Perturbative treatment of three-nucleon force contact terms in three-nucleon Faddeev equations
aa r X i v : . [ nu c l - t h ] M a r Noname manuscript No. (will be inserted by the editor)
Perturbative treatment of three-nucleon force contactterms in three-nucleon Faddeev equations
H. Wita la, J. Golak, R. Skibi´nski,K. Topolnicki
Received: date / Accepted: date
Abstract
We present a perturbative approach to solving the three-nucleon con-tinuum Faddeev equation. This approach is particularly well suited to dealing withvariable strengths of contact terms in a chiral three-nucleon force. We use examplesof observables in the elastic nucleon-deuteron scattering as well as in the deuteronbreakup reaction to demonstrate high precision of the proposed procedure andits capability to reproduce exact results. A significant reduction of computer timeachieved by the perturbative approach in comparison to exact treatment makesthis approach valuable for fine-tuning of the three-nucleon Hamiltonian parame-ters.Special Issue: “Celebrating 30 years of Steven Weinberg’s papers on Nuclear Forcesfrom Chiral Lagrangians”
The nuclear force problem is the basic one for understanding nuclear phenomena.Since the birth of nuclear physics it has been at the centre of experimental andtheoretical studies. Extensive efforts based on purely phenomenological approachesor incorporating the meson-exchange picture have led to numerous nucleon-nucleon(NN) potentials, able to describe with high precision a vast amount of available
H. Wita laM. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krak´ow, PolandE-mail: [email protected]. GolakM. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krak´ow, PolandR. Skibi´nskiM. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krak´ow, PolandK. TopolnickiM. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krak´ow, Poland H. Wita la, J. Golak, R. Skibi´nski, K. Topolnicki data [1]. In spite of the enormous progress in understanding properties of thetwo-nucleon interaction, applications of these ideas to the many-nucleon forcesencountered consistency problems and called for a more systematic framework.The advent of QCD gave a new impetus to the derivation of nuclear forces anda major breakthrough occurred with the emergence of the effective field theory(EFT) concept and publishing of S. Weinberg’s seminal paper [2]. It paved theway for developing accurate and precise nuclear forces within the EFT framework[3, 4, 5, 6, 7].The progress in constructing nuclear forces within EFT approach is presentlydocumented by availability of numerous high precision NN potentials which pro-vide satisfactory description of NN data in a wide energy range. Recently a newgeneration of chiral NN potentials was introduced and developed up to the fifthorder (N LO) of chiral expansion by the Bochum-Bonn [8, 9] and Idaho-Salamanca[10] groups. While in the Idaho-Salamanca force the nonlocal momentum spaceregularization is applied directly in momentum space by introducing a cutoff pa-rameter Λ , the one-pion and two-pion exchange contributions in the Bochum-Bonnpotential are regularized in coordinate space using a cutoff parameter R and thentransformed to momentum space. For the contact interactions the Bochum-BonnNN force employs a simple Gaussian nonlocal momentum-space regulator with thecutoff Λ = 2 R − . Potentials of both groups are available for a number of regu-larization parameters. They provide a very good description of the NN data set(Idaho-Salamanca) or the phase shifts and mixing angles of the Nijmegen partialwave analysis [11] (Bochum-Bonn), used to fix the low-energy constants accom-panying the NN contact interactions. The latest and most precise EFT-based NNinteraction is the so called semilocal momentum-space (SMS) regularized chiralpotential of the Bochum-Bonn group [12], developed up to N LO and even in-cluding some terms from the next order of chiral expansion (N LO + ). In thispotential a new momentum-space regularization scheme has been employed forthe long-range contributions and a nonlocal Gaussian regulator has been appliedto the minimal set of independent contact interactions. This new approach can bestraightforwardly