# Elastic scattering of 3He + 4He with SONIK

S. N. Paneru<EMAIL_ADDRESS>Department of Physics & Astronomy, Ohio
University, Athens, Ohio 45701, USA  C. R. Brune  Department of Physics &
Astronomy, Ohio University, Athens, Ohio 45701, USA  D. Connolly  TRIUMF,
Vancouver, British Columbia, Canada  D. Odell  Department of Physics &
Astronomy, Ohio University, Athens, Ohio 45701, USA  M. Poudel  Department of
Physics & Astronomy, Ohio University, Athens, Ohio 45701, USA  D. R. Phillips
Department of Physics & Astronomy, Ohio University, Athens, Ohio 45701, USA
J. Karpesky  Department of Physics, Colorado School of Mines, Golden, Colorado
80401, USA  B. Davids  TRIUMF, Vancouver, British Columbia, Canada  Physics
Department, Simon Fraser University, Burnaby, BC, Canada  C. Ruiz  TRIUMF,
Vancouver, British Columbia, Canada  A. Lennarz  TRIUMF, Vancouver, British
Columbia, Canada  U. Greife  Department of Physics, Colorado School of Mines,
Golden, Colorado 80401, USA  M. Alcorta  TRIUMF, Vancouver, British Columbia,
Canada  R. Giri  Department of Physics & Astronomy, Ohio University, Athens,
Ohio 45701, USA  M. Lovely  Department of Physics, Colorado School of Mines,
Golden, Colorado 80401, USA  M. Bowry  TRIUMF, Vancouver, British Columbia,
Canada  M. Delgado  TRIUMF, Vancouver, British Columbia, Canada  Instituto de
Física Teórica IFT-UAM/CSIC, Madrid, Spain  N. E. Esker  TRIUMF, Vancouver,
British Columbia, Canada  A. Garnsworthy  TRIUMF, Vancouver, British Columbia,
Canada  C. Seeman  TRIUMF, Vancouver, British Columbia, Canada  P. Machule
TRIUMF, Vancouver, British Columbia, Canada  J. Fallis  North Island College,
British Columbia, Canada  A. A. Chen  McMaster University, Ontario, Canada  F.
Laddaran  University of British Columbia, British Columbia, Canada  A. Firmino
University of Alberta, Edmonton, Alberta, Canada  C. Weinerman  McGill
University, Montreal, Quebec, Canada

###### Abstract

Measurements of the elastic scattering cross section of 3He and 4He are
important in order to improve constraints on theoretical models of ${}^{4}\rm
He(^{3}\rm He,\gamma)^{7}\rm Be$, a key reaction in Big Bang nucleosynthesis
and solar neutrino production. The astrophysical $S$-factor for this reaction
is a significant source of uncertainty in the standard-solar-model prediction
of the 7Be and 8B solar neutrino fluxes. The elastic scattering measurements
reported in the literature do not extend to low energies and lack proper
uncertainty quantification. A new measurement of the 4He(3He,3He)4He reaction
has been made at center-of-mass energies $E_{\text{c.m.}}=0.38-3.13$ MeV using
the Scattering of Nuclei in Inverse Kinematics (SONIK) scattering chamber: a
windowless, extended gas target surrounded by an array of 30 collimated
silicon charged particle detectors situated at TRIUMF. This is the first
elastic scattering measurement of 3He+4He made below 500 keV and it has
greater angular range and better precision than previous measurements. The
elastic scattering data were analyzed using both $R$-matrix and Halo Effective
Field Theory (Halo EFT) frameworks, and values of the $s$-wave scattering
length and effective range were extracted. The resulting improvement in
knowledge of the $s$-wave effective-range function at low energies will reduce
the overall uncertainty in $S_{34}$ at solar energies.

###### pacs:

##  I INTRODUCTION

The reaction 4He(3He,$\gamma$)7Be is of critical importance for the production
of high energy neutrinos during pp-chain burning in low mass stars like our
sun. The 7Be produced by this reaction undergoes electron capture to produce
7Be neutrinos in the pp-II chain. In the pp-III chain, the 7Be undergoes a
radiative proton capture reaction to form 8B which subsequently $\beta^{+}$
decays to produce 8B neutrinos. The total active flux of these 7Be and 8B
neutrinos has been measured by the Borexino and SNO detectors with
uncertainties of $\pm 3\%$ and $\pm 4\%$, respectively Agostini  _et al._
(2019); Aharmim  _et al._ (2013). The Super Kamiokande experiment also
reported a measurement of the 8B neutrino flux with an uncertainty of $\pm
3\%$ Abe  _et al._ (2016). However, the predicted 7Be and 8B neutrino fluxes
from the calculations of the standard solar model (SSM) have uncertainties of
$\pm$6$\%$ and $\pm$12$\%$, respectively Vinyoles  _et al._ (2017). The low
energy astrophysical $S$ factor for the 4He(3He,$\gamma$)7Be radiative capture
reaction, $S_{34}$($E$), is respectively the first and second most uncertain
nuclear input in the SSM prediction of the 7Be and 8B neutrino fluxes Vinyoles
_et al._ (2017). It must be known at or near the Gamow peak energy of $\sim$18
keV, which is experimentally inaccessible due to Coulomb barrier suppression.
The cross sections are unmeasurably small at these energies, so available data
starting around $E_{\text{c.m.}}=100$ keV must be extrapolated to solar
energies with the aid of theoretical models.

Several different theoretical approaches used to calculate the
4He(3He,$\gamma$)7Be reaction cross section at these energies are summarized
in the “Solar Fusion II” review Adelberger  _et al._ (2011). In that work
these approaches were sifted and a subset of them used to extrapolate the
experimental capture data available in 2011. The resulting recommended zero-
energy astrophysical $S$-factor for the 4He(3He,$\gamma$)7Be reaction is
$S_{34}(0)$=0.56$\pm$0.02 (expt) $\pm$ 0.02 (theory) keV b.

The 4He(3He,$\gamma$)7Be reaction also plays a key role in determining the
quantity of ${}^{7}{\rm Li}$ produced by Big-Bang Nucleosynthesis (BBN) Cyburt
_et al._ (2016). In this case, the important energy range is approximately
$100\lesssim E_{\text{c.m.}}\lesssim 600$ keV Nollett and Burles (2000);
Iliadis and Coc (2020), a region where the cross section can be measured
directly by experiment. Currently, the BBN predictions for the ${}^{7}{\rm
Li}$ abundance are about a factor of three higher than observations, which is
far too large of a difference to be explained by uncertainties in the
4He(3He,$\gamma$)7Be reaction. Since the 4He(3He,$\gamma$)7Be reaction rate at
BBN temperatures can be determined directly from experimental data, i.e.,
without extrapolation or a detailed model, we do not focus on this application
in the present work.

The capture reaction 4He(3He,$\gamma$)7Be proceeds dominantly through a non-
resonant direct capture mechanism into the ground and first excited state of
7Be. Measurements of the 4He(3He,$\gamma$)7Be cross section have been made by
detecting the prompt $\gamma$-rays, the 7Be activity, and the 7Be recoils,
which could be broadly categorized into prompt and activation measurements.
The results from these two types of measurements were not in agreement until
1998, as summarized in Solar Fusion I Adelberger  _et al._ (1998). With the
advancement in measurement techniques, the recent results from both types of
measurement are remarkably consistent, as pointed out in Ref. deBoer  _et al._
(2014). The cross section for 4He(3He,$\gamma$)7Be has been measured for
center-of-mass energies from 90 keV to 3.2 MeV.

4He+3He elastic scattering is an important constraint on theoretical models
and phenomenological descriptions of the 4He(3He,$\gamma$)7Be reaction: any
model of the capture reaction should also be able to describe low-energy
elastic scattering. In phenomenological descriptions such as $R$-matrix,
4He(3He,3He)4He is an open channel and it affects the extrapolation of
4He(3He,$\gamma$)7Be cross-section data to solar energies. In theoretical
models the quality of scattering wave functions that are input to the
calculation of the 4He(3He,$\gamma$)7Be reaction cross section can be assessed
by the models’ ability to describe the elastic scattering cross section.

A comprehensive $R$-matrix analysis of 4He(3He,$\gamma$)7Be reaction by deBoer
et al. deBoer  _et al._ (2014) studied the effects of elastic scattering data
on the inferred astrophysical $S$-factor values at solar energies and reported
$S_{34}$(0)=0.542 $\pm$ 0.023 keV b—a central value about 3$\%$ lower than is
recommended in Solar Fusion II. The authors of that study emphasized the need
for a new study of 4He(3He,3He)4He covering a wide angular range with detailed
uncertainty estimates. This conclusion is bolstered by recent ab initio and
Halo Effective Field Theory (EFT) calculations. In Ref. Dohet-Eraly  _et al._
(2016) the no-core shell model with continuum (NCSMC) Dohet-Eraly  _et al._
(2016) was used to compute the 4He(3He,$\gamma$)7Be reaction ab initio. Dohet-
Eraly et al. found discrepancies between the elastic scattering phase shifts
they predict and experimental observations. Iliadis et al. Iliadis  _et al._
(2016) performed a global Bayesian estimate based on microscopic models and ab
initio methods. Scaling the model calculations to fit the capture data, they
reported $S_{34}$(0)=0.572 $\pm$ 0.012 keV b. This central value is 2$\%$
higher than Solar Fusion II and 6$\%$ higher than that of de Boer et al..
Meanwhile, two recent Halo EFT calculations that used 3He and 4He as degrees
of freedom showed the strong connection between the $s$-wave scattering length
and effective range and the shape of the capture-reaction $S$-factor at low
energies Higa  _et al._ (2018); Zhang  _et al._ (2020a). Ref. Zhang  _et al._
(2020a) recommended $S_{34}(0)=0.577^{+0.015}_{-0.016}$ keV b based on a
Bayesian analysis of radiative capture data.

It is thus clear that the 4He(3He,3He)4He elastic scattering is important both
in astrophysics and few-body nuclear theory. However, there are only a few
experimental studies of this reaction at the low energies where the
information is most pertinent to solar fusion. Most of the experiments
reported in the literature are motivated to understand the structure of 7Be
and consequently are focused on high resonance energies Miller and Phillips
(1958); Tombrello and Parker (1963); Barnard  _et al._ (1964); Spiger and
Tombrello (1967); Ivanovich  _et al._ (1968); Chuang (1971); Hardy  _et al._
(1972); Boykin  _et al._ (1972). The only measurement extending to low
energies reached $E[^{3}$He]=1.2 MeV, but lacks error estimates Mohr  _et al._
(1993).

This paper describes a new measurement at TRIUMF of the elastic scattering
cross section of the 4He(3He,3He)4He reaction. The measurement was carried out
at incident beam energies as low as $E$[3He]=0.721 MeV. The experimental
method used to measure the elastic scattering is explained in Sec. II. In Sec.
III we discuss the details of the data analysis and the calculation of the
differential scattering cross section. In Sec. IV the differential scattering
cross sections from this measurement are compared to existing measurements
from literature. In this paper the $s$-wave scattering length for the 3He+4He
system is determined using both a multilevel $R$-matrix approach and Halo EFT
to simultaneously analyze the new elastic scattering data from this work and
the data of Ref. Barnard  _et al._ (1964). These analyses are described in
Secs. V and VI, respectively. The results of the analyses are presented in
Sec. VII, which also contains a comparison with previous results from the
literature. We conclude in Sec. VIII.

##  II EXPERIMENT

The elastic scattering measurement of 4He(3He,3He)4He was performed at TRIUMF.
A 3He beam in the 1+ charge state was produced using the TRIUMF Off-Line Ion
Source (OLIS) Jayamanna (2014). The beam was accelerated using the Isotope
Separator and Accelerator-I (ISAC-I) facility and delivered to the the
Scattering of Nuclei in Inverse Kinematics (SONIK) Connolly (2015) apparatus
with an intensity of about 1012 $\text{s}^{-1}$. SONIK was filled with 4He gas
maintained at a typical pressure of 5 Torr.

