# On the Transfer of Polarization from the Initial to the Final Proton
in the Elastic Process $e\vec{p}\to e\vec{p}$

M. V<EMAIL_ADDRESS>Joint Institute for Power and
Nuclear Research – Sosny, National Academy of Sciences of Belarus, 220109
Minsk, Belarus

###### Abstract

The $Q^{2}$ dependence of the ratio of the cross sections with and without
proton spin flip, as well as the polarization asymmetry in the process
$e\vec{p}\to e\vec{p}$ has been numerically analyzed using the results of
JLab’s polarization experiments on the measurements of the ratio of the Sachs
form factors in the $\vec{e}p\to e\vec{p}$ process. The calculations have been
made for the case where the initial (at rest) and final protons are fully
polarized and have a common spin quantization axis, which coincides with the
direction of motion of the final proton. The longitudinal polarization
transfer to the proton has been calculated in the case of the partially
polarized initial proton for a kinematics used in the experiment reported in
[A. Liyanage et al. (SANE Collaboration), Phys. Rev. C 101, 035206 (2020)],
where the double spin asymmetry was measured in the $\vec{e}\vec{p}\to ep$
process. A noticeable sensitivity of the polarization transfer to the proton
to the form of the $Q^{2}$ dependence of the ratio $\mu_{p}G_{E}/G_{M}$ has
been found. This sensitivity may be used to conduct a new independent
experiment to measure this dependence in the $e\vec{p}\to e\vec{p}$ process. A
criterion to assess the reliability of measurements of the ratio of Sachs form
factors using the Rosenbluth technique has been proposed and used to analyze
the results of two experiments.

###### pacs:

13.40.Gp; 13.60.Fz; 13.88.+e; 29.25.Pj

Introduction. The electric $G_{E}$ and magnetic $G_{M}$ form factors, the so-
called Sachs form factors (SFFs), have been experimentally studied since the
mid-1950s in the elastic scattering of unpolarized electrons off a proton. All
experimental data on the behavior of the SFFs have been obtained by applying
the Rosenbluth technique (RT), which is based on using the Rosenbluth cross
section (in the one-photon approximation) for the $ep\to ep$ process in the
rest frame of the initial proton Rosen :

$\displaystyle\sigma=\frac{d\sigma}{d\Omega_{e}}=\frac{\alpha^{2}E_{2}\cos^{2}(\theta_{e}/2)}{4E_{1}^{3}\sin^{4}(\theta_{e}/2)}\frac{1}{1+\tau_{p}}\left(G_{E}^{2}+\frac{\tau_{p}}{\varepsilon}G_{M}^{2}\right).$
(1)

Here, $\tau_{p}=Q^{2}/4M^{2},Q^{2}=-q^{2}=4E_{1}E_{2}\sin^{2}(\theta_{e}/2)$
is the square of the momentum transfer to the proton; $M$ is the mass of the
proton; $E_{1}$ and $E_{2}$, are the energies of the initial and final
electrons, respectively; $\theta_{e}$ is the electron scattering angle;
$\varepsilon=[1+2(1+\tau_{p})\tan^{2}(\theta_{e}/2)]^{-1}$ is the degree of
linear polarization of the virtual photon Dombey ; Rekalo74 ; AR ; GL97  that
varies in the range $0\leqslant\varepsilon\leqslant 1$; and $\alpha=1/137$ is
the fine structure constant.

Formula (1) shows that the main contribution to the cross section for the
process $ep\to ep$ at high $Q^{2}$ values comes from the term proportional to
$G_{M}^{\,2}$. Because of this, extracting the contribution of $G_{E}^{\,2}$
even at $Q^{2}\geqslant 1$ GeV2 becomes a challenging problem ETG15 ;
Punjabi2015 . The RT was used to determine the experimental dependence of SFFs
on $Q^{2}$, which is described up to $Q^{2}\approx 6$ GeV2 by the dipole
approximation; their ratio

$R\equiv\mu_{p}G_{E}/G_{M}$ (2)

is quite accurately approximated by the equality $R\approx 1$, where
$\mu_{p}$=2.79 is the magnetic moment of the proton.

Akhiezer and Rekalo Rekalo74  proposed a method to measure the ratio $R$ that
is based on the polarization transfer from the initial electron to the final
proton in the process $\vec{e}p\to e\vec{p}$. Precision experiments based on
this method conducted at JLab Jones00 ; Gay01 ; Gay02  revealed that the ratio
$R$ decreases rapidly with increasing $Q^{2}$, which indicates that the dipole
dependence (scaling) of the SFFs is violated. This decrease proved to be
linear in the range $0.4\,\mbox{GeV}^{2}\leqslant Q^{2}\,\leqslant 5.6$ GeV2.

