1 Introduction

The last thirty years have been a period of many discoveries in the study of nuclear potentials and several-body processes. First, the new generation of semi-phenomenological potentials, such as the Argonne V18 [1] force or the CD Bonn [2] interaction, led to a significant improvement in the description of experimental data. This situation was further improved thanks to nuclear potential models derived from the Chiral Effective Field Theory. The most important advantages of the latter include (a) direct connection to QCD and other processes, like pion-nucleon scattering, (b) the possibility of deriving nuclear forces in a perturbative expansion, which allows assigning a physical sub-process to a specific order of expansion, (c) the possibility of deriving consistent two-nucleon, three-nucleon forces, etc. and (d) the hierarchy of nuclear forces naturally resulting from the model, justifying the treatment of many-body forces as corrections to the dominant two-nucleon interaction. Theoretical work has resulted in several chiral models of nuclear forces [3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Among them, the results obtained by E. Epelbaum’s group occupy a special place. This group has, over the years, provided several chiral force models, the most recent of which is the SMS model that uses the semi-local regularization. Within this model, complete two-body forces up to the fifth order of chiral expansion (\(\hbox {N}^4\hbox {LO}\)) have been derived to date. Further, some F-wave contributions from the next order are also known, and combined with the \(\hbox {N}^4\hbox {LO}\) SMS potential create the \(\hbox {N}^4\hbox {LO}^+\) version of the model. That version of the interaction gives remarkably high quality of the two-nucleon data description, what can be quantified by \(\chi ^2/\textrm{data} \approx 1.06\) for neutron-proton data in the energy range 0–300 MeV [10].

On the other hand, the work of experimenters provides increasingly more accurate experimental results, facilitating verification of theoretical models. In the three-nucleon sector, very precise data are currently available for the cross-sections in elastic nucleon-deuteron scattering and in the deuteron breakup reaction induced by a nucleon [17,18,19,20]. The latter has been measured practically over the entire range of angles defining the momenta of free nucleons in the final state. These cross section data are accompanied by results for a number of polarization observables, with dominating measurements of the analyzing powers, i.e. for reactions with one polarized particle in the initial channel, see e.g. [21, 22]. Some efforts were also made to measure more complex polarization observables, which resulted in, among others, experiments performed at the IUCF at energies of 135 and 200 MeV [23]. Due to the fact that the correct description of polarization observables is a more demanding task than the description of cross sections, further experiments in this field are currently planned. One of them is the spin correlation coefficients experiment planned at RIKEN [24, 25].

With this in mind, in the present work we focus on theoretical predictions describing the impact of the short-range \(\hbox {N}^4\hbox {LO}\) three-nucleon force on a complete set of spin correlation coefficients. To present a systematic picture, we show results for five reaction energies, that is for five neutron kinetic energies in the laboratory frame. In following we stick to the nonrelativistic formalism, described in Sect. 2. Our results are given in Sect. 3 and we conclude in Sect. 4.

2 Formalism

In order to obtain the spin correlation coefficients a two-step procedure has been applied. First, the 13 free parameters of the 3NF have been found. They are \(c_D\) and \(c_E\) which occur at \(\hbox {N}^2\hbox {LO}\) and \(c_i\), \(i=1,\dots ,11\), present in the \(\hbox {N}^4\hbox {LO}\) short-range 3NF. To this end the emulator [26,27,28] was used to effectively perform the \(\chi ^2\)/data minimization fit, which in turn allowed us to prepare combined 3NF at \(\hbox {N}^2\hbox {LO}\) and \(\hbox {N}^4\hbox {LO}\). In the second step, the obtained forces were used to solve the Faddeev equation, which enabled the determination of the transition amplitudes and observables [29]. Both the emulator and the solution of the full Faddeev equation were performed in momentum space and, in fact, were done within the same formalism.

The above-mentioned emulator is dedicated to the problem of adjusting the free parameters of the three-body force. It bases on several observations that allow us to introduce approximations to the Faddev’s equations. Namely, we take advantage of the fact that the 3NF components associated with free parameters act only in the few lowest partial waves. Additionally, as described below, the second-order terms in the iterative equation can be neglected.

