PHYSICAL REVIEW C VOLUME 13, NUMBER 6 JUNE 1976

Suppression effects in two-channel separable 'Q AX potentials

Patrick J. Hooyman and L. H. SchickUniversity of Wyoming, Laramie, Wyoming 82071

(Received 31 March 1975)

A two-channel YN potential with a single nonlocal separable form in each of the four potential matrix ele-ments is used to represent the low-energy '$0 AN interaction. An attempt is made to determine the param-eters of this model by matching its prediction for the AN scattering length and effective range to the valuesgiven by the 1972 Brown, Downs, and Iddings (BDI2) meson-theoretic potential with the AN ~ZN and theZN ~ZN interactions unsuppressed and completely suppressed, It is shown that independent of the shapesassumed for the separable forms this cannot be done. With only AN ~ZN suppression used in the determin-ation of the AN scattering parameters, it is found that, as in earlier works, an unwanted AN resonance occurs.By varying one of these parameters away from the BDI2 value, this resonance is eliminated, and the shapedependence of the AN phase shift is investigated for this case.

NUCLEAR REACTIONS YN potential, A%scattering length and effective ranges,two-c&~~~el separable potential, AN ZN suppression.

I. INTRODUCTION

Schick and Tyagi (ST)' and Damle and Schick(DS)' investigated the possibility of using a phe-nomenological two-channel YN potential with asingle nonlocal separable (NLS) form in each offour potential matrix elements to represent thelow- energy S-wave AN interaction. They foundthat fitting the 1970 low-energy scattering param-eters taken from the meson-theoretic potential(MTP) of Brown, Downs, and Iddings' an unwanted'S, resonance below the Z-channel threshold wasalways present. ' This present work is an exten-sion of the work of ST and DS utilizing a revisedset of potential parameters obtained from an MTPby BDI in 1972.'

The 2 x 2 coupled channel potential energy oper-ator used in BDI1 and BDI2 took the form

«hC

~Vch yVcc

where V» represents the interaction YN-X¹Here z is a cross-channel suppression parametervarying from 0 to 1. E = 1 gives no suppression ofthe Z charnel while & =0 gives complete suppres-sion. A number of works have shown that in somehypernuclear systems the AN —ZN transition maybe at least partially suppressed. ' The parameterg is used to represent this suppression. The pa-rameter y was introduced in BDI2 and is used tomeasure how important the contribution of V« isto the low-energy AN potential. We shall refer theuse of values of & and y other than one as & sup-pression and y suppression, respectively.

Following ST and DS we write V as

V ~hh hvh Ckhcvhvc

E'Xchv g vh yXccvcvc

vh ' Xhh Xh c vh

0 vc l c~ch yacc 0 vc

where each X„ is a strength parameter, and v„ isa shape function. In a relative momentum spacerepresentation each matrix element Vx~ takes theform

V»lor) = X»vx(u'r)vr(k„) . (3)

For convenience we defined X~ = g~X»/2v, Zc= yc&«/2v, and Ax'= p~ pcX~c'/4w', where p„ isthe YN reduced mass. We used 939, 1115, and1193 MeV for the masses of the nucleon, A, and Z,respectively. We have set I= c = 1 throughout thiswork.

We are interested in producing a simple phe-nomenological AN potential for use in binding en-ergy calculations of hypernuclei. Hence it makesmore sense to determine the potential parametersin V by fitting the BDI2 AN scattering lengths andeffective ranges for various values of & and y thanto fit the BDI2 higher-energy phase shifts. Fur-thermore, since the MTP described in BDI2 in-cluded a tensor part while the NLS potential weused did not, we restricted our analysis to the'So AN interaction. The values of the 'S, AN low-energy scattering parameters for different & and

y as given in BDI2 are reproduced here in Table I.Using the potential given in Eq. (3) therefore, we

solved, as in ST, the two-channel Lippmann-

2454

13 SUPPRESSION EFFECTS IN TWO- CHANNEL SEPARABLE. . . 2455

fp

Suppression of VA~ with V&& at the full model value

1.0 1.00.750.500.250.0

1077-1.60-1.49-1.44-1.42

3.183.373.513.603.64

Suppression of V&& with VA& at the full model value

1.00.750.500.250.0

1.0 -1.77-1.77-1.78-1.78-1.78

3 ' 183.183.183.173.17

Suppression of VA& with V&& completely suppressed

0.0 1.00.750.500.250.0

-1.78-1.61-1.50-1.44-1.42

3.173.363.513.603.64

TABLE I. The BDI2 Ap ~SO scattering length and ef-fective range as functions of e and y. All lengths arein fm.

