# Numerical values of $f^{F}$, $f^{D}$, $f^{S}$ coupling constants in $SU(3)$
invariant interaction Lagrangian of vector-meson nonet with $1/2^{+}$ octet
baryons

Cyril Adamuscin  Institute of Physics, Slovak Academy of Sciences, Bratislava
Erik Bartos  Institute of Physics, Slovak Academy of Sciences, Bratislava
Stanislav Dubnicka  Institute of Physics, Slovak Academy of Sciences,
Bratislava  Anna Zuzana Dubnickova  Department of Theoretical Physics,
Comenius University, Bratislava

###### Abstract

It is demonstrated how an utilization of all existing experimental information
on electric and magnetic nucleon form factors, to be described by the Unitary
and Analytic (U&A) nucleon electromagnetic structure model in space-like and
time-like regions simultaneously, can provide numerical values of $f^{F}$,
$f^{D}$, $f^{S}$ coupling constants in $SU(3)$ invariant interaction
Lagrangian of the vector-meson nonet with $1/2^{+}$ octet baryons. The latter,
together with universal vector-meson coupling constants $f_{V}$, play an
essential role in a prediction of $1/2^{+}$ octet hyperon electromagnetic form
factors behaviors.

octet baryons, electromagnetic structure, form factors, electric and magnetic
moments

###### pacs:

13.40, 14.20

##  I Introduction

Recently the BESIII Collaboration has published ablikim  new results on the
proton electromagnetic (EM) form factors (FFs) in the time-like region by
measuring the process $e^{+}e^{-}\to p\bar{p}$ with very high precision at 12
different center-of-mass energies from $4.9836$ GeV2 to $13.4762$ GeV2. In
such way the obtained results enriched a set of already existing number of
experimental points on proton EM FFs in the space-like and time-like regions
substantially. Moreover, the BESIII Collab. and before also the BaBar Collab.
lees , measuring the polar angular distribution $F(\cos\theta_{p})$ of created
protons at few different energies, have been able to determine a separate
information on the proton electric and on the proton magnetic FFs in time-like
region for the first time.

On the other hand a very successful microscopic model for the proton EM FFs in
the near-threshold time-like region has been elaborated xi  to be based on the
assumption that the behavior of the EM FFs as a function of the energy is
given by the initial or final state interaction between proton and antiproton
in the processes $e^{+}e^{-}\leftrightarrow p\bar{p}$. In this approach the
antinucleon-nucleon potential is constructed in the framework of chiral
effective field theory xiwu  and fitted to results of partial-wave analysis of
existing $\bar{p}p$ scattering data. As a result predictions of the proton EM
FFs $G_{E}^{p}(t)$ and $G_{M}^{p}(t)$, and also their ratios are in good
agreement with existing data up to $t=3.986\;{\rm GeV}^{2}$, i.e. at the
region of a validity of the model under consideration.

Next intentions of the BESIII Collab. are to extend such data measurements for
$e^{+}e^{-}\to Y\bar{Y}$ processes, where $Y$ are hyperons to be members of
the $1/2^{+}$ octet baryons

$p$, $n$, $\Lambda$, $\Sigma^{+}$, $\Sigma^{0}$, $\Sigma^{-}$, $\Xi^{0}$,
$\Xi^{-}$.

In a preparation of such a program it can be useful to dispose with some model
predictions for hyperon EM FF behaviors and subsequently also of the
corresponding differential and total cross-sections.

But the model presented in xi  is not applicable for such model predictions of
hyperon EM FFs as it is based on the existence of baryon-antibaryon scattering
data, what is unattainable in the case of hyperons.

In this paper we demonstrate a scheme in the framework of which one can do
such predictions. As one can see further, an important role in the prediction
of hyperon EM FFs play the advanced Unitary and Analytic ($U\&A$) nucleon EM
structure model, respecting the $SU(3)$ symmetry and OZI rule okubo -iizuka
violation, then a knowledge of the universal vector-meson coupling constants
$f_{V}$ to be extracted from the experimental values of vector-meson lepton
widths

$\Gamma(V\to
e^{+}e^{-})=\frac{\alpha^{2}m_{V}}{3}\bigg{(}\frac{f_{V}^{2}}{4\pi}\bigg{)}^{-1}$
(1)

with $\alpha=\frac{1}{137}$ to be the QED fine structure coupling constant and
$m_{V}$ the vector-meson mass, and a knowledge of the numerical values of
$f^{F}$, $f^{D}$, $f^{S}$ coupling constants in $SU(3)$ invariant Lagrangian

$\displaystyle
L_{VB\bar{B}}=\frac{i}{\sqrt{2}}f^{F}[\bar{B}^{\alpha}_{\beta}\gamma_{\mu}B^{\beta}_{\gamma}-\bar{B}^{\beta}_{\gamma}\gamma_{\mu}B^{\alpha}_{\beta}](V_{\mu})^{\gamma}_{\alpha}+$
(2)
$\displaystyle+\frac{i}{\sqrt{2}}f^{D}[\bar{B}^{\beta}_{\gamma}\gamma_{\mu}B^{\alpha}_{\beta}+\bar{B}^{\alpha}_{\gamma}\gamma_{\mu}B^{\beta}_{\gamma}](V_{\mu})^{\gamma}_{\alpha}+$
$\displaystyle+\frac{i}{\sqrt{2}}f^{S}\bar{B}^{\alpha}_{\beta}\gamma_{\mu}B^{\beta}_{\alpha}\omega^{0}_{\mu}$

describing strong interactions of the nonet of vector-mesons with $1/2^{+}$
octet baryons, where $B,\bar{B}$ and $V$ are baryon, anti-baryon and vector-
meson octuplet matrices and $\omega^{0}_{\mu}$ is omega-meson singlet.

The advanced $U\&A$ nucleon EM structure model represents a harmonious
unification of the vector-meson pole contributions and cut structure of EM
FFs, which represent the so-called continua contributions in nucleon EM FFs.

The nucleon EM FFs are analytic functions in the whole complex plane of their
variable $t$, besides cuts on the positive real axis from the lowest branch
point $t_{0}$ to $+\infty$.

The shape of nucleon EM FFs is directly related with an existence of complex
conjugate pairs of poles on unphysical sheets of the Riemann surface in
$t$-variable, corresponding to unstable true neutral vector-mesons olive
$\rho(770)$, $\omega(782)$, $\phi(1020)$; $\rho^{\prime}(1450)$,
$\omega^{\prime}(1420)$, $\phi^{\prime}(1680)$; $\rho^{\prime\prime}(1700)$,
$\omega^{\prime\prime}(1650)$, $\phi^{\prime\prime}(2170)$ with the quantum
numbers of the photon to be revealed explicitly in the process of electron-
positron annihilation into hadrons.

As a result the complex nature of nucleon EM FFs for $t>t_{0}$ in time-like
region is secured by imaginary parts of the vector-meson poles on unphysical
sheets of the Riemann surface in $t$-variable and the cuts on the positive
real axis of the first, so-called physical sheet, of the Riemann surface,
whereby imaginary part of the EM FFs are given by a difference of the FF
values on the upper boundary of the cuts and the FF values on the lower
boundary of the cuts.

Every nucleon electric FF is canonically normalized to the nucleon electric
charge and every nucleon magnetic FF is normalized to the nucleon magnetic
moment.

All these nucleon EM FFs govern the asymptotic behaviors as predicted by the
quark model of hadrons to be proven also in the framework of the QCD
brodskylepage .

The advanced $U\&A$ nucleon EM structure model depends on the coupling
constants ratios $(f_{VNN}/f_{V})$ to be defined for every of nine above-
mentioned unstable true neutral vector-mesons in VMD model. However, not all
nine of them appear explicitly in the advanced $U\&A$ nucleon EM structure
model as free parameters. In the process of a construction of the VMD model to
be automatically normalized and governing the correct asymptotic behavior,
some of these coupling constants ratios can be expressed through the table
mass values of the true neutral vector-mesons under consideration and other
coupling constant ratios. And just here some freedom exists in the choice,
which of the coupling constants ratios $(f_{VNN}/f_{V})$ will be left as free
parameters of the model.

The problem is that not for all nine true neutral vector-mesons under
consideration an experimental information on the lepton width $\Gamma(V\to
e^{+}e^{-})$ exists. The latter is known olive  only for ground state vector-
mesons $\rho(770)$, $\omega(782)$ and $\phi(1020)$. For all the first and
second excited states the Rev. of Part. Physics olive  declares, that their
lepton decays are seen, but they are not precisely measured experimentally up
to now.

Just for this reason in the present advanced nucleon EM structure model we
keep the coupling constant ratios $(f_{\rho NN}/f_{\rho})$, $(f_{\omega
NN}/f_{\omega})$, $(f_{\phi NN}/f_{\phi})$ and also
$(f_{\omega^{\prime}NN}/f_{\omega^{\prime}})$,
$(f_{\phi^{\prime}NN}/f_{\phi^{\prime}})$, for which some model estimations of
$f_{\omega^{\prime}}$ and $f_{\phi^{\prime}}$ exist donnachieclegg , as free
parameters to be determined in comparison of the present advanced nucleon EM
structure model with all existing data on nucleon EM FFs in space-like and
time-like regions simultaneously.

The estimation of $f_{\omega^{\prime}}$, $f_{\phi^{\prime}}$ (and also
$f_{\rho^{\prime}}$ for a determination of $f_{\rho^{\prime}NN}$ from
($f_{\rho^{\prime}NN}/f_{\rho^{\prime}}$) to be completely given by the table
mass values of vector-mesons under consideration) in donnachieclegg  will be
only a model admixture in subsequent determination of the $1/2^{+}$ octet
hyperon EM FF behaviors.