The starting point for the emulator used is the observation that the \(V^{(1)}\) part of a 3NF, that is the part of 3NF symmetric under the exchange of particles 2 and 3, can be written as

$$\begin{aligned} V^{(1)}= & {} V(\theta _0) + \sum _{i=1}^N c_i \Delta V_i \equiv V(\theta _0) + \Delta V(\theta ) ~, \end{aligned}$$
(1)

with a parameter-free term \(V(\theta _0)\) and a sum of N parameter-dependent terms \(\Delta V_i\) multiplied by strengths parameters \(c_i\), collected in the \(\Delta V(\theta )\) term.

If all the \(c_i\) parameters are known and full 3N Hamiltonian \(H=H_0+V^{2N}+V^{3N}\) does not contain any unknown parameters, one can proceed to the exact solution of the Faddeev equation

$$\begin{aligned} T\vert \phi \rangle= & {} t P \vert \phi \rangle + (1+tG_0)V^{(1)}(1+P)\vert \phi \rangle + t P G_0 T \vert \phi \rangle \nonumber \\{} & {} + (1+tG_0)V^{(1)}(1+P)G_0T \vert \phi \rangle \,, \end{aligned}$$
(2)

where \(G_0\) is the free three-nucleon propagator, t is a solution of the Lippmann-Schwinger equation with the 2N potential \(V^{2N}\), and P is a permutation operator. The initial state \(\vert \phi \rangle \equiv \vert \textbf{q}_0 \rangle \vert \phi _d \rangle \) describes the free motion of the nucleon (neutron) with the relative momentum \(\textbf{q}_0\) and the internal deuteron wave function \(\vert \phi _d \rangle \). The elastic scattering transition amplitude leading to the final neutron-deuteron state \(\vert \phi ' \rangle \) is then given by [29, 30]

$$\begin{aligned} \langle \phi ' \vert U \vert \phi \rangle= & {} \langle \phi ' \vert PG_0^{-1} \vert \phi \rangle + \langle \phi '\vert V^{(1)}(1+P)\vert \phi \rangle \nonumber \\{} & {} + \langle \phi ' \vert V^{(1)}(1+P)G_0T\vert \phi \rangle + \langle \phi ' \vert PT \vert \phi \rangle ~. \end{aligned}$$
(3)

However, if the \(c_i\) parameters are unknown, one must first determine their values. We used the emulator [26,27,28] for this purpose. The splitting in (1) entails a similar splitting for the T operator \(T = T(\theta _0) + \sum _{i=1}^N c_i \Delta T_i \equiv T(\theta _0) + \Delta T(\theta )\). Inserting both sums in the Faddeev equation (2) and neglecting, due to the smallness of the \(\Delta V(\theta )\), the second-order terms, i.e. terms proportional to \(\Delta V(\theta ) (1+P)G_0 \Delta T(\theta )\) allows us to end with a sequence of equations to be solved: one standard Faddeev equation for \(T(\theta _0)\) dependent on \(V(\theta _0)\) only, and N equations for \(\Delta T_i\), each of which depends only on one term \(\Delta V_i\) and the parameter-free \(V(\theta _0)\) part of 3NF. A few further steps, discussed in [27, 28] lead to the elastic scattering amplitude. Its dependence on the \(c_i\) parameters is following:

$$\begin{aligned} \langle \phi ' \vert U \vert \phi \rangle= & {} \langle \phi ' \vert U_0 \vert \phi \rangle + \sum _i c_i \langle \phi ' \vert U_i \vert \phi \rangle + \sum _{i,k} c_i c_k \langle \phi ' \vert U_{ik} \vert \phi \rangle ~, \end{aligned}$$
(4)

and is a convenient relation for practical use in \(\chi ^2\)/data minimization. In Eq. 4, the \(U_0\), \(U_i\), and \(U_{ik}\) are parts of the transition amplitude which depend neither on \(\Delta V_i\) nor on \(\Delta T_i\), depend on \(\Delta V_i\) or \(\Delta T_i\), or depend on \(\Delta V_i\) and \(\Delta T_i\), respectively. The quality of this approximation was tested in [27], where it was shown that it is sufficient to realistically estimate the \(c_i\) values.

