# A symmetry preserving contact interaction treatment of the kaon

Zanbin Xing<EMAIL_ADDRESS>Lei Chang<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071, China

###### Abstract

A symmetry-preserving regularization procedure for dealing with the contact
interaction model is proposed in this work. This regularization procedure
follows a series of consistency conditions which are necessary to maintain
gauge symmetry. Under this regularization, proofs for the preservation of the
Ward-Takahashi identities are given and the loop integrals in the contact
interaction model are systematically computed. As an application example, the
kaon electromagnetic form factor and $K_{l3}$ transition form factor are
computed and self-consistent results are obtained. Since the proposed
regularization properly handles the divergence, one is freed from the
inconsistencies caused by the regularization and can concentrate more on the
physical discussion.

##  I introduction

Over the years, the Dyson-Schwinger Equations(DSEs) have proved to be a
powerful tool for studying non-pertubative nature of Quantum
Chromodynamics(QCD) in the continuum[1, 2, 3]. Self-consistent treatments of
the quark gap equation and bound state equations, such as Bethe-Salpeter
equation(BSE) and Faddeev equation, creates a bridge connecting the hadrons to
the fundamental degrees of freedom of QCD, quarks and gluons[4, 5].

Within the framework of DSEs, a vector-vector contact interaction
approximation was proposed in Ref. [6]. Despite its simplicity of the contact
interaction model in describing the real world, it has been used to calculate
a wide range of hadron properties, including mass spectrum, various decay
processes, electromagnetic form factors and transition form factors, and the
parton distributions, see Ref. [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21]. However, due to the non-renormalizable nature of the contact
interaction, the regularization scheme becomes a crucial part in the practical
calculation and a good regularization scheme should properly characterise the
divergence structure of the theory. It is worth noting that symmetries, and in
particular the Ward-Takahashi identities(WTIs), provide a strong constraint
that must be preserved during the regularization process.

The regularization procedures performed in previous studies of contact model
have been unsatisfactory. Firstly, WTIs do not naturally hold under the
previous regularization procedure. Moreover, there are cases where
inconsistent results occur in the calculation of kaon electromagnetic form
factor, when the principle of charge conservation is violated [10]. The main
reason for these problems is that previous regularization procedure fails to
properly deal with the quadratic and logarithmic divergent integrals.

Inspired by Ref. [22] we presented a new regularization procedure in this
work. This proper regularization meets many interesting properties of the
dimensional regularization without changing the space-time dimension. One of
the most fascinating properties is that gauge symmetries are preserved under
this regularization.

This paper is organized as follows, Sec. II introduces the new symmetry
preserving regularization that properly handle the divergent integrals. Sec.
III discusses this new regularization in the contact interaction model,
incorporating the preserving of (axial-)vector WTIs, and gives steps for the
systematic calculations with contact model. Sec. IV provides results of the
kaon electromagnetic form factor and $K_{l3}$ form factor under the new
regularizaiton, and the last section gives a brief summary.

##  II symmetry preserving regularization

Before discussing the regularization procedures in detail, it is helpful to
introduce the so-called one-fold irreducible loop integrals(ILIs) in Ref.
[22].

$\displaystyle I_{-2\alpha}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{q}\frac{1}{(q^{2}+\mathcal{M}^{2})^{\alpha+2}}\,,$
$\displaystyle I^{\mu\nu}_{-2\alpha}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{q}\frac{q_{\mu}q_{\nu}}{(q^{2}+\mathcal{M}^{2})^{\alpha+3}}\,,$
$\displaystyle I^{\mu\nu\rho\sigma}_{-2\alpha}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\int_{q}\frac{q_{\mu}q_{\nu}q_{\rho}q_{\sigma}}{(q^{2}+\mathcal{M}^{2})^{\alpha+4}}$
(1)

with $\int_{q}\doteq\int\frac{d^{4}q}{(2\pi)^{4}}$ and $\alpha=-1,0,1,\cdots$.
Here $\alpha=-1$ represents quadratically divergent integrals and $\alpha=0$
represents logarithmically divergent integrals. With the help of Feynman
parametrezation, it is straightforward to conclude that all one loop integrals
can be expressed in terms of these integrals. Where $\mathcal{M}$ is a
function of Feynman parameters, external momenta and the corresponding mass
scales. A regularization procedure can be implemented after rearranging one
loop integrals into these ILIs.

In Ref. [22], a loop regularization is proposed which simulates in many
interesting features to the momentum cutoff, Pauli-Villars and dimensional
regularization without modifying the original Lagrangian formalism, and is
directly performed in the space-time dimension of original theory. The loop
regularization is equivalent to introducing a weight function to regularize
the proper-time variable $\tau$ integration[23],for example,

$\displaystyle
I^{LR}_{-2\alpha}(\mathcal{M}^{2})=\lim_{N\to\infty}\int_{q}\int_{0}^{\infty}$
$\displaystyle d\tau\mathcal{W}_{N}(\tau,M_{c},\mu_{s})$ (2)
$\displaystyle\times\frac{\tau^{\alpha+1}}{\Gamma(\alpha+2)}e^{-\tau(q^{2}+\mathcal{M}^{2})}\,,$

where the superscript $LR$ denotes the loop regularization and $\Gamma(n)$ is
the gamma function. An explicit form of the weight function is

$\mathcal{W}_{N}(\tau,M_{c},\mu_{s})=e^{-\tau\mu_{s}^{2}}(1-e^{-\tau
M_{R}^{2}})^{N}\,,$ (3)

with $M_{R}^{2}=M_{c}^{2}h_{w}(N)lnN$, $h_{w}(N)\gtrsim 1$ and
$h_{w}(N\rightarrow\infty)=1$. The two energy scale $M_{c}$ and $\mu_{s}$
serve as ultraviolet(UV) and infrared(IR) cutoff, respectively. It is worth
noting that when $N\rightarrow\infty$ and $\mu_{s}=0$ the weight function
becomes

$\lim_{N\to\infty}\mathcal{W}_{N}(\tau,M_{c},\mu_{s}=0)=\theta(\tau
M_{c}^{2}-1)\,.$ (4)

Thus the weight function regularizes the proper-time integral just as
regularizes it with a hard UV cutoff $1/M_{c}^{2}$.

We introduce a regularization procedure which is based on the Schwinger’s
proper-time method. The regularization procedure for the scalar type ILIs is

$\displaystyle I_{-2\alpha}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{q}\frac{1}{(q^{2}+\mathcal{M}^{2})^{\alpha+2}}$
$\displaystyle=$
$\displaystyle\int_{q}\int_{0}^{\infty}d\tau\frac{\tau^{\alpha+1}}{\Gamma(\alpha+2)}e^{-\tau(q^{2}+\mathcal{M}^{2})}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+2)}\frac{e^{-\tau\mathcal{M}^{2}}}{16\pi^{2}}$
$\displaystyle\rightarrow I_{-2\alpha R}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{\tau_{uv}^{2}}^{\tau_{ir}^{2}}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+2)}\frac{e^{-\tau\mathcal{M}^{2}}}{16\pi^{2}}\,.$
(5)

The label $R$ in the subscript denotes the regularized integrals. It is
already seen that a hard UV cutoff $\tau_{uv}=1/M_{c}$ is equivalent to the
loop regularization with $\mu_{s}=0$. However, instead of the sliding energy
scale $\mu_{s}$ in the loop regularization, we introduce an hard IR cutoff
$\tau_{ir}$ to implement confinement, as proposed in Ref. [24]. This way of
dealing with the IR cutoff matches the regulators in previous contact model
studies, which, as we shall see, can also maintain gauge symmetries if the
tensor type ILIs are properly regularized. Before proceeding, it is noted that
when integer $\alpha<-1$, the loop integral vanishes under Eq. (5), which
happens to be the same property of dimensional regularization.