Measurements of $R$, which were repeated with a higher accuracy Pun05 ;
Puckett10 ; Puckett12 ; Puckett17 ; Qattan2005  in a broad $Q^{2}$ range up to
8.5 GeV2 using both the Akhiezer – Rekalo method Rekalo74  and the RT, only
confirmed these disagreements.

Experimental values of $R$ Liyanage2020  were obtained by the SANE
collaboration using the third approach. Namely, they were extracted from the
measurements of the double spin asymmetry in the process $\vec{e}\vec{p}\to
ep$ in the case where the electron beam and the proton target are partially
polarized. The degree of polarization of the proton target was $P_{t}=(70\pm
5)$%. The experiment was carried out at two electron beam energies
$E_{1}=5.895$ and 4.725 GeV and two values of $Q^{2}=2.06$ and 5.66 GeV2. The
values of $R$ obtained in Liyanage2020  are in good agreement with the results
of previous polarization experiments carried out in Jones00 ; Gay01 ; Gay02 ;
Pun05 ; Puckett10 ; Puckett12 ; Puckett17 .

The fourth method was proposed in JETPL18 . This method allows extracting
$G_{E}^{\,2}$ and $G_{M}^{\,2}$ from the direct measurements of cross sections
with and without proton spin flip in the elastic process with the polarization
transfer from the initial to the final proton:

$\displaystyle e(p_{1})+\vec{p}\,(q_{1},s_{1})\to
e(p_{2})+\vec{p}\,(q_{2},s_{2}),$ (3)

when the initial (at rest) proton is fully polarized along the direction of
motion of the detected final recoil proton. This method, which is also
applicable in the two-photon exchange (TPE) approximation, enables measurement
of the squares of absolute values of the generalized SFFs in a similar way
JETPL19 .

In this study, we use the results of JLab’s polarization experiments, where
the ratio $R$ was measured in the process $\vec{e}p\to e\vec{p}$, to
numerically analyze the $Q^{2}$ dependence of the ratio of the cross sections
with and without proton spin flip and the polarization asymmetry in the
process $e\vec{p}\to e\vec{p}$ in the case where the initial (at rest) and
final protons are fully polarized and have a common spin quantization axis
that coincides with the direction of motion of the detected final recoil
proton. In the case of the partially polarized initial proton, the
longitudinal polarization transfer to the proton was calculated in the
kinematics of the experiment Liyanage2020 . A criterion to assess the
reliability of the measured $R$ ratio using the RT is proposed, which is used
to analyze the measurements made in two well-known experiments Andivahis1994 ;
Qattan2005 .

Cross section for the process $e\vec{p}\to e\vec{p}$ in the rest frame of the
initial proton. Let us consider spin 4-vectors $s_{1}$ and $s_{2}$ of the
initial and final protons with 4-momenta $q_{1}$ and $q_{2}$, respectively, in
process (3) in an arbitrary reference frame. The conditions of orthogonality
($s_{i}q_{i}=0$) and normalization ($s_{i}^{2}=-1$) unambiguously provide the
following expressions for the time ($s_{i0}$) and space
($\mbox{\boldmath$s$}_{i}$) components of these spin 4-vectors
$s_{i}=(s_{i0},\mbox{\boldmath$s$}_{i})$ in terms of their 4-velocities
$v_{i}=q_{i}/M$ ($i=1,2$):

$\displaystyle
s_{i}=(s_{i0},\mbox{\boldmath$s$}_{i}),\;s_{i0}=\mbox{\boldmath$v$}_{i}\,\mbox{\boldmath$c$}_{i},\;\mbox{\boldmath$s$}_{i}=\mbox{\boldmath$c$}_{i}+\frac{(\mbox{\boldmath$c$}_{i}\mbox{\boldmath$v$}_{i})\,\mbox{\boldmath$v$}_{i}}{1+v_{i0}}\;,$
(4)

where the unit 3-vectors $\mbox{\boldmath$c$}_{i}$
($\mbox{\boldmath$c$}_{i}^{2}=1$) specify the spin projection (quantization)
axes.

