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Faddeev equation approach for three-cluster nuclear reactions A.Deltuva,A.C.Fonseca,andR.Lazauskas 2 1 0 2 n a J Abstract In this lecture we aim to present a formalism based on Faddeev-like 4 equations for describing nuclear three-cluster reactions that include elastic, trans- 2 ferandbreakupchannels.Twodifferenttechniquesbasedonmomentum-spaceand ] configuration-spacerepresentationsareexplainedindetail. Animportantnewfea- h tureofthesemethodsisthepossibilitytoaccountfortherepulsiveCoulombinter- t - actionbetweentwoofthethreeclustersinallchannels.Comparisonwithprevious l c calculationsbasedonapproximatemethodsusedinnuclearreactiontheoryisalso u discussed. n [ 1 v 1 Introduction 9 7 9 Nuclear collision experiments, performedat ion accelerators, are a very powerful 4 tooltostudynuclearpropertiesatlowandintermediateenergies.Inordertointer- . pretaccumulatedexperimentaldata appropriatetheoreticalmethodsare necessary 1 0 enabling the simultaneous description of the available elastic, rearrangement and 2 breakupreactions. 1 Regardlessofitsimportance,thetheoreticaldescriptionofquantum-mechanical : v collisionsturnsouttobeoneofthemostcomplexandslowlyadvancingproblemsin i theoreticalphysics.Ifduringthelastdecadeaccuratesolutionsforthenuclearbound X r a A.Deltuva CentrodeF´ısicaNuclear daUniversidade deLisboa, P-1649-003 Lisboa, Portugal,e-mail:del- [email protected] A.C.Fonseca CentrodeF´ısicaNucleardaUniversidadedeLisboa,P-1649-003Lisboa,Portugal R.Lazauskas IPHC,IN2P3-CNRS/Universite´LouisPasteurBP28,F-67037StrasbourgCedex2,France,e-mail: [email protected] 1 2 A.Deltuva,A.C.Fonseca,andR.Lazauskas stateproblembecameavailable,fullsolutionofthescatteringproblem(containing elastic,rearrangementandbreakupchannels)remainslimitedtothethree-bodycase. Themaindifficultyisrelatedtothefactthat,unliketheboundstatewavefunc- tions, scattering wave functions are not localized. In configuration space one is obliged to solve multidimensional differential equations with extremely complex boundary conditions; by formulating the quantum-mechanicalscattering problem in momentum space one has to deal with non-trivial singularities in the kernel of multivariableintegralequations. A rigorous mathematical formulation of the quantum mechanical three-body problemintheframeworkofnonrelativisticdynamicshasbeenintroducedbyFad- deevintheearlysixties[19],inthecontextofthethree-nucleonsystemwithshort rangeinteractions.Inmomentumspacetheseequationsmightbeslightlymodified byformulatingthemintermsofthree-particletransitionoperatorsthataresmoother functionscomparedtothesystemwavefunctions.Suchamodificationwasproposed byAlt,Grassberger,andSandhas[1](AGS). Solutions of the AGS equations with short range interactions were readily obtained in the early seventies. As large computers became available progress fol- lowedleading,bytheendeighties,tofullyconvergedsolutionsofthese equations forneutron-deuteron(n-d)elasticscatteringandbreakupusingrealisticshortrange nucleon-nucleon (N-N) interactions. Nevertheless the inclusion of the long range Coulomb force in momentum space calculations of proton-deuteron(p-d) elastic scatteringandbreakupwiththesamenumericalreliabilityascalculationswithshort rangeinteractionsalone,onlybecomepossibleinthelastdecade. Significantprogresshasbeenachieved[16,17]bydevelopingthescreeningand renormalizationprocedurefortheCoulombinteractioninmomentumspaceusinga smoothbutatthesametimesufficientlyrapidscreening.Thistechniquepermittedto extendthecalculationstothesystemsofthree-particleswitharbitrarymassesabove thebreakupthreshold[11,18]. Howeverithastakensometimetoformulatetheappropriateboundaryconditions in configurationspace for the