i(Qi, Q2)ep2(0i)ep2(02) x i=35 x E ePi(0i,Ci)(ePi(02,P%i + ^i2) + ePi(02,P%i — Ci2))- (5.19) i=0 The term of the sum given as (ePi(02, pii + Pi2) + ePi(02, pii — Pi2)) is taking into account common treatment of the mirror configurations tp2 = Pi ± Pi2. Moreover, one has to realize that depending on the energy of the protons, efficiency of the Pizza 2 detector can contribute or not to the total efficiency e(S, Qi, Q2), what leads to the following scenarios: 1. proton number 1 was registered in Pizza 2, whereas proton number 2 reached only Pizza 1 detector: e(S, Qi, Q2) = e^(Qi, Q2) ■ eP>i(Qi, Q2)eP2(0i) x i=35 x E eP i(0i , Pi)(eP i (02, Pi + Pi2) + eP 1(02, Pi — Pi2)) • (5.20) i=0 2. proton number 2 was registered in Pizza 2, whereas proton number 1 reached only Pizza 1 detector: e(S, Qi, Q2) = e^(Qi, Q2) ■ ecP>i(Qi, Q2)eP2(02) x i=35 x E eP i(0i , Pi)(eP i (02, Pi + Pi2) + eP i (02, Pi — Pi2)) • (5-21) i=0 3. both of the detected protons reached only Pizza 1 detector: e(S, Qi, Q2) = e^(Qi, Q2) ■ eP>i(Qi, Q2) x i=35 x E eP i(0i , Pi)(eP i (02, Pi + Pi2) + eP i (02, Pi — Pi2)) • (5.22) i=0 5.10. EVALUATION OF THE BREAKUP OBSERVABLES 59 4. both of the detected protons reached Pizza 2 detector (this situation never occurs): e(S, Qi, Q2) = £q(Qi, Q2) • epi(Qi, Q2)£p2(di)£p2(02) X i=35 X £P 1(01. Pi)(eP 1(02, Pi + Pi2) + eP 1(02, Pi — Pi2))- (5.23) i=0 Depending on the situation, one has to apply one of the above formulae. The C(t) is the factor related to the luminosity, which depends on the total beam current I0, the density gt and the thickness Axt of the target, as well as the electronic dead time t. Therefore, the above parameter could be written as follows: C(t) = (1 - t)IoQtAxt. (5.24) In the similar way one can write the number of the elastic scattering events, Nei(Qf ), with the deuteron registered at the angle Qf (what defines the proton detection angle Q2l), which is expressed by: i=35 Nei(Q1l) = £ Nd(tf,e\) i=0 dn = dQf (0i) • AQ1l • C(t) • e(Q?E), (5.25) where the dyv is the elastic scattering cross section, and the other factors are analogous to the ones defined above. The e(Qi,Ef) is related to the efficiencies and correction factors obtained with respect to the single-track events and is expressed as: i=35 e(Q?,Ef) = e%(0?)£P2«) £ ep 1(0?,\)ep,(0f. Pi + n), (5.26) i=0 Finally, the breakup cross section could be expressed in terms of the elastic scattering cross section due to the fact that both of these reactions were measured simultaneously. Therefore, the factor C(t) is the same in the Eqs. 5.18 and 5.25. Taking advantage of the above conclusion, the final formula for the differential breakup cross section for a given angular configuration can be expressed as follows: d"n (S0 0 p ) = dn± (0el ) Nbr (S. Q1 . Q2) * dQidQ2dS( ’ 1 2,Pi2) dQi( 1 ) Nfl(Q|l) AQ|l_ e(Qi ,Ef ) AQ1AQ2AS e(S, Qi, Q2) ' (.J The elastic scattering cross section was taken from [42]. Fig. 5.31 shows a sample breakup cross section distribution for a chosen kinematical configuration, together with the full set of theoretical calculations. More results are presented in Appendix A. 60 CHAPTER 5. DATA ANALYSIS Figure 5.31: Example of the differential breakup cross section for the angular configuration specified in the picture. Theoretical predictions are shown as bands and lines, as specified in the legend. 5.10.3 Vector analyzing powers of the breakup process In the case of the vector analyzing powers analysis the protons grouped around the kine-matical curve were divided into AS bins having width of 16 MeV and projected on the central kinematics. For a given S — bin in the selected configuration and assumed D — bin of 16 MeV (see also previous Section), the numbers of the breakup events as a function of the azimuthal angle pi have been counted. The width of pi bin was chosen to be 20o. The obtained numbers of events were normalized to the beam current, corrected for the dead time and scaled by an adequate trigger factor. In the next step, the ratios, similar to the ones introduced in the elastic scattering analysis (see Sec. 5.8.2), were built for each S bin: f(pi) = Npl{N—N°(iPi), (5.28) No( Pi) where (C, pi) defines a given kinematical point (0i,92, pi2,S, pi) and N^ol(pi), NO(pi) denote the numbers of events for polarized (Pz = — |, Pzz = 0) and unpolarized (Pz = 0, Pzz = 0) beam states, respectively. For the breakup process f^(pi) is in general expressed by the formula: 3 3 f( p i) = — 2sin Pipz Az(C) + 2cos Pipz Ay(C ) +2 sin2 pi Pzz AXX(C) + 2 cos2 pi Pzz Ayy (C ) — cos pi sin pi Pzz Axy (C), (5.29) where Ax and Ay denote vector analyzing powers, whereas Axx, Ayy and Axy are the 5.10. EVALUATION OF THE BREAKUP OBSERVABLES 61 Figure 5.32: A sample dependency of f £ combination from Eq. 5.32 vs. cos c1 for 16 MeV wide S-bin centered at 88 MeV for the chosen kinematical configuration specified in the picture. The solid line represents fit of the linear function 5.32 to the data points with experimental value of Pz taken from Tab. 5.2. The resulting from the fit value of Ay is displayed in the panel. tensor analyzing powers. The f £ (c1) dependency for pure vector polarized beam state simplifies to: 3 fî(Ci) = 2Pz(Ay(£) cos Ci — Ax(0 sin Ci)• (5.30) Thus, the values of Ax and Ay can be extracted in a very simple way, if one computes combinations of f£ (c1) obtained separately for +Ci2 and — Ci2 (taking advantage of the parity restrictions, see Sec. 2.3): f t=(e1,e2,+‘pi2,s) (ci) — f^^^Ci ) = 3pz Ax(£)sin Ci, (5.31) f(ci) + f(ci) = 3PzAy(£) cos Cn (5.32) Using the beam polarization