# Proton charge radius extraction from muon scattering at MUSE
using dispersively improved chiral effective field theory

F. Gil-Domínguez<EMAIL_ADDRESS>Instituto de Física Corpuscular
(IFIC) (centro mixto CSIC-UV), Institutos de Investigación de Paterna,
C/Catedrático José Beltrán 2, 46980 Paterna, Valencia, Spain  J. M. Alarcón
<EMAIL_ADDRESS>Universidad de Alcalá, Grupo de Física Nuclear y de
Partículas, Departamento de Física y Matemáticas, 28805 Alcalá de Henares
(Madrid), Spain  C. Weiss<EMAIL_ADDRESS>Theory Center, Jefferson Lab,
Newport News, VA 23606, USA

###### Abstract

The MUSE experiment at Paul Scherrer Institute will perform the first
measurement of low-energy muon-proton elastic scattering (muon lab momenta
115–210 MeV) with the aim of determining the proton charge radius. We study
the prospects for the proton radius extraction using the theoretical framework
of Dispersively Improved Chiral Effective Field Theory (DI$\chi$EFT). It
connects the proton radii with the finite-$Q^{2}$ behavior of the form factors
through complex analyticity and enables the use of data up to $Q^{2}\sim$ 0.1
GeV2 for radius extraction. We quantify the sensitivity of the $\mu p$ cross
section to the proton charge radius, the theoretical uncertainty of the cross
section predictions, and the size of two-photon exchange corrections. We find
that the optimal kinematics for radius extraction at MUSE is at momenta 210
MeV and $Q^{2}\sim$ 0.05–0.08 GeV2. We compare the performance of electron and
muon scattering in the same kinematics. As a byproduct, we obtain explicit
predictions for the $\mu p$ and $ep$ cross sections at MUSE as functions of
the assumed value of the proton radius.

††preprint: JLAB-THY-23-3834

##  I Introduction

The electromagnetic size is a fundamental characteristic of the proton
observed in nuclear and atomic physics. It is quantified by the root-mean-
squared radii $r_{E}\equiv\sqrt{\langle r^{2}\rangle_{E}}$ and
$r_{M}\equiv\sqrt{\langle r^{2}\rangle_{M}}$, defined by the derivatives of
the electric and magnetic form factors (FFs), $G_{E}$ and $G_{M}$, at momentum
transfer $Q^{2}=0$, see Ref. [1] for a review. The radii can be determined
experimentally either from elastic electron-proton ($ep$) or muon-proton ($\mu
p$) scattering or from the nuclear corrections to the energy levels of
electronic or muonic hydrogen atoms.

The proton charge radius has been the object of extensive studies in the last
decade. The extraction from muonic hydrogen measurements in 2010,
$r_{E}=0.84184(67)$ fm [2], differed by 5$\sigma$ from the CODATA value
accepted at the time, $r_{E}=0.8768(69)$ fm [3], obtained from electronic
hydrogen and electron scattering data (“proton radius puzzle”). The
discrepancy motivated experimental and theoretical efforts aiming to improve
the extraction methods, quantify the uncertainties, and reconcile the results;
see Refs. [4, 5] for reviews. The questions raised include the performance of
various methods for extraction of the radius from scattering data, the
comparison of scattering and atomic results, and potential differences between
electron and muon interactions. The cumulative results from these studies tend
to favor the “smaller” charge radius. However, one essential piece is still
missing – the extraction of the radius from low-energy $\mu p$ elastic
scattering.

The MUSE experiment at Paul Scherrer Institute aims to perform the first
precise determination of the proton charge radius from $\mu p$ elastic
scattering at muon lab momenta 115–210 MeV [6]. Good understanding of the
theoretical uncertainties is needed in order to optimize the extraction
procedure and assess the final error in the radius. Important questions are
the sensitivity of the experimental observables to the proton radius, the
theoretical uncertainty in the relation between the proton radius and the
finite-$Q^{2}$ FFs, the size and uncertainty of two-photon exchange (TPE)
corrections, and the optimal muon energy and $Q^{2}$ range for constraining
the radius.

The analytic properties of the proton FF play an essential role in the radius
extraction from scattering data. As a function of complex $Q^{2}$, the form
factor has singularities at $Q^{2}<0$, resulting from the $t$-channel exchange
of hadronic states (pions, resonances) between the electromagnetic current and
the proton. These singularities govern the behavior of the FF at $Q^{2}>0$,
where it is measured in scattering experiments. This structure implies a
correlation between the derivative of the FF at $Q^{2}=0$ and its values at
finite $Q^{2}$, which is essential for the radius extraction and must be
implemented in the theoretical analysis.

The recently developed method of Dispersively Improved Chiral Effective Field
Theory (DI$\chi$EFT) [7, 8] combines dispersion relations with dynamical input
from chiral EFT to describe the nucleon FFs at low $Q^{2}$ from first
principles. It generates FFs with correct analytic properties (position of
singularities) and realistic quantitative behavior (strength of
singularities), which provide an excellent description of scattering data up
to $Q^{2}\sim$ 1 GeV2 [9]. It also quantifies the theoretical uncertainty of
the FF calculations. A special feature of this method is that it generates FF
predictions that depend on the assumed proton radius as a parameter. As such
it explicitly realizes the correlations between the proton radius and the
finite-$Q^{2}$ behavior of the FF. It permits the use of finite-$Q^{2}$ data
for the radius extraction with controlled uncertainties, which has many
experimental and theoretical advantages. The method has been used successfully
for the extraction of the proton electric and magntic radii from electron
scattering data [10, 9].

In this work we use DI$\chi$EFT to study the prospects for proton radius
determination in $\mu p$ elastic scattering at MUSE and optimize the
extraction procedure. We compute the $\mu p$ cross section with the
DI$\chi$EFT FFs, quantify the theoretical uncertainties and TPE corrections,
and evaluate the sensitivity to the proton radius. Specifically, we attempt to
answer the following questions:

  1. 1.

What is the theoretical sensitivity of the $\mu p$ cross section in MUSE
kinematics to the proton radius?

  2. 2.

What are the theoretical uncertainties in the $\mu p$ cross section resulting
from the DI$\chi$EFT FF predictions and from TPE corrections?

  3. 3.

What kinematic range in beam energy and $Q^{2}$ has the most impact on the
radius extraction?

  4. 4.

What are the differences between $ep$ and $\mu p$ scattering in radius
extraction in MUSE kinematics?

We demonstrate that the radius extraction is characterized by a trade-off
between several effects – the sensitivity of the cross section to the radius,
the theoretical uncertainty in the FF predictions for a given radius, and the
size and kinematic dependence of TPE corrections [10, 9]. We determine the
optimal kinematics for radius extraction at MUSE based on these
considerations. In addition, we provide predictions of the expected $\mu p$
and $ep$ cross sections for the nominal value of the proton radius.