In the laboratory reference frame, where $q_{1}=(M,\mbox{\boldmath$0$})$ and
$q_{2}=(q_{20},\mbox{\boldmath$q$}_{2})$, the spin projection axes
$\mbox{\boldmath$c$}_{1}$ and $\mbox{\boldmath$c$}_{2}$ are chosen such that
they coincide with the direction of motion of the final proton:

$\displaystyle\mbox{\boldmath$c$}=\mbox{\boldmath$c$}_{1}=\mbox{\boldmath$c$}_{2}=\mbox{\boldmath$n$}_{2}=\mbox{\boldmath$q_{2}$}/|\mbox{\boldmath$q$}_{2}|\,.$
(5)

Then the spin 4-vectors $s_{1}$ and $s_{2}$ of the initial and final protons,
respectively, in the laboratory reference frame have the form

$\displaystyle
s_{1}=(0,\mbox{\boldmath$n$}_{2})\,,\;s_{2}=(|\mbox{\boldmath$v$}_{2}|,v_{20}\,\mbox{\boldmath$n_{2}$})\,,\,\mbox{\boldmath$n$}_{2}=\mbox{\boldmath$q_{2}$}/|\mbox{\boldmath$q$}_{2}|\,.$
(6)

Method JETPL18  is based on the formula for differential cross section for
process (3) in the laboratory reference frame in the case where the initial
and final protons are polarized and have a common axis of the spin projections
$c$ given by Eq. (5):

$\displaystyle\frac{d\sigma_{\delta_{1},\delta_{2}}}{d\Omega_{e}}$
$\displaystyle=$
$\displaystyle\omega_{+}\sigma^{\uparrow\uparrow}+\omega_{-}\sigma^{\downarrow\uparrow}\,,$
(7) $\displaystyle\sigma^{\uparrow\uparrow}$ $\displaystyle=$
$\displaystyle\sigma_{M}\,G^{2}_{E},\;\;\sigma^{\downarrow\uparrow}=\sigma_{M}\frac{\tau_{p}}{\varepsilon}\,G^{2}_{M}\,,$
(8) $\displaystyle\sigma_{M}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}E_{2}\cos^{2}(\theta_{e}/2)}{4E_{1}^{\,3}\sin^{4}(\theta_{e}/2)}\frac{1}{1+\tau_{p}}\,.$
(9)

Here, $\omega_{\pm}$ are the polarization factors specified by the formulas

$\displaystyle\omega_{+}=(1+\delta_{1}\delta_{2})/2,\,\,\omega_{-}=(1-\delta_{1}\delta_{2})/2\,,$
(10)

where $\delta_{1,2}$ are the double projections of the spins of the initial
and final protons on the common axis of spin projections $c$ (5). It should be
noted that Eq. (7) is valid at $-1\leqslant\delta_{1,2}\leqslant 1$.

The corresponding experiment to measure the squares of the SFFs in the
processes with and without proton spin flip can be implemented in the
following way. The initial proton at rest should be fully polarized along the
direction of motion of the detected final recoil proton. By measuring the
differential cross sections $\sigma^{\uparrow\uparrow}$ and
$\sigma^{\downarrow\uparrow}$ (8) as functions of $Q^{2}$, it is possible to
determine the $Q^{2}$ dependence of $G_{E}^{2}$ and $G^{2}_{M}$ and to measure
in this way these form factors.

It is noteworthy that Eq. (7), similar to Eq. (1), is the sum of two terms one
of which contains only $G_{E}^{\,2}$ and the second contains only
$G_{M}^{\,2}$. By averaging and summing Eq. (7) over the polarizations of the
initial and final protons, the Rosenbluth cross section given by Eq. (1) and
denoted as $\sigma_{R}$ can be represented in the alternative form JETPL18

$\displaystyle\sigma_{R}=\sigma^{\uparrow\uparrow}+\sigma^{\downarrow\uparrow}\,.$
(11)

Consequently, the two terms one of which contains only $G_{E}^{~{}2}$ and the
second contains only $G_{M}^{~{}2}$ in the representation of Rosenbluth
formula (1) in the form of Eq. (11) physically mean the cross sections with
and without proton spin flip, respectively, in the case where the initial
proton at rest is fully polarized along the direction of motion of the final
proton.

It is often asserted in publications, including textbooks on elementary
particle physics, that the SFFs are employed simply because of convenience
since they enable the representation of the Rosenbluth formula in a
straightforward and concise form. Since such formal conclusions regarding
advantages of using the form factors are also contained in popular monographs
published many years ago AB ; BLP , they are not subject to criticism and are
still reproduced in the literature; see, e.g., Paket2015 .