Pz obtained from the elastic scattering analysis (see Sec. 5.8.2), the breakup vector analyzing powers were evaluated from linear fits of the above combinations of f as functions of the sine or cosine of the first proton azimuthal angle Ci, respectively. A sample dependency of f£ combination from Eq. 5.32 vs. cos ci for one configuration is illustrated in Fig. 5.32. In Fig. 5.33 sample results for the two vector analyzing powers, Ax and Ay, are shown for one chosen kinematical configuration. The full sets of the theoretical calculations are also presented. More results are given in Appendix B. 62 CHAPTER 5. DATA ANALYSIS Figure 5.33: Examples of the d — p breakup vector analyzing powers Ax and Ay for one kinematical configuration of the two coincident protons, specified in the legend. Theoretical predictions are shown as bands and lines, as specified in the legend. 5.11 Discussion of possible sources of uncertainties The observables obtained in the analysis i.e. cross sections and analyzing powers are affected by statistical and systematic uncertainties in a slightly different way. This section contains a brief discussion of the potential sources of the above errors. 5.11.1 Statistical uncertainties The statistical error of the number n of independent events is in this case given by the Poisson distribution: An = jn. (5.33) For this experiment the number of collected events n was downscaled by an appropriate factor related to the trigger type (see Sec. 4.3.5). Thus, the real number of events together with its statistical error can be written as: N = n ■ 2x, (5.34) AN = jn ■ 2x = vN ■ 2x, (5.35) where 2x is the downscaling factor, which for the single (elastic) events is equal 4, whereas for the coincident (breakup) ones is equal to 1. Statistical uncertainties of the measured cross section distributions comprise the error of the measured number of the breakup coincidences, as well as statistical uncertainties of all quantities used in the normalization. These quantities are: • number of the elastic scattering events, • efficiencies of the Pizza 1 (obtained from the coincident events) and Pizza 2 detectors (obtained from the elastic events) - see Sec. 5.9.2. The estimated value of the statistical errors are around 2% and 0.1%, respectively, 5.11. DISCUSSION OF POSSIBLE SOURCES OF UNCERTAINTIES 63 • values of the elastic scattering cross sections given by Shimizu et al. [42] and used in normalization. The value of this errors is about 1%. In the case of the vector analyzing powers, the statistical uncertainties have following sources: • the fit procedure, which are the main contributions. The values vary depending on configuration. • statistical error of the vector beam polarization Pz, which also contains the statistical error of the experimentally obtained value of the iTn analyzing power of the elastic scattering. The mentioned above values introduced the uncertainties of about 3% and 1% [72], respectively. See Sec. 5.8.2 for more details, • statistical error of the normalization factor k. The estimated value of the statistical error is about 0.2%, see also Sec. 5.8.2. 5.11.2 Systematic uncertainties Influence of the systematic errors was reduced significantly by detailed study of the setup geometry and the detection efficiency. Potential sources of the systematic uncertainties and their influences on the observables are listed below: 1. Luminosity. The luminosity (total beam current) is different for the polarized Ipoi and unpolarized I° states, what is reflected in the normalization factor k = . It has been checked that its value is stable in time and does not depend on the downscaling factors applied to the triggers. k is obtained in the analysis as a parameter of the fit (Eq. 5.11) to the elastic scattering data and is directly applicable for the breakup reaction. Normalization to the luminosity is not expected to cause any systematic uncertainty on the analyzing power results. 2. Reconstruction of the particle emission angles. The errors due to the reconstruction of the particle emission angles 9 and p originate from the finite thickness of the target and size of the beam spot on the target. Moreover, they depend on angular granulation related to the overlapping front and rear spirals of the Quirl detector, what gives 20000 pixels of the size from 0.011 mm2 to 0.1 mm2 (see Sec. 4.3.4). The determination of the distance between the target and Quirl also influences the accuracy of the angle reconstruction. This kind of uncertainties can affects the results only when the reconstructed angles are shifted from the real ones. The analysis of the elastic scattering events (see Sec. 5.3) allowed for an accurate verification of the measured target-Quirl distance and confirmed that there is no systematic shift of the reconstructed polar angles. In conclusion, the accuracy of determination of the azimuthal angle p is connected to the number of spirals of the Quirl detector (200 spirals on the both sides) and is given by the formula 2°° ~0.0314[rad]~1.8°. In case of the solid angle, the accuracy vary from 10_6 to 10_5 depending on the size of the pixels. Only systematic uncertainties originating from determination of the azimuthal angle p can affect the analyzing power results and are estimated to be below 1%. 