II. e AND p SUPPRESSION

Because we want to work with different values ofy, rather than use the notation of the Appendixwhere this parameter is suppressed, we shallwrite the Ap scattering length and effective rangeas a(e, y) =—I/n(e, y) and r, (&, y}, respectively. Fore = 1 and y arbitrary Eqs. (A8) and (A9) may be re-written as

t(r)[n(I, r) —n(o, I}lie&.h, '/g, .']u, [n(0, 1)+A,][n(l, y)+A, ]

(4)

if the choice of fitting the BDI2 data at g =0 isj. j. 3changed to &=&, &, or —,.

In the remainder of Sec. III we changed the BDI2parameters for q = 0 to values for which the un-wanted resonance was eliminated. For two suchcases the dependence of our results on the shapesv~ and v~ is described. In particular we presentresults showing that our method of determining thepotential parameters is really different from fittingthe AN phase shift for energies up to the ZNthreshold.

Schwinger equation for the AN 'Sp scattering prob-lem. For an arbitrary value of y we then assumedthat the AN scattering length and effective rangeat & =p and & = q were known and we obtained onecondition on the potential shape v~(h„) and threeother equations giving, respectively, X„, X~, and

Xx in terms of the potential shapes and the"known" scattering parameters. These four equa-tions are given in the Appendix. Using a no- reso-nance criterion developed by SD we could checkfor the presence of a resonance below the ZNthreshold.

In Sec. 0 we tried to match both the & and the ysuppression results of BDI2 in the following way.We took the "known" 'S, AN scattering lengths andeffective ranges' to be those given in Table I for&=0,' &=1, y=1, and &=1, y=v1. We foundin-dependent of the shapes v„and vc that with theseBDI2 values we could not satisfy Xr'&0 (unitarity).

In Sec; III we tried to match only the E suppres-sion results of BDI2 in the following way. We sety = 1 and took the "known" 'S, AN scattering lengthsand effective ranges to be those given in Table I for& = 1 and g = 0. We chose to fit at & = 1 since thisshould be appropriate for most light hypernucleiand by definition it holds for Ap scattering. Thechoice of E =0 was based on simplicity. We pre-sent results showing that for at least eight differ-ent one-parameter shapes of v~, independent ofthe shape v~, there is still an AN resonance belowthe ZN threshold. We then show this is the case

[n(1,y)+A, ]h, '[n(0, 1)+A,][u,—}t(y)]

'

respectively, where

[ro(1 y) ro(0, 1)]2[n(1, y) —n(0, 1)]

(6)

Since from Eq. (6) }t(v) and }t(1) may be calculateddirectly from the BDI2 results it follows from thislast equation that u„may be so calculated also.

If we were to consider only & suppression, thenwe would set y=1 in these last three equations andproceed to investigate specific potential shapes aswe do in the next section. Here, however, we wantto include y suppression as well. Not only do weinsist our model reproduce the BDI2 values ofn(0, 1), x,(0, 1), n(1, 1), and r, (1,1), but also theBDI2 values for n(1, v) and r, (1, v) where v issome specific value of y other than 1. This meansthe last three equations are to hold both for y= 1and y= v when the BDI2 values are inserted intheir respective right hand sides. But the lefthand side of Eq. (4) is according to our model aconstant, not dependent ony. We set Xr'(y= 1)= Xx (y = v) and using Eqs. (5) and (6) obtain

[n(1, 1} n(0, 1)][n(1, 1) —n(1, v)]

[n(1, v) n(0, 1)]-XI[n(1, 1) —n(1, v)]

