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EPJ manuscript No. (will be inserted by the editor) A = 3 A model for antinuclei production in proton-nucleus collisions R.P. Duperraya, K.V. Protasovb, L. Deromec, and M. Buénerdd 3 Institut des Sciences Nucléaires, IN2P3-CNRS,UJFG, 53, Avenuedes Martyrs, F-38026 Grenoble Cedex, France 0 0 2 Received: date/ Revised version: date n a Abstract. Asimplecoalescencemodelbasedonthesamediagrammaticapproachofantimatterproduction J in hadronic collisions as used previously for antideuterons is used here for the hadroproduction of mass 3 1 antinuclei. It is shown that the model is able to reproduce the existing experimental data on t and 3He 3 production without any additional parameter. 1 PACS. 24.10.-i Nuclear-reaction models and methods v 3 0 1 Introduction oldeffectsandtheanisotropyoftheangulardistributions. 1 This approach can reproduce most existing data with- 1 The increasinginterestin the study of productionoflight out any additional parameter in energy domains where 0 antinuclei in proton-proton and proton-nucleus collisions theinclusiveantiprotonproductioncrosssectionsarewell 3 0 is motivated by the presence of anti-nuclei in cosmic rays known. / whichhaspotentiallyimportantimplicationsonthematter- h antimatter asymmetry of the universe.From this point of Thisarticlereportsontheapplicationofthisapproach t - view, it is important to determine the amount of anti- totheproductionofA=3antinuclei.Itisthefirstmicro- l c matter which can be produced in the galaxy through the scopic calculation of this cross section to the knowledge u interaction of high-energy protons with the interstellar of the authors. In [7], the 3He production cross section in n gas.Anewgenerationofexperiments(AMS[1],PAMELA proton-protoncollisionswascalculatedusingthestandard : v [2])shouldbe able to measurethe flux ofanti-matterin a coalescence model, with the parameter p0 taken from the i near future. d production data. X Thecalculationsofthetand3Heproductioncrosssec- ar tions reported here are based on the same diagrammatic Unfortunately, the experimental data required to be approach to the coalescence model as used recently [3] to compare to the calculations are limited. Only two sets describe the d production in proton-proton and proton- of experiments have measured the production of mass nucleus collisions. 3 antinuclei in proton-nucleus collisions. t and 3He were The coalescence model [4] is based on the simple hy- discovered at IHEP (Serpukhov), with one experimental pothesis that the nucleons, produced during the collision points measured for t and one for 3He [8,9], while in the of a beam and a target, fuse into light nuclei whenever CERNexperiment(SPS,WA33)[10,11],fourexperimen- the momentum of their relative motion is smaller than a talpointsweremeasuredfortandeightfor3He.Forthese coalescence radius p in the momentum space, which is 0 latter data however, the t and 3He production cross sec- a free parameter of the model, usually fit to the experi- tions were measured with respect to the pion production mentaldata(see [5]for example).A simple diagrammatic cross section at the same momentum. This requires the approach to the coalescence model developed in [6] pro- correspondingexperimentalvalues ofthe pionproduction vided a microscopic basis to the model. In this approach, cross section to be known to extract the values of the t the parameter p is expressed in terms of the slope pa- 0 and 3He production cross sections. rameteroftheinclusivenucleonproductionspectrumand of the wave function of the produced nucleus. The article is organized as follows. The main ideas of This diagrammatic approach has been generalized in the theoretical approach are described in section 2. The [3]toantideuteronproductionbytakingintoaccountthresh- formalism is generalized to the case of A = 3 antinuclei a [email protected] productioninsection3.Section4isdevotedtotheresults b [email protected] andthecomparisontotheexperimentaldata.Abriefsum- c [email protected] maryoftheworkisprovidedbeforetheworkisconcluded d [email protected] in the last section. 