Comprehensive Insight into Regular Damped Oscillatory Structures from Effective Electromagnetic Form Factor Data of Some Mesons and Nucleons

Abstract

Regular damped oscillatory structures from the “effective” electromagnetic form factors of the hadrons $h={\pi }^{\pm },K^{\pm },K^0,p,n$ were investigated. The “effective” electromagnetic form factor behaviors were calculated from the experimental data on the total cross-sections ${\sigma }_{tot}(e^{+}{e}^{-}\to h\overline{h})$ with errors. The apparent oscillations were observed for the first time for the proton, and we show, also taking other hadrons into consideration, that they are an arbitrary artifact resulting from a very simplistic theoretical description based on an elementary three-parameter model. If the data are described by a more appropriate and physically well-founded Unitary and Analytic model, then the oscillations disappear. In spite of this, if the three-parameter model is used to describe the “effective” electromagnetic form factor data, an interesting phenomenon is observed. The oscillations are opposite for particles which form an isospin doublet. By using the physically well-founded Unitary and Analytic model, it is demonstrated that this feature originates from the special transformation properties of the electromagnetic current of the corresponding particles in the isotopic space.

1. Introduction

The electromagnetic (EM) structure of any hadron is completely described by the EM form factors (FFs), the number of which depends on the spin of the considered hadron. Their behaviors in the timelike region can also be obtained from the measured total cross-sections ${\sigma }_{tot}(e^{+}{e}^{-}\to h\overline{h})$. However, the procedure is not straightforward and depends on the spin of a considered particle.

There is an essential difference between the charged pion, the charged and neutral K-mesons with the spin “0”, the proton and neutron with the spin “1/2”, and the deuteron with the spin “1”. The EM structure of the mesons with the spin “0” is completely described by one EM FF, and in the case of the octet of baryons, like the proton and neutron, the complete description of their EM structure requires two different EM FFs, the electric $G_E\left(s\right)$ and the magnetic $G_M\left(s\right)$. In the case of the deuteron, three FFs are needed, the electric, the magnetic and the quadrupole FF.

If several form factors are involved, their realistic value cannot be calculated from only one value of the measured total cross-section ${\sigma }_{tot}(e^{+}{e}^{-}\to h\overline{h})$ at the concrete total c.m. energy squared “s”.

Almost one decade ago, a new phenomenon appeared [1] in elementary particle physics, the so-called “regular damped oscillatory structures” (RDOSs) from the ”effective” proton EM FF data obtained from the data on the total cross-section ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$. After a couple of years, new data on the process $e^{+}{e}^{-}\to n\overline{n}$ with neutrons were measured [2] in a rather broad region of energies, and in this case, RDOSs were revealed but with the opposite behavior. There are conjectures [3,4] that the origin of RDOSs is in the quark gluon structure of the protons and neutrons.

The latter prompted us to also investigate RDOSs from existing data on the EM structure of the charged pion [5] and the charged and neutral K-mesons [6], as they are also compounds of the quarks and gluons. We did not investigate the RDOSs of the deuteron in the six-quark state here, as the experimental data on the total cross-section ${\sigma }_{tot}(e^{+}{e}^{-}\to d\overline{d})$ are still unavailable.

Further, investigations of damped oscillatory regular structures from the “effective” electromagnetic form factors of the hadrons $h={\pi }^{\pm },K^{\pm },K^0,p,n$ were carried out from a uniform point of view.

We started with the investigations of the proton and neutron with 1/2 spin, which is completely described by two EM FFs, the electric $G_E^N\left(s\right)$ and the magnetic $G_M^N\left(s\right)$. In this case, one is unable to determine the values of these FFs from one value of the measured total cross-section

$$\begin{array}{c}\hfill {\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})={\displaystyle \frac{4\pi {\alpha }^2{C}_p{\beta }_p\left(s\right)}{3s}}\left[|G_M^p{\left(s\right)|}^2+{\displaystyle \frac{2m_p^2}{s}}{\left|G_E^p\left(s\right)\right|}^2\right],\end{array}$$

(1)

or

$$\begin{array}{c}\hfill {\sigma }_{tot}(e^{+}{e}^{-}\to n\overline{n})={\displaystyle \frac{4\pi {\alpha }^2{\beta }_n\left(s\right)}{3s}}\left[|G_M^n{\left(s\right)|}^2+{\displaystyle \frac{2m_n^2}{s}}{|G_E^n\left(s\right)|}^2\right],\end{array}$$

(2)

where ${\beta }_N\left(s\right)=\sqrt{1-{\displaystyle \frac{4m_N^2}{s}}}$ is the velocity of the outgoing nucleon in the c.m. system, $\alpha$ = 1/137 and $C_p={\displaystyle \frac{\pi \alpha /{\beta }_p\left(s\right)}{1-exp(-\pi \alpha /{\beta }_p\left(s\right))}}$ is the so-called Sommerfeld–Gamov–Sakharov Coulomb enhancement factor [7]. Therefore, the new concept of the “effective” EM FF of the nucleons

$$\begin{array}{c}\hfill G_{eff}^p\left(s\right)=\sqrt{{\displaystyle \frac{{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to p\overline{p})}{\frac{4\pi {\alpha }^2{C}_p{\beta }_p\left(s\right)}{3s}\left(1+\frac{2m_p^2}{s}\right)}}},\end{array}$$

(3)

and

$$\begin{array}{c}\hfill G_{eff}^n\left(s\right)=\sqrt{{\displaystyle \frac{{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to n\overline{n})}{\frac{4\pi {\alpha }^2{\beta }_n\left(s\right)}{3s}\left(1+\frac{2m_n^2}{s}\right)}}},\end{array}$$

(4)

has been introduced by a few of experimental groups in a somewhat unnatural way by the requirement of equality $G_E^N\left(s\right)=G_M^N\left(s\right)$, which is exactly valid only at the threshold, for all “s” up to $+\infty$, with the hope of obtaining at least some information on the EM structure of the investigated object.

2. Regular Damped Oscillatory Structures from the “Effective” Proton EM FF

Immediately after publishing the first “effective” proton EM FF data [8,9], obtained by the relation (3) from the measured total cross-section (1) by the initial state radiation (ISR) technique, the authors of the paper [1] described them with the three-parameter formula [10]

$$\begin{array}{c}\hfill G_{eff}\left(s\right)={\displaystyle \frac{A}{(1+\frac{s}{m_a^2}){(1-\frac{s}{0.71})}^2}}.\end{array}$$

(5)

Then, by a subtraction of the fitted curve from these data, taking errors into account, they revealed the RDOSs for the first time.

We have repeated the latter procedure, collecting all existing data on the “effective” proton FF [8,9,11,12,13,14], as presented in the left side of Figure 1a, and describing them by the three-parameter Formula (5) (see Figure 1b right) with parameter values $A=9.02\pm 0.40$ and $m_a^2=8.52\pm 0.97$ GeV² and ${\chi }_1^2$/ndf = 4.61.

