THE USE OF ANALYTICITY INNUCLEAR PHYSICS

M.P. LOCHER

SwissInstitutefor NuclearResearch,Villigen, Switzerland

and

T. MIZUTANI

SwissInstitutefor Nuclear Research,Villigen, SwitzerlandandInstitut dephysique,Universitéde Liege,Belgium

INORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICS REPORTS (Review Section of PhysicsLetters)46. No. 2)1978)43—92.North-Holland PublishingCompan~

THE USE OF ANALYTICITY IN NUCLEAR PHYSICS

M.P. LOCHER

Swiss institute br Nuclear Research. Villigen, Switzerland

and

T. MIZUTANI

Swi,s,s institute for Nuclear Reseurch,Villiqeii. Switzerland

and Inst itut de physique.Unirersitd de Liege,Belqiutn5

ReceivedFebruary1978

Contents:

1. Introduction 45 4. Surveyofresults 632. Singularitiesof scatteringamplitudes 46 4.1. Couplingconstantsin few nucleonsystems 65

2.1. Poles,coupling constantsandasymptoticnorma- 4.2. Pionnuclearcouplingconstants 75

lisations 48 Acknowledgement 7822. Multiparticle singularities 50 Appendices

3. Analytic extrapolations 54 A. Residuesfor thegeneralspin case 783.1. Forwarddispersionrelationsandthediscrepancy B. Antisymmetrisationfactors 84

function 54 C. Standardexamples 87

3.2. Coulombmodilications 57 D. Estimateofu- and i-channelanomalouscuts 893.3. Angular crosssectionextrapolations 59 References 903.4. Conformalmapping 60

Abstract:

The generalanalyticpropertiesof nuclearscatteringamplitudesarediscussedby meansof simpleexamples.Analyticextrapolationsto thepolesaresurveyed.The methodsusedareforward amplitude dispersionrelationsin theenergyvariable,angularextrapolationsin the differential crosssection,conformalmapping techniques,and combinationsthereof. A unified notation is explained in detail

in order to relatepole parametersto the asymptoticnormalisationof wave functions in conventionalnuclearphysics. The existingresultsfor vertexconstantsin fewnucleonsystemsarecompiledandcritically discussed.In particularorigin andstrengthofbackgroundandtheextrapolationerrorsareconsidered.Similarly, theinformationon theeffectivepion nuclearcoupling constantsis reviewed.

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M.P. Locherand T.Mizutani, The useofanalyticity in nuclearphysics 45

1. Introduction

The useof analytic propertiesof scatteringamplitudesis well establishedin particle physicsand of increasingimportancein nuclearphysicsaswell. Ratherwide-spread,however,is the feelingamongstnuclear physicists that dispersionrelations (and extrapolationsinto the unphysicalregion) are conceptuallydifficult and technically complicated.The aim of the presentreview isthereforetwofold. First it is intendedas an introductionto non-experts.The specialaspectsarisingfrom the scatteringon weaklyboundnucleartargetsarestressed.The maintext is keptclear fromall technicalcomplicationswhich arealmostentirelydueto thespin andthe chargeof the collidingparticles.In this way the dynamicalimportanceof the singularitiesin the unphysicalregion forthe scatteringprocessat physicalenergiesandanglesis madeevident.

Our secondpoint is the presentationof the considerablebody of resultsobtainedin hadron—nucleusscattering’. In the presentreviewweconcentrateon thenuclearquantitiesdeterminedfromanalyticextrapolations.They form an essentialconstraintto the few-bodyscatteringproblemasa whole. We havemadeacarefulcomparisonof the correspondingresults on nuclearcouplingconstantsusingaunified notation,andwe commenton the reliability of the numbersgiven in thetablesby consideringthe relativemeritsof the methodsusedandthe natureof the experimentalinput.

In nuclearscatteringthe analyticpropertiesof the scatteringamplitudeneededfor mostof theapplicationsdiscussedbelowcanonly be conjectured.We recall that in particlephysicsanalyticity(or equivalentlythe validity of dispersionrelations)hasbeenstudiedfor many years.Analyticitycan be provenby the methodsof local relativistic field theory*, but only for forwardandnear-forwardxN, KN and(unmeasurable)irir andKK elastic scattering.The fundamentalcaseof NNelasticscatteringis not in this list.

We remark in passingthatamain ingredientin the proofs of analyticity both in the classicaland in the field-theoreticalcontext is causality.This strengthensour belief thata high degreeofanalyticity of the scatteringamplitude is fundamentallyrequired,whereasthe actual proofs willdependon specific(possiblytoo restrictive)formulationsof causalityandon somefurtherstandardassumptionsof field theory.We shall thereforeassumethat the limitations mentionedabovearetechnicalin natureand that all the analyticity requiredin nuclearapplicationsis present.In parti-cular,the nearbysingularitieswill be given by the exchangeof oneor severalparticlesfollowingthe establishedrules of field theory. In fact, the more recentconformalmappingtechniquesarelessexactingin this respect.They allow to exploitanalyticity in finite domainswhich is establishedmore readily. The useof pole extrapolationsin nuclear reactionsdatesback to the analysisofstrippingby Amado[3] and,moreimplicitly, to ChristianandGammel[27] for nucleon—deuteronscattering.A first applicationof forwarddispersionrelationsto the samereaction,by Blanken-becleret al. [17], sufferedfrom the lack of adequateexperimentaldata.The generalstructureofanalyticity in nuclearreactionshasbeendiscussedby Shapiro[98] andSchnitzer [96, 97]. Apartfrom theseprecursoryattempts,a sustainedeffort of applyinganalyticity (anddispersionrelationsin particular)to nuclearreactionsextendsbackoveraperiodof aboutnineyearsonly. The resultsof theseeffortsarereportedin this review.

In section 2 we introducethe basic terminology.The singularitiesof the scatteringamplitudeare discussedby meansof simpleexamples.Importantpointsarethe relationbetweenthe residues

Theclosingdatefor thematerialcoveredis December1977.* The readerinterestedin theprinciple is referredto ref. [99], wheresucha proof is shownfor a simpleexample.

46 M.P. Locherand T. Mizutani, Theuseof analyticity in nuclear physics

of bound-statepoles and the asymptoticnormalizationof wave functions, as well as the role ofnuclear form factors. Typical competing singularitiesare due to multinucleonexchange.Theproceduresusedto performanalyticextrapolationsare discussedin section3. The contributionsfrom the unphysicalregionare projectedout by meansof the discrepancyfunctionmethodwhichis basedon dispersionrelationsin the energyvariableof the forwardamplitude(FDRs).Dispersionrelationsin theangularvariablefor thepurposeof extrapolationshavenot beentried for nuclearreactionsyet. However, numerousstraight pole extrapolationsof the differential crosssectionin the angularvariablehavebeenperformed(the classificationas dispersionrelationis erroneousin this context).Morerecentlyextrapolationsaremadein anewvariable,obtainedby a conformalmapping technique.In this way the assumedanalyticity in the energyand angularvariablesisexploited more fully. The current way of handling Coulomb corrections for charged-particlescatteringis briefly discussedin this sectionas well. Section4 reviews the results for nucleoniccouplingconstantsin light nuclearsystems.Somecrossreferencesto informationon theseconstantsfrom conventionalnuclearphysicsare made.The presentknowledgeof the effectivestrengthofpion nuclearcouplingis alsoreviewed. Appendix A introducesthenotation for the generalspincase.The antisymmetrisationfactors neededto connect residueswith nuclear wave functioninformationare given in appendixB. Sincethe generalcaseis somewhatinvolvedwe list the resultsfor the simplestcasesin appendixC. Appendix D containsestimatesof triangle singularities.

2. Singularitiesof scattering amplitudes

We shall illustrate the main pointsconsideringthe simplestnuclearexample,namelyelasticnd scattering,see Blankenbecleret al. [17] andLocher [73]. Similar notationand further back-groundcanbe found in the paperby EricsonandLocher [50]. For an earlydiscussionof nuclearsingularitiesand the connectionsto potential scatteringsee refs. [96—98].

The elastic scatteringamplitude is a functionof two variableswhich we take as the lab energyw = m + E andthe scatteringangle ~. The correspondingrelativistic notationsare theinvariantMandeistamvariabless andt: the energys,

s = m2 + M2 + 2wM (lab) (la)

s = W2 = (m + M + CE)2 (cm), (lb)

andthe momentumtransfert,

t = —2K~(1— cos0) (cm), (2)

whereK and0 are the cm momentumand scatteringanglerespectively(lab quantitiesfor t arelessconvenient).The third variable,the“crossedmomentumtransfer” u, is not independent:

u=2m2-i-2M2—s—t. (3)

For furthernotation seefig. 1.Neglectingspin for the moment the lab scatteringamplitude,f’(E, ~)for positive energiesis

directly relatedto the differentialcrosssectionas follows

do’/dC~= f(E,~2, E >0 (lab) (4)

M.P. Locherand T. Mizutani, The useof analyticityin nuclearphysics 47

l~

IdI (dl

Fig. 1. Generaldiagramfor theelasticscatteringof two particleswith massesm (neutron)andM (deuteron).

and the samerelation holdsin the cm systemwith cm quantitieson bothsides.We shalldenotethe cm amplitudeby Cf(CE 0).

We know thatthe nd systemhasaboundstate,the triton. Is the crosssectionin the scatteringregion,the so-calledphysicalregion,at all relatedto the existenceof aboundstate?The bindingof thetwo particlesoccursat negativeenergyE~,in the“unphysical”region.However,the scatteringamplitude in the physicalregion E ~ m getsacontribution,the pole term

ftriton pole = r/(E1 — E) (lab) (5)

forming part of the directly observablecross section.The pole position is determinedby the

bindingenergyof the neutronin the triton [50],

(6)

suchthat

E~= (1 + m/M)e = —9.39 MeV or CE = —e. (7)

Generalisationsto other particlesare obvious.In the invariant notation of eq. (1) the poletermis at s”

2 = m1. The nonrelativisticconnectionis

s — m~= (W + mj(W — m~)~ 2mt(CE + ~)~ 2M(E — E~) (8)

sincefrom eq. (1):CE ~ ME/(M + m) = (~u/m)E, with ~ = mM/(m + M). (9)

NotealsothatCf = (M/\/~)f~ (~u/m)f, (10)

wherethe last approximationis valid nonrelativistically.The poletermis convenientlyvisualisedby agraph,seefig. 2. No perturbationtheoryis implied,

however!Fig. 2 is a Feynmangraph in its kinematical aspectsonly. In particular the residue ris a renormalisedquantity*, directly relatedto its observablecontributionin the scatteringampli-tude.Of course,if an explicit lagrangianis an adequatedescriptionthe residueis expressibleintermsof a triton—deuteron—neutroncoupling constant,g~0.For that reasonthe residuer itself

Fig. 2. The triton pole in nd scatteringasanexamplefor a direct (or s-channel)intermediatestate.* Theresidueis energyindependantby definition,

48 M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics

is sometimescalled tdn couplingconstant.The relation is discussedby, e.g., Ericsonand Locher

[50].The connectionof a boundstateto a pole in the scatteringamplitude is valid in field theory.

In the following, however,we shall discussit in Schrodingertheory andsimultaneouslyobtainan interpretationof the pole residuein termsof a property of the nuclearwave function, namelyits asymptotic(large r) normalisation.This interpretationis of courselimited to a situationwherethe conceptsof wave functionsand potentialsmake sensewhereasthe primary definition of r

as the boundstatepoleresidueis independentof that.

2.1. Poles,coupling constantsand asymptoticnormalisations

Let

q~(r)= R(r) ~m(~) = Y1~fr) (11)

be a solution of theSchrodingerequation.The radial part is

1 d~ 1(1+1)(—~~ + V(r) + 2jir

2 — CE)Yi(r) = 0, (12)

with V beinga potential of short rangeR, ji denotingthe reducedmassand CE the total kineticcm energy.For an s-wave bound-statesolution we have CE = —c wherec is given in eq. (6) forthetriton. Theradial part outsidethe rangeof V obeys

1 d~— cy

0(r) = 0, r > R. (13)21u dr

For large r the bound-statesolution of the Schrodingerequationlocalised in spaceis therefore4(r)ljmr>>R = NY

0e~r/r = ~ CY0e”7r, (14)

where ic2 = 2iw. (The asymptoticwave function is normalisedto onefor C = 1, a limit which is

never realised,however,cf. eq. (A-b’) of appendixA.) The asymptoticnormalisationconstantN (or C) reflects the dynamicsat short distancesthrough overall normalisation(henceC � 1).Borrowing the known relation of wave function to scatteringamplitude,see, e.g., ref. [50], weobtain for the bound-statecontributionto the cm amplitude

Cfiriton = — (~/2x)CE~C ~ + CE) <K, ~>2 / ~ + CE) (15)

which is apoleof the scatteringamplitudehavingthe residue

p = [M/(m + M)]2 r = (R/21r) lirn (c + CE)2 <K, ~ (16)

[The relationto the residuer of the lab amplitude(5) follows from (9) and(10).]The Fouriertransformof j(r), eq. (14), is:

<K, ~>= J~(r)e1Krd3r= NY04m/(K

2+ K2) = N(~4~/2~)(CE+ c)t (17)

Becauseof(16) the only contributionto the residuecomesfrom the solutionsingularat CE = —t;

M.P. Locherand T. Mizutani, The useof analyticityin nuclearphysics 49

andwe geta oneto one relationbetweenthe pole residuep andthe asymptoticnormalisationofthe neutronwave wavefunction in the triton. Collectingthe factorswe have(without spin) for thecm residue

p = (ji/2ir)N2 41r/(2JL)2 = N2(2~u)’’. (18a)

For generalI onefinds:

p = (— 1)’(21 + 1)N2(2~z)”’. (18b)

The signchangein eq. (18b) is very important. It is relatedto the knownparity (— 1)’ of thewavefunction ~, see,e.g.,ref. [50]. It is actuallythe reasonthat the potentialscorrespondingto particleexchangein the driving channelarerepulsivefor odd 1 (photonexchangefor equalcharges)andattractivefor even1 (gravitation).By meansof the forwarddispersionrelationsto be discussedinsection3 the sign of a residuecan be determinedexperimentally.

Other diagramscan be drawn for nd scatteringwhich correspondto particleexchange.Thesimplestone is protonexchange,fig. 3. Our customaryway of readingdiagram3 is from left toright correspondingto nd —+ nd elastic scattering(direct or s-channel).But we can readit alsofrom the bottom to the top correspondingto tid —~ dti scattering(exchangeor u-channel).Thisis the channelwherethe proton appearsasa direct intermediatestate.Just as therewas a polein s for thetriton, cf. eq. (8), thereis nowapoleat theinvariantenergyu112 = mp(sandu exchangetheir role for an exchangepole)

cc 1/(u — mt). (19)

Fig. 3. The protonexchangepolediagramin nd scatteringservingasanexamplefor anexchangechannel(or u-channel)intermediatestate.

Consultingeqs. (1), (2) and (3) whereu is expressedin the direct-channelvariableswehave

u — m~= (M — rn)2 — 2’~E(m+ M) — CE2 + 2K2(1 — cos 0) — m~ (cm) (20)

~ —2me~— 2K2(5/4 + cos0), with cd = rn + rnp — M.

One seesthat the position of the exchangepole(19) dependsboth on direct-channelenergyandangle. For the forwardamplitude the pole is locatedat, cf. eqs. (1), (3) and(9),

CE = — [(M — m)/(M + m)]sd = (~u/m)E~. (20a)

In the variablez = cos0, the pole occursat

z~,= — [(m2 + M2) (2mM) 1 + ic~/K2], ,c~= med (cm), (20b)

wheres~,K2 4 m2 hasbeenneglected.Generalisationsto other exchangeprocessesare againobvious(cf. appendicesA to C).

At this point we want to stressthat the exchangepole(19) is only namedafter the exchange

50 M.P. Locherand T. Mizutani. The useofanalvticitv in nuclearphysics

channeltid -+ dti whereit is a direct intermediatestate.This doesby no meansimply that thecontributionof diagram3 is arelativistic effect in thes-channelnd —+ nd! After all exchangepolesarisenaturally in properlyantisymmetriseddirect-reactiontheory,see appendixB.

It is importantto note that the dpn vertex occurringin the exchangediagram3 appearsalsoin otherreactions.The simplestoneis the direct deuteronpolein triplet np scattering.Fig. 4 leadsto a pole in the np relativeenergyand formula (15) applies with obvious modifications. ThenormalisationNdPfl can be determinedvery accurately from low-energy triplet np scatteringin the effectiverangeapproximation.

Fig. 4. The deuteron as a direct channel pole in triplet np Fig. 5. An examplefor a strippingreaction:the Born approxi-scattering. mation in i

2C(d,p)13C* (3.09 MeV) due to neutronexchange(from ref. [3]).