three-bodyproblem[32, 33, 36] and even longerto reformulatetheoriginalFaddeevequationstoallowtheincorporationoflong-range Coulomb like interactions [34, 35]. Rigorous solution of the three-body problem withshortrangeinteractionshasbeenachievedjustafterthesetheoreticaldevelop- ments, both below and abovebreakupthreshold.On the otherhand the numerical solution for the three-body problem including charged particles above the three- particle breakupthreshold has been achievedonly recently.First it has been done byusingapproximateMerkurievboundaryconditionsin configurationspace [27]. Neverthelessthisapproachprovedtobearathercomplextasknumerically,remain- ingunexploredbeyondthe p-dscatteringcase,butnotyetforthe p-dbreakup. Finally,veryrecentlyconfigurationspacemethodbasedoncomplexscalinghave beendevelopedandappliedfor p-d scattering[31].Thismethodallowstotreatthe scattering problem using very simple boundary conditions, equivalentto the ones employedtosolvethebound-stateproblem. Faddeevequationapproachforthree-clusternuclearreactions 3 The aim of this lecture is to present these two recently developed techniques, namely the momentum-space method based on screening and renormalization as wellastheconfiguration-spacecomplexscalingmethod.Thislectureisstructured as follows:the first partservesto introducetheoreticalformalismsfor momentum spaceandconfigurationspacecalculations;inthesecondpartwepresentsomese- lectedcalculationswithanaimtotesttheperformanceandvalidityofthetwopre- sentedmethods. 2 Momentum-space description ofthree-particle scattering Wedescribethescatteringprocessinasystemofthree-particlesinteractingviapair- wiseshort-rangepotentialsva ,a =1,2,3;weusetheodd-man-outnotation,thatis, v is the potential between particles 2 and 3. In the framework of nonrelativistic 1 quantummechanicsthecenter-of-mass(c.m.)andtheinternalmotioncanbesepa- ratedbyintroducingJacobimomenta mgkb mb kg pa = − , (1) mb +mg ma (kb +kg) (mb +mg)ka qa = − , (2) ma +mb +mg with(abg )beingcyclicpermutationsof(123);ka andma aretheindividualparticle momentaandmasses,respectively.Thec.m.motionisfreeandinthefollowingwe consideronlytheinternalmotion;thecorrespondingkineticenergyoperatorisH 0 whilethefullHamiltonianis 3 (cid:229) H =H0+ va . (3) a =1 2.1 Alt, Grassberger,and Sandhas equations Weconsidertheparticlea scatteringfromthepaira thatisboundwithenergyea . Theinitialchannelstate ba qa istheproductoftheboundstatewavefunction ba forthepaira andaplan|ewaveiwiththerelativeparticle-paira momentumqa ;|thei dependenceonthediscretequantumnumbersissuppressedinournotation. ba qa | i istheeigenstateofthecorrespondingchannelHamiltonianHa =H0+va withthe energyeigenvalueE=ea +qa2/2Ma whereMa istheparticle-paira reducedmass. Thefinalchannelstateistheparticle-pairstateinthesameordifferentconfiguration bb qb inthecaseofelasticandrearrangementscatteringor,inthecaseofbreakup, | i it is the state of three free particles pgqg with the same energy E = pg2/2m g + | i 4 A.Deltuva,A.C.Fonseca,andR.Lazauskas q2g/2Mg andpairg reducedmassm g;anysetofJacobimomentacanbeusedequally wellforthebreakupstate. Thestationaryscatteringstates[42,22]correspondingtotheabovechannelstates are eigenstates of the full Hamiltonian; they are obtained from the channel states usingthefullresolventG=(E+i0 H) 1,i.e., − − ba qa (+) = i0Gba qa , (4) | i | i pa qa (+) = i0Gpa qa . (5) | i | i The full resolvent G may be decomposed into the channel resolvents Gb =(E+ i0 Hb )−1 and/orfreeresolventG0=(E+i0 H0)−1 as − − G=Gb +Gb v¯b G, (6) withb =0,1,2,3andv¯b =(cid:229) g3=1d¯bg vg whered¯bg =1−d bg .Furthermore,thechan- nelresolvents Gb =G0+G0Tb G0, (7) canberelatedtothecorrespondingtwo-particletransitionoperators Tb =vb +vb G0Tb , (8) embeddedintothree-particleHilbertspace.UsingthesedefinitionsEqs.