##  II Methods

###  II.1 Lepton-proton elastic scattering

The elastic lepton-proton scattering process $l(k)+p(p)\rightarrow
l(k^{\prime})+p(p^{\prime})$, where $l=\mu^{\mp}$ or $e^{\mp}$, is described
by the invariant variables

$\displaystyle s\equiv(k+p)^{2},\hskip
20.00003ptQ^{2}=-t\equiv-(k-k^{\prime})^{2}.$ (1)

In the initial proton rest frame (lab frame), the initial and final muon
momenta are $\bm{k}$ and $\bm{k}^{\prime}$, the energies are
$\omega\equiv\sqrt{|\bm{k}|^{2}+m^{2}}$ and
$\omega^{\prime}\equiv\sqrt{|\bm{k}^{\prime}|^{2}+m^{2}}$, and the invariants
are given by

$\displaystyle s=M^{2}+2M\omega+m^{2},\hskip
20.00003ptQ^{2}=2M(\omega-\omega^{\prime}),$ (2)

where $m$ is the lepton mass and $M$ the proton mass. The scattering angle
$\theta_{\rm lab}=\textrm{angle}(\bm{k}^{\prime},\bm{k})$ is related to the
final lepton energy and momentum by

$\displaystyle\cos\theta_{\rm
lab}=\frac{\omega\omega^{\prime}-m^{2}-M(\omega-\omega^{\prime})}{|\bm{k}||\bm{k}^{\prime}|}.$
(3)

The kinematic range of the momentum transfer accessible at a given initial
lepton momentum is

$\displaystyle 0\;\leq\;Q^{2}\;\leq\;\frac{4M^{2}|\bm{k}|^{2}}{s}\equiv
Q^{2}_{\rm max}.$ (4)

In the one-photon-exchange approximation, the differential cross section for
unpolarized scattering is given by (see e.g. Ref. [11])

$\displaystyle\frac{d\sigma_{1\gamma}}{dQ^{2}}$
$\displaystyle=\frac{\pi\alpha^{2}}{2M^{2}|\bm{k}|^{2}}\frac{(\epsilon/\tau_{P})\,G_{E}^{2}+G_{M}^{2}}{1-\epsilon_{T}}.$
(5)

Here $\alpha$ is the fine structure constant, and $G_{E,M}\equiv
G_{E,M}(Q^{2})$ are the electric and magnetic Sachs FFs of the proton.
$\epsilon$ is the virtual photon polarization parameter and given by

$\displaystyle\epsilon=\frac{\displaystyle Q^{2}_{\rm
max}-Q^{2}+\frac{m^{2}}{s}(4M^{2}+Q^{2})}{\displaystyle Q^{2}_{\rm
max}-Q^{2}+\frac{Q^{2}}{2s}(4M^{2}+Q^{2})},$ (6)

and $\tau_{P}\equiv Q^{2}/(4M^{2})$. $\epsilon/\tau_{P}$ is the ratio of the
fluxes of longitudinal and transverse polarized photons in the one-photon-
exchange approximation. $\epsilon_{T}$ is the degree of linear polarization of
the transverse photons,

$\displaystyle\epsilon_{T}=\frac{\displaystyle Q^{2}_{\rm
max}-Q^{2}}{\displaystyle Q^{2}_{\rm
max}-Q^{2}+\frac{Q^{2}}{2s}(4M^{2}+Q^{2})},$ (7)

and is bounded by $0\leq\epsilon_{T}<1$. In the case of zero lepton mass (as
usually assumed in electron scattering) $\epsilon=\epsilon_{T}$, but for non-
zero lepton mass (muon scattering) there are important differences. $\epsilon$
attains values $>1$ at $Q^{2}=0$, and remains nonzero at $Q^{2}=Q^{2}_{\rm
max}$,

$\displaystyle\epsilon(Q^{2}=0)$
$\displaystyle=\frac{\omega^{2}}{|\bm{k}|^{2}}\;>\;1,$ (8)
$\displaystyle\epsilon(Q^{2}=Q^{2}_{\rm max})$
$\displaystyle=\frac{m^{2}s}{2M^{2}|\bm{k}|^{2}}\;>\;0.$ (9)

Two-photon exchange (TPE) corrections play an important role in the analysis
of low-energy lepton-proton elastic scattering, see Refs. [12, 13] for a
review. At order $\alpha^{3}$, the correction arises from the interference
between the two-photon and one-photon exchange amplitudes and is usually
included through a multiplicative factor modifying the one-photon exchange
cross section,

$\displaystyle\frac{d\sigma}{dQ^{2}}\approx\frac{d\sigma_{1\gamma}}{dQ^{2}}\,\left(1+\delta_{2\gamma}\right).$
(10)

The correction $\delta_{2\gamma}$ for $\mu p$ scattering has been computed in
several theoretical approaches, such as dispersion theory [14, 11] and chiral
effective field theory [15, 16, 17]. In this work we use the results of Ref.
[14], which give corrections $\delta_{2\gamma}\lesssim 0.5\%$ in the kinematic
range of the MUSE experiment. While in Ref. [14] the inelastic contribution to
the dispersion integral for $\delta_{2\gamma}$ was computed in forward
kinematics, the analysis of Ref. [11] showed that this approximation is
accurate within $10\%$.

###  II.2 DI$\chi$EFT representation of form factors

Figure 1: Flowchart of the DI$\chi$EFT description of the nucleon FFs. Shown
is the case of the proton electric FF $G_{E}$; the same flow applies to the
neutron electric FF. In the case of the magnetic FF $G_{M}$, the right-hand
side of the dispersive sum rules is given by the magnetic moment $\mu$ and the
magnetic radius $r^{2}_{M}$.

The foundations of the DI$\chi$EFT method and its applications are described
in detail in Refs. [18, 7, 8]. Here we provide a brief summary, emphasizing
the information flow (parameters) and other features relevant to proton radius
extraction.

DI$\chi$EFT is based on dispersion theory, in which the FFs $G_{E,M}(t)$ at
spacelike momentum transfer $t<0$ are represented as integrals over their
imaginary parts $\textrm{Im}\,G_{E,M}(t)$ on the cut at timelike $t>0$, the
so-called spectral functions. The main steps in the DI$\chi$EFT construction
of the spectral functions and the FFs are summarized in Fig. 1. In the
following we describe the steps for $G_{E}$; the ones for $G_{M}$ are similar.

In the first step, one constructs the spectral function. The proton FF has an
isovector and isoscalar component, $G_{E}\equiv G_{E}^{V}+G_{E}^{S}$. The
isovector FF $G_{E}^{V}$ has the two-pion cut at $t>t_{\rm thr}=4M_{\pi}^{2}$.
The isovector spectral function is represented as the sum of two parts,

$\displaystyle\text{Im}\,G_{E}^{V}(t)$
$\displaystyle=\text{Im}\,G_{E}^{V}(t)[\pi\pi]+\text{Im}\,G_{E}^{V}(t)[\textrm{high-
mass}].$ (11)

The $\pi\pi$ part covers the region $4M_{\pi}^{2}\leq t<t_{\rm max}\approx 1$
GeV2 and is computed theoretically, using the elastic unitarity relation in
the $\pi\pi$ channel and relativistic chiral EFT for the $\pi N$ amplitudes.
This approach includes the $\rho$ resonance in the $\pi\pi$ channel and
delivers realistic $\pi\pi$ spectral functions, which is essential for proton
radius extraction (see below). The free parameters entering in this part are
the low-energy constants of the chiral EFT calculation; in the present partial
N2LO implementation this is one parameter, $\lambda$ [8]. The high-mass part
of the spectral function covers the region $t>1$ GeV2 and is parametrized by a
single effective pole $\pi a_{1}\delta(t-t_{1})$; this form has been shown to
be sufficient for the dispersive analysis of low-$t$ FFs, which see only the
overall spectral strength in the high-mass region [8]. The free parameters
entering in this part are the pole strength $a_{1}$ and pole position $t_{1}$.
The isoscalar FF $G_{E}^{S}$ has a three-pion cut, and the spectral function
is represented as

$\displaystyle\text{Im}\,G_{E}^{S}(t)$
$\displaystyle=\text{Im}\,G_{E}^{S}(t)[\pi\pi\pi]+\text{Im}\,G_{E}^{S}(t)[\textrm{high-
mass}].$ (12)

The $\pi\pi\pi$ part is overwhelmingly concentrated in the $\omega$ resonance
and parametrized by a pole $\pi a_{\omega}\delta(t-M_{\omega}^{2})$. The high-
mass part is parametrized by an effective pole, whose position can be taken as
the $\phi$ mass, $\pi a_{\phi}\delta(t-M_{\phi}^{2})$. The free parameters
entering in the isoscalar spectral function are the pole strengths
$a_{\omega}$ and $a_{\phi}$. Altogether, this step results in a theoretical
parametrization of the spectral function

$\displaystyle\text{Im}\,G_{E}(t|\Lambda)=\text{Im}\,G_{E}^{V}(t)+\text{Im}\,G_{E}^{S}(t),$
(13)

where $\\{\Lambda\\}$ collectively denotes the free parameters; in the present
analysis $\\{\Lambda\\}=\\{\lambda,a_{1},t_{1};a_{\omega},a_{\phi}\\}$.

