Nucleon-nucleon dynamics at medium energies (II). Results for NN phase parameters

Nuclear Physics A338 (1980) 317-331© North-Hoftand Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permiuion from the publisher

NUCLEON-NUCLEON DYNAMICS AT MEDIUM ENERGIES (II) . Reealts for NN phase parameters RICHARD R. SILBAR"

T7uontical Dioision, Los Alamar Scicntifrc Laboratory, Univcraity of California, Los Alamos, New Mexico 87S4S and

W. M. IüAE1~" Department of Physics and Astronomy, Rufgers Unitxrsity, New Brtrnswick, Newlersty 08903 Received 14 August 1979 A6abrsct. We present predictions for nucleon-nucleon elastic scattering phase parameters based on a

unitary, relativistic, one-pion-exchange model, which takes single-pion-production inelasticity into account. The agreement of the high-L phase shifts with data is considerably improved at inter mediate energies by inclusion of the Nd inelastic channel. Our predicted inelasticities are in generally good agreement with the data, but are smaller than the predictions of Green and Sainio . The Argand plotsof the tDz, 3F3,'Pl, and'G, allshow counterclockwise motion resulting from the onset of inelastic channels .

1. Introdaclion In the first paper') in this series,we presented a detailed account ofthe three-body model we are using in an extensive program of calculations for the coupled NN and NNTr channels. With this model we calculate the coupled "isobar amplitudes" NN-> NN' andNN-> Nd, where N' and d are interacting ~rN states ("quasiparticles") in the (I, J) _ (i, ~) and (~, i) states, respectively. The calculation is relativistic, unitary, and takes all spin complications into account. For the moment, the only forces are those generated by one-pion exchange (OPE), but we hope to include short-range heavy-boson exchange in the near future . In the present paper, we give the results of the OPE calculation for the elastic NN phase parameters. These parameters are obtained from the NN-~ NN' amplitude by requiring the N' gtiasiparticle to be on-shell, i.e., have the mass of the nucleon. With the inclusion of the d-quasiparticle, the predicted elastic phase shifts, Silt, are quite different from conventional OPE calculations, particularly in the intermediateenergy region . In fact, the Nd channel considerably improves agreement with the high-L partial-wave phase shifts . What is even more interesting are the predictions for the inelasticity parameters, pLSt and ah which come out of the calculation in a " Work supported by the US Department of Energy.

"" Work supported in part by the National Science Fou»dation . 317

318

R. R. SILBAR AND W. M. KLOET

natural way because of the unitary coupling to the NN~r channel. Since single-pion production is the most important inelasticity up to about 2 GeV kinetic energy, we present our predictions over the range T,,,, = 0 to 2.5 GeV. The model does a good job predicting the energy dependence and magnitude of the spin-averaged inelastic cross section over this range z'3). That is, the inelasticity at these energies appears to be well-described by OPE forces . We expect that the prsr and c~, presented here will not be significantly changed by the inclusionof short-range forces. As such, they may be useful as starting values for phase shift analyses of elastic data in this energy region where the inelasticity essentially doubles the number of phase parameters. Where possible we have for comparison included inour figures the results of recent phase-shift analyses'). Further, in the graphs of the I =1 prs~, we compare with the predictions of the coupled-channel two-body calculation of Green and Sainio S). We also comment about what may be expected to change as our model is further refined. Of much current interest are the Argand plots of certain NN partial waves which are alleged to contain "dibaryon resonances". Such plotswill be discussed separately in sect . 4. 2. I =1 NN phase parameters For an uncoupled NN channel, such as amplitude f = (rl

'So,

ez~a _

3 Po, 3P,, 'Dz, etc., the partial-wave

1)~2i

(2.1)

is related to the calculated amplitude TN for NN -> NN according to eq . (3.5) of ref. '). For coupled channels, such as 3Pz and 3Fz, we use the parameterization of the NN S-matrix given by Arndt 6), +a 2 _ rcos p l cos 2eez~ 1 i sin 2ee'ca 1 +° l ( ) S 2.2 +a~ ces oos 2sez~J' - Li sin 2se'c a,+a s Pz

which reduces to the usual Stapp nulcear-bar phases below the pion-production threshold (,~, = cos p, -> 1, ~ -~ 0). In the conventions of ref. 1) the off-diagonal t-matrix element, e.g., (3Fz~ T~3P~, has a sign opposite to the usualchoice . (The factor of i~ in the angular momentum decomposition of a plane wave was not included.) Thus we introduce an additional minus sign so that the mixing parameter s,, has the conventional sign.

