---
abstract: 'Large discrepancies between quasi-free neutron-neutron (nn) cross section data from neutron-deuteron (nd) breakup and theoretical predictions based on standard nucleon-nucleon (NN) and three-nucleon (3N) forces are pointed out. The nn $^1S_0$ interaction is shown to be dominant in that configuration and has to be increased to bring theory and data into agreement. Using the next-to-leading order (NLO) $^1S_0$ interaction of chiral perturbation theory ($\chi$PT) we demonstrate that the nn QFS cross section only slightly depends on changes of the nn scattering length but is very sensitive to variations of the effective range parameter. In order to account for the reported discrepancies one must decrease the nn effective range parameter by about $\approx 12 \% $ from its value implied by charge symmetry and charge independence of nuclear forces.'
author:
- 'H. Wita[ł]{}a'
- 'W. Glöckle'
title: 'The nn quasi-free nd breakup cross section: discrepancies to theory and implications on the $^1S_0$ nn force'
---

Introduction {#one}
============

The knowledge of the NN interaction is fundamental for interpreting nuclear phenomena. The proton-proton (pp) experiments provide a solid data basis [@nijm90; @nijm93], which restricts theoretical assumptions about the strong part of the pp force. In case of the neutron-proton (np) system this is only true to a smaller extent. The partial wave analysis of the np data [@nijm93] relies on the assumption that the isospin $ t=1 $ piece can be taken over from the pp system and only the $ t=0 $ part is free in the adjustment to the data. The lack of a free neutron target forbids neutron-neutron (nn) experiments, therefore the information on the nn interaction can be deduced only in an indirect way. To that aim the best tool seems to be the study of the three-nucleon (3N) system composed from two neutrons and the proton. It is simple enough to allow a rigorous theoretical treatment, e.g. in the framework of Faddeev equations [@physrep96]. The neutron-deuteron elastic scattering together with the neutron induced deuteron breakup, supplemented with the triton properties, offer a data basis which can be used to test properties of the nn force. Especially the nd breakup process with its rich set of configurations for free three outgoing nucleons seems to be a powerfull tool to test the nuclear Hamiltonian. By comparing theoretical predicitons to the nd breakup data in different configurations not only can the present day models of two-nucleon (2N) interactions be tested, but also effects of three-nucleon forces (3NF’s) can be studied.

The nn quasi-free scattering (QFS) refers to a situation where the outgoing proton is at rest in the laboratory system. In the nd breakup also np QFS is possible. Here one of the neutrons is at rest while the second neutron together with the proton form a quasi-freely scattered pair.

The reported nn QFS cross sections taken at $E_n^{lab}=26$ MeV [@siepe2] and at $E_n^{lab}=25$ MeV [@ruan1] overestimate the nd theory by $\approx 18 \%$. Surprisingly, when instead of the nn pair the np pair is quasi-freely scattered, the theory follows nicely the np QFS cross section data taken in the $E_n^{lab}=26$ MeV nd breakup measurement [@siepe2]. That good description of the np QFS cross section contrasts with the drastic discrepancy between the theory and the nn QFS cross section data taken in the same experiment [@siepe2].

We do not expect surprises in case of the pp QFS data [@raup1; @przyb1; @zejma1], since the information of the rich set of pp data has been incorporated into the pp forces. In fact a recent analysis [@wit_coul] including the Coulomb force to pp QFS data lead to a nice agreement, while in previous analysis [@raup1; @przyb1; @zejma1] the Coulomb force was not yet included. Additional theoretical efforts to include all effects of the Coulomb force beyond the ones in [@wit_coul] are underway.

In section \[two\] we exemplify the stability of the QFS cross sections against changes of modern nuclear forces. We also demonstrate that below $\approx 30$ MeV the $^1S_0$ and $^3S_1$-$^3D_1$ NN force components dominate the QFS cross sections. In section \[three\] we analyse the np as well as the nn QFS data from [@siepe2] in terms of rigorous solutions of the 3N Faddeev equation and discuss necessary changes in the $^1S_0$ nn force component to remove the discrepancies in the nn QFS cross section. Thereby a detailed study is performed using the next-to-leading order (NLO) chiral NN force, composed of contact interactions and the one-pion exchange potential. It reveals that the effective range parameter is decisive to reconcile theory and data. The outcome is discussed in section \[four\] and further experimental insights on the nn force are proposed. Finally we summarize in section \[five\].

