#  Feasibility study for precisely measuring the EIC 3He beam polarization
with the Polarized Atomic Hydrogen Gas Jet Target polarimeter at RHIC

A. A. Poblaguev<EMAIL_ADDRESS>Brookhaven National Laboratory, Upton, New
York 11973, USA

(December 12, 2022)

###### Abstract

The Polarized Atomic Hydrogen Gas Jet Target polarimeter (HJET) is used to
measure the absolute proton beam polarization,
$\sigma_{P}^{\text{syst}}/P\\!\lesssim\\!0.5\%$, at the Relativistic Heavy Ion
Collider. Here I consider the possibility of employing HJET to measure the
${}^{3}\text{He}$ ($h$) beam polarization at the Electron-Ion Collider (EIC).
The dominant contribution to the ratio of the $h^{\uparrow}{p}$ and
$p^{\uparrow}{h}$ analyzing powers, which is needed for such measurements, can
be easily calculated using well-known values of the proton and helion magnetic
moments, but some corrections should be applied to achieve the required
accuracy. It was found that corrections due to absorption and
${}^{3}\text{He}$ breakup effectively cancel in the ratio and a correction due
to hadronic spin-flip amplitudes can be derived from the proton beam
measurements. As a result, the anticipated systematic uncertainty in the
measured ${}^{3}\text{He}$ beam polarization can satisfy the EIC requirement
$\sigma_{P}^{\text{syst}}/P\\!\lesssim\\!1\%$.

##  I Introduction

High energy, 41–275 GeV, polarized proton and helion [${}^{3}\text{He}$
($A_{h}\\!=\\!3$, $Z_{h}\\!=\\!2$)] beams are proposed for the future
Electron-Ion Collider (EIC) [1]. The EIC physics program is being designed
assuming precise knowledge of the beam polarization [2]

$\sigma_{P}^{\text{syst}}/P\lesssim 1\%.$ (1)

Since hadron beams at EIC will be similar to those at the Relativistic Heavy
Ion Collider (RHIC), experience with absolute proton beam polarization
measurements at RHIC is very important for the development of hadronic
polarimetry at EIC.

At RHIC, the absolute vertical proton beam polarization is determined by the
Atomic Polarized Hydrogen Gas Jet Target (HJET) [3]. A critically important
feature of the HJET polarimeter is a relatively low-density gas jet target
with no walls or windows which allows one to make continuous during the RHIC
store measurements in the Coulomb-nuclear interference (CNI) region.

At HJET, the polarized proton beam is scattered off the hydrogen target and
recoil protons are counted in the left/right symmetric recoil spectrometer
detectors near $90^{\circ}$ to the beam direction [4]. The measured beam spin-
correlated asymmetry, $a_{\text{N}}$, is proportional to the vertical
polarization $P$ of the beam

$a_{\text{N}}(T_{R})=A_{\text{N}}(t)P,$ (2)

where $t\\!=\\!-2m_{p}T_{R}$ is momentum transfer squared, $T_{R}$ is the
recoil proton kinetic energy, and $m_{p}$ is a proton mass. The analyzing
power $A_{\text{N}}(t)$ is dominantly defined by the interference of the
electromagnetic spin-flip and hadronic nonflip amplitudes. In this
approximation and for the CNI scattering, a simple expression for
$A_{\text{N}}(t)$ was obtained in Ref. [5] (see Fig. 2). In the precision
measurement of $A_{\text{N}}(t)$ carried out at HJET [4], it was found that
analyzing power for the forward elastic $p^{\uparrow}{p}$ scattering was
predicted in Ref. [5] with several percent accuracy.

Due to the spin-flipping polarized target, HJET can be self-calibrated. For
the scattering of the identical particles, since the beam and jet spin
correlated asymmetries are concurrently measured using the same recoil
protons, the beam polarization can be related via

$P_{\text{beam}}=P_{\text{jet}}\times
a_{\text{N}}^{\text{beam}}(T_{R})/a_{\text{N}}^{\text{jet}}(T_{R})$ (3)

to the jet one $P_{\text{jet}}\\!\approx\\!0.96\\!\pm\\!0.001$, which is
monitored by the Breit-Rabi polarimeter [3].

In previous RHIC polarized proton Runs 15
($E_{\text{beam}}\\!=\\!100\,\text{GeV}$) and 17 (255 GeV), the beam
polarization of about $P_{\text{beam}}\\!\sim\\!55\%$ was measured with a
systematic error of $\sigma^{\text{syst}}_{P}/P\\!\lesssim\\!0.5\%$ and a
typical statistical uncertainty of $\sigma_{P}^{\text{stat}}\\!\approx\\!2\%$
per 8 h RHIC store [4]. Also, the elastic ${pp}$ analyzing power was precisely
determined at both beam energies [6].

For EIC ${}^{3}\text{He}$ beam polarimetry, an obvious suggestion is to employ
a polarized ${}^{3}\text{He}$ gas target (He3J) and, thus, utilize the
advantage of the HJET method (3), which does not require detailed knowledge of
the analyzing power. However, to implement this idea, new experimental
techniques should be developed for He3J to allow measurements in the CNI
region as well as for precision monitoring, $\sigma_{P}/P\\!\ll\\!1\%$, of the
helion target polarization. Also, backgrounds in He3J may significantly differ
from those of HJET. In other words, it is not proved yet that the low
systematic uncertainties achieved at HJET would apply in the He3J case.

In this paper, the feasibility of precisely measuring the EIC
${}^{3}\text{He}$ beam polarization using HJET is investigated.

Since 2015, HJET routinely operates in the RHIC ion beams. The recoil
spectrometer performance was found to be very stable for a wide range of ion
species ($d$, O, Al, Zr, Ru, Au) and over the beam energies used (3.8–100
GeV/nucleon) [4].

Generally, $A_{\text{N}}^{\text{beam}}\neq A_{\text{N}}^{\text{jet}}$ for
nonidentical beam and target (jet) particles. Therefore, for $hp$ scattering,
Eq. (3) should be adjusted by the analyzing power ratio:

$\displaystyle
P_{\text{beam}}=P_{\text{jet}}\,a_{\text{N}}^{\text{beam}}/a_{\text{N}}^{\text{jet}}\times{\cal
R},$ (4) $\displaystyle{\cal
R}=\frac{\mu_{p}-1}{\mu_{h}/Z_{h}-m_{p}/m_{h}}\times\left[1+\text{corr}\right].$
(5)

In the leading order approximation, the ratio can be expressed [7] via
magnetic moments of a proton, $\mu_{p}$, and a helion, $\mu_{h}$. However,
anticipated corrections, for example, due to the hadronic spin-flip amplitude
and due to a possible breakup of the beam ${}^{3}\text{He}$ should be
considered.

In this paper, numerical analysis of the corrections was done for 100
GeV/nucleon ${}^{3}\text{He}$ beam scattering off proton target in CNI region,
$-t\\!<\\!0.02\,\text{GeV}^{2}$. It was found that the corrections are small
and well controllable, which suggests that the EIC requirement (1) can be
satisfied in the HJET measurements.