2.1 . PHASE SHIFIS, I =1

The I =1 phase shifts predicted by our three-body OPE model are shown in fig. 1 as the solid lines. We also show predictions of the modelwhen no couplings to the Nd channel are allowed ("NN-only", the chain-dash line) and three different phase shift analyses : (a) the energy-dependent I =1 fit of Arndt and Roper') (dashed line), (b)

NUCLEON-NUCLEON DYNAMICS (In

31 9

the BASQUE phase shifts e) (open bows), and (c) the analyses at higher energies by Hoshizaki 9) (open circles). As a general comment, note that in each partial wave the inclusion of the Nd channel (the "Nd box potential") gives a more positive phase shift below the "threshold" for Nd at around 600 MeV. (Since d for us means a resonating ~rN system with a width, the "threshold" is not sharply defined.) This attractive force due to closed inelastic channels is a well-known effect in particle and nuclear physics lo) ; it may provide sufficient attraction, even, to create a resonant NNstate. (We return to this later, in sect . 4.) We now comment on each of the partial-wave phase shifts, referring in every case to the graphs in fig. 1. 1 So: Although the 1So phase shift climbs rapidly to a positive value of about 30°, the driving term in the Blankenbecler-Sugar integral equation, BOPE~ [see ref. 1)] is in fact always repulsive. This is due (in s-waves only) to the "constant term" in Bow, which in coordinate space gives the well-known repulsive S-function singularity at r = 0. (The usual notion that VoPS is an attractive potential in 1 S o is because, in coordinate space, this S-function is often ignored.) In the present model, the vertex functions tend to smear out the singularity at r = 0, and our "NN-only" result (the chain-dash curve) is qualitatively similar to the OPE result in fig. 5.1 of Jackson, Riska, and Verwest "), (henceforth JRV) . The agreement of our full 1 S o calculation with the phase-shift data is fortuitous at this point. The s-wave phase shifts will be much affected by whatever short-range forces we will eventually include. The rise in the 1 S o phase shift seen in both the Arndt and Roper and Hoshizaki phases between 0.6 and 1.0 GeV may well be real and the result of attraction due to the NN*(1470) threshold 12). The Roper resonance, N*(1470), is not included in our model [see subsect. 4.4 of ref. 1)]. 3 : The similarity of our "full" and "NN-only" curves in this partial wave reflects Po a weak coupling to the Nd channel. We are uncertain as to the origin of this. The small Nd contribution gives a small predicted inelasticity in this partial wave (as discussed below) . The OPE result of JRV is much larger than ours ("NN-only"), probably reflecting the sensitivity of a J = 0 partial wave to short-range behavior induced by the form factors. In both cases the data lie rather lower. JRV achieve a fit when they include two-pion, p, and ~ exchange contributions. This indicates the need for the spin-orbit NN potential (repulsive in this partial wave) and for a repulsive central core, both of which can come from vector meson exchange . 3P1 : Our curves follow the low-energy data to about 50 MeV, then rise due to the Nd box attraction . Here, as in all the remaining partial waves, our "NN-only" curve and JRV's "OPE" curve are qualitatively very similar. In this case the L ~ S potential will be repulsive, but presumably the repulsive central . core will be more important for lowering the predicted S.

T,~ b (GeV)

â, a

tDz

T,ab

(GeV)

0

3Fz -eo

~o

3F 3

Tt,b (GeV)

3 F4

Tt,b (GeV)

Fg. 1 . I =1 NN phase shifts, 8~, in degrees . The solid line gives our model's predictions, the chain-dash line its predictions with the NN channeh only . Phase-shift analyses are : short-dash line, Arndt and Roper, ref.') ; open squares, BASQUE, ref . s) ; open circles, Hoshizati, ref. ~ .

