Three-nucleon system: Irreducible and reducible contributions of the three-nucleon force

There are different strategies for setting up the quantum-mechanical equations for obtaining the three-nucleon bound and scattering states. We work in momentum-space representation and employ integral equations of the Alt, Grassberger, and Sandhas (AGS) type alt:67a . There are three pairwise potentials $v_{\alpha}$, i.e.,

The strategy for setting up the integral equations works with the two-particle part of the potential summed into the transition matrix $T_{\alpha}(z)$, but keeping the three-particle part in its potential form $V_{(3)}$ uniterated. The basis of states, employed for calculations in the three-nucleon system, is, in the purely nucleonic Hilbert sector, the direct product of an antisymmetrized nucleon pair and a third nucleon without antisymmetrization; thus, the basis is overcomplete. In the Hilbert sector with a $\Delta$ isobar the same pair symmetrization is used, since the $\Delta$ isobar can be created from any of the three nucleons in a symmetric fashion. The isospin formalism is used for distinguishing protons and neutrons including both total isospin $\frac{1}{2}$ and $\frac{3}{2}$ states. The results of the integral equations are bound-state amplitudes $|\psi_{\alpha}\rangle$ and multichannel transition amplitudes $U_{\beta\alpha}$, connecting the initial two-body channel with the non-interacting particle of label $\alpha$ to the final two-body channel with the non-interacting particle of label $\beta$, $\beta=0$ being the final three-particle breakup channel. All multichannel transition amplitudes depend on the available energy $E+i0$; that dependence is notationally suppressed. The derivation of the equations is given in Ref. deltuva:09e ; the fact, that Ref. deltuva:09e describes the three-nucleon system in terms of a completely nucleonic Hilbert space is immaterial for the derivation of equations; the resulting equations of Ref. deltuva:09e can therefore be taken over without any change.

We take over the potential CD Bonn + $\Delta$ deltuva:03c as coupled-channel two-baryon potential, keeping its parameters unchanged in this paper. It is based on the exchange of $\pi$, $\rho$, $\omega$, and $\sigma$ mesons. We remind that this potential is tuned to the deuteron and to two-nucleon elastic scattering below 350 MeV lab energy, and that the potential parts, connected with the $\Delta$ isobar, are therefore underdetermined by that procedure. Those potential parts would be physically better determined by a simultaneous fit to inelastic two-nucleon scattering, i.e., to the data of channels with a single $\pi$. However, at present such a tuning procedure is too demanding for us; it would also require the addition of a Hilbert sector with a single $\pi$ and the corresponding additions to the Hamiltonian part of baryon number 2.

We add an irreducible 3N potential to the Hamiltonian. Among the multitude of possible three-baryon potentials, illustrated in the lower row of Fig. 2, the 3N one was argued by us in Section I to be most important.

The $A_{2\pi}$ and $C_{2\pi}$ terms in $u_{\gamma}$ and $V_{3}$ are related to $2\pi$-meson exchange; the Fujita-Miyazawa mechanism, illustrated in its reducible form in Fig. 3, makes an important contribution to the 3N potential, requiring the parameter combination $C_{2\pi}=\frac{1}{4}A_{2\pi}$, as chosen in Ref. urbana9 . Since that dynamic part is provided by the employed coupled-channel two-baryon potential CD Bonn+$\Delta$ in a reducible form, the Fujita-Miyazawa contribution has to be taken out from the parameters in our reparametrization of an irreducible 3N potential in the Urbana-like fashion. We note in passing, that the reducible Fujita-Miyazawa process, provided by the coupled-channel two-baryon potential CD Bonn + $\Delta$, also includes the $\rho$-meson exchange and is therefore richer than the original Fujita-Miyazawa 3N force of Ref. fujita:57a . Compared to the Urbana IX 3N force the strength of the potential should be reduced and deviations from the ratio $C_{2\pi}/A_{2\pi}=\frac{1}{4}$ are acceptable. In fact, that ratio $C_{2\pi}/A_{2\pi}$ got increased in Ref. stadler:92a when the Fujita-Miyazawa process was removed there from the Tucson-Melbourne 3N potential coon:79 . We therefore have to expect a similar increase of that ratio $C_{2\pi}/A_{2\pi}$ in our reparametrization of an irreducible 3N potential in the Urbana-like fashion.