Turning now to the regularization of the tensor type ILIs, the regularization
procedure is

$\displaystyle I^{\mu\nu}_{-2\alpha}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{q}\frac{q_{\mu}q_{\nu}}{(q^{2}+\mathcal{M}^{2})^{\alpha+3}}$
$\displaystyle=$ $\displaystyle\int_{q}\int_{0}^{\infty}d\tau
q_{\mu}q_{\nu}\frac{\tau^{\alpha+2}}{\Gamma(\alpha+3)}e^{-\tau(q^{2}+\mathcal{M}^{2})}$
$\displaystyle=$
$\displaystyle\int_{q}\int_{0}^{\infty}d\tau\delta_{\mu\nu}\frac{q^{2}}{4}\frac{\tau^{\alpha+2}}{\Gamma(\alpha+3)}e^{-\tau(q^{2}+\mathcal{M}^{2})}$
$\displaystyle=$
$\displaystyle\delta_{\mu\nu}\int_{0}^{\infty}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+3)}\frac{e^{-\tau\mathcal{M}^{2}}}{32\pi^{2}}$
$\displaystyle\rightarrow I^{\mu\nu}_{-2\alpha R}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\delta_{\mu\nu}\int_{\tau_{uv}^{2}}^{\tau_{ir}^{2}}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+3)}\frac{e^{-\tau\mathcal{M}^{2}}}{32\pi^{2}}\,,$
(6)

and

$\displaystyle I^{\mu\nu\rho\sigma}_{-2\alpha}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\int_{q}\frac{q_{\mu}q_{\nu}q_{\rho}q_{\sigma}}{(q^{2}+\mathcal{M}^{2})^{\alpha+4}}$
$\displaystyle=$ $\displaystyle\int_{q}\int_{0}^{\infty}d\tau
q_{\mu}q_{\nu}q_{\rho}q_{\sigma}\frac{\tau^{\alpha+3}}{\Gamma(\alpha+4)}e^{-\tau(q^{2}+\mathcal{M}^{2})}$
$\displaystyle=$ $\displaystyle\int_{q}\int_{0}^{\infty}d\tau
S_{\mu\nu\rho\sigma}\frac{q^{4}}{24}\frac{\tau^{\alpha+3}}{\Gamma(\alpha+4)}e^{-\tau(q^{2}+\mathcal{M}^{2})}$
$\displaystyle=$ $\displaystyle
S_{\mu\nu\rho\sigma}\int_{0}^{\infty}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+4)}\frac{e^{-\tau\mathcal{M}^{2}}}{64\pi^{2}}$
$\displaystyle\rightarrow I^{\mu\nu\rho\sigma}_{-2\alpha R}(\mathcal{M}^{2})$
$\displaystyle=$ $\displaystyle
S_{\mu\nu\rho\sigma}\int_{\tau_{uv}^{2}}^{\tau_{ir}^{2}}d\tau\frac{\tau^{\alpha-1}}{\Gamma(\alpha+4)}\frac{e^{-\tau\mathcal{M}^{2}}}{64\pi^{2}}\,,$
(7)

where
$S_{\mu\nu\rho\sigma}=\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\sigma\nu}+\delta_{\mu\sigma}\delta_{\nu\rho}$
is the total symmetric tensor. It is obvious that the regularized tensor type
ILIs and scalar type ILIs are related as follows

$\displaystyle I^{\mu\nu}_{-2\alpha R}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\alpha+2)}{2\Gamma(\alpha+3)}\delta_{\mu\nu}I_{-2\alpha
R}(\mathcal{M}^{2})\,,$ (8) $\displaystyle I^{\mu\nu\rho\sigma}_{-2\alpha
R}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\alpha+2)}{4\Gamma(\alpha+4)}S_{\mu\nu\rho\sigma}I_{-2\alpha
R}(\mathcal{M}^{2})\,.$ (9)

These relations are precisely the so-called consistency conditions of gauge
symmetry in Ref. [22, 23], which are independent of regularization and are
necessary for preserving the gauge invariance of theories. It is noted that
ILIs under dimensional regularization also satisfy these conditions. These
consistency conditions connect tensor type ILIs and scalar type ILIs and then
any gauge invariant theories can be properly described in terms of the
regularized scalar type ILIs. In fact, there are a series of consistency
conditions for ILIs with even more Lorentz index which are rarely to
encountered and are therefore not presented here.

We now consider the regularization procedure in previous contact model
studies, such as in Ref. [6]. For example, the quadratic divergent tensor type
ILI is regularized as follows,

$\displaystyle I^{\mu\nu}_{2}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int_{q}\frac{q^{\mu}q^{\nu}}{(q^{2}+\mathcal{M}^{2})^{2}}$
$\displaystyle\rightarrow$
$\displaystyle\int_{q}\frac{\delta_{\mu\nu}}{4}\left(\frac{1}{q^{2}+\mathcal{M}^{2}}-\frac{\mathcal{M}^{2}}{(q^{2}+\mathcal{M}^{2})^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{q}\int_{0}^{\infty}d\tau\frac{\delta_{\mu\nu}}{4}e^{-\tau(q^{2}+\mathcal{M}^{2})}(1-\tau\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\delta_{\mu\nu}\int_{0}^{\infty}d\tau\frac{e^{-\tau\mathcal{M}^{2}}}{64\pi^{2}\tau^{2}}(1-\tau\mathcal{M}^{2})$
$\displaystyle\rightarrow I^{\mu\nu}_{2R^{\prime}}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\delta_{\mu\nu}\int_{\tau_{uv}^{2}}^{\tau_{ir}^{2}}d\tau\frac{e^{-\tau\mathcal{M}^{2}}}{64\pi^{2}\tau^{2}}(1-\tau\mathcal{M}^{2})\,,$
(10)

where $R^{\prime}$ denotes the regularization in previous contact model
studies. Apparently, this way of regularization does not satisfy the
consistency conditions and hence breaks the gauge symmetries, especially the
WTIs. This inappropriate regularization of the tensor type ILIs is also
responsible for the inconsistent results that appear in the more complex
computations as the case of the triangle diagrams. The key difference between
the two regularizations $R$ and $R^{\prime}$ is which step to do the symmetry
analysis, i.e., the substitution like

$\displaystyle q_{\mu}q_{\nu}\rightarrow\delta_{\mu\nu}\frac{q^{2}}{4}\,.$
(11)

In Eq. (6), one first perform the proper-time method so that the divergent
momentum integrals become well-defined, followed by the substitution. In this
way, the tensor type integrals are properly regularized and gauge symmetry is
preserved. On the contrary, in Eq. (10) the substitution is made while the
integrals are still divergent, then the divergent structures of theories are
destroyed and leads to possible inconsistency in further calculations. It is
well known that the substitution is only valid when the loop integral is
convergent. In principle, if we push the regulators $\tau_{uv}\to 0$ and
$\tau_{ir}\to\infty$, which restores the original ILIs, the two
regularizations $R$ and $R^{\prime}$ lead to the same results for convergent
integrals. However, we keep the two regulators $\tau_{uv}$ and $\tau_{ir}$
even when the ILIs are convergent, so that the regularized ILIs still retain
the following relation to the original ILIs

$I_{-2(\alpha+1)R}(\mathcal{M}^{2})=-\frac{1}{\alpha+2}\frac{d}{d\mathcal{M}^{2}}I_{-2\alpha
R}(\mathcal{M}^{2})\,.$ (12)

In the following sections, we will adopt the new regularization $R$ to
illustrate how these consistency conditions keep gauge symmetries and then
recalculate the kaon electromagnetic and transition form factors. For
simplicity, the label $R$ is suppressed thereafter.