In the second step, one imposes the dispersive sum rules for the proton charge
and radius

$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\frac{\text{Im}\,G_{E}(t|\Lambda)}{t}=Q_{E},$ (14)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\frac{\text{Im}\,G_{E}(t|\Lambda)}{t^{2}}=\frac{r^{2}_{E}}{6},$
(15)

where $Q_{E}=1$ is the proton charge and $r_{E}^{2}>0$ the proton charge
radius squared. (The same relations are imposed for the neutron electric FF,
in which case $Q_{E}=0$ and $r_{E}^{2}<0$ is the negative neutron charge
radius squared.) These relations express the FF at $t=0$ and its derivative as
integrals over the spectral function. One uses them to constrain the
parameters in the spectral function. In particular, Eq. (15) is valid for any
assumed value of the proton charge radius $r_{E}$, and one can use it to
express one of (or a combination of) the original parameters in terms of the
radius, i.e., to introduce the radius as a parameter:

$\displaystyle\\{\Lambda\\}\rightarrow\\{r_{E},\Lambda^{\prime}\\}.$ (16)

In this way one obtains a set of spectral functions that depend explicitly on
the assumed radius, as well as on the remaining parameters $\Lambda^{\prime}$

$\displaystyle\text{Im}\,G_{E}(t|r_{E},\Lambda^{\prime}).$ (17)

In the present analysis we use Eqs. (14) and (15) for the proton spectral
function (and the same relations for the neutron) to fix the chiral low-energy
constant $\lambda$ and the effective pole strengths
$a_{1},a_{\omega},a_{\phi}$, retaining the isovector effective pole position
$t_{1}$ as the only undetermined parameter.

In the third step, one computes the spacelike FFs ($t<0$, or $Q^{2}>0$) as the
dispersion integral with the spectral function,

$\displaystyle G_{E}(t|r_{E},\Lambda^{\prime})=\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt^{\prime}\frac{\text{Im}\,G_{E}(t^{\prime}|r_{E},\Lambda^{\prime})}{t^{\prime}-t}.$
(18)

The FF thus obtained depends on the assumed radius $r_{E}$ and the
undetermined parameters $\Lambda^{\prime}$. The nominal prediction for the FF
with assumed radius $r_{E}$ is obtained with the nominal values of
$\Lambda^{\prime}$,

$\displaystyle G_{E}(t|r_{E})=G_{E}(t|r_{E},\Lambda^{\prime}_{\rm nom}).$ (19)

In the last step, the theoretical uncertainty of the FF with assumed radius
$r_{E}$ is estimated by varying $\Lambda^{\prime}$ over a plausible range,

$\displaystyle\delta
G_{E}(t|r_{E})=\textrm{var}_{\Lambda^{\prime}}\;G_{E}(t|r_{E},\Lambda^{\prime}).$
(20)

In this way one obtains a nominal prediction and a theoretical uncertainty
estimate for the form factor with any given assumed radius. In the present
analysis, the undetermined parameter is the position of the isovector high-
mass pole; its nominal value is $t_{1}=$ 2.1 GeV2, and the plausible range of
variation for the uncertainty estimate is $t_{1}=$ 1.4–2.8 GeV2 [10].

The magnetic FF $G_{M}^{V}$ and its uncertainty are constructed by an
analogous procedure. The dispersive sum rules for $G_{M}$ now involve the
magnetic moment $\mu$ and the magnetic radius $r_{M}^{2}$ see Ref. [8] for
details. A computer code generating the radius-dependent DI$\chi$EFT FFs
$G_{E,M}$ used in the present analysis is available in the supplemental
materials of Ref. [9].

###  II.3 Proton radius extraction

Figure 2: Illustration of the correlation between the proton radius and the
spacelike FF, resulting from analyticity and the information flow in
DI$\chi$EFT. The assumed value of the radius constrains the spectral function
through the sum rule Eq. (15). The corresponding FF at $Q^{2}>0$ is produced
by the dispersion integral Eq. (18). Variation of the radius causes variation
of the spectral function and the corresponding FF. (The graph shows only the
isovector part of the spectral function at $t>0$.)

The DI$\chi$EFT representation of the FFs enables a new theory-guided method
of proton radius extraction (see Fig. 2) [10, 9]. For each assumed value of
the proton radius, the theory generates a spectral function whose features
(height of the $\rho$ resonance peak, strength of effective poles)
quantitatively depend on the value of the radius. The dispersion integral
projects these features into the spacelike region, up to spacelike momentum
transfers of the order $Q^{2}\sim M_{\rho}^{2}$ and beyond. This effectively
correlates the assumed value of the radius with the behavior of the spacelike
FF at finite $Q^{2}$ of this order. The correlation described here is based on
complex analyticity and the particular information flow in the DI$\chi$EFT
calculation and extends far beyond what one could infer from the series
expansion in $Q^{2}$ with a given first derivative. Altogether, this allows
one to recruit FF data of the order $Q^{2}\sim M_{\rho}^{2}$ and beyond for
constraining the proton radii.

Radius extraction using DI$\chi$EFT proceeds as follows. For a range of
assumed radii, one generates the DI$\chi$EFT FFs as functions of $Q^{2}$,
including their theoretical uncertainties resulting from the undetermined
parameters, Eqs. (19) and (20). From these FFs one predicts the cross section
for the given assumed radius, including its theoretical uncertainty from the
FFs and two-photon exchange corrections. For a given experimental setup
(kinematic coverage, statistical and systematic errors) one can then assess
how data in a given range of energies and $Q^{2}$ can constrain the radii. The
optimal range is determined by a trade-off between the sensitivity of the
DI$\chi$EFT FFs to the value of the radius, the theoretical uncertainty of the
DI$\chi$EFT FFs, the two-photon exchange effects, and the precision of the
data [10, 9]. The actual radius can then be determined by a fit in this
optimal range, taking into account all the uncertainties. In the following we
apply this method to $\mu p$ scattering at MUSE and discuss the prospects for
proton radius extraction.

The DI$\chi$EFT method offers several advantages compared to other methods of
proton radius extraction. Compared to empirical fits (polynomials, splines),
the DI$\chi$EFT method incorporates the analytic structure of the FFs, which
includes both the position of the singularities at $t>0$ and the quantitative
distribution of strength in the spectral function. The analytic structure
governs the global behavior of the FF, which is difficult to implement in
approaches based on polynomial expansions because of strong correlations
between higher-order coefficients (analyticity effectively controls the
“collective behavior” of higher derivatives of the FF at $Q^{2}=0$ [7]).
Compared to traditional dispersion analysis [19, 20, 21, 22], the DI$\chi$EFT
method allows the strength of the isovector spectral function in the $\rho$
resonance region to vary with the proton radius in a theoretically controlled
manner, providing critical flexibility for fitting the spacelike FF data and
recruiting them for radius determination. In traditional dispersive fits the
$\pi\pi$ part of the isovector spectral function is completely fixed by
theory, and the spacelike FF data only constrain the high-mass part of the
spectral function, which restricts the interplay of the FF data with the
proton radius (the $\rho$ resonance region of the spectral function accounts
for about half the value of the proton radius in Eq. (15) [7]).