64 CHAPTER 5. DATA ANALYSIS 3. Calibration and energy correction procedures. The errors are related to the determination of the peak centers in the ADC spectra obtained in the calibration run (see Sec. 5.6) for each GeWall detector. Further uncertainties could also originate from the determination of the peak centers, when checking the correctness of the calibration i.e. obtaining the kinematical relations of the deposited energy vs. 9 (polar angle) for the elastic scattering events. This procedure provided the scaling factors for the energy losses correction - see Sec. 5.6. The obtained uncertainties of the calibration correction procedure were established to be about 3%. In conclusion, their influence on the breakup cross section distributions is less than 1%. 4. Energy cuts. Selection of the interesting events (protons and deuterons) is connected with imposing different energy cuts or applying defined gates on the AE — E spectra. Thus, “leaking” of interesting particles to and from the identification ranges can be a source of uncertainties. It has been checked for the identification gate of protons, that slight changing of the defined region borders introduces the uncertainty of at most 1%. 5. Background subtraction. The background subtraction procedure is used for determination of the vector beam polarization Pz, as well as in normalization of the breakup cross section. In case of the elastic deuterons, the errors originate from the gaussian extrapolation of the background below the elastic peak (see Sec. 5.8.1, Fig. 5.18). For the elastic protons, the linear fit was assumed as the background model and the parameters obtained in this procedure are the sources of uncertainties (see Sec. 5.8.1, Fig. 5.18). Moreover, the limits of integrating of the breakup peak (see Sec. 5.10.1, Fig. 5.30) for separate slices along S, were chosen arbitrarily, what also can cause additional errors. The estimated value of uncertainty originating from the background subtraction procedure is about 1.5%. 6. Efficiencies. The calculated efficiencies of the detectors are a source of possible errors. The assumed model (with its parameters) for calculating the factors describing the contribution of different effects to the total efficiency causes very significant systematic errors, what affects the cross section distributions in a serious way. In order to estimate their values the calculations were performed for additional two sets of the parameters Rq (radius of the spot characterizing the charge splitting on the spirals) and Rp1 (radius of the spot in case of the Pizza 1 sectors) - see Sec. 5.9 for more details. The parameters in this auxiliary calculations were assumed to be Rq = 0.22 mm, RP1 = 0.6 mm and Rq = 0.18 mm, RP1 = 0.4 mm. The estimated values of the uncertainty obtained for the relative azimuthal angles between the two protons p12 > 120° is about 4^8%. The evaluated values of the systematic errors for p12 < 120° are quite large and affect the cross section distributions in a serious way, what is demonstrated in Fig. 5.34. The model of charge distribution over spirals in the Quirl detector does not completely reproduce, crucial for the efficiency analysis, geometrical effects connected with the ’’overlapping clusters“. This simple model needs further developments to be able to produce correction factors in a proper way. 5.11. DISCUSSION OF POSSIBLE SOURCES OF UNCERTAINTIES 65 Therefore, the results of the Quirl efficiency resulting from that model are biased with rather large systematic uncertainties, which are estimated to be as large as 60% (for 13 configurations) and 35% (for another 8 configurations). Within the discussed range of ¥12 < 1200 only a fraction of geometries is affected, corresponding to the lay out of the spirals. It has been established that the strongest effect is observed for configurations fulfilling the condition ^——"q^ = 10 (60%), and smaller for a few neighbouring geometries with p12 = 60° and described by the formula AÛ+sign(’ifi22 —60°)*2 = 10, where A9 = l 91 - 91 1 (35%o) (see Table 5'3)- A9 = | 91 - 92 | ¥>12[°] 0 2 4 6 8 20 60% 60% 35% - 35% 40 - - 35% 35% - 60 - - - 35% - 80 - - - 35% 60% 100 - - - - 35% Table 5.3: Systematic uncertainties for configurations very strongly affected by calculations of the detection system efficiencies. For majority of configurations systematic uncertainties due to that effect are around 4-8%. Figure 5.34: Example of the breakup cross section distribution for the configuration characterized by very small value of the relative azimuthal angle tp12 = 20°. The red points are the experimental data obtained with the model parameters Rq = 0.2 mm, Rp 1 = 0.5 mm. The blue and green points refer to the same data, but with parameters Rq = 0.4 mm, Rp 1 = 0.18 mm and Rq = 0.6 mm, Rp 1 = 0.22 mm, respectively. The dashed line represents the cross section calculated on the basis of CDB+A+C potential. 66 CHAPTER 5. DATA ANALYSIS 7. Normalization to the elastic scattering. The breakup cross sections were normalized to the elastic scattering data in order to obtain the absolute values. Thus, the obtained results are affected by the error of 1.6% quoted by Shimizu et al. [42]. In determination of the absolute values of the cross section the most significant systematic uncertainty is due to efficiencies. The other uncertainties discussed above also contribute, but they are much less important. Therefore, the overall systematic error of the breakup cross section for a majority of the studied configurations is established to be 5-10%, depending on the geometry. Analysis of the polarization observables relies on the determination of rates measured with the polarized and unpolarized beams. Thus, the advantage of this approach is obvious i.e. the experimental factors, which appear in the