SONIK is a windowless, extended gas target surrounded by an array of 30
collimated silicon charged particle detectors. The chamber was commissioned
with two separate measurements, 4He(3He,3He)4He elastic scattering and
7Li($p$, $p$)7Li elastic scattering. The charged particle detectors are
mounted in an assembly referred to as the detector telescope hereafter. The
design details of SONIK and the detector telescopes are shown in Fig. 1.

Figure 1: (Color online) a) 3-D model of SONIK. b) SONIK design details. The
3He beam traverses the 4He gas target from left to right in the figure. c) A
detector telescope assembly. The dimensions in the figures are in mm.

Each detector telescope is collimated by a 2.0-mm-wide rectangular slit
aperture at the telescope’s interface with the gas volume and a 1.0-mm-
diameter circular aperture in front of the silicon charged particle detectors.
The two apertures are separated by a distance of 11.0 cm. The distance from
the front aperture to the observation point on the beam axis in the gas target
is 6.0 cm. The detectors are placed at a distance of 17.0 cm from the center
of the beamline, measured along the axis of the telescope, and at observation
angles ranging from 22.5∘ to 135∘ in the laboratory frame. The beam delivered
to SONIK enters the windowless gas target through a 6-mm diameter aperture and
exits through an 8-mm-diameter aperture; they are separated by 23.98 cm. A
constant pressure is achieved along the extended gas target by using the
Detector of Recoils and Gammas of Nuclear Reactions (DRAGON) differential
pumping system. Helium gas was cleaned by continuous recirculation through a
LN2 cooled zeolite trap. The detector telescopes are arranged such that they
observe three different points, termed interaction regions, in the gas target
along the beam direction ($z$-direction). Since each interaction region has a
different $z$-coordinate, the bombarding energy and therefore the scattering
energy varies slightly by interaction region. This arrangement of three
interaction regions is highly beneficial for inverse kinematics experiments
with radioactive beams, where the beam time is limited and there are narrow
resonances to be studied. Additional details of the experimental setup are
given in Refs. Connolly (2015); Paneru (2020).

4He(3He,3He)4He elastic scattering was measured at 9 energies corresponding to
3He beam energies of $E[^{3}$He]= 0.721, 0.878, 1.303, 1.767, 2.145, 2.633,
3.608, 4.347, and 5.490 MeV. This is the first ever measurement made below
$E_{\text{c.m.}}$= 0.50 MeV of 4He(3He,3He)4He elastic scattering. Since the
projectile and target masses are comparable, we observed both the recoils and
ejectiles from the elastic scattering in our detectors. A typical raw spectrum
for two incident beam energies from the experiment is shown in Fig. 2.

Figure 2: Typical spectrum from the experiment. The red and blue histograms
represent the spectra obtained at $\theta_{\text{lab}}=40^{\circ}$ for
$E$[3He]=1.303 MeV and 3.608 MeV, respectively. The 3He and 4He peaks are
resolved at the higher incident energy but not at the lower beam energy.

The 3He and 4He peaks are well resolved at high incident beam energies. We
aimed for 1000 counts in the 90∘ detector for each incident beam energy before
changing to the next energy. For low incident beam energies, we couldn’t
observe the 3He ejectiles at 90∘, so the next detector to observe was then at
75∘. The 120∘ and 135∘ spectra were not used for the analysis because of their
limited utility.

##  III DATA ANALYSIS

The differential elastic scattering cross section in the laboratory frame of
reference at bombarding energy $E_{0}$ and scattering angle $\theta_{0}$ is
given by:

$\frac{d\sigma}{d\Omega}(E_{0},\theta_{0})=\frac{N_{\rm
det}\sin{\theta_{0}}}{nN_{\rm inc}G\epsilon},$ (1)

where $N_{\text{det}}$ is the number of detected particles, $n$ is the target
density, $N_{\text{inc}}$ is the number of incident beam particles, $G$ is the
$G$ factor Silverstein (1959) discussed in Section III.5, and the quantity
$\epsilon$ is the beam transmission through an empty gas target.

###  III.1 Beam Energy Determination

The beam energies were measured using the DRAGON facility according to the
procedure given in Ref. Hutcheon  _et al._ (2012). The 3He beam in charge
state $q$ was centered on a 2-mm slit downstream of DRAGON’s first magnetic
dipole MD1, and the measured field value $B$ was converted to energy per
nucleon $E/A$ by using the relation

$\frac{E}{A}=C_{\text{mag}}\left(\frac{qB}{A}\right)^{2}-\frac{1}{2\mu
c^{2}}\left(\frac{E}{A}\right)^{2}$ (2)

where $E$ is the kinetic energy of 3He in MeV, $C_{\text{mag}}$=48.15(7)
MeVT-2 Hutcheon  _et al._ (2012), $A=3.016$ is the mass of 3He in atomic mass
units, and $\mu$ is one atomic mass unit in $\text{MeV}/c^{2}$.

With DRAGON, we are able to make direct stopping power measurements. The beam
passing through the gas target at varying pressures is deflected by the first
bending magnet (MD1) downstream of the target and centered onto the charge
slit. The magnetic field strength required to transmit the beam through the
charge slit is measured. The stopping power measurements for SONIK were
performed at $E$[3He]=1.767 MeV with pinhole apertures (1.5 mm diameter) at
the beam entrance and beam exit positions as well as with the standard (6 and
8mm) apertures in order to measure the effective length of the target. The
target areal density is determined from the target pressure using the ideal
gas law. The beam energy is plotted as a function of target density which
yields the linear relationship shown in Fig. 3. The slope of this line is the
stopping power for 3He in 4He gas. At $E$[3He]=1.767 MeV the stopping power
obtained via this approach is 11.97$\pm$0.53 eV/(1015atoms/cm2). Meanwhile,
that obtained from a Stopping and Range of Ions in Matter (SRIM) Ziegler  _et
al._ (2010) calculation is 11.08 eV/(1015atoms/cm2). The central value of the
experimentally measured stopping power differs from the SRIM prediction by
$8.0\%$. The measured stopping power value is consistent within 1-$\sigma$
error bars if the uncertainty in the SRIM predictions of 4.3$\%$ SRI  is taken
into consideration. The length of the gas target with standard apertures (i.e.
beam entrance and beam exit apertures with diameters of 6 mm and 8 mm,
respectively) is termed the effective length in our experiment. The effective
length differs from the physical length as the gas in the differential pumping
system diffuses outwards, thereby increasing the length of the gas target with
which the beam interacts. With the assumption of linear energy loss, the
effective target length can be determined via

$l_{\text{eff}}=\frac{\Delta E}{n\mathcal{S}},$ (3)

where $\Delta E$ is the energy loss in the target, $\mathcal{S}$ is the
stopping power of 3He in 4He, and $n$ is the target number density. The
effective length for the gas target was calculated to be
$l_{\text{eff}}=24.61\pm 1.09$ cm.

Figure 3: Determination of stopping power for $E$[3He]=1.767 MeV. The error
bars are entirely due to the systematic error associated with the constant
$C_{\text{mag}}$ in Eq. (2).

The stopping power measurement was only performed for $E$[3He]=1.767 MeV,
where an 8% difference from the SRIM calculation was found. Assuming the same
difference for other incident energies, the SRIM-calculated stopping powers
were increased by 8$\%$ to obtain the stopping power of 3He in 4He gas. With
the assumption of linear energy loss, the effective beam energy $E_{i}$ at
each interaction region $i$ in the gas target is calculated by

$E_{i}=E_{\text{in}}-\mathcal{S}t_{i},~{}~{}~{}\text{with}~{}~{}~{}~{}~{}i=1,2,3$
(4)

where $E_{\text{in}}$ is the incident beam energy, $\mathcal{S}$ is the scaled
stopping power and $t_{i}$ is the target areal density for interaction region
$i$. The errors on the incident energy, effective length, measured stopping
power, temperature, and pressure are propagated to calculate the error on the
energy at each interaction region. The error was calculated to be 6.6$\%$ of
the energy loss, which amounts to a maximum of 3 keV in $E[^{3}\text{He}]$ for
the lowest 3He incident beam energy at interaction region III.

###  III.2 Transmission

The incident beam currents are measured upstream and downstream of the SONIK
chamber using Faraday cups FC4 and FC1, respectively. The ratio of FC4 to FC1
gives us the measure of beam transmission through the target. With no gas in
the target, the ratio FC4/FC1 should be ideally 1, which is not the case
practically. The exercise of measuring the beam transmission was done at the
start of each energy change but was not measured for each individual run. It
is believed to be a constant factor for all runs for a given incident beam
energy. The empty gas beam transmission for $E$[3He]=5.490, 3.608, 2.633, and
1.303 MeV were measured again before making an energy change. The mean of the
difference of the transmission measured at the start and at the end of run
before the energy change for these energies was found to be 3$\%$. Therefore,
for all energies, an additional 3$\%$ uncertainty was added in quadrature with
the counting errors in the empty gas beam transmission measurement.

###  III.3 Beam Normalization

The FC4 Faraday cup reading is used to determine the number of incident beam
particles $N_{\text{inc}}$. The target density of 4He, $n$, is determined from
the pressure and temperature of the gas target. The product of the number of
incident beam particles, target density, and the empty gas beam transmission
$\epsilon$ gives the normalization for each energy measurement. The beam
normalization error for each energy is the error associated with the
normalization and is a common mode error for all data points for a given
energy. There was a change in incident beam intensity in between the runs
while acquiring data for$E[^{3}$He]= 2.633 MeV resulting in a different
normalization. So, there are two set of runs of data for $E[^{3}$He]= 2.633
MeV. The common mode errors for the different energies lie between 3.7 and
9.6$\%$ and are listed in Table VII.1.

###  III.4 Yield Measurement

The individual peak yields from the raw spectrum were extracted after
background subtraction. The yields for both the 3He and 4He peaks were
calculated whenever possible. The low-energy feature seen in the typical
spectrum from the experiment shown in Figure 2 is background due to the
contribution of detector noise and electrons produced from in-target
scattering which tail off gradually with increasing energy. Beside these
sources, a contribution to the background also arises from energy-degraded
ions from beam scattering off the aperture edges upstream of the SONIK
interaction regions, the energy degraded scattered particles from the edges of
the Si detector collimators, and particles backscattered out of the Si
detector before depositing all of their energy. The last two sources of
background were included in the Geant4 simulation, but the effects were too
small to explain the observed tails on the 3He and 4He peaks. When the peaks
are fully resolved, polynomials were used to characterize the background,
which then were integrated analytically to estimate the background
contribution to the peak yields. When the peaks overlap, two Gaussian
functions with a common exponential or polynomial background function are used
to fit the spectrum and extract the peak yields. The random errors for each
peak yield were determined considering the errors due to the choice of fit
parameters, the use of a Gaussian function to fit the peak, and calculating
yields for each one hour spectrum versus the yields for the summed spectrum
for a given beam energy. The random background error estimates from each of
these three components were added in quadrature to calculate the total random
background error. The random background error was estimated for each peak for
all energies and was added in quadrature with the statistical error to obtain
the reported point-to-point error in the differential scattering cross
section.

###  III.5 G factor

For charged particle scattering experiments with a gas target and collimated
detector telescopes, the relationship between the number of detected
particles, $N_{\text{det}}$, and the differential scattering cross section,
$\frac{\text{d}\sigma}{\text{d}\Omega}$, is often expressed as

$N_{\text{det}}=\frac{n~{}N_{\text{inc}}~{}G~{}}{\sin{\theta_{0}}}\frac{d\sigma}{d\Omega}(E_{0},\theta_{0}),$
(5)

where $n$ is the areal density of target nuclei, $N_{\text{inc}}$ is the
number of incident beam particles, $E_{0}$ is the beam energy, $\theta_{0}$ is
the central angle subtended by the detector system, and the acceptance of the
detector is given by the $G$ factor instead of the usual solid angle.The
inclusion of the sine term makes $G$ independent of $\theta_{0}$ at leading
order.