##  III Analysis

###  III.1 Sensitivity of $\mu p$ cross section to proton radius

We now apply the DI$\chi$EFT framework to study the prospects for proton
radius extraction at MUSE. In the first step, we study the sensitivity of the
$\mu p$ elastic scattering cross section to the proton electric radius and
compare it with the theoretical uncertainties resulting from the DI$\chi$EFT
FF predictions, from two-photon exchange corrections, and from the magnetic FF
contributions.

To exhibit the various effects, we generate a set of DI$\chi$EFT FF
predictions by varying the proton electric radius over the range $r_{E}=$
0.83–0.88 fm in steps $\Delta r_{E}=$ 0.01 fm, and evaluate the $\mu p$
elastic scattering cross section with each of these FFs (the magnetic radius
is kept at its nominal value; the role of the magnetic FF in the cross section
is discussed below). We include in the cross section the TPE correction of
Ref. [14]. Figure 3 shows the predicted cross sections for various incident
muon momenta $k\equiv|\bm{k}|$, as functions of $Q^{2}$. The lines show the
cross section obtained with the nominal DI$\chi$EFT FF predictions for each
value of the radius, Eq. (19); the associated bands show the variation due to
the theoretical uncertainty of the DI$\chi$EFT FF predictions for the given
radius, Eq. (20). The bands at the bottom of the plots show the absolute size
of the TPE correction in the cross section predictions (note that this is the
overall size of the TPE correction, not its theoretical uncertainty). The
standard dipole cross section ($\sigma_{SD}$) used for normalization is the
one-photon exchange cross section evaluated assuming the standard dipole
$Q^{2}$-dependence $\propto(1+Q^{2}/0.71\,\textrm{GeV}^{2})^{-2}$ for both FFs
$G_{E,M}$.

Figure 3: DI$\chi$EFT predictions for the $\mu^{-}p$ differential cross
section at MUSE for several assumed values of the proton radius. The cross
section predictions include the TPE correction, Eq. (10), and are normalized
to the standard dipole cross section without TPE effects. Lines (solid,
dashed, dotted, dahed-dotted): Nominal DI$\chi$EFT predictions for the assumed
value of the proton radius (see legend). Shaded bands around lines: Intrinsic
theoretical uncertainty of DI$\chi$EFT predictions, unrelated to assumed
proton radius. Shaded band at bottom: TPE contribution to cross section [14].

One observes: (a) The sensitivity of the cross section to the proton radius
increases with $Q^{2}$ and with the beam momentum $k$, because the separation
of the FF predictions with different radii increases with $Q^{2}$ [10, 9]. At
the highest beam momentum, $k=$ 210 MeV, the relative variation of the cross
section reaches $\Delta\sigma/\sigma\sim$ 1% for $Q^{2}$ at the upper end of
the range shown here. (b) The theoretical uncertainty of the cross section
predictions for given radius also increases with $Q^{2}$ [10, 9]. Overall, the
theoretical uncertainty is substantially smaller than the relative variation
of the cross section for $\Delta r_{E}$ = 0.01 fm over the kinematic range
shown here. (c) The magnitude of the TPE correction does not vary strongly
with $Q^{2}$ and $k$ over the range covered here. At the upper end of the
$Q^{2}$ range, the magnitude of the TPE correction is comparable to the
relative variation of the cross section with $\Delta r_{E}$ = 0.01 fm. This
clearly shows the importance of the TPE correction for radius extraction.

Figure 4: Ratio of magnetic and electric contributions to the $\mu p$ elastic
scattering cross section, $(\tau_{P}/\epsilon)G_{M}^{2}/G_{E}^{2}$, in MUSE
kinematics. The vertical dotted lines represent the kinematic upper limits of
$Q^{2}$ at the given beam momentum $k$, Eq. (4).

We also need to consider the uncertainties resulting from the contribution of
the magnetic FF to the $\mu p$ elastic scattering cross section. This is
particularly important, as with the DI$\chi$EFT framework we can recruit data
at higher $Q^{2}$ for radius extraction, comparable to $Q^{2}_{\rm max}$ at
the given $k$. Figure 4 shows the ratio of magnetic and electric contributions
to the one-photon exchange cross section,
$(\tau_{P}/\epsilon)G_{M}^{2}/G_{E}^{2}$, in MUSE kinematics. One sees that
the ratio depends mainly on $Q^{2}$, having values $\sim$0.1 at $Q^{2}$ = 0.04
GeV2 and reaching $\sim$0.4 at $Q^{2}$ = 0.08 GeV2. Overall, the magnetic
contributions to the cross section are limited in all kinematic settings.
Using the DI$\chi$EFT framework and the results of the analysis of $ep$
scattering data of Ref. [9], we have computed the effect of the experimental
uncertainties of $G_{M}$ on the $\mu p$ cross section predictions in MUSE
kinematics. We observe a maximum variation in the cross sections of the order
of $0.04\%$ at the highest $Q^{2}$, which is small compared to the variation
of $\sim$1% resulting from a change of the electric radius $\Delta
r_{E}/r_{E}$ = 1%. We conclude then that the current experimental
uncertainties in $G_{M}$ do not limit the extraction of the proton electric
radius from the $\mu p$ scattering data at the accuracy considered here.

###  III.2 Optimal kinematics for proton radius extraction

In the second step, we discuss the optimal kinematic range for the radius
extraction at MUSE. It is determined by the trade-off between the sensitivity
of the cross section to the radius, the theoretical uncertainties of the
DI$\chi$EFT FF predictions and the TPE corrections, and the experimental
errors of the cross section measurement. While the experimental errors can
only be estimated at present, some interesting conclusions can already be
obtained at this stage.

Figure 5: Estimated accuracy of $\mu^{-}p$ cross section measurement required
to discriminate between different values of the proton radius. Lines:
Differences between DI$\chi$EFT cross section predictions for proton radii
differing by $\Delta r_{E}$ (values see legend). Shaded band at bottom: Size
of the TPE contribution [14].

To make this assessment, we use the difference between the cross section
predictions for different radii in Fig. 3 as an estimate of the experimental
accuracy required to discriminate between these values of the radii. At each
value of $Q^{2}$ in Fig. 3, we compute the minimal difference between the
cross section predictions for radii differing by a given $\Delta r_{E}$,
taking the minimum over all pairs of radii with the given $\Delta r_{E}$, and
taking into account their theoretical uncertainties (i.e., computing the
minimal gap between the theoretical uncertainty bands of the cross section
predictions for a given $\Delta r_{E}$). The minimal cross section difference
computed in this way is independent of the nominal value of $r_{E}$. Figure 5
shows the minimal cross section differences obtained in this way, for radius
differences $\Delta r_{E}=$ 0.01, 0.02, 0.03 and 0.04 fm, as functions of
$Q^{2}$. One observes: (a) The cross section differences depend strongly on
$Q^{2}$ for fixed $k$. They depend relatively weakly on $k$ for fixed $Q^{2}$
(when comparing them at a fixed $Q^{2}$ that is kinematically accessible at
multiple values of $k$). The main role of $k$ is to define the kinematically
accessible range of $Q^{2}$. (b) At low values of $Q^{2}$, high experimental
precision is needed for radius determination. At $Q^{2}\lesssim$ 0.015 GeV2, a
relative accuracy $\leq 0.2\%$ is needed for $\Delta r_{E}=$ 0.01 fm,
independently of $k$. (c) The demands on the experimental accuracy decrease at
higher $Q^{2}$. At $Q^{2}\sim$ 0.05 GeV2, a relative accuracy $\sim 0.5\%$ is
needed for $\Delta r_{E}=$ 0.01 fm.

