---

Centro de Fisica Nuclear da Universidade de Lisboa, P-1699 Lisboa Codex, Portugal

arXiv:nucl-th/9305014v1 20 May 1993Three-Nucleon Forceandthe ∆-Mechanism for Pion Production and Pion AbsorptionM. T. Pe˜naCentro de Fisica Nuclear da Universidade de Lisboa, P-1699 Lisboa Codex, PortugalandContinuous Electron Beam Accelerator Facility, Newport News, Virginia 23606, U.S.A.P.

U. SauerInstitut f¨ur Theoretische Physik, Universit¨at Hannover, D-3000 Hannover 1, GermanyA. StadlerInstitut f¨ur Theoretische Physik, Universit¨at Hannover, D-3000 Hannover 1, GermanyandDepartment of Physics, College of William and Mary, Williamsburg, Virginia 23185, U.S.A.∗G.

KortemeyerInstitut f¨ur Theoretische Physik, Universit¨at Hannover, D-3000 Hannover 1, Germany(March 21, 2018)AbstractThe description of the three-nucleon system in terms of nucleon and ∆de-grees of freedom is extended to allow for explicit pion production (absorption)from single dynamic ∆de-excitation (excitation) processes. This mechanismyields an energy dependent effective three-body hamiltonean.

The Faddeevequations for the trinucleon bound state are solved with a force model that hasalready been tested in the two-nucleon system above pion-production thresh-old.The binding energy and other bound state properties are calculated.The contribution to the effective three-nucleon force arising from the pionicdegrees of freedom is evaluated. The validity of previous coupled-channel cal-culations with explicit but stable ∆isobar components in the wavefunction isstudied.21.30.+y, 21.10.Dr, 21.45+v, 27.10.+hTypeset using REVTEX1

I. INTRODUCTIONTwo-nucleon and three-nucleon forces are effective interactions [1]. A truely microscopicdescription of the nucleus in terms of quantum chromodynamics with quark and gluondegrees of freedom would avoid the notion of two-nucleon and three-nucleon forces altogether.If one views the nucleus in the more traditional hadronic picture as a system of nucleons,isobars, their corresponding antiparticles and mesons, one encounters interactions in formof baryon-meson vertices, but again no two-nucleon and three-nucleon forces.

Two-nucleonand three-nucleon forces arise when subnucleonic degrees of freedom are frozen. They aretherefore artifacts of theoreticians who choose to work in a Hilbert space with a restrictednumber of degrees of freedom.At intermediate energies the pion and ∆-isobar degrees of freedom become active innuclear reactions.

Any efficient description of nuclear phenomena at intermediate energieshas to treat the pion and ∆-isobar degrees of freedom explicitly besides the nucleon one.A Hilbert space doing so is illustrated in Fig. 1.

It contains – besides the purely nucleonicsector HN – a sector H∆in which one nucleon is replaced by a ∆-isobar and a sector Hπin which one pion is added to the nucleons. That extended, though still rather restrictedHilbert space is motivated in Ref.

[2]. A force model acting in that Hilbert space is developedin Refs.

[2] and [3]. Its hamiltonian is diagrammatically defined in Fig.

2. It is not covariant.Its unlinked one-baryon processes (e) and (f) can fully account for pion-nucleon scattering upto 300 MeV pion lab energy.

Those processes are to be parametrized consistently with data.The ∆-isobar of the force model is a bare particle which only by its coupling to pion-nucleonstates describes the physical P33 resonance. The force model builds up the mechanism forpion production or pion absorption as a two-step process, i.e.,• by the excitation of a nucleon to or by the deexcitation of a ∆-isobar through theinstantaneous transition potential (b) and• by the subsequent decay of that ∆-isobar into or by its formation from pion-nucleonstates through the pion-nucleon-∆vertex (e).The force model is tested in the two-nucleon system above pion-threshold, e.g., in Refs.

[4]and [5] for all reactions with at most one pion, coupled by unitarity. It is not tuned yetand therefore fails to account for many observables as any nucleon-nucleon potential wouldwithout a proper fit.

However, the force model is constructed to account for nucleon-nucleonscattering below pion-threshold with satisfactory quality. A comparison with other forcemodels of similar structure is given in Ref.

[6].The force model of Fig. 2 is not only meant to account for the two-nucleon system belowand above pion threshold.In heavier nuclei it can also provide a microscopic basis fordescribing reactions at intermediate energies, and the force model has been employed thatway [7].

When applied to nuclear structure problems, the force model yields corrections [8]for the picture of the nucleus as a system of nucleons only interacting through two-nucleonforces with each other and through single-nucleon currents with external probes. E.g., whenapplied to the trinucleon bound state, it yields corrections to the effective three-nucleonforce and to the effective two- and three-nucleon currents [9,10].

Characteristic examples forcontributions to the three-nucleon force are illustrated in Fig. 3.

It has been argued [11,12]2

and shown [13] that the contributions arising from the ∆-isobar and the pion degrees offreedom are most important in a full and realistic three-nucleon force. Subject of this paperis the relation between the three-nucleon force and the force model of Fig.

2 with ∆-isobarand pion degrees of freedom.Preliminary discussions of that theme have been given inRefs. [14] and [15].Sect.

II recalls the precise definition of the force model of Fig. 2.

It applies the forcemodel to the trinucleon bound state. The resulting set of equations is very close to thoseof a coupled-channel treatment [9,10] for nuclear bound states, in which the ∆-isobar isconsidered a stable particle with fixed mass.

Sect. III describes the actual calculations carriedout in this paper for the full force model and for approximated variants of it.

The calculationsare meant to explore the validity of the corresponding coupled-channel description of nuclearbound states. Sect.

IV presents the obtained results. Sect.

V discusses conclusions.II. APPLICATION OF THE FORCE MODEL WITH ∆-ISOBAR AND PIONDEGREES OF FREEDOM TO THE TRINUCLEON BOUND STATEThe hamiltonian H of the force model acts in the three sectors of Hilbert space andcouples them.

The projectors on the two baryonic sectors HN and H∆are denoted by PNand P∆, respectively, with the abbreviation P = PN + P∆, the projector on the sector Hπwith a pion by Q. Thus, PN + P∆+ Q = 1.

The kinetic part H0 of the hamiltonian definesthe Hilbert sectors and commutes with the projectors. It consists of the individual baryoniccontributions h0(i), i being the baryon label; in the Hilbert sector Hπ it has the additionalpion contribution h0(π).

It includes rest masses. The mass m0∆of the bare ∆-isobar used inthe definitions of the Hilbert sector H∆and of the force model is unobservable.

The single-particle momentum of a nucleon, a ∆-isobar and a pion is kN, k∆and kπ, respectively.The kinetic energies of the nucleon and of the ∆-isobar are taken to be nonrelativistic, i.e.,εN(kN) = mN + k2N/2mN and ε∆(k∆) = m0∆+ k2∆/2m0∆, the kinetic energy of the pion tobe relativistic, i.e., ωπ(kπ) =qm2π + k2π. The interaction part H1 of the hamiltonian is builtfrom instantaneous unretarded potentials.

