# Calculation of the chiral Lagrangian coefficients with light vector mesons

Zi-Kan<EMAIL_ADDRESS>Qing Wang1,2222Corresponding
author<EMAIL_ADDRESS>
1Department of Physics, Tsinghua University, Beijing 100084, People’s Republic
of China
2Center for High Energy Physics, Tsinghua University, Beijing 100084, People’s
Republic of China

###### Abstract

We calculate the coefficients in the effective chiral Lagrangian from QCD,
which includes pseudo-scalar mesons and vector mesons (with hidden symmetry),
up to $\mathcal{O}(p^{4})$. This encompasses both the normal and anomalous
parts. Our work builds on a previous study that derived the chiral Lagrangian
from first principles of QCD, where the low-energy coefficients are defined in
terms of specific Green’s functions in QCD. This research extends our earlier
efforts that focused on calculating the low-energy coefficients of the chiral
Lagrangian for pure pseudo-scalar mesons. This marks the first calculation of
chiral Lagrangian coefficients for vector mesons from QCD, particularly for
the important parameters $a$ and $g$, which are typically considered inputs in
existing literature. Notably, the regularization method used previously is
inadequate for this broader scope. We find that cut-off regularization yields
reasonable results for both pseudo-scalar mesons and vector mesons, though it
has certain limitations. Finally, we demonstrate that our method aligns with
the Weinberg sum rules.

##  I Introduction

The chiral Lagrangian for low lying pseudo-scalar mesons Weinberg (1979);
Gasser and Leutwyler (1984, 1985) stands as the most successful effective
field theory, having been extended to $\mathcal{O}(p^{8})$ Bijnens et al.
(2019) for higher precision in the low-energy expansion. To extend the energy
reach of chiral perturbation theory (ChPT), more hadrons can be included as
active degrees of freedom. Beyond the low-lying pseudo-scalar mesons, the next
mesons in the hadron spectrum are the light vector mesons, the $\rho$ meson
and its flavor partners. Several phenomenological models describe these vector
mesons, including the matter field model Ecker et al. (1989a), the massive
Yang-Mills model Schwinger (1967); Kaymakcalan and Schechter (1985); Meissner
(1988), the hidden local symmetry (HLS) model Bando et al. (1988); Harada and
Yamawaki (2003) and the anti-symmetric tensor field model Ecker et al.
(1989b), among others. It has been shown that these models are equivalent at
least at the tree level, and the relation of low energy coefficients (LECs)
across different models have been studied Ecker et al. (1989a); Tanabashi
(1996). In the chiral Lagrangian, the number of the LECs increases rapidly
with higher orders of the low-energy expansion or when more hadrons are
included. For example in the three-flavor case, the
$\mathcal{O}(p^{2}),\mathcal{O}(p^{4}),\mathcal{O}(p^{6})$ and
$\mathcal{O}(p^{8})$ chiral Lagrangians for low lying pseudo-scalar mesons
have 2, 12, 117 and 1959 LECs respectively Graf et al. (2021). In the anti-
symmetric tensor field model incorporating light vector mesons, the
$\mathcal{O}(p^{2}),\mathcal{O}(p^{4})$ and $\mathcal{O}(p^{6})$ chiral
Lagrangians for the three-flavor case have 4, 70 and 846 LECs respectively Guo
et al. (2020). Meanwhile, in the HLS model, the
$\mathcal{O}(p^{2}),\mathcal{O}(p^{4})$ and $\mathcal{O}(p^{6})$ chiral
Lagrangians for the three-flavor case have 4, 31 and 495 LECs respectively Guo
et al. (2020). Given the large number of LECs, it is challenging to fix them
all from experimental data, making the calculation of LECs crucial for
enhancing the predictive power of the chiral Lagrangian.

There are various methods to obtain LECs, with some of the more common ones
being resonance chiral theory Ecker et al. (1989b); Amoros et al. (2000);
Knecht and Nyffeler (2001), sum rules Golterman et al. (2014), lattice QCD
Necco (2008); Aoki et al. (2017), holographic theory Harada et al. (2006,
2010); Ma et al. (2013), QCD Wang et al. (2000); Yang et al. (2002); Jiang et
al. (2010); Jiang and Wang (2010) and global fit Bijnens and Jemos (2012).
Some methods provide values for individual LECs, while others yield values for
combinations of LECs. Each method has its own advantages and disadvantages.
The most systematic and reliable approach involves studying the relationship
between the chiral Lagrangian and the fundamental principles of QCD. In
previous papers Wang et al. (2000); Wang and Wang (2000), we derived the
effective chiral Lagrangian for pseudo-scalar mesons and vector mesons (with
hidden symmetry) from the first principles of QCD without approximation. All
the LECs are expressed in terms of specific Green’s functions in QCD, which
can be considered the fundamental QCD definitions of these coefficients.
However, solving these expressions strictly is challenging. Therefore, we
developed a systematic numerical method to calculate the LECs, which are
expressed in terms of the quark self-energy $\Sigma(k^{2})$ under certain
approximations, such as the large-$N_{c}$ limit, the leading order in
dynamical perturbation theory and the improved ladder approximation. By
solving the Schwinger-Dyson equation (SDE) for $\Sigma(k^{2})$, we obtain the
approximate LECs for pseudo-scalar mesons up to the $\mathcal{O}(p^{6})$ Yang
et al. (2002); Jiang et al. (2010); Jiang and Wang (2010), which are
consistent with the experimental values. Compared to the similar studies which
rely on certain a priori assumptions or approximations from the outset Balog
(1984); Andrianov (1985); Klevansky (1992); Hatsuda and Kunihiro (1994);
Bijnens (1996); Holdom (1992), the systematic errors in our calculations can
be compensated step by step Jiang et al. (2015).

Although our calculation for the LECs of pseudo-scalar mesons was successful,
extending these calculations to vector mesons is nontrivial. The key
difference lies in the introduction of HLS and the unique properties of vector
mesons. Pseudo-scalar mesons are treated as Goldstone bosons following
spontaneous chiral symmetry breaking. In the HLS model, vector mesons are
considered gauge fields, with their masses arising from the Higgs mechanism.
In previous papers Yang et al. (2002); Lu et al. (2002), we used Schwinger
proper time regularization to calculate the LECs for pseudo-scalar mesons,
yielding results that matched those in the literature Espriu et al. (1990) and
appeared reasonable Yang et al. (2002); Jiang et al. (2010). However, when
this regularization was applied to the HLS model to calculate the LECs for
vector mesons, the decay constant for vector mesons significantly deviated
from experimental values. We explored the impact of different regularizations
on the $\mathcal{O}(p^{2})$ LECs and found that the cut-off regularization
produced LECs for vector mesons consistent with experimental values, albeit at
a cost. Notably, $\mathcal{O}(p^{4})$ LECs are insensitive to the
regularizations, and there was minimal change in the LECs for pseudo-scalar
mesons. References Chen et al. (2020, 2021) also used cut-off regularization
to calculate the LECs in the chiral Lagrangian for heavy-light mesons,
obtaining the reasonable results. This suggests that cut-off regularization
may be suitable for higher energy regions. In summary, the
$\mathcal{O}(p^{2})$ LECs are sensitive to regularizations, and Schwinger
proper time regularization, previously used, is inadequate for vector mesons.
While the new regularization method aligns with empirical data, it still
presents some issues. Nonetheless, we demonstrate that our method is
consistent with the Weinberg sum rules Weinberg (1967) at the end of this
article.

It is worth noting that no new parameters are introduced for vector mesons in
this paper, even though the number of LECs is larger than those for pure
pseudo-scalar mesons. The parameters in the calculation are entirely fixed by
pseudo-scalar mesons, as in Refs. Yang et al. (2002); Jiang et al. (2010). For
the most important coefficients $a$ and $g$, this is the first time that they
have been obtained from theoretical calculation, whereas they are typically
treated as inputs in the literature.

The paper is organized as follows. In Sec. II, we provide a brief review about
the $\mathcal{O}(p^{2})$ Lagrangian in HLS model. In Sec. III, we review our
previous derivation of the effective chiral Lagrangian with vector mesons from
QCD. In Sec. IV, we perform the momentum expansion and express the LECs in
term of the quark’s self-energy. In Sec. V, we present and discuss the
numerical results. We derive the Weinberg sum rules using our method in Sec.
VI, and summarize in the end. The $\mathcal{O}(p^{4})$ Lagrangian, including
the normal and anomalous parts in HLS model, along with the analytical
expressions for corresponding LECs, are included in the Appendix.

##  II The hidden local symmetry model of Vector Mesons

The hidden local symmetry model describing the vector mesons is based on the
$G_{\text{global}}\times H_{\text{local}}$ symmetry, where $G=SU(3)_{L}\times
SU(3)_{R}$ is the global chiral symmetry of the ChPT and $H=SU(3)_{V}$ is the
HLS. The entire symmetry is spontaneously broken down to a diagonal sum $H$
which is the flavor symmetry. The gauge bosons of HLS acquire masses by the
Higgs mechanism, and are then identified as the light vector mesons, $\rho$
meson and its flavor partners.

In the HLS model, the building block $U$ in the ChPT is split into two
variables $\xi_{L}$ and $\xi_{R}$:

$U=\xi_{R}^{\dagger}\xi_{L}.$ (1)

The ambiguity in this split is known as the hidden local symmetry,
$H_{local}$. Under the chiral symmetry and the HLS, these two variables
transfrom as

$\xi_{L,R}\to\xi^{\prime}_{L,R}=h(x)\cdot\xi_{L,R}(x)\cdot g^{\dagger}_{L,R},$
(2)

where $h(x)\in[SU(3)_{V}]_{local},~{}g_{L,R}\in[SU(3)_{L}\times
SU(3)_{R}]_{global}.$ These variables are parameterized as

$\xi_{L,R}=\mathrm{e}^{\mathrm{i}\sigma/F_{\sigma}}\mathrm{e}^{\pm\mathrm{i}\pi/F_{\pi}},~{}~{}~{}\pi=\pi^{a}T_{a},~{}\sigma=\sigma^{a}T_{a},$
(3)

where $\pi$ denote the Goldstone bosons associated with the spontaneous chiral
symmetry breaking and $\sigma$ denote the Goldstone bosons associated with the
spontaneous breaking of HLS which are absorbed into the gauge bosons.

When gauging the chiral symmetry, we can construct two important building
blocks that are chiral invariant and covariant under HLS:

$\displaystyle\hat{\alpha}_{\bot\mu}$
$\displaystyle=(D_{\mu}\xi_{R}\cdot\xi_{R}^{\dagger}-D_{\mu}\xi_{L}\cdot\xi_{L}^{\dagger})/(2\mathrm{i}),$
(4) $\displaystyle\hat{\alpha}_{\parallel\mu}$
$\displaystyle=(D_{\mu}\xi_{R}\cdot\xi_{R}^{\dagger}+D_{\mu}\xi_{L}\cdot\xi_{L}^{\dagger})/(2\mathrm{i}),$
$\displaystyle\hat{\alpha}_{\bot\mu}$ $\displaystyle\to
h(x)\hat{\alpha}_{\bot\mu}h^{\dagger}(x),~{}\hat{\alpha}_{\parallel\mu}\to
h(x)\hat{\alpha}_{\parallel\mu}h^{\dagger}(x).$

The covariant derivatives are defined as

$\displaystyle D_{\mu}\xi_{L}$
$\displaystyle=\partial_{\mu}\xi_{L}-\mathrm{i}V_{\mu}\xi_{L}+\mathrm{i}\xi_{L}\mathcal{L}_{\mu},$
(5) $\displaystyle D_{\mu}\xi_{R}$
$\displaystyle=\partial_{\mu}\xi_{R}-\mathrm{i}V_{\mu}\xi_{R}+\mathrm{i}\xi_{R}\mathcal{R}_{\mu},$

where $V_{\mu}=g\rho_{\mu}$ are the gauge fields of HLS, $g$ is the coupling
of HLS, and $\mathcal{L}_{\mu},\mathcal{R}_{\mu}$ are the external gauge
fields of the chiral symmetry. These gauge fields transform as

$\displaystyle V_{\mu}$ $\displaystyle\to
h(x)(V_{\mu}+\mathrm{i}\partial_{\mu})h^{\dagger}(x),$ (6)
$\displaystyle\mathcal{L}_{\mu}$ $\displaystyle\to
g_{L}(\mathcal{L}_{\mu}+\mathrm{i}\partial_{\mu})g^{\dagger}_{L},$
$\displaystyle\mathcal{R}_{\mu}$ $\displaystyle\to
g_{R}(\mathcal{R}_{\mu}+\mathrm{i}\partial_{\mu})g^{\dagger}_{R}.$

The order assignment of the new variables is

$\xi_{L}\sim\xi_{R}\sim\mathcal{O}(1),~{}~{}~{}V_{\mu}\sim
g\sim\mathcal{O}(p).$ (7)

The leading order Lagrangian is given by Harada and Yamawaki (2003)

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{A}+\mathcal{L}_{\chi}+a\mathcal{L}_{V}+\mathcal{L}_{\text{kin}}(V_{\mu})$
(8)
$\displaystyle=F^{2}_{\pi}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}]+\frac{F_{\pi}^{2}}{4}\mathrm{tr}[\hat{\chi}+\hat{\chi}^{\dagger}]+F^{2}_{\sigma}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}^{\mu}_{\parallel}]-\frac{1}{2g^{2}}\mathrm{tr}[V_{\mu\nu}V^{\mu\nu}],$

where $\hat{\chi}$ is the adjoint representation of the HLS for external
source field $\chi$ using the ”converters” $\xi_{L}$ and $\xi_{R}$ as
$\hat{\chi}=\xi_{R}\chi\xi_{L}^{\dagger}=2B_{0}\xi_{R}(s+\mathrm{i}p)\xi_{L}^{\dagger}$,
and $V_{\mu\nu}$ is the gauge field strength of the HLS gauge boson defined by

$V_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}-\mathrm{i}[V_{\mu},V_{\nu}].$
(9)

When the parameter $a\equiv F_{\sigma}^{2}/F_{\pi}^{2}$ is specifically chosen
as 2, the above Lagrangian can reproduce several phenomenological facts, such
as the universality of the $\rho$-coupling $g_{\rho\pi\pi}=g$, the $\rho$
meson dominance of the pion’s electromagnetic form factor $g_{\gamma\pi\pi}=0$
and the KSRF formulas Kawarabayashi and Suzuki (1966); Riazuddin and
Fayyazuddin (1966):

$g_{\rho}=2F_{\pi}^{2}g_{\rho\pi\pi},~{}~{}~{}m_{\rho}^{2}=2g^{2}_{\rho\pi\pi}F_{\pi}^{2}.$
(10)

Similarly to the square of the pseudoscalar meson mass, HLS model assigns
$\mathcal{O}(p^{2})$ to the square of the vector meson mass:
$m_{\rho}^{2}\sim\mathcal{O}(p^{2})$, which is inconsistent with experimental
values. However, it is emphasized in the Section 4.1 of Ref. Harada and
Yamawaki (2003) that, “thanks to the gauge invariance, the HLS makes possible
the systematic expansion including the vector meson loops, particularly when
the vector meson mass is light. It turns out that such a limit can actually be
realized in QCD when the number of massless flavors $n_{f}$ becomes large as
was demonstrated in Refs. Harada and Yamawaki (1999); Harada and Yamawaki
(2001a). Then one can perform the derivative expansion with including the
vector mesons under such an extreme condition where the vector meson masses
are small, and extrapolate the results to the real world $n_{f}=3$ where the
vector meson masses take the experimental values.” The detailed discussion on
the power counting in the HLS model can be found in Ref. Harada and Yamawaki
(2003). The $\mathcal{O}(p^{4})$ Lagrangian in HLS model can be found in the
Appendix A.