The fitting procedure was performed using a set of 786 data points, see Tab.2 of [28] at energies \(\hbox {E}=10\), 70 and 135 MeV. The resulting values of \(c_i\) parameters are given in Tab.3 of [28] and are also used to obtain the results presented in the next section. Note, that the two free parameters, of the \(\hbox {N}^2\hbox {LO}\) 3NF, \(c_D\) and \(c_E\), have been fixed separately for calculations which take or do not take the short-range \(\hbox {N}^4\hbox {LO}\) 3NF into account.

We solve Eqs. (2) and (3) in the momentum-space partial-wave basis \(\vert p q \alpha \rangle \), defined by the magnitudes of the relative Jacobi momenta p and q and a set of discrete quantum numbers \(\alpha \) comprising angular momenta, spins and isospins in the j-I coupling. We take into account all the 3N partial wave states up to the 2N angular momentum \(j_{max}=5\) and the 3N angular momentum \(J_{max}=\frac{25}{2}\). We restrict the 3NF to act in the partial waves with the total 3N angular momentum \(J \le 7/2\). For details of our numerical performance see [29].

Fig. 1
figure 1

The spin correlation coefficient \(C_{x,x}\) for neutron-deuteron (nd) scattering at four laboratory energies of the incoming nucleon: a) \(\hbox {E}=10~\hbox {MeV}\), b) \(\hbox {E}=135~\hbox {MeV}\), c) \(\hbox {E}=190~\hbox {MeV}\), and d) \(\hbox {E}=250~\hbox {MeV}\). The predictions based on the 2N SMS \(\hbox {N}^4\hbox {LO}^+\) interaction are represented by the black dotted curve. The red dashed curve shows predictions of the SMS model with 2N \(\hbox {N}^4\hbox {LO}^+\) supplemented by the \(\hbox {N}^2\hbox {LO}\) 3NF and the black solid curve represents the \(\hbox {N}^4\hbox {LO}^+\) + \(\hbox {N}^2\hbox {LO}\) 3NF + short-range \(\hbox {N}^4\hbox {LO}\) 3NF results

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3 Results

There are 13 non-zero spin correlation coefficients [29], with the deuteron being in vector or tensor polarization state. In the following we stick to the standard convention on defining polarization axes [29, 31]. The complete set of spin correlation coefficients is shown in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 as a function of the centre-of-mass scattering angle \(\theta \). In each figure we show predictions at four neutron beam laboratory energies: \(\hbox {E}=10~\hbox {MeV}\) (left top), \(\hbox {E}=135~\hbox {MeV}\) (right top), \(\hbox {E}=190~\hbox {MeV}\) (left bottom) and \(\hbox {E}=250~\hbox {MeV}\) (right bottom). Note that the two first energies are the ones used for fixing the \(\hbox {N}^2\hbox {LO}+\hbox {N}^4\hbox {LO}\) 3NF parameters, thus strictly speaking among results based on \(\hbox {N}^4\hbox {LO}\) 3NF only the ones given in the bottom row are explicit predictions. All panels show calculations based on the 2N SMS \(\hbox {N}^4\hbox {LO}^+\) only (black dotted curve), on the 2N SMS \(\hbox {N}^4\hbox {LO}^+\) force combined with the complete \(\hbox {N}^2\hbox {LO}\) 3NF (red dashed curve), and on the later interaction supplemented by the \(\hbox {N}^4\hbox {LO}\) short-range 3NF (black solid curve). Finally, in some of the plots available data at \(\hbox {E}=135~\hbox {MeV}\) from the IUCF experiment [23] are shown.