Another exampleis the stripping reaction 12C(d,p)13C* which has beendiscussedvery earlyin our contextby Amado [3]. The correspondingdiagram,fig. 5, leadsto a neutronpole whoseresidueis proportional to the normalisationsNdPflNI3C*12cfl. The secondfactor is related,to theasymptoticnormalisationand spectroscopicfactor of 13C* by simple numericalconstants(seeappendicesA and B). The stripping reaction hasa pole in cbs0 for fixed energy,its position isgiven in eq. (A-iS). The information obtainedfrom an extrapolationto the pole is discussedinsection4.1 below.

2.2. Multiparticle singularities

If there were no other singularitiesthan direct and exchangepoles in the unphysicalregion(E <0, t > 0 or cos0~> 1) onecould simply look for their contributionsto the crosssectionin the physicalregion.In afewcases(like np elasticscattering,fig. 4) the poledominatessostrongly,that suchastraightforwardprocedureis possible.Generally,the signal from the pole hasto fighttwo differentkinds of backgroundin the regionwherethe measurementsaremade.The first oneare the contributions from the physicalregion itself. They can be subtractedout by meansofdispersionrelationsin the energyvariable for the specialcaseof the forwa

1d amplitude,seesec-tion 3.1. Thesecondkind of backgroundarethe contributionsfrom cutsin the unphysicalregion.Thosecomefrom multiparticleexchange.They canbe very strongandcloseto the poleswhich weareinterestedin. The situationin thenuclearcaseis outlined in the following subsections.

2.2.1. Formfactorsand anomalouscuts

It is clear that the singularity structureof nuclearsystemsmustreflect the fact thatnucleiareweakly bound (e ~ m). As far as possiblewe discussthe main featureson the simple nd case.The scatteringfrom a deuteronshowsa steeplyfalling cross sectionin the momentumtransfervariable,governedby a form factor in conventionalterminology.The characteristicslope is thedeuterondimension K

1, where K2 = me = 2/1~~,C= (46 MeV/c)2. We have to find the graphswhich havea rangeof order K ~. In dispersiontheory the rangeof asingularity is definedas the

M.P. Locherand T. Mizutani,The useof analyticity in nuclearph3’sics 51

inverseof its typical distancefrom the physicalregion. In fact we know alreadythat the protonexchangepole, fig. 3, is such agraph.The forwardamplitudehasapole at CE = —e/3 andthesingularityin cos0 is given in eq. (20b). Thedistanceis againcontrolledby m~= ic~.Moreconven-tionally the rangeof thisgraphcanbedefinedby consideringthe Fouriertransform*of the protonexchangepole [22 + 2K2(1 + cos0)] 1, where 22 = 2rne + K2/2. This leads to an exchangepotential I’~ e~’/r where the range parameteris approximately 22 ~ 2me = 2ic~.Weakbindingdeterminesthereforethe rangeof proton exchangein any formulation.

If we iterate the proton exchangewe get the triangulargraph of fig. 6. This is the dominantt-channelsingularitywe werelooking for. The natureof the two-nucleoninteractionneednot bespecified.As the two nucleonsin the intermediatestatecanhaverelativemomentum,the diagrampresentedin fig. 6 leadsnot to a pole but to asingularity spreadout overa region in t, abranch

-channel

s-channel:III:I::E~E~::IIIII:IFig. 6. The triangulardiagramin ndscatteringleadingto thedominantsingularity in themomentumtransferort-channel.The nuclearpart correspondsto thedeuteronform factor(anomalouscut), seealsoappendixD.

cut. We expect,in this case,that the branchpoint should be locatedatadistancecorrespondingto the replacementKd —~ 2Kd or ta 8ic~in the invariant momentumtransfervariable. Such abranchcut is called “anomalous”(subscripta) but wehaveseenthat it arisesnaturally from weakbinding. Our intuitive estimatefor ta is almostright. Applying the standardLandaurules [68],to determinethe singularitiesof the diagramgiven in fig. 6, cf. ref. [50] for details,oneobtains

ta = 16ic~~ 1.7 m~, ,c~= mc. (21)

The range of this diagram is therefore(ta)~”2 = (4lcdY”. As t < 0 for physicalnd scattering,cf. eq. (2), the branchpoint (21) is very close to the s-channelphysical region andgovernsthenuclearangulardistribution. In fact the branchpoint (21) is the closestsingularity of t-channelnaturein this example.The next onecomesfrom 27t exchange,fig. 7, leadingto a branchpointat t

2,, = 4m~> ta. The situationin the t-planeis summarisedin fig. 8. It is typicalfor anynuclear

2~~~/ \ ~ s-channel criomalous cut (form bztor}/ \ scatterrig ( 2i~cut

i~~T-~ S1UMNMflNNI 1.7rn~ 4m~

Fig. 7. Theexchangeof two pions in the momentumtransfer Fig. 8. The singularities of the nd scattering amplitude inor t-channelfor nd scattering.(Becauseof isospinconservation the complex t-plane.The plane is cut to the right starting atthe intermediatetwo nucleonsmust have T = 1.) This is the theanomalousthresholdat t, = 1.7 m,~.This situationis typicalsecond nearest singularity but it is much weaker than the for nuclearscattering.i-triangle offig. 6. (Onepion exchangeis forbidden by isospin.)

* with respectto theexchange-channelmomentumtransfer(k + k’)2 = 2K2 (1 + cos6).

57 34.?. Locherand T. Mizutani. Theuseof analyticity in nuclearphysics

scattering.We stressagainthat the closenessof the branchpoint ta andits veryexistenceis exclu-sively dueto weak bindinge 4 m.

Evidently there are many more multiparticle exchangegraphs than those of figs. 6 and 7.A furtherexampleis given in fig. 9. We shallcall it the trianglegraphin the exchangeor u-channel.

Fig. 9. Triangulardiagramdue to (pit0) exchangein nd scattering.The correspondingsingularity in theexchangeor u-channel is

alsoanomalousbut weak.

The weakly bounddpn vertex occursonly once.The “anomaly” is thereforeonly weak and thecorrespondingbranchcut not soclose to physicalnd scatteringas the t-channeltriangle. In factwe can obtain the branchpoint for the (pit°)exchangeby the simplesubstitution Kd —+ lCd + m~in eq. (20a) and(20b) for the protonexchangepole. In thisway oneobtains

E~~0= — (lCd + m~o)

2/2m= — 17.3 MeV (lab) (22a)

for thebranchpoint in theenergyvariableof the forwardnd amplitude.Eq. (22a)andthecorres-pondingexpressionfor theangularbranchpoint

= — [(m2.+ M2) (2mM) ‘ + (,~+ m~o)2/K2] (22b)

remainaccuratewhenderivedfrom Landaurules.The u-channeltriangleis not only relatively far (cf. fig. 13 below) but it is alsoweak. To show

this we haveestimatedthe contributionsfrom the t-triangle and from the u-triangleassumingconstant,spin-independentvertex functions, i.e., we calculatethe effective long-rangepart. Therelevantformulae are collected in appendix D. For the ratio of amplitudesor T-matricesweobtainfrom (D-1) and(D-2)

R = T/T V/~NNNd ~ ~(K, 0), (23)

Nd TNN..NN

where

~(K, 0) = A—1 arcsin[A(~2 + A2)— 12] ( f)’2arcsin [—t( —f + l6K~Li]1:2

with f from (2), A2 = K2(5/4 + cos 0) and ~ = lCd + m,~.Observethat ~ is a slowly varyingtrigonometricfunctionof order one,equalto 4Kd(icd + m~) 1 at threshold.For E~= 6 MeV (lab)we seefrom table5 in the appendixD that R ~ 1/20 with no significantdependenceon scatteringangle. The ratio R is smallbecausethe pion—nucleoncoupling constantis relatively weak (firstfactor) andbecausethe it-productionamplitude is small comparedto nucleonelastic scattering(secondfactor). Sincewe tookthevalueat the 33 resonancefor the off-shell it-productionamplitude7,,~ (6 MeV) this estimateis conservativeand the u-channeltrianglecan safely be neglected.

Anomalousor nuclear structurecutsare thereforeimportantin the t-channelonly. They area dominant featurefor any angularextrapolation.Their contributionto the forward amplitudeis large but energy independentby definition. Since in all applicationsthe forward amplitudes

M.P. LocherandT. Mizutani, The useofanalyticity in nuclearphysics 53

are normalizedto experiment(subtractionconstant),anomalouscuts or nuclearform factorsappearnowhereexplicitly in energyextrapolationsat fixed angle. We concludethis section byremarkingthat the situationfor the deuteronwith respectto t-channelandu-channelanomalouscutsis not special.With the appropriatechangesin massesandbindingenergiesour resultscanbe carriedover to other light nucleiandremainqualitativelyunchanged.

2.2.2. Further multinucleonsingularitiesWe havediscussedbriefly all the relevant singularitieswhich determinenon relativistic nd

scattering.However, our model reactionis somewhatspecialas the numberof nucleonsin theexchangeor u-channelis limited to one. For the elastic scatteringof anucleonon a target withatomicnumberA, the exchangeof (A — 1) baryonsbecomespossible,fig. 10. The (A — 1) systemmaybeboundleadingto an exchangepole,or unboundleadingto an exchangecut. The positionof the exchangepole is given by eq. (20) with the replacementsm —~ mN, M —~ mA, mp m,4_,

andc = ~ + mAl — m,~.Similarly the branchpoint for the continuumexchangeobtainsfromthe sameformulawith mAl beingthemassof the unboundsystematzerokineticenergy.Whereasthe residueof the bound(A — 1) systemis often the quantityof interest,the unboundexchangeforms an unwantedbackground.Contrary to the u-channeltriangle arising from the exchangeof an extrapion, the unboundmultinucleonexchangeis close(asillustratedin fig. ii for nt andn

4Hescattering).Furthermoreit is generallystrong,asis known from theaccumulatedevidence.The presenceof thesecutsformsaseverelimitation for theextractionof individualnuclearcouplingconstantsboth for angular-variableandfor energy-variableextrapolations.

j3.13~4.2f2O

Fig. 10. Generaldiagramfor theexchangeof A — 1 nucleons Fig. 11. Explicit examplesof exchange,or u-channelsingulari-in theexchangeoru-channel,The (A — 1) systemmaybebound ties for elastic fit and n—4He scattering.The numbersin theleadingto anexchangepoleorunbound,leadingto anexchange graphsare the cm energies(MeY) of the correspondingpolescut in theNA elasticscatteringamplitude, and branchpoints for the forward amplitude.Similarly, poles

andcutsareinducedin thescatteringangle.Theirlocationin thecos0-planedepei~dson theenergy.

Of coursein a generalnuclearreactiondirect channelboundstatesandresonancesmay occurbelow thresholdas well, seefig. 12. The triton was an examplefor the nd case.In the caseof

Fig. 12. Generalgraph describingdirect channelintermediatestatesfor elastic nucleon-nucleusscattering.Particle stablestatesofthe(A + 1) systembelowtheelasticscatteringthresholdlead to polesof theamplitude.Above threshold we have theunitarity cut[including resonancesin the (A + 1) system].

54 M.P. Locher and T. Mizutani. The use0/analyticity in nuclearphysics

elastic nucleon—nucleusscatteringthe pole position is given by eq. (6) with m —* rnN, M —s rn.4,

m1 —* m4~1 andhencee = m~+ mA — m.1~1.The pole is thenatCE = —e = A(A + i)’E4~1. (24)

Thesedirect-channelpoles can introduceasubstantialamount of further structureinto the un-physicalregion as hasbeendiscussedfor the caseof n’

2C scatteringalreadyby, e.g.,EricsonandLocher [50].

There are of coursealsodirect-channelresonancescorrespondingto fig. 12 above the elasticscatteringthreshold.In the caseof forwarddispersionrelations(seebelow) theseresonancesaresubtractedout when forming the dispersionintegralover the physicalregionanddo not interferewith the extrapolation.Sufficiently precisedataare availablefor A ~ 14 at present.For angularpole extrapolationsin the differential cross section,resonancescan form a severebackgroundmaskingthe signal from the unphysicalregion.

3. Analyticextrapolations

We aregoing to makeextrapolationsto the poles,therebyexploiting the analyticity off(E, ~).

In the first subsectionwe shalldeal with the forwardamplitudef(E,~9= 0) which is independentof the more complicatedsingularitiesin the angularvariable t, since t = const. = 0. Extrapola-tions in the angularvariablefollow in subsection3.3.

3.1. Forward dispersionrelations (FDR) and the discrepancyfunction

The analyticstructureof the forwardlab amplitude*f(E) for nd scatteringis shownin fig. 13.The poles and the exchangecut have beendiscussedin section2. The cutsmarked0nd and a,~d

originatefrom intermediatestatesin the physicalregionof nd and tid scattering,respectively(seefig. 14). Thesecuts are completelydeterminedby the optical theoremwhich follows from theunitarity of the S matrix andrelatestheimaginarypart of the forwardelasticscatteringamplitudeto the total cross sectionin the physicalregion of nd and tid scattering,respectively,

(p/4ir)a(w) = Imf(w), w > m, (25a)

(p/4ir)ö’(ai) = Imf(~), a > m, p2 = — m2 and öi = —w, (25b)

____n P,,,,~_

triton O’nd E _XF~X~__x~xT~-173 -9./. -11 MeV d d d d

Fig. 13. The singularity structureof the forward nd amplitude Fig. 14. Graphs symbolising the unitarity relation )opticalin the complexenergyplane. The position of thedirect triton theorem) for nd and lid scattering.The lines betweenthe cutpoleandtheprotonexchangepole in thelabenergyE is indicat- circles stand for physical )on-shell) intermediatestates.Theed. Thepit°cut correspondsto theweak anomaloustriangleof numberof particles is only limited by selectionrules and thefig. 9. The imaginary partsin thephysicalregionsareexpressible availableenergy.throughnd and Pd total crosssectionsby meansof the opticaltheorem.

~ The cm amplitudehasundesiredkinematicalbranchcuts in ihe relativistic context.

M.P. Locherand T. Mizutani,The useof analyticityin nuclearphysics 55

with w = m + E, andp = lab momentum.Barredand unbarredquantitiesrefer to physicaltidandphysicalnd scattering,respectively.

Eq. (25b) for the antiparticleamplitude is not yet in auseful form for the dispersionrelation.It hasto be convertedby meansof thecrossingrelation:

f(w) = [f(_w)]*, with ä = —w. (26)

To makeeq. (26) plausiblewe note thatanygraph,in particularthe protonexchangeof fig. 3 orthe triton pole of fig. 2, leads to acontribution for both the amplitudef (nd scattering)andf(tid scattering).All onehasto do formally, is to invert the neutronfour-vectorin any oneof theexpressions.This substitutionrule leadsto the crossingrelationgraph by graph.In field theory,however,one proves that relation (26) is valid independentof perturbationtheory. Note thateq. (26) is not a crossingsymmetryas it relatesphysically different amplitudesto each other*.Using (26) we can expressIm f for negativeenergiesin terms of measurableantinucleoncrosssections:

Imf(w) = —p(4mY~(—w), w < —m. (27)

The equationsaboveare valid for any elastic scattering.For most of the applicationsdiscussedfurtherbeloweq. (27) canbe usedto positively showthat contributionsfrom co < —m arenegli-gible.

We are now ready to introducedispersionrelations. We needone further assumption:f(E)hasno othersingularitiesthancutsandpoleson the realaxis, as is shownin fig. 13 for our exampleof nd scattering(we recall that thishasbeenprovenfor, e.g., the irN amplitude).WecanthenapplyCauchy’sformula for a largesemicircleC as in fig. 15:

f(w + ic) = ~ , ~ . dw’. (28)—09—iC

C

1mw

Re Ca

Fig. 15. The path for theCauchyintegral in thecomplexenergyplaneneededfor thederivationof dispersionrelations,

If the contribution from the upper border of the semicirclevanishesin the limit of infinite radiusone obtainsthe dispersionrelation

Ref(w) = P JImf(w’)dw’ (29)

* Symmetricand antisymmetricunder crossingare the linear combinationsf(~ (w) = 4[ f (co) ±f(co)] obeyingf(±)(.~) =

±[f~(w)]*,

56 M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics

Similar relationsfor fixed energycanbe written in theangularvariablest or cos0. Slightlymodifiedrelations,“subtracted”relations,hold for the caseof badconvergenceat infinity:

Ref(w) = Re f(w~)+ (w — ~ J ~~w’)~• (30)it — ((.0 — (Oo) (w — w)

Eq. (30) obtainsby applyingCauchy’sformulato the moreconvergentexpression[1(w) — f(w0)]x (w — ~~)_1 or by formally subtractingf(w0) from (29). The price for improvedconvergence

is the knowledgeof Ref(~~)at somearbitraryenergywo (usuallythe elasticscatteringthreshold).In many casesthe experimentalnormalization of the amplitude is a desirablefeature even ifconvergencein the form (29) is expected.The subtractionprocedurecan be repeatedif necessary.Eq. (29), or (30), form the basisof manyapplications.The dispersionrelationsallow to calculatethe real partof theamplitudefrom analyticity if only the spectrumIm f(w) is givenfor all energies.In the discussionfollowing fig. 13 we haveseenthat Imf is measurablefor w~> m, see(25) to (27).For w~< rn, Imf(w) is not obtainabledirectly from a cross-sectionmeasurement.We can,however,turn the questionaroundandlearnsomethingabout the unphysicalspectrumfrom thedispersionrelation. In facteqs. (29) or (30) areamuchsuperiorframeworkfor poleextrapolationsthansimplepoledominance.