##  III chiral effective Lagrangian including light vector mesons from QCD

In this section, we review our previous derivation of the effective chiral
Lagrangian with vector mesons from QCD Wang and Wang (2000). The start point
of the derivation is the generating functional of QCD, with an external source
$J(x)$ introduced for the composite light quark fields, given in Ref.Wang et
al. (2000):

$\displaystyle Z[J]$
$\displaystyle=\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\exp\left\\{\mathrm{i}\int
d^{4}x[\bar{\psi}(i\not{\partial}+J)\psi]\right\\}\int\mathcal{D}\Psi\mathcal{D}\bar{\Psi}\mathcal{D}A_{\mu}\Delta_{F}(A_{\mu})$
(11) $\displaystyle\times\exp\left\\{\mathrm{i}\int
d^{4}x\left[\mathcal{L}_{QCD}(A)-\frac{1}{2\xi}[F^{i}(A_{\mu})]^{2}-g\mathcal{I}_{i}^{\mu}A_{\mu}^{i}+\bar{\Psi}(\mathrm{i}\not{\partial}-M-g\not{A})\Psi\right]\right\\},$

The definitions and conventions for fields in Eq. (11) are the same as those
in Ref.Wang et al. (2000). After integrating out the quark and gluon fields,
and integrating in the bilocal auxiliary field
$\Phi^{\sigma\rho}(x,y)\sim(1/N_{c})\bar{\psi}_{\alpha}^{\sigma}(x)\psi_{\alpha}^{\rho}(y)$,
the result is

$Z[J]=\int\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{0}[J,\Phi,\Pi]},$
(12)

where

$\displaystyle\Gamma_{0}[J,\Phi,\Pi]$
$\displaystyle=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+J-\Pi]+N_{c}\int
d^{4}xd^{4}x^{\prime}\Phi^{\sigma\rho}(x,x^{\prime})\Pi^{\sigma\rho}(x,x^{\prime})$
(13) $\displaystyle+N_{c}\sum_{n=2}^{\infty}\int d^{4}x_{1}\cdots
d^{4}x_{n}d^{4}x^{\prime}_{1}\cdots
d^{4}x^{\prime}_{n}\frac{(-\mathrm{i})^{n}(N_{c}g^{2})^{n-1}}{n!}$
$\displaystyle\times\bar{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})\Phi^{\sigma_{1}\rho_{1}}(x_{1},x^{\prime}_{1})\cdots\Phi^{\sigma_{n}\rho_{n}}(x_{n},x^{\prime}_{n}).$

The detail of this derivation can be found in Ref.Wang et al. (2000).

Further by integrating in the local pseudo-scalar mesons and vector meson
fields, and integrating in the corresponding auxiliary field, the result can
be expressed as

$Z=\int\mathcal{D}\xi_{R}\mathcal{D}\xi_{L}\mathcal{D}V\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\delta(\xi_{L}^{\dagger}\xi_{L}-1)\delta(\xi_{R}^{\dagger}\xi_{R}-1)\delta(\det\xi_{R}-\det\xi_{L})\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]},$
(14)

where

$\displaystyle\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]=\Gamma_{0}[J,\Phi,\Pi]+\mathrm{i}\Gamma_{I}[\Phi]$
(15) $\displaystyle+$ $\displaystyle\mathrm{i}N_{c}\int
d^{4}x\mathrm{tr}_{f}\left\\{\Xi\left[\mathrm{e}^{-\mathrm{i}\frac{\theta}{N_{f}}}\xi_{R}\mathrm{tr}_{l}(P_{R}\Phi^{T})\xi_{L}^{\dagger}-\mathrm{e}^{\mathrm{i}\frac{\theta}{N_{f}}}\xi_{L}\mathrm{tr}_{l}(P_{L}\Phi^{T})\xi_{R}^{\dagger}\right]\right\\}$
$\displaystyle+$ $\displaystyle\mathrm{i}N_{c}\int
d^{4}x\left\\{V_{\mu}^{ab}+\left[(\xi_{L}P_{R}+\xi_{R}P_{L})\left(\frac{\Pi}{4{\mu}^{4}}-\frac{\mathrm{i}\not{\partial}}{4}\right)(\xi^{\dagger}_{R}P_{R}+\xi^{\dagger}_{L}P_{L})\right]^{(a\xi)(b\zeta)}\gamma_{\mu}^{\zeta\xi}\right\\}\tilde{V}^{\mu,ba}.$

After formally finishing the integration over the auxiliary fields
$\Phi,\Pi,\Xi$ and $\tilde{V}$, we obtain

$Z=\int\mathcal{D}\xi_{R}\mathcal{D}\xi_{L}\mathcal{D}V\delta(\xi_{L}^{\dagger}\xi_{L}-1)\delta(\xi_{R}^{\dagger}\xi_{R}-1)\delta(\det\xi_{R}-\det\xi_{L})\mathrm{e}^{\mathrm{i}S_{\text{eff}}[\xi_{R},\xi_{L},V,J,\Xi_{c},\tilde{V}_{c},\Phi_{c},\Pi_{c}]},$
(16)

where

$\mathrm{e}^{\mathrm{i}S_{\text{eff}}[\xi_{R},\xi_{L},V,J,\Xi_{c},\tilde{V}_{c},\Phi_{c},\Pi_{c}]}=\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}.$
(17)

The fields $\Xi_{c},\tilde{V}_{c},\Phi_{c},\Pi_{c}$ are the corresponding
classical field defined by

$\displaystyle\Phi_{c}$
$\displaystyle=\frac{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi~{}\Phi~{}\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}}{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}},$
(18) $\displaystyle\Pi_{c}$
$\displaystyle=\frac{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi~{}\Pi~{}\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}}{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}},$
$\displaystyle\Xi_{c}$
$\displaystyle=\frac{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi~{}\Xi~{}\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}}{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}},$
$\displaystyle\tilde{V}^{\mu}_{c}$
$\displaystyle=\frac{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi~{}\tilde{V}^{\mu}~{}\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}}{\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi\mathcal{D}\Pi\mathrm{e}^{\mathrm{i}\Gamma_{2}[\xi_{R},\xi_{L},V,J,\Xi,\tilde{V},\Phi,\Pi]}},$

which satisfy the classical field equations:

$\frac{\partial S_{\text{eff}}}{\partial\Phi^{\sigma\rho}_{c}}=\frac{\partial
S_{\text{eff}}}{\partial\Pi^{\sigma\rho}_{c}}=\frac{\partial
S_{\text{eff}}}{\partial\Xi^{\sigma\rho}_{c}}=\frac{\partial
S_{\text{eff}}}{\partial\tilde{V}^{\mu}_{c}}=0.$ (19)

To simplify the result, we define the rotated source and fields denoted with
subscript $\xi$:

$\displaystyle J_{\xi}(x)$
$\displaystyle\equiv[\xi_{L}(x)P_{R}+\xi_{R}(x)P_{L}][J(x)+\mathrm{i}\not{\partial}][\xi^{\dagger}_{R}(x)P_{R}+\xi^{\dagger}_{L}(x)P_{L}],$
(20) $\displaystyle\Phi^{T}_{\xi}(x,y)$
$\displaystyle\equiv[\xi_{R}(x)P_{R}+\xi_{L}(x)P_{L}]\Phi^{T}(x,y)[\xi^{\dagger}_{L}(y)P_{R}+\xi^{\dagger}_{R}(y)P_{L}],$
$\displaystyle\Pi^{T}_{\xi}(x,y)$
$\displaystyle\equiv[\xi_{L}(x)P_{R}+\xi_{R}(x)P_{L}]\Pi^{T}(x,y)[\xi^{\dagger}_{R}(y)P_{R}+\xi^{\dagger}_{L}(y)P_{L}],$
$\displaystyle V^{\mu}_{\xi}$ $\displaystyle\equiv
V^{\mu}-\frac{1}{2}[\xi_{R}\mathrm{i}\partial^{\mu}\xi_{R}^{\dagger}+\xi_{L}\mathrm{i}\partial^{\mu}\xi^{\dagger}_{L}].$

After this rotation, there is no explicit $\xi_{L},\xi_{R}$-dependence in Eq.
(15), and all $\xi_{L},\xi_{R}$-dependence is absorbed into variables with
subscript $\xi$. The final expression for the generating functional becomes

$\displaystyle Z$
$\displaystyle=\int\mathcal{D}\xi_{R}\mathcal{D}\xi_{L}\mathcal{D}V\delta(\xi_{L}^{\dagger}\xi_{L}-1)\delta(\xi_{R}^{\dagger}\xi_{R}-1)\delta(\det\xi_{R}-\det\xi_{L})$
(21)
$\displaystyle\times\exp\\{\mathrm{i}S_{\text{eff}}[1,1,V_{\xi},J_{\xi},\Xi_{c},\tilde{V}_{c},\Phi_{\xi
c},\Pi_{\xi c}]\\},$

where

$\mathrm{e}^{\mathrm{i}S_{\text{eff}}[1,1,V_{\xi},J_{\xi},\Xi_{c},\tilde{V}_{c},\Phi_{\xi
c},\Pi_{\xi
c}]}=\int\mathcal{D}\Xi\mathcal{D}\tilde{V}\mathcal{D}\Phi_{\xi}\mathcal{D}\Pi_{\xi}\mathrm{e}^{\mathrm{i}\Gamma_{2}[1,1,V_{\xi},J_{\xi},\Xi,\tilde{V},\Phi_{\xi},\Pi_{\xi}]},$
(22)

and

$\displaystyle\Gamma_{2}[1,1,V_{\xi},J_{\xi},\Xi,\tilde{V},\Phi_{\xi},\Pi_{\xi}]=$
$\displaystyle\Gamma_{0}[J_{\xi},\Phi_{\xi},\Pi_{\xi}]+\mathrm{i}\Gamma_{I}[\Phi_{\xi}]$
(23) $\displaystyle+$ $\displaystyle\mathrm{i}N_{c}\int
d^{4}x\mathrm{tr}_{f}\left\\{\Xi\left[\mathrm{e}^{-\mathrm{i}\frac{\theta}{N_{f}}}\mathrm{tr}_{l}(P_{R}\Phi^{T}_{\xi})-\mathrm{e}^{\mathrm{i}\frac{\theta}{N_{f}}}\mathrm{tr}_{l}(P_{L}\Phi^{T}_{\xi})\right]\right\\}$
$\displaystyle+$ $\displaystyle\mathrm{i}N_{c}\int
d^{4}x\left[V_{\xi\mu}^{ab}+\frac{1}{4{\mu}^{4}}\Pi_{\xi}^{(a\xi)(b\zeta)}\gamma_{\mu}^{\zeta\xi}\right]\tilde{V}^{\mu,ba}.$

The measure of light quark field $\mathcal{D}\bar{\psi}\mathcal{D}\psi$ is not
invariant under this rotation, and therefore the term
$\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+\not{J}-\Pi]$ in $\Gamma_{0}$ becomes

$\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+\not{J}-\Pi]=\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+\not{J}_{\xi}-\Pi_{\xi}]+\Gamma_{WZ}[\xi_{R}^{\dagger}\xi_{L},l,r].$
(24)

where $\Gamma_{WZ}[\xi_{R}^{\dagger}\xi_{L},l,r]$ is the well-known Wess-
Zumino term Harada and Yamawaki (2003).

The details of this derivation can be found in Ref. Wang and Wang (2000),
where the definition of the vector mesons was given by

$V_{\mu}^{ab}=-\left[(\xi_{L}P_{R}+\xi_{R}P_{L})\left(\frac{\Pi}{4{\mu}^{4}}\right)(\xi^{\dagger}_{R}P_{R}+\xi^{\dagger}_{L}P_{L})\right]^{(a\xi)(b\zeta)}\gamma_{\mu}^{\zeta\xi}.$
(25)

This transforms homogeneously under the HLS, indicating that it is not the
gauge boson. In fact, the rotated vector external source $v_{\xi}$ acts as the
HLS gauge boson because of the definition of the rotated external source
$J_{\xi}$:

$J_{\xi}(x)\equiv[\xi_{L}(x)P_{R}+\xi_{R}(x)P_{L}][J(x)+\mathrm{i}\not{\partial}][\xi^{\dagger}_{R}(x)P_{R}+\xi^{\dagger}_{L}(x)P_{L}].$
(26)

In Ref.Wang and Wang (2000), an attempt to compensate the above mismatch by
adding an extra term in $V_{\mu}$ resulted in the non-equivalence between the
derived chiral Lagrangian and those in the HLS model. Therefore, in this
paper, the definition of the vector mesons is revised to

$V_{\mu}^{ab}=-\left[(\xi_{L}P_{R}+\xi_{R}P_{L})\left(\frac{\Pi}{4{\mu}^{4}}-\frac{\mathrm{i}\not{\partial}}{4}\right)(\xi^{\dagger}_{R}P_{R}+\xi^{\dagger}_{L}P_{L})\right]^{(a\xi)(b\zeta)}\gamma_{\mu}^{\zeta\xi},$
(27)

which transforms inhomogeneously under the HLS, confirming that it is indeed
the corresponding gauge boson.

##  IV the calculations of the $p^{2}$ order and $p^{4}$ order LECs

Under the large-$N_{c}$ limit, the field integrations in Eq.(22) can be
carried out using the saddle point approximation with the classical field
equations:

$\displaystyle\frac{1}{4\mu^{4}}\delta^{(4)}(x-y)\widetilde{\not{V}}_{c}^{\sigma\rho}(x)+\Phi^{\sigma\rho}_{\xi
c}(x,y)=-\mathrm{i}[(\mathrm{i}\not{\partial}+J_{\xi}-\Pi_{\xi
c})^{-1}]^{\rho\sigma}(y,x),$ (28)
$\displaystyle\widetilde{\Xi}^{\sigma\rho}(x)\delta^{(4)}(x-y)+\Pi^{\sigma\rho}_{\xi
c}(x,y)+\sum_{n=1}^{\infty}\int d^{4}x_{1}\cdots
d^{4}x_{n}d^{4}x^{\prime}_{1}\cdots d^{4}x^{\prime}_{n}$
$\displaystyle\times\frac{(-\mathrm{i})^{n+1}(N_{c}g^{2})^{n}}{n!}\bar{G}(x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})\Phi_{\xi
c}(x_{1},x^{\prime}_{1})\cdots\Phi_{\xi c}(x_{n},x^{\prime}_{n})=0,$
$\displaystyle\mathrm{tr}_{l}\left[\mathrm{e}^{-\mathrm{i}\frac{\theta}{N_{f}}}P_{R}\Phi^{T}_{\xi
c}-\mathrm{e}^{\mathrm{i}\frac{\theta}{N_{f}}}P_{L}\Phi^{T}_{\xi c}\right]=0,$
$\displaystyle\mathrm{tr}_{l}\left[V_{\xi,\mu}^{ab}(x)+\frac{1}{4{\mu}^{4}}\Pi_{\xi
c}(x,x)^{(a\xi)(b\zeta)}(\gamma_{\mu})^{\zeta\xi}\right]=0.$

These equations indicate that $\Phi_{\xi c}$ and $\Pi_{\xi c}$ serve as the
quark propagator and the quark self-energy with the external sources $J_{\xi}$
respectively. Under the large-$N_{c}$ limit, the effective action is

$\displaystyle S_{\text{eff}}$
$\displaystyle=\Gamma_{2}[1,1,V_{\xi},J_{\xi},\Xi_{c},\tilde{V}_{c},\Phi_{\xi
c},\Pi_{\xi c}]=\Gamma_{0}[J_{\xi},\Phi_{\xi c},\Pi_{\xi c}]$ (29)
$\displaystyle=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+J_{\xi}-\Pi_{\xi
c}]+N_{c}\int d^{4}xd^{4}x^{\prime}\Phi_{\xi
c}^{\sigma\rho}(x,x^{\prime})\Pi_{\xi c}^{\sigma\rho}(x,x^{\prime})$
$\displaystyle+N_{c}\sum_{n=2}^{\infty}\int d^{4}x_{1}\cdots
d^{4}x_{n}d^{4}x^{\prime}_{1}\cdots
d^{4}x^{\prime}_{n}\frac{(-\mathrm{i})^{n}(N_{c}g^{2})^{n-1}}{n!}$
$\displaystyle\times\bar{G}^{\sigma_{1}\cdots\sigma_{n}}_{\rho_{1}\cdots\rho_{n}}(x_{1},x^{\prime}_{1},\cdots,x_{n},x^{\prime}_{n})\Phi_{\xi
c}^{\sigma_{1}\rho_{1}}(x_{1},x^{\prime}_{1})\cdots\Phi_{\xi
c}^{\sigma_{n}\rho_{n}}(x_{n},x^{\prime}_{n}).$