Fig. 2
figure 2

The same as in Fig. 1 but for the \(C_{y,y}\) coefficient

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Fig. 3
figure 3

The same as in Fig. 1 but for the \(C_{z,z}\) coefficient

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Fig. 4
figure 4

The same as in Fig. 1 but for the \(C_{x,z}\) coefficient

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Fig. 5
figure 5

The same as in Fig. 1 but for the \(C_{z,x}\) coefficient

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Fig. 6
figure 6

The same as in Fig. 1 but for the \(C_{xx,y}\) coefficient

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Fig. 7
figure 7

The same as in Fig. 1 but for the \(C_{xy,x}\) coefficient

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Fig. 8
figure 8

The same as in Fig. 1 but for the \(C_{xy,z}\) coefficient

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Fig. 9
figure 9

The same as in Fig. 1 but for the \(C_{xz,y}\) coefficient

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Fig. 10
figure 10

The same as in Fig. 1 but for the \(C_{yy,y}\) coefficient

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Fig. 11
figure 11

The same as in Fig. 1 but for the \(C_{yz,x}\) coefficient

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Fig. 12
figure 12

The same as in Fig. 1 but for the \(C_{yz,z}\) coefficient

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Fig. 13
figure 13

The same as in Fig. 1 but for the \(C_{zz,y}\) coefficient

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Even a quick look at the figures reveals that inclusion of the \(\hbox {N}^4\hbox {LO}\) short-range 3NF in most cases changes the predictions significantly. This is observed already at the lowest energy \(\hbox {E}=10~\hbox {MeV}\), where introducing the \(\hbox {N}^4\hbox {LO}\) shifts the prediction by a few percent. To be more quantitative let us define the relative difference

$$\begin{aligned} \Delta (C_{\alpha ,\beta },\theta ) \equiv \frac{C_{\alpha ,\beta }^{full}-C_{\alpha ,\beta }}{0.5(C_{\alpha ,\beta }^{full}+C_{\alpha ,\beta })}, \end{aligned}$$
(5)

with \(C_{\alpha ,\beta }^{full}\) being the spin correlation coefficient with neutron and deuteron spin polarizations denoted by \(\alpha \) and \(\beta \) obtained with inclusion of the 3NF \(\hbox {N}^4\hbox {LO}\) terms. The \(C_{\alpha ,\beta }\) values are calculated only with the \(\hbox {N}^2\hbox {LO}\) 3NF. Obviously both spin correlation coefficients entering formula (5) have to be taken at the same scattering angle \(\theta \).

Exemplary values of \(\Delta (C_{\alpha ,\beta })\) at \(\hbox {E}=10\,\hbox {MeV}\) are: \(\Delta (C_{y,y},60^{\circ })=6.05\)% and \(\Delta (C_{y,y},115^{\circ })\approx 220\)% with the latter angle chosen at the minimum of the \(C_{y,y}\) where its value is close to zero, what explains big value of \(\Delta (C_{y,y},115^{\circ })\). Strong influence of the \(\hbox {N}^4\hbox {LO}\) 3NF is also seen for \(C_{xy,x}\) in Fig. 7 (\(\Delta (C_{xy,x},80^{\circ })=116\)%) or \(C_{zz,y}\) in Fig. 13 (\(\Delta (C_{zz,y},115^{\circ })=-36\)%). It is also worth noting that the spin correlation coefficient at \(\hbox {E}=10~\hbox {MeV}\), in the case of the deuteron tensor polarization, takes on small values, what makes it currently useless in experimental tests.

At \(\hbox {E}=135~\hbox {MeV}\) the role of the \(\hbox {N}^4\hbox {LO}\) short-range 3NF remains significant for \(C_{y,y}\) (Fig. 2) at the central scattering angles, \(C_{x,z}\) (\(C_{z,x}\)) at the maximum (minimum) at backward scattering angles (Figs. 4 and 5), and for \(C_{yz,z}\) (Fig. 12) above \(\theta >90^{\circ }\). For these observables the relative difference reaches \(\Delta (C_{y,y},90^{\circ })=41\)%, \(\Delta (C_{z,x},150^{\circ })=30\)%, and \(\Delta (C_{yz,z},130^{\circ })=167\)%. As for the \(\hbox {E}=10~\hbox {MeV}\), at \(\hbox {E}=135~\hbox {MeV}\) the chiral \(\hbox {N}^2\hbox {LO}\) is practically meaningless, and only slightly modifies predictions based on pure 2N force. Comparison with the data reveals clear discrepancies for many of the spin correlation coefficients. These discrepancies are already seen for predictions based on the 2N foce only or on the 2N force combined with \(\hbox {N}^2\hbox {LO}\) 3N interaction. Inclusion of the short-range \(\hbox {N}^4\hbox {LO}\) components of the 3NF either does not change the picture, as e.g. for \(C_{x,x}\) (Fig. 1), \(C_{z,z}\) (Fig. 3), or \(C_{xy,x}\) (Fig. 7), or makes the discrepancy even worse, see e.g. \(C_{y,y}\) (Fig 2) and \(C_{zz,y}\) (Fig. 13).

