A phase-shift analysis of elastic pion-deuteron scattering

Nuclear Physics A350 (1980) 205-226 @ North-lowland Publishing Co., Amsterdam

Notto be reproduced by photoprint or microfilm without written permission from the publisher

A PHASE-SHUT ANALYSIS OF ELASTIC PION-DEUTERON SCATTERING J. ARVIEUX Laboratoire National Satwtte, F-91 190 Gif-sur-Yvette, France and Institut des Sciences Nucleates F-38026 Grenoble, France and A.S. RINAT CEN Saclay DPhT, F-91190 Gif-sur-Yvette, France and Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel Received 23 July 1980 Abshcad: Recent data on differential and total cross sections in the region 82 < T, (MeV) < 292 from SIN and CERN and an isolated polarization measurement formed the material for a wd phase-shift analysis. For virtually all cases x2 < 1 could be reached, but extracted phase parameters are not unique. We emphasize the strong constraints polarization data would exert on presently ambiguous phase parameters. We present our results in Argand plots and briefly discuss their behaviour.

1. Introduction When considering data on NN, ?rN or KN scattering it is the ‘elementarity’ of the involved particles which underlies the wish to parametrize the data in terms of phase shifts or helicity amplitudes. As a rule there is less incentive to treat, for instance, elastic particle-nucleus scattering data in a similar fashion. Indeed in standard calculations one directly describes the measured observables and not the usually non-unique extracted phases. There the basic tool is the off-shell projectile-nucleon scattering matrix which is only constrained by the on-shell elementary phase-shifts. Whereas the above may be said to hold in general it may, nevertheless, in some cases be useful to study projectile-nucleus phase shifts, for instance in the case of a multiparticle resonance. If such a resonance is either near-elastic and/or sufficiently narrow, total or differential cross sections themselves provide clear signals. If, however, the compound system has a resonance with a large width, the behaviour of a phase, rather than the one of observables directly measured, may provide necessary (but not su~cient~ evidence for its existence. Selected pion elastic scattering from some simple nuclear targets like 4He [ref. I)], *‘C [ref. 2)], 160 [ref. 3)] h ave thus been analyzed [see ref. 4, for a parameterization of 205 December

1980

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J. Arvieux, A.S. Rinat / Phase-shift analysis

total rather than of partial wave amplitudes]. When displayed in Argand plots, several low J” partial wave amplitudes for all cases showed quasi-resonance behaviour with Re f(E,,) = 0 at E,, somewhat below the TN resonance energy. A further broadening is shown by r-nucleus total cross sections 5). Many years ago Landau et al. 6> emphasized that in all likelihood one does not deal with nearly overlapping resonances in various partial waves, but rather with the manifested motion of a A within a nucleus. Thus the r-nuclear optical potential for incident energies around T, - 180 MeV is to lowest order the resonating TN amplitude smeared over the Fermi motion. The latter immediately explains why, for instance, in the Born approximation (TVA- VT*), the A as seen in PA scattering appears to be distributed over several partial waves (see for instance ref. 7, for a display in the case of 7r160 scattering). For two reasons the case of rd scattering deserves particular attention. With only a single spectator at the time, the phenomenon just discussed should be particularly pronounced. Indeed, taking the shape of the total cross section as a standard, out of all CT:;, a:: shows the sharpest peak. The second reason relates to structures seen in CT: for various nucleon polarization states “) as well as to the interpretation of extracted NN phases ‘). Indeed, if as claimed, some of the structures in the region s”* - 2.1-2.3 GeV (S = total energy squared) are inelastic di-baryon resonances, these must also couple to the 7rd and 7rNN channels. We wish to postpone the question of what experiments may help to decide between genuine resonances and resonance-like behaviour (pseudoresonances), but it will undoubtedly be necessary to have available a phase-shift parametrization of the elastic 7rd data. Before presenting our results, we mention that the data available till were too imprecise to enable a phase-shift analysis from experirecently 10*12*13) mental data. A first step in that direction has been made where calculated Glauber amplitudes, approximately fitting the older data, produced theoretical Argand plots 14). These exhibited the expected loops which were then suspected to reflect pseudo-resonances, Also other theoretical tests pointed in this direction. We present a phase-shift analysis based on differential and total cross sections (and on the only polarization datum available). Using theoretical predictions as starting values one reaches nearly always x2 s 1, but the extracted phase parameters, as expected, are not unique. Those parameters are displayed in Argand diagrams which show characteristic pseudo-resonance behaviour. We emphasize the importance of even moderately accurate polarization measurements which will sharply limit ambiguities. 2. Parametrization