##  II Helicity amplitudes for forward elastic scattering

Polarization phenomena in the CNI elastic scattering of the transversely
polarized spin-1/2 beam particle off the spin-1/2 target (generally non-
identical) is greatly predetermined by two helicity amplitudes [8, 9,
*PhysRevD.35.407, 5], nonflip

$\phi_{+}(s,t)=\frac{\langle+\\!+|{\cal M}|+\\!+\rangle\>+\>\langle+\\!-|{\cal
M}|+\\!-\rangle}{2}$ (6)

and spin-flip

$\phi_{5}(s,t)=\langle+\\!+|{\cal M}|+\\!-\rangle.$ (7)

The subscripts used follow the notations of Ref. [8]. Both amplitudes can be
decomposed to the hadronic and electromagnetic parts

$\phi(s,t)\to\phi^{\text{had}}(s,t)+\phi^{\text{em}}(s,t)\times
e^{i\delta_{C}(s,t)},$ (8)

where $\delta_{C}$ is the Coulomb phase [11, 12], and $s$ and $t$ are center
of mass energy squared and momentum transfer squared, respectively.

The optical theorem fixes the imaginary part of the forward, $t\\!\to\\!0$,
hadronic nonflip amplitude:

$\phi^{\text{had}}_{+}(s,t)=\left[i+\rho^{bt}(s)\right]\times
s\sigma^{bt}_{\text{tot}}(s)e^{B^{bt}(s)t/2}/8\pi,$ (9)

where $\sigma_{\text{tot}}$ is the total cross section, $B$ is the “slope,”and
indexes $b$ and $t$ specify beam and target particles, respectively. The
hadronic spin-flip amplitude is commonly referred to by a dimensionless
complex parameter $r_{5}\\!=\\!R_{5}\\!+\\!iI_{5}$ [8] defined as

$\phi_{5}^{\text{had}}(s,t)/\text{Im}\,\phi_{+}^{\text{had}}(s,t)=(\sqrt{-t}/m_{p})\times
r_{5}^{bt}(s).$ (10)

Without losing generality, it will be assumed that only the beam particle is
polarized. For immediate estimates, $r_{5}$ can be approached [6] by

$|r_{5}|\approx 0.02.$ (11)

The electromagnetic amplitudes were calculated in Ref. [9, *PhysRevD.35.407].
Neglecting Dirac and Pauli form factors, they can be written [7, 13] as

$\displaystyle\phi_{+}^{\text{em}}(s,t)/\text{Im}\,\phi_{+}^{\text{had}}(s,t)$
$\displaystyle=t_{c}/t+{\cal B}^{\text{nf}},$ (12)
$\displaystyle\phi_{5}^{\text{em}}(s,t)/\text{Im}\,\phi_{+}^{\text{had}}(s,t)$
$\displaystyle=\left(t_{c}/t+\cal{B}^{\text{sf}}\right)\,\varkappa_{b}^{\prime}\sqrt{-t}/m_{p}$
(13) $\displaystyle t_{c}$ $\displaystyle=-8\pi\alpha
Z_{b}Z_{t}/\sigma_{\text{tot}}^{bt},$ (14)
$\displaystyle\varkappa^{\prime}_{b}$
$\displaystyle=\varkappa_{b}-2m_{b}^{2}/s\approx\varkappa_{b}-m_{b}/m_{t}\times
m_{p}/E_{\text{beam}},$ (15) $\displaystyle\varkappa_{b}$
$\displaystyle=\mu_{b}/Z_{b}-m_{p}/m_{b}.$ (16)

Here, $\mu_{b}$ is a magnetic moment in nuclear magnetons of the beam particle
($\mu_{p}\\!=\\!2.793$ for a proton and $\mu_{h}\\!=\\!-2.128$ for a helion)
and $E_{\text{beam}}$ is the beam energy per nucleon. $\cal{B}^{\text{nf}}$
reflects the difference between Coulomb $F_{\text{em}}^{\text{nf}}(t)$ and
hadronic $F_{+}(t)$ form factors and can be approximated as

${\cal
B}^{\text{nf}}/t_{c}=\frac{d}{dt}\left[\frac{F_{\text{em}}^{\text{nf}}(t)}{F_{+}(t)}\right]_{t=0}=\frac{{\cal
B}_{a}^{\text{nf}}}{t_{c}}+\frac{r_{b}^{2}+r_{t}^{2}}{6}-\frac{B^{bt}}{2},$
(17)

where $r_{b}$ and $r_{t}$ are rms charge radii of the beam and target
particles ($r_{p}\\!=\\!0.841(1)\,\text{fm}$ [14],
$r_{h}\\!=\\!1.96(1)\,\text{fm}$ [15, *Morton:2006zz]) and ${\cal
B}_{a}^{\text{nf}}$ parametrize the absorptive (i.e., due to the initial and
final state inelastic hadronic interactions) correction [17],
$F_{\text{em}}(t)\\!\to\\!F_{\text{em}}(t)\exp{\left({\cal
B}_{a}t/t_{c}\right)}$, to the nonflip electromagnetic form factor. To
evaluate spin-flip ${\cal B}^{\text{sf}}$ for the $p^{\uparrow}{h}$
scattering, one can use, in Eq. (17), ${\cal
B}_{a}^{\text{sf}}\approx\alpha/2$ [18] and, also, the proton charge radius
$r_{b}\\!=\\!r_{p}$ should be replaced by the magnetic one
$r_{p}^{\text{M}}\\!=\\!0.851(26)\,\text{fm}$ [19]. Since helion spin is
mostly carried by the neutron [20], the magnetic radius of a neutron
$r_{n}^{\text{M}}\\!=\\!0.873(11)\,\text{fm}$ [21] is about the same as for a
proton, and a ${}^{3}\text{He}$ is dominantly in a space symmetric state, one
may not distinguish between values of ${\cal B}^{\text{sf}}$ for
$p^{\uparrow}{h}$ and $h^{\uparrow}{p}$ scattering.

Figure 1:  Coulomb phase in elastic proton and/or helion scattering at
$E_{\text{beam}}\\!=\\!100\,\text{GeV/nucleon}$.

Coulomb phases for ${pp}$, ${ph}$, and ${hh}$ scattering calculated, in
accordance with Ref. [8, 18], as

$\delta_{C}(t)=\alpha
Z_{b}Z_{t}\,\left[\ln{\left|\left(\frac{B^{bt}}{2}\\!+\\!\frac{r_{b}^{2}\\!+\\!r_{t}^{2}}{6}\right)t\right|}\\!+\\!0.5772\right]$
(18)

are displayed in Fig. 1.

