First results for the triton binding energy obtained from the relativistic
spectator or Gross equation are reported. The Dirac structure of the nucleons
is taken into account. Numerical results are presented for a family of
realistic OBE models with off-shell scalar couplings. It is shown that these
off-shell couplings improve both the fits to the two-body data and the
predictions for the binding energy.Comment: 5 pages, RevTeX 3.0, 1 figure (uses epsfig.sty

A. Stadler

Alfred Stadler

B. S. Pudliner

F. Gross

F. Sammarruca

Franz Gross

G. C. Wick

G. Rupp

J. A. Tjon

J. Carlson

J. L. Forest

W. Glöckle

English

arXiv

First results for the triton binding energy obtained from the relativistic spectator or Gross equation are reported. The Dirac structure of the nucleons is taken into account. Numerical results are presented for a family of realistic OBE models with off-shell scalar couplings. It is shown that these off-shell couplings improve both the fits to the two-body data and the predictions for the binding energy

Stadler, Alfred

Gross, Franz

UNT (University of North Texas) Digital Library

arXiv:nucl-th/9607012   9 Jul 1996Relativistic calculation of the triton binding energy and its implicationsAlfred Stadler1,2∗ and Franz Gross1,31College of William and Mary, Williamsburg, VA 231852Centro de F́ısica Nuclear da Universidade de Lisboa, 1699 Lisboa Codex, Portugal3Thomas Jefferson National Accelerator Facility, Newport News, VA 23606(June 15, 2000)First results for the triton binding energy obtained from the relativistic spectator or Grossequation are reported. The Dirac structure of the nucleons is taken into account. Numerical resultsare presented for a family of realistic OBE models with off-shell scalar couplings. It is shown thatthese off-shell couplings improve both the fits to the two-body data and the predictions for thebinding energy.21.10.Dr,21.45.+v,21.30.-x,13.75.C,11.10.StPreprint # WM–96–109, TJNAF-TH-96-13, CFNUL-96-51The first realistic nonrelativistic calculations of thetriton binding energy were completed in the 1970’s [1].Later it was shown that different methods gave the sameresults, and that the binding energy could be calculatedto an accuracy of a few keV by considering all nucleon-nucleon (NN) partial waves up to j = 4 [2]. Today,if three-body forces (3BFs) are not considered, a smalldiscrepancy of about 0.5-1.0 MeV remains between theexperimentally observed value of −8.48 MeV and val-ues obtained from realistic nonrelativisticNN potentials.State-of-the art calculations now include sophisticated3BFs, and when their strength is adjusted to give thecorrect triton binding energy, an excellent value is alsoobtained for the 4He binding energy (and to a lesser ex-tent other light nuclei up to A ' 7) [3].However, relativistic effects should make a contribu-tion to the binding energy at the level of several hundredkeV. Using a mean momentum of about 200 MeV (con-sistent with nonrelativistic estimates) we expect to seecorrections of the order of (v/c)2 ' (p/m)2 ' 4%. If thisis 4% of the binding energy, then it amounts to about300 keV. However, if relativity has a greater effect on theattractive σ exchange part of the force (as it does in nu-clear matter calculations using the Walecka model) thenwe might obtain an effect 10 times larger.The importance of this problem has been recognized,and relativistic effects have been estimated using a sepa-rable kernel in the Bethe-Salpeter equation [4], assumingminimal relativity in the Blankenbecler-Sugar equation[5], and by adding corrections of first order in (v/c)2 tothe Schrödinger equation [6]. All of these calculationsinclude some contributions coming from relativistic kine-matics, but none treats the full Dirac structure of thenucleons, or investigates effects which might arise from arealistic relativistic treatment of the NN dynamics.The purpose of this letter is to present the first numer-ical calculations of the triton binding energy obtainedfrom the manifestly covariant three-body spectator (orGross) equations for three identical spin 1/2 particles,and to discuss the