Finally, subtracting the fitted curve in the right side of Figure 1b from the proton’s “effective” EM FF data with errors, RDOSs are confirmed more expressively in Figure 2, as they were first revealed in [1].

3. Regular Damped Oscillatory Structures from the Neutron “Effective” EM FF

After publishing the first neutron “effective” EM FF data [2] in the left side of Figure 3, obtained by the relation (4) from the measured total cross-section (2), our description of these data by the three-parametric Formula (5) of [10] has not been successful.

In the description of the neutron “effective” FF data from the paper [2] by the three-parameter function (5) of the authors [10], the parameter $m_a^2$ has been growing to boundless values, indicating its unimportant role for a satisfactory description of the data under consideration. Therefore, we have excluded the term $(1+s/m_a^2)$ from (5) and in the remaining expression only the parameters A(1) = A, and A(3) = 0.71 have been left to be free in the fitting procedure. Then, the values $A\left(1\right)=0.0698\pm 0.0060$ and $A\left(3\right)=2.6079\pm 0.0493$ GeV² lead to the description of the data on the neutron “effective” EM FF data in the left side of Figure 3a with ${\chi }^2/ndf$ = 482/16, as presented by the curve in the right side of Figure 3b.

Then, the subtraction of the fitted curve from the data in the left side of Figure 3a has revealed the RDOSs from the neutron “effective” EM FF data, as presented in Figure 4. It indicates just the opposite behavior to the RDOSs from the proton “effective” EM FF data in Figure 2, e.g., the peaks at $4.5$ GeV² and 6 GeV² are in the opposite directions. This phenomenon seems to be interesting and it will be elucidated later on.

4. Search for Regular Damped Oscillatory Structures from Charged Pion EM FF Data

The charged pion EM FF $F_{\pi }^c\left(s\right)$ data in the timelike region with errors can be in principle calculated from the measured total cross-section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-})$. In this procedure, no unphysical demands are needed (contrary to the the nucleon “effective” FF data) because there is only one function $|F_{\pi }^c\left(s\right)|$ completely describing the measured total cross-section, which can be identified with the pion “effective” EM FF.

However, another problem appears here. The isovector charged pion EM FF $F_{\pi }^c\left(s\right)$, represented by the $\gamma {\pi }^{+}{\pi }^{-}$ vertex, is generated by the strong interactions, but not all of the ${\pi }^{+}{\pi }^{-}$ pairs in the measured total cross-section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-})$ have a strong interaction origin. Some portion of them is generated by the electromagnetic isospin violating decay of $\omega \left(782\right)\to {\pi }^{+}{\pi }^{-}$ and also by the $m_d$-$m_u$ quark mass difference, creating a deformation of the right wing of the $\rho \left(770\right)$ meson peak, to be known as the $\rho -\omega$ interference effect. Since we are searching for damped oscillatory regular structures from the charged pion EM FF, one has to eliminate the contribution of $\omega \left(782\right)\to {\pi }^{+}{\pi }^{-}$, which cannot be achieved by experimental physicists.

For the measurements [15,16,17] of the total cross-section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\left(\gamma \right))$, the following procedure to eliminate the $\omega \left(782\right)\to {\pi }^{+}{\pi }^{-}$ contribution has been carried out [5].

The total cross-section of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ process is expressed by the sum of $F_{\pi }^c\left(s\right)$ and the $\omega \left(782\right)\to {\pi }^{+}{\pi }^{-}$ decay contribution (further denoted by $F_{\pi }^{\prime }\left(s\right)$) in the form

$${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-})={\displaystyle \frac{\pi {\alpha }^2{C}_{{\pi }^c}{\beta }_{\pi }^3\left(s\right)}{3s}}|F_{\pi }^c\left(s\right)+R{e}^{i\varphi }{\displaystyle \frac{m_{\omega }^2}{m_{\omega }^2-s-i{m}_{\omega }{\Gamma }_{\omega }}}{|}^2,$$

(6)

where $F_{\pi }^c\left(s\right)$ is the isovector charged pion EM FF. The latter can be expressed by the Unitary and Analytic (U&A) model given by the formula (3.66) from [18].

$$\begin{array}{c}\hfill F_{\pi }^{th}\left[W\left(s\right)\right]={({\displaystyle \frac{1-W^2}{1-W_N^2}})}^2{\displaystyle \frac{(W-W_Z)(W_N-W_P)}{(W_N-W_Z)(W-W_P)}}\\ \hfill \times [{\displaystyle \frac{(W_N-W_{\rho })(W_N-W_{\rho }^{*})(W_N-1/W_{\rho })(W_N-1/W_{\rho }^{*})}{(W-W_{\rho })(W-W_{\rho }^{*})(W-1/W_{\rho })(W-1/W_{\rho }^{*})}}({\displaystyle \frac{f_{\rho \pi \pi }}{f_{\rho }}})\\ \hfill +\sum _{v={\rho }^{\prime },{\rho }^{″},{\rho }^{‴}}{\displaystyle \frac{(W_N-W_v)(W_N-W_v^{*})(W_N+W_v)(W_N+W_v^{*})}{(W-W_v)(W-W_v^{*})(W+W_v)(W+W_v^{*})}}({\displaystyle \frac{f_{v\pi \pi }}{f_v}})].\end{array}$$

(7)

The latter respects all known properties of the isovector EM FF of the charged pion: the analytic properties with two square-root-type branch points. The first one at $s_0=4m_{\pi }^2$ and the second one is $s_{in}$, which effectively represents all higher contributions from inelastic processes. This branch point is left to be a free parameter of the model, numerically fixed in a fitting procedure of existing data.

$$W\left(s\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+{\left(\frac{s-s_0}{s_0}\right)}^{1/2}}-\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-{\left(\frac{s-s_0}{s_0}\right)}^{1/2}}}{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+{\left(\frac{s-s_0}{s_0}\right)}^{1/2}}+\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-{\left(\frac{s-s_0}{s_0}\right)}^{1/2}}}}$$

(8)

is the conformal mapping of the four-sheeted Riemann surface in the s variable into one W-plane,

$$W_N=W\left(0\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+i}-\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-i}}{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+i}+\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-i}}}$$

(9)

is the normalization point in the W-plane, and finally,

$$W_v=W\left(s_v\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+{\left(\frac{s_v-s_0}{s_0}\right)}^{1/2}}-\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-{\left(\frac{s_v-s_0}{s_0}\right)}^{1/2}}}{\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}+{\left(\frac{s_v-s_0}{s_0}\right)}^{1/2}}+\sqrt{{\left(\frac{s_{in}-s_0}{s_0}\right)}^{1/2}-{\left(\frac{s_v-s_0}{s_0}\right)}^{1/2}}}}$$

(10)

are pole positions generated by all isovector vector meson resonances forming the U&A model of the charged pion EM structure (7).