##  III Analyzing power

Figure 2:  The elastic $p$ and ${}^{3}\text{He}$ analyzing powers calculated
[Eq. (20)] for
$\sigma_{\text{tot}}^{pp}\\!=\\!\sigma_{\text{tot}}^{ph}/0.9A_{h}\\!=\\!\sigma_{\text{tot}}^{hh}/0.8A_{h}^{2}\\!=\\!39\,\text{mb}$.
$T_{R}$ is the recoil (target) particle kinetic energy.

In the considered approximation, the analyzing power is given by

$A_{\text{N}}(s,t)={-2\text{Im}\left(\phi_{5}^{*}\phi_{+}\right)}/{|\phi_{+}|^{2}}.$
(19)

The dominant component [5],

$A_{\text{N}}(t)=\frac{\varkappa_{b}\sqrt{-t_{c}}}{m_{p}}\times\frac{(t_{c}/t)^{1/2}}{(t_{c}/t)^{2}+1},$
(20)

can be easily calculated if the total cross section $\sigma_{\text{tot}}^{bt}$
is known. In this approach, $|A_{\text{N}}(t)|$ reaches maximum at
$t\\!=\\!\sqrt{3}t_{c}$. For the proton and/or helion beam and target, the
$A_{\text{N}}(t)$ dependence on the experimentally measured recoil energy
$T_{R}\\!=\\!-t/2m_{t}$ is depicted in Fig. 2.

Analyzing power uncertainties in Eq. (20) are about 5–10% [4]. To meet EIC
requirement (1), corrections to Eq. (20) should be considered:

$\displaystyle A_{\text{N}}(t)$
$\displaystyle=A_{\text{N}}^{(0)}(t)\times(1+\xi_{0}+\xi_{1}t/t_{c}),$ (21)
$\displaystyle A_{\text{N}}^{(0)}(t)$
$\displaystyle=\varkappa_{b}^{\prime}\sqrt{-t}/m_{p}F_{\text{cs}}(t),$ (22)
$\displaystyle F_{\text{cs}}(t)$
$\displaystyle=t_{c}/t+2Z_{b}Z_{t}\alpha\ln{(t_{c}/t)}$
$\displaystyle+\beta_{0}+(1\\!+\\!\beta_{1})\,t/t_{c}+\beta_{2}\left(t/t_{c}\right)^{2},$
(23)

where, omitting some small terms,

$\displaystyle\xi_{0}$
$\displaystyle=-\rho\delta_{C}(t_{c})-2I_{5}/\varkappa^{\prime}_{b},~{}~{}~{}~{}~{}\xi_{1}={\cal
B}^{\text{sf}}-2R_{5}/\varkappa^{\prime}_{b},$ (24) $\displaystyle\beta_{0}$
$\displaystyle=-2\left[\rho+\delta_{C}(t_{c})-{\cal
B^{\text{nf}}}\right],~{}~{}\beta_{1}=\rho^{2},~{}~{}~{}~{}\beta_{2}=0.$ (25)

Here, $A_{\text{N}}^{(0)}(t)$ is the analyzing power neglecting some small
corrections, in particular, due to the hadronic spin-flip amplitude $r_{5}$.
In Eq. (23), the term $\beta_{2}$ is reserved for a possible correction due to
the beam ${}^{3}\text{He}$ breakup.

It should be noted [13] that $\rho^{\text{meas}}\\!=\\!\rho-{\cal
B}_{a}^{\text{nf}}$ is actually measured in an experimental study of
$d\sigma/dt$ (if absorptive corrections are not considered in the data
analysis). Thus, if such a determined $\rho^{\text{meas}}$ is used to
calculate $\beta_{0}$ then one should set ${\cal B}_{a}^{\text{nf}}\\!=\\!0$
in Eq. (17).

Similarly, the spin-flip absorptive correction leads to an effective
replacement

$r_{5}\to r_{5}^{\text{meas}}=r_{5}-\varkappa_{b}{\cal
B}_{a}^{\text{sf}}/2,\qquad{\cal B}_{a}^{\text{sf}}\approx\alpha/2.$ (26)

It is assumed that $A_{\text{N}}^{(0)}(s,t)$ can be unambiguously derived from
standalone measurements of the unpolarized elastic cross section
$d\sigma/dt\propto|\phi_{+}(s,t)|^{2}$. The hadronic spin-flip amplitude
contributes only to $\xi_{0}$ and $\xi_{1}$.

If the beam polarization $P_{\text{beam}}$ is known, one can determine
hadronic spin flip amplitude parameter $r_{5}$ from the measured spin
asymmetry $a(T_{R})$ dependence on $T_{R}$:

$\xi_{0}+\xi_{1}\,T_{R}/T_{c}=a(T_{R})/P_{\text{beam}}A_{\text{N}}^{(0)}(-2m_{p}T_{R})-1.$
(27)

Alternatively, if $r_{5}$ is predetermined, the beam polarization can be
found. For that, it may be helpful to measure beam polarization as function of
$T_{R}$,

$\displaystyle P_{\text{meas}}(T_{R})$ $\displaystyle=$ $\displaystyle
a(T_{R})/A_{\text{N}}(T_{R})$ (28) $\displaystyle\approx$ $\displaystyle
P_{\text{beam}}\times\left(1-\delta\xi_{0}-\delta\xi_{1}\,T_{R}/T_{c}\right)$

where $\delta\xi_{0}$ and $\delta\xi_{1}$ denote systematic errors in the
values of $\xi_{0}$ and $\xi_{1}$, respectively. Since $\delta\xi_{1}$ can be
determined directly in the linear fit of Eq. (28), the systematic uncertainty
in the measured beam polarization is actually predefined by $\delta\xi_{0}$
only:

$\delta^{\text{syst}}P_{\text{beam}}/P_{\text{beam}}=-\delta\xi_{0}.$ (29)

##  IV Systematic errors due to uncertainties in $A_{\text{N}}^{(0)}(t)$

Uncertainties in the predetermined $A_{\text{N}}^{(0)}(t)$ and, consequently,
in data fits (27) or (28) may depend on a possible error in the value of total
cross section
$\delta_{\sigma}\\!=\\!\delta\sigma_{\text{tot}}/\sigma_{\text{tot}}\\!=\\!-\delta
t_{c}/t_{c}$ as well as on errors $\delta\beta_{i}$ in the $d\sigma/dt$
parametrization. In RHIC Run 15 (100 GeV), the resulting effective alterations
of the spin-flip corrections $\Delta\xi_{0}$ and $\Delta\xi_{1}$ were
estimated as

$\displaystyle\\!\\!\\!\Delta\xi_{0}$ $\displaystyle=$ $\displaystyle
0.22\,\delta_{\sigma}+0.49\,\delta\beta_{0}+0.58\,\delta\beta_{1}-0.82\,\delta\beta_{2},$
(30) $\displaystyle\\!\\!\\!\Delta\xi_{1}$ $\displaystyle=$ $\displaystyle
0.09\,\delta_{\sigma}-0.05\,\delta\beta_{0}+0.08\,\delta\beta_{1}-0.20\,\delta\beta_{2}.$
(31)

It should be understood that linearized parametrizations given in Eqs. (30)
and (31) may strongly depend on the event selection cuts used.