##  III symmetry preserving regularization in the contact model

The quark gap equation in the contact model can be written as

$S^{-1}_{f}(p)=S_{f0}^{-1}(p)+\frac{4}{3m_{g}^{2}}\int_{q}{\gamma_{\alpha}S_{f}(q)\gamma_{\alpha}}\,,$
(13)

where $S^{-1}_{f}(p)$ is the inverse of the dressed quark propagator with
flavor $f$ and momentum $p$, which has the general form

$S^{-1}_{f}(p)=i\not{p}+M_{f}\,,$ (14)

where $M_{f}$ is independent of momentum due to the features of contact
interaction. $S^{-1}_{f0}=i\not{p}+m_{f}$ is the inverse of the bare quark
propagator with $m_{f}$ being the current quark mass.

In terms of the ILIs, the gap equation becomes

$\hat{M}_{f}=\frac{16M_{f}}{3m_{g}^{2}}I_{2}(M_{f}^{2}),$ (15)

where $\hat{M}_{f}=M_{f}-m_{f}$.

In the contact model, the meson’s Bethe-Salpeter amplitude(BSA) depends only
on meson’s total momentum Q, thus the most general form of the
pseudoscalar(PS) meson BSA with a outgoing $a$ quark and an incoming $b$ quark
can be expressed as

$\Gamma^{ab}_{PS}(Q)=i\gamma_{5}E_{PS}(Q)+\frac{\gamma_{5}\not{Q}}{M_{ab}}F_{PS}(Q)\,,$
(16)

where we have introduced the usual reduced mass
$M_{ab}=\frac{2M_{a}M_{b}}{M_{a}+M_{b}}$. The BSAs satisfy the following
homogeneous Bethe-Salpeter equation

$\displaystyle\Gamma^{ab}_{PS}(Q)$ $\displaystyle=$
$\displaystyle-\frac{4}{3m_{G}^{2}}\int_{q}\gamma_{\alpha}\chi^{ab}_{PS}(q,q_{-})\gamma_{\alpha}\,,$
(17)

where $q_{-}=q-Q$ and
$\chi^{ab}_{PS}(q,q_{-})=S_{a}(q)\Gamma^{ab}_{PS}(Q)S_{b}(q_{-})$ is the
corresponding Bethe-Salpeter wave function.

In the following, we will show how the new regularization procedure preserves
the axial-vector WTI exactly. The axial-vector WTIs states

$\displaystyle Q_{\mu}\Gamma^{ab}_{5\mu}(Q)=S^{-1}_{a}(q)i\gamma_{5}$
$\displaystyle+$ $\displaystyle i\gamma_{5}S^{-1}_{b}(q_{-})$ (18)
$\displaystyle-$ $\displaystyle i(m_{a}+m_{b})\Gamma^{ab}_{5}(Q)\,.$

The axial-vector vertex $\Gamma^{ab}_{5\mu}(Q)$ satisfies an inhomogeneous BSE

$\Gamma^{ab}_{5\mu}(Q)=\gamma_{5}\gamma_{\mu}-\frac{4}{3m_{G}^{2}}\int_{q}\gamma_{\alpha}\chi^{ab}_{5\mu}(Q)\gamma_{\alpha}\,,$
(19)

with $\chi^{ab}_{5\mu}(Q)=S_{a}(q)\Gamma^{ab}_{5\mu}(Q)S_{b}(q_{-})$. The
pseudoscalar vertex $\Gamma_{5}(Q)$ satisfies the equation in analogy with the
axial-vector vertex by replacing inhomogeneous term $\gamma_{5}\gamma_{\mu}$
with $\gamma_{5}$. The axial-vector WTI connects the vertex BSEs and the gap
equation, leading to the following two identities

$\displaystyle\hat{M}_{a}+\hat{M}_{b}=\frac{4}{3m_{G}^{2}}\int_{q}\left(\frac{4M_{a}}{q^{2}+M_{a}^{2}}+\frac{4M_{b}}{q_{-}^{2}+M_{b}^{2}}\right)\,,$
$\displaystyle 0=\frac{4}{3m_{G}^{2}}\int_{q}\left(\frac{2Q\cdot
q}{q^{2}+M_{a}^{2}}-\frac{2Q\cdot q_{-}}{q_{-}^{2}+M_{b}^{2}}\right)\,.$ (20)

By analysing the integrals with Feynman parametrization, one can rearrange the
expressions in terms of ILIs and arrive at

$\displaystyle\hat{M}_{a}+\hat{M}_{b}$ $\displaystyle=$
$\displaystyle\frac{16(M_{a}+M_{b})}{3m_{G}^{2}}\int_{0}^{1}du\\{I_{2}(\omega_{2})+\frac{(uM_{b}-\bar{u}M_{a})}{M_{a}+M_{b}}$
(21) $\displaystyle\times
I_{0}(\omega_{2})((2u-1)Q^{2}-M_{b}^{2}+M_{a}^{2})\\}\,,$

and

$0=\frac{8}{3m_{G}^{2}}\int_{0}^{1}du\\{Q^{2}I_{2}(\omega_{2})-2Q_{\mu}Q_{\nu}I_{2}^{\mu\nu}(\omega_{2})\\}\,,$
(22)

where $\bar{u}=1-u$ and
$\omega_{2}=uM_{b}^{2}+\bar{u}M_{a}^{2}+u\bar{u}Q^{2}$.

Consider the following integral

$H_{AV}=\frac{16(M_{a}+M_{b})}{3m_{G}^{2}}\int_{0}^{1}g_{AV}(u,M_{a},M_{b},Q^{2})du\,,$
(23)

with

$g_{AV}(u,M_{a},M_{b},Q^{2})=\frac{d}{du}\left[I_{2}(\omega_{2})\frac{(uM_{b}-\bar{u}M_{a})}{M_{a}+M_{b}}\right]\,.$
(24)

On the one hand, one can evaluate this integral directly

$\displaystyle H_{AV}$ $\displaystyle=$
$\displaystyle\frac{16(M_{a}+M_{b})}{3m_{G}^{2}}\left.I_{2}(\omega_{2})\frac{(uM_{b}-\bar{u}M_{a})}{M_{a}+M_{b}}\right|^{1}_{0}$
(25) $\displaystyle=$
$\displaystyle\frac{16}{3m_{G}^{2}}(M_{a}I_{2}(M_{a}^{2})+M_{b}I_{2}(M_{b}^{2}))\,.$

On the other hand, one can split the total derivative before integration, and
it is easy to see that(with the help of the properties in Eq. (12))

$\displaystyle H_{AV}$ $\displaystyle=$
$\displaystyle\frac{16(M_{a}+M_{b})}{3m_{G}^{2}}\int_{0}^{1}du\\{I_{2}(\omega_{2})+\frac{(uM_{b}-\bar{u}M_{a})}{M_{a}+M_{b}}$
(26) $\displaystyle\times
I_{0}(\omega_{2})((2u-1)Q^{2}-M_{b}^{2}+M_{a}^{2})\\}\,,$

which is precisely the right hand side Eq. (21). Then the Eq. (21)is nothing
but the sum of two quark gap equations.

The right hand side of Eq. (22) is exactly zero due to the consistent
conditions contained in the new regularization $R$. While in Ref. [6], the
vanish of the right hand side of Eq. (22) is imposed by hand. The reason is
that the previous regularization $R^{\prime}$ fails to fulfill the consistency
conditions and thus gauge symmetry is explicitly broken.