 Stability and sensitivity studies {#two}
==================================

It is known that nd scattering theory provides QFS cross sections which are highly independent from the realistic NN potential used in the calculations and that they practically do not change when any of the present day 3NF’s is included [@physrep96; @kur2002; @wit2010]. We exemplify it in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] for the nn and np QFS geometries of ref. [@siepe2]. There results of 3N Faddeev calculations [@physrep96] based on different high precision NN forces (CD Bonn [@cdbonn], Nijm I and Nijm II [@nijm]) alone or combined with the TM99 3NF [@TM; @tm99] are shown.

The sensitivity study performed in [@wit2010] revealed that at energies below $\approx 30$ MeV the $^1S_0$ and $^3S_1$-$^3D_1$ NN force components provide the most dominant contribution to the QFS cross sections with much less contributions of higher partial waves. Specifically, in the np QFS geometries the $^3S_1$-$^3D_1$ is the dominant force component while for nn QFS it is the $^1S_0$ force which contributes decisively. Again we exemplify it for nn and np QFS geometries of ref. [@siepe2] in Fig. \[fig\_sensiv\_pw\_qfsnnnp\].

Such a dominance for the QFS peak is understandable since the QFS cross sections are practically insensitive to the action of the presently available 3NF. Then at low energy the largest contribution should be provided by the S-wave components of the NN potential. In case of free np and nn scattering these are the $^1S_0$(np)+$^3S_1-^3D_1$ and $^1S_0$(nn) contributions, respectively. In the simple minded spirit that under QFS condition one of the three nucleons (at rest in the lab system) is just a spectator such a dominance of a two-nucleon encounter is to be expected. In reality, however, the projectile nucleon also interacts with that “ spectator” particle and the three nucleons at low energies undergo higher order rescatterings [@physrep96; @hwit89]. Thus the scattering to the final nn (np) QFS configuration also receives contributions from the np $ ^3S_1-^3D_1 $ (nn $ ^1S_0 $) interaction. Despite of all that the numerical results clearly reveal that for the np QFS configuration the $ ^3S_1-^3D_1 $ force is the most dominant contribution and for the nn QFS it is the $  ^1S_0 $ force (for free nn scattering there is no $ ^3S_1-^3D_1 $ interaction possible). This implies that the nn QFS is a powerful tool to study the $  ^1S_0 $ nn force component.

That extreme sensitivity of the nn QFS cross section to the $^1S_0 $ nn force component is demonstrated in Fig. \[fig\_1s0nn\_changes\_qfs\] for the QFS geometries of ref. [@siepe2]. To that aim we multiplied the $  ^1S_0 $ nn matrix element of the CD Bonn potential by a factor $ \lambda$. The result is, that the nn QFS cross section undergoes significant variations while the np QFS cross section is practically unchanged. The displayed $\lambda$-parameters include also the value $ \lambda=1.08 $ which is necessary to get agreement with the nn QFS data of ref.[@siepe2].

While both, $^1S_0$ and $^3S_1-^3D_1$, np forces are well determined by np scattering data (with the restrictions mentioned above) and by the deuteron properties, the $^1S_0$ nn force is determined up to now only indirectly due to lack of free nn data. The disagreement between data and theory in the nn QFS peak points to the possibility of a flaw in the nn $^1S_0$ force. It was shown in [@wit2010] that in order to remove the $\approx 18 \%$ discrepancy found in [@siepe2] for the nn QFS cross section required an increased strength of the $^1S_0$ nn interaction which when given in terms of a factor $\lambda$ amounts to $\lambda \approx 1.08$. In Fig. \[ann\_reff\_fac\] we show the effect of the $ \lambda$-modification for the nn scattering length $a_{nn}$ and for the effective range parameter $r_{eff}$, and in Fig. \[ebind1s0\_fac\] for the binding energy of two neutrons in the $^1S_0$ state. It is seen that taking $\lambda =1.08$ leads to a nearly bound state of two neutrons.