(4)and(5) canbewrittenastriadsofLippmann-Schwingerequations ba qa (+) = d ba ba qa +Gb v¯b ba qa (+), (9) | i | i | i pa qa (+) = (1+G0Tb )pa qa +Gb v¯b pa qa (+), (10) | i | i | i with a being fixed and b =1,2,3; they are necessary and sufficient to define the states ba qa (+) and pa qa (+) uniquely.However,in scattering problemsit may | i | i bemoreconvenienttoworkwiththemultichanneltransitionoperatorsUba defined suchthattheiron-shellelementsyieldscatteringamplitudes,i.e., Uba ba qa =v¯b ba qa (+). (11) | i | i OurcalculationsarebasedontheAGSversion[1]ofthree-particlescatteringtheory. InaccordancewithEq.(11)itdefinesthemultichanneltransitionoperatorsUba by thedecompositionofthefullresolventGintochanneland/orfreeresolventsas G=d ba Ga +Gb Uba Ga . (12) ThemultichanneltransitionoperatorsUba withfixeda andb =1,2,3aresolutions ofthreecoupledintegralequations 3 Uba =d¯ba G−01+(cid:229) d¯bg TgG0Uga . (13) g=1 Faddeevequationapproachforthree-clusternuclearreactions 5 The transition matrixU0a to final states with three free particles can be obtained fromthesolutionsofEq.(13)byquadrature,i.e., 3 U0a =G−01+(cid:229) TgG0Uga . (14) g=1 Theon-shellmatrixelements bb qb′ Uba ba qa areamplitudes(uptoafactor) h | | i forelastic(b =a )andrearrangement(b =a )scattering.Forexample,thediffer- entialcrosssectionforthea +(bg ) b +6 (ga )reactioninthec.m.systemisgiven → by dds Wa →b b =(2p )4Ma Mb qqab′ |hbb qb′ |Uba |ba qa i|2. (15) The cross section for the breakup is determined by the on-shell matrix elements p′gq′g U0a ba qa . Thus, in the AGS framework all elastic, rearrangement, and h | | i breakupreactionsarecalculatedonthesamefooting. FinallywenotethattheAGSequationscanbeextendedtoincludealsothethree- bodyforcesasdoneinRef.[12]. 2.2 InclusionoftheCoulomb interaction The Coulomb potential w , due to its long range, does not satisfy the mathemat- C ical properties required for the formulation of standard scattering theory as given in the previoussubsection for short-range interactions va . However, in nature the Coulomb potential is always screened at large distances. The comparison of the datafromtypicalnuclearphysicsexperimentsandtheoreticalpredictionswithfull Coulombis meaningfulonly if the full and screenedCoulomb becomephysically indistinguishable.ThiswasprovedinRefs.[44,43]wherethescreeningandrenor- malizationmethodforthescatteringoftwochargedparticleswasproposed.Webase ourtreatmentoftheCoulombinteractiononthatidea. Although we use momentum-space framework, we first choose the screened Coulombpotentialinconfiguration-spacerepresentationas w (r)=w (r)e (r/R)n, (16) R C − andthentransformittomomentum-space.HereRisthescreeningradiusandncon- trols the smoothness of the screening. The standard scattering theory is formally applicable to the screened Coulomb potential w , i.e., the Lippmann-Schwinger R equationyieldsthetwo-particletransitionmatrix t =w +w g t , (17) R R R 0R where g is the two-particle free resolvent. It was proven in Ref. [44] that in the 0 limitofinfinitescreeningradiusRtheon-shellscreenedCoulombtransitionmatrix 6 A.Deltuva,A.C.Fonseca,andR.Lazauskas (screenedCoulombscatteringamplitude) p t p with p =p,renormalizedbyan ′ R ′ infinitely oscillating phase factor z−R1(p)h=e|2if|R(ip), approaches the full Coulomb amplitude p t p in general as a distribution. The convergence in the sense of ′ C h | | i distributionsissufficientforthedescriptionofphysicalobservablesinarealexper- imentwheretheincomingbeamisnotaplanewavebutwavepacketandtherefore the crosssection is determinednotdirectly by the scatteringamplitude butby the outgoingwavepacket,i.e.,bythescatteringamplitudeaveragedovertheinitialstate