Requirement of the normalization of the model to the electric charge reduces the number of free coupling constant ratios $(f_{v\pi \pi }/f_v)$ in (7). The isovector nature of $F_{\pi }^c\left(s\right)$ implies that only the rho-meson and its excited states $\rho \left(770\right),{\rho }^{\prime }\left(1450\right),{\rho }^{″}\left(1700\right),{\rho }^{‴}\left(2150\right)$ [19] (as revealed in [20]) contribute to the FF behavior to cover the energetic region of the data up to 9 GeV². The fulfillment of the reality condition $F_{\pi }^{*}\left(s\right)=F_{\pi }\left(s^{*}\right)$ leads to the appearance of two complex conjugate rho-meson poles on unphysical sheets. The behavior on the left-hand cut of the second Riemann sheet given by the analytic continuation of the elastic FF unitarity condition [21] is approximated using a Padé approximant with one pole $W_P$ and one zero $W_Z$, which are considered as free parameters.

In (6), $\varphi =arctan{\displaystyle \frac{m_{\omega }{\Gamma }_{\omega }}{m_{\rho }^2-m_{\omega }^2}}$ is the $\rho -\omega$ interference phase and R is the $\rho -\omega$ interference amplitude to be real.

The optimal parameter values have been found in the analysis of existing data [15,16,17] on $|F_{\pi }^{\prime }{\left(s\right)|}^2$ and the results are presented in

Table 1. Parameter values of the analysis of data in [15,16,17] with minimum of ${\chi }^2/ndf=0.988$.

$s_{in}=1.2730\pm 0.0130$ GeV2	$m_{\rho }=0.7620\pm 0.0080$ GeV	${\Gamma }_{\rho }=0.1442\pm 0.0014$ GeV
$(f_{{\rho }^{\prime }\pi \pi }/f_{{\rho }^{\prime }})=-0.0706\pm 0.0012$	$m_{{\rho }^{\prime }}=1.3500\pm 0.0110$ GeV	${\Gamma }_{{\rho }^{\prime }}=0.3320\pm 0.0033$ GeV
$(f_{{\rho }^{″}\pi \pi }/f_{{\rho }^{″}})=0.0580\pm 0.0010$	$m_{{\rho }^{″}}=1.7690\pm 0.0180$ GeV	${\Gamma }_{{\rho }^{″}}=0.2531\pm 0.0025$ GeV
$(f_{{\rho }^{‴}\pi \pi }/f_{{\rho }^{‴}})=0.0021\pm 0.0005$	$m_{{\rho }^{‴}}=2.2470\pm 0.0110$ GeV	${\Gamma }_{{\rho }^{‴}}=0.0700\pm 0.0007$ GeV
$R=0.0113\pm 0.0002$	$W_Z=0.2845\pm 0.0033$	$W_P=0.3830\pm 0.0060$

as taken from [5].

Now, the pure isovector charged pion EM FF $F_{\pi }^c\left(s\right)$ is determined by writing the absolute value squared of (6) in the form of a product of the complex and the complex conjugate terms

$$|F_{\pi }^c\left(s\right)+R{e}^{i\varphi }{\displaystyle \frac{m_{\omega }^2}{m_{\omega }^2-s-i{m}_{\omega }{\Gamma }_{\omega }}}{|}^2=\{F_{\pi }^c\left(s\right)+R{e}^{i\varphi }{\displaystyle \frac{m_{\omega }^2}{m_{\omega }^2-s-i{m}_{\omega }{\Gamma }_{\omega }}}\}.\{F_{\pi }^{c*}\left(s\right)+R{e}^{-i\varphi }{\displaystyle \frac{m_{\omega }^2}{m_{\omega }^2-s+i{m}_{\omega }{\Gamma }_{\omega }}}\},$$

where expressions $F_{\pi }^c\left(s\right)=\left|F_{\pi }^c\left(s\right)\right|e^{i{\delta }_{\pi }}$, $F_{\pi }^{c*}\left(s\right)=\left|F_{\pi }^c\left(s\right)\right|e^{-i{\delta }_{\pi }}$ have been substituted taking into account the identity between the pion EM FF phase and the P-wave isovector $\pi \pi$-phase shift ${\delta }_{\pi }\left(s\right)$ = ${\delta }_1^1\left(s\right)$, which follows from the charge pion EM FF elastic unitarity condition, practically considered to be valid up to 1 GeV. As a result, the quadratic equation for the absolute value of the pure isovector charged pion EM FF $|F_{\pi }^c\left(s\right)|$ has been found [5].

$$\begin{array}{c}\hfill |F_{\pi }^c{\left(s\right)|}^2+\left|F_{\pi }^c\left(s\right)\right|{\displaystyle \frac{2R{m}_{\omega }^2}{{(m_{\omega }^2-s)}^2+m_{\omega }^2{\Gamma }_{\omega }^2}}[(m_{\omega }^2-s)cos({\delta }_1^1-\varphi )+m_{\omega }{\Gamma }_{\omega }sin({\delta }_1^1-\varphi )]\\ \hfill +{\displaystyle \frac{R^2{m}_{\omega }^4}{{(m_{\omega }^2-s)}^2+m_{\omega }^2{\Gamma }_{\omega }^2}}-{\displaystyle \frac{3s}{\pi {\alpha }^2{\beta }_{\pi }^3\left(s\right)}}{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-})=0.\end{array}$$

(11)

The solution of the latter has given the relation with two signs

$$\begin{array}{c}\hfill |F_{\pi }^c\left(s\right)|=-{\displaystyle \frac{R{m}_{\omega }^2}{{(m_{\omega }^2-s)}^2+m_{\omega }^2{\Gamma }_{\omega }^2}}[(m_{\omega }^2-s)cos({\delta }_1^1-\varphi )+m_{\omega }{\Gamma }_{\omega }sin({\delta }_1^1-\varphi )]\\ \hfill \pm \{{\displaystyle \frac{R^2{m}_{\omega }^4}{{[{(m_{\omega }^2-s)}^2+m_{\omega }^2{\Gamma }_{\omega }^2]}^2}}{[(m_{\omega }^2-s)cos({\delta }_1^1-\varphi )+m_{\omega }{\Gamma }_{\omega }sin({\delta }_1^1-\varphi )]}^2\\ \hfill -{\displaystyle \frac{R^2{m}_{\omega }^4}{{(m_{\omega }^2-s)}^2+m_{\omega }^2{\Gamma }_{\omega }^2}}+{\displaystyle \frac{3s}{\pi {\alpha }^2{\beta }_{\pi }^3\left(s\right)}}{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}){\}}^{1/2},\end{array}$$

(12)

whereby a physical solution is with the “+” sign of the second term.

The most accurate existing ${\delta }_1^1\left(s\right)$ data from [22] have been described by a model-independent parameterization [23] with $q={[(s-4m_{\pi }^2)/4]}^{1/2}$ leading to numerical values of parameters in Table 1, which provide the best description of ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-})$ as measured in [15,16,17]. In this way, information on the $|F_{\pi }^c\left(s\right)|$ of the pure isovector EM FF of the charged pion has been extracted with errors; see Tables 2–4 of [5], respectively.

All obtained data on the $|F_{\pi }^c\left(s\right)|$ of the pure isovector EM FF of the charged pion as a function of s from the threshold up to 9 GeV², as given in Tables 2–4 of [5], are graphically presented in the left side of Figure 5a.