It connects the pionic sector only to the one witha ∆-isobar, thus, PNH1Q = QH1PN = 0. The interaction part of the hamiltonian takes theformH1 = (PN + P∆)H1(PN + P∆) +P∆H1Q + QH1P∆+ QH1Q .

(2.1)The Schr¨odinger equation for the trinucleon bound-state energy EB and for the wavefunction |ΨB⟩[H0 + H1]|ΨB⟩= EB|ΨB⟩(2.2)is projected onto the baryonic and pionic sectors of Hilbert space, i.e.,[P(H0 + δH0(EB))P + P(H1 + δH1(EB))P]P|ΨB⟩= EBP|ΨB⟩,(2.3a)3

Q|ΨB⟩=QEB −QHQQH1P P|ΨB⟩,(2.3b)⟨ΨB|P|ΨB⟩+ ⟨ΨB|Q|ΨB⟩= 1 . (2.3c)The triton binding energy ET without rest masses, i.e., ET = EB −3mN, and the baryoniccomponents (PN +P∆)|ΨB⟩of the trinucleon wave function follow from solving the projectedequation (2.3a).

The pionic component Q|ΨB⟩of the wave function is obtained from its ∆-component P∆|ΨB⟩according to Eq. (2.3b) by quadrature.

The projected equation (2.3a)contains the energy-dependent parts δH0(EB) and δH1(EB), i.e.,PδH0(z)P ="PH1QQz −QHQQH1P#disconnected,(2.4a)PδH1(z)P ="PH1QQz −QHQQH1P#connected. (2.4b)The disconnected part δH0(z) is of one-baryon nature, the connected part δH1(z) containstwo-baryon and possibly three-baryon pieces according to Fig.

4. Both parts are only definedin the baryonic sectors of Hilbert space and are – in the considered force model – nonzero onlyin the one with a ∆-isobar, i.e., PδH0(z)PN = PNδH0(z)P = PδH1(z)PN = PNδH1(z)P =0.

They show an energy dependence, though the original hamiltonian acts instantaneouslywithout time delay. That energy dependence arises from projecting the pionic component outfrom the wave function.

However, by that energy dependence the pionic component preservesits active presence. Thus, the energy dependence in δH0(z) and δH1(z) is necessary and itis well regulated, always prescribed without any arbitrariness in all applications.

This paperonly deals with the trinucleon bound state; but clearly the same set of equations (2.3) holdfor scattering problems: Due to the energy dependence of the effective baryonic hamiltonianP(H0+δH0(z)+H1+δH1(z))P the wave function components of bound and scattering states,projected onto the baryonic sectors, are not orthogonal, though they belong to states ofdifferent three-nucleon energy. However, the formalism naturally restores the orthogonalityfor the full states which include their pionic components.

In the same way only the fullbound-state wave function is to be normalized according to Eq. (2.3c).

The controlled energydependence of the effective baryonic hamiltonian is in contrast to a phenomenologicallychosen energy dependence of the two-nucleon potential as used once in a while [16]; in thelatter case there are no rules which affect the change in the energy dependence to be adoptedfor different applications.Despite the energy dependence of the effective hamiltonian P(H0 + δH0(z) + H1 +δH1(z))P the projected baryonic equation (2.3a) can be decomposed into a set of equa-tions for Faddeev amplitudes in the standard way. We introduce the notationGP0 (z) =Pz −P[H0 + δH0(z)]P(2.5)for the effective resolvent andP[H1 + δH1(z)]P =Xivi(z) +XiWi(z)(2.6)4

for the effective baryonic interaction with vi(z) denoting the two-baryon interaction betweenthe pair (jk), (ijk) cyclic, and with Wi(z) denoting the three-baryon interaction, particlei being the ∆-isobar in the force before interaction. The effective two-baryon interactionvi(z) has instantaneous contributions arising from H1 in and between both baryonic sectorsand energy-dependent contributions arising from δH1(z) in the Hilbert sector H∆with a∆-isobar.

The effective three-baryon interaction Wi(z) of the discussed force model onlyhas energy-dependent contributions arising from δH1(z) and consequently it is nonzero onlyin the baryonic sector H∆. Process (e) of Fig.

4 depends on the coordinates of all threebaryons, all three of them interact, it therefore yields a true three-baryon force Wi(z).However, it is a singular one through the δ-function for the ∆-isobar momentum; in fact,it is unlinked, though it is derived from Eq. (2.4b) and labelled there otherwise.

Thus, ithas the mathematical structure of a two-body interaction vi(z) in a three-particle Hilbertspace. The Appendix A derives the mathematical structure of that particular contributionand shows how it is to be combined with the two-baryon interactions in the calculationaltreatment.Using the two-baryon interaction vi(z) for defining the transition matrixTi(z) = vi(z)h1 + GP0 (z)iTi(z)(2.7)in the three-baryon space and introducing baryonic Faddeev amplitudes P|ψi⟩, i.e.,P|ψi⟩= GP0 (EB) [vi(EB) + Wi(EB)] P|ΨB⟩,(2.8)the effective Schr¨odinger equation (2.3a) gets equivalent to the set of Faddeev equationsP|ψi⟩= GP0 (EB){Ti(EB)[Pijk + Pikj]+[1 + Ti(EB)GP0 (EB)]Wi(EB)×[1 + Pijk + Pikj]}P|ψi⟩,(2.9a)P|ΨB⟩= N[1 + Pijk + Pikj]P|ψi⟩.

(2.9b)The particular form (2.9a) of Faddeev equations is introduced in Ref. [17] for the case inwhich explicit and irreducible three-body forces are present.

The Faddeev equations (2.9a)greatly simplify, once three-body forces are absent. In Eqs.

(2.9) Pijk and Pikj are cyclic andanticyclic permutation operators of three particles (ijk). In Eq.

(2.9b) N is a normalizationconstant which only the normalization condition (2.3c) for the full wave function determines.III. CALCULATIONAL APPARATUSCalculations of the trinucleon bound state are carried out for the force model of Fig.

2and for variants of it. It has been tested in the two-nucleon system above pion thresholdby Refs.

[4] and [5]. The force model is used there and for this paper in the approximationQH1Q = 0, which neglets all interactions in the pionic sector Hπ of the Hilbert space.Since QH1Q = 0, all energy-dependent three-baryon contributions to the effective bary-onic interaction (2.6) disappear and the Faddeev equations (2.9a) simplify to those withtwo-baryon forces only, i.e.,5

P|ψi⟩= GP0 (EB)Ti(EB)[Pijk + Pikj]P|ψi⟩. (3.1)The only energy-dependent contributions to the effective baryonic hamiltonian (2.3a) whichsurvive are the processes (a) and (b) of Fig.

4. Both are determined by the pion-nucleon-∆vertex QH1P∆which is calibrated through pion-nucleon scattering in the P33 partialwave.

The single-baryon nature of the vertex is made explicit by the notation QH1P∆=Pi Qh1(i)P∆, i being the label of the baryon which is transformed from a ∆-isobar to apion-nucleon state. Process (a) yields the ∆-isobar self-energy correction P∆δH0(z)P∆inthe three-baryon resolvent, i.e.,P∆δH0(z)P∆=XiP∆h1(i)QQ[z −Qh0(j)Q −Qh0(k)Q] −Qh0(i)Q −Qh0(π)QQh1(i)P∆,(3.2a)process (b) the retarded one-pion exchange P∆δH1(z)P∆in the effective two-baryon inter-action, i.e.,P∆δH1(z)P∆=Xj̸=kP∆h1(k)QQ[z −Qh0(i)Q] −Qh0(j)Q −Qh0(k)Q −Qh0(π)QQh1(j)P∆.