NUCLEON-NUCLEON DYNAMICS (In

32 1

0

3H4 _S

Fg. 1-continued

3P2: This partial wave, which mixes with the s Fz channel, definitely needs a repulsive central core, from w-exchange, since the L " S potential here will be attractive . 'D2 : Our predicted peak in the phase shift at the opening of the Nd "threshold" is too sharp and too high. Perhaps the too-strong coupling to the SSZ Nd channel, mostly through the tensor part of the OPE force, would be reduced if the p-exchange contribution is included in the NN ~ Nd amplitude. (In the NN -> NN case, tensor p-exchange has the opposite sign to tensor ~r-exchange.) Another way to reduce the peak would be to include central repulsion from w-exchange . 3F2: Although already a "large partial wave", the coupling to 3PZ may be responsible for the poor agreement of our curve with the phase-shift data . This partial wave will be helped with repulsive central and L ~ S potentials . 3F3: By now L is large enough that our OPE calculation is in good agreement with the phase shift data up to 700 MeV or so. Note the importance of the Nd channel in assuring this ; the "NN-only" curve misses most of the data above 200 MeV. However, our calculation by no means agrees with the sharp drop in S(3F3) found by Hoshizaki. s : Agreement here is fairly reasonable, but it would be helpful to have more Fa attraction at lower energies . Perhaps intermediate-range "a-exchange" or the (here

322

R. R. SILHAR AND W. M. KLOET

attractive) L ~ S potential might help. Again note the importance of the Nd box attraction in achieving this agreement. 1 Ga : There is good agreement with the data when the Nd channel is included. 3 H4 : Agreement with the BASQUE phases is good, but our predicted curves do not follow Hoshizaki's downward trend. This partial wave couples with the 3F4 channel, but that may not much affect our future predictions. 3Hs : Our predictions follow the general trend of the phase-shift data, including the rise around 500 MeV seen by Arndt and Roper, but there is too much scatter in the data points at the present to claim good agreement. 3H6 : While the graph is different from that for 3Hs, the same comments apply. In summary, the OPE model predicts phase shifts in reasonable agreement with the phase shift data up to 600 MeV for L ? 3 and in poor agreement for L ~ 2. At higher energies we find our predictions often disagree with Hoshizaki's results, even for some of the higher L-values. The predictions of our model as given in fig. 1 are for the gaussian form of the NN~r cutoff function, vx [see eq . (4.31a) of ref.')] . The Yamaguchi form of vN [eq. (4.31b)] has also been used, and the predicted phases are not much different. For example, the 1 So phase shift for the Yamaguchi vertex is slightly flatter with energy than thegaussian case shown in fig. la. At 25 MeV SY is 2.2°smaller than Sa, they are about equal to 1400 MeV, and S, r is 1 .5° larger than Sa at 2500 MeV. 2.2. INELASTICTTY PARAMETERS, I =1

For ease of showing small inelasticities we plot, in fig. 2, the angle p instead of the more usual ~ = cos p defined in eq . (2.1). In addition to our predictions (including both NN' and Nd channels), shown as the solid curves, we give the inelasticities found in three phase shift analyses''e'9) included in fig. 1, with the same notation . Furthermore, the predictions of the coupled two-body channel calculations of Green and Sainio S) are shown as the long-dash curves . In general the agreement of our p's with the phase-shift analyses is quite good, in line with the idea that most of the inelasticity in this energy range is associated with peripheral pion production through the d-isobar 2). We make the following comments about each of the partial-wave inelasticities, always referring to the graphs in fig. 2 . 1 So : The only phase shift analysis so far that includes inelasticity in this partial wave is Hoshizaki's, and the scatter of the points is too big to say there is anything more than "reasonable agreement" . The fast rise of the Green and Sainio inelasticity at threshold is difficult for us to understand, since inelasticity through the Nd channel must involve an Nd state in a relative d-wave and that through d~rr must involve an p-wave . (Green and Sainio include the da channel, but we do not.) 3Po: Some recent phase shift analyses'' 9) find very large inelasticity in this partial wave already at 500 MeV, in contrast with our prediction. Again, unless this reflects qualitatively new physics, we find this difficult to understand in terms of the angular

NUCLEON-NUCLEON DYNAMICS (In

32 3

momentum threshold barriers involved . If inelasticity is produced via the Nd and NN' isobars, the isobar-nucleon angular momentum is p-wave . The d~r channel does not couple to 3Po at all. One possible inelastic channel which involves a relative s-wave is a pion and an NN-quasiparticle with I =1, Jp = 0+ (d*) . If this were an important source of inelasticity, however, we would expect to see a comparably large (or larger) inelasticity in the 3Pi partial wave, since that couples to an s-wave d~r channel. This is not the case, as çan be seen from Hoshizaki's values of p near 600 MeV: p(3P1 ) ~ zp(3Po). A d*~r channel has not been included in our model (nor in most other isobar-model calculations). 3P, : Reasonable agreement at energies to 800 MeV is seen . (The BASQUE analysis at 325 MeV assumed all inelasticity to be due to the pp -> dTr reaction with d and ~ in a relative s-wave .) Hoshizaki's S( 3P~) and p( 3Pi) show rather more structure around 1 GeV than our predictions . 3Pz : Our prediction lies below that of Green and Sainio and the Hoshizaki phase-shift analysis. Other phase-shift analyses do not fit for inelasticity in this channel. 1D2: The rise of our curve is faster and the peak value is higher than indicated by the phase-shift analyses . This is probably closely connected to the overshoot found and discussed for S('D Z). 3Fz : Good agreement.