Another factor to consider is the uncertainty in the theoretical calculation
of the TPE correction [14]. Depending on the kinematics, this contribution to
the cross section can be crucial for determining the radius with the necessary
precision. Figure 5 compares the value of the TPE correction with the
predicted cross section differences for a given radius difference (note that
the plots show the estimated total value of the TPE correction, not its
uncertainty). The theoretical uncertainty of the TPE correction is not well
known; however, we can assess how an assumed theoretical uncertainty of the
TPE correction would impact on the overall uncertainty of the radius
extraction. For the lowest beam momentum, $k=$ 115 MeV, the size of the TPE
correction is larger than the variation of the cross section prediction for
$\Delta r_{E}$ = 0.01 fm. The TPE correction thus has a decisive influence on
the radius extraction in this kinematics. The situation becomes more favorable
at higher beam momenta, where the TPE correction is comparable or smaller than
the cross section variation for $\Delta r_{E}$ = 0.01 fm.

Overall, our analysis suggests that the optimal kinematics for proton radius
determination at MUSE with the DI$\chi$EFT method is at the highest beam
momentum, $k=$ 210 MeV, using momentum transfers $Q^{2}\sim$ 0.05–0.08 GeV2,
at the upper end of the experimentally accessible range. In this setting, the
experimental precision required for radius determination with $\Delta r_{E}$ =
0.01 fm is estimated at $\approx 0.8\%$. The final uncertainty of the radius
extraction depends on the theoretical uncertainty of the TPE correction, which
is not known at present.

Figure 6: DI$\chi$EFT predictions for the average of $\mu^{+}$ and $\mu^{-}$
elastic scattering cross sections Eq. (21) in MUSE kinematics, normalized to
the standard dipole. Lines, shaded bands: Same notation as in Fig. 3.

An alternative method for proton radius extraction with $\mu p$ scattering
uses the average of $\mu^{+}$ and $\mu^{-}$ cross sections,

$\displaystyle\bar{\sigma}\equiv[\sigma(\mu^{+}p)+\sigma(\mu^{-}p)]/2,$ (21)

in which the TPE correction cancels due to its charge dependence. The same
DI$\chi$EFT analysis of radius sensitivity and optimal kinematics as above can
be performed in this case. The cross section prediction is now given by the
one-photon exchange cross section Eq. (5). Figure 6 shows the predicted cross
section and its theoretical uncertainty. The assessment of the optimal $Q^{2}$
values is the same as for $\mu^{+}p$ above.

###  III.3 Cross section prediction for nominal radius

Figure 7: DI$\chi$EFT predictions for $\mu p$ elastic scattering cross at MUSE
for the nominal proton charge radius $r_{E}=0.842(2)~{}\text{fm}$. Dashed line
and shaded band (blue): $\mu^{-}p$ cross section, including one- and two-
photon exchange contributions. Solid line and shaded band (red): Average of
$\mu^{+}p$ and $\mu^{-}p$ cross sections, given by the one-photon exchange
contribution.

The present study focuses on the prospects for extracting the proton radius
from $\mu p$ scattering experiments at MUSE. The proton radius can also be
extracted from atomic spectroscopy and $ep$ scattering experiments. In this
context we can use DI$\chi$EFT to predict the $\mu p$ cross section expected
for a given value of the radius and its theoretical uncertainty. For
reference, we give here the prediction for the $\mu p$ cross section with the
proton charge radius obtained in the previous DI$\chi$EFT analysis of $ep$
scattering results [10, 9]

$\displaystyle r_{E}=0.842(2)~{}\text{fm}.$ (22)

Figure 7 shows the predicted $\mu^{-}p$ cross section in MUSE kinematics, Eq.
(10), which includes the TPE correction; and the charge-averaged cross
section, Eq. (21) in which the TPE correction cancels.

###  III.4 Comparison of $ep$ and $\mu p$ scattering

Figure 8: DI$\chi$EFT predictions for the differential cross section of $ep$
scattering for several assumed value of the proton radius (compare with Fig. 3
for $\mu p$ scattering). The cross section predictions include the TPE
correction and are normalized by the standard dipole cross section without TPE
effects. Solid lines: Nominal DI$\chi$EFT predictions for the assumed value of
the proton radius. Bands around solid lines: Intrinsic theoretical uncertainty
of DI$\chi$EFT prediction (unrelated to assumed proton radius). Blue band at
bottom: TPE contribution to cross section [23].

It is interesting to compare the prospects for proton radius extraction in
$ep$ and $\mu p$ scattering in the same kinematics. The MUSE experiment will
measure both $ep$ and $\mu p$ scattering, and methods for proton radius
extraction were studied intensively in earlier $ep$ scattering experiments.
Characteristic differences between $ep$ and $\mu p$ occur in the TPE effects
[14, 23] and in the role of the magnetic FF. We exhibit them by repeating the
DI$\chi$EFT analysis for $ep$ scattering and comparing with the $\mu p$
results.

Figure 8 shows the DI$\chi$EFT predictions for the $ep$ cross section for a
range of assumed values of the proton radius, in the same style as Fig. 3 for
$\mu p$. One observes: (a) The TPE corrections have different kinematic
dependence in $ep$ than in $\mu p$ scattering [14, 23]. In $ep$ they increase
strongly with $Q^{2}$ at fixed $k$, and decrease with $k$ at fixed $Q^{2}$ In
$\mu p$ the dependencies are much weaker. (b) The size of the TPE corrections
relative to the variation of the cross section with the radius is much larger
in $ep$ than in $\mu p$, especially at low beam momenta. At $k=$ 115 MeV and
$Q^{2}=0.02$ GeV2, the size of TPE correction amounts to a change of the
radius $\Delta r_{E}\approx$ 0.03 fm in $ep$ scattering, compared to $\Delta
r_{E}\approx$ 0.015 fm in $\mu p$ scattering in the same kinematics. (c)
Overall, the different size and kinematic dependence of the TPE corrections
causes a different $Q^{2}$-dependence of the cross section for $ep$ and $\mu
p$ scattering at low $Q^{2}$.

Figure 9: Estimated accuracy of $e^{-}p$ cross section measurements required
to discriminate between different values of the proton radius (compare with
Fig. 5 for $\mu p$ scattering). Lines: Differences between DI$\chi$EFT cross
section predictions for proton radii differing by $\Delta r_{E}$ (values see
legend). Shaded band at bottom: Size of the TPE contribution [23].

Figure 9 shows the estimated accuracy of the $e^{-}p$ cross section
measurement required for discriminating between different values of the
radius, in the same style as Fig. 5 for $\mu^{-}p$. The optimal kinematics for
radius extraction in $ep$ scattering at MUSE can be determined in the same way
as for $\mu p$. The results of Figure 9 show that the experimental accuracy
required for radius extraction from $ep$ scattering is least at the highest
beam momentum $k=210$ MeV. The optimal $Q^{2}$ value determined by the trade-
off between radius sensitivity and theoretical uncertainty is at $Q^{2}\sim$
0.065 GeV2, slightly below the kinematic limit.

Figure 10: DI$\chi$EFT predictions of the ratio of $\mu p$ to $ep$ elastic
scattering cross sections $\sigma$, Eq. (10), including the TPE corrections,
for several assumed values of the proton radius.

