Spin-isospin excitation of ${}^3$He with three-proton final state

Spin-isospin excitation of ${}^3$He nucleus by proton-induced charge-exchange reaction, ${}^3\mathrm{He}(p,n)ppp$, at forward neutron scattering angle is studied in a plane wave impulse approximation (PWIA). In PWIA, cross sections of the reaction is written in terms of proton-neutron scattering amplitudes and response functions of the transition from ${}^{3}$He to three-proton state by spin-isospin transition operators. The response functions are calculated with realistic nucleon-nucleon potential models using a Faddeev three-body method. Calculated cross sections agree with available experimental data in substance. Possible effects arising from the uncertainty of proton-neutron amplitudes and three-nucleon interactions in three-proton system are examined.


Introduction
Three-nucleon (3N) systems: 3 H, 3 He, nucleon-deuteron elastic and breakup reactions, etc., have been playing important roles in the quest for the details of interactions among nucleons. These systems are essentially total isospin T = 1 2 states. (Although the breaking of charge symmetry in nuclear interaction and the Coulomb interaction allow a mixture of T = 3 2 components, its percentage is quite small [1].) On the other hand, knowledge of interactions among three nucleons, especially of three-nucleon interactions, in T = 3 2 states is expected for studies on heavier nuclei, neutron-rich nuclei, neutron-star matter, etc. Since there is no bound state of three-neutron (3n) and three-proton (3p) systems, which are typical T = 3 2 states [2], observables related these systems may be obtained from nuclear reactions that produce them as final continuum states. Reaction mechanism of such reaction needs to be simple as possible to reduce ambiguity in extracting information on the nuclear interaction.
In the present paper, I will study a charge-exchange reaction: 3 He(p, n)ppp reaction at incident proton energies of several hundreds MeV and the reaction angle θ n = 0 • . Although this is a four-body reaction that is still difficult to perform rigorous calculations at high energies, the cross section of the reaction in PWIA is written in terms of n(p, n)p (pn) scattering amplitudes and response functions of 3N system. The former can be taken from nucleon-nucleon (NN) databases [3,4]. The latter corresponds to a transition from the initial 3 He bound state to final 3p continuum states, in which one needs to solve three-body problem.
The present author has developed a method to solve the quantum mechanical three-body problem applying the Faddeev method [5]. This method is based on solving the Faddeev equation as integral equations in coordinate space, which even includes long range Coulomb force effects [6,7], and has been successfully applied for the proton-deuteron systems [8] and three alpha-particles systems [9]. In this paper, this method will be applied for calculating the response functions of 3p final states.
In Ref. [10], the cross section I(0 • ) for the 3 He(p, n)ppp reaction at the incident proton energy T p = 200 MeV was measured. In Ref. [11], the polarization transfer coefficient in the transverse direction, D N N (0 • ), and that in the longitudinal direction, D LL (0 • ), as well as I(0 • ) were measured at T p = 346 MeV. One of the measured polarization transfer coefficients, D N N (0 • ), is consistent with the corresponding pn values. However, the other one, D LL (0 • ), deviates from the pn values. The authors of Ref. [11] show that this discrepancy may be attributed to a 3p resonance with spin-parity 1 2 − .
Existence of resonant states in multi-neutron or multi-proton states has been a longstanding problem in nuclear physics. Recent compilation of the mass number A = 3 systems [2] reports negatively for the existing of A = 3 resonance. In Ref. [12], a possibility of existing of four-neutron (tetraneutron, 4n) resonant state was reported. In Ref. [13], it was shown that the existing of the 4n resonant state demands an attractive T = 3 2 three-nucleon potential (3NP) that is tremendously strong. Effects of such a 3NP on the 3p system will be studied.
In Sec. 2, I will summarize the formalisms to calculate the response functions and then observables in 3 He(p, n)ppp (θ n = 0 • ) reaction. In Sec. 3, I will show some results of calculations and compare them with available experimental data. Summary will be given in Sec.