##  II Existing experimental information on nucleon EM FFs

The EM structure of the nucleons (iso-doublet compound of the proton and
neutron), as revealed experimentally for the first time in elastic unpolarized
electron-proton scattering in the middle of the last century, is completely
described by four independent scalar functions, the electric
$G_{Ep}(t)$,$G_{En}(t)$ and the magnetic $G_{Mp}(t)$, $G_{Mn}(t)$ FFs,
dependent of one variable to be chosen as the squared momentum transferred
$t=-Q^{2}$ of the virtual photon $\gamma^{*}$. The experimental information on
these functions consists of 11 following different sets of data on:

  * •

the ratio $\mu_{p}G^{p}_{E}(t)/G^{p}_{M}(t)$ in the space-like $(t<0)$ region
from polarization experiments jones -puckett

  * •

$G^{p}_{E}(t)$ in the space-like $(t<0)$ region bernauer

  * •

$|G^{p}_{E}(t)|$ in the time-like $t>0$ region; only from exp. in which
$|G^{p}_{E}(t)|$=$|G^{p}_{M}(t)|$ is assumed

  * •

$G^{p}_{M}t)$ in the space-like $(t<0)$ region bernauer -andivakis

  * •

$|G^{p}_{M}(t)|$ in the time-like $t>0$ region ablikim ,lees ,bassompierre
-aubert

  * •

$|G^{p}_{E}(t)/G^{p}_{M}(t)|$ in the time-like $t>0$ region ablikim ,lees

  * •

$G^{n}_{E}(t)$ in the space-like $(t<0)$ region hanson -golak

  * •

$|G^{n}_{E}(t)|$ in the time-like $t>0$ region; only from exp. in which
$|G^{n}_{E}(t)|$=$|G^{n}_{M}(t)|$ is assumed

  * •

$G^{n}_{M}(t)$ in the space-like $(t<0)$ region hanson ,rock -anderson

  * •

$|G^{n}_{M}(t)|$ in the time-like $t>0$ region antonelli

  * •

the ratio $\mu_{n}G^{n}_{E}(t)/G^{n}_{M}(t)$ in the space-like $(t<0)$ region
from polarization experiments on the light nuclei plaster ,riordan

from which not all are equally trustworthy.

The most reliable, from all above-mentioned data, are considered to be the
experimental points on the ratio $\mu_{p}G^{p}_{E}(t)/G^{p}_{M}(t)$ jones
-puckett  in the space-like region, which have been extracted from
simultaneous measurement of the transverse component

$P_{t}=\frac{h}{I_{0}}(-2)\sqrt{\tau(1+\tau)}G_{Ep}G_{Mp}\tan{\theta/2}$ (3)

and the longitudinal component

$P_{l}=\frac{h(E_{e}+E_{e^{\prime}})}{I_{0}m_{p}}\sqrt{\tau(1+\tau)}G_{Mp}^{2}\tan^{2}{\theta/2}$
(4)

of the recoil proton’s polarization in the polarized electron scattering plane
of the polarization transfer process $\overrightarrow{e}p\to
e\overrightarrow{p}$, where $h$ is the electron beam helicity, $I_{0}$ is the
unpolarized cross-section excluding $\sigma_{Mott}$ and
$\tau=-\frac{t}{4m^{2}_{p}}$, by means of the relation

$\mu_{p}\frac{G_{Ep}}{G_{Mp}}=-\frac{P_{t}}{P_{l}}\frac{(E_{e}+E_{e^{\prime}})}{2m_{p}}\tan{\theta/2}.$
(5)

The data jones -puckett  clearly demonstrate that a general belief in the
dipole behavior of the proton electric FF $G^{p}_{E}(t)$ in the space-like
region to be obtained by the Rosenbluth method from the process of the elastic
scattering of unpolarized electrons on unpolarized protons $e^{-}p\to e^{-}p$
described by the differential cross-section in the laboratory system

$\displaystyle\frac{d\sigma^{lab}(e^{-}p\to
e^{-}p)}{d\Omega}=\frac{\alpha^{2}}{4E^{2}}\frac{\cos^{2}(\theta/2)}{\sin^{4}(\theta/2)}\frac{1}{1+(\frac{2E}{m_{B}})\sin^{2}(\theta/2)}$
(6)
$\displaystyle.\bigg{[}\frac{G_{E}^{2}(t)-\frac{t}{4m_{B}^{2}}G_{M}^{2}(t)}{1-\frac{t}{4m_{B}^{2}}}-2\frac{t}{4m_{B}^{2}}G_{M}^{2}(t)\tan^{2}(\theta/2)\bigg{]}$

with the QED the fine structure constant $\alpha=1/137$, the incident electron
energy $E$ and the scattering angle $\theta$, before the polarization
experiments jones -puckett  have been carried out, is no more valid.

This is a reason, why further we ignore all older data on $G^{p}_{E}(t)$ in
the space-like region and we take into account only the newest data from MAMI
bernauer  at the region $-0.5524GeV^{2}<t<-0.0152GeV^{2}$, where the
application of the Rosenbluth method is still justified.

On the second place to be concerned of the reliability are the data on the
proton magnetic FF $G^{p}_{M}(t)$ bernauer -andivakis  in the space-like
region, which have been obtained by the Rosenbluth method from experimental
information on the differential cross-section describing elastic scattering of
unpolarized electrons on unpolarized protons. As one can see from the
expression of the differential cross-section (6), with increased negative
values of $t$, the proton magnetic FF is dominant in comparison with the
proton electric FF and so, the data on $G^{p}_{M}(t)$ extracted by the
Rosenbluth method are reliable.

Less reliable are data ablikim ,lees ,bassompierre -aubert  on absolute value
of the magnetic FF $|G^{p}_{M}(t)|$ in the time-like $t>0$ region (the same is
concerned also of the neutron magnetic FF $|G^{n}_{M}(t)|$ in the time-like
$t>0$ region) as the most of them are extracted from the total cross-section
of the electron-positron annihilation process into proton-antiproton pair

$\sigma_{tot}^{c.m.}(e^{+}e^{-}\to
p\bar{p})=\frac{4\pi\alpha^{2}\beta_{p}}{3t}[\mid
G^{p}_{M}(t)\mid^{2}+\frac{2m_{p}^{2}}{t}\mid G^{p}_{E}(t)\mid^{2}]$ (7)

with $\beta_{p}=\sqrt{1-\frac{4m_{p}^{2}}{t}}$, or from the total cross-
section of the antiproton-proton annihilation into electron-positron pair

$\sigma_{tot}^{c.m.}(\bar{p}p\to
e^{+}e^{-})=\frac{2\pi\alpha^{2}}{3p_{c.m.}\sqrt{t}}[\mid
G^{p}_{M}(t)\mid^{2}+\frac{2m_{p}^{2}}{t}\mid G^{p}_{E}(t)\mid^{2}]$ (8)

with $p_{c.m.}$ to be the antiproton momentum in the c.m. system, by and
assumption $|G^{p}_{E}(t)|$=$|G^{p}_{M}(t)|$, which is exactly valid only at
the proton-antiproton threshold following directly from the definition of both
FFs, or by an assumption that for high values of $t$ $|G^{p}_{E}(t)|$=0 and
then the total cross-sections under consideration give information only on
$|G^{p}_{M}(t)|$, what is not well-founded.

Very promising method of a determination of $|G^{p}_{E}(t)|$ and
$|G^{p}_{M}(t)|$ is demonstrated in ablikim ,lees  by a measurement of the
distribution of $\theta_{p}$, the angle between the proton momentum in the
$p\bar{p}$ rest frame , and the momentum of the $p\bar{p}$ system in the
$e^{+}e^{-}$ c.m. frame and by a subsequent fitting of the obtained
experimental points by the expression

$F(\cos\theta_{p})=N_{norm}\bigg{[}1+\cos^{2}\theta_{p}+\frac{4m^{2}_{p}}{t}R^{2}(1-\cos^{2}\theta_{p})\bigg{]}$
(9)

to be obtained from

$\frac{d\sigma(t)}{d\Omega}=\frac{\alpha^{2}\beta_{p}C}{4t}\bigg{[}|G^{p}_{M}(t)|^{2}(1+\cos^{2}\theta_{p})+\frac{4m^{2}_{p}}{t}|G^{p}_{E}(t)|^{2}\sin^{2}\theta_{p}\bigg{]}$
(10)

with the Coulomb correction factor $C$, where

$R=\frac{|G^{p}_{E}(t)|}{|G^{p}_{M}(t)|}$ (11)

and

$N_{norm}=\frac{2\pi\alpha^{2}\beta
L}{4t}\bigg{[}1.94+5.04\frac{m^{2}_{p}}{t}R^{2}|G^{p}_{M}(t)|^{2}\bigg{]}$
(12)

is the overall normalization factor and $L$ is the integrated luminosity.

However, the latter method is in the phase of a development and the ratios $R$
with $|G^{p}_{M}(t)|$ at only several values of the energy $t$ have been
determined ablikim ,lees  up to now.

The existing data on $G^{n}_{E}(t)$,$G^{n}_{M}(t)$ in the space-like region
and the data on $\mu_{n}G^{n}_{E}(t)/G^{n}_{M}(t)$ are even more model
dependent as they have been extracted from the scattering processes of
electrons on light nuclei by various theoretical model ingredients.

All these data are graphically presented in Figs. (1-11).

Figure 1: Data on the ratio $\mu_{p}G^{p}_{E}(t)/G^{p}_{M}(t)$ in the space-
like $(t<0)$ region from polarization experiments jones -puckett  Figure 2:
Proton electric FF data in the space-like $(t<0)$ region from MAMI bernauer
Figure 3: Proton electric FF data in the time-like $(t>0)$ region from exps.
in which $|G^{p}_{E}(t)|$=$|G^{p}_{M}(t)|$ has been assumed Figure 4: Proton
magnetic FF data in the space-like $(t<0)$ region bernauer -andivakis  Figure
5: Proton magnetic FF data in the time-like $(t>0)$ region ablikim ,lees
,bassompierre -aubert  Figure 6: Data on the ratio
$|G^{p}_{E}(t)/G^{p}_{M}(t)|$ in the time-like $(t>0)$ region ablikim ,lees
Figure 7: Neutron electric FF data in the space-like $(t<0)$ region hanson
-golak  Figure 8: Neutron electric FF data in the time-like $(t>0)$ region
from an assumption $|G^{n}_{E}(t)|$=$|G^{n}_{M}(t)|$ Figure 9: Neutron
magnetic negative FF data in the space-like $(t<0)$ region hanson ,rock
-anderson  Figure 10: Neutron magnetic FF data in the time-like $(t>0)$ region
antonelli  Figure 11: Data on the ratio $\mu_{n}G^{n}_{E}(t)/G^{n}_{M}(t)$ in
the space-like $(t<0)$ region from polarization experiments on the light
nuclei plaster ,riordan

Despite of all their shortcomings they will be further analyzed by one
advanced $U\&A$ nucleon EM structure model simultaneously.

