New Insight on Damped Oscillating Regular Structures from “Effective” Proton Electromagnetic Form Factor Data

Abstract

An objective existence of damped oscillation regular structures from the proton “effective” electromagnetic form factor data, appearing when these data are described by a two parametric Tomasi-Gustafsson-Rekalo function, is investigated. If the same data are described with an improved precision by the function dependent on twelve physical parameters, obtained from the total cross section ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ expression through the proton electromagnetic form factors represented by the Unitary and Analytic model, no damped oscillation regular structures are observed. As a result it seems, the damped oscillation regular structures are created from the proton “effective” electromagnetic form factor data description by the two-parametric formula of Tomasi-Gustafsson-Rekalo, which is unable to describe the proton “effective” electromagnetic form factor data with a sufficient high accuracy.

1. Introduction

Every strongly interacting particle possess the electromagnetic structure, which theoretically is described by the corresponding electromagnetic (EM) form factors (FFs). They are defined as coefficients in the matrix elements of current operators decomposition into maximal number of independent relativistic covariants constructed from momenta and spin parameters. They describe the electromagnetic structure of the hadron in one-photon exchange approximation, which is broken down at the next orders of perturbation theory. The EM FFs possess definite analytic properties derived from the causality in the quantum field theory.

The number of EM FFs is given by the spin of the considered hadron. The EM structure of the proton with the spin S = 1/2 is completely described by two EM FFs, the electric $G_E^p\left(s\right)$ and magnetic $G_M^p\left(s\right)$. Their experimental behavior in the space-like region is obtained by the scattering of the unpolarized, or polarized electrons on protons. In the time-like region the electric $G_E^p\left(s\right)$ and magnetic $G_M^p\left(s\right)$ FFs are complex functions and an information on the behavior of their absolute values nowadays is obtained from the measurement of the 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)

with the velocities ${\beta }_p\left(s\right)=\sqrt{1-{\displaystyle \frac{4m_p^2}{s}}}$ of the outgoing protons and antiprotons in (c.m.) system, $\alpha$ = 1/137 and the so-called Sommerfeld-Gamov-Sakharov Coulomb enhancement factor [1] $C_p={\displaystyle \frac{\pi \alpha /{\beta }_p\left(s\right)}{1-exp(-\pi \alpha /{\beta }_p\left(s\right))}}$, which accounts for the EM interaction between the outgoing $p\overline{p}$ pairs.

The functions $G_E^p\left(s\right)$ and $G_M^p\left(s\right)$ in Equation (1) are the Sachs electric and magnetic form factors (FFs), respectively. These two functions have the same values at the proton-antiproton threshold, as it follows from their definition in terms of the Dirac $F_1^p\left(s\right)$ and the Pauli $F_2^p\left(s\right)$ FFs

$$\begin{array}{cc}\hfill G_E^p\left(s\right)& =F_1^p\left(s\right)+{\displaystyle \frac{s}{4m_p^2}}{F}_2^p\left(s\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill G_M^p\left(s\right)& =F_1^p\left(s\right)+F_2^p\left(s\right).\hfill \end{array}$$

(3)

Because it is not possible to determine both EM FFs $G_E\left(s\right)$ and $G_M\left(s\right)$ from single experimental value on ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to p\overline{p})$, without additional assumptions, experimental groups in [2,3,4,5,6,7], and also in [8] extended a validity of the threshold identity $|G_E^p\left(4m_p^2\right)| \equiv |G_M^p\left(4m_p^2\right)|$ in Equation (1) to all s-values above the threshold and the data with errors obtained by means of the consequent expression

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

(4)

has been defined as the proton “effective” EM FF. It is a starting quantity in investigation of the “proton damped oscillation regular structures”.

It is clear from the previous considerations that the concept of “the proton ‘effective’ EM FF” in the elementary particle physics is temporary and it will survive only until the moment when sufficiently high luminosity of the $e^{+}{e}^{-}$ colliders is achieved, allowing the precise determination of the separate absolute values of the electric and magnetic timelike proton FFs from angular distribution of the reactions $e^{+}{e}^{-}\to p\overline{p}$, $p\overline{p}\to e^{+}{e}^{-}$, or possibly from measurements of the vector and tensor polarization components of the final protons and antiprotons in the $e^{+}{e}^{-}\to p\overline{p}$, process [9].