 Implications on the $^1S_0$ nn effective range parameter {#three}
=========================================================

Since the multiplication of the $^1S_0$ potential matrix element by a factor $\lambda$ induces changes in the effective range as well as in the scattering length the question arises, which from both effects is more important for the nn QFS cross section variations ? To answer that question we performed 3N Faddeev calculations based on the next-to-leading (NLO) order $\chi$PT potential [@epel2000; @epel] including all np and nn forces up to the total angular momentum $j_{max}=3$ in the two-nucleon subsystem. The $^1S_0$ component of that interaction is composed of the one-pion exchange potential and contact interactions parametrized by two parameters $\tilde C_{^1S_0}$ and $C_{^1S_0}$ $$V(^1S_0) = \tilde C_{^1S_0} + C_{^1S_0} (p^2 + p'^2) ~.$$ Standard values are $\tilde C_{^1S_0} =-0.1557374 *
10000$ GeV$^{-2}$ and $C_{^1S_0} = 1.5075220 * 10000$ GeV$^{-4}$ for cut-off combinations $\{\Lambda, \tilde {\Lambda} \} = \{450$ MeV , $500$ MeV} [@epel].

Multiplying $\tilde C_{^1S_0}$ by a factor $C_2(^1S_0)$ and $C_{^1S_0}$ by a factor $C_1(^1S_0)$ one can induce changes of the nn $^1S_0$ interaction. Requiring either the value of the scattering length $a_{nn}$ or the value of the effective range parameter $r_{eff}$ to be constant correlates the $C_1(^1S_0)$ and $C_2(^1S_0)$ factors.

Changing $C_1({^1S_0})$ and $C_2(^1S_0)$ in such a way that the scattering length is kept constant and equal $a_{nn}=-17.6$ fm leads to changes of the effective range $r_{eff}$ shown in Fig. \[reff\_from\_c1\_c2\_new\]. The resulting changes of the nn and np QFS cross sections for geometries of ref. [@siepe2] are shown in Fig. \[e26p0\_qfs\_NLO\_reff\] for five sets of $C_1(^1S_0)$ and $C_2(^1S_0)$ factors with different nn $^1S_0$ effective range parameters ranging from $r_{eff}=2.03$ fm to $r_{eff}=3.07$ fm; one of them corresponding to the value required by the data.

Similarily, changing $C_1(^1S_0)$ and $C_2(^1S_0)$ while keeping the effective range constant to $r_{eff}=2.75$ fm, leads to changes of the nn $^1S_0$ scattering length $a_{nn}$ shown in Fig. \[ann\_from\_c1\_c2\_new\]. The resulting changes of the nn and np QFS cross sections are presented in Fig. \[e26p0\_qfs\_NLO\_ann\] for four values of the nn $^1S_0$ scattering length ranging from $a_{nn}=-10.9$ fm to $a_{nn}=-75.9$ fm. It is clearly seen that the nn QFS cross sections depend only slightly on a change of the scattering length. The variations of the QFS cross section maximum stays below $\approx \pm 4 \%$. On the other side much stronger variations of the nn QFS cross sections result from changes of the effective range (see Fig. \[e26p0\_qfs\_NLO\_reff\]).

Thus we can conclude that the $\lambda$-enhancement mechanism for the $^1S_0$ nn force studied in [@wit2010] acts mainly through the change of the effective range parameter. Thus in order to remove the discrepancies found in [@siepe2] and [@ruan1] for the nn QFS cross section a change of the nn $^1S_0$ effective range parameter is required. Its value taken under the assumption of charge symmetry and charge independence of nuclear forces is $r_{eff}=2.75$ fm and it has to be changed to $r_{eff} \approx
  2.41$ fm. That implies a large charge symmetry and charge independence breaking effect of about $\approx 12 \%$ for that parameter.

We would like to add that the discussed changes of $r_{eff}$ did not affect the elastic nd cross section nor vector or tensor analysing powers to a measurable extent. Only more complicated spin observables in elastic nd scattering are affected but the present day experimental errors are much larger than those changes.

 Discussion and further experimental information {#four}
================================================

Is such a large isospin breaking effect at all possible in view of the present understanding of nuclear forces?

First of all it seems unprobable that only the effective range would reveal large isospin breaking and the scattering length will be left unaffected. In $\chi$PT the leading isospin breaking contribution is provided by isospin breaking contact interaction without derivatives [@epel_privat]. It turns out that the effective range parameter is quite insensitive to that isospin breaking contact force and typical isospin breaking effects for $r_{eff}$ are small and under $\approx 1 \%$ [@epel_privat].

The reported discrepancies for nn QFS require however a much larger effect for $r_{eff}$ of the order $\approx 12 \%$. Only when the contact terms in next orders would be unnaturally large one could expect larger isospin breaking effects for $r_{eff}$. Assuming naturalness it seems rather unprobable.