of data

We start with a brief summary of a standard parametrization spin-one scattering 15) which will be applied to the pion-deuteron

of spin-zero on case.

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We start with partial-waves characterized by a total momentum J, parity T and channel angular momentum L. With odd and even intrinsic parities T for pions and deuterons, respectively, the total parity T = n,,~~(-)~ = (-)L+l. Consider first the partial-wave representation for r*d, neglecting electromagnetic (EM) effects. One defines the S-matrix, partial-wave amplitudes f and the strong phases S by (k is the c.m. momentum) S=1+2ikf=e’“.

(1)

If J = L, all quantities are diagonal and we write & = S& = SJ. In a11other cases (L = J rt 1) a diagonalization is required to obtain the strong eigenphases 8: (i.e. Si*i). Since already at 2.226 MeV above threshold inelastic channels open, phases are complex and we define e2i8J

z: 77J

e2it3L = rl;

e2i

Re8J

,

e2iReSi,

where the parameters qJ are standard inelasticities which must be < 1 by unitarity. Well-known di~culties are met when one wishes to relate the physical partialwave amplitudes for ?r*d scattering including EM effects, to the strong phases. We thus use the standard approximation f TC - f LALLg+ eziuEf&., .

(3)

In (3) f”’ is the ‘strong’, Coulomb corrected, amplitude, f” is the partial-wave Rutherford amplitude, ALL!is the Kronecker symbol, uL is the Coulomb phase and f” is the ‘strong’ amplitude where additional EM effects have been neglected. In the Blatt-Biedenharn convention of eigenphases 16) we then find for given J, the following relations *‘) between partial wave amplitudes f”, Coulomb phases uL, strong eigenphases S’, SC and mixing parameters E’ 1 + 2i exp (-2ic7f)fJ = exp (2&V) , 1 + 2i exp (-2ia,_1)fJ-

= cos2 .5’ exp (2isi) +sin2 E’ exp (28:),

1+2i exp (-2irJ+1)fJ+ = sin2 eJ exp (28:) 2i exp (-icrJ-1 -iaJ+l)f$_

+cos2 .T~exp (2i&),

(4)

= sin cJ cos eJ [exp (2iaJ) -exp (2&)],

where f’ is written for f”’ (L = L’ = J), f: for f”’ (L = L = J* l), and f:_ for f”’ (L=L’1:2=J*l). Pion-deuteron observables are actually most conveniently expressed with helicity amplitudes FAA‘ (>.,A’=&l, 0) of which only FI1, FIO, FI_I and FOo are independent 18).The remaining 5 amplitudes are obtained when using time-reversal

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and parity-conservation.

One thus finds

T*o(kq g (f3)= 3J2(lF1,12l&l2 + l&II2 - IKw12) ,

(5)

du/dR (8) is the cross section for scattering on an unpolarized target; iT1, is the deuteron vector analyzing power and T 20, Tzl, T2* are the tensor analyzing powers. The total and total reaction cross section are, in terms of helicity or eigen partial wave parameters, u tOt= -g

[Im F1r(O”) + Im F,,(O”)]

(6)

3. Phase-shift 3.1. SEARCH

analysis

METHOD

We use the code MINUITS

from CERN

19) which minimizes

a x2 per degree

of

freedom (7) where N,, is the total number of data points analyzed, N,,, is the number of free observable with experimental parameters (15 at most), 07” is an experimental uncertainty A6Pp and 0:” is the theoretical value for this observable. Although no constraints were placed on inelasticity parameters 7, the experimental values were always < 1; only when mixing parameters E were also varied, was unitarity, in some cases, broken on a level of a few percent. The search routine MINUITS incorporates different minimization methods. For a a minimizasmall number of parameters (N,,, c 6) we have mainly used MIGRAD,