The uncertainties in $A_{\text{N}}^{(0)}(t)$ may also lead to a non-linearity
of Eq. (27). Considering the effective variation of $\beta_{0}$ as a measure
of the non-linearity, one finds (for Run 15 data)

$\Delta\beta_{0}=4.20\,\delta_{\sigma}-1.00\,\delta\beta_{0}+2.10\,\delta\beta_{1}-2.92\,\delta\beta_{2},$
(32)

where $\Delta\beta_{0}$ is a correction to the value of $\beta_{0}$ found in
the fit with $\beta_{0}$ considered as a free parameter. In the HJET data
analysis [4], this effect was used for an independent experimental evaluation
of $\rho$. In terms of measured correction,
$\Delta\rho\\!=-\\!\Delta\beta_{0}/2$, to the value of $\rho$ assumed, the
results can be presented as

$\Delta\rho=0.029(25)|_{\text{100\,GeV}},~{}\Delta\rho=-0.019(18)|_{\text{255\,GeV}}.$
(33)

Thus, the value of $\beta_{0}$ can be used as a free parameter in the spin-
correlated asymmetry fit. However, the subsequent contribution of about 2% to
the polarization uncertainty (29), in this case, does not allow one to meet
the EIC requirement (1).

To parametrize $A_{\text{N}}^{(0)}(t)$, one can fit a measured differential
cross section at low $t$,

$d\sigma/dt=\sigma_{\text{tot}}^{2}(1+\rho^{2})e^{Bt}/16\pi\times
F_{\text{cs}}(t)\,t_{c}/t.$ (34)

In an experimental study of $d\sigma/dt$, the luminosity calibration is
critically important for the precision determination of $\sigma_{\text{tot}}$,
while parameters $\beta_{i}$ can be extracted from the $t$-dependence fit of
Eq. (34). For elastic ${pp}$ scattering, the fit is significantly simplified
because $\beta_{2}\\!=\\!0$ and the value of $\beta_{1}\\!=\\!\rho^{2}$ is
small.

For $E_{\text{beam}}\\!=\\!100\,\text{GeV}$ elastic proton-proton scattering,
$B^{pp}\\!=\\!11.2(2)\,\text{GeV}^{-2}$ [22, *Bartenev:1973kk],
$\sigma_{\text{tot}}^{pp}\\!=\\!38.39(2)\,\text{mb}$ and
$\rho^{pp}\\!=\\!-0.079(1)$. The values of $\sigma_{\text{tot}}^{pp}$ and
$\rho^{pp}$ are taken from a global fit [24] of the ${pp}$ and ${p\bar{p}}$
data. Consequently,
$t_{c}^{pp}\\!=\\!-1.86\\!\times\\!10^{-3}\,\text{GeV}^{2}$ which corresponds
to the proton recoil energy of $T_{c}^{pp}\\!\approx\\!1\,\text{MeV}$.

In the absolute majority of experimental evaluations of the proton-proton
forward real to imaginary ratio $\rho^{pp}$ at RHIC/EIC energies, the data
were analyzed using the electromagnetic form factor in the dipole
approximation, $(1-t/\Lambda^{2})^{-2}$ with
$\Lambda^{2}=0.71\,\text{GeV}^{2}$ [25], and disregarding absorptive
correction, ${\cal{B}}_{a}^{\text{nf}}\\!=\\!0$. Since actually
$\rho^{pp}_{\text{exp}}\\!=\\!\rho^{pp}\\!-\\!{\cal
B}_{a}^{pp}\\!-\\!(r_{p}^{2}/3\\!-\\!4/\Lambda^{2})t_{c}$ was measured in
these experiments, the expression for $\beta_{0}$ in (25) should be replaced
by

$\\!\\!\beta_{0}^{pp}=-2\left[\rho^{pp}_{\text{exp}}+\delta^{pp}_{C}(t_{c})+\left(r_{p}^{2}/3-4/\Lambda^{2}\right)t_{c}\right]\\!.$
(35)

After that, for the ${pp}$ scattering, $d\sigma/dt$ related uncertainties in
$A_{\text{N}}^{(0)}(t)$ and, consequently, in values of $\xi_{0}$ and
$\xi_{1}$ can be neglected. Nonetheless, $\text{Re}\,r_{5}$, determined in a
hadronic spin-flip amplitude fit (27), is a little ($\approx\\!0.003$) altered
by the spin-flip absorptive correction [see Eq. (26)].

For the helion-proton scattering at the EIC energies, a precision experimental
study of the forward $d\sigma^{hp}/dt$ was not done yet.

For a light nucleus $A$, the total $pA$ cross section can be related to the
${pp}$ one as

$\sigma_{\text{tot}}^{p\text{A}}=A\sigma_{\text{tot}}^{pp}\times\left(1-\delta_{\text{A}}\right).$
(36)

To evaluate the screening parameter $\delta_{\text{A}}$ [26], one can
approximate the $pA$ forward scattering amplitude $f_{\text{A}}$ by

$f_{\text{A}}=\sum_{i}{f_{i}}+i{\langle r^{-2}\rangle}_{\text{A}}\sum_{i\neq
j}{f_{i}f_{j}/2}+\dots,$ (37)

where indexes $i$ and $j$ denote the constituent neutrons and protons, and it
is assumed that forward ${pp}$ and ${pn}$ amplitudes are equal,
$f_{n}\\!=\\!f_{p}\\!=\\!(i+\rho^{pp})\sigma_{\text{tot}}^{pp}/4\pi$.
Following Ref. [27], one can estimate $\delta_{d}\\!=\\!0.051$,
$\delta_{h}\\!=\\!0.119$, and $\delta_{\alpha}\\!=\\!0.246$ for deuteron,
helion, and alpha, respectively.

The result for the deuteron is in good agreement with the value of
$\delta_{d}^{\text{exp}}\\!=\\!0.047(5)$ derived from the
$\sigma_{\text{tot}}$ measurement [28, *Carroll:1975xf] at 100 GeV, but there
is a noticeable discrepancy for ${}^{4}\text{He}$,
$\delta_{\alpha}^{\text{exp}}\\!=\\!0.166(6)$ [30]. Therefore, it was assumed
that

$\delta_{h}=0.1\times\left[1\pm(20\text{--}30)\%\right]$ (38)

to depict $A_{\text{N}}(t)$ in Fig. 2.

EIC requirement (1) implies the following constraints (considered separately)
on accuracy of the $d\sigma^{hp}/dt$ parametrization:

$|\delta_{\sigma}^{hp}|\\!<\\!5\%,\,\>|\delta{\beta_{0}^{hp}}|\\!<\\!2\%,\,\>|\delta{\beta_{1}^{hp}}|\\!<\\!2\%,\,\>|\delta{\beta_{2}^{hp}}|\\!<\\!1\%.$
(39)

It may be noted that uncertainty in value of the ${hp}$ cross-section followed
from Eq. (38) is consistent with the constraint on $\delta_{\sigma}^{hp}$.