utilised to regularize also three-nucleon (3N) forces. That newfamily of semilocal chiral potentials provides an outstanding description of the NNdata and is available up to N LO + for five values of the cutoff Λ .Applications of EFT approach in the form of chiral perturbation theory (ChPT)have resulted not only in the theoretically well grounded NN potentials but alsofor the first time have given a possibility to apply in practical calculations NNforces augmented with consistent 3N interactions, both derived within the sameformalism. Understanding of nuclear spectra and reactions based on these con-sistent chiral two- and many-body forces has become a hot topic of present dayfew-body studies and is also the main aim of the Low Energy Nuclear PhysicsInternational Collaboration (LENPIC) [13].The first nonvanishing contributions to the 3N force (3NF) appear at next-to-next-to-leading order of chiral expansion (N LO) [3, 14] and comprise in additionto the 2 π -exchange term two contact contributions with strength parameters c D and c E [15]. The difficult task to derive the chiral 3NF at next-to-next-to-next-to-leading order (N LO) has been performed in [16, 17]. At that order five differenttopologies contribute to 3NF. Three of them are of long-range character [16] andare given by two-pion (2 π ) exchange graphs, by two-pion-one-pion (2 π − π ) ex-change graphs, and by the so-called ring diagrams. They are supplemented by the erturbative treatment of three-nucleon force contact .... 3 short-range two-pion-exchange-contact (2 π -contact) term and by the leading rela-tivistic corrections to 3NF [17]. The 3NF at N LO order does not involve any newunknown low-energy constants (LECs) and depends only on two parameters, c D and c E that parameterize the leading one-pion-contact term and the 3N contactterm present already at N LO. The c D and c E values need to be then fixed at thisorder, as at N LO, from a fit to few-nucleon data. At the higher order, N LO, inaddition to long- and intermediate-range interactions generated by pion-exchangediagrams [18, 19], the chiral N LO 3NF involves thirteen purely short-range oper-ators, which have been worked out in [20].Since the advent of numerically exact three-nucleon continuum Faddeev cal-culations the elastic nucleon-deuteron (Nd) scattering and the deuteron breakupreaction have become a powerful tool to test modern models of the nuclear forces[21, 22, 23]. With the appearance of high precision (semi)phenomenological NN po-tentials and first models of 3NF the question about the importance of 3NF hasdeveloped into the main topic of 3N system studies. That issue has been givena new impetus by ChPT-based approaches and the possibility to apply consis-tent two- and many-body nuclear forces, derived within this framework, in 3Ncontinuum calculations.First applications of (semi)phenomenological NN and 3N forces to elastic Ndscattering and to the nucleon-induced deuteron breakup reaction revealed inter-esting cases of discrepancies between pure two-nucleon (2N) theory and data,indicating possibly large 3NF effects [24, 25].Using chiral 3NF’s in 3N continuum requires numerous time consuming com-putations with varying strengths of the contact terms in order to establish theirvalues. They can be determined for example from the H binding energy andthe minimum of the elastic Nd scattering differential cross section at the energy( E lab ≈
70 MeV), where the effects of 3NF start to emerge in elastic Nd scattering[24, 26]. Specifically at N LO, after establishing the so-called ( c D , c E ) correlationline, which for a particular chiral NN potential combined with a N LO 3NF givespairs of ( c D , c E ) values reproducing the H binding energy, a fit to experimentaldata for the elastic Nd cross section is performed to determine the c D and c E strengths. Fine-tuning of the 3N Hamiltonian parameters requires an extensiveanalysis of available 3N elastic Nd scattering and breakup data. That ambitiousgoal calls for a significant reduction of computer time necessary to solve the 3NFaddeev equations. Thus finding an efficient emulator for exact solutions of the3N Faddeev equations seems to be essential and of high priority.In