physicalwavepacket.Inpracticalcalculations[2,16]thisaveragingiscarriedout implicitly, replacing the renormalizedscreened Coulomb amplitude in the R ¥ → limitbythefullone,i.e., Rlim¥ z−R1(p)hp′|tR|pi→hp′|tC|pi. (18) → Sincez 1(p)isonlyaphasefactor,theaboverelationsindeeddemonstratethatthe −R physicalobservablesbecomeinsensitivetoscreeningprovidedittakesplaceatsuf- ficientlylargedistancesRand,intheR ¥ limit,coincidewiththecorresponding → 1 quantitiesreferringto the full Coulomb.Furthermore,renormalizationby z−R2(pi) intheR ¥ limitrelatesalsothescreenedandfullCoulombwavefunctions[23], → i.e., Rlim¥ (1+g0tR)|piz−R12(p)=|y C(+)(p)i. (19) → Thescreeningandrenormalizationmethodbasedontheaboverelationscanbe extendedto more complicatedsystems, albeit with some limitations. We consider thesystemofthree-particleswithchargesza ofequalsigninteractingviapairwise strong short-range and screened Coulomb potentials va +wa R with a being 1, 2, or3.Thecorrespondingtwo-particletransitionmatricesarecalculatedwiththefull channelinteraction (R) (R) Ta =(va +wa R)+(va +wa R)G0Ta , (20) (R) and the multichanneltransition operatorsU for elastic and rearrangementscat- ba teringaresolutionsoftheAGSequation 3 Uba(R)=d¯ba G−01+(cid:229) d¯bg Tg(R)G0Uga(R); (21) g=1 alloperatorsdependparametricallyontheCoulombscreeningradiusR. InordertoisolatethescreenedCoulombcontributionstothetransitionamplitude thatdivergeintheinfiniteRlimitweintroduceanauxiliaryscreenedCoulombpo- tentialWc.m. betweentheparticlea andthecenterofmass(c.m.)oftheremaining a R pair.ThesamescreeningfunctionhastobeusedforbothCoulombpotentialswa R andWc.m..Thecorrespondingtransitionmatrix a R Tac.Rm.=Wac.Rm.+Wac.Rm.G(aR)Tac.Rm., (22) Faddeevequationapproachforthree-clusternuclearreactions 7 withG(aR)=(E+i0 H0 va wa R)−1 isatwo-body-likeoperatorandtherefore itson-shellandhalf-s−hellb−ehav−iorinthelimitR ¥ isgivenbyEqs.(18)and(19). → AsderivedinRef.[16],thethree-particletransitionoperatorsmaybedecomposed as Uba(R) = d ba Tac.Rm.+[1+Tbc.Rm.G(bR)]U˜ba(R)[1+Ga(R)Tac.Rm.] (23) = d ba Tac.Rm.+(Uba(R)−d ba Tac.Rm.). (24) where the auxiliary operatorU˜(R) is of short range when calculated between on- ba (R) shellscreenedCoulombstates.Thus,thethree-particletransitionoperatorU has ba along-rangepartd ba Tac.Rm. whereastheremainderUba(R)−d ba Tac.Rm. isashort-range operatorthatisexternallydistorteddueto thescreenedCoulombwavesgenerated by [1+G(aR)Tac.Rm.]. On-shell, both parts do not have a proper limit as R→¥ but the limit exists after renormalization by an appropriate phase factor, yielding the transitionamplitudeforfullCoulomb hbb q′b |Uba(C)|ba qa i=d ba hba q′b |Tac.Cm.|ba qa i +Rlim¥ [Zb−R21(qb′ )hbb q′b |(Uba(R)−d ba Tac.Rm.)|ba qa iZa−R12(qa )]. (25) → Thefirsttermontheright-handsideofEq.(25)isknownanalytically[44];itcor- respondstotheparticle-paira fullCoulombtransitionamplitudethatresultsfrom theimplicitrenormalizationofTc.m.accordingtoEq.(18).TheR ¥ limitforthe a R → remainingpart(Uba(R)−d ba Tac.Rm.)ofthemultichanneltransitionmatrixisperformed numerically; due to the short-range nature of this term the convergence with the increasing screening radius R is fast and the limit is reached with sufficient accu- racyatfiniteR;furthermore,itcanbecalculatedusingthepartial-waveexpansion. We emphasizethatEq. (25)is by no meansan approximationsince it is based on theobviouslyexactidentity(24)wheretheR ¥ limitforeachtermexistsandis → calculatedseparately. TherenormalizationfactorforR ¥ isadivergingphasefactor → Za R(qa )=e−2iF a R(qa ), (26) whereF a R(qa ),thoughindependentoftheparticle-pairrelativeangularmomentum la intheinfiniteRlimit,mayberealizedby F a R(qa )=s laa (qa )−h laa R(qa ), (27) with the divergingscreened Coulombphase shift h a (qa ) correspondingto standard boundary conditions and the proper Coulombla Rone s a (qa ) referring to the la logarithmically distorted proper Coulomb boundary conditions. For the screened CoulombpotentialofEq.