The practicalmethodis the so-calleddiscrepancyfunction method. It projectsout the contri-butionsfrom theunphysicalregion (polesand cuts).The procedureconsistsin the following threesteps:

(1) Calculatethe imaginarypartfor w > m from total crosssectionsandcomputethe truncatedintegral

ReJ~hYS(w)= PJ1mf(w~)do~ (31)

(Contributionsfrom w < —m aresmallandcanbe addedif needed.)(2) CalculateRef(w) for a rangem < w <Wmax from phaseshifts or Coulomb interference,

i.e., independentlyof dispersionrelations.(3) Formthe discrepancyfunction

A(w) = Re j’(w) — Re~ (32)

The function A is completelycalculablefrom experimentalinput. It is, by its very constructionthe total contribution from the unphysicalregion in the dispersionrelation (29). (In the caseofnd scatteringthesearethe protonand triton polesand the weak pit°cut of fig. 13.) The advantageof working with FDR and the discrepancymethodis clearly the model-independentisolationofthe unphysicalregion. Ideally, the contributions from the physicalregion cancelcompletelyinthe discrepancyfunction within the accuracyof the experimentalinput. This doesof coursenotyet solve the problemof isolatingacertainpolecontributionfrom other,competingsingularitiesin the unphysicalregion.However,if a singlepoledoesdominatethe unphysicalregion(which isfar less exactingthanoutright pole dominance)this can be testeddirectly by plotting

1/A = r1(Epoie — E), (33)

cf. eq. (5). The plot must be linear andextrapolateto the correctpoleposition Epote. The residue

M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics 57

is thengiven by the slope of (33). Suchapole extrapolationis shownin section4, fig. 24, for anactualexample.

The problemof isolatingcompetingsingularitieswithin the unphysicalregion remains.Com-peting cutscan be deemphasisedto someextent,by the conformal mappingmethoddescribedin section3.4. It is anon-rigorousmethodjustified ultimately by its success.

3.2. CoulombmodUlcations

So far we haveonly consideredneutralparticlescatteringby a short-rangeinteraction.Sincethere are no neutralnucleiwe haveto considerelectromagneticeffects (for a generaldiscussionseeHamiltonetal. [58]). Themethodsdiscussedbelowhavebeenusedboth for energyandangularextrapolations.Themodificationsdueto thelong-rangeelectromagneticinteractionareparticular-ly importantnearforward directionandnearthreshold.The leadingmodificationsareshowninthe graphsof fig. 16.

x. >(><

Fig. 16. Leadingelectromagneticsingularities:one photonexchangeand thedistortionin theinitial and final state.The doublelinedenotesthenucleusA.

Variousprescriptionshavebeenusedto define amodified amplitudewhich is approximatelyfree of electromagneticsingularities.The minimal prescriptionfor elastic scatteringis [105]:

C!(CE 0) = [Cf(CE 0) — Cf0(CE 0)] (34)

o(1)

whereC0 is thes waveCoulombpenetrationfactor(distinguishC0 from the asymptoticnormalisa-

tion C!),C~(~)= e”~

2F(1+ i~2= 2iu~/[exp(2in~)— 1], (35)

with

= ZaZA e2 ji5~]K= c/K, K

2 = 2I1aAE, (36)

andCf0 is the pureCoulombamplitude.The amplitude Cf is the full physicalscatteringamplitude.

For the caseof a reactionprocessonedefinesaccordingly(see,e.g., refs. [82, 101])

C(CE 0) — Cf(CE 0) (37

— e~2F(1— u7~)e~”2F(1+ i~~)’

where,j~and?Jf arethe appropriateCoulombparametersfor the initial andfinal channels,respect-ively.

By constructionthe modified amplitudef is free from the essentialsingularity at CE = 0, hasno polesat CE = — c~2(21L

5An2)~anymore(n = 1,2, 3,. . .) andis not singularat0 = 0. It is expected

to obeya normaldispersionrelation,similar to unchargedparticles.Near adirect-channelpolethe Coulombmodifiedelastic amplitudebehavesas [105]

58 M.P.Locherand T. Mizutani, The useof analyticity in nuclearphysics

C!(CE 0) ~2Cfpoie(CE, 0), (38)

with

= ~/K, K2 = 2/AaAL

and ‘fpoie is the nuclearpole contribution to the amplitude ‘f in eq. (34). The asymptoticboundstatewavefunction for the chargedclustersaandA is not of the Yukawatype but hasthe formof aWhittaker function

lirn ~~r) = ~ C(n, Sl) W_~,1.4.112(2Kr)/r. (39)

Eq. (39) replacesthe definition of the asymptoticnormalisationC, eq. (A-b), in the chargedcase.The residuep of the amplitudeeq. (A-5) is obtainedby p —~ [‘(1 + y)

2p accordingly.In the caseof an exchangepole an additionalcomplicationarises.Insertion of a photon line

in the (u-channel)exchangegraph of fig. 3 leads to an electromagneticcut which startsdirectlyat the exchangepole, since the photonhasmass zero,see fig. 17. The modification (34) of theamplitudehasto begeneralisedto include this cut. Simpleexpressionsexist only for zeroorbitalangularmomentumof the boundstate.For detailsthe readeris referredto Ter-Martirosian[101]andMorinigo [82]. This treatmenthasbeenusedfor an angularpoleextrapolationin the p3Hecrosssectionby KisslingerandNichols [64]. The occurrenceof interferencetermsrequiresa fullset of phaseshiftsas input. The exchangecut modificationhasbeenappliedto the exchangepolesin forward dispersionrelationsby Bornandet al. [24]. The correctionis particularlylargefor aweaklyboundsystem(separationenergy~ Coulombenergy).It reducesthe residueof n-exchangein pd scatteringby abouta factor 2 (Bornandet a!. [24]) whereasthe triton residuein p4Hescatteringis reducedby about17~ (Plattner[89]).

Clearly, the list of graphsconsideredso far is far from complete.We haveavoidedintroducinggraphswherephoton lines connectdirectly to the particle producingthe pole, see fig. 18. Suchgraphslead to vertex correctionswhich could only be meaningfullydiscussedin the context ofthe correspondingradiativecorrectionsandthe appropriate renormalisationprocedures.Thisis impossiblewithout somemodeltheory for stronginteractions.In the absenceof suchaprogramthe nuclearvertices determinedby analyticextrapolationsfrom somemodified amplitudef willstill containsomeelectromagneticcontributions.

* and

Fig. 17. The exchangeof(A — I) nucleonsandanextraphoton Fig. 18. Typical vertexcorrections)one endof the photon linein the u-channel.This leadsto an electromagneticcut which connectsto an internal nucleonline).startsat thenuclearexchangesingularity.

The modification (34) and its generalisationare prescriptionsfor fixed energyE. Shifts in theenergypositionof nuclearpolesdueto the Coulombrepulsionof the constituentprotonscannotbe describedin sucha framework. An alternativeto eq. (34) is theapplicationof standardouterCoulombcorrections.One assumesthe interactionto be purely stronginside the region r < R

M.P. Locherand T. Mizutani, The useof analyticityin nuclearphysics 59

and purely Coulomboutside.The wave function correspondingto the observedphaseshifts isthenmatchedto the free-spacesolutionatr = R. To the extent that the inner Coulomb correctionsfor r < R can be neglectedone obtains in this way a purely strong interaction amplitude. Poleenergiesare shifted in this recipe. An interestingnuclearexample,wherethis methodhas beenapplied [52], is ~ scattering.Elasticc~scatteringhasthe 8Be groundstateasa resonancebarelyabove threshold. Removing the Coulomb repulsionby meansof outer Coulomb corrections,~& becomesanuclearboundstate.For the resultsof this FDR analysiswe refer to section4.1.

The angularextrapolationof the stripping reaction d(d,n)3He by Borbély [20] usesvertexcorrections[44] of thetypein fig. 18 exclusively.The neglect,ofCoulombdistortion in theentrancechannel(fig. 16) is not warranted.

3.3. Angularcross sectionextrapolations

Analytic propertiesin the angularvariablescos0 or t are less readily exploited. Dispersionrelationsin t arefar moredifficult from a practicalpoint of view alreadyin particlephysics,sincethe opticaltheorem,whichplayedacrucialrole in determiningtheinput for theforwarddispersionrelations,hasno simpleanaloguefor scatteringanglesdifferentfrom zero.Moreover,in thenuclearcase,the spectrumin the t and u-channel is more complicatedas discussedin subsection2.2.It is thereforenot surprising thatnuclearapplicationsare limited to the simplestpossiblecases,namely to partial wave dispersionrelationsin itd, nd and pd elastic scattering,see SchiffandTranThanh Van [95], Avishai et al. [7], andEbenhöhet al. [46]. The aim of theseapplicationsis the calculationof the amplitude in the physicalregion assumingthe driving termsin theun-physicalregion to be known. Since weareconcentratingon analyticextrapolationsin this reportwe shallnot follow this line further.

Due to the difficulties mentioned, no nuclear extrapolations in combination with dispersionrelations in the angularvariable havebeenmade.There are however,numerousapplicationswherepole extrapolationsin cos0 are attempted directly in the differential cross section.Thecrosssectionas the absolutesquareof theamplitudeis not an analyticfunction. It is likely, how-ever,thatIm f(w) itself is ananalyticfunction, sincethiscanbeshownin the field theoreticcontextfor favorablecases.Im f(w) is analyticin the largeLehmannellipse,whereasf(w)itself is analyticin the small Lehmannellipse, see,e.g., ref. [45]. We shall thereforeassumethat analyticextra-polationsin the crosssectionmakesensein the nuclearcasealso. Fromwhat weknow we cannothopethat simpleangularpole extrapolationsto, e.g.,the nucleonexchangepolein nd scattering,canbe veryreliable,sincethereis the strongbackgroundfrom the form factor or t-channelano-malouscut (section 2.2.1) and moreover from multinucleon exchangein the u-channelif thetargetis heavierthanthe deuteron(section2.2.2). Morerecentpoleextrapolationsthereforemakebetteruseof theassumedanalyticity in cos0by usingaconformallymappedvariable.Polynomialexpansionsin the new variable have better convergenceproperties,at least mathematically.This conformal mappingtechniqueis explainedin the next subsection.The practicalbenefits(andtheir limitations) are discussedin the contextof the extrapolationerror of the resultsin theintroduction of section4.

60 M.P. Locherand T. Mizutani, The useof analyticitv in nuclearphysics

3.4. Conformalmapping

The conformalmappingtechniquehasbeenintroducedinto particlephysicsin the earlysixties(Ciulli andFischer[30], Frazer[53], Atkinson [5]). The conventionalenergyor angularvariablesare mappedconformallyfirst andexpansionsmadein the new variables.For a reviewof methodandresultsin particlephysicsseeref. [31].

To illustrate the principle we borrow an instructive examplefrom Ciubli and Fischer [30].It showsthatmappingcan bea necessityto achieveformal convergencein the first place.Assumethat pp scatteringhadno other singularity than it° exchange(seefig. 19). Accordingto our rulesthe amplitudewill haveapoleat t = m~or, usingeq. (2), at

cos0~= 1 + rn~/2K2 r. (40)

p~—~-p

p~p

Fig. 19. ir°exchangein pp scattering.

Becauseof the identity of the two protonsasymmetrisedamplitudemust be used.Disregardingnormalisationthis is

A(z) = ~ + z + ~ — z = i2 z2’ (41)

wherez = cos0 andthe physical region is 0 <z2 < 1. Considernow a Taylor series around= 1. It will convergein the circle of fig. 20 touchingthe pole at z2 =

This circle vanishesfor E —÷ ~, seeeq. (40). A Taylor seriesaroundz2 = 1 will thereforedivergeat z2 = 0 beyonda certaincritical energy(which is E~b~ 50 MeV in our example).

A simple conformalmappingremediesthis situation:

w(z~)=22l1Z z

2~ (42)

It mapsthe halfplaneto the left of the shadedline in fig. 20 into the unit circle of fig. 21.

Irnz2z plane

w- plane

K: ________ I7~.~ Re z2 -1~ 0 [2r2-1]~1(z2=r2lV~ (z2*1) (z2*0l~

Fig. 20. The radiusof convergencefor a Taylor seriesaround Fig. 21. The situation of fig. 20 after the conformal mappingz2 = 1 (where z = cos0). The point z2 = is theposition of z2 —. 0(z2), eq. (42). The numbersin parenthesesarethecorrcs-the it0 exchangepole in pp scattering. pondingpoints in thez2-plane.

M.P.Locherand T.Mizutani, The useofanalyticityin nuclearphysics 61

The mapping(42) achievestwo things:(a) the Taylor seriesaroundw = 0 in thenew variableconvergesindependentof energyin the

wholephysicalregion,and(b) the convergencein the newvariableis faster.

Table 1 belowillustratesbothproperties.In thefirst line the exacts-waveprojection is given fortwo energies.Obviouslyif the functionalform is known,as in eq. (41) above,that is the endof it:the bestexpansionsystemof a function is still the function itself. Suppose,however,we did notknow eq. (41) but hadsomemeansof obtainingthe first few Taylor coefficients.The lower linesshowthe s-waveprojectionsfor an increasingnumberof terms in the Taylor seriesfor (41).

Table 1Exact and approximates-wave projectionsof A(z), eq. (41), from

Ciulli and Fischer[30]

EnergyE~b: 49.4 MeV 72 MeV

Exact: 1.386 2.10

Expansion in (z2 — 1) in w in (z2 — 1) in w

one term 1.87 1.87 4 4two terms 1.17 1.30 —0.16 1.4threeterms 1.49 1.403 5 2.4four terms 1.33 1.383 —2 1.99

The lower energyshowsconvergencein bothvariables,but the expansionin w convergesfaster.For the higherenergythe expansionin (z2 — 1) divergesbut convergesin w. The resultsfor thissimpleexampleare typical.

In a generalcaseand in amore physicalsituationwe are given a small numberof valuesof afunction (with errors!)overan interval ratherthanthe functionandsomederivativesin apoint.To be specific let usagainconsiderthe cos0 planefor nd scattering,seefig. 22. The functionF(z)is given on the experimentaldomain —1 <z < 1, whereit is supposedto be real [F(z) standsfor an amplitudeor eventhe differentialcrosssection].In suchasituationaconvenientmethodis to expandF(z) into a systemof orthogonalpolynomials.A practicalchoicearethe Legendrepolynomials which minimisethe L2 norm

J dz [F(z) — FN(z)]2, (43)

cosO plane

— — — ~ —

exchangeCut / exp. domain ~ onom.cut (form factor I______ 10 DIWXXXMMMI ~_____________

‘.1 -

pTr (‘~\.~ 1~1

p-pole -————

Fig. 22. The complexcos0-planefor nd scattering.The ellipse denotesthe domain of convergencefor a Legendreexpansionbasedon theexperimentalsegment(— 1. 1).

62 M.P. Locherand T. Mizutani, Theuseofanalvticity in nuclearphysics

where

FN(z) = v~Oa0P5(z). (44)

With this choiceand

F(z) = FN(z) + RN(z) (45)

wehavein particularthe usualstatisticalx2 interpretation.Different normsimply differentortho-

gonal polynomials,but the remarkswhich follow are valid more generally,cf. Ciulli et ab. [31].Supposewe want to extrapolateinto thez-plane;it is known that the expansion(44) converges

to F(z) for N —+ ~ only within an ellipse touchingthe closestsingularity andhaving foci at + Iand — 1. As in the previousexample, cf. fig. 20, the region of convergence(the ellipse of fig. 22)is thusmuchsmallerthan theregionof analyticity. As before,a conformal mappingprovesbene-ficial: it is possibleto mapthe whole region of anabyticity of fig. 22 into the inside of the ellipsein fig. 23 of the w-plane(isolatedpolesare removedby appropriatefactorsfirst).

cuts w — plane

~pit branch p-pale anom~branchpoint exp domain ~/poInt

ii H H H H ii IS IS I

IJJ~‘I-I-Cus

Fig. 23. The situationoffig. 22 after theconformalmapping w —* ii )cos0) suggestedby Cutkoskyand Deo.The region ofanalyticit\coincideswith theellipse of convergence.

Suchaconformalmapping

w = w(z) (46)

is givenexplicitly by CutkoskyandDeo [33]. It is notjustanimprovement,as ourpreviousexamplewould suggestalready,but it is in a well-definedsensethe optimalsolution to the problem.Onecan show [31, 33] that the remainderRN in eq. (45) is asymptotically(N —* cx~)smallest for themapping(46). Any mappingthatdoesnot transformthefull regionof analyticity into the domainof convergencein the new variableis (asymptotically)inferior.