Afterwards, these data have been, as much as possible, described by a similar formula to (5); the nucleon “magic” number 0.71 had been, however, substituted by a third free parameter $A_3$.

The best description of the data in Figure 5 has been achieved with parameter values $A=3.9888\pm 0.0061$, $m_a^2=5.5647\pm 0.1915$ GeV² and the charged pion “magic” number $A_3=-5.5647\pm 0.1037$ GeV². The result is graphically presented in the right side of Figure 5b by the dashed line.

If the dashed line values are subtracted from the experimental values of $|F_{\pi }^c\left(s\right)|$ (Tables 2–4 of [5]), damped oscillatory structures from the charged pion “effective” EM FF data appear, as presented in Figure 6.

5. Search for Damped Oscillatory Structures from Charged K-Meson EM FF Data

The data on the charged K-meson EM structure are contained in the total cross-section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to K^{+}{K}^{-})$. To extract these data, no additional unphysical assumptions are needed [6], as $|F_K^{Ch}\left(s\right)|$ is the only function completely describing the measured total cross-section of the electron–positron annihilation into the $K^{+}{K}^{-}$ pair. The $|F_K^{Ch}\left(s\right)|$, with errors understood as the charged kaon “effective” EM FF, has been calculated by means of the following relation:

$$\begin{array}{c}\hfill G_{eff}^{K^{Ch}}\left(s\right)=\left|F_{K^{Ch}}\left(s\right)\right|=\sqrt{{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to K^{+}{K}^{-}){\displaystyle \frac{3s}{\pi {\alpha }^2{C_K}^{Ch}{{\beta }_K^3}^{Ch}}}},\end{array}$$

(13)

where ${{\beta }_K}^{Ch}\left(s\right)=\sqrt{1-{\displaystyle \frac{4{m_K^2}^{Ch}}{s}}}$, $\alpha$ = 1/137 and ${C_K}^{Ch}={\displaystyle \frac{\pi \alpha /{{\beta }_K}^{Ch}\left(s\right)}{1-exp(-\pi \alpha /{{\beta }_K}^{Ch}\left(s\right))}}$ is the so-called Sommerfeld–Gamov–Sakharov Coulomb enhancement factor [7] of charged kaons, which accounts for the EM interaction between the outgoing $K^{+}{K}^{-}$. The total cross-section data used in (13) have been, in [6], taken from two recent ISR measurements of the process $e^{+}{e}^{-}\to K^{+}{K}^{-}\left(\gamma \right)$, one from [24] for $s<25$ GeV², and another from [25] in the range 6.76 GeV² $<s<$ 64 GeV². These two sets of data are together graphically presented in Figure 7 and the region (2–7) GeV² is shown in more detail in Figure 8, all taken from [6].

It can be seen from Figure 7 that the data in [24] above 6.5 GeV² are, in some points, inconsistent. As the same experimental BABAR group in the paper [25] repeated measurements of the $e^{+}{e}^{-}\to K^{+}{K}^{-}$ process from 6.76 GeV² to 64 GeV² and obtained more precise data, we have excluded in [6] all data from [24] in the energy range 6.76–25 GeV² and substituted them with precise data from [25].

To make the oscillatory structure appear from the charged K-meson EM FF timelike data by using the same procedure as for the proton, the modification of the Formula (5) has been carried out in [6], in the sense that the magic nucleon number 0.71 GeV² was left as a free parameter A3 in our analysis. The best description of the data in Figure 7 has been achieved with A = $5.14773\pm 0.0013$, $m_a^2$ = $0.2400\pm 0.0709$ GeV² and A3 = $0.8403\pm 0.0024$ GeV², as is graphically presented in Figure 9 by the dashed line. If dashed curve data are subtracted from selected charged K-meson FF data, RDOSs are observed around the line crossing the zero, as seen in Figure 10, to be taken from [6].

6. Search for Damped Oscillatory Structures from Neutral K-Meson EM FF Data

The neutral K-meson EM FF $|F_{K^0}\left(s\right)|$ data in the timelike region could be obtained in [6] from the measured total cross-section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to K_S{K}_L)$.

Nevertheless, there are no data on the function $|F_{K^0}\left(s\right)|$ with errors published during the last decade; therefore, we have calculated them in the paper [6] by means of the relation

$$\begin{array}{c}\hfill G_{eff}^{K^0}\left(s\right)=\left|F_{K^0}\left(s\right)\right|=\sqrt{{\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to K_S{K}_L){\displaystyle \frac{3s}{\pi {\alpha }^2{\beta }_{K^0}^3}}},\end{array}$$

(14)

with ${\beta }_{K^0}\left(s\right)=\sqrt{1-{\displaystyle \frac{4m_{K^0}^2}{s}}}$, $\alpha$ = 1/137, from one recent measurement [26] of the process $e^{+}{e}^{-}\to K_S{K}_L\left(\gamma \right)$ by the ISR technique in the interval of energy values s (1.1664–4.84) GeV² and from two measurements [17,27] by the scan method, the first one in the $\varphi$-resonance region (1.0080–1.1236) GeV² and the second in the range of energies (4.0000–9.4864) GeV². Their numerical values with errors taken from [6] are given in

Table 2. The data on $|F_{K^0}\left(s\right)|$ with errors.