Cross section (7) can be represented in the form

$\displaystyle
d\sigma_{\delta_{1},\delta_{2}}/d\Omega_{e}=(1+\delta_{2}\delta_{f})(\sigma^{\uparrow\uparrow}+\sigma^{\downarrow\uparrow}),$
(12) $\displaystyle\delta_{f}=\delta_{1}(R_{\sigma}-1)/(R_{\sigma}+1),$ (13)
$\displaystyle
R_{\sigma}=\sigma^{\uparrow\uparrow}/\sigma^{\downarrow\uparrow},$ (14)

where $\delta_{f}$ is the degree of the longitudinal polarization of the final
proton. If the initial proton is fully polarized ($\delta_{1}=1$),
$\delta_{f}$ coincides with the standard definition of polarization asymmetry

$\displaystyle A=(R_{\sigma}-1)/(R_{\sigma}+1)\;.$ (15)

As follows from Eq. (8), the ratio of the cross sections with and without
proton spin flip $R_{\sigma}$ (14) can be represented in terms of the
experimentally measurable quantity $R\equiv\mu_{p}\,G_{E}/G_{M}$:

$\displaystyle
R_{\sigma}=\frac{\sigma^{\uparrow\uparrow}}{\sigma^{\downarrow\uparrow}}=\frac{\varepsilon}{\tau_{p}}\,\frac{G_{E}^{2}}{G_{M}^{2}}=\frac{\varepsilon}{\tau_{p}}\frac{R^{2}}{\mu_{p}^{2}}.$
(16)

The expression on the right-hand side of Eq. (16) for $R_{\sigma}$ is quite
frequently used in publications. For example, the authors of Qattan2015
reported two formulas for the reduced cross sections for the $ep\to ep$
process that include $R_{\sigma}$; however, they seem to be unaware of its
physical meaning.

Using the standard notation and replacing $\delta_{f}$ with $P_{r}$ and
$\delta_{1}$ with $P_{t}$, we rewrite Eq. (13) for the degree of longitudinal
polarization of the final proton in the alternative form

$\displaystyle P_{r}=P_{t}(R_{\sigma}-1)/(R_{\sigma}+1).$ (17)

Inverting the connection in (17), we obtain an expression for $R^{2}$ as a
function of $P_{r}/P_{t}$:

$\displaystyle
R^{2}=\mu_{p}^{2}\,\frac{\tau_{p}}{\varepsilon}\,\frac{1+R_{p}}{1-R_{p}},\;\;R_{p}=\frac{P_{r}}{P_{t}}\,,$
(18)

which can be used to extract the ratio $R$ in the proposed method of the
polarization transfer from the initial to the final proton.

We calculate the $Q^{2}$ dependence of the polarization asymmetry $A$ (15),
the ratio of the cross sections $R_{\sigma}$ (16), and the polarization
transfer to the proton $P_{r}$ (17) for both cases where the dipole dependence
is valid ($R=R_{d}$) and is violated ($R=R_{j}$):

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{d}=1,$
(20) $\displaystyle
R_{j}=\frac{1}{1+0.1430Q^{2}-0.0086Q^{4}+0.0072Q^{6}}.~{}~{}~{}~{}$

The formula for $R_{j}$ is taken from Qattan2015 ; Kelly’s parametrization
Kelly2004  can be used instead.

Numerical results and discussion. To identify general characteristics, Figs. 1
and 2 show the $x=Q^{2}$ dependencies of the ratio of cross sections
$R_{\sigma}$ (16) and polarization asymmetry $A$ (15), respectively,
calculated numerically with (lines $Rd1,Rd2,\ldots,Rd6$ and
$Ad1,Ad2,\ldots,Ad6)$ $R=R_{d}$ given by Eq. (20) and (lines
$Rj1,Rj2,\ldots,Rj6$ and $Aj1,Aj2,\ldots,Aj6$) $R=R_{j}$ given by Eq. (20) for
the electron-beam energies $E_{1}=1,2,\ldots,6$ GeV, respectively.

The plots displayed in Fig. 1 show that the ratio of the cross sections with
and without proton spin flip $R_{\sigma}$ (16) for all electron beam energies
decreases with increasing $Q^{2}$. However, this decrease at $R=R_{j}$ is
faster than in the case of dipole dependence ($R=R_{d}$) owing to the presence
of the denominator in Eq. (20) for $R_{J}$. It should also be noted that the
difference in the behavior of the ratio $R_{\sigma}$ (16) calculated with
$R=R_{d}$ and $R_{j}$ is insignificant at a low electron-beam energy.

Figure 1:  Ratio $R_{\sigma}$ (16) calculated with (lines $Rd1$,
$Rd2$,…,$Rd6$) $R=R_{d}$ given by Eq. (20) and (lines $Rj1$, $Rj2$, …,$Rj6$)
$R=R_{j}$ given by Eq. (20) versus $Q^{2}$ (GeV2) for the energies
$E_{1}=1,2,...,6$ GeV, respectively.

Figure 2:  Polarization asymmetry $A$ (15) calculated with (lines $Ad1$,
$Ad2$,…,$Ad6$) $R=R_{d}$ given by Eq. (20) and (lines $Aj1$, $Aj2$,…,$Aj6$)
$R=R_{j}$ given by Eq. (20) versus $Q^{2}$ (GeV2) for the energies
$E_{1}=1,2,...,6$ GeV, respectively.