##  III Advanced Unitary and Analytic nucleon EM structure model

First we shall demonstrate, how it is possible in the advanced nucleon EM
structure model to keep the coupling constants ratios $(f_{\rho
NN}/f_{\rho}),(f_{\omega NN}/f_{\omega}),(f_{\phi NN}/f_{\phi})$ and also
$(f_{\omega^{\prime}NN}/f_{\omega^{\prime}}),(f_{\phi^{\prime}NN}/f_{\phi^{\prime}})$
as free parameters of the model to be then subsequently determined numerically
in comparison of the advanced $U\&A$ model with all existing data
simultaneously.

Nevertheless, before that we would like to note that the nucleon EM FFs
$G^{p}_{E}(t),G^{p}_{M}(t),G^{n}_{E}(t),G^{n}_{M}(t)$ are very suitable for
extraction of an experimental information on the nucleon EM structure from the
earlier mentioned physical quantities (3)-(9). But for a construction of
various nucleon EM structure models the flavour-independent iso-scalar and
iso-vector parts $F^{N}_{1s}(t),F^{N}_{1v}(t),F^{N}_{2s}(t),F^{N}_{2v}(t)$ of
the Dirac and Pauli FFs to be defined by a parametrization of the matrix
element of the nucleon EM current

$<N|J^{EM}_{\mu}|N>=e\bar{u}(p^{\prime})\\{\gamma_{\mu}F^{N}_{1}(t)+\frac{i}{2m_{N}}\sigma_{\mu\nu}(p^{\prime}-p)_{\nu}F^{N}_{2}(t)\\}u(p)$
(13)

are more suitable.

Both sets of these FFs are related as follows

$\displaystyle G^{p}_{E}(t)$ $\displaystyle=$
$\displaystyle[F^{N}_{1s}(t)+F^{N}_{1v}(t)]+\frac{t}{4m^{2}_{p}}[F^{N}_{2s}(t)+F^{N}_{2v}(t)]$
(14) $\displaystyle G^{p}_{M}(t)$ $\displaystyle=$
$\displaystyle[F^{N}_{1s}(t)+F^{N}_{1v}(t)]+[F^{N}_{2s}(t)+F^{N}_{2v}(t)]$
$\displaystyle G^{n}_{E}(t)$ $\displaystyle=$
$\displaystyle[F^{N}_{1s}(t)-F^{N}_{1v}(t)]+\frac{t}{4m^{2}_{n}}[F^{N}_{2s}(t)-F^{N}_{2v}(t)]$
$\displaystyle G^{n}_{M}(t)$ $\displaystyle=$
$\displaystyle[F^{N}_{1s}(t)-F^{N}_{1v}(t)]+[F^{N}_{2s}(t)-F^{N}_{2v}(t)],$

whereby experimental fact of a creation of true neutral vector-meson
resonances with quantum numbers of the photon in $e^{+}e^{-}\to hadrons$ is in
the first approximation taken into account by a saturation of $F^{N}_{1s}(t)$,
$F^{N}_{2s}(t)$ with neutral iso-scalar vector mesons $\omega(782)$,
$\phi(1020)$, $\omega^{\prime}(1420)$, $\phi^{\prime}(1680)$,
$\omega^{\prime\prime}(1650)$, $\phi^{\prime\prime}(2170)$ and
$F^{N}_{1v}(t)$, $F^{N}_{2v}(t)$ with neutral iso-vector vector-meson
resonances $\rho(770)$, $\rho^{\prime}(1450)$, $\rho^{\prime\prime}(1700)$ in
the corresponding VMD FF parametrization in the zero-width approximation.

For the sake of a generality let us consider FF $F(t)$ with a normalization
$F(0)=F_{0}$, the asymptotic behavior $F(t)_{t\to\infty}\sim 1/t^{m}$ and to
be saturated with $n$ \- true neutral vector mesons $V_{n}$. Then in the
framework of the standard $VMD$ model in the zero-width approximation one can
write

$F(t)=\frac{m^{2}_{1}}{m^{2}_{1}-t}a_{1}+\frac{m^{2}_{2}}{m^{2}_{2}-t}a_{2}+...+\frac{m^{2}_{n}}{m^{2}_{n}-t}a_{n},$
(15)

where $a_{n}=(f_{V_{n}NN}/f_{V_{n}})$, $f_{V_{n}NN}$ is the coupling constant
of an interaction of the $n$-th vector-meson with nucleons and $f_{V_{n}}$ is
the universal vector-meson coupling constant to be determined numerically from
the lepton width of the $n$-th vector-meson (1). Now requiring the
normalization of (15) at $t=0$ one gets the following equation for coupling
constant ratios $(f_{iNN}/f_{i})$

$\sum_{i=1}^{n}(f_{iNN}/f_{i})=F_{0}.$ (16)

Then transforming (15) into a common denominator one gets in the numerator
polynomial of $t^{n-1}$ degree. In order to achieve the required asymptotic
behavior of $F(t)$ one has to put some coefficients in the polynomial of the
numerator , starting from the highest degree $t^{n-1}$, to be zero, and one
comes to another $m-1$ equations

$\sum_{i=1}^{n}m^{2r}_{i}(f_{iNN}/f_{i})=0;r=1,2,...,m-1$ (17)

for $n$-coupling constant ratios. As a result a solution of the system of
equations (16),(17) will lead $m$ coupling constant ratios to be given through
the table masses of vector mesons and all additional couplig constant ratios
$(f_{m+1}NN/f_{m+1}),...,(f_{n}NN/f_{n})$, which will be considered to be free
parameters of the model.

The general solution of the system of eqs. (16),(17) for $n>m$ leads dubnicka
to the FF to be saturated by $n$-vector-meson resonances in the form suitable
for the unitarization

$\displaystyle F(t)$ $\displaystyle=$ $\displaystyle
F_{0}\frac{\prod_{j=1}^{m}m^{2}_{j}}{\prod_{j=1}^{m}(m^{2}_{j}-t)}+$
$\displaystyle+$
$\displaystyle\sum_{k=m+1}^{n}\bigg{\\{}\sum_{j=1}^{m}\frac{m^{2}_{k}}{(m^{2}_{k}-t)}\frac{\prod^{m}_{j\neq
i,j=1}m^{2}_{j}}{\prod^{m}_{j\neq k,j=1}(m^{2}_{j}-t)}\frac{\prod^{m}_{j\neq
i,j=1}(m^{2}_{j}-m^{2}_{k})}{\prod^{m}_{j\neq i,j=1}(m^{2}_{j}-m^{2}_{i})}-$
$\displaystyle-$
$\displaystyle\frac{\prod_{j=1}^{m}m^{2}_{j}}{\prod_{j=1}^{m}(m^{2}_{j}-t)}\bigg{\\}}(f_{kNN}/f_{k})$

and for $n=m$

$F(t)=F_{0}\frac{\prod_{j=1}^{m}m^{2}_{j}}{\prod_{j=1}^{m}(m^{2}_{j}-t)}$ (19)

for which the required asymptotic behavior $F(t)_{t\to\infty}\sim 1/t^{m}$ and
for $t=0$ the normalization $F(0)=F_{0}$ are fulfilled automatically.

Now these general results will be applied to the flavor-independent iso-scalar
and iso-vector parts of the Dirac and Pauli nucleon FFs $F^{N}_{1s}(t)$,
$F^{N}_{1v}(t)$, $F^{N}_{2s}(t)$, $F^{N}_{2v}(t)$ by means of which all
required properties of the nucleon $U\&A$ electromagnetic structure model like

  * •

experimental fact of a creation of unstable vector-meson resonances in
$e^{+}e^{-}$ annihilation processes into hadrons

  * •

the analytic properties

  * •

the reality conditions

  * •

the unitarity conditions

  * •

normalizations

  * •

asymptotic behaviors

are achieved.

However, before the latter we have to specify normalizations and asymptotic
behaviors of flavor-independent iso-scalar and iso-vector parts of the Dirac
and Pauli nucleon FFs.

The nucleon EM FFs are normalized as follows

$G^{p}_{E}(0)=1;G^{p}_{M}(0)=\mu_{p};G^{n}_{E}(0)=0;G^{n}_{M}(0)=\mu_{n}$ (20)

with $\mu_{p}$ and $\mu_{n}$ to be proton and neutron magnetic moments,
respectively.

The asymptotic behaviors of nucleon EM FFs are

$G^{p}_{E}(t)_{\mid_{t\to\infty}}=G^{p}_{M}(t)_{\mid_{t\to\infty}}=G^{n}_{E}(t)_{\mid_{t\to\infty}}=G^{n}_{M}(t)_{\mid_{t\to\infty}}\sim\frac{1}{t^{2}}.$
(21)

Then from (14) one obtains the normalizations of iso-scalar and iso-vector
parts of the Dirac and Pauli nucleon FFs

$F^{N}_{1s}(0)=F^{N}_{1v}(0)=\frac{1}{2};F^{N}_{2s}(0)=\frac{1}{2}(\mu_{p}+\mu_{n}-1);F^{N}_{2v}(0)=\frac{1}{2}(\mu_{p}-\mu_{n}-1)$
(22)

and their asymptotic behaviors are

$F^{N}_{1s}(t)_{\mid_{t\to\infty}}=F^{N}_{1v}(t)_{\mid_{t\to\infty}}\sim\frac{1}{t^{2}};F^{N}_{2s}(t)_{\mid_{t\to\infty}}=F^{N}_{2v}(t)_{\mid_{t\to\infty}}\sim\frac{1}{t^{3}}.$
(23)

If we apply (III) to $F^{N}_{1s}(t)$ with $m=2$ and $n=\omega^{\prime\prime}$,
$\phi^{\prime\prime}$, $\omega^{\prime}$, $\phi^{\prime}$, $\omega$, $\phi$
one obtains