3 F3: Good agreement with the general trend of the data. However, much of the reason for a rapid counterclockwise circling in Hoshizaki's Argand plot of this partial-wave amplitude is the sharp drop in p which he finds at 830 MeV and which we do not have . 3Fa : Our inelasticities are somewhat lower than the others . ' G4: Good agreement. 3 H4 : Good agreement. 3Hs : Good agreement. 3He : Good agreement. To summarize, the agreement with the phase shift analyses' data is good in high partial waves, even at higher energies . It is in fact in reasonable agreement in many of the low partial waves as well, with the nôtable exception of 3Po. We do not predict the structure found by Hoshizaki in 3P1 and 3F3. In general, the inelasticities predicted by our model are smaller than those of Green and Sainio. 2.3 . MDQNG PARAI~TERS, I =1 The I =1 NN channels which couple through the OPE tensor force are 3Pz3F2, 3F4'3H4, etc . Graphs of our predictions for the mixing parameters e,r, a, [see eq . (2 .2)], together with the phase shift data as above, are shown in fig. 3. The agreement for ea, involving L ? 3, is rather better than for e2, which presumably requires short-range modifications of the tensor force (as, for example, from p-exchange).

324

R. R . SILBAR AND W. M. KLOET

so t

s0 3Po

0

sp ~

3Pz

IA

Ib

T,ab (GeV)

T,ab (GeV)

'D z

0

3F4

0

0

3F3 IA

LS

T~,b (GeV)

20

26

o

.S 0

.

LO .

. Ib .

-

.

T,ab (GeV)

Fig . 2. I =1 NN inelasticity parameters, pisr~ in degrees . Notation is as in fi~. l, except that the long-dash lines are predictions by Grcen and Sainio, ref . ) .

NUCLEON-NUCLEON DYNAIrIICS (II)

sH

4 f

m

3

Hs

eo

3

Hs

0

f

0

QS

IA

1b

20

so 0

T,~ b (GeV)

Ob

IA

T,,, b

Ib

20

(GeV)

Fig . 2.~ontinued

lo 5 0

0

-s

-15 2

EB 0

_2

T,ab (GeV)

T,ab (GeV)

Fig . 3. I =1 NN mixing parameters, sJ and a~ in degrees. Notation is as in fig. 1 .

25

326

R. R. SILBAR AND W. M. KLOET

Fig. 3 also shows our predictions for the off-diagonal phases a~ associated with the inelasticity . From experiment very little is known about these parameters . No phase-shift analysis yet has varied these parameters to fit the data ; they have always s) been set equal to zero . Green and Sainio have made some qualitative predictions of a2, assuming an experimentally known behavior for e2. If e2 changes sign around 600 MeV, as is indicated in fig. 3, this may affect the predicted behavior of a2 . It is worth commenting briefly at this point on the numerical accuracy of the phase-parameter predictions given here . The half-off-shell partial-wave amplitudes (p', a'L'S'J~T~p, aLSJ) are, for a given set of mesh points {pr}~ required to satisfy the coupled integral equations to better than 1% for each p' = p r . The extracted p-parameters, however, for p=0, may have larger errors, since dp,=d~Su~/sin p. Likewise, the a-parameters can also have larger errors, particularly when the corresponding e is small, because da =d~S12~/sin 2e. 3. I = 0 NN phase parameters The I = 0 phase shifts have only recently become well-known up to about 600 MeV. Fig. 4 shows our predictions compared with several recent phase shift analyses'3 " 1`.ls). Once again, the agreement of our OPE model phase shifts with the data is poor for the low partial waves and becomes good for partial waves with L z 3. The I = 0 inelasticities p, and the mixing parameters e and a, are shown in figs . 5 and 6, respectively . The inelasticity in all partial waves is small, in accord with the small inelasticity in our model from the N' isobar. (Isospin forbids Nd excitation .) It is interesting that the predicted inelasticities in the spin-singlet partial waves are quite small oompared,with the triplet inelasticities . Further, we predict more inelasticity in the J =L -1 triplet waves than in the J = L and J = L + 1 waves. We were at first somewhat chagrined to have predicted a 3S1 3D1 mixing parameter e t which is too large and, worse yet, of the wrong sign. However; the sign and magnitude of et are very closelyrelated to S (3S1) . Thisisapparent, forexample, in the curves of JRV's fig. 5 .1 . Our predicted S(3S1) is much like their curve labeled "N", and their corresponding "N"curve for et is also large and negative, like ours. Thus, we have some hope that, when short-range repulsive forces are incorporated and the S(3S,) reproduces the low-energy phase-shift data, e t will be in better agreement. 4. Argand diagrams As mentioned in'the introduction, there is currently much interest in the possible existenceof both I =1 and I = 0 dibaryon resonances . These aresuggested mainly by structure in the energy dependence of differences of spin-dependent total cross sections such as d~L = ~( ~) - v(~) [spins aligned along the beam direction,