As seen in Fig. 1, the $Q^{2}$ dependence of $R_{\sigma}$ for each electron-
beam energy has a sharp boundary at $Q^{2}_{max}$, which is the maximum
possible value of $Q^{2}$ that corresponds to the backscattering of the
electron, i.e., scattering at $180^{\circ}$. The values $Q^{2}_{max}$ for each
electron-beam energy $E_{1}=1,2,...,6$ GeV are presented in Table 1, which
shows that $Q^{2}_{max}$ does not exceed 10.45 GeV2 for all energies under
consideration.

Table 1:  Values of $Q^{2}_{max}$ that set the boundaries of the $Q^{2}$ spectra of $R_{\sigma}$ and the values of $(Q_{0})_{\\{d,j\\}}^{2}$ at which $\sigma^{\uparrow\uparrow}=\sigma^{\downarrow\uparrow}$; the polarization asymmetry (15) is zero in this case  $E_{1}$ (GeV) | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0
---|---|---|---|---|---|---
$Q^{2}_{max}$ (GeV2) | 1.277 | 3.040 | 4.868 | 6.718 | 8.578 | 10.443
$(Q_{0}^{2})_{d}$ (GeV2) | 0.358 | 0.424 | 0.435 | 0.446 | 0.446 | 0.446
$(Q_{0}^{2})_{j}$ (GeV2) | 0.336 | 0.380 | 0.391 | 0.402 | 0.402 | 0.402

Table 1 also displays the values of $(Q_{0}^{2})_{\\{d,j\\}}$ that correspond
to the coinciding cross sections with and without proton spin flip; in this
case, their ratio is $R_{\sigma}=1$, and the polarization asymmetry is zero.
In the case of dipole dependence, $(Q_{0}^{2})_{d}\approx M^{2}/2$, where $M$
is the mass of the proton. If the dipole dependence is violated,
$(Q_{0}^{2})_{j}\approx 0.40$ GeV2; i.e., the cross sections
$\sigma^{\uparrow\uparrow}$ and $\sigma^{\downarrow\uparrow}$ become equal at
approximately the same point where the linear decrease in the ratio $R$
begins. Thus, the points where $Q^{2}=Q_{0}^{2}$ are in a certain sense
specific.

If $Q^{2}>Q_{0}^{2}$, the spin-flip cross section
$\sigma^{\downarrow\uparrow}$ exceeds the cross section without spin flip
$\sigma^{\uparrow\uparrow}$, and their ratio is then $R_{\sigma}<1$. As a
result, the helicity carried away by the recoil proton becomes negative. Its
absolute value reaches a maximum value of 1 for the backscattered electron.

The calculations represented in Fig. 1 make it possible to understand why
measurements of the ratio $R$ using the RT at high $Q^{2}$ values are
difficult. They should be conducted in a kinematics where the relative
contribution of the $\sigma^{\uparrow\uparrow}$ term to the cross section
$\sigma_{R}$ (11) exceeds the accuracy $\Delta_{0}$ with which the Rosenbluth
cross section is measured in this experiment:

$\displaystyle\frac{\sigma^{\uparrow\uparrow}}{\sigma^{\uparrow\uparrow}+\sigma^{\downarrow\uparrow}}=\frac{R_{\sigma}}{(1+R_{\sigma})}>\Delta_{0}.$
(21)

The restrictions on the kinematics of experiments where the RT is used have
not yet been considered in publications, including Egle_2018 ; Gramolin2016 ;
Blunden2020 . This problem is nevertheless of importance and deserves special
attention.

According to Eq. (21), the kinematics of the conducted experiment should
satisfy the condition

$\displaystyle R_{\sigma}>\Delta_{0},$ (22)

which can be considered as a necessary condition for reliable measurements and
can be used as a criterion for the reliability of measurements.