J. Arvieux, A.S. Rinat / Phase-shift analysis

209

tion subroutine based on a variable metric method. For iV,,, > 6 we used SIMPLEX which is faster than MIGRAD. However, it only gives some estimate of the diagonal elements of the error matrix and disregards error correlations. Finally we also ran SEEK, a Monte Carlo subroutine, to find out some best starting values for the phase parameters without using any theoretical input. This method proved inadequate, the resulting amplitudes showing no sensible pattern as function of energy. 3.2. THEORETICAL

INPUT

Since realistic models for the description of rd scattering exist, it is natural to use as starting values for amplitudes those calculated with the best founded model. There is a consensus that the desired model should describe rN and NN scattering as well as pion absorption (and emission) rr + N *N, without overcounting ‘O). Till today only one actual calculation of elastic 7rd scattering including all these features has been completed ‘l). In this calculation the nucleon pole in the Pii VN channel which is responsible for the absorption-emission mechanism is retained but it is suspected that inclusion of the hitherto neglected PI1 background will reduce the absorption effects due to the nucleon pole alone 22)t. In view of this uncertainty we shall take as input, amplitudes calculated with a theory where the absorption channel is neglected altogether. Such theory comes close to a covariant relativistic potential theory and we judged the corresponding amplitudes to be the best starting point. Two sets of results, obtained with slightly different parameters are available 21*23,24). For reference below we observe that, due to angular momentum barrier effects, the dominant amplitudes have L = L’ = J - 1 followed by those with L = L’ = J, The ones with L = L’ = J l 1 are the smallest and of the same order of magnitude as the non-diagonal elements. All amplitudes grouped thus decrease with L with the exception of the dominant f?i amplitude. 3.3. COMMENTS

ON DATA

For the seven pion lab energies around T,,.= 82, 116, 142, 180, 217, 254 and 292 MeV, we analyzed the following (partially preliminary) data sets: (a) Total cross sections from SIN 25). (b) Elastic 7r+d differential cross sections in the angular range 8 - 20”-140”, also from SIN 26). (c) Elastic 7r+d differential at 141, 177 and 260MeV in the angular range 0 = 132.5”-172.5” from CERN 27) (we disregard the slight differences in those and corresponding SIN energies). (d) 180” elastic cross sections at 180, 220, 255 and 280 MeV from SIN 28). An interpolation has been made at 217 MeV, with some increased uncertainty in order t Footnote added in proof: Pion-deuteron calculations including the effects of both the pole (absorption) and non-pole parts of the pion-nucleon Pll partial wave have recently been published 33).

J. Arvieux, AS.

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Rinat / Phase-shift analysis

to take into account the scatter of the experimental data in this region of energy, and the trend of the excitation function cr(T,,, 180”) has been extrapolated up to 292 MeV. (e) 180” elastic scattering cross section and tensor analyzing power TZO at 140 MeV from LAMPF 29). (f) Finally we have used the preliminary results from an experiment of T+ scattered by a vector-polarized deuteron target at 142 MeV, done at SIN 30).

254

217

142

82

0.21

’

’

20

’

40

6

‘0 do A

8 Fig. 1. Differential cross sections for rr+d elastic scattering for pion kinetic energies in the lab system T,, = 82 to 292 MeV from refs. 26-28). The horizontal error bars represent,angular bins. When not indicated by vertical error bars the uncertainties are of the order of the symbol size. The curves are fits for various phase-shift analyses solutions. Notice Coulomb interference effects at small angles for T,, = 82 and 116 MeV. At 217,254 and 292 MeV solution are drawn which include (solid lime) or do not include (dotted line) the 180” data points in the analysis.