(3.2b)In the single-baryon part P∆δH0(z)P∆the operator [z −Qh0(j)Q −Qh0(k)Q] of the three-baryon resolvent reflects the fact that at the available energy z the noninteracting nucleonsj and k are present and propagate besides the ∆-isobar i; that operator becomes a c-numberparameter in a three-baryon momentum-space basis. In the two-baryon part P∆δH1(z)P∆the operator [z −Qh0(i)Q] of the resolvent reflects the fact that at the available energy zthe noninteracting nucleon i is present and propagates beside the interacting nucleon-∆pair(jk); that operator becomes a c-number parameter in a three-baryon momentum-space basis.Both retarded contributions P∆δH0(z)P∆and P∆δH1(z)P∆are defined in Eqs.

(3.2) for thethree-baryon system.However, they notice the presence and absence of noninteractingparticles through their energy dependence, they are therefore different, e.g., in one-baryonand two-baryon systems. The structure of both parts P∆δH0(z)P∆and P∆δH1(z)P∆andtheir use in the trinucleon bound-state calculation are now discussed.A.

Three-Baryon Basis StatesThe three-baryon basis states |p1q1ν1⟩, required for the calculation in the Hilbert sectorsHN and H∆are diagramatically defined in Fig. 5, (p1q1) are the magnitudes of the Jacobimomenta and ν abbreviates all discrete quantum numbers.

The calculation is done in thetrinucleon c.m. system, thus, the total momentum k is zero.

In the purely nucleonic Hilbertsector HN the definition of the Jacobi momenta is standard. In the Hilbert sector H∆theJacobi momenta are nonrelativistically defined with the mass mN for nucleons and m0∆forthe ∆-isobar.

The momenta (p1q1) are denoted by (p∆q∆) when the ∆-isobar is the spectatorparticle 1, i.e.,6

p∆= mNkN2 −mNkN32mN,(3.3a)q∆= m0∆(kN2 + kN3) −2mNk∆m0∆+ 2mN,(3.3b)k = k∆+ kN2 + kN3 = 0 ,(3.3c)H0|p∆q∆ν∆⟩1 ="2mN + p2∆mN+q2∆4mN+ m0∆+ q2∆2m0∆#|p∆q∆ν∆⟩1(3.4)with k∆, kN2 and kN3 being single-particle momenta of the baryons. The momenta (p1q1)are denoted by (pNqN) when a nucleon is the spectator particle 1, i.e.,pN = m0∆kN2 −mNk∆mN + m0∆,(3.5a)qN = mN(kN2 + k∆) −(mN + m0∆)kN12mN + m0∆,(3.5b)k = kN1 + kN2 + k∆= 0 ,(3.5c)H0|pNqNνN⟩1 ="mN + m0∆+ p2N2 1mN+1m0∆!+q2N2(mN + m0∆)+mN + q2N2mN#|pNqNνN⟩1(3.6)The basis states are antisymmetrized with respect to particles 2 and 3.

In the latter case|pNqNνN⟩1 the ∆-isobar is taken to be particle 3 before antisymmetrization.The basisstates |p∆q∆ν∆⟩1 and |pNqNνN⟩1 are orthogonal to each other; they are different states inthe complete set of states describing two nucleons and one ∆-isobar.B. The ∆-Isobar Self-Energy Correction P∆δH0(z)P∆and the Effective Three-BaryonResolventThe effective three-baryon resolvent (2.5) is illustrated in Fig.

6.It is trivial in theHilbert sector HN, but receives the pionic correction P∆δH0(z)P∆in the sector H∆withone ∆-isobar. That pionic correction is also seen in P33 pion-nucleon scattering as illustratedin Fig.

7, and it is calibrated there.The pion-nucleon transition matrix in the P33 partial wave ist(z∆, k∆) = Qh1(i)P∆×P∆z∆−m0∆−k2∆2m0∆−P∆h1(i)QQz∆−Qh0(i)Q −Qh0(π)QQh1(i)P∆P∆h1(i)Q ,7

(3.7a)t(z∆, k∆) = |f⟩1z∆−M∆(z∆, k∆) −k2∆2m0∆+ i2Γ∆(z∆, k∆)⟨f|(3.7b)with Qh1(i)P∆= |f⟩, P∆h1(i)Q = ⟨f|. Pion-nucleon relative and c.m.

momenta, i.e., π andk∆, are introduced byπ = ωπ(kπ)kN −mNkπmN + ωπ(kπ),(3.8a)k∆= kN + kπ . (3.8b)Since the pion is treated relativistically and the nucleon nonrelativistically, the reduction ofoperators from a many-baryon to a single-baryon form and, conversely, the embedding of asingle-baryon operator in many-baryon systems can often be done only approximately.

Weuse the approximationQh0(i)Q + Qh0(π)Q= mN + k2N2mN+qm2π + k2π≈mN +π22mN+qm2π + π2 +k2∆2(mN +qm2π + π2)= QhπN0 rel(i)Q +k2∆2(mN +qm2π + π2),(3.8c)which avoids angles between the three-momenta π and k∆in the kinetic energy operator; itis employed in the step from Eq. (3.7a) to Eq.

(3.7b), and it is believed to be quite accurate.The kinetic energy operator of relative pion-nucleon motion QhπN0 rel(i)Q is introduced. Thus,the effective mass M∆(z∆, k∆) and the effective width Γ∆(z∆, k∆) of the ∆-isobar with mo-mentum k∆yield that pionic correction P∆δH0(z)P∆, once the propagation of two additionalnucleons in P∆δH0(z)P∆is taken into account according to Eq.

(3.2a). In a three-baryonsystem with single ∆-isobar excitationP∆[H0 + δH0(z)]P∆|p∆q∆ν∆⟩1 ="2mN + p2∆mN+ q2∆4mN+ M∆z −2mN −p2∆mN −q2∆4mN , q∆+ q2∆2m0∆−i2Γz −2mN −p2∆mN −q2∆4mN , q∆#|p∆q∆ν∆⟩1 .

(3.9a)This paper carries out a trinucleon bound-state calculation only, thus, the required three-baryon available energy z is always smaller than 3mN. As a consequence, the single-baryonavailable energy in the effective mass and width of the ∆-isobar according to Eq.

(3.9a) iswith z −2mN −p2∆/mN −q2∆/4mN < mN < mN +mπ. The width Γ∆(z∆, k∆) therefore doesnot contribute to the effective three-baryon resolvent (2.5), which in the Hilbert sector H∆takes the form8

1⟨p′∆q′∆ν′∆|GP0 (z)|p∆q∆ν∆⟩1 = δ(p′∆−p∆)p2∆δ(q′∆−q∆)q2∆δν′∆ν∆×z −2mN −p2∆mN −q2∆212mN +1m0∆−M∆z −2mN −p2∆mN −q2∆4mN , q∆−1(3.9b)for the considered available energies z and for the basis states |p∆q∆ν∆⟩1 of Eq. (3.4).The trinucleon bound-state calculation requires the operators in the Hilbert sector H∆also in the basis |pNqNνN⟩1 of Eq.