Table 2:  Values of $R_{\sigma}$ (16) calculated with $R=R_{d}$ given by Eq. (20) for the electron-beam energies $E_{1}=1,2,...,6$ GeV and $Q^{2}=1,2,...,9$ GeV2 $Q^{2}$ (GeV2) | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0
---|---|---|---|---|---|---|---|---|---
$Rd6$, 6 GeV | 0.444 | 0.215 | 0.136 | 0.095 | 0.068 | 0.049 | 0.034 | 0.022 | 0.012
$Rd5$, 5 GeV | 0.440 | 0.209 | 0.129 | 0.086 | 0.057 | 0.036 | 0.020 | 0.006 |
$Rd4$, 4 GeV | 0.432 | 0.199 | 0.115 | 0.068 | 0.037 | 0.013 |  |  |
$Rd3$, 3 GeV | 0.415 | 0.175 | 0.084 | 0.031 |  |  |  |  |
$Rd2$, 2 GeV | 0.365 | 0.105 |  |  |  |  |  |  |
$Rd1$, 1 GeV | 0.114 |  |  |  |  |  |  |  |
Table 3:  Values of $R_{\sigma}$ (16) calculated with $R=R_{d}$ given by Eq. (20) for $E_{1}$ and $Q^{2}$ values used in the measurements in Andivahis1994  Q2 (GeV2) | 1.75 | 2.50 | 3.25 | 4.00 | 5.00 | 6.00 | 7.00 | 8.83
---|---|---|---|---|---|---|---|---
$Rd9$, 9.800 GeV |  |  |  | 0.107 | 0.083 | 0.067 | 0.055 |
$Rd8$, 5.507 GeV | 0.246 | 0.165 | 0.120 | 0.091 | 0.064 |  |  | 0.006
$Rd7$, 4.507 GeV |  |  |  | 0.079 | 0.049 |  | 0.009 |
$Rd6$, 3.956 GeV |  | 0.157 | 0.100 | 0.067 | 0.035 | 0.012 |  |
$Rd5$, 3.400 GeV |  | 0.136 | 0.085 | 0.049 | 0.016 |  |  |
$Rd4$, 2.837 GeV |  | 0.114 | 0.059 | 0.022 |  |  |  |
$Rd3$, 2.407 GeV | 0.182 | 0.087 | 0.029 |  |  |  |  |
$Rd2$, 1.968 GeV |  | 0.041 |  |  |  |  |  |
$Rd1$, 1.511 GeV | 0.065 |  |  |  |  |  |  |

Table 2 summarizes the values of $R_{\sigma}$ (16) calculated with $R=R_{d}$
for $E_{1}=1,2,\ldots,6$ GeV and $Q^{2}=1,2,\ldots,9$ GeV2 corresponding to
the $Rd1,Rd2,...,Rd6$ plots in Fig. 1. All $R_{\sigma}$ values in Table 2 for
$Q^{2}=7.0$ and 8.0 GeV2, except one where $R_{\sigma}=0.006$, satisfy the
inequality $R_{\sigma}\geqslant 0.020$. Using criterion (22), we conclude that
the RT-based measurements at $Q^{2}=7.0$ GeV2 should be carried out with an
accuracy of no less than 1.9 %, while the measurements made at $Q^{2}=8.0$
GeV2 require an accuracy of $0.3\div 0.5$ %. Thus, RT measurements of the
ratio $R$ at high $Q^{2}$ values is difficult because the relative
contribution of the $\sigma^{\uparrow\uparrow}$ term to the Rosenbluth cross
section (11) decreases, and the accuracy of measuring this quantity should be
correspondingly increased. It is noteworthy that measuring the Rosenbluth
cross sections with an accuracy better than 2 % using the RT technique was an
unrealistic task in old experiments for many reasons Bernauer2014 .

The criterion (22) for the reliability of data does not specify the meaning of
the accuracy $\Delta_{0}$ of the measurement of the Rosenbluth cross sections
and specific inaccuracies (statistical, systematic, or normalization)
determining this accuracy. Analyzing the reliability of the experiment
reported Andivahis1994  using results obtained in Blunden2020 , we show below
that $\Delta_{0}$ is determined by the normalization uncertainty. After that,
the same approach will be applied to analyze the measurements reported in
Qattan2005 .

Analysis of the reliability of two well-known experiments. To analyze the
reliability of the ratio $R$ measured in the experiment in Andivahis1994 , we
calculate the ratio $R_{\sigma}$ (16) for all electron beam energies $E_{1}$
and squares of the momentum transfer to the proton $Q^{2}$ used in this
experiment. The corresponding results are presented in Table 3. Empty cells in
Table 3 indicate that no measurements were made at the corresponding $E_{1}$
and $Q^{2}$ values.

The values displayed in bold in Table 3 for $Q^{2}\geqslant 5.0$ GeV2 are
classified as unreliable results. To be confident in this conclusion, we
address study Blunden2020  where experiments Qattan2005 ; Andivahis1994  were
reanalyzed taking into account the contribution from the TPE. Figure 15b in
Blunden2020  shows that measurements at $Q^{2}<5.0$ GeV2 in Andivahis1994
with the added TPE contribution agree well with results from Puckett17 ;
however, even the inclusion of the TPE fails to remove disagreements at
$Q^{2}=5.0$ GeV2. For this reason, the lower value in the column for
$Q^{2}=5.0$ GeV2 in Table 3 is classified as unreliable, i.e., having
insufficient accuracy. Table 3 and criterion (22) show that the accuracy of
measurements in Andivahis1994  was about $1.7\div 2.1$ %. The normalization
uncertainty in measurements of the Rosenbluth cross section, which at all
$Q^{2}$ values in Andivahis1994  was 1.77 % (see Andivahis1994 ; Gramolin2016
; Bernauer2014 ), falls into the same range. Thus, the accuracy of
measurements $\Delta_{0}$ appearing in Eq. (22) should be identified with the
normalization uncertainty. Given this accuracy (1.77 %), reliability criterion
(22) fails for all values on the diagonal of Table 3 at $Q^{2}\geqslant 5.0$
GeV2.