$\displaystyle
F^{N}_{1s}(t)=\frac{1}{2}\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}+$
(24)
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime}})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})}+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\omega^{\prime\prime}}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega^{\prime}})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\Bigg{\\}}(f^{(1)}_{{\omega^{\prime}}NN}/f_{\omega^{\prime}})+$
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\phi^{\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi^{\prime}})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})}+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime}}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime}})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\Bigg{\\}}(f^{(1)}_{{\phi^{\prime}}NN}/f_{\phi^{\prime}})+$
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega}}{(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})}+\frac{m^{2}_{\omega}m^{2}_{\omega^{\prime\prime}}}{(m^{2}_{\omega}-t)(m^{2}_{\omega^{\prime\prime}}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\Bigg{\\}}(f^{(1)}_{{\omega}NN}/f_{\omega})+$
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\phi}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})}+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)}\Bigg{\\}}(f^{(1)}_{{\phi}NN}/f_{\phi}).$

to $F^{N}_{1v}(t)$ with $m=2$ and $n=\rho^{\prime\prime}$, $\rho^{\prime}$,
$\rho$ one obtains

$\displaystyle
F^{N}_{1v}(t)=\frac{1}{2}\frac{m^{2}_{\rho^{\prime\prime}}m^{2}_{\rho^{\prime}}}{(m^{2}_{\rho^{\prime\prime}}-t)(m^{2}_{\rho^{\prime}}-t)}+$
(25)
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\rho}m^{2}_{\rho^{\prime}}}{(m^{2}_{\rho}-t)(m^{2}_{\rho^{\prime}}-t)}\frac{(m^{2}_{\rho^{\prime}}-m^{2}_{\rho})}{(m^{2}_{\rho^{\prime}}-m^{2}_{\rho^{\prime\prime}})}+\frac{m^{2}_{\rho}m^{2}_{\rho^{\prime\prime}}}{(m^{2}_{\rho}-t)(m^{2}_{\rho^{\prime\prime}}-t)}\frac{(m^{2}_{\rho^{\prime\prime}}-m^{2}_{\rho})}{(m^{2}_{\rho^{\prime\prime}}-m^{2}_{\rho^{\prime}})}-$
$\displaystyle-\frac{m^{2}_{\rho^{\prime\prime}}m^{2}_{\rho^{\prime}}}{(m^{2}_{\rho^{\prime\prime}}-t)(m^{2}_{\rho^{\prime}}-t)}\Bigg{\\}}(f^{(1)}_{{\rho}NN}/f_{\rho}).$

to $F^{N}_{2s}(t)$ with $m=3$ and
$n=\omega^{\prime\prime},\phi^{\prime\prime},\omega^{\prime},\phi^{\prime},\omega,\phi$
one obtains

$\displaystyle
F^{N}_{2s}(t)=\frac{1}{2}(\mu_{p}+\mu_{n}-1)\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}+$
(26)
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime\prime}}m^{2}_{\phi^{\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\phi^{\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi^{\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi^{\prime}})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\omega^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\omega^{\prime}}m^{2}_{\phi^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\phi^{\prime}}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi^{\prime}})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\phi^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\phi^{\prime}}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime}})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi^{\prime}})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega^{\prime}})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}\Bigg{\\}}(f^{(2)}_{{\phi^{\prime}}NN}/f_{\phi^{\prime}})+$
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}m^{2}_{\omega}}{(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\omega}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega})(m^{2}_{\omega^{\prime}}-m^{2}_{\omega})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\omega^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\omega^{\prime}}m^{2}_{\omega}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\omega}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega})(m^{2}_{\omega^{\prime}}-m^{2}_{\omega})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega^{\prime}})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}\Bigg{\\}}(f^{(2)}_{{\omega}NN}/f_{\omega})+$
$\displaystyle+\Bigg{\\{}\frac{m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}m^{2}_{\phi}}{(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\phi}-t)}\frac{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi})}{(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\omega^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\omega^{\prime}}m^{2}_{\phi}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)(m^{2}_{\phi}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi^{\prime\prime}})(m^{2}_{\omega^{\prime}}-m^{2}_{\phi^{\prime\prime}})}+$
$\displaystyle+\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\phi}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\phi}-t)}\frac{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\phi})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\phi})}{(m^{2}_{\omega^{\prime\prime}}-m^{2}_{\omega^{\prime}})(m^{2}_{\phi^{\prime\prime}}-m^{2}_{\omega^{\prime}})}-$
$\displaystyle-\frac{m^{2}_{\omega^{\prime\prime}}m^{2}_{\phi^{\prime\prime}}m^{2}_{\omega^{\prime}}}{(m^{2}_{\omega^{\prime\prime}}-t)(m^{2}_{\phi^{\prime\prime}}-t)(m^{2}_{\omega^{\prime}}-t)}\Bigg{\\}}(f^{(2)}_{{\phi}NN}/f_{\phi}).$

Application of (19) to $F^{N}_{2v}(t)$ with $m=3$ and
$n=\rho^{\prime\prime},\rho^{\prime},\rho$ leads to

$F^{N}_{2v}(t)=\frac{1}{2}(\mu_{p}-\mu_{n}-1)\frac{m^{2}_{\rho^{\prime\prime}}m^{2}_{\rho^{\prime}}m^{2}_{\rho}}{(m^{2}_{\rho^{\prime\prime}}-t)(m^{2}_{\rho^{\prime}}-t)(m^{2}_{\rho}-t)}.$
(27)

The expressions (24)-(27) are automatically normalized and they govern the
asymptotic behaviors as predicted by the quark model of hadrons.

An unitarization of the model, i.e. an incorporation of the correct analytic
properties of the nucleon EM FFs, is achieved by the non-linear
transformations

$\displaystyle t$ $\displaystyle=$ $\displaystyle
t^{s}_{0}+\frac{4(t^{1s}_{in}-t^{s}_{0})}{[1/V(t)-V(t)]^{2}};\quad
t=t^{v}_{0}+\frac{4(t^{1v}_{in}-t^{v}_{0})}{[1/W(t)-W(t)]^{2}};$ (28)
$\displaystyle t$ $\displaystyle=$ $\displaystyle
t^{s}_{0}+\frac{4(t^{2s}_{in}-t^{s}_{0})}{[1/U(t)-U(t)]^{2}};\quad
t=t^{v}_{0}+\frac{4(t^{2v}_{in}-t^{v}_{0})}{[1/X(t)-X(t)]^{2}},$

respectively and a subsequent inclusion of the nonzero values of vector-meson
widths.

In non-linear transformations $t^{s}_{0}=9m^{2}_{\pi}$,
$t^{v}_{0}=4m^{2}_{\pi}$, $t^{1s}_{in},t^{1v}_{in},t^{2s}_{in},t^{2v}_{in}$
are the square-root branch points, as it is transparent from the inverse
transformations

$V(t)=i\frac{\sqrt{(\frac{t^{1s}_{in}-t^{s}_{0}}{t^{s}_{0}})^{1/2}+(\frac{t-t^{s}_{0}}{t^{s}_{0}})^{1/2}}-\sqrt{(\frac{t^{1s}_{in}-t^{s}_{0}}{t^{s}_{0}})^{1/2}-(\frac{t-t^{s}_{0}}{t^{s}_{0}})^{1/2}}}{\sqrt{(\frac{t^{1s}_{in}-t^{s}_{0}}{t^{s}_{0}})^{1/2}+(\frac{t-t^{s}_{0}}{t^{s}_{0}})^{1/2}}+\sqrt{(\frac{t^{1s}_{in}-t^{s}_{0}}{t^{s}_{0}})^{1/2}-(\frac{t-t^{s}_{0}}{t^{s}_{0}})^{1/2}}}$
(29)

and similarly for $W(t)$, $U(t)$, and $X(t)$, which map the corresponding
four-sheeted Riemann surfaces always into one $V-,W-,U-,X-$ plane.

Practically let us demonstrate the unitarization on the case of the Dirac iso-
scalar FF (24). The non-linear transformation
$t=t^{s}_{0}+\frac{4(t^{1s}_{in}-t^{s}_{0})}{[1/V(t)-V(t)]^{2}}$ implies also
the relations
$m^{2}_{r}=t^{s}_{0}+\frac{4(t^{1s}_{in}-t^{s}_{0})}{[1/V_{r0}-V_{r0}]^{2}}$
and $0=t^{s}_{0}+\frac{4(t^{1s}_{in}-t^{s}_{0})}{[1/V_{N}-V_{N}]^{2}}.$

Then every term

$\frac{m^{2}_{r}}{m^{2}_{r}-t}\equiv\frac{m^{2}_{r}-0}{m^{2}_{r}-t}=\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}\frac{(V_{N}-V_{r0})(V_{N}+V_{r0})(V_{N}-1/V_{r0})(V_{N}+1/V_{r0})}{(V-V_{r0})(V+V_{r0})(V-1/V_{r0})(V+1/V_{r0})}$
(30)

in (24) is factorized into asymptotic term and on the so-called finite-energy
term (for $\mid t\mid\to\infty$ it turns out to be a real constant) giving a
resonant behavior around $t=m^{2}_{r}$.