The situation at \(\hbox {E}=190~\hbox {MeV}\) resembles that at \(\hbox {E}=135~\hbox {MeV}\). The \(\hbox {N}^4\hbox {LO}\) terms are important for most of the coefficients and scattering angle’s ranges. For \(C_{y,y}\) (Fig. 2) \(\Delta (C_{y,y},80^{\circ })=65\)%, for \(C_{yz,x}\) (Fig. 11) \(\Delta (C_{yz,x},125^{\circ })=-92\)%, and for \(C_{yz,z}\) (Fig. 12) \(\Delta (C_{yz,z},130^{\circ })=182\)%. Only for \(C_{x,x}\) (Fig. 1), \(C_{xx,y}\) (Fig. 6), and \(C_{x,zy}\) (Fig. 9) the \(\hbox {N}^4\hbox {LO}\) components of the 3NF have little impact on the results.

Also at the highest considered energy, \(\hbox {E}=250~\hbox {MeV}\), we do not observe many changes. In most cases the angular dependence of \(C_{\alpha ,\beta }\) is similar to that for \(\hbox {E}=190~\hbox {MeV}\). The \(\hbox {N}^4\hbox {LO}\) 3NF yields more or less similar \(\Delta \)’s leading to \(\Delta (C_{y,y},80^{\circ })=66\)%, \(\Delta (C_{yz,x},125^{\circ })=-79\)%, and \(\Delta (C_{yz,z},130^{\circ })=113\)% for the same cases as shown above for \(\hbox {E}=190~\hbox {MeV}\).

4 Conclusions

We have presented a complete set of the spin correlation coefficients for the neutron-deuteron elastic scattering in the energy range 10–250 MeV. That choice was partly dictated by the planned future experiments. Specifically, we investigated the significance of the whole set of short-range components of the \(\hbox {N}^4\hbox {LO}\) three-nucleon interaction [12,13,14]. That force has been combined with the SMS \(\hbox {N}^4\hbox {LO}^+\) two-nucleon [10] and the \(\hbox {N}^2\hbox {LO}\) [7] three-nucleon potentials. The 13 strength parameters of the 3NF were found by performing a \(\chi ^2\)-minimization to three-nucleon data. It was possible thanks to the use of the emulator of Faddeev equations proposed in our earlier works [26,27,28].

We found that the included \(\hbox {N}^4\hbox {LO}\) 3NF has a significant impact on the predicted spin correlation coefficients for most of them. Depending on the reaction energy, the scattering angle and specific choice of the polarization observable, the \(\hbox {N}^4\hbox {LO}\) 3NF changes the magnitude of some spin correlation coefficients even by several dozen percent. In general, the \(C_{yz,z}\) and \(C_{yz,x}\) at higher energies seem to be the most sensitive to that new 3NF components and thus the most interesting coefficients to be compared with future experimental data. On the other hand, the \(\hbox {N}^4\hbox {LO}\) 3NF contributions have only tiny effect on the \(C_{xz,y}\) and \(C_{xx,y}\) spin correlation coefficients.

A comparison with the available data [23] at \(\hbox {E}=135~\hbox {MeV}\) reveals two cases. For some observables, like \(C_{x,x}\) or \(C_{z,z}\), there is no improvement in data description. However, we found also coefficients, e.g. \(C_{y,y}\) and \(C_{zz,y}\) for which the \(\hbox {N}^4\hbox {LO}\) short-range 3NF moves theoretical predictions away from the data.

Due to the missing \(\hbox {N}^3\hbox {LO}\) 3NF components our study should be regarded as preliminary. Results with such an unexpectedly big role of \(\hbox {N}^4\hbox {LO}\) 3NF indicate that it is necessary to include first the three-body force at \(\hbox {N}^3\hbox {LO}\) in the analysis. Note that this will require re-adjusting the values of the free parameters \(c_D, c_E\) in the \(\hbox {N}^2\hbox {LO}\) and \(c_i\) in the \(\hbox {N}^4\hbox {LO}\) 3NF. There are no new free parameters of 3NF at \(\hbox {N}^3\hbox {LO}\) [32, 33]. The magnitude of the observed discrepancies between the data and the current predictions is one more argument for an important role of the \(\hbox {N}^3\hbox {LO}\) 3NF contributions. The work on derivation and partial wave decomposition of the \(\hbox {N}^3\hbox {LO}\) 3NF is ongoing.