# The time-like electromagnetic kaon form factor

A.S. Miramontes<EMAIL_ADDRESS>Instituto de Física y Matemáticas,
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040,
Mexico  Adnan Bashir<EMAIL_ADDRESS>Instituto de Física y Matemáticas,
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040,
Mexico

###### Abstract

We compute the electromagnetic charged kaon form factor in the time-like
region by employing a Poincaré covariant formalism of the Bethe-Salpeter
equation to study quark-antiquark bound states in conjunction with the
Schwinger-Dyson equation for the quark propagator. Following a recent kindred
calculation of the time-like electromagnetic pion form factor, we include the
most relevant intermediate composite particles permitted by their quantum
numbers in the interaction kernel to allow for a decay mechanism for the
resonances involved. This term augments the usual gluon mediated interaction
between quarks. For a sufficiently low energy time-like probing photon, the
electromagnetic form factor is saturated by the $\rho(770)$ and $\phi(1020)$
resonances. We assume $SU(2)$ isospin symmetry throughout. Our results for the
absolute value squared of the electromagnetic form factor agree qualitatively
rather well and quantitatively moderately so with available experimental data.

##  I Introduction

The light pseudoscalar mesons, pion and kaons, play a crucial role in the
description of low-energy dynamics of the strong interactions where emergent
phenomena like confinement and dynamical chiral symmetry breaking (DCSB)
become dominant. Analyzing the internal structure of these Goldstone modes
associated with DCSB is of foremost importance to obtain a better
understanding of non-perturbative QCD as well as formation and properties of
hadrons. Extensive experimental measurements have been reported on the pion
electromagnetic form factor for space-like as well as time-like photons [1, 2,
3, 4, 5, 6, 7, 8]. In the latter case, it is dominated by the $\rho(770)$
resonance. In a conspicuous contrast, the existing data on the kaon
electromagnetic form factor is scarce. For space-like photons, only a few data
points have been reported at low $Q^{2}>0$ [9, 10], $Q$ being the four-
momentum of the probing photon. For time-like photons, existing measurements
lie solely in the close vicinity of the $\phi$ resonance [11, 12, 13] whereas
the region $-1<Q^{2}<0$ remains hitherto unexplored.

Nevertheless, the Electron Ion Collider (EIC) plans to generate a large amount
of high precision data on the kaon electromagnetic form factor both for space-
like and time-like photon momenta [14, 15]. Additionally, data in the space-
like domain is currently being analysed at the Jefferson Laboratory 12 GeV
(upgraded) Experiment E12-09-011 [16]. Moreover, with increasing effort and
precision in experimental measurements of time-like form factors in the
upcoming PANDA experiment [17], understanding of the QCD dynamics in this
intriguing domain is of paramount interest.

On theoretical side, the space-like electromagnetic pion form factor has been
explored in lattice QCD [18, 19, 20], a combined analysis of Schwinger-Dyson
(SDE) and Bethe-Salpeter equations (BSE) [21, 22], holographic QCD [23] and
front form dynamics [24] among other models. The functional approach of SDE
and BSE has only recently been extended to evaluate the time-like pion form
factor [25, 26]. Similarly, the kaon form factor has also been examined within
different frameworks, for example, lattice QCD [19], chiral perturbation
theory [27], holografic QCD [28], SDE and BSE [29, 21], etc. For the time-like
kaon form factor, only limited research is available, for example in
dispersion theory [30] and vector meson dominance models [31].

We study the time-like electromagnetic kaon form factor using a coupled
formalism of SDE and BSE which captures the non-perturbative character of QCD
excellently well and has produced plenty of hadron physics predictions. In
particular, it has been successfully applied to describe electromagnetic,
axial and transition form factors [32, 33, 34, 22, 35, 36, 37, 38, 25]. These
form factors probe the implications of confinement and DCSB in the infrared
and help us understand how a transition to the perturbative realm of QCD takes
place by matching onto the predictions of asymptotic QCD within the grasp of
the same experiment. These form factors also play a pivotal role in testing
the very limits of the celebrated standard model of particle physics put
forward by Glashow, Weinberg and Salam in precision measurements and
calculations of physical observables such as the anomalous magnetic moment of
muon, see for example [39, 40, 41, 42] and the latest reviews [43, 44] on the
subject.

Figure 1: IA diagrams representing the coupling of the external field to the
valence quarks. In the case of the electromagnetic charged kaon form factor,
the external field couples to the $u$ (left diagram) and $s$ (right diagram)
quarks.

In this article, we exploit the same strategy as developed for a recent
calculation of the time-like electromagnetic pion form factor [25] and study
the electromagnetic $K^{\pm}$ form factor. In this case, the low energy time-
like region is saturated not only by the $\rho$ resonance but also by the
$\phi$ resonance around $m_{\phi}=1.019$ GeV. Therefore, in order to capture
the correct behaviour of the kaon form factor in this domain, the decay
mechanism $\phi\rightarrow KK$ must incorporate additional features. To this
end, we include explicit pions and kaon contributions into the Bethe-Salpeter
interaction kernel and the quark propagators.