s [GeV2]	$|{\mathit{F}}_{{\mathit{K}}^0}\left(\mathit{s}\right)|$ ± err.	s [GeV2]	$|{\mathit{F}}_{{\mathit{K}}^0}\left(\mathit{s}\right)|$ ± err.	s [GeV2]	$|{\mathit{F}}_{{\mathit{K}}^0}\left(\mathit{s}\right)|$ ± err.	s [GeV2]	$|{\mathit{F}}_{{\mathit{K}}^0}\left(\mathit{s}\right)|$ ± err.
$1.0080$	$11.4683\pm 0.3506$	$1.0588$	$16.7599\pm 0.1729$	$1.9600$	$0.2369\pm 0.0592$	$4.2025$	$0.1120\pm 0.0496$
$1.0201$	$19.6100\pm 0.1093$	$1.0692$	$10.9758\pm 0.1380$	$2.0736$	$0.2821\pm 0.0529$	$4.4100$	$0.0829\pm 0.0004$
$1.0241$	$27.1841\pm 0.1405$	$1.0816$	$07.8863\pm 0.1274$	$2.1904$	$0.2968\pm 0.0453$	$4.4944$	$0.0713\pm 0.0035$
$1.0302$	$36.4652\pm 0.4169$	$1.1025$	$05.0943\pm 0.0752$	$2.3104$	$0.3999\pm 0.0394$	$4.6225$	$0.0653\pm 0.0136$
$1.0323$	$45.4557\pm 0.2066$	$1.1046$	$05.1158\pm 0.1376$	$2.4336$	$0.4022\pm 0.0390$	$4.7306$	$0.0748\pm 0.0075$
$1.0343$	$57.6351\pm 0.1353$	$1.1236$	$03.8750\pm 0.1138$	$2.5600$	$0.4505\pm 0.0322$	$4.8400$	$0.0738\pm 0.0057$
$1.0363$	$70.4012\pm 0.1847$	$1.1664$	$02.4831\pm 0.0778$	$2.6896$	$0.4488\pm 0.0351$	$4.9836$	$0.0876\pm 0.0666$
$1.0384$	$81.3097\pm 0.1731$	$1.2544$	$01.2826\pm 0.0687$	$2.8224$	$0.3531\pm 0.0381$	$5.3333$	$0.0716\pm 0.0048$
$1.0404$	$77.0351\pm 0.1361$	$1.3456$	$00.8939\pm 0.0603$	$2.9584$	$0.2534\pm 0.0536$	$5.6949$	$0.0342\pm 0.0055$
$1.0424$	$59.0975\pm 0.1734$	$1.4400$	$00.7436\pm 0.0578$	$3.0976$	$0.1659\pm 0.0528$	$5.7408$	$0.0367\pm 0.0038$
$1.0445$	$48.1169\pm 0.1663$	$1.5376$	$00.5341\pm 0.0553$	$3.2400$	$0.0872\pm 0.0727$	$6.9929$	$0.0437\pm 0.0050$
$1.0465$	$37.7921\pm 0.2656$	$1.6384$	$00.4013\pm 0.0520$	$3.3856$	$0.1015\pm 0.0508$	$7.0034$	$0.0455\pm 0.0043$
$1.0506$	$26.1639\pm 0.3264$	$1.7424$	$00.3877\pm 0.0538$	$3.5344$	$0.0724\pm 0.0724$	$8.4100$	$0.0253\pm 0.0036$
$1.0568$	$18.6920\pm 0.1371$	$1.8496$	$00.3253\pm 0.0529$	$4.0000$	$0.1223\pm 0.0663$	$9.4864$	$0.0257\pm 0.0030$

and are graphically presented in Figure 11.

In the study of damped oscillatory structures from the neutral K-meson EM FF timelike data, the modified Formula (5) with 0.71 GeV² becoming a free parameter A3 has been used again for fitting the data in Figure 11. The best description has been achieved with parameter values A = $0.4729\pm 0.0061$, $m_a^2$ = $-0.9267\pm 0.0016$ GeV² and A3 = $0.9269\pm 0.0003$ GeV² and the result is graphically presented in Figure 12 (also taken from [6]) by the dashed line.

If the dashed line is subtracted from the neutral K-meson FF data in Figure 11, taking errors into account, damped oscillatory structures are observed around the line crossing the zero, as depicted in Figure 13.

By a comparison of the oscillations depicted in Figure 10 and Figure 13 with those of the nucleons in Figure 2 and Figure 4, one finds that the damped oscillation regular structures of isodoublets have exactly opposite behaviors.

This interesting feature has to be explained by some serious physical arguments.

7. Damped Oscillation Structures from the Possible Deuteron “Effective” EM FF Data

The deuteron EM structure is completely described by three independent EM FFs, usually chosen to be the charge $G_C\left(s\right)$, the magnetic $G_M\left(s\right)$ and the quadrupole $G_Q\left(s\right)$ FF. Then, the total cross-section of the electron–positron annihilation into the deuteron–antideuteron pair is expressed as function of EM FFs:

$$\begin{array}{c}\hfill {\sigma }_{tot}(e^{+}{e}^{-}\to d\overline{d})={\displaystyle \frac{\pi {\alpha }^2{C}_D{\beta }_d^3\left(s\right)}{3s}}({\displaystyle \frac{s}{m_d^2}}{|G_C\left(s\right)+G_M\left(s\right)+G_Q\left(s\right)|}^2+\\ \hfill +[2|G_C\left(s\right)+{\displaystyle \frac{s}{2m_d^2}}{G}_Q\left(s\right){|}^2+|G_C\left(s\right)+{\displaystyle \frac{s}{2m_d^2}}{G}_M\left(s\right){|}^2]).\end{array}$$

(15)

Again, it is not a simple task to obtain any experimental information on the corresponding deuteron EM FFs in the $s>4m_d^2$ region. With the aim of obtaining at least some information on the EM structure of the deuteron, one can define the “effective” deuteron EM FF by a requirement of the equality $G_C\left(s\right)=G_M\left(s\right)=G_Q\left(s\right)$ for all $s>4m_d^2$ up to $+\infty$, like in the case of the nucleons, and in this way obtain data on

$$\begin{array}{c}\hfill G_{eff}^d\left(s\right)=\sqrt{{\displaystyle \frac{s{\sigma }_{tot}(e^{+}{e}^{-}\to d\overline{d})}{\pi {\alpha }^2{C}_D{\beta }_d^3\left(s\right)(\frac{3s}{m_d^2}+|1+\frac{3}{2m_d^2}{|}^2)}}}\end{array}$$

(16)

from experimental data on the ${\sigma }_{tot}(e^{+}{e}^{-}\to d\overline{d})$. However, the latter are still missing.

8. Origin of the Opposite-Behavior Phenomenon in RDOSs of Isodoublets

Investigating the RDOSs of nucleons charged and neutral K-mesons, we have observed that the RDOSs of the proton are exactly opposite to the RDOSs of the neutron, and the same phenomenon has been observed in the isodoublet of K-mesons.

Next we investigate the latter phenomenon on the isodoublet of the proton and neutron, and the same procedure can also be repeated in the case of the charged and neutral K-mesons.

The idea consists in a creation of artificial neutron “effective” EM FF data from the proton “effective” EM FF data only, by application of the physically well-founded U&A model of the nucleon EM structure, together with the transformation properties of the nucleon EM current in the isotopic space and then by their description using a Tomasi–Gustafsson–Rekalo three-parameter function in order to produce artificial regular neutron damped oscillation structures.

The proton and neutron EM FFs in the total cross-sections (1) and (2), respectively, and consequently also in (3) and (4), are the Sachs proton electric and proton magnetic FFs, which can be expressed through the Dirac and Pauli EM FFs, to be defined by the parameterization of the matrix element of the nucleon EM current

$$\begin{array}{c}\hfill <N|J_{\mu }^{EM}\left(0\right)|N>=e\overline{u}\left(p^{\prime }\right)[{\gamma }_{\mu }{F}_1^N\left(t\right)+{\displaystyle \frac{i}{2m_N}}{\sigma }_{\mu \nu }{(p^{\prime }-p)}^{\nu }{F}_2^N\left(t\right)]u\left(p\right),\end{array}$$

(17)

and the latter, taking into account the special transformation properties of the nucleon EM current in the isotopic space, are further split into the same isoscalar and isovector parts for both nucleons, with the “+” sign for protons and with the “−” sign for neutrons, as follows:

$$\begin{array}{c}\hfill G_E^p\left(t\right)=F_1^p\left(s\right)+{\displaystyle \frac{s}{4m_p^2}}{F}_2^p\left(s\right)=[F_{1s}^N\left(t\right)+F_{1v}^N\left(t\right)]+{\displaystyle \frac{t}{4m_p^2}}[F_{2s}^N\left(t\right)+F_{2v}^N\left(t\right)],\\ \hfill G_M^p\left(t\right)=F_1^p\left(s\right)+F_2^p\left(s\right)=[F_{1s}^N\left(t\right)+F_{1v}^N\left(t\right)]+[F_{2s}^N\left(t\right)+F_{2v}^N\left(t\right)],\end{array}$$