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We start the discussion with du/dR (fig. 1). At 82 and 116 MeV only one set of data exists. At 142 MeV the 180”point from Holt et al. 29) is in good agreement with an extrapolation of the smaller angle data from Gabathuler et al. 26) and with the CERN data “). However, at 180 MeV the CERN data definitely fall below the large-angle SIN data (up to 135”) and the 180” SIN point 28). Allowing for renormalization, a factor 1.23 somewhat larger than the normalization uncertainty given by the authors *‘) gives the best fit for all 8. At T,, = 254 MeV the CERN data (shown as open rectangles in fig. 1) continue smoothly the SIN data but they extrapolate towards 180” to a value smaller by a factor -2 than the 180” measurement. The disagreement cannot simply be overcome by renormalizing the CERN data as suggested for 180 MeV. Since there is as yet no way to decide which are the best data we have analyzed the two sets separately together with the SIN data at 8 = 20”-140”. The same phenomenon happens at 292 MeV but there the disagreement between the extrapolated value and the 180” data point is at most 30%. The rd total cross sections, which have very small uncertainties, are shown in fig. 2. Concerning the analyzing powers we shall only discuss iTI and T20 which are currently measured.

4. Results 4.1. DISCUSSION

In the range T,, = 142-256 MeV, for which two independent theoretical calculations have been published, we have used both as starting values. The resulting x2 are similar in both cases with some slight preference for the calculations of the Lyon group 24) which extend from 70 to 320 MeV and have a smoother behaviour in energy. Theoretical amplitudes up to J = 8 are included. The resulting experimental amplitudes are the same within the stated uncertainties for both sets of starting values (we shall call “experimental” amplitudes, those resulting from a fit of the experimental data, although they are not free from some theoretical prejudice). In a first round of calculations we have varied phase parameters at random. As expected we found out that the amplitudes &i,,_, are most sensitive to the experimental data (see remark at the end of subsect. 3.2). In a second round we searched for amplitudes other than the dominant ones which would bring some decisive improvement to the fits. The following comments can be made, energy by energy: 82 and 116MeV. Excellent fits (x2 = 0.22) are obtained by varying the partial wave phases (Sic)SF1 or S322.The calculated TzOis very close to the theoretical value (fig. 3a, b). For iTI (fig. 4a, b) the shapes are the same but the calculated values are -50% larger than given by theory.

J. Arvieux, A.S. Rinat / Phase-shift analysis

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Fig. 2. Black dots represent experimental total cross sections ar as function of the incident pion kinetic energy in the lab system T,.,.The solid line is the fit resulting from the phase-shift analysis. The crosses are the resulting reaction cross section (the dotted line is a visual guide).

ICI-

I

I

I

a

82 MeV

05 ,-

I-

E cl-

-0.5 ,-

-1.cI-

I

I

50

loo

I

150

8 Fig. 3a. Tensor analyzing power Tzc for T,, = 82 MeV. The solid line is a theoretical prediction by Giraud et al. 24) for +’ - d scattering; the dashed line (TH) is obtained for rr+d scattering using the theoretical amplitudes; the shaded area is the envelope of phase-shift analysis calculations using diverse combinations of phases &, ST,, 8:~ and Sir. The angle 8 (in the c.m.s.) is in degrees.

X Arvieux, A.S. Rind / Phase-shift analysis

IO-

I

1

213

I

b

116 MeV

0.5

$3

I-

O

-0.5

Fig. 3b, Same as fig. 3a for T, = 116 MeV.

I

I

I

142 MeV

C

Fig. 3~ TZOfor T, = 142 MeV. The solid line (TH) is the resutt of a theoretical calculation for w”-d scattering; the shaded area is the envelope of calculations using diverse ~mbinations of phases Sk,, S:,, S&, Sfs and S: 1 ; the dashed line which goes through the 180” point of HoIt ef af.“) is obtained by varying S& and all J = Z phases and mixing parameter.

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I

I

I

50

100

150

J

e Fig. 3d. Same as fig. 3c but for T,, = 180 MeV.

Fig. 3e. TZOfor T, = 217 Me?‘; the dashed line corresponds to the best solution with S&,, S:, and S& variable and 8;~ kept fixed at theoretical value; for the shaded area a431 is also taken as free parameter (in combination or not with S:,).