The repulsion with the strength parameter $B_{0}$ in the Urbana IX 3N force was required to prevent overbinding of nuclear matter. However, that argument is not relevant in the present context any longer. The coupled-channel two-baryon potential provides repulsion due to the two-nucleon dispersion. That fact was confirmed for nuclear matter by Ref. manzke:78 . We therefore expect that our reparametrization of the irreducible 3N potential should have a smaller or even a vanishing $B_{0}$ value compared to the Urbana IX 3N force. We therefore set $B_{0}=0$ in our first try and retune $A_{2\pi}$ and $C_{2\pi}$ having in mind that the resulting dynamic model has to account for the ${}^{3}\rm H$ binding energy $|E_{{}^{3}\rm H}|$ and attempting also to fit the neutron-deuteron doublet scattering length $a_{2}$. The potential CD Bonn + $\Delta$ misses $|E_{{}^{3}\rm H}|$ only by 176 keV and with $a_{2}=0.695$ fm is already pretty close to the experimental value of 0.65$\pm$0.04 fm dilg:71 . Thus, the required parameters in the added modified Urbana-like 3N potential should be quite mild ones.

However, with the coupled-channel two-baryon potential CD Bonn + $\Delta$ and the modified Urbana IX 3N force as irreducible 3N force in addition, we are unable to fit the ${}^{3}\rm H$ binding energy $|E_{{}^{3}\rm H}|$ and the neutron-deuteron doublet scattering length $a_{2}$ simultaneously with weak $A_{2\pi}$ and $C_{2\pi}$ parameters. We therefore included also the $B_{0}$ term and tried a wide range of parameters, using a rather coarse raster, but staying within sensible limits. In the end we chose two groups of modifications only, whose parameters are listed in the Table 1 and which appear to us characteristic for our parameter search with Urbana-like 3N potentials. In the first group the ratio $C_{2\pi}/A_{2\pi}$ was set larger than $\frac{1}{4}$ by hand and not subjected to the fit.

The first group of parametrizations consists of the 3N potentials (U1, U2, U3). The two terms augmented by the parameters $A_{2\pi}$ and $C_{2\pi}$ yield both attraction; if repulsion is needed, it is provided by the term with the parameter $B_{0}$. All models fit $|E_{{}^{3}\rm H}|$ perfectly and, except for U3, underpredict the neutron-deuteron doublet scattering length $a_{2}$. Running down this list of three modifications (U1, U2, U3) the individual strength parameters for attraction and repulsion become increasingly larger; when accounting for the neutron-deuteron doublet scattering length $a_{2}$ as in U3, the three parameters are individually largest. Furthermore, we are disturbed that the $A_{2\pi}$ and $C_{2\pi}$ parts of the 3N potential U3 make a massive contribution of about 5 MeV to the triton binding $|E_{{}^{3}\rm H}|$, which has to be balanced by a corresponding comparable repulsion due to the $B_{0}$ term, just to squeeze out the required small additional binding of 176 keV. Thus, the simultaneous account of the ${}^{3}\rm H$ binding $|E_{{}^{3}\rm H}|$ and of the neutron-deuteron doublet scattering length $a_{2}$ appears difficult. We are only able to achieve that goal with the irreducible 3N potential U3 of questionable parametrization, i.e., the attraction and repulsion required for balancing each other, are unnaturally large. However, this tuning difficulty appears not to be a feature characteristic for our special 3N force model. Indeed, in Ref. kievsky:10a another study was performed using the purely nucleonic Argonne V18 potential wiringa:95a in combination with various 3N forces; reproducing experimental values of the neutron-deuteron doublet scattering length $a_{2}$ and of 3N and 4N binding energies required a balance of attractive and repulsive 3N force contributions, individually even stronger than in the model CD Bonn + $\Delta$ + U3 of this paper. Furthermore, Ref. epelbaum:14 used purely nucleonic 2N and 3N potentials consistently derived from $\chi$EFT, thus, it uses rather different dynamics; nevertheless, the simultaneous account of the ${}^{3}\rm H$ binding $|E_{{}^{3}\rm H}|$ and of the neutron-deuteron doublet scattering length $a_{2}$ was also either impossible at all or led to unnaturally large coupling constants, quite similar to our finding. In addition, we note that all realistic purely nucleonic potentials combined with standard phenomenological 3N forces fitted to $|E_{{}^{3}\rm H}|$ are underpredicting $a_{2}$ by a comparable amount witala:03a .

In some parameter domains we noted a surprising sensitivity for the nucleon analyzing power $A_{y}$ of elastic nucleon-deuteron scattering. We therefore created a second group of parametrizations consisting of the 3N potentials (U1, U4, U5). The repulsive term with the parameter $B_{0}$ is left out for them; attraction and, if needed, repulsion are both provided by the $A_{2\pi}$ and $C_{2\pi}$ terms. In the 3N potentials U4 and U5 the two parameters have opposite signs. We remind that also the 3N potential U1 has $B_{0}=0$, whereas the two other parameters yield attraction; a 3N potential with both parameters $A_{2\pi}$ and $C_{2\pi}$ yielding repulsion is unable to account for the ${}^{3}\rm H$ binding $|E_{{}^{3}\rm H}|$. None of the parametrizations of the second group (U1, U4, U5) is able to account for the neutron-deuteron doublet scattering length $a_{2}$, but U5 is roughly adjusted to the neutron analyzing power $A_{y}$ of elastic neutron-deuteron scattering at 10 MeV neutron energy.

