Nuclear Phyalcs A272 (1976) 327-337 ; © North-RollandPwbIWtinp Co., AnraterdartNot to be reproduced bY photoprlnt or microfilm withoat wrlttaa permlw{on from the publi.her

APPLICATION OF THE 1VIETHOD OF INTEGRAL EQUATIONSFOR CALCULATING THE VERTEX CONSTANTS

FOR AN a-PARTICLE

A. G. BARYSHNIKOV and L . D. BLOKHINTSEV

Institute of Nuclear Physics, Moscow State University, Mosrnw, USSR

and

I. M . NARODETSKYInstitute for Theoretical and Experimental Physics, Moscow, USSR

Received 10 October 1975(Revised 12 August 1976)

Abstract : The vertex constants Gr:, .�, and G~ for the virtual decays a ~~ t(s)+p(n) and a y d+d arecalculated by solving the Faddeev-Yakubovslry equations for four nucleons . A spin-dependrntseparable potential with the Hulthén form factor is used a8 the NN interaction. The resonantHilbert-Schmidt expansion is applied to solve the integral equations. The values obtainedG,?,.�, = 17.9 t 1 .7 fm and G~ = 18 .1 t 1 .3 fm are compared with the phenomenological valuesextracted from áata on nuclear reactions.

1. IntroductionIn recent years the vertex constants characterising the virtual decay a -" b+c

ofthe nucleus into two fragments have found increasing application in nuclear physics[see, for example, refs.'-`)] . The vertex constants, coinciding (apart from kinemat-ical factors) with the matrix element for the decay a -~ b+c on the mass shell, arethe model-independent characteristics of the given process, being similar in theirmeaning to the coupling constants in elementary-particle physics. Therefore vertexconstants appear in a natural way in various dispersion approaches to nuclearreaction theory t - 8) . Furthermore, a knowledge ofthe appropriate vertex constanfsallöws one, as a rule, to calculate the partial wave amplitudes for large orbital angularmomentum. The vertex constants define the distribution density of the fragmentsbandc at large distances and, therefore, can also be calculated from the known formof a wave function . Comparison of the theoretical values of the vertex constantscalculated from the microscopic nuclear structure theory with the phenomenologicalvalue obtained from the analysis of the experimental data on nuclear reactionsenables one to extract valuable information on the nucleon-nucleon interaction .So far microscopic calculations ofvertex constants have been made only for the

simplest two- and throe-nucleon systems 9- tt) . The present paper gives the results327

328

A. G. BARYSHNIKOV et á,

of the first calculation of these constants for the virtual decays a -. t(s)+p(n) anda -" d+d t, where the labels t and s denote, respectively, the fragments 3H and3He tt .Our calculations are based on solving the Faddeev-Yakubovsky equations t')

for four nucleons. The possibility of the practical solution of these equations hasrecently been demonstrated in refs . t°-ts) . We use the spin~ependent separableNN potential ofYamaguchi type . Theparameters ofthe potential used can be foundin ref. te).The present paper is organized as follows. Sect . 2 contains the definitions and

some general relations for the vertex constants. The results of the calculation of thevertex constants are presented in sect . 3. Sect . 4 contains a comparison with empiricaldata and discussion .

2. Some general relations for the vertex constantsThe mass, kinetic energy, momentum and c.m . radius vector of the particle i,

as well as its spin and spin projection will be labelled m,, E,, pt, r t, J, and M, respec-tively . Let then F~,k = mtm~/(m,+m,~, q �~ _ (mtpk-mt,p,)l(m,+m,~ be the reducedmass and the momentum ofrelative motion ofthe particles i, kttt,

Et~ = m t+m k-m�and trt = E, - prl~~-The nonrelativistic matrix element for the virtual decay a ~ b+ccan be written as

-~Ka~r~,r~ =

4n ~ G;û~(6., Q~ Q~xJ~M~~M~ISmsximrSmsIJ.MJYu~,(4u~~

(1)ts~ �.~ .

where (apaß~ B ~yu~) are the Clebsch-Gordan coefficients, Y,��(¢) is the sphericalfunction (Y,*�,(¢) _ (-)~'Y,_m,(~)) and ~ = qlq . The value of the invariant formfactor Gb~(tr� trb, tr~) on the mass shell, i.e . at Q, = Qb = Q~ = 0 will be referredto as the vertex constant G;~. The quantities G,~ have the dimension of (length)and will be expressed in units of fm~.The vertex constant G;b~ is connected with residue of the elastic scattering am-

plitude of the fragments .b and c taken at the pole with respect to the variableEb~= q~,f2~ corresponding to the intermediate nucleus a as follows

