Effects of chiral three-nucleon forces on $^{4}$He-nucleus scattering in a wide range of incident energies

It is a current important subject to clarify properties of chiral three-nucleon forces (3NFs) not only in nuclear matter but also in scattering between finite-size nuclei. Particularly for the elastic scattering, this study has just started and the properties are not understood in a wide range of incident energies ($E_{\rm in}$). We investigate basic properties of chiral 3NFs in nuclear matter with positive energies by using the Brueckner-Hartree-Fock method with chiral two-nucleon forces of N$^{3}$LO and 3NFs of NNLO, and analyze effects of chiral 3NFs on $^{4}$He elastic scattering from targets $^{208}$Pb, $^{58}$Ni and $^{40}$Ca over a wide range of $30 \lesssim E_{\rm in}/A_{\rm P} \lesssim 200$ MeV by using the $g$-matrix folding model, where $A_{\rm P}$ is the mass number of projectile. In symmetric nuclear matter with positive energies, chiral 3NFs make the single-particle potential less attractive and more absorptive. The effects mainly come from the Fujita-Miyazawa 2$\pi$-exchange 3NF and slightly become larger as $E_{\rm in}$ increases. These effects persist in the optical potentials of $^{4}$He scattering. As for the differential cross sections of $^{4}$He scattering, chiral-3NF effects are large in $E_{\rm in}/A_{\rm P} \gtrsim 60$ MeV and improve the agreement of the theoretical results with the measured ones. Particularly in $E_{\rm in}/A_{\rm P} \gtrsim 100$ MeV, the folding model reproduces measured differential cross sections pretty well. Cutoff ($\Lambda$) dependence is investigated for both nuclear matter and $^{4}$He scattering by considering two cases of $\Lambda=450$ and $550$ MeV. The uncertainty coming from the dependence is smaller than chiral-3NF effects even at $E_{\rm in}/A_{\rm P}=175$ MeV.


I. INTRODUCTION
How do three-nucleon forces (3NFs) work in nuclear many-body systems? This is an important subject to be answered in nuclear physics. Even if 3NFs do not exist on a fundamental level, they come out in effective theories with a finite momentum cutoff Λ by renormalizing the degrees of freedom present above Λ. The representative example is the 2π-exchange process with intermediate nucleon excited states, typically the ∆(1232) isobar. It is now called the Fujita-Miyazawa 3NF [1]. As a phenomenological approach, attractive 3NFs were introduced to reproduce the binding energies for light nuclei [2], whereas repulsive 3NFs were used to explain the empirical saturation properties in symmetric nuclear matter [3].
Essential progress on this subject was made by chiral effective field theory (EFT) [4,5] based on chiral perturbation theory. The theory provides a low-momentum expansion of two-nucleon force (2NF), 3NF and manynucleon forces, and makes it possible to define the forces systematically. Figure 1 shows chiral 3NFs in the nextto-next-to-leading order (NNLO). Diagram (a) corresponds to the Fujita-Miyazawa 2π-exchange 3NF [1], and diagrams (b) and (c) mean 1π-exchange and con- * toyokawa@phys.kyushu-u.ac.jp tact 3NFs, respectively. The filled-square vertex has a strength c D in the diagram (b) and c E in the diagram (c). Quantitative roles of chiral 3NFs were extensively investigated, particularly for light nuclei and nuclear matter [6]; more precisely, see Ref. [7] for light nuclei, Refs. [8,9] for ab initio nuclear-structure calculations in lighter nuclei and Refs. [10][11][12][13][14][15][16] for nuclear matter. In addition, effects of chiral four-nucleon forces were found to be small in nuclear matter [17,18]. The chiral g matrix, calculated from chiral 2NF+3NF with the Brueckner-Hartree-Fock (BHF) method, yields a reasonable nuclear matter sat-uration curve for symmetric nuclear matter, when the parameters, c D and c E , of NNLO 3NFs are tuned [13]. Nuclear scattering is another place to investigate 3NF effects. The theoretical description of N +d scattering has been naturally associated with the necessity of 3NFs [7,19], when the theory starts with sophisticated 2NFs determined from the experiments. Microscopic evaluation of nuclear optical potentials for nucleon-nucleus (NA) and nucleus-nucleus (AA) elastic scattering has a long history. The g-matrix folding model [20][21][22][23][24][25] is a standard method for deriving the optical potentials of NA and AA elastic scattering microscopically. In fact, the potentials have been used to analyze various kinds of nuclear reactions in many papers. In the model, the optical potentials were obtained by folding the g matrix [20][21][22][23][24][25] with the projectile (P) density ρ P and the target (T) one ρ T . This description has been quite successful in explaining many elastic scattering. At first, the effects of 3NFs were phenomenologically investigated in Ref. [23] for NA elastic scattering and in Refs. [22,26] for NA and AA elastic scattering. The 3NFs reduce differential cross section and improve the agreement with measured vector analyzing powers. However, the role of 3NFs has not been clarified quantitatively, because the folding potential is adjusted to measured cross sections.