The article is organized as follows, in Section II we summarize the main
elements of the SDE and BSE approach i.e, the quark propagators, the Bethe-
Salpeter amplitudes (BSA), the quark-photon vertex (QPV) and the truncation
scheme which goes beyond the usual rainbow-ladder (BRL) approximation employed
for similar calculations. In section III, we present our numerical results for
the kaon form factor in the time-like region using two different sets of
parameters in the effective interaction. Finally, in section IV, we present
our conclusions and scope for future work.

##  II Electromagnetic form factors in the SDE/BSE formalism

The interaction of a virtual photon with a pseudoscalar meson $M$ is described
by a single form factor, $F_{M}(Q^{2})$, and is conveniently written as:

$\langle\textbf{P}(p_{1})|J^{\mu}|\textbf{P}(p_{2})\rangle=e(p_{1}+p_{2})^{\mu}F_{M}(Q^{2})\;,$
(1)

where $Q=p_{1}-p_{2}$ is the probing photon four-momentum and $e$ is the
elementary electromagnetic charge. On the other hand, the electromagnetic
current, which describes the coupling of a single photon with a quark-
antiquark system, can be expressed as

$J^{\mu}=\bar{\Psi}_{\textbf{P}}^{f}G_{0}(\mathbf{\Gamma}^{\mu}-K^{\mu})G_{0}{\Psi_{\textbf{P}}}^{i}~{},$
(2)

where $\Psi_{\textbf{P}}^{i}$ and $\Psi_{\textbf{P}}^{i}$ are the BSA of the
incoming and outgoing meson $M$, respectively, $G_{0}$ includes the
appropriate product of dressed quark propagators and the term
$\mathbf{\Gamma}^{\mu}$ represents the impulse approximation (IA) diagrams
which represent the coupling of the photons to the dressed valence quarks (see
Fig. 1),

$\mathbf{\Gamma}^{\mu}=\left(S^{-1}\otimes
S^{-1}\right)^{\mu}=\Gamma^{\mu}\otimes S^{-1}+S^{-1}\otimes\Gamma^{\mu}\;,$
(3)

where $S^{-1}$ is the inverse quark propagator. Effects beyond the IA terms
are encoded in the second term involving $K^{\mu}$ which characterizes the
interaction of the photon with the Bethe-Salpeter kernel describing the two-
body interaction [45, 25]. Both the terms in Eq. (2) need to be included in
order to preserve charge conservation precisely. To compute the expression in
Eq. (2), we require the knowledge of the quark propagators involved, the meson
BSA, the QPV and the corresponding interaction kernels. To render this article
self-contained, we summarise the key elements of the SDE/BSE approach in the
next subsection, see recent and relevant reviews [46, 47, 48, 49, 50] for
further details.

Figure 2: Beyond IA diagrams necessary for a consistent calculation when we
include meson exchange into the interaction kernel. The left diagram
corresponds to the IA diagram, the middle diagram represents the coupling of
the photon to the meson and the last diagram the coupling of the photon with a
meson-quark vertex. For this work, we have included only the first IA diagram.

###  II.1 Elements of the SDE/BSE approach

The coupled SDE/BSE formalism to calculate form factors needs basic
ingredients which we discuss below. Note that all computation which follows is
carried out in the Euclidean momentum space.

The quark-photon vertex: Due to the fortunate coincidence that the strongly
interacting quarks are also electrically charged, the internal structure of a
meson can be probed through a photon interacting with its constituents. The
fully-dressed QPV, $\Gamma^{\mu}$, which characterizes the interaction between
quarks and photons in quantum electrodynamics, can be described via the
following inhomogeneous BSE:

$\displaystyle\left(\Gamma^{\mu}\right)_{a\alpha,b\beta}\left(p,Q\right)$
$\displaystyle=$ $\displaystyle
Z_{2}\left(\gamma^{\mu}\right)_{ab}t_{\alpha\beta}$ $\displaystyle+$
$\displaystyle\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)\,S_{r\rho,e\epsilon}\left(k_{1}\right)$
$\displaystyle\times$
$\displaystyle\left(\Gamma^{i,\mu}\right)_{e\epsilon,n\nu}\left(Q,q\right)S_{n\nu,s\sigma}\left(k_{2}\right)\,,$

where $Z_{2}$ is the quark wave-function renormalization constant and $S(k)$
is the fully dressed quark propagator. Among the kinematic factors, $Q$ is the
photon momentum, $p$ and $q$ are the external and internal relative momenta
between the quark and the antiquark, respectively, while the internal quark
and antiquark momenta are defined as $k_{1}=q+Q/2$ and $k_{2}=q-Q/2$ such that
$Q=k_{1}-k_{2}$ and $q=(k_{1}+k_{2})/2$. The Latin letters represent Dirac
indices while the Greek letters represent flavour indices. The isospin
structure of the vertex is given by
$t_{\alpha\beta}=\textrm{diag}\left(\nicefrac{{2}}{{3}},\nicefrac{{-1}}{{3}},\nicefrac{{-1}}{{3}}\right)$.
The QPV can be decomposed in a basis composed of eight vectors transverse to
the photon momentum and four non-transverse ones, all of which are included in
our calculation.