Let’s now focus on the vector vertex which satisfies the vector WTI

$\displaystyle Q_{\mu}i\Gamma^{ab}_{\mu}(Q)=S^{-1}_{a}(q)$ $\displaystyle-$
$\displaystyle S^{-1}_{b}(q_{-})$ (27) $\displaystyle-$
$\displaystyle(m_{a}-m_{b})\Gamma^{ab}_{I}(Q)\,,$

and follows the inhomogeneous BSE

$\displaystyle\Gamma^{ab}_{\mu}(Q)=\gamma_{\mu}-\frac{4}{3m_{G}^{2}}\int_{q}\gamma_{\alpha}\chi^{ab}_{\mu}(Q)\gamma_{\alpha}\,,$
(28)

with $\chi^{ab}_{\mu}(Q)=S^{a}(q)\Gamma^{ab}_{\mu}(Q)S^{b}(q_{-})$. The scalar
vertex $\Gamma^{ab}_{I}(Q)$ follows a similar equation with inhomogeneous term
being the identity matrix $I_{D}$. Under the contact interaction , the general
form of the vector and scalar vertexes can be written as

$\displaystyle\Gamma^{ab}_{\mu}(Q)$ $\displaystyle=$
$\displaystyle\gamma_{\mu}^{L}V^{ab}_{L}(Q^{2})+\gamma_{\mu}^{T}V^{ab}_{T}(Q^{2})$
(29) $\displaystyle+I_{D}(-iQ_{\mu})V^{ab}_{S}(Q^{2})\,,$
$\displaystyle\Gamma^{ab}_{I}(Q)$ $\displaystyle=$ $\displaystyle
I_{D}S^{ab}_{1}(Q^{2})-\frac{i\not{Q}}{M_{ab}}S^{ab}_{2}(Q^{2})\,,$ (30)

where $\gamma_{\mu}^{L}=\frac{Q_{\mu}\not{Q}}{Q^{2}}$,
$\gamma_{\mu}^{T}=\gamma_{\mu}-\gamma_{\mu}^{L}$. In Ref. [21], an additional
interaction kernel is proposed which maintains the WTIs. This additional
kernel can automatically generate a quark anomalous moment term
$\sigma_{\mu\nu}Q_{\nu}$ in the vector vertex, which has significant impact on
vector meson. However, since the additional kernel does not influence the
discussion in this paper, we will not adopt it for simplicity.

The most straightforward way of solving the BSE equations is to compute the
momentum integrals in the BSE directly. However, this method is
computationally cumbersome when evaluating the $\gamma$ matrix. Thus we solve
the vector and scalar vertex BSEs by projecting the BSE with the various
projecting operators. However, it is stressed that the Lorentz index of vector
or tensor objects is important and should be retained before regularization.
As an example, we illustrate this requirement through the solution of vector
vertex,

$\displaystyle V^{ab}_{T}(Q^{2})=\frac{1}{1-f^{ab}_{T}(Q^{2})}\,,$ (31)

with

$f^{ab}_{T}(Q^{2})=-\frac{4}{3m_{G}^{2}}\text{tr}_{D}\delta_{\mu\nu}\int_{q}P^{T}_{\nu}\gamma_{\alpha}S^{a}(q)\gamma^{T}_{\mu}S^{b}(q_{-})\gamma_{\alpha}\,,$
(32)

by using the projection operator
$P^{T}_{\nu}=\frac{\gamma^{T}_{\nu}}{\text{tr}_{D}(\gamma^{T}_{\mu}\gamma^{T}_{\mu})}$.
Here we express

$P^{T}_{\mu}\otimes\gamma^{T}_{\mu}\to\delta_{\mu\nu}P^{T}_{\nu}\otimes\gamma^{T}_{\mu}\,,$
(33)

which is used to retain the Lorentz index of the vector vertex
$\Gamma^{ab}_{\mu}(Q)$ herein. Analyzing the integral with Feynman
parametrization and rearranging it into ILIs, and then regularizing the ILIs
with $\delta_{\mu\nu}$ uncontracted, yields the following expression

$\displaystyle f^{ab}_{T}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{4}{3m_{G}^{2}}\delta_{\mu\nu}\int_{0}^{1}du\frac{2}{3}(\frac{Q_{\mu}Q_{\nu}}{Q^{2}}-\delta_{\mu\nu})I_{0}(\omega_{2})$
(34)
$\displaystyle\times(uM_{b}^{2}-M_{b}M_{a}+\bar{u}M_{a}^{2}+2u\bar{u}Q^{2})\,.$

It can be seen that the transverse structure is exactly maintained with the
index of vector vertex uncontracted before regularization. If the contraction
is performed before regularization, the transverse structure may be destroyed.

By solving the corresponding BSE, the rest of the dressing functions in these
two vertices are

$\displaystyle V^{ab}_{L}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1+4f^{ab}_{1}(Q^{2})}{1-f^{ab}_{L}(Q^{2})}\,,$ (35)
$\displaystyle V^{ab}_{S}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{-4f^{ab}_{2}(Q^{2})}{1-f^{ab}_{L}(Q^{2})}\,,$ (36)
$\displaystyle S^{ab}_{1}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1+2(M_{b}-M_{a})f^{ab}_{2}(Q^{2})}{1-f^{ab}_{L}(Q^{2})}\,,$
(37) $\displaystyle S^{ab}_{2}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{2M_{ab}f^{ab}_{2}(Q^{2})}{1-f^{ab}_{L}(Q^{2})}\,,$ (38)

where

$\displaystyle f^{ab}_{L}(Q^{2})$ $\displaystyle=$ $\displaystyle
8Q^{2}(f^{ab}_{2}(Q^{2}))^{2}-2(M_{b}-M_{a})f^{ab}_{2}(Q^{2})$ (39)
$\displaystyle-4f^{ab}_{1}(Q^{2})(1+2(M_{b}-M_{a})f^{ab}_{2}(Q^{2}))\,,$
$\displaystyle f^{ab}_{1}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{4}{3m_{g}^{2}}\int_{0}^{1}[I_{0}(\omega_{2})(uM_{b}^{2}+\bar{u}M_{a}^{2}$
(40) $\displaystyle+M_{b}M_{a}+2u\bar{u}Q^{2})-I_{2}(\omega_{2})]\,,$
$\displaystyle f^{ab}_{2}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{4}{3m_{g}^{2}}\int_{0}^{1}I_{0}(\omega_{2})(uM_{b}-\bar{u}M_{a})\,.$
(41)

Plugging the solved vector and scalar vertices into the vector WTI Eq. (27),
one finds the following equation

$\hat{M}_{b}-\hat{M}_{a}=-4(M_{b}-M_{a})f^{ab}_{1}(Q^{2})+4Q^{2}f^{ab}_{2}(Q^{2})\,.$
(42)

This equation can be proved in analogy with Eq. (23) by introducing

$H_{V}=\frac{16}{3m_{G}^{2}}\int_{0}^{1}g_{V}(u,M_{a},M_{b},Q^{2})du\,,$ (43)

where

$g_{V}(u,M_{a},M_{b},P^{2})=\frac{d}{du}\left[I_{2}(\omega_{2})(uM_{b}+\bar{u}M_{a})\right]\,.$
(44)

One finds Eq. (42) is exactly the difference between the two quark gap
equations.

In the end of this section, we list the systematic procedures in regularizing
the 1-loop integrals in the contact model.

  * 1.

First, one uses Feynman parametrization to analyze the integral and then
rearranges the expressions in terms of the ILIs;

  * 2.

Regularize the ILIs through the new regularization and express all regularized
tensor type ILIs in terms of scalar type ones through the consistent
conditions;

  * 3.

When evaluating formulae that contain vector or tensor objects, their Lorentz
index should be retained rather than contracted before the tensor type ILIs
are regularized.