J. Art&xx, A.S. Rittat/ Phase-shift analysis

I

I

I 254 MeV

21.5

f

0.0

-05

-1.0

Fig. 3f. T20for T,, = 254 MeV; the dashed curves represent diverse solutions obtained with ~mbjnations of ~$0, S:,, S& and S& as free variables; the dot-dashed curves represent equivalent solutions with S:, added as free parameter.

8 Fig. 3g. Same as fig 3f but for T,, = 292 MeV.

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J. Arvieux, A.S. Rinat J Phase-shift analysis

IO

-I

50

100

150

8 Fig. 4a. Vector analyzing power iTI for T, = 82 MeV; see caption of fig. 3a.

1.0

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I

b

116 MeV

0.5

e.-

00

-05

-1.0

50

100

150

Fig. 4b. Same as fig. 4a but for T,, = 116 MeV.

J. Amieux, AS. R&tat / Pease-Swift analysis I

I

217

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C

142 MeV

05

g

0.

-0.5

Fig. 4~. Same as fig. 4a but for T,, = 142 MeV.

05

3

t-=0

.-

-0.5

1

I

508 ‘O”

I

150

Fig. 4d. iTzl for T = 180 MeV. Some solutions for Re 8’00 > 0 and Re sf < 0 are shown for different combinations of phase parameters a&-,,ST,, S& and S:,.

J. Aruieux, A.S. Rinat j Phase-shiftanalysis

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I .o

I

I

I 217 MeV

-,u/i

50

100

e

150

8 Fig. 4s. iTI for T, = 217 P&Q; the dashed curve is the best fit with St,, S:,, and S& variable and S$ kept tied at theoretical value; for the shaded area S& is also taken as free parameter in combination with or without S:,.

I

I t

254 MeV

-0.5

-1.0

-

I 50

8

I 100

I 150

Fig. 4f. iTI for T, = 254 MeV; see caption of fig, 3f.

J. Arvieux, AS.

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Rinat J Phase-shift analysis

I

219

I

50 8 ‘O” ‘50 Fig. 4g. iTll for T,, = 292 MeV; see

captionof

fig. 3f.

142MeV. Reasonably good fits (x2 = 1.05) are obtained with S& and S:, ; no improvement with Sz2 (x2 = 1.1). If S:, is allowed to vary, one finds two discrete solutions. In solution I (x2 = 0.73) the splitting of the P-waves follows a natural order S& > S:, > Si2 as given by theory and iTl1 is predicted to be positive at all angles (-sin 8 shape), the maximum being determined by the size of the splitting. In solution II (x2 = 0.70) Re S:, is negative; hence S?, > Si2 > S:,, and iT1, exhibits a large positive maximum around 60”, changes sign around 90” and exhibits a large negative maximum around 120” (-sin 28 shape). Preliminary results from an experiment done at SIN with a vector-polarized deuteron target 30)indicate positive iTI1 between 8 = 70” and 140”; although not yet final, these results rule out solution II. The iT1, angular distribution calculated for solution I is compared to the theoretical one in fig. 4c. A substantial part of the X*-value is due to a bad fit of the only polarization datum published till now, namely r2, (140 MeV; 180”) = -0.23 k 0.15 while the theoretical values computed with a Yamaguchi form factor (PD = 6.7%) are -0.77 [ref. *‘)I or -0.74 [ref. *“)I. The calculated values after searches with Sio, S:,, Sz2 and S:, range from -0.61 to -0.70 (fig. 3~). By varying S A0a value of -0.45 can be found but the resulting amplitude fho - 0, which is difficult to reconcile with non-vanishing values at adjacent energies. In order to fit perfectly the T20 (142, 180”) point, one must vary small L = L’= J + 1 phases and/or mixing parameters. One such solution (x*=0.52) obtained with SAo, ST,, St3 and .5f3 as free parameters is shown in fig. 3c, but again the experimental amplitudes are difficult to reconcile with the