The Bethe-Salpeter Equation: The description of mesons as bound states of two
quarks in our framework is effectuated through the homogeneous BSE

$\displaystyle\left(\Gamma\right)_{a\alpha,b\beta}\left(p,P\right)=\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(P,p,q\right)\times
S_{r\rho,e\epsilon}\left(k_{1}\right)$
$\displaystyle\left(\Gamma\right)_{e\epsilon,n\nu}\left(q,P\right)S_{n\nu,s\sigma}\left(k_{2}\right)~{},$
(5)

where $P$ is the total momentum of the meson. For pseudoscalar mesons,
abbreviated as PS in the subscripts below, the Dirac part of the BSA $\Gamma$
can be expanded in a tensorial basis with four elements:

$\displaystyle\Gamma^{i}_{PS}(p,P)=\tau^{i}~{}\gamma_{5}\\{E_{PS}(p,P)-i\not{P}F_{PS}(p,P)$
$\displaystyle\hskip 34.1433pt-i\not{p}(p\cdot
P)G_{PS}(p,P)-\left[\not{P},\not{p}\right]H_{PS}(p,P)\\}\,,~{}$ (6)

where $E_{PS},F_{PS},G_{PS}$ and $H_{PS}$ are four independent dressing
functions. The two-body interaction in Eq. (II.1) is represented by
$[K(P,p,q)]$; it corresponds to two-particle irreducible quark/antiquark
scattering kernel, which contains all possible interactions between the quark
and the antiquark within the bound state.

The quark propagator: The dressed quark propagator can be obtained by solving
the corresponding SDE, which reads

$S^{-1}=S_{0}^{-1}-Z_{1f}\int_{q}\gamma_{\mu}S(q)\Gamma^{qgl}_{\nu}(q,k)D_{\mu\nu}(k)~{},$
(7)

where $S_{0}^{-1}(p)$ the renormalized bare propagator,

$S_{0}^{-1}(p)=Z_{2}\left(i\not{p}+Z_{m}m\right)~{},$ (8)

and $Z_{1f}$, $Z_{2}$ and $Z_{m}$ are renormalization constants associated
with the quark-gluon vertex, the quark propagator and the quark mass,
respectively. Moreover, $m$ is the current quark mass, $\Gamma^{qgl}$ is the
full quark-gluon vertex and $D_{\mu\nu}$ is the full gluon propagator which,
in the Landau gauge acquires the following form

$D_{\mu\nu}(k)=\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\frac{Z(k^{2})}{k^{2}}~{},$
(9)

with $Z(p^{2})$ being the gluon dressing function. For the simplicity of
notation, we have suppressed the color indices.

###  II.2 Rainbow ladder truncation and beyond

In the SDE/BSE formalism, one of the simplest non-perturbative approximation
that preserves the axial-vector (AxWTI) and vector Ward-Takahashi identities
(VWTI) is the so called rainbow-ladder (RL) truncation, where the BSE
interaction kernel is simplified to a vector-vector gluon exchange with an
effective coupling $\alpha\left(k^{2}\right)$. Omitting colour indices, this
kernel can be written as

$K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)=\alpha\left(k^{2}\right)\gamma^{\mu}_{ar}\gamma^{\nu}_{sb}D^{\mu\nu}\left(k\right)\delta^{\alpha\rho}\delta^{\sigma\beta}~{},$
(10)

where $k=p-q$ is the momentum flowing through the gluon propagator. On the
other hand, the truncated quark SDE in the RL truncation reads,

$Z_{1f}\gamma_{\mu}Z(k^{2})\Gamma_{\nu}^{\text{qgl}}(q,p)\rightarrow
Z_{2}^{2}\gamma_{\mu}4\pi\alpha(k^{2})\gamma_{\nu}\,.$ (11)

Note that the effective coupling $\alpha(k^{2})$ describes the strength of the
quark-antiquark interaction. There exist primarily two models in the
literature to parameterize it: the Maris-Tandy (MT) model [51, 52] and the
Qin-Chang (QC) model [53].111It is worth mentioning that both the models (MT
and QC) for the effective coupling $\alpha(k^{2})$ yield very similar results
for hadron spectroscopy and structure. We choose to employ the MT model in
order to make consistent comparison with the work reported earlier on the
time-like pion electromagnetic form factor [25]. For the purpose of this
article, we employ the MT model. It reads as follows