In ${pp}$ forward elastic Re/Im ratio experiments [14], the typical
statistical accuracy of a measurement was $|\delta\rho|\gtrsim 0.01$ or
$|\delta{\beta_{0}^{pp}}|\\!\gtrsim\\!2\%$. (However, combined analysis [31]
of the all measurements significantly improves accuracy of the determination
of $\beta_{0}^{pp}$.) For ${hp}$ scattering, the experimental evaluation of
$\beta_{0}$ is expected to be less accurate,
$|\delta{\beta_{0}^{hp}}|\\!>\\!|\delta{\beta_{0}^{pp}}|$ due to the breakup
related uncertainties in $\beta_{1}$ and $\beta_{2}$.

A theoretical extrapolation of the ${pp}$ values of $\sigma_{\text{tot}}$,
$\rho$, and $B$ to those of ${hp}$, if this is possible to within the required
accuracy, may also be complicated by the necessity to precisely evaluate the
nonflip absorptive correction to a $hp$ electromagnetic amplitude ${\cal
B}_{a}^{hp}$ and the ${}^{3}\text{He}$ breakup contributions to $\beta_{0}$,
$\beta_{1}$, and $\beta_{2}$.

A detailed investigation of the possibility to improve $d\sigma^{hp}/dt$
parametrization in a theoretical analysis and/or in a dedicated experimental
study is beyond the scope of this paper.

##  V ${}^{3}\text{He}$ beam polarization measurement using polarized hydrogen
jet target

For ${}^{3}\text{He}^{\uparrow}$ beam scattering off the polarized hydrogen
jet, the anticipated problem with the cross section related uncertainties can
be eliminated by concurrent measurement of the beam and jet spin asymmetries:

$\frac{a_{\text{N}}^{h}(T_{R})}{a_{\text{N}}^{p}(T_{R})}=\frac{P_{\text{beam}}}{P_{\text{jet}}}\times\frac{\varkappa_{h}^{\prime}(1\\!+\\!\omega_{\varkappa})-2I_{5}^{hp}-2R_{5}^{hp}\,T_{R}/T_{c}}{\varkappa_{p}^{\prime}(1\\!+\\!\omega_{\varkappa})-2I_{5}^{ph}-2R_{5}^{ph}\,T_{R}/T_{c}}$
(40)

Here, indexes ${ph}$ and ${hp}$ relate to the $p^{\uparrow}h$ and
$h^{\uparrow}{p}$ hadronic spin-flip amplitudes, respectively, and

$\omega_{\varkappa}(T_{R})=-\rho\delta_{C}-{\cal
B}^{\text{sf}}T_{R}/T_{c}+b(T_{R})$ (41)

where $b(T_{R})$ is the effective correction to $\phi_{+}^{\text{had}}$ due to
the ${}^{3}\text{He}$ breakup. Since both $r_{5}^{ph}$ and $r_{5}^{hp}$ are
expected to be small (11), $\omega_{\varkappa}$ can be omitted in Eq. (40) if
$|b(T_{R})|\\!<5\%$. According to the estimate in Ref. [32], this condition
for $b^{hp}(T_{R})$ is fulfilled. Thus, the helion beam polarization derived
from Eq. (40) can be approximated as

$P_{\text{meas}}(T_{R})=P_{\text{beam}}\times\left(1+\delta\xi_{0}+\delta\xi_{1}\,T_{R}/T_{c}\right)$
(42)

where the method dependent systematic errors,

$\displaystyle\delta\xi_{0}$ $\displaystyle=$ $\displaystyle
2\delta{I_{5}^{hp}}/\varkappa_{h}^{\prime}-2\delta{I_{5}^{ph}}/\varkappa_{p}^{\prime},$
(43) $\displaystyle\delta\xi_{1}$ $\displaystyle=$ $\displaystyle
2\delta{R_{5}^{hp}}/\varkappa_{h}^{\prime}-2\delta{R_{5}^{ph}}/\varkappa_{p}^{\prime},$
(44)

are predefined by uncertainties in the values of $r_{5}^{ph}$ and
$r_{5}^{hp}$.

###  V.1 Hadronic spin-flip amplitudes in elastic $h^{\uparrow}p$ and
$p^{\uparrow}h$ scattering

Currently, theory cannot predict $r_{5}^{hp}$ from first principles. However,
precision measurements of the proton-proton $r_{5}$ at HJET [6]

100 GeV: $\displaystyle
R_{5}^{pp}=\left(-16.4\\!\pm\\!0.8_{\textrm{stat}}\\!\pm\\!1.5_{\textrm{syst}}\right)\\!\times\\!10^{-3},$
(45) $\displaystyle
I_{5}^{pp}\,=\left(\;\;-5.3\\!\pm\\!2.9_{\textrm{stat}}\\!\pm\\!4.7_{\textrm{syst}}\right)\\!\times\\!10^{-3},$
(46) 255 GeV: $\displaystyle
R_{5}^{pp}=\left(\;\;-7.9\\!\pm\\!0.5_{\textrm{stat}}\\!\pm\\!0.8_{\textrm{syst}}\right)\\!\times\\!10^{-3},$
(47) $\displaystyle
I_{5}^{pp}\,=\left(\phantom{-}19.4\\!\pm\\!2.5_{\textrm{stat}}\\!\pm\\!2.5_{\textrm{syst}}\right)\\!\times\\!10^{-3},$
(48)

allow one to consider the possibility of extrapolating
$r_{5}^{pp}\\!\to\\!r_{5}^{hp}$ and $r_{5}^{pp}\\!\to\\!r_{5}^{ph}$.

It was shown in Ref. [33] that at high energy, under a wide range of
assumptions, the ratio of the spin-flip to the nonflip parts of a CNI elastic
proton-nucleus amplitude is equal to that of proton-proton scattering at the
same beam energy (per nucleon). In particular,

$r_{5}^{ph}/r_{5}^{pp}=1\pm(\lesssim 5\%).$ (49)

This statement can be qualitatively understood in the $pA$ amplitude approach
given by Eq. (37). Since the proton-nucleon ratio of the spin-flip and nonflip
amplitudes

$\displaystyle\eta$ $\displaystyle=f^{\prime}_{n}/f_{n}=f^{\prime}_{p}/f_{p}$
$\displaystyle=\left(1+\rho^{pp}\right)r_{5}^{pp}\,\sqrt{-t}/m_{p}\approx
0.002$ (50)

is small for the HJET $t$ range, the expression for the spin-flip amplitude
$f^{\prime}_{\text{A}}$ can be derived by replacing
$f_{i}\\!\to\\!\eta{f_{i}}$ in (only) one amplitude in each term of sum (37).
The obvious result, $f^{\prime}_{\text{A}}\\!=\\!\eta f_{\text{A}}$, can be
interpreted as

$r_{5}^{ph}/r_{5}^{pp}=\left(i+\rho^{pp}\right)/\left(i+\rho^{ph}\right)\approx
1-i\rho^{pp}\delta_{\text{h}}.$ (51)

This estimate can be altered by a possible difference between a spin-flip and
nonflip values of $\langle{r^{-2}}\rangle_{\text{A}}$. However [33], the
subsequent correction to $r_{5}$, which was included in uncertainty in Eq.
(49), is small and nearly $A$ independent.