(16)theinfiniteRlimitofF a R(qa )isknownanalytically, 8 A.Deltuva,A.C.Fonseca,andR.Lazauskas F a R(qa )=Ka (qa )[ln(2qa R) C/n], (28) − whereC 0.5772156649istheEulernumberandKa (qa )=a e.m.za (cid:229) g d¯ga zgMa /qa is the Co≈ulomb parameter with a 1/137. The form of the renormalization e.m. phase F a R(qa ) to be used in the actu≈al calculations with finite screening radii R is notunique,butthe convergedresultsshow independenceof the chosenformof F a R(qa ). For breakup reactionswe follow a similar strategy. However, the proper three- bodyCoulombwavefunctionanditsrelationtothethree-bodyscreenedCoulomb wavefunctionis,ingeneral,unknown.Thispreventstheapplicationofthescreening andrenormalizationmethodto the reactionsinvolvingthreefreechargedparticles (nucleonsornuclei)inthefinalstate.However,inthesystemoftwochargedparti- clesandaneutralonewithzr =0,thefinal-stateCoulombdistortionbecomesagain atwo-bodyproblemwiththescreenedCoulombtransitionmatrix Tr R=wr R+wr RG0Tr R. (29) Thismakesthechannelr ,correspondingtothecorrelatedpairofchargedparticles, themostconvenientchoiceforthedescriptionofthefinalbreakupstate.Asshown inRef.[17],theAGSbreakupoperator 3 U0(aR)=G−01+(cid:229) Tg(R)G0Uga(R), (30) g=1 canbedecomposedas U0(aR)=(1+Tr RG0)U˜0(aR)(1+Ga(R)Tac.Rm.), (31) wherethereducedoperatorU˜(R)(Z)calculatedbetweenscreenedCoulombdistorted 0a (R) initialandfinalstatesisoffiniterange.InthefullbreakupoperatorU (Z)theex- 0a ternaldistortionsshowupinscreenedCoulombwavesgeneratedby(1+G(aR)Tac.Rm.) intheinitialstateandby(1+Tr RG0)inthefinalstate;bothwavefunctionsdonot haveproperlimitsasR ¥ .Thereforethefullbreakuptransitionamplitudeinthe → caseoftheunscreenedCoulombpotentialisobtainedviatherenormalizationofthe (R) on-shellbreakuptransitionmatrixU intheinfiniteRlimit 0a hpr′ q′r |U0(Ca )|ba qa i=Rlim¥ [z−r R21(pr′ )hp′r q′r |U0(aR)|ba qa iZa−R12(qa )], (32) → wherep′r istherelativemomentumbetweenthechargedparticlesinthefinalstate, q′r thecorrespondingparticle-pairrelativemomentum,and zr R(p′r )=e−2ikr (p′r )[ln(2p′r R)−C/n], (33) Faddeevequationapproachforthree-clusternuclearreactions 9 the final-state renormalization factor with the Coulomb parameter k r (pr′ ) for the pairr .ThelimitinEq.(32)hastobeperformednumerically,but,duetotheshort- range nature of the breakup operator, the convergence with increasing screening radiusRisfastandthelimitisreachedwithsufficientaccuracyatfiniteR.Thus,to includetheCoulombinteractionviathescreeningandrenormalizationmethodone onlyneedstosolvestandardscatteringtheoryequations. 2.3 Practicalrealization We calculate the short-range part of the elastic, rearrangement,and breakup scat- teringamplitudes(25)and(32)bysolvingstandardscatteringequations(21),(22), and(30)withafiniteCoulombscreeningradiusR.Weworkinthemomentum-space partial-wavebasis[10],i.e.,weusethreesets pa qa n a pa qa (la [La (sb sg)Sa ]Ia sa Ka )JM with(a ,b ,g )beingcyclicper- |mutationsio≡f|(1,2,3).H{eresa isthespino}fparticleia ,La andla aretheorbitalan- gularmomentaassociatedwith pa andqa respectively,whereasSa ,Ia ,andKa are intermediateangularmomentathatarecoupledtoatotalangularmomentumJwith projectionM.Alldiscretequantumnumbersareabbreviatedbyn a .Theintegration overthemomentumvariablesisdiscretizedusingGaussianquadraturerulesthereby convertingasystemofintegralequationsforeachJ andparityP =( )La +la into − averylargesystemoflinearalgebraicequations.Duetothehugedimensionthose linearsystemscannotbesolveddirectly.InsteadweexpandtheAGStransitionop- erators(21)intothecorrespondingNeumannseries 3 3 3 Uba(R)=d¯ba G−01+(cid:229) d¯bg Tg(R)d¯ga +(cid:229) d¯bg Tg(R)G0 (cid:229) d¯gs Ts(R)d¯sa +···, (34) g=1 g=1 s =1 thataresummedupbytheiterativePademethod[5];ityieldsanaccuratesolution of Eq. (21) even when the Neumann series (34) diverges. Each two-particle tran- sitionoperatorTg(R) isevaluatedinitsproperbasis pgqgn g ,thus,transformations | i betweenallthreebasesareneeded.Thecalculationoftheinvolvedoverlapfunctions pb qb n b pa qa n a followscloselythecalculationofthree-nucleonpermutationop- h | i eratorsdiscussedinRefs.