For the SONIK chamber, we have the front aperture as a vertical slit of width
2$b$, where the slit is perpendicular to the plane defined by the beam axis
and the center line of the detector system. The rear aperture is circular with
radius $a$. The distance between the apertures is $h$ and the distance from
the central interaction point on the $z$-axis (beam axis) to the rear aperture
is $R_{0}$. A schematic diagram of SONIK doubly-collimated apertures is shown
in Fig. 4.

Figure 4: A detector telescope of SONIK.

Assuming $a,b\ll R_{0}$ and $a,b\ll h$, the analytical expression for the $G$
factor for a collimated detector like that of SONIK is given by Silverstein
Silverstein (1959) as

$G=G_{00}(1+\Delta_{0}),$ (6)

where

$G_{00}=\frac{2\pi a^{2}b}{R_{0}h},$ (7)

and

$\Delta_{0}=\frac{a^{2}\cos^{2}{\theta_{0}}}{4R^{2}_{0}\sin^{2}\theta_{0}}-\frac{b^{2}}{2h^{2}}-\frac{3a^{2}}{8}\left[\frac{1}{h^{2}}+\frac{1}{R^{2}_{0}}\right].$
(8)

The quantity $\Delta_{0}$ incorporates second order corrections in the ratio
of $a$ and $b$ to either $R_{0}$ or $h$. Eq. (6) includes the geometrical
effects on the acceptance of the detector system, however the acceptance also
depends on slit edge scattering, multiple scattering, beam size, beam
divergence, etc. A charged particle traversing the gas target undergoes
numerous small angle collisions (screened elastic scattering) with the gaseous
atoms termed “multiple scattering”. In our experiment, the effect of multiple
scattering could be visualized in two processes. First, the incident beam
particles undergo multiple scattering, effectively increasing the beam
diameter. Second, a charged particle from the elastic scattering undergoes
multiple scattering before it is detected in the Si detector, changing the
acceptance of the detector. The elastic scattering of 3He+4He was measured as
low as $E$[3He]=721 keV in this experiment, at which the effect of multiple
scattering is expected to be the greatest. The multiple scattering of the
incident beam particle would affect the overall acceptance. To account for
these effects, particularly multiple scattering, a Monte Carlo simulation was
developed in the Geant4 Agostinelli  _et al._ (2003) framework to calculate
the $G$ factor for SONIK. The $G$ factor calculated from the simulation also
includes the effects of the energy and angular spreads of the beam.

The Geant4 simulation was performed in two steps. First, the 3He beam
particles were introduced along the $z$-direction through the 4He gas target
kept at a temperature $T=30^{\circ}$C and pressure $P=5$ Torr. The trajectory
of each beam particle is stored. We used the G4Urban Msc-Model Urbán (2006) to
simulate the multiple scattering effects of the 3He particles in the 4He gas
target. The 3He beam introduced in the 4He gas target loses energy as it
traverses the target. The stopping power of 3He in the 4He gas target
calculated from the simulation was in good agreement with SRIM calculations
Ziegler  _et al._ (2010). Second, the scattered events were generated using
the information from the stored tracks and the scattered particles detected by
the Si detectors. The tracks are chosen randomly from the stored beam particle
trajectories. The scattered particle properties such as position and energy
are extracted from the chosen track. For a line beam without multiple
scattering and assuming an energy- and angle-independent differential cross
section, i.e., $\frac{d\sigma}{d\Omega}=\frac{\sigma}{4\pi}$, the product
$nN_{\text{inc}}\sigma$ in Eq. (5) gives the number of reactions per unit
length along the beam path, which provides a link between the simulation and
the $G$ factor. A Monte Carlo simulation is implemented by generating events
randomly from a uniform distribution along the length $\Delta
z=z_{\text{max}}-z_{\text{min}}$, along the beam axis and randomly from a
uniform distribution into a solid angle $\Delta\Omega=(\rm
cos~{}\theta_{\text{min}}-\rm
cos~{}\theta_{\text{max}})(\phi_{\text{max}}-\phi_{\text{min}})$. The
parameters $z_{\text{min}}$, $z_{\text{max}}$, $\theta_{\text{min}}$,
$\theta_{\text{max}}$, $\phi_{\text{min}}$, and $\phi_{\text{max}}$ are
determined using the geometry as in Ref. Silverstein (1959). The energy of the
generated scattering event $E$ is randomized within the energy limit,
$E_{\text{min}}$ and $E_{\text{max}}$, using the inverse transform method
assuming the probability of scattering is inversely proportional to the square
of the energy. The energy limits $E_{\text{min}}$ and $E_{\text{max}}$ were
determined from the stored tracks and correspond to the energy of the beam
particle at $z_{\text{min}}$ and $z_{\text{max}}$, respectively. For a given
energy, the corresponding position $\vec{r}$ of the scattering event was
obtained from the stored tracks. The generated event with co-ordinates
($\vec{r}$,$\theta$,$\phi$,$E$) was accepted based on the acceptance-rejection
method. Let $N_{\text{ev}}$ be the number of events generated along $\Delta z$
into the solid angle $\Delta\Omega$ and $N_{\text{det}}$ be the number of
events detected in the Si detector in the Geant4 simulation, given as

$N_{\text{ev}}=nN_{inc}\frac{d\sigma}{d\Omega}(E_{0},\theta_{0})\Delta
z\Delta\Omega.$ (9)

From Eq. (5) and Eq. (9), the $G$ factor can be computed as

$G=\frac{N_{\text{det}}}{N_{\text{ev}}}\rm sin\theta_{0}\Delta z\Delta\Omega.$
(10)

The simulation was run for $N_{\text{ev}}$=106 events. The plot of the $G$
factor as a function of energy of the scattered particle is shown in Fig. 5.
The $G$ factors for both 3He and 4He particles show the same behaviour as a
function of energy.

Figure 5: The $G$ factor as a function of kinetic energy of the scattered
particle. The blue circles and green squares are the computed $G$ factor from
the Geant4 simulation for 3He and 4He, respectively. The red-dashed line is
the $G$ factor calculated using Eq. (6). The dotted line is explained in the
text.

The calculated $G$ factor from the simulation is in good agreement with the
values obtained with the analytic expression, Eq. (6), starting around
particle energies above 1 MeV. However, if we go further down in particle
energy, the differences between the $G$ factor from the simulation and the
analytic expression increase and become significant for energies below 500 keV
as represented by the dotted line in Fig. 5. The lowest particle energy for
which the $G$ factor from the simulation is used is 0.24 MeV, for which the
$G$ factor is 80.6 nm. Note that our results are consistent with Eq. (6) if we
do not introduce the multiple scattering effects in our simulation. These
results with multiple scattering switched off were a benchmark for our
simluation, and they are shown in Fig. 6.

Figure 6: The $G$ factor as a function of scattering angle in laboratory frame
of reference for $E$[3He]=721 keV. The filled (open) points represent the $G$
factor obtained from a Geant4 simulation with (without) introducing the
multiple scattering model in the simulation. The red-dashed line is the $G$
factor calculated using Eq. (6).

The choice of multiple scattering model is one of the sources of systematic
uncertainty in the $G$ factor derived from the Monte Carlo simulation. We used
two models, namely the G4Urban Msc-model Urbán (2006) and the Wentzel-VI Msc-
model Ivanchenko  _et al._ (2010) to simulate the multiple scattering. The
systematic uncertainty in the $G$ factor due to the choice of multiple
scattering model in this work was approximately $\pm 1\%$.

It is often the case that the geometrical components of SONIK, such as the
apertures, will have some errors associated with their dimensions when made in
the machine shop. Measurements of the rectangular apertures and circular
apertures in each of the detector telescopes were made with an optical
comparator or shadow graph. The rectangular aperture dimensions, $b$, are on
average 0.1$\%$ larger than the specified value of 1.0 mm and have a standard
deviation of 0.4$\%$ around the mean. The circular aperture radii, $a$, are on
average 2.2$\%$ larger than the specified value of 0.5 mm and have a standard
deviation of 1.6$\%$ about the mean. The error on $R_{0}$ and $h$ is
negligible compared to the error on $a$. In Sec. V.1 the variation in the
aperture dimensions of each detector is modeled using a detector-dependent
normalization, to which we assign a Gaussian prior with a mean of 0.96 and
standard deviation of 0.032 (cf. Eq. (7)). This implies an additional 27
normalization factors, $c_{j}$, corresponding to the 27 detectors that were
actually used to collect the data.

###  III.6 Error Budget

This section describes the systematic error for all measurements made in the
experiment independent of angle and energy. In our experiment, the beam is
tuned through the SONIK chamber using the CCD camera and the beam profile
monitors. However, the beam position might change during the run period, which
changes the acceptance of the detector or $G$ factor. The effect of a change
in beam position on the calculated differential elastic scattering cross
section was estimated to be $\pm$1$\%$, as explained in detail in Ref. Paneru
(2020). The systematic uncertainties due to various other factors are
presented in Table 1. The individual systematic uncertainties are added in
quadrature to report the total systematic uncertainty for all measurements
from this experiment. The total systematic uncertainty for all measurements is
estimated to be within 2.0$\%$.

Table 1: Estimation of systematic uncertainties. Source of Error | Value
---|---
Target pressure and temperature | 1$\%$
Beam intensity | 1$\%$
Beam position | 1$\%$
Model Uncertainty in Geant4 | 1$\%$
Total | 2.0$\%$

The differential scattering cross sections from this work are shown in Figs.
10, 11, and 12. The red circles and purple squares represent the differential
scattering cross sections calculated using 3He and 4He ejectiles,
respectively.

##  IV Comparison with previous data

To compare our result with previous measurements of elastic 3He-4He scattering
whose energy range overlaps that of our experiment we use the ratio of the
experimental differential scattering cross section to the cross section
calculated using the $R$-matrix parameters determined in Ref. deBoer  _et al._
(2014). That ratio is plotted in Fig. 7 (beam energies of 5.490 MeV, 4.347
MeV, and 3.608 MeV) and Fig. 8 (beam energies of 2.633 and 1.767 MeV). Overall
the results from this work are consistent with previous determinations but
have better precision. The data from Spiger and Tombrello Spiger and Tombrello
(1967) shown in the top panel of Fig. 7 are from a measurement at
$E$[3He]=5.438 MeV, a slightly different energy compared to our measurement.
The lowest panel in Fig. 7 shows that the data from this work are in good
agreement with those of Barnard et al. Barnard  _et al._ (1964). Turning to
Fig. 8, the Mohr et al. Mohr  _et al._ (1993) measurements at $E$[3He]=2.6 MeV
and $E$[3He]=1.7 MeV show jumps in between the data points, whereas our result
corresponds to a smooth angular distribution. The lower panel of Fig. 8 shows
that the present result is in fair agreement with data from Chuang Chuang
(1971) at forward angles, although three backward-angle data points from Ref.
Chuang (1971) disagree.

The Spiger and Tombrello Spiger and Tombrello (1967), Tombrello and Parker
Tombrello and Parker (1963), and Chuang Chuang (1971) measurements were made
using a gas cell with foil entrance and exit windows. The energy loss
corrections for the charged particles at the entrance and exit windows of the
gas cell introduce additional systematic uncertainties in the energy
determinations for these older measurements. The use of a windowless gas
target in our work avoids the need for the energy loss corrections. This could
account for discrepancies seen in the cross section at backward angles between
our results and the past measurements. The Mohr et al. Mohr  _et al._ (1993)
measurement was performed with a jet gas target and the scattered particles
were detected using 10 surface barrier detectors placed at fixed positions.
20Ne was mixed with the 4He gas in the target for the normalization purposes.
This measurement does not specify its systematic uncertainty, so the
differences seen in Fig. 8 with respect to our new data are hard to assess.