2456 PATRICK J. HOOYMAN AND L. H. SCHICK

On the other hand from Eq. (2) it is clear that wemust have Xx'&0. From Eqs. (A7) and (A6) it fol-lows that go, &0, while from Eq. (A5) it followsthat uo&0. Equations (A4) through (A6) implyA, &0, while from Table I a(0, 1)&0, a(l, y) &0,and a(1, y) —a(0, 1)&0 for ye l. We obtain thusfrom Eq. (4) and X»2&0 the condition $(y) &0,which from Eq. (5) and some of the same inequal-ities just used, leads to

(6)

From Eq. (6) and Table I we find in units of fm'

)((y) =- 1.652, —1.652, —1.615, —1.650, and—1.650 for y=1, 4, &, 4, and 0, respectively. Inthese same units we find from Eq. (7), Table I,and the just quoted results for y(v), ve 1 that u,= —3.267, —1.726, and —1.726 for v=&, 4, and 0,respectively. No matter what suppressed value ofy= v we choose9 we cannot satisfy Eq. (8).'o

Our result then is that independent of the shapesv~ and v~ we cannot use an NLS potential V of theform given in Eq. (2) to reproduce the z and y sup-pression effects present in the AN potential ofBDI2 near zero energy, i.e. , forcing V to repro-duce a(0, 1), r, (0, 1), a(1, 1), r,(1,1), a(I, v), andro(1, y) for any y& 1, results in X»2&0 which is anonphysical condition.

On the other hand from an examination of TableI for a given &, both a and r, are almost entirelyinsensitive to y. We turn back therefore to seeingwhat can be learned about the feasibility of usingsome particular one parameter shapes for v~ orv~ when we insist on an NLS potential reproducingonly the & suppression effects obtained in BDI2near zero energy.

III. e SUPPRESSION ONLY

As indicated in the Introduction the 2 x 2 phenom-enological potential under consideration is thatgiven in Eq. (2). Using the BDII data and simpleYamaguchi shapes, this potential yielded ST andDS a resonance. We found using the newer BDI2data and Yamaguchi shapes that a resonance isstill present.

The simplicity of this model in the Faddeevequations and the sensitivity of the resonance toslight changes in the data as shown by ST and DSled us to hope that a change in the single-channelshapes would eliminate the resonance. With thisin mind a total of eight potentials were employedin the model; they took the forms:

1v»(k»)

(k 2+ p 2)0/2

("generalized Yamaguchi, "p = 1,2, 2, 4, 5, 6),

6(r- Rx)4m

("5 shell" defined as p= 6 for convenience)

(4»r) ' r&R»v»r =

0 r&R

("cut off" defined as P = 7 for convenience),

(10)

where r is the relative hyperon-nucleon distance.To facilitate comparison with p ~6 shapes we de-fined P»=1/R» for P=7, 8. For each of theseshapes p~ was chosen as a free parameter and theother four parameters were directly calculated byinverting the equations for the effective ranges andscattering lengths at & = 0 and at z = 1, y = 1. Thedetails of this may be found in the Appendix.

A. Calculations with SDI2 data

TABLE II. $ and pA for various vh's using the BDI2scattering lengths and effective ranges.

PA gm)

p=l2345678

0.8088520.7264860.702508

-0.841556-0.669444-0.440842-0.510858

0.640600

1.0470.72280.58500.72990.67080.65123.8161.706

In ST and DS it was shown that a necessary con-dition for no resonance is

(n, +A,)(a, —n, )

k,'(a, +A,)[(roo- r„)/2+u„(a, —n, )]

(12)

where a, =—1/n, and r«are the scattering lengthand effective range, respectively, for & =i andy=1; i.e. , in the notation of Sec. II a, =1/a, (i, 1)and r„—=r, (i, 1). In addition k, is the value of k~ atthe Z-channel threshold, and A, and u„are depen-dent on A-channel parameters as given in theAppendix. When ( is calculated for the eight po-tentials using the data of BDI2, it is found to al-ways be less than 1, as shown in Table II. Therange parameter, P~"', of the A- channel potentialsas given in Table II are reasonable with the excep-tion of P~