In Refs. [27,28], as the first attempt, we made qualitative discussion for chiral-3NF effects on elastic scattering by using the hybrid method in which the existing local version of Melbourne g matrix [21] was modified on the basis of the chiral g matrix constructed from chiral 2NFs and 3NFs. The work showed that chiral-3NF effects are small for NA elastic scattering, but important for AA elastic scattering. Recently, we directly parameterized the chiral g matrix as a local potential based on chiral 2NF+3NF, as briefly reported in Ref. [25]. In this paper, we present a full understanding of chiral-3NF effects on 4 He elastic scattering over a wide range of 30 < ∼ E in /A P < ∼ 200 MeV by using the local version of the chiral g matrix , where E in stands for an incident energy in the laboratory system and A P is the mass number of projectile.
The g matrices calculated so far are provided by a local potential with Yukawa or Gaussian form, since this procedure makes the folding calculation much easier.
Investigation of chiral-3NF effects on NA and AA elastic scattering has just started with lower incident energies per nucleon such as E in /A P ≈ 70 MeV by using the gmatrix folding model [25,27,28], since chiral EFT is more reliable for lower incident energies. As mentioned above, the folding potentials were recently calculated from the local version of chiral g matrix in Ref. [25]. The chiral g-matrix folding model accounts for experimental data considerably well on NA scattering at E in = 65 MeV and 4 He+ 58 Ni scattering at E in /A P = 72 MeV. This model also showed that chiral-3NF effects are small for NA elastic scattering, but sizable for 4 He elastic scattering.
In our previous studies for 4 He elastic scattering, we used the Melbourne g matrix in Ref. [29] and the chiral g matrices based on chiral 2NF and chiral 2NF+3NF in Ref. [25]. After Ref. [25] was published, we found some numerical errors in our nuclear-matter calculations including chiral 3NFs; see Ref. [30] for the details. In the present work, we then adopt the corrected version of chiral g-matrix; see Appendix for the matrix. Further discussion will be made later in Sec. II B.
In this paper, we first investigate basic properties of chiral 3NFs in symmetric nuclear matter for positive energies up to 200 MeV by using the BHF method with chiral 2NFs of N 3 LO and chiral 3NFs of NNLO. We show that chiral-3NF effects provide density-dependent repulsive and absorptive corrections to the single-particle potential and that the effects slightly become larger as the energy increases. We also point out that the corrections mainly come from the Fujita-Miyazawa 2π-exchange 3NF of diagram (a).
Second, we analyze chiral-3NF effects on 4 He scattering from various targets in a wide range of incident energies by using the chiral g-matrix folding model. In order to make our discussion clear, we take 4 He scattering as AA scattering, since the g-matrix folding model is confirmed to work well for 4 He scattering in virtue of negligibly small projectile-breakup effects [29,31]; see Sec. II D for further discussion. In addition, as targets we take heavier nuclei, 208 Pb, 58 Ni and 40 Ca, since the g matrix is evaluated in nuclear matter and is considered to be more suitable for heavier targets. For the targets, the experimental data are available in a wide range of In the present paper, we mostly consider the case of the cutoff scale Λ = 550 MeV. As the third subject, Λ dependence is investigated for nuclear matter with positive energies and 4 He elastic scattering by taking two other cases of Λ = 450 and 550 MeV.