1fie partial wave amplitudes in eq . (2) are normalised so that the dif%rential crosssection for unpolarized particles is

2

(ZJ~+ 1x2Jb+1) ss-,~~~

f Some preliminary results sre presented in ref. 1 ~).rT In what followsweassume the validity of isobaric invariance and usethe label N for the nucleonspand

n and the label T for the three-particle fragments 'H and 'He.r~r Hr~fter we put A = c = 1 .

where

~9~S~mslF~I9~Smsi = 4n

~

(ImtSms~J11~~eru ".u.~

vERTIX CONSTANTS

329

x(I~míS~ms1~111~<JI~S~IF1xIJIS~ Y~*�(~~)3i".~~ " (4í, .)~

In eq . (4) the quantities 1 and S are the orbital angular momentum and the spin ofthe channel . If 1 and S are conserved then

<J1S~F~~JIS) _ - 2n exp (iS~ts) sin S,,ts

( )5

4n.where 8�s is the scattering phase shift.The vertex constant G~ can also be related to the coefficient in the asymptotics

of the overlap integral I,~(r) of the normalized wave functions >/~� ~b and ~. ofthe nuclei a, b and c [ref. `)].

Lbc(r) = J~b(Tt,À`~(T.~.(TtP T~,r~TbdT.

_ ~ til;n.(rxJt,Mt,~.M.ISmsx~rSmsIJ~Yr.~~(P~fs~ .,.,

where Tb, T~ are the sets of coordinates (including spin and isospin) of the nuclei band c, and r = rb-r~. When r-. oo we have I;~(r)~ C,~exp(-x,~r)/r t, whenx;~ = 2~e,~ and C,~ is the real constant related to Gb~ by

In eq . (~ the factor N~ is due to the identity of nucleons . In the isospin formalismthis factor is N~ _ (Ab+A~!/Ab!A~1, where Ab and A~ are the mass numbers ofthe b- and o-nuclei . If the antisymmetrization is carried out separately for neutronsand protons, then Nb~ _ (Nb+NJ !(Zn+Z~!/Nb!N~ 1Zb !Z~ l, where Nb(Zb) andN~(Z~) are the number of neutrons (protons) in the nuclei b and c +t .

In the three-dimensional Schrödinger formalism, which is used in the presentpaper, the form factors in eq . (1) are functions of one invariant variable qn~ only :G;~(Qar Qb+ Q~ = Gbc(gba) and the vertex constant is

G:~ = G.û.(ix.u.)~

r This formula u valid in most cases including that ofa being an a-perfide, though in some cases theasymptotic behaviour ofthe overlap integral may be different .rr In numerical calculations, model wave functions for the a-nucleus are sometimes used, which totally

ignore the identity ofthenucleons including in theb-fragmentwith the nucleons included inthe o-fragment,i .e . use is made of the functions f4, whicó are not antisymmetrized with respect to the permutation ofthenucleons between the b- and o-fragments (the cluster model of 6 Li may serve as an example) . In this caseN~ = 1 .

330

A. G. BARYSHNIKOV et a1.

In this formalism from eqs. (1) and (6) it is easy to derive the equivalent expressions

Gabc(gbc) _ (- ~(4nN~)~

~,

(JeM~oMoISms)~agaa111b11[o

or

x (im tSms~J,MJJ1!(gbcr)YIn ~(P)~b(zb)~ó(tic)Vbo~a(TÓ+ za r~tbds,d 3r

/ z

lGaba(gno) _ -(4nN~)} 12~ +E,~ /

Jol~4~r)1~(r)r2~,(10)

where V~is theinteractionbetween th\e b- and o-fragments andjt(q~r) is the spherical

Basel function . Accordingto eq . (8) the vertex constant Gam, is determined by expres-sions (9) or (10) at q~ = ix,b , and for eq . (10) one should use the limiting proceduresince its right-hand side involves an uncertainty of the 0 x oo type when qb~ -. irc,b~ .The vertex constants are related to the reduced width 9~ and the spectroscopic

factors S, s by

S,s = N~Z,~~,

Gdnv --_ (G°droN)s = ~al~(Ei- E~Frr>(Ei~

Goeaa _ (Go~z _ $~~(Ez -Ea~aa(Ex~

where h~I~ is the spherical Hankel function of the first kind and ro is the channelradius.