(3.6). Compared with Eq.

(3.9b) the three-baryon resol-vent is more complicated, since nondiagonal, in this basis. In order to simplify calculations,we opted for an approximation which also makes 1⟨p′Nq′Nν′N|GP0 (z)|pNqNν⟩1 diagonal in themomenta and channels of the basis states |pNqNνN⟩1.

The approximation is best seen in thequantity1z −QH0QQH1P∆|pNqNνN⟩1=Q[z −Qh0(1)Q −Qh0(2)Q] −Qh0(3)Q −Qh0(π)QQh1(3)P∆|pNqNνN⟩1=1z −2mN −k2N12mN −k2N22mN −k2∆2m0∆−Qh0(3)Q −Qh0(π)Q −k2∆2m0∆Qh1(3)P∆|pNqNνN⟩1≈1z −2mN −k2N12mN −k2N22mN −k2∆2m0∆−QhπN0 rel(3)QQh1(3)P∆|pNqNνN⟩1 . (3.10)The ∆-isobar carries the baryon label 3.

The baryon kinetic energy operator k2N1/2mN +k2N2/2mN+k2∆/2m0∆without rest masses is rewritten in terms of the Jacobi momenta (pNqN)of Eq. (3.5), whereas [Qh0(3)Q −Qh0(π)Q −k2∆/2m0∆] is approximated in the last step ofEq.

(3.10) by the pion-nucleon relative kinetic energy QhπN0 rel(3)Q including rest masses.According to Eq. (3.8c) Qh0(3)Q+Qh0(π)Q has a c.m.

contribution k2∆/2mN +qm2π + π2with k2∆not being a function of the magnitudes pN and qN of the Jacobi momenta only, butalso of the angle between them; the c.m. dependence couples partial waves νN in a nontrivialway.

Once that c.m. contribution is accounted for by k2∆/2m0∆with sufficient accuracy, thebasis states |pNqNνN⟩1 become true eigenstates of the single-baryon part H0 +δH0(z) in theeffective hamiltonian, i.e.,P∆[H0 + δH0(z)] P∆|pNqNνN⟩1 ="2mN + p2N2 1mN+1m0∆!+ q2N2 1mN + m0∆+1mN!+ M∆ z −2mN −p2N2 1mN+1m0∆!−q2N2 1mN + m0∆+1mN!, 0!−i2Γ∆ z −2mN −p2N2 1mN+1m0∆!−q2N2 1mN + m0∆+1mN!, 0!#|pNqNνN⟩1 .

(3.11a)The approximation (3.10) employs the effective mass M∆(z∆, k∆) and the effective widthΓ∆(z∆, k∆) of the ∆-isobar at the momentum k∆= 0, though the pion-nucleon system is9

not at rest in a three-baryon system; the reason is that the approximation (3.10) works withthe pion-nucleon relative kinetic energy operator QhπN0 relQ and pushes the dependence onthe moving pion-nucleon c.m. into an appropriate available energy z∆.

The approximation(3.10) also assumes baryon 3 to be the ∆-isobar; however, the result (3.11a) is symmetricin baryons 2 and 3 and therefore applies to the basis state |pNqNνN⟩1 antisymmetrized withrespect to baryons 2 and 3. Using the result (3.11a) the effective three-baryon resolvent (2.5)becomes diagonal also in the basis states |pNqNνN⟩1 of the Hilbert sector H∆and takes theform1⟨p′Nq′Nν′N|GP0 (z)|pNqNνN⟩1 = δ(p′N −pN)p2Nδ(q′N −qN)q2Nδν′NνN×z −2mN −p2N21mN +1m0∆−q2N21mN+m0∆+1mN−M∆z −2mN −p2N21mN +1m0∆−q2N21mN+m0∆+1mN, 0−1(3.11b)for the available energy z considered in the trinucleon bound state.Due to its energy dependence, the ∆-isobar self-energy correction P∆δH0(z)P∆is differ-ent in a two-baryon system: It notices the absence of the third baryon.

When described bythe basis states |pNνN⟩for particles 2 and 3 in the basis of Eq. (3.6), it becomesP∆hH0 + δH[2]0 (zN∆)i|pNνN⟩="mN + p2N2 1mN+1m0∆!+ M∆ zN∆−mN −p2N2 1mN+1m0∆!, 0!−i2Γ∆ zN∆−mN −p2N2 1mN+1m0∆!, 0!#|pNνN⟩(3.12)The approximation of Eq.

(3.10) on the pion-nucleon c.m. kinetic energy is used accordingly.The superscript [2] indicates that the self-energy correction P∆δH[2]0 (zN∆)P∆refers to a two-baryon c.m.

system; it will be needed for the actual realization of the employed force modelin Subsect. D.The validity of the approximation (3.10), important for the three-baryon resolvent (3.11b)and for the self-energy correction (3.12) in the two-baryon system, is proven as follows: Theapproximating step in Eq.

(3.10) is not carried out, [Qh0(3)Q + Qh0(π)Q −k2∆/(2m0∆)] isused full, the dependence of k∆on the angle between pN and qN is kept; however, that angleis assumed to be fixed at values 0 or π/2 or π, respectively, thus, k∆again remains effectivelydependent on the magnitudes of the Jacobi momenta pN and qN only and does not yieldany channel coupling. Using approximation (3.10) and the three different approximationsindicated in this paragraph in trinucleon calculations, the trinucleon binding energy varies byless than 1 keV, i.e., within the numerical accuracy; the approximation (3.10) is henceforthconsidered quite satisfactory.10

C. Retarded One-Pion Exchange P∆δH1(z)P∆The retarded one-pion exchange P∆δH1(z)P∆of the effective baryonic hamiltonian inEq. (2.3a) is illustrated by process (b) of Fig.

4. It is defined in Eq.

(3.2b) for the three-baryon system. Refs [4,5] test the employed force model in the two-nucleon system abovepion threshold.

Thus, the retarded one-pion exchange P∆δH1(z)P∆is also needed in thetwo-baryon c.m. system.

It is different there, since it notices the absence of the third non-interacting nucleon. It is notationally differentiated as P∆δH[2]1 (zN∆)P∆by the superscript[2].

It takes the formP∆δH[2]1 (zN∆)P∆= vN∆→∆N(zN∆) ,(3.13a)vN∆→∆N(zN∆) =Xj,k=2,3j̸=kP∆h1(k)Q×QzN∆−Qh0(2)Q + Qh0(3)Q + Qh0(π)Q −(kN2+kN3+kπ)22mN +√m2π+k2πQh1(j)P∆. (3.13b)It can be calculated in the basis |pNνN⟩used in Eq.

(3.12) for the corresponding ∆-isobarself-energy correction P∆δH[2]0 (zN∆)P∆in the two-baryon system; the explicit form of itsmatrix elements is given in Ref. [14].

When embedding the retarded one-pion exchange intoa three-baryon system, the approximation mN +qmπ + k2π ≈m0∆is used as in Eq. (3.10)for the total mass of the interacting pion-nucleon system.

