Volume 74B, number 4, 5 PHYSICS LETTERS 17 April 1978

UNIQUE ISOSPIN-ZERO PHASE-SHIFT SOLUTION FOR

NUCLEON-NUCLEON SCATTERING NEAR 325 MeV

Ronald BRYAN, Robert B. CLARK and Bruce VERWEST Department o f Physics, Texas A & M University, College Station, TX 77843, USA

Received 17 October 1977 Revised manuscript received 18 January 1978

The recent measurement ofD t in np scattering at 325 MeV by the BASQUE group permits a unique ! = 0 phase shift so- lution to be determined, provided that the triplet G-wave phase shifts are constrained by theory.

In this letter we present a unique determination of the (isospin) 1= 0 nucleon nucleon (NN) scattering matrix at 325 MeV, made possible by the recent mea- surement o f D t (depolarization transfer) in np scatter- ing at TRIUMF by the BASQUE group [1 ] * 1. This Tlab type no. of data exp. norm. ref. group also provides new high-statistics measurements (MeV) and error

[1,2] of np polarization (P)which help to sharpen the analysis. The other world np data [3-10] used in the 278.4 a 1 [3]

300 da/d~2 17 floated [4] phase shift analysis are listed in table 1 and consist of 307 P 8 1.00 + 0.03 [5] total cross section (o), differential cross section 309.6 do/d~2 11 1.00 + 0.10 [6] (do/dr2) and polarization measurements. There still 310 P 19 1.00 +_ 0.07 [7] does not exist a sufficient variety of experiments to 310.8 a 1 [3] permit a determination of the scattering matrix from 325 P 42 } norm.,C°mm°n [1,2]

np data alone, so we assume charge independence and 325 Dt 8 floated [1] adopt the pp I = 1 325 MeV phase shifts which are 343.8 do/dI2 11 1.00 -+ 0.10 [6] relatively well determined by the world pp data. We 344.3 a t [3] have chosen a recent set of pp phase parameters due 350 da/d~2 17 floated [8] to Arndt, Hackman and Roper (AHR) [11]. (These 350 /' 12 1.00 +- 0.10 [91

351.5 a I [10l phase shifts are very similar to their published set 378.9 a l [3] [12] .) We also adopt these authors' phase-parameter 150 data + 5 norm. data derivatives (d6/dTlab) for I = 1 and 0 states, to accom- modate data over the range of energies in table 1. (Phase shift analyses do not depend critically on the slopes, and AHR provide a convenient set.)

We carried out extensive investigations of the I = 0 phase parameters fitting two data bases: data base B, which includes nearly all the world np data over the 278 to 379 MeV range (table 1), and data base A,

,1 Amsler et al. [1 ] present essentially the same solution.

Table 1 Neutron-proton measurements selected for phase shift analy- ses reported in text (data selection B). Data selection A con- sists of these same data less D t.

which is data base B less the D t measurements. Data base A was studied to determine the effect of adding Dt, and to account for the many (seemingly inconsist- ent) I = 0 phase shift analyses reported in the litera- ture [13-16] prior to the BASQUE measurements.

In our analyses we initially set the l/> 5 partial- wave phase parameters equal to the one-pion-exchange contribution (OPEC) and searched on the lower partial-

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wave phase parameters to minimize the goodness-of- fit parameter, X 2, as is standard practice. However, for both the data base A and data base B analyses, one of the l = 4 phase shifts, namely ~(3G4), searched to values several degrees lower than any reasonable theoretical model will permit ,2. Currently the most reasonable estimates for the I = 4 phase shifts at 325 MeV are most likely those of the n + 2n + • disper- sion-theoretical NN potential models developed by the Paris group [18] and by the Stony Brook group [19]. Their predictions for 8(3G3), 8(3G4), and 8(3G5) are listed in table 2 along with the OPEC pre- dictions. As one may see, OPEC dominates and the 27r + w corrections amount to less than 1.5 °. In the phase shift analyses, on the other hand, 8(3G 4) searches to 4 ° for data base A and 2.7 ° for data base B, well below the theoretical average value of 8.4 °.