The term $\Gamma_{I}$ is neglected because it is an $\mathcal{O}(1)$ term, and
the last two terms in Eq.(15) vanish due to Eq.(28). The second and third
terms in Eq.(29) are still too complex to calculate directly. Therefore, we
consider only the leading order in dynamical perturbation, accounting for the
QCD interaction in the SDE that lead to the nonperturbative solution of chiral
symmetry breaking, while neglecting other QCD corrections in positive powers
of $g$ (perturbative). This approximation yields the well-known Pagels-Stokar
formula for the pion decay constant $f_{\pi}$ Pagels and Stokar (1979). At the
leading order in dynamical perturbation, the third terms in Eq.(29) can be
neglected due to their dependence on positive powers of $g$. The second terms
in Eq.(29) can also be neglected because they are of the same order as the
third term, as shown in Eq.(28). Thus, the effective action simplifies to

$S_{\text{eff}}=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+J_{\xi}-\Pi_{\xi
c}].$ (30)

We define a new external source $J^{\prime}_{\xi}(x)$ and a auxiliary field
$\Pi^{\prime}_{\xi c}$ as

$\displaystyle J^{\prime}_{\xi}(x)$ $\displaystyle\equiv
J_{\xi}(x)-[\xi_{L}(x)P_{R}+\xi_{R}(x)P_{L}]\mathrm{i}\not{\partial}[\xi^{\dagger}_{R}(x)P_{R}+\xi^{\dagger}_{L}(x)P_{L}],$
(31) $\displaystyle\Pi^{\prime}_{\xi c}$ $\displaystyle\equiv\Pi_{\xi
c}+\not{V}-[\xi_{L}(x)P_{R}+\xi_{R}(x)P_{L}]\mathrm{i}\not{\partial}[\xi^{\dagger}_{R}(x)P_{R}+\xi^{\dagger}_{L}(x)P_{L}].$

They both transform homogeneously under the HLS. After this rearrangement, the
effective action becomes

$S_{\text{eff}}=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+\not{V}+J^{\prime}_{\xi}-\Pi^{\prime}_{\xi
c}].$ (32)

Eq.(28) indicates that $\Pi_{\xi c}$ functions as the quark self-energy, and
we maintain the identification from Ref.Yang et al. (2002):

${\Pi^{\prime}}_{\xi
c}^{\sigma\rho}(x,y)=[\Sigma(\bar{\nabla}_{x}^{2})]^{\sigma\rho}\delta^{4}(x-y),$
(33)

where $\bar{\nabla}_{x}^{\mu}=\partial^{\mu}_{x}-\mathrm{i}V^{\mu}$ is the
covariant derivative ensuring the correct homogeneous transformation of
${\Pi^{\prime}}_{\xi c}^{\sigma\rho}$ under the HLS. The effective action can
then be written as

$S_{\text{eff}}=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\bar{\not{\nabla}}+J^{\prime}_{\xi}-\Sigma(\bar{\nabla}_{x}^{2})].$
(34)

When the vector meson field vanishs, $\Sigma(-p^{2})$ is the conventional
quark self-energy obtained by solving the SDE. Retaining only the leading
order in dynamical perturbation and applying the improved ladder
approximation, the SDE in the Euclidean space-time is

$\Sigma(p^{2})-\frac{3N_{c}}{2}\int\frac{\mathrm{d}^{4}p}{4\pi^{3}}\frac{\alpha_{s}[(p-q)]}{(p-q)^{2}}\frac{\Sigma(q^{2})}{q^{2}+\Sigma^{2}(q^{2})}=0.$
(35)

To calculate the LECs, we expand the effective action as a Taylor series:

$\displaystyle S_{\text{eff}}$
$\displaystyle=-\mathrm{i}N_{c}\int\mathrm{d}^{4}x\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{tr}\ln[\not{k}+\mathrm{i}\bar{\not{\nabla}}+J^{\prime}_{\xi}-\Sigma((-\mathrm{i}k+\bar{\nabla})^{2})]$
$\displaystyle=-\mathrm{i}N_{c}\int\mathrm{d}^{4}x\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{tr}\ln[\not{k}-\Sigma(-k^{2})+\mathrm{i}\bar{\not{\nabla}}+J^{\prime}_{\xi}-\Sigma_{1}(\bar{\nabla})]$
$\displaystyle=-\mathrm{i}N_{c}\int\mathrm{d}^{4}x\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\left\\{\mathrm{tr}\ln[\not{k}-\Sigma]+\mathrm{tr}\ln[1+(\mathrm{i}\bar{\not{\nabla}}+J^{\prime}_{\xi}-\Sigma_{1})(\not{k}+\Sigma)X]\right\\}$
$\displaystyle\simeq-\mathrm{i}N_{c}\int\mathrm{d}^{4}x\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{tr}\left[\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}[(\mathrm{i}\bar{\not{\nabla}}+J^{\prime}_{\xi}-\Sigma_{1})(\not{k}+\Sigma)X]^{n}\right],$
(36)

where ‘tr’ includes the traces over spinor and flavour spacec,
$J^{\prime}_{\xi}=\not{v}^{\prime}_{\xi}+\not{a}_{\xi}\gamma_{5}-s_{\xi}+\mathrm{i}p_{\xi}\gamma_{5}$,
$X=1/(k^{2}-\Sigma^{2})$, $\Sigma=\Sigma(-k^{2})$, and
$\Sigma_{1}(\bar{\nabla})\equiv\Sigma((-\mathrm{i}k+\bar{\nabla})^{2})-\Sigma(-k^{2})$.

After making the momentum expansion, the $\mathcal{O}(p^{2})$ Lagrangian is

$\mathcal{L}_{(2)}=F^{2}_{0}\mathrm{tr}[a_{\xi}^{2}]+F^{2}_{0}B_{0}\mathrm{tr}[s_{\xi}]+F^{2}_{\sigma}\mathrm{tr}[{v^{\prime}}_{\xi}^{2}].$
(37)

where $F_{0}$ is the pion decay constant in the chiral limit, and these
coefficients are expressed as the functions of the quark self-energy in the
Minkowski space-time:

$\displaystyle F_{0}^{2}B_{0}$
$\displaystyle=\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{4\Sigma(-k^{2})}{k^{2}-\Sigma^{2}(-k^{2})},$
(38) $\displaystyle F_{0}^{2}$
$\displaystyle=-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k^{2}+2\Sigma^{2}(-k^{2})}{[k^{2}-\Sigma^{2}(-k^{2})]^{2}},$
$\displaystyle F_{\sigma}^{2}$
$\displaystyle=-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k^{2}-2\Sigma^{2}(-k^{2})}{[k^{2}-\Sigma^{2}(-k^{2})]^{2}}.$

The expression for $F_{0}^{2}B_{0}$ in Eq.(38) aligns with those in Refs. Aoki
et al. (1990); Holdom et al. (1990); Roberts and Williams (1994). The
expression for $F_{0}^{2}$ is consistent with the well-known Pagels-Stokar
formula, up to a total derivative term Wang et al. (2000). Utilizing the
following identities,

$a_{\xi}^{\mu}=\hat{\alpha}_{\bot}^{\mu},~{}~{}{v^{\prime}}_{\xi}^{\mu}=\hat{\alpha}_{\parallel}^{\mu},~{}~{}s_{\xi}=\frac{1}{4B_{0}}(\hat{\chi}+\hat{\chi}^{\dagger}),$
(39)

we can reproduce the leading order Lagrangian, excluding the kinetic energy
term of the vector meson in Eq.(8). The kinetic energy term of the vector
meson belongs to $\mathcal{O}(p^{4})$ in our calculation.

After making the momentum expansion, the $\mathcal{O}(p^{4})$ Lagrangian is

$\displaystyle\mathcal{L}_{(4)}$
$\displaystyle=\mathrm{tr}_{f}[\mathcal{C}_{2}(\mathrm{d}_{\mu}a^{\mu})^{2}+\mathcal{C}_{3}(\mathrm{d}^{\mu}a^{\nu}-\mathrm{d}^{\nu}a^{\mu})(\mathrm{d}_{\mu}a_{\nu}-\mathrm{d}_{\nu}a_{\mu})+\mathcal{C}_{4}a^{4}+\mathcal{C}_{5}a^{\mu}a^{\nu}a_{\mu}a_{\nu}$
(40)
$\displaystyle+\mathcal{C}_{6}s^{2}+\mathcal{C}_{7}p^{2}+\mathcal{C}_{8}sa^{2}+\mathcal{C}_{9}V^{\mu\nu}V_{\mu\nu}-\mathrm{i}\mathcal{C}_{10}V^{\mu\nu}a_{\mu}a_{\nu}+\mathcal{C}_{11}p\mathrm{d}_{\mu}a^{\mu}$
$\displaystyle+\mathcal{C}_{12}(\mathrm{d}_{\mu}{v^{\prime}}^{\mu})^{2}+\mathcal{C}_{13}(\mathrm{d}^{\mu}{v^{\prime}}^{\nu}-\mathrm{d}^{\nu}{v^{\prime}}^{\mu})(\mathrm{d}_{\mu}{v^{\prime}}_{\nu}-\mathrm{d}_{\nu}{v^{\prime}}_{\mu})+\mathcal{C}_{14}{v^{\prime}}^{4}$
$\displaystyle+\mathcal{C}_{15}{v^{\prime}}^{\mu}{v^{\prime}}^{\nu}{v^{\prime}}_{\mu}{v^{\prime}}_{\nu}+\mathcal{C}_{16}s{v^{\prime}}^{2}-\mathrm{i}\mathcal{C}_{17}V^{\mu\nu}{v^{\prime}}_{\mu}{v^{\prime}}_{\nu}+\mathcal{C}_{18}s\mathrm{d}_{\mu}{v^{\prime}}^{\mu}$
$\displaystyle+\mathcal{C}_{19}a^{2}{v^{\prime}}^{2}+\mathcal{C}_{20}a_{\mu}a_{\nu}{v^{\prime}}^{\mu}{v^{\prime}}^{\nu}+\mathcal{C}_{21}a_{\mu}a_{\nu}{v^{\prime}}^{\nu}{v^{\prime}}^{\mu}+\mathcal{C}_{22}[(a^{\mu}{v^{\prime}}_{\mu})^{2}+({v^{\prime}}^{\mu}a_{\mu})^{2}]$
$\displaystyle+\mathcal{C}_{23}a_{\mu}{v^{\prime}}_{\nu}a^{\mu}{v^{\prime}}^{\nu}+\mathcal{C}_{24}[(a^{\mu}{v^{\prime}}_{\mu})^{2}-({v^{\prime}}^{\mu}a_{\mu})^{2}]+\mathcal{C}_{25}p(a^{\mu}{v^{\prime}}_{\mu}-{v^{\prime}}^{\mu}a_{\mu})$
$\displaystyle+\mathcal{C}_{26}p(a^{\mu}{v^{\prime}}_{\mu}+{v^{\prime}}^{\mu}a_{\mu})+\mathcal{C}_{27}(\mathrm{d}^{\mu}{v^{\prime}}^{\nu}-\mathrm{d}^{\nu}{v^{\prime}}^{\mu})a_{\mu}a_{\nu}+\mathcal{C}_{28}(\mathrm{d}^{\mu}{v^{\prime}}^{\nu}-\mathrm{d}^{\nu}{v^{\prime}}^{\mu})V_{\mu\nu}$
$\displaystyle+\mathcal{C}_{29}(\mathrm{d}^{\mu}a^{\nu}-\mathrm{d}^{\nu}a^{\mu})(a_{\mu}{v^{\prime}}_{\nu}+{v^{\prime}}_{\mu}a_{\nu})+\mathcal{C}_{30}(\mathrm{d}^{\mu}a^{\nu}-\mathrm{d}^{\nu}a^{\mu})(a_{\mu}{v^{\prime}}_{\nu}-{v^{\prime}}_{\mu}a_{\nu})$
$\displaystyle+\mathcal{C}_{31}\mathrm{d}_{\mu}a^{\mu}(a_{\nu}{v^{\prime}}^{\nu}-{v^{\prime}}^{\nu}a_{\nu})],$

where the subscript $\xi$ of rotated external sources has been omitted for
simplicity, and the coefficients $\mathcal{C}_{i}$ are expressed as the
functions of the quark self-energy, detailed in the Appendix B. The
corresponding items for $\mathcal{C}_{2}\sim\mathcal{C}_{11}$ are the terms
for pure pseudo-scalar mesons, as in Refs.Yang et al. (2002); Jiang et al.
(2010). The corresponding items for
$\mathcal{C}_{18},\mathcal{C}_{24},\mathcal{C}_{26},\mathcal{C}_{30}$ are not
invariant under charge conjugation, and thus should not appear which is an
oversight in Ref.Wang and Wang (2000). As will be seen later, the results for
these four coefficients are zero, confirming the accuracy of our calculation.
Other terms account for vector mesons. By applying the following identities:

$\displaystyle a_{\xi}^{\mu}$
$\displaystyle=\hat{\alpha}_{\bot}^{\mu},~{}~{}{v^{\prime}}_{\xi}^{\mu}=\hat{\alpha}_{\parallel}^{\mu},$
(41) $\displaystyle s_{\xi}$
$\displaystyle=\frac{1}{4B_{0}}(\hat{\chi}+\hat{\chi}^{\dagger}),~{}~{}\mathrm{i}p_{\xi}=\frac{1}{4B_{0}}(\hat{\chi}-\hat{\chi}^{\dagger}),$
$\displaystyle\mathrm{d}^{\mu}{v^{\prime}}^{\nu}_{\xi}-\mathrm{d}^{\nu}{v^{\prime}}^{\mu}_{\xi}$
$\displaystyle=\hat{\mathcal{V}}^{\mu\nu}-V^{\mu\nu}+\mathrm{i}[{v^{\prime}}_{\xi}^{\mu},{v^{\prime}}_{\xi}^{\nu}]+\mathrm{i}[a_{\xi}^{\mu},a_{\xi}^{\nu}]$
$\displaystyle\mathrm{d}^{\mu}a^{\nu}_{\xi}-\mathrm{d}^{\nu}a^{\mu}_{\xi}$
$\displaystyle=\hat{\mathcal{A}}^{\mu\nu}+\mathrm{i}[{v^{\prime}}_{\xi}^{\mu},a_{\xi}^{\nu}]+\mathrm{i}[a_{\xi}^{\mu},{v^{\prime}}_{\xi}^{\nu}],$

and the equations of fields:

$\displaystyle\mathrm{d}_{\mu}a_{\xi}^{\mu}$
$\displaystyle=-\mathrm{i}(a-1)[v^{\prime}_{\xi,\mu},a_{\xi}^{\mu}]+B_{0}\left(p_{\xi}-\frac{1}{N_{f}}\mathrm{tr}[p_{\xi}]\right)+\mathcal{O}(p^{4}),$
(42) $\displaystyle\mathrm{d}_{\mu}{v^{\prime}}_{\xi}^{\mu}$
$\displaystyle=\mathcal{O}(p^{4}),$

we can reproduce the kinetic energy term of the vector meson and the next
leading order Lagrangian in the HLS model:

$\mathcal{L}_{(4)}=-\frac{1}{2g^{2}}\mathrm{tr}[V_{\mu\nu}V^{\mu\nu}]+\mathcal{L}_{(4)y}+\mathcal{L}_{(4)\omega}+\mathcal{L}_{(4)z},$
(43)