Additional modifications with further intermediate parameters appear unnecessary to be accounted in the paper, since the two groups of models for an irreducible 3N force suffice to describe the observed trends.

In this subsection, we compare the predictions of interactions with the added irreducible 3N potentials, listed in the Table 1, for selected three-nucleon scattering observables; we do so separately for the two distinct groups (U1, U2, U3) and (U1, U4, U5). We choose selected observables which best illustrate the effects of the added 3N potentials in either group, desired and undesired, i.e., the neutron analyzing power $A_{y}$ of neutron-deuteron elastic scattering at 10 MeV neutron energy, the differential cross section $d\sigma/d\Omega$ and the deuteron analyzing power $A_{y}(d)$ of proton-deuteron elastic scattering at 135 MeV proton energy, and the differential cross section for neutron-deuteron breakup at 13 MeV neutron lab energy in the space star configuration.

First, results for group (U1, U2, U3) are illustrated in Figs. 4 and 5. The general trend is that 3N force effects due to the irreducible U1 are insignificant, even smaller than those obtained from the reducible 3N force contributions due to the explicit $\Delta$-isobar excitation; they improve the agreement with the experimental data howell:87 ; ermisch:05a ; ramazani:08a ; sekiguchi:02a only slightly. In contrast, the effects due to the 3N potential U3 are larger than those due to the explicit $\Delta$-isobar excitation, but not beneficial for the description of the data. At 135 MeV proton-deuteron elastic scattering, they move the theoretical predictions into the right direction, but they are far too strong, often dramatically overshooting the discrepancy with data. This somehow uncontrolled behavior is not unexpected, given the individually very strong repulsive and attractive terms in U3. The results for U2 are intermediate between those for U1 and U3. We conclude that the irreducible 3N potential U1, being the softest one, appears to make most sense physically, even though it predicts $a_{2}=0.559$ fm and we have to give up our ambition to account for the experimental $a_{2}$ value precisely, when choosing U1 for further studies.

Concerning the potential CD Bonn + $\Delta$ + U3 that is forced to fit the neutron-deuteron doublet scattering length $a_{2}$, we observe another interesting similarity in Fig. 4 with the results of Ref. epelbaum:14 , which are obtained from $\chi$EFT 2N and 3N forces, also fitted to the ${}^{3}\rm H$ binding $|E_{{}^{3}\rm H}|$ and to the neutron-deuteron doublet scattering length $a_{2}$. In addition to the similarity found in Subsection II.B, we observe that in both cases, based on completely different dynamic models, the neutron analyzing power $A_{y}$ in neutron-deuteron elastic scattering at 10 MeV is increased at angles $\Theta_{\mathrm{c\!\>\!.m\!\>\!.}}<100^{\circ}$, but decreased $\Theta_{\mathrm{c\!\>\!.m\!\>\!.}}>110^{\circ}$, moving theoretical predictions away from the experimental data. Thus, this additional undesired effect of irreducible 3N forces on the analyzing power $A_{y}$ appears to be a general feature of presently available force models that fit 2N data, $|E_{{}^{3}\rm H}|$ and $a_{2}$ simultaneously and points to a further aspect of the long-standing $A_{y}$ puzzle in low-energy 3N scattering. In Ref. kievsky:10a the analyzing power was studied only at 3 MeV, but the conclusions are essentially the same, i.e., improving $a_{2}$ increases the discrepancy for $A_{y}$, except when using the Argonne V18 potential supplemented with a particular 3N force of $\chi$EFT.

Second, the above conjecture on the general difficulty of accounting simultaneously for characteristic three-nucleon data, seen for the potential group (U1, U2, U3), is supported also by our results for the potential group (U1, U4, U5) as shown in Fig. 6. The model CD Bonn + $\Delta$ + U5 is adjusted to reproduce the experimental data for the neutron analyzing power $A_{y}$ at 10 MeV, seemingly overcoming the long-standing $A_{y}$ puzzle. The price for this fit, however, is an increased discrepancy for the neutron-deuteron doublet scattering length $a_{2}$, as can be seen in Table 1. The 3N potential U4 that improves $a_{2}$ compared to U1, increases the discrepancy for $A_{y}$ at the same time, much like U2 and U3 do. Thus again, the simultaneous account of 2N data, $|E_{{}^{3}\rm H}|$, $a_{2}$, and $A_{y}$ appears to be impossible with the presently available force models. The models CD Bonn + $\Delta$ + U4 and CD Bonn + $\Delta$ + U5, although the latter is successful for low-energy $A_{y}$, contain unusually large parameters $A_{2\pi}$ and $C_{2\pi}$, which make the 3N potentials U4 and U5 as questionable as U3 before. As shown in Fig. 7 for proton-deuteron elastic scattering at 135 MeV, especially U5 yields a very strong effect that destroys the reasonable agreement with data achieved by the potentials CD Bonn + $\Delta$ or CD Bonn + $\Delta$ + U1.