Its relation to the same processin the two-nucleon system can then be given, i.e.,1⟨p′Nq′Nν′N|P∆δH1(z)P∆|pNqNνN⟩1 = δ(q′N −qN)q2N× ⟨p′Nν′N|vN∆→∆Nz −mN −q2N21mN +1mN+m∆|pNνN⟩. (3.14)The three-baryon basis |pNqNνN⟩1 of Eq.

(3.5) is the appropriate one for a nucleon-∆in-teraction in a three-baryon system. We note that the energy dependence in the retardedone-pion exchange P∆δH1(z)P∆has a precise meaning and changes the retarded interactionin a controlled way depending on the many-baryon system into which it is embedded.D.

Parametrization of the Interaction Hamiltonian (2.1)The interaction hamiltonian is the force model of Fig. 2 with active pion and ∆-isobardegrees of freedom.

Those degrees of freedom only become active in isospin-triplet partialwaves; in isospin-singlet partial waves the interaction is purely nucleonic and representedsolely by process (a) of Fig. 2.

In this paper it is assumed that the interaction hamiltonianH1 vanishes in the pionic sector, i.e., QH1Q = 0. In general, that is a physically severeassumption, employed already in Ref.

[4]: E.g., in the presence of a pion two nucleonscannot be bound; thus, all pion-deuteron processes are not described internally consistentunder such an assumption.11

Two distinct parametrizations, labelled by H(1) and S(1) in the tables and the resultsection, are chosen for the baryonic interaction, which differ by their forms of the pion-and rho-exchange transition potential P∆H1PN: The parametrization with the transitionpotential of Ref. [10] used there for the force model is labelled H here, since it is based onrather hard form factors, whereas that with the transition potential of Refs.

[4,5] based onrather soft form factors is labelled by S.The instantaneous nucleon-∆potential P∆H1P∆is that of Refs. [4,18]; its exchange partillustrated in Fig.

2(c) is based on pion and rho exchange; only half of the full pion exchangeis kept in P∆H1P∆, since the other half, denoted by 12πR in Table I, is generated explicitly bythe force model as P∆δH1(z)P∆in a retarded fashion according to Eq. (3.14); the subscriptR in the notation 12πR indicates its retardation.

The unretarded half of the pion exchange,kept in P∆H1P∆, is identified with the on-shell form vN∆→∆N(zN∆on) of Eq. (3.13a) as inRefs.

[14,4]; it is denoted by 12πS in Table I, the subscript S in the notation 12πS indicatesthat it is unretarded, but based on the soft form factors of the retarded pion-exchangeP∆δH1(z)P∆.In contrast to Refs. [4,5] the nucleonic part of the interaction is chosen asPNH1PN = VNN −PNH1P∆P∆2mN −P∆hH0 + δH[2]0 (2mN) + H1 + δH[2]1 (2mN)iP∆P∆H1PN(3.15)The choice (3.15) yields exact phase equivalence at zero kinetic energy and approximatephase equivalence at low kinetic energies between the full force model and a realistic, butpurely nucleonic reference potential VNN.

The Paris Potential [19] is chosen as referencepotential VNN.That reference potential is employed in all isospin-singlet partial waves.The choice (3.15) is a conceptual improvement compared to Ref. [4].

The improved phaseequivalence is documented in Ref. [20] which also demonstrates that that improvement isquantitatively irrelevant for observables of the two-nucleon system above pion threshold.E.

Solution of Trinucleon EquationsThe Faddeev equations (3.1) are solved in momentum space using the technical apparatusof Refs. [10] and [13].

The two-baryon interaction is assumed to act in all partial waves up tototal pair angular momentum I = 2. 18 purely nucleonic Faddeev amplitudes PN|ψi⟩arisein the partial-wave decomposition defined by the Jacobi coordinates and discrete quantumnumbers of Fig.

5; in addition, 14 Faddeev amplitudes P∆|ψi⟩with a single ∆-isobar in thepair and one with the ∆-isobar as spectator are taken into account as in Refs. [10,13].

Theemployed discretization of the equations (3.1) is the one of Ref. [13].

The triton bindingenergy ET, and the baryonic components P|ΨB⟩of the wave function according to Eq. (2.9b),are obtained from such a calculation for the force model defined in Subsect.

D. The pioniccomponent Q|ΨB⟩of the wave function can in principle be gotten from the baryonic onesP|ΨB⟩by Eq. (2.3b); however, this paper only computes its weight ⟨ΨB|Q|ΨB⟩in thetrinucleon wave function according to12

⟨ΨB|Q|ΨB⟩= ⟨ΨB|P∆H1QQ(EB −QH0Q)2QH1P∆|ΨB⟩= ⟨ΨB|P∆ −∂∂z! [δH0(z) + δH1(z)] P∆|ΨB⟩z=EB(3.16a)⟨ΨB|Q|ΨB⟩= 3XνZp2∆dp∆q2∆dq∆⟨ΨB|p∆q∆ν∆⟩1× −∂∂z∆!M∆(z∆, q∆) 1⟨p∆q∆ν∆|ΨB⟩z∆=ET +mN−p2∆mN −q2∆4mN+3Xν′νZp′2Ndp′Np2NdpNq2NdqN⟨ΨB|p′Nq′Nν′N⟩1× −∂∂zN∆!⟨p′Nν′N|vN∆→∆N(zN∆)|pNνN⟩1⟨pNqNνN|ΨB⟩zN∆=ET +2mN−q2N21mN +1mN +m0∆ .

(3.16b)F. Comparison with Coupled-Channel CalculationsThe force model of this paper, defined in Fig.

2 and employed in the calculation of thetrinucleon bound state, is used in two parametrizations as Subsect. D describes.

In the forcemodel, the bare ∆-isobar is dynamically coupled to pion-nucleon states and builds up, bythat coupling, the physical P33 resonance in the nuclear medium. We say the force modelis based on a dynamic ∆-isobar.

The calculation of that full force model is compared withcoupled-channel calculations of the trinucleon bound state in which the ∆-isobar does notcouple to pion-nucleon states, keeps a fixed mass m0∆without any pionic correction, i.e.,P∆δH0(z)P∆= 0, and interacts with a nucleon only through unretarded potentials, i.e.,P∆δH1(z)P∆= 0. In those coupled-channel calculations QH1P∆= P∆H1Q = 0, thus, thetrinucleon bound state does not have any pionic components either, i.e., Q|ΨB⟩= 0.

We saythose coupled-channel variants of the full force model are based on a stable ∆-isobar. Thetechnical apparatus which this paper borrows from Refs.

[10,13] was originally designed forthose coupled-channel calculations.The aim of this paper is to find the validity of coupled-channel approximations to thefull force model. This is the reason why various choices for the stable ∆-isobar mass m0∆and for the instantaneous pion exchange between the ∆-isobar and a nucleon are tried out.The various choices are listed in Table I and are described – together with the trinucleonresults – in Sect.

IV.IV. RESULTSCalculations of the form which Refs.