implications of these calculations.Some preliminary results were reported in conference pro-ceedings [7].The three-body spectator equations were first intro-duced and applied to scalar particles in 1982 [8], andthen extended to the case of three spin 1/2 particles inlectures given at the University of Hannover soon after-ward. Recently, a more tractable form for the equationshas been developed, and a full derivation of the equationswill be published elsewhere [9]. In this letter we describeonly a few of their features briefly.In the absence of 3BFs the three-body scattering am-plitude is obtained from a sum of all successive two-bodyscatterings. Because the three particles are identical,each two-body scattering differs from the others only bya permutation, and they can therefore all be summed byone operator equation of the form|Γ1〉 = 2M1G1P12|Γ1〉 , (1)where |Γ1〉 is a vertex function describing the contribu-tion to the bound state from all processes in which the 23pair was the last to interact (with particle 1 a spectator),the two-body amplitude M1 describes the scattering ofthe 23 pair, G1 is the propagator for the 23 pair, and P12is a permutation operator interchanging particles 1 and2. (The factor of 2 comes from the contribution of P13which equals the one of P12.)The three-body spectator equations have the samestructure as (1), but incorporate the additional featurethat the spectator is restricted to its positive energymass-shell in all intermediate states. With the conven-tions implied above, consistency also requires that par-ticle 2 be on-shell, so that two particles are always on-shell. We think of these constraints as a reorganizationof Eq. (1) which will, in some cases, improve its conver-gence. The constraints are manifestly covariant, and leadto the following equation|Γ12〉 = 2M122G12P12|Γ12〉 , (2)where the lower index labels the second on-shell particle.Hence only particle 3, the (unique) particle which hasjust left one interaction and is about to enter anotherone, is off-shell in Eq. (2).To reduce Eq. (2) to a practical form, we take ma-trix elements of the operators using three-particle helic-ity states similar to those defined by Wick [10]. Both ρ-spin states (where ρ = + is the u spinor positive energystate and ρ = − is the v spinor negative energy state)of the off-shell particle must be treated. The three-bodystates will be written in the abbreviated form |J1(23)ρ〉,where J is the total angular momentum of the state, ρthe ρ-spin of the off shell particle, 1 = {q, λ1} (where qand λ1 are the magnitude of the three momentum andthe helicity of the spectator in the three-body c.m.), and(23) = {p̃, j, mj, λ2, λ3} (where p̃ is the magnitude of therelative three momentum of the 23 system, j and mj arethe angular momentum of the pair and its projection inthe direction of q, and λ2 and λ3 are the helicities of par-ticles 2 and 3, all defined in the rest frame of the 23 pair).We will suppress all isospin indices. Using this notation,the final form of the three-body spectator equation forΓ1 is〈J1(23)ρ|Γ1〉 =∑j′m′∑λ′′2λ′′3ρ′′λ′1λ′2λ′3ρ′∫ qcrit0q′2dq′mEq′∫ π0dχ sinχ×〈j(23)ρ|M1|j(2′′3′′)ρ′′〉 mEp̃′′gρ′′(q, p̃′′)×Pρ′′ρ′12 [1(2′′3′′), 1′(2′3′)]mEp̃′〈J ′1′(2′3′)ρ′|Γ1〉 , (3)where Pρ′′ρ′12 [1(2′′3′′), 1′(2′3′)] is the matrix element of thepermutation operator, given below, and gρ(q, p̃) the prop-agator of the off-shell particle in different ρ-spin states2g+(q, p̃) =12Ep̃ −Wq, g−(q, p̃) = − 1Wq. (4)Because four-momentum is conserved in the relativisticformalism, the mass Wq of the 23 pair depends on q,W 2q = M2t +m2 − 2MtEq , (5)with Eq =√m2 + q2. Note that Eq. (3) includes asum over intermediate helicities and angular momentumquantum numbers, and an integration over the internalspectator momentum q′ and the angle χ between the di-rections of q′ and q. The integration over q′ has beenlimited to the finite interval [0, qcrit], where qcrit is theroot of the equation Wqcrit = 0. At this critical spec-tator momentum (equal to ' 4m/3 ' 1200 MeV), thetwo-body subsystem is recoiling at the speed of light andthe relativistic effects are enormous! Contributions