For an unpolarized proton scattering off a nucleus with only one polarized
nucleon, one can readily find $r_{5}^{Ap}\\!=\\!r_{5}^{pp}/A$ if
$\langle{r^{-2}}\rangle_{\text{A}}$ is the same for each pair of the
constituent nucleons. This is expected to be a good approximation for
$h^{\uparrow}{p}$ scattering, i.e., $r_{5}^{hp}=r_{5}^{pp}/3$ [34], since a
${}^{3}\text{He}$ is dominantly in a space symmetric $S$ state, in which the
constituent protons are restricted by the Pauli principle to be a spin singlet
and, thus, the helion spin is carried by the neutron only. In a more detailed
analysis of a helion wave function, it was found that the polarization of the
neutron in a fully polarized ${}^{3}\text{He}$ is 87%, while protons should
have a slight residual polarization of $-2.7\%$ [20]. So,
$r_{5}^{hp}/r_{5}^{pp}\\!=\\!0.27\\!\pm\\!0.06$, where the uncertainty
reflects the discrepancy between two estimates.

Here, the following relations between helion-proton and proton-proton hadronic
spin-flip amplitudes will be assumed

$\displaystyle r_{5}^{ph}$ $\displaystyle=\phantom{0.27}\,r_{5}^{pp}\pm
0.001\pm i\,0.001,$ (52) $\displaystyle r_{5}^{hp}$
$\displaystyle=0.27\,r_{5}^{pp}\pm 0.001\pm i\,0.001.$ (53)

###  V.2 Evaluation of the method dependent uncertainties in the
${}^{3}\text{He}$ beam polarization measurement

The nonidentical particle factor in the asymmetries ratio (40) is mainly
defined by the well-known magnetic moment based ratio
$\varkappa_{h}/\varkappa_{p}\\!=\\!-0.7797$. For the 100 GeV/nucleon beam, the
hadronic spin-flip amplitude correction to such measured beam polarization is
about $-0.8\%$ and the correction due to the $m^{2}/s$ term in (15) is about
$+0.8\%$.

The hadronic spin-flip dependent uncertainties in the beam polarization
measurement may be evaluated as

$\displaystyle\delta\xi_{0}$ $\displaystyle=\pm
0.1\,I_{5}^{pp}+G_{\varkappa}\,\delta{I_{5}^{pp}}\approx\pm 0.5\%,$ (54)
$\displaystyle G_{\varkappa}$
$\displaystyle=2\times\left[0.27/\varkappa_{h}^{\prime}-1/\varkappa_{p}^{\prime}\right]=-1.50.$
(55)

The numerical estimate of $\delta\xi_{0}$ follows from the quadratic sum of
the uncertainties in the ratios $r_{5}^{hp,ph}/r_{5}^{pp}$ and in the value of
$I_{5}^{pp}$. For the calculation, Run 15 value of $I_{5}^{pp}\\!=\\!-0.0053$
but Run 17 evaluation of $\delta{I_{5}^{pp}}\\!=\\!\pm 0.0035$ were used.
Since some essential sources of systematic errors were eliminated after Run 15
(100 GeV), the 255 GeV value of $\delta{I_{5}^{pp}}$ is more relevant in this
paper. However, it is assumed that better measurements of $I_{5}$ at 100 GeV
will be done in a future RHIC Run 24 or at EIC.

Pedantically, $I_{5}$ and $R_{5}$ in Eq. (40) should be replaced by [8]

$I_{5}\to I_{5}-\delta_{C}R_{5}\quad\text{and}\quad R_{5}\to R_{5}-\rho
I_{5}.$ (56)

However, such corrections are inessential in the context of the EIC
requirement (1).

A small correction to Eq. (13) due to the difference between Dirac and Pauli
form factors [13] may be included in $\omega_{\varkappa}(T_{R})$ [Eq. (41)]
and, thus, does not require special consideration.

Estimating $r_{5}^{ph}$ (52) and $r_{5}^{hp}$ (53), it was assumed that
hadronic spin-flip parameters $r_{5}^{pn}\\!=r_{5}^{np}\\!=\\!r_{5}^{pp}$ are
the same for proton-proton and proton-neutron scattering. This equity can be
violated by the isovector component of the scattering amplitude. For the
C-odd, J=1 Regge pole exchange, the coupling ratio for ${np}$ and ${pp}$ for
nonflip amplitudes is [35]

$f^{-}_{np}/f^{-}_{pp}=0.901\pm 0.023.$ (57)

Assuming the same ratio for the spin-flip amplitudes and using the
experimentally evaluated Reggeon spin-flip couplings [6], one finds

$\delta\xi_{0}^{pn}\approx-0.22\%,\qquad\delta\xi_{1}^{pn}\approx-0.19\%.$
(58)

The correction to $\delta\xi_{0}$ is small compared to the uncertainty given
in Eq. (54) and, thus, can be discarded.

At HJET, systematic uncertainty in the recoil proton energy calibration was
evaluated [4] to be

$\displaystyle\delta^{\text{cal}}T_{R}=$
$\displaystyle\delta{c_{0}}+\delta{c_{1}}\,T_{R},$ (59)
$\displaystyle\delta{c_{0}}=\pm 15\,\text{keV},~{}~{}\delta{c_{1}}=\pm 0.01$
(60)

The constant term can contribute to the $\sigma_{P}^{\text{syst}}/P$, but the
resulting value,

$\displaystyle\delta\xi_{0}^{T_{R}}=$ $\displaystyle
G_{\varkappa}R_{5}^{pp}\delta{c_{0}}/T_{c}^{hp}\approx\pm 0.06\%,$ (61)

can be neglected.

##  VI Summary

I have studied feasibility of a polarized hydrogen gas jet target to be used
for precision determination of the ${}^{3}\text{He}^{\uparrow}$ beam
polarization at EIC. The numerical estimates were done for 100 GeV/nucleon
beam energy.