(18)

and

$$\begin{array}{c}\hfill G_E^n\left(t\right)=F_1^n\left(s\right)+{\displaystyle \frac{s}{4m_p^2}}{F}_2^n\left(s\right)=[F_{1s}^N\left(t\right)-F_{1v}^N\left(t\right)]+{\displaystyle \frac{t}{4m_n^2}}[F_{2s}^N\left(t\right)-F_{2v}^N\left(t\right)],\\ \hfill G_M^n\left(t\right)=F_1^n\left(s\right)+F_2^n\left(s\right)=[F_{1s}^N\left(t\right)-F_{1v}^N\left(t\right)]+[F_{2s}^N\left(t\right)-F_{2v}^N\left(t\right)].\end{array}$$

(19)

Here, we would like to stress that as a result, both the proton EM FFs (18) and the neutron EM FFs (19) depend on the same physically interpretable free parameters. However, they are determined by fitting the existing data on the ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ only, using the advanced 9-vector-meson resonance U&A model [28] of the nucleon EM structure presented in Appendix A.

Its derivation can be found in detail in the paper [18].

One can find the results of the fit of ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ data [8,9,10,11,12,13,14] by means of the U&A model for the proton EM FFs in

Table 3. Values of free parameters of the proton EM structure $U\&A$ model (A1)–(A12) from optimal description of ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ data [8,9,10,11,12,13,14] only.

$s_{in}^{1s}=1.4279\pm 0.0196{\mathrm{GeV}}^2;$	$s_{in}^{1v}=2.9240\pm 0.0784{\mathrm{GeV}}^2;$
$s_{in}^{2s}=1.8304\pm 0.1764{\mathrm{GeV}}^2;$	$s_{in}^{2v}=2.6873\pm 0.0784{\mathrm{GeV}}^2;$
$(f_{{\omega }^{\prime }NN}^{\left(1\right)}/f_{{\omega }^{\prime }})=-0.0309\pm 0.0084;$	$(f_{{\varphi }^{\prime }NN}^{\left(1\right)}/f_{{\varphi }^{\prime }})=-0.6423\pm 0.0061;$
$(f_{\omega NN}^{\left(1\right)}/f_{\omega })=0.9970\pm 0.0026;$	$(f_{\varphi NN}^{\left(1\right)}/f_{\varphi })=-0.0315\pm 0.0010;$
$(f_{{\varphi }^{\prime }NN}^{\left(2\right)}/f_{{\varphi }^{\prime }})=0.7848\pm 0.0183;$	$(f_{\omega NN}^{\left(2\right)}/f_{\omega })=-0.4967\pm 0.1835;$
$(f_{\varphi NN}^{\left(2\right)}/f_{\varphi })=0.1490\pm 0.0022;$	$(f_{\rho NN}^{\left(1\right)}/f_{\rho })=0.1845\pm 0.0037$.

.

With numerical values of the free parameters in Table 3, one can describe well all existing data on ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ (see Figure 14a left), and also the proton “effective” EM FF data in the right side of Figure 1b. And by exchanging the EM FFs of the proton (18) in the U&A model with the neutron EM FFs (19), the curve for the neutron “effective” EM FF behavior in the right side of Figure 14b is predicted theoretically too.

The crucial moment in our investigations is the fact that the artificial neutron “effective” EM FF data in the right side of Figure 14b right are obtained from the data on ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ only, in the following way. First, we evaluate deviations of the proton “effective” EM FF data from the curve describing them in the right side of Figure 1b, by a subtraction of the curve from existing data. Then, the obtained deviations are added to the theoretically predicted curve for the neutron “effective” EM FF in the right side of Figure 14b at the same energy value “s”. As a result, one obtains the 46 artificial points on the neutron “effective” EM FF with errors of the proton “effective” EM FF data scattered around the theoretically predicted curve in the right side of Figure 14b, which, moreover, perfectly describes them.

When such artificially created data on the neutron “effective” EM FF are described by the three-parameter function (5) with values of the parameters $A\left(1\right)=1.0971\pm 0.0879$, $m_a^2=6.1112\pm 0.6670$ GeV², $A\left(3\right)=2.5321\pm 9,9162$ GeV², and this curve is subsequently subtracted from artificial data with errors, the RDOSs appear as seen in Figure 15. One can see that they are exactly opposite to the proton RDOSs in Figure 2 to be obtained from the proton “effective” EM FF data in the left side of Figure 1a.

As the artificial pseudo-data on the neutron “effective” EM FF have been obtained from the proton “effective” EM FF data only, by the U&A analytic model (the same for the proton and neutron) through the relations (18) and (19), respectively, which reflect explicitly the special transformation of the nucleon EM current in the isospin space, we come to the conclusion that the origin of the phenomenon of the opposite RDOS behaviors for protons and neutrons is in the special transformations of the nucleon EM current in the isotopic space.

9. Description of Hadron “Effective” FF Data by Unitary and Analytic Approach

If the same data on hadron “effective” FFs are described by slightly more complicated but physically well-founded universal U&A models [18] of their EM structure, no damped oscillation regular structures appear. As the data of all considered hadrons are not of the same quality, the corresponding U&A models are also specific from one case to the other.

9.1. Description of Charged Pion “Effective” EM FF Data by U&A Model

The application of the U&A model (Equations (7)–(10), with parameters from Table 1) to the data on $G_{eff}^{{\pi }^{\pm }}\left(s\right)=\left|F_{\pi }^c\left(s\right)\right|$, summarized in Tables 2–4 of [5], leads to a perfect description of the latter, as shown by the full line in Figure 16. If these full-line data in Figure 16 are subtracted from the data in Tables 2–4 of [5], no oscillatory regular structures appear, as can be clearly seen in Figure 17.

9.2. Description of Proton “Effective” EM FF Data by U&A Model

To describe the “effective” proton FF by the U&A model, we substitute the explicit form of the proton total cross-section (1) into the expression (3). Then, the relation between the absolute values of the complex proton EM FFs squared, represented by means of the U&A model and the proton “effective” EM FF (3), is found

$$\begin{array}{c}\hfill G_{eff}^p\left(s\right)=\sqrt{{\displaystyle \frac{\left[|G_M^p{\left(s\right)|}^2+\frac{2m_p^2}{s}{\left|G_E^p\left(s\right)\right|}^2\right]}{\left(1+\frac{2m_p^2}{s}\right)}}}.\end{array}$$

(20)

So, the description of the data on the proton “effective” FF data with ${\chi }^2/ndf=1.85$ (see Figure 18) is achieved.

Then, the subtraction of the curve of Figure 18 from the proton’s “effective” FF data with errors demonstrates no RDOSs, as clearly seen in Figure 19.