In the case of an a-particle, from the angular momentum and parity conservationlaws it follows that for thevertex a ~T+N l = S =~ 0and for the vertex a -. d+d,1 = S = 0 or 1 = S = 2. Confining ourselves to the main component with1 = S = 0 and denoting the amplitudes of the TN and dd scattering in astate with1 = S = T = 0 (T being the isospin) by F1.N(El ) and F~(Ez), we obtain fromeq. (2)

(12)

(13)

where El = E, .e, +ET, EZ = E~+ 2Ed, E,.r,(Ea~ is the kinetic energy of relativemotion of T and N(oftwo deuterons), Ed < 0, ET < 0 and E, < 0 are the bindingenergies of a deuteron, triton and an a-particle, respedively, and the factor ~ ineq . (12) is due to the isos~tin kinematics . The normalization of the amplitudes F,.,~

and Fdd in eqs. (12) and (13) is consistent with eq . (5) :

VERTEX CONSTANTS

331

8n

sin &,.

2~c

sin ST

d

where m is the nucleon mass, KT = ~mF, .,.,,,, Ká = 2mEda , and ST and Sd are theappropriate scattering phase shifts .We note that our constant G,zt,.i,, is related in the following manner to the dimension-

less quantities (g~rn ;o)z , ro, RN and Cz used, respectively, in refs . z " 19-z~)

GaN = ~z~~~: o) z = ~atNro = ~rt~tNRN = (Go)ZCZ ,

(l5)where (Go)z = 9~(KdI'N~N)~N, with ~N = 0.210 fm the Compton wavelength of anucleon .

3. Calcalxtion of the vertex constantsIn this section we present results for the vertex constants Ga2t.�� , Ga",.,a and G~,

where a* denotes the excited 0+ state of 4He. Our starting point is the coupled setof one-dimensional integral equations, which can be found from the Faddeev-Yakubovsky equations by using Hilbert-Schmidt (HS) representation for 3+ 1 and2+2 subamplitudes'6 " zz) t

m

~z

~. �(q+ z) = e. t(z) ~JD: "a(4~ 4 ; z)~� . Cz- ~~A.a~4~ ; z)q~ zd9~,

where z is the total energy ofthe system and e,(z) are the four-body eigenvalues tt_

The generalized potentials C,'�, . and D,'m. and the generalized propagators ~� and$,', are defined in ref.'s) ttt" TyeeigenfundionsA.~(q;z) andB,t.(q;z) are normalized as

( qz

z1~0, 1

Making use of the HS expansions for the four-body amplitudes and taking intof Equations similar to ours have been obtainod by Alt et al . ") .tt The ~~, levels E, sre determined by the equation e,(E,) = 1 .ttt Note the misprint in ref.' 6) : the sums over k in the r .h.s . ofeqs . (1~ and (25) should be muhipliedby 2.

332

A. G. BARYSHNIKOV et af.

account the definitions (12), (13) we arrive at the following expressions for the vertexconstants

with

8nxG?rn =

LAi t(~dt1v ; EJ~x,3fY,,Y.

8axGá"r~ =

~

LAxa(iKa "~; Ea ")~x,

(18)3V'Ye%é

Kx1nv = 2m(ET - EJ,

Ká"rN = 2m(ET- Ea.),

xmaa = 2m(2Ed -EJ,

Ká"aa = ~(2Ea-Ea") .

(19)

Ineq . (18) Y~ _ (fit/~)~=s.+ Y. _ (fix/~)~-g." ~ Y,, _ (~ü/~)s-s*~ Y; _ (fit/~~-xs,where q and ~ are the eigenvalues for the 3 + 1 and 2 + 2 subsystems.As the first step we have to solve the eigenvalue equations (16), (17) for positive

values of qx . The number of integral equations depends on the numbers N� and N,',of3 + 1 and2+2 eigenfunctions which are kept in the four-body equations. In ref. ' e)the values Ea = -45.73 MeV and E~. _ -11 .69 MeV were found for N,~,+~ = 4,N<~ 1 = 3, N° = N,' = 4 where N~t' are thè number of 3 + 1 eigenfunctions corre-sponding to the positive and negative eigenvalues q . Hereafter, unless otherwisestated, we use the approximation N~* ~ = 2, Nt~ ' = 1, N° = N~ = 2 to whichcorrespond the following values of the a-particle ground and excited state bindingenergies : Ea = -45.69 MeV, E,. _ -11 .33 MeV. Use of this approximation doesnot practically affect the accuracy of calculations but reduces substantially thecomputer time required for calculations .For the direct .calculation of the vertex constants we need the values of the funo-

tions A,t(q ; E,), g.°i(q ; E,) (r = 1, 2; E t = Ea , EZ = E,r") at the imaginary valuesq = irc wherex is determined by eqs. (19) t : á.r�, =1 .671 fm-x , xx".rn, = 0.0145 fm - x,ááa = 1 .989 fm -x , x~ = 0.331 fm- x .For the analytic continuation of the functions A,t(q ; E,) and E.°i(q ; E,) from the

region of positive momentum values qz we use eqs. (16) in which the values of thefunctionsA.~(q; E,) (1 ~ n 5 N�) and B'�(q ; E,) (1 ~ n S N�) for qx > 0, obtainedfrom the direct solution of integral equations, are takenas the input data and the

r We use, naturally, the calculated values Er ~ -11.03 MeV, E, a -45.69 MeV and E, " ~ - 11 .33MeV corresponding to the chosen NN interaction.