2 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 2 Diagrammatic approach to the coalescence since they cancel in further calculations. Using the con- model ventional graph technique [13], the expression for M can be written in the form: The main ideas of the diagrammatic approach of the co- d4p d4p d4p 1 2 3 alescence model for nuclear fragment production are re- M = minded here for the reader’s convenience [6]. The sim- Z (2π)4 Z (2π)4 Z (2π)4 plest Feynman diagram of Fig. 1 corresponding to fusion 2mp 2mp 2mp of three nucleons is considered as a basis for the coales- m2 p2 i0m2 p2 i0m2 p2 i0 p− 1− p− 2− p− 3− cencemodel.Herethesymbolf designatesthestateofall i(2π)4δ4(p +p +p p ) particlesbutnucleons1,2and3whichformthetritiumor 1 2 3 t − the helium3nucleusproducedinthe finalstate (specified M(1,2,3→t)MA (3) by the t symbol on the graph). where M(1,2,3→t) is the vertex of coalescence of 1,2,3 into t (proportional to the three-nucleon wave function in the momentum space in the nonrelativistic approximation), m the nucleon mass, the three fractions being the in- 1 p dividual nucleon propagators of 1, 2 and 3. The inte- grals have to be performed over energies and momenta of 3 t the (virtual) particles.The delta functions ensureenergy- 2 momentum conservation at the t vertex, pt = (pt,Et), A with pt = p1 +p2 +p3 being the momentum of t, Et its energy. The dependence of the amplitude M on its A variables (the particle momenta) is also needed explicitly for the calculations. In lack of a reliable theoretical form, f this can be done in a ”minimal” way, by using empirical shapes. The inclusive nucleon spectra usually have a de- creasing form which can be approximated by a Gaussian function in the center of mass frame: Fig. 1. Thesimplest Feynmandiagram correspondingtocoa- lescence of three nucleonsinto a tritium or 3He. E d3σp exp p2/Q2 , (4) p dp3 ∝ − p p (cid:0) (cid:1) The physicalpicture behindthis diagramisquite sim- where Q defines the slope parameter of the momentum ple: the nucleons produced in a collision (block A) are distribution.Accordingly,the amplitude M canbe writ- A slightly virtual and can fuse without any further inter- ten in the following way: action with the nuclear field. This diagram is not the onlypossiblecontributiontothefulltransitionamplitude. p2+p2+p2 However, mutual cancellations of a number of other con- MA =Cexp − 1 2Q22 3 tributing diagrams result in the diagram of Fig. 1 being (cid:18) (cid:19) dominant [12]. This diagram can be calculated using the =Cexp p2t exp q2 exp 3p2 ,(5) technique developed in [13]. −6Q2 −Q2 −4Q2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The probability for three nucleon coalescence is given by: where d3W = M 2 mt d3pt , (1) pt = p1+p2+p3, t | | Et (2π)3 p= 1 (p p ), 1 2 √3 − wheremtinthemassofthetfragmentandEtitsenergyin 1 q= (p +p 2p ). (6) the (beam)nucleon-(target)nucleon center of mass system 1 2 3 2√3 − of the colliding nuclei. The probability for three nucleon production is then: Assuming a statistical independence in the three nucleon productionprocess,the inclusive productioncrosssection d9W = M 2 mp d3p1 mp d3p2 mp d3p3 , (2) can be written as the product of the three independent 123 | A| E1 (2π)3 E2 (2π)3 E3 (2π)3 probabilities: whereMA isthe amplitude correspondingtothe block A, d9W123 = 1 d3W1d3W2d3W3, (7) i.e.,accountingfor the inclusive productionofnucleons1, dp3dp3dp3 σ2 dp3 dp3 dp3 1 2 3 inel 1 2 3 2 and 3 and other particles in the final state f. To avoid cumbersomeexpressionsinequations(1)and(2),the fac- where σ is the total reaction cross-section of the col- inel tors corresponding to colliding nuclei have been omitted liding particles. Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 3 After integration of (3), taking into account (5) and (7), and dividing by the incident particle flux, the t pro- 1,0 duction cross section takes the form: d3σ 96π6 d3σ d3σ d3σ 0,8 E t = S 2E 1E 2E 3, (8) t dp3 m2σ2 | | 1 dp3 2 dp3 3 dp3 t p inel 1 2 3 0,6 with p1 =p2 =p3, pt =3p1 and R(x) 0,4 q2 3 p2 d3p d3q S = exp Ψ (p,q) . (9) −Q2 − 4Q2 t (2π)3(2π)3 Z (cid:18) (cid:19) 0,2 Where Ψt(p,q) M123→t is the wave function of the t (or 3He) system n∝ormalized by the condition 0,0 0 20 40 60 80 100 120 140 x(GeV) d3p d3q Ψ (p,q)2 =1. (10) | t | (2π)3(2π)3 Fig. 2. Dependence of the threshold factor R(x), with x as Z definedin thetext. The factor 1/2, accounts for nucleons and A = 3 nuclei spins, is included in (8). The three-nucleon wave function Thecrosssectionfortproductioncanthenbewritten(see is needed at sufficiently large momenta to compute the 8): amplitude. The wave function of [14] has been used (see appendix for discussion). d3σ 96π6 The structure of (8) is the same as that of the coales- Et dp3t = m2σ2 M1(p1)M2(p2)M3(p3) cence model and the S integral in 8 can be straightfor- t p inel (cid:20)Z wardly related to the coalescence momentum: d3p d3q 2 Ψ (p,q) . (13) d3p d3q t (2π)3(2π)3# p3 =18√3π2 0 Z (2π)3(2π)3 This model could be practically used directly to describe q2 3 p2 the production of t. The production threshold of the an- exp Ψ (p,q). (11) −Q2 − 4Q2 t tiparticlehastobetakenintoaccounthoweverinthecross (cid:18) (cid:19) section calculation. The same procedure to evaluate the Thus, in the approach based on the diagram of Fig. 1 cross section near threshold as that used in [3] will be and within the approximations made above, the coales- applied here. In nucleon-nucleon collisions, the main re- cence momentum p is not an adjustable parameter any- action producing a t particle is NN t+5N. Near the 0 → more, but it is determined by the inclusive proton spec- threshold of this reaction, the energy dependence of the trum and by the trinucleon wave function. Note that in t production cross section is mostly governed by the five that case,p0 depends on the momentum distribution and nucleons phase space: Φ s+m2t −2√sEt;5mp , should thus be energy and system dependent. (cid:16)p (cid:17) d3σ E t Φ s+m2 2√sE ;5m . (14) t dp3 ∝ t − t p 3 Application to three-antinuclei production t (cid:18)q (cid:19) The phase space Φ for n particles with masses, momenta andenergies,m ,p ,andE respectively,isdefinedinthe In order to generalize the diagrammatic approach of the i i i usual way (in the center of mass) coalescence model to the production of A = 3 antinuclei (noted t further below), two effects have to be taken Φ(√s;m ,m ,...m )= into account: the anisotropy of angular distributions and 1 2 n the thresholdeffects [3].Isotropicangulardependence are n 1 d3p n n iδ3 p δ E √s . frequently assumed in nonrelativistic collisions. However, (2π)3 2Ei i! i− ! in relativistic collisions, the momentum distributions are Yi=1 Xi=1 Xi=1 strongly anisotropic and the low energy approximation ItwascalculatedherebyusingthestandardCERNlibrary cannot be used. To take this into account, formula (8) program (W515, subroutine GENBOD) [15]. √s is the can be easily generalized to any angular dependence. As- totalenergyofthenparticlesinthecenterofmasssystem. suming the inclusive nucleon production cross section to A phenomenological correction factor R can thus be be given by the (parameterized) amplitude M1(p1): introduced in formulae (13) which then reads: d3σ Φ(x;5m ) E 1 = M (p )2, (12) R(x)= p , (15) 1 dp3 | 1 1 | Φ(x;5 0) 1 × 4 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle wherex= s+m2 2√sE ,andwherethedenominator 4.2 Proton Aluminium collision data at 70GeV/c t − t containsthehighenergylimitofthephasespacetoensure p R to be dimensionless and to do not change the value of TheantinucleiproducedintheSerpukhovexperiments[8, the cross section out of the space phase boundary. The 9] were obtained from a 70 GeV/c proton beam incident limits of R are thus: onanaluminiumtargetat27 mradscatteringanglesand 20GeV/cfor3He,and0and25GeV/cfort.Theinclusive R 0, E Emax = s+m2t −(5mp)2 m , antiprotoncrosssectionswereavailablefrom[18]and[19] → t → t 2√s − t! in the same kinematical conditions. In Fig. 3 the p cross section data from [8] are com- R 1, √s . pared with the results of fits using a functional form [20]. → →∞ The solid curve corresponds to a fit of a large sample of If p2 (√s E )2, the expression s+m2 2√sE t ≪ − t t − t p+A p data from 12 up to 400 GeV incident energies canbereplacedby√s E .Thissameapproximationwas → − t p not including those from reference [8] which were found made in [3]. The functional dependence of R(x) is shown notto be compatible with the othersets ofdata [20]. The in Fig. 2. calculatedvaluesareinfairagreementwiththetwolowest momentum data points (which were obtained by extrap- olation from measurements at other angles). They over- 4 Results on antinuclei production data estimate the other data points by a factor of 2 to 4. The dotted curve is a renormalization of the solid curve by 4.1 Status of the data a factor 2.5 to fit these latter points, while the dashed ≈ curvecorrespondstothefitofthesinglesetofdatapoints This section is introduced with a brief overview of the shownonthefigure,whichparametershowevergivequite current experimental situation on the antinuclei produc- poor agreement with the other sets of data [20]. tion relevant to the present study, i.e., in proton-proton and proton-nucleus collisions. The antinuclei production in ion-ion collisions will be quoted only for completeness. – As mentioned in the introduction, the experimental data on the production of mass 3 antinuclei are ex- tremelyscarce,andmuchlessinformativethanthaton antideuteronproduction,withonlytwoexperimentsor setsofexperimentsreportingonmass3antinucleipro- duction in proton-nucleus collisions [8,9,10,11]. Note that there are no experimental data available on the production of these antinuclei in proton-proton collisions. The production of 3He has been observed recently in various heavy ion studies like Pb+Pb collisions at ultra relativistic incident energies [16]. These data are out of the scope of the present work. They will not be discussed here (see [3] for a discussion). – Coalescence calculations require the antiproton production cross section to be known for antiproton momenta equal to approximately one third of the A = 3 antinuclei momenta. Unfortunately, in most experi- mentsthedifferentialcrosssectionsforantiprotonand A = 3 antinuclei productions were not measured at thismomentum.Thepcrosssectionthushadtobeex- trapolatedtotheappropriatekinematicalregionwhen Fig. 3. Inclusive differential cross section for antiproton pro- no other data were available, which of course, intro- ductioninp+Alcollisionsasafunctionofthetotalmomentum