this paper we propose such an emulator and test its efficiency as well as abil-ity to accurately reproduce exact solutions of 3N Faddeev equations. In Sec. 2 webriefly describe the formalism of 3N continuum Faddeev calculations and presentthe new scheme which is well-suited to fast calculations with varying strengths ofthe contact terms in a chiral 3NF. Tests and an evaluation of that new approachbased on example observables in elastic Nd scattering and in selected breakupconfigurations are presented in Sec. 3. We summarize and conclude in Sec. 4. Theoretical predictions shown in the present paper were obtained within the 3NFaddeev formalism using chiral two- and three-body forces. The formalism itself
H. Wita la, J. Golak, R. Skibi´nski, K. Topolnicki and numerical performance were presented in numerous publications so we onlybriefly summarize the formalism and for details refer to [21, 27, 28, 29].Neutron-deuteron scattering with nucleons interacting via NN interactions v NN and a 3NF V = V (1) + V (2) + V (3) , is described in terms of a breakupoperator T satisfying the Faddeev-type integral equation [21, 27, 28] T | φ i = tP | φ i + (1 + tG ) V (1) (1 + P ) | φ i + tP G T | φ i + (1 + tG ) V (1) (1 + P ) G T | φ i . (1)The 2N t -matrix t is the solution of the Lippmann-Schwinger equation with theinteraction v NN . V (1) is the part of a 3NF which is symmetric under the in-terchange of nucleons 2 and 3: V = V (1) (1 + P ). The permutation operator P = P P + P P is given in terms of the transposition operators, P ij , whichinterchange nucleons i and j . The initial state | φ i = | q i| φ d i describes the free mo-tion of the neutron and the deuteron with the relative momentum q and containsthe internal deuteron wave function | φ d i . Finally, G is the resolvent of the three-body center-of-mass kinetic energy. The amplitude for elastic scattering leading tothe two-body final state | φ ′ i is then given by [21, 28] h φ ′ | U | φ i = h φ ′ | P G − | φ i + h φ ′ | P T | φ i + h φ ′ | V (1) (1 + P ) | φ i + h φ ′ | V (1) (1 + P ) G T | φ i , (2)while the corresponding amplitude for the breakup reaction reads h ~p~q | U | φ i = h ~p~q | (1 + P ) T | φ i , (3)where the free three-body breakup channel state | ~p~q i is defined in terms of the twoJacobi (relative) momentum vectors ~p and ~q .We solve Eq. (1) in the momentum-space partial wave basis | pqα i , determinedby the magnitudes of the 3N Jacobi momenta p and q together with the angularmomenta and isospin quantum numbers α containing the 2N subsystem spin, or-bital and total angular momenta s, l and j , the spectator nucleon orbital and totalangular momenta with respect to the center of mass (c.m.) of the 2N subsystem, λ and I : | pqα i ≡ | pq ( ls ) j ( λ
12 ) I ( jI ) J ( t T i . (4)The total 2N and spectator angular momenta j and I as well as isospins t and , arefinally coupled to the total angular momentum J and isospin T of the 3N system.In practice a converged solution of Eq. (1) using partial wave decomposition inmomentum space at a given energy E requires taking all 3N partial wave statesup to the 2N angular momentum j max = 5 and the 3NF force acting up to the 3Ntotal angular momentum J = 7 /
2. The number of resulting partial waves (equalto the number of coupled integral equations in two continuous variables p and q )amounts to 142. The required computer time to get one solution on a personalcomputer is about ≈ LO has one parameter-free term erturbative treatment of three-nucleon force contact .... 5 (2 π -exchange contribution) and two contact terms with strength parameters c D and c E . At N LO there are more several parameter-free parts but again only twocontact terms. At N LO again, parameter free contributions are supplemented byfifteen contact terms with strengths: c D , c E , c E , ..., c E . All these contact termsare restricted to small 3N total angular momenta and to only few partial wavestates for a given total 3N angular momentum J and parity π . For example for J π = 7 / ± all the matrix elements < pqα | V (1) | p ′ q ′ α ′ > proportional to c E and c E vanish, while the c D and c E terms are nonzero only for a restricted numberof α, α ′ pairs (mostly these