Finally, we provide the local version of chiral g matrix including chiral-3NF effects with a 3-range Gaussian form for the case of E in /A P = 75 MeV. This may strongly encourage the application of the chiral g matrix for studying various kinds of nuclear reactions. This local version of chiral g matrix is referred to as "Kyushu chiral g matrix" in this paper.
In Sec. II, we present the theoretical framework composed of the BHF method and the folding model, and show some basic results of BHF calculations for chiral 2NF+3NF. In Sec. III, the results of the chiral g-matrix folding model are shown for 4 He elastic scattering. Section IV is devoted to a summary.

A. BHF equation for 2NF+3NF
We first recapitulate the BHF method for 2NF+3NF, following Ref. [12]. Because it is not easy to treat a 3NF V 123 even in nuclear matter, we introduce an effective 2NF V eff 12 by applying the mean-field approximation, or the normal ordering prescription, to the 3NF: where A means the antisymmetrization and k i corresponds to quantum numbers of the i-th nucleon. Equation (1) leads where V 12 (3) is defined by summing up 3NF V 123 over the third nucleon in the Fermi sea: with assuming the center-of-mass (c.m.) frame: Note the factor 1/3 in Eq. (2). The g matrix g 12 is a solution to the BHF equation where G 0 is the nucleon propagator with the Pauli exclusion operator in the numerator and with the singleparticle energy of the nucleon having a momentum k in the denominator. Here T is the standard kinetic-energy operator of nucleon, and the single-particle potential U(k) is defined by [12] with the effective g matrix, so-calledg matrix, including additional rearrangement terms of the 3NF origin: Note that k is related to the incident energy E in as E in = ( k) 2 /(2m) + Re [U]. The present formulation is consistent with the second-order perturbation of Ref. [32], because of the factor 1/6 in Eq. (7). For the symmetric nuclear matter where the proton density ρ p agrees with the neutron one ρ n , the Fermi momentum k F is related to the matter density ρ = ρ p + ρ n as k 3 F = 3π 2 ρ/2, so that the normal density ρ = ρ 0 = 0.17 fm −3 is realized at k F = 1.35 fm −1 .

B. Some basic results of BHF calculations
Theg matrix is calculated from chiral 2NF of N 3 LO and chiral 3NF of NNLO by using the BHF method. In BHF calculations, the form factor exp{−(q ′ /Λ) 6 − (q/Λ) 6 } is introduced for both V 12 and V 12 (3) . We mainly consider the case of Λ = 550 MeV, and take another case Λ = 450 MeV when Λ dependence of physical quantities is estimated. The low-energy constants relevant for 3NFs are (c 1 , c 3 , c 4 ) = (−0.81, −3.4, 3.4) [33] in units of GeV −1 .
As noted earlier, some errors were found in nuclearmatter calculations with chiral 3NFs of Ref. [12], after Ref. [25] was published. Although the qualitative importance of chiral 3NFs for improving nuclear matter saturation properties does not change, the saturation curve is changed by the corrections. To restore reasonable nuclear saturation properties, which are basically important for further application for microscopic derivation of nuclear optical potentials, the remaining two parameters c D and c E are tuned [30]. In consideration of the uncertainty that the c D and c E terms yield almost identical contributions when c D ≃ 4c E , c D is determined as −2.5 by setting c E = 0 for Λ = 450 MeV and next c E is fixed as 0.25 for Λ = 550 MeV with keeping c D = −2.5. These values are somewhat different from those determined in few-body systems within continuous uncertainties. It has been recognized [9], however, that low-energy-constants fixed solely in few-body systems are not adequate in heavier systems. In this article, we use the corrected version of the chiral g matrix.