duces additional uncertainty in the calculations. inthelaboratoryframefrom[8],comparedtocalculatedvalues as discussed in thetext. The three nucleonwavefunctions neededin the calculations are much less well know than the deuteron wave function.Inaddition,thesamewavefunctionwillbeused Itmustbeemphasizedthatthelowmomentumregion, for 3He and t nuclei (see appendix). The inaccuracy on say p < 10 GeV/c, which is the useful region for the lab the tri-nucleon wave functions is thus another source of coalescence calculations, with p p /3 is particularly p ≈ t uncertainty. important here, with unfortunately no data point from The total reaction cross-section used in the calcula- direct measurement available over the relevant range. tionswasdescribedbymeansoftheparameterizationpro- Fig.4comparesthecalculationsfortheA=3produc- posed in [17]. tion cross section for the three parameterizations shown Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 5 parameter C1 C2 C3 C4 value 0.94 1.88 7.05 1.69 3c) 1E-7 2V/ Table 1. Values of the parameters of relation 16 obtained by e G fittingtheπ−productioncrosssectionsfor200and300GeV/c mb/ 1E-8 protonson Beryllium. ( 3p d s/ 3Ed 1E-9 experiment. The measured distributions have been fit by means of the following functional form, inspired from ref [22]: 1E-10 d3σ E π− =C σ (1 x)C2e−C3xe−C5p⊥, (16) dp3 1 in − 1E-11 (cid:0) (cid:1) 15 20 25 30 35 40 45 50 55 − ∗ ∗ where x=E /E (E is the total energy of the inclu- p (GeV/c) max lab sive particle in the center of mass frame, σin is the total reactioncrosssectionforthesystemincollision,√sisthe Fig. 4. The inclusive differential cross-section for ¯t [8] (open totalenergy of the system and p⊥ the transversemomen- circles) and 3He[9] (black circles) production by 70 GeV pro- tum ofthe emitted particle.The values of the parameters tons on Al target as a function of the total momentum in the obtainedaregiveninTable1andtheresultsofthisfitare laboratory frame compared to calculations using the micro- presented in Fig. 5. The parameterization (16) has been scopic model of coalescence and the same three different parameterizations of the antiproton production cross-section as shown on figure3, with the same graphical conventions. on Fig. 3 with the experimental data. The calculations using the global fit renormalized to the 70 GeV/c data (seeFig.3)givebyfarthebestagreementwiththe tdata (dotted curve). The other two sets of p cross section pa- rametersoverestimatethedatabyasoundorderofmagni- tude. This is apparentlyconsistentwith the largerp cross section predicted by these two sets of parameters for low p momentum region to which the t cross section is most sensitive. The factor of about 2 between the p cross sec- tionspredictedbythetwogroupsofparameterstranslates into a factor of about 10 for the t cross section because of the approximately cubic dependence of the latter on the p cross section. However it is somewhat puzzling that this agreementis obtainedwith parameterswhicharenot consistent with the whole body of p data [20]. 4.3 Proton beryllium collision data at 200 GeV/c Fig.5.Experimentalinclusivedifferentialcross-sectionforπ− production in p+Be collisions [21] (symbols) at 200 GeV/c In the CERN experiments [10,11], p, t and 3He were pro- (fullcircles)and300GeV/c(fulltriangles)comparedwiththe duced in proton-beryllium collisions at 200, 210, and 240 functional form 16. GeV/c and detected at 0 degree scattering angle [11], while p were measured at 200 GeV/c [10] on the same targets.For these data however,the productioncrosssec- used to extract the experimental p production cross sections were measured as the