We would like to note that in such case the investigation of the objective existence of damped oscillation regular structures from the separate absolute values of the nucleon electric and the nucleon magnetic FFs data will be still interesting and the results may be different from conclusions of this paper pertinent of the proton “effective” EM FF.

Despite of the promising future experiments, investigating the first proton “effective” EM FF data [2,3], the following new phenomenon has been observed in the paper [10]. The authors of this paper, first describing the data on the “effective” proton EM FF by a two free parametric formula

$$\begin{array}{c}\hfill G_{eff}^p={\displaystyle \frac{A}{(1+s/m_a^2){(1-s/0.71{\mathrm{GeV}}^2)}^2}},\end{array}$$

(5)

from [11] with the values of parameters A = 7.7 and $m_a^2=14.8{\mathrm{GeV}}^2$, then subtracting the obtained curve from the proton “effective” EM FF data, revealed the “proton damped oscillation regular structures” as it is demonstrated in Figure 1.

In the next Section all other new proton “effective” EM FF data are collected, then repeating the procedure of the paper [10], results of the authors of the paper [10] are confirmed even more expressively. However, further we bring some doubt in their objective existence.

2. Proton Damped Oscillation Regular Structures Seem to Have No Objective Existence

Collecting also other existing data on the proton “effective” FF measured till now [4,5,6,7] together with [2,3], then again describing them by a two free parametric formula [11], utilizing the least squares method given by the CERN programm MINUIT, however, now with slightly different values of parameters $A=8.9\pm 0.3$ and $m_a^2=9.2\pm 0.8{\mathrm{GeV}}^2$, finally subtracting the obtained fitted curve from all completed proton “effective” EM FF data, the result of the paper [10] is confirmed in Figure 2.

Here we would like to note, that despite of the latter discussion on the proton “effective” EM FF data, also the separate data on $|G_E^p\left(s\right)|$ and $|G_M^p\left(s\right)|$ exist, which, however, are considered to be not fully credible, because they have been extracted from ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ under some simplified assumptions [12,13,14,15,16,17,18,19]. Like that the $|G_E^p\left(s\right)|$ and $|G_M^p\left(s\right)|$ are roughly equal if measurements of the ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ are carried out not very far above the $p\overline{p}$ threshold, or that $|G_E^p\left(s\right)|\approx 0$ at the energies enough far away from it, as it follows directly from the explicit form of the ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ in (1).

These data in combination with the most reliable proton electric FF spacelike data [20,21,22,23,24,25] given by the ratio $G_E^p\left(s\right)/G_M^p\left(s\right)$ and measured by the Akhiezer-Rekalo polarization method [26,27], and with the proton magnetic FF spacelike data [28,29,30,31,32,33,34], evaluated by the Rosenbluth technique from the differential cross section of the elastic scattering of electrons on unpolarized protons, are presented in Figure 3.

Despite the inaccuracy of the data in Figure 3, we have described them by the Unitary and Analytic (U&A) model [35] of the proton EM structure, derived in the paper [36] in detail, being very productive in investigation of various problems of the electromagnetic structure of hadrons.

The model is formulated in the language of flavour independent the isoscalar $F_{1s}^N\left(s\right),F_{2s}^N\left(s\right)$ and isovector $F_{1v}^N\left(s\right),F_{2v}^N\left(s\right)$ parts of the Dirac and Pauli FFs, which appear, if the mixed transformation properties of the EM current $J_{\mu }^{EM}\left(0\right)$ under the rotation in the isospin space are utilized. The latter consist in the fact, that one part of $J_{\mu }^{EM}\left(0\right)$ transforms like the isoscalar and another part like the third component of the isovector. As a result of these transformation properties there are splittings of the proton Dirac and Pauli EM FFs to flavour independent isoscalars $F_{1s}^N\left(s\right),F_{2s}^N\left(s\right)$ and isovectors $F_{1v}^N\left(s\right),F_{2v}^N\left(s\right)$ FFs as follows

$$\begin{array}{c}\hfill F_1^p\left(s\right)=[F_{1s}^N\left(s\right)+F_{1v}^N\left(s\right)]\\ \hfill F_2^p\left(s\right)=[F_{2s}^N\left(s\right)+F_{2v}^N\left(s\right)]\end{array}$$

(6)

whereby the sign between them is specified by the sign of the third component of the proton isospin to be $+1/2$.