Thereare of courseotherrealisationsof the optimal mappingthan fig. 23. A mathematicallyequivalentformulation has beengiven by Ciulli [28] wherethe region of analyticity is mappedinto annularrings. The experimentalpointsarethengiven on theinner circle of the ring which isperhapslessconvenientthanin the Cutkosky—Deomappingof fig. 23 wherethe endpointsof theexperimentaldomainare invariant under the transformation.

Although the mapping is optimal in the well-defined mathematicalsensedescribedabove,we must not forget thatanypracticalapplicationis limited to a few termsin the series(44) or inthe correspondingseriesin the new variablew. Apart from solving anyconvergenceproblemsofprinciple, the usefulnessof the mapping(46) can only be demonstratedby explicit applications

M.P. Locherand T. Mizutani, The useof analyticityin nuclearphysics 63

for which crosschecksexist. The experienceacquiredtoday shows that some accelerationofconvergenceanda reductionof backgrounddueto competingcutscanbe achievedfor asmallnumberof termsalready.An instructivecaseis againour standardnd example,seefig. 22. In thevariablez the form factorcut is closeandmakesextrapolationto the proton poleby Legendreexpansionformally impossibleaboveacertainenergy.In the new variable w the cutshavebeenmoved as “far away” as possibleand polynomial expansionbecomesformally justified. Sucha“conformal poleextrapolation”hasbeendoneby Kisslinger [62] (notethat thisis not adispersionrelationin cos0). The nd caseis particularlyfavorablesince the u-channelbackgroundconsistsof the weak(pit°)exchange(recall the resultsof section2.2.1). For 3He or 4Heasa targetthereisthe continuumcontributionfrom the exchangeof severalnucleons,fig. 11, which is both closeto theexchangepolesandstrong.(Seesection4.1 for adiscussionof the resultsin thoseandsimilarcases.)In such a situationthe suppressionof backgroundby conformal mapping aloneis not‘sufficient at all as hasbeenverified on modelamplitudesby LocherandMizutani [76], contraryto the claimsin the literature(comparethe error discussionin the introductionto section4.)

We concludewith one further comment.The asymptotic rate of convergenceof Legendreexpansions,eq.(44), is known.With someadditionalassumptionsonecanthereforetry to estimatethe (mapped)remainderRN of eq. (45). Explicit prescriptionshavebeengiven by CutkoskyandDeo [33]. Such systematicerrors due to truncation terms are usually neglectedaltogetherinstandardChew—Low—Goebelextrapolationswithout mapping. However, theseconsiderationsarenot of practicalimportanceso far. The main limitation of anyextrapolationin our experienceis dueto the statisticalnatureof the data.Theseerrorshavea tendencyto get out of control veryquickly when extrapolatingbeyondthe experimentalsupport, long before more sophisticatedcriteria start to matter. Analyticity alone cannot control propagationof extrapolationerrors.Stabilisationcanbe achievedin principle by imposingboundsin the regionof analyticity and/oron its boundary.We refer to the existingreview literature(Pi~ut[87, 88], Ciulli et al. [31]). Thepractical useof stabilising conditionsfor extrapolationsinto the domain of analyticity hasbeendiscussedby Ciulli andCiulli [29] andCaprini et a!. [26].

4. Surveyof results

Analytic extrapolationshavebeenperformedfor a variety of nuclearreactions.With nucleonsor light nuclei servingasprojectilestherangeof targetnucleiextendsto atomic numberA ~ 14.A detailedsummaryof the resulting few nucleoncoupling constantsis presentedin section4.1.In section4.2 the situationregardingpion nuclearcoupling constantsis reviewed.

The systematicreliability for the various types of extrapolationsis not the same.From thediscussionin section3.1 we expectthatextrapolationsusingthe discrepancyfunction A shouldbeparticularlyreliablesincetheuseof dispersionrelationsin theforwardamplitudeallowsto subtractout all contributionsfrom the physicalregion (A hasno right-handcut). As anillustrationwe showin fig. 24 the cleanextrapolationto the.6Li pole in dtx elasticscattering(from Plattneret a!. [90]).The extrapolationgoesright through the 6Li pole position. For an amplitudenormalised(sub-tracted)at threshold,E/A hasto be plottedinsteadof 1/A, eq. (33). Direct angularextrapolationsofthe differentialcrosssectionsufferfrom thepresenceof thet-channelcontributions(sections2.2.1and3.3). In bothcases,however,the nature,strengthanddistanceof competingsingularitiesmust

64 M.P. Locherand T. Mizutani, The useof analvticitv in nuclear physics

.6E/~([)

lMeV/fm) /

6~(~’~’ IEd IMeVI

Fig. 24. The inverse discrepancyfunctionplot E/A for daelastic scatteringderivedfrom FDR by Plattneret al. [90]. [& standsfortheCoulombcorrecteddiscrepancyfunction, cp. eq. (34).] The black dot denotesthe known

6Li-pole position.Typical errorsare In-

dicatedby error bars.

be discussedfor eachreactionindividually. This is essentialfor anymeaningfulerror assignmentto nuclearcouplingconstants.

In deducingpoleparametersby anymethodthefirst stepis an interpolationof experimentaldataover some physical region. The experimentalerror of differential cross sectionsand forwardamplitudesbeing typically 5 % in favorablecases,the interpolationerror, computedby the usualstatisticalcriteria, will be of similar order. Many papersquotedin section4.1 assignthis inter-polationerror to their results.A reasonableerrorestimate,however,mustincludethe extrapolationerror aswell. Without somestabibisingprinciples [26, 29] which mustbe imposeda priori in ourcontext, this is an ill-defined problem.Apart from theseaspectsof statisticsandstability, theindividual singularity structurementionedfurther aboveis of crucial importancefor the back-groundproblem.To geta feeling for realisticordersof magnitudeLocherandMizutani [76] havemadeextrapolationson artificial data for anumberof few-bodyreactionsfor which modernpoleextrapolationsexist (earliertestsarequotedin thatpapersee[43], e.g.).Both extrapolationsin thedifferentialcrosssectionand in thediscrepancyfunctionhavebeendiscussed.The testshavebeenmadewith andwithout mappingtechniques.Modelamplitudeshavingthe appropriatesingularitystructurewereusedto systematicallydiscussthe extrapolationerrorasafunctionof poledistance,polestrengthandstrengthof competingbackground.Spin hasbeentreatedin an effectiveway andonly neutralprojectileswereconsidered.It is, however,plausiblethatmorecomplicateddynamicsonly will increasethe extrapolationerror in general.The results can be summarisedas follows:For FDRs the extrapolationerror is small (about 3 %) only in favorablecaseslike nd scattering,wherethe protonexchangepole is well isolatedandthebackgroundis weak(cf. section2.2.1 andfig. 13). The ratio of backgrounddistanceto extrapolationdistanceis the most importantpara-meter.In typical few-bodyreactionslike nt or nc~elastic scattering(seefig. 11) this ratio is about4/3 and the resulting extrapolationerror about 30 %, unlessthe backgroundis parametrisedexplicitly and subtracted(as has beendone in most FDR applications).For straight angularextrapolationsin the differential crosssection theseerrorsare somewhatlarger,as is expected.They areabout5 % for proton exchangein nd scatteringand 50% for deuteronexchangein nt

M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics 65

scattering.The benefitof mappingtechniquesis sizableonly in thosecaseswherethe pole is wellisolated already in the conventionalvariables(nd case).No qualitative improvementresults ifthe backgroundis close(asin the nt case),see[76]. Sinceanumberof angularpoleextrapolationsreportedbelowrely exclusivelyon mappingin suchcritical cases,the error given in thesepapersmustbe drasticallyincreased.The errorsquotedabovehaveto be consideredas typical andcon-servative.We shallindicatefor the individual resultsin sections4.1 and4.2 whetherthe singularitystructureis favorableor not.

4.1. Couplingconstantsin few nucleonsystems

In this section we report the results from analytic pole extrapolationsin purely nucleonicsystems.The resultsfrom extrapolationsusingthediscrepancyfunctionmethod(FDR) andfromangularextrapolationsin mappedvariablesspanthe rangeA ~ 15. Somefurtherresults,obtainedfrom (implicit or explicit) polemodels,falling into this rangearealsoreported.

The primequantitydeterminedis the poleresiduer or p, eq. (5) and (16). Equivalentlywe shalluse G2, the squareof the vertex constant.In the spinlesscasethe relationfor an s-waveboundstatein thedirect or s-channelis just

p=JL(21r)’G2. (47)

The determinationof the residuep (or G2) by analyticextrapolationis direct andhighly model-independentas the requiredanalyticity does not dependon any particular assumptionsaboutthe dynamics.The methodis basicallyrelativistic andit doesnot requirethe existenceof anuclearpotentialor anyotherknowledgeof off-shell properties.

It is of coursedesirableto comparethe vertex constantsfrom analyticextrapolationschemeswith wavefunctioninformationfrom conventionalnuclearphysics.Thecrosslink is theasymptoticnormalisation[eq. (14)] of the nuclearwavefunction. This comparisonthenwill dependon theconceptof apotential,on the explicit radialdependenceof nuclearwavefunctionsandon amodelfor the correctinclusion of geometricfactors(dueto spin andantisymmetrisation)togetherwitha knowledgeof the cluster probability of the constituentclustersforming the bound state.Thecombination of the geometricfactorsand the cluster probability is the spectroscopicfactor S[seeappendixB, eq. (B-6)].

At this point it is worth mentioningthatmany of the commonly usednuclearwavefunctionshavea wrong asymptoticbehaviour.Consideringthe spinlessI = 0 casefor simplicity, the nuclearwave functionrelatesto the vertexfunction G(K2)by

G(K2) = Jd3r~1k r V(r) ~(r), (48)

whereV is the potentialactingbetweenthe clustersformingthe bound-stateclusterwavefunction~ andr is the interclusterdistance.The vertexconstantintroducedin (47) furtheraboveis just

G G( — K2). (49)

Using the Schrodingerequation,eq. (48) can be rewritten as

G(K2) = —[(K2 + K2)/2j~]Jd3re~~(r) = —(s + CE) <K, ~>. (50)

66 M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics

Clearly, only thosenuclearwavefunctions which haveasingularity at CE = —r lead to a non-vanishingresiduep = G2(2~t)~ In particular shell-modelwavefunctionsdo not havethe correctYukawatail, exp(— Kr)/r, correspondingto the bound-statemomentumK. ForsuchwavefunctionsoneextrapolatesG2(K2 = 0) � 0 to the pole G2(K2 = — K2) by somerecipe,an examplebeinggiven by Goldfarb et al. [57].

Systematiccomparisonsof coupling constantsfrom analyticity with nuclearwave functionsdeducedfrom basic NN interactions,from Faddeevcalculations,or empirically from electronscatteringexist in the literature. Keeping in mind the model characterof this material and thesystematicerrors inherentto anyanalyticextrapolationschemesthe agreementis generallyverygood.We shallcommenton andreferto suchcomparisonsatthe appropriateplacesin the detaileddiscussionbelow.

Table 2 sumniarisesthe results. For reasonsof direct comparisonand becauseof numerousmistakesin theliteraturewehaveintroduceda un~JIednotationwhich is explainedin theappendices.AppendixA definesthe vertexfunction G(k)for arbitrary spin. The relation of the vertexconstantto the poleresiduep (or r) and to the asymptoticnormalisationC is givenin eqs.(A-5) and(A-12).In appendixB we deducethe necessarygeneralisationsdue to isospin and antisymmetrisation.

The quantity~contains the clusterprobability,whereasC denotesthe pureasymptoticnormalisa-tion henceforth,seeeq. (B-7). The spectroscopicfactorS is definedin (B-6). The generalformulaeare unavoidablysomewhatinvolved. For the convenienceof the readerwe thereforecollectedthe relevantresultsfor the simplestfew-body situationsin appendixC.

We comment on the entriesin table 2 in somedetailmaking comparisonwith relevantinforma-tion from othersourceswhenavailable.

The triplet np effective rangeapproximationto np scattering,although usually not listed inour context, is a very old and particularlyreliablepole extrapolationsince the deuteronpole isvery closeandcompletelydominateslow-energynp scattering.The asymptoticdpn normalisationC is relatedto the triplet effectiveranger~in our notation by (cf. GoldbergerandWatson[56])

C2(dpn)= (1 — Kr~) 1, (51)

whereK2 = me, with m being the nucleonmassande the deuteronbindingenergy.(In this casewe haveC2 = C2) The valuefor r~quotedin the tableis from LomonandWilson [77]. Withoutusingtheoreticalconstraintsits statisticalerror is 1 %. Since icr~is closeto onehalf, theerror ofC2 is about2 %. We have markedwith an asteriskthoseentries in table 2 which we think areparticularly reliableandwe haveindicateda realisticerror estimate.

The protonexchangepole in elastic nd scatteringoffers the possibility to determinethe dpnvertex constantfrom a different scatteringprocess.Forward amplitudeextrapolationshave beenperformedwith andwithout conformalmappingprocedures(FDR andFDRM). For the relevantspin andisospinfactorswe refer to example(i) of appendixC. The statisticalerror of G2(dpn) is~ 5 %. An extrapolationerror of about 5% has to be addedaccordingto the error discussionfurther above.(Locherand Mizutani give 3 % assuminga 1 % quality of the data.)The mappedresult[75] andthe unmappedresult [73] shouldnot becompareddirectlysincethe poleregularisa-tion is madedifferently. However,the resultsagreewithin ~heerrorsspecifiedabove.Theagreementof the dispersionrelationresultswith the effectiverangevalueis agratifyingconsistencytestof themethodas the experimentalinput is entirelydifferent.

The samequantity, G2(dpn), can alsobe deducedfrom neutronexchangein proton—deuteronscattering.The correspondingforwardamplitudeextrapolationhasbeenmadeby Bornandet al.

M.P. Locherand T. Mizutani,The useof analyticity in nuclearphysics 67

[24]. Theycorrectedthe quartetamplitude for the s-waveCoulomb penetrationfactor and forthe u-channelphotonexchangecut of fig. 17 (seesection3.2). The agreementis againexcellent.

The vertex constantsdeducedso far include the deuteronD-stateeffectively. Denotingtheasymptoticratio of D- to S-statein the deuteronwave function by R onewould expectorder-R(~ 3 %) D-statecontributionsto the forward amplitude.In fact, at low energies,whereorbitalangularmomentumand spin decouple,thereis no contributionlinear in R. SinceR2 ~ iO~,the D-statecontributionis negligible [23]. For theunpolariseddifferentialcrosssectionthe D-statecontributestrivially orderR2 andhenceis negligibleas well [21]. In orderto determinethe asymp-totic D-statenormalisationfrom differentialcrosssections,onehasto analysethe tensorpolarisa-donin pd scatteringwhich is linear in R. However,presentdatado not allow extrapolationto theneutronexchangepole with satisfactoryaccuracy[4] in order to determineR directly.

Angular extrapolationsof the unpolarisednd differential crosssectioncan thereforebe usedto determinethe S-statedpn vertex independently[62]. The mappingtechniqueis applied tosuppressbackground.The valuesdeducedby Kisslingerat differentenergiesscatterconsiderably.The averagevalueis statedto be 25% abovethe effectiverangepredictionandquotedin table2.Thedeviation,althoughsomewhatlarge,is consistentwith thelimited effectivenessof the mappingmethodto reducebackgroundwhichwas mentionedatthe beginningof section4. This is confirmedby the averagevalue of Dubni&a andDumbrajs[39] andasimilar value from pd scatteringbyBorbélyandNichitiu [22] whichboth aresomewhattoo low. In bothcasesangularextrapolationsin amappedvariablewereused.

The situation regardingthetdn vertex constant is similarly favorable.Therearethreeforwardamplitude extrapolationsfor nd scatteringin table2 (seeexample(i), appendixC for notation)which all agreeto within 10 %. The most accurateevaluationof the dispersionintegral hasbeenmadeby Bornand[23] (hencetheasteriskin the table).As in the caseof the deuteron,configura-tions different from the symmetricS-statetriton wavefunctionareincludedeffectively in thevertexconstants.Their contributionis expectedto be small [21, 23]. In termsof backgroundthe tritonpole is still ratherwell isolated.Oncethe protonexchangepole is subtractedfrom the forwardamplitudethe only perturbingsingularityis from the weak(N + it) exchangecut (seesection2.2.1and fig. 13). Forward amplitude extrapolationsthereforedeterminethe tdn vertex almost asreliablyas the dpnvertex.All thevertexconstantslisted furtherbelowin table2 haveconsiderablylargerbackgroundandextrapolationerrors.

For the tdn vertex we alsolist the resultsby Bower [25] which are the by-productof amuchmoreambitiousattemptof calculatingnd partial wavesfrom dispersionrelations.Variousinputsfor the inelasticitiesandbackgroundcuts give the rangeof G2(tdn) indicated in the table (hiscoupling constantg~relatesby g~= 8j2fldG2(tdn)/9 to our notation). A calculation similar inmethodby Rinatetal. [94] treatsthetdn strengthasaparameterwhichis controlledby demandingthe correcttriton bindingenergy.Due to certainsimplifying assumptionsit is hardto assesstheerror.