Figure 7: The ratio of the experimental differential scattering cross section
to the cross section calculated using $R$-matrix parameters from Ref. deBoer
_et al._ (2014) at beam energies of 5.490 MeV, 4.347 MeV, and 3.608 MeV. The
interaction region III measurements of this work, represented by the blue
points, are, in general, consistent with previous determinations but are more
precise. Only interaction region III data are shown for these comparisons.
Figure 8: The ratio of the experimental differential scattering cross section
to the cross section calculated using $R$-matrix parameters from Ref. deBoer
_et al._ (2014) at beam energies of 2.633 MeV and 1.767 MeV. The discrepancies
between the interaction region III measurements of this work, represented by
the blue points, and previous data are discussed in the text.

##  V R-MATRIX ANALYSIS

In this section we present an analysis of the differential cross section for
4He(3He,3He)4He elastic scattering using $R$-matrix theory Lane and Thomas
(1958). The phenomenological $R$-matrix code Azure2 Azuma  _et al._ (2010) is
used to analyze the elastic scattering data from this experiment and from
Barnard et al. Barnard  _et al._ (1964). We adopt the alternative
parametrization of $R$-matrix theory presented in Ref. Brune (2002), so the
$R$-matrix parameters are expressed in terms of the observed resonance energy
$\tilde{E}$ and the observed reduced width amplitude $\tilde{\gamma}$. The
channel radius is fixed at 4.2 fm. A channel radius of 4.3 fm was adopted in
Ref. deBoer  _et al._ (2014).

Most of the experiments reported in the literature are studies of the
structure of 7Be and consequently are focused on higher energies than the
present work Miller and Phillips (1958); Tombrello and Parker (1963); Barnard
_et al._ (1964); Spiger and Tombrello (1967); Ivanovich  _et al._ (1968);
Chuang (1971); Hardy  _et al._ (1972); Boykin  _et al._ (1972). The data of
Barnard et al. Barnard  _et al._ (1964) were found to contain the most
complete uncertainty information. The data of Spiger and Tombrello Spiger and
Tombrello (1967) are reported to have a systematic uncertainty as low as
1.1$\%$ and a maximum relative error of 9$\%$. The Spiger and Tombrello
measurement only extends as low as $E$ [3He]=4.7 MeV. The only measurement
extending to lower energies, that of P. Mohr et al. Mohr  _et al._ (1993),
does not quantify systematic uncertainties and, as shown in Fig. 8, has
unexplained systematic variations of the cross section with angle. For these
reasons, only the data of Barnard et al. Barnard  _et al._ (1964) and the data
reported in this work were used in the $R$-matrix analysis. All data included
in the analysis were taken at energies below the proton emission threshold.

The ground state spins and parities of 4He and 3He are 0+ and 1/2+,
respectively. Restricting our calculations to orbital angular momentum
$\ell\leq 3$, the allowed total angular momentum and parities in 7Be are 1/2+,
1/2-, 3/2-, 5/2+, 3/2+, 7/2-, and 5/2-. The level diagram of the compound
nucleus 7Be is shown in Fig. 9, with the energies of the levels taken from
Ref. Tilley  _et al._ (2002a). The energy range covered in this experiment,
$0.38\leq E_{\text{c.m.}}\leq 3.13$ MeV, is highlighted. The $R$-matrix
analysis was started with the states of 7Be shown in Fig. 9. But, within the
experimental energy range, the states 1/2+, 5/2+, and 3/2+ are not identified
in the literature. Therefore, these channels are introduced into the analysis
via background levels. We also add background levels in the 1/2- and 3/2-
channels, in addition to the levels that represent the bound states of 7Be
which exist in those channels. The background levels for the 5/2+, and 3/2+
states are introduced at an excitation energy of 12.0 MeV. The excitation
energies of the 1/2+, 1/2-, and 3/2- background levels are fixed at 14.0,
21.6, and 21.6 MeV, in order to reproduce the trend of the Spiger and
Tombrello Spiger and Tombrello (1967) phase shifts for $s$\- and $p$-waves at
high energies.

Similarly, the alpha width of the 5/2- state is fixed at 1.9 MeV to reproduce
the trend of the Spiger and Tombrello Spiger and Tombrello (1967) 5/2- phase
shift data up to $E$[3He]=9 MeV.

Figure 9: The levels and separation energies introduced in the $R$-matrix fit
were taken from Ref. Tilley  _et al._ (2002a). The energy range covered in the
present measurement is represented by the curly brace. All energies are in
MeV.

The asymptotic normalization constants (ANCs) used for the sub-threshold 3/2-
and 1/2- states are fixed at $C_{3/2}$=3.7 fm-1/2 and $C_{1/2}$=3.6 fm-1/2
deBoer  _et al._ (2014), respectively. Including radiative capture data
provides better constraints on the ANCs than can be obtained from scattering
data alone, so we chose to fix the ANCs.

###  V.1 Bayesian Analysis

In what follows we did not use AZURE2’s built-in $\chi^{2}$ analysis, instead
employing a Markov Chain Monte Carlo (MCMC) analysis to obtain probability
distributions and study how parameter uncertainties propagate to extrapolated
quantities. The goal of this MCMC analysis is to approximate the Bayesian
posterior distribution function for the $R$-matrix parameters, which we
collect into a vector $\vec{\theta}$ 111We use $\vec{\theta}$ to represent the
multi-dimensional parameter vector, trusting the reader to distinguish this
from a scattering angle using context.. By Bayes’s theorem that posterior can
be obtained as

$p(\vec{\theta}|D)=\frac{p(D|\vec{\theta})p(\vec{\theta})}{p(D)}~{},$ (11)

where the likelihood, $p(D|\vec{\theta})$, is chosen to be

$p(D|\vec{\theta})=\prod_{i,j}\frac{1}{\sqrt{2\pi}\sigma_{i,j}}e^{-\frac{1}{2}\chi^{2}_{i,j}}~{},$
(12)

where

$\chi^{2}_{i,j}=\frac{(f(x_{i,j})-\tilde{c}_{i,j}y_{i,j})^{2}}{(\tilde{c}_{i,j}\sigma_{i,j})^{2}},$
(13)

where 222The following description applies only to the SONIK data, the Barnard
data is treated via one overall normalization factor. For Barnard data, the
overall systematic normalization, $c_{\text{Barnard}}$, the prior we set is a
Gaussian with mean 1 and standard deviation 0.05, in accord with Ref. Barnard
_et al._ (1964). $i$ indexes the beam energy, $j$ indexes the detector,
$f(x_{i,j})$ is the differential scattering cross section from the $R$-matrix,
$y_{i,j}$ is the data point value, $\sigma_{i,j}$ is the statistical
uncertainty of the data point, and $\tilde{c}_{i,j}$ is a composite
normalization factor defined by Eq. (15) below.

The prior distribution function

$p(\vec{\theta})=\prod_{i=1}^{N_{p}}p(\theta_{i})~{},$ (14)

is a product of each parameter’s prior distribution, with $N_{p}$ representing
the number of sampled parameters. Typically, with $\chi^{2}$ minimization,
each parameter is allowed to move freely in an unbounded space. The analyses
presented here mimic that freedom by imposing uniform priors with generous
upper and lower bounds. Those bounds are shown in Table 2; these priors are
quite similar to the ones adopted in Ref. Odell  _et al._ (2021). Since the
analysis we perform here only considers elastic scattering data, the sign of
the reduced width amplitudes cannot be uniquely determined. We obtain a
solution where all partial widths have the same signs found in Ref. Odell  _et
al._ (2021), where capture data was also included.

Parameter | Lower Bound | Upper Bound
---|---|---
$\Gamma_{\alpha}^{(1/2-)}$ | -150 MeV | 150 MeV
$\Gamma_{\alpha}^{(1/2+)}$ | 0 MeV | 100 MeV
$\Gamma_{\alpha}^{(3/2-)}$ | -100 MeV | 100 MeV
$\Gamma_{\alpha}^{(3/2+)}$ | 0 MeV | 100 MeV
$\Gamma_{\alpha}^{(5/2+)}$ | 0 MeV | 100 MeV
$E_{x}^{(7/2-)}$ | 2 MeV | 10 MeV
$\Gamma_{\alpha}^{(7/2-)}$ | 0 | 10 MeV
Table 2: Limits of uniform prior distributions set for the $R$-matrix
parameters that were sampled in the MCMC analysis.

The normalizations applied to the data, $c_{i,j}$, are, for each data point, a
product of three different effects: $c_{\rm SONIK}$, $c_{i}$, and $c_{j}$. The
overall systematic uncertainty is accounted for with $c_{\rm SONIK}$. This
factor is applied to all data points. An energy-dependent systematic
uncertainty is accounted for with $c_{i}$. Each energy bin has its own
associated $c_{i}$, which applies to the data from all three interaction
regions at a given energy. A detector-specific systematic uncertainty is
accounted for with $c_{j}$. All points measured with the same detector are
adjusted by $c_{j}$. There are 27 detectors in total. The resulting
normalization adjustment of the data is then

$\tilde{c}_{i,j}=c_{\rm SONIK}c_{i}c_{j},$ (15)

where $c_{i}$ and $c_{j}$ depend on the energy bin and detector used to
measure $y_{i,j}$. We set Gaussian priors for each of three types of
normalizations. For the overall systematic normalization, $c_{\rm SONIK}$, the
prior we set is a Gaussian of mean 1 and standard deviation 0.02, in accord
with the error budget of Table 1. For the energy bin normalizations, $c_{i}$,
the prior chosen is a Gaussian of mean 1 and standard deviation
$\sigma_{E_{i}}$, with the $\sigma_{E_{i}}$’s are tabulated in the fourth
column in table VII.1. Finally, the detector-specific normalizations, $c_{j}$,
are each assigned a Gaussian prior of mean 0.96 and standard deviation 0.032,
to account for variances in the aperture dimensions, cf. Sec. III.5.

Since the priors on the $R$-matrix parameters are intentionally left very
broad, our Bayesian posterior can be well approximated as a likelihood
$e^{-\chi^{2}_{\rm aug}/2}$, where the chi-squared is:

$\chi^{2}=\sum_{i}\left(\sum_{j}\frac{(f(x_{i,j})-\tilde{c}_{i,j}y_{i,j})^{2}}{(\tilde{c}_{i,j}\sigma_{i,j})^{2}}\right)$
(16)

and the augmented chi-squared is:

$\chi_{\rm
aug}^{2}=\chi^{2}+\sum_{i}\left(\frac{c_{i}-1}{\sigma_{E_{i}}}\right)^{2}+\sum_{j}\left(\frac{c_{j}-0.96}{0.032}\right)^{2}+\left(\frac{c_{\rm
SONIK}-1}{0.02}\right)^{2}+\left(\frac{c_{\rm
Barnard}-1}{0.05}\right)^{2}~{}.$ (17)

The MCMC analysis was performed with a publicly available ensemble sampler,
emcee Foreman-Mackey  _et al._ (2013), and the previously mentioned $R$-matrix
code, AZURE2. The pairing was enabled by a publicly available Python layer,
BRICK Odell  _et al._ (2021).

##  VI ANALYSIS USING HALO EFFECTIVE FIELD THEORY

An EFT is a controlled expansion in a ratio
$Q\equiv\frac{p_{\text{typ}}}{\Lambda}$, where $p_{\text{typ}}$ is the low-
momentum scale that typifies the scattering and $\Lambda$ is the momentum
scale at which the theory breaks down—see, e.g., Ref. Kaplan (1995) for an
introduction. The EFT expansion of an observable $y$ in powers of $Q$ can be
written as Weinberg (1979); Epelbaum  _et al._ (2009); Wesolowski  _et al._
(2019)

$y(p,\theta)=y_{\text{ref}}(p,\theta)\sum_{\nu}c_{\nu}(p,\theta)\ Q^{\nu}.$
(18)

Here $\nu$ indexes the order of different contributions. We denote $\nu=0$ as
leading order (LO), $\nu=1$ next-to-leading order (NLO), and $\nu=2$ as next-
to-next-to-leading order (NNLO).