Then the relation between the proton EM FFs and the flavour-independent isoscalar and isovector parts of the proton Dirac and Pauli FFs is

$$\begin{array}{c}\hfill G_E^p\left(s\right)=[F_{1s}^N\left(s\right)+F_{1v}^N\left(s\right)]+s/4m_p^2[F_{2s}^N\left(s\right)+F_{2v}^N\left(s\right)]\\ \hfill G_M^p\left(s\right)=[F_{1s}^N\left(s\right)+F_{1v}^N\left(s\right)]+[F_{2s}^N\left(s\right)+F_{2v}^N\left(s\right)]\end{array}$$

(7)

with normalizations

$$\begin{array}{cc}\hfill G_E^p\left(0\right)& =1\hfill \\ \hfill G_M^p\left(0\right)& ={\mu }_p\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill F_{1s}^N\left(0\right)& =F_{1v}^N\left(0\right)=1/2\hfill \\ \hfill F_{2s}^N\left(0\right)& ={\displaystyle \frac{1}{2}}({\mu }_p+{\mu }_n-1)\hfill \\ \hfill F_{2v}^N\left(0\right)& ={\displaystyle \frac{1}{2}}({\mu }_p-{\mu }_n-1),\hfill \end{array}$$

(9)

where ${\mu }_p=2.79284{\mu }_N$ and ${\mu }_n=-1.91304{\mu }_N$ are the magnetic moments of the proton and neutron, respectively, which are measured in the nucleon magneton ${\mu }_N$ units.

So, an explicit form of $F_{1s}^N\left(s\right)$, $F_{1v}^N\left(s\right)$, $F_{2s}^N\left(s\right)$, $F_{2v}^N\left(s\right)$ leads to a description of the proton EM FFs $G_E^p\left(s\right)$ and $G_M^p\left(s\right)$ in the whole region of their definition.

However, there is no theory predicting behaviors of the functions $F_{1s}^N\left(s\right)$, $F_{1v}^N\left(s\right)$, $F_{2s}^N\left(s\right)$, $F_{2v}^N\left(s\right)$ and we know only some of their properties, like their analyticity, fulfilling the reality condition $F^{*}\left(s\right)=F\left(s^{*}\right)$, normalization conditions (9) and the asymptotic behaviors,

$$\begin{array}{cc}\hfill F_{1s}^N{\left(s\right)}_{\left|s\right|\to \infty }& =F_{1v}^N{\left(s\right)}_{\left|s\right|\to \infty }\sim {\displaystyle \frac{1}{s^2}}\hfill \\ \hfill F_{2s}^N{\left(s\right)}_{\left|s\right|\to \infty }& =F_{2v}^N{\left(s\right)}_{\left|s\right|\to \infty }\sim {\displaystyle \frac{1}{s^3}}\hfill \end{array}$$

(10)

following from QCD up to the logarithmic corrections [37] for $G_E^p\left(s\right)$ and $G_M^p\left(s\right)$ EM FFs.

All above mentioned properties are contained just in the constructed proton EM structure U&A model, which is defined on the four-sheeted Riemann surface, generated by two square root branch points. One $s_0$ representing two-pion, or three-pion threshold, dependent if it is concerned of the isovector parts, or the isoscalar parts of the Dirac and Pauli FFs, and another, the inelastic effective threshold $s_{in}$, approximating contributions of all other inelastic thresholds generated by the particles in the intermediate states of the unitarity condition, therefore in a description of the data it is left as a free parameter of the model and its numerical value is found by an optimal description of existing data.

The four-sheeted Riemann surfaces, on which the U&A models of $F_{1s}^N\left(s\right)$, $F_{1v}^N\left(s\right),F_{2s}^N\left(s\right)$ and $F_{2v}^N\left(s\right)$ are defined, are mapped by the transformation

$$\begin{array}{c}\hfill V\left(s\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{s-s_0^s}{s_0^s}\right)}^{1/2}}-\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{s-s_0^s}{s_0^s}\right)}^{1/2}}}{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{s-s_0^s}{s_0^s}\right)}^{1/2}}+\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{s-s_0^s}{s_0^s}\right)}^{1/2}}}},\end{array}$$

(11)

into one V-plane with unit disc, whereby the first physical sheet of the Riemann surface is completely mapped into left half disc, the second sheet is mapped into right half disc, the third sheet into left half plane besides the left half disc and finally the fourth sheet is mapped into right half plane besides the right half disc.