One can prove

  * 1.

if $m^{2}_{r}-\Gamma^{2}_{r}/4<t_{in}\Rightarrow V_{r0}=-V^{*}_{r0}$

  * 2.

if $m^{2}_{r}-\Gamma^{2}_{r}/4>t_{in}\Rightarrow V_{r0}=1/V^{*}_{r0}$

which lead in the case 1. to the expression

$\frac{m^{2}_{r}}{m^{2}_{r}-t}=\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}\frac{(V_{N}-V_{r0})(V_{N}-V^{*}_{r0})(V_{N}-1/V_{r0})(V_{N}-1/V^{*}_{r0})}{(V-V_{r0})(V-V^{*}_{r0})(V-1/V_{r0})(V-1/V^{*}_{r0})}$
(31)

and in the case 2. to the following expression

$\frac{m^{2}_{r}}{m^{2}_{r}-t}=\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}\frac{(V_{N}-V_{r0})(V_{N}-V^{*}_{r0})(V_{N}+V_{r0})(V_{N}+V^{*}_{r0})}{(V-V_{r0})(V-V^{*}_{r0})(V+V_{r0})(V+V^{*}_{r0})}.$
(32)

Lastly, introducing the non-zero width of the resonance by a substitution

$m^{2}_{r}\rightarrow(m_{r}-i\Gamma_{r}/2)^{2}$ (33)

i.e. simply one has to rid of ”0” in sub-indices of the previous two
expressions, one gets:

in the 1. case

$\displaystyle\frac{m^{2}_{r}}{m^{2}_{r}-t}$ $\displaystyle\rightarrow$
$\displaystyle\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}\frac{(V_{N}-V_{r})(V_{N}-V^{*}_{r})(V_{N}-1/V_{r})(V_{N}-1/V^{*}_{r})}{(V-V_{r})(V-V^{*}_{r})(V-1/V_{r})(V-1/V^{*}_{r})}=$
$\displaystyle=$
$\displaystyle\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}L_{r}(V)$

and in the 2. case

$\displaystyle\frac{m^{2}_{r}}{m^{2}_{r}-t}$ $\displaystyle\rightarrow$
$\displaystyle\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}\frac{(V_{N}-V_{r})(V_{N}-V^{*}_{r})(V_{N}+V_{r})(V_{N}+V^{*}_{r})}{(V-V_{r})(V-V^{*}_{r})(V+V_{r})(V+V^{*}_{r})}=$
$\displaystyle=$
$\displaystyle\bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\bigg{)}^{2}H_{r}(V).$

Then for every iso-scalar and iso-vector Dirac and Pauli FF one obtains just
one analytic and smooth from $-\infty$ to $+\infty$ function in the forms

$\displaystyle
F^{N}_{1s}[V(t)]=\Bigg{(}\frac{1-V^{2}}{1-V^{2}_{N}}\Bigg{)}^{4}\Bigg{\\{}\frac{1}{2}H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime\prime}}(V)+$
(36)
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(V)H_{\omega^{\prime}}(V)\frac{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega^{\prime}})}{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega^{\prime\prime}})}+H_{\omega^{\prime\prime}}(V)H_{\omega^{\prime}}(V)\frac{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\omega^{\prime}})}{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi^{\prime\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime\prime}}(V)\Bigg{]}(f^{(1)}_{\omega^{\prime}NN}/f_{\omega^{\prime}})+$
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(V)H_{\phi^{\prime}}(V)\frac{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\phi^{\prime}})}{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega^{\prime\prime}})}+H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime}}(V)\frac{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi^{\prime}})}{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi^{\prime\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime\prime}}(V)\Bigg{]}(f^{(1)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})+$
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(V)L_{\omega}(V)\frac{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega})}{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega^{\prime\prime}})}+H_{\omega^{\prime\prime}}(V)L_{\omega}(V)\frac{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\omega})}{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi^{\prime\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime\prime}}(V)\Bigg{]}(f^{(1)}_{\omega
NN}/f_{\omega})+$
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(V)L_{\phi}(V)\frac{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\phi})}{(C^{1s}_{\phi^{\prime\prime}}-C^{1s}_{\omega^{\prime\prime}})}+H_{\omega^{\prime\prime}}(V)L_{\phi}(V)\frac{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi})}{(C^{1s}_{\omega^{\prime\prime}}-C^{1s}_{\phi^{\prime\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(V)H_{\phi^{\prime\prime}}(V)\Bigg{]}(f^{(1)}_{\phi
NN}/f_{\phi})\Bigg{\\}}$

dependent on 5 free physically interpretable parameters,

$(f^{(1)}_{\omega^{\prime}NN}/f_{\omega^{\prime}})$,
$(f^{(1)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})$, $(f^{(1)}_{\omega
NN}/f_{\omega})$, $(f^{(1)}_{\phi NN}/f_{\phi})$, $t^{1s}_{in}$

$\displaystyle
F^{N}_{1v}[W(t)]=\Bigg{(}\frac{1-W^{2}}{1-W^{2}_{N}}\Bigg{)}^{4}\Bigg{\\{}\frac{1}{2}L_{\rho}(W)L_{\rho^{\prime}}(W)+$
(37)
$\displaystyle+\Bigg{[}L_{\rho^{\prime}}(W)L_{\rho^{\prime\prime}}(W)\frac{(C^{1v}_{\rho^{\prime}}-C^{1v}_{\rho^{\prime\prime}})}{(C^{1v}_{\rho^{\prime}}-C^{1v}_{\rho})}+L_{\rho}(W)L_{\rho^{\prime\prime}}(W)\frac{(C^{1v}_{\rho}-C^{1v}_{\rho^{\prime\prime}})}{(C^{1v}_{\rho}-C^{1v}_{\rho^{\prime}})}-$
$\displaystyle-L_{\rho}(W)L_{\rho^{\prime}}(W)\Bigg{]}(f^{(1)}_{\rho
NN}/f_{\rho})\Bigg{\\}}$

dependent on 2 free physically interpretable parameters $(f^{(1)}_{\rho
NN}/f_{\rho})$ and $t^{1v}_{in}$

$\displaystyle
F^{N}_{2s}[U(t)]=\Bigg{(}\frac{1-U^{2}}{1-U^{2}_{N}}\Bigg{)}^{6}\Bigg{\\{}\frac{1}{2}(\mu_{p}+\mu_{n}-1)H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)+$
(38)
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)H_{\phi^{\prime}}(U)\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi^{\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi^{\prime}})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\omega^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\omega^{\prime}}(U)H_{\phi^{\prime}}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi^{\prime}})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)H_{\phi^{\prime}}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime}})(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi^{\prime}})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)\Bigg{]}(f^{(2)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})+$
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)L_{\omega}(U)\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\omega})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\omega^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\omega^{\prime}}(U)L_{\omega}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\omega})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)L_{\omega}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega})(C^{2s}_{\phi^{\prime}}-C^{2s}_{\omega})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)\Bigg{]}(f^{(2)}_{\omega
NN}/f_{\omega})+$
$\displaystyle+\Bigg{[}H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)L_{\phi}(U)\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\omega^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\omega^{\prime}}(U)L_{\phi}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime\prime}})(C^{2s}_{\omega^{\prime}}-C^{2s}_{\phi^{\prime\prime}})}+$
$\displaystyle+H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)L_{\phi}(U)\frac{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi})(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi})}{(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}-$
$\displaystyle-
H_{\omega^{\prime\prime}}(U)H_{\phi^{\prime\prime}}(U)H_{\omega^{\prime}}(U)\Bigg{]}(f^{(2)}_{\phi
NN}/f_{\phi})\Bigg{\\}}$

dependent on 4 free physically interpretable parameters

$(f^{(2)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})$, $(f^{(2)}_{\omega
NN}/f_{\omega})$, $(f^{(2)}_{\phi NN}/f_{\phi}),t^{2s}_{in}$

$\displaystyle
F^{N}_{2v}[X(t)]=\Bigg{(}\frac{1-X^{2}}{1-X^{2}_{N}}\Bigg{)}^{6}\Bigg{\\{}\frac{1}{2}(\mu_{p}-\mu_{n}-1)L_{\rho}(U)L_{\rho^{\prime}}(U)H_{\rho^{\prime\prime}}(U)\Bigg{\\}}$
(39)

dependent on 1 free physically interpretable parameter $t^{2v}_{in}$, where

$\displaystyle
L_{r}(V)=\frac{(V_{N}-V_{r})(V_{N}-V^{*}_{r})(V_{N}-1/V_{r})(V_{N}-1/V^{*}_{r})}{(V-V_{r})(V-V^{*}_{r})(V-1/V_{r})(V-1/V^{*}_{r})},$
(40) $\displaystyle
C^{1s}_{r}=\frac{(V_{N}-V_{r})(V_{N}-V^{*}_{r})(V_{N}-1/V_{r})(V_{N}-1/V^{*}_{r})}{-(V_{r}-1/V_{r})(V_{r}-1/V^{*}_{r})},r=\omega,\phi$

$\displaystyle
H_{l}(V)=\frac{(V_{N}-V_{l})(V_{N}-V^{*}_{l})(V_{N}+V_{l})(V_{N}+V^{*}_{l})}{(V-V_{l})(V-V^{*}_{l})(V+V_{l})(V+V^{*}_{l})},$
(41) $\displaystyle
C^{1s}_{l}=\frac{(V_{N}-V_{l})(V_{N}-V^{*}_{l})(V_{N}+V_{l})(V_{N}+V^{*}_{l})}{-(V_{l}-1/V_{l})(V_{l}-1/V^{*}_{l})},l=\omega^{\prime\prime},\phi^{\prime\prime},\omega^{\prime},\phi^{\prime}$

$\displaystyle
L_{k}(W)=\frac{(W_{N}-W_{k})(W_{N}-W^{*}_{k})(W_{N}-1/W_{k})(W_{N}-1/W^{*}_{k})}{(W-W_{k})(W-W^{*}_{k})(W-1/W_{k})(W-1/W^{*}_{k})},$
(42) $\displaystyle
C^{1v}_{k}=\frac{(W_{N}-W_{k})(W_{N}-W^{*}_{k})(W_{N}-1/W_{k})(W_{N}-1/W^{*}_{k})}{-(W_{k}-1/W_{k})(W_{k}-1/W^{*}_{k})},k=\rho^{\prime\prime},\rho^{\prime},\rho$

$\displaystyle
L_{r}(U)=\frac{(U_{N}-U_{r})(U_{N}-U^{*}_{r})(U_{N}-1/U_{r})(U_{N}-1/U^{*}_{r})}{(U-U_{r})(U-U^{*}_{r})(U-1/U_{r})(U-1/U^{*}_{r})},$
(43) $\displaystyle
C^{2s}_{r}=\frac{(U_{N}-U_{r})(U_{N}-U^{*}_{r})(U_{N}-1/U_{r})(U_{N}-1/U^{*}_{r})}{-(U_{r}-1/U_{r})(U_{r}-1/U^{*}_{r})},r=\omega,\phi$

$\displaystyle
H_{l}(U)=\frac{(U_{N}-U_{l})(U_{N}-U^{*}_{l})(U_{N}+U_{l})(U_{N}+U^{*}_{l})}{(U-U_{l})(U-U^{*}_{l})(U+U_{l})(U+U^{*}_{l})},$
(44) $\displaystyle
C^{2s}_{l}=\frac{(U_{N}-U_{l})(U_{N}-U^{*}_{l})(U_{N}+U_{l})(U_{N}+U^{*}_{l})}{-(U_{l}-1/U_{l})(U_{l}-1/U^{*}_{l})},l=\omega^{\prime\prime},\phi^{\prime\prime},\omega^{\prime},\phi^{\prime}$