$\displaystyle\alpha(q^{2})$ $\displaystyle=$
$\displaystyle\pi\eta^{7}\left(\frac{q^{2}}{\Lambda^{2}}\right)^{2}e^{-\eta^{2}\frac{q^{2}}{\Lambda^{2}}}$
(12) $\displaystyle+$
$\displaystyle\frac{2\pi\gamma_{m}\big{(}1-e^{-q^{2}/\Lambda_{t}^{2}}\big{)}}{\textnormal{ln}[e^{2}-1+(1+q^{2}/\Lambda_{QCD}^{2})^{2}]}\,,$

which includes a Gaussian term, dominant in the infrared, to provide enough
interaction strength for DCSB to take place and an ultraviolet dominant term
which reproduces the one-loop QCD behavior of the quark propagator at large
momenta. The free parameters of the MT model are $\Lambda$ and $\eta$. These
are usually fitted to reproduce the pion mass and the its weak decay constant.
In the SDE/BSE framework, the running quark masses $m_{u}$, $m_{d}$ and
$m_{s}$ are also introduced as input parameters. The scale $\Lambda_{t}=1$ GeV
is brought in for technical reasons alone and has no bearing on the computed
observables. The anomalous dimension is $\gamma_{m}=12/(11N_{c}-2N_{f})=12/25$
with $N_{f}=4$ flavours and $N_{c}=3$ colours. For the QCD mass scale, we use
$\Lambda_{QCD}=0.234$ GeV.

In the RL truncation, the solutions of the homogeneous BSE are bound sates
which cannot develop a decay width. Note that the bound states, determined as
solutions of BSEs, appear as poles in Green functions with the corresponding
quantum numbers. In particular, when we compute the QPV in the RL truncation,
electrically neutral vector mesons manifest themselves as poles located on the
negative real axis of $Q^{2}$. As a consequence, the electromagnetic form
factor develops the same pole on the real axis of the time-like region.
However, in general, any dynamics that is correlated with the presence of
virtual intermediate particles, e.g., decays, will be absent from the results
of any calculation using only the RL truncation. Therefore, in order to shift
the poles from the negative real axis to the complex plane to incorporate
decay mechanisms, we employ the BRL truncation. It is enforced by
incorporating the presence of intermediate particles in the interaction
kernel, leading to an adequate description of the time-like form factors.

In Refs. [54, 55, 56], the authors implemented the appearance of intermediate
particles by introducing explicit pionic degrees of freedom and pion-quark
interactions both into the truncated SDE and in the interaction kernel. The
pion-quark interaction vertex, represented by the pion BSA is calculated via a
consistently truncated BSE. The explicit and illustrative diagrammatic
modification of the quark SDE as well as the interaction kernel are presented
in Appendix A. In addition to the gluon exchange, this BRL truncation contains
a $t-$channel meson exchange (second term in lowest row of Fig. 5) and an
$s-,u-$channel meson exchange (third and fourth terms in the lowest row of
Fig. 5). At this point, it is informative to recall that the authors in [54,
55, 56] only take into account the gluon exchange plus the $t-$channel meson
exchange diagrams. The implementation of the $s-,u-$channel meson exchange
into the interaction kernel leads to several numerical difficulties, such as
contour deformation integration to calculate the BSE, which are detailed in
Refs. [45, 57]. Our BRL truncation yields a correct description of the QPV
which develops a multi-particle branch cut along the negative real
$Q^{2}$-axis, starting at the two-pion production threshold as expected [45].
Additionally, it allows for the description of the $\rho$-meson as a finite-
width resonance which is primarily due to the intermediate process
$\rho\rightarrow\pi\pi$ [57]. Subsequently, the electromagnetic pion form
factor was calculated in [25]. The absolute value and the phase of this
physical observable reported there compare favorably to the available
experimental data for the space-like and the time-like momenta, developing the
expected “bump” on the negative real $Q^{2}$-axis. In [42], we calculated pion
and kaon space-like electromagnetic form factors and computed box
contributions to the hadronic light-by-light piece of the muon’s anomalous
magnetic moment $a_{\mu}$, where our results are compatible with the ones
reported using other reliable contemporary frameworks. Finally, in [58], this
truncation was employed to study isospin symmetry breaking and its effect on
the light meson masses.

For the study of the electromagnetic charged kaon form factor, we follow the
same strategy as the one adopted in the previous work on the electromagnetic
pion form factor [25]. The mass difference between $u$ and $s$ quarks, which
compose the charged kaon, leads to an interference between different vector
meson resonances. In this case, in addition to the $\rho$ meson resonance, a
description of the $\phi$ meson and its decay $\phi\rightarrow KK$ needs to be
properly implemented into the interaction kernel. To include the decay
mechanism, kaon emission and absorption is included, together with a kaon-
quark interaction, given by the full kaon BSA, into the strange quark SDE. The
interaction kernel now includes the exchange of kaons in the $t-,s-$ and
$u-$channels (see Fig. 5). The IA diagrams necessary to calculate the
electromagnetic current are depicted in Fig. 2. While the first diagram (the
coupling of the photon with the $u$ quark) encodes the decay
$\rho\rightarrow\pi\pi$ via pionic correction in the up-quark SDE and pion
exchange in different channels in the interaction kernel, the second diagram
(the coupling of the photon to the $s$ quark) includes the decay
$\phi\rightarrow KK$ via kaonic corrections in the strange-quark SDE and kaon
exchange in different channels in the interaction kernel.