The first step is trivial, but provides a convenient way to manage the
expressions in calculation. We can then easily implement regularization in
step 2. We emphasis again a proper regularization should let the regularized
tensor type ILIs satisfy the consistent conditions, otherwise gauge symmetries
will be broken. The third rule concerns vector and tensor objects, whose
Lorentz metrics are important. To be clear, this rule charges not only for
solving the BSEs, but also for any physical quantity that includes vector and
tensor objects. An easy way of implementing this rule can be achieved by, say,
Eq. (33), which will generate the following typical integrals

$\delta_{\mu\nu}\int_{q}\frac{q_{\mu}q_{\nu}}{(q^{2}+\mathcal{M}^{2})^{\alpha+2}}\,,$
(45)

where $\delta_{\mu\nu}$ comes from Eq. (33). It is apparent that different
orders of regularization and contraction lead to different results, take
$\alpha=-1$ for example:

$\displaystyle\delta_{\mu\nu}\int_{q}\frac{q_{\mu}q_{\nu}}{(q^{2}+\mathcal{M}^{2})^{2}}\to\delta_{\mu\nu}I_{2R}^{\mu\nu}=2I_{2R}\,,$
(46)
$\displaystyle\delta_{\mu\nu}\int_{q}\frac{q_{\mu}q_{\nu}}{(q^{2}+\mathcal{M}^{2})^{2}}=I_{2}-\mathcal{M}^{2}I_{0}\to
I_{2R}-\mathcal{M}^{2}I_{0R}\,.$

Here arrows represent operations of regularization and equal signs represent
operations of contraction. And we explicitly use the label $R$ to distinguish
the regularized ILIs from the orignal ones. It is observed that if one
contracts the index before regularization, the unregularized tensor ILIs are
turned into the unregularized scalar ILIs, which incorrectly characterises the
divergences of theories, and then some fundamental properties are lost, such
as the current conservation.

##  IV Kaon form factors

As the simplest bound state with strangeness, kaon has attracted much
attention since its discovery in the middle of the last century[25] and has
led to many important researches in standard model, such as the introduction
of strangeness[26], the violation of parity[27], quark mixing and the CKM
matrix[28, 29], CP violation[30] and GIM mechanism[31]. Recently, CERN has
approved a world-unique QCD facility where highly intense and energetic kaon
beams are used to map out the complete spectrum of excited kaons with an
unprecedented precision[32]. This approval of CERN injects vitality into a
wide range nowadays studies on low energy kaon phenomenology and high energy
processes including kaon excitations, and opens a completely new horizon in
kaon physics.

The kaon electromagnetic and transition form factor have also attracts
fruitful studies[33, 34, 35, 36, 37, 38, 39]. In previous contact model
computation of these two form factors[10], inconsistency results occurred due
to the failure of the regularization to properly handle the divergent
integrals.

To calculate the kaon electromagnetic form factor and the $K_{l3}$ transition
form factor, three elements is required. The quark propagator and vector
vertex are given in Sec. III(in the following, $\Gamma_{\mu}^{ff}$ denotes the
quark-photon vertex and $\Gamma_{\mu}^{su}$ denotes the quark–W boson vertex).
The remaining ingredients are the pion and kaon BSAs, which are solved from
the pseudoscalar BSEs. In solving this equation, we follow Ref. [10], the only
difference being the regularization. Even though we get the same kernel as in
Ref. [10], it is worth nothing that the following relation holds,

$K_{FF}=-\frac{(M_{s}+M_{u})^{2}}{2M_{s}M_{u}}K_{FE}\,.$ (48)

This is a natural result since the axial-vector WTI is automatically satisfied
under the new regularization. Since the meson BSE is a homogeneous equation,
normalization of the BSA is need for the computation of physical quantities.
The canonical normalisation of pseudoscalar meson reads

$Q_{\mu}=N_{c}\text{tr}_{D}\int_{q}\Gamma^{ab}_{PS}(-Q)S(q)\Gamma^{ab}_{PS}(Q)\frac{\partial}{\partial
Q_{\mu}}S(q_{-})\,,$ (49)

and the leptonic decay constant reads

$f^{ab}_{PS}Q_{\mu}=\frac{1}{\sqrt{2}}N_{c}\text{tr}_{D}\int_{q}\gamma_{5}\gamma_{\mu}\chi^{ab}_{PS}(q,q_{-})\,.$
(50)

The parameter are fixed to obtain the the following physical observables,
$m_{u}=0.007GeV$, $m_{s}=0.17{\rm GeV}$, $m_{G}=0.132{\rm GeV}$,
$\tau_{uv}=1/0.905{\rm GeV}^{-1}$, $\tau_{ir}=1/0.24{\rm GeV}^{-1}$.

Table 1:  masses and decay constants, the unit is GeV $M_{u}$ | $M_{s}$ | $m_{\pi}$ | $m_{K}$ | $f_{\pi}$ | $f_{K}$
---|---|---|---|---|---
$0.368$ | 0.533 | 0.140 | 0.499 | 0.101 | 0.106

###  IV.1 Kaon electromagnetic form factor

In the impulse approximation, the kaon electromagnetic form factor is

$F^{em}_{K^{+}}(Q^{2})=e_{u}F^{u}_{K^{+}}(Q^{2})+e_{s}F^{s}_{K^{+}}(Q^{2})\,,$
(51)

where

$\displaystyle 2P_{\mu}F^{u}_{K^{+}}(Q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\text{tr}_{D}\int_{q}S^{s}(q)i\Gamma^{K^{+}}(-p)S^{u}(q+p)$ (52)
$\displaystyle\times i\Gamma^{uu}_{\mu}(Q)S^{u}(q+k)i\Gamma^{K^{+}}(k)\,,$
$\displaystyle 2P_{\mu}F^{s}_{K^{+}}(Q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\text{tr}_{D}\int_{q}S^{s}(q-p)i\Gamma^{K^{+}}(-p)S^{u}(q)$ (53)
$\displaystyle\times i\Gamma^{K^{+}}(k)S^{s}(q-k)i\Gamma^{ss}_{\mu}(Q)\,,$

here $p=P+Q/2$, $k=P-Q/2$ with $k$ the incoming kaon momentum and $p$ the out
going kaon momentum. The on-shell condition entails $P\cdot Q=0$ and
$P^{2}=-m_{K}^{2}-Q^{2}/4$. We express the form factor formula as follows

$\displaystyle F^{u}_{K^{+}}(Q^{2})$ $\displaystyle=$
$\displaystyle(F^{u}_{EE}(Q^{2})E_{K^{+}}E_{K^{+}}+F^{u}_{EF}(Q^{2})E_{K^{+}}F_{K^{+}}$
(54) $\displaystyle+F^{u}_{FF}(Q^{2})F_{K^{+}}F_{K^{+}})V^{uu}_{T}(Q^{2})\,,$

where $E_{K^{+}},F_{K^{+}}$ is kaon’s amplitude and the corresponding
functions can be found by the new regularization approach as

$\displaystyle F^{u}_{EE}(Q^{2})$ $\displaystyle=$ $\displaystyle
2N_{c}\left(\int_{x}I_{0}(\omega_{K2})-\int_{xy}A_{EE}I_{-2}(\omega_{K3})\right)$
$\displaystyle F^{u}_{EF}(Q^{2})$ $\displaystyle=$
$\displaystyle-2\frac{M_{u}+M_{s}}{M_{us}}F^{u}_{EE}(Q^{2})+N_{c}\int_{x}A_{EF}I_{0}(\omega_{K2})$
$\displaystyle F^{u}_{FF}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{M_{u}+M_{s}}{2M_{us}}F^{u}_{EF}(Q^{2})+N_{c}\int_{x}A_{FF}I_{0}(\omega_{K2})$
(55) $\displaystyle+N_{c}\int_{x}B_{FF}I_{0}(M_{u}^{2}+x(1-x)Q^{2})\,,$

where

$\displaystyle A_{EE}=2((M_{s}-M_{u})^{2}-m_{K}^{2})(1-x-y)+Q^{2}(x+y)\,,$
$\displaystyle A_{EF}=\frac{4(M_{s}(1-x)+M_{u}x)}{M_{us}}\,,$ $\displaystyle
A_{FF}=\frac{-2(M_{s}^{2}(1-x)+M_{u}^{2}x+M_{u}M_{s})}{M^{2}_{us}}\,,$
$\displaystyle B_{FF}=\frac{4Q^{2}x(1-x)}{M_{us}^{2}}\,,$ (56)

and $\int_{x}=\int_{0}^{1}dx$, $\int_{xy}=\int_{0}^{1}dx\int_{0}^{1-x}dy$,
$\omega_{K2}=xM_{u}^{2}+(1-x)M_{s}^{2}-x(1-x)M_{K}^{2}$,
$\omega_{K3}=(x+y)M_{u}^{2}+(1-x-y)M_{s}^{2}-m_{K}^{2}(x+y)(1-x-y)+Q^{2}xy$.