' for p=7.We then calculated in the a, —r«plane the (= 1

curves for the various potentials and plotted theresults as solid lines in Fig. i. The values of (below the ( = 1 line are less than one, while above

SUPPRESSION EFFECTS IN TWO-CHANNEL SEPARABLE. . . 245V

- I.20

—I.50

C

—1.40D

-1.50

-1.603.4

8 321

4.0

able complication of the form of the equations butthey still retain the useful property of $...„&1being necessary for no resonance. This propertyis retained because of the fact that the equation forX~ is identical to the equation obtained by matchinga and ~p at & =0, 1 with $ being replaced by g, , ,

Table III shows the results of our calculations ofThe fitting at & = 1 was always done since this

should be the value of & for most hypernuclei aswell as for free Ap scattering. &' was varied from0 (full suppression) through 4, &, and -„ in thehope that less suppression would eliminate theresonance. As can be seen from Table IG thisattempt failed although for those potentials withpositive (... an increase of $,„toward unity isseen when E'= a.

r (frn)

FIG. 1. The $ =1 (solid lines) and $ =+~ (dashed lines)curves in the ao-rop plane for the shapes vh withp =1through 8 as indicated. For all cases a& = —1.77 fm and

rpj = 3 18 fm, the BDI2 values ~

the (= 1 line g increases until it crosses fromplus to minus infinity. The dashed lines are the

$ =+~ lines. The g=+~ curves occur in the sameorder and have the same general shape as the $=1 curves. That is, if the )=1 curve of P, isabove (below) the g = 1 curve of p„ then the $ =+ ~curve p, will be above (below) the t = a~ curve ofp, . For this reason we do not include all $ = + ~curves in the figure. As can be readily seen fromthe figure the behavior is similar for two groupsof potentials. For P = 1, 2, 3, and 8 the BDI2 a,and ro are close to the $ =1 values, while for P= 4,5, 6, and V the BDI values have crossed the f=+~line and $ is therefore negative.

We also determined the equations of this modelfor arbitrary &'s. That is, rather than deriveequations to fit the data at & = 0 and & = 1, we de-rived them to fit at & = &' and & = c . As can beseen in the Appendix, this results in a consider-

TABLE III. $« for various vA's using the BDI2 datafor various values of e.

B. Calculations with a(0,1) and ro(0, 1) in the no-resonance

region

In as attempt to explore the properties of thismodel, we allowed first a(0, 1) and then ro(0, 1) tovary from the BDI2 values. Specifically we choseap or xpp such that they were in a no- resonance re-gion of the ao- r~ plane for as many potentials aspossible. Two sets of a, and ~~ were tried:

ao = —1.2V5 fm and t'oo = 3 64 fm

ao=- 1.42 fm and rpp 3 50 fm.

a, and rpg are held at their BDI2 singlet values:

1 VV fm and roc =3.18 fm

In each case one of the parameters is a BDI2 pa-rameter and the other is chosen to insure $&1 forthe four potentials p = 1, 2, 3, and 8 as may beseen in Table IV. As can be seen from Fig. 1 it ispossible to fit at most the four potentials men-tioned. This is a particularly fortuitous set as itincluded the standard Yamaguchi (p = 2) shape

TABLZ IV. $ and PA' for various vA's using a& and

rpf equal to their BDI2 values always, and using two dif-ferent sets of "no resonance" a o and rpp.

Pg, (fm)

8 p— 1 275 frn and rop

——3.64 fm

p = 1 0.808852 0.726483 0.702514 -0.841565 -0.669446 -0.440847 -0.510868 0.64060

i4

0.844320.754340.72830

-0.79025-0.63449-0.42288-0.48821

0.66156

12

0.937500.824420.79232

-0.68222-0.55747-0.38016-0.43572

0.71187

34

0.865510.759720.72992

-0.66397-0.53644-0.35898-0.413870.65697

P =1238

20.706.194.963.05

1.0140.7010.5671.655

p =1238

4.052.742.461.88

1.0190.7030.5691.658

ao=-1.42 fm and roo=3 5 fm

2458 PATRICK J. HOOYMAN AND L. H. SCHICK

TABLE V. The phase shift in degrees for various vA's

as functions of EA using two different sets of "no reso-nance" ao and rpp, ~Q is a Yamaguchi shape with P~ '=1.0 fm.