In this section we briefly describe the EFT we use to calculate the
3He-$\alpha$ scattering reaction. Full details of the EFT can be found in Ref.
Poudel and Phillips (2022).

This EFT is built on the scale separation between the large de Broglie
wavelength of the quantum-mechanical scattering process and the small size of
the 3He and $\alpha$ nuclei. It is thus an example of “Halo EFT”; see Ref.
Hammer  _et al._ (2017, 2020) for recent review. In this approach 7Be is a
bound state of 3He and $\alpha$ nuclei. Such a description is accurate because
the energies by which the ground- and excited-state of 7Be are below the
3He-$\alpha$ scattering threshold are 1.6 and 1.2 MeV Tilley  _et al._ (2002b)
respectively. These are small compared to the energy scales at which 3He and
4He can be broken up into smaller constituents. These energy scales, as well
as the sizes of the two helium isotopes, yield an EFT breakdown momentum of
$\Lambda\approx 200$ MeV/c Zhang  _et al._ (2020b).

We take the typical momentum of the collision to be $p_{\rm
typ}=\max\\{q,p\\}$, where $q=2p\sin(\theta/2)$ is the momentum transfer of
the scattering reaction. The bulk of the SONIK data were taken for $p$ between
60 MeV/c and 90 MeV/c. In this energy range Halo EFT has been successfully
applied to the 3He($\alpha$,$\gamma$)7Be reaction Higa  _et al._ (2018); Zhang
_et al._ (2020b); Premarathna and Rupak (2020) and used to fit scattering
phase shifts Higa  _et al._ (2018); Premarathna and Rupak (2020). In Ref.
Poudel and Phillips (2022) the choice $\Lambda=200$ MeV/c was validated by
showing that it leads to an EFT with a regular convergence pattern, i.e., once
$Q$ is chosen in this way the coefficients $c_{0}$, $c_{1}$, and $c_{2}$ in
Eq. (18) have roughly the same size.

The EFT contains only minimal assumptions about the 3He-$\alpha$ dynamics:
rotational invariance, unitarity, analyticity of the amplitude, and the
presence of a short-range strong interaction as well as a long-range Coulomb
potential. Since the Coulomb-modified effective-range expansion (CM-ERE) is
based on the same set of assumptions the Halo EFT $t$-matrix has the same form
as that obtained in the CM-ERE Kong and Ravndal (2000); Higa  _et al._ (2008);
Poudel and Phillips (2022). The EFT Lagrangian is expressed as an expansion in
powers of $p^{2}$, so, at a given order in the EFT, the CM-ERE is reproduced
up to the corresponding order of $p^{2}$.

The CM-ERE amplitude associated with 3He-$\alpha$ scattering in the
$(l,J=l\pm\frac{1}{2})^{\text{th}}$ channel, $T_{l}^{\pm}$, takes the form
Bethe (1949); Hamilton  _et al._ (1973):

$T^{\pm}_{l}(E+i\epsilon)=-(2l+1)\frac{2\pi}{\mu}\frac{\Bigg{[}\frac{\Gamma(2l+2)}{2^{l}\Gamma(l+1)}\Bigg{]}^{2}C_{l}^{2}(\eta)\text{e}^{2i\sigma_{l}}p^{2l}P_{l}(\cos\theta)}{\Bigg{[}\frac{\Gamma(2l+2)}{2^{l}\Gamma(l+1)}\Bigg{]}^{2}C_{l}^{2}(\eta)p^{2l+1}(\cot\delta^{\pm}_{l}-i)},$
(19)

where the quantities $\delta^{\pm}_{l}$ are the phase shifts for the channels
$l^{\pm}$. The phase shift for the $\pm$ channels in the $l$th partial wave
are, in turn, given by

$\Bigg{[}\frac{\Gamma(2l+2)}{2^{l}\Gamma(l+1)}\Bigg{]}^{2}C^{2}_{l}(\eta)p^{2l+1}(\cot\delta^{\pm}_{l}-i)=2k^{2l+1}_{c}K^{\pm}_{l}(E)-\frac{2k_{c}p^{2l}}{(\Gamma(l+1))^{2}}\frac{\Gamma(1+l+i\eta)\Gamma(1+l-i\eta)}{\Gamma(1+i\eta)\Gamma(1-i\eta)}H(\eta),$
(20)

where

$K^{\pm}_{l}=\frac{1}{2k_{c}^{2l+1}}\Bigg{(}-\frac{1}{a^{\pm}_{l}}+\frac{1}{2}r^{\pm}_{l}p^{2}+\frac{1}{4}P^{\pm}_{l}p^{4}+O((p^{2})^{3})\Bigg{)}$
(21)

is the effective-range function. Equations (20) and (21) relate the phase
shifts to the coefficients of powers of $p^{2}$ in the expansion of the
function $K$. $K(E)$ is analytic in $p^{2}$ for $|p|<1/R$, where $R$ is the
range of the strong interaction. The coefficients of a Taylor series expansion
of $K$ in $p^{2}$ are (apart from numerical factors) the effective range
parameters (ERPs). To obtain phase shifts from ERPs, the polynomial
$K$-function is truncated at a suitable order. In equations (19) and (20),
$P_{l}(\cos\theta)$ is the $l$th Legendre polynomial calculated at the cosine
of scattering angle $\theta$ and the quantities $C_{l}(\eta)$ and
$e^{2i\sigma_{l}}$ are given respectively by

$\text{e}^{2{\rm i}\sigma_{l}}=\frac{\Gamma(1+l+{\rm i}\eta)}{\Gamma(1+l-{\rm
i}\eta)}$ (22)

and

$C_{l}(\eta)=\frac{2^{l}}{\Gamma(2l+2)}\exp{(-\pi\eta/2)}\ |\Gamma(1+l+{\rm
i}\eta)|.$ (23)

Table 3: Hierarchy of power counting in our EFT: | s-wave | p-wave | $\nu$
---|---|---|---
LO | - | - | 0
NLO | $r_{0}$ |  $r^{+}_{1}$, $P^{\pm}_{1}$ | 1
NNLO | $\frac{1}{a_{0}}$ | $\frac{1}{a^{\pm}_{1}},r^{-}_{1}$ | 2

The EFT power counting is a particular assignment of the terms that should
appear in the $K$-function at a given order $\nu$. That assignment is chosen
to ensure that the pattern (18) is satisfied. In Ref. Poudel and Phillips
(2022) an assignment that achieves this was found although it should be clear
that such an assignment depends on having some knowledge of the size of the
ERPs themselves, and so can only be accomplished in the light of at least some
data on the system.

The organization of Ref. Poudel and Phillips (2022) has at LO only the
contributions proportional to the $H$-function for both $s$-and $p$-wave
channels, i.e., it takes $K=0$. We say this piece of the inverse amplitude is
of order $p$ for $s$-waves and of order $p^{3}$ for $p$-waves. (We assume
$\eta\sim 1$.) The terms proportional to ERPs that appear in $K$ are
corrections to this limit.

For $s$-waves, the term $\frac{1}{a_{0}}$ is a very small momentum and we take
it to be $\sim p_{\text{t}yp}^{3}/\Lambda^{2}$. Meanwhile, the $s$-wave
effective range $r_{0}$ scales naturally $\sim\frac{1}{\Lambda}$. Therefore
for $s$-waves, we include the term proportional to $r_{0}$ in the effective-
range expansion at NLO (its effect is $\sim p_{\text{t}yp}^{2}/\Lambda$) and
that from $\frac{1}{a_{0}}$ at NNLO. Regarding $p$-waves, we consider both
$p$-wave shape-parameter terms, $\frac{1}{4}P^{\pm}_{1}p^{4}$, at NLO, as the
shape parameters are natural ($\sim 1/\Lambda$) and so this term is $\sim
p_{\text{typ}}^{4}/\Lambda$. The other $p$-wave ERPs are unnaturally small, so
in the $J=\frac{3}{2}^{-}$ channel, we take the contribution from
$\frac{1}{2}r_{1}^{+}p^{2}$ at NLO and consider that of $\frac{1}{a_{1}^{+}}$
only at NNLO. Both $\frac{1}{a_{1}^{-}}$ and $\frac{1}{2}r_{1}^{-}p^{2}$ are
considered to be NNLO effects. This organization is summarized in Table 3. All
results presented below are computed at NNLO.

However, even at NNLO, if we are to describe the higher-energy portion of the
SONIK data at the required accuracy, we must include the ${\frac{7}{2}}^{-}$
partial wave in the analysis. To account for the impact of the
${\frac{7}{2}}^{-}$ 7Be level at $E_{\rm x}=4.57$ MeV Piluso  _et al._ (1971)
on observables in the energy range of interest Ref. Poudel and Phillips (2022)
employed a phenomenological treatment of it, based on R-matrix theory Lane and
Thomas (1958). The focus both in Ref. Poudel and Phillips (2022) and here is
not on the resonance itself. The goal of phenomenologically adding its
amplitude to the EFT analysis is solely to stop it contaminating the
extraction of the ERPs. Since Ref.Poudel and Phillips (2022) points out that
the inclusion of contributions to the amplitude from the $\frac{5}{2}^{-}$
partial wave is also essential for a consistent analysis of the data, we also
include them in the EFT analysis presented here. In adding these contributions
to the scattering amplitude from $f$-waves, we employ the following resonance
energies and physical widths:

$\displaystyle E^{\frac{7}{2}^{-}}_{R}$ $\displaystyle=$ $\displaystyle 5.22\
{\rm MeV\ in\ lab},$ $\displaystyle E^{\frac{5}{2}^{-}}_{R}$ $\displaystyle=$
$\displaystyle 9.02\ {\rm MeV\ in\ lab},$
$\displaystyle\Gamma^{\frac{7}{2}^{-}}$ $\displaystyle=$ $\displaystyle 0.159\
{\rm MeV},$ $\displaystyle{\rm and}\ \Gamma^{\frac{5}{2}^{-}}$
$\displaystyle=$ $\displaystyle 1.8\ {\rm MeV}.$

which are used to generate the $\frac{7}{2}^{-}$ and $\frac{5}{2}^{-}$ phase
shifts using $l=3$ in the following formula:

$\delta_{l^{\pm}}=-\phi_{l}+\tan^{-1}\frac{P_{l}(E,\rho)}{P_{l}(E^{c}_{R},\rho)}\frac{\frac{1}{2}\Gamma^{c}}{E^{c}_{R}-E}.$
(24)

In Eq. (24),

$\phi_{l}=\tan^{-1}\frac{F_{l}(\eta,\rho)}{G_{l}(\eta,\rho)}$ (25)

and

$P_{l}(E,\rho)=\rho/(F^{2}_{l}+G^{2}_{l})$ (26)

where $F_{l}$ and $G_{l}$ are the usual regular and irregular Coulomb
functions Abramowitz and Stegun (1964) and $\rho=pr$ where $r$ is the channel
radius, here taken to be $4.2$ fm. The channels are represented with
superscript $c$; for $l=3$, $c=\frac{7}{2}^{-},\frac{5}{2}^{-}$. The resonance
energies and physical widths that we adopt mimic the f-wave phase shifts
produced by Bayesian R-matrix analysis.