The value in Table 3 for $Q^{2}=8.83$ GeV2 and $E_{1}=5.507$ GeV corresponds
to $R_{\sigma}=0.006$, a value that requires the accuracy of measurements of
$0.3\div 0.5$ %. However, this accuracy was attained in the experiment
reported in Bernauer2010-1  and performed only in 2010 and, moreover, in the
region $Q^{2}<1$ GeV2. It should be noted that the statistical, systematic,
and normalization uncertainties of measurements at $Q^{2}=8.83$ GeV2 in
Andivahis1994  are 3.89, 1.12, and 1.77 %, respectively Gramolin2016 . They
significantly exceed the measurement accuracy required at $Q^{2}=8.83$ GeV2.
In addition, the procedure of RT based measurements was violated at
$Q^{2}=8.83$ GeV2, since measurements should be performed for each $Q^{2}$
value for at least two or better three energies of the electron beam
Bernauer2010 .

The results of the reanalysis of the experiment Qattan2005  with the added TPE
contribution were also shown in Fig.15 b in Blunden2020 . They are
systematically located above the green stripe in that figure that corresponds
to the results of polarization experiments in Puckett17 , which at first
glance is an indication that measurements Qattan2005  are unreliable. However,
this is not true. To analyze the reliability of the RT-based measurements of
the $R$ ratio in the experiment reported in Qattan2005 , the ratio
$R_{\sigma}$ (16) was calculated for all energies of the electron beam $E_{1}$
and squares of the momentum transfer to the proton $Q^{2}$ used in the
experiment. The corresponding results are presented in Table 4.

Table 4:  Values of $R_{\sigma}$ (16) calculated with $R=R_{d}$ given by Eq. (20) for $E_{1}$ and $Q^{2}$ values used in the measurements in Qattan2005  $Q^{2}$ (GeV2) | 2.64 | 3.20 | 4.10
---|---|---|---
$Rd5$, $E_{1}=4.702$ GeV | 0.148 | 0.115 | 0.078
$Rd4$, $E_{1}=3.772$ GeV | 0.134 | 0.098 | 0.058
$Rd3$, $E_{1}=2.842$ GeV | 0.102 | 0.063 | 0.018
$Rd2$, $E_{1}=2.262$ GeV | 0.061 | 0.018 |
$Rd1$, $E_{1}=1.912$ GeV | 0.020 |  |

The accuracy of measurements required for the values given in bold in Table 4
proves to correspond to the normalization accuracy in Qattan2005  equal to 1.7
% Bernauer2014 , thus indicating their reliability. This conclusion is based
on the necessity of using the same approach to analyze measurements in
Qattan2005  and Andivahis1994 , where $\Delta_{0}$ is determined by the
normalization uncertainty. Since measurements in Qattan2005  are reliable, a
more accurate reanalysis is required to remove the remaining disagreements
between “Qattan2005  \+ TPE” and Puckett17  discovered in Blunden2020 .

On the feasibility of the measurement of the ratio of SFFs in the $e\vec{p}\to
e\vec{p}$ process. The method to measure squares of SFFs in the processes with
and without proton spin flip proposed in JETPL18  requires a fully polarized
proton target, which may be obtained in the distant future. As stated above,
it may be considered in a broader sense as a technique based on the
polarization transfer from the initial to the final proton. Generally, if the
initial proton is partially polarized, the degree of the longitudinal
polarization transfer to the recoil proton is given by Eq. (17). An experiment
to measure this quantity is currently quite realistic, since a partially
polarized proton target with a high degree of polarization $P_{t}=(70\pm 5)$ %
was already used in Liyanage2020 . For this reason, it would be the most
expedient to carry out the proposed experiment at the facility used by the
SANE collaboration Liyanage2020  at the same value of $P_{t}=0.70$, electron
beam energies $E_{1}=4.725$ and 5.895 GeV, and the same values of the square
of the momentum transfer to the proton $Q^{2}=2.06$ and 5.66 GeV2. The
difference between the proposed experiment and that performed in Liyanage2020
is that the electron beam should be unpolarized, while the detected recoil
proton should move exactly along the direction of quantization of the proton
target spin. The degrees of the longitudinal and transverse polarizations of
the final proton were measured in Jones00 ; Gay01 ; Gay02 ; Pun05 ; Puckett10
; Puckett12 ; Puckett17 . In the proposed experiment, the degree of the
longitudinal polarization of the recoil proton alone should be measured, which
is an advantage compared to the technique proposed in Rekalo74 .