We wish to reemphasize here, as already discussed in [25], this BRL truncation
does not satisfy the AxWTI and VWTI simultaneously. We can choose to satisfy
one of the two identities while mildly violating the other one. The AxWTI
identity ensures that pion is a massless bound state in the chiral limit,
while the VWTI leads to charge conservation in electromagnetic processes.
Since we focus on electromagnetic form factors, we choose to preserve the VWTI
and ensure charge conservation. Nevertheless, we observe that the violation of
the AxWTI leads only to a small and insignificant error in relevant physical
observables.

It is also worth mentioning that in the previous calculation of the
electromagnetic pion form factor, the meson vertices appearing in the
interaction kernel, Eqs. (13-16), were approximated by the leading pion BSA
proportional to $\gamma_{5}$, whereas in this current calculation for the kaon
electromagnetic form factor, the full BSA has been considered, leading to a
better description of the $\rho$ resonance and its decay width, as detailed in
the following section.

##  III Numerical Results

Figure 3: Absolute value squared of the electromagnetic kaon form factor on
the time-like $Q^{2}<0$ region. We use the parameters $\eta=1.6$ and
$\Lambda=0.74$ GeV. We provide a comparison with experimental measurements
extracted from [13]. Both resonances ($\rho$ and $\phi$) are reproduced in our
calculation.

Following the SDE/BSE framework described in the previous section, we
calculate the electromagnetic charged kaon form factor in the time-like range
$Q^{2}<0$. We work in the $SU(2)$ isospin symmetric limit, where mass of the
up and down quarks are taken to be equal. We adjust the free parameters $\eta$
and $\Lambda$ in the effective MT interaction, together with the quark masses
$m_{u}$, $m_{d}$ and $m_{s}$, in order to reproduce the experimental values of
the mass and decay constant for the pion and kaon mesons. We employ two
different sets of parameters in the MT interaction:

  * •

BRL-I: $\eta=1.5$ $\Lambda=0.78$ GeV.

  * •

BRL-II: $\eta=1.6$, $\Lambda=0.74$ GeV.

  * •

For the current quark masses, we employ the values $m_{u/d}=6.8$ MeV and
$m_{s}=85.0$ MeV at the renormalization scale $\mu=19$ GeV.

It is important to bear in mind that in order to have a fully consistent
computational framework, the electromagnetic current has to be calculated in
the beyond-IA scheme employed, including diagrams of the photon coupling with
the corresponding meson and with a meson-quark vertex (see Fig. 2).
Nevertheless, to alleviate cumbersome numerical effort, we compute only the IA
diagrams.

In Table I, we collect the fitted masses for the pion and the kaon as well as
the predicted masses for the $\rho$ and the $\phi$ mesons, taking into account
only the RL truncation plus the $t$-channel meson exchange in the interaction
kernel (i.e. without the $s-$ and the $u-$-exchange channels). Additionally,
we include the results for different masses calculated by employing only the
RL truncation with the values $\eta=1.8$ and $\Lambda=0.72$ for comparison.
Note that for each case, we also report the weak decay constants. From Table
I, we can readily extract informative conclusions:

  * •

Employing only the RL truncation with the parameters that reproduce the
correct values of the pion and kaon masses, the $\phi$ meson is around 50 MeV
above the experimental value, while the $\rho$ meson is around 45 MeV below
the experimental value. It seems to suggest that it is impossible to obtain
the correct $\rho$ and $\phi$ masses simultaneously with one set of parameters
in the MT interaction in our approach.

  * •

Nevertheless, in this article we aim to describe primarily the qualitative
behaviour of the electromagnetic form factor in the time-like region.
Therefore, we will consider a 5$\%$ deviation from the experimental data
sufficiently acceptable in one of the first efforts made in this direction
through the SDE/BSE formalism.

  * •

The set BRL-I reproduces a better value for the $\rho$ mass, but at the cost
of a larger value for the $\phi$ meson mass. The set BRL-II is adopted in
order to get a more acceptable result for the $\phi$ mass but this is achieved
at the expense of an undesirable reduction in the masses of the $\pi$, $K$ and
$\rho$ mesons. The upshot of this exercise is that we seek the best compromise
with the truncation schemes hitherto available to us.

| RL | BRL I | BRL II | Exp.
---|---|---|---|---
$m_{\pi}$ | 0.140 | 0.139 | 0.128 | 0.139
$f_{\pi}$ | 0.131 | 0.138 | 0.135 | 0.131
$m_{{}_{k}}$ | 0.493 | 0.493 | 0.480 | 0.493
$f_{k}$ | 0.155 | 0.159 | 0.157 | 0.156
$m_{\rho}$ | 0.730 | 0.762 | 0.737 | 0.775
$f_{\rho}$ | 0.215 | 0.225 | 0.220 | 0.216
$m_{\phi}$ | 1.070 | 1.077 | 1.036 | 1.019
$f_{\phi}$ | 0.259 | 0.265 | 0.260 | 0.236
$M_{\rho}$ | 0.739 | 0.752 | 0.728 | 0.775
$\Gamma_{\rho}$ | 0.0 | 0.122 | 0.123 | 0.145
$M_{\phi}$ | 1.070 | 1.077 | 1.030 | 1.019
$\Gamma_{\phi}$ | 0.0 | 0.006 | 0.005 | 0.004
Table 1: Numerical results for the masses ad decay constants. In the RL we use
$\eta=1.8$ and $\Lambda=0.72$ GeV, for BRL 1 we employ $\eta=1.5$ and
$\Lambda=0.78$ GeV, and for BRL II $\eta=1.6$ and $\Lambda=0.74$ GeV. The
masses $M_{\rho}$, $M_{\phi}$ and the respective decay widths have been
extracted from our resulting time-like form factor using Pade approximant. All
mass-dimensioned quantities are expressed in GeV. The experimental values have
been extracted from Ref. [59].