Consider the interchange $s\leftrightarrow u$ and $p\leftrightarrow-k$ in Eq.
(53) and comparing it with Eq. (52), one finds that $F^{s}_{K^{+}}(Q^{2})$ is
obtained from $-F^{u}_{K^{+}}(Q^{2})$ by the interchange of $s\leftrightarrow
u$. From the explicit expressions of $F^{u}_{EF}(Q^{2})$, and some symmetries
of the Feynman parameters, one gets the following relation

$\displaystyle F^{u}_{EF}(Q^{2}=0)+F^{s}_{EF}(Q^{2}=0)=0\,.$ (57)

Once again we see that the symmetry preserving regularization procedure leads
to the desired result, Eq. (57), which should always hold, whereas in previous
contact interaction studies this relation is broken because of improper
regularization procedure. The result for the kaon electromagnetic form factor
is presented in Fig. 1, which, as expected, is hard due to the nature of
contact interaction.

Figure 1: The momentum dependence of kaon electromagnetic form factors
$F_{u/s}(t=-Q^{2})$.

###  IV.2 $K_{l3}$ transition form factors

The impulse approximation result for the $K_{l3}$ transition amplitude is

$\displaystyle M^{K_{l3}}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}N_{c}\text{tr}_{D}\int_{q}\\{S^{u}(q-p)\Gamma^{\pi}(-p)S^{u}(q)$
(58)
$\displaystyle\times\Gamma^{K^{+}}(k)S^{s}(q-k)i\Gamma^{su}_{\mu}(Q)\\}\,,$

where $k$ the incoming kaon momentum and $p$ the out going pion momentum. The
on-shell conditions entails $P\cdot Q=(m_{K}^{2}-m_{\pi}^{2})/2$ and
$P^{2}=-(m_{K}^{2}+m_{\pi}^{2})/2-Q^{2}/4$. The amplitude can be expressed as

$M^{K_{l3}}_{\mu}=\frac{1}{\sqrt{2}}(2P_{\mu}f_{+}(Q^{2})-Q_{\mu}f_{-}(Q^{2}))\,.$
(59)

In addition to the primary and secondary transition form factor $f_{+}(Q^{2})$
and $f_{-}(Q^{2})$, it is helpful to characterize the transitions by the
following function

$f_{0}(Q^{2})=f_{+}(Q^{2})-Q^{2}f_{-}(Q^{2})/(m_{K}^{2}-m_{\pi}^{2})\,.$ (60)

Since the expressions for $f_{+}(Q^{2})$ and $f_{-}(Q^{2})$ are complicated,
we fit these two functions by interpolating in the domain $-2<t/{\rm
GeV}^{2}<0.5$.

$\displaystyle f_{+}(t)$ $\displaystyle=$ $\displaystyle
f_{+}(0)\frac{1-0.7397t+0.1159t^{2}}{1-1.4312t+0.4574t^{2}}\,,$ $\displaystyle
f_{-}(t)$ $\displaystyle=$ $\displaystyle
f_{-}(0)\frac{1-1.2922t+0.0933t^{2}}{1-2.0758t+1.0671t^{2}}\,,$ (61)

with $t=-Q^{2}$ and in the following we use $t$ as variable of $f_{+,-,0}$.
And in the physical interesting domain $m_{l}^{2}<t/{\rm GeV}^{2}<t_{m}$,
where $t_{m}=(M_{K}-m_{\pi})^{2}\sim 0.13{\rm GeV}^{2}$, a linear fit is
performed,

$\displaystyle f_{+}(t)$ $\displaystyle=$ $\displaystyle 0.9838+0.7751t\,,$
$\displaystyle-f_{-}(t)$ $\displaystyle=$ $\displaystyle 0.1128+0.0992t\,.$
(62)

The slopes of transition form factors are less deep than the results reported
in Ref. [36]. The results for the FFs $f_{+}$ and $f_{-}$ are plotted in Fig.
2.

Figure 2: Results for $f_{+/-}(t)$. In upper panel, the filled circles and
squares are results of the calculation, the solid and dashed lines are linear
fits in Eq. (IV.2). In lower panel, the solid and dashed lines are the results
of the calculation.

Here we list in Table. 2 a range of quantities that used to characterise the
kaon semileptonic decays. The difference between the results of previous work
[10] and the present results comes from the regularization procedure.

Table 2:  quantities derived from the kaon transition form factors | $f_{+}(0)$ | $f_{+}(t_{m})$ | $-f_{-}(0)$ | $-f_{-}(t_{m})$ | $f_{0}(t_{\Delta})$
---|---|---|---|---|---
here | 0.98 | 1.09 | 0.11 | 0.12 | 1.05
Ref. [10] | 0.98 | 1.07 | 0.087 | 0.096 | 1.06
Ref. [36] | 0.96 | 1.13 | 0.10 | 0.11 | 1.18
Ref. [40, 41, 42] | 0.96 | 1.16 | 0.12 |  |

Current algebra predicts[43] that

$f_{0}(t_{\Delta}=m_{K}^{2}-m_{\pi}^{2})=f_{K}/f_{\pi}+\Delta_{CT}\,,$ (63)

where $f_{\pi,K}$ is the leptonic decay constant of pion and kaon, and the
correction $\Delta_{CT}$ is of $\mathcal{O}(m_{u},m_{d})$, which is usually
negligible. In our calculations, we obtain
$\Delta_{CT}/f_{0}(t_{\Delta})=3.7\times 10^{-4}$. Actually, $\Delta_{CT}=0$
if one works in the chiral limit, i.e., $m_{u}=0=m_{\pi}$. It is known that in
the solution of Eq. (19) has the form in the neighbourhood of $P^{2}=0$(with
$a,b=u$)

$\Gamma_{5\mu}^{uu}(P)=\frac{P_{\mu}}{P^{2}}\sqrt{2}f_{\pi}\Gamma^{\pi}(P)+\gamma_{5}\gamma_{\mu}F_{R}(P)\,,$
(64)

where $F_{R}$ is the regular part of vertex at $P^{2}=0$. Plugging Eq. (64)
into the axial vector WTI Eq. (18), one obtains the following generalised
Goldberger-Treiman relations[44]

$\displaystyle\sqrt{2}E_{\pi}(P)f_{\pi}=2M_{u}\,,$ (65)
$\displaystyle\frac{2F_{\pi}(P)}{E_{\pi}(P)}+F_{R}(P)=1\,.$ (66)

Then pion BSA can be expressed as follows

$\Gamma^{\pi}(P)=\frac{S^{-1}(q+P)i\gamma_{5}+i\gamma_{5}S^{-1}(q)-\gamma_{5}\not{P}F_{R}(P)}{\sqrt{2}f_{\pi}}\,.$
(67)

Substitute Eq. (67) into the $K_{l3}$ transition amplitude Eq. (58), one
arrives at the following expressions for the transition form factors

$\displaystyle f^{cl}_{+}(t)$ $\displaystyle=$
$\displaystyle\frac{f_{K}}{2f_{\pi}}-\frac{E_{\pi}-2F_{\pi}}{2}(f_{cl+}^{FE}(t)E_{K}+f_{cl+}^{FF}(t)F_{K})\,,$
$\displaystyle f^{cl}_{-}(t)$ $\displaystyle=$
$\displaystyle\frac{f_{K}}{2f_{\pi}}-\frac{E_{\pi}-2F_{\pi}}{2}(f_{cl-}^{FE}(t)E_{K}+f_{cl-}^{FF}(t)F_{K})\,,$

where $f_{+/-}^{FX}(t)$ with $X=E,F$ comes from the following way of
expressing the transition form factor in analogy with Eq. (54)