220

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R&tat/ Phase-shift analysts

corresponding theoretical ones at neighbouring energies. Yet this solution cannot be rejected on these grounds alone: it is conceivable that future polarization data at other energies will produce experimental amplitudes again consistent with a smooth energy dependence. The conclusion of the extensive study at 142 MeV is that polarization data are sensitive to small amplitudes which have negligible effects on the differential and total cross sections alone. More data are needed to sort out non-dominant amplitudes which, in turn, are most sensitive to the deuteron wave function 31) and/or details of the absorption mechanism. 18OMeV. Very good fits (x2 = 0.73) can be obtained varying only 2 parameters: Im Sk and Im & but the theoretical value of Re S&, is negative in disagreement with experimental amplitudes at higher energies. Good fits (x2 = 0.4-0.6) having Re S&, > 0 can be obtained when starting the search with values opposite to the theoretical one. In conclusion, at 180 MeV there is a low sensitivity of the cross section data alone to the real parts of the phase shifts, resulting in large uncertainties for these parameters. Polarization data are much more sensitive and this is demonstrated in fig. 3d for TzOand fig. 4d for iT1l. 217MeV. With &, S:i and Si2 as free parameters, a good fit (x2= 1.17) is obtained which improves slightly (x2= 1.00) when S433 is included in the variation (experimental backward rise better reproduced). No improvement is obtained if 6 :r is included in the search. Moreover Re Sir is positive (kf:r lies outside the theoretical Argand path) in contrast with results at all other energies where Sir brings some slight improvements and the final value lies always inside the theoretical Argand diagram (see fig. 5b, below). The variation of S & has an especially large.effect on TzO at backward angles (fig. 3e) and S:, has strong effects on the angular distribution of iTI (fig. 4e). 254 MeK The energy T, -254 MeV is of particular interest in view of the inability of the theory to predict the deep minimum around 8 = 100” reported by Gabathuler et al. 13). The new SIN/CERN data come closer to the predictions but theory still overshoots the data by a factor of -2 at the minimum and beyond. If only S&, S:, and S,“, are varied x2 = 1.25 (SIN + CERN data) or 1.70 (SIN + 180” data). The backward rise is better reproduced by varying also S& (,y’ = 0.4 to OS), and no further improvement is brought about by varying additional parameters. Good fits can also be obtained with the following combination of free parameters: I%, S:,, S& +S& or S& or Sg3 resulting in very different tensor T20 (fig. 3f) and vector i?‘rr (fig. 4g) analyzing powers. In all cases the calculated f:, and $2 amplitudes depart strongly from the theoretical value (see fig. 5c, d) and are mainly responsible for the disagreement between experiment and theory. 292MeV. Very large x2 (4.4) if only &, ST, and Sz2 are varied freely. No improvement (x2 = 3.6-4.4) when S*22 or Sz3 or Sg3 or S& are also varied. Spectacular improvement (x2 = 0.3-0.4) when S z3 is introduced in the search. Predictions for T20 (fig. 3g) and iTI (fig. 4g) for different combinations of parameters are given.

1. Arvieux, AS. Rinat / Phase-shift analysis

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We conclude this subsection by showing the calculated da/da (fig. 1) where different analysis of the same data sets are undistinguishable. Also shown in fig. 2 is the perfect fit to the total cross section together with the calculated reaction cross section.

4.2. ARGAND

DIAGRAMS

In figs. 5a-e are shown the Argand diagrams (i.e. Im k& as a function of Re k&) for J” = l-, l+, 2’, 3- and 4+ together with the theoretical predictions (solid line). Regions drawn around marked pion lab energies represent envelopes of all solutions having x2< 1, including some obtained by varying other than the five dominant amplitudes. These regions take into account phase parameter uncertainties as calculated by SIMPLEX (see subsect. 3.1). Due to the multiplicity of good solutions we can actually only trace Argand bands without a prescription of how to link points between the marked regions. & (fig. 5a): large uncertainties on the sign of Re 6: at 180 MeV; f& departs from the theoretical value from 217 MeV up. fir (fig. 5b): when added as a free parameter does not bring sizable improvements to the x2 but induces a smoother behaviour with energy for other amplitudes. The experimental Argand path tends to be inside the theoretical one. 1

I

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a

f'

00

0.6

-

I

I

273

,

--\k)l

nn

Re kf Fig. 5a. Argand plots giving Im kf as function of Re kf for the amplitude &. The solid line curve represents the theoretical predictions at 82 (*I, 116 (Cl), 142 (V), 180 (Xl, 21’7 f+), 254 (9) and 292 (A) MeV. The regions drawn for each of these energies represent all solutions having a x2 s 1. For 254 MeV the solid line corresponds to the analysis of the SIN+ 180” data, the dotted line to the analysis of the SIN + CERN data.