Since it seems unlikely that isospin breaking effects will show up, if at all, in the effective range parameter alone without affecting simultaneously the nn scattering length, the question of a possible existence of a bound state of two neutrons reappears.

Present day NN interactions allow only one bound state of two nucleons, namely the deuteron, where the neutron and the proton are interacting in a state with angular momenta $l=0$ or $2$, total spin $s=1$, and total angular momentum $j=1$. When the neutron and proton are interacting with the $^1S_0$ force no bound state exists and only a virtual resonant state occurs as documented by the negative scattering length $a_{np}=-21.73$ fm. Also the data for the proton-proton system exclude a $^1S_0$ pp bound state; however in this case the nuclear force is overpowered by the strong pp Coulomb repulsion. Assuming charge-independence and charge-symmetry of strong interactions also the two neutrons should not bind in the $^1S_0$ state.

It also seems that modern nuclear forces do not allow for the 3n and 4n systems to be bound [@pieper]. However, in view of the strong discrepancies between theory and data found in the nd breakup measurements for the nn QFS geometry, which cannot be explained by present day nuclear forces, it appears reasonable to check experimentally the possibility of two neutrons being bound.

There are reactions which provide conditions advantegous for a hypothetical di-neutron bound state. Such conditions can be found e.g. when two neutrons are moving with equal momenta and with relative energy close to zero. That occurs in the so called final-state-interaction (FSI) geometry of the nd breakup. Incomplete nd breakup measurements have been performed in the past to study properties of the $^1S_0$ nn force [@tornow96]. Even a dedicated experiment was performed in order to look for a hypothetical $^1S_0$ nn bound state [@vwitsch2006] in which the spectrum of the proton going in forward direction has been measured with the aim of a precise determination of its high energy region. The negative result of [@vwitsch2006] showed that the nd reaction is not suitable for such a study.

It seems that much more appropriate would be reactions in which from the begining two neutrons occupy a configuration advantegous for their binding.

It is known [@blank; @friar] that $^3$He is predominantely a spatially symmetric S state with its two protons mainly in opposite spin states. This component amounts for $\approx 90 \%$ of the $^3$He wave function. Similarily, the two neutrons in $^3$H are restricted to be in a spin-singlet state. That makes the triton target a very suitable tool to look for a nn bound state in $\gamma$ induced breakup of $^3$H. The idea is to measure the spectra of the outgoing protons in such a reaction. The two-neutron bound state, if existant, should reveal itself as a peak above the highest available proton energy from the 3-body decay of $^3$H. We show in Figs. \[fig1\] to \[fig2\] the outgoing proton spectra from the $\gamma(^3H,p)nn$ reaction for a number of $\gamma$ energies and angles of the outgoing protons. The big advantage of that reaction is that $\gamma$ interacts predominantely with the proton.

Also other reactions, such as e.g. $^3$H(n,d)nn and $^3$H(d,$^3$He)nn, provide conditions advantegous for two neutrons to bind. They are complimentary and independent from the $^3$H($\gamma$,p)nn reaction and the data from all three processes should provide an answer to the question whether two neutrons can form a bound state. The reaction $^3$H(d,$^3$He)nn cannot presently be treated in a theoretically rigorous manner, however with the rapid increase in computer power such a treatment based on Fadeev-Yakubovsky equations can be expected in the near future.

Summary {#five}
=======

The strong discrepancy in the nn QFS nd break up configuration found in [@siepe2; @ruan1] is reconsidered. It is documented again that at low energies (below $\approx 30$ MeV) the nn (np) QFS cross section depends dominantly only on the $^1S_0$ ($^3S_1-^3D_1$) NN force component and higher partial wave contributions are quite small. Furthermore the theoretical results are quite stable under exchange of the standard nuclear forces. Also the present day available 3N forces have negligible effect on the QFS configurations. Since no direct measurement of the nn force is available there is the possibility that the properties of the nn force are still unsettled. Thus simply multiplying the nn $ ^1S_0$ force matrix element by a factor $ \lambda=1.08$ one can perfectly well reconcile theory and data. In addition we performed a more detailed study using the NLO chiral potential, which is composed of the one-pion exchange and contact interactions depending on two parameters. That dependence allowed us to study separately variations in the scattering length $ a_{nn}$ leaving the effective range parameter $ r_{eff}$ constant and vice versa. Thereby it turned out that the nn QFS peak height is very sensitive to $ r_{eff}$ and hardly sensitive to $ a_{nn}$. The outcome for an agreement with the data is the requirement that $ r_{eff}$ decreases from the value $r_{eff} =2.75$ fm to a significantly smaller one, $r_{eff} = 2.41$ fm. That strongly breaks charge symmetry and charge independence and is not supported by present day chiral potential theory.