The relations between the coefficients $\mathcal{C}_{i}$ and the LECs of HLS
model are given by

$\displaystyle-\frac{1}{2g^{2}}$
$\displaystyle=\mathcal{C}_{9}+\mathcal{C}_{13}-\mathcal{C}_{28},$
$\displaystyle y_{1}$
$\displaystyle=\mathcal{C}_{4}+2\mathcal{C}_{13}-\mathrm{i}\mathcal{C}_{27},~{}~{}y_{2}=\mathcal{C}_{5}-2\mathcal{C}_{13}+\mathrm{i}\mathcal{C}_{27},~{}~{}y_{3}=2\mathcal{C}_{13}+\mathcal{C}_{14},$
$\displaystyle y_{4}$
$\displaystyle=-2\mathcal{C}_{13}+\mathcal{C}_{15},~{}~{}y_{5}=4\mathcal{C}_{3}+\mathcal{C}_{19}-2\mathrm{i}\mathcal{C}_{29},$
$\displaystyle y_{6}$
$\displaystyle=4\mathcal{C}_{3}-2\mathcal{C}_{13}+\mathcal{C}_{20}+\mathrm{i}\mathcal{C}_{27}+2\mathrm{i}\mathcal{C}_{29},$
$\displaystyle y_{7}$
$\displaystyle=2\mathcal{C}_{13}+[\mathcal{C}_{21}+2(a-1)^{2}\mathcal{C}_{2}-2\mathrm{i}(a-1)\mathcal{C}_{31}]-\mathrm{i}\mathcal{C}_{27},$
$\displaystyle y_{8}$
$\displaystyle=-2\mathcal{C}_{3}+[\mathcal{C}_{22}-(a-1)^{2}\mathcal{C}_{2}+\mathrm{i}(a-1)\mathcal{C}_{31}]-\mathrm{i}\mathcal{C}_{29},$
$\displaystyle y_{9}$
$\displaystyle=-4\mathcal{C}_{3}+\mathcal{C}_{23}+2\mathrm{i}\mathcal{C}_{29},~{}~{}y_{10}=0,~{}~{}y_{11}=0,~{}~{}y_{12}=0,~{}~{}y_{13}=0,$
$\displaystyle y_{14}$
$\displaystyle=0,~{}~{}y_{15}=0,~{}~{}y_{16}=0,~{}~{}y_{17}=0,~{}~{}y_{18}=0,$
$\displaystyle\omega_{1}$
$\displaystyle=\frac{\mathcal{C}_{8}}{4B_{0}},~{}~{}\omega_{2}=0,~{}~{}\omega_{3}=\frac{\mathcal{C}_{16}}{4B_{0}},~{}~{}\omega_{4}=0,$
$\displaystyle\omega_{5}$
$\displaystyle=\frac{\mathrm{i}}{4B_{0}}\left[\mathcal{C}_{25}+\mathrm{i}(a-1)B_{0}\mathcal{C}_{2}+\mathrm{i}(a-1)\mathcal{C}_{11}+B_{0}\mathcal{C}_{31}\right],$
$\displaystyle\omega_{6}$
$\displaystyle=\frac{\mathcal{C}_{6}}{16B_{0}^{2}},~{}~{}\omega_{7}=0,~{}~{}\omega_{8}=-\frac{1}{16B_{0}^{2}}\left[\mathcal{C}_{7}+B_{0}^{2}\mathcal{C}_{2}+B_{0}\mathcal{C}_{11}\right],$
$\displaystyle\omega_{9}$
$\displaystyle=\frac{1}{16B_{0}^{2}}\left[\frac{1}{N_{f}}B_{0}^{2}\mathcal{C}_{2}+\frac{1}{N_{f}}B_{0}\mathcal{C}_{11}\right],$
$\displaystyle z_{1}$
$\displaystyle=\mathcal{C}_{13},~{}~{}z_{2}=\mathcal{C}_{3},~{}~{}z_{3}=-2\mathcal{C}_{13}+\mathcal{C}_{28},$
$\displaystyle z_{4}$
$\displaystyle=-\mathcal{C}_{10}-4\mathcal{C}_{13}+\mathrm{i}\mathcal{C}_{27}+2\mathcal{C}_{28},~{}~{}z_{5}=-\mathcal{C}_{17}-4\mathcal{C}_{13}+2\mathcal{C}_{28},$
$\displaystyle z_{6}$
$\displaystyle=4\mathcal{C}_{13}-\mathrm{i}\mathcal{C}_{27},~{}~{}z_{7}=4\mathcal{C}_{13},~{}~{}z_{8}=-4\mathcal{C}_{3}+\mathrm{i}\mathcal{C}_{29}.$
(44)

For the derivation of the Wess-Zumino anomalous term from QCD, refer to
Refs.Ma and Wang (2003); Jiang and Wang (2010). Besides the Wess-Zumino term,
the $\mathcal{O}(p^{4})$ anomalous Lagrangian can also emerge from the
momentum expansion of the effective action:

$\displaystyle\mathcal{L}_{A,(4)}=$ $\displaystyle
b_{1}\mathrm{i}\epsilon^{\mu\nu\sigma\rho}\mathrm{tr}_{f}[a_{\mu}a_{\nu}a_{\sigma}v_{\rho}]+b_{2}\epsilon^{\mu\nu\sigma\rho}\mathrm{tr}_{f}[a_{\mu}v_{\nu}(\mathrm{d}_{\sigma}{v^{\prime}}_{\rho}-\mathrm{d}_{\rho}{v^{\prime}}_{\sigma})$
(45)
$\displaystyle+a_{\mu}(\mathrm{d}_{\sigma}{v^{\prime}}_{\rho}-\mathrm{d}_{\rho}{v^{\prime}}_{\sigma})v_{\nu}]+b_{3}\epsilon^{\mu\nu\sigma\rho}\mathrm{tr}_{f}[a_{\mu}v_{\nu}V_{\sigma\rho}+a_{\mu}V_{\sigma\rho}v_{\nu}]$
$\displaystyle+b_{4}\mathrm{i}\epsilon^{\mu\nu\sigma\rho}\mathrm{tr}_{f}[a_{\mu}v_{\nu}v_{\sigma}v_{\rho}],$

where the coefficients $b_{i}$ are expressed as the functions of the quark
self-energy in the Minkowski space-time:

$\displaystyle b_{1}$
$\displaystyle=-4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}(\Sigma^{4}+\Sigma^{2}k^{2}),$
(46) $\displaystyle b_{2}$
$\displaystyle=\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{3}\Sigma^{2},$
$\displaystyle b_{3}$
$\displaystyle=2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{3}(\Sigma\Sigma^{\prime}k^{2}+\Sigma^{2}),$
$\displaystyle b_{4}$
$\displaystyle=-4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{3}\Sigma^{2},$

where $X=1/(k^{2}-\Sigma^{2})$. Utilizing Eq.(41) and the following
identities:

$\displaystyle\hat{\alpha}_{L}^{\mu}=v_{\xi}^{\mu}-a_{\xi}^{\mu},~{}~{}\hat{\alpha}_{R}^{\mu}=v_{\xi}^{\mu}+a_{\xi}^{\mu},~{}~{}\epsilon_{\mu\nu\sigma\rho}F_{V}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}V^{\mu\nu},$
(47)
$\displaystyle\epsilon_{\mu\nu\sigma\rho}\hat{F}_{L}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}(\mathcal{V}^{\mu\nu}-\mathcal{A}^{\mu\nu}),~{}~{}\epsilon_{\mu\nu\sigma\rho}\hat{F}_{R}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}(\mathcal{V}^{\mu\nu}+\mathcal{A}^{\mu\nu}),$

we can reproduce the $\mathcal{O}(p^{4})$ anomalous Lagrangian in Eq.(88)
except for the Wess-Zumino term. The relations between the coefficients
$b_{i}$ and the LECs $c_{i}$ in Eq.(88) are given by

$\displaystyle c_{1}=$
$\displaystyle-\frac{2\pi^{2}}{N_{c}}(b_{1}+4b_{2}+b_{4}),~{}~{}c_{2}=\frac{2\pi^{2}}{N_{c}}(b_{1}-b_{4}),$
(48) $\displaystyle c_{3}=$
$\displaystyle-\frac{16\pi^{2}}{N_{c}}(b_{3}-b_{2}),~{}~{}c_{4}=-\frac{16\pi^{2}}{N_{c}}b_{2}.$

##  V numerical results and discussion

We choose the same running coupling constant $\alpha_{s}(p^{2})$ as the ones
in Refs.Yang et al. (2002); Jiang et al. (2010):

$\alpha_{s}(k^{2})=\frac{12\pi}{33-2n_{f}}\times\left\\{\begin{array}[]{cc}7&\ln(k^{2}/\Lambda^{2}_{\text{QCD}})\leq-2\\\
7-4[2+\ln(k^{2}/\Lambda^{2}_{\text{QCD}})]^{2}/5&-2\leq\ln(k^{2}/\Lambda^{2}_{\text{QCD}})\leq
0.5\\\
1/\ln(k^{2}/\Lambda^{2}_{\text{QCD}})&\ln(k^{2}/\Lambda^{2}_{\text{QCD}})\geq
0.5\end{array}\right.$

which is plotted in FIG. 1. It has the correct asymptotic behavior:

$\alpha_{s}(k^{2})\stackrel{{\scriptstyle
k^{2}\to\infty}}{{\longrightarrow}}\frac{12\pi}{33-2n_{f}}\frac{1}{\ln(k^{2}/\Lambda^{2}_{\text{QCD}})}.$
(49)

and there is only one parameter $\Lambda_{\text{QCD}}$.

Figure 1: The running coupling constant $\alpha_{s}(p^{2})$.

With this running coupling constant, we solve the SDE for the quark self-
energy, Eq.(35), numerically, and the details can be found in Ref.Yang et al.
(2002). The obtained self-energy and its derivatives are plotted in FIG. 2,
where the solid line represents the quark self-energy
$\Sigma[k^{2}]/\Lambda_{\text{QCD}}$, the dashed line denotes the first
derivative of quark self-energy $\Lambda_{\text{QCD}}\Sigma^{\prime}[k^{2}]$,
the dash-dotted line represents the second derivative
$\Lambda^{3}_{\text{QCD}}\Sigma^{\prime\prime}[k^{2}]$ and the dotted line
denotes the third derivative
$\Lambda^{5}_{\text{QCD}}\Sigma^{\prime\prime\prime}[k^{2}]$. The higher-order
derivatives of the quark self-energy are discontinuous, because the running
coupling constant $\alpha_{s}(p^{2})$ used in this paper is approximate. These
discontinuities may introduce additional errors, so we retain only up to the
second derivative of the self-energy and discard the higher order derivatives:

$\Sigma_{1}(\bar{\nabla})\simeq(-2\mathrm{i}k\cdot\bar{\nabla}+\bar{\nabla}^{2})\Sigma^{\prime}(-k^{2})+\frac{1}{2}(-2\mathrm{i}k\cdot\bar{\nabla}+\bar{\nabla}^{2})^{2}\Sigma^{\prime\prime}(-k^{2}).$
(50)

This approximation will not affect the results for the $\mathcal{O}(p^{2})$
coefficients.

Figure 2: The obtained self-energy and its derivatives, where the solid line
denotes the quark self-energy $\Sigma[k^{2}]/\Lambda_{\text{QCD}}$, the dashed
line denotes the first derivative of quark self-energy
$\Lambda_{\text{QCD}}\Sigma^{\prime}[k^{2}]$, the dash-dotted line denotes the
second derivative $\Lambda^{3}_{\text{QCD}}\Sigma^{\prime\prime}[k^{2}]$ and
the dotted line denotes the third derivative
$\Lambda^{5}_{\text{QCD}}\Sigma^{\prime\prime\prime}[k^{2}]$.

By making use of the wick rotation $k^{0}\to\mathrm{i}k^{4}$, the $p^{2}$
order LECs can be expressed as the functions of the quark self-energy in the
Euclidean space-time:

$\displaystyle F_{0}^{2}B_{0}$
$\displaystyle=N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{4\Sigma(k^{2})}{k^{2}+\Sigma^{2}(k^{2})},$
(51) $\displaystyle F_{0}^{2}$
$\displaystyle=-N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k^{2}-2\Sigma^{2}(k^{2})}{[k^{2}+\Sigma^{2}(k^{2})]^{2}},$
$\displaystyle F_{\sigma}^{2}$
$\displaystyle=-N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k^{2}+2\Sigma^{2}(k^{2})}{[k^{2}+\Sigma^{2}(k^{2})]^{2}}.$

The asymptotic behavior of $\Sigma(k^{2})$ reflecting chiral symmetry breaking
is

$\Sigma(k^{2})\stackrel{{\scriptstyle
k^{2}\to\infty}}{{\longrightarrow}}\frac{\ln^{\gamma-1}(k^{2}/\Lambda^{2}_{\text{QCD}})}{k^{2}},$
(52)

where $\gamma=9N_{c}/[2(33-2n_{f})]$. With this asymptotic behavior of
$\Sigma(k^{2})$, it is evident that $F_{0}^{2}B_{0}$ exhibits logarithmic
divergence, while both $F_{0}^{2}$ and $F_{\sigma}^{2}$ show quadratic
divergence. Due to the lack of an analytical solution for the quark self-
energy, the momentum integral in Eq.(51) can not be performed analytically.
This renders the commonly used regularizations in quantum field theory, such
as dimensional regularization, inapplicable. In the literature, several
regularizations are employed, including $\zeta$-function regularization Espriu
et al. (1990), Schwinger proper time regularization Bijnens (1996); Lu et al.
(2002); Yang et al. (2002); Jiang et al. (2010) and cut-off regularization
Chen et al. (2020, 2021). These regularizations involve a physical ultraviolet
cut-off $\Lambda$ which should be on the order of the chiral symmetry breaking
scale. If we assume $\Sigma(p^{2})$ to be a constant mass $m$ as in Refs.
Espriu et al. (1990); Bijnens (1996), the momentum integral in Eq.(51) can be
evaluated analytically, and the results under different regularizations are
listed in TABLE 1.

The relation between the quark condensate $\langle\bar{\psi}\psi\rangle$ and
$F_{0}^{2}B_{0}$ is given by

$\langle\bar{\psi}\psi\rangle=-n_{f}F_{0}^{2}B_{0}.$ (53)

The chiral symmetry is spontaneously broken only if $F_{0}^{2}B_{0}$ is
positive. In the $\zeta$-function regularization, which was the earliest
regularization used in Ref. Espriu et al. (1990), $F_{0}^{2}B_{0}$ is
logarithmically divergent but negative, making this regularization
inappropriate. It was replaced with Schwinger proper time regularization in
Ref. Bijnens (1996). In the Schwinger proper time regularization, the
logarithmic divergence in $F_{0}^{2}B_{0}$ is same as the ones in
$\zeta$-function regularization, but the quadratic divergence in
$F_{0}^{2}B_{0}$ is positive and dominant. As emphasized in the Chapter 6 of
Ref. Bijnens (1996), “The chiral symmetry is spontaneously broken by the
quadratic divergence. In a regulator that does not have the quadratic
divergence, like dimensional regularization, one always works in the phase
where chiral symmetry is explicitly realized in the spectrum. In the ENJL
model this means that we treat it as being in a phase with weakly interacting
massive quarks.” This discussion applies to QCD as well. The Schwinger proper
time regularization, whether assuming a constant quark mass Bijnens (1996) or
using a dynamical quark self-energy Yang et al. (2002); Jiang et al. (2010)
has indeed provided reasonable LECs for pseudo-scalar mesons.