In Figs. 3 and 4 we show our results for the kaon electromagnetic form factor
for a time-like ($Q^{2}<0$) photon. As already discussed in the text, there
are two main approximations within the scope of our calculation.

  * •

The first is the IA, only taking into account the diagrams where the photon
couples to the $u$ and $s$ quarks (Fig. 1), neglecting further couplings. It
leads to a deviation from $F_{k}(Q^{2}=0)=1$ of around only 1$\%$.

  * •

As already mentioned earlier in the text, the second approximation relates to
including intermediate resonances into the interaction kernel. It entails that
one must choose whether the AxWTI or the VWTI is preserved, while the other is
violated. Since we are interested in the electromagnetic form factor, the
symmetry we naturally want is the conservation of charge. Therefore, we choose
to ensure the VWTI is satisfied throughout.

As already demonstrated in [45], the inclusion of different meson exchange
channels into the interaction kernel leads to a branch cut which lies along
the real axis for $Q^{2}<0$, starting at the two-particles production
threshold $Q^{2}=-4m_{\pi,K}^{2}$ 222As detailed in [45], in addition to the
multiparticle branch cuts, the quark propagators also manifest themselves as
branch cuts that we have to deal with while integrating to solve for the QPV.
For the two different sets of MT parameters we are able to calculate the form
factor up to around -1.5 GeV2. Beyond this value, the branch cuts from the
pions/kaons and the branch cuts from the quark propagators begin to overlap
and the contour deformation is no longer possible for the required
integration.
. The time-like electromagnetic form factors thus acquire the same branch
cuts. Therefore, they are defined as $F(Q^{2}+i\epsilon)$. For the present
calculation we use $\epsilon=0.0001$ GeV2. The inclusion of the decay kernels
moves the $\rho$ and $\phi$ resonances poles from the real axis to the complex
plane in the QPV, as is manifest in Figs. 3 and 4. In Fig. 3, we depict our
results for the kaon electromagnetic form factor with the BRL-1 truncation
from 0.0 GeV2 to -1.2 GeV2. The extracted $\phi$ resonance pole in this case
lies at the mass value $M=1.030$ GeV and has the decay width of
$\Gamma_{\phi}=0.005$ GeV. Additionally, the corresponding kaon charge radii
obtained are $\sqrt{r^{2}_{K}}=0.592$ fm for BRL-I and
$\sqrt{r^{2}_{K}}=0.590$ for BRL-II.

In the inset of Fig. 3 we can observe that the $\rho$ pole is also present
even though it is not apparently visible on the main plot 333It is important
to note that the radially excited states such as $\rho(1400)$ are also encoded
in our truncation. Nevertheless, the corresponding resonance poles appear
outside the range of our calculation. Results on the radially excited states
employing the SDE/BSE formalism will be published elsewhere.. In Fig. 4 we
show the electromagnetic kaon form factor in the region $[-1,-1.2]$ GeV2,
where we can notice that the resonance pole position is sensitive to the
parameters which define the MT interaction. We wish to emphasize again that
including only RL plus the t-channel mesons exchange into the interaction
kernel implies that the $\rho$ and $\phi$ mesons are stable bound states and
do not develop a decay width, i,e. the poles are located in the negative real
axis of the time-like form factor. It is the inclusion of both the $s-$ and
$u-$channels meson exchange, with not widely disparate contributions, which
shifts the resonance poles from the real axis to the complex plane, describing
the correct behaviour of the form factor. We parameterize the solution of the
electromagnetic kaon form factor employing a Padé fit in order to extract the
pole position as $M^{2}_{{\rm pole}}=-Q^{2}+iM_{{\rm pole}}\Gamma$, where
$M_{{\rm pole}}$ is the mass of the meson and $\Gamma$ is the corresponding
decay width. In Table I, we collect the values of the extracted masses and
decay widths for the $\rho$ and $\phi$ mesons. In our previous calculation of
the electromagnetic pion form factor in Ref. [25] we observed that using only
the leading component $\gamma_{5}$ of the pion BSA into the vertices of the
interaction kernel, underestimates the decay width of the $\rho$ to
$\Gamma=0.105$ GeV. Using the full BSA in the vertices, leads to a much better
result of $\Gamma=0.123$ GeV.

Additionally we explicitly compute the separate contributions arising from the
$s-$ and $u-$ meson exchange channels on the total quantitative outcome. We
conclude that the $s-$ channel is relatively more dominant. It contributes
around 65% to the decay widths, while the remaining contribution stems from
the $u-$ meson exchange channel. Therefore none can be ignored in terms of
their quantitative impact. Moreover, it is worth emphasizing that both the
channels must be included in order to satisfy the Ward identities.