$\displaystyle
L_{k}(X)=\frac{(X_{N}-X_{k})(X_{N}-X^{*}_{k})(X_{N}-1/X_{k})(X_{N}-1/X^{*}_{k})}{(X-X_{k})(X-X^{*}_{k})(X-1/X_{k})(X-1/X^{*}_{k})},$
(45) $\displaystyle
C^{2v}_{k}=\frac{(X_{N}-X_{k})(X_{N}-X^{*}_{k})(X_{N}-1/X_{k})(X_{N}-1/X^{*}_{k})}{-(X_{k}-1/X_{k})(X_{k}-1/X^{*}_{k})},k=\rho^{\prime},\rho$

$\displaystyle
H_{\rho^{\prime\prime}}(X)=\frac{(X_{N}-X_{\rho^{\prime\prime}})(X_{N}-X^{*}_{\rho^{\prime\prime}})(X_{N}+X_{\rho^{\prime\prime}})(X_{N}+X^{*}_{\rho^{\prime\prime}})}{(X-X_{\rho^{\prime\prime}})(X-X^{*}_{\rho^{\prime\prime}})(X+X_{\rho^{\prime\prime}})(X+X^{*}_{\rho^{\prime\prime}})},$
(46) $\displaystyle
C^{2v}_{\rho^{\prime\prime}}=\frac{(X_{N}-X_{\rho^{\prime\prime}})(X_{N}-X^{*}_{\rho^{\prime\prime}})(X_{N}+X_{\rho^{\prime\prime}})(W_{X}+X^{*}_{\rho^{\prime\prime}})}{-(X_{\rho^{\prime\prime}}-1/X_{\rho^{\prime\prime}})(X_{\rho^{\prime\prime}}-1/X^{*}_{\rho^{\prime\prime}})}.$

since in a fitting procedure of all existing data by means of this version of
the $U\&A$ nucleon EM structure model simultaneously one finds

$\displaystyle(m^{2}_{\omega}-\Gamma^{2}_{\omega}/4)<t^{1s}_{in};(m^{2}_{\phi}-\Gamma^{2}_{\phi}/4)<t^{1s}_{in};$
(47)
$\displaystyle(m^{2}_{\omega^{\prime}}-\Gamma^{2}_{\omega^{\prime}}/4)>t^{1s}_{in};(m^{2}_{\phi^{\prime}}-\Gamma^{2}_{\phi^{\prime}}/4)>t^{1s}_{in};$
$\displaystyle(m^{2}_{\omega^{\prime\prime}}-\Gamma^{2}_{\omega^{\prime\prime}}/4)>t^{1s}_{in};(m^{2}_{\phi^{\prime\prime}}-\Gamma^{2}_{\phi^{\prime\prime}}/4)>t^{1s}_{in};$

$\displaystyle(m^{2}_{\rho}-\Gamma^{2}_{\rho}/4)<t^{1v}_{in};(m^{2}_{\rho^{\prime}}-\Gamma^{2}_{\rho^{\prime}}/4)<t^{1v}_{in};(m^{2}_{\rho^{\prime\prime}}-\Gamma^{2}_{\rho^{\prime\prime}}/4)<t^{1v}_{in};$
(48)

$\displaystyle(m^{2}_{\omega}-\Gamma^{2}_{\omega}/4)<t^{2s}_{in};(m^{2}_{\phi}-\Gamma^{2}_{\phi}/4)<t^{2s}_{in};$
(49)
$\displaystyle(m^{2}_{\omega^{\prime}}-\Gamma^{2}_{\omega^{\prime}}/4)>t^{2s}_{in};(m^{2}_{\phi^{\prime}}-\Gamma^{2}_{\phi^{\prime}}/4)>t^{2s}_{in};$
$\displaystyle(m^{2}_{\omega^{\prime\prime}}-\Gamma^{2}_{\omega^{\prime\prime}}/4)>t^{2s}_{in};(m^{2}_{\phi^{\prime\prime}}-\Gamma^{2}_{\phi^{\prime\prime}}/4)>t^{2s}_{in};$

$\displaystyle(m^{2}_{\rho}-\Gamma^{2}_{\rho}/4)<t^{2v}_{in};(m^{2}_{\rho^{\prime}}-\Gamma^{2}_{\rho^{\prime}}/4)<t^{2v}_{in};(m^{2}_{\rho^{\prime\prime}}-\Gamma^{2}_{\rho^{\prime\prime}}/4)>t^{2v}_{in}.$
(50)

##  IV Results of the analysis of existing nucleon EM FF data

We have collected 534 reliable experimental points on the nucleon EM structure
from more than 40 independent experiments as they are represented graphically
in Figs. 1-11. They have been analysed simultaneously by means of the 9
resonance $U\&A$ model of the nucleon EM structure as formulated in the
previous Section. The minimum of the $\chi^{2}$=2 214 has been achieved with
the values of 12 free parameters of the model with a clear physical meaning as
they are presented in Table 1.,

Table 1. The numerical values of the free parameters of the nucleon $U\&A$ EM
structure model, respecting SU(3) symmetry, formulated in the previous section

$\displaystyle t^{1s}_{in}$ $\displaystyle=$ $\displaystyle(1.0442\pm
0.0200)GeV^{2}$ $\displaystyle t^{2s}_{in}$ $\displaystyle=$
$\displaystyle(1.0460\pm 0.1399)GeV^{2}$ $\displaystyle t^{1v}_{in}$
$\displaystyle=$ $\displaystyle(2.9506\pm 0.5326)GeV^{2}$ $\displaystyle
t^{2v}_{in}$ $\displaystyle=$ $\displaystyle(2.3449\pm 0.7656)GeV^{2}$
$\displaystyle(f^{(1)}_{\omega NN}/f_{\omega})$ $\displaystyle=$
$\displaystyle(1.5717\pm 0.0022)$ $\displaystyle(f^{(1)}_{\phi NN}/f_{\phi})$
$\displaystyle=$ $\displaystyle(-1.1247\pm 0.0011)$
$\displaystyle(f^{(1)}_{\omega^{\prime}NN}/f_{\omega^{\prime}})$
$\displaystyle=$ $\displaystyle(0.0418\pm 0.0065)$
$\displaystyle(f^{(1)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})$ $\displaystyle=$
$\displaystyle(0.1879\pm 0.0010)$ $\displaystyle(f^{(2)}_{\omega
NN}/f_{\omega})$ $\displaystyle=$ $\displaystyle(-0.2096\pm 0.0067)$
$\displaystyle(f^{(2)}_{\phi NN}/f_{\phi})$ $\displaystyle=$
$\displaystyle(0.2657\pm 0.0067)$
$\displaystyle(f^{(2)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})$ $\displaystyle=$
$\displaystyle(0.1781\pm 0.0029)$ $\displaystyle(f^{(1)}_{\rho NN}/f_{\rho})$
$\displaystyle=$ $\displaystyle(0.3747\pm 0.0022)$

whereby the results are not very sensitive on the position of the effective
inelastic thresholds $t^{1s}_{in}$, $t^{2s}_{in}$, $t^{1v}_{in}$,
$t^{2v}_{in}$.

The corresponding description of the data by means of this $U\&A$ model with
numerical values of parameters given in Table 1. is graphically shown in Figs.
12-19.

Of course one could not expect that a description of such gigantic set of data
to be obtained from so much independent experiments, every of them to be
charged with corresponding statistical and systematical errors, will be
consistent with rules of a standard statistics. We have been able to reduce
the total $\chi^{2}$ on 522 degrees of freedom only to the value 4.24.

The results of the analysis can be summarized as follows:

  * •

a perfect description (see Fig. 12) of the most reliable nucleon EM structure
data, i.e. the data on the ratio $\mu_{p}G^{p}_{E}(t)/G^{p}_{M}(t)$ in the
space-like region to be obtained in polarization experiments, is achieved.

  * •

description of all other existing data (see Figs. 13-19) is quite reasonable
too, besides an inconsistency of the data on neutron EM FFs in the time-like
region with all other data on nucleon EM FFs, indicating that the total-cross
section of $e^{+}e^{-}\to n\bar{n}$ is considerably larger than it has been
found in FENICE experiment antonelli  in Frascati. So, we are coming to the
same conclusions as it was pointed out by one of the the authors in papers
dubnicka2 , dubnicka3  published more than 25 years ago.

  * •

again the existence of the zero of the proton electric FF $G^{p}_{E}(t)$
approximately at $t_{z}=-13$ GeV2 is confirmed, which has been predicted in
the paper adamuscin  for the first time.

  * •

electric and magnetic mean square charge radii of the nucleons are determined
to be $<r^{2}_{Ep}>=(0.7182\pm 0.0369)$ fm2, $<r^{2}_{Mp}>=(0.7573\pm 0.0133)$
fm2, $<r^{2}_{En}>=(-0.1162\pm 0.0369)$ fm2, $<r^{2}_{Mn}>=(0.8312\pm 0.0195)$
fm2, whereby the obtained electric root mean square charge radius of the
proton is consistent with the value $<r_{Ep}>=0.84184\pm 0.00067$ fm
determined in the muon hydrogen atom spectroscopy pohl  and in this way the
electric proton charge radius puzzle is definitely solved. Recently to the
same conclusions came also Ulf.-G.Meissner with collaborators in meissner .

  * •

electric mean square charge radius of the neutron is found to be almost
identical with the value given by Rev. of Part. Physics olive .

Figure 12: Prediction of proton electric to magnetic FFs ratio behavior in
space-like region by $U\&A$ model respecting $SU(3)$ symmetry and its
comparison with existing data

.

Figure 13: Prediction of proton electric FF behavior by $U\&A$ model
respecting $SU(3)$ symmetry and its comparison with existing data. Figure 14:
Prediction of proton magnetic FF behavior by $U\&A$ model respecting $SU(3)$
symmetry and its comparison with existing data. Figure 15: Prediction of the
absolute value of proton electric to magnetic FFs ratio behavior in time-like
region by $U\&A$ model respecting $SU(3)$ symmetry and its comparison with
existing data. Figure 16: Prediction of the absolute value of neutron electric
to magnetic FFs ratio behavior in time-like region by $U\&A$ model respecting
$SU(3)$ symmetry. Figure 17: Prediction of neutron electric FF behavior by
$U\&A$ model respecting $SU(3)$ symmetry and its comparison with existing the
data. Figure 18: Prediction of neutron magnetic FF behavior by $U\&A$ model
respecting $SU(3)$ symmetry and its comparison with existing data. Figure 19:
Prediction of neutron electric to magnetic FFs ratio behavior in space-like
region by $U\&A$ model respecting $SU(3)$ symmetry and its comparison with
existing data.