Now, a phase shift representation of the data with unphysical G-wave phase shifts is useless, because the incorrect l = 4 phase shifts throw off the lower phase shifts as well. And the possibility that the strongly non-OPEC 8(3G4) phase shift may be right after all may be considered as remote, say, as the possibility that g2 is actually 10, or that mrrc 2 is actually 180

MeV. Accordingly we carried out phase shift analyses

with the l = 4 phase shifts set to an average of the theoretical models' predictions (table 2), the 1/> 5 phase parameters set to OPEC, and the 0 < l ~< 3 phase shifts searched. To discover as many solutions as possible, we carried out extensive "parameter stud-

,2 See ref. [17] for a discussion on the range of values for 6 (3G 4) predicted by several meson-theoretical and purely

pheno menological NN models.

Table 2 Triplet G-wave nuclear-bar phase shifts at 330 MeV due to the Paris and the Stony Brook potentials. The average values, inter- polated to 325 MeV, are also listed along with the OPEC pre- dictions (g2 n = 15.0, m n = 135.04 MeV/c 2) at 325 MeV.

OPEC Paris Stony Average Brook

6(3G3) - 4 . 0 2 ° -5 .48 ° - 5 . 49 ° - 5 . 38 °

8 (3 G4) 7.98° 8.88° 8.06° 8.35 °

8(3G5) - 1 . 5 7 ° - 0 . 1 9 ° - 0 . 6 5 ° - 0 . 4 2 °

ies", wherein one phase parameter is stepped off at regular intervals over a predetermined range, and the remaining free phase shifts are searched each time for minimum X 2. Parameter studies were carried out for all 0 ~< l ~< 2, I = 0 phase shifts. The most solutions in one sweep appeared in the 5(3D 1) study. Here four solutions showed up in the data base A searches, and four in the data base B searches, as illustrated in fig. 1. We found five solutions in all in the data selection A studies with X 2 less than 300, which we labelled A1 through A5; their ×2 are 190, 193,231,280, and 294, respectively. From a statistical standpoint, only solu- tions A1 and A2 need be taken seriously; A1 is slightly favored. The phase shifts for these two solutions are listed in table 3.

Physically, solutions A1 and A2 are quite different. The 3S 1 phase shifts differ by over 15 °, and the 3D 2 phase shifts by almost 17 °. Solution A1 is in reasona- ble accord with most theoretical models, including its 6(381) and e 1 (see ref. [15]), while solution A2, in

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Table 3 325 MeV I = 0 nuclear-bar phase-parameter solutions for data base A (solutions A1 and A2), and for data base B (solution B1); in these analyses the l t> 5 partial-wave phase parameters are set to OPEC (g~ = 15.0, mrrC 2 = 135.04 MeV) and the l = 4 phase shifts to the theoretical values listed under "aver- age" in table 2.

A1 A2 B1

x 2 189.6 193.2 224.6 6(351) -1.5 ° + 2.0 ° 14.0 ° + 1.0 ° -0.5 ° _+ 1.9 ° e 1 11.8 -+ 1.1 5.9 -+ 1.3 8.4 +0.7 ~5(1pI) -29.9 -+ 1.1 -30.1 -+ 1.1 -28.0 + 1.1 6(3D1) -23.7 ± 1.2 -32.9 -+ 1.1 -25.6 +-1.0 6(3D~) 23.0 +- 0.8 6.1 -+ 1.7 24.2 -+ 0.8 6(3D3) 3.7 -+0.7 -1.6 -+0.6 2.8 +-0.7 e 3 7.0 -+ 0.5 11.6 -+ 0.3 7.3 -+ 0.5 8(1F3) -4.6 -+0.6 -5.6 +-0.5 -6.1 -+0.5

our opinion, will be impossible to fit with any ordinary meson-theoretical potential because its d(3D2) is so violently non-OPEC. Moreover the A1 phase shifts in- terpolate nicely between values at 200 MeV and 425 MeV found by AHR [12], whereas solution A2's 3S1, 3D1, and 3D 2 phase shifts interpolate very poorly.