The behavior of FFs under consideration is practically formed by the experimentally confirmed isoscalar and isovector vector meson resonances with quantum numbers of the photon $J^{PC}=1^{--}$ and the isospin values 1 and 0, which one finds in the Review of Particle properties, published by the Particle Data Group always with all new achievements every even year.

Every of these vector meson resonances with the mass $m_r^2$ (their width is in the first step ignored) is mapped into four poles, either on the imaginary axis (two complex conjugate inside of the unit disc and two complex conjugate out of the unit disc) by means of the relation

$$\begin{array}{c}\hfill V\left(m_r^2\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}-\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}}{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}+\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}}},\end{array}$$

(12)

if the mass of the resonance fulfils the condition $s_0^s<m_r^2<s_{in}$, or into four symmetric poles on the unit circle by means of the following relation

$$\begin{array}{c}\hfill V\left(m_r^2\right)=i{\displaystyle \frac{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}+\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}}{\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}+{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}-\sqrt{{\left(\frac{s_{in}^{1s}-s_0^s}{s_0^s}\right)}^{1/2}-{\left(\frac{m_r^2-s_0^s}{s_0^s}\right)}^{1/2}}}},\end{array}$$

(13)

if the mass of the resonance fulfils the condition $m_r^2>s_{in}$.

Only with such forms of transformations then in explicit expressions of $F_{1s}^N\left(s\right)$, $F_{1v}^N\left(s\right)$, $F_{2s}^N\left(s\right)$, $F_{2v}^N\left(s\right)$, after substituting the instability of the resonance by means of the Breit-Wigner relation $m_r^2\to {(m_r-i{\displaystyle \frac{\mathsf{\Gamma }}{2}})}^2$, all considered resonances from the imaginary axis, and from the unit circle, are shifted exclusively into the unphysical sheets of the Riemann surface and in this manner the correct analyticity and unitarity of the model on the first sheet of the considered FFs is not violated.

Further a construction of explicit forms of the flavour-independent $F_{1s}^N\left(s\right)$, $F_{1v}^N\left(s\right)$, $F_{2s}^N\left(s\right)$, $F_{2v}^N\left(s\right)$ functions is demonstrated in detail, with all above discussed properties, however, depending on some free parameters to be numerically evaluated in comparison of the model with experimental data on the proton EM FFs in Figure 3.

We start from the naive vector-meson-dominance (VMD) model representation

$$\begin{array}{c}\hfill F\left(s\right)=\sum _{j=1}^n{\displaystyle \frac{m_j^2}{m_j^2-s}}(f_{jhh}/f_j),\end{array}$$

(14)

which does not reflect any of the properties formulated above, besides the vector meson resonances in the $e^{+}{e}^{-}$ annihilation processes into hadron-antihadron pairs, considering them first as the stable particles. The $f_{jhh}$ and $f_j$ are the vector-meson coupling constants of a vector meson interaction with hadron h and the universal vector meson coupling constant describing the charged lepton decay of the considered resonance, respectively.

Each term in (14) behaves like

$$\begin{array}{c}\hfill F{\left(s\right)}_{\left|s\right|\to \infty }\sim s^{-1},\end{array}$$

(15)

however, from a destructive interference of ground and excited state vector mesons, one can enforce the sum to behave like

$$\begin{array}{c}\hfill F{\left(s\right)}_{\left|s\right|\to \infty }\sim s^{-m},\end{array}$$

(16)

where $1<m\le n$ and “n” is a number of considered vector mesons in (14).

In detail, the procedure one finds in [38], where first (14) is transformed into a rational function of “s”, creating a common denominator and obtaining the polynomial of (n − 1) degree in the numerator. Afterwards requiring the first (m − 1) coefficients from the highest powers of “s” to be zero, a complicated system of (m − 1) linear homogeneous algebraic equations for coupling constant ratios $a_j=(f_{jhh}/f_j)$ is found, from the solution of which, together with the normalization conditions (9), one could find explicit forms of $F_{1s}^N\left(s\right),F_{1v}^N\left(s\right),F_{2s}^N\left(s\right),F_{2v}^N\left(s\right)$ functions.

Nevertheless, there is a way to simplify the latter complicated system of linear homogeneous algebraic equations.