However if the calculation for pseudo-scalar mesons in Ref. Yang et al.
(2002); Jiang et al. (2010) is generalized to vector mesons, the obtained
parameter $a\equiv F_{\sigma}^{2}/F_{0}^{2}\simeq 0.35$. This result is
significantly smaller than the phenomenological value $a_{\text{ph}}\sim 2$,
and violates many phenomenological facts discussed in Sec. II. The Schwinger
proper time regularization with a constant quark mass even yields a zero
result for $F_{\sigma}^{2}$, as shown in TABLE 1. Thus, Schwinger proper time
regularization previously used does not work for vector mesons. In cut-off
regularization, the divergence in $F_{0}^{2}B_{0}$ and the logarithmic
divergence in $F_{0}^{2}$ are same as in the Schwinger proper time
regularization, but the dominant contribution for $F_{0}^{2}$ and
$F_{\sigma}^{2}$ becomes the quadratic divergence, which is absent in other
regularizations. In this case, the obtained values of $F_{0}^{2}$ and
$F_{\sigma}^{2}$ are reasonable, although their signs are negative. As
emphasized in the Section 4.5 of Ref. Harada and Yamawaki (2003), “In the
usual phenomenological study in the ChPT of the pseudoscalar mesons as well as
the calculations in the early stage of the ChPT with HLS, only the logarithmic
divergence was included. As far as the bare theory is not referred to, the
quadratic divergence is simply absorbed into redefinitions of the parameters.
$\cdots$ However, as was shown in Refs. Harada and Yamawaki (1999); Harada and
Yamawaki (2001a, b), the inclusion of the quadratic divergence is essential to
studying the phase transition with referring to the bare theory. Moreover, it
was shown Harada and Yamawaki (2001c) that inclusion of the quadratic
divergence is needed to match the HLS with the underlying QCD even for
phenomenological reason.” Quadratic divergence also appear in the bare
parameters determined by Wilsonian matching of the EFT with the underlying QCD
in Ref. Harada and Yamawaki (2001c). Specifically, the $\mathcal{O}(p^{2})$
LECs are sensitive to the regularizations used. This suggests that the current
handling of high-energy degrees of freedom in such research is still
relatively crude. Most $\mathcal{O}(p^{4})$ LECs, on the other hand, diverge
logarithmically, and are therefore less sensitive to regularizations.

Table 1: The $\mathcal{O}(p^{2})$ order coefficients under different regularizations when assuming $\Sigma(p^{2})$ as a constant mass $m$ | $F_{0}^{2}B_{0}$ | $F_{0}^{2}$ | $F_{\sigma}^{2}$
---|---|---|---
cut-off regularizationChen et al. (2020, 2021) | $\frac{N_{c}}{4\pi^{2}}m\left[\Lambda^{2}-m^{2}\ln\frac{\Lambda^{2}}{m^{2}}\right]$ | $-\frac{N_{c}}{16\pi^{2}}\left[\Lambda^{2}-4m^{2}\ln\frac{\Lambda^{2}}{m^{2}}\right]$ | $-\frac{N_{c}}{16\pi^{2}}\Lambda^{2}$
Schwinger proper time regularizationBijnens (1996); Lu et al. (2002) | $\frac{N_{c}}{4\pi^{2}}m\left[\Lambda^{2}-m^{2}\ln\frac{\Lambda^{2}}{m^{2}}\right]$ | $\frac{N_{c}}{4\pi^{2}}m^{2}\ln\frac{\Lambda^{2}}{m^{2}}$ | 0
$\zeta$-function regularizationEspriu et al. (1990) | $-\frac{N_{c}}{4\pi^{2}}m^{3}\ln\frac{\Lambda^{2}}{m^{2}}$ | $\frac{N_{c}}{4\pi^{2}}m^{2}\ln\frac{\Lambda^{2}}{m^{2}}$ |

In this paper, we adopt cut-off regularization, with the physical ultraviolet
cutoff chosen as $\Lambda=1000^{+100}_{-100}$MeV. The pion decay constant in
the chiral limit $|F_{0}^{2}|=(87\text{MeV})^{2}$ is used to determine the
parameter $\Lambda_{\text{QCD}}$ in the running coupling constant, resulting
in $\Lambda_{\text{QCD}}=537_{-94}^{+89}$MeV. The obtained quark condensate
$\langle\bar{\psi}\psi\rangle$ is

$\langle\bar{\psi}\psi\rangle=-n_{f}F_{0}^{2}B_{0}=-(336_{-51}^{+48}~{}\text{MeV})^{3},$
(54)

which compares with the experimentally determined value
$\langle\bar{\psi}\psi\rangle_{\text{exp}}=-(250~{}\text{MeV})^{3}$ from the
QCD sum rule at the typical hadronic mass scale Shifman et al. (1979). The
important parameter $a$ in HLS model is calculated as

$a\equiv\frac{F_{\sigma}^{2}}{F_{0}^{2}}=1.97_{-0.30}^{+0.34},$ (55)

which is consistent with the phenomenological value $a_{\text{ph}}=2$. As
discussed in Sec. II, this result supports many phenomenological facts, for
example the universality of the $\rho$-coupling $g_{\rho\pi\pi}=g$, the $\rho$
meson dominance of the pion’s electromagnetic form factor $g_{\gamma\pi\pi}=0$
and the KSRF formulas.

The obtained coefficients $\mathcal{C}_{i}$ in Eq.(40) are listed in TABLE 2.
The value $\equiv 0$ means that the corresponding item breaks charge
conjugation invariance, indicating the correctness of our calculation. The
coupling constant for HLS can be calculated as

$g=\left(-\frac{1}{2(\mathcal{C}_{9}+\mathcal{C}_{13}-\mathcal{C}_{28})}\right)^{\frac{1}{2}}=7.45_{-0.05}^{+0.08}.$
(56)

This is the first calculation of important parameters $a$ and $g$ from the
first principles of QCD, rather than treating them as input values as in the
literature. Using these, we obtain the vector meson mass, the $\rho\pi\pi$
coupling and the $\rho-\gamma$ mixing strength Harada and Yamawaki (2003) as

$\displaystyle M_{V}^{2}$
$\displaystyle=ag^{2}|F_{0}^{2}|=(910_{-79}^{+85}\text{MeV})^{2},$ (57)
$\displaystyle g_{\rho\pi\pi}$
$\displaystyle=\frac{1}{2}ag=7.35_{-1.17}^{+1.34},$ $\displaystyle g_{\rho}$
$\displaystyle=ag|F^{2}_{0}|=0.111_{-0.018}^{+0.020}\text{GeV}^{2}.$

Considering the large theoretical uncertainty in this calculation, the
obtained coupling for $\rho\pi\pi$ and the mixing strength for $\rho-\gamma$
are consistent with the experimental values Harada and Yamawaki (2003),

$g_{\rho\pi\pi}|_{\text{exp}}=6.00\pm
0.01,~{}~{}g_{\rho}|_{\text{exp}}=0.119\pm 0.001\text{GeV}^{2}.$ (58)

The obtained vector meson mass is slightly larger than the experimental value
$m_{\rho}|_{\text{exp}}=775$MeV Workman et al. (2022), because the obtained
coupling constant $g$ for HLS slightly larger than the phenomenological value
$g\sim 6$. We look forward to higher precision calculations like in Ref.Jiang
et al. (2015) to address this discrepancy.

Table 2: The obtained coefficients $\mathcal{C}_{i}$ in Eq.(40). $\mathcal{C}_{6},\mathcal{C}_{7}$ are in units of $10^{-3}$GeV2, $\mathcal{C}_{8},\mathcal{C}_{11},\mathcal{C}_{16},\mathcal{C}_{25}$ are in units of $10^{-3}$GeV, the rest of coefficients are in units of $10^{-3}$. The resulting LECs are taken as the values at $\Lambda=1000$MeV with the superscript the difference caused at $\Lambda=1100$MeV and the subscript the difference caused at $\Lambda=900$MeV. The value $\equiv 0$ means that the corresponding item breaks the invariance under charge conjugation. $i$ | $\mathcal{C}_{i}$ | $j$ | $\mathcal{C}_{j}$ | $k$ | $\mathcal{C}_{k}$
---|---|---|---|---|---
2 | $3.37_{+0.03}^{-0.04}$ | 12 | $10.0_{+0.6}^{-0.3}$ | 22 | $-15.2_{-2.2}^{+1.5}$
3 | $-5.23_{-1.12}^{+0.76}$ | 13 | $-9.45_{-1.10}^{+0.74}$ | 23 | $21.3_{+4.3}^{-2.8}$
4 | $-1.91_{-2.06}^{+1.30}$ | 14 | $-15.2_{-2.2}^{+1.5}$ | 24 | $\equiv 0$
5 | $7.35_{+2.13}^{-1.38}$ | 15 | $14.0_{+2.2}^{-1.5}$ | 25 | $19.2_{-2.5}^{+2.3}\mathrm{i}$
6 | $18.8_{-1.2}^{+1.3}$ | 16 | $5.63_{-0.98}^{+0.93}$ | 26 | $\equiv 0$
7 | $26.2_{-3.5}^{+3.8}$ | 17 | $29.1_{+2.8}^{-2.0}$ | 27 | $9.69_{+4.31}^{-2.78}\mathrm{i}$
8 | $21.5_{-2.1}^{+1.8}$ | 18 | $\equiv 0$ | 28 | $-21.7_{-2.2}^{+1.6}$
9 | $-21.2_{-1.3}^{+0.9}$ | 19 | $-20.8_{-4.4}^{+2.9}$ | 29 | $22.1_{+4.4}^{-2.9}\mathrm{i}$
10 | $-18.2_{-4.4}^{+2.9}$ | 20 | $33.1_{+8.6}^{-5.7}$ | 30 | $\equiv 0$
11 | $32.0_{-4.2}^{+3.7}$ | 21 | $-7.48_{-4.26}^{+2.76}$ | 31 | $15.2_{+0.1}^{-0.1}\mathrm{i}$

The obtained values of $\mathcal{O}(p^{4})$ LECs in HLS model are listed in
TABLE 3. Few studies provide these values. Ref.Ma et al. (2013) calculated
$\mathcal{O}(p^{4})$ LECs in HLS model using two holographic QCD models: the
Sakai-Sugimoto model and the BPS model. The resulting values are listed in
TABLE 4. The outcomes from these two holographic QCD models show significant
differences, and there is also a notable discrepancy compared to our results.
We anticipate more experimental values or model calculations for the
$\mathcal{O}(p^{4})$ LECs of HLS model in the future.

In HLS model, the pion electromagnetic form factor $F_{V}^{\pi^{\pm}}$ is
constructed from two contributions, one from $\gamma\pi\pi$ direct coupling
diagram, another from $\rho$-mediated diagram:

$F_{V}^{\pi^{\pm}}(Q^{2})=g_{\gamma\pi\pi}(Q^{2})+\frac{g_{\rho}(Q^{2})g_{\rho\pi\pi}(Q^{2})}{m_{\rho}^{2}+Q^{2}}.$
(59)

By using the Lagrangian given in Eq. (8) and Eq. (85), the above expression
can be rewritten as Harada et al. (2010)

$F_{V}^{\pi^{\pm}}(Q^{2})=\left(1-\frac{\tilde{a}}{2}\right)+\tilde{z}\frac{Q^{2}}{m_{\rho}^{2}}+\frac{\tilde{a}}{2}\frac{m_{\rho}^{2}}{m_{\rho}^{2}+Q^{2}},$
(60)

where

$\displaystyle\tilde{a}$
$\displaystyle=a\left(1-\frac{g^{2}z_{4}}{2}-g^{2}z_{3}+\frac{(g^{2}z_{3})(g^{2}z_{4})}{2}\right),$
(61) $\displaystyle\tilde{z}$
$\displaystyle=\frac{1}{4}a(g^{2}z_{6}+(g^{2}z_{3})(g^{2}z_{4})).$

Using the results provided in Eq. (55), Eq. (56) and TABLE 3, the obtained
values are
$\tilde{a}=2.09_{-0.31}^{+0.35},~{}~{}\tilde{z}=-0.781_{+0.123}^{-0.148}$. In
Ref.Harada et al. (2010), $\tilde{a}$ and $\tilde{z}$ are determined in the
holographic QCD model (Sakai-Sugimoto model) as $\tilde{a}_{\text{SS}}\simeq
2.62,~{}\tilde{z}_{\text{SS}}\simeq 0.08$. The values fitting the experimental
data are $\tilde{a}_{\text{fit}}=2.44,~{}\tilde{z}_{\text{fit}}=0.08$. By
comparison, our calculated $\tilde{a}$ is consistent with other results within
the error range, but our $\tilde{z}$ shows some deviation from other results.

The relations between the coefficients in the HLS model and the anti-symmetric
tensor field model are given by Tanabashi (1996)

$a=\frac{F_{V}^{2}}{F_{0}^{2}},~{}~{}g=\frac{M_{V}}{F_{V}},~{}~{}z_{4}=\frac{2F_{V}(F_{V}-2G_{V})}{M_{V}^{2}}.$
(62)

With these relations, we obtain the coefficients $F_{V}$ and $G_{V}$ in the
anti-symmetric tensor field model:

$F_{V}=122^{+10}_{-10}\text{MeV},~{}~{}G_{V}=56.0^{+4.5}_{-4.3}\text{MeV}.$
(63)

Considering the pion decay constant in this calculation is taken in the chiral
limit, these coefficients are consistent with the experimental values
$F_{V}|_{\text{exp}}=154$MeV, $G_{V}|_{\text{exp}}=69$MeV Ecker et al.
(1989b).

Table 3: The obtained values of $\mathcal{O}(p^{4})$ LECs. The LECs are in units of $10^{-3}$. The resulting LECs are taken as the values at $\Lambda=1000$MeV with the superscript the difference caused at $\Lambda=1100$MeV and the subscript the difference caused at $\Lambda=900$MeV. The value $\equiv 0$ means that the constants vanish at the large-$N_{c}$ limit. $i$ | $y_{i}$ | $\omega_{i}$ | $z_{i}$
---|---|---|---
1 | $-11.1_{-0.1}^{+0.1}$ | $3.22_{+0.29}^{-0.23}$ | $-9.44_{-1.11}^{+0.73}$
2 | $16.5_{+0.1}^{-0.2}$ | $\equiv 0$ | $-5.23_{-1.12}^{+0.76}$
3 | $-34.1_{-4.4}^{+2.9}$ | $0.844_{+0.001}^{-0.002}$ | $-2.77_{-0.05}^{+0.06}$
4 | $32.9_{+4.4}^{-2.9}$ | $\equiv 0$ | $2.97_{-0.02}^{+0.01}$
5 | $2.57_{-0.10}^{+0.10}$ | $-12.2_{-0.4}^{+0.4}$ | $-34.6_{-3.0}^{+2.0}$
6 | $-22.9_{-6.6}^{+4.4}$ | $0.422_{+0.159}^{-0.091}$ | $-28.1_{-0.2}^{+0.2}$
7 | $19.2_{-2.1}^{+1.3}$ | $\equiv 0$ | $-37.8_{-4.4}^{+2.9}$
8 | $-0.567_{+4.37}^{-2.88}$ | $0.401_{-0.101}^{+0.043}$ | $-1.21_{+0.09}^{-0.11}$
9 | $-1.98_{+0.02}^{-0.04}$ | $-0.329_{-0.020}^{+0.016}$ |
10$\sim$18 | $\equiv 0$ |  |
Table 4: The obtained values of $\mathcal{O}(p^{4})$ LECs from two holographic QCD modelsMa et al. (2013). The LECs are in units of $10^{-3}$. $i$ |  $y_{i}$(SS) |  $y_{i}$(BPS) |  $z_{i}$(SS) |  $z_{i}$(BPS)
---|---|---|---|---
1 | $-1.10$ | $-71.9$ |  |
2 | $1.10$ | $71.9$ |  |
3 | $-2.83$ | $-154$ |  |
4 | $2.83$ | $154$ | $10.8$ | $90.3$
5 | $-15.9$ | $-12.3$ | $-7.33$ | $-131$
6 | $13.7$ | $-197$ |  |
7 | $2.21$ | $209$ |  |
8 | $-7.96$ | $-6.14$ |  |
9 | $15.9$ | $12.3$ |  |

Using of Eq.(87), we obtain the contributions of vector mesons to the
$\mathcal{O}(p^{4})$ Lagrangian for pure pseudo-scalar mesons. The obtained
values of $L_{i}$ and their comparison with values in the literature are
listed in TABLE 5. Our results are consistent with those in the literature.