It is known that chiral 3NFs make repulsive corrections to the binding energy of symmetric nuclear matter [12]. What happens in positive energy? Figure 2 shows E in dependence of U for the case of k F = 1.2 fm −1 for the cutoff Λ = 550 MeV. This density is realized in the peripheral region of a target nucleus and hence important for elastic scattering. Filled (open) circles denote the results of BHF calculations with (without) chiral 3NFs. One can see that chiral 3NFs make U less attractive and more absorptive. The 3NF corrections slightly increase as E in goes up. Our results are consistent with the second-order perturbation calculation by Holt et. al. [32]. Figure 3 shows U as a function of E in at k F = 1.2 fm −1 , but two cases of Λ = 450 and 550 MeV are taken in BHF calculations to see the uncertainty coming from Λ dependence on U. The Λ dependence is plotted as an error bar. The error bar plotted by a solid (dashed) line denotes the results of BHF calculations with (without) chiral 3NFs; note that panels (a) and (b) correspond to the real and imaginary parts of U. Particularly for BHF calculations with chiral 3NFs, there is a tendency that the uncertainty become larger as E in increases from 80 MeV. Even at E in = 175 MeV, however, chiral 3NF effects are larger than the uncertainty. This enables us to make reliable discussion on chiral-3NF effects.
In order to obtain deeper understanding of the properties of chiral 3NFs, we classifyg(k F , E in ) with the total spin S and isospin T of the interacting two-nucleon system. The total single-particle potential U is obtained by the single-particle potential U ST in each (S, T ) channel as where U ST is defined by Eq. (6) withg replaced byg ST . Figure 4 shows Here we do the following three kinds of BHF calculations: I. All kinds of chiral 3NFs, i.e., diagrams (a)-(c) in Fig. 1  calculations show that chiral 3NF effects are significant for 3 O (S = 1, T = 1) and 3 E (S = 1, T = 0) channel and the real part of 1 E (S = 0, T = 1) channel. Small circles (squares) represent the real (imaginary) part of U ST for calculation III. For 3 E and 3 O, one can see from calculations II and III that chiral 3NF effects mainly come from the Fujita-Miyazawa 2π-exchange 3NF of diagram (a). For the real part of 1 E (S = 0, T = 1) channel, the effect of diagram (a) is sizable, but it is considerably reduced by the effects of diagram (b) and (c). As a net effect of these properties, chiral 3NFs make U less attractive and more absorptive, and the repulsion mainly stems from diagram (a) in its 3 O component and the absorption does from diagram (a) in its 3 O and 3 E components. The chiral-3NF effects become more significant at larger incident energies. One can easily expect that these properties persist also in the optical potentials of 4 He scattering, since U plays a role of "optical potential" of nucleon scattering in nuclear matter. This point will be discussed later in Sec. III.   Theg matrixg(k F , E in ) of Eq. (7) is a nonlocal potential depending on k F and E in , being calculated in symmetric nuclear matter. In addition, it is obtained numerically. These properties are quite inconvenient in various applications. In order to circumvent the problem, the Melbourne group showed that elastic scattering are determined by the on-shell and near-on-shell components of g matrix [21], and provided a local version of g matrix in which the potential parameters are so determined as to reproduce the relevant components [21,34,35]. The Melbourne g matrix thus obtained well accounts for NN scattering in free space that corresponds to the limit of ρ = 0, and the Melbourne g-matrix folding model reproduces NA scattering, as already mentioned in Sec. I.
