# Experimental studies of the three nucleon system dynamics in the proton
induced deuteron breakup at 108 MeV

A. Łobejko1,2, E. Stephan1, I. Ciepał3, J. Golak4, N. Kalantar–Nayestanaki5,
S. Kistryn4, B. Kłos1, A. Kozela3, P. Kulessa3, W. Parol3, R. Skibiński4, I.
Skwira-Chalot6, A. Szadziński1, A. Wilczek1, H. Witała4, B. Włoch3,7, J.
Zejma4 1Institute of Physics, University of Silesia, PL-41500 Chorzów, Poland
2Institute of Experimental Physics, Faculty of Mathematics, Physics and
informatics, University of Gdańsk, PL-80308 Gdańsk, Poland
3Institute of Nuclear Physics, PAS, PL-31342 Kraków, Poland
4Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian
University, PL-30348 Kraków, Poland
5ESRIG, University of Groningen, 9747 AA, Groningen, The Netherlands
6Faculty of Physics, University of Warsaw, PL-02093 Warszawa, Poland
7Université de Bordeaux, CNRS, LP2I Bordeaux, FR-33170 Gradignan, France

###### Abstract

The differential cross sections for the 2H(p,pp)n reaction have been measured
for 84 angular configurations of the outgoing protons in the range of polar
angles from 13 to 33 degrees with a proton beam of 108 MeV. Data have been
collected in the Cyclotron Center Bronowice (CCB) at the Institute of Nuclear
Physics PAS in Cracow, using the BINA detector setup. Analysis leading to
determination of the breakup cross section values is described. Absolute
normalization is obtained by normalization to the simultaneously measured
2H(p,d)p scattering events.Experimental results are compared to the state-of-
the-art theoretical calculations. Global analysis shows significant influence
of the Coulomb interaction and small effects of three nucleon force in the
studied phase space region.

††preprint: APS/123-QED

##  I Introduction

Scattering experiments in systems of three nucleons provide particularly rich
and sensitive data for testing the state-of-the-art potentials of nuclear
interactions, such as realistic potentials or potentials stemming from Chiral
Effective Field Theory (ChEFT). In view of the neglect of the nucleons’
internal degrees of freedom, applying the realistic potentials to describe
systems beyond two nucleons requires complementing them with many-body forces,
in particular a so-called 3-nucleon force (3NF). Several models of such an
interaction have emerged [1, 2, 3, 4]. Alternatively, the explicit
$\Delta$-isobar has been introduced in the coupled channel approach,
generating 3NF in systems of three and more nucleons [5, 6]. In ChEFT [7], the
3NF naturally appears in the next-to-next-to-leading order (NNLO). The 3NF
proved essential for the proper description of the nuclear matter, among
others, for exact calculation of binding energies and low-lying excited states
of light nuclei, correct determination of the neutron drip-line and saturation
in nuclear matter [8, 9, 10, 11, 12].

In the proton-deuteron elastic scattering the importance of 3NF contribution
to the differential cross section grows fast with energy [13]. The residual
discrepancy between data and theory, which is significantly reduced when 3NF
is taken into account [14], also shows a systematic increase with rising
center-of-mass energy. The first attempts at relativistic calculations predict
influence on the elastic scattering cross section much too small to be
responsible for the observed discrepancy and playing a role in different
kinematic regions [15, 16, 17]. The impact of Coulomb interaction on elastic
scattering cross section at medium energies is very small.

The deuteron breakup in collision with a proton is very interesting as
complementary field of studies, since the three-body final state is rich in
kinematic configurations revealing diverse sensitivity to specific effects.
The sensitivity to 3NF was shown for a deuteron beam energy of 65 MeV/nucleon
and 80 MeV/nucleon [18, 19, 20]. Moreover, a predicted huge influence of the
Coulomb interaction [21, 22] was experimentally confirmed [23, 24, 25], even
at energies as high as 170 MeV/nucleon [26]. According to the theoretical
prediction, the Coulomb effects barely depend on beam energy, and they are
very significant for $pp$ pair final state interaction (FSI). The cross-
section’s description is improved by extending theoretical models with both
3NF and Coulomb interaction [27]. For deuteron beam energy of 130 MeV (65
MeV/nucleon), the improvement of results in the whole phase space of a
detector was observed [24].

Both reaction channels in proton-deuteron collisions: elastic scattering and
deuteron breakup, are not only a validation field for the quality of nuclear
interaction models, but have been considered for constraining the parameters
of ChEFT potentials [28, 29]. So far, there are no calculations including
simultaneously all the dynamic ingredients (3NF, Coulomb interactions,
relativistic effects), therefore the data base covering large reaction phase
space and wide range of beam energies is necessary to single out the
particular effects in the range of their maximum visibility. Large detectors
combined with modern data acquisition systems facilitate experiments covering
large phase space regions. The Cyclotron Center Bronowice (CCB) at Institute
of Nuclear Physics PAS, Kraków, provides unique opportunity to continue the
studies of few-nucleon systems with the use of proton beams in the energy
range of interest for this research, i.e. high enough to expect a significant
effect of the 3NF force, and below the threshold for pion production. At 108
MeV, there exist data for the elastic scattering, in good agreement with the
calculations including 3NF [30]. The main goal was to determine the
differential cross section for the breakup reaction normalized relatively to
the elastic scattering.

##  II Experiment

The experiment was carried out at CCB with the unpolarized proton beam
provided by isochronous cyclotron Proteus C-235. Protons at the kinetic energy
of 108 MeV were impinging on the liquid deuterium ($LD_{2}$) target
111https://experimentsccb.ifj.edu.pl. The reaction products were measured
using the BINA (Big Instrument for Nuclear polarization Analysis) detection
system, designed to study the few-nucleon reactions at intermediate energies.