The calculated $Q^{2}$ dependence of the polarization transfer to the final
proton $P_{r}$ (17) in the kinematics of the experiment reported Liyanage2020
is plotted in Fig. 3, where the lines $Pd5$ ($Pd4$) and $Pj5$ ($Pj4$) are
obtained with $R=R_{d}$ given by Eq. (20) and with $R=R_{j}$ given by Eq. (20)
for the electron beam energy $E_{1}=5.895$ (4.725) GeV, respectively. For all
lines in Fig. 3, the degree of polarization of the proton target is
$P_{t}=0.70$.

Figure 3:  Longitudinal polarization of the recoil proton $P_{r}$ (17)
calculated with (lines $Pd5$ and $Pd4$) $R=R_{d}$ given by Eq. (20) and (lines
$Pj5$ and $Pj4$) $R=R_{j}$ given by Eq. (20) versus the square of the momentum
transfer to the proton $Q^{2}$ (GeV2) for the values $E_{1}$ and $P_{t}$ used
in Liyanage2020 .

The plots in Fig. 3 show that the polarization transfer to the final proton
significantly depends on the form of the $Q^{2}$ dependence of $R$. The
absolute value of this polarization transfer in the case of the violation of
the SFFs scaling, i.e., in the case $R=R_{j}$, is significantly greater than
that in the case $R=R_{d}$. A quantitative estimate of this difference is
shown in Table 5, where the longitudinal polarizations of the recoil proton
$Pj5$, $Pd5$, $Pj4$, and $Pd4$, as well as their relative differences
$\Delta_{dj5}=(Pj5-Pd5)/Pj5$ and $\Delta_{dj4}=(Pj4-Pd4)/Pj4$ in percent, are
displayed for two electron-beam energies of 5.895 and 4.725 GeV and two values
$Q^{2}=2.06$ and 5.66 GeV2.

Table 5:  Longitudinal polarization of the recoil proton $P_{r}$ (17) for electron-beam energies of 5.895 and 4.725 GeV and $Q^{2}=2.06$ and 5.66 GeV2 $Q^{2}$ (GeV2) | $Pd5$ | $Pj5$ | $Pd4$ | $Pj4$ |  $\Delta_{dj5}$, % |  $\Delta_{dj4}$, %
---|---|---|---|---|---|---
2.06 | –0.46 | –0.55 | –0.47 | –0.56 | 16.6 | 16.1
5.66 | –0.63 | –0.69 | –0.65 | –0.69 | 9.1 | 6.4

Table 5 shows that the relative differences between $Pj5$ and $Pd5$ and
between $Pj4$ and $Pd4$ at $Q^{2}=2.06$ GeV2 are 16.6 and 16.1 %,
respectively, and decrease to 9.1 and 6.4 %, respectively, at $Q^{2}=5.66$
GeV2.

Conclusions. Using the results of the JLab’s polarization experiments, where
the ratio $R$ was measured in the $\vec{e}p\to e\vec{p}$ process, we have
numerically analyzed the ratio of cross sections with and without proton spin
flip and the polarization asymmetry as functions of the square of the momentum
transfer to the proton in this process for the case where the initial (at
rest) proton and the final proton are fully polarized and have a common spin
quantization axis that coincides with the direction of motion of the detected
recoil proton. If the initial proton is partially polarized, the longitudinal
polarization transfer to the proton is calculated for the kinematics used by
the SANE collaboration Liyanage2020  in the experiments to measure double spin
asymmetry in the $\vec{e}\vec{p}\to ep$ process. A noticeable sensitivity of
the polarization transfer to the proton to the form of the $Q^{2}$ dependence
of the ratio $R$ has been found. This sensitivity may be used to conduct a new
independent experiment to measure this dependence in the $e\vec{p}\to
e\vec{p}$ process. A criterion for the reliability of measurements of $R$
using the Rosenbluth technique has been proposed and used to analyze results
of two experiments reported in Andivahis1994 ; Qattan2005 . The results of the
analysis may be used to identify the reasons for the disagreements that remain
between the results of measurements Qattan2005  with the added TPE
contribution and polarization experiments Puckett17 , which were discovered in
Blunden2020 .

Acknowledgments. I am grateful to R. Lednicky for his interest in the study
and helpful discussions.

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