[4,5] and this paper report on have three physicsobjectives in mind:13

1. The full force model of Fig.

2 should be tuned in the two-nucleon system above pionthreshold. In particular, that tuning process should fix the strength and shape of thetwo-baryon transition potential P∆H1PN from two-nucleon to nucleon-∆states and ofthe nucleon-∆potential P∆H1P∆, potentials on which one lacks detailed informationotherwise.2.

The calculation of the trinucleon bound state should determine the amount of three-nucleon force arising from the explicit excitation of a ∆-isobar and from the explicitproduction of a pion in the trinucleon bound state.3. The conditions under which the simpler coupled-channel calculations for the trinucleonbound state approximate the results of the full force model with a dynamic ∆-resonanceare to be found.This paper follows that ambitious program, though it is unable to carry it through tofull satisfaction.

It uses two versions of the force model of Fig. 2 with the approximationQH1Q = 0.

That approximation is an inconsequential one for the trinucleon bound state, asthe results will prove, but a fatal one, if serious tuning of the force model to all observablesin the two-nucleon system above pion threshold were attempted.The version S based on soft form factors in the transition potential is tested for manyobservables of the two-nucleon system above pion threshold and its successes and failures arewell documented in Refs. [4,5]; version S is a moderatly realistic force model.

In contrast,the version H based on hard form factors in the transition potential has not been tested yet;its realistic nature above pion threshold is doubtful; version H is used in this paper, sincethe original coupled-channel calculations [9,10] for the trinucleon bound state were basedon its transition potential. Thus, item (1) of the program list is not carried out with anysatisfaction.

The two versions are labelled H(1) and S(1) in Table I which summarizes theirdefining properties and in Table II which collects their predictions for the triton bindingenergy ET, for the effective two-nucleon and three-nucleon contributions of the ∆-isobarand the pion to the binding, i.e., ∆E2 and ∆E3 according to the technique of Ref. [10], andfor the nucleonic, ∆-isobar and pionic probabilites in the triton bound state, i.e., PL, P∆and Pπ.

The effect of the explicit ∆-isobar and pion degrees of freedom on the trinucleonbound state properties are for the chosen versions of the full force model well isolated. Thus,item (2) of the program list is carried out.The results derived from the full force models H(1) and S(1) are compared with those ofcorresponding standard coupled-channel calculations without explicit pion degrees of free-dom.

In those coupled-channel calculations the effective ∆-mass M∆(z∆, k∆) is taken to beconstant and is equated to a stable mass m0∆, its standard value being 1232 MeV, the reso-nance position in P33 pion-nucleon scattering. Furthermore, the pionic nucleon-∆exchangepotential becomes instantaneous.

In the standard coupled-channel calculation of Ref. [18]that pionic contribution is used in a local form with hard form factors, denoted by πH inTable I.

The two standard coupled-channel force models are labelled H(4) and S(4), H(4)is identical with the force model A3 of Ref. [18].

Table I summarizes the defining propertiesof the full force model in the parametrizations H(1) and S(1) and three coupled-channelvariants of the full force model, i.e., H(2), H(3), H(4) and S(2), S(3), S(4); the variants(2) and (3) interpolate between the full force models (1) and the standard coupled-channel14

models (4): Variant (2) works with a mass of 1290 MeV for the stable ∆-isobar which islarger than the resonance value 1232 MeV; the effective ∆-mass M∆(z∆, k∆) is documentedin Ref. [2], it is not shown again in this paper; in bound-state problems the available energyz∆of the effective ∆-mass is smaller than mN according to Sect.

IIIB and then the effective∆-mass becomes larger than its resonance value 1232 MeV and approaches the bare massof 1315 MeV; thus, the value of 1290 MeV chosen as stable ∆-mass in variant (2) shouldapproximate the effective ∆-mass M∆(z∆, k∆) rather well; variant (2) also works with theinstantaneous limit πS for the pion-exchange nucleon-∆potential; variant (2) should repro-duce the results of the full force models H(1) and S(1) best. Variant (3) works with a stable∆-isobar mass of 1232 MeV, the resonance value, but preserves πS for the pion-exchangenucleon-∆potential.

Table II summarizes the respective trinucleon results. It is also worthnoticing at this point that in models (1), based on a dynamic ∆isobar, from the two contri-butions to the pionic probability in Eq.

(3.16), the self-energy contribution δH0 dominatesover the retardation term δH1, which is found to be one order of magnitude lower.The main result of this paper – in answer to item (3) of the program list – is: Thetrinucleon properties derived from the full force model, defined in Fig. 2 and parametrized inSect.

III.D as H(1) and S(1), are well approximated by those of the corresponding standardcoupled-channel models H(4) and S(4) with a stable ∆-isobar.The standard coupled-channel models account for all corrections of trinucleon properties due to the explicit ∆-isobar and pion degrees of freedom within 90%. The quality of the approximation can be readof from Table II where also the results of the Paris potential, the purely nucleonic referencepotential for all considered force models, are listed.

The quality of the approximation evenimproves when the parametrization of the coupled-channel model is better tuned to the fullforce model as for example in variant (2). On the other hand the fact that the standardcoupled-channel models H(4) and S(4) approximate some trinucleon properties of theircorresponding full force models even more successfully than their seemingly better tunedvariants (3) appears to be accidental.As expected, the force model H(1) based on hard form factors in the transition potentialP∆H1PN to nucleon-∆states yields larger probabilities P∆and Pπ for the ∆-isobar andthe pion in the trinucleon bound state than the force model S(1) does with its softer formfactors.

In both cases, however, the probability Pπ of the pionic components in the wavefunction is extremely small; thus, the simplifying approximation QH1Q = 0 which neglectsall interactions in the pionic sector Hπ of the Hilbert space is well justified.V. CONCLUSIONSFor the first time, this paper carries out the conceptual idea underlying the previouscoupled-channel calculations of Refs.

[9], [10] and [18]:A contribution to the three-nucleon force arises from the mechanism for pion productionand pion absorption; that mechanism is seen in the two-nucleon system above threshold.This paper makes the step from two-nucleon reactions without and with a pion to trinucleonproperties and isolates effects related to the explicit ∆-isobar and pion degrees of freedom.Furthermore, this paper justifies the general use of coupled-channel calculations withstable ∆-isobars and indicates ways for improving their simulations of the full force model.15

ACKNOWLEDGMENTSM.T.P. thanks the Theory Group at CEBAF for the kind hospitality granted during herstay there.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) undercontract No.Sa 247/7-2 and Sa 247/7-3, by the Deutscher Akademischer Austauschdi-enst (DAAD) under Contract No. 322-inida-dr, by the DOE under Grant No.

DE-FG05-88ER40435, and by JNICT under Contract No. PBIC/C/CEN/1094/92.

The calculationsfor this paper were performed at Regionales Rechenzentrum f¨ur Niedersachsen (Hannover),at Continuous Electron Beam Accelerator Facility (Newport News), and at National EnergyResearch Supercomputer Center (Livermore).16

REFERENCES∗Present address. [1] W. Gl¨ockle and P. U. Sauer, Europhysics News 15, 5 (1984).

[2] H. P¨opping, P. U. Sauer and Zhang Xi-Zhen, Nucl. Phys.

A474, 557 (1987), and Erra-tum, Nucl. Phys.

A550, 563 (1992). [3] T.-S. H. Lee, Phys.

Rev. Lett.