Angular extrapolationswith mappingare alsolisted. For the resultsby Dubni~kaet al. [42]the averageover the energies1, 2, and3.5 MeV is given (at 6 MeV the extrapolationfails forunknownreasons).The correctiongiven in ref. [39] is includedin table 2. We haveconvertedtheir residuesr into G2 andC2 by meansof our eq. (C-3) and (C-5’). The conversionof r intoC2 asgivenin eq. (A-16) of ref. [42] is not correct.

The angulardependenceof the d(d, p)t stripping reactionhas beenusedby Borbély [19, 20]to deducethe tdn vertex constant.The known value of the dpn vertexis input (for the tablewe

Table2Couplingconstantsin few-nucleonsystems

Coupling Process Method Ref. G2 )fm) Remarks

dpn np(D) 0ER GoldbergerandWatson. 0.432)±2°) 1.68) = C2) triplet effectiverange1964 [56] r = 1.75 fm

nd)E) ~FDR Locher.1970 [73]: Bornand. 0.4271±7°,) 1.661 = (~

1977 [23]FDRM Locher, 1975 [75] 0.401 .56) = C2) samedataas Locher.

1970 [73]AEM Kisslinger 1972 [62] 0.540 2.10 ( = C2) averageAEM Dubni~kaand Dumbrajs, 0.381 1.48)= C2) average

1974 [39]

pd(E) 5FDRC Bornandet al., 1976 [24] 0.427)±7~,) 1.66 (= C2)AEM Borbély andNichitiu. 0.40 1.55)= (2)

1975 [22]

tdn nd(D) FDR Locher.1970 [73] 1.01 2.40FDRM Locher, 1975 [75] 0.925 2.20

*FDR Bornand,1977 [23] l.09)±15°j 2.59(1 ~l ~87~

DR Bower, 1972 [25] 1093 2,2o~ range given for dif-ferent inelasticitiesandbackgroundcuts

PWDR Rinat et al., 1972 [94] 1.42 3.37 from table I of Rinalet al.

nt(E) AEM Dubniëkaet al., 1973 [42] 0.99 2.37 averageover firstDubniékaand Dumbrajs, threeenergies1974 [39]

d(d, p)t AEM Borbely,l974, l975 [19,20] 1.23 2.92 ~ = 0.43 fm

(d, t), (d, p) PM Borbélyand Dolinsky. 1.30 3.00 G~5~= 0.43 fm

on light nuclei 1970 [21]3Hedp pd(D) *FDRC Plattneret al.. 1977 [91] 1.38(±IS ~) 3.50

Bornand,1977 [23]

p3He(E) FDRC Plaflner et al., 1977 [91] 1.32 3.35AEM Kisslinger,1973 [63] ] normalisedtoAEMC KisslingerandNichols, ~ 0.755 1.90 standardconventions

1975 [64]MPSC Bolsterli and Hale, 1972 1.10 2.80 normalisedto

[18] standardconventions

d(d.n)3He AEMC Borbely.1974, l975[19.20] 1.00 2.54 = 0.43 fm

(3He,d) and(d,p1 PM Borbélyand Dolinsky, 0.88 2.23 G~5~= 0.43 fm

on light nuclei 1970 [21]

(3Hcdp) p3He(E) FDRC Plattneret al.. 1977 [9l] 0.98 5.85 continuumcut

effective included

s3Hen nz(E) FDR Ericsonand Locher, 4.6 eff. 5.40 continuumcui1968 [49] included

FDR Ericsonand Locher. 11.0 23.3 12.9 27.4 var. positions for ef-1970 [50] fective background

pole5FDR Locher, 1972 [74] 11.3 ) ±15 °~ 13.3 lefthand cut approxi-

matedby two poles0FDR PlaMneret al.. 1973 [93] 10.5 (±l5’~) 12.4 lefthand cut approxi-

matedby momentsFDRM Locher,1975 [75] 14.5 17.0 samedata as Locher.

1972 [74]

a(p,d)3He PM Dolinsky and Turevtsev. 9.95 11.7 G]5, = (1.43 fm

1969 [36]

Table2 (cont’d)

Coupling Process Method Ref. G2 (fm) C2 Remarks

na(E) AEM Kisslinger,1972 [62] -.-8.5 10.0 effectiveUEPA Baryshnikovetal.,1974[14] 8.0 8.5 9.4 10.0 effective

atp pa(E) FDRC Plattneret al., 1973 [93] 13.4 16.0

~FDRC(M) Plattner,1977 [89] 11.1 (±15%) 13.3

UEPA Baryshnikovetal.,1974[14] —7.0 “-8.4 effective

at(E) UEPA Baryshnikovand 10.0 12.0Blokhintsev,1975 [9]

add da(E) AEM(C) Dubni~kaand Dumbrajs, —0.6 —0.7 effective,far away1975 [40]

12C(d,a)’°B), PM Baryshnikovand —25 —.28 effectivei4N(d a)’2CJ Blokhintsev,1971 [8]

6Liad da(D) *FDRC Plattneretal., 1976 [90] ~ 022 46Bornand,1977 [23] 5

a6Li(E) PMC Trudl andBierter, 1969 0.04 0.8 correctedvalue[102]

UEPA Baryshnikovand 0.05 1.0Blokshintsev,1976 [10]

a 6Li(E),6Li(p,3He)a PM Dolinskyetal.,1972/74[37] -.0.06 —.1.2d6Li(E) PM Dolinskyet al.,1972/74[37] 0.22 4.6

8Beaa aa(D) FDRC Fang-LandauandLocher, 0.013 1.82(mC2) outerCoulomb1973 [52] corrections

‘2C”Cn ni2C(E) AEM DubniékaandDumbrajs, —23.5-. —6.9 (36 <C2 < S,~= 2.61

1974 [39] < 106) poleveryfar

‘3C(3.09) ‘2C(d,p)iSC* AE Amado, 1959 [3] 0.085 0.91 5 = 0.9‘2Cn (C2 = 1.01)

i3Ni2Cp pi2C(D) FDRC MeyerandPlattner, —0.301 3.13 S = 0.53

1977 [81] (C2 = 5.91)

i2C(d n)’3N PM BorbélyandDolinsky, —0.12 1.24

1970 [21]

14N’3Nn niSN(E) AEM DubniékaandDumbrajs, —0.86 —0.46 2.1 3.9 S~= 0.721974 [39] (2.9 < C2 < 5.4) poleveryfar

1st column — Typeof coupling.2nd column — Processforming thebasisfor theextrapolation:

ab — elasticabscattering,(D) meansdirect or s-channelpole,(E) meansexchangeor u-channelpole,A(a,b)B — reactiona + A —‘ b + B,(a,b) — reactiona + A -. b + B for severaltargetsA andproductsB.

3rd column — Methodofextraction(anasterisk* denotesparticularlyreliableresults):AE: angularvariableextrapolationin differentialcrosssections;AEM: AE plusconformalmapping;AEMC: AEM plusCoulombcorrection;DR: dispersionrelations;ER: effectiverangetheory; FDR: forward dispersionrelations(discre-pancyfunction method);FDRM: FDR plus mapping;FDRC: FDR plus Coulombcorrection [FDRC(M): mappingtested];MPSAC: modified phaseshift analysisplus Coulombcorrection;PM: polemodel (peripheralmodel);PMC:polemodelplus Coulombcorrection;PWDR.: partial wavedispersionrelations;UEPA: unitarisedexchangepoleampli-tudes(K-matrix partial waveanalysis).

4th column — References:author(s),year ofpublication,andreferencenumber.5th column — VertexconstantsG2 in fm, seeeq. (B.5) of appendixB.6th column — AsymptoticwavefunctionnormalisationC2 (and,whenavailable,alsoC2);C2 = f32C2where132 is theclusterprobability,

seeeq. (B-5) and(B-7). C2 is dimensionless.

70 lkI.P. Locher and T. Mizutani, The useof analyticitv in nuclearphysics

useG2(dpn) = 0.43 fm for this purpose).Similar resultsfrom polemodelfits to strippingreactionson light nuclei are also given (Borbély and Dolinsky [21]). Thesestripping reactionshave theadvantageto avoid multinucleonexchange(cf. section2.2.2)and thereforesomeof thebackgroundproblems.

Similar remarksapply for the 3Hedp vertex constant which can be determinedfrom elasticproton—deuteronscattering(direct pole), proton—3He scattering(exchangepole), or from strippingreactions.We considerthe forward amplitudeextrapolationsto be particularlyreliable (Plattneret al. [91]). We prefer the value from elastic pd scatteringwherethe direct pole is well isolatedfrom backgroundonce the neutron exchangeis subtracted.The deuteronexchangein p—3Hescatteringsuffersfrom nearbynp continuumexchange.Nevertheless(makinguseof spin informa-tion), the contributionof unboundtriplet np exchangehasbeenfoundto besmall [91] andthe twodeterminationsagreewith each other, well within our estimatederror of l5~.The dominantCoulombcorrectionis the S-wavepenetrationfactorof eq. (34), the u-channelCoulombcorrection(fig. 17) being only a few percentfor the strongly bound 3He system(cf. section3.2). Sincenofurther correctionsare madethe residueslisted still containelectromagneticeffects and shouldnot be comparedimmediatelyto the tdn vertexconstant.

Usingappropriatespininformationthe 3He dpvertex(whereddenotesthevirtual singletdeute-ron state)hasalsobeendeterminedby Plattneret al. [91]. The effectiveconstantlistedin the tableincludesthe singlet np continuumcontributionwhich is largejudging from the interceptof theinversediscrepancyfunction plot. Cp. the theoreticalresult for G2(3Hedp) of Belyaevet al. [15].

Angular extrapolationsof p3He differentialcrosssectionswith mappinghavebeenperformedby Kisslinger [63] andKisslingerandNichols [64] (including Coulombcorrections).We converttheir valueC~Nfor the 3Hedp vertex into our standardnotation by going backto their primarydefinitions,eqs.(16) and(17) of ref. [64], relatingC~Nto thedifferentialcrosssection.By comparisonwith our eq. (C-8) onefinds (2 = (2/3)C~N.The value deducedby Kim andTubis [61] by extra-polating the wavefunction from a Faddeevcalculationwith Reidsoft-corepotentialto the poleis(2 = 2.86. A similar three-bodycalculationby Barysnikovet al. [11] gives (2 ~ 4.6. Since Kimand Tubis usestandardnotation (as do Plattneret a!. [91]) Kisslinger and Nichols differ by

35 %. This is howeverconsistentwith the estimateof the extrapolationerror for the caseof astrongand close backgroundcut (np exchange)as discussedin the introduction to section 4.We recall thatmappingaloneis an inadequatemethodto suppressbackgroundin sucha situation(cf. LocherandMizutani [76]).

A different type of angularextrapolationfor p3He scatteringhasbeenmadeby Boisterli andHale [18]. They calculatethe peripheralpartial waves from deuteronexchange,iteratedin aSchrodingerequation.The peripheralwavesare proportional to the deuteronvertex parameterC~Hwhich is fitted simultaneouslywith the low partial wave. Singletnp exchangeis assumedtohavethe samestrengthas d-exchange(which is plausiblefrom the analysisby Plattneret a!. [91]).The value quoted in table 2 is obtainedfrom (2 = ~C~H = 2.8, deduciblefrom the primarydefinition of the T-matrix and the spin factorsin ref. [18]. This conversionfactorwhich hasbeenoverlookedin referencesto Bolsterli andHale’spaperleadsto completeconsistencywith Kim andTubis. A conservativeerror estimateof C~His 300/s, consideringthe indirect way of the deter-mination, the angulartype of extrapolationand the effective inclusion of the triplet np cut intothedeuteronpole.

Stripping reactionsas mentionedin the contextof the tdn vertex offer anotherpossibility todetermineC2(3Hedp). Results for angularextrapolationsusing a straight pole model (Borbély

M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics 71

andDolinsky [21]) for anumberof light nucleartargets from 9Be to ‘9F are given in the table(renormalisedto our standardC2(dpn) = 0.43 fm wherenecessary).Coulombcorrections,includ-ing the u-channelphotonexchangecut were applied to d(d, n)3He by Borbély [19,20], leadingto consistentvaluesfor C2(3Hedp)within therather largeerrorsexpectedfor suchangularextra-polations.To obtainC2 [eq. (B-7)] from the valuesfor (2 in table2, the clusterprobability fl2 = 0.9for thetdn (or 3Hedp)vertexby Werbyetal. [106] canbeused.

The availabletheoreticalandexperimentalinformationon three-bodycouplingconstantsfromvariousothersourceshasbeendiscussedrepeatedlyin the literature.In particularwe refer to thecompilationsby Kok andRinat [65], Goldfarbet al. [57] andLim [69]. Photodisintegrationhasbeendiscussedmorerecentlyby GibsonandLehman [54].

The four-nucleon coupling constants form the nextsetof entriesin table2. The 4He 3He n vertexconstant was determinedfirst by forward amplitudeextrapolationusingthe inversediscrepancyfunction plot. The valuededucedby EricsonandLocher [49] includedthenearbythree-nucleonexchangecut. Using a one-poleparametrisationof this backgroundthe permissiblerange forG2(~3Hen)was given by Ericsonand Locher [50]. Higher energyconstraintsandexplicit back-groundparametrisationswereusedlaterby Locher [74] to narrowtherangestill further.Theuse-fulnessof mappingin the energyvariablewas testedby Locher(ref. [75]). The 1975 valuein table2is withoutanyallowancefor background.Sincethe cut is too closeto rely on backgroundsuppres-sion by mappingalone(seeLocherandMizutani [76] andthe generalerror discussionprecedingthissubsection)the 1972valuemustbepreferred.An independentforwardamplitudeextrapolationwith backgroundparametrisationby Plattner et al. [93] confirms this conclusion.The valuededucedfor C2(c~Hen) by Dolinsky andTurovtsev [36] from a neutronpick-up reaction bystraight angularextrapolationis surprisinglyclose. In their procedurethe low partial wavesareassumedto be totally absorbedfor 1 < L. The cut-off parameterL is fitted andenergydependent(L ~ 4).The successof the pick-upreactionis presumablyagaindueto theabsenceof multinucleonexchanges.

Angularextrapolationswith mappinghavebeenperformedby Kisslinger [62] on nci~differentialcrosssections.In quoting Kisslinger’s resultwe haveto rely on his conversioninto the notationof EricsonandLocher [50]. The valuefor G2(ot3Hen) obtainedmustbe consideredas aneffectivevaluewhich includescontinuumthree-nucleonexchange(seefig. 11). Mappingaloneis certainlyinsufficient to separateout the 3He exchangepolesincewe know from theFDR analysisthat thecontinuumexchangeis strong(cf. alsothe extrapolationerror discussionat the beginningof sec-tion 4).

A partial waveanalysisusingaK-matrix formalismto enforceunitarity hasbeenperformedbyBaryshnikovet al. [14]. Again their valuesfor G2(ix3Hen) and G2(c~tp)haveto beconsideredaseffectivevaluesincludingcontinuumexchange.

From pc~elastic scatteringthe isospin relatedvertex constantG2(xtp) has beendeterminedrepeatedly.The forward amplitudeextrapolationby Plattneret al. [93] correctsjust for the Cou-lomb penetrationfactorof eq.(34). Usingthe mostrecentdataandcorrectingalsofor the u-channelphotonexchangecut of fig. 17 one obtains* G2(~tp)= 11.1 fm (Plattner[89]) very closeto the

n valuet. This value is also in agreementwith the most recentexperimentalresults from

* we thankG.R. Plattnerfor communicatingthis resultprior to publication.

Recall that thetdnand 3Hedpverticesdiffer by 20%.

72 M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics

electronscattering,i.e.,with the normalisationof thechargedparticleasymptoticwave functionofeq. (39) (Plattner[89]).

The partialwaveanalysiswith K-matrix unitarisationof theexchangepoleby Baryshnikovet al.[14] leadsto an effectivecttp vertexas was alreadymentionedin the x3Hen case.A similar analysisof c~telasticscatteringby BaryshnikovandBlokhintsov[9] yields avaluefor G2(jxtp) muchcloserto the FDR results.This might be dueto the fact thatthe protonexchangepolefor this reactionhasno backgroundfrom multinucleonexchange.

The generalnuclearphysicsinformation on the x3Hen vertexhasbeencompiledby Lim [70].Note that C2 in Lim’s table 1 is relatedto our standardnotation by C2(Lim) = 2(2. The nor-malisationsC2 from variousmodelsfor the 4He wavefunction tendto besomewhatsmallerthanthe FDR results (correctingby handfor the wrong asymptoticbehaviouras needed).We alsomentionthata theoreticalvalueG2(~tn)~ G2(tx3Hep) 18 fm hasbeendeducedfrom Faddeev—Yakubovskyequationsby Baryshnikovet al. [13].