$\displaystyle f_{+/-}(t)$ $\displaystyle=$
$\displaystyle(f_{+/-}^{EE}(t)E_{\pi}E_{K}+f_{+/-}^{EF}(t)E_{\pi}F_{K}$ (69)
$\displaystyle+f_{+/-}^{FE}(t)F_{\pi}E_{K}+f_{+/-}^{FF}(t)F_{\pi}F_{K})\,,$

and $f^{FX}_{cl+/-}(t)=f^{FX}_{+/-}(t,m_{\pi}=0)$. Furthermore, one can
extract the vector vertex ingredients,

$\displaystyle f_{+}^{FX}(t)$ $\displaystyle=$ $\displaystyle
f_{+}^{FX,T}(t)V_{T}^{su}(t)\,,$ (70) $\displaystyle f_{-}^{FX}(t)$
$\displaystyle=$ $\displaystyle
f_{-}^{FX,T}(t)V_{T}^{su}(t)+f_{-}^{FX,L}(t)V_{L}^{su}(t)$ (71)
$\displaystyle+f_{-}^{FX,S}(t)V_{S}^{su}(t)\,.$

Our computation indicates the following relations in the chiral limit

$\displaystyle f_{cl+}^{FX,T}(t)+\frac{t}{m_{K}^{2}}f_{c-}^{FX,T}(t)=0\,,$
(72) $\displaystyle f_{cl-}^{FX,L}(t)\sim\mathcal{O}(t-m_{K}^{2})\,,$ (73)
$\displaystyle f_{cl-}^{FX,S}(t)\sim\mathcal{O}(t-m_{K}^{2})\,,$ (74)

which leads to

$\displaystyle
f^{cl}_{0}(t_{\Delta})=f^{cl}_{+}(t_{\Delta})+f^{cl}_{-}(t_{\Delta})=\frac{f_{K}}{f_{\pi}}\,.$
(75)

Hence $\Delta_{CT}$ vanishes in the chiral limit.

##  V summary

In this paper, we introduce a symmetry preserving regularization based on
Schwinger’s proper-time method. This regularization meets many good features
of dimensional regularization without changing space-time dimensions. One of
the most important properties is that it ensures the regularized ILIs satisfy
a series of consistency conditions which are necessary to preserving gauge
symmetries. In particular, we show that gauge symmetries are preserved by
proving that the WTIs hold exactly after the regularization. We also show why
the regularization procedure in previous studies of the contact model breaks
WTIs. Systematic steps for regularizing the contact model are presented at the
end of Sec. III.

As an application example, we recalculated kaon form factors which exhibits
inconsistency in Ref. [10] because of the improperly regularized ILIs, whereas
under the new regularization the results are self-consistent. The present
regularization would also rescue the flaw in the calculation of other form
factors, such as heavy-light mesons semileptonic transitions(see [45] for some
discussion).

With this symmetry preserving regularization, the hadron properties computed
from contact model are valid, as potential inconsistencies from regularization
are eliminated. It is hope that the contact model studies can provide more
inspiring results that may shed lights on the realistic DSE studies.

##  VI acknowledgements

Work supported by National Natural Science Foundation of China (grant no.
12135007).