containing S and S − D quantum numbers) [14,15]. Bearing that in mind and taking into account the fact that contact termsyield a small contribution to the 3N potential energy compared to the leading,parameter-free part, it is possible to apply a perturbative approach in order toinclude the contact terms.Let us split the V (1) part of 3NF into a parameter-free term V ( θ ) and a sumof contact terms ∆V ( θ ): h pqα | V (1) (1 + P ) | p ′ q ′ α ′ i = h pqα | V ( θ )(1 + P ) | p ′ q ′ α ′ i + h pqα | ∆V ( θ )(1 + P ) | p ′ q ′ α ′ i , (5)with θ = ( c D = 0 , c E = 0 , c E i = 0) and θ = ( c D , c E , c E i ) the set of values forcontact terms for which we would like to find solution of 3N Faddeev equations.Then we divide the 3N partial wave states into two sets: β and the remainingone, α . The β set is defined by nonvanishing matrix elements of ∆V ( θ ). FromEq. (1) one obtains (omitting the Jacobi momenta in notation of partial wavestates): h α | T ( θ ) | φ i = h α | tP | φ i + h α | (1 + tG )[ V ( θ ) + ∆V ( θ )](1 + P ) | φ i + h α | tP G T ( θ ) | φ i + h α | (1 + tG )[ V ( θ ) + ∆V ( θ )](1 + P ) G T ( θ ) | φ ih β | T ( θ ) | φ i = h β | tP | φ i + h β | (1 + tG )[ V ( θ ) + ∆V ( θ )](1 + P ) | φ i + h β | tP G T ( θ ) | φ i + h β | (1 + tG )[ V ( θ ) + ∆V ( θ )](1 + P ) G T ( θ ) | φ i . (6)Introducing T ( θ ) and ∆T ( θ ) such that T ( θ ) = T ( θ ) + ∆T ( θ ), one gets: h α | T ( θ ) | φ i + h α | ∆T ( θ ) | φ i = h α | tP | φ i + h α | (1 + tG ) V ( θ )(1 + P ) | φ i + h α | (1 + tG ) ∆V ( θ )(1 + P ) | φ i + h α | tP G T ( θ ) | φ i + h α | tP G ∆T ( θ ) | φ i + h α | (1 + tG ) V ( θ )(1 + P ) G T ( θ ) | φ i + h α | (1 + tG ) V ( θ )(1 + P ) G ∆T ( θ ) | φ i + h α | (1 + tG ) ∆V ( θ )(1 + P ) G T ( θ ) | φ i + h α | (1 + tG ) ∆V ( θ )(1 + P ) G ∆T ( θ ) | φ i . (7)Since ∆V ( θ ) has nonvanishing elements only for channels | β i then it followsthat h α | (1 + tG ) ∆V ( θ )(1 + P ) | φ i = 0 h α | (1 + tG ) ∆V ( θ )(1 + P ) G T ( θ ) | φ i = 0 h α | (1 + tG ) ∆V ( θ )(1 + P ) G ∆T ( θ ) | φ i = 0 , (8)and Eq. (7) can be written as two separate equations for h α | T ( θ ) | φ i and h α | ∆T ( θ ) | φ i : h α | T ( θ ) | φ i = h α | tP | φ i + h α | (1 + tG ) V ( θ )(1 + P ) | φ i + h α | tP G T ( θ ) | φ i H. Wita la, J. Golak, R. Skibi´nski, K. Topolnicki + h α | (1 + tG ) V ( θ )(1 + P ) G T ( θ ) | φ ih α | ∆T ( θ ) | φ i = h α | tP G ∆T ( θ ) | φ i + h α | (1 + tG ) V ( θ )(1 + P ) G ∆T ( θ ) | φ i . (9)Inserting the decomposition of T ( θ ) into the second equation in (6) for channels | β i one obtains: h β | T ( θ ) | φ i = h β | tP | φ i + h β | (1 + tG ) V ( θ )(1 + P ) | φ i + h β | tP G T ( θ ) | φ i + h β | (1 + tG ) V ( θ )(1 + P ) G T ( θ ) | φ ih β | ∆T ( θ ) | φ i = h β | (1 + tG ) ∆V ( θ )(1 + P ) | φ i + h β | (1 + tG ) ∆V ( θ )(1 + P ) G T ( θ ) | φ i + h β | (1 + tG )[ V ( θ ) + ∆V ( θ )](1 + P ) G ∆T ( θ ) | φ i + h β | tP G ∆T ( θ ) | φ i . (10)The first equations in (9) and (10) are the Faddeev equations (1) for T ( θ ).Since the two leading terms for h β | ∆T ( θ ) | φ i in (10) are of the order of ∆V ( θ )then h α | ∆T ( θ ) | φ i ≈ T ( θ ) is found (it is independent from parameters θ ). In the next stepa solution of second equation in the set (10) for h β | ∆T ( θ ) | φ i is obtained and fromthat h α | ∆T ( θ ) | φ i is calculated by: h α | ∆T ( θ ) | φ i = h α | tP G X β Z p ′ q ′ | p ′ q ′ β ih p ′ q ′ β | ∆T ( θ ) | φ i + h α | (1 + tG ) V ( θ )(1 + P ) G X β Z p ′ q ′ | p ′ q ′ β ih p ′ q ′ β | ∆T ( θ ) | φ i . (11)Finally, T ( θ ) is calculated as h α | T ( θ ) | φ i = h α | T ( θ ) | φ i + h α | ∆T ( θ ) | φ ih β | T ( θ ) | φ i = h β | T ( θ ) | φ i + h β | ∆T ( θ ) | φ i (12) To check the quality of the proposed scheme we have chosen one NN potential fromamong available chiral NN interactions, namely the SMS N LO + chiral potentialof the Bochum-Bonn group [12] with the regularization cutoff Λ = 450 MeV, andcombined it with the chiral N LO 3NF augmented by one out of thirteen contactterms from the N LO 3NF, that is E . The low-energy constants of the contactinteractions in that 3NF were adjusted to the triton binding energy and we usedthe set of strengths θ = ( c D = − . , c E = − . , c E = 2 .