The addition of the BASQUE D t measurements to data base A causes every solution except A1 to either disappear or have drastically increased X 2. Fig. 1 shows the effect of adding the D t data in our ~5(3D1) parameter study. Most important ly , solution A2 disap- pears. The B1, B3, and B5 phase shifts generally re- semble the A1, A3, and A5 phase shifts, respectively. Again statistically speaking, only solution B1 need be considered any further. Thus a unique solution for the 325 MeV data has emerged.

Solution B1 agrees with modern NN potential-mod- el predictions even better than does A1; e.g. the B1 e l , 3D 1, and 3D 2 phase parameters come closer to the Paris model predictions than do the corresponding A1 phase parameters. Other phase shifts fit about equally well. However, the X 2 of B1 is much higher than is expected from adding eight D t points to data base A; X 2 increases by 35.

The reason may be found in the predict ion of solu- tion A1 for D t, i l lustrated in fig. 2. This predict ion does not agree very well with the BASQUE findings; X 2 for the fit to the 8 data is 96. Moderate readjust- ment of the A1 phase parameters yields the much

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Fig. 2. Predictions for D t by phase shift solutions A1, A2, and B1 (defined in text).

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better solution-B1 fit to Dr; X 2 is now 16. However, on a statistical basis, the fit is still poor. The reason is that, with the triplet G-wave phase shifts fixed at the theoretical values, the D t data and the da/d~2 data at 300 MeV[4] are somewhat incompatible. Specifically, B1 predicts a lower d o / d ~ than De Pangher measured in the 45 ° to 105 ° range; the 7 da/d~ data contr ibute an average x2/datum = 2.8. We would recommend that the do/dr2 experiment be repeated, particularly since it was a cloud chamber experiment performed over 20 years ago with a beam spread of 140 MeV.

After carrying out parameter studies on the lower partial-wave phase shifts for data bases A and B, we re- peated the process for 6(3G3), 6(3G4), and 6(3G5). We found a low-x 2 region in multi-dimensional phase- parameter space centered near solution A1 (B1) for data base A (B), over which ~(3G3) and ~(3G5) did not vary too much, but wherein ~(3G4) ranged some 12 °. We indicate this in fig. 3 by plott ing X 2 versus 6(3G4) for data bases A and B. One may regard this low X 2 region as evidence that the world np data are in rough agreement with one-pion exchange, as the J = 3, 4, and 5 triplet G-wave phase shifts bear the OPEC signature ( - , +, - ) . Furthermore, even though 6(3G4) ~ 3 ° at the X 2 minima, this is not definitive evidence against the magnitude of OPEC, as the X 2

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Volume 74B, number 4, 5 PHYSICS LETTERS 17 April 1978

280

260

240 X 2

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8( 3G 4) Fig. 3. X 2 versus 6 (3G4) at 325 MeV, with the remaining 0 ~< l ~< 4, I = 0 phase parameters searched to minimize ×2 for data selections A and B. Various published solutions indicated according to (approximate) data base and value of 6(3G4). SH1 and SH2: ref. [15], solutions 1 and 2; MAW-IX E1 and ED: ref. [13], energy-independent and (0-400 MeV) energy- dependent solutions at 330 MeV; MAW-X EI and ED: ref. [14], energy-independent and (constrained) energy-depend- ent solutions at 330 MeV; AHR III: ref. [16], solution SE(A); BASQUE, ref. [1]; A1 and BI: table 2.

versus 6(3G4) curves also have shoulders near 11 °. A somewhat different selection or t reatment of the data might well turn the shoulder into a local minimum - i.e., produce a new solution (see below). We conclude that it is still necessary to supplement the world data with theory for G-waves.