##  IV Conclusions and scope

In this article, we present an exploratory study of the electromagnetic
charged kaon form factor in the time-like region employing the formalism of
SDE/BSE. We focus on the effects of intermediate mesons (pions and kaons) in
the interaction kernel. The main approximations we employ in this calculation
are the IA and the small and numerically insignificant violation of the AxWTI.
Notwithstanding, the results obtained are in fairly satisfactory agreement
with the experimental observations. Our results are consistent with the fact
that the kaon form factor is not only saturated by the $\rho(770)$ resonance,
but mainly by the $\phi(1020)$ resonance in the appropriate low energy regime.
As can be deciphered in the plots illustrated in the previous section, our
results develop two bumps close to the experimental mass values for the $\rho$
and $\phi$ mesons. The exact pole position is sensitive to the choice of the
free parameters in the MT interaction. The decay width extracted from our
results for the case of the $\rho$ shows a clear improvement on our previous
calculation. In the present case, we have employed the full pion BSA into the
pion vertices appearing in the system of SDE/BSE. On the other hand, our
result for the decay width for the $\phi$ meson is very close to experimental
results. It is worth mentioning that our calculation has been carried out in
the isospin symmetric limit. Thus it lacks $\omega-\rho$ mixing effects. These
effects might play a noticeable role in capturing the correct behaviour of the
time-like form factor, especially close to the resonances. First results on
isospin breaking effects on the splitting of light mesons have been explored
using the SDE/BSE framework in [58]. In the future, we would like to implement
this effects into our calculations of form factors.

Figure 4: Absolute value squared of the time like kaon form factor using
different values for the free parameters in the MT interaction, the upper
panel: $\eta=1.5$, $\Lambda=0.78$ and the bottom panel: $\eta=1.6$,
$\Lambda=0.74$

###### Acknowledgements.

A. S. Miramontes acknowledges Consejo Nacional de Ciencia y Teconología
(CONACyT), Mexico, for the financial support provided to him through the
program “Postdoctorados Nacionales por México”. A. Bashir is grateful to the
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico, for the
Coordinación de la Investigación Científica (CIC) project grant no. 4.10.

##  Appendix A Interaction kernels

Figure 5: Truncations used herein for the BSE interaction kernel $K$ (lower
diagram) and the quark SDE one (upper diagram). In the lower diagram, the
terms on the right-hand side correspond to the rainbow-ladder, pion/kaon
exchange, and the $s$\- and the $u$-channel pion/kaon decay contributions to
the truncation, respectively. The $s$\- and the $u$-channel pion/kaon decay
terms do not contribute to the quark SDE.

The Bethe-Salpeter interaction kernel which represents the exchange of
explicit mesonic degrees of freedom as defined in [54, 55] reads as follows,

$\displaystyle K^{(t)~{}ut}_{rs}(q,p;P)=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\frac{C}{4}[\Gamma_{\textbf{P}}^{j}]_{ru}\left(\frac{p+q-P}{2};p-q\right)[Z_{2}\gamma^{5}]_{ts}D_{\textbf{P}}(p-q)$
$\displaystyle~{}~{}~{}+\frac{C}{4}[\Gamma_{\textbf{P}}^{j}]_{ru}\left(\frac{p+q-P}{2};q-p\right)[Z_{2}\gamma^{5}]_{ts}D_{\textbf{P}}(p-q)$
$\displaystyle~{}~{}~{}+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\textbf{P}}^{j}]_{ts}\left(\frac{p+q+P}{2};p-q\right)D_{\textbf{P}}(p-q)$
$\displaystyle~{}~{}~{}+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\textbf{P}}^{j}]_{ts}\left(\frac{p+q+P}{2};q-p\right)D_{\textbf{P}}(p-q)~{},$
(13)

in combination with the following truncation of the quark SDE

$\displaystyle S^{-1}(p)$ $\displaystyle=$ $\displaystyle
S^{-1}(p)^{RL}-\frac{3}{2}\int_{q}\Bigg{[}Z_{2}\gamma_{5}S(q)\Gamma_{\pi}\left(\frac{p+q}{2},q-p\right)$
(14)
$\displaystyle+Z_{2}\gamma_{5}S(q)\Gamma_{\textbf{P}}\left(\frac{p+q}{2},p-q\right)\Bigg{]}\frac{D_{\pi}(k)}{2}~{},$

with P the corresponding $\pi$, $K$ meson and $S^{-1}(p)^{RL}$ being the
right-hand-side of the quark SDE in the RL truncation with the gluon-mediated
interaction as described in Sect. II.2. In Eqs. (13) and (14) the meson
propagator is taken as $D_{\pi}(k)={(k^{2}+m_{\pi}^{2})^{-1}}$.