TABLE VI. The phase shift in degrees for variousvA's) as a function of EA with &~ =—0 and a& and ro& equalto their BDI2 values for two difference sets of "no reso-nance" ao and rpo, &~ is a Yamaguchi shape.

3.070 (deg) at hA (MeV)10.30 30.0 60.0 77.998 3.07

6 (deg) at EA (MeV)10.30 30.0 60.0 77.998

ao =-1.275 fm and rpo =3 64 fm a0=-1.275 fm and rop=3

p=1238

22.26 27.6022.20 27.1022.18 26.9322.13 26.38

26.08 22.3624.00 18.4823.23 16.9520.33 10.36

20.7216.2114.416.18

p=1238

22.2622.2022.1822.13

27.60 26.09 22.3827.10 24.01 18.5526.93 23.25 17.0226.39 20.35 10.42

20.8116.4914.726.44

a& =-1.42 fm and roo =3.50 fm a& =-1.42 fm and roo =3.50 fm

P = I 22.26 27.622 22.20 27.113 22.19 26.948 22.13 26.39

26.2124.0923.3120.39

22.7018.7717.2310.56

21.4317.0015.236.96

p=1238

22.2622.2022.1822.13

27.63 26.22 22.7927.12 24.10 18.8626.94 23.32 17.3026.39 20.39 10.58

21.8717.5915.837.25

which most of the previous work in this field hasutilized.

In Table V we look at the Ap phase shifts (8) asfunctions of k„ for four v~(p=1, 2, 3, 8). For theupper half of this table a, =- 1.275 fm and happ

= 3.64 fm were used, while in the lower half ap= —1.42 fm and happ 3.50 fm were used, a, and spy

being kept at the BDI2 values throughout. In allcases shown v~ was assumed to be a Yamaguchishape with P~ '= 1.0 fm. Other values of P~

'ranging from 0.5 to 2.5 fm were also used, but theresults obtained varied at most by only a few per-cent from those given in Table V; i.e. , for a givenvalue of p there is almost a total lack of P~ depen-dence in 5. Not surprisingly, for low energies((30 MeV) there is very little dependence on theshape v~.

At higher energies the shape dependence becomesquite noticeable with 5 decreasing with increasingP. In fact, just below threshold 5 changes by 14',from a maximum at P = 1 (8 =20') to a minimum atp=8 (8=8'). This can be explained as a conse-quence of the fact that the higher the value of p,the more quickly the potential falls. Furthermore,this indicates that if we had chosen to fit phaseshifts, we could not have reproduced the & depen-dence of BDI2. That is, we have eliminated theshape dependence in the energy region most sig-nificant for hypernuclear calculations by fitting theBDI2 low- energy data rather than by fitting phaseshifts.

As an additional model for investigation, we setX~ = 0 and recalculated the phase shifts as above.The results may be seen in Table VI. Here P~ isnot varied; i.e. , with X~ =0 the four parametersP~, P» X~, and Xx are determined by matchingthe four BDI2 parameters a, a„happ, and rpy Forthese same shapes we calculated P~

' and it was

TABLE VII. The phase shift in degrees (for variousv&'s) as a function of EA with A,x=0 and a& and

rpg equal to their BDI2 values for a& and roo equal to twodifferent "no resonance" sets. vA is a Yamaguchi shape.

6 (deg) at EA (MeV)3.07 10.30 30.0 60.0 77.998

0 0= I 275 fm and ~oo =3-64 fm

P=1238

22.20 27.1022.20 27.1022.20 27.1022.20 27.10

24.0224.0127.0127.01

18.5718.5518.5418.54

16.6016.4916.4716.46

a 0 =-1.42 fm and r&p =3 50 fm

P =I238

22.20 27.1222.20 27.1222.20 27.1122.20 27.12

24.1224.1024.1024.10

19.0118.8618.8418.87

19.2517.5917.4617.61

always less than P~' (which is the same as when

Xc 40). Physically one would expect vc to be oflonger range than v~ so this model is not a goodone. Again the same v~ shape dependence is ob-served as above when X~ 40 and the same conclu-sions may be drawn.