forq′ > qcrit are very small, and come from two-body stateswith spacelike four-momenta. It seems sensible to sim-ply neglect the region q′ ≥ qcrit and set the three-bodyamplitudes to zero there. As it turns out, the solutionsgo smoothly to zero as q → qcrit anyway, so we may im-pose the condition that they are zero beyond this pointwithout making the amplitudes discontinuous in q.Exchanging particles 1 and 2 implies that particle 2becomes the spectator and now its momentum and helic-ity must be expressed in the c.m. frame of the three-bodysystem, while the variables of particles 1 and 3 must beexpressed in the rest frame of the 13 pair. Boosting fromone frame to another introduces Wigner rotations of boththe single particle and two-body helicities. The final re-sult for the permutation operator isPρ′′ρ′12 [1(2′′3′′), 1′(2′3′)] = (−1)m−λ1+λ′3√2j + 1√2j′ + 1×d(J)m−λ1,m′−λ′2(χ)d(j)m,λ′′2−λ′′3(θ̃′′) d(j′)m′,λ′1−λ′3(θ̃′)× d(1/2)λ1λ′1 (β1) d(1/2)λ′′2λ′2(−β2)N ρ′′ρ′λ′′3λ′3(q, q′, χ) , (6)where the functions d(1/2)m1,m2(β) are the Wigner rota-tion matrices, and N ρ′′ρ′λ′′3λ′3(q, q′, χ) describes exactly theWigner rotations of the off-shell particle 3, as well asthe nontrivial matrix elements between the different ρ-spinors u and v of particle 3 as they appear in the restframes of the 23 pair and the 13 pair.We have solved Eq. (3) numerically for a variety ofrealistic NN models. The two-body amplitudes obtainedfor all of these models result from an exact solution of thetwo-body spectator equation, as described in Ref. [11],and are therefore fully consistent with the three-bodyequations.These models will be described in detail elsewhere.Briefly, they are all one-boson exchange (OBE) modelswith a kernel composed of the exchange of 6 commonlyused bosons: the π, η, σ, δ, ω, and ρ. The parameters ofeach model were determined by fitting to the NN phaseshifts below 350 MeV and to deuteron properties.In all cases the following pion coupling was used:gπΛπ = gπ[γ5 − νπ2m[(m− \p′) γ5 + γ5 (m− \p)]]= gπ[(1 − νπ)γ5 +νπ2mγ5 \q], (7)where p and p′ are the four momenta of the incoming andoutgoing nucleons, and the couplings proportional to νπdo not contribute if the nucleons are on-shell. In thisfamily, we fixed g2π/4π = 13.34 and chose νπ = 1, givingthe conventional pseudovector pion coupling with largeoff-shell effects.A particular feature of these models, and a centralpoint of this letter, is that they also include phenomeno-logical scalar σ (with I = 0) and δ (I = 1) exchangeswith off-shell scalar-nucleon-nucleon (sNN) couplings ofthe formgsΛs(p′, p) = gs[1− νs2m(m− \p′ +m− \p)+κs4m2(m− \p′) (m− \p)]. (8)The vertex Λs(p′, p) given in Eq. (8) is the most generalform the sNN vertex can take, but as far as we knowthe off-shell scalar couplings which depend on νs and κshave never been studied previously. The family of modelsdiscussed here has κs = 0 and values of ν varying from0→ 2.6, whereνσ = −0.75 ν νδ = 2.60 ν . (9)We will see that these couplings proportional to ν areextremely important.The results of our calculations are summarized inFig. 1. The lower panel of Fig. 1 shows how χ2 for the fitsto the two-body data varies with ν for this family of OBEmodels. Fits were done for values of ν = 0, 0.5, 1.0, 1.6,1.8, 1.9, 2.0, 2.2, and 2.6, and the dashed curve smoothlyinterpolates these individual cases. We emphasize thateach of these models with different values of ν are realis-tic in the sense that for each case OBE parameters (13 inall) were adjusted to give the best possible fit to the NNdata below 350 MeV. Although the 13 parameters differonly slightly from case to case, the models are not quiteequivalent. The figure shows that there is a significantvariation in the quality of the fit; the best models lie inthe region 1.5 ≤ ν ≤ 2.0 (all with χ2 ≤ 2.45 as shownin the figure). This probably rules out the model withν = 0 (for example). We conclude that the introductionof these ν-dependent couplings significantly improves thefit to the two-body data and that the implicit choice ofν = 0 made in previous work is not optimal.The upper panel of Fig. 1 shows the variation of thethree-body binding energy with ν. The rapid dependenceof the binding energy on ν is rather striking. An increasein ν from 0 to 1.6 changes our prediction from −6.24 to30.0 0.5 1.0 1.5 2.0 2.52.53.02 /datum-9.0-8.0-7.0-6.0ET[MeV]654321experimentalvalueFIG. 