BINA was previously used for such measurements at KVI, Groningen, The
Netherlands, by a Dutch-Polish collaboration, providing a rich database for
proton-deuteron and deuteron-deuteron collisions [24, 32, 33, 34, 35, 36, 20].
In 2012, the BINA setup was installed at CCB. In addition to mechanical
adaptations and a review of the MWPC electronics etc., the new slow control
system was introduced and the decision was made to cut the two middle bars of
the $E$-detector in half.

The BINA detector (see Fig. 1) consist of two parts: a forward Wall and a
central-backward part - Ball. It allows measurement of charged particle
coincidences over a wide angular range, providing access to an almost full
phase-space of few-body scattering reaction. During the first experiment at
CCB, dedicated also to the detector commissioning on the new beam line, only
the Wall detectors worked optimally, so this paper presents data which were
collected in the forward Wall.

Figure 1: A schematic side view of the BINA Detection System. The figure is
adopted from [37]

.

Figure 2: Projection of 24 vertical $\Delta E$ strips and 10 horizontal $E$
bars on the plane perpendicular to the beam axis. The polar angles ($\theta$)
are represented by circles. An inner dashed-line circle represents the beam
pipe. Numbering of detector elements and quarters is given. A rectangle marked
with a dotted line shows approximate limits of the MWPC acceptance.

The Wall part consists of a Multi-Wire Proportional Chamber (MWPC) and two
layers of scintillator hodoscopes forming a two-dimensional array of $\Delta
E-E$ telescopes for particle identification and setting the trigger conditions
(Fig. 2). Particles scattered in the forward direction were measured with the
polar angles $\theta$ between $13^{\circ}-28^{\circ}$ in a whole azimuthal
angle $\varphi$. Additionally, the polar angle was extended up to $35^{\circ}$
at the corners of a square-shaped MWPC region with only limited coverage of
the azimuthal angles. The exact values of the angular acceptance varied
slightly between experimental runs due to certain freedom of relative
positioning of the Wall-Ball system. The MWPC detector consists of three anode
planes with wires oriented vertically (X), horizontally (Y) and diagonally
(U). The first layer of the scintillator hodoscopes consists of 24 vertically-
placed thin transmission-$\Delta E$ strips. The second one ($E$ detector) is
composed of 10 horizontally-placed thick stopping-$E$ bars read from both ends
by PMT’s. They are forming an arch thus reducing the contribution of so-called
"cross-over" events - particles producing a detectable signal simultaneously
in two adjacent bars. Two central scintillators of $E$ detector (No 5 and 6 in
Fig. 2) were shaped to make possible passage of the beam pipe, such that only
a narrow linking between two sides remained. As mentioned earlier, the
decision was taken to cut and optically isolate left and right parts of these
detectors.

The central-backward part of the detector serves also as a vacuum chamber
where the target system is located. During the experiment, we used two kinds
of targets. The first one, an aluminum target (Al) with a thin zinc sulfide
($ZnS$) layer, was prepared to optimize the beam position and was also used
during the calibration runs. The second, the liquid deuterium target
($LD_{2}$), was for studying the reactions of interest.

The trigger system was designed to detect two kinds of processes: the elastic
scattering registered as a single charged particle and the breakup reaction -
coincidences of two protons. The individual prescaling factors were applied to
obtain representative samples of both event types. For more information on the
electronics, readout and data acquisition systems, see Ref. [38]. The triggers
were determined by logic conditions based on: signals from $\Delta E$
detectors, left PMT’s of $E$ detector ($E_{L}$), and right PMT’s of $E$
detector ($E_{R}$). The logic conditions of the triggers are presented in
Table 1. Trigger T1 was a minimum bias trigger, with at least one charged
particle in the whole detection setup (single particle). The T2 trigger was
based on the multiplicity hits, and it was used to select the coincidences of
two particles registered in the Wall detector (Wall-Wall coincidence).

Table 1: The logic conditions of the triggers. MULT denotes condition on hit multiplicity. |  |  Down-
---|---|---
Symbol |  Logic notation |  scaling factor
T1 |  (OR $E_{L}$) OR (OR $E_{R}$) OR (OR $\Delta E$) |  $2^{4}$
T2 |  (MULT $E_{L}\geq 2$) OR (MULT $E_{R}\geq 2$) |  $2^{0}$

##  III Data analysis

Data analysis started with selection of events time-correlated with the
trigger and identification of reaction channels of interest. The next steps
included energy calibration, determination of detection efficiency and
determination of luminosity for the cross section normalization.

###  III.1 Event selection and reconstruction

The track reconstruction was based on hits in the MWPC planes. The first step
of the track reconstruction procedure (described in details in Ref. [20]) was
to project the coordinates of hits in X and U planes onto a central Y plane.
The point of passage of the particle was determined as the center of the
circle inscribed in the right angled triangle formed by the hit wires (see
Fig. 3). The hits were treated as correlated (produced by the same particle)
when the distance $d$ between the XY intersection and the active wire U was
less than 7 mm. Assuming the reaction point in the target center, we
calculated polar ($\theta$) and azimuthal ($\varphi$) scattering angles in the
laboratory frame.

Figure 3: A graphic represents three hit wires X, Y, and U, which are the
basis of the track reconstruction procedure. Center of the circle inscribed in
the XYU triangle represents position assigned to the particle track.

The analysis focused only on tracks with MWPC hits matched with the $\Delta E$
and $E$ hodoscopes. The particle identification (PID) of both particle types
was based on the $\Delta E-E$ technique and an example of such a
identification spectrum is presented in Fig. 4. Proton and deuteron
distributions can be well distinguished, thus simple graphical cuts (red and
black contours in Fig. 4) were implemented. The gates were wide enough to
avoid a significant loss of particles, and a slight overlap of them was
allowed, see Fig. 4.