9.3. Description of Neutron “Effective” FF Data by U&A Model

Similarly to the proton, for the neutron, one can also derive the relation

$$\begin{array}{c}\hfill G_{eff}^n\left(s\right)=\sqrt{{\displaystyle \frac{\left[|G_M^n{\left(s\right)|}^2+\frac{2m_n^2}{s}{\left|G_E^n\left(s\right)\right|}^2\right]}{\left(1+\frac{2m_n^2}{s}\right)}}}\end{array}$$

(21)

between the neutron EM FFs represented by means of the U&A model and the neutron “effective” FF (4). An accurate description of the neutron “effective” FF data is achieved with parameters given in

Table 4. Values of free parameters of the neutron EM structure U&A model (A1)–(A12) from optimal description of the neutron “effective” FF data in the right side of Figure 3a.

$s_{in}^{1s}=2.2188\pm 0.0053{\mathrm{GeV}}^2;$	$s_{in}^{1v}=6.0930\pm 0.0002{\mathrm{GeV}}^2;$
$s_{in}^{2s}=3.2748\pm 0.0009{\mathrm{GeV}}^2;$	$s_{in}^{2v}=5.6194\pm 0.0067{\mathrm{GeV}}^2;$
$(f_{{\omega }^{\prime }NN}^{\left(1\right)}/f_{{\omega }^{\prime }})=2.2045\pm 0.0172;$	$(f_{{\varphi }^{\prime }NN}^{\left(1\right)}/f_{{\varphi }^{\prime }})=-0.5700\pm 0.0028;$
$(f_{\omega NN}^{\left(1\right)}/f_{\omega })=0.10903\pm 0.0022;$	$(f_{\varphi NN}^{\left(1\right)}/f_{\varphi })=0.17613\pm 0.0001;$
$(f_{{\varphi }^{\prime }NN}^{\left(2\right)}/f_{{\varphi }^{\prime }})=-0.7089\pm 0.0002;$	$(f_{\omega NN}^{\left(2\right)}/f_{\omega })=0.0032\pm 0.0001;$
$(f_{\varphi NN}^{\left(2\right)}/f_{\varphi })=0.0001\pm 0.0001;$	$(f_{\rho NN}^{\left(1\right)}/f_{\rho })=0.1314\pm 0.0001$.

, as shown in the left side of Figure 3. Subtracting then the fitted curve from the data on the neutron “effective” FF, one finds some structures (see Figure 20), from which, however, one cannot draw any conclusions about the (non)existence of RDOSs. In order to obtain a definitive statement, more precise data on $e^{+}{e}^{-}\to n\overline{n}$ are needed.

Figure 20. The result of a subtraction of the curve in Figure 21 from the neutron “effective” FF data with errors indicates no damped oscillation regular structures.

Figure 20. The result of a subtraction of the curve in Figure 21 from the neutron “effective” FF data with errors indicates no damped oscillation regular structures.

9.4. Description of Charged K-Meson “Effective” EM FF Data by U&A Model

Here, we demonstrate that when charged K-meson data in Figure 7 are accurately described by a physically well-founded U&A model of the K-meson EM structure (see Appendix B), no damped oscillatory structures are observed.

We use the model from the paper [18] as the universal U&A model of the EM structure of all hadrons. It unifies the experimental observation of unstable vector-meson resonances (see [19]), mainly identified in the electron–positron annihilation processes into hadrons, the presence of two square root branch cut approximations of the analytic properties of FFs in the complex plane of c.m. energy squared s and the asymptotic behavior of FFs, as predicted by the quark model of hadrons.

In order to describe the selected data in Figure 7 in the framework of the U&A approach [18], we first split the charged K-meson FF in accordance with the special transformation properties of the K-meson EM current in the isospin space, into a sum of the isoscalar and isovector parts

$$\begin{array}{c}\hfill F_{K^{Ch}}\left(s\right)=F_K^s\left(s\right)+F_K^v\left(s\right)\end{array}$$

(22)

with their norms

$$\begin{array}{c}\hfill F_K^s\left(0\right)=F_K^v\left(0\right)={\displaystyle \frac{1}{2}}.\end{array}$$

(23)

Then, the question of what isoscalar and isovector resonances, experimentally confirmed in [19], should saturate the isoscalar and isovector parts of the charged kaon EM FF has to be solved. Practically, one cannot consider all experimentally confirmed resonances with isospin $I=0$ and isospin $I=1$, as given in [19], because they will produce 25 free parameters, and insufficient and inconsistent up-to-date experimental information is available, so we are unable to determine reasonable values of their parameters. Consequently, a selection of contributing resonances has to be carried out as follows.

First, we consider the three existing $\varphi$ isoscalar resonances. The K-mesons also consist of strange quarks; therefore, one can expect that in a description of the selected data in Figure 7 and Figure 8, all three $\varphi$ resonances with the isospin $I=0$ will be dominant, and therefore, the masses and widths of $\varphi \left(1020\right)$ and ${\varphi }^{\prime }\left(1680\right)$ will be left as free parameters of the model, and the parameters of $\varphi \left(2170\right)$ will be fixed at the PDG values, since, in the corresponding energy region, the analyzed data are insufficient for their evaluation. From Figure 7 (in more detail in Figure 8), between 2.0 GeV² and 7.0 GeV², one finds two bumps corresponding approximately to ${\varphi }^{\prime }\left(1680\right)$ and ${\varphi }^{″}\left(2170\right)$. The first bump could also contain a contribution from another resonance with $I=0$, ${\omega }^{″}\left(1650\right)$, which is therefore not excluded in our considerations. There is no indication in existing data of a contribution of ${\omega }^{\prime }\left(1420\right)$; therefore, it is not taken into account at all.

On the other hand, from the experience of our previous analysis, one cannot achieve a satisfactory description of the existing data without the inclusion of the ground state resonances $\rho \left(770\right)$ and $\omega \left(782\right)$. Further, because contributions of the isovector part of the K-meson FF, though not dominant, cannot be ignored, we include contributions of all three $\rho$-mesons, with fixed parameters from the paper [23]. There, one can also find reasons for why not to use their parameters from [19]. The $\omega \left(782\right)$ and ${\omega }^{″}\left(1650\right)$ masses and widths are also fixed at the PDG values.

The U&A model of the K-meson EM structure has the form presented in Appendix B. As a result, the U&A model of the K-meson EM structure depends altogether on 14 free parameters, whose numerical values (see

Table 5. Parameter values of the U&A model in the analysis of selected data on $|F_{K^{\pm }}\left(s\right)|$ with minimum of ${\chi }^2/ndf=1.79$.