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Rinat / Phase-shift analysis I

I

I

b

Re kf

fit.

Fig. Sb. Same as fig. Sa but for amplitude

ffI (fig. 5~): begins to depart from the theoretical value already at 82 MeV. Describes an Argand plot with a radius smaller than the theoretical one which crosses the real axis around 180 MeV in the lab system. The important departure from the theoretical value occurs for a total energy Js= 2.18 GeV at which a ID2 structure has been observed in a-@ scattering ‘) (same J but L,, = L,d + 1). I

I

I

I

I

I C

fi

06-

I

I

-0.2

I

I

0.0

I

0.2

Re kf Fig. SC. Same as fig. Sa but for amplitude fTl.

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J. Arvieux, A.S. Rinat / Phase-shift analysis

Re

223

kf f&.

Fig. Sd. Same as fig. 5a but for amplitude

& (fig. 5d): at low energy (82-180 MeV) good fits can be obtained with theoretical values. Begins to strongly depart from theory at 217 MeV. Describes a sharp loop between 180 and 292 MeV (sharper for the SIN+ 180” data than for the SIN+ CERN data) crossing the real axis slightly above 217 MeV, in good agreement with a 3F3 structure at Js = 2.22-2.26 GeV observed in fi-fi scattering a). I

I

I

e

Re kf Fig. Se. Same as fig. 5a but for amplitude

fi3.

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J. Arvieux, AS.

Rinat / phase-shift analysts

f& (fig. Se); good fits with theoretical values up to 2.17 MeV but departs markedly at 254 and 292 MeV. This amplitude tends to lie outside the theoretical Argand diagram. All other amplitudes: can be changed by large amounts without affecting strongly the preceding results, but effects on deuteron analyzing powers are important and one must await the availability of polarization data at different energies to draw meaningful conclusions on these parameters.

5. Discussion and summa~ We presented above a first determination of pion-deuteron phase shifts for 82 < T,,(MeV) < 292. Our analysis employed two sources of information. On the one hand we used new and accurate experimental information on differential and total cross sections and we also discussed an isolated polarization datum. On the other hand we trusted theories for the 7rd (rrNN) system and chose theoretical partial wave amplitudes as starting values. With about 20 relevant complex parameters per energy one clearly has to select a smaller number to be varied. Fortunately only a very limited number is needed in order to obtain a x2 < 1 and, not surprisingly, these phase parameters are also the dominant ones like S’& = S:,, Sk,, &, S:,. Statistically it makes then little sense to include more parameters, yet we sometimes did so. Those trials obviously had little to do with a better fit for a given energy. Instead we could achieve that certain phase parameters interpolate more smoothly between similar ones for neighbouring energies. Clearly such a procedure is only satisfactory when dominant phase parameters for other energies are not affected in the same measure when less important phases are allowed to be free. The thus derived experimental phase parameters are not necessarily very close to the theoretical ones, not even when the latter produce satisfactory agreement with the data. As an example we give the & amplitude for T,, = 142 MeV which are experimentally (0.17 f 0.05) + (0.40* 0.02)i and theoretically (0.25 + 0.33)i. These differences are of particular interest for T, = 256 MeV, for which virtually all calculations failed to describe the minimum observed by Gabathuler et al. 13). Even when taking the most conservative estimates for the uncertainties, the fitted f$ and & amplitudes appear to deviate most significantly from the theoretical predictions. For f:, the experimental value is (-0.06&0.03) +(0.22*0.02)i against a theoretical value of -0.06+0.31i. For & these are (-0.03*0.01)+(0.14*0.02)i against 0.07 + 0.12i. Even without intrinsic ambiguities and those incurred by truncating the partial wave series 32), we are left with those due to imperfect or still missing data. We thus showed that preliminary results for the sign of iTI, (141 MeV, 8) already arbiter between two solutions both fitting equally well dr/dLS.