As a byproduct of our 6(3G4) parameter study on data base A, we discovered that nearly all the previous- ly published 325 MeV solutions are related ,3 . We plotted the 0 ~< l ~< 4 phase parameters versus 6(3G4) that emerged from the 3G 4 parameter study. The so- lutions agreed with our solutions fairly closely for matching values of 6(3G4). Limited space prevents showing these graphs, but we do indicate in fig. 3 where each published solution falls along the ×2 curve opposite its value of 8(3G4). Different handling of the

:1:3 High-energy constraints on the 0-750 MeV energy-de- pendent MAW-IX analysis [13] put this solution some- what outside the common range at 325 MeV.

data alters the shape of the curve - e.g., Signell and

Holdeman [15] find local minima at 6(3G4) = 1 ° and 10 ° - but the solutions all belong to one family.

Interestingly enough, we find that when the world np data are supplemented with theoretical G-wave phase shifts, the lower partial-wave phase parameters are quite stable to variations in the data base. In one case we removed the De Pangher do/dg2 experiment - thus leaving no dcr/dg2 data whatsoever from 53 ° to 114 ° - yet the B1 phase parameters varied by less than 2 standard deviations (s.d.). As a less drastic ex- ample, the BASQUE phase shifts [1] agree with the B1 phase shifts to within 2 (of our) s.d.

Doubling the quoted s.d. for solution B 1 may there fore be a realistic estimate of the absolute error. If so, then one finds that the Paris NN potential predicts 6(3D1) 4 ° to 8 ° below experiment, and 6(3D2) 5 ° to 8 ° above experiment at 325 MeV, confirming trends set at lower energies. This is indicative of too much tensor splitting and perhaps signals the neglect of a p ion -nuc leon form factor and/or (other) three-pion- exchange effects in the intermediate range of the po- tential (1 .4 -2 .2 fro).

We wish to thank Professor P.S. Signell for provid- ing us with a copy of the phase shift analysis program used in much of this research. We also would like to thank Professor R.A. Arndt for help during the early stages of the project, and Professor D.V. Bugg for helpful conversations and use of the BASQUE data prior to publication. This work was supported in part by the U.S. Department of Energy, under Contract EY-76-S-05-5223.

References

[1] C. Amsler et al., Phys. Lett. 69B (1977) 419. [2] D.V. Bugg, private communication. [3] T.J. Devlin et al., Phys. Rev. D8 (1973) 136. [4] J. De Pangher, Phys. Rev. 99 (1955) 1447. [5] D. Cheng, B. Macdonald, J.A. Helland and P.M. Ogden,

Phys. Rev. 163 (1967) 1470. [6] AJ.Bersbach, R.E. Mischke and T.J. Devlin, Phys. Rev.

D13 (1976) 535. [7] O. Chamberlain et al., Phys. Rev. 105 (1957) 288. [8] A. Ashmore et al., Nucl. Phys. 36 (1962) 258. [9] R.T. Siegel, A.J. Hartzler and W.A. Love, Phys. Rev, 101

(1956) 838. [10] A. Ashmore, R.G. Jarvis, D.S. Mather and S.K. Sen,

Proc. Phys. Soc. (London) A70 (1957) 745.

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[11 ] R.A. Arndt, private communication. [12] R.A. Arndt, R.H. Hackman and L.D. Roper, Phys. Rev.

C15 (1977) 1002. [13] M.H. MacGregor, R.A. Arndt and R.M. Wright, Phys.

Rev. 173 (1968) 1272. [14] M.H. MacGregor, R.A. Arndt and R.M. Wright, Phys.

Rev. 182 (1969) 1714.

[15] P. Signell and J. Holdeman Jr., Phys. Rev. Lett. 27 (1971) 1393.

[16] R.A. Arndt, R.H. Hackman and L.D. Roper, Phys. Rev. C15 (1977) 1021.

[17] R. Bryan, Phys, Rev. Lett. 35 (1975) 967. [18] M. Lacombe et al., Phys. Rev. D12 (1975) 1495. [19] A.D. Jackson, D.O. Riska and B. VerWest, Nucl. Phys.

A249 (1975) 397.

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