ratios to the π− production tions[10].Theresultingcrosssectionvaluesarecompared cross sections at the same momentum. The knowledge of in Fig. 6 with the results of the fit ofa functional form to the corresponding experimental π− production cross sec- a large sample of p+A p+X data from 12 GeV/c up → tion, or a good parameterisation of the latter, is thus re- to 400 GeV/c incident momenta [20]. It is seen that the quired in order to allow the values of the p, t and 3He data points derived previously and the calculated values production cross sections to be calculated. are in fair agreement. This consistency gives confidence Fortunately, the p+Be π− +X cross section has to the following steps of the analysis for the evaluation beenmeasuredat200and30→0GeV/cincidentmomentum of the t and 3He production cross sections. In Fig. 7, in [21] in similar kinematical conditions as in the CERN the t and 3He production cross sections are compared to 6 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 23V/c) 1E-6 p+Al-200GeV/c Ge mb/ ( s3/dp 3Ed 1E-7 10 20 30 40 50 p (GeV/c) lab 1E-6 23V/c) p+Be-200GeV/c Ge mb/ ( s3/dp 1E-7 3Ed Fig. 6. Experimentalinclusive differentialcross-sections for p production in p+Al collision (circles), andin p+Be collision (×10−1,triangles),[18]comparedwiththeresultsoffitsusing 1E-8 a functional form (curves) [20]. 10 15 20 25 30 35 40 45 50 calculations using the microscopic model of coalescence. plab(GeV/c) The agreement between experimental and calculated val- uanesdvcaarliceuslfartoimonpsoaorretwoigthoiond.oOnenotrhdeearvoeframgaeghnoiwtuedvee.rTdahtias 23GeV/c) p+Be-210GeV/c result should be considered as a success in account of the mb/ numerous sources of uncertainties of the calculations and s3/dp( 1E-7 of the limited accuracy of the measurements. Note also 3Ed that refs [10] and [11] report experimental values in dis- agreement by a factor of 2. Furthermore the t and 3He production cross section, measured at the same momen- 1E-8 tum,shouldbeinprinciplyclosetoeachother(thisfactis clearlyseeninthesameexperimentfortand3Heproduc- 5 10 15 20 25 30 35 40 45 50 tion)whereas,inthisexperiment,theyarequitedifferent. p (GeV/c) lab 23V/c) p+Be-240GeV/c 5 Conclusion Ge mb/ ( It has been shown in this work that the diagrammatic s3/dp 1E-7 approach to the coalescence model developed previously 3Ed previously can successfully account for the mass 3 antinuclei production cross section in proton-nucleus collisions 1E-8 over wide kinematical conditions without any additional parameter.Thesecalculationsrequireagoodknowledgeof the antiproton production cross-section and of the three- 10 15 20 25 30 35 40 45 50 nucleon wave function. These results would be further plab(GeV/c) used to calculate t and 3He flux in cosmic rays. Fig. 7. Experimental inclusive differential cross-section for t (fullcircles)and3He(opencircles)productioninp+Alandp+ Appendix Be collisions, compared to calculations using the microscopic model of coalescence (solid line). In this appendix, we briefly remind how the wave functions of the trinucleon is written in [14], while in this paper slightly different definitions have been used. A useful function is obtained from solving the Faddeev equation analytical parameterizationof the bound trinucleon wave with the Reid soft-core potential. Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 7 The total wave function of the triton is written as a References sum of 3 Faddeev components 1. SeesectionIVinNucl.Phys.B(Proc.Suppl.)113(2002). 