q^{2}(2s_{1}+s_{2})+s_{1}(s_{1}-s_{2})\right)$
$\displaystyle+q^{2}\left(q^{4}-2q^{2}(s_{1}+s_{2})+s_{1}^{2}+s_{1}s_{2}+s_{2}^{2}\right)]$
$\displaystyle\times\frac{s_{2}^{(2)}-s_{2}^{(1)}}{|s_{2}^{(2)}-s_{2}^{(1)}|}[\delta(s_{2}-s_{2}^{(1)})-\delta(s_{2}-s_{2}^{(2)})]|_{M_{2}^{2}=m_{s}^{2}}$
$\displaystyle-\frac{\langle
GG\rangle}{24\left((-q^{2}+s_{1}+s_{2})^{2}-4s_{1}s_{2}\right)^{3/2}}\frac{\partial}{\partial
M_{1}^{2}}$
$\displaystyle\times[M_{2}^{2}(q^{2}+s_{1}-s_{2})+m_{s}^{2}(q^{2}-s_{1}+s_{2})+q^{2}(-q^{2}+s_{1}+s_{2})]$
$\displaystyle\times\frac{s_{1}^{(2)}-s_{1}^{(1)}}{|s_{1}^{(2)}-s_{1}^{(1)}|}[\delta(s_{1}-s_{1}^{(1)})-\delta(s_{1}-s_{1}^{(2)})]|_{M_{2}^{2}=m_{s}^{2}},$
(59)

where

$\displaystyle s_{1}^{(1)}$
$\displaystyle=\frac{m_{1}^{2}\left(m_{2}^{2}+s_{2}\right)+\left(m_{2}^{2}-s_{2}\right)\left(\sqrt{m_{1}^{4}-2m_{1}^{2}\left(m_{2}^{2}+q^{2}\right)+\left(m_{2}^{2}-q^{2}\right)^{2}}-m_{2}^{2}+q^{2}\right)}{2m_{2}^{2}},$
$\displaystyle s_{1}^{(2)}$
$\displaystyle=\frac{m_{1}^{2}\left(m_{2}^{2}+s_{2}\right)-\left(m_{2}^{2}-s_{2}\right)\left(\sqrt{m_{1}^{4}-2m_{1}^{2}\left(m_{2}^{2}+q^{2}\right)+\left(m_{2}^{2}-q^{2}\right)^{2}}+m_{2}^{2}-q^{2}\right)}{2m_{2}^{2}},$
$\displaystyle s_{2}^{(1)}$
$\displaystyle=-\frac{m_{1}^{4}-m_{1}^{2}m_{2}^{2}-m_{1}^{2}q^{2}-m_{1}^{2}s_{1}+\left(s_{1}-m_{1}^{2}\right)\sqrt{m_{1}^{4}-2m_{1}^{2}\left(m_{2}^{2}+q^{2}\right)+\left(m_{2}^{2}-q^{2}\right)^{2}}-m_{2}^{2}s_{1}+q^{2}s_{1}}{2m_{1}^{2}},$
$\displaystyle s_{2}^{(2)}$
$\displaystyle=-\frac{m_{1}^{4}-m_{1}^{2}m_{2}^{2}-m_{1}^{2}q^{2}-m_{1}^{2}s_{1}+\left(m_{1}^{2}-s_{1}\right)\sqrt{m_{1}^{4}-2m_{1}^{2}\left(m_{2}^{2}+q^{2}\right)+\left(m_{2}^{2}-q^{2}\right)^{2}}-m_{2}^{2}s_{1}+q^{2}s_{1}}{2m_{1}^{2}}.$
(60)

It should be noted that for the $GG(a)$ diagram, before doing derivatives on
$M_{1}^{2}$ and $M_{2}^{2}$, one must set the mass of the $k_{1},k_{2},k$
lines as $M_{1},M_{2},0$. For the $GG(b)$ and $GG(c)$ diagrams, before doing
derivatives, one must set the mass of the $k_{1},k_{2},k$ lines as
$M_{1},m_{s},0$ and $m_{s},M_{2},0$, respectively.

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