The ERPs—$a_{0},r_{0},a^{+}_{1},r^{+}_{1},P^{+}_{1},a^{-}_{1},r^{-}_{1}\
\text{and}\ P^{-}_{1}$—span an 8-dimensional parameter space. Using
relationships between the effective-range amplitude and bound-state properties
Zhang  _et al._ (2020a); Poudel and Phillips (2022), we reparametrize the
space in terms of the ANCs, replacing $r^{\pm}_{1}$ by $C^{\pm}_{1}$ using
equation (43) from Ref. Poudel and Phillips (2022). The ANCs are fixed at
$C^{+}_{1}=C_{3/2}=3.7~{}{\rm fm}^{-1/2}$ and $C^{-}_{1}=C_{1/2}=3.6~{}{\rm
fm}^{-1/2}$—as was done in the $R$-matrix analysis. We also determine
$a^{\pm}_{1}$ from the location of the two 7Be bound states using equation
(39) from Ref. Poudel and Phillips (2022). These two constraints reduce the
eight-dimensional ERP space to a four-dimensional one.

To compute the posterior of the EFT parameters we employ Bayes’s theorem, Eq.
(11), as was done in Sec. V.1 for the $R$-matrix analysis. However, for the
EFT analysis we employ a $\chi^{2}$ function that is different to the standard
one, Eq. 16. Truncation of the EFT series at order $\nu_{max}$ induces an
error in the observable $y$ Furnstahl  _et al._ (2015) at data point $j$ in
data set $i$ of:

$\Delta y(x_{i,j})=y_{\rm ref}(x_{i,j})c^{\rm rms}Q_{i,j}^{\nu_{\rm max}+1},$
(27)

where $c^{\rm rms}$ represents the rms value of the EFT coefficients defined
in Eq. (18) and $x_{i,j}$ is a kinematic point $(p_{j},\theta_{j})$ in dataset
$i$. We therefore use a modified $\chi^{2}$ Wesolowski  _et al._ (2019):

$\chi^{2}_{{\rm EFT},i}=[\vec{r}^{\,T}(\sigma^{\rm expt}+\sigma^{\rm
th})^{-1}\vec{r}]_{i}$ (28)

where $\chi^{2}_{{\rm EFT},i}$ is the modified $\chi^{2}$ of dataset $i$. The
matrix elements of the theory covariance matrix in dataset $i$ are

$\sigma^{\text{th}}_{jk}=(y_{\text{ref}})_{j}(y_{\text{ref}})_{k}(c^{\rm
rms})^{2}Q^{\nu_{\rm max}+1}_{j}Q^{\nu_{\rm max}+1}_{k}.$ (29)

It accounts for the error due to omitted higher-order terms in the EFT. In
this analysis we take that error to be completely correlated across the
kinematic space, see Ref. Melendez  _et al._ (2019) for a more advanced
treatment. We take the experimental covariance matrix to be diagonal

$\sigma^{\rm expt}_{jk}=\tilde{c}_{i,j}^{2}\sigma^{2}_{j}\delta_{jk},$ (30)

while the entries of the residual vector $\vec{r}$ for dataset $i$ are defined
by

$r_{j}=f(x_{i,j})-\tilde{c}_{i,j}y_{i,j}.$ (31)

The inclusion of the truncation errors modifies the likelihood to

$p(D|\vec{\theta})=\frac{1}{\sqrt{(2\pi)^{N}{\rm det}(\sigma^{\rm
expt}+\sigma^{\rm th})}}e^{-\frac{1}{2}\sum_{i}\chi^{2}_{{\rm EFT},i}}.$ (32)

In Eq. 32 $N$ is the total number of data points and det means the matrix
determinant.

In addition to the ERPs, Eq. (28) includes as parameters the normalizations
$\tilde{c}_{i,j}$ of the differential cross-section data, each of which is a
product of the three different normalization factors, see Sec. V.1 for
details. We adopt the priors for the overall, energy-dependent, and detector-
dependent normalization factors specified in Sec. V.1.

To construct the EFT error model we take $\nu_{max}=2$, since the calculation
is carried out to NNLO. Meanwhile, $(y_{\text{ref}})_{j}$ is taken to be the
LO cross section at data point $j$. $\bar{c}$ is then estimated from the size
of the shifts from LO to NLO and NLO to NNLO to be $c^{\rm rms}=0.70$, as
described in Ref. Poudel and Phillips (2022). Further details regarding the
Bayesian analysis of the NNLO Halo EFT calculation can also be found in that
work.

We observe that the EFT has an expansion parameter of 0.2 at forward angles in
the lowest SONIK energy bin, but $Q$ approaches one for the backward-angle
data at the highest SONIK $E_{\rm c.m.}$ of 3.1 MeV. We therefore do not
expect Halo EFT to accurately describe all the data collected in this
experiment. That, after all, is why Poudel and Phillips truncated their
analysis at $E_{\rm c.m.}=2.5$ MeV Poudel and Phillips (2022). The inclusion
of the truncation error in the likelihood ameliorates the theory’s failure to
describe higher-$Q$ data, since it decreases the statistical weight of data
for which $Q$ is larger. Nevertheless, the sensitivity of the inference to
assumptions regarding the nature of the truncation error becomes quite severe
as $Q\rightarrow 1$. In spite of this, we will include all the SONIK data in
our Halo EFT analysis, so that we can make a direct comparison with the
$R$-matrix analysis.

##  VII RESULTS

###  VII.1 R-matrix results

The results from the simultaneous fitting of the elastic scattering data of
the current measurement and the data of Barnard et al. Barnard  _et al._
(1964) are shown in Figs. 10, 11, 12, and 13. The blue bands in the figures
correspond to the $R$-matrix analysis, and green bands correspond to the Halo
EFT analysis. The medians of the normalization factors, $\tilde{c}_{i,j}$ for
the SONIK data and $c_{\rm Barnard}$ for the Barnard data, have been applied
to the data in the figures. Although the total $\chi^{2}$ from both analyses
are comparable at the lowest three energies, as seen in Fig. 10 and with
comparable $\chi^{2}$ and $\chi^{2}_{\rm EFT}$ values from Table VII.1, the
two analyses differ in the angular distribution. The two analyses also differ
significantly in terms of $\chi^{2}$ at the three highest SONIK energies, cf.
Table VII.1 and discussion in Sec. VII.2. The two analyses produce similar
fits at the intermediate SONIK energies.

The $\chi^{2}$ values for each data segment are presented in Table VII.1 and
the best fit $R$-matrix parameters are presented in Table 5. A total of 1097
data points were fitted simultaneously with 46 free $R$-matrix parameters. The
fits to the whole data set have a minimum reduced $\chi^{2}$ of 1.85. This
value is calculated at the point in parameter space that maximizes the
posterior and therefore at the best values of the normalization parameters
found by the sampler. It cannot be straightforwardly interpreted as a measure
of the quality of the $R$-matrix fit, since the uncertainties of the
normalization factors are not accounted for in the covariance matrix used to
compute this standard $\chi^{2}$. The dominant contribution to $\chi^{2}$
comes from the differential scattering cross section data points at forward
angles (22.5${}^{\circ}\leq\theta_{\text{lab}}\leq 35^{\circ}$) in the
laboratory frame of reference. The width of the 5/2- level introduces a
significant change in the total $\chi^{2}$ for the $R$-matrix fit to the
differential scattering cross section results at $E[^{3}\text{He}]=5.490$ MeV.
The width of the 7/2- level is reported to be 175 $\pm$ 7 keV Piluso  _et al._
(1971) which is consistent with the width obtained fitting the data from this
work alone. However, the width of the 7/2- level obtained from the
simultaneous fit of the data of this work and Barnard et al. Barnard  _et al._
(1964) is significantly lower than the value reported in Ref. Piluso  _et al._
(1971). In fact, if the stated energy uncertainty of $\pm$40 keV for data
above 4 MeV in Barnard et al. Barnard  _et al._ (1964) is accounted for in the
fit the central value of both $\tilde{E}_{7/2-}$ and $\tilde{\gamma}$ change
by more than the uncertainty quoted in Table 5. The difference in the alpha
widths of the 7/2- resonance needs to be resolved by future experiments.

At low energies, the 4He(3He,$\gamma$)7Be reaction primarily occurs through
$E1$ external $s$-wave capture contributions Christy and Duck (1961). However,
$d$-wave capture and internal contributions must also be considered. The
internal $s$-wave part comes from the $J=1/2^{+}$ background level, which
interferes with the external contribution to produce the large capture cross
section at low energies deBoer  _et al._ (2014). Therefore, the $s$-wave
scattering length, $a_{0}$, is of particular importance. It is related to the
$R$-matrix parameters via Paneru  _et al._ (2019)

$a_{0}=-a\left[\frac{M_{cc}}{x^{2}K_{1}^{2}(x)}-\frac{2I_{1}(x)}{x^{2}K_{1}(x)}\right],$
(33)

where
$M_{cc}=\mathbf{\tilde{\gamma}}^{T}_{c}\mathbf{\tilde{A}}\mathbf{\tilde{\gamma}}_{c}$,
${\mathbf{\tilde{A}}}$ is the level matrix as defined in Ref. Brune (2002),
$c$ is the channel index, $a$ is the channel radius, $I_{1}(x)$ and $K_{1}(x)$
are modified Bessel functions with $x=\left(8Z_{1}Z_{2}e^{2}\mu
a/\hbar^{2}\right)^{1/2}$, $Z_{1}e$ and $Z_{2}e$ are the nuclear charges,
$\hbar$ is the reduced Planck’s constant, and $\mu$ is the reduced mass.

Figure 10: Differential elastic scattering cross sections, relative to
Rutherford’s prediction, as measured with SONIK. Results are obtained from an
MCMC analysis of SONIK and Barnard data. Bands encompass the 16th to 84th
percentile of the inferred probabibilty distributions. Blue bands correspond
to the $R$-matrix analysis, and green bands correspond to the Halo EFT
analysis. Red circles with error bars indicate ${}^{3}{\rm He}$ peaks. Purple
squares with error bars indicate ${}^{4}{\rm He}$ peaks. The three panels
along a row for a given $E[^{3}$He] beam energy are from interaction regions
I, II, and III, respectively. Figure 11: Differential elastic scattering cross
sections, relative to Rutherford’s prediction, as measured with SONIK. Colors
and symbols are as described in Fig. 10. The additional yellow band in the
bottom row corresponds to the second run at $E[{}^{3}{\rm He}]=2.633{\rm MeV}$
discussed in III.3. Figure 12: Differential elastic scattering cross sections,
relative to Rutherford’s prediction, as measured with SONIK. Colors and
symbols are as described in Fig. 10. Figure 13: Differential elastic
scattering cross sections, relative to Rutherford’s prediction, as reported in
Ref. Barnard  _et al._ (1964). Results are obtained from an MCMC analysis of
SONIK and Barnard data. Bands are as in Fig. 10 and grey circles represent the
data from Ref. Barnard  _et al._ (1964). Total $\chi^{2}$ at maximum posterior
probability for the $R$-matrix fit is 1098.56, resulting in $\chi^{2}/{\rm
datum}$ = 1.70. The total $\chi^{2}$ for the EFT fit to these data is 1996.24.
Table 4: 3He beam energies, corresponding center-of-mass energy range, angular
range (in the c.m. frame), normalization uncertainty, denoted
$\sigma_{E_{i}}$, at each energy, normalization factors $c_{i}$ obtained in
the $R$-matrix fit, $\chi^{2}$ from both $R$-matrix and Halo EFT, and number
of data points $N$ of the angular distributions from the SONIK experiment
reported in this work (part A) and the excitation function of Barnard et al.
Barnard  _et al._ (1964) (part B). The $\chi^{2}$ and $\chi^{2}_{\rm EFT}$ per
degree of freedom from $R$-matrix and Halo EFT analyses were found to be 1.85
and 3.15, respectively.