0) (according to thenotation of Refs. [14, 15]).We solved the 3N Faddeev equation (1) exactly at two incoming neutron ener-gies E = 70 and 190 MeV with such a choice of NN and 3N forces as well as withthe same NN potential combined with the 3NF restricted only to the parameterfree 2 π -exchange N LO term (set θ = ( c D = 0 . , c E = 0 . , c E = 0 . θ set forms a starting point inthe proposed perturbative treatment of Eqs. (9)-(12) and has to be calculated onlyonce, regardless of how many variations of strength parameters are required. In thenext step, we performed, at the same energies, our perturbative treatment, solving erturbative treatment of three-nucleon force contact .... 7 first the second equation in set (10). Having determined h α | ∆T ( θ ) | φ i from Eq.(11)we calculated our emulator solution of Eq. (12), with two sets of 3N channels | β i ,one comprising the 2N subsystem states S + S − D and second includingall 2N states with the total angular momentum j ≤
2. The number of 3N partialwaves diminishes from 142 to 34 in the second set of states and to 20 in the firstone. Such a smaller set of 3N channels leads to a reduction (by a factor of ≈
4) ofthe computer time required in the perturbative approach as compared to the exactcomputation (one run in the exact approach requires ≈
30 minutes of a personalcomputer time, provided that the V ( θ )(1 + P ) and V ( θ i )(1 + P ) kernels, actingin (1 + tG ) V ( θ )(1 + P ) G T ( θ ) | φ i term of Eq. (1), are calculated in advance (withthe strength of the contact term i : c i and θ i = ( c i = 1 . , c k = i = 0 . ≈ S + S − D choice of | β i channels ((magenta) dash-double-dotted line) follows quite well theexact predictions ((orange) long dashed line). The results for the choice j ≤ S state. We show in Fig. 4a and b the cross section for FSI(1-2) inthe d(n,nn)p breakup reaction exactly at the FSI condition as a function of thelaboratory angle of one of the two FSI interacting nucleons. For one detectionangle we show in Fig. 4c also the FSI cross section along the arc-length parameterS, which for the fixed angles defines unambiguously the energies of all the threeoutgoing nucleons.Under the symmetrical-space-star condition the momenta of the free outgoingnucleons have the same magnitudes in the 3N c.m. frame and form a three-pointed”Mercedes-Benz” star in the plane symmetrical with respect to the incoming nu-cleon momentum (the plane is bent at the angle θ c.m.plane with respect to the incomingnucleon momentum). We show the SST cross section as a function of that anglein Fig. 4e and f as well as for the case of θ c.m.plane = 90 o along the arc-length param-eter S in Fig. 4d. The results are similar to the elastic scattering case. Again theperturbative treatment reproduces quite well cross sections calculated with exactsolutions. Only at 190 MeV in the region of FSI peak the precision is reduced andamounts to ≈ − H. Wita la, J. Golak, R. Skibi´nski, K. Topolnicki
We presented an approximate approach which enables us to take into accountcontact terms of a chiral 3NF in the 3N continuum Faddeev calculations. Suchcontact terms are short ranged and thus act only in the partial waves with lowangular momenta. That reduction of 3N partial wave states together with smallmagnitudes of the contact contributions as compared with the leading NN po-tential and parameter-free terms in a 3NF enable us to treat the contact termsperturbatively. The proposed perturbative approach allows one to reduce by thefactor of about four the required computation time and is thus especially suited torepeated calculations with varying strengths of contact terms. We checked that theproposed treatment allows us to reproduce surprisingly well the exact predictionsfor nd elastic scattering as well as for nd breakup observables. It is conceivablethat with the help of the constructed emulator of the exact solutions of 3N Fad-deev equations fine tuning of a 3N Hamiltonian parameters based on available 3Nscattering data is feasible.
Acknowledgements
This study has been performed within Low Energy Nuclear Physics In-ternational Collaboration (LENPIC) project and was supported by the Polish National ScienceCenter under Grant No. 2016/22/M/ST2/00173. The numerical calculations were performedon the supercomputer cluster of the JSC, J¨ulich, Germany. We would like to thank A. Nogga,K. Hebeler, and P. Reinert for providing us with matrix elements of chiral 3NF’s.