4. In appendix, some kinematical values related to the reaction will be summarized.

Theoretical background
In this section, I will consider the charge-exchange reaction, 3 He( p, n)ppp (θ n = 0 • ) by PWIA, in which the n( p, n)p (θ n = 0 • ) scattering amplitude and response functions for the transition from 3 He to 3p continuum states are the basic elements. (See Refs. [14,15], e.g., for the general formalism of PWIA.) Kinematics of the reaction is characterized by the incident proton energy in the laboratory (Lab.) system T p , and the energy transfer in Lab. system ω Lab defined by Eq. (A2a) in Appendix. The direction of the incident proton and thus of the outgoing neutron is taken to be z-axis.
First, I introduce the 3p Hamiltonian in the center of mass (c.m.) system, whereĤ 0 is the kinetic energy operator of the three-body system, andV is an interaction potential, which consists of two-nucleon potentials (2NPs) and 3NPs.
Let |Ψ (±) m 1 m 2 m 3 (q, p) be an eigenstate of the HamiltonianĤ 3p associated with an asymptotic 3p-state, in which the relative momentum between two protons is q, the momentum of the third proton with respect to c.m. of the proton-pair is p, and the spin projection of the proton i is m i . The superscript (±) expresses the outgoing (+) or incoming (−) boundary condition.
The eigenvalue problems is written aŝ with where m p is the mass of the proton.
The pn observables, differential cross section, polarization transfer coefficients, are given as follows: In the process considered, there are three operators corresponding to each term of Eq. (9): the isovector spin-scalar operatorÔ c , the isovector spin-longitudinal operatorÔ L , and the isovector spin-transverse operatorÔ T , which are defined bŷ where Q c.m. is the momentum transfer, Eq. (A8b) in Appendix, t (+) i an isospin operator that transforms the neutron i in 3 He to proton i in the final 3p state, r i (σ i ) the coordinate vector in the 3N-c.m. system (the Pauli spin matrix) of the particle i. The corresponding response functions will be denoted as R c (E), R L (E), and R T (E), respectively, The unpolarized differential cross section I(0 • ) and the polarization transfer coefficients, where a kinematical factor N K is given in Eq. (A10) in Appendix.

Results and discussion
In this section, calculations of the observables for the reactions, 3 He(p, n)ppp (θ n = 0 • ), at T p = 346 MeV and 200 MeV will be presented and compared with available experimental data.
In solving three-body equations, 3N partial wave states for which 2NPs and 3NPs are active, are restricted to those with total NN angular momenta J ≤ 6 for bound state calculations, and J ≤ 4 for continuum state calculations. For continuum state calculations, 3N states with total angular momenta J 0 = 1 2 and 3 2 are taken into account. An error of these truncating procedures is estimated to be at most 2 % from comparisons of results with J ≤ 4 NN states and those with J ≤ 3 ones, and contributions from J 0 = 5 2 states, which demonstrates that it is good enough for the purposes of the present work.
The NN scattering length parameters of these models for 1 S 0 states: pp, nn, and pn, are compared with empirical values [21] in Table 1. As this table shows, AV8', AV14, and dTRS models are charge independent. In this work, charge-dependent version of AV14 and dTRS in Table 1 are introduced by adding potentials that break the charge independence as done in Ref. [22]. Such potentials for AV14 and dTRS are denoted by AV14(CD) and dTRS(CD), respectively.
The 3 He wave function is calculated using each 2NP model with the Brazil model of the two-pion exchange three-nucleon potential given in Ref. [23], whose cutoff mass parameter of the πNN-vertex Λ π is tuned to reproduce the empirical binding energy [8].  are compared with the experimental data of Ref. [11]. In Fig. 2, calculated values of I(0 • ) for T p = 200 MeV are compared with the experimental data [10]. In both figures, calculations of all 2NP models in Table 2 fall within narrow bands, which demonstrates small pp-2NP dependency of the observables.  Table 1 are shown by bands (light magenta). The experimental data (black points and histogram) are taken from Ref. [11]. Dashed horizontal lines (green) in (b) and (c) are the corresponding pn values in Table 2. the existence of a 3p resonance in 1 2 − state centered at ω r = 16 ± 1 MeV with the width of Γ = 11 ± 3 MeV.