Figure 4: Two-dimensional spectrum built for one $\Delta E-E$ telescope
($\Delta E=2$, $E=0$); good separation of protons (lower band; red gate) and
deuterons (upper spot, black gate) is visible.

In order to perform the energy calibration, dedicated runs were carried out
with proton beams at energies of 70, 83, 97, 108 and 120 MeV impinging on an
aluminum target. The registered particles were sorted by the side s (s = right
/ left), the detector number n (n = 1, 2, …, 10), and polar angle $\theta$
binned by 4 degrees. The signals $c_{1}$ and $c_{2}$ registered by
photomultiplier tubes on both ends of each $E$ detector were combined in the
geometrical average ($C=\sqrt{c_{1}\cdot c_{2}}$) in order to suppress the
effects of light attenuation along the bar. Due to the cutting of the two
central scintillators, in their case the geometrical average was replaced by a
signal of one side ($C=c_{1}$ or $C=c_{2}$). The spectra of protons scattered
on the Al target, were analyzed for a chosen set of parameters: beam energies,
$s,~{}n$ and $\theta$. On the basis of the Monte Carlo simulation performed in
Geant4, the actual energy deposited by protons, $E_{dep}$, was determined. The
relation between $C$ and $E_{dep}$ is described by linear calibration above
approximately 50 MeV. Below this value, the light quenching effect is expected
due to the large stopping power [39, 40]. The calibration curve fitted to the
entire range of $C$ is described by the non-linear equation [41]:

$E_{dep}\ (s,n,\theta)=a\ (s,n,\theta)\cdot C+b\ (s,n,\theta)\cdot\sqrt{C};$
(1)

where: $a\ (s,n,\theta)$ and $b\ (s,n,\theta)$ denote the fit parameters.

Having reconstructed particle trajectory, a series of simulations were
performed to determine the particle’s energy at the point of reaction.
Finally, we obtained the initial ($E_{initial}$) versus deposited energy
($E_{dep}$) spectra for each angular bin, for details see Ref. [41].

###  III.2 Corrections for detection efficiency and hadronic interactions

In the case of elastic scattering, a single particle was recorded in the Wall.
Therefore, $1$-dimensional efficiency as a function of the $\theta$ angle was
used. The breakup reaction is reconstructed from coincidences of two protons
and thus characterized by four non-separable variables: two polar and two
azimuthal angles. Consequently, a $2$-dimensional ($\theta$, $\varphi$)
efficiency map was constructed for this purpose and employed for the event-by-
event correction. In addition, we considered the configuration efficiency of
the system related to the geometrical acceptance for registering two
coincident protons in separate detector elements. The total efficiency is
defined as a product:

$\varepsilon_{total}=\varepsilon_{MWPC}\cdot\varepsilon_{\Delta
E}\cdot\varepsilon_{E}\cdot{\varepsilon}_{conf},$ (2)

where: $\varepsilon_{MWPC},\ \varepsilon_{\Delta E},\ \varepsilon_{E},\
\varepsilon_{conf}$ are the efficiencies of the Multiwire Proportional
Chamber, $\Delta E$, $E$ detectors, and configurational efficiency (only in
the case of the breakup reaction), respectively. To avoid ambiguity, only
single-particle events recorded in the $\Delta E$ and $E$ hodoscopes were
considered to construct the map of MWPC efficiency. The MWPC efficiency, see
Fig. 5, upper panel, is a product of efficiencies of all its three planes:
$\varepsilon_{MWPC}\ (\theta,\varphi)=\varepsilon_{x}\
(\theta,\varphi)\cdot\varepsilon_{y}\ (\theta,\varphi)\cdot\varepsilon_{u}\
(\theta,\varphi)$, calculated as follows:

$\varepsilon_{x}\ (\theta,\varphi)=\frac{N_{xyu}\ (\theta,\varphi)}{N_{yu}\
(\theta,\varphi)}$ (3)

where:

  * •

$N_{xyu}\ (\theta,\varphi)$ is a number of counts registered in the selected
angular segment, satisfying the condition of one hit (or one cluster of hits)
per the $X$, $Y$, and $U$ planes;

  * •

$N_{yu}\ (\theta,\varphi)$ is a number of counts registered in the same
angular segment with at least one hit (or cluster) per the $Y$ and $U$ planes.

The $\Delta E$ efficiency (see Fig. 5, lower panel) was calculated similarly,
by counting the ratio of the number ($N_{\Delta E}$) of particles
reconstructed based on signals from MWPC, $\Delta E$ and $E$, to those
($N_{all}$) registered regardless of the presence of information from $\Delta
E$ :

$\varepsilon_{\Delta E}\ (\theta,\varphi)=\frac{N_{\Delta E}\
(\theta,\varphi)}{N_{all}\ (\theta,\varphi)}$ (4)

Figure 5: The $2$-dimensional efficiency maps for MWPC (upper panel) and
$\Delta E$ detector (lower panel).

Scintillator elements of the $E$ detector are thick enough to stop all the
charged particles emitted at the beam energy of 108 MeV. The gaps between the
scintillator blocks have the thickness of the foil wrapped around the
detectors and their influence on efficiency was considered to be negligible.