Finally, Fig. 8 studies another famous 3N puzzle, as bothersome as the $A_{y}$ problem, i.e., the fivefold differential cross section $d^{5}\sigma/dSd\Omega_{1}d\Omega_{2}$ of the nucleon-deuteron breakup at 13 MeV in the space star configuration characterized by the polar and azimuthal angles of two detected nucleons $(\theta_{1},\theta_{2},\varphi_{2}-\varphi_{1})=(50.5^{\circ},50.5^{\circ},120.0^{\circ})$. This observable is known to be quite insensitive to changes in the Hamiltonian witala:10a ; ishikawa:09a . Indeed, even the effect of the very strong 3N forces U3 and U5 is insignificant and much smaller than the discrepancy between the predictions and experimental data strate:89 ; setze:05a ; rauprich:91 . Fig. 8 presents theoretical results for neutron-deuteron scattering; the corresponding theoretical results for proton-deuteron breakup also strongly disagree with the data, despite the inclusion of Coulomb in the calculations; the proton-deuteron results overpredict the data. Thus, none of the considered 3N force models studied by us is able to cure the space star anomaly.

We choose the irreducible 3N potential U1 as the most reasonable one for a further comparative study of irreducible and reducible contributions to the 3N force; the addition of U1 accounts for the binding of ${}^{3}\rm H$ by construction, but it also yields the ${}^{3}\rm H$ - ${}^{3}\rm He$ mass difference perfectly, i.e., the binding of ${}^{3}\rm He$ becomes 7.718 MeV, its experimental value also being 7.718 MeV. In this subsection for a wide range of 3N scattering observables we compare the predictions of the potentials CD Bonn without any 3N force, CD Bonn + $\Delta$ with reducible contributions to the 3N force, and of CD Bonn + $\Delta$ + U1 with reducible and irreducible contributions to the 3N force.

In Fig. 9 we show the differential cross section and various spin observables for elastic proton-deuteron scattering at 135 MeV, in Fig. 10 the differential cross section and nucleon-to-nucleon spin transfer coefficients for elastic proton-deuteron scattering at 250 MeV, and in Fig. 11 the fivefold differential cross section and nucleon analyzing powers for proton-deuteron breakup at 135 MeV. In contrast to Subsection III.A, we carry out the calculations for proton-deuteron scattering fully and include the proton-proton Coulomb force using the method of screening and renormalization taylor:74a ; alt:80a ; deltuva:05a ; deltuva:05d , though we do not show the explicit Coulomb effects separately in the plots. In elastic scattering at these energies the Coulomb effect is irrelevant, except for small angles deltuva:05a ; it is seen in Figs. 9 and Fig. 10 as wiggles in all observables below $\Theta_{\mathrm{c\!\>\!.m\!\>\!.}}<10^{\circ}$. In breakup the Coulomb effect is most visible in configurations with low relative proton-proton energy deltuva:05d , mostly affecting the differential cross section. For example, in the configuration $(28^{\circ},28^{\circ},20^{\circ})$ of Fig. 11 the proton-proton Coulomb force reduces the cross section at the peak by about 25%.

The reducible contributions to the 3N force, i.e., the explicit $\Delta$-isobar excitation, create the largest effects in the studied observables. The additional effects of the irreducible 3N force U1 are minor even at higher energies, and they do not destroy any achieved account of the experimental data. In proton-deuteron elastic scattering, the reducible and irreducible 3N-force contributions move the predictions quite often into the same direction such that the inclusion of irreducible 3N force U1 even slightly improves the description of the experimental data. The most prominent examples are the differential cross section around the minimum and at backward angles, the nucleon analyzing power $A_{y}$ at 135 MeV around the minimum and the nucleon-to-nucleon spin transfer coefficient $K_{y}^{y}$ at 250 MeV, as well as the deuteron vector analyzing power $A_{y}(d)$ at 135 MeV shown in Fig. 5. In the breakup reaction the reducible and irreducible 3N forces may have effects of the same or of opposite signs as illustrated in Fig. 11, depending on the kinematical configuration and depending on the specific observable. In the considered example the Coulomb repulsion between the protons is important for the achieved gross agreement with the cross section data, but neither 3N force contribution is beneficial in accounting for the experimental $A_{y}$ data of Ref. eslami:10a , especially in the $(28^{\circ},28^{\circ},20^{\circ})$ configuration.