Table 5: The values of $L_{i}$ obtained by Eq.(87) and the values in the literature, which are in units of $10^{-3}$. The value $\equiv 0$ means that the constants vanish at the large-$N_{c}$ limit. | This paper | Ref.Jiang et al. (2010) | Ref.Gasser and Leutwyler (1985) | Ref.Pich (2008) | Ref.Bijnens and Jemos (2012)
---|---|---|---|---|---
$L_{1}$ | $0.987_{+0.011}^{-0.013}$ | $1.23^{+0.03}_{-0.04}$ | 0.9$\pm$0.3 | 0.4$\pm$0.3 | 0.88$\pm$0.09
$L_{2}$ | $1.97_{+0.03}^{-0.02}$ | $2.46^{+0.05}_{-0.08}$ | 1.7$\pm$0.7 | 1.4$\pm$0.3 | 0.61$\pm$0.20
$L_{3}$ | $-5.58_{-0.06}^{+0.07}$ | $-6.85^{-0.14}_{+0.21}$ | -4.4$\pm$2.5 | -3.5$\pm$1.1 | -3.04$\pm$0.43
$L_{4}$ | $\equiv 0$ | $\equiv 0$ | 0$\pm$0.5 | -0.3$\pm$0.5 | 0.75$\pm$0.75
$L_{5}$ | $0.806_{+0.072}^{-0.060}$ | $1.48^{-0.01}_{-0.03}$ | 2.2$\pm$0.5 | 1.4$\pm$0.5 | 0.58$\pm$0.13
$L_{6}$ | $\equiv 0$ | $\equiv 0$ | 0$\pm$0.3 | -0.2$\pm$0.3 | 0.29$\pm$0.85
$L_{7}$ | $-0.329_{-0.020}^{+0.017}$ | $-0.51^{+0.05}_{-0.06}$ | -0.4$\pm$0.15 | -0.4$\pm$0.2 | -0.11$\pm$0.15
$L_{8}$ | $0.823_{+0.058}^{-0.048}$ | $1.02^{-0.06}_{+0.06}$ | 1.1$\pm$0.3 | 0.9$\pm$0.3 | 0.18$\pm$0.18
$L_{9}$ | $8.33_{+0.10}^{-0.12}$ | $8.86^{+0.24}_{-0.37}$ | 7.4$\pm$0.7 | 6.9$\pm$0.7 | 5.93$\pm$4.30
$L_{10}$ | $-8.00_{-0.08}^{+0.11}$ | $-7.40^{-0.29}_{+0.44}$ | -6.0$\pm$0.7 | -5.5$\pm$0.7 | -4.06$\pm$0.39

The obtained values of $\mathcal{O}(p^{4})$ anomalous LECs in HLS model and
their comparison with values in the literature are listed in TABLE 6. The
results from the two holographic QCD models differ significantly, and there is
also a significant difference compared to our results. By using the Lagrangian
given in Eq. (8), Eq. (85) and Eq. (88), effective $\omega\pi^{0}\gamma$
vertex function can be written as Harada et al. (2010)

$\Gamma^{\mu\nu}[\omega_{\mu}(p),\pi^{0},\gamma^{*}_{\nu}(k)]=\frac{eN_{c}}{8\pi^{2}F_{\pi}}\epsilon^{\mu\nu\alpha\beta}p_{\alpha}k_{\beta}\cdot[A^{\omega\pi\gamma}+B^{\omega\pi\gamma}D_{\rho}(k^{2})],$
(64)

where

$\displaystyle A^{\omega\pi\gamma}=$
$\displaystyle\frac{1}{2}g[(c_{4}+c_{3}\cdot
g^{2}z_{3})-c_{3}(1-g^{2}z_{3})],$ (65) $\displaystyle B^{\omega\pi\gamma}=$
$\displaystyle
gc_{3}(1-g^{2}z_{3}),~{}~{}D_{\rho}(k^{2})=\frac{m^{2}_{\rho}}{m^{2}_{\rho}-p^{2}}.$

The $\omega-\pi^{0}$ transition form factor $F_{\omega\pi^{0}}$ can be
extracted from the $\omega\to\pi^{0}l^{+}l^{-}(l^{\pm}=e^{\pm},\mu^{\pm})$
decay width Harada et al. (2010):

$F_{\omega\pi^{0}}(q^{2})=(1-\tilde{r})+\tilde{r}D_{\rho}(q^{2}),$ (66)

where
$\tilde{r}=B^{\omega\pi\gamma}/(A^{\omega\pi\gamma}+B^{\omega\pi\gamma})$.
Using the results given in Eq. (56), TABLE 3 and TABLE 6, the obtained value
is $\tilde{r}=1.77$. In Ref.Harada et al. (2010), $\tilde{r}$ is determined in
the holographic QCD model (Sakai-Sugimoto model) as
$\tilde{r}_{\text{SS}}\simeq 1.53$, and the value by fitting to the
experimental data is $\tilde{r}_{\text{fit}}=2.08$. By comparison, our
calculated $\tilde{r}$ is consistent with other results.

Table 6: The obtained values of $\mathcal{O}(p^{4})$ anomalous LECs and the values in the literature. | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$
---|---|---|---|---
This paper | 0.0389 | 0.190 | -1.50 | -0.458
SS modelMa et al. (2013) | 0.382 | -0.130 | 0.767 | –
BPS modelMa et al. (2013) | -0.207 | 3.03 | 1.47 | –

##  VI Weinberg sum rules

Although the signs of obtained $F_{0}^{2}$ and $F_{\sigma}^{2}$ are negative,
our method is consistent with the Weinberg sum rules Weinberg (1967) which
relate the spectral functions of the vector and axial-vector currents. These
spectral functions, $\rho_{V}(p^{2})$ and $\rho_{A}(p^{2})$, are defined by
the quark current Green functions:

$\displaystyle\langle V_{i}^{\mu}(x)V_{j}^{\nu}(0)\rangle$
$\displaystyle=(2\pi)^{-3}\delta_{ij}\int\mathrm{d}^{4}p\theta(p^{0})\mathrm{e}^{-\mathrm{i}p\cdot
x}\rho_{V}(p^{2})[-g^{\mu\nu}+p^{\mu}p^{\nu}/p^{2}],$ (67)
$\displaystyle\langle A_{i}^{\mu}(x)A_{j}^{\nu}(0)\rangle$
$\displaystyle=(2\pi)^{-3}\delta_{ij}\int\mathrm{d}^{4}p\theta(p^{0})\mathrm{e}^{-\mathrm{i}p\cdot
x}\\{\rho_{A}(p^{2})[-g^{\mu\nu}+p^{\mu}p^{\nu}/p^{2}]+F_{\pi}^{2}\delta(p^{2})p^{\mu}p^{\nu}\\},$

where $V_{i}^{\mu}(x)$ and $A_{i}^{\mu}(x)$ are vector and axial-vector
currents respectively. The first and second Weinberg sum rules are given by

$\displaystyle\int_{0}^{\infty}[\rho_{V}(\mu^{2})-\rho_{A}(\mu^{2})]\mu^{-2}d\mu^{2}=F_{\pi}^{2},$
(68)
$\displaystyle\int_{0}^{\infty}[\rho_{V}(\mu^{2})-\rho_{A}(\mu^{2})]d\mu^{2}=0,$

respectively, where $p^{2}=\mu^{2}$.

In this section, we will derive the Weinberg sum rules using our method. We
start with Eq.(13), which can be regarded as the deformed action of QCD after
integrating out the quark and gluon fields, and integrating in the bilocal
auxiliary fields $\Phi^{\sigma\rho}(x,y)$ and $\Pi^{\sigma\rho}(x,y)$. In this
case, there is no need to integrate in the local pseudo-scalar mesons and
vector meson fields. After formally completing the integration over the
auxiliary fields $\Phi,\Pi$, taking the large-$N_{c}$ limit and the leading
order in dynamical perturbation, the generating functional of quark current
Green functions simplifies to

$W[J]=-\mathrm{i}N_{c}\mathrm{Tr}\ln[\mathrm{i}\not{\partial}+J-\Pi_{c}],$
(69)

where $J(x)=\not{v}(x)+\not{a}(x)\gamma^{5}-s(x)+\mathrm{i}p(x)\gamma^{5}$ is
the external source for quark current, and $\Pi_{c}$ is the classical field of
auxiliary field $\Pi^{\sigma\rho}(x,y)$ which satisfies the classical field
equation:

$\Phi^{\sigma\rho}_{c}(x,y)=-\mathrm{i}[(\mathrm{i}\not{\partial}+J-\Pi_{c})^{-1}]^{\rho\sigma}(y,x).$
(70)

Firstly, we calculte the vector current Green function:

$\displaystyle\langle V_{i}^{\mu}(x)V_{j}^{\nu}(y)\rangle$
$\displaystyle=\left.\frac{\delta^{2}W}{\delta v^{i}_{\mu}(x)\delta
v^{j}_{\nu}(y)}\right|_{J=0}$ (71)
$\displaystyle=-\mathrm{i}N_{c}\mathrm{d}^{4}z\mathrm{d}^{4}z^{\prime}\mathrm{tr}_{lf}\left.\left[\gamma^{\mu}\lambda^{i}\Phi_{c}(x,z)\frac{\delta\Phi_{c}^{-1}(z,z^{\prime})}{\delta
v^{j}_{\nu}(y)}\Phi_{c}(z^{\prime},x)\right]\right|_{J=0}$
$\displaystyle=-\mathrm{i}N_{c}\mathrm{d}^{4}z\mathrm{d}^{4}z^{\prime}\mathrm{tr}_{lf}\left[\gamma^{\mu}\lambda^{i}\Phi_{c}(x,z)\delta(z-z^{\prime})\delta(z-y)\gamma^{\nu}\lambda^{j}\Phi_{c}(z^{\prime},x)\right]|_{J=0}$
$\displaystyle=-\mathrm{i}N_{c}\mathrm{tr}_{lf}\left[\gamma^{\mu}\lambda^{i}\Phi_{c}(x,y)\gamma^{\nu}\lambda^{j}\Phi_{c}(y,x)\right]|_{J=0},$

where $\lambda^{i}$ is the Gell-Mann matrix in flavor space. As in section IV,
$\Pi_{c}$ can be further approximated as the quark self-energy:

$\Pi_{c}(x,y)=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\mathrm{e}^{-\mathrm{i}p\cdot(x-y)}\Sigma(-p^{2}).$
(72)

Then the vector current Green function is

$\displaystyle\langle V_{i}^{\mu}(x)V_{j}^{\nu}(y)\rangle$ (73)
$\displaystyle=-\mathrm{i}N_{c}\delta^{ij}\int\frac{\mathrm{d}^{4}p\mathrm{d}^{4}q}{(2\pi)^{8}}\mathrm{e}^{-\mathrm{i}(p-q)\cdot(x-y)}\mathrm{tr}_{l}\left[\gamma^{\mu}\frac{1}{\not{p}-\Sigma(-p^{2})}\gamma^{\nu}\frac{1}{\not{q}-\Sigma(-q^{2})}\right]$
$\displaystyle=-\mathrm{i}N_{c}\delta^{ij}\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\mathrm{e}^{-\mathrm{i}p\cdot(x-y)}\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\mathrm{tr}_{l}\left[\gamma^{\mu}\frac{1}{\not{p}+\not{q}-\Sigma(-(p+q)^{2})}\gamma^{\nu}\frac{1}{\not{q}-\Sigma(-q^{2})}\right]$
$\displaystyle=4\mathrm{i}N_{c}\delta^{ij}\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\mathrm{e}^{-\mathrm{i}p\cdot(x-y)}\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}$
$\displaystyle\times\left[-g^{\mu\nu}\Sigma(-(p+q)^{2})\Sigma(-q^{2})+g^{\mu\nu}p\cdot
q+g^{\mu\nu}q^{2}-q^{\mu}p^{\nu}-p^{\mu}q^{\nu}-2q^{\mu}q^{\nu}\right]f(p\cdot
q,q^{2},p^{2}),$

where we shift integral momentum $p^{\mu}$ in the second equal sign, and

$f(p\cdot
q,q^{2},p^{2})\equiv\frac{1}{(p+q)^{2}-\Sigma^{2}(-(p+q)^{2})}\frac{1}{q^{2}-\Sigma^{2}(-q^{2})}.$
(74)

We define some integral functions as

$\displaystyle\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\Sigma(-(p+q)^{2})\Sigma(-q^{2})f(p\cdot
q,q^{2},p^{2})$ $\displaystyle\equiv F_{1}(p^{2}),$ (75)
$\displaystyle\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}q^{\mu}f(p\cdot
q,q^{2},p^{2})$ $\displaystyle\equiv p^{\mu}F_{2}(p^{2}),$
$\displaystyle\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}q^{\mu}q^{\nu}f(p\cdot
q,q^{2},p^{2})$ $\displaystyle\equiv
g^{\mu\nu}G_{1}(p^{2})+p^{\mu}p^{\nu}G_{2}(p^{2}).$

Finally the vector current Green function becomes

$\langle
V_{i}^{\mu}(x)V_{j}^{\nu}(y)\rangle=4\mathrm{i}N_{c}\delta^{ij}\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\mathrm{e}^{-\mathrm{i}p\cdot(x-y)}T^{\mu\nu},$
(76)

where

$T^{\mu\nu}\equiv\left[g^{\mu\nu}(-F_{1}+p^{2}F_{2}+2G_{1}+p^{2}G_{2})-p^{\mu}p^{\nu}(2F_{2}+2G_{2})\right].$
(77)

The conservation of vector current requires the following subtraction, as in
Ref. Bijnens (1996):

$T^{{}^{\prime}\mu\nu}=T^{\mu\nu}-g^{\mu\nu}T^{\rho\sigma}\frac{p_{\rho}p_{\sigma}}{p^{2}}$
(78)

Then the conservation of vector current is satisfied:
$p_{\mu}T^{{}^{\prime}\mu\nu}=0$. Compared to Eq.(67), we obtain the spectral
function $\rho_{V}(p^{2})$:

$\rho_{V}=-\frac{2\mathrm{i}N_{c}}{\pi}p^{2}(2F_{2}+2G_{2}).$ (79)

Similarly, we can obtain the spectral function $\rho_{A}(p^{2})$ as

$\displaystyle\rho_{A}=$
$\displaystyle-\frac{2\mathrm{i}N_{c}}{\pi}(F_{1}+p^{2}F_{2}+2G_{1}+p^{2}G_{2}),$
(80) $\displaystyle\frac{\rho_{A}}{p^{2}}+F_{\pi}^{2}\delta(p^{2})=$
$\displaystyle-\frac{2\mathrm{i}N_{c}}{\pi}(2F_{2}+2G_{2})$

There is a relation between these two spectral functions:

$\frac{\rho_{V}}{\mu^{2}}-\frac{\rho_{A}}{\mu^{2}}=F_{\pi}^{2}\delta(\mu^{2}).$
(81)

where $p^{2}=\mu^{2}$. After performing the integral, we obtain the Weinberg
sum rules

$\displaystyle\int_{0^{-}}^{\infty}[\rho_{V}(\mu^{2})-\rho_{A}(\mu^{2})]\mu^{-2}d\mu^{2}=\int_{0^{-}}^{\infty}F_{\pi}^{2}\delta(\mu^{2})d\mu^{2}=F_{\pi}^{2},$
(82)
$\displaystyle\int_{0^{-}}^{\infty}[\rho_{V}(\mu^{2})-\rho_{A}(\mu^{2})]d\mu^{2}=\int_{0^{-}}^{\infty}F_{\pi}^{2}\mu^{2}\delta(\mu^{2})d\mu^{2}=0.$

##  VII Conclusion

In this paper, we review our original derivation of effective chiral
Lagrangian involving pseudo-scalar mesons and vector mesons (with hidden
symmetry) from QCD. With proper approximations, we successfully obtain the
analytical expressions for the $\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$
LECs in both the normal and anomalous parts of the chiral Lagrangian based on
the quark self-energy. Using $|F_{0}^{2}|=(87\text{MeV})^{2}$ to fix the
$\Lambda_{\text{QCD}}$ in the running coupling constant and choosing a cutoff
$\Lambda=1000^{+100}_{-100}$MeV, we calculate the LECs numerically up to
$\mathcal{O}(p^{4})$. This marks the first time that LECs for vector mesons
have been calculated directly from QCD, particularly for the critical
parameters $a$ and $g$, which are typically treated as input values in the
literature. The obtained $a$ is consistent with the phenomenological value,
allowing us to reproduce many phenomenological results in our calculation. The
obtained $g$ is slightly larger than the phenomenological value, but the non-
Trln term in the action might compensate for this discrepancy Jiang et al.
(2015). This aspect of the study is ongoing. The $\mathcal{O}(p^{4})$ LECs we
derive are comparable to those found in the literature, and we anticipate more
experiment values or model calculation for the $\mathcal{O}(p^{4})$ LECs of
HLS model in the future.