[10,22].Aspecialtreatment[5,10]isneededforthein- tegrablesingularitiesarising fromthe pair boundstate polesin Tg(R) andfromG0. Furthermore,wehavetomakesurethatRislargeenoughtoachieve(afterrenormal- ization)theR-independenceoftheresultsuptoadesiredaccuracy.However,those R values are larger than the range of the nuclear interaction resulting in a slower convergenceof the partial-wave expansion. As we found in Ref. [16], the practi- calsuccessofthescreeningandrenormalizationmethoddependsverymuchonthe choiceofthescreeningfunction,inourcaseonthepowerninEq.(16).Wewantto ensurethatthescreenedCoulombpotentialw approximateswellthetrueCoulomb R onew fordistancesr<Randsimultaneouslyvanishesrapidlyforr>R,provid- C ing a comparativelyfast convergenceof the partial-wave expansion. As shown in 10 A.Deltuva,A.C.Fonseca,andR.Lazauskas Fig. 1 Differential cross without Coulomb section and deuteron vec- R = 5 fm toranalyzingpoweriT11 of sr) 300 R = 10 fm thea d elasticscatteringat mb/ R = 15 fm 4.81MeVdeuteronlaben- ( 200 ergyasfunctionsofthec.m. Wd scatteringangle.Convergence / sd withthescreeningradiusR 100 usedtocalculatetheshort- rangepartoftheamplitudes 0.4 d + a fi d + a isstudied:R=5fm(dotted E = 4.81 MeV curves), R=10fm(dash- d dottedcurves),andR=15fm 1 (solidcurves).Resultswithout T1 0.0 i Coulombaregivenbydashed curves.Theexperimentaldata arefromRefs.[4,30]. -0.4 0 60 120 180 Q (deg) c.m. Ref.[16],thisisnotthecaseforsimpleexponentialscreening(n=1)whereasthe sharpcutoff(n ¥ )yieldsslowoscillatingconvergencewiththescreeningradius → R. However,we foundthat valuesof 3 n 8 providea sufficiently smooth and ≤ ≤ rapid screening around r =R. The screening functions for different n values are comparedinRef.[16]togetherwiththeresultsdemonstratingthesuperiorityofour optimal choice: using 3 n 8 the convergence with the screening radius R, at ≤ ≤ which the short range part of the amplitudes was calculated, is fast enough such that the convergenceof the partial-wave expansion, though being slower than for the nuclear interaction alone, can be achieved and there is no need to work in a plane-wavebasis.Hereweusen=4andshowinFigs.1and2fewexamplesforthe R-convergenceofthea -deuteronscatteringobservablescalculatedinathree-body model (a ,p,n); the nuclear interaction is taken from Ref. [11]. The convergence with R is impressively fast for both a -deuteron elastic scattering and breakup. In addition we note that the Coulomb effect is very large and clearly improves the descriptionoftheexperimentaldata,especiallyforthedifferentialcrosssectionin a -deuteronbreakupreaction.This is dueto the shiftof the a p P-wave resonance positionwhenthea pCoulombrepulsionisincludedthatleadstothecorresponding changesinthestructureoftheobservables. In addition to the internal reliability criterion of the screening and renormalization method — the convergencewith R — we note that our results for proton- deuteron elastic scattering [15] agree well over a broad energy range with those ofRef.[28]obtainedfromthevariationalconfiguration-spacesolutionofthethree- nucleonSchro¨dingerequationwithunscreenedCoulombpotentialandimposingthe properCoulombboundaryconditionsexplicitly.