We also calculated a, and r„at & = &, &, and &

for 4 v~'s (P=1,2, 3, 8). The Z-channel parametersdo not enter into the calculation of a, and r~. Whencompared with the BDI2 parameters, the resultswere quite impressive. If either of the end points(a, and a,) correspond to the end points of BDI2,then the e dependence (of the parameter whose end

points were fit) matches the e dependence of BDI2almost exactly. This is true even though the otherparameter's dependence is by necessity quite dif-ferent. In other words, for this model, it doesn' tmake any difference whether we choose to fit the

SUPPRESSION EFFECTS IN TWO-CHANNEL SEPARABLE. . . 2459

data at & = 1 and 0 or & = 1 and &', we automaticallyfit all &'s. It should be noted that this was true forfour different vh's.

In an attempt to examine the shape dependence ofthe phase shift (6) on vc, we set Xc again equal tozero and tried four vc's (p=1, 2, 8, 8). As can beseen from Table VII, there was very little vcshape dependence found, and a Yamaguchi vc ap-pears to be a reasonable choice. At this point, itis worth noting that Xc = 0 insures no Z- channelresonance, and so it would appear that the logicalthing to do is to use the BDI2 data with Ac=0. Un-fortunately, there are no free parameters for thiscase, Pc being determined by the equation for Xc:

—1 —$(gcp/gcp) 0 e

8'co

APPENDIX

where

(kh)

Ah

(1+A~g»)

(Al)

A =X — ")

=1/L~ ~x gcc1+y)Lpgcc

But, leaving the y dependence implicit, kh cot5h=5k~ + (1/f») =- a, + pr„k~'+, so expanding1/f» and equating powers of k~':

Following ST, for the NLS potential given in Eq.(3) we arrive at an expression for the AN scatter-ing amplitude given by

This means Pc must satisfy:

gcp(f E)gc2(l c)

a, =-Ao+ —,Qo

(1/2)r„=A,

(A2)

(A3)

For the potentials considered here this impliesthat at least $ & 1 for Pc & 0. In fact for P = 1, $ & 2for Pc&0. And so, for physical results, the noresonance condition of ST and DS is necessaryeven when the Z channel is missing.

In conclusion, the model investigated here inSec. GI yields Ap phase shifts below the Z-channelthreshold that for a given one-parameter shape vcshow very little dependence on pc, and which forXc = 0 show very little dependence on which one-parameter shape vc is used. However, at energiesabove 30 MeV, a significant vh shape dependencewas noted. This indicates that fitting phase shiftswould not be an acceptable method of solution ifthis model is to be used in binding energy calcula-tions as it would not produce the & dependence ofBDI2 which the model reproduces remarkably well.

There are two obvious ways of trying to eliminatethe unwanted resonance while fitting the BDI2 val-ues for the four parameters a„a„r~, and t'„.Both of these involve putting into Eq. (2) somethingthat will play the role of the strong repulsive corepresent in the BDI2 MTP. The first would be touse a multipararneter shape for vh. The secondwould be to replace the single term NLS potentialfor Vhh by a sum of NLS potentials. Both of theseare being investigated.

where

z o r"hhh, g +gg &+ ~ ~ . (A4)

vg (kg) = up+ upkg + ' ' '

—=up(1+uP~'+ . ),&h = &o+ &a&h'+ ' '

2 v~'(k)k'dkghh + y2 y 2 r l r

o h

2 vc (k)k'dkw k'+r'(k '-k ') '

(A5)

(A6)

(AV)

1+ykcgcoXg (1+

yacc

gcp) —4 X„gop

E X 'gogc~X,(1+yX,g,) —e'X,'gcp '

(A8)

and g~~ is the Cauchy principal value of the inte-gral given in Eq. (A4). Equations (A2) and (AS)may be inverted for the potential parameters.Fitting at & =P and & = q we find

ko is the value of the AN relative momentum at theZN threshold, and r' is the ratio of the ZN to theAN reduced mass:

2gcc g EO gc2~h

P2 q2

p —q up(a +Ap) up(ap+Ap)

p'[A, —prp, +u, (a, +Ap)](ap+Ap)' =q'[A, ——r p+up„(pa~+A )]p(a, +Ap)',

1 —$.,(gcpkp'/gcp)0

gco

(A9)

(A10)

(A11)

2460 PATRICK J. HOOYMAN AND L. H. SCHICK 13

5.,(a, —a.)rcPe(p' —q')u, (a, +A,) (av+ A,)gc,'

[p'(av+A, )' —q'(a, +A,)'](a~- a,)k,'(p'- q')(as+As) (a~+A,)[s(r~- r,~)+ u, (av - a,)J

'

(A12)

(A13)

Again, the y dependence of (,~ has been suppressed.In addition for a fixed y we may substitute the

results of Eqs. (A9) through (A12) into Eqs. (A2)and (A3) to obtain

with

D =P'(av+A, )' —q'(a, +A,)'

a, =N, /D, ,

where

N, =p'(a~+A, ) —q'(a, +A,) —s'(av a,)

D, =P'a, (a~+A, ) —q'a~(a, +A,)

+ s Ao(av —aq)

r„+N,

q'(a, +A,)' 2M„vt'OP+

N,

(A14) M,~= (e, —q')(a, +A,)x [s(r(&v rs, )(a +As)(p q )

+u„(a~- a,)'(p'- e')).In particular, for p=O and q=1 (or q=O and p=1)the above equations reduce to the usual equationsof ST.

The no Z-channel induced resonance condition ofST and DS is retained since for y =1 the dependenceof f~a on A.c is identical to that of ST and the equa-tion for Xn (All) is changed only in that the $~ isnow a slightly more complicated function of theA- clemnel parameters. This means that our noresonance condition is still $~, & 1 as shown by DS.

iL. H. Schick and ¹ K. Tyagi, Phys. Rev. D 5, 1794(1972), hereafter referred to as ST.

2P. S. Damle and L. H. Schick, Phys. Rev. D 5, 1801(1972), hereafter referred to as DS.

~J. T. Brown, B. W. Downs, and C. K. Iddings, Ann.Phys. (N.Y.) 60, 148 (1970), hereafter referred to asBDI1.

4The resonance is "unwanted" because it is not presentin the meson-theoretic potential that the NLS potentialof ST and DS is supposed to represent.

SJ. T. Brown, B. W. Downs, and C. K. Iddings, Nucl.Phys. B47, 138 (1972), hereafter referred to as BDI2.

J. Dabrowski and E. Fedoryn'ska, Nucl. Phys. A210,509 (1973); A. R. Bodmer, Phys. Rev. 141, 1387 (1966);B. F. Gibson, A. Goldberg, and M. S. Weiss, Phys.Rev. C 6, 741 (1972); J. Law and T. D. Nguyen, Nucl.Phys. 824, 579 (1970).

VAlthough in the phenomenological model we are usingwe have referred to the hN potential. without distinguish-ing between the Ap and An systems, it is to be noted

that the BDI2 parameters we are using as input werecalculated for the AP system. Because of the existenceof charge symmetry breaking A-nucleon forces, theanalogous parameters for the An system would be dif-ferent from those given in Table I. See, for example,Ref. 5.

SFor e = 0, of course, the AV channel decouples from theM channel and as far as AN scattering is concernedthe value of y is immaterial. We have set y= 1 in thiscase, merely for convenience.

9The BDI2 values of c(1,y) and ro(1, y) given in Table Ifor y= 1 and y=$ are identical. Our analysis in thissection hoMs for y = 1 and y = &&1, but it actually can-not he applied for the BD12 itnput when v =$.

~For a similar analysis of the 3S&- D& AN interactionusing the appropriate BDI2 input see L. H. Schick, inI'ere Body Problems in Nuclear end Purticle Physics(Les Presses De L'Universite Laval, Quebec, 1975),p. 541.