The proper registration of two charged particles in the BINA Wall required two
reconstructed hits in all the MWPC planes and signals from two various
elements of $\Delta E$ detector, and two elements of the $E$ detector. The
probability of losses due to two protons impinging the same detector element
strongly depends on the angular configuration. There were also event losses
due to inefficiency of the coincidence trigger. The coincidence trigger (T2)
condition described in Table. 1 requires two signals in $E$ detector’s left or
right side. Since the central detectors were cut in the middle, the PMT’s
signal from those detectors was always registered from one (left or right)
side only and for certain event geometries the trigger condition was not
fulfilled. All these effects were accounted by the total configurational
efficiency obtained by analysis of Geant4 Monte Carlo simulations. The
efficiency was calculated for each analyzed breakup configuration as a ratio
of the number of events with complete information from all the detectors and
the geometry compatible with the T2 trigger condition ($N_{g}$) to the total
number of simulated events ($N_{all}$):

$\bm{\varepsilon}_{conf}(\theta_{1},\theta_{2},\varphi_{12})=\frac{N_{g}(\Delta
E,E)}{N_{all}(\Delta E,E)}.$ (5)

Three sets of results obtained for one chosen combination of polar angles
$\theta_{1}$ and $\theta_{2}$ are presented in function of $\varphi_{12}$ in
Fig. 6. As expected, the configurations with the smallest relative angles
$\varphi_{12}$ are particularly prone to configurational efficiency losses.
The coplanar ones ($\varphi_{12}=180^{\circ}$) are affected by the trigger
inefficiency.

Figure 6:  Configurational efficiency for selected $\theta_{1}$, $\theta_{2}$
and all $\varphi_{12}$, obtained in simulation. The blue squares and green
triangles refer to the efficiency calculated separately for $E$ and $\Delta
E$. The red points represent the total configurational efficiency for the BINA
detection setup.

A set of simulations was carried out to calculate the fraction of particles
undergoing hadronic interaction [33]. For deuterons, losses due to hadronic
interactions have also been estimated from the experimental data [42, 41]. The
difference between the measured and simulated contribution of hadronic
interactions is of around 12% relatively, corresponding to a 2% systematic
error of the integrated luminosity.

###  III.3 Determination of integrated luminosity

The integrated luminosity was calculated based on the number of the
elastically scattered particles $N^{el}(\theta)$, and the known elastic
scattering cross section $\sigma\ ^{el}_{LAB}\ (\theta)$ [30]. Elastically
scattered particles, protons or deuterons, were registered in the forward Wall
as single tracks since their coincident particles were emitted towards Ball.
In the proton energy spectra, elastically scattered protons were contaminated
with protons from the breakup reaction, and there was also small contribution
of protons scattered from the target frame. Therefore, the luminosity was
determined based on the analysis of much cleaner deuteron spectra.

Figure 7:  Spectra of deuterons registered at $\theta=20^{\circ}$ obtained
with two PID cuts: a broad gate (black line) and a narrow gate (red line);
blue line represents a difference of these spectra.

Deuterons were sorted according to their polar angles. The energy spectrum of
deuterons elastically scattered at $\theta=20^{\circ}\pm 0.5^{\circ}$ is
presented in Fig. 7. Narrowing the deuteron PID gate (so as to eliminate its
overlap with the proton gate, see Fig. 4) we observed a significant reduction
of the background, thus the protons were the main contributor. This is
confirmed by the presence of a peak in the difference between the spectra for
the wider and narrower gates (blue line in Fig. 7), which extends above the
deuteron energies. After subtracting the background approximated by straight
lines, the sums under the peaks for both gate widths are very similar, with
slightly larger values for the wide gate. Finally, for the data analysis, a
wide gate was chosen to avoid deuteron losses. The resulting $N^{el}(\theta)$
values were corrected for the detection efficiency
$\varepsilon_{total}(\theta)$, hadronic interactions and a solid angle
$\Delta\Omega$. Due to the square shape of our detection setup, starting from
a certain theta angle the limited azimuthal acceptance should be taken into
account. For the two largest analysed polar angles, limits on the MWPC
corners, $\varphi=45^{\circ}\pm 10^{\circ},135^{\circ}\pm
10^{\circ},225^{\circ}\pm 10^{\circ},315^{\circ}\pm 10^{\circ}$, were applied.

We normalized our data to the differential cross section data for pd elastic
scattering at 108 MeV that were measured at KVI with the use of the Big Bite
Spectrometer [30]. The cross-section error given in Ref. [30] ranges from
$4.4\%$ for the highest $\theta$ angles to $6.5\%$ for the lower ones. The
second source of systematic uncertainty is related to our measurement and
analysis. As a measure of it, we took the variations of the count’s rates in
the four quarters of the detector [41]. The total systematic uncertainty
varies between $6.8\%$ and 15.4% depending on the $\theta$ angle. The
statistical errors are minor and range between $0.78-0.85\%$. Fig. 8 shows the
resulting luminosity values with their systematic uncertainties.

Figure 8: The luminosity value obtained for the set of $\theta$ angles,
calculated as a mean weighted by a systematic error represented by the hatched
boxes (more details in text). The statistical errors are smaller than the
point’s size.

The average luminosity is shown in Fig. 8 as a red horizontal line. The points
in the $\theta$ range between $14^{\circ}-29^{\circ}$ are consistent within
the systematic uncertainty represented by the black dashed lines, while the
distance of the last point corresponding to $\theta=32^{\circ}$ goes beyond
the range of error. It is on the limit of the geometrical acceptance of the
Wall detector, but the acceptance losses have been taken into account, thus
indicating small shape difference between our results and the KVI data [30].
The influence of this point on the average value is insignificant (within
systematic uncertainty of the average).