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Predictions of currentmodels. Phys. Rev. C , 024003-1-15 (2002)26. Wita la H., Gl¨ockle W., H¨uber D., Golak J., Kamada H.: Cross Section Minima in ElasticNd Scattering: Possible Evidence for Three-Nucleon Force Effects. Phys. Rev. Lett. ,1183-1186 (1998)27. Wita la H., Cornelius T. and Gl¨ockle W.: Elastic scattering and break-up processes in then-d system. Few-Body Syst. , 123-134 (1988)28. H¨uber D., Kamada H., Wita la H., and Gl¨ockle W.: How to include a three-nucleon forceinto Faddeev equations for the 3N continuum: a new form. Acta Physica Polonica B28 ,1677-1685 (1997)29. Gl¨ockle W.: The Quantum Mechanical Few-Body Problem. Springer-Verlag 1983. . d σ / d Ω [ m b s r - ] y θ c.m. [deg]-0.4-0.20.00.2iT θ c.m. [deg] -0.4-0.20.00.20.4E=190 MeVE=70 MeV E=190 MeVE=70 MeVE=70 MeV E=190 MeV Fig. 1 (color online) The angular distribution dσdΩ , the neutron analyzing power A y , andthe deuteron vector analyzing power iT in elastic nd scattering at the incoming neutronlaboratory energy E = 70 and 190 MeV. The different curves are predictions of the chiralSMS N LO + NN potential with regulator Λ = 450 MeV alone ((grey) solid line) or combinedwith the N LO 3NF comprising also one contact term E from N LO. The exact predictionfor that combination with strength parameters ( c D = − . , c E = − . , c E = 2 . LO 3NF (set of parameters ( c D = 0 . , c E = 0 . , c E = 0 .
0) is shown by (maroon) shortdashed line. The predictions based on approximate treatment with channels | β i restricted toonly S + S − D and to all j max = 2 channels, are represented by (magenta) dash-double-dotted and (black) dotted lines, respectively.erturbative treatment of three-nucleon force contact .... 11 -0.4-0.20.00.2T -0.8-0.40.0-0.6-0.4-0.20.0-0.2-0.10.0T θ c.m. [deg]-0.3-0.2-0.10.0T θ c.m. [deg] -0.3-0.2-0.10.0E=70 MeV E=190 MeVE=70 MeV E=190 MeVE=70 MeV E=190 MeV Fig. 2 (color online) The same as in Fig. 1 but for deuteron tensor analyzing powers T , T ,and T .2 H. Wita la, J. Golak, R. Skibi´nski, K. Topolnicki y,y yy’ (N-N) 0 60 120 θ c.m. [deg]0.20.40.6K yy’ (d-N) 0 60 120 180 θ c.m. [deg] 0.00.20.40.6E=190 MeVE=70 MeVE=70 MeV E=190 MeVE=70 MeV E=190 MeV Fig. 3 (color online) The same as in Fig. 1 but for the spin correlation C y,y , polarizationtransfer coefficient from nucleon to nucleon K y ′ y ( N − N ), and polarization transfer coefficientfrom deuteron to nucleon K y ′ y ( d − N ).erturbative treatment of three-nucleon force contact .... 13 θ [deg]0.11.0 d σ / d Ω d Ω d S [ m b s r - M e V - ] θ [deg] 0.11.00 50 100 150S [MeV] 0.0160 80 100S [MeV]0.010.10 d σ / d Ω d Ω d S [ m b s r - M e V - ] θ planec.m. [deg]0.010.101.00 d σ / d Ω d Ω d S [ m b s r - M e V - ] θ planec.m. [deg] 0.010.10 d(n,nn)pE=70 MeV a) b)c) d) d(n,nn)pE=190 MeVd(n,nn)p E=190 MeV d(n,nn)p E=190 MeV d(n,nn)p E=70 MeV d(n,nn)pE=190 MeV e) f)
FSI(nn) FSI(nn)SST FSI(nn)SST( θ planec.m. =90 o )SST θ =θ =35 ο φ =0 ο θ =θ =54.5 ο φ =120 ο Fig. 4 (color online) The exclusive breakup d(n,nn)p cross section d σ/dΩ dΩ dS at theincoming neutron lab. energy E = 70 and 190 MeV for the neutron-neutron FSI condition( ~p = ~p ) as a function of the lab. production angle of the outgoing neutrons θ lab = θ lab and φ = 0 o : a), b), and for the SST condition as a function of the c.m. angle between plane, inwhich in the 3N c.m. frame momenta of three outgoing nucleons are placed, and the incomingnucleon momentum: e), f). In c) the cross sections for a particular FSI(nn) configuration fromb) produced at θ lab = θ lab = 35 o and in d) for the SST from f) with θ c.m.plane = 90 oo