It was shown that concurrent measurement of the ${}^{3}\text{He}$ beam
$a_{\text{N}}^{h}(T_{R})$ and hydrogen jet target $a_{\text{N}}^{p}(T_{R})$
spin correlated asymmetries dependence on the recoil proton kinetic energy
$T_{R}$ allows one to precisely determine the beam polarization. For that, the
measured, almost linear, function of $T_{R}$, defined as

$\displaystyle
P_{\text{meas}}(T_{R})=P_{\text{jet}}\frac{a_{\text{N}}^{h}(T_{R})}{a_{\text{N}}^{p}(T_{R})}$
$\displaystyle\quad\;\;\times\frac{\varkappa_{p}-m_{p}^{2}/m_{h}E_{\text{beam}}-2I_{5}-2R_{5}\,T_{R}/T_{c}}{\varkappa_{h}-m_{h}/E_{\text{beam}}-0.54I_{5}-0.54R_{5}\,T_{R}/T_{c}},$
(62)

should be extrapolated to $T_{R}\\!\to\\!0$:

$\\!\\!\\!P_{\text{beam}}\\!=\\!P_{\text{meas}}(0)\times\left[1\pm
0.006_{\text{syst}}\pm 0.005_{r_{5}}\pm 0.002_{\text{mod}}\right]\\!.$ (63)

Here, $\varkappa_{p}\\!=\\!1.793$, $\varkappa_{h}\\!=\\!-1.398$,
$E_{\text{beam}}$ is the helion beam energy per nucleon,
$T_{c}\\!\approx\\!0.7\,\text{MeV}$ (for 100 GeV/nucleon helion beam), and
$r_{5}\\!=\\!R_{5}\\!+\\!iI_{5}$ is the elastic $\mathit{pp}$ hadronic spin-
flip amplitude parameter determined at $s\\!=\\!2m_{p}E_{\text{beam}}$.

For the measurement-dependent systematic error in Eq. (63), I used results of
emulation of HJET performance at EIC [36], the value of $r_{5}^{pp}$ related
uncertainty is based on Run 17 (255 GeV) proton beam measurements at RHIC [6],
and the model-dependent uncertainties in the $r_{5}^{pp}$ extrapolation to
$r_{5}^{ph},\,r_{5}^{hp}$ were estimated in this paper.

The evaluated accuracy of the beam polarization measurement meets the EIC
requirement $\sigma_{P}^{\text{syst}}/P\\!\lesssim\\!1\%$ for hadronic
polarimetry. Remarkably that breakup correction $b^{hp}(T_{R})$, which,
potentially, may give an inappropriately (for the polarimetry) large
contribution to $A_{\text{N}}$, cancels in Eq. (62).

Possible errors in predetermined values of the total cross section
$\delta\sigma_{\text{tot}}^{hp}$, energy calibration $\delta{c_{1}}$ (59), and
absorptive correction $\delta{\cal B}_{a}^{\text{sf}}$ (26) may result in
$P_{\text{meas}}(T_{R})$ dependence on $T_{R}$

$\displaystyle T_{c}\,dP_{\text{meas}}$ $\displaystyle(T_{R})/{dT_{R}}=$
$\displaystyle
G_{\varkappa}\\!\times\\!\left[{\delta\sigma_{\text{tot}}^{hp}}/{\sigma_{\text{tot}}^{hp}}+\delta{c_{1}}+\delta{\cal
B}_{a}^{\text{sf}}\varkappa_{p}/2\right]\\!.$ (64)

So, the sum of errors in values of $\sigma_{\text{tot}}^{hp}$, $c_{1}$, and
${\cal B}_{a}^{\text{sf}}$ defined in Eq. (64) can be experimentally evaluated
with an accuracy of ${\cal O}(10^{-3})$ directly in the linear fit of
$P_{\text{beam}}^{h}(T_{R})$.

The HJET (with possible improvements if required) is being considered for the
measurement of the proton beam absolute polarization at EIC. According to
estimates done in this paper, exactly that polarimeter also has the ability to
precisely measure helion beam polarization (if proton-proton $r_{5}$ for the
beam energy used is already known). Therefore, some common potential problems
for the EIC proton and helion beam polarimetry, e.g., due to much shorter
bunch spacing of 10 ns at EIC compared to 107 ns at RHIC [36], were not
considered here.

The ${}^{3}\text{He}$ beam polarization uncertainty dependence on $r_{5}^{pp}$
can be reduced by a factor of about 4 if an unpolarized hydrogen jet target
will be used. In this case, also, event statistics can be significantly
increased snd some essential systematic uncertainties in the measurements can
be eliminated. However, to implement this method much better knowledge of the
$d\sigma^{hp}/dt$ parametrization (including the beam ${}^{3}\text{He}$
breakup effect) is needed.

It should be also noted that developing alternative approaches to
${}^{3}\text{He}$ beam polarimetry at EIC, e.g., polarized ${}^{3}\text{He}$
target, is important to obtain a confident result.

###### Acknowledgements.

The author would like to thank N. H. Buttimore, B. Z. Kopeliovich, and A.
Zelenski for useful discussions and valuable comments, and acknowledges
support from the Office of Nuclear Physics in the Office of Science of the US
Department of Energy. This work is authored by employees of Brookhaven Science
Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of
Energy.