In Figs. 20-25 it is clearly demonstrated that the nucleon EM FFs represented
by the $U\&A$ model indeed fulfil the reality condition $G^{*}(t)=G(t^{*})$,
i.e. they all are real functions from $-\infty$ up to the lowest branch point
$t_{0}=4m^{2}_{\pi}$ on the positive real axis.

Figure 20: Prediction of real (solid line) and imaginary (dashed line) parts
of the proton electric FF by $U\&A$ model respecting $SU(3)$ symmetry. Figure
21: Prediction of real (solid line) and imaginary (dashed line) parts of the
proton magnetic FF by $U\&A$ model respecting $SU(3)$ symmetry. Figure 22:
Prediction of a phase difference of the proton electric and magnetic FFs by
$U\&A$ model respecting $SU(3)$ symmetry. Figure 23: Prediction of real (solid
line) and imaginary (dashed line) parts of the neutron electric FF by $U\&A$
model respecting $SU(3)$ symmetry. Figure 24: Prediction of real (solid line)
and imaginary (dashed line) parts of the neutron magnetic FF by $U\&A$ model
respecting $SU(3)$ symmetry. Figure 25: Prediction of a phase difference of
the neutron electric and magnetic FFs by $U\&A$ model respecting $SU(3)$
symmetry.

The imaginary parts of all nucleon EM FFs are different from zero just only
from the lowest branch point at $t=0.0784GeV^{2}$ to $+\infty$ and their
behaviors are given by the unitarity conditions of the corresponding FFs.

##  V Numerical values of the $f^{F}$, $f^{D}$, $f^{S}$ and $f^{F^{\prime}}$,
$f^{D^{\prime}}$, $f^{S^{\prime}}$ coupling constants

The $SU(3)$ invariant Lagrangian (2) of vector-meson nonet $\rho^{-}$,
$\rho^{0}$, $\rho^{+}$, $K^{*-}$, $K^{*0}$, $\bar{K}^{*0}$, $K^{*+}$,
$\omega$, $\phi$ with $1/2^{+}$ octet baryons $p$, $n$, $\Lambda$,
$\Sigma^{+}$, $\Sigma^{0}$, $\Sigma^{-}$, $\Xi^{0}$, $\Xi^{-}$ provides the
following expressions

$\displaystyle f^{(1)}_{\rho NN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(f^{D}_{1}+f^{F}_{1})$ (51) $\displaystyle
f^{(1)}_{\omega NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S}_{1}\cos\theta-\frac{1}{2\sqrt{3}}(3f^{F}_{1}-f^{D}_{1})\sin\theta$
$\displaystyle f^{(1)}_{\phi NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S}_{1}\sin\theta+\frac{1}{2\sqrt{3}}(3f^{F}_{1}-f^{D}_{1})\cos\theta$

and

$\displaystyle f^{(2)}_{\rho NN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(f^{D}_{2}+f^{F}_{2})$ (52) $\displaystyle
f^{(2)}_{\omega NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S}_{2}\cos\theta-\frac{1}{2\sqrt{3}}(3f^{F}_{2}-f^{D}_{2})\sin\theta$
$\displaystyle f^{(2)}_{\phi NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S}_{2}\sin\theta+\frac{1}{2\sqrt{3}}(3f^{F}_{2}-f^{D}_{2})\cos\theta$

where angle $\theta=43.8^{0}$ and it is given by the Gell-Mann-Okubo quadratic
mass formula

$\displaystyle
m^{2}_{\phi(1020)}\cos^{2}\theta+m^{2}_{\omega(782)}\sin^{2}\theta=\frac{4m^{2}_{K^{*}(892)}-m^{2}_{\rho(770)}}{3}.$
(53)

From the $SU(3)$ invariant Lagrangian of the first excited vector-meson nonet
$\rho^{-^{\prime}}$, $\rho^{0^{\prime}}$, $\rho^{+^{\prime}}$,
$K^{*-^{\prime}}$, $K^{*0^{\prime}}$, $\bar{K}^{*0^{\prime}}$,
$K^{*+^{\prime}}$, $\omega^{\prime}$, $\phi^{\prime}$ with $1/2^{+}$ octet
baryons $p$, $n$, $\Lambda$, $\Sigma^{+}$, $\Sigma^{0}$, $\Sigma^{-}$,
$\Xi^{0}$, $\Xi^{-}$

$\displaystyle
L_{V^{\prime}B\bar{B}}=\frac{i}{\sqrt{2}}f^{F}[\bar{B}^{\alpha}_{\beta}\gamma_{\mu}B^{\beta}_{\gamma}-\bar{B}^{\beta}_{\gamma}\gamma_{\mu}B^{\alpha}_{\beta}](V^{\prime}_{\mu})^{\gamma}_{\alpha}+$
(54)
$\displaystyle+\frac{i}{\sqrt{2}}f^{D}[\bar{B}^{\beta}_{\gamma}\gamma_{\mu}B^{\alpha}_{\beta}+\bar{B}^{\alpha}_{\gamma}\gamma_{\mu}B^{\beta}_{\gamma}](V^{\prime}_{\mu})^{\gamma}_{\alpha}+$
$\displaystyle+\frac{i}{\sqrt{2}}f^{S}\bar{B}^{\alpha}_{\beta}\gamma_{\mu}B^{\beta}_{\alpha}\omega^{0^{\prime}}_{\mu}$

another two systems of expressions

$\displaystyle f^{(1)}_{\rho^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(f^{D^{\prime}}_{1}+f^{F^{\prime}}_{1})$ (55)
$\displaystyle f^{(1)}_{\omega^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S^{\prime}}_{1}\cos\theta^{\prime}-\frac{1}{2\sqrt{3}}(3f^{F^{\prime}}_{1}-f^{D^{\prime}}_{1})\sin\theta^{\prime}$
$\displaystyle f^{(1)}_{\phi^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S^{\prime}}_{1}\sin\theta^{\prime}+\frac{1}{2\sqrt{3}}(3f^{F^{\prime}}_{1}-f^{D^{\prime}}_{1})\cos\theta^{\prime}$

and

$\displaystyle f^{(2)}_{\rho^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(f^{D^{\prime}}_{2}+f^{F^{\prime}}_{2})$ (56)
$\displaystyle f^{(2)}_{\omega^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S^{\prime}}_{2}\cos\theta^{\prime}-\frac{1}{2\sqrt{3}}(3f^{F^{\prime}}_{2}-f^{D^{\prime}}_{2})\sin\theta^{\prime}$
$\displaystyle f^{(2)}_{\phi^{\prime}NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}f^{S^{\prime}}_{2}\sin\theta^{\prime}+\frac{1}{2\sqrt{3}}(3f^{F^{\prime}}_{2}-f^{D^{\prime}}_{2})\cos\theta^{\prime}$

are obtained, where angle $\theta^{\prime}=50.3^{0}$ and it is again given by
the Gell-Mann-Okubo quadratic mass formula

$\displaystyle
m^{2}_{\phi^{\prime}(1680)}\cos^{2}\theta^{\prime}+m^{2}_{\omega^{\prime}(1420)}\sin^{2}\theta^{\prime}=\frac{4m^{2}_{K^{*^{\prime}}(1680)}-m^{2}_{\rho^{\prime}(1450)}}{3},$
(57)

however, with the first excited states of the corresponding particles

So, if one knows numerical values of the coupling constants on the left-hand
side of the expressions (51),(52) and (55),(56) one can find numerical values
of all $f^{F}$, $f^{D}$, $f^{S}$ and $f^{F^{\prime}}$, $f^{D^{\prime}}$,
$f^{S^{\prime}}$ coupling constants in both $SU(3)$ invariant Lagrangians (I)
and (54), needed for a prediction of EM FFs behaviors of all $1/2$ octet
hyperons $\Lambda$, $\Sigma^{+}$, $\Sigma^{0}$, $\Sigma^{-}$, $\Xi^{0}$,
$\Xi^{-}$.

For (51) and (52) it is straightforward to determine them from the numerical
values in Table 1., besides $(f^{(2)}_{\rho NN}/f_{\rho})$ which is calculated
by means of the relation

$\displaystyle(f^{(2)}_{\rho
NN}/f_{\rho})=\frac{1}{2}(\mu_{p}-\mu_{n}-1)\frac{C^{2v}_{\rho^{\prime\prime}}C^{2v}_{\rho^{\prime}}}{(C^{2v}_{\rho^{\prime}}-C^{2v}_{\rho})(C^{2v}_{\rho^{\prime\prime}}-C^{2v}_{\rho})}=2.8956,$
(58)

and the values $f_{\rho}=4.9582$, $f_{\omega}=17.0620$, $f_{\phi}=-13.4428$ to
be calculated from existing data olive  on lepton width $\Gamma(V\to
e^{+}e^{-})$ by means of the relation $\Gamma(V\to
e^{+}e^{-})=\frac{\alpha^{2}m_{V}}{3}{(\frac{f^{2}_{V}}{4\pi})}^{-1}$.