,o

0

m 0

-eo

eb

LO

T,~b

Ib

(GeV)

20

2b

Ob

lb

20

lb

se

T,~b (GeV)

e 4 0 -4

0

_z

'H 5

_g

T, ab

(GeV)

T,ab (GeV)

Fig . 4. I ~ 0 NN phase shifts, 8r sa in degrees . Solid curves are predictions of our model. Phase-shift analyses are : short-dash line, Arndt and Roper, ref. l3 ) p open circles, Texas A & M, ref. 14) ~ open equates, !s) . BASQUE, ref .

328

R . R . SILBAR AND W . M . KLOET

1s

eo

'H s

ts

IS

0

Ob

IA

T,~b

16

20

26

(GeV)

0

Ob

3 I5 ID

15

2A

T,~b (GeV)

Fig. 5 . I = 0 NN inelasticity parameters, ô. ~R in degrees . All curves are predictions of our model.

m lo

Et

0

_m lo

e 6 4 2 0

OS

IA

T>ab

Ib

(GeV)

20

2S

Ttab

(GeV)

Fig . 6. I = 0 NN mixing parameters, a~ and a~, in degrees . Notation is as in fig. 4 .

yS

NUCLEON-NUCLEON DYNAMICS (In

32 9

parallel and antiparalle1 16)], C~(LL ; 00) near e~m. = 90° [ref l')], and the polarization of recoil protons in photodisintegration of the deuteron'$). Possible resonances have been suggested as occurring in various partial waves, in particular, 1DZ , 3F3 , 1G4 , and possibly 3Po. It is therefore interesting to show Argand plots in fig. 7 for a number of selected partial waves using our predicted phase parameters.

Fig . 7 . Argand plots for selected I =1 partial waves as predicted by our OPE model with unitary coupling to inelaadc channels. Numbers indicate laboratory tinedc energies, in MeV.

The first thing to note is that many of our predicted partial waves do show the counterclockwise motion on the Argand plot that is typical of a resonance. This is particularly true for the 1D2 partial wave and to a lesser extent also for 3 F3 , 3P t , and 1G4. All of these loops reflect considerable inelasticity, i.e., the onset of excitation of the Nd channels. We have not studied the behavior of these partial wave amplitudes in the complexenergy plane. Thus we cannot make statements about the existence of resonances, i.e., the presence of poles on the second sheet. The'DZ case involves Nd in a relative s-wave, SS 2 , and the greatest speed on the Argand plot is around 600 MeV ( W= 2100 MeV) near the Nd "threshold". The 3F 3 and 3P, curves are fastest athigher energies, around 900 MeV ( W= 2250 MeV), because of the p-wave angular momenttun barrier in the Nd channel. Likewise the (small) 'G4 wave is fastest at yet higher energy, reflecting the Nd d-wave . However, the predicted tD2 , 3F3, etc., amplitudes are not suf&dent Z) to produce the structure observed in, say, d~L . For example, to obtain the sharp dip in dvL near 800 MeV, Hoshizaki fits the 3F3 partial wave with an Argand behavior similar in shape to ours, but which courses the loop much more quickly.