## References

  * Accardi  _et al._ [2016] A. Accardi, L. Albacete, J., M. Anselmino, _et al._ , Electron Ion Collider: The Next QCD Frontier, Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex] .
  * Abdul Khalek  _et al._ [2022] R. Abdul Khalek _et al._ , Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, Nucl. Phys. A 1026, 122447 (2022), arXiv:2103.05419 [physics.ins-det] .
  * Zelenski  _et al._ [2005] A. Zelenski, A. Bravar, D. Graham, W. Haeberli, S. Kokhanovski, Y. Makdisi, G. Mahler, A. Nass, J. Ritter, T. Wise, and V. Zubets, Absolute polarized H-jet polarimeter development, for RHIC, Nucl. Instrum. Meth. A 536, 248 (2005).
  * Poblaguev  _et al._ [2020a] A. A. Poblaguev, A. Zelenski, G. Atoian, Y. Makdisi, and J. Ritter, Systematic error analysis in the absolute hydrogen gas jet polarimeter at RHIC, Nucl. Instrum. Meth. A 976, 164261 (2020a), arXiv:2006.08393 [physics.ins-det] .
  * Kopeliovich and Lapidus [1974] B. Z. Kopeliovich and L. I. Lapidus, On the necessity of polarization experiments in colliding $pp$ and $\overline{p}p$ beams, Yad. Fiz. 19, 218 (1974), [Sov. J. Nucl. Phys. 19, 114 (1974)].
  * Poblaguev  _et al._ [2019] A. A. Poblaguev, A. Zelenski, E. Aschenauer, G. Atoian, K. O. Eyser, H. Huang, Y. Makdisi, W. B. Schmidke, I. Alekseev, D. Svirida, and N. H. Buttimore, Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter, Phys. Rev. Lett. 123, 162001 (2019), arXiv:1909.11135 [hep-ex] .
  * Buttimore [2009] N. H. Buttimore, Forward helion scattering and neutron polarization, AIP Conf. Proc. 1105, 189 (2009).
  * Buttimore  _et al._ [1999] N. H. Buttimore, B. Z. Kopeliovich, E. Leader, J. Soffer, and T. L. Trueman, The spin dependence of high-energy proton scattering, Phys. Rev. D 59, 114010 (1999), arXiv:hep-ph/9901339 .
  * Buttimore  _et al._ [1978] N. H. Buttimore, E. Gotsman, and E. Leader, Spin-dependent phenomena induced by electromagnetic-hadronic interference at high energies, Phys. Rev. D 18, 694 (1978).
  * Buttimore  _et al._ [1987] N. H. Buttimore, E. Gotsman, and E. Leader, Erratum: Spin-dependent phenomena induced by electromagnetic-hadronic interference at high energies, Phys. Rev. D 35, 407 (1987).
  * Cahn [1982] R. Cahn, Coulombic-Hadronic Interference in an Eikonal Model, Z. Phys. C 15, 253 (1982).
  * Kopeliovich and Tarasov [2001] B. Z. Kopeliovich and A. V. Tarasov, The Coulomb phase revisited, Phys. Lett. B 497, 44 (2001), arXiv:hep-ph/0010062 .
  * Poblaguev [2019] A. A. Poblaguev, Corrections to the Elastic Proton-Proton Analyzing Power Parametrization at High Energies, Phys. Rev. D 100, 116017 (2019), arXiv:1910.02563 [hep-ph] .
  * Zyla  _et al._ [2020] P. A. Zyla _et al._ (Particle Data Group), Review of Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * Shiner  _et al._ [1995] D. Shiner, R. Dixson, and V. Vedantham, Three-Nucleon Charge Radius: A Precise Laser Determination Using 3He, Phys. Rev. Lett. 74, 3553 (1995).
  * Morton  _et al._ [2006] D. C. Morton, Q. Wu, and G. W. F. Drake, Nuclear charge radius for 3He, Phys. Rev. A 73, 034502 (2006).
  * Kopeliovich  _et al._ [2021] B. Z. Kopeliovich, M. Krelina, and I. K. Potashnikova, Probing the Pomeron spin structure with Coulomb-nuclear interference, Phys. Lett. B 816, 136262 (2021), arXiv:2102.01595 [hep-ph] .
  * Poblaguev [2022a] A. A. Poblaguev, Coulomb phase corrections to the transverse analyzing power $A_{N}(t)$ in high energy forward proton-proton scattering, Phys. Rev. D 105, 096039 (2022a), arXiv:2111.01696 [hep-ph] .
  * Lee  _et al._ [2015] G. Lee, J. R. Arrington, and R. J. Hill, Extraction of the proton radius from electron-proton scattering data, Phys. Rev. D 92, 013013 (2015), arXiv:1505.01489 [hep-ph] .
  * Friar  _et al._ [1990] J. L. Friar, B. F. Gibson, G. L. Payne, A. M. Bernstein, and T. E. Chupp, Neutron polarization in polarized 3He targets, Phys. Rev. C 42, 2310 (1990).
  * Kubon  _et al._ [2002] G. Kubon _et al._ , Precise neutron magnetic form-factors, Phys. Lett. B 524, 26 (2002), arXiv:nucl-ex/0107016 .
  * Bartenev  _et al._ [1973a] V. Bartenev _et al._ , Measurement of the Slope of the Diffraction Peak for Elastic $p$-$p$ Scattering from 8 to 400 GeV., Phys. Rev. Lett. 31, 1088 (1973a).
  * Bartenev  _et al._ [1973b] V. Bartenev _et al._ , Real Part of the Proton-Proton Forward-Scattering Amplitude from 50 to 400 GeV, Phys. Rev. Lett. 31, 1367 (1973b).
  * Fagundes  _et al._ [2017] D. A. Fagundes, M. J. Menon, and P. V. R. G. Silva, Leading components in forward elastic hadron scattering: Derivative dispersion relations and asymptotic uniqueness, Int. J. Mod. Phys. A 32, 1750184 (2017), arXiv:1705.01504 [hep-ph] .
  * Chan  _et al._ [1966] L. H. Chan, K. W. Chen, J. R. Dunning, N. F. Ramsey, J. K. Walker, and R. Wilson, Nucleon Form Factors and Their Interpretation, Phys. Rev. 141, 1298 (1966).
  * Glauber [1955] R. J. Glauber, Cross Sections in Deuterium at High Energies, Phys. Rev. 100, 242 (1955).
  * Gross  _et al._ [1978] D. Gross, E. Jenkins, A. Kuznetsov, E. Malamud, B. Morozov, V. Nikitin, P. Nomokonov, Y. Pilipenko, A. Sandacz, V. Smirnov, and R. Yamada, Real part of the $p$-$p$, $p$-$d$, and $p$-$n$ forward scattering amplitudes from 50 to 400 gev, Phys. Rev. Lett. 41, 217 (1978).
  * Carroll  _et al._ [1974] A. S. Carroll _et al._ , Total Cross-Sections of $p$ and $\overline{p}$ on Protons and Deuterons Between 50 and 200 GeV/c, Phys. Rev. Lett. 33, 928 (1974).
  * Carroll  _et al._ [1976] A. S. Carroll _et al._ , Total Cross-Sections of $\pi^{\pm}$, ${K}^{\pm}$, $p$, and $\overline{p}$ on Protons and Deuterons Between 23 and 280 GeV/c, Phys. Lett. B 61, 303 (1976).
  * Burq  _et al._ [1981] J. P. Burq _et al._ , Measurements of the Total $\pi$-He and $p$-He Cross-sections and the Slopes of the Forward Diffraction Peak at Energies From 50 to 300 GeV, Nucl. Phys. B 187, 205 (1981).
  * Cudell  _et al._ [2002] J. R. Cudell, V. V. Ezhela, P. Gauron, K. Kang, Y. V. Kuyanov, S. B. Lugovsky, E. Martynov, B. Nicolescu, E. A. Razuvaev, and N. P. Tkachenko (COMPETE), Benchmarks for the forward observables at RHIC, the Tevatron Run II and the LHC, Phys. Rev. Lett. 89, 201801 (2002), arXiv:hep-ph/0206172 .
  * Poblaguev [2022b] A. A. Poblaguev, Breakup corrections to the 3He beam polarization measurements at the future BNL Electron Ion Collider, Phys. Rev. C 106, 065203 (2022b), arXiv:2207.06999 [hep-ph] .
  * Kopeliovich and Trueman [2001] B. Z. Kopeliovich and T. L. Trueman, Polarized proton nucleus scattering, Phys. Rev. D 64, 034004 (2001), arXiv:hep-ph/0012091 .
  * Buttimore  _et al._ [2001] N. H. Buttimore, E. Leader, and T. L. Trueman, An absolute polarimeter for high-energy protons, Phys. Rev. D 64, 094021 (2001), arXiv:hep-ph/0107013 .
  * Patrignani  _et al._ [2016] C. Patrignani _et al._ (Particle Data Group), Review of Particle Physics, Chin. Phys. C 40, 100001 (2016), Plots of cross sections and related quantaties (rev.).
  * Poblaguev  _et al._ [2020b] A. A. Poblaguev, A. Zelenski, and G. Atoian, The prospects on the absolute proton beam polarimetry at EIC, PoS PSTP2019, 007 (2020b).