Figure 10 shows the ratio of the $\mu p$ to $ep$ cross sections (including the
TPE corrections) for the same set of assumed proton radii. The ratio directly
expresses the different $Q^{2}$-dependence of the $ep$ and $\mu p$ cross
sections. Its magnitude and $Q^{2}$-dependence are determined by the kinematic
factors in the one-photon-exchange cross section Eq. (5) et seq. One observes
that the ratio is remarkably insensitive to the proton charge radius in this
kinematic regime, especially at the lower values of $k$. Only at $k=$ 210 MeV
the differences between the radii become visible at the largest $Q^{2}$
values.

The results of Fig. 8 show that the TPE corrections play a much larger role in
proton radius extraction from $ep$ scattering than $\mu p$ scattering, and
that they limit the theoretical uncertainty of the extracted radius. With the
DI$\chi$EFT method, the influence of TPE corrections in $ep$ can be minimized
by using the data at the highest beam momentum $k=$ 210 MeV and momentum
transfers in the range $Q^{2}\sim$ 0.03–0.08 GeV2 for radius extraction. In
this kinematics the cross section shows good sensitivity to the proton charge
radius, the theoretical uncertainty of the DI$\chi$EFT predictions is small,
and the size of the TPE corrections amounts to a shift of the radius $\Delta
r_{E}\sim$ 0.01 fm (see Fig. 8). The ability to recruit higher-$Q^{2}$ data
for radius extraction with DI$\chi$EFT is thus even more advantageous in $ep$
than in $\mu p$ scattering.

Figure 11: Ratio of magnetic and electric contributions to the $ep$ elastic
scattering cross section, $(\tau_{P}/\epsilon)G_{M}^{2}/G_{E}^{2}$, in MUSE
kinematics (compare with Fig. 4 for $\mu p$ scattering). The vertical dotted
lines represent the kinematic upper limits of $Q^{2}$ at the given beam
momentum $k$, Eq. (4).

An important difference between $ep$ and $\mu p$ scattering appears in the
contribution of the magnetic FF at large momentum transfers, at the upper end
of the allowed kinematic range. In $ep$ scattering $\epsilon=0$ at $Q^{2}\sim
Q^{2}_{\rm max}$ (if one neglects the electron mass), while in $\mu p$
scattering $\epsilon$ attains a finite value, see Eq. (9). In $ep$ scattering
the one-photon exchange cross section for $Q^{2}\rightarrow Q^{2}_{\rm max}$
is therefore dominated by $G_{M}$. Figure 11 shows the ratio of magnetic and
electric contributions to the one-photon-exchange cross section for $ep$
scattering, in the same way as Fig. 4 for $\mu p$. One sees that the magnetic
contribution to the cross section is substantially larger in $ep$ than $\mu p$
already for $Q^{2}$ in the middle of the kinematic range. This circumstance
must be taken into account when assessing the sensitivity of the cross section
to $r_{E}$ in DI$\chi$EFT, and one should remain in the region where the cross
section is not dominated by $G_{M}$. We have quantified the impact of the
uncertainty in $G_{M}$ on the proton radius extraction from $ep$ scattering in
the same way as for $\mu p$ (see Sec. III.1), using the DI$\chi$EFT framework
and the experimental uncertainty of $G_{M}$ obtained in an earlier analysis
[9]. We find that the current experimental uncertainty of $G_{M}$ produces a
relative variation of the cross section of at most $\sim 0.05\%$ in the range
covered by Fig. 8. The uncertainty from $G_{M}$ is thus not a limiting factor
of the DI$\chi$EFT-based radius extraction from $ep$ scattering data in MUSE
kinematics.

Figure 12: DI$\chi$EFT predictions for the average of $e^{+}$ and $e^{-}$
elastic scattering cross sections, as in Eq. (21) for $\mu^{+}p$ and
$\mu^{-}p$, in MUSE kinematics (compare with Fig. 6 for $\mu p$ scattering).

Because of the large TPE corrections in $ep$ scattering, it would be an
attractive option to perform the proton radius extraction with the average of
$e^{-}p$ and $e^{+}p$ cross sections [see Eq. (21) for $\mu^{\pm}p$], in which
the TPE effects cancel. In this case the cross section is accurately described
by the one-photon-exchange formula, and the analysis greatly simplifies.
Figure 12 shows the DI$\chi$EFT predictions for the charge radius dependence
of the charge-averages cross section $\bar{\sigma}$ in $e^{\pm}p$ scattering
in MUSE kinematics.

##  IV Conclusions

In this work we have used the DI$\chi$EFT framework to study the prospects for
the proton radius extraction from $\mu p$ scattering at MUSE. The principal
conclusions are:

(i) When extracting the radius from fits to cross section data at low momentum
transfers $Q^{2}<$ 0.01 GeV2, the TPE corrections need to be included with
high precision. At $Q^{2}<$ 0.01 GeV2, the estimated absolute size of TPE
correction amounts to a shift of the extracted radius by 0.03–0.04 fm. Any
theoretical uncertainty of the TPE correction will influence the extracted
radius proportionally.

(ii) The DI$\chi$EFT method allows one to extract the radius from fits to
cross section data at higher momentum transfers $Q^{2}\sim$ few times 0.01
GeV2 in the MUSE kinematic range. This is advantageous experimentally, because
the higher sensitivity of the cross section to the radius lowers the demands
on the experimental precision of the cross section measurement. It is also
advantageous theoretically, as it reduces the influence of the TPE correction
on the radius extraction.

(iii) The optimal kinematics for the DI$\chi$EFT-based radius extraction at
MUSE is $k=210$ MeV and $Q^{2}\sim$ 0.05–0.08 GeV2, at the upper end of the
kinematic coverage. It is determined by the trade-off between theoretical
effects – the sensitivity of the cross section to the radius, the uncertainty
of the DI$\chi$EFT FF predictions, and the TPE correction. In this kinematics,
even a 100% uncertainty of the TPE correction would shift the extracted radius
only by 0.01 fm. An experimental precision of $\leq$ 0.5% is required for
determining the radius with 0.01 fm accuracy. Such accuracy would be
sufficient for solving the proton radius puzzle.

(iv) In $ep$ scattering in MUSE kinematics, the TPE corrections are generally
larger that in $\mu p$, and the advantages of using the DI$\chi$EFT method
with higher-$Q^{2}$ data for radius extraction are even more compelling. The
ratio of same-charge $ep$ and $\mu p$ cross sections is predicted to be
practically independent of the proton radius and can be used for validation of
the analysis.

Our findings affirm the need for accurate theoretical estimates of the TPE
corrections in elastic $\mu p$ and $ep$ scattering. If the radius extraction
is performed using the DI$\chi$EFT framework and data at finite momentum
transfers $Q^{2}\sim$ 0.05–0.08 GeV2, as recommended here, efforts should
focus on improving the TPE estimates in this kinematic region. At these finite
values of $Q^{2}$ the constraints on the TPE amplitude from the limit of
forward scattering ($Q^{2}=0$) are less restrictive, and the calculations
become more dependent on dynamical assumptions [14, 23]. Methods based on the
$1/N_{c}$ expansion of QCD could enable systematic calculations of TPE effects
with controlled theoretical uncertainties; see Refs. [24, 25] for recent
developments.

The DI$\chi$EFT framework used in the present study is a general method that
could be improved through further development. In particular, the theoretical
uncertainties of the FF predictions could be reduced by using a more flexible
parametrization of the high-mass states in the spectral function, Eq. (11),
and constraining it with spacelike nucleon FF data. In the analysis here we
have used the version of Ref. [8], where the high-mass states are described by
a single effective pole, which permits simple uncertainty estimates and is
sufficient for the MUSE $Q^{2}$-range. A version with a more elaborate
description of the high-mass states, using multiple poles with randomized
parameters for the uncertainty estimates, was described in Ref. [26] and could
be employed for elastic scattering analysis and radius extraction at higher
$Q^{2}\lesssim$ 1 GeV2.

