Amplitude-phase patterns: A new look at strong interactions

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A578 (1994) 441-470

Amplitude-phase patterns: A new look at strong interactions G a r y R. G o l d s t e i n

a

F i r o o z A r a s h b,c, M i c h a e l . J. M o r a v c s i k d,,

a Physics Department, Tufts University, Medford, MA 02155, USA b Center for Theoretical Physics and Mathematics, AEOI, P.O. Box 11365-8486, Tehran, Iran c Physics Department, Amirkabir University, Hafez Avenue, Tehran 15914, Iran d Institute of Theoretical Science and Physics Department, Eugene, OR 97403, USA

Received 16 November 1993

Abstract The phases of complex spin-dependent scattering amplitudes for elastic processes NN, ~-N, ~-d, along with pp ~ d~-+, are analyzed in various frames of reference for spin quantization. When all available energies and angles are compiled it is seen that the "phase histograms" for each reaction have remarkably simple properties in one choice of optimal frame; the phases tend to be integer multiples of 90 °, within existing uncertainties. A two-component model for ~-N is presented that reproduces the striking pattern of phases and its generalization is discussed.

I. Introduction

In a series of papers [1-9] over the past few years, we reported the apparent existence of a rather striking feature of a n u m b e r of strong-interaction reactions in a wide range of energies and angles. The purpose of the present p a p e r is twofold. First, we want to summarize the evidence for this feature by displaying the whole wide assortment of information we have on this feature, most of which was not shown in detail in the previous brief publications. In fact, much of this information was generated only very recently, since the publication of most of the previous notes. The information displays the dependence of the feature, if any, on various p a r a m e t e r s that may be thought to be relevant. The conclusion that emerges is that the feature exists in a broad range of the presumably relevant

* Deceased. Elsevier Science B.V. SSDI 0375-9474(94)00185-P

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parameters, although uncertainties in the data and theoretical analyses allow some doubt and should encourage more precise measurements. Second, we would like to suggest some possible interpretations of this feature. Specifically, we explore a way of thinking about strong-interaction reactions which, in some ways, is radically different from the approaches used up to now, and in fact is in certain respects opposite to them. We will show, with a simple model, that the new interpretation of strong-interaction dynamics is, at least, a quantitatively possible one. We will also suggest some advantages the new interpretation may have in allowing us to carry out reliable and relatively simple calculations of the experimental observables in strong-interaction reactions, perhaps circumventing some of the longstanding difficulties in particle physics in this respect.

2. The phenomenological status of amplitude-phase patterns The feature noted in the above quoted references was stated as follows. First, the feature was noted in four types of strong-interaction reactions: (a) elastic nucleon-nucleon scattering [2-4,7], (b) elastic pion-nucleon scattering [5], (c) elastic p i o n - d e u t e r o n scattering [8], and (d) the reaction [9] pp --* d~ -+. Second, the feature pertained to reaction spin amplitudes at various angles and energies. These amplitudes are complex, and their number depends on the spins of the particles participating in the reaction and on the symmetries that constrain the reactions (i.e. the conservation laws that hold for those reactions). The number is, however, the same at all energies and at all angles except for some special geometrical configurations such as collinear reactions and (in some cases) reaction angles of 90 °. Such reaction amplitudes can be specified in an infinite number of different systems [10], since we can choose an infinite number of different sets of basis vectors spanning the spin space of a reaction. A subset of this infinite set of sets of basis vectors is the one in which the relationship between the experimental observables and the bilinear products of amplitudes is the simplest. Such a subset of bases is called the optimal formalism [10]. This subset of optimal formalisms contains an infinite number of different formalisms; while the format of the matrices is specified in the optimal formalisms, the directions of the quantization axes of the particles remain arbitrary except to the extent to which they are linked and specified by conservation laws. Even the most stringent set of conservation laws, however, leaves us with an infinite set of formalisms. In particular, if parity conservation holds (which is the case for strong-interaction reactions), the quantization direction of particles must be either normal to the reaction plane or in the reaction plane. If the quantization direction of each particle is normal to the reaction plane, we have a pure transversity formalism, while if it is in the reaction plane, we have a pure planar formalism. A special case of the pure planar system, which has been used extensively, is the helicity formalism in which the quantization direction of each particle is in the direction of its momentum. There are, in general, also hybrid formalisms in which

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some particles are in the planar system and some in the transversity system, unless conservation laws (such as time-reversal invariance and identical particles) prohibit the existence of such a system. In the present paper we will use only three such optimal systems: The pure transversity formalism, the pure helicity formalism, and the planar-transverse formalism in which the quantization direction of each particle is perpendicular to both the normal to the reaction plane and to the helicity direction. In the previous discussions the feature was reported to exist in the p l a n a r transverse system. What is this feature that has been reported? It pertains to the phase differences in