Really, the utilization of the superconvergent sum rules for the imaginary parts of FFs and their approximation by the $\delta$-function one finds a more simple system of (m − 1) linear homogeneous algebraic equations, which is proved in [38] to be equivalent with the previous one.

Finally, their solution together with the normalization of FFs (9) at length given in [39], gives a general form of $F_{1s}^N\left(s\right),F_{1v}^N\left(s\right),F_{2s}^N\left(s\right),F_{2v}^N\left(s\right)$ FFs. They are expressed through the masses of considered vector meson resonances, possess the asymptotic behaviors (16) and normalization (9), and are ready for the unitarization.

Such well physically founded U&A model is formulated in the language of the isoscalar $F_{1s}^N\left(t\right),F_{2s}^N\left(t\right)$ and isovector $F_{1v}^N\left(t\right),F_{2v}^N\left(t\right)$ parts of the proton Dirac and Pauli EM FFs, in which all 9 experimentally confirmed [40], 3 isovector and 6 isoscalar, vector mesons, $\rho \left(770\right)$, $\omega \left(782\right)$, $\varphi \left(1020\right)$, ${\omega }^{\prime }\left(1420\right)$, ${\rho }^{\prime }\left(1450\right)$, ${\omega }^{″}\left(1650\right)$, ${\varphi }^{\prime }\left(1680\right)$, ${\rho }^{″}\left(1700\right)$, ${\varphi }^{″}\left(2170\right)$ with quantum numbers of the photon $\left(1^{--}\right)$ are taken into account. Then, if their masses and widths are fixed at the values from the PDG Table [40], the U&A model, derived in [36], depends on 12 free parameters numerically evaluated (see

Table 1. Values of free parameters of the proton EM structure $U\&A$ model evaluated in the analysis of the proton EM FFs data with graphic results in Figure 4.

List of parameters

${\chi }^2/ndf=1.74;$$s_{in}^{1s}=(1.675\pm 0.036){\mathrm{GeV}}^2$; $s_{in}^{1v}=(2.968\pm 0.009){\mathrm{GeV}}^2;$

$s_{in}^{2s}=(1.859\pm 0.002){\mathrm{GeV}}^2$; $s_{in}^{2v}=(2.443\pm 0.021){\mathrm{GeV}}^2;$

$(f_{{\omega }^{\prime }pp}^{\left(1s\right)}/f_{{\omega }^{\prime }})=-0.294\pm 0.002$; $(f_{{\varphi }^{\prime }pp}^{\left(1s\right)}/f_{{\varphi }^{\prime }})=-0.530\pm 0.003;$

$(f_{\omega pp}^{\left(1s\right)}/f_{\omega })=0.638\pm 0.003$; $(f_{\varphi pp}^{\left(1s\right)}/f_{\varphi })=-0.027\pm 0.001;$

$(f_{{\varphi }^{\prime }pp}^{\left(2s\right)}/f_{{\varphi }^{\prime }})=0.308\pm 0.016$; $(f_{\omega pp}^{\left(2s\right)}/f_{\omega })=0.168\pm 0.038;$

$(f_{\varphi pp}^{\left(2s\right)}/f_{\varphi })=0.123\pm 0.004$; $(f_{\rho pp}^{\left(1v\right)}/f_{\rho })=-0.080\pm 0.001$.

) in a fitting algorithm (least squares utilizing the CERN program MINUIT) of data in Figure 3.

The explicit form of the U&A model is given in Appendix A.

A graphical presentation of the quality of description of data with ${\chi }^2/ndf=1.74$ is demonstrated in Figure 4. It appeared however that using the parameters in Table 1, the $U\&A$ model disagrees with the published data on ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$, strongly underestimating it at s > 5 GeV² (Figure 5). Since the cross section at large s is dominated by the proton magnetic form factor squared, the underestimation is obviously caused by a similar underestimation of this form factor seen in Figure 4.

Therefore the parameter values in Table 1 have been corrected to the values fairly describing the ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ experimental data by a direct fit of them by the cross section to be expressed through $G_E^p\left(s\right)$ and $G_M^p\left(s\right)$ with starting numerical values of parameters from the Table 1.

As a result one expects that then also the proton “effective” EM FF data through the relation (4) will be more precisely reproduced.