50, 1571 (1983);T.-S. H. Lee and A. Matsuyama, Phys. Rev.

C 32, 516 (1985); Phys. Rev.

C 36, 1459(1987). [4] M. T. Pe˜na, H. Garcilazo, U. Oelfke and P. U. Sauer, Phys.

Rev. C 45, 1487 (1992).

[5] A. Bulla and P. U. Sauer, Few-Body Syst. 12, 141 (1992).

[6] H. Garcilazo and T. Mizutani, πNN Systems, World Scientific, Singapore, 1990. [7] T.-S. H. Lee and K. Ohta, Phys.

Rev. C 25, 3043 (1982).

[8] P. U. Sauer, Prog. Nucl.

Part. Phys.

16, 35 (1986). [9] Ch.

Hajduk and P. U. Sauer, Nucl. Phys A322, 329 (1979).

[10] Ch. Hajduk, P. U. Sauer and W. Strueve, Nucl.

Phys. A405, 581 (1983).

[11] A. M. Green, Rep. Prog. Phys.

39, 1109 (1976). [12] B. H. J. McKellar, in Few Body Dynamics, ed.

A. N. Mitra, I. Slaus, V. S. Bhasin andV. Y. Gupta (North Holland, Amsterdam, 1976) p.

508. [13] A. Stadler and P. U. Sauer, Phys.

Rev. C 46, 64 (1992).

[14] K. Dreissigacker, S. Furui, Ch. Hajduk, P. U. Sauer and R. Machleidt, Nucl.

Phys.A375, 334 (1982). [15] P. U. Sauer, Nucl.

Phys. A543, 291c (1992).

[16] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, P. Pir`es and R. de Tourreil,Phys. Rev.

D 12, 1495 (1975). [17] A. Stadler, W. Gl¨ockle and P. U. Sauer, Phys.

Rev. C 44, 2319 (1991).

[18] M. T. Pe˜na, H. Henning and P. U. Sauer, Phys. Rev.

C 42, 855 (1990). [19] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cˆot´e, P. Pir`es and R. deTourreil, Phys.

Rev. C 21, 861 (1980).

[20] A. Valcarce, F. Fern´andez, H. Garcilazo, M. T. P˜ena and P. U. Sauer, Hannover-Salamanca Preprint, 1992. [21] M. Stingl and A.T. Stelbovics, J. Phys.

G 4, 1371 (1978); J. Phys. G 4, 1389 (1978);Nucl.

Phys. A294, 391 (1978).APPENDIX A: CALCULATION OF THE EFFECTIVE SINGULARTHREE-BARYON FORCE ARISING FROM PROCESSES OF THE TYPEFIG.

4(E)The process of Fig. 4(e) is redrawn in Fig.

8; a characteristic process of higher order inpotentials is also shown there. Characteristic for both processes is that they are unlinkedin a particular way: The two nucleons unconnected with the ∆-isobar can interact upto infinite order of the two-nucleon potential; the nucleon produced by the decay of the∆-isobar interacts with the simultaneously created pion also up to infinite order in thepion-nucleon potential, however, it does not interact with either of the other two nucleons.17

The processes of Fig. 8 depend on the coordinates of all three baryons.

They thereforeyield irreducible contributions to the effective three-baryon force.However, due to thatparticular disconnectedness those contributions are singular in the same way as a two-baryoninteraction is singular in a three-baryon Hilbert space. The disconnectedness problem, whichthe process of Fig.

4(e) yields resembles the one encountered in two-nucleon scatteringwithin the framework of πNN dynamics by Ref. [21].

This paper chooses QH1Q = 0, thus,that singular three-baryon force does not arise in the actual calculation. Nevertheless, itsfunctional form is given in this appendix for conceptual completeness.The interaction QH1Q in the Hilbert sector Hπ with a pion is decomposed as follows,i.e.,QH1Q =XivNNi+XivπNi.

(A1)In the two-nucleon potential vNNithe subscript i denotes the spectating nucleon, in thepion-nucleon potential vπNithe subscript i denotes the nucleon interacting with the pion.All potentials in the interaction QH1Q are instantaneous. Processes up to infinite order inthe potentials contribute; the potentials are resummed into transition matrices, i.e.,tNN1 z −h0(1) −h0(π) −(kN2 + kN3)24mN!= vNN11 +1hz −h0(1) −h0(π) −(kN2+kN3)24mNi−h0(2) + h0(3) −(kN2+kN3)24mN× tNN1 z −h0(1) −h0(π) −(kN2 + kN3)24mN!,(A2a)tπN1(z −h0(2) −h0(3))= vπN1"1 +1[z −h0(2) −h0(3)] −h0(1) −h0(π) tπN1(z −h0(2) −h0(3))#.

(A2b)The two-nucleon transition matrix tNN1is defined in the c.m.system of nucleons 2and 3; in Eq. (A2a) the pion-nucleon kinetic energy will be approximately split in theform h0(1) + h0(π) = hπN0 rel(1) + k2∆/2m0∆as in the context of Eq.

(3.10). In contrast thepion-nucleon transition matrix tπN1is defined for a moving pion-nucleon system as for thetransition matrix in Eq.

(3.7); however different symbols are used for the pion-nucleontransition matrices of Eqs. (A2b) and (3.7), since their dynamic content is different.According to Eq.

(2.9a) only the part W1 of the considered three-baryon force is neededfor determining the Faddeev amplitude P|ψ1⟩; that part is calculated in the chosen basisof the Hilbert sector H∆. However, the matrix elements 1⟨p′Nq′Nν′N|W1(z)|pNqNνN⟩1 andthe nondiagonal matrix elements 1⟨p∆q∆ν∆|W1(z)|pNqNνN⟩1 are identically zero, only thematrix elements for the basis states |p∆q∆ν∆⟩1 are nonzero and have to be computed.

Theoperator W1 of the considered three-baryon force is build up by simpler quantities, i.e., thetwo-nucleon and pion-nucleon transition matrizes tNN1and tπN1 . Those transition matricesact in the Hilbert sector Hπ, they do not act on the three-baryon basis states |p∆q∆ν∆⟩1,but these outside basis states simplify those transition matrices inside W1 by allowing thereplacement of some operators through their corresponding eigenvalues, e.g.,18

tNN1 z −h0(1) −h0(π) −(kN2 + kN3)24mN!. .

. |p∆q∆ν∆⟩1= tNN z −hπN0 rel(1) −q2∆2 12mN+1m0∆!!.

. .

|p∆q∆ν∆⟩1 ,(A3a)tπN1(z −h0(2) −h0(3)) . .

. |p∆q∆ν∆⟩1 = tπN z −2mN −p2∆mN−q2∆4mN, q∆!.

. .

|p∆q∆ν∆⟩1 . (A3b)The dots in both equations indicate that the transition matrices do not act directly onthe basis states |p∆q∆ν∆⟩1.

Both transition matrices remain operators with respect to therelative pion-nucleon motion.The pion-nucleon transition matrix tπN of Eq. (A3b) sums up the nonresonant part ofthe pion-nucleon interaction in the P33 partial wave.

The nonresonant part is weak. Thus,only contributions of first order in the pion-nucleon transition matrix tπN are considered.The three arising contributions are shown in Fig.