## Acknowledgments

We thank A. Gramolin for making available a computer code that was used to
validate the numerical calculation of the one-photon exchange cross section,
and P. Blunden and W. Melnitchouk for helpful correspondence about two-photon
exchange effects.

J.M.A. acknowledges support from the Spanish MICINN grant
PID2019-106080GB-C21. This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics under
contract DE-AC05-06OR23177.

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# Precision small scattering angle measurements of proton-proton and proton-
nucleus analyzing powers at the RHIC hydrogen jet polarimeter††thanks:
Presented at “Diffraction and Low-$x$ 2022”, Corigliano Calabro (Italy),
September 24-30, 2022.

A. A. Poblaguev  A. Zelenski  E. Aschenauer  G. Atoian  K. O. Eyser  H. Huang
W. B. Schmidke
I. Alekseev, D. Svirida
N. H. Buttimore  Brookhaven National Laboratory, Upton, New York 11973, USA
Alikhanov Institute for Theoretical and Experimental Physics, 117218, Moscow,
Russia  School of Mathematics, Trinity College, Dublin 2, Ireland

(November 30, 2022)

###### Abstract

At RHIC, the hydrogen jet target polarimeter (HJET) is used to measure proton
beam polarization with accuracy
$\sigma_{P}^{\text{syst}}/P\\!\lesssim\\!0.5\%$ by counting low energy (1–10
MeV) recoil protons in left-right symmetric detectors. The HJET performance
also allowed us to precisely measure $pp$ and $p$A (where A is any ion stored
at RHIC) analyzing powers in the CNI region. The results of the measurements
are discussed.

##  1 Introduction

The Polarized Atomic Hydrogen Gas Jet Target (HJET) [1] is employed to measure
absolute vertical polarization of the high energy ($\sim\\!100\,\text{GeV}$)
proton beams at the Relativistic Heavy Ion Collider (RHIC). The jet
polarization is about $P_{j}\\!\sim\\!96\%$ and is monitored with accuracy of
about $0.1\%$.

Figure 1:  A schematic view of the HJET recoil spectrometer.

Figure 2:  Measured $z_{R}$ vs $\sqrt{T_{R}}$ for the 100 GeV proton beam.

The recoil protons from the RHIC beam scattering off the jet are counted in
the left-right symmetric Si strip detectors depicted in Fig. 2. For each
proton detected, time of flight (ToF), kinetic energy $T_{R}$, and
$z_{R}$-coordinate (along the beam) in the detector discriminated by the strip
width of 3.75 mm are determined. Detailed description of the measurements is
given in Ref. [2].

The following kinematical relations are important for the data analysis

$t=-2m_{p}T_{R},$ (1)
$\frac{z_{R}-z_{\text{jet}}}{L}=\sqrt{\frac{T_{R}}{2m_{p}}}\times\left[1+\frac{m_{p}}{E_{\text{beam}}}\left(\frac{m_{p}}{M}+\frac{\Delta}{T_{R}}\right)\right],$
(2)

where $t$ is the momentum transfer squared, $m_{p}$ is a proton mass,
$E_{\text{beam}}$ is the beam energy per nucleon, $M$ is the beam particle
mass, $\Delta\\!=\\!M_{X}\\!-\\!M$ with $M_{X}$ being the effective scattered
mass, and $z_{\text{jet}}$ ($\langle{z_{\text{jet}}}\rangle\\!=\\!0$,
$\langle{z_{\text{jet}}^{2}}\rangle^{1/2}\\!\approx\\!2.5\,\text{mm}$) is the
coordinate of the scattering point.

The HJET detector geometry allows us to make measurements only in the Coulomb-
nuclear interference (CNI) momentum transfer range
$0.0013\\!<\\!-t\\!<\\!0.018\,\text{GeV}^{2}$
($0.6\\!<\\!T_{R}\\!<\\!10\,\text{MeV})$, which is nearly independent of the
beam particle mass and energy.

For recoil protons, measured ToF and $T_{R}$ must be kinematically consistent.
Considering the correlation shown in Fig. 2, one can easily identify the
elastic events ($\Delta\\!=\\!0$). The background rate (as a function of
$T_{R}$) can be interpolated [2], with a relative accuracy of about 3%, to the
elastic values of $z_{R}(T_{R})$. Thus, the background can be accurately
subtracted from the elastic data, which allows a low,
$\sigma_{P}^{\text{syst}}/P\\!\lesssim\\!0.5\%$, systematic uncertainty in the
beam polarization measurement.

##  2 Spin asymmetries measured with the HJET

Figure 3: Single spin-flip elastic $pp$ $A_{\text{N}}(t)$. Filled areas show
experimental uncertainties scaled by a factor of 50. Violet is for stat+syst
and orange is for syst only.

Figure 4: Double spin-flip forward elastic $pp$ analyzing power
$A_{\text{NN}}(t)$.

For vertically polarized beam ($b$) and target ($j$), the recoil proton
azimuthal angle distribution can be written as

$\frac{d^{2}\sigma}{dtd\varphi}\\!=\\!\frac{d\sigma}{2\pi
dt}\,\Big{[}1+\left(A_{\text{N}}^{j}P_{j}\\!+\\!A_{\text{N}}^{b}P_{b}\right)\\!\sin\;\\!\\!{\varphi}+\left(A_{\text{NN}}\sin^{2}\>\\!\\!\\!{\varphi}\\!+\\!A_{\text{SS}}\cos^{2}\>\\!\\!\\!{\varphi}\right)\\!P_{b}P_{j}\Big{]}.$
(3)

Since $\cos{\varphi}\\!\approx\\!0$ at HJET, the measurements are insensitive
to $A_{\text{SS}}(t)$. For elastic $p^{\uparrow}p^{\uparrow}$ scattering,
$A_{N}^{b}=A_{N}^{j}=A_{N}(t)$ and the asymmetries,
$a_{\text{N}}^{b,j}(T_{R})=A_{\text{N}}^{b,j}(t)P_{b,j}$,
$a_{\text{NN}}(T_{R})=A_{\text{NN}}(t)P_{b}P_{j}$ concurrently measured [2]
using the same events allows one to determine $P_{b}$, $A_{N}(t)$, and
$A_{\text{NN}}(t)$.

Theoretical parametrization of the forward elastic proton-proton analyzing
powers was developed in Refs. [3, 4, 5]. In a simplified form,

$A_{\text{N}}(t)=\frac{\sqrt{-t}}{m_{p}}\,\frac{(\kappa_{p}-2\text{Im}\,r_{5})\,t_{c}/t-2\text{Re}\,r_{5}}{(t_{c}/t)^{2}-2(\rho+\delta_{C})\,t_{c}/t+1},$
(4)

where $\kappa_{p}\\!=\\!1.792$ is the anomalous magnetic moment of a proton,
$t_{c}\\!=\\!-8\pi\alpha/\sigma_{\text{tot}}$, $\sigma_{\text{tot}}$ is the
total $pp$ cross section, $\rho$ is the Re/Im amplitude ratio,
$\delta_{C}\\!\approx\\!\alpha\ln{t_{c}/t}\\!+\\!0.024$ is the Coulomb phase,
and $|r_{5}|\\!\sim\\!0.02$ is the hadronic single spin-flip amplitude
parameter. A more accurate expression discussed in [6, 7] includes small but
essential corrections to meet the experimental precision achieved at HJET.

