Proton-${}^{3}\rm He$ elastic scattering at intermediate energies

One of the open questions in nuclear physics nowadays is a complete knowledge of the interactions acting among nucleons. Modern nucleon-nucleon ($NN$) potentials reproducing the $NN$ observables up to 350 MeV with very high precision do not describe that well various nuclear phenomena, e.g., some few-nucleon scattering observables, nuclear binding energies, and nuclear matter properties Glöckle, et al. (1996); Carlson, et al. (2015). The three-nucleon forces (3$N$Fs), arising naturally in the standard meson-exchange picture Fujita and Miyazawa (1957) as well as in the chiral effective field theory ($\chi$EFT) van Kolck (1994); Epelbaum, et al. (2009), have been suggested as possible candidates to improve the situation. Few-nucleon reactions offer good opportunities to investigate the nature of the nuclear interactions since rigorous numerical calculations with two-nucleon ($2N$) and $3N$ forces and high-precision experiments are feasible.

Nucleon-deuteron $(Nd)$ elastic scattering at energies above $\approx 60~{}\rm MeV/nucleon$ has been considered as a solid basis to explore the nuclear interactions focusing on the 3$N$Fs. Sizable discrepancies between the data and rigorous numerical calculations with realistic $NN$ potentials were found in the cross section minimum Sakamoto, et al. (1996). They were successfully explained by inclusion of the two-$\pi$ exchange $3N$F models that reproduce the binding energies of ${}^{3}\rm H$ and ${}^{3}\rm He$ Witała, et al. (1998); Sakai, et al. (2000); Witała, et al. (2001), or substantially reduced by calculations in an extended Hilbert space which allow the explicit excitation of a nucleon to a $\Delta$ isobar, yielding an effective $3N$F Nemoto, et al. (1998); Deltuva, et al. (2003). It has been recently reported that the deuteron-proton ($d$-$p$) elastic cross section at 70 MeV/nucleon Sekiguchi, et al. (2002) constrains low-energy constants of 3$N$Fs in $\chi$EFT Epelbaum, et al. (2019).

The four-nucleon ($4N$) system has also become a test field for modern nuclear forces. It is the simplest system of investigating the nuclear interactions in $3N$ subsystems with the total isospin $T=3/2$, whose importance is suggested in asymmetric nuclear systems, e.g., neutron-rich nuclei Pieper, et al. (2001) and pure neutron matter Gandolfi, et al. (2014); Lynn, et al. (2016). In recent years remarkable theoretical studies in solving the $4N$ scattering problem with realistic Hamiltonians have been reported Deltuva and Fonseca (2007); Lazauskas (2009); Viviani, et al. (2013) even above the $4N$ breakup threshold Deltuva and Fonseca (2013); Fonseca and Deltuva (2017), opening new possibilities for nuclear force study in the $4N$ system at intermediate energies.

Thanks to developments in technology for high quality polarized and unpolarized proton beams together with sophisticated techniques for the polarized ${}^{3}\rm He$ target system, $p$-${}^{3}\rm He$ scattering has an experimental advantage that allows high precision measurements of the cross section and a variety of spin observables. Indeed, at proton energy below $50~{}\rm MeV$ rich data sets for $p$-${}^{3}\rm He$ elastic scattering are available, covering the cross section Lovberg (1956); Famularo, et al. (1954); Brolley, et al. (1960); Clegg, et al. (1964); McDonald, et al. (1964); Kim, et al. (1964); Drigo, et al. (1966); Harbison, et al. (1970); Hutson, et al. (1971); Murdoch, et al. (1984); Viviani, et al. (2001); Fisher, et al. (2006), proton analyzing power McDonald, et al. (1964); Harbison, et al. (1970); Baker, et al. (1971); Jarmie and Jett (1974); Birchall, et al. (1984); Alley and Knutson (1993); Viviani, et al. (2001); Fisher, et al. (2006), $\rm{}^{3}He$ analyzing power Baker, et al. (1971); Alley and Knutson (1993); Müller, et al. (1978); McCamis, et al. (1985); Daniels, et al. (2010), and spin-correlation coefficients Baker, et al. (1971); Alley and Knutson (1993); Daniels, et al. (2010). However, the existing data basis is rather poor at higher energies, mostly limited to the cross section Votta, et al. (1974); Goldstein, et al. (1970); Wesick, et al. (1985); Langevin-Joliot, et al. (1970) and the proton analyzing power Wesick, et al. (1985). Few data exist for the ${}^{3}\rm He$ analyzing power and the spin correlation coefficients Shimizu, et al. (2007).

As for the theoretical descriptions of $p$-${}^{3}\rm He$ elastic scattering, calculations taking into account the $2N$ and $3N$ forces are reported in the framework of the Kohn variational approach up to 5.54 MeV (below $3N$ breakup threshold)  Viviani, et al. (2013). At these lower energies, the data are well explained by the calculations with the $2N$ interactions. The exception is the proton analyzing power $A_{y}$, for which the so-called $A_{y}$ puzzle exists as seen in the $Nd$ elastic scattering. At higher energies, calculations in the framework of the Alt-Grassberger-Sandhas (AGS) equation are presented using various realistic $NN$ potentials up to 35 MeV Deltuva and Fonseca (2013); Fonseca and Deltuva (2017). Although calculations including $3N$Fs have not been done for energies above the breakup threshold so far, these works performed coupled-channel calculations with the $\Delta$-isobar degree of freedom as an alternative source of $3N$ and $4N$ forces. At incident energies above 20 MeV, the calculations with realistic $NN$ potentials underpredict the cross section data in the minimum, as observed also in the $Nd$ elastic scattering but at higher center-of-mass energy. The $\Delta$-isobar effects slightly improve the agreement with the data for the cross section. In line with this feature it would be interesting to see how the theoretical calculations based on realistic nuclear potentials explain the data for $p$-${}^{3}\rm He$ elastic scattering at intermediate energies.

In this paper we present the first precise data set for $p$-${}^{3}\rm He$ elastic scattering at intermediate energies, the cross section $d\sigma/d\Omega$ and the proton analyzing power $A_{y}$ at 65 MeV, and the ${}^{3}\rm He$ analyzing power $A_{0y}$ at 70 MeV spanning a wide angular range. In addition, we present the spin correlation coefficient $C_{y,y}$ at 65 MeV at the angles of $46.6^{\circ}$, $89.0^{\circ}$, $133.2^{\circ}$ in the center-of-mass system. The data are compared with rigorous numerical calculations for $4N$-scattering based on various realistic $NN$ potentials as well as with the $\Delta$-isobar excitation, in order to explore possibilities of $p$-${}^{3}\rm He$ elastic scattering as a tool to study nuclear interactions. However, the Coulomb force is omitted this time, thus the theoretical predictions of the present work in fact refer to the mirror reaction $n$-${}^{3}\rm H$.

In Sec. II, we describe the experimental procedure and the data analysis. Section III presents a comparison between the experimental data and the theoretical predictions, and discussion follows in Sec. IV. Finally, we summarize and conclude in Sec. V.

The measured cross section $d\sigma/d\Omega$, the proton analyzing power $A_{y}$, the ${}^{3}\rm He$ analyzing power $A_{0y}$, and the spin correlation coefficient $C_{y,y}$ are shown in Fig. 4 as a function of the center-of-mass (c.m.) scattering angle $\theta_{\rm c.m.}$ together with the theoretical calculations. The observables for the $p$-${}^{3}\rm He$ elastic scattering were calculated from the solutions of exact AGS equations as given in Refs. Deltuva and Fonseca (2013); Fonseca and Deltuva (2017) using a number of $NN$ potentials: the Argonne $v_{18}$ (AV18) Wiringa, et al. (1995), the CD Bonn Machleidt (2001), and the INOY04 Doleschall (2004). The calculations based on two semilocal momentum space regularized chiral $NN$ potentials of the fifth order ($\rm N^{4}LO$) with the cutoff parameters $\Lambda=400~{}{\rm MeV}/c$ (SMS400) and $\Lambda=500~{}{\rm MeV}/c$ (SMS500) Reinert, et al. (2018) are also presented. In addition, to test the importance of $3N$ and $4N$ forces in the $p$-${}^{3}\rm He$ elastic scattering, the calculations based on the CD Bonn$+\Delta$ model Deltuva, et al. (2003), which allows an excitation of a nucleon to a $\Delta$ isobar and thereby yields effective $3N$Fs and $4N$Fs, are presented.

The AGS equations for $4N$ transition operators are solved in the momentum-space partial wave representation Deltuva and Fonseca (2013) including $NN$ waves with total angular momentum below 4. Since the rigorous treatment of the Coulomb force requires the inclusion of much higher partial waves, the Coulomb force is omitted in the present study. Given relatively high energy, it is expected to be significant at small angles up to $\theta_{\rm c.m.}\sim 40^{\circ}$ only.

As shown in Fig. 4, the calculations with the $NN$ forces underestimate $d\sigma/d\Omega$ at the backward angles $\theta_{\rm c.m.}\gtrsim 80^{\circ}$. It is also found that there is a large sensitivity of the calculations to the input $NN$ forces at the minimum region. INOY04, which is fitted to reproduce the ${}^{3}\rm He$ binding energy, provides a better description of the data, but it still underestimates the data. In addition, $\Delta$-isobar contributions in the $p$-${}^{3}\rm He$ elastic scattering, which are estimated by the difference between the CD Bonn+$\Delta$ and CD Bonn calculations, are clearly seen to the limited angles $\theta_{\rm c.m.}\lesssim 110^{\circ}$. In the $Nd$ elastic scattering, the $\Delta$-isobar effects increase $d\sigma/d\Omega$ to reduce the discrepancy from the data over all angles Deltuva, et al. (2003).

The calculated $A_{y}$ has a relatively small sensitivity to $NN$ forces, and the description of $A_{y}$ is moderate. Small but visible effects of the $\Delta$ isobar are predicted by CD Bonn+$\Delta$, which leads to a better agreement with the data depending on the measured angles.

As for $A_{0y}$, the calculations based on the $NN$ potentials are close to each other. The $A_{0y}$ data deviate largely from the $NN$ force calculations at the minimum $\theta_{\rm c.m.}\sim 90^{\circ}$ as well as the maximum $\theta_{\rm c.m.}\sim 140^{\circ}$, which was not seen at lower energies Viviani, et al. (2013); Deltuva and Fonseca (2013); Fonseca and Deltuva (2017). The $\Delta$-isobar effects shift the calculated results slightly but in the wrong direction at $\theta_{\rm c.m.}\sim 100^{\circ}$.

For $C_{y,y}$, the angular dependence looks quite different from that at lower energies Viviani, et al. (2013); Deltuva and Fonseca (2013); Fonseca and Deltuva (2017), and large $\Delta$-isobar effects are predicted at the angles $\theta_{\rm c.m.}=100^{\circ}$–$140^{\circ}$. The data at the limited angles have moderate agreements to all the calculations with no definite conclusions for the $\Delta$-isobar effects.

In Refs. Deltuva and Fonseca (2013); Fonseca and Deltuva (2017), it was found that $d\sigma/d\Omega$ for the $p$-${}^{3}\rm He$ elastic scattering calculated with the AV18, CD Bonn, and INOY04 $NN$ potentials at lower energies of 7–35 MeV scale with the binding energy (B.E.) of ³He. Inspired by this fact we investigate the scaling relation between B.E.(³He) and the cross section at 65 MeV in panels (a)–(c) of Fig. 5. In the minimum region of angles $\theta_{\rm c.m.}=80^{\circ}$–$150^{\circ}$ the calculated $d\sigma/d\Omega$, normalized by the corresponding experimental data, are plotted as a function of B.E.(³He). Linear correlations, shown as red straight lines in the figure, exist for the calculations based on the $NN$ potentials including the two chiral $NN$ potentials, i.e., SMS400 and SMS500. From the correlation lines one can predict the $d\sigma/d\Omega$ corresponding to the $NN$ potential that reproduces the experimental B.E.(${}^{3}\rm He$), which almost coincides with the INOY04 result. As shown in the figure, the predictions underestimate the experimental $d\sigma/d\Omega$ by 20–30%. Note that the calculations with CD Bonn+$\Delta$, that will be discussed later, are not included in the fitting of the correlation lines.

In panels (d)–(f) of Fig. 5, we also demonstrate the scaling relation between B.E.(${}^{3}\rm H$) and the cross section for the deuteron-proton ($d$-$p$) elastic scattering at 70 MeV/nucleon. The calculated $d\sigma/d\Omega$ at 70 MeV/nucleon, normalized by the experimental data of the $d$-$p$ elastic scattering Sekiguchi, et al. (2002), are plotted as a function of B.E.(${}^{3}\rm H$) Deltuva ; Witała, et al. (2003); the Coulomb interaction is not taken into account. A similar scaling also exists in the $d$-$p$ elastic scattering as seen in the $p$-${}^{3}\rm He$ scattering. The red straight lines are obtained by fitting the calculations with $NN$ potentials: AV18, CD Bonn, Nijmegen I, II Stoks, et al. (1994), and INOY04. These lines, as in the case of the $p$-${}^{3}\rm He$ scattering, allow us to predict $d\sigma/d\Omega$ with a $NN$ potential that reproduces the experimental B.E.(${}^{3}\rm H$). The panels (d)–(f) of Fig. 5 show that the predictions underestimate the experimental $d\sigma/d\Omega$ by 10–20%. In the figure, the calculations taking into account the Tucson-Melbourne’99 2$\pi$-exchange $3N$F (TM99-$3N$F) Coon and Han (2001) in which the cutoff parameter is fitted to reproduce the experimental B.E.(${}^{3}\rm H$) for each combined $NN$ potential, i.e., AV18, CD Bonn, Nijmegen I and II, are presented Witała, et al. (2003); Sekiguchi, et al. (2017). In addition, calculations including an irreducible $3N$ potential contribution to CD Bonn+$\Delta$ which reproduces the experimental B.E.(${}^{3}\rm H$), the model U2 of Ref. Deltuva and Sauer (2015), are also shown. The calculated results provide good agreements with the experimental data, indicating strong evidence for the need to include the $3N$Fs. The discrepancy between the data and the predictions with a $NN$ potential that reproduces the experimental B.E.($3N$) for the $p$–${}^{3}\rm He$ elastic scattering at the cross section minimum angles is similar in size or even larger than that for the $d$-$p$ elastic scattering at a similar incident energy. It should be interesting to see whether the combinations of $2N$ and $3N$ forces, that give good descriptions of the $d$-$p$ scattering cross section, explain the data for the $p$-${}^{3}\rm He$ scattering.

An interesting feature found in the above mentioned correlations is that dependence of the calculated $d\sigma/d\Omega$ on B.E.($3N$) for the $d$-$p$ scattering is smaller than that for the $p$-${}^{3}\rm He$ scattering. It is quantified by the gradients of the correlation lines: $\sim 0.1/\rm MeV$ for the $d$-$p$ scattering and $\sim 0.3/\rm MeV$ for the $p$-${}^{3}\rm He$ scattering. There is a speculation that a weaker sensitivity for the $d$-$p$ elastic cross section is related to dominance of the total $3N$ spin $S=3/2$ state (quartet state) in the nucleon-deuteron elastic scattering Koike and Taniguchi (1986). As for the neutron-deuteron $s$-wave scattering length, the quartet state is insensitive to the difference of $NN$ potentials because of the Pauli principle, which prevents two neutrons getting close to each other Bahethi and Fuda (1972); Witała, et al. (2003). Therefore it is expected that relatively large dependence seen in the $p$-${}^{3}\rm He$ elastic cross sections is a reflection of medium- and short-range details of the nuclear interactions including $3N$Fs.

In the following, we discuss the $\Delta$-isobar effects. As shown in panels (a)–(c) of Fig. 5, the calculations of the CD Bonn+$\Delta$ model for the $p$-${}^{3}\rm He$ elastic scattering are off from the correlation lines at backward angles and move in a direction opposite to the experimental data. Thus, the $\Delta$-isobar effects do not improve the agreement with the data. Meanwhile, as shown in panels (d)–(f) of Fig. 5, the calculations of the CD Bonn+$\Delta$ model for the $Nd$ elastic scattering are off from the correlation lines, but the $\Delta$-isobar effects provide a better agreement with the data.

It is known that effective $3N$ and $4N$ forces due to the excitation of a nucleon to a $\Delta$ isobar are often partially canceled by $\Delta$-isobar effects of $2N$ nature, the so-called $2N$ dispersion Nemoto, et al. (1998); Deltuva, et al. (2003, 2008); Nemoto (1999). To study $\Delta$-isobar effects more in detail, the effects of the $2N$ dispersion, and those of $3N$ and $4N$ forces, are singled out separately as in Ref. Deltuva and Fonseca (2007); Deltuva, et al. (2008). The results for the cross sections are shown in Fig. 6(a) as a ratio to the calculation based on the CD Bonn potential. At the minimum angles, large contributions of the $\Delta$-generated $3N$ and $4N$ forces increase the cross section values. However, together with this, there is a strong dispersive $\Delta$-isobar effect, which is opposite to the $3N$F and $4N$F effects. As a result, the net effects of the $\Delta$ isobar are small, and their effects are even reversed at $\theta_{\rm c.m.}\approx 140^{\circ}$. In the $Nd$ elastic scattering the dispersive $\Delta$-isobar effect is smaller than that of the $\Delta$-generated $3N$Fs, and then the net contributions of the $\Delta$ isobar increase the cross section Nemoto, et al. (1998); Nemoto (1999). Since the calculated $3N$ binding energy with CD Bonn+$\Delta$ is still smaller than the experimental value by about $0.2~{}\rm MeV$ Deltuva, et al. (2003, 2008), further attractive effects attributed to the irreducible $3N$Fs in $NNN$-$NN$$\Delta$ model space Deltuva and Sauer (2015) should be considered. It will be interesting to study in the future how such attractive contributions affect the cross sections for the $p$-${}^{3}\rm He$ scattering.

Regarding the spin observables, large $\Delta$-isobar effects are predicted for the spin correlation coefficient $C_{y,y}$ (see Fig. 4). Interestingly, Fig. 6(b) shows that the predicted $\Delta$-isobar effects are mainly due to the $2N$ dispersion. Experimental data of the spin correlation coefficient $C_{y,y}$ in a wide angular range are needed for a detailed discussion of the $\Delta$-isobar effects.

We have reported the precise data set for $p$-${}^{3}\rm He$ elastic scattering at intermediate energies: $d\sigma/d\Omega$ and the proton analyzing power $A_{y}$ taken at 65 MeV in the angular regime $\theta_{\rm c.m.}=26.9^{\circ}$–$170.1^{\circ}$; the ${}^{3}\rm He$ analyzing power $A_{0y}$ at $70~{}\rm MeV$ in the angular regime $\theta_{\rm c.m.}=46.6^{\circ}$–$141.4^{\circ}$, and the spin correlation coefficient $C_{y,y}$ at $65~{}\rm MeV$ for $\theta_{\rm c.m.}=46.6^{\circ}$, $89.0^{\circ}$, and $133.2^{\circ}$.

For the cross section the statistical error is better than $\pm 2\%$ and the systematic uncertainties are estimated to be 3%. The absolute values of the cross section were deduced by normalizing the data to the $p$-$p$ scattering cross section given by the phase-shift analysis program SAID. For the proton and ${}^{3}\rm He$ analyzing powers the statistical uncertainties are less than 0.02. They are 0.03–0.06 for the spin correlation coefficient $C_{y,y}$. The systematic uncertainties for all the measured spin observables do not exceed the statistical ones.

The data are compared with rigorous $4N$-scattering calculations based on various realistic $NN$ nuclear potentials without the Coulomb force. Clear discrepancies have been found for some of the measured observables, especially in the angular regime around the $d\sigma/d\Omega$ minimum. Linear correlations exist between the calculations of the ${}^{3}\rm He$ binding energy and those of $d\sigma/d\Omega$, which enables us to evaluate $d\sigma/d\Omega$ with a $NN$ potential that reproduces B.E.(${}^{3}\rm He$). Predicted values of $d\sigma/d\Omega$ in the minimum region clearly underestimate the data. A similar tendency is obtained in the $Nd$ elastic scattering, where discrepancies are largely resolved by incorporating $3N$Fs. The $NN$ potential dependence for the cross section in the $p$-${}^{3}\rm He$ scattering is found to be larger than that in the $d$-$p$ elastic scattering, which could allow us to anticipate a wealth of information on the nuclear interactions from further investigation of the $p$-${}^{3}\rm He$ scattering at these energies.

The $\Delta$-isobar effects in the $p$-${}^{3}\rm He$ observables are estimated by the $NN$+$N\Delta$ coupled-channel approach. They do not always remedy the difference between the data and the calculations based on the $NN$ potentials. In the case of $d\sigma/d\Omega$, large contributions of the effective $3N$ and $4N$ forces are largely canceled by the dispersive $\Delta$-isobar effect, that leads to a rather small total $\Delta$-isobar effect. The results are in contrast to those in the $d$-$p$ scattering, where the cancellation is less pronounced. Since this approach still misses the $3N$ binding energies, its extensions, e.g., the irreducible 3$N$ potential combined with the $NN$+$\Delta$ model Deltuva and Sauer (2015), are needed. Together with this, large dispersive $\Delta$-isobar effects predicted in the spin correlation coefficient $C_{y,y}$ should be investigated experimentally in the future.

From these obtained results we conclude that $p$-${}^{3}\rm He$ elastic scattering at intermediate energies is an excellent tool to explore the nuclear interactions including $3N$Fs that could not be accessible in $3N$ scattering. Recent study of the $\chi$EFT nuclear potentials intends to use the $d$-$p$ scattering data at intermediate energies to derive the higher-order $3N$Fs Epelbaum, et al. (2020). It would be interesting to see how the predictions with such $3N$Fs explain the data for the $p$-${}^{3}\rm He$ elastic scattering, which will enable us to perform detailed discussions of the effects of $3N$Fs including the $T=3/2$ isospin channels.