##  IV Determination of experimental differential cross section

The differential cross section for the 2H(p,pp)n reaction was analyzed for a
set of kinematic configurations within the angular acceptance of the Wall. The
breakup reaction kinematics is determined by the momenta of protons
$\vec{p_{1}}$ and $\vec{p_{2}}$. The geometrical configuration is defined by
polar angles $\theta_{1},\theta_{2}$ of the momentum vectors and their
relative azimuthal angle $\varphi_{12}$. The events identified as proton-
proton coincidences were corrected by efficiency factors and sorted according
to their angular configuration ($\theta_{1},\theta_{2},\varphi_{12}$). The
order of protons in the case of symmetric configurations was randomized,
otherwise, the proton with a lower polar angle ($\theta$) was chosen as the
first one. The range of the polar angles $\theta_{1}$ and $\theta_{2}$ from
$13^{\circ}$ to $33^{\circ}$ has been divided into 4-degree wide bins, and the
relative azimuthal angle $\varphi_{12}$ was sorted into bins of 20 degrees.
Configurations characterized with the lowest azimuthal angles were a subject
to large losses of configurational efficiency, thus they required a specific
approach to the analysis. The results presented here are limited to
$\varphi_{12}\geq 50^{\circ}$. We also considered the acceptance losses due to
the square shape of the detector. Similarly to the elastic scattering, safe
limits on the azimuthal angle: $\pm 10^{\circ}$ around each diagonal were set.
Since the bins cover the range
$\theta\in(\theta_{c}-2^{\circ},\theta_{c}+2^{\circ})$ around the central
$\theta_{c}$ value, the acceptance limits already affected the configuration
with $\theta_{c}=27^{\circ}$. Since setting the limits on azimuthal angles of
both protons would also affect $\varphi_{12}$, the configurations where both
the coincident protons fulfilled the condition $\theta_{c}\geq 27^{\circ}$
were excluded from the analysis.

Figure 9: Upper Panel: $E_{2}$ vs. $E_{1}$ spectrum of the proton-proton
coincidences obtained for a selected angular configuration
$\theta_{1}=15^{\circ},\theta_{2}=19^{\circ},\varphi_{12}=160^{\circ}$. The
calculated kinematics is presented as a solid, black line. The arrows show new
variables: distance ($D$) and arc length ($S$), described in the text. Lower
Panel: The $E_{2}$ vs. $E_{1}$ spectrum transformed into $S$ vs. $D$
variables. The red box represents one slice of $S$ equal to $\Delta S=$ 8 MeV,
projected onto the $D$ axis in the next step.

Data correctly classified to the angular configuration should group around the
kinematics in the $E_{2}$ vs. $E_{1}$ plane of proton energies (see Fig. 9,
upper panel). Such a spectrum can be transformed into two other kinematic
variables. The $S$ variable denotes the arc length value measured along the
kinematic curve (see Fig. 9, upper panel) with the starting point $S=0$ chosen
arbitrarily at the point where the energy of the second proton reaches the
minimum. In this analysis, the kinematic curve was sliced along its length
into the segments of $\Delta S=$ 8 MeV. The $D$ variable is the distance of
each data point from the theoretical kinematic curve in the $E_{2}-E_{1}$
plane. The example of the $E_{2}$ vs. $E_{1}$ histogram transformed into the
$S$ vs. $D$ spectrum is presented in Fig. 9, lower panel. For each $\Delta S$
slice, the events were projected into the $D$ axis to perform the background
subtraction. The background was estimated by a linear function between limits
of integration ($E_{a},E_{b}$) corresponding to a distance of $\pm 3\sigma$
from the peak position. Events below the linear function were subtracted. The
number of events, $N_{br}(\theta_{1},\theta_{2},\varphi_{12},S)$ was summed up
within the $S$-bin and normalized to the integrated luminosity. For each
angular configuration, the $S$ distribution of the cross section was obtained,
see examples in Fig. 13.

##  V Experimental uncertainties and data consistency checks

The experimental uncertainties are summarized in Table 2. The statistical
uncertainties were calculated for each point separately. The systematic errors
are estimated as global uncertainties, i.e. the same for all experimental
points except for the uncertainty on configurational efficiency, which is
calculated for each angular configuration separately.

Table 2: Summary of uncertainties. Sources of errors | The impact on breakup
---|---
cross section [%]
Statistical uncertainties | $2.3-11.1\%$
Total systematic error | $3.9-8.3\%$
1\. Normalization | $3.5\%$
2\. Particle identification | $1\%$
3\. Configurational efficiency | $0.01-6.8\%$
4\. Energy calibration |
\+ angle reconstruction |
\+ detector efficiency | $1\%$
5\. Trigger efficiency | $-0\%,+3\%$
6\. Hadronic interactions | $1\%$

The potential sources of the systematic uncertainties and estimation of their
effect on the cross section value are briefly described below.

  1. 1.

Normalization

The systematic error of luminosity value is 3.5%, see discussion in Sec.
III.3.

  2. 2.

Particle identification

To estimate the impact of PID cuts we analyzed the events regardless of the
particle type. The distribution of breakup events for both cases (with the
presence (left panel) and absence (right panel) of an identification gate) for
the same $S$-variable and kinematic configuration are presented in Fig. 10.
The slight increase of the cross section value calculated without the
identification gate is mainly caused by a higher number of random coincidences
and the contribution of hadronic events. After background subtraction, the
impact of the identification procedure on the cross section value was
estimated at 1%.

Figure 10: Example of $D$ distribution of proton pairs from the breakup
reaction obtained with (left panel) and without (right panel) the PID cut;
data correspond to one $\Delta S$ bin of the angular configuration
$\theta_{1}=18^{\circ},\theta_{2}=20^{\circ},\varphi_{12}=120^{\circ}$.

  3. 3.

Configurational efficiency

Configurational efficiency is closely related to the geometry of the detection
system. The geometry of the Wall was studied on the basis of events
particularly sensitive to its details, registered on the edges of $E$-detector
bars, see Ref. [43]. According to the results, the simulated setup was
modified by introducing a 1 degree inclination of the $E$ detector. The
simulation results obtained in both approaches (for nominal and modified
geometries) for one combination of $\theta_{1}=15^{\circ}$ and
$\theta_{2}=19^{\circ}$ angles are compared in Fig. 11. For all the breakup
configurations the relative difference varies between 0.01% and 6.8%, strongly
depending on the relative azimuthal angle $\varphi_{12}$. Conservatively, we
treat this difference as the systematic uncertainty of the configurational
efficiency and apply individually to each configuration.