1. Triton binding energy Et for specific values of jmax(upper panel) and χ2 for the fits to the two-body data (lowerpanel) versus the scalar meson off-shell parameter ν defined inthe text. The curves in both panels are smooth interpolationsthrough the actual calculations. The lower panel also includesthe line χ2 = 2.45 for reference.−8.76 MeV, and a value in good agreement with experi-ment would be obtained for ν ' 1.5, still in the range ofν’s which give the best fit to the two-body data.The panel and Table 1 (for the cases with smaller ν)also show how the binding energy converges as the num-ber of three-body partial waves, characterized by thehighest included pair angular momentum jmax, increases.Because of the large increase in the predicted values asthe number of channels increases from 28 (jmax = 1) to52 (jmax = 2), we were concerned about the convergenceof the three-body calculations and studied it in detail.We carried the calculations all the way to jmax = 6 with148 channels. We find that the individual contributionsfrom channels with odd j tend to cancel while those fromchannels with even j are all attractive. Thus, the stepsfrom even to odd jmax are small compared to those fromodd to even jmax. From a detailed study of the individualcontributions we estimate that the results are convergedto about 1 keV for ν = 0 and to about 5 keV for ν = 1.6.We conclude that the best of the two-body models ex-amined so far yield a three-body binding energy fromabout −8.5 to −9.5 MeV. In subsequent work we will dis-play the dependence of these results on the boosts, thenegative ρ-spin states, and other relativisitc effects, andwe will study additional two-body models. Here we willdiscuss the origin and implications of the ν dependencewhich we have observed.To understand why the binding energy is so sensitiveto ν, we may look at the half off-shell Born amplitudefor scalar exchange (i.e. the amplitude with one nucleonoff-shell). For the positive ρ-spin sector, we haveTABLE I. Absolute values of the triton binding energies inMeV. The first row is the result when only 1S0 and3S1-3D1positive energy channels are included. The other rows showresults obtained when all channels with two-body angularmomentum j ≤ jmax are included. The total number ofthree-body channels in each case is N.jmax N coupling parameter ν0.0 0.5 1.0 1.61+ 5 6.003 6.345 6.850 7.7691 28 5.963 6.318 6.812 7.6522 52 6.180 6.639 7.299 8.4413 76 6.214 6.695 7.393 8.6154 100 6.232 6.726 7.452 8.7405 124 6.233 6.726 7.452 8.7366 148 6.235 6.731 7.461 8.757Vs=g2s{u(p′)[1− νs2m(m− \p)]u(p)}{u(−p′)u(−p)}m2s − (p′ − p)2' Vs[1− νs2Ep −W2m]= CsVs , (10)where Vs is the usual scalar potential obtained from sucha reduction when νs = 0, p = (W − Ep,p) is the mo-mentum of the off-shell particle, W is the energy ofthe two-body system in its rest frame, and we have ig-nored the lower components of the Dirac spinors in car-rying out the reduction. The effect of the νs depen-dence is to multiply the scalar potential by the factorCs = [1 − νs(2Ep − W )/2m]. In applications to two-body scattering, the ν-dependent term is a small cor-rection with a sign depending on the energy, but in thethree-body bound state it is always positive. Assumingan average nucleon momentum of about 200 MeV givesroughly a 10% variation over the range of ν from 0 to 2.The observed variation of about 4 MeV over this rangeis explained therefore if the average strength of the σ-exchange potential is about 40 MeV, which is the rightorder of magnitude. This shows how the large variationin binding energy which we observe can be explained bya “small” relativistic effect.An OBE model with off-shell couplings has a very richstructure. For example, consider two successive interac-tions of a scalar meson with a single nucleon. The vertexfunction (8) contains the operator m−\p which is