9. They have the following analytic form1⟨p′∆q′∆ν′∆|W1|p∆q∆ν∆⟩1 =δ(q′∆−q∆)q2∆⟨f|1z −2mN −p′2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)×(⟨p′∆ν′∆|tNN z −q2∆2 12mN+1m0∆!−hπN0 rel(1)!|p∆ν∆⟩×1z −2mN −p2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)×tπNz −2mN −p2∆mN −q2∆212mN +1m0∆, q∆+Xν′′∆Zp′′2∆dp′′∆⟨p′∆ν′∆|tNNz −q2∆212mN +1m0∆−hπN0 rel(1)|p′′∆ν′′∆⟩×1z −2mN −p′′2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)×tπNz −2mN −p′′2∆mN −q2∆212mN +1m0∆, q∆×1z −2mN −p′′2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)×⟨p′′∆ν′′∆|tNNz −q2∆212mN +1m0∆−hπN0 rel(1)|p∆ν∆⟩+tπNz −2mN −p′2∆mN −q2∆212mN +1m0∆, q∆×1z −2mN −p′2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)19

× ⟨p′∆ν′∆|tNNz −q2∆212mN +1m0∆−hπN0 rel(1)|p∆ν∆⟩×1z −2mN −p2∆mN −q2∆212mN +1m0∆−hπN0 rel(1)|f⟩. (A4)The δ-function δ(q′∆−q∆)/q2∆yields the singular structure of the three-baryon force Wi(z).In calculations with QH1Q ̸= 0 the effective three-baryon force Wi(z) arises and hasthe discussed singular part of Eq.

(A4). That singular part has to be combined with thetwo-baryon interaction vi(z) of same singularity structure.20

FIGURESFIG. 1.

Hilbert space for a many-nucleon system. Besides the purely nucleonic sector HN thereis the sector H∆in which one nucleon is turned into a ∆-isobar and the sector Hπ in which a singlepion is added.FIG.

2. Building blocks of the force model with ∆-isobar and pion degrees of freedom.

Thehermitian adjoint pieces corresponding to the processes (b) and (e) are not shown. The ∆-isobar isa bare particle; process (e) yields the physical P33 pion-nucleon resonance by iteration; process (f)stands for the nonresonant pion-nucleon interactions; in general, it could also have contributions inP33; none of those possible P33 background contributions is indicated in the following Figs.

3 and4. The extended force model acts in isospin-triplet partial waves only.

In isospin-singlet partialwaves the force model is purely nucleonic and reduces to process (a).FIG. 3.

Examples for contributions to the effective three-nucleon force arising in a three-nucleonsystem from the force model of Fig. 2.

The contributions are irreducible in the purely nucleonicHilbert sector HN. Selected contributions up to fourth order in two-particle potentials are shown.With respect to the pion-nucleon interaction, possible P33 background contributions are not con-sidered in this figure.FIG.

4. Energy-dependent contributions to the effective hamiltonian of Eq.

(2.3a) arising fromprojecting out the pionic component from the trinucleon wave function.All energy-dependentcontributions act in the baryonic Hilbert sector H∆with a ∆-isobar. In the top row the onlycontribution of one-baryon nature PδH0(z)P is shown.

Row two (three) gives characteristic ex-amples of two-(three-)baryon nature in PδH1(z)P. With respect to the pion-nucleon interaction,possible P33 background contributions are not considered in this figure. The contributions (c) – (g)disappear, once interactions in the Hilbert sector Hπ are not taken into account, i.e., QH1Q = 0.FIG.

5. Three-body Jacobi coordinates.

The magnitude of the corresponding momenta are p1and q1. In the momentum-space basis states |p1q1ν1⟩1 the antisymmetrized state of pair 2 and 3and the spectator state are coupled with respect to their angular momenta I and j and isospin Tand t1 i.e., |p1q1[(LS)I(ls1)j]J J z(Tt1)T T z⟩1.

The quantum numbers L(l) and S(s1) refer to theorbital angular momentum and spin of the pair (spectator), J (Jz) and T (Tz) are total angularmomentum (projection) and total isospin (projection) of the three-body bound state.FIG. 6.

Effective three-baryon resolvent (2.5). Its form in the Hilbert sector HN is diagramat-ically shown on the left side and its form in the Hilbert sector H∆on the right side.FIG.

7. Characteristic contribution to the pion-nucleon transition matrix in the P33 partialwave.The force model of Fig.

2 does not have any additional background potential QH1Q inthat partial wave. An effective propagation of the ∆-isobar can be read offfrom the transitionmatrix and reoccurs – together with the propagation of two additional nucleons – in the effectivethree-baryon resolvent (2.5) of Fig.

6.21

FIG. 8.

Characteristic contributions to the effective three-nucleon force of the type to be cal-culated in this appendix. The process of Fig.

4(e) is redrawn; a characteristic process of higherorder in the potentials, i.e., of third order in the two-nucleon potential and of second order in thepion-nucleon potential, is also shown.FIG. 9.

The three contributions of first order in the nonresonant pion-nucleon transition ma-trix to the effective three-nucleon force, calculated in this appendix. The shaded boxes denotenucleon-nucleon and pion-nucleon transition matrices, they can be differentiated by their externallegs.22

TABLESTABLE I. Employed Force Models with ∆-Isobar and Pion Degrees of FreedomBare ∆-MassEffective ∆-MassP∆H1PNP∆[H1 + δH1(z)]P∆π-Exchange, Fig.

2(c)m0∆[MeV]H(1)1315M∆(z∆, k∆)H[10]12πR + 12πS[4]H(2)1290m0∆H[10]πS[4]H(3)1232m0∆H[10]πS[4]H(4)1232m0∆H[10]πH[18]S(1)1315M∆(z∆, k∆)S [4]12πR + 12πS[4]S(2)1290m0∆S [4]πS[4]S(3)1232m0∆S [4]πS[4]S(4)1232m0∆S [4]πH[18]TABLE II. Results for some trinucleon bound state properties.

The computed binding energiesare correct within 10 keV only. Thus, the last digit in rows ET , ∆E2 and ∆E3 of this table are notsignificant on an absolute scale.

The last digit is, however, significant for relative changes, and thisis the reason why it is quoted – against our practice in other papers. The nucleonic probabilitiesPL in the wave function are split up according to total orbital angular momentum L and for L = 0also according to the symmetry properties of the orbital wave function components in the standardway.ParisH(1)H(2)H(3)H(4)S(1)S(2)S(3)S(4)ET [MeV]-7.381-7.849-7.866-7.885-7.912-7.627-7.636-7.643-7.667∆E2[MeV]-0.4560.4250.4940.4600.2480.2270.2720.258∆E3[MeV]--0.924-0.910-0.998-0.991-0.494-0.482-0.534-0.544PS[%]90.1388.2388.3588.0688.2089.0589.1388.9589.00PS′[%]1.401.241.231.231.221.311.311.301.30PP [%]0.060.080.080.090.080.080.080.080.08PD[%]8.418.688.698.708.718.618.618.628.63P∆[%]-1.711.641.931.790.920.871.051.00Pπ[%]-0.06---0.04---23

---

출처: arXiv:9305.014 • 원문 보기