$E~{}[^{3}\text{He}]$ (MeV) |  $E_{\text{c.m.}}$ (MeV) |  $\theta_{\text{c.m.}}$ (degrees) |  Normalization uncertainty |  $c_{i}$ from |  |  |
---|---|---|---|---|---|---|---
R-matrix |  |  |  |  |  |  |
| $\chi^{2}_{{\rm R},i}$ | $\chi^{2}_{{\rm EFT},i}$ | N | analysis |  |  |
(A) This work |  |  |  |  |  |
$5.490$ | $3.122-3.127$ | $30.00-138.90$ | 8.7$\%$ | $1.023^{+0.005}_{-0.005}$ | 171.00 | 214.23 | 53
$4.347$ | $2.470-2.476$ | $39.26-135.00$ | 6.0$\%$ | $0.975^{+0.004}_{-0.004}$ | 74.71 | 373.85 | 53
$3.608$ | $2.045-2.052$ | $39.26-135.00$ | 7.5$\%$ | $0.992^{+0.004}_{0.004}$ | 49.28 | 253.89 | 52
$2.633$ | $1.488-1.496$ | $30.00-138.90$ | 3.7$\%$ | $0.987^{+0.003}_{-0.003}$ | 96.76 | 82.59 | 52
$2.633$ | $1.488-1.496$ | $39.26-135.00$ | 5.9$\%$ | $0.995^{+0.004}_{-0.004}$ | 91.57 | 99.95 | 52
$2.145$ | $1.209-1.219$ | $39.26-135.00$ | 4.1$\%$ | $0.983^{+0.003}_{-0.003}$ | 99.3 | 94.56 | 52
$1.767$ | $0.992-1.003$ | $39.26-135.00$ | 5.4$\%$ | $0.988^{+0.004}_{-0.004}$ | 111.17 | 72.95 | 46
$1.303$ | $0.724-0.737$ | $39.26-135.00$ | 9.6$\%$ | $0.931^{+0.004}_{-0.003}$ | 112.96 | 87.30 | 45
$0.878$ | $0.479-0.495$ | $60.61-110.00$ | 7.4$\%$ | $1.077^{+0.006}_{0.006}$ | 28.2 | 29.31 | 29
$0.721$ | $0.385-0.403$ | $68.97-108.07$ | 6.1$\%$ | $1.025^{+0.007}_{-0.007}$ | 14.41 | 15.13 | 17
SONIK Total |  |  |  |  | 849.35 | 1323.77 | 451
(B) Barnard et al. Barnard  _et al._ (1964) |  |  |  |  |  |
$2.454-5.737$ | $1.39-3.27$ | $54.77-140.80$ | 5$\%$ | $1.010^{+0.002}_{-0.002}$ | 1098.56 | 1996.24 | 646
Total |  |  |  |  | 1947.92 | 3320.01 | 1097

Table 5:  The observed energies $\tilde{E}_{\text{x}}$ and reduced width amplitudes $\tilde{\gamma}$ obtained from the best R-matrix fit with channel radius set at 4.2 fm. States in parentheses are introduced as background levels. The parameters in bold were treated as fit parameters and all others were held constant. Jπ | $l$ |  $\tilde{E}_{\text{x}}$ (MeV) |  $\tilde{\gamma}$ (MeV${}^{1/2})$
---|---|---|---
3/2- | 1 | 0.000 | 0.931
1/2- | 1 | 0.429 | 1.151
7/2- | 3 | $\mathbf{4.5639_{-0.0003}^{+0.0003}}$ | $\mathbf{0.924^{+0.003}_{-0.003}}$
5/2- | 3 | 6.730 | 1.767
1/2+ | 0 | (14.000) | $\mathbf{1.683_{-0.004}^{+0.004}}$
1/2- | 1 | (21.600) | $\mathbf{-2.939_{-0.036}^{+0.038}}$
3/2+ | 2 | (12.000) | $\mathbf{1.224_{-0.013}^{+0.013}}$
5/2+ | 2 | (12.000) | $\mathbf{1.155_{-0.012}^{+0.012}}$
3/2- | 1 | (21.600) | $\mathbf{-2.300_{-0.018}^{+0.019}}$

Using the MCMC-generated chain of $R$-matrix parameters and Eq. (33), the
$s$-wave scattering length $a_{0}$ was calculated to be $33.10$ fm. The
uncertainty from the MCMC analysis amounts to an uncertainty of $\pm 0.13$ fm
in the $s$-wave scattering length. Likewise, the MCMC results were used to
calculate the effective range function $K_{L}$ at $E$=0 and small positive
energies. The effective range is then obtained by numerical differentiation.
The $s$-wave effective range $r_{0}$ was determined to be 1.009 fm. The
uncertainty from the MCMC analysis yields an error bar of $\pm 0.001$ fm.

The sensitivity of the $s$-wave scattering parameters to the choice of
excitation energy of the 1/2+ level was studied at a fixed channel radius of
4.2 fm. A separate $R$-matrix fit in which the excitation energy for the 1/2+
state was allowed to vary was conducted using BRICK; this analysis is denoted
SB+ in the text hereafter. The excitation energy of the 1/2+ background level
resulting in the minimum total $\chi^{2}$ was found to be 9.22 MeV. However,
the trend of the experimental 1/2+ phase shift at higher energies determined
by Spiger and Tombrello Spiger and Tombrello (1967) is then not explained. The
$a_{0}$ and $r_{0}$ values obtained from the $SB^{+}$ analysis are
35.82$\pm$0.13 fm and 1.098$\pm$0.008 fm, respectively

The $s$-wave scattering parameters remain fairly constant with the choice of
channel radius. The channel radius was varied between 3.8–4.6 fm keeping other
parameters fixed, which resulted in $a_{0}$ and $r_{0}$ changing by 0.8 fm and
0.008 fm, respectively, from their values at a channel radius of 4.2 fm Paneru
(2020).

The results quoted so far were obtained with the ANCs fixed to the same values
that were used in Ref. deBoer  _et al._ (2014). We studied the implications of
varying the ANCs for the inferred $a_{0}$ and $r_{0}$ parameters by
considering the three sets of ANCs listed in Table I of Ref. Zhang  _et al._
(2020a). All three produce a change in the inferred $a_{0}$ and $r_{0}$
(relative to the SB analysis above) of $<1$ fm and $<0.01$ fm for $a_{0}$ and
$r_{0}$, respectively. Adopting the ANCs quoted from a recent measurement of
Kiss et al. Kiss  _et al._ (2020) yields a change of +1.4 fm and +0.01 fm for
$a_{0}$ and $r_{0}$.

We also studied the sensitivity of the scattering parameters to the choice of
data sets and the energy range of the data set. The SONIK and Barnard et al.
data sets are represented by S and B, respectively. The sensitivity of the
scattering parameters excluding the data above $E$[3He]$<$4 MeV was studied;
the analyses using these energy-truncated data sets are represented by the
superscript $(t)$. The SB+ model is a superset of SB. In the SB analysis, the
background 1/2+ level is fixed at $E_{x}^{(1/2+)}=14$ MeV. With SB+, we allow
that parameter to vary between 2 and 20 MeV.

The results for $a_{0}$ and $r_{0}$ for several different $R$-matrix analyses
and the EFT analysis described in this work are depicted in Fig. 14 and
summarized in Table 6. Fig. 14 reveals several interesting points. First, of
the seven different data models studied with $R$-matrix theory, six of them
exhibit the same $a_{0}$-$r_{0}$ correlation. Only the SB+ model breaks this
consistency. The additional freedom in the 1/2+ channel changes the
correlation between $a_{0}$ and $r_{0}$ entirely. Second, the EFT analysis
displays a very different different correlation structure from all of the
$R$-matrix analyses. Finally, none of the truncated-data analyses encapsulate
their associated complete-data analysis. As more data is included, one expects
a refinement of the previous result. In this case, it is particularly striking
that the inclusion of higher energy data significantly changes the extracted
low-energy scattering parameters.

Figure 14: (Color online) Scattering length and effective range posterior
probability distributions from each of the seven $R$-matrix analyses and the
EFT analysis.

Data Model |  $a_{0}$ (fm) |  $r_{0}$ (fm)
---|---|---
SB | $33.10_{-0.13}^{+0.13}$ | $1.009_{-0.001}^{+0.001}$
S | $36.67_{-0.36}^{+0.37}$ | $1.033_{-0.002}^{+0.002}$
B | $32.97_{-0.15}^{+0.16}$ | $1.009_{-0.001}^{+0.001}$
SB(t) | $36.36_{-0.60}^{+0.61}$ | $1.031_{-0.004}^{+0.004}$
S(t) | $40.10_{-0.74}^{+0.64}$ | $1.050_{-0.003}^{+0.003}$
B(t) | $30.90_{-0.96}^{+0.95}$ | $0.993_{-0.008}^{+0.007}$
SB+ | $35.82_{-0.13}^{+0.13}$ | $1.098_{-0.008}^{+0.008}$
Halo EFT | $41.89_{-0.89}^{+0.90}$ | $0.994_{-0.005}^{+0.006}$
Table 6: Extracted scattering lengths and effective ranges— with
uncertainties— from eight different data analyses. The first seven invoke
different choices of $R$-matrix parameters and/or different data sets. The
eighth is the analysis using Halo EFT that is described in the next section.

The $R$-matrix result for $a_{0}$, and $r_{0}$ is presented in the last line
of Table 7. It is obtained from a simultaneous $R$-matrix fit of all data from
Barnard et al. and the data of this work. This model, termed as SB, is our
preferred model. The lower and upper limits in the $R$-matrix-extracted
$a_{0}$ and $r_{0}$ values determined from the sensitivity studies listed in
Table 6 and the variation of the ANCs are accounted for through an additional
“analysis error.” This error is estimated as ${}^{+7.5}_{-3.0}$ fm, and
${}^{+0.096}_{-0.023}$ fm, respectively.

$s$-wave scattering parameters published in the literature are also presented
in Table 7. J. Dohet Eraly et al. Dohet-Eraly  _et al._ (2016) used the chiral
nucleon–nucleon interaction within the ab initio no-core shell model with
continuum (NCSMC) to calculate the 3He(4He, $\gamma$)7Be astrophysical
$S$-factor and deduced the $s$-wave scattering length. The scattering
parameters have also been calculated using a microscopic cluster model Kamouni
and Baye (2007). The scattering parameters for the 3He+4He system have been
extracted from a Bayesian analysis of the capture data below 2 MeV that used
Halo EFT Zhang  _et al._ (2020a). Premarathna and Rupak also performed a
Bayesian analysis of the capture data and the phase shifts from Boykin et al
Boykin  _et al._ (1972) to infer the scattering parameters Premarathna and
Rupak (2020). The $R$-matrix fit to the SONIK and Barnard data yields an
$s$-wave scattering length and $s$-wave effective range in fair agreement
(within 1.5$\sigma$) with all but one of these values previously published in
the literature, provided the dispersion of $a_{0}$, and $r_{0}$ values with
respect to the choice and energy range of data sets included in the analysis
is considered. The exception is the NSCMC calculation of Ref. Dohet-Eraly  _et
al._ (2016) which obtained a much smaller scattering length than was found in
any of the data analyses or in the microscopic cluster model.

Table 7: $s$-wave scattering parameters for the 3He+4He system. $a_{0}$ (fm) |  $r_{0}$ (fm) | Method | Reference
---|---|---|---
7.7 | - | NCSMC | Dohet-Eraly  _et al._ (2016)
41.06 | 1.01 | Microscopic | Kamouni and Baye (2007)
|  | Cluster Model |
40${}^{+5}_{-6}$ | 1.09${}^{+0.09}_{-0.1}$ | Halo EFT | Premarathna and Rupak (2020)
$50^{+7}_{-6}$ | 0.97$\pm$ 0.03 | Halo EFT | Zhang  _et al._ (2020a)
$42_{-1}^{+1}$ | $0.994^{+0.006}_{-0.005}$ | ${\rm Halo~{}EFT}$ | This Work
$33.10\pm 0.13(\text{stat})^{+7.5}_{-3}(\text{analysis})$ | $1.009\pm 0.001(\text{stat})^{+0.096}_{-0.023}(\text{analysis})$ |  $R$-matrix | This Work

The results for the different normalization factors applied to the SONIK data
in the SB analysis are summarized in Fig. 15. The results from the $R$-matrix
(blue) and EFT (green) analyses of the SB data model are in good agreement at
low energies (small values of the data point index). While significant effort
was put into accounting for detector-specific systematic effects, the overall
result, $\tilde{c}_{i,j}$ clearly shows that the energy-dependent systematics
dominate in both analyses. While the EFT normalization factors tend toward
lower values than the $R$-matrix ones as the energy increases (larger data
point indices), the “bunching” of $\tilde{c}_{i,j}$ with respect to the energy
bins — indicated by vertical, colored bands — is consistent between both
theories. These results are only for the SONIK data. The Barnard data set
lacked the necessary uncertainty information to apply such a detailed
treatment of its systematic error.