The results are

$\displaystyle f^{(1)}_{\rho NN}$ $\displaystyle=$ $\displaystyle 1.8578$ (59)
$\displaystyle f^{(1)}_{\omega NN}$ $\displaystyle=$ $\displaystyle 26.8163$
$\displaystyle f^{(1)}_{\phi NN}$ $\displaystyle=$ $\displaystyle 15.1191$

and

$\displaystyle f^{(2)}_{\rho NN}$ $\displaystyle=$ $\displaystyle 14.3570$
(60) $\displaystyle f^{(2)}_{\omega NN}$ $\displaystyle=$
$\displaystyle-3.5762$ $\displaystyle f^{(2)}_{\phi NN}$ $\displaystyle=$
$\displaystyle-3.5718$

However, for excited vector meson coupling constants in (55) and (56) a big
problem appeared. There are no data (see olive ) on $\Gamma(V\to e^{+}e^{-})$
for $\omega^{\prime}(1420),\rho^{\prime}(1450),\phi^{\prime}(1680)$ in order
to determine $f_{\rho^{\prime}}$, $f_{\omega^{\prime}}$ and
$f_{\phi^{\prime}}$. The first two are calculated from lepton widths estimated
by Donnachie and Clegg donnachieclegg  $f_{\rho^{\prime}}=13.6491$,
$f_{\omega^{\prime}}=47.6022$ and the third one for $\phi^{\prime}$ is
determined to be $f_{\phi^{\prime}}=-33.6598$ from the relations
$f^{2}_{\rho^{\prime}}:f^{2}_{\omega^{\prime}}:f^{2}_{\phi^{\prime}}=\frac{1}{9}:1:\frac{1}{2}$
following (see bramon ) from the quark structure of the corresponding vector
mesons and the electric charges of the three constituent quarks from which
these vector mesons are compound.

Having numerical values of
$f_{\rho^{\prime}},f_{\omega^{\prime}},f_{\phi^{\prime}}$ and calculating
missing in Table 1. coupling constant ratios for excited vector mesons by the
relations

$\displaystyle(f^{(1)}_{\rho^{\prime}NN}/f_{\rho^{\prime}})=\frac{1}{2}\frac{C^{1v}_{\rho^{\prime\prime}}}{(C^{1v}_{\rho^{\prime\prime}}-C^{1v}_{\rho^{\prime}})}-\frac{(C^{1v}_{\rho^{\prime\prime}}-C^{1v}_{\rho})}{(C^{1v}_{\rho^{\prime\prime}}-C^{1v}_{\rho^{\prime}})}(f^{(1)}_{\rho
NN}/f_{\rho})=0.7635$ (61)

$\displaystyle(f^{(2)}_{\rho^{\prime}NN}/f_{\rho^{\prime}})=-\frac{1}{2}(\mu_{p}-\mu_{n}-1)\frac{C^{2v}_{\rho^{\prime\prime}}C^{2v}_{\rho}}{(C^{2v}_{\rho^{\prime\prime}}-C^{2v}_{\rho^{\prime}})(C^{2v}_{\rho^{\prime}}-C^{2v}_{\rho})}=-1.3086,$
(62)

$\displaystyle(f^{(2)}_{\omega^{\prime}NN}/f_{\omega^{\prime}})=\frac{1}{2}(\mu_{p}+\mu_{n}-1)\frac{C^{2s}_{\omega^{\prime\prime}}C^{2s}_{\phi^{\prime\prime}}}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}-$
(63)
$\displaystyle-\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}(f^{(2)}_{\omega
NN}/f_{\omega})-$
$\displaystyle-\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}(f^{(2)}_{\phi
NN}/f_{\phi})-$
$\displaystyle-\frac{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\phi^{\prime}})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\phi^{\prime}})}{(C^{2s}_{\phi^{\prime\prime}}-C^{2s}_{\omega^{\prime}})(C^{2s}_{\omega^{\prime\prime}}-C^{2s}_{\omega^{\prime}})}(f^{(2)}_{\phi^{\prime}NN}/f_{\phi^{\prime}})=-0.5771$

one obtains the results

$\displaystyle f^{(1)}_{\rho^{\prime}NN}$ $\displaystyle=$ $\displaystyle
10.4211$ (64) $\displaystyle f^{(1)}_{\omega^{\prime}NN}$ $\displaystyle=$
$\displaystyle 1.9900$ $\displaystyle f^{(1)}_{\phi^{\prime}NN}$
$\displaystyle=$ $\displaystyle-6.3247$

and

$\displaystyle f^{(2)}_{\rho^{\prime}NN}$ $\displaystyle=$
$\displaystyle-17.8612$ (65) $\displaystyle f^{(2)}_{\omega^{\prime}NN}$
$\displaystyle=$ $\displaystyle-27.4712$ $\displaystyle
f^{(2)}_{\phi^{\prime}NN}$ $\displaystyle=$ $\displaystyle-5.9948$

Now, by a solution of the systems of algebraic equations (51), (52), (55) and
(56) according to three unknown constants $f^{F},f^{D},f^{S}$ one comes to the
following expressions

$\displaystyle f_{1}^{F}=\frac{1}{2}\left[\sqrt{3}\left(f_{\phi
NN}^{(1)}\cos\theta-f_{\omega NN}^{(1)}\sin\theta\right)+f_{\rho
NN}^{(1)}\right]$ (66) $\displaystyle f_{1}^{D}=\frac{1}{2}\left[3f_{\rho
NN}^{(1)}-\sqrt{3}\left(f_{\phi NN}^{(1)}\cos\theta-f_{\omega
NN}^{(1)}\sin\theta\right)\right]$ $\displaystyle
f_{1}^{S}=\sqrt{2}\left(f_{\omega NN}^{(1)}\cos\theta+f_{\phi
NN}^{(1)}\sin\theta\right)$

$\displaystyle f_{2}^{F}=\frac{1}{2}\left[\sqrt{3}\left(f_{\phi
NN}^{(2)}\cos\theta-f_{\omega NN}^{(2)}\sin\theta\right)+f_{\rho
NN}^{(2)}\right]$ (67) $\displaystyle f_{2}^{D}=\frac{1}{2}\left[3f_{\rho
NN}^{(2)}-\sqrt{3}\left(f_{\phi NN}^{(2)}\cos\theta-f_{\omega
NN}^{(2)}\sin\theta\right)\right]$ $\displaystyle
f_{2}^{S}=\sqrt{2}\left(f_{\omega NN}^{(2)}\cos\theta+f_{\phi
NN}^{(2)}\sin\theta\right)$

$\displaystyle
f_{1}^{F^{\prime}}=\frac{1}{2}\left[\sqrt{3}\left(f_{\phi^{\prime}NN}^{(1)}\cos\theta^{\prime}-f_{\omega^{\prime}NN}^{(1)}\sin\theta^{\prime}\right)+f_{\rho^{\prime}NN}^{(1)}\right]$
(68) $\displaystyle
f_{1}^{D^{\prime}}=\frac{1}{2}\left[3f_{\rho^{\prime}NN}^{(1)}-\sqrt{3}\left(f_{\phi^{\prime}NN}^{(1)}\cos\theta^{\prime}-f_{\omega^{\prime}NN}^{(1)}\sin\theta^{\prime}\right)\right]$
$\displaystyle
f_{1}^{S^{\prime}}=\sqrt{2}\left(f_{\omega^{\prime}NN}^{(1)}\cos\theta^{\prime}+f_{\phi^{\prime}NN}^{(1)}\sin\theta^{\prime}\right)$

$\displaystyle
f_{2}^{F^{\prime}}=\frac{1}{2}\left[\sqrt{3}\left(f_{\phi^{\prime}NN}^{(2)}\cos\theta^{\prime}-f_{\omega^{\prime}NN}^{(2)}\sin\theta^{\prime}\right)+f_{\rho^{\prime}NN}^{(2)}\right]$
(69) $\displaystyle
f_{2}^{D^{\prime}}=\frac{1}{2}\left[3f_{\rho^{\prime}NN}^{(2)}-\sqrt{3}\left(f_{\phi^{\prime}NN}^{(2)}\cos\theta^{\prime}-f_{\omega^{\prime}NN}^{(2)}\sin\theta^{\prime}\right)\right]$
$\displaystyle
f_{2}^{S^{\prime}}=\sqrt{2}\left(f_{\omega^{\prime}NN}^{(2)}\cos\theta^{\prime}+f_{\phi^{\prime}NN}^{(2)}\sin\theta^{\prime}\right)$

and by a substitution of numerical values (59), (60), (64), (65),
respectively, one finds the numerical values

$\displaystyle f_{1}^{F}=-5.69470$ (70) $\displaystyle f_{1}^{D}=9.4103$
$\displaystyle f_{1}^{S}=42.1706$

$\displaystyle f_{2}^{F}=7.08952$ (71) $\displaystyle f_{2}^{D}=21.6245$
$\displaystyle f_{2}^{S}=-7.1465$

$\displaystyle f_{1}^{F^{\prime}}=0.38580$ (72) $\displaystyle
f_{1}^{D^{\prime}}=20.45639$ $\displaystyle f_{1}^{S^{\prime}}=-5.0842$

$\displaystyle f_{2}^{F^{\prime}}=6.0577$ (73) $\displaystyle
f_{2}^{D^{\prime}}=-41.7801$ $\displaystyle f_{2}^{S^{\prime}}=-31.3390$

of all scrutinized coupling constants in SU(3) invariant interaction
Lagrangians of vector-meson nonets with $1/2^{+}$ octet baryons, which allow
us to predict behaviors of EM FFs of all $1/2^{+}$ octet hyperons in space-
like and time-like regions, including their real and imaginary parts, phase
differences of electric and magnetic FFs and corresponding differential and
total cross-sections, in which EM structure of hyperons is planned to be
measured. Predictions on all above mentioned quantities will be given in the
next paper which will be published elsewhere.

##  VI Conclusions

All existing 11 sets of data on nucleon EM FFs in space-like and time-like
region from more than 40 different experiments has been analysed, from which
534 reliable experimental points are here reasonably described by 9 resonance
$U\&A$ model of nucleon EM structure, which respects the SU(3) symmetry and
OZI rule violation. The analysis revealed the following results:

  * •

an existence of the zero of $G_{Ep}(t)$ around $t_{z}=13\;\rm{GeV}^{2}$ is
again confirmed

  * •

the value of the proton charge rms radius coincides with the value obtained in
the muon hydrogen atom spectroscopy experiment and in this way the existing
puzzle is definitively removed

  * •

the value of the neutron charge mean squared radius is identical with the
value given by Rev. of Part. Physics

  * •

the coupling constants in the SU(3) invariant Lagrangian of the vector meson
nonet interaction with $1/2^{+}$ octet baryons $f^{F},f^{D},f^{S}$ are
determined numerically, which allow to predict all quantities describing the
EM structure of $1/2^{+}$ octet hyperons.

The work was supported by Slovak Grant Agency for Science VEGA, gr. No.
1/0158/13 and by Slovak Research and Development Agency APVV, gr. No.
APVV-0163-12.

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