Differential Cross Section for Proton Induced Deuteron Breakup at 108 MeV

The differential cross sections for the proton-deuteron breakup reaction have been measured for more than 200 angular configurations of outgoing protons in the range of polar angles from 13 to 27 degrees with a proton beam at 108 MeV. The paper presents the experimental results of the selected configurations, which are compared to state-of-the-art theoretical calculations. In some regions of the phase space, a strong influence of the Coulomb interaction is observed.


Motivation
Deuteron breakup in collision with a proton is a tool for testing modern calculations describing nuclear interactions between three nucleons [1,2,3].Recent progress in the theoretical approaches to such systems allowed for the first time to account for the essential parts of the dynamics after the leading nucleon-nucleon interaction.This includes three-nucleon force and Coulomb force effects and calculations performed within a proper relativistic formalism [4,5,6].The theoretical calculations had to be confronted with a rich set of measurements.For this purpose, the BINA (Big Instrument for Nuclear-polarization Analysis) detector setup has been installed at the Cyclotron Center Bronowice (CCB) in Kraków.The combination of the large phase space coverage of the BINA system and the wide range of accessible beam energies provides a unique possibility to study the dynamics in a three-nucleon system.In the following we present the first data set collected for the 2 H(p, pp)n reaction at 108 MeV.

Experimental setup
The BINA detection setup comprises two main parts -the forward Wall and the central-backward part called Ball.The Wall consists of a three-plane Multi-Wire Proportional Chamber (MWPC) and an array of almost square ∆E − E telescopes formed by two layers of scintillators.Particles scattered in the forward direction can be measured with the polar angles θ between 13 • − 29 • in a whole azimuthal angle φ.The BINA Ball is built of 149 phoswich detectors covering the angles 40 • − 160 • and also serves as a vacuum chamber where the liquid deuterium target is located.More information about the details of the BINA detector is presented in the papers [7,8].

Breakup reaction analysis
Several data analysis methods were applied to determine the differential cross-sections for the breakup reaction.In the first step, the experimental data were pre-selected, and then the angles of the charged particles were reconstructed.Registered particles were identified based on the ∆E − E technique [8].The efficiency of the Wall detector is excellent, mostly reaching over 97%.Besides the corrections for the Wall efficiency, the correction for the edge events, the configurational efficiency, and the efficiency connected with the hadronic interactions were taken into account.A set of simulations has been carried out to calculate the number of particles undergoing hadronic interaction.The simulations have been performed for proton and deuteron energies ranging from 20 to 160 MeV [9].Moreover, elastically scattered deuterons from the experiment were also used to estimate the losses associated with the hadronic interaction.The Gaussian function was fitted to the deuteron energy spectrum at the selected θ angle, which is presented in Fig. 1, left panel.The number of particles lost (k loss ) due to the hadronic interaction was estimated by the ratio of particles' count in the tail (N tail ) to all particles (N all ) for given energy distribution for experimental data and simulations (k loss = N tail /N all ).The N tail value has been determined as the sum of the events' number below the green line corresponding with the value of the −3σ from the peak position, see Fig. 1, left panel.The N all is defined as the number of all particles counted in the whole histogram.In Fig. 1, right panel, we present the results of simulations for deuterons and protons and data analysis for deuterons.The final correction applied in the further analysis is: The breakup reaction kinematics is determined by the momenta of protons ⃗ p 1 and ⃗ p 2 .The events identified as proton-proton coincidences were analyzed event-by-event and sorted according to angular configuration (θ 1 , θ 2 , φ 12 ).Data correctly classified to the angular configuration should group around the central line of the kinematical curve in the E 2 vs. E 1 plane of proton energies (see, Fig. 2, left panel ).Such a spectrum can be transformed into  two other kinematical variables.The S variable denotes the arc length value measured along the kinematic curve 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 kinematical curve was sliced along its length into the segments of ∆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. 2, middle panel.For each ∆S slice, the events are projected into the D axis to perform the background subtraction.The background is estimated by a linear function between limits of integration (E a , E b ) corresponding to a distance of ±3σ from the peak position.Events below the linear function are subtracted, and the D-projected distribution after the background subtraction has a Gaussian shape (Fig. 2, right panel ).To normalize our breakup cross section, we calculated luminosity based on the number of elastically scattered deuterons for the known proton-deuteron elastic cross section measured at KVI [10].

Experimental results and comparison with theory
Differential cross-sections have been obtained for more than 200 angular configurations.The experimental results have been compared with the state-of-the-art theoretical calculations based on the CD-Bonn potential (CDB), considering the 3NF effects introduced in two ways (TM99 and ∆-isobar models) and/or Coulomb interactions between protons (C).The examples of the differential cross-section for two different configurations are shown in Fig. 3.The first one is asymmetrical (left panel ) with the polar angles θ 1 = 16 • , θ 2 = 20 • , and coplanar (φ 12 = 180 • ).
The second one (right panel ) is symmetrical and characterised by a low azimuthal angle:

Conclusions
Finally, we can conclude that the contribution of the Coulomb force in the analyzed range of protons energies and angular configurations is very significant.On the contrary, the effect of the three-nucleon force is negligible.However, we have studied only a part of the phase space.Configurations close to the so-called neutron-proton final state interaction (FSI) are kinematically similar to the elastic scattering and can reveal stronger sensitivity to 3NF.Finally, we intend to compare the experimental results with other theoretical predictions, including the Coulomb interaction, based on the different NN potential (Av18) and the Urbana-Illinois X (UIX) 3NF model.

Figure 1 .
Figure1.Left panel: The experimental deuteron energy distribution spectrum at the selected polar angle θ = 20 • with the fit of Gaussian function.The green line corresponds with the value of the −3σ from the peak position.The events summed up below this line constitute the N tail value.Right panel : The simulated and experimental results of energy loss (k loss ) due to hadronic interaction in the E detector presented as an initial energy function.The blue squares and the magenta triangles represent simulated deuterons and protons, respectively.The experimental results are presented as green points with corresponding statistical errors smaller than the size of the points.

Figure 2 .
Figure 2. Left Panel : E 2 vs. E 1 spectrum of the proton-proton coincidences obtained for a chosen angular configuration θ 1 = 20 • , θ 2 = 24 • , φ 12 = 160 • .The arrows show the calculated value's direction: distance (D) and arc length (S) described in the text.Middle Panel : The E 2 vs. E 1 spectrum transformed into S vs. D variables.The red box represents one slice of S equal to ∆S = 8 MeV, projected onto the D axis in the next step.Right panel : The number of events obtained after the background subtraction.

1 θFigure 3 .
Figure 3. Examples of the differential cross section for pd breakup reaction for two angular configurations -asymmetrical (left panel ), θ 1 = 16 • , θ 2 = 20 • , and φ 12 = 180 • , and symmetrical one (right panel ) θ 1 = 14 • , θ 2 = 14 • , and φ 12 = 20 • .Black points represent the experimental data with statistical errors, and gray bands illustrate the systematic uncertainties.The available theoretical calculations are shown as colored lines, as the legend describes, common for both panels.First four theories listed in the legend correspond to calculations of Deltuva et al., the last two -of Wita la et al.CDB is a CD-Bonn potential, TM99 and ∆ mean two different description of 3NF; C -Coulomb interaction between two protons.
•, where the Coulomb effect is strongly dominant.As described in the legend, the available theoretical calculations are shown as colored lines common for both panels.As we know from experiments at other beam energies, this region is particularly sensitive to Coulomb interactions.The theoretical calculations neglecting the Coulomb repulsion between protons fail in describing the data.Adding the Coulomb interaction to calculations dramatically improves the situation.