330

R. R SII.BAR AND W. M. KLOET

5. Condoeions

In this paper we have presented results for the NN phase parameters obtained from our unitary model of elastic and inelastic scattering. The driving forces in the calculations here are given by one-pion exchange . In comparing with the results of recent phase-shift analyses, we find : (i) The phase parameters generally agree well with the data for high-L partial waves. The coupling to the Nd channel improves the agreement for the 1=1 phase shifts Sam, with data considerably, particularly at intermediate energies . This coupling is also the most important one for the inelasticity parameters pLS,. (ü) The model, as expected, does not give good predictions for the low-L partial waves. This situation is expected to be remedied when short-range forces are included in the model (in a way which takes advantage of the flexibility of the choice of léft-hand-cut contributions) . (üi) The disparity between our predictions and certain of the I =1 phase parameters suggests that a number of partial waves could bear experimental re-examination . In particular we mention the inelasticity parameters for 3Po and 3F3. As regards the I = 0 partial waves, the lack of any higher energy data precludes making any definite statements beyond the general comments in (i) and (ü) above. (iv) The model predictions of the inelasticity parameters p and a could be useful in reducing the number of free parameters in future phase shift analyses . In this regard, however, we note the sometimes large differences between our results and those of Green and Sainio s). In general, there ismore inelasticity in their model than in ours. (v) The Argand plots of a number of inelastic I =1 partial wave amplitudes (1Dz, 3F3, 3P1, and 1Ga) show an interesting counterclockwise motion as energy increases. This behavior in our model is similar to that found by Hoahizaki 9) in some of the partial waves, but usually involves a slower motion . This behavior in our model is associated with the Nd inelasticity and is not necessarily to be interpreted as due to resonances . (vi) The model in its present form reproduces the general features of the total spin-dependent cross sections s) but fails to predict the details. A future version of this model including short-range forces may be able to show whether new phenomena like dibaryon resonances are really necessary to explain the recent data . For example, if the structure found by Hoshizaki in both S and ~ for the 3F3 partial wave holds up in future analyses, we may see that we need also to include narrow resonances in the model to give the observed sharp dips and rises superimposed on the generally correct, smooth behavior of our predictions. We wish to acknowledge helpful conversions with R. A. Bryan, J. F. Dubach, L. Heller, M. B . Johnson, E. Lomon, Yu. A. Simonov, and B . J. Verwest. In addition, we thank R. A. Bryan for making available to us his compilation of recent phase shifts a) .

NUCLEON-NUCLEON DYNAMICS (II)

33 1

References 1) W. M. Kloet and R. R. Silber, Nud. Phys . A338 (1980) 281 2) W. M. Kloet, R. R. Silber, R. Aaron and R. D. Amado, Phys. Rev. Lett . 39 (1977) 1643 3) W. M. Kloet and R. R. Silbér, in Few-body systems and nuclear forces, Proc. Graz Conf., 1978 (Springer, Berlin, 1978) vol. I, p. 119 4) R. A. Bryan, in Few-body systems and nuclear forces, Pros . Graz Conf., 1978 (Springer, Berlin, 1978) vol. II, p. 2 5) A. M. Green and M. E. Sainio, J. of Phys . G S (1979) 503 6) R. A. Arndt, Rev. Mod. Phys. 39 (1967)710 7) R. A. Arndt and L. D. Roper, private communication to R. A. Bryan (1978) 8) D. V. Bugg,J. A. Edgington, C. Amsler, R. C. Brown, C.J. Oram, K. Shakardû, N. M. Stewart, Ci . A. Ludgate, A. S. Clough, D. Axer, S. Jaocard and J. Vâvra, J. of Phys . G 4 (1978) 1025 ; and private communication to R. A. Bryan 9) N. Hoshizaki, Prog . Theor. Phys. 60 (1978) 1996 ; and private communication to R. A. Bryan 10) In the NN case, most recently by E. L. Lomon, Few-body systems and nuclear forces, Pros. Graz Conf., 1978 (Springer, Berlin, 1978) vol. I, p. 9; the earliest work known to us is R. A. Arndt, Phya. Rev. 16S (1968) 1834 11) A. D. Jackson, D. O. Risks and B. J. Verwest, Nud. Phys. A249 (1975) 397 12) E. L. Lomon, private communication ; and ref.' 13) R. A. Arrdt and L. D. Roper, Phya . Rev. C1S (1977) 1002 ; and private communication to R. A. Bryan (1978) 14) R. A. Bryan, R. B. Clarkand B. J. Verwest, Phys. Rev. C 18 (1978) 371; Phys. Lett . 74B (1978) 321 15) D. V. Bugg et al., private communication to R. A. Bryan (1978) 16) I. P. Auer et aL, Phys. Rev. Lett. 41(1978) 354 17) I. P. Auer et aL, Phya. Rev. Lett. 41 (1978) 1436 18) T. Kamee et aL, Phya. Rw . Lett . 38 (1977) 468 and 471