just theinverse of the nucleon propagator, so that it can removethe nucleon propagator and contract the two interactionvertices to a single vertex describing the emission of twomesons from a single point. If the two mesons emergingfrom this point are coupled to a second nucleon they gen-erate a triangle or a bubble diagram. These diagrams aretwo-boson exchange terms similar to those (involving pi-ons) which would emerge from a nonlinear sigma model.Alternatively, if these two mesons couple to two differentnucleons, they generate diagrams usually associated withthree-body forces. It is easy to generalize this result: an4OBE model with off-shell couplings is equivalent to an-other OBE model without these couplings, but with anadditional specific family of N -boson exchange diagramsand N -body forces.We conclude with two observations. The discoverythat off-shell scalar couplings play an important role inboth improving the description of two-body data and inpredicting three-body binding energies would only havebeen possible in the context of a relativistic formalismclosely connected to an effective field theory. The factthat these couplings are equivalent to a strong energydependence in the context of nonrelativistic theory isprecisely the reason they could not have been discov-ered there; nonrelativistic potentials are supposed to beenergy independent. In the context of an effective fieldtheory, however, these are a natural and legitimate exten-sion of the simplest assumption about the spin structureof the sNN vertex. The most general sNN vertex wasgiven in Eq. (8) above, and can have only three differentspin couplings. Once the third term depending on κs in(8) is studied, all of the possibilities will have been ex-hausted. In this way an effective field theory is tightlyconstrained, even if some of its interactions are stronglyenergy dependent in a nonrelativistic context.We believe that this way of looking at dynamics mayvery well be the most significant contribution to comefrom relativistic methods. The traditional argumentssuggesting that relativistic effects are very small refer torelativistic kinematics only. As Eq. (8) illustrates, rel-ativistic dynamics provides a new way to study nuclei,even at low energies.One of us (FG) acknowledges the hospitality of PeterSauer and the University of Hannover, where the initialidea to persue this program began, and the theory groupat CFNUL where this letter was written. Early work waspartially supported by a grant from NATO. The majorpart of this work could not have been done without thesupport of the Department of Energy under grant #DE-FG05-88ER40435 which is gratefully acknowledged. Inaddition, one of us (AS) thanks JNICT for support un-der contract # BCC/4394/94. Numerical calculationswere performed at NERSC (Livermore), TJNAF, and atCFNUL.∗ Present address.[1] J. A. Tjon, B. F. Gibson, J. S. O’Connell, Phys. Rev.Lett. 25, 540 (1970).[2] W. Glöckle, H. Wita la, H. Kamada, D. Hüber, J. Go-lak, AIP Conf. Proc. 334, 45 (1995); B.F. Gibson, Nucl.Phys. A543, 1c (1992); Few-Body Systems, Suppl. 7, 80(1994).[3] B. S. Pudliner, V. R. Pandharipande, J. Carlson, and R.B. Wiringa, Phys. Rev. Lett. 74, 4396 (1995).[4] G. Rupp and J. A. Tjon, Rhys. Rev. C 45, 2133 (1992).[5] F. Sammarruca, D. P. Xu, and R. Machleidt, Phys. Rev.C 46, 1636 (1992).[6] J. Carlson, V. R. Pandharipande, and R. Schiavilla,Phys. Rev. C 47, 484 (1993); J. L. Forest, V. R. Pand-haripande, and J. L. Friar, Phys. Rev. C 52, 568 (1995);J. L. Forest, V. R. Pandharipande, J. Carlson, and R.Schiavilla, Phys. Rev. C 52, 576 (1995).[7] A. Stadler and F. Gross, AIP Conf. Proc. #334 (AIPN.Y.), p. 867 (1995); F. Gross, A. Stadler, J. W. VanOrden, and N. Devine, Few-Body Sys. 8, 269 (1995);F. Gross, Electromagnetic Studies of the Deuteron(NIKHEF, Amsterdam), p. 1 (1996).[8] F. Gross, Phys. Rev. C 26, 2226 (1982).[9] A. Stadler, F. Gross, and M. Frank, in preparation.[10] G. C. Wick, Ann. of Phys. (N.Y.) 18, 65 (1962).[11] F. Gross, J. W. Van Orden, and K. Holinde, Phys. Rev.C 45, 2094 (1992).5

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