Figure 15: Normalization factor results at each SONIK data point are shown as
a product of the three different systematic effects described in subsection
V.1. The results are shown in increasing energy (from left to right) as a
function of data point index. Each vertical, shaded region corresponds to a
different energy bin. $R$-matrix results are shown as blue circles with error
bars. EFT results are shown as green squares with error bars.

###  VII.2 Results from Halo EFT

Table 7 also includes the $s$-wave scattering parameters from a NNLO Halo EFT
analysis of the same data set as was used in the $R$-matrix analysis. This fit
reproduces the SONIK data well, especially for c.m. energies below 2 MeV. The
band of cross-sections generated from the posterior samples obtained from
sampling the EFT likelihood, Eq. (32), for each SONIK energy bin and Barnard
angular bin are shown respectively in Figures 10, 11, 12, and 13. The
$\chi^{2}_{\rm EFT}$ values obtained from this analysis for each energy bin
are provided in Table VII.1. Note that $\chi^{2}_{\rm EFT}$ is generically
less than the standard $\chi^{2}$, because it includes a theory-error piece of
the covariance matrix (see Eq. 28). For comparison the total $\chi^{2}$ for
the EFT fit to the SONIK data is 2165, as compared to $\chi^{2}_{\rm
EFT}=1324$. Most of the difference between $\chi^{2}_{\rm EFT}$ and the
standard $\chi^{2}$ accumulates above $E~{}[{}^{3}{\rm He}]=1.767$ MeV.

The EFT analysis also accumulates large $\chi^{2}_{\rm EFT}$ in the higher
SONIK energy bins (especially at backward angles) and for the portion of the
Barnard data in and beyond the ${\frac{7}{2}}^{-}$ resonance. For the latter
data set the largest $\chi^{2}$ contribution comes from the $140.80^{\circ}$
bin. The large value of $\chi^{2}$ around the ${\frac{7}{2}}^{-}$ resonance,
especially at backward angles, suggests that the approach of Ref. Poudel and
Phillips (2022) does not adequately describe that resonance. (In fact, the
analysis of Ref. Poudel and Phillips (2022) did not include the data in the
highest SONIK energy bin, because the EFT is not tailored to that region.). If
we choose to sample the width of the resonance, $\Gamma^{{\frac{7}{2}}^{-}}$,
in the EFT calculation we obtain $151$ keV, not the $159$ keV used to produce
the results presented here. The sizable $\chi^{2}_{\rm EFT}$ in the EFT fit in
the vicinity of the resonance could likely be improved by better parameter
estimation or a better model of the resonance. The inference of the $s$-wave
parameters is also surprisingly sensitive to the description of the
${\frac{5}{2}}^{-}$ phase shift. If the EFT analysis is performed with
$\Gamma^{{\frac{5}{2}}^{-}}=1.9$ MeV a larger $a_{0}$ (and a smaller $r_{0}$)
are obtained. The effect of such changes in the $f$-wave phase shifts on the
$s$-wave ERPs is not quantified in the uncertainties we provide here. The
median values of the posterior samples for effective range parameters (ERPs)
from this analysis are as follows: $a_{0}=42^{+1}_{-1}~{}{\rm
fm},r_{0}=0.994^{+0.006}_{-0.005}~{}{\rm
fm},P_{1^{+}}=1.681^{+0.005}_{-0.005}~{}{\rm
fm},P_{1^{-}}=1.810^{+0.009}_{-0.010}~{}{\rm fm}$. This analysis finds that
$a_{0}$ and $r_{0}$ are anti-correlated, with a correlation coefficient of
-0.92.

Returning to Table 7, the central value of $a_{0}$ from the Halo EFT analysis
of SONIK and Barnard data is very different from that predicted in the NCSMC
(ab initio) calculation of Ref. Dohet-Eraly  _et al._ (2016) while consistent
with the result of the microscopic calculation of Ref. Kamouni and Baye
(2007). Compared to the other Halo EFT analyses of data listed in Table 7, the
68% interval for $a_{0}$ found in this analysis completely falls within the
distribution of $a_{0}$ obtained in Ref. Premarathna and Rupak (2020). But our
68% $a_{0}$ interval does not overlap the one obtained from
3He($\alpha$,$\gamma$) data in Ref. Zhang  _et al._ (2020a). Meanwhile, the
$r_{0}$ values are consistent between the microscopic prediction of Ref.
Kamouni and Baye (2007) and the Halo EFT analyses of Refs. Premarathna and
Rupak (2020); Zhang  _et al._ (2020a). Also, the $r_{0}$ value reported here
is consistent with the natural scale of $\frac{1}{\Lambda}$ assigned by EFT.

##  VIII CONCLUSIONS

Figure 16: Top: The scattering phase shifts for $\ell=0$ are shown in
comparison to the analyses in Ref. Boykin  _et al._ (1972) and Ref. Spiger and
Tombrello (1967). The solid, blue line represents the median calculated in the
SB $R$-matrix analysis. The dashed, green line represents the median
calculated in the EFT analysis. The red, shaded region indicates the energy
range over which the SONIK measurements were carried out. White squares with
error bars and grey x’s indicate the analyses of Boykin  _et al._ (1972) and
Spiger and Tombrello (1967), respectively. Bottom: The effective range
function, $K(E)$, is plotted as a function of the center of mass energy.

The elastic scattering reaction 4He(3He,3He)4He was measured at 9 different
energies from $E_{\text{c.m.}}=0.38-3.13$ MeV. This data set includes the
first measurement of elastic scattering in the 3He+4He system below
$E_{\text{c.m.}}=500$ keV. The angular range covered is
$30^{\circ}<\theta_{\text{c.m.}}<139^{\circ}$, a wider range than in previous
measurements. This elastic scattering measurement of 4He(3He,3He)4He is the
first scientific measurement made using SONIK. Its success validates the use
of SONIK for charged particle scattering measurements. The resulting data is
presented in this paper, together with detailed error estimates which are
lacking in previous measurements of 3He + 4He elastic scattering. They are
consistent with previous experimental measurements and have better precision.

The extraction of $s$-wave effective-range parameters for the 3He+4He system
from these data was carried out using both an $R$-matrix and a Halo EFT
analysis. We used the Bayesian $R$-matrix Inference Code Kit BRICK Odell  _et
al._ (2021) to calibrate the $R$-matrix model against the data of this work
and the elastic scattering data of Ref. Barnard  _et al._ (1964). The
$R$-matrix parameter posteriors were then employed to calculate the $s$-wave
scattering length and effective range, fully propagating the model
uncertainties to these extracted quantities. This yields $a_{0}=33.10\pm
0.13~{}(\text{stat})^{+7.5}_{-3}(\text{analysis})$ fm. The same combined SONIK
+ Barnard scattering data set was analyzed using Halo EFT at NNLO—also with
full uncertainty quantification. The result $a_{0}=42\pm 1$ fm is obtained in
that approach. The two analyses thus yield discrepant values for $a_{0}$, with
a concomitant discrepancy in their results for $r_{0}$. The $s$-wave
scattering length from the $R$-matrix analysis is in fair agreement with the
prediction of the microscopic cluster model Kamouni and Baye (2007), and
previous inferences from data using Halo EFT Premarathna and Rupak (2020);
Zhang  _et al._ (2020a).

The discrepancies in the inferred $a_{0}$ and $r_{0}$ values naturally suggest
an examination of the $s$-wave phase shifts. The $s$-wave phase shifts from
the $R$-matrix and Halo EFT analyses are compared in the upper panel of Fig.
16. The two analyses agree with each other over most of the energy range of
the SONIK data. At the energies of the Boykin et al. Boykin  _et al._ (1972)
phase shifts both analyses yield lower phase shifts than were reported in that
work. The phase shifts inferred using $R$-matrix and Halo EFT begin to diverge
a little at the upper end of the SONIK energy range. This is related to the
fact that, at slightly higher energies, the $R$-matrix analysis describes the
phase shifts from Ref. Spiger and Tombrello (1967) much better than the Halo
EFT result does. These differences in the phase shift at high energy then
affect the behavior at low energy, as is evident from the $s$-wave effective
range function shown in the lower panel of Fig. 16.

The EFT of Ref. Poudel and Phillips (2022) that was constructed to describe
the elastic scattering reaction measured in this experiment breaks down at
backward angles for the higher-energy data bins. In Ref. Poudel and Phillips
(2022) Poudel and Phillips attempted to mitigate this via phenomenological
inclusion of the $7/2-$ resonance. Even though that was done here too, the
$\chi^{2}$ of the EFT fit to data is strikingly large for $E_{\rm c.m.}>2$
MeV, in spite of the addition of a theory component of the errors in the
$\chi^{2}$. Future work to build a better EFT description through the $7/2-$
resonance (cf. Ref. Higa  _et al._ (2022) for the case of the 7Be-proton
system) is needed. The quality of the $R$-matrix fit also deteriorates in the
final energy bin of the data set described here, and the value of $a_{0}$
inferred is higher if only data below $E_{\rm c.m.}=2.2$ MeV is used, see Fig.
14. Future studies should address whether Halo EFT and $R$-matrix agree if
only low-energy data is used.

A better measurement of elastic 3He-4He scattering in the vicinity of the 7/2-
resonance, as well as an accurate determination of the position of this
resonance and its $\alpha$-width may also help resolve the discrepancy between
the Halo EFT and $R$-matrix analyses of the SONIK data. The inference of
$s$-wave parameters is surprisingly sensitive to the description of the 5/2-
phase shift and the width of 5/2- level. This issue could be explored further
by a similar analysis with the addition of scattering data above the proton
separation energy in 7Be.

Ultimately, smaller uncertainties in the scattering parameters will reduce the
overall uncertainty in $S_{34}(0)$, just as smaller uncertainties in the
$s$-wave scattering lengths for 7Be+$p$ from Ref. Paneru  _et al._ (2019) led
to reduced uncertainty in $S_{17}(0)$ presented in Ref. Higa  _et al._ (2020).
The data from this measurement should be used in global $R$-matrix and Halo
EFT analyses of 4He(3He,$\gamma$)7Be data, in order to resolve disagreements
between previous analyses regarding $S_{34}(0)$ and reduce the extrapolation
error therein.

## Acknowledgements

We are grateful to R. J. deBoer for his expertise on ${}^{3}{\rm
He}+{}^{4}{\rm He}$ scattering and reactions and assistance with Azure2. We
are indebted to Keerthi Jayamana and the TRIUMF OLIS staff for providing the
intense and pure 3He beam, and we are grateful to the TRIUMF operations staff
whose hard work made the experiment possible. This work was supported in part
by the U.S. Department of Energy under Grants No. DE-NA0003883, and No. DE-
FG02-93ER40789. The Canadian authors are supported by the Natural Sciences and
Engineering Research Council of Canada under grants: SAPPJ-2016-00029 and
SAPPJ-2019-00039. TRIUMF receives federal funding through a contribution
agreement with the National Research Council of Canada. This research was
supported in part by the Notre Dame Center for Research Computing.

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