3 2. O.Adraniet al., Nucl.Instr. Meth. A478, 114 (2002). Ψ = Ψi(q ,p ). t i i 3. R.P. Duperray, K.V. Protasov, A.Yu. Voronin, Eur. Xi=1 Phys.J., A16, 27 (2003) In (9)-(13), we make use of only one Faddeev component 4. S.T. Butler and C.A. Pearson, Phys. Rev. Lett. 7, 69 Ψ , while due the exchange symmetry of two-nucleons in (1961); Phys. Lett. 1, 77 (1962); Phys. Rev. 129, 836 t (1963); thetriton,allthethreeFaddeevcomponentsareidentical. A.Schwarzschild and C. Zupancic, Phys.Rev. 129, 854 Furthermore, Faddeev components Ψ are decomposed in t (1963); terms of their partial wave components with respect to L.P. Csernai and J.I. Kapusta, Phys. Rep. 131, 223 the spin-isospin and angular momentum basis φ (pˆ,qˆ) . α (1985). 5. See for example the classical paper S. Nagamyia et al., Ψ (p,q)= ψ (p,q)φ (pˆ,qˆ), t α α Phys.Rev.C24971(1981)and Nucl.Phys.A661(1999) α X for recent references. with, if p (i=1,2,3) is the nucleon momenta 6. V.M. Kolybasov, Yu.N. Sokol’skikh, Phys. Lett. B225, i 31 (1989); Sov. J. Nucl. Phys.55, 1148 (1992). p= 1(p p ),q= 1 (p +p 2p ). 7. P. Chardonnet, J. Orloff, P. Salati, Phys. Lett. B409, 1 2 1 2 3 2 − 2√3 − 313 (1997). 8. Y.M. Antipovet al., Phys. Lett. B34, 164(1971). The following normalization is used 9. N.K. Vishnevskii et al., Sov. J. Nucl. Phys. 20, 371 (1974). Ψt(p,q)2d3pd3q= dpdqp2q2 ψα(p,q)2 =1. 10. W.Bozzoli et al., Nucl. Phys. B144, 317(1978). | | | | Z α Z 11. A.Bussière et al., Nucl. Phys. B174, 1(1980). X 12. M.A.BraunandV.V.Vechernin,Sov.J.Nucl.Phys. 44, Note that, in these expressions, the definition of p and 506 (1986); 36, 357 (1982). the normalization differ from (6) and (10). Ψ is a sum of t 13. I.S.Shapiro,Dispersiontheoryofdirectnuclearreactions, the partial wave state α which is a label for the following in Selected Topics in Nuclear Theory, Ed. F. Janouch, physical quantities: IAEA, Vienna, 1963; I.S. Shapiro, Usp. Fiz. Nauk. 92, – L,theangularmomentumofthepairofnucleons(1-2). 549 (1967) [ Sov. Phys.Usp. 10, 515 (1968)]. – l,theangularmomentumofnucleon3accordingtothe 14. Muslim and Y.E.kim Nucl. Phys. A427, 235 (1984). center of the mass of the pair of nucleons (1-2). 15. F.James,MonteCarloPhaseSpace,CERN68-15(1968); – , the total angular momentum of the triton. http://wwwinfo.cern.ch/asdoc/shortwrupsdir/w515/top.html – Ls, the spin of the pair of nucleons (1-2). 16. G. Appelquist et al., Phys. Lett. B376, 245(1996), see – , the total spin of the triton. Nucl.PhysA661,1999 for more on this issue. – ST, the isospin of the pair of nucleons (1-2). 17. J.R. Letaw, R. Silberberg, and C.H. Tsao, Ap.J.Suppl. 51 271 (1983) Onlytwocomponentslabelαweretakenintoaccount, 18. F. Binon et al., Phys.Lett. 30B, 510(1969). α=1,2. 19. Yu.B.Bushninetal., Sov.J.Nucl.Phys.10,337(1970). – For α=1, L=l= =0, s=1, =1/2 and T =0. 20. R.P. Duperray, C.Y. Huang, K.V. Protasov and – For α=2, L=l=L=s=0, =S1/2 and T =1. M.Buénerd, in preparation. L S 21. W.F. Baker et al., Phys. Lett. 51B 303 (1974) Ofcourse,the facttoconsideronlytwopartialwavestate 22. A.N. Kalinovskii, N.V. Mokhov and Yu.P. Nikitin, Pas- isanapproximationwhichgivestheprobabilityof89.25% sage of high-energy particules through matter, American oftrinucleonbeinginthepartialwavestateα.In[14],the Instituteof Physics, New York (1989). parameterization for ψ (p,q) is given by α 3 ψ (p,q) =pLpl p2+Ω2 −1 q2+Ω2 −1 α p1 qm m=1 (cid:0) (cid:1) Y (cid:0) (cid:1) 6 6 C ij , (p2+µ2) q2+ν2 i=1j=1 i j XX withΩ ,Ω ,µ ,ν andC a(cid:0)lldepend(cid:1)ingonthepartial p1 qm i j ij wavelabelα.Thenumericalvaluesofthesecoefficientscan be found in [14]. 3He(ppn)and t(pnn) are considered to have the same wave function. Altough, because of the presence of the Coulombinteraction,thesetwowavefunctionsareslightly different,thisdifferenceisnegligiblecomparedtotheother uncertainties of present calculations.