In our previous paper [25], following the Melbournegroup procedure [21,34,35], we succeeded in parameterizing a local version of chiralg matrix in a 3-range Gaussian form for each of the central, spin-orbit and tensor components. The Gaussian form makes various kinds of numerical calculations efficient. The range and strength parameters were so determined as to reproduce the onshell and near-on-shell matrix elements of the original g matrix for each spin-isospin channel, k F and E in . As for the central part, the range parameters obtained were (0.4, 0.9, 2.5) in units of fm. In this paper, we repeated this procedure for E in up to 200 MeV and parameterized a local version of chiralg matrix with good accuracy, as shown below. Since the analysis was already made at E in = 65 MeV in Ref. [25], we make the same analysis for higher energies, say E in = 150 MeV, in this paper. Whenever we have to distinguish the two types of g matrices, we call the local version ofg matrix "Kyushu chiral g matrix" and the original nonlocalg matrix "original chiral g matrix". For the case of E in = 75 MeV as an example, we present the parameter set of Kyushu chiral g matrix in Appendix A. Figure 5 shows differential cross sections as a function of c.m. scattering angle θ c.m. for p+n scattering at E in = 150 MeV in free space, i.e., in the limit of ρ = 0. The solid and dashed lines denote the results of original and Kyushu chiral t matrices, respectively; note that the g matrix is reduced to the t matrix in the limit of ρ = 0. The Kyushu chiral t matrix reproduces the result of original chiral t matrix well. Figure 6 shows k F dependence of U ST at E in = 150 MeV. Both 2NF and 3NF are taken into account in BHF calculations. The filled circles (squares) denote the results of the real (imaginary) part of original chiral g matrix, whereas the solid (dashed) lines correspond to the real (imaginary) part of Kyushu chiral g matrix. The range k F < ∼ 1.35 fm −1 (ρ < ∼ ρ 0 ) contributes to the optical potentials of 4 He scattering, when the potentials are constructed by the folding model explained in Sec. II D. In particular, the Fermi momentum k F ≈ 1.2 fm −1 , corresponding to the peripheral region of the optical potentials, is important for the elastic scattering. The Kyushu

D. Folding model
In this paper, the optical potentials are derived by folding Kyushu chiral g matrix with ρ P and ρ T for 4 He scattering on 208 Pb, 58 Ni and 40 Ca targets. In general, the folding potential is referred to as a double-folding (DF) model for AA scattering, while it is called a single-folding (SF) model for NA scattering.
In the g-matrix SF model for NA elastic scattering, the so-called local-density approximation is taken, that is, the value of ρ in g(ρ) is identified with the value of ρ T at the midpoint r m of interacting two nucleons: ρ = ρ T (r m ). Target-excitation effects on the elastic scattering are well taken into account by this framework. In fact, the Melbourne g-matrix SF model succeeded in reproducing NA scattering [21]. In our previous work [25], furthermore, we showed that the Kyushu chiral g-matrix SF model also well accounted for proton scattering at E in = 65 MeV and chiral-3NF effects are small there.
The g-matrix DF model for AA scattering had a problem to be settled. In order to obtain the g matrix applicable for AA scattering, in principle, we have to consider two Fermi spheres in nuclear-matter calculations and solve a collision between a nucleon in the first Fermi sphere and a nucleon in the second one [37,38]. However, actual calculations are not feasible. In fact, all the g matrices provided so far were obtained by assuming a single Fermi sphere and solving nucleon scattering on the Fermi sphere. For consistency with the nuclear-matter calculation, we assumed ρ = ρ T (r m ) in g(ρ) and applied the framework to 3,4 He scattering in a wide energy range of 30 < ∼ E in /A P < ∼ 180 MeV [29,31]  g-matrix DF model based on the target-density approximation (TDA) well accounted for 3,4 He scattering, particularly for forward differential cross sections where 3NF effects are considered to be negligible [22,23,26]. In our previous analysis [25], the DF-TDA model based on Kyushu chiral g matrix well explained 4 He scattering at E in /A P ≈ 72 MeV. We then take the DF-TDA model for 4 He scattering in this paper throughout all the incident energies 30 < ∼ E in /A P < ∼ 180 MeV where the experimental data are available. The DF model naturally treats both the direct and knock-on exchange processes [38][39][40]. In the latter process, interacting two nucleons are exchanged and thereby the potential becomes nonlocal. However, the nonlocality can be localized with high accuracy by the local momentum approximation [20], as proven in Refs. [41,42]. The folding potential U (R) thus obtained is a function of the distance R between P and T; where the indices µ and ν are the isospin of corresponding nucleon and s = r T − r P − R is the coordinate between interacting two nucleons. The densities ρ P(T) andρ P(T) represent the one-body and mixed densities of P (T); The Fermi momentum k P(T) F is related to the density ρ P(T) . The direct (exchange) term of g-matrixg g DR(EX) pn,np = 1 8 (g 00 ±g 01 ± 3g 10 + 3g 11 ) .