So, what might be a solution to remove the discrepancy?

If the data are taken for granted there remains the possibility that a di-neutron exists. We propose additional experimental investigations, like the $ ^3H( \gamma,p)nn$ process and evaluated the proton spectra at various emission angles emphasizing its high energy region.

The direct inclusion of $\Delta$-degrees of freedom into $\chi$PT allows for a rich set of additional NN and 3N force diagrams which are presently under investigation [@epel_privat1]. This might reconcile theory and data also for the space-star discrepancy [@physrep96] in the nd breakup process.

Right now the situation is unsettled.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work was supported by the Polish 2008-2011 science funds as the research project No. N N202 077435. It was also partially supported by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD”(VH-VI-231) and by the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. H.W. would like to thank the Kyushu University and Triangle Universities Nuclear Laboratory for hospitality and support during his stay in both institutions. The numerical calculations have been performed on the supercomputer cluster of the JSC, Jülich, Germany.

[99]{}

J.R. Bergervoet et al., Phys. Rev. C[**[41]{}**]{}, 1435 (1990). V. G. J. Stoks [*et al.*]{}, Phys. Rev. C [**48**]{}, 792 (1993). W. Glöckle, H. Wita[ł]{}a, D. Hüber, H. Kamada, J. Golak, Phys. Rep. [**[274]{}**]{}, 107 (1996). A. Siepe et al., Phys. Rev. C[**[65]{}**]{}, 034010 (2002). X.C. Ruan et al., Phys. Rev. C[**[75]{}**]{}, 057001 (2002). G. Rauprich et al., Nucl. Phys. [**[A535]{}**]{}, 313 (1991). H. Patberg et al., Phys. Rev. C[**[53]{}**]{}, 1497 (1996). J. Zejma et al., Phys. Rev. C[**[55]{}**]{}, 42 (1997). H. Wita[ł]{}a, R. Skibiński, J. Golak, and W. Glöckle, Eur. Phys. Journal A [**[41]{}**]{}, 385 (2009).

H. Wita[ł]{}a and W. Glöckle, J. Phys. G: Nucl. Part. Phys. [**37**]{}, 064003 (2010). J. Kuroś-Żo[ł]{}nierczuk, H. Wita[ł]{}a, J. Golak, H.Kamada, A. Nogga, R. Skibński, W. Glöckle, Phys. Rev. C[**[66]{}**]{}, 024004 (2002). H. Wita[ł]{}a, W. Glöckle, and Th. Cornelius, Few-Body Systems [**[6]{}**]{}, 79 (1989). R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C[**53**]{}, R1483 (1996). V. G. J. Stoks [*et al.*]{}, Phys. Rev. C [**49**]{}, 2950 (1994). S.A. Coon [*et al.*]{}, Nucl. Phys. [**A317**]{}, 242 (1979); S. A. Coon, W. Glöckle, Phys. Rev. C[**23**]{}, 1970 (1981). S. A. Coon and H. K. Han, Few Body Syst., [**30**]{}, 131 (2001). E. Epelbaum et al., Nucl. Phys. [**295**]{}, 671 (2000). E. Epelbaum, Prog. Part. Nucl. Phys. [**57**]{}, 654 (2006). E. Epelbaum, private communication. S.C. Pieper, Phys. Rev. Lett. [**90**]{}, 252501 (2003). W. Tornow, R.T. Braun, H. Wita[ł]{}a, N. Koori, Phys. Rev. C [**54**]{}, 42 (1996). W. von Witsch, X. Ruan, and H. Wita[ł]{}a Phys. Rev. C [**74**]{}, 014001 (2006). B. Blankleider and R.M. Woloshyn, Phys. Rev. C [**29**]{}, 538 (1984). J.L. Friar, B.F. Gibson, G.L. Payne, A.M. Bernstein, and T.E. Chupp, Phys. Rev. C [**42**]{}, 2310 (1990). R. B. Wiringa, V. G. J. Stoks, R. Schiavilla, Phys. Rev. C[**51**]{}, 38 (1995). J. Golak, R. Skibiński, H. Wita[ł]{}a, W. Glöckle, A. Nogga, H. Kamada, Phys. Rep. [**[415]{}**]{}, 89 (2005). E. Epelbaum, private communication.