Dubni~kaand Dumbrajs [40] have tried to determinethe ~ddvertex constant by angularextrapolationto the deuteronexchangepole in dcx scattering.For the table, we take only theresultsfor thoseenergieswherethe backwardcrosssectionis reasonablysmoothandwell deter-mined.AveragingoverE~= 6.3, 7.5and37.3 MeV weget r = —0.15 for the residueof theforwardlab amplitudef(d-pole)= —r(E~— EI~~l.We converted*r into our standardvertexconstantG2 by

r = —mN(61r) G2. (52)

(Note that F(aA) = 2 and v(aA) = ~/~/2 using the n—p formalism, see appendix B.) To judgethe natureof the extrapolationwe mention that coscO(dpole) = —4.85 andcosCO(np) = — 5.15at E~= 7.5 MeV. Note that the extrapolationdistanceis very largeproducinguncontrollableextrapolationerrors.Furthermore,over such adistance,the np continuumcontributionis indis-tinguishablefrom thedeuteronpole.Accordingto ourgeneralerrordiscussionthisis not remediedby mappingandthe valuededucedis at bestan estimateof an effectiveconstant.A polemodelanalysisof deuteronpick-up reactionsby Baryshnikovand Blokhintsev [8] yields an order ofmagnitudelarger result.The methodis somewhatindirectand the valueis alsoeffective. We notethat Bornand[23] givesan upper limit (2(cxdd) <0.7(in our standardnotation) from dct forwarddispersionrelations(the valueincludesunboundnp exchangeas well), whichappearsto confirmthe smallestimateby Dubni~kaandDumbrajs[40]. We remarkthatthe asymptoticnormalisationin Lim’s tabulation [72] is very large(C2 is of order 50 from nuclearwave function information).Similarly the calculatedvertex constantby Baryshnikov et al. [13] is large(G2 ~ 18 fm). Since(2 = f32(cdd)C2[seeeq. (B-7)], the clusterprobability /32 would haveto be verysmall if Bornand’slimit holds, andquite large(/32 ~ 0.5) if the value (2 ~ 25 from the pick-up reactionis correct.An independentdeterminationof G2(dc~)would be very welcomein order to resolvethis question.

The most reliable value for the 6Li dtx vertex constant comes from the forward dispersionrelationsfor dc~scatteringby Plattneret al. [90]. In this reactionthe 6Li ground stateis adirectpole just below thresholdandwell isolated from the next singularity (d-exchange).There is nodetectablecontribution from the d-exchangein the inversediscrepancyplot, see fig. 24. The

* Lim’s 1976 tabulation [72] quoteserroneouslyC~Bs= ~2 29 (insteadof 0.6) for the Dubnic’ka and Dumbrajs result [40].

Lim [72] usesour standardnotationandtheconversionformula is correct,theerror is dueto a misprint in thereviewpaperby Dub-niékaand Dumbrajs[41]. we thank G.R. Plattnerfor information on this point.

M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics 73

tabulatedvalueis correctedfor the Coulombpenetrationfactor.The analysisof spin dependencein dix scattering[23] allows to deducea limit on the D-stateadmixturein 6Li which is foundto benegligible.

Earlier informationon the 6Li dix vertex is tabulatedby Lim [71] [note C~bS(Lim)= C2], theaveragevaluebeing(2 ~ 4.0. However,thisincludesC2 = 4.8 from Truöl andBierter [102] (fromtwo-nucleontransferin x6Li scattering)which in fact is much lower dueto a numericalmistake(P. Truöl, private communication).The relation to our notation is G2 = itg2(TB)/m~withg2(TB) = 0.12 (formerly 1.52)leadingto the correctedvalue(2 0.8 for Truöl andBierter [102].This is againan effective constantincluding deuteronandunboundtriplet np exchange(singletnp exchangeis isospinforbidden,as in thecaseof elasticdix scatteringfurtherabove).Similarvalues(2 ~ 1.0 havebeenfound from pole model fits two two-nucleontransferreactionson 6Li byBaryshnikovandBlokhintsev [10] and by Dolinsky et al. [37]. The disadvantageof thesetwo-nucleontransfer reactionsin terms of backgroundproblemsand extrapolation distancewasalreadystressed.The situationin a four-nucleonexchangereaction(like d6Li elastic scattering)is slightly morefavorable.Becauseof the high bindingenergyof the ct-particle3Hen andtp un-boundexchangecutsarebetterseparatedfrom the tx-pole (Dolinsky etal. [38]). The correspondingvaluefor G2(6Li ctd) deducedby Dolinskyet al. [37] is indeedlargerandcoincideswith the FDRCresult.

The ~& ~ vertex is aparticularlyinterestingtestcasefor electromagneticcorrections.The 8Beresonanceis barelyabovethreshold(at E~~ 184 keV), its extremelynarrowwidth is entirelydueto electromagneticeffects. The forwarddispersionanalysisin termsof the discrepancyfunctionwas madeby Fang-LandauandLocher [52]. Coulomb repulsionpreventing8Be to be bound,it seemedinappropriateto correct for the Coulombpenetrationfactor only, aprocedurewhichdoesnot alterpolepositions.Experimentalphaseswere thereforeconvertedinto “purely” nuclearphasesby meansof outerCoulombcorrections.In this way the resonancebecomesaboundstate(at E~B~~ — 100 keV). Its lab residuer can be determinedfrom the discrepancyfunction ratheraccuratelyto be r = 0.08 sincethe poleis well seperatedfrom background(2it exchangestartsat

—10 MeV, the dominantanomalouscut at EL ~ —30 MeV). The vertexconstantG2 obtainsfrom the observedlab residuer of the 8Be-poleby

G2 = irr(4m~)”’, (53)

cf. eq. (C-6). The correspondingnormalisation (2 is obtainedby observingthat Fv = (3/2)1/2

in the np formalism,cf. appendixB, eq. (B-5):

(2 = 2m~G2(n,cF2v2Y~’. (54)

Since the clusteringprobability is almostunity (Kurath [67]), (2 ~ c2.Finally we commenton vertex constantsinvolving morecomplex nuclei, ‘2C and heavier.

A typical exampleis the ‘2C1 ‘C n vertex deducedfrom straight angular extrapolationof the crosssectionafter mapping(Dubni~kaand Dumbrajs [39]). The extrapolationdistanceis enormous(cos0 = —24 atthe ‘1C poleafter mapping,for E~= 3.76 MeV), backgroundis closeandaccord-ing to our generalerror discussionandthe testsmadeby LocherandMizutani [76] this situationshould make us very cautious.For completenesswe quote the resultingrangeof valuesfor thevertex constantwhich hasbeenconvertedinto (2 by using 5exp = 2.61 from Nair et al. [83].Similarly largevalues havebeenobtainedin a pole model fit to 12C(p,d)11C by Dolinsky andandTurovtsev[36] resultingin — 5.5 > G2 > — 50.

74 M.P. Locherand T. Mizutani,The useofanalyticity in nuclearphysics

In this contextweremark that largeresiduesareto beexpectedfrom the exponentialenhance-ment factor expKR (iK is the bound state momentum,R a typical radius) for strongly boundnuclearsystems(cf. eq. (29) and p. 51 of EricsonandLocher [50]). Forangularmomentum1 = 1,a reducedwidth @2(12C11Cn) ~ @2(’2C’1B p) = 0.26, and R = 4.3 fm from MacFarlaneandFrench[79] we indeedobtain

G2 = —3mO2(~11R)’KRh~’

1(iKR)~2

~ —31 fm, for t2Ct1Cn. (55)

(The reducedwidth @2 containsthe spectroscopicfactor S, it should not be confusedwith @~.

The relation is @2 = S�, see MacFarlaneandFrench[79].)The stripping reaction ‘2C(d, p)13C* (3.09 MeV) was analysedby oneof the first angularpole

extrapolations(Amado[3]). Usingtheaboverelationto G2 for 1=0with Amado’svalues®2 = 0.056

andR = 4.2 fm, we obtainthe value quotedin table2. The spectroscopicfactor S = 0.9 which isneededto obtainC2 is from amorerecentPWBA analysisby GloverandJones[55]. Theirresultsfor the 13C*12C n vertex at 12 MeV are G2 = 0.28fm and C2 = 2.6. Thereis obviously astrongenergydependence.

The analysisof elasticpt2Cscatteringby forwarddispersionrelations(Meyer andPlattner[81])is a particularly clean case as the t3N ground state is the only direct channel pole, atE”’(13N) = —2.11 MeV, relatively well separatedfrom heavyparticleexchange(11B andits excitedstates),EL(llBgs) = —14.6 MeV. Yet, to determinethe 13N residueit turned out to be useful toconsiderthe spin flip amplitudeg separately(afterdivision by sin 0), a procedurewhich suppressessomeof the heavy particleexchange.(The methodhasbeentestedon the pp spin-flip amplitudewhereit yielded the correct ir°ppcoupling constant,Viollier et al. [104]. It hasalso beenusedtodeducean upperlimit for the D-stateadmixturein 4Hefrom theanalysisof elasticp4Hescattering,Plattneret al. [92].) The resultingvaluesfor the 13N’2Cp vertex,G2 andC2, arelisted in table2.*Theconversioninto C2 is from an optical modelcalculationby MeyerandPlattner[81]. It corres-ponds to a spectroscopicfactor S = 0.53 which is consistentwith the literature, see CohenandKurath [32], VermaandGoidhammer[103] andNair etal. [83].

The single-particletransferreactionsdiscussedby BorbélyandDolinsky [21] which we men-tioned alreadyin the contextof the tdn vertex leadto a reasonablemagnitudefor the 13N12Cpvertexas well. Furthervertex constantsup to atomicnumberA = 19 werealsodeterminedin thispaper.

Thelastentryis the 14N13Nn vertexconstantdeducedfrom angularextrapolationaftermapping(Dubni~kaandDumbrajs [39]). The largeuncertaintiesmentionedfor the 12C’ tCn vertex arealso presentin this case.From the pick-up reaction t4N(n, d)13C, Borbély and Dolinsky [21]deducedthe relatedvertexconstantG2(’4Nt3Cp) = 0.3 fm which is in conflict with the reducedwidth from MacFarlaneand French[79] leading to G2 = 4.3 fm.

A considerableamountof information on nuclearverticesbasedon analyticity hasbeencom-piled in table 2. In particularwe stressthe resultsfor the few-bodysystemsproper(A ~ 6). Thereliability, the model independenceandthe considerableaccuracyachievedmakethesevertexconstantsan important constraint to few-body wavefunctionsandavaluablepieceof input forthe driving termsin Faddeevcalculations.The increasinguseof spin informationwill allow higher

Similarly, valuesfor the(effective) 2C p1 1B vertexhavebeendeterminedin ref. [SI].

M.P. Locherand T.Mizutani, The useofanalyticity in nuclearphysics 75

accuraciesandthe extractionof moresophisticatedquantities.Thisis alsotruefor theintermediaterange(6 ~ A ~ 20) wherethe increasingcomplexity hasnot preventedto obtain similar vertexinformationin recentyears.It is clearthat the increasingquality of datawill openup new areasof applications.

4.2. Pion nuclearcouplingconstants

The methodsdiscussedfornucleon—nuclearscatteringarealsoapplicableto thenuclearscatter-ing of pions.The main differencestemsfrom the fact thatpions canbe absorbed.Sinceno pionnuclearboundstatesdueto stronginteractionareknown,the polesof the 3~±scatteringamplitudeon thenucleusX = (Z,N) correspondto the groundstateandthe excitedstatesX’ of the (Z ±1,N ~ 1)nuclearsystems.It is importantto notethatall thesepoles(andthecorrespondingexchangepoles)are locatedaboutonepion massm~belowthe elastic scatteringthresholdaround* co ~ 0sincenuclearbindingenergiesaresmallon apion massscale,cf. eq. (58) below.A pion massis alargeextrapolationdistanceandit is thereforeobviousthat we shall not be able to resolvetheindividual polecontributions.The 3tXX’ coupling constantwill thereforebe an effectivecouplingsummedover all thecontributingintermediatenuclearstates.

Theanalytic structureand formalism for ivX scatteringis discussedin detail by EricsonandLocher [50]. We repeathereonly the formulaeneededfor the poleextrapolations.Theapplicationof dispersionrelationsfor physicalenergiesof the elasticscatteringamplitudeis atopic in its ownright andnot discussedin this report.

For the time beingonly the forwardamplitudedispersionrelationshaveallowed to extract it

nuclear coupling constantswith somereliability.** The analytic structurefor the forward labamplitudet

f~(w) = ~[f,r+~(W) —f,~-~(w)] (56)

is shownin fig. 25. To avoid overloadingthe figure the mirrored part of the spectrumwhich isobtainedfrom crossingantisymmetry

f~(—co)= _[f(~(w)]* (57)

is not shown.

ca -picine

rn,~

Fig. 25. The analyticstructureofftt(w) for aX scattering.Crossesdenotenuclearpoles.The cut from w~to w = m,, is dueto un-

physicalpion absorption.For co > m,, theimaginarypart obtainsfromtotal crosssectionsIm f~(w)= 1[o,,+~ — a,,-~]p(4sz)1,

The nuclearpolesmentionedfurtherabovearelocatedat

~ —m~(2M)~+ C. (58)

* We use thenotation o = E + in,, for thetotal laboratoryenergyof thepion.** Attemptsto determinetheeffectivepion coupling to the ‘He—3H systemfrom angulardistributions havefailed, see Dubnh~kaetal, [42]. Compare,however,thenote on page78.

we assumeX to haveneutronexcess.

76 M.P. Locherand T. Mizutani, Theuseof analyticity in nuclearphysics

(with ~ *~ m~)where M is the target massand the mass of the intermediatenuclear state i isM + C

1. Unboundintermediatenuclear stateswithout pions lead to an unphysicalabsorptivecut from (d0 ~ 0 to w = m~.

We haveintroduceddirectly the iso-antisymmetricamplitude (56) which is projectingout thedesiredpoletermsdueto the trivial kinematicfactorw in eq. (59) below,whereasthecorrespondingsymmetric combinationof poles is multiplied by w1 0 and hencestrongly suppressed.Thecorrespondingdispersionrelation for f~ reads

Ref~(w)= ~ +~pJ~Irn~~ (59)

O)()

where we haveused that ~ is odd under crossing.Neglectingw~/in~we introduce

~ /i~xxi/(w~— w~)~ ~ /,~xx,/w2 /2/w2 (60)

andevaluateeq. (59) at threshold:

Ieff = ~m~Ref~(m~) — in~Pjlmf~(w’)d~ (61)

it (I) —mi,

obtainingthe sum rule for the pion nuclearcouplingconstant.

Results:We discussthe resultsobtainedfrom the sumrule (61) for threeisospin-l/2nucleishownin table 3. We startwith

9Be and7Li since the experimentalinput, the scatteringlengthsandtotalcrosssectionsneededfor the dispersionintegralaremore completefor thesenuclei. Historically,the first examplewas 9Be (Ericson et al. [51]). The 2p—ls strong-interactionlevel-shift AE

1~in the ir

9Be atom was known at the time. It is relatedto the itX scatteringlength a by*

~ AEj~(2m~Z3tx3Y’. (62)

Table 3

The a-nucleuseffectivecoupling constant/,,~deducedfrom forward dispersionrelations,see eq. (60).and thesumrule eq. (61). We use~ = 0.08

Reaction Reference i. =

it 3He Nichitiu et al., 1977 [84] 1.5

it ~Li Squieret al., 1973 [100] 0.75PilkuhnCt al.. 1976 [86] 0.S7

it 9Be T.E.O.EricsonCt al., 1967 [51] 075T.E.O.Ericsonand Locher, 1970 [50] 5

M. Ericson,1971 [47] 0.75Osland, 1973 [85] 0.50Squieret al., 1973 [100] 0.63Ericsonand Krell, 1975 [48] 0.75Pilkuhnet al., 1976 [86] 0.73

* Negativescatteringlengths arerepulsive in our notation.