See Refs. [26,31,43,44] for the detail of the formulation of the DF model. The S matrices for 4 He elastic scattering are obtained by solving the one-body Schrödinger equation with U (R). For the targets 208 Pb and 58 Ni, the matter densities ρ T are evaluated by the spherical Hartree-Fock (HF) method based on the Gogny-D1S interaction [45], where the spurious c.m. motions are removed with the standard manner [46]. For the projectile 4 He and the target 40 Ca, we take the phenomenological proton-density determined from electron scattering [47]; here the finite-size effect of proton charge is unfolded with the standard procedure [48], and the neutron density is assumed to have the same geometry as the proton one, since the difference between the neutron root-mean-square radius and the proton one is only 1% in spherical HF calculations.

III. RESULTS
Now we analyze 4 He elastic scattering on nuclei systematically in a wide range E in /A P = 26-175 MeV. Here heavier targets 208 Pb, 58 Ni and 40 Ca are considered, because the g matrix is calculated in nuclear matter and thereby the g-matrix DF model is expected to be more reliable for heavier targets. Figure 7 shows differential cross sections dσ/dΩ as a function of transfer momentum q for 4 He scattering from a 208 Pb target in E in /A P = 26-175 MeV where the experimental data are available. The solid and dashed lines stand for the results of the Kyushu chiral g-matrix DF model with and without 3NF effects, respectively. Chiral 3NFs improve the agreement of the theoretical results with the experimental data. Particularly for E in /A P > ∼ 100 MeV, the agreement is pretty good. We can observe the same features also for 58 Ni and 40 Ca targets, as shown in Figs. 8 and 9, although there is a tendency that the agreement becomes better as the target mass increases. Now we analyze effects of Fujita-Miyazawa 2πexchange 3NF on differential cross sections dσ/dΩ for 4 He+ 58 Ni scattering. In Fig 10, the solid, dashed and dot-dashed lines denote the results of calculations I, II and III, respectively; see Sec. II B for the definition of gmatrix calculations. The difference between calculations I and II means effects of all 3NFs, and that between calculations II and III corresponds to effects of Fujita-Miyazawa 2π-exchange 3NF. The resultant cross sections show that the Fujita-Miyazawa 2π-exchange 3NF is the main contribution of chiral-3NF effects on 4 He scattering. Figure 11 shows the R dependence of the optical potentials U (R) for 4 He elastic scattering from a 58 Ni target at E in /A P =26, 60 and 175 MeV. The solid and dashed lines represent the U (R) with and without chiral-3NF effects; note that only the central potential is generated by the DF-TDA model. As expected, chiral-3NF effects make repulsive and absorptive corrections to the optical potentials, and the corrections slightly increase as E in goes up; note that the effects hardly depend on E in in the peripheral region, R ≈ 6 fm, that is important for the elastic scattering. As already mentioned in Sec. II B, the repulsive correction mainly comes from the Fujita-Miyazawa 2π-exchange 3NF in its 3 O component, and the absorptive correction stems from the 3 E and 3 O components of Fujita-Miyazawa 2π-exchange 3NF. Figure 12 shows the uncertainty coming from Λ dependence of differential cross sections dσ/dΩ for 4 He+ 58 Ni elastic scattering. Here two cases of Λ = 550 and 450 MeV are considered. Λ dependence is shown by a hatching for each of 2NF and 2NF+3NF calculations; note that the hatching region surrounded by solid (dashed)     lines means the uncertainty coming from Λ dependence for 2NF+3NF (2NF) calculations. As expected, Λ dependence becomes larger as E in increases, but the uncertainty coming from Λ dependence is still smaller than chiral-3NF effects, even at E in /A P = 175 MeV.