Such task from a mathematical point of view is rigorously unsolvable, however, practically one could realize it, because the form of functions $G_E^p\left(s\right)$ and $G_M^p\left(s\right)$ is by the U&A model strongly restricted. They represent a well-matched unification of pole contribution of unstable vector mesons with a cut structure in the complex s plane, where the cuts represent the so-called continua contributions generated by the exchange of more than one particle in the corresponding Feynman diagrams. Moreover they ensure that the FFs imaginary parts are nonzero only above the lowest possible branch point on the positive real axis, as it is required by the FFs unitarity conditions.

Besides the analyticity, FFs fulfill the reality condition $F^{*}\left(s\right)=F\left(s^{*}\right)$, which guarantees FFs to be real on the real axis below their the lowest branch point and generate always two complex conjugate poles exclusively on unphysical sheets for each vector meson. All FFs are normalized, possess the asymptotic behavior predicted by the QCD up to the logarithmic corrections and depend on reasonable number of free parameters as given in Table 1.

The corrected values by the procedure discussed above are presented in

Table 2. Corrected values of free parameters in Table 1 by a direct description of the data on ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ in Figure 5.

List of parameters

${\chi }^2/ndf=1.85;$$s_{in}^{1s}=(1.428\pm 0.020){\mathrm{GeV}}^2$; $s_{in}^{1v}=(2.924\pm 0.078){\mathrm{GeV}}^2;$

$s_{in}^{2s}=(1.830\pm 0.176){\mathrm{GeV}}^2$; $s_{in}^{2v}=(2.687\pm 0.078){\mathrm{GeV}}^2;$

$(f_{{\omega }^{\prime }pp}^{\left(1s\right)}/f_{{\omega }^{\prime }})=-0.031\pm 0.008$; $(f_{{\varphi }^{\prime }pp}^{\left(1s\right)}/f_{{\varphi }^{\prime }})=-0.642\pm 0.006;$

$(f_{\omega pp}^{\left(1s\right)}/f_{\omega })=0.997\pm 0.003$; $(f_{\varphi pp}^{\left(1s\right)}/f_{\varphi })=-0.032\pm 0.001;$

$(f_{{\varphi }^{\prime }pp}^{\left(2s\right)}/f_{{\varphi }^{\prime }})=0.785\pm 0.018$; $(f_{\omega pp}^{\left(2s\right)}/f_{\omega })=-0.497\pm 0.184;$

$(f_{\varphi pp}^{\left(2s\right)}/f_{\varphi })=0.149\pm 0.002$; $(f_{\rho pp}^{\left(1v\right)}/f_{\rho })=0.185\pm 0.004$.

and by means of them a perfect agreement of the calculated ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ with ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ data is achieved, as one can see in Figure 6.

Having the curve of ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ in Figure 6, one calculates a behavior of the proton “effective” EM FF in Figure 7, employing the relation (4), which is in agreement with proton “effective” EM FF data.

Finally, subtraction of the curve in Figure 7, obtained by means of the relation (4), from existing experimental data on the proton “effective” EM FF, indicates in Figure 8 (compare it with Figure 1 and Figure 2) no objective existence of the “proton damped oscillation regular structures”.

To a similar conclusion came also the authors of the paper [41] on the base of results obtained by an application of the timelike nucleon EM FFs constructed within the simple extended vector meson dominance model.

3. Conclusions and Discussion

In this paper an experimental data on the proton EM FFs, as obtained from experiments on the $e^{+}{e}^{-}$ annihilation into proton-antiproton pair, are investigated in more detail. First they have been described by well physically founded Unitary and Analytic model of the electromagnetic structure of hadrons. Then the results are confronted with the data on the corresponding total cross section and a transparent disagreement has been found (see Figure 5). Therefore the same model for the total cross section has been corrected by parameters in Table 2, obtained in a direct fit of experimental total cross section data with errors. The obtained improved agreement of the behavior of ${\sigma }_{tot}(e^{+}{e}^{-}\to p\overline{p})$ with the data (see Figure 6) finally has been used to calculate an improved behavior of the proton “effective” EM FF in Figure 7, which then was subtracted from existing data with errors. The obtained result is presented in Figure 8, which indicates (compare it with Figure 1 and Figure 2) no objective existence of the “proton damped oscillation regular structures”, first revealed in the paper [10]. In our opinion, an explanation of the latter result is probably in a description of the proton “effective” EM FF data by two free parametric formula of [11], which is unable to reproduce the data with the adequate accuracy.