$s_{in}^s=1.0984\pm 0.2292$ GeV2;

$m_{\varphi }=1019.298\pm 0.063$ MeV;  ${\Gamma }_{\varphi }=4.304\pm 0.083$ MeV;  $(f_{\varphi KK}/f_{\varphi })=0.331\pm 0.063$;

$m_{{\varphi }^{\prime }}=1656.620\pm 4.969$ MeV;  ${\Gamma }_{{\varphi }^{\prime }}=356.860\pm 4.444$ MeV;  $(f_{{\varphi }^{\prime }KK}/f_{{\varphi }^{\prime }})=-0.568\pm 0.102$;

$m_{{\varphi }^{″}}=2001.300\pm 22.817$ MeV;  ${\Gamma }_{{\varphi }^{″}}=530.502\pm 34.300$ MeV;

$(f_{{\varphi }^{″}KK}/f_{{\varphi }^{″}})=1/2-(f_{\omega KK}/f_{\omega })-(f_{{\omega }^{″}KK}/f_{{\omega }^{″}})-(f_{\varphi KK}/f_{\varphi })-(f_{{\varphi }^{\prime }KK}/f_{{\varphi }^{\prime }})$;

$(f_{\omega KK}/f_{\omega })=0.273\pm 0.044$;  $(f_{{\omega }^{″}KK}/f_{{\omega }^{″}})=0.354\pm 0.103$;

$s_{in}^v=1.6765\pm 0.1337$ GeV2;

$(f_{{\rho }^{\prime }KK}/f_{{\rho }^{\prime }})=1/2-(f_{\rho KK}/f_{\rho })-(f_{{\rho }^{″}KK}/f_{{\rho }^{″}})$;

$(f_{\rho KK}/f_{\rho })=0.440\pm 0.017$;  $(f_{{\rho }^{″}KK}/f_{{\rho }^{″}})=0.036\pm 0.005$;

, taken from [6]) have been evaluated in the analysis of selected data from Figure 7.

A description of these charged K-meson EM FF data is presented in Figure 22 by the full line. One can find the description of the resonant region between 2.0 GeV² and 7.0 GeV² in more detail in Figure 23.

If the full line in Figure 22 is subtracted from selected charged K-meson FF data in Figure 7 with errors, no damped oscillatory structures are observed around the line crossing the zero, as shown in Figure 24.

9.5. Description of Neutral K-Meson “Effective” EM FF Data by U&A Model

Here, we demonstrate that if neutral K-meson EM FF data in Figure 25 are described by a proper physically well-founded U&A model, no damped oscillatory structures are revealed.

The simplest way to obtain a correct description of the data in Table 2 is to exploit the special transformation properties of the K-meson EM current in the isospin space and to express the neutral K-meson EM FF as the difference of the isoscalar and isovector parts

$$\begin{array}{c}\hfill F_{K^0}\left(s\right)=F_K^s\left(s\right)-F_K^v\left(s\right),\end{array}$$

(24)

where $F_K^s\left(s\right)$ and $F_K^v\left(s\right)$ are the same as those in (22). Then, the charged and neutral K-meson EM FFs should depend on the same numerical values of 14 parameters of the U&A model.

A direct fitting procedure of them by the K-meson EM FF U&A model (A13)–(A16) provides 14 parameter values, as presented in

Table 6. Parameter values from the analysis of selected data on $|F_{K^0}\left(s\right)|$ by the K-meson EM FF U&A model (A13)–(A16) with minimum of ${\chi }^2/ndf=4.37$.

$s_{in}^s=2.0728\pm 0.0196$ GeV2;

$m_{\varphi }=1019.158\pm 0.176$ MeV;  ${\Gamma }_{\varphi }=4.214\pm 0.030$ MeV;  $(f_{\varphi KK}/f_{\varphi })=0.336\pm 0.001$;

$m_{{\varphi }^{\prime }}=1649.760\pm 5.356$ MeV;  ${\Gamma }_{{\varphi }^{\prime }}=340.032\pm 12.438$ MeV;  $(f_{{\varphi }^{\prime }KK}/f_{{\varphi }^{\prime }})=-0.217\pm 0.001$;

$m_{{\varphi }^{″}}=2028.740\pm 46.556$ MeV;  ${\Gamma }_{{\varphi }^{″}}=383.418\pm 45.489$ MeV;

$(f_{{\varphi }^{″}KK}/f_{{\varphi }^{″}})=1/2-(f_{\omega KK}/f_{\omega })-(f_{{\omega }^{″}KK}/f_{{\omega }^{″}})-(f_{\varphi KK}/f_{\varphi })-(f_{{\varphi }^{\prime }KK}/f_{{\varphi }^{\prime }})$;

$(f_{\omega KK}/f_{\omega })=0.278\pm 0.001$;  $(f_{{\omega }^{″}KK}/f_{{\omega }^{″}})=0.088\pm 0.001$;

$s_{in}^v=2.0077\pm 0.0785$ GeV2;

$(f_{{\rho }^{\prime }KK}/f_{{\rho }^{\prime }})=1/2-(f_{\rho KK}/f_{\rho })-(f_{{\rho }^{″}KK}/f_{{\rho }^{″}})$;

$(f_{\rho KK}/f_{\rho })=0.606\pm 0.004$;  $(f_{{\rho }^{″}KK}/f_{{\rho }^{″}})=-0.044\pm 0.010$;

. These parameters, however, differ from parameter values in Table 5. The latter may indicate that the data on the charged K-meson EM FF and the data on the neutral K-meson EM FF are inconsistent, or the conservation of the isospin in the EM interactions is not exact. Nevertheless, a solution of this problem is not the subject of this paper.

The accurate description of the $|F_{K^0}\left(s\right)|$ data corresponding to the parameters in Table 6 is graphically presented in Figure 25 by the full line.

The dotted line is a prediction of the $|F_{K^0}\left(s\right)|$ behavior from the data on the charged K-meson FF by the U&A model, indicating the mentioned inconsistency of existing data in K-meson FFs, or the inexact conservation of the isospin in EM interactions.

If the full line in Figure 25 is subtracted from neutral K-meson EM FF data in Table 2 with errors, one obtains points with uncertainties around the line crossing the zero (Figure 26), which clearly demonstrates the absence of RDOSs.

10. Conclusions and Discussion

The RDOSs from the proton “effective” form factor data in [1] sparked an interest in studying the damped oscillatory structures from the EM FFs data of other hadrons, for which solid data, together with a physically well-founded model for their accurate description, exist.

Here, the existence of the RDOSs of the charged pion and the charged and neutral K-meson EM FF data and also data on the neutron EM FF were investigated by using the same procedure as the one used in the case of the proton in [1].

When the “effective” data of the considered hadrons are described by the three-parameter function of [10], the regular damped oscillatory structures appear. However, if, for a description of the same data, a more physically well-founded U&A model of the EM structure of hadrons is applied, no regular damped oscillatory structures are observed.

So, the results of all our investigations indicate that there is no objective existence of RDOSs from the “effective” hadron EM FF data and their appearance is due to the application of the three-parameter formula [10] without any physical background, and this cannot describe the latter data with adequate accuracy.

Recently, a few works appeared in the literature [29,30,31] that study the electromagnetic form factors of the nucleons in the time-like region and try to understand their oscillating features within the proposed theoretical models. From these results, it seems that the existence of RDOSs is not confirmed, at least in the case of nucleon “effective” FFs. The final verification of this fact will require further investigations.