analysis, like efficiency of the detection system or identification methods (applying different identification cuts in the AE — E spectra), as well as uncertainties connected with determination of solid angles are cancelled. In the ratio given by Eq. 5.28 in Sec. 5.10.3 only corrections connected with the background subtraction play a significant part and are included in denominator of the ratio. The overall value of the systematic errors for Ax and Ay is estimated to be around 1-2%. Chapter 6 Results 6.1 Experimental results Experimental results of the cross section values are obtained for 135 kinematical configurations of the two breakup protons. Polar angles 91 and 92 of the two protons are changing between 5° and 14° with the step of 2° and their relative azimuthal angle ¥12 is taken in the range from 20° to 180°, with the step of 20°. The experimental results were integrated (see Sec.5.10.3 ) within the ranges of A91 = A92 = 2° and A^12 = 10° for each combination of the central values 91, 92 and ¥12. The bin size along the kinematic curve S was chosen to be 8 MeV. In the case of vector analyzing powers Ax and Ay, the experimental results were obtained in 24 kinematical configurations for each observable. In the analysis the event integration limits were chosen to be A91 = A92 = 3° and A^12=20°. The bin size of the S value was 16 MeV. The choice of event integration limits enabled to reach quite sufficient statistical accuracy enabling comparisons with different theoretical predictions. Sample results of the cross section and analyzing powers Ax and Ay for chosen kinemat-ical configurations were presented in Sec. 5.10 in Figs. 5.31 and 5.33, together with the full set of the theoretical calculations. The bulk of such individual comparisons for all the evaluated configurations for cross section and vector analyzing powers is collected in Appendix A and Appendix B, respectively. 6.2 Averaging of the theoretical predictions over the integration limits The chosen angular ranges which define the geometrical configurations of the breakup protons are wide enough to obtain quite good statistical accuracy, however, the obtained results are very sensitive to the averaging effects. Thus, in order to perform reasonable comparisons of the data with the theoretical models, the averaging over the same limits had to be applied to the calculated values of the vector analyzing powers and the cross sections. For that purpose, for each configuration given by the central values of angles 91, 92, ¥12, the analyzing powers and cross section a° values have been calculated for all combinations of angles 91 ± 1A91, 922 ± 1A92, ^î2 ± 1 A^12 and the central values, with the step of 1 MeV in variable S. There was, however, one exception for the configurations 67 68 CHAPTER 6. RESULTS defined by the central values of the polar angles of the two protons equal to 91 = 130, 92 = 130. Due to the detection system acceptance (the highest available values of the polar angle is 13.50) the calculations for this geometry were performed within the ranges of the polar angles 91 = 13+°05 , 92 = 13+°05 . It is important to remember that S is defined individually for each kinematical curve, thus the same values of S for two different combinations of angles are related to two different pairs of proton energies (E1; E2 ). Figure 6.1: Figures present the results of the theoretical calculations with the AV18 NN potential combined with the Urbana IX 3NF and with the Coulomb interaction included (AV18+UIX+C) for two chosen configurations described in the picture. The theoretical curves marked with red, dark blue and magenta colours refer to various combinations of the 9i, 92 and <^12 (specified in the legend) within the integration limits, which were taken into account in the averaging procedure. The thick dark blue lines represent the calculations for the central configurations, whereas the black ones present the final results of the averaging approach. The differences between the thick dark blue lines and the black ones are clearly seen. 6.3. COMPARISONS OF THE RESULTS WITH THEORY 69 Analyzing power values obtained for a given configuration and S were weighted with a product of a° and the solid angle factor, whereas the cross section values were only corrected with the the solid angle factor. Finally, the evaluated data were placed on the E2 vs. E1 plane and projected onto the curve corresponding to the relativistic kinematics, calculated for the central geometry (91, 92, <^12). The importance of the averaging procedure is demonstrated in Fig. 6.1. This approach is similar to the analysis of the experimental data, therefore it assures that averaging of the theoretically calculated vector analyzing powers and cross sections is equivalent to event integration within the ranges accepted in the analysis. The applied procedure also projects the results of non-relativistic calculations onto relativistic kinematics. In this way they can be directly compared to the S distributions of the data, without necessity to correct for difference of arc-lengths calculated along relativistic and non-relativistic kinematic curves. Moreover, it has been checked that employing of more dense grid of angles for averaging has no influence on the results. 