Figure 11: The configurational efficiency simulated for an nominal geometry
(black triangles) and the one determined after rotating the $E$ detector (red
triangles). Red triangles have been slightly shifted horizontally for better
presentation of the results.

  4. 4.

Energy calibration

The total impact of energy calibration, angle reconstruction, and detector
efficiency correction on systematic uncertainty was estimated at 1%. More
details can be found in Ref. [41].

  5. 5.

Trigger efficiency

The breakup events were collected with the coincidence trigger, T2 (see Table
1). On the other hand, we normalize breakup to the elastic events collected
with a minimum bias trigger T1. Any differences in the trigger efficiencies
will not be canceled in the ratio. In order to check the relative difference
of trigger efficiencies, breakup events were analyzed with a single trigger.
The number of proton pairs obtained for a wide angular range
$\theta_{1}\in(14^{\circ},26^{\circ})$,
$\theta_{2}\in(14^{\circ},26^{\circ})$, and
$\varphi_{12}\in(60^{\circ},120^{\circ})$ with a single trigger has been
adequately scaled ($T_{1}=2^{4}$) and compared with the number of events
collected with the coincidence trigger. The accuracy of this comparison is
limited by the statistics, and the estimated effect is about 3%. Since the
coincidence trigger provides a lower number of counts, the uncertainty was
implemented as an asymmetric error.

  6. 6.

Hadronic interactions

The correction of the losses due to hadronic interactions of protons and
deuterons was calculated based on simulations. The estimation of these losses
based on experimental data shows that both for breakup protons [20] and
elastically scattered deuterons (this experiment, see Ref. [41]), the
simulated corrections are lower than the experimental ones by about 10-12%,
which translates to the 2% of the corrected value. If indeed the simulation
slightly underestimates the hadronic interaction losses, it happens for both
protons and deuterons and the impact on the cross section result is partially
canceled in the normalization and the final uncertainty was estimated at 1%.

##  VI Experimental results and comparison with theory

The differential cross section for the 2H(p,pp)n breakup reaction at 108 MeV
was analyzed for 84 angular configurations, which corresponds to over 500 data
points. The cross section distributions in function of $S$-variable are
presented in Appendix A.

The data are compared with a set of theoretical predictions based on two
realistic potentials of the nucleon-nucleon (NN) interactions, CD-Bonn and
Argonne V18. Calculations of Witała et al.[14] were performed with the CD-Bonn
potential, either alone (later referred to as CDB Witala) or augmented with
the Tucsone-Melbourne 99 3NF model (CDB+TM99). Deltuva’s calculations [21]
with the CD-Bonn potential (CDB Deltuva) use the approach developed by the
Lisbon-Hanover-Vilnius group to implement 3NF by adding $\Delta$-isobar
excitation using the Coupled-Channel approach (CDB+$\Delta$). We also compare
our data with his calculations based on the Argonne V18 potentials (AV18), and
implementing the Urbana-Illinois IX 3NF model (AV18+UIX). Moreover, the set of
calculations by A. Deltuva included also the Coulomb interaction
(correspondingly CDB+C, CDB+$\Delta$+C, AV18+C and AV18+UIX+C). The two
theoretical groups applied different assumptions on isospin structure. The
calculations of Witała et al. were performed with $np$ interaction in the
1$s_{0}$ wave, while calculations of Deltuva were performed using both $pp$
and $np$ interactions in all isospin triplet waves, including 1/2 and 3/2
total 3N isospin components.

Our data analysis accepted breakup events falling into certain angular ranges
around the central configuration: 4∘ and 20∘ wide bins for the polar and
azimuthal angles, respectively. Due to significant variations of the cross
section within these ranges, averaging of the theoretical values over the same
range was necessary. Therefore, the theoretical cross-sections have been
calculated for the central configuration and all the combinations:
$\theta_{1}\pm 2^{\circ}$, $\theta_{2}\pm 2^{\circ}$, and $\varphi_{12}\pm
10^{\circ}$. Since all the calculations were non-relativistic, and a different
kinematic curve characterizes each configuration, therefore we projected all
the theoretical distributions (in the $E_{1}$ vs $E_{2}$ plane) onto a common,
central relativistic kinematics. Afterwards, we determined the average of the
theoretical cross section values, weighted with the solid angle of a
particular configuration for each step of the $S$-variable and averaged the
cross section over the $S$ bin ($\pm$ 4 MeV). Finally, we obtained the
averaged theoretical distribution to be compared with the experimental
results. Examples of calculations with the AV18 potential for two chosen
configurations is shown in Fig. 12. We present two configurations, one with a
typical and the other with a pronounced averaging effect. The cross section
distributions obtained for central kinematics (dashed lines) are compared to
the results of the above described procedure of averaging and projecting onto
relativistic kinematics (solid lines). The effect of averaging is small but
not negligible.

Figure 12: Theoretical calculations for the AV18 potential for two selected
angular configurations: $\theta_{1}=15^{\circ}$, $\theta_{2}=19^{\circ}$,
$\varphi_{12}=160^{\circ}$ and $\theta_{1}=23^{\circ}$,
$\theta_{2}=27^{\circ}$, $\varphi_{12}=140^{\circ}$ (green and blue dashed
lines, respectively). Black and magenta solid lines show the calculations
after averaging them over acceptance ranges (see text).

Figure 13: Examples of the differential cross section distributions for the
$pd$ breakup reaction at 108 MeV. Black points represent experimental data
with statistical errors, and gray bands illustrate their systematic
uncertainties. The theoretical calculations are shown as lines, see the legend
in the top panel. The pure NN potentials are depicted by dashed lines, while
their combinations with 3NF are shown as dotted lines. The calculations
including Coulomb interaction are shown as solid lines.