M.P. Locherand T.Mizutani,The useofanalyticity in nuclearphysics 77

The sum rule,however,calls for

Ref~(m~)= ~(a÷ — a4 = a~— a, (63)

wherea.5, the it + X scatteringlength,is not knownfrom mesicatoms.Onethereforehasusedthesecondequality of eq. (63) andhasscaledthe isosymmetricscatteringlength a~~ = ~(a.. + a)from known level shifts in neighbouringisospinzeronuclei. (For

9Be we havea - —0.28m~anda~~ ~ —0.23m 1)

The values for Ref - ~(ma)usedin table 3 show little variation.They range from 0.048m~(Ericsonet al. [51]) to 0.058m; 1 (Pilkuhn Ct al. [86]). Half of this contributesto /e2ff accordingto eq. (61). Thedispersionintegralwhich is dominatedby the 3,3 resonanceregion,was estimatedto contribute0.04 to ~ by Ericsonet al. [51]. Measuredir crosssectionson 9Beandscaledir~crosssectionson 12C nearresonancewereused.The integralwas reevaluatedby MagdaEricson[47]. Fromtheresonanceregionsheobtained0.03.Thisreductionwascompensatedby acalculatedthresholdcontributionof 0.01 which is amany-bodyeffect.The low energycontributionhasbeenconfirmed by an optical model calculation (M. Ericsonand Krell [48]). In 1973 Osland [85]determinedthe dispersionintegralfrom calculatedGlauberinput (with irN phaseshifts from [2])for resonanceandhigherenergies,resultingin avalueof 0.015. Sincehe neglectedthe thresholdregion this explainshis low valuefor /~f. By 1973 experimentalresultsfor total crosssectionsofbothpion chargesbecameavailablein the resonanceregion,establishingthe resonancecontribu-tion as0.03. The correspondingvaluefor /~by Squieret al. [100] is for thesamescatteringlengthas usedby Ericsonet al. [51]. A similar analysisby Pilkuhn et al. [86] yields aconsistentresult.For nucleirangingfrom d to 9Be theseauthorstried to deduceapion nuclearcouplingconstantfrom the iso-symmetricamplitude as well. However, pole contributions in the iso-symmetricamplitudearefirst order suppressedby thekinematicalfactor w~(~0)mentionedin the contextof eq. (59). For thesamereasonsthedefinitionof thestrengthof theeffectivepoledependscriticallyon the averagepoleposition.The iso-symmetricamplitude,therefore,cannotbe reliably usedforthe determinationof pion nuclearcouplingconstants.The total contributionfrom theunphysicalregion as determinedby Pilkuhn et al. hasreasonablemagnitudeandA-dependence,however.In this contextwe mention a discrepancybetweenpionic forward dispersionrelationson theiso-symmetricnucleus4He and real partsfrom Coulombinterference(Binon et al. [16]) in thephysical regionat co ~ 400 MeV. The origin of this discrepancy,shouldbe explained.

Thesumrule (61) leadsto couplingconstantsfor 7Li of similar originandreliability asdiscussedfor the caseof 9Be.

Apart from uncertaintiesin the scatteringlength and incompleteexperimentalinput for thedispersionintegralin thephysicalregion* aprincipal uncertaintyis dueto thedispersionintegraleq. (61) over the unphysicalpionabsorptioncut, seefig. 25. It is difficult to reliably estimatethiscontribution. As far as it extendsinto the regionof poles aroundco ~ 0 (seeeq. (58) andfig. 25)its contributionis containedanyhowin the definition of the effectivecouplingconstantin eq. (60).Note, however,that the contributionfrom the regionaroundw ~ 0 should be small, sincethenon-polepartsoff ~(co) shouldvanishin thesoft pionlimit co —* 0. A furtherestimatecanbemadeby extrapolatingImf(co) due to absorptionbelow threshold. Using specific parametrisationsthe unphysicalregion has beenfound to contributenegligibly in isospin symmetricdispersion

~Energiesabovethe resonancecontributelitte to thedispersionintegral in eq. (61). The corresponding error in thesumrule is

undercontrol,

78 M.P. Locherand T.Mizutani, The useof analyticity in nuclearphysics

relations. It has thereforebeenconjecturedthat the unphysicalcontribution to the couplingconstantshould not exceed 10% (Ericson et al. [51]). Similarly, the unphysicalregion in theevaluationby Pilkuhn et al. [86] appearsto contributelittle. As the error from the subtractionconstantand thephysicalregion in eq. (61) is of order 10% we believethatthe overall error of theeffective couplingconstantfor 9Be and7Li in table 3 is about 15%.

Thesumrule (61) hasalsobeenexploitedfor the caseof 3He by Nichitiu andSapozhnikov[84].The ir 3He scatteringlength hasrecentlybeenconstrainedby the accuratemeasurementof thestrong interactionlevel shift by Abela et al. [I] (leading to a

1~ ~He = 0.06m~’).The scatteringlengthfor ir~

3He could be obtainedfrom ir ~H by isospininversion.In the absenceof anexperi-mental number Nichitiu and Sapozhnikov have used an empirical extrapolation: a~+~= —0.2m j. Thedispersionintegralis determinedby experimentin the resonanceregionwhereasbelow 100MeV an optical model was used. The resultingvalue for /~ quoted in the table 3appearsto be large. Reductionsmight comefrom varioussources.The value usedfor a~+‘He ~

about twice the coherentsum of nucleonpoles (this alone would reduce/e2~fby 0.025). Opticalcrosssectionsbelow100 MeV tendto betoo highwhich could affect the dispersionintegral.

An earlier attempt to determinelit 3He3H by a Chew—Low extrapolation(Mach and Nichitiu[80]) is evenmoremodeldependent.In particularthe sensitivity to the three-nucleonform factorproducesa factor-twouncertaintyin the resultt.

A completelydifferent approachrelatespion nuclearcoupling constantsto nuclear /3-decaymatrix elementsfor X’ —~Xev~in thesoftpion limit. In thisway Kim andPrimakoff[60] obtained~lit 3He 3H = 1.05/.~ whereasfor A around 10 the ratio is about0.5 for ground statetransitions,see alsoref. [66].

The renormalisationof the pion nuclearcouplingconstanthasalsobeendiscussedby Delormeet al. [35] andDelormeandFigureau[34] in the contextof nuclearpolarisationeffects.For 9Betheyobtain). ~ 0.75 similar to the valuesquotedin our table 3.

Acknowledgement

We are grateful to our theoreticalandexperimentalcolleaguesfor numerousclarifying dis-cussions.Thanksaredueto Miss A. Ebnerfor the speedanddiligence in editingadifficult manu-script.

Appendix A. Residuesfor the general spin case

In this appendixwe extendthe notation in the main text to arbitraryspin in binary reactions.The aim is to relatepoleresiduesto the normalisationof asymptoticwave functionsin the generalcase.Nucleon antisymmetrisationis discussedin appendixB. For Coulombmodificationsseesection3.2 in the main text. We usenon-relativistickinematicsthroughout.

We first considerelastic scatteringa+ A -+ a + A through the direct (or s-channel)bound

Note added in proof: Recentangular extrapolationsgive /2 = 0.045 from p 3H —* n 3He and~2 = 0.055 from 3He 3H —* 3He ‘H

differentialcrosssectionswith about30~/~error (0. Dumbrajs.preprints,Helsinki 1978).* Sincethereis little structure in theA = 3 systemthis resultshouldbedirectlycomparableto theeffectivevaluequotedin table3.

For A around 10 excitedstatesareexpectedto contributesignificantly to theeffectivecoupling constant.

M.P.Locherand T. Mizutani, The useof analyticity in nuclearphysics 79

jaAa jaAa

JAAA

Fig. 26. Direct channelpolefor elasticscatteringwith arbitrary spin.Theparticles label thespinj and thethird component2. Theassignmentsof cm momentaarealsoindicated.

staten (for the spinlesscaseseesection 2.1). The pole contributionto the cm amplitude Cf fromfig. 26 is

cf~(kF, k; CE) = — PaA 1 G~jAA~(k~)*G~jAAA, (A-i)2ir ~ C,~+

where

/2aA = mamA(ma + mA), E~ ~ = ma + mA — m~,k(k’) initial (final) cm momenta, K = = k’I, CE = K2(2!.taA)~1,

Ja,JA spin of particlesaandA, respectively,

2 = (ia, 2A), 2’ = (2~,2~) third componentsof spin,

= (2ir)353(k’ — k) or = ~

In non-relativistic scattering theory the vertex function G(k) at the naA vertex is related to thepotential “~AbetweenparticlesaandA by

~aAaJA)~A(’~) = (/a2JA2A; k~“aA Lin~n>= ~ + CE) <ia~a/A~Ak ~ (A-2)

wherein the last stepwe haveusedto Schrödingerequationfor the boundstaten. To relate thepoleresiduein (A-i) to theboundstatewavefunction<r~j~2~>‘P~,,,~~(r)wefirst makeadecom-position into channelspin statesS. 2~andorbital angularmomentumstates1, 2,:

~ = sJ~ ~ <Ja~aiA~AS2~><S2512,~ yl(j~)GJ~~,(K) (A-3)

SA~,lAi

wherek = k/K. Comparingwith (A-i) we definetheresiduefunction

R~(k’,k) = ~ ~~k’)*~.~(k) (A-4)

andobtainin channelspin representation

R~.A.SsAS(k’,k) = ~ <s’1~l’2;~ <s2~l2,j~2~>~ (A-4’)2ir j2,l’2l,~

The poleresidueis thelimit K —~ i~ç:

p~(Ji~’,1T) = lirn R~(k’,k), (A-5)K- iic,,

whereK~=

2I1aACn~

With the sign conventionin (A-4), the residueis positivefor an S-waveboundstateof spinlessparticles(this conventionis the mostnaturalchoice in the contextof dispersionrelationsand is

80 M.P. Locherand T. Mizutani, The useof analyticity in nucearphysics

usedby EricsonandLocher [50], Locher [73], Plattneret al. [93], etc.).In the contextof forward dispersionrelationsone definesa “spin averaged” residue as

= I ~ PAA~, (A-6)

[Ia/A] A(1a.AA)

or expressedby channelspin vertexconstants

n — /2aA [In] ~ (~‘j,,( ~2

1) — . . .L_1’. Si~.”~nJJ‘ —2ir [/aIA] ~i

where

D021m] (2j~ + I)(2j2 + l)...(21m + 1).

The combinationof spin weightsin (A-6) or (A-7) arisesnaturally for the imaginarypart deducedfrom unpolarizedtotal crosssectionsby the optical theorem.It doesnot occurdirectly in this wayin unpolarizeddifferential crosssections.

The overlapintegral in (A-2) is expressiblein termsof wave functionsas

<1a2JA2A; k ~ = Je’~ (ra)~7AAA(rA)~jA(ra,rA, r) d3ra d3rA d3r. (A-8)

wherer is the coordinatebetweenclustersa andA. Recouplingangularmomenta

<Ia~JA~A;k In~n>= ~ <Ia~.J~S~~><S2~l2,jn~n>

SA51A,

X Y,~(k)flsi(naA)JIi(Kr)~(r)r2 dr, (A-8’)

wherer4~(r)is a normalizedradial wave function*, $~[4~(r)]2r2 dr = 1, describingthe relative

motion of the clusters.Thequantity /3~,(naA)is theclusterprobability for the boundstaten beingformedby thenucleiaandA with total angularmomentum(S ® 1) = I,,. If thetotal wave functionis a pureconfigurationof the clustersaandA in this angularmomentumstatewehave

5l’I~A~(rS,rA, r) = [{‘f~~(r5) ®

tI’JAAA(rA)}s ® yi(r)]I~(r) (A-9)

with ~ = 1.

For ashort rangenuclearpotential V~,Awe maywrite= ~ C(n, SI) e- ~/r + g~(r), (A- 10)

whereg~(r)decreasesfaster than the Yukawa term. At largedistanceswe havethe asymptoticform

lirn 4~’j(r)= ~ C(n, Sl)e””T/r. (A-to’)

For azero-rangepotential the dimensionlessconstantC is unity due to the normalisationcon-

* Note Y04~0(r)= f(r) of eq. (14) in themain text.

M.P.Locherand T.Mizutani, The useof analyticityin nuclearphysics 81

dition. However,eq~(A-10’)is not apermissablewavefunctionat theorigin as it is notasolutionof the Schrodingerequationthere.Theshort-rangebehaviourof I’~Afor r ~ l/K~makesC differentfrom one.Another definition of the asymptoticnormalisationwhich is frequently usedis

N(n, Sl) = ~ C(n, Sl). (A-il)

The relation betweenC andthe vertexconstantGobtainsfrom eqs.(A-2), (A-3), (A-8’) and(A-b)as

(G~)2= —~— /3~,(naA)2ic~(—)‘C2(n, Sl) (A-12)I1aA

whereG G(K = iic~)andthespin averagedresiduebecomes

p5 = ~ -~- ~ (— )‘/3~1(naA)C

2(n,Sl). (A-7’)[/a/A] /~aA Si

As anextstepwe discussthe residuesassociatedwith particle exchangepoles.Although our maininterest is in the pole singularitiesof elastic amplitudes,it is convenientto study the generalbinary reaction a + A —~b + B of fig. 27 (cm system).If particlea is the projectile the figuredescribesthepick-upreaction(a, B) on nucleusA. If particleA is theprojectilewehavethestrippingreaction(A, b) on nucleusa. For a = b andA = B the graph of fig. 27 describeselasticscatteringthroughthe exchangeof particlee.

jaAa

JA a JbXb

Fig. 27. Notationfor a generalgraphdue to theexchangeof particlee.

The T-matrix correspondingto fig. 27 reads

T~(k’,k) = 5(aeA)~ ~B q’)*G.~A~tA~(q)(CE — — — (k ~k’)2 — CAbe) -‘ 1

(A-13)with

CE = K2(2/1aA)~1, CAbe = mb + me — mA, K ~ K’ =

5(aeA) is the signatureconnectedwith nucleonantisymmetrisation(seeappendixB), relevantfor elasticscatteringonly,

q = k’ ±ILbCk/m~,q’ = ±k + /2aek’/me, relativemomentain the vertexfunctionsG.

The lower signsin (A-13) andfurtherbelowapply for the caseof elasticscattering(a = b, A = B).

The energydenominatorin (A-13)maybe reexpressedas

—(q2 + !~be)(2/Lbe)~= —(q’2 + 1C~ae)(2/~aeY’1, (A-14)

82 M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics

with

= 2~z1~e~11,cf. eq. (A-i).

The vanishingof the denominatorin (A-13), andthus(A-14), determinesthe pole position. For agiven physicalenergy(momentum)the pole appearsin z cos0 (0 = cm scatteringangle)at

ze cos0~= ±mC(2/IbCKK’) 1 [K’2 + KAbe + (/Abe/me)2K2]

= ±me(2~.taeKK’Y’[K2 + KHae + (iiae/me)2K’2], (A-IS)

wherethe equality of the two expressionsis trivially establishedby energyconservation:

K2(2lUaA)~1— CAbe = K’2(2RbE)~— ~Bae~ (A-l6)

It is also easyto show that Ze~> 1, i.e., the pole is always in the unphysicalregion.In order to obtain the residuefor the angularextrapolationof the differentialcrosssection,

it is convenientto write the G’s in the angularmomentumreducedform, cf. eq. (A-3). Then thenumeratorof eq. (A- 13) reads

V (JB~8 ( ‘\*CJA1A I

L.~ J,,A,,JeA,,~.q ~= 4it ~ <Ia2a/A2A~X2X> KX2xu2~~J2~><Jb/tbIB2B~Y~Y>

XYuvJ

X KY2yv2~~.J~J> yA~~(/~’)*Y~”(ic)Ty~~~(K’,K), (A 17)

whereX(Y) is the initial (final) channelspin andJ is the total angularmomentum.The expressionfor TjJ~~~(K’,K) is given by

T~ K’ K — V ~so+i~ ~ 5 [21]! [21’]! 11/2Yv,Xu( , — ( ‘ S+ flU Y~ri l(2~)!(2fl) !(2y) !(2r) ! JilL

x be/me)Y(+~ae/me) K~(0 y u)(~~ v){fl y u}

X [iftJBS’51’~v]~2G~j(q’)G~~(q)q- ‘q’ —

x ~(_~[Xy]h/2 [AL]~5Ja3 JB ~lalA X~A 1’ Y~Y v jA LS lb Aj U’ A S’j LI A S J LL X 1 J Lu X L

(A-18)

with

R=2jb+2jA+X+Y+l+l’+2S+25’—A+L—J.

Thecontributionto the unpolariseddifferentialcrosssectionfrom the exchangeof particlee nowobtainsas

da iIaAILbBme I W(z)+ B ÷— a + A) = 4it2K’K3 (z — Ze)2 [JalA]’ (A-19)

M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics 83

where

W(z) = [uu’vv’]112[ghJJ’]( ~)X+ Y+g+h—u—u’—v—v’—2).jX YJJ’uuvv’glt.tj

(u u’ g\(v’ v h~\( J’ J g\( J’ J h\ ~J’ J g p J’ h~oo o)~o0 o)~_2~2~o)~_2~2~0)lu u’ Xflv’ v yf h(z)

x Tji~~~(K’,K)Tj!...~~.(K’,K). (A-20)

The poleresiduein the extrapolationusuallyis definedthrough

urn K’K3(z — Ze)2 ~(b + B a + A) (A2l)

— I1aAitbBme uJI \ (A II’— 2 ~ kZe),4ir [ia/A]

wherethe substitution

q ~Abe’ q’ ‘~ ‘KBae,

should be madein W(Ze). Further reductionof ~2 in termsof C’s areof coursepossiblethrough(A-12). We note that for 1(1’) having the value zero in eq. (A-3), the direct evaluation ofda/d1~(b+ B ~—a+ A) dueto the.exchangeof particlee is far easierthanusing(A-18) to (A-20).

For a fixed physicalscatteringanglethe exchangeprocessconsideredhereappearsasa poleof the amplitude at unphysicalenergy,cf. eqs. (A-l3) and (20). For the caseof forward elasticscattering,a+ A —s a + A, the polepositionin the cm systemis at

mCE = — ma +~mft CAae,

andthecorrespondingresiduereads

I.taA me= 5(aeA) ~ ma + mA ~ (A22)

with— (\L+2jAl’j~+).s~ ..~ vv’1112— ‘. I .~tS J J A

jj’L

~ (~ ~‘ ~ ~ ~ ~‘ ~}{~~ O3f(lKAac)~’(~KAae), (A23)

whereS(S’) is the initial (final) channelspin 2~(2k) its third component,f = Je ® 1, J’ = ./e ® t~G~f(q)is relatedto G~,~(q)of eq. (A-3) through

= ~(_)2i~+i+io+l[aJl1I2{ Ia ~ a}Gif(q) (A-24)

For simpleexamplesseeappendixC.