6.3 Comparisons of the results with theory The obtained cross section values and the vector analyzing power data for the d-p breakup reaction at 130 MeV were compared to the state-of-the-art theoretical calculations, described briefly in Sec. 3. The theoretical predictions were obtained with realistic NN potentials (refered to as 2N), with the NN forces combined with the TM99 3NF (2N+TM99), as well as with the AV18 potential combined with the Urbana IX 3NF (AV18+UIX) and also with the Coulomb interaction included (AV18+UIX+C). In addition calculations within the coupled-channel approach with the CD Bonn + A potential (CDB+A) and with the Coulomb force included (CDB+A+C) were also used in comparisons. Moreover, the data were confronted with the prediction based on the ChPT framework at two orders: N2LO and incomplete N3LO. In the figures the results are presented as bands reflecting the spread of the results of calculations obtained with the following realistic potentials: CD Bonn, AV18, Nijm I and Nijm II. In a similar way the above 2N potentials complemented with the TM99 3NF are presented. The ChPT results are also shown as bands, but in this case width of the band represents theoretical uncertainty estimated along with the calculations. The remaining approaches are shown as lines. A sample set of the obtained cross section values and the analyzing powers were presented in Figs. 5.31 and 5.33, whereas the whole bulk of the evaluated data is displayed in Appendices A and B, respectively. In order to quantitatively inspect the description of the whole data set provided by various models and to identify regions where some interesting effects or problems exist, the value of x2 per degree of freedom have been calculated. This kind of calculations was performed for each observable i.e. Ax, Ay and the cross section, for each type of the theoretical prediction. In case of the theoretical results presented in the figures as bands, the X2 values were calculated with respect to the center of the band. Table 6.1 presents the global x2 per degree of freedom for the whole data sets of the differential cross section and of the polarization observables Ax and Ay. These results are also shown graphically in Figs. 6.2 and 6.3. Based on the above information one can draw some global conclusions concerning the 70 CHAPTER 6. RESULTS THEORY OBSERVABLE da5 dQidQydS Ax Ay 2N 26.0 0.75 1.57 2N+TM99 26.5 0.75 1.59 ChPT N2LO 18.6 0.76 1.60 ChPT N3LO 17.0 0.75 1.55 CDB+A 18.0 0.75 1.56 CDB+A+C 3.0 0.74 1.55 AV18+UIX 17.3 0.74 1.57 AV18+UIX+C 3.2 0.74 1.54 Table 6.1: Global x2 per degree of freedom for the experimental cross section and analyzing powers values with respect to different theoretical predictions. In the case of cross section important are the differences between various model approaches. Figure 6.2: Global x2/d.o.f. for the cross section data with respect to various model predictions presented as histogram. See remark in Tab. 6.1 caption. theoretical description of the data. In case of Ax, all values of X2/d.o.f agree with each other and are smallest than 1, what can be due to the overestimated statistical errors. For Ay the obtained values of x2/d.o.f are higher, around 1.55, independently on the considered theoretical prediction. This fact indicates that the calculational approaches in this case are less successful in describing the data. In general one can conclude that none of the two vector analyzing powers, Ax and Ay, reveal any significant sensitivity to the dynamical effects and that they are quite well reproduced by the state-of-the-art calculations. The calculations predict the small values (in the investigated region) of Ax and Ay correctly over the whole data set and, in contrast to the obtained cross section values, no sensitivity to the Coulomb interaction is visible. The global features of the cross section data were also investigated. Quality of these data strongly depends on the obtained correction factors related to the efficiencies of 6.3. COMPARISONS OF THE RESULTS WITH THEORY 71 Figure 6.3: Global x2/d.o.f. results from Table 6.1 presented as histogram and grouped with respect to the type of the theoretical model, with colours differentiating between the Ax and Ay observables. the detection system. As it was mentioned in Sec. 5.9.1 and discussed in detail in Sec. 5.11, the efficiency analysis provided very high systematic errors, which affect several breakup configurations characterized by rather small values of the relative azimuthal angles <^i2 < 1200. Size of this effect depends on the kinematical configuration within the mentioned above range of <^12. The configurations in which the obtained breakup cross sections are biased with the highest systematic uncertainties are listed in Table 6.2. The systematic errors connected with these particular configurations are presented in Sec. 5.11, in Table 5.3. In the x2/d.o.f. analysis the experimental points belonging to the “unreliable” configurations were biased with their systematic uncertainties instead of the statistical ones. The obtained global value of x2/d.o.f. with respect to the theories with the Coulomb interaction included is about 3. The values of x2/d.o.f. obtained with respect to the theories without Coulomb interaction included are about six (N2LO, N3LO, CDB+A, AV18+UIX) and nine (2N, 2N+TM99) times larger than for the calculations based on the CDB+A+C, AV18+UIX+C potentials. Ad = 9i - d2 [0] Ai2 H 0 2 4 6 8 20 7/7, 9/9,11/11 5/7, 7/9, 9/11,11/13 5/9, 7/11, 9/13 5/13 40 5/9, 7/11, 9/13 5/11, 7/13 60 5/11, 7/13 80 5/11, 7/13 5/13 100 5/13 Table 6.2: The configurations very strongly affected by calculations of the detection system efficiencies. The table lists pairs of 01/02 angles (in degrees). 