1

VERTEX CONSTANTS

333

kernels ~~.(q, q' ; E,), D:� .(q, q' ; E,) are calculated at a purely imaginary value ofone of the arguments q = ix . The values of the functions A~ t(q ; E,) and B°t(q ; Et)thus obtained are represented by the section ofthe solid curve on the left ofthe pointqi = 0 in figs . 1-3. Unfortunately, a direct analytic continuation ofthe vertex func-tions is feasible only for q2 > - qó where qó = 1 .32 fin -2 for the decay a -. T+N,qó = 1 .10 fin -s for the decay a -" d+d and qû = 0.25 fm_a for the decay

-.~�� -~-o

~

2

a

~

s

e

Fig. 1 . The vertex ßmction A 1 ,(q ; EJ for the virtual decay a ~ T+N. Curve 1 is obtainod in theapproximation N~+ ~ = 2, N;- ~ = 1, N° = N; = 2. Gtirve 2 represents the values N~+~ = 1, N~~ ~ 0,N° = N; = 1 . The approximation N~+~ = 2, N~ ~ = 1, N° = 1V,' = 1 leads to a curvewhich practicallyresembles curve 1 and therefore it is deleted from the ßgure . The solid portions ofthe curvy are obtainedfmm the solution ofeq . (16) (q' > 0) or from the calculation of the integral: on the right-head side of

e9 . (l~ (9 = < 0). The dashed portions are plotted using the extrapolation procedure .

~s

Fig . 2. The vertex function ~,(q ; E, for the virtual decay a y d+d . The slid portion of the curve isplotted as indintod in the caption offig. 1 ; the dashedportion isobtained by theextrapolation.

334

A. G. BARYSHNIKOV et al.

Fig . 3 . The vertex function .lz,(q ; E~,) for the virtual process a' -+ T+N.

a* -" d+d. This comes from the fact that at imaginary q for calculating the kernelsC and D one has to use integral expressions te) ; when q2 5 - qó the singularitiesof the integrands in these expressions cross the contour ofintegration . Therefore wecalculate the values ofthe vertex functions in the region qi < - qó using the analyticapproximation and the extrapolation proof dare . The various methods ofthe analyticapproximation, which well describe the vertex functions in the region qZ > -qólead to similar values for the quantities A 11(itca.rn, ; EJ and B°1(itc~ ; EJ and,respectively, to similar values of the constants G,Z~. rr, and G~. The values of thefunctions A tt(q ; EJ and B°t(q ; EJ derived by extrapolation are shown by dashedcurves in figs . 1 and 2. For the decay a* ~ T+N the extrapolation described isnot required since, due to the similarity of the values E,, and E,. the value of thefunction AZt(iKa " , .N ; E~.) can be calculated directly from eq . (16). Unfortunately,for the virtual decay a* -. d+dthe various methods of the polynomial approxima-tion lead to significantly different values ~t (itc~.~; E~,) . Therefore we have beenunable to determine reliably the value ofthe vertex constant G~~for the calculationof which one has apparently to use more refined methods.Our results obtained by averaging over different variants of the analytic approx-

imation are presented in table 1 along with the values of the rms deviations T . Inthe calculations we have used valued ofthe constants y, and yc occurring in eqs. (18) .

f The value G~ = ig .l t1.3 6n is refined as compered with that given in ref. '_) and is obtained byimproving the extrapolation procedure .

x= (fm-~)

1 .671

1 .989

0.0145.l, 1 (ix ; E,) (~si~)

4.806f 0.113

0.9699~,(ix ; E~ (~3n)

3.757f0.132G~ (fm)

17.9

f1.7

18.1

f1 .3

0.059

VERTEX CONSTANTS

335

TABLE 1

The results ofthe calculation of the vertex constants for an a-partide

a -~ T+N

a ~ d+d

a " -. T+N

TABLE 2Nustration ofthe convergence ofthe HS method

which are equal, respectively, to 0.035 MeV-1 and 0.1fÓ MeV-1 . These values werederived from the solution of the 3+ 1 and 2+2 problems .To conclude, we consider the convergence of the HS method in the problem of

calculation of the vertex constant G;.N. Table 2 gives the results of the calculationfor several values N~ and N� . The results of this table show that even a few firstapproximations ofour method give the value ofthe required constant to a reasonableaccuracy . A further increase of the number of terms in the HS expansion leads tocorrections which are much smaller that the error determined by our procedure ofthe analytic extrapolation .The good convergence of the HS expansion also follows from the comparison of

curves 1 and 2 in fig. 1 .