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Also the only published tensor polarization data point TZO(140; 180”) is very restrictive. It is so remote from predictions that, all by itself, it significantly raises the x2 for that energy. The quest for polarization data hardly needs further emphasis. Simultaneously it will be desirable to use in a future analysis theoretical input with reliable absorption corrections. Finally, we presented the resulting amplitudes in Argand plots. A non-unique analysis does not produce curves for Im kfiL, against Re kf&+, but rather bands connecting regions which for each energy solutions have x2 < 1. Even without the ability to trace Argand plots one infers that all are counter clock-wise traversed when T, increases. In several publications the question has been asked whether the plots fake resonances or not. We cannot forego the observation that Im f: and Im fz are maximal for 4s - 2.18 GeV and -2.24 GeV respectively. These energies are conspicuously close to the positions where structures in (~~~~~r(i), 8) [ref. *)I have been seen and which have been conjectured to be di-baryon resonance in some phase shift analysis 9). We shall return to these questions elsewhere. One of the authors (ASR) thanks the Theory Group at Saclay (DPhT) for their hospitality in the autumn of 1979. Both appreciated allocation of computer time and assistance. Dr. Stanovnik and his colleagues have been kind enough to supply their data when still preliminary. Likewise we wish to thank Drs. Fayard and Lamot for supplying us with their amplitudes. References 1) Yu.A. Shcherbakov, T. Angelescu, I.V. Falomkin, MM. Kulyukin, V.I. Lyashenko, R. Mach, A. Mihul, N.M. Kao, F. Nichitu, G.B. Pontecorvo, V.K. Sarychieva, M.G. Sapozhnikov, M. Semerdjieva, T.M. Troshev, N.I. Trosheva, F. Balestra, L. Busso, G. Garfagnini and G. Piragino, Nuovo Cim. 31A (1976) 249 2) J. Beiner, Nucl. Phys. B53 (1973) 349 3) J.P. Albanese, J. Arvieux, J. BoIger, E. Boschitz, C.H.Q. Ingram, J. Jansen and J. Zichy, Nucl. Phys. A, to be pubfished 4) A. Gersten, Nucl. Phys. A219 (1974) 317; J.F. Germond and C. Wilkin, Nucl. Phys. A287 (1975) 477 5) A.S. Ciough, G.K. Turner, B.W. Allardyce, C.J. Batty, D.J. Baugh, M.J. McDonald, R.A.J. Riddle, L.H. Watson, M.E. Cage, G.J. Pyle and G.T.A. Squier, Nucl. Phys. B76 (1974) 15; A.S. Carroll, I.H. Chiang, C.B. Dover, T.F. Kycia, K.K. Li, P.O. Mazur, D.N. Michael, P.M. Mockett, D.C. Rahm and R. Rubinstein, Phys. Rev. Cl4 (1976) 635 6) R.H. Landau, S.C. Pathak and F. Tabakin, Ann. of Phys. 70 (1973) 299 7) M. Hirata, J.H. Koch, F. Lenz and E.J. Moniz, Ann. of Phys. 120 (1979) 205 8) I.P. Auer, A. Beretras, E. Colton, D. Hill, K. Nield, H. Spinka, D. Underwood, Y. Watanabe and A. Yokosawa, Phys. Lett. 70B (1977) 475 9) N. Hoshizaki, Prog Theor. Phys. 58 (1977) 716 10) E.G. Pewitt, T.H. Fields, G.B. Yodh, J.G. Fetkovitch and M. Derrick, Phys. Rev. 131(1963) 1826 11) J. Norem, Nucl. Phys. B33 (1971) 512 12) R.H. Cole, J.S. McCarthy, R.C. Minehardt and E. Wadlinger, Phys. Rev. Cl7 (1978) 681

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