The scattering amplitude can be decomposed into the near-and far-side components [60]. As illustrated in Fig.  13, these components are well defined, when outgoing waves are generated only in the peripheral region of T. 4 He scattering on a heavier target is a good case. The absorptive correction of chiral-3NF effects makes the decomposition more applicable. The decomposition is a convenient tool for investigating the interplay between differential cross sections dσ/dΩ and the real part of U (R). The near-side (far-side) outgoing waves are mainly induced by repulsive Coulomb (attractive nuclear) force, so that very-forward-angle (middle-angle) scattering are dominated by the near-side (far-side) components. As a consequence of this property, a large interference pattern appears in differential cross sections at the forward angles where the two components become comparable, and the far-side dominance is realized at middle angles af-ter the interference pattern. In the middle angle region, any repulsive correction to U (R) reduces differential cross sections. Figure 14 shows the near/far decomposition of differential cross sections dσ/dΩ for 4 He+ 58 Ni scattering at E in /A P = 72 MeV. The dotted and dashed lines represent the near-and far-side cross sections, respectively, and the solid line denotes differential cross sections before the near/far decomposition; here chiral-3NF effects are taken into account. The solid line shows a large interference pattern at θ c.m.   Finally, we comment on chiral-3NF effects on total reaction cross sections σ R briefly. Radii of stable and unstable nuclei are often determined from measured σ R with the folding model and/or the Glauber model. Figure 15 shows σ R as a function of E in /A P for 4 He scattering on 58 Ni and 208 Pb targets. Closed circles (squares) mean the results of Kyushu chiral g matrix with (without) 3NF effects. The two kinds of results are close to each other, indicating that chiral-3NF effects are negligible for σ R . The fact ensures that the determination of nuclear radii from measured σ R is reliable.

IV. SUMMARY
We investigated basic properties of chiral 3NFs in symmetric nuclear matter with positive energies up to 200 MeV by using the BHF method with chiral 2NFs of N 3 LO and chiral 3NFs of NNLO in the Bochum-Bonn-Jülich [33], parameterization, and analyzed chiral-3NF effects on 4 He elastic scattering from heavier targets 208 Pb, 58 Ni and 40 Ca over a wide incident-energy range of 30 < ∼ E in /A P < ∼ 200 MeV by the Kyushu chiral g-matrix folding model.
First, we summarize the basic properties of chiral 3NFs in symmetric nuclear matter with positive energies E in up to 200 MeV: (1) Chiral 3NFs make the single-particle potential U less attractive and more absorptive.
(2) The repulsive and absorptive corrections slightly increase as E in goes up.
(3) Chiral 3NF effects on U mainly come from the Fujita-Miyazawa 2π-exchange 3NF (diagram (a) in Fig. 1). More precisely, the repulsion mainly stems from the 3 O component of the diagram (a) and the absorption does from the 3 O and 3 E components of the diagram (a).
Properties (1)-(3) persist in the optical potential of 4 He scattering. This is natural, since the single-particle potential plays a role of the optical potential in nuclear matter. However, it should be noted that chiral-3NF effects depend little on E in in the peripheral region that is important for the elastic scattering.
Chiral-3NF effects are evident for 4 He scattering in E in /A P > ∼ 60 MeV at the middle angles where the cross sections are dominated by the far-side component of the scattering amplitude. The repulsive correction of chiral 3NFs reduces the far-side component and thereby yields better agreement with the experimental data. Eventually, the Kyushu chiral g-matrix DF model reproduces measured differential cross sections pretty well, particularly for 4 He scattering at E in /A P > ∼ 100 MeV.
All the analyses mentioned above were made with Λ = 550 MeV. In order to investigate Λ dependence in nuclear-matter and 4 He-scattering calculations, we take Λ = 450 MeV in addition to Λ = 550 MeV. The uncertainty coming from Λ dependence is smaller than chiral-3NF effects. There is a tendency that the uncertainty becomes larger as E in increases, but it is still smaller than chiral-3NF effects even at E in = 175 MeV.