4. Results and àiscuesion

Due to the absence of other microscopic calculations of the vertex constantsGdZrr, and G~ we can compare the obtained values of these constants only withthe phenomenological values extracted from the data on nuclear reactions .The values G~ and G,~~, obtained with different methods are presented in table 3.

The letters tp or ~cn in the second column indicate to which of these two constantsa given value belongs. We see that the spread of the phenomenological values G,Zrrr,is rather great ; as it was to be expected, G~ G~. Some overestimation of theobtained value G,?t�, = 17.9f 1 .7 fm compared with the phenomenological valuescan be due to a marked overbinding of an a-particle for the NN potential used .There are very sparse data on the constant Gam. An analysis of the elastic da

N~ N° N;, Z(MeV)

Y,(MeV- ')

~nv(fm-2)

A~i(ix rn ; E~(fm'~2)

3GsT1+(fm)

1 1 -44.29 0.0147 1 .604 4.132 13.03 1 1 -45.42 0.0145 1 .658 4.679 16.93 2 2 -45.69 0.0144 1 .671 4.806 17.9

33 6 A. Ci . BARYSHNIKOV et al.

TABLE 3The values of G~ and G,~t, obtained with different methods

') We notethat the sign ofthe pole contribution to the K-matrix in ref. ~°) is in error. This mistake doesnot affect the value G,~,p = 10.0 fm obtained in the pure pole approximation for the K-matrix. On theother hand, the value ofG~ obtained in ref. ~°) by solving the integral equation for theKmatrix shouldbe changed from 12 .2 fm to 7.1 fm .

scattering and the (d, a) reaction in the frame of the peripheral model ze) yieldsG~ = 12-30 fm, which does not contradict our value 18 .1 f1 .3 fm.

It is interesting to note that the value G~Gd~trt x 1 which follows from ourresults agrees with the predictions based on eqs. (2) and (3) of ref. ze) t .

Microscopic calculations of the vertex constants are more tedious than calcula-tions of binding energies since they require the knowledge of wave functions atimaginary momentum vahus (in momentum representation) or at largeinterfragmentdistances (in coordinate representation). On the other hand, as follows from thecalculations of the vertex constant for the decay t -. d+n [refs. 9-tt )], the vertexconstants are rather sensitive to the type ofNN interaction and can supply valuableinformation on it . In addition knowledge of the vertex wnstants is essential fornuclear reaction theory . Therefore we consider the vertex constants to be importantnuclear characteristics which should be calculated microscopically along with suchquantities as binding energies, rms radii, magnetic and quadrupole moments, etc.

Note added in proof Recently a paper has appearod [S . Dubni~ka and O. Dum-brajs, Phys . Lett.~57B (1975) 327] in which the addcoupling constant has been foundby extrapolating the elastic da scattering differential cross section to the deuteronexdumge pole. Täe value G~ x 0.4 im which can be inferrod from that paper isin sharp disagreement both with our theoretical value 18.1 fm and with thephenomenological values obtained from nuclear reaction data .

t We take the opportunity to note that the minus sign should be substituted for the plus sign on theright-hand aides of eqa. (2) 'and (3) of ref. 36)

Method G;~ or G;s (fm) Refs.

Peripheral model, reactions s(d, p1a, a(P, d}r, a(p, a)p 7.3 t 0.4 (tp, rn) 14-I6)Dispersion relations for forward scattering amplitude, 11 .4f 1 .4 (rn) 19)

na scatteringDispersion relations for forward scattering amplitude, 13 .4 t l .l (tp) so)

pa and na scattering 10.6t1 .1 (rn)Method of tonformal mapping, na scattering 8.S t 1 .3 (sn) )Parametriration of relative motion wave function oft 6.7 (tp) II)andporsandn 7.1 (sn)

Dispersion Kmatrix approach, pa and na 7.0 (tp) 7)scattering 8.0-8 .5 (sn)

Disperaion K-matrix approach, to scattering 7.1-10.0 (tp) ~°) .)Exchange mechanism of sa scattering 17 .5 (rn) i9)Microscopic calculations 17.9t 1.7 (tp, sn) present paper

VERTEX CONSTANTS

337

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