##  3 Forward elastic proton-proton analyzing powers

During the RHIC Runs in 2015 and 2017 with beam energies of 100 and 255 GeV,
respectively, the forward elastic proton-proton $A_{\text{N}}(t)$ (Fig. 4) and
$A_{\text{NN}}(t)$ (Fig. 4) were precisely measured [8] at HJET.

Figure 5: Experimental 1-sigma contours (stat+syst) for $r_{5}$. The HJET
results are marked “1”($\sqrt{s}\\!=\\!13.76$ GeV) and “2”(21.92 GeV). The
Regge fit extrapolations to 200 GeV are “3”for Froissaron and “4”for simple
pole Pomeron. “5”and “6”are the STAR value (200 GeV) [12] and the same after
applying (in this analysis) the absorption and $r_{p}$ related corrections.

The hadronic single spin-flip amplitude ($r_{5}$), was clearly isolated as
shown in Fig. 5. Here, compared to the published [8] results, we applied a
correction
$\text{Re}\,r_{5}=\text{Re}\,r_{5}^{\cite[cite]{[\@@bibref{}{Poblaguev:2019saw}{}{}]}}+(3.1_{\text{abs}}+0.8_{r_{p}})\times
10^{-3}$ due to the absorption [7] and updated value of the proton charge
radius $r_{p}\\!=\\!0.841\,\text{fm}$ [9].

To find the spin-flip amplitude dependence on the center of mass energy
squared, $s=2m_{p}(m_{b}+E_{\text{beam}})$, the following parametrization was
used

$\sigma_{\text{tot}}(s)\,r_{5}(s)=f_{5}^{+}R^{+}(s)+f_{5}^{-}R^{-}(s)+f_{5}^{P}P(s),$
(5)

where couplings $f_{5}^{\pm,P}$ are free parameters in the fit and Reggeon
pole $R^{\pm}(s)$ and Pomeron $P(s)$ (in Froissaron approximation) are
functions found in a fit [11]
$\sigma_{\text{tot}}(s)\left[i+\rho(s)\right]\\!=\\!R^{+}(s)\\!+\\!R^{-}(s)\\!+\\!P(s)$
of the unpolarized $pp$ data.

The $r_{5}$ fit result, $f_{5}^{P}=0.054\pm 0.002_{\text{stat}}\pm
0.003_{\text{syst}}$, suggests non-vanishing hadronic spin-flip amplitude at
very high energies. The extrapolation of the HJET values of $r_{5}$ to
$\sqrt{s}=200\,\text{GeV}$ can be compared with the STAR measurement [12].

The correction applied to $r_{5}$ improves the agreement of the HJET results
with the parametrisation in Eq. (5), since $\chi^{2}\\!=\\!2.2\\!\to\\!0.7$
(ndf=1). Although consistency with STAR value of $r_{5}$ dis-improves, the
discrepancy is not critically significant, $\chi^{2}/\text{ndf}\\!=\\!4.8/3$
(which is statistically equivalent to about 1.8 standard deviations).

For the $A_{\text{NN}}(t)$, the hadronic double spin-flip amplitude ($r_{2}$)
is also non-zero and the Regge fit gives
$f_{2}^{P}\\!=\\!-0.0020\\!\pm\\!0.0002$ [8].

Figure 6: Shown are elastic and inelastic events for the 255 GeV proton beam,
left to right, normalized event rate
$R(\sqrt{T_{R}},z_{R})\\!=\\!N_{\text{bin}}(\sqrt{T_{R}},z_{R})/N_{\text{max}}^{\text{el}}$,
and the beam $A_{\text{N}}^{b}(t,\Delta)$ and target
$A_{\text{N}}^{j}(t,\Delta)$ analyzing powers. A cutoff of $R>0.4\%$ was used.

Figure 7: Same as Fig. 6, but for the 100 GeV proton beam and $R>0.1\%$
cutoff.

##  4 Inelastic proton-proton scattering

With the HJET detectors, the inelastic events $p_{b}\,p_{j}\\!\to\\!(\pi
X)_{b}\,p_{j}$ can be separated (due to relatively large
$\Delta\\!\geq\\!m_{\pi}$) from the elastic ones and the inelastic analyzing
powers for the beam $A_{\text{N}}^{b}(t,\Delta)$ and target
$A_{\text{N}}^{j}(t,\Delta)$ spins can be evaluated.

Preliminary results for the 255 GeV proton beam are shown in Fig. 6. Only bins
with event rate (after background subtraction) $R>0.4\%$ relative to the
elastic maximum were analyzed. The inelastic events ($R$ up to 5%) are well
identified in the upper left corner of the histograms.

One can see that
$A_{\text{N}}^{j}\\!<\\!A_{\text{N}}^{\text{elastic}}\\!<\\!A_{\text{N}}^{b}$.
The inelastic analyzing power grows with decreasing of $\Delta$. For
$A_{\text{N}}^{b}(t,\Delta)$, values of about 20% are observed in the data.

For the 100 GeV beam (Fig. 7), the detected inelastic rate is much lower,
$R\lesssim 0.5\%$. Nonetheless, results for the analyzing powers are,
qualitatively, about the same as for 255 GeV. A 100 GeV beam spin inelastic
analyzing power up to 35% was observed.

##  5 Elastic proton-nucleus analyzing power

Figure 8:  The dependence of the proton-nucleus elastic $A_{\text{N}}^{pA}(t)$
on the beam ion (left) and the beam energy (right). The measured analyzing
powers are normalized by the ${pp}$ one calculated for
$E_{\text{beam}}\\!=\\!100\,\text{GeV}$ and no hadronic single spin-flip,
$r_{5}\\!=\\!0$.

Since 2015, HJET was routinely operated in RHIC Heavy Ion Runs. It was found
that HJET performance in an ion beam is as good as in a proton one.
Consequently, the proton-nucleus analyzing power $A_{\text{N}}^{pA}(t)$ can be
precisely measured. The study have been performed for six ions [2H ($d$), 16O,
27Al, 96Zr, 96Ru, and 197Au]. Also, the energy scans were done for Au and $d$.

Some preliminary results for normalized,
$A_{\text{N}}^{\text{norm}}(t)\\!=\\!A_{\text{N}}^{pA}(t)/A_{\text{N}}^{pp}(t)$,
analyzing powers are shown in Fig. 8. Systematic uncertainties in the
measurements were not considered. The beam nucleus breakup fraction in the
elastic data is expected to be small, $\lesssim\\!1\%$ [10].

For 100 GeV/nucleon Au beam, the experimental data was compared with
theoretical predictions in Ref. [13]. It was found that absorption corrections
are very important for calculation of $A_{\text{N}}^{p\text{Au}}(t)$. However,
not all essential discrepancies between the data and theory were eliminated.

##  6 Summary

The HJET, which was designed to measure absolute proton beam polarization at
RHIC, can also be considered as a standalone fixed target experiment to
precisely measure proton-proton and proton-nucleus analyzing powers in the CNI
region.

The measurements of elastic $pp$ $A_{\text{N}}(t)$ and $A_{\text{NN}}(t)$
resulted in finding non-zero single and double spin Pomeron couplings.

Preliminary results for inelastic $pp$ $A_{\text{N}}^{b}(t,\Delta)$ and
$A_{\text{N}}^{j}(t,\Delta)$ for 100 and 255 GeV, and elastic $p\text{A}$
analyzing powers, in wide range of $1\\!<\\!\text{A}\\!<\\!200$ (for
$E_{\text{beam}}\\!=\\!100\,\text{GeV}$) and
$3.8\\!<E_{\text{beam}}\\!<\\!100\,\text{GeV}$ (for Au), were obtained.
However, to properly understand these measurements, an appropriate theoretical
description of these analyzing powers is needed.

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