![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2]. QFS nn refers to the angles of the two neutrons: $\theta_1=\theta_2=42^o$ and QFS np refers to the angle $\theta_1=39^o$ of the detected neutron and $\theta_2=42^o$ for the proton. In both cases $\phi_{12}=180^o$. The (practically overlapping) lines correspond to different underlying dynamics: CD Bonn [@cdbonn] - dashed (blue), Nijm I - dotted (black), Nijm II [@nijm] - dashed-dotted (green), CD Bonn+TM99 - solid (red), Nijm I +TM99 [@TM; @tm99] - dashed-double-dotted (orange), Nijm II + TM99 - double-dashed-dotted (maroon). All partial waves with 2N total angular momenta up to $j_{max}=5$ have been included. []{data-label="fig_qfs_nn_np_indep_NN_3NF"}](e26p0_qfs_S_diff_pot_fig2.eps)

![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2] specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\]. The different lines show contributions from different NN force components. The solid (red) line is the full result based on the CD Bonn potential [@cdbonn] and all partial waves with 2N total angular momenta up to $j_{max}=5$ included. The dotted (black), dashed-dotted (green), and dashed (blue) lines result when only contributions from $^1S_0$, $^3S_1-^3D_1$, and $^1S_0+^3S_1-^3D_1$ are kept calculating the cross sections. The dashed-double-dotted (brown) line presents the contribution of all partial waves with the exception of $^1S_0$ and $^3S_1-^3D_1$. []{data-label="fig_sensiv_pw_qfsnnnp"}](e26p0_qfs_S_alfa_fig4.eps)

![(color online) The cross section $d^5\sigma/d\Omega_1d\Omega_2dS$ for the $E_n^{lab}=26$ MeV nd breakup reaction $d(n,nn)p$ (upper panel) and $d(n,np)n$ (lower panel) as a function of the S-curve length for two complete configurations of Ref. [@siepe2] specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\]. The lines show sensitivity of the QFS cross sections to the changes of the nn $^1S_0$ force component. Those changes were induced by multiplying the $^1S_0$ nn matrix element of the CD Bonn potential by a factor $\lambda$. The solid (red) line is the full result based on the original CD Bonn potential [@cdbonn] ($a_{nn}=-18.8$ fm, $r_{eff}=2.79$ fm) and all partial waves with 2N total angular momenta up to $j_{max}=5$ included. The dashed (blue), dotted (black), and dashed-dotted (green) lines correspond to $\lambda=0.9$ ($a_{nn}=-8.3$ fm, $r_{eff}=3.12$ fm), $0.95$ ($a_{nn}=-11.7$ fm, $r_{eff}=2.96$ fm), and $1.05$ ($a_{nn}=-42.0$ fm, $r_{eff}=2.66$ fm), respectively. The double-dashed-dotted (violet) line shows cross sections obtained with $\lambda=1.08$ ($a_{nn}=-134.7$ fm, $r_{eff}=2.61$ fm), which factor is required to get agreement with nn QFS data of ref. [@siepe2]. []{data-label="fig_1s0nn_changes_qfs"}](e26p0_qfs_S_fac_fig6.eps)

![(color online) The changes of the nn scattering length $a_{nn}$ and the effective range parameter $r_{eff}$ with factor $\lambda$ by which the $^1S_0$ nn matrix element of the CD Bonn potential is multiplied: $V_{nn}(^1S_0)=\lambda * V_{CD~Bonn}(^1S_0)$. []{data-label="ann_reff_fac"}](ann_reff_fac.eps)

![(color online) The range of $\lambda$ values by which the $^1S_0$ nn matrix element of the CD Bonn potential is multiplied ($V_{nn}(^1S_0)=\lambda *
V_{CD~Bonn}(^1S_0)$), for which the two neutrons form a bound state with the binding energy $E_b$. []{data-label="ebind1s0_fac"}](ebind1s0_fac.eps)