## References

  * Roberts and Williams [1994] Craig D. Roberts and Anthony G. Williams.  Dyson-Schwinger equations and their application to hadronic physics.  _Prog. Part. Nucl. Phys._ , 33:477–575, 1994.  doi: 10.1016/0146-6410(94)90049-3.
  * Maris and Roberts [2003] Pieter Maris and Craig D. Roberts.  Dyson-Schwinger equations: A Tool for hadron physics.  _Int. J. Mod. Phys. E_ , 12:297–365, 2003.  doi: 10.1142/S0218301303001326.
  * Fischer [2019] Christian S. Fischer.  QCD at finite temperature and chemical potential from Dyson–Schwinger equations.  _Prog. Part. Nucl. Phys._ , 105:1–60, 2019.  doi: 10.1016/j.ppnp.2019.01.002.
  * Eichmann [2009] Gernot Eichmann.  _Hadron properties from QCD bound-state equations_.  PhD thesis, Graz U., 2009.
  * Eichmann et al. [2016] Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams, Reinhard Alkofer, and Christian S. Fischer.  Baryons as relativistic three-quark bound states.  _Prog. Part. Nucl. Phys._ , 91:1–100, 2016.  doi: 10.1016/j.ppnp.2016.07.001.
  * Gutierrez-Guerrero et al. [2010] L. X. Gutierrez-Guerrero, A. Bashir, I. C. Cloet, and C. D. Roberts.  Pion form factor from a contact interaction.  _Phys. Rev. C_ , 81:065202, 2010.  doi: 10.1103/PhysRevC.81.065202.
  * Roberts et al. [2011a] Hannes L. L. Roberts, Lei Chang, Ian C. Cloet, and Craig D. Roberts.  Masses of ground and excited-state hadrons.  _Few Body Syst._ , 51:1–25, 2011a.  doi: 10.1007/s00601-011-0225-x.
  * Roberts et al. [2011b] H. L. L. Roberts, A. Bashir, L. X. Gutierrez-Guerrero, C. D. Roberts, and D. J. Wilson.  pi- and rho-mesons, and their diquark partners, from a contact interaction.  _Phys. Rev. C_ , 83:065206, 2011b.  doi: 10.1103/PhysRevC.83.065206.
  * Chen et al. [2012] Chen Chen, Lei Chang, Craig D. Roberts, Shaolong Wan, and David J. Wilson.  Spectrum of hadrons with strangeness.  _Few Body Syst._ , 53:293–326, 2012.  doi: 10.1007/s00601-012-0466-3.
  * Chen et al. [2013] Chen Chen, Lei Chang, Craig D. Roberts, Sebastian M. Schmidt, Shaolong Wan, and David J. Wilson.  Features and flaws of a contact interaction treatment of the kaon.  _Phys. Rev. C_ , 87:045207, 2013.  doi: 10.1103/PhysRevC.87.045207.
  * Segovia et al. [2014a] Jorge Segovia, Chen Chen, Ian C. Cloët, Craig D. Roberts, Sebastian M. Schmidt, and Shaolong Wan.  Elastic and Transition Form Factors of the $\Delta(1232)$.  _Few Body Syst._ , 55:1–33, 2014a.  doi: 10.1007/s00601-013-0734-x.
  * Segovia et al. [2014b] Jorge Segovia, Ian C. Cloet, Craig D. Roberts, and Sebastian M. Schmidt.  Nucleon and $\Delta$ elastic and transition form factors.  _Few Body Syst._ , 55:1185–1222, 2014b.  doi: 10.1007/s00601-014-0907-2.
  * Xu et al. [2015] Shu-Sheng Xu, Chen Chen, Ian C. Cloet, Craig D. Roberts, Jorge Segovia, and Hong-Shi Zong.  Contact-interaction Faddeev equation and, inter alia , proton tensor charges.  _Phys. Rev. D_ , 92(11):114034, 2015.  doi: 10.1103/PhysRevD.92.114034.
  * Bedolla et al. [2015] Marco A. Bedolla, J. J. Cobos-Martínez, and Adnan Bashir.  Charmonia in a contact interaction.  _Phys. Rev. D_ , 92(5):054031, 2015.  doi: 10.1103/PhysRevD.92.054031.
  * Serna et al. [2016] F. E. Serna, M. A. Brito, and G. Krein.  Symmetry-preserving contact interaction model for heavy-light mesons.  _AIP Conf. Proc._ , 1701(1):100018, 2016.  doi: 10.1063/1.4938727.
  * Lu et al. [2017] Ya Lu, Chen Chen, Craig D. Roberts, Jorge Segovia, Shu-Sheng Xu, and Hong-Shi Zong.  Parity partners in the baryon resonance spectrum.  _Phys. Rev. C_ , 96(1):015208, 2017.  doi: 10.1103/PhysRevC.96.015208.
  * Serna et al. [2017] Fernando E. Serna, Bruno El-Bennich, and Gastão Krein.  Charmed mesons with a symmetry-preserving contact interaction.  _Phys. Rev. D_ , 96(1):014013, 2017.  doi: 10.1103/PhysRevD.96.014013.
  * Gutiérrez-Guerrero et al. [2019] L. X. Gutiérrez-Guerrero, Adnan Bashir, Marco A. Bedolla, and E. Santopinto.  Masses of Light and Heavy Mesons and Baryons: A Unified Picture.  _Phys. Rev. D_ , 100(11):114032, 2019.  doi: 10.1103/PhysRevD.100.114032.
  * Zhang et al. [2021] Jin-Li Zhang, Zhu-Fang Cui, Jialun Ping, and Craig D Roberts.  Contact interaction analysis of pion GTMDs.  _Eur. Phys. J. C_ , 81(1):6, 2021.  doi: 10.1140/epjc/s10052-020-08791-1.
  * Gutiérrez-Guerrero et al. [2021] L. X. Gutiérrez-Guerrero, G. Paredes-Torres, and A. Bashir.  Mesons and baryons: Parity partners.  _Phys. Rev. D_ , 104(9):094013, 2021.  doi: 10.1103/PhysRevD.104.094013.
  * Xing et al. [2021] Zanbin Xing, Khépani Raya, and Lei Chang.  Quark anomalous magnetic moment and its effects on the $\rho$ meson properties.  _Phys. Rev. D_ , 104(5):054038, 2021.  doi: 10.1103/PhysRevD.104.054038.
  * Wu [2003] Yue-Liang Wu.  Symmetry principle preserving and infinity free regularization and renormalization of quantum field theories and the mass gap.  _Int. J. Mod. Phys. A_ , 18:5363–5420, 2003.  doi: 10.1142/S0217751X03015222.
  * Wu [2004] Yue-Liang Wu.  Symmetry preserving loop regularization and renormalization of QFTs.  _Mod. Phys. Lett. A_ , 19:2191–2204, 2004.  doi: 10.1142/S0217732304015361.
  * Ebert et al. [1996] Dietmar Ebert, Thorsten Feldmann, and Hugo Reinhardt.  Extended NJL model for light and heavy mesons without q - anti-q thresholds.  _Phys. Lett. B_ , 388:154–160, 1996.  doi: 10.1016/0370-2693(96)01158-6.
  * Rochester and Butler [1947] G. D. Rochester and C. C. Butler.  Evidence for the Existence of New Unstable Elementary Particles.  _Nature_ , 160:855–857, 1947.  doi: 10.1038/160855a0.
  * Gell-Mann [1953] M. Gell-Mann.  Isotopic Spin and New Unstable Particles.  _Phys. Rev._ , 92:833–834, 1953.  doi: 10.1103/PhysRev.92.833.
  * Lee and Yang [1956] T. D. Lee and Chen-Ning Yang.  Question of Parity Conservation in Weak Interactions.  _Phys. Rev._ , 104:254–258, 1956.  doi: 10.1103/PhysRev.104.254.
  * Cabibbo [1963] Nicola Cabibbo.  Unitary Symmetry and Leptonic Decays.  _Phys. Rev. Lett._ , 10:531–533, 1963.  doi: 10.1103/PhysRevLett.10.531.
  * Kobayashi and Maskawa [1973] Makoto Kobayashi and Toshihide Maskawa.  CP Violation in the Renormalizable Theory of Weak Interaction.  _Prog. Theor. Phys._ , 49:652–657, 1973.  doi: 10.1143/PTP.49.652.
  * Christenson et al. [1964] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay.  Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson.  _Phys. Rev. Lett._ , 13:138–140, 1964.  doi: 10.1103/PhysRevLett.13.138.
  * Glashow et al. [1970] S. L. Glashow, J. Iliopoulos, and L. Maiani.  Weak Interactions with Lepton-Hadron Symmetry.  _Phys. Rev. D_ , 2:1285–1292, 1970.  doi: 10.1103/PhysRevD.2.1285.
  * Taboada-Nieto et al. [2022] U. Taboada-Nieto, P. G. Ortega, D. R. Entem, F. Fernández, and J. Segovia.  Kaon spectrum revisited.  9 2022.
  * Maris and Tandy [2000] Pieter Maris and Peter C. Tandy.  The pi, K+, and K0 electromagnetic form-factors.  _Phys. Rev. C_ , 62:055204, 2000.  doi: 10.1103/PhysRevC.62.055204.
  * Xiao et al. [2002] Bo-Wen Xiao, Xin Qian, and Bo-Qiang Ma.  The Kaon form-factor in the light cone quark model.  _Eur. Phys. J. A_ , 15:523–527, 2002.  doi: 10.1140/epja/i2002-10059-y.
  * Stamen et al. [2022] Dominik Stamen, Deepti Hariharan, Martin Hoferichter, Bastian Kubis, and Peter Stoffer.  Kaon electromagnetic form factors in dispersion theory.  _Eur. Phys. J. C_ , 82(5):432, 2022.  doi: 10.1140/epjc/s10052-022-10348-3.
  * Ji and Maris [2001] Chueng-Ryong Ji and Pieter Maris.  K(l3) transition form-factors.  _Phys. Rev. D_ , 64:014032, 2001.  doi: 10.1103/PhysRevD.64.014032.
  * Bazavov et al. [2013] A. Bazavov et al.  Kaon semileptonic vector form factor and determination of $|V_{us}|$ using staggered fermions.  _Phys. Rev. D_ , 87:073012, 2013.  doi: 10.1103/PhysRevD.87.073012.
  * Boyle et al. [2015] Peter A. Boyle et al.  The kaon semileptonic form factor in Nf = 2 + 1 domain wall lattice QCD with physical light quark masses.  _JHEP_ , 06:164, 2015.  doi: 10.1007/JHEP06(2015)164.
  * Troitsky and Troitsky [2021] S. V. Troitsky and V. E. Troitsky.  K0 and K+-meson electromagnetic form factors: A nonperturbative relativistic quark model versus experimental, perturbative, and lattice quantum-chromodynamics results.  _Phys. Rev. D_ , 104(3):034015, 2021.  doi: 10.1103/PhysRevD.104.034015.
  * Sciascia [2008] B. Sciascia.  Precision tests of the SM with leptonic and semileptonic kaon decays.  _Nucl. Phys. B Proc. Suppl._ , 181-182:83–88, 2008.  doi: 10.1016/j.nuclphysbps.2008.09.008.
  * Beringer et al. [2012] J. Beringer et al.  Review of Particle Physics (RPP).  _Phys. Rev. D_ , 86:010001, 2012.  doi: 10.1103/PhysRevD.86.010001.
  * Cirigliano et al. [2012] Vincenzo Cirigliano, Gerhard Ecker, Helmut Neufeld, Antonio Pich, and Jorge Portoles.  Kaon Decays in the Standard Model.  _Rev. Mod. Phys._ , 84:399, 2012.  doi: 10.1103/RevModPhys.84.399.
  * Callan and Treiman [1966] C. G. Callan and S. B. Treiman.  Equal Time Commutators and K Meson Decays.  _Phys. Rev. Lett._ , 16:153–157, 1966.  doi: 10.1103/PhysRevLett.16.153.
  * Maris et al. [1998] Pieter Maris, Craig D. Roberts, and Peter C. Tandy.  Pion mass and decay constant.  _Phys. Lett. B_ , 420:267–273, 1998.  doi: 10.1016/S0370-2693(97)01535-9.
  * Xu et al. [2021] Zhen-Ni Xu, Zhu-Fang Cui, Craig D. Roberts, and Chang Xu.  Heavy + light pseudoscalar meson semileptonic transitions.  _Eur. Phys. J. C_ , 81(12):1105, 2021.  doi: 10.1140/epjc/s10052-021-09898-9.