Herein, the quark-meson vertex $\Gamma_{\pi}$ is taken to be the full
pion/kaon BSA. The exchange of the mesons in the kernel can also appear in the
$s$\- and the $u$-channels [54]. They read:

$\displaystyle K^{(s)~{}he}_{da}(q,p,r;P)=~{}$
$\displaystyle\frac{C}{2}~{}D_{\textbf{P}}\left(\frac{P+r}{2}\right)D_{\textbf{P}}\left(\frac{P-r}{2}\right)~{}\left[[Z_{2}\gamma_{5}]_{dc}S_{cb}\left(p-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{ba}\right.$
$\displaystyle~{}~{}~{}~{}\times[\Gamma_{\textbf{P}}^{j}]_{hg}\left(q-\frac{P}{4}-\frac{r}{4};\frac{r-P}{2}\right)S_{gf}\left(q-\frac{r}{2}\right)[\Gamma_{\textbf{P}}^{j}]_{fe}\left(q+\frac{P}{4}-\frac{r}{4};-\frac{P+r}{2}\right)$
$\displaystyle+~{}[\Gamma_{\textbf{P}}^{j}]_{dc}\left(p+\frac{P}{4}-\frac{r}{4};\frac{P+r}{2}\right)S_{cb}\left(p-\frac{r}{2}\right)[\Gamma_{\textbf{P}}^{j}]_{ba}\left(p-\frac{P}{4}-\frac{r}{4};\frac{P-r}{2}\right)$
$\displaystyle~{}~{}~{}~{}\left.\times[Z_{2}\gamma_{5}]_{hg}S_{gf}\left(q-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{fe}\right]~{},$
(15)

and

$\displaystyle K^{(u)~{}he}_{da}(q,p,r;P)=~{}$
$\displaystyle\frac{C}{2}~{}D_{{\textbf{P}}}\left(\frac{P+r}{2}\right)D_{{\textbf{P}}}\left(\frac{P-r}{2}\right)~{}\left[[Z_{2}\gamma_{5}]_{dc}S_{cb}\left(p+\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{ba}\right.$
$\displaystyle~{}~{}~{}~{}\times[\Gamma_{\textbf{P}}^{j}]_{hg}\left(q-\frac{P}{4}-\frac{r}{4};\frac{r-P}{2}\right)S_{gf}\left(q-\frac{r}{2}\right)[\Gamma_{\textbf{P}}^{j}]_{fe}\left(q+\frac{P}{4}-\frac{r}{4};-\frac{P+r}{2}\right)$
$\displaystyle+~{}[\Gamma_{\textbf{P}}^{j}]_{dc}\left(p+\frac{P}{4}+\frac{r}{4};\frac{P-r}{2}\right)S_{cb}\left(p+\frac{r}{2}\right)[\Gamma_{\textbf{P}}^{j}]_{ba}\left(p-\frac{P}{4}+\frac{r}{4};\frac{P+r}{2}\right)$
$\displaystyle~{}~{}~{}~{}\left.\times[Z_{2}\gamma_{5}]_{hg}S_{gf}\left(q-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{fe}\right]~{},$
(16)

where $r$ is an additional integration momentum in the BSE (cf. Eqs. (II.1) or
(II.1)). The resulting truncation of the BSE kernel as well as the quark SDE
are depicted in Fig. 5. The inclusion of the $s$ and $u$-channel meson
exchange into the interaction kernel given in equations (15) and (16)
generates a highly non-trivial analytic structure of the integrals in the BSE.
It is induced by the intermediate pions/kaons potentially going on-shell as
well as by singularities in the quark propagators, see Ref. [45] for details.
The techniques for finding viable contour deformations for performing the
numerical integration in a mathematically correct way are also detailed
therein.

##  Appendix B Poles of the quark propagator

It is well known that the truncated quark SDE employed in this article
features pairs of complex conjugate poles in the complex plane (see, e.g.,
Ref. [60]). In order to facilitate the use of the quark propagators and easily
identify the analytic structures generated by those poles in the form factor
as well as vertex calculations, it is useful to parameterize the quark
propagator simply as a sum of two terms containing complex conjugate poles
(see, e.g., Ref. [61])

$\displaystyle S(p)$ $\displaystyle=$
$\displaystyle-i\not{p}\sigma_{v}(p^{2})+\sigma_{s}(p^{2})\,,$
$\displaystyle\sigma_{v}(p^{2})$ $\displaystyle=$
$\displaystyle\sum_{i}^{n}\left[\frac{\alpha_{i}}{p^{2}+m_{i}}+\frac{\alpha_{i}^{\ast}}{p^{2}+m_{i}^{\ast}}\right]\,,$
$\displaystyle\sigma_{s}(p^{2})$ $\displaystyle=$
$\displaystyle\sum_{i}^{n}\left[\frac{\beta_{i}}{p^{2}+m_{i}}+\frac{\beta_{i}^{\ast}}{p^{2}+m_{i}^{\ast}}\right]~{},$
(17)

where the parameters $m_{i}$, $\alpha_{i}$, $\beta_{i}$ can be obtained by
fitting the corresponding quark SDE solution along the $p^{2}$ real axis or,
alternatively, on a parabola in the complex plane that does not enclose the
poles. We use two pairs of complex conjugate poles as these are enough to
provide a sufficiently precise fit for the quark propagator.

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