![(color online) Changes of the effective range parameter $r_{eff}$ in the $^1S_0$ partial wave caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ as shown in bottom part of this figure. This correlation between the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ corresponds to a constant value of the scattering length $a_{nn}=-17.6$ fm. []{data-label="reff_from_c1_c2_new"}](reff_from_c1_c2_new.eps)

![(color online) Changes of QFS cross sections for configurations specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] caused by correlated change of factors $C_1(^1S_0)$ and $C_2(^1S_0)$ shown in Fig. \[reff\_from\_c1\_c2\_new\]. All lines show results of Faddeev calculations based on LO $\chi$PT potential and all partial waves with 2N total angular momenta up to $j_{max}=3$ included. They differ in the nn $^1S_0$ force which was obtained keeping constant scattering length $a_{nn}=-17.6$ fm and changing constants $C_1(^1S_0)$ and $C_2(^1S_0)$ to get different effective ranges which are: solid (red line) - $C_1(^1S_0)=1.0$, $C_2(^1S_0)=1.0$, $r_{eff}=2.75$ fm, dashed (blue line) - $C_1(^1S_0)=1.5$, $C_2(^1S_0)=1.0615$, $r_{eff}=3.07$ fm, dotted (black line) - $C_1(^1S_0)=0.8$, $C_2(^1S_0)=0.9275$, $r_{eff}=2.54$ fm, dashed-dotted (green line) - $C_1(^1S_0)=0.5$, $C_2(^1S_0)=0.7675$, $r_{eff}=2.03$ fm. The double-dashed-dotted (violet) line shows cross sections obtained with $C_1(^1S_0)=0.7064$, $C_2(^1S_0)=0.8842$, $r_{eff}=2.41$ fm, which are required to get agreement with nn QFS data of ref.[@siepe2]. []{data-label="e26p0_qfs_NLO_reff"}](e26p0_qfs_NLO_reff.eps)

![(color online) Changes of the nn scattering length $a_{nn}$ in the $^1S_0$ partial wave caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ as shown in the bottom part of this figure. This correlation between the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ corresponds to a constant value of the effective range parameter $r_{eff}=2.75$ fm. []{data-label="ann_from_c1_c2_new"}](ann_from_c1_c2_new.eps)

![(color online) Changes of QFS cross sections for configurations specified in Fig. \[fig\_qfs\_nn\_np\_indep\_NN\_3NF\] caused by a correlated change of the factors $C_1(^1S_0)$ and $C_2(^1S_0)$ shown in Fig. \[ann\_from\_c1\_c2\_new\]. All lines show results of Faddeev calculations based on the NLO $\chi$PT potential and all partial waves with 2N total angular momenta up to $j_{max}=3$ included. They differ in the nn $^1S_0$ force which was obtained keeping the effective range parameter $r_{eff}=2.75$ fm constant and changing the constants $C_1(^1S_0)$ and $C_2(^1S_0)$ to get different scattering lengths which are: solid (red line) - $C_1(^1S_0)=1.0$, $C_2(^1S_0)=1.0$, $a_{nn}=-17.6$ fm, dashed (blue line) - $C_1(^1S_0)=0.8$, $C_2(^1S_0)=0.8953$, $a_{nn}=-10.9$ fm, dotted (black line) - $C_1(^1S_0)=1.3$, $C_2(^1S_0)=1.1139$, $a_{nn}=-45.3$ fm, dashed-dotted (green line) - $C_1(^1S_0)=1.4$, $C_2(^1S_0)=1.1410$, $a_{nn}=-76.0$ fm. []{data-label="e26p0_qfs_NLO_ann"}](e26p0_qfs_NLO_ann.eps)

![(color online) The spectra of the outgoing proton from the reaction $^3H(\gamma,p)nn$ with $E_{\gamma}=10$ MeV at different lab. angles of the proton. They have been calculated using the AV18 [@av18] NN interaction and the current composed of single nucleon and meson exchange currents [@physrep2005]. []{data-label="fig1"}](3Hincl_w10_th_0_to_75.eps)

![(color online) The spectra of the outgoing proton from the reaction $^3H(\gamma,p)nn$ with $E_{\gamma}=10$ MeV at different lab. angles of the proton. They have been calculated using the AV18 [@av18] NN interaction and the current composed of single nucleon and meson exchange currents [@physrep2005]. []{data-label="fig2"}](3Hincl_w10_th_90_to_180.eps)