The methodology for deriving the chiral effective Lagrangian of vector mesons
and calculating the LECs from QCD can be straightforwardly extended to axial-
vector mesons, and such studies are also in progress.

In the studies focused on calculating the LECs in ChPT from underlying theory
or corresponding approximate models, there are regularization dependence and
sign issues for the $\mathcal{O}(p^{2})$ LECs. The choice of regularization is
generally guided by phenomenological reasons. Despite these challenges, we
have demonstrated that our method is consistent with Weinberg sum rules.
Further research is needed to address these issues comprehensively.

##  VIII Acknowledgments

We would like to thank Shao-Zhou Jiang and Yong-Liang Ma for useful
discussions.

##  Appendix A The $\mathcal{O}(p^{4})$ Lagrangian in HLS model

The $\mathcal{O}(p^{4})$ Lagrangian for general $N_{f}$ are given by Harada
and Yamawaki (2003)

$\mathcal{L}_{(4)}=\mathcal{L}_{(4)y}+\mathcal{L}_{(4)\omega}+\mathcal{L}_{(4)z},$

where

$\displaystyle\mathcal{L}_{(4)y}$
$\displaystyle=y_{1}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}_{\bot\nu}\hat{\alpha}^{\nu}_{\bot}]+y_{2}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\bot\nu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\bot}]+y_{3}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\nu}_{\parallel}]$
(83)
$\displaystyle+y_{4}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]+y_{5}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\nu}_{\parallel}]+y_{6}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\bot\nu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]$
$\displaystyle+y_{7}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\bot\nu}\hat{\alpha}^{\nu}_{\parallel}\hat{\alpha}^{\mu}_{\parallel}]+y_{8}\\{\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}_{\bot\nu}\hat{\alpha}^{\nu}_{\parallel}]+\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\nu}_{\bot}\hat{\alpha}^{\mu}_{\parallel}]\\}$
$\displaystyle+y_{9}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\parallel}]+y_{10}(\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}])^{2}+y_{11}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\bot\nu}]\mathrm{tr}[\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\bot}]$
$\displaystyle+y_{12}(\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}^{\mu}_{\parallel}])^{2}+y_{13}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}_{\parallel\nu}]\mathrm{tr}[\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]+y_{14}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}]\mathrm{tr}[\hat{\alpha}_{\parallel\nu}\hat{\alpha}^{\nu}_{\parallel}]$
$\displaystyle+y_{15}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\bot\nu}]\mathrm{tr}[\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]+y_{16}(\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\parallel}])^{2}+y_{17}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\parallel\nu}]\mathrm{tr}[\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\parallel}]$
$\displaystyle+y_{18}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}_{\parallel\nu}]\mathrm{tr}[\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\bot}],$
$\displaystyle\mathcal{L}_{(4)\omega}$
$\displaystyle=\omega_{1}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}(\hat{\chi}+\hat{\chi}^{\dagger})]+\omega_{2}\mathrm{tr}[\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\bot}]\mathrm{tr}[\hat{\chi}+\hat{\chi}^{\dagger}]$
(84)
$\displaystyle+\omega_{3}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}^{\mu}_{\parallel}(\hat{\chi}+\hat{\chi}^{\dagger})]+\omega_{4}\mathrm{tr}[\hat{\alpha}_{\parallel\mu}\hat{\alpha}^{\mu}_{\parallel}]\mathrm{tr}[\hat{\chi}+\hat{\chi}^{\dagger}]$
$\displaystyle+\omega_{5}\mathrm{tr}[(\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}_{\bot\mu}-\hat{\alpha}_{\bot\mu}\hat{\alpha}^{\mu}_{\parallel})(\hat{\chi}-\hat{\chi}^{\dagger})]$
$\displaystyle+\omega_{6}\mathrm{tr}[(\hat{\chi}+\hat{\chi}^{\dagger})^{2}]+\omega_{7}(\mathrm{tr}[\hat{\chi}+\hat{\chi}^{\dagger}])^{2}$
$\displaystyle+\omega_{8}\mathrm{tr}[(\hat{\chi}-\hat{\chi}^{\dagger})^{2}]+\omega_{9}(\mathrm{tr}[\hat{\chi}-\hat{\chi}^{\dagger}])^{2},$
$\displaystyle\mathcal{L}_{(4)z}$
$\displaystyle=z_{1}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}\hat{\mathcal{V}}^{\mu\nu}]+z_{2}\mathrm{tr}[\hat{\mathcal{A}}_{\mu\nu}\hat{\mathcal{A}}^{\mu\nu}]+z_{3}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}V^{\mu\nu}]+\mathrm{i}z_{4}\mathrm{tr}[V_{\mu\nu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\bot}]$
(85)
$\displaystyle+\mathrm{i}z_{5}\mathrm{tr}[V_{\mu\nu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]+\mathrm{i}z_{6}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\bot}]+\mathrm{i}z_{7}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\parallel}]-\mathrm{i}z_{8}\mathrm{tr}[\hat{\mathcal{A}}_{\mu\nu}(\hat{\alpha}^{\mu}_{\bot}\hat{\alpha}^{\nu}_{\parallel}+\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}^{\nu}_{\bot})]$
$\displaystyle=z_{1}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}\hat{\mathcal{V}}^{\mu\nu}]+z_{2}\mathrm{tr}[\hat{\mathcal{A}}_{\mu\nu}\hat{\mathcal{A}}^{\mu\nu}]+z_{3}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}V^{\mu\nu}]+\mathrm{i}z_{4}\mathrm{tr}[V_{\mu\nu}a^{\mu}_{\xi}a^{\nu}_{\xi}]$
$\displaystyle+\mathrm{i}z_{5}\mathrm{tr}[V_{\mu\nu}{v^{\prime}}^{\mu}_{\xi}{v^{\prime}}^{\nu}_{\xi}]+\mathrm{i}z_{6}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}a^{\mu}_{\xi}a^{\nu}_{\xi}]+\mathrm{i}z_{7}\mathrm{tr}[\hat{\mathcal{V}}_{\mu\nu}{v^{\prime}}^{\mu}_{\xi}{v^{\prime}}^{\nu}_{\xi}]-\mathrm{i}z_{8}\mathrm{tr}[\hat{\mathcal{A}}_{\mu\nu}(a^{\mu}_{\xi}{v^{\prime}}^{\nu}_{\xi}+{v^{\prime}}^{\mu}_{\xi}a^{\nu}_{\xi})],$

where
$\hat{\mathcal{V}}_{\mu\nu}\equiv\frac{1}{2}[\hat{\mathcal{R}}_{\mu\nu}+\hat{\mathcal{L}}_{\mu\nu}],~{}~{}\hat{\mathcal{A}}_{\mu\nu}\equiv\frac{1}{2}[\hat{\mathcal{R}}_{\mu\nu}-\hat{\mathcal{L}}_{\mu\nu}]$,
and $\hat{\mathcal{R}}_{\mu\nu},\hat{\mathcal{L}}_{\mu\nu}$ are ”converted”
field strengths of the external gauge fields
$\mathcal{R}_{\mu\nu},\mathcal{L}_{\mu\nu}$ respectively:

$\displaystyle\hat{\mathcal{R}}_{\mu\nu}$
$\displaystyle=\xi_{R}\mathcal{R}_{\mu\nu}\xi_{R}^{\dagger},~{}\hat{\mathcal{L}}_{\mu\nu}=\xi_{L}\mathcal{L}_{\mu\nu}\xi_{L}^{\dagger}.$
(86)

Integrating out the vector mesons by use of its equation of motion in the
Lagrangian of the HLS, we obtain the $\mathcal{O}(p^{4})$ Lagrangian for
pseudoscalar mesons Gasser and Leutwyler (1984, 1985) where the coefficients
$L_{i}$ are expressed as the combinations of the coefficients in HLS model:

$\displaystyle L_{1}$
$\displaystyle=\frac{1}{32g^{2}}-\frac{1}{32}z_{4}+\frac{1}{32}y_{2}+\frac{1}{16}y_{10},$
(87) $\displaystyle L_{2}$
$\displaystyle=\frac{1}{16g^{2}}-\frac{1}{16}z_{4}+\frac{1}{16}y_{2}+\frac{1}{16}y_{11},$
$\displaystyle L_{3}$
$\displaystyle=-\frac{3}{16g^{2}}+\frac{3}{16}z_{4}+\frac{1}{16}y_{1}-\frac{1}{8}y_{2},$
$\displaystyle L_{4}$
$\displaystyle=\frac{1}{4}\omega_{2},~{}~{}~{}L_{5}=\frac{1}{4}\omega_{1},~{}~{}~{}L_{6}=\omega_{7},$
$\displaystyle L_{7}$
$\displaystyle=\omega_{9},~{}~{}~{}L_{8}=\omega_{6}+\omega_{8},$
$\displaystyle L_{9}$
$\displaystyle=\frac{1}{4}\left(\frac{1}{g^{2}}-z_{3}\right)-\frac{1}{8}(z_{4}+z_{6}),$
$\displaystyle L_{10}$
$\displaystyle=-\frac{1}{4g^{2}}+\frac{1}{2}(z_{3}-z_{2}+z_{1}),$
$\displaystyle H_{1}$
$\displaystyle=-\frac{1}{8g^{2}}+\frac{1}{4}(z_{3}+z_{2}+z_{1}),$
$\displaystyle H_{2}$ $\displaystyle=2(\omega_{6}-\omega_{8}).$

The $\mathcal{O}(p^{4})$ anomalous Lagrangian in HLS model is Harada and
Yamawaki (2003)

$\Gamma[\xi^{\dagger}_{R}\xi_{L},V,l,r]=\Gamma_{\text{WZ}}[\xi^{\dagger}_{R}\xi_{L},l,r]+\frac{N_{c}}{16\pi^{2}}\int\mathrm{d}^{4}x\sum^{4}_{i=1}c_{i}\mathcal{L}_{i},$
(88)

where $\Gamma_{\text{WZ}}$ is the well-known Wess-Zumino term, and
$\mathcal{L}_{i}$ are gauge invariant terms which conserve parity and charge
conjugation but violate the intrinsic parity:

$\displaystyle\mathcal{L}_{1}$
$\displaystyle=\mathrm{i}\epsilon_{\mu\nu\sigma\rho}\mathrm{tr}_{f}[\hat{\alpha}_{L}^{\mu}\hat{\alpha}_{L}^{\nu}\hat{\alpha}_{L}^{\sigma}\hat{\alpha}_{R}^{\rho}-\hat{\alpha}_{R}^{\mu}\hat{\alpha}_{R}^{\nu}\hat{\alpha}_{R}^{\sigma}\hat{\alpha}_{L}^{\rho}],$
(89) $\displaystyle\mathcal{L}_{2}$
$\displaystyle=\mathrm{i}\epsilon_{\mu\nu\sigma\rho}\mathrm{tr}_{f}[\hat{\alpha}_{L}^{\mu}\hat{\alpha}_{R}^{\nu}\hat{\alpha}_{L}^{\sigma}\hat{\alpha}_{R}^{\rho}],$
$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\epsilon_{\mu\nu\sigma\rho}\mathrm{tr}_{f}[F_{V}^{\mu\nu}(\hat{\alpha}_{L}^{\sigma}\hat{\alpha}_{R}^{\rho}-\hat{\alpha}_{R}^{\sigma}\hat{\alpha}_{L}^{\rho})],$
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}\mathrm{tr}_{f}[\hat{F}_{L}^{\mu\nu}(\hat{\alpha}_{L}^{\sigma}\hat{\alpha}_{R}^{\rho}-\hat{\alpha}_{R}^{\sigma}\hat{\alpha}_{L}^{\rho})-\hat{F}_{R}^{\mu\nu}(\hat{\alpha}_{R}^{\sigma}\hat{\alpha}_{L}^{\rho}-\hat{\alpha}_{L}^{\sigma}\hat{\alpha}_{R}^{\rho})],$

where the building blocks are given by

$\displaystyle\hat{\alpha}_{L}^{\mu}=\hat{\alpha}^{\mu}_{\parallel}-\hat{\alpha}^{\mu}_{\bot},~{}~{}\hat{\alpha}_{R}^{\mu}=\hat{\alpha}^{\mu}_{\parallel}+\hat{\alpha}^{\mu}_{\bot},~{}~{}\epsilon_{\mu\nu\sigma\rho}F_{V}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}V^{\mu\nu},$
(90)
$\displaystyle\epsilon_{\mu\nu\sigma\rho}\hat{F}_{L}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}(\hat{\mathcal{V}}^{\mu\nu}-\hat{\mathcal{A}}^{\mu\nu}),~{}~{}\epsilon_{\mu\nu\sigma\rho}\hat{F}_{R}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}(\hat{\mathcal{V}}^{\mu\nu}+\hat{\mathcal{A}}^{\mu\nu}).$