84 M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics

Appendix B. Antisymmetrisation factors

In this sectionwegeneralizethe resultsof appendixA. To includenucleonantisymmetrisationtwo differentformalismscanbeused.In the isospinformalismthe neutronandprotonareconsider-ed as the two componentsof oneparticle, the isospindoublet N = (n, p) (generalisedPauli prin-ciple). In this case,the vertexfunctionG~(K)of eq. (A-3) obtainsan extraClebsch-Gordancoeffi-cient

d(naA) <1a1a1AtALmntn>, (B-I)

whereI denotesthe isospin andi the third component.For later conveniencewe shall considerG~(K)to includethe isospinfactor (B-i) such that the relation (A-3) remainsformally unchanged.

Alternatively, the treatmentof neutronsand protonsas two different kinds of particlesmayprove more convenient.We shall namethis approachnp formalism. In this casewe shall taked(naA) 1 in theformulaebelow.Togetherwith appropriatechangesin the antisymmetrisationfactorsthe final result will of coursebe the sameas in the isospin formalism.

The antisymmetrisationof the nucleonsinside the nucleusaffects the results of appendixAin two ways: (i) the relation betweenG and C (A-12) is modified, and (ii) the signature~(aeA)in eq. (A-i3) arises.

(i) Antisymmetrisationfactors in the vertexfunction:(a) The definition of the vertex function (A-2) is replacedby, seeAustern[6]

~aJA~tA(’~) d<ja2jA2A; k~~A~ln~n> = —(Es + CE) .r/Ja~aiA~A k~j~2~>, (B-2)

where ia~aiA2A; k>d is totally antisymmetricwith respectto the transpositionof any pair ofnucleonsin thea + A system(in thenp formalismonly theantisymmetrybetweenpairsof neutronsandprotonsseparatelyis enforced).The internalwave functionsof the nucleiaandA areassumedto be totally antisymmetricalready.Denotingthenumbersof nucleonsin the nucleiby theparticlelabelsaandA we have

la~a/A~A;k)’d = PQAIJ5~JA2A;k>, (B-3)

where

~aA = .~i~i~)? I ~ (i — a ~ (see,e.g.,Austernref. [6]) (B-4)

with p~1being the transpositionoperatorfor nucleonsi andj. We rewrite eq. (B-2) as

= —(C5 + CE)F(aA) <Ja

2aiA2A k~j525>, (B-2’)

whereF(aA) is just anumericalfactor. For particleabeing a single nucleonwe have

F(aA) =

In thenp formalismthe antisymmetriserPaA is a direct product ~aA = ~aA (protons) x ~aA (neu-trons)actingon protonsandneutronsseparately,leadingto FaA = FaA (protons) x FaA (neutrons).

(b) Yet anotherfactor due to antisymmetrisationhasto be includedinto the definition of thevertexfunctionG in eq. (A-3). This is the so-calledcoefficientof fractionalparentage(c.f.p.) which

M.P. Locherand T. Mizutani, The useofanalyticity in nuclearphysics 85

we shalldenoteby

v(j5, SI).

For its definition see,e.g.,McCarthy[78]. Roughlyspeakingv is the weight for thespecificclusterpartition which forms the boundstatej,, out of the internally antisymmetricclustersaandA inthe spin stateS andtheangularmomentumstate1.

We confineourselvesto give somesimple examplesfor the factorsd(naA), F(aA) andv(j~,Si)in table4.

Table4Isospinandantisymmetrisationfactors

dpn vertexfor S-statedeuteron

d(aA) F(aA) v(j~,SI)

isospinformalism 1/\/~ ~ 1np formalism 1 1 1

tdn (or3Hedp)vertex(S-wavedeuteronandtritons)

d(aA) F(aA) v(j,, Sl)~

isospinformalism 1 ~ —i~~/~= [(1/2)2(01)1/2 1/2 1/2~}(l/2)~[10]l/2 1/2]

np formalism I ,,,/~ —\/3/2 = _[l]i/2[O]i/2w(l/

2 1/2 1/2 1/2,01)3Hed’pvertex(d’ = singletdeuteron,3He in S-state)

d(aA) F(aA) v(j,,Sl)’~

isospin formalism 1/,,/~ ,,,/~ 1/\/~= [(1/2)2(10)1/2 1/2 1/21} (1/2)~{10] 1/2 1/2]

npformalism 1 ,.,,/~ 1/2 = —[0] W(1/21/2 1/2 1/2,00)

~ SeeDe-Shalitand Talmi,NuclearShell Theory,1963, p. 533.

In the examplesgiven in the tableoneverifies explicitly that theproductd(aA)F(aA)v(j5,SI) is

independentof the choiceof formalism.To sumit all up the relationbetweenGandtheasymptoticnormalisationC of eq. (A-12) just obtainsthe threefactors d, F andv

((~)2 = (~ )i [d(aA)F(aA)v(j5,Sl)/351(naA)C(n,SI)]2. (B-5)

I~aA

Weremarkthat thequantity

S(j5,SI) [d(aA)F(aA)v(j5,Sl)/3(naA)]

2 (B-6)

is calledthe spectroscopicfactor. It is independentof the choiceof the two formalismsmentionedabove.Note, however,that S is sometimesdefinedwithout the isospin factor d(aA).

Whencomparingvertex constantsandasymptoticnormalisationsthe definition of the factorsin (B-5), (A-i) and (A-2) is of crucial importance.We use

G~ G~q(iic5)or simply G, C(n,Si) or C, C(n,SI) or C f351(naA)C(n,Si),

I~(n,Si) ~ C(n, SI), (~J~~)2= (~ )i S(j~,Si) C2(n,Si). (B-7)

/1aA

86 M.P. Locherand T. Mizutani, The useofanalyticity in nuclear physics

The relationsof thesewith otherfrequently occurringnotationsby different authorsare

G2(polemodel) = ±G2(ours) Borbély,Dolinsky, Baryshnikov,etc.

~(Rinat etal.) = (2ir)312 G(ours)

D(Goldfarbetal.) = (dFv) 1 G(ours) (B-8)

C(Plattneret al.) = C(Kim—Tubis) = C(ours)

N(Plattneret al.) = 51/2\/’5~ C(ours).

(ii) Thesignatureö(aeA):We havementionedin appendixA thattheelasticamplitudea+ A—sa + Adueto the exchangeof particlee, eq. (A-i3), hasasignaturefactor 5(aeA). In the contextof fieldtheory this signatureessentiallyreflects the propertiesof the amplitude under the parity andcrossingoperation(see appendixA of, e.g., EricsonandLocher [50]). In this section,however,we explainits origin in theframework of direct nuclearreactiontheory.

The Born term for the T-matrix with properly antisymmetrisedplane-wavestates(a � A) is

T = d<jS2~jA2~k’~~ k>d = <Ja~aJA’~A;k’~PaAJ’~A~la).jA2A;k>, (B-9)

where

‘~aA = >.I:o~p (B-4’)

is the operatorantisymmetrisingthe nucleonsin particlea with thosein particleA, P and~ beingapermutationandits signature,respectively.The summationis over (A + a) !(a !A !) 1 permuta-tions. Eq. (B-9) thereforecontainsadirect term andvariouskinds of exchangeterms

T = <direct> + ~.f(n)~~ <exchange~n>. (B-b)

The summationin (B-b) extendsover all classesn wheren is the numberof transpositionscharac-teristic for that class.The numberf(n) is the multiplicity of class n, and ö,, the classsignature.The exchangeprocessof fig. 27 (exchangeof particlee) belongsto oneand only oneclassn. Ifthis classresultsfrom an evennumberof pair transpositions:

ö(aeA) = 55(e) = + 1,

for an oddnumberwe haveö(aeA) = — I.

To illustrate this procedurewe considerdeuteron—tritonscatteringvia neutron exchange.

The Born T-matrix (in isospin formalism) readsT = <d(12),t(345);k’~PdtVdt(l

2, 345)~d(12),t(345);k>, (B-l I)

with

127 5 ‘\

‘~dt= j~I( i — ~ pJ. (B-4”)j=i+1 /

The decompositionof the T-matrix is showndiagrammaticallyin fig. 28. The third graph is theclass of exchangeterms correspondingto neutron exchange:f(n) = 3 and ~(dnt) = + I. The

M.P. Locherand T.Mizutani, The useof analyticityin nuclearphysics 87

~0RNx -6 +3direct exchange exchange

(I trxnspxa~tixx( (2 tranapxsitions(

Fig. 28. Symmetrystructureofdirect andexchangegraphsfor dt elasticscattering.

correspondingcontributionto the T-matrix is

T(n-exch.)= 3 <d(45),t(123);k’~Vdt(l2,345)Id(12),t(345);k>. (B-12)

Utilising the Schrödingerequationdescribingthe binding of nd to form t by the potential T’~d

it is possibleto transform(B-12) intoT(n-exch.)=<t(i23)~J<~d(12’3)~d(l2),n(3);q’>~~<d(4S),n(3); q~~‘,d(45,3)~t(345)>

K2 (k + k’)2 K’2x~ E——————— ~ , (B-12)

2md 2m~ 2md j

where

q = k’ + m5k(in0 + md) ~, q’ = k + m5k’(m0 + md) 1, and CE = K

2(2~dt) ~.

Observethat the numeratoris just a productof vertex functions Gtdfl definedin eq. (B-2). Eq.(B-12’) is thereforeequivalentto the desiredexpression(A-l3) for elasticscattering.

Generalisationsof eq. (B-12’) to otherexchangeprocessesareobvious.In particularoneobtainsthe signatureö(ndt) = — 1 for deuteronexchangein nt elastic scattering, 5(ddc~)= +1 for d-exchangein dx scattering,etc.

Appendix C. Standard exemples

As an illustration to appendicesA andB we give explicit formulaefor nd andnt elasticscatter-ing.

(i) Poleresiduesin theforward nd amplitude.In order that the unpolarisedtotal crosssectioncanbe usedfor the dispersionintegralin the evaluationof the discrepancyfunction thespin averagedamplitude is introduced,cf. section3.1 andeq. (A-6):

Cf(CE) = . ~. ~ ~~A~E) (cm). (C-i)[mid] A~A~

Note that Im Cj(CE) = Ka(CE)(4ir) 1 in termsof the unpolarisedtotal crosssection.At low energyCj(CE) [1/2] Cf l/2(CE) + [3/2] Cf3/2(CE) = lCfl/2(CE) + ~f3/2(CE), (C-2)

nfd nid

where Cf l/2(CE) and Cf 312(cE) are doublet and quartetforward amplitudes,respectively.Cf(CE)

hastwo poles,thedirect triton poleandtheprotonexchangepole.The triton pole residueis obtainedfrom eqs. (A-5) to (A-7). Neglecting the triton D-state it

88 M.P. Locherand T. Mizutani, The useof analyticity in nuclearphysics

occursonly in the doubletamplitude.The residuein the spin averagedamplitude(C-2) is

= ~i~4~G)2 (cm), (C-3)

whereG1 G11~

0(iicj, cf. eq. (A-3).The proton exchangeresidueis foundusingeqs. (A-22) to (A-24) wherewe neglectthe deuteron

D-state.The pole appearsboth in the doubletand in thequartetamplitudes.The residuein thespin averagedamplitude(C-2) is

p(p-exch.)= ~‘m~ [~-(—~G~~0)+ ~G~pd]= (cm), (C-4)

with

Gd~fl G~o(iKd).

To relate the aboveresiduesto the asymptoticnormalisationconstantswe use (B-6), (B-7),

andthevaluesof d, F, v in table4 to find

= —~— C~, p(p-exch.)= ~ (~ (cm). (C-5)Pnp

In the literature the lab residuesr~and r1 are often used.Writing

cf(CE) ~ (cm),pole E

f(E) (lab),Epoie E

for the pole contributions [compare with eqs. (9), (10) and (16) main text] the relation for anykind of polesis

r = (ma + mA~2 (C-6)

\ m~ /

where ina, mA aremassesof projectileandtarget,respectively.In our case

= = 27k1 ~ r(p-exch.)= ~d/~nd ~ = -~- C~. (C-S’)

l6mN 8Rnp

(Note that mixed notationslike Cf(E) using cm amplitudesand lab energiesalso occur, cf. e.g.,Plattneret al. [91]. Eq. (C-6) must be modified accordingly.)

(ii) Deuteronexchangeresiduein the unpoiarisednt differential crosssection:Neglectingthe tritonD-statewe apply eqs. (A-18) to (A-21) andwe get

~ K)~dpoIe= 2G~(— )2X {i ~ ~ (C7)

M.P. Locherand T.Mizutani, The useof analyticity in nuclearphysics 89

whereG~is the sameasin (C-3) andthe residuein (A-21) becomes

2 2rr1 .1. i~2 (.1. 1 ii21—2 ~ufl~mdIJ2 2 ‘( ~ 1( 4Pd-exeh. — A 2 I xi ~ j. + 3~~ ~ G~

-t7t L~2 2 ‘-‘) (2 ~ ~‘) J

= ~‘~‘ [~+ 3(l/36)]G~= 3~2’°I~= 2.l36I~= 8.543,c~0C~. (C-8)

Appendix D. Estimate of u- and t-channel anomalouscuts

We are calculatingthe t-channeltriangle, fig. 6, assumingconstantvertices.Spin is treatedeffectively by relating the parametersto observedcross sections.The contribution to the ndT-matrix is~’

1(K, 0) = — 8~N~m~‘aNN( — t) 1/2 arcsin~/~t( — t + l6ic2) 1, (D.i)

with= mNCd

wheret = — 2K2(1 — cos0) is the momentumtransferandaNN = — mNTNN(41r) 1 is the effectivespin averagedNN amplitude in its cm system.For thedeuteronvertexwe shall usethe HulthénvalueN~= 2Kf3(ic + /3)(fl — 2 = 155 MeV. Thenuclearpartin eq.(D-b) is identicalto thelongrange expansionof a Hulthén form factor. For the purposeof illustration we shall chooseLE = 6 MeV wherethe effectiveamplitude hasbeendeterminedas aNN = — 3.25 fm from a fitto experiment(LocherandMizutani [76]). Thedefinition of aNN includesafactor two for thecohe-rent scatteringoff thetwo targetnucleons.

By asimilar treatmentof the (pn°)exchangewe obtain for the u-channeltriangle of fig. 9 theT-matrix

T~i(K,0) = 2\/ /RNNNdl~,P~.dl~Aarcsin[A(c~2+ A2)~2], (D-2)

where

A2 = K2(~+ cosO), = ,c + m~.

Thequantity A2 is the invariantmomentumtransferin the exchangechannelup to aconstant,seeeq. (20).

In the following we calculate 7~and L, for LE = 6 MeV which is a typical energyfor poleextrapolations.We takeas generousupperlimit thecrosssectionfor pp —+ dir~at the A resonance

= (mNmR)112(27tY‘Qa~7,p—xda~2 (D-3)

(Q,~is thepion momentumin theird cmsystem),to obtainan upperlimit for the effectivex-produc-

tion T-matrix:~,p..di~6MeV < 3.i x iO~MeV2. (D-4)

* Recall our normalisationCf = 4fld(2~) T = —,nN(3n)~T.

9)) MY. Locherand T. Mizutani. The useof analyticity in nuclearphysics

In table5 we showthe results.For forwarddispersionrelationsthe u-channeltriangleis a back-ground for the proton exchangeand the triton pole. It is found to be negligiblealsoin this com-parison. Note that the values for the u-triangle are a generousupper limit, whereasall othercontributionsshownhavetheir realisticmagnitude.Two-pion exchangein the t-channelis negli-gible andnot listed in thetable.

Table5Contributionsof varioussingularitiesto theelas-tic nd scattering T-matrix at LE~= 6 MeV in

units lo~ MeV2

cmangle 5’ 90 75

triton pole 5.0 5.0 5.0p-exchangepole 1.5 2.4 6.0t-triangle 3.6 3.3 3.)u-triangle 0.18 0.18 0.l9

We concludeby giving the expressionfor the triton pole and the proton exchangepolewhichhavebeenusedin the table. For the triton pole we havelistedthe full doublet T~matrix:*

= — -~ ~—--~--- (cm), (D-6)/1nd t~

whereCE = — 6.26 MeV and r, = 0.382 from ref. [73]. For the proton exchangeT-matrix wewrite

T~(K,0)= — ~-~4(~+ cos0+ K2/K2)~. (D-7)1~NK

Eq. (D-7) is for spinlessparticles.For the table we haveusedthe deuteronvaluesN~= 155 MeVand K = 46 MeV/c. To obtain the contribution to the doublet andquartet amplitude multiplyby —4 and 1, respectively.

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