72 CHAPTER 6. RESULTS Figure 6.4: Example of a map of x2/d.o.f values, calculated for the measured distributions of the vector analyzing power Ay compared to predictions of the AV18+UIX+C. Cells of the map refer to individual configurations defined by combination of angles given on the axes. The same convention is in force for the calculated x2/d.o.f values of the cross sections but on a grid used in that case. In conclusion, only the theories with implemented electromagnetic interaction reproduce the experimental data of the cross sections in a reasonable way. Moreover, the obtained results of x2/d.o.f. for 2N potentials and for the AV18+UIX predictions can indicate the importance of the 3NF for the description of the data. In contrary to the analyzing power results, here the dynamical effects are significant. These global features can be investigated more carefully with the use of the maps of X2/d.o.f values, which were calculated for each individual geometrical configuration of the two outgoing protons. An example of such a map is shown in Fig. 6.4. The obtained values of x2/d.o.f for all configurations are plotted on the 9i,92 (polar angles of the two protons) vs. pi2 (the relative azimuthal angle) plane as colour boxes. Colour of each individual box is related to the x2/d.o.f value for the whole distribution of the scrutinized observable (summed along S), calculated with respect to a particular theory. The full set of such maps for cross sections and polarization observables is presented in the next two subsections. 6.3.1 The vector analyzing powers - individual configurations Figs. 6.5 and 6.6 present the full set of the obtained X2/d.o.f values for each individual configuration. Based on this maps few conclusions can be drawn. A quite good description of the vector analyzing power Ax data is confirmed in practically the full studied range of the phase-space, except two configurations with the uncommonly high value of the X2/d.o.f., close to 1.4 for each considered theory. In the case of the vector analyzing power Ay one can notice that in the majority of the configurations value of the x2/d.o.f. is higher than 1, what is true for all used calculations. 6.3. COMPARISONS OF THE RESULTS WITH THEORY 73 Figure 6.5: Set of x2/d.o.f. maps, analogical to the example presented in Fig. 6.4, obtained for vector analyzing powers Ax compared to various theoretical predictions. Ax for <^12 = 1800 is required (parity constraints) to be zero. Figure 6.6: Set of x2/d.o.f. maps, analogical to the example presented in Fig. 6.4, obtained for vector analyzing powers Ay compared to various theoretical predictions. 74 CHAPTER 6. RESULTS Nevertheless, there are certain configurations with the x2/d.o.f. smaller than 1, and only one with extremely high value of the x2/d.o.f. close to 4.5, what is observed for all theoretical predictions. One can conclude that there is evidently a problem with the description of Ay by the presently available theoretical approaches. The obtained values of the Ax and Ay analyzing powers are very small and as it was mentioned above, they do not reveal any sensitivity to neither the 3NF nor Coulomb effects in the studied part of the phase-space. Apart from the studies of the global x2 and examinations of the individual kinematical configurations, the analysis with respect to other kinematical variables was performed. For that purpose the variables were chosen to be: the energy of the relative motion of the two protons E12, relative azimuthal angle <^12 of the breakup protons, pair of the polar angles of the two protons d1, d2. In the first case (Fig. 6.7) the x2/d.o.f. values were calculated for all experimental points grouped with respect to the energy of the relative motion of the two protons. The results obtained for Ax presented in Fig. 6.7 are distributed randomly with respect to the X2/d.o.f. values, which are smaller than 1 for the majority of the data points. There is no significant differences observed between the used for comparisons theoretical predictions. In the case of Ay the majority of the points have x2/d.o.f. around 1.5. At the highest E12 values some systematic increase of the x2/d.o.f. values can be observed, what can be a hint to certain dynamical origin, missing in the theories. The data sorted according to the <^12 values are presented in Fig. 6.8. The results are quite consistent with each other considering various theoretical predictions for both observables. For Ay the distribution is rather random, whereas for Ax the experimental points agree with each other. Figure 6.7: Quality of description of the vector analyzing power data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on the relative energy of the two breakup protons. 6.3. COMPARISONS OF THE RESULTS WITH THEORY 75 Figure 6.8: Quality of description of the vector analyzing power data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on the relative azimuthal angle <^12 of the two breakup protons. Figure 6.9: Quality of description of the vector analyzing power data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on combination of the proton emission polar angles. 76 CHAPTER 6. RESULTS Examining the distributions shown in Fig. 6.9 one can notice that various theoretical approaches agree with each other, except the one point described with d1 = 90, d2 = 90. Here also random pattern of x2/d.o.f. distributions is observed. In conclusion, the investigated region of the phase-space does not reveal any interesting effects connected with the 3N dynamics and in general the data confirm all theoretical models. 6.3.2 The cross section values - individual configurations The conclusion of the importance of the Coulomb force for the description of the experimental data is also confirmed when studying x2/d.o.f. for the individual configurations. Such results are presented in Fig. 6.3.2. As one can notice, the smallest values of x2/d.o.f. are obtained when the data are compared with the CDB+A+C and AV18+UIX+C approaches and, in general, the same pattern is observed for both of the predictions. However, there exist configurations with quite large values of x2/d.o.f., which appear for the +12 < 1200 and can indicate indigences in the model providing the efficiency correction factors for the cross section data. In the map one can also observe areas characterized by the values of x2/d.o.f < 2, which are present in all panels related to different theories. In these particular configurations different theoretical predictions agree with each other and do not reveal sensitivity to any dynamical effect. In case of the calculations which do not take into account the Coulomb force the obtained values of x2/d.o.f are generally very high and can even achieve values of x2/d.o.f~300. This fact confirms that these models fail to reproduce the experimental data in the majority of the investigated configurations, what is reflected in the pattern of the maps related to these theories. Importance of the Coulomb force for proper description of the experimental data also demonstrates itself in the analysis with respect to the kinematical variables like relative azimuthal angle +12 of the breakup protons, pair of the polar angles of the two protons d1, d2 and the energy of the relative motion of the two protons E12. These dependencies are presented in Figs. 6.11 - 6.13. In the first case (see Fig. 6.11) the obtained values of X2/d.o.f for Coulomb-containing predictions are between 2 and 4. One can observe that for <^12 = 80° the inclusion of the electromagnetic interaction almost does not change the cross sections. In case of the rest of the theoretical predictions one can distinguish two groups (2N, 2N+TM99, N2LO and CDB+A, AV18+UIX, N3LO), which provide slightly different, but generally large values of x2/d.o.f. Better description is obtained for the models of the second group. Moreover, quite significant 3NF effects are also visible when comparing the red and black circles (about 10%), as well as the cyan squares and black circles. The obtained dependency of x2/d.o.f. on different combinations of d1;d2 (see Fig. 6.12) can provide some interesting conclusion. With increasing Ad =| d1 — d2 | value the Coulomb effects play less important part in reproducing the experimental data and for high Ad all the theories predict almost the same values of the cross sections. The results obtained for calculations with respect to the energy of the relative motion of the two protons E12 (see Fig. 6.13) stay in agreement with the previous conclusions. In the case of small values of E12 the Coulomb effects are extremely high and disagreement between the experimental data and theoretical predictions decreases with the increasing relative energy. For very large E12 > 5.5 MeV the discrepancies are still present, but become much smaller. This range of E12 corresponds to a quick separation of the two Figure 6.10: Set of \2/d.o.f. maps, analogical to the example presented in Fig.6.4, obtained for the differential cross sections and compared to various theoretical predictions. 6.3. COMPARISONS OF THE RESULTS WITH THEORY 78 CHAPTER 6. RESULTS Figure 6.11: Quality of description of the cross section data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on the relative azimuthal angle <^12. Points with very large x2/d.o.f. values are scaled down by factors indicated in the panels. Figure 6.12: Quality of description of the cross section data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on the combination of the proton emission polar angles. Points with very large x2/d.o.f. values are scaled down by factors indicated in the panels. 6.3. COMPARISONS OF THE RESULTS WITH THEORY 79 Figure 6.13: Quality of description of the cross section data with various theoretical predictions (defined in the legend), expressed as dependence of x2/d.o.f. on the relative energy of the two breakup protons E12. Points with very large x2/d.o.f. values are scaled down by factors indicated in the panels. protons, and therefore is less sensitive to the electromagnetic interaction. In general, one can conclude that the Coulomb force is a very important ingredient in the 3N system dynamics. The inclusion of the electromagnetic interaction in the theories which model the 3N system properties dramatically changes the quality of the data descriptions. 80 CHAPTER 6. RESULTS Chapter 7 Summary and conclusions The goal of this dissertation was the experimental investigation of the 1H(d,pp)n breakup reaction at 130 MeV in the forward angular region. The measurement was performed on one of the external beam lines of the COSY accelerator in the Research Center Jülich. The Germanium Wall detection system used in this experiment covered a very narrow range of the forward angles for the breakup process. As the result of the data analysis , 5 the vector analyzing powers Ax and Ay, as well as the differential cross sections düld[çl2dS of the investigated reaction have been obtained. The vector analyzing powers have been evaluated for 42 kinematical configurations resulting in about 300 data points. The values of cross sections have been analyzed for 145 kinematical configurations, what forms a data base of about 2700 data points. In addition, as a byproduct of the beam polarization studies, angular distributions for the iTu analyzing power of the d — p elastic scattering process at the same energy have been obtained. These results complemented the existing data base in the angular area not covered by other experiments. The results have been compared to the theoretical predictions which model the nuclear dynamics in various ways. They comprise approaches based on the purely realistic NN (CD Bonn, AV18, Nijm I, Nijm II) potentials and combining them with the TM99 3NF and Urbana IX models. Moreover, the data are confronted with the calculations of the coupled-channel approach obtained with the CD Bonn + A potential and the similar predictions but including the Coulomb interaction. Finally, they are compared to the results obtained within the ChPT framework at N2LO including full dynamics and, currently not complete, at N3LO. For the vector analyzing powers all the theoretical predictions agree with each other and do not reveal any sensitivity to the dynamical effects like 3NF or Coulomb interaction. In the case of Ax the theoretical models quite well describe the experimental data, whereas they fail to reproduce Ay equally well. From the theoretical point of view, the situation is totally different for the predicted values of the differential cross sections in the context of the Coulomb force. Only the models with the Coulomb interaction included (i.e. CD Bonn+A+C, AV18+UIX+C) not only stay in good agreement with each other, but also reproduce the evaluated data in a consistent way. These general conclusions can be drawn already for the configurations described with p12 > 1200. The most sensitive region of the phase-space characterized by the geometries with very small relative azimuthal angles p12 turned out to be very challenging for investigations of such effects. The assumed model which provides correction factors of the detection system efficiencies to the cross section data, what is crucial in this 81 82 CHAPTER 7. SUMMARY AND CONCLUSIONS analysis, fails to completely reproduce the overlapping clusters effects of the Quirl detector and needs further developments. Fortunately, even in the region tp12 < 1200 there is a large number of configurations which are not much affected by the efficiency corrections and for which absolute values of the cross sections are not biased by large systematic uncertainties. The results obtained in the x2/d.o.f analysis confirm the necessity of inclusion of the electromagnetic interaction into the calculations in order to assure reasonable agreement between the theoretical conclusions and the experimental data. Influences of the 3NF effects on the cross section data are rather small, but still visible, in the studied region of the phase-space. Appendix A Breakup Cross Sections Results This Appendix contains experimental results of the differential cross sections for the reaction 1H(d,pp)n at 130 MeV. The results are obtained for the two protons registered at given 91, 92 and p12 angles, with the event integration ranges of A91 = A92 = 20 and Ap12 = 10°. The evaluated data are compared with a set of theoretical predictions, presented in the figures as colour bands and lines listed in the included legend. The experimental results and theoretical calculations are presented as a function of the arc-length S along the kinematical curve. In the figures the error bars represent the statistical uncertainties only. Configurations biased with large systematic uncertainties appear in the sets of figures as the ones with red frames. 83 84 Appendix A Figure 7.1: The differential breakup cross sections for the same relative azimuthal angle p12 = 20°. Cyan and magenta bands represent calculations based on realistic potentials (with and without 3NF included, respectively), green and orange - chiral theories (at N2LO and N3LO, respectively), black lines - calculations of the coupled channel approach (solid with and dashed without Coulomb interaction included), dashed maroon line -calculations based on the realistic AV18 potential combined with the Urbana IX 3NF and dotted violet line - the same predictions, but with the Coulomb force implemented. Appendix A 85 Figure 7.2: Same as in Fig. 7.1, but for p12 = 400. 86 Appendix A Figure 7.3: Same as in Fig. 7.1, but for <^l2 = 60°. Appendix A 87 Figure 7.4: Same as in Fig. 7.1, but for p12 = 80°. 88 Appendix A Figure 7.5: Same as in Fig. 7.1, but for p12 = 1000. Appendix A 89 Figure 7.6: Same as in Fig. 7.1, but for = 1200. 90 Appendix A Figure 7.7: Same as in Fig. 7.1, but for pi2 = 1400. Appendix A 91 Figure 7.8: Same as in Fig. 7.1, but for c12 = 1600. 92 Appendix A Figure 7.9: Same as in Fig. 7.1, but for <^12 = 1800. Appendix B Breakup Vector Analyzing Powers Results This Appendix contains experimental results of the vector analyzing powers for the reaction 1H(d,pp)n at 130 MeV. The results are obtained for the two protons registered at given 91, 92 and p12 angles, with the ranges of A91 = A92 = 30 and Ap12 = 400. The evaluated data are compared with a set of theoretical predictions, which are presented in the figures as colour bands and lines listed in the included legend. The experimental results and theoretical calculations are presented as a function of the arc-length S along the kinematical curve. In the figures the error bars represent the statistical uncertainties only. 93 94 Appendix B Figure 7.10: The vector analyzing powers Ax for the same relative azimuthal angle