##  Appendix B The analytical expressions for $p^{4}$ order LECs

$\displaystyle\mathcal{C}_{2}=$ $\displaystyle
2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\Sigma^{2}k^{2}-2\Sigma^{3}{\Sigma^{\prime}}k^{2}+\frac{2}{3}\Sigma{\Sigma^{\prime}}k^{4}-\Sigma^{4}-\frac{1}{3}k^{4}-\frac{4}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}]$
(91)
$\displaystyle+\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}+\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{14}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+4\Sigma^{3}{\Sigma^{\prime}}k^{2}+\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}+{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}]$
$\displaystyle+\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[-\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-5\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-4\Sigma^{3}{\Sigma^{\prime}}k^{2}-\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}-{\Sigma^{\prime}}^{2}k^{6}$
$\displaystyle-{\Sigma}^{4}+\frac{1}{3}k^{4}+\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}+(k^{2}-\Sigma^{2})(2\Sigma^{\prime}\Sigma
k^{2}+\Sigma^{\prime}\Sigma^{3}-\frac{7}{6}\Sigma^{\prime\prime}\Sigma
k^{4}-\frac{1}{2}\Sigma^{\prime\prime}\Sigma^{3}k^{2})],$
$\displaystyle\mathcal{C}_{3}=$
$\displaystyle\frac{\mathrm{i}N_{c}}{4}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}+\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{14}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+4\Sigma^{3}{\Sigma^{\prime}}k^{2}+\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}+{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}]$
(92)
$\displaystyle+\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[-\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-5\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-4\Sigma^{3}{\Sigma^{\prime}}k^{2}-\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}-{\Sigma^{\prime}}^{2}k^{6}$
$\displaystyle-{\Sigma}^{4}+\frac{1}{3}k^{4}+\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}+(k^{2}-\Sigma^{2})(2\Sigma^{\prime}\Sigma
k^{2}+\Sigma^{\prime}\Sigma^{3}-\frac{7}{6}\Sigma^{\prime\prime}\Sigma
k^{4}-\frac{1}{2}\Sigma^{\prime\prime}\Sigma^{3}k^{2})],$
$\displaystyle\mathcal{C}_{4}$
$\displaystyle=2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\Sigma^{4}+3\Sigma^{2}k^{2}+\frac{1}{3}k^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
(93) $\displaystyle\mathcal{C}_{5}$
$\displaystyle=-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\Sigma^{4}+2\Sigma^{2}k^{2}+\frac{2}{3}k^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
$\displaystyle\mathcal{C}_{6}$
$\displaystyle=2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k^{2}+\Sigma^{2}}{[k^{2}-\Sigma^{2}(-k^{2})]^{2}},$
$\displaystyle\mathcal{C}_{7}$
$\displaystyle=2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}-\Sigma^{2}(-k^{2})},$
$\displaystyle\mathcal{C}_{8}$
$\displaystyle=-4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{[2k^{2}+\Sigma^{2}]\Sigma}{[k^{2}-\Sigma^{2}(-k^{2})]^{3}},$
$\displaystyle\mathcal{C}_{9}=$
$\displaystyle\frac{\mathrm{i}N_{c}}{4}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\\{[2\Sigma^{\prime\prime}\Sigma
X]-4X^{2}\Sigma^{\prime\prime}k^{2}(k^{2}\Sigma^{\prime}+\Sigma^{2}\Sigma^{\prime}+\Sigma)$
$\displaystyle+2X^{2}({\Sigma^{\prime}}^{2}-\Sigma^{\prime}\Sigma^{\prime\prime}k^{2}+\Sigma^{\prime\prime}\Sigma^{\prime\prime}k^{4}/3)(k^{2}+\Sigma^{2})\\}+2X^{3}\\{-2[\Sigma^{\prime}\Sigma^{3}+\Sigma^{3}{\Sigma^{\prime}}^{3}k^{2}$
$\displaystyle+3\Sigma^{2}{\Sigma^{\prime}}^{2}k^{2}+3\Sigma{\Sigma^{\prime}}^{3}k^{4}+{\Sigma^{\prime}}^{2}k^{4}]$
$\displaystyle+[\Sigma^{3}\Sigma^{\prime\prime}k^{2}-\Sigma\Sigma^{\prime\prime}k^{4}]+\frac{4}{3}\Sigma^{\prime\prime}k^{4}[\Sigma^{\prime}k^{2}(3\Sigma\Sigma^{\prime}+1)+\Sigma(\Sigma^{2}{\Sigma^{\prime}}^{2}+3\Sigma\Sigma^{\prime}+1)]\\}$
$\displaystyle-X^{4}[-2\Sigma^{4}-4{\Sigma^{\prime}}^{2}\Sigma^{4}k^{2}-\frac{4}{3}{\Sigma^{\prime}}^{4}\Sigma^{4}k^{4}-8\Sigma^{\prime}\Sigma^{3}k^{2}-\frac{32}{3}{\Sigma^{\prime}}^{3}\Sigma^{3}k^{4}-2\Sigma^{2}k^{2}$
$\displaystyle-\frac{40}{3}{\Sigma^{\prime}}^{2}\Sigma^{2}k^{4}+\Sigma^{2}k^{2}-8{\Sigma^{\prime}}^{4}\Sigma^{2}k^{6}-\frac{16}{3}\Sigma^{\prime}\Sigma
k^{4}+8\Sigma^{\prime}\Sigma k^{4}-\frac{32}{3}{\Sigma^{\prime}}^{3}\Sigma
k^{6}$
$\displaystyle-2k^{4}+4{\Sigma^{\prime}}^{2}k^{6}-\frac{4}{3}{\Sigma^{\prime}}^{4}k^{8}-\frac{2}{3}k^{4}+2k^{4}-\frac{8}{3}{\Sigma^{\prime}}^{2}k^{6}]\\}$
$\displaystyle-\frac{\mathrm{i}N_{c}}{12}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\\{X^{2}(\Sigma^{\prime\prime}\Sigma^{\prime\prime}k^{4})(k^{2}+\Sigma^{2})+4X^{3}\Sigma^{\prime\prime}k^{4}[\Sigma^{\prime}k^{2}(3\Sigma\Sigma^{\prime}+1)$
$\displaystyle+\Sigma(\Sigma^{2}{\Sigma^{\prime}}^{2}+3\Sigma\Sigma^{\prime}+1)]-3X^{4}[\Sigma^{4}-2{\Sigma^{\prime}}^{2}\Sigma^{4}k^{2}-\frac{2}{3}{\Sigma^{\prime}}^{4}\Sigma^{4}k^{4}-\frac{16}{3}{\Sigma^{\prime}}^{3}\Sigma^{3}k^{4}$
$\displaystyle-\frac{20}{3}{\Sigma^{\prime}}^{2}\Sigma^{2}k^{4}-\frac{1}{2}\Sigma^{2}k^{2}+4{\Sigma^{\prime}}^{2}\Sigma^{2}k^{4}-4{\Sigma^{\prime}}^{4}\Sigma^{2}k^{6}-\frac{8}{3}\Sigma^{\prime}\Sigma
k^{4}-\frac{16}{3}{\Sigma^{\prime}}^{3}\Sigma k^{6}$
$\displaystyle+k^{4}-2{\Sigma^{\prime}}^{2}k^{6}-\frac{2}{3}{\Sigma^{\prime}}^{4}k^{8}-\frac{1}{3}k^{4}-\frac{4}{3}{\Sigma^{\prime}}^{2}k^{6}]\\},$
(94) $\displaystyle\mathcal{C}_{10}=$
$\displaystyle-4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[-\frac{2}{3}k^{4}+2\Sigma^{3}{\Sigma^{\prime}}k^{2}-\frac{2}{3}\Sigma{\Sigma^{\prime}}k^{4}+{\Sigma}^{4}+\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}$
(95) $\displaystyle+\frac{1}{3}\Sigma^{\prime\prime}\Sigma
k^{6}-\frac{1}{3}\Sigma^{\prime\prime}\Sigma^{3}k^{4}]$
$\displaystyle-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}+\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{14}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+4\Sigma^{3}{\Sigma^{\prime}}k^{2}+\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}+{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}]$
$\displaystyle-2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[-\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-5\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-4\Sigma^{3}{\Sigma^{\prime}}k^{2}-\frac{8}{3}\Sigma{\Sigma^{\prime}}k^{4}-{\Sigma^{\prime}}^{2}k^{6}$
$\displaystyle-{\Sigma}^{4}+\frac{1}{3}k^{4}+\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}+(k^{2}-\Sigma^{2})(2\Sigma^{\prime}\Sigma
k^{2}+\Sigma^{\prime}\Sigma^{3}-\frac{7}{6}\Sigma^{\prime\prime}\Sigma
k^{4}-\frac{1}{2}\Sigma^{\prime\prime}\Sigma^{3}k^{2})],$
$\mathcal{C}_{11}=-2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{2\Sigma+k^{2}\Sigma^{\prime}}{[k^{2}-\Sigma^{2}(-k^{2})]^{2}},$
(96) $\displaystyle\mathcal{C}_{12}=$ $\displaystyle
2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}\\{[\Sigma^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{2}{3}\Sigma\Sigma^{\prime}k^{4}-\Sigma^{4}-\frac{1}{3}k^{4}-\frac{4}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}]$
(97) $\displaystyle+$
$\displaystyle\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}-2\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{2}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}-\frac{4}{3}\Sigma{\Sigma^{\prime}}k^{4}]$
$\displaystyle-\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{2}{3}\Sigma\Sigma^{\prime}k^{4}+\frac{2}{3}{\Sigma^{\prime}}^{2}k^{6}-\Sigma^{4}$
$\displaystyle-\frac{1}{3}k^{4}-\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+(k^{2}-\Sigma^{2})(\Sigma^{3}\Sigma^{\prime}+\frac{1}{2}\Sigma\Sigma^{\prime\prime}k^{4}-\frac{1}{2}\Sigma^{3}\Sigma^{\prime\prime}k^{2}-\frac{1}{3}\Sigma\Sigma^{\prime\prime}k^{4})],$
$\displaystyle\mathcal{C}_{13}=$
$\displaystyle\frac{\mathrm{i}N_{c}}{4}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}-2\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{2}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}-\frac{4}{3}\Sigma{\Sigma^{\prime}}k^{4}]$
(98)
$\displaystyle-\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{2}{3}\Sigma\Sigma^{\prime}k^{4}+\frac{2}{3}{\Sigma^{\prime}}^{2}k^{6}-\Sigma^{4}$
$\displaystyle-\frac{1}{3}k^{4}-\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+(k^{2}-\Sigma^{2})(\Sigma^{3}\Sigma^{\prime}+\frac{1}{2}\Sigma\Sigma^{\prime\prime}k^{4}-\frac{1}{2}\Sigma^{3}\Sigma^{\prime\prime}k^{2}-\frac{1}{3}\Sigma\Sigma^{\prime\prime}k^{4})],$
$\displaystyle\mathcal{C}_{14}$
$\displaystyle=2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{-\Sigma^{2}k^{2}+\Sigma^{4}+\frac{1}{3}k^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
(99) $\displaystyle\mathcal{C}_{15}$
$\displaystyle=-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\frac{2}{3}k^{4}-2\Sigma^{2}k^{2}+\Sigma^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
(100) $\displaystyle\mathcal{C}_{16}$
$\displaystyle=4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\Sigma^{3}}{[k^{2}-\Sigma^{2}(-k^{2})]^{3}},$
(101) $\displaystyle\mathcal{C}_{17}=$
$\displaystyle-\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}-2\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}+\frac{2}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-{\Sigma^{\prime}}^{2}k^{6}+\Sigma^{4}-\frac{4}{3}\Sigma{\Sigma^{\prime}}k^{4}]$
(102)
$\displaystyle-\frac{\mathrm{i}N_{c}}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\Sigma^{2}k^{2}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{2}{3}\Sigma\Sigma^{\prime}k^{4}+\frac{2}{3}{\Sigma^{\prime}}^{2}k^{6}-\Sigma^{4}$
$\displaystyle-\frac{1}{3}k^{4}-\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}+(k^{2}-\Sigma^{2})(\Sigma^{3}\Sigma^{\prime}+\frac{1}{2}\Sigma\Sigma^{\prime\prime}k^{4}-\frac{1}{2}\Sigma^{3}\Sigma^{\prime\prime}k^{2}-\frac{1}{3}\Sigma\Sigma^{\prime\prime}k^{4})]$
$\displaystyle+4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\frac{2}{3}k^{4}-2\Sigma^{2}k^{2}+\Sigma^{4}-\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}-\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}$
$\displaystyle-(k^{2}-\Sigma^{2})(\frac{1}{3}\Sigma\Sigma^{\prime\prime}k^{4})],$

$\displaystyle\mathcal{C}_{19}$
$\displaystyle=4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{-\Sigma^{2}k^{2}-\Sigma^{4}+\frac{1}{3}k^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
(103) $\displaystyle\mathcal{C}_{20}$
$\displaystyle=4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{-\frac{2}{3}k^{4}+\Sigma^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
$\displaystyle\mathcal{C}_{21}$
$\displaystyle=4\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\Sigma^{2}k^{2}-\Sigma^{4}+\frac{1}{3}k^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
$\displaystyle\mathcal{C}_{22}$
$\displaystyle=-2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{-\frac{1}{3}k^{4}+\Sigma^{2}k^{2}-\Sigma^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
$\displaystyle\mathcal{C}_{23}$
$\displaystyle=-2\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\frac{2}{3}k^{4}+\Sigma^{4}}{[k^{2}-\Sigma^{2}(-k^{2})]^{4}},$
$\displaystyle\mathcal{C}_{25}$
$\displaystyle=4N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{\Sigma}{[k^{2}-\Sigma^{2}(-k^{2})]^{2}},$
$\displaystyle\mathcal{C}_{27}=$
$\displaystyle-4N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[-\frac{2}{3}k^{4}+\Sigma^{3}\Sigma^{\prime}k^{2}-\frac{1}{3}\Sigma\Sigma^{\prime}k^{4}+\Sigma^{4}]$
(104)
$\displaystyle-2N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}\\{[-\Sigma^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}-\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}-\Sigma^{4}+\frac{1}{3}k^{4}]$
$\displaystyle+\frac{1}{2}\times[\frac{2}{3}k^{4}+2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}+\Sigma^{4}]\\},$
$\displaystyle\mathcal{C}_{28}=$
$\displaystyle\mathrm{i}N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}\\{-1\times[-\frac{1}{3}k^{4}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-\frac{4}{3}\Sigma^{3}{\Sigma^{\prime}}^{3}k^{4}-3\Sigma^{3}\Sigma^{\prime}k^{2}-\frac{10}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}$
$\displaystyle-\frac{4}{3}\Sigma{\Sigma^{\prime}}^{3}k^{6}+\frac{1}{3}{\Sigma^{\prime}}^{2}k^{6}+\Sigma\Sigma^{\prime}k^{4}+\Sigma^{2}k^{2}-\Sigma^{4}+(k^{2}-\Sigma^{2})(\Sigma^{3}\Sigma^{\prime}+\frac{3}{2}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{2}$
$\displaystyle+\frac{1}{2}{\Sigma^{\prime}}^{2}k^{4}-\frac{1}{6}\Sigma\Sigma^{\prime\prime}k^{4}-\frac{1}{2}\Sigma^{3}\Sigma^{\prime\prime}k^{2}-\Sigma^{2}\Sigma^{\prime}\Sigma^{\prime\prime}k^{4}-\frac{1}{3}\Sigma^{\prime}\Sigma^{\prime\prime}k^{6})+\frac{1}{2}\Sigma\Sigma^{\prime\prime}(k^{2}-\Sigma^{2})^{2}]$
$\displaystyle+\frac{1}{2}\times[\frac{2}{3}k^{4}-\Sigma^{4}{\Sigma^{\prime}}^{2}k^{2}-\frac{4}{3}\Sigma^{3}{\Sigma^{\prime}}^{3}k^{4}-\frac{4}{3}\Sigma^{2}{\Sigma^{\prime}}^{2}k^{4}-\frac{5}{3}{\Sigma^{\prime}}^{2}k^{6}-2\Sigma\Sigma^{\prime}k^{4}$
$\displaystyle-\frac{4}{3}\Sigma{\Sigma^{\prime}}^{3}k^{6}-2\Sigma^{2}k^{2}+\Sigma^{4}-(k^{2}-\Sigma^{2})(\Sigma^{2}\Sigma^{\prime}\Sigma^{\prime\prime}k^{4}+\frac{2}{3}\Sigma\Sigma^{\prime\prime}k^{4}+\frac{1}{3}\Sigma^{\prime}\Sigma^{\prime\prime}k^{6})],$
(105) $\displaystyle\mathcal{C}_{29}=$ $\displaystyle
2N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}\\{[-\Sigma^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}-\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}-\Sigma^{4}+\frac{1}{3}k^{4}]$
(106)
$\displaystyle+\frac{1}{2}\times[\frac{2}{3}k^{4}+2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}+\Sigma^{4}]\\},$
$\displaystyle\mathcal{C}_{31}=$
$\displaystyle-4N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}[\Sigma^{2}k^{2}-\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{1}{3}\Sigma\Sigma^{\prime}k^{4}-\Sigma^{4}-\frac{1}{3}k^{4}]$
(107)
$\displaystyle-2N_{c}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}X^{4}\\{[-\Sigma^{2}k^{2}-2\Sigma^{3}\Sigma^{\prime}k^{2}-\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}-\Sigma^{4}+\frac{1}{3}k^{4}]$
$\displaystyle+\frac{1}{2}\times[\frac{2}{3}k^{4}+2\Sigma^{3}\Sigma^{\prime}k^{2}+\frac{4}{3}\Sigma\Sigma^{\prime}k^{4}+\Sigma^{4}]\\}.$

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