Examples of the experimental differential cross section for the $pd$ breakup
reaction, compared to the theoretical calculations, are presented in Fig. 13.
The top two panels, (a) and (b), show configurations with large $\varphi_{12}$
values. They are characterized with relatively large cross sections, thus
systematic uncertainties dominate over statistical ones. The bottom two
panels, (c) and (d), present configurations with $\varphi_{12}=60^{\circ}$,
characterized by an order of magnitude lower cross section and thus dominating
statistical errors. Generally, in regions of relatively low cross section, we
observe a significant sensitivity of the result to the details of the dynamics
beyond the leading NN terms, which is reflected in a wide range of theoretical
values. At the largest studied polar angles (see panel (d)), discrepancy
between the data and all the theories is observed.

###  VI.1 Reduced chi-squared analysis

A quantitative comparison between the experimental data and theoretical
predictions was performed on the basis of a reduced chi-square, redefined as
follows:

$\chi^{2}_{red}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\sigma_{i}^{exp}-\sigma_{i}^{th}}{\Delta\sigma_{i}^{tot}}\right)^{2},$
(6)

where $\sigma_{i}^{exp}$ and $\sigma_{i}^{th}$ were the experimental and
theoretical values of the cross section given for each $i-th$ point,
respectively, while the $\Delta\sigma_{i}^{tot}$ was the total uncertainty of
each experimental point, calculated as
$\Delta\sigma_{i}^{tot}=\sqrt{(\sigma_{stat}^{2}+\sigma_{syst}^{2})}$, with
$\sigma_{stat}$ \- a statistical uncertainty, and $\sigma_{syst}$ \- a
systematic uncertainty. $\chi^{2}$ per degree of freedom was determined
globally and for specific data subsets.

Figure 14: The global $\chi^{2}_{red}$ derived for all data points,
individually for each theory.

At the outset $\chi^{2}_{red}$ has been computed as a global factor, where $N$
corresponds to the total number of data points. In Fig. 14 we observe clear
dependence of the data description on the theoretical model, which is
reflected in a global $\chi^{2}_{red}$ varying from approximately 4 to 12. The
highest values correspond to calculations performed by Witala et al.
Admittedly, adding the TM99 three-nucleon force improves the $\chi^{2}_{red}$
compared to the pure CD-Bonn potential, but the discrepancy remains. There are
also four results grouped around $\chi^{2}_{red}\approx 8$ corresponding of
the theoretical calculations performed by Deltuva. Here, the effect of adding
the $\Delta$-isobar to the CD-Bonn potential and the Urbana IX 3NF to the AV18
potential is very small, respectively slightly increasing or decreasing the
global $\chi^{2}_{red}$. The global $\chi^{2}_{red}$ analysis clearly shows a
significant improvement in the description of the cross section data when the
Coulomb force is taken into account. Not only the $\chi^{2}_{red}$ is reduced
by a factor of two, but also the effect of Urbana IX 3NF is revealed. However,
the size of this effect is similar to the difference in description given by
calculations with two different NN potentials.

In the next step, the quality of the description was checked locally.
$\chi^{2}_{red}$ was calculated individually for each angular configuration,
where $N$ is the number of experimental points in a given configuration. The
results are presented as 2-dimensional maps, plotted for each theoretical
calculation separately, see Figs. 15-17.

Figure 15: The $\chi^{2}_{red}$ results obtained individually for each angular
configurations, presented as a 2-dimensional map defined by combination of
polar and relative azimuthal angles. The data are compared with two selected
theories: CDB (Witala) and CDB+TM99.

The $\chi^{2}_{red}$ maps obtained for theories without Coulomb (Fig. 15, and
the two left panels in Figs. 16 and 17) demonstrate deficiency of the data
description in particular regions. These areas are mainly located at
intermediate $\varphi_{12}$ and also in the upper part corresponding to the
higher polar angles $\theta_{1}$ and $\theta_{2}$.

The maps enable us to observe the quality of the theoretical description for
each of the configurations separately, but for better visibility of the
effects they have been truncated to a maximal value of 10. The full range of
$\chi^{2}_{red}$ can be observed separately for the relative azimuthal angles
$\varphi_{12}$ (Fig. 18, upper panel) and polar angles $\theta_{1}$ and
$\theta_{2}$ (Fig. 18, lower panel). For calculations without Coulomb (blue
points and squares, green shaded bars), the greatest discrepancy is for
$\varphi_{12}=140^{\circ}$ and maximal polar angles.

Figure 16: Similar as in Fig. 15, but for different set of theoretical
calculations: CDB (Deltuva), CDB+$\Delta$, CDB+C, CDB+$\Delta$+C.

Figure 17: Similar as in Fig. 15, but for different set of theoretical
calculations: AV18, AV18+UIX, AV18+C, AV18+UIX+C.

Adding the Coulomb interaction substantially improves the situation, as shown
in two right panels in Figs. 16 and 17. Additionally, we observe that all the
calculations including Coulomb interaction (red and magenta shaded bars) reach
minimum of $\chi^{2}_{red}$ in a similar region ($\varphi_{12}=100^{\circ}$,
$160^{\circ}$ and $180^{\circ}$, see Fig. 18, upper panel). Lower panel in
Fig. 18 shows a minimum of $\chi^{2}_{red}$ between
$(\theta_{1}+\theta_{2})/2=15^{\circ}$ and
$(\theta_{1}+\theta_{2})/2=21^{\circ}$, observed for all the theoretical
calculations including Coulomb interaction.

Figure 18: The $\chi^{2}_{red}$ results presented as a function of the
relative azimuthal angle $\Delta\varphi_{12}$ (upper panel), and with respect
to the average of $\theta_{1}$ and $\theta_{2}$ (lower panel). The legend is
common for both panels.

Even calculations neglecting Coulomb indicate certain improvement of
description when TM99 or Urbana IX 3NF are added, in particular at the bottom-
left corners of the maps in Figs. 15-17. This region corresponds to the lowest
$\varphi_{12}$ and $\theta$’s. Once the Coulomb interaction is introduced, the
possible effects of 3NF can be reliably investigated in the whole studied
phase space. The overall picture is not so clear, but locally the improvement
can be seen, especially at low $\varphi_{12}$, see Fig. 18 upper panel and an
example in Fig. 13(c). It is also visible in the maps, in their bottom-left
corners, where low $\varphi_{12}$ is combined with the lowest polar angles.
This confirms the observation for theories without Coulomb.

The remaining discrepancy between data and calculations is visible at low
$\varphi_{12}$ and large $\theta$’s, i.e. in left-top corner of the maps, see
also the example in Fig. 13(d).

##  VII Summary

The differential cross sections for the 2H(p,pp)n reaction at beam energy of
108 MeV have been measured with the forward Wall of the BINA detector. The
data, collected in the range of polar angles of outgoing protons from 13 to 33
degrees, were sorted into 84 configurations, corresponding to various
combinations of polar and relative azimuthal angles. Taking into account
binning in the energy variable $S$, over 500 data points were obtained.

The experimental results have been compared to the state-of-the-art
theoretical calculations including contributions from two-nucleon interaction
combined or not with the 3NF or Coulomb force. The quality of the data
description was studied by performing a reduced chi-square ($\chi^{2}_{red}$)
analysis for each of the available models.

The investigated region of the phase space is characterized by significant
variation in the differential cross section values. Regions with the lower
cross-section reveal greater sensitivity to the dynamic effects. In almost the
entire studied phase space, the Coulomb effects are present. Among all the
compared theoretical models, the best agreement is obtained for the
calculations including Coulomb interactions (CDB+C, CDB+$\Delta$+C AV18+C,
AV18+UIX+C). The results show importance of the Coulomb effects at the studied
energy and lead to the conclusion that the Coulomb interaction has to be
necessarily included into the theoretical description. It is important also in
the context of future comparisons of the data with the intensively developed
calculations within the Chiral Effective Field Theory.

In the studied part of the breakup phase space, the effects of three-nucleon
force are moderate or negligible. For the most sensitive configurations, we
observe a larger influence of the TM99 and UrbanaIX forces than of the
$\Delta$-isobar. There is also a small kinematical region where all the
theories underestimate the data. There, the 3N forces (TM99 and UIX) lower the
cross section values, thus improve the description, but by far not enough to
eliminate the discrepancy.

In the future, the results presented here will be extended by analyzing data
from the subsequent experimental run covering a wider angular range.
Configurations close to the so-called neutron-proton final state interaction
are kinematically similar to the elastic scattering and can reveal stronger
sensitivity to 3NF.

###### Acknowledgements.

This work was supported by the Polish National Science Centre under Grants No.
2020/37/N/ST2/02360, 2016/23/D/ST2/01703 and 2012/05/B/ST2/02556. This
research was also supported in part by the Excellence Initiative – Research
University Program at the Jagiellonian University in Kraków. The numerical
calculations were performed on the supercomputer cluster of the JSC, Jülich,
Germany. We wish to express our gratitude to the PROTEUS cyclotron crew at the
CCB for providing us with an excellent and stable beam during the experiment.

## Appendix A

Appendix A contains all the differential cross section distributions for the
2H(p,pp)n breakup reaction at the beam energy of 108 MeV obtained in the
analysis described in this paper. Each figure presents 7 configurations
corresponding to individual azimuthal angles $\varphi_{12}$ from $60^{\circ}$
to $180^{\circ}$ and one combination of $\theta_{1}$ and $\theta_{2}$
specified in the caption. The legend is presented in panel 8th.

Figure 19: The differential cross section for $pd$ breakup reaction obtained
for polar angles $\theta_{1}=15^{\circ}$ and $\theta_{2}=15^{\circ}$ and for a
set of azimuthal angles $\varphi_{12}$ specified in the panels. Black points
represent the experimental data with statistical errors, and gray bands
illustrate the systematic uncertainties. The available theoretical
calculations are shown as lines, listed in the legend. In one panel the
results are scaled by 0.5 to fit the common vertical axis, as indicated in the
panel. Figure 20: The same as Fig. 19 but for polar angles
$\theta_{1}=15^{\circ}$ and $\theta_{2}=19^{\circ}$. Figure 21: The same as
Fig. 19 but for polar angles $\theta_{1}=15^{\circ}$ and
$\theta_{2}=23^{\circ}$. Figure 22: The same as Fig. 19 but for polar angles
$\theta_{1}=15^{\circ}$ and $\theta_{2}=27^{\circ}$. Figure 23: The same as
Fig. 19 but for polar angles $\theta_{1}=15^{\circ}$ and
$\theta_{2}=31^{\circ}$. Figure 24: The same as Fig. 19 but for polar angles
$\theta_{1}=19^{\circ}$ and $\theta_{2}=19^{\circ}$. Figure 25: The same as
Fig. 19 but for polar angles $\theta_{1}=19^{\circ}$ and
$\theta_{2}=23^{\circ}$. Figure 26: The same as Fig. 19 but for polar angles
$\theta_{1}=19^{\circ}$ and $\theta_{2}=27^{\circ}$. Figure 27: The same as
Fig. 19 but for polar angles $\theta_{1}=19^{\circ}$ and
$\theta_{2}=31^{\circ}$. Figure 28: The same as Fig. 19 but for polar angles
$\theta_{1}=23^{\circ}$ and $\theta_{2}=23^{\circ}$. Figure 29: The same as
Fig. 19 but for polar angles $\theta_{1}=23^{\circ}$ and
$\theta_{2}=27^{\circ}$. Figure 30: The same as Fig. 19 but for polar angles
$\theta_{1}=23^{\circ}$ and $\theta_{2}=31^{\circ}$.

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