Using three observables measured in Ref. [11] along with the pn amplitudes in Table 2, the response functions, R c , R L , and R T , are calculated by Eqs. (12a) -(12c). Thus obtained response functions are compared with the calculated ones with AV18 in Fig. 3. The figure shows that the extracted R L and R T have similar shape and magnitude as the calculations, but the extracted R c is a few times larger than the calculation. The resonance-like behavior in D LL (0 • ) as a function of ω Lab is reflected in R c , but not in R L and R T . However, the calculations are not able to reproduce this tendency.
The observables in this work largely owe to the pn amplitudes, which are related pn observables as Eqs.   statistical ones with respect to the averaging procedure, and effects of the experimental error are not taken into account. Fig. 4 shows the calculated observables with the above fitted pn-amplitudes (only the central values are used.), which gives a better agreement with the data, although the rapid dependence of D LL (0 • ) is not reproduced.
Next, I will study effects of 3NP on the observables. Recently, the possibility of a resonant 4n state at low energy is indicated experimentally in Ref. [12]. In Ref. [13], effects of T = 3 2 3NP on 4n-system as well as 3n-system are studied. The functional form of 3NP used in Ref. [13] is as follows.
where T = 1 2 or 3 2 , r ij is the distance between the i-th and j-th nucleons, and P ijk (T ) is a projection operator on the 3N isospin T state.
The range parameters used in Refs. [13,26] are b 1 = 4.0 fm and b 2 = 0.75 fm. The strength parameters of the shorter range term W 2 (T ) for both of T = 1 2 and T = 3 2 are fixed to be 35.0 MeV in Ref. [13], and also in this work. The required value of the strength parameter for the longer range term W 1 ( 3 2 ) for J π = 0 + 4n-state to bind as the lower bound of the experimental value [12] is −36.14 MeV [13]. This value contrasts with W 1 ( 1 2 ) = −2.04 MeV, which is determined to reproduce the binding energies of 3 H, 3 He, and 4 He in combination with Argonne V 8 ' (AV8') NN potential [18].
In the following, I will use the AV18, which is more repulsive than AV8' in 3N(T = 1 2 ) bound state. As a consequence of this, a more attractive value: W 1 ( 1 2 ) = −2.55 MeV, is used to reproduce 3 He binding energy. However, this difference may not be essential in the present case.
In Fig. 5, calculated values with V 3NP ( 3 2 ) taking W 1 ( 3 2 ) = −36.0 MeV are compared with the AV18 calculations. The introduction of the V 3NP ( 3 2 ) shifts the peak of the cross section to higher in the magnitude and lower in the position, which makes the agreement with the experimental data worse than the AV18 calculation. On the other hand, effects of the 3NP on the D N N (0 • ) and D LL (0 • ) are quite small. These rather small effects of the 3NP on the 3p system in spite of the large value of W 1 ( 3 2 ) are however consistent with the analysis of 3n systems in Ref. [13], and should be due to a large separation among three protons by the Pauli principle.
In Ref. [13], dependence of the strength parameters in the 3NP on the total angular momentum and parity J π 0 is not considered for simplicity. Here, I will examine the J π 0dependence of the parameter W 1 ( 3 2 ). In Table 3  The calculations have little NN potential dependence, and show a reasonable agreement with available experimental data, except that the energy-transfer dependence of D LL (0 • ) is much smoother than the data. Introductions of the attractive 3NP for the 3N(T = 3 2 ) state suggested from the analysis of the 4n state as well as further strength enhanced 3NPs for J π 0 = 1 2 − state so as to produce a 3p resonance state are examined. But they cannot resolve the discrepancy.
These results suggest that the curious energy-transfer dependence of the experimental D LL (0 • ), which was the basis of the existence of the 3p resonance [11], is not consistent with conventional models of the nuclear interaction, which indicates the need for further experimental studies of the reaction.
Also, a need for good knowledge of observables in n( p, n)p at the very forward angles is stressed to reduce ambiguity in the calculation.
Finally, it is remarked that precise calculations of observables related to 3n-or 3p-system with theoretical models of the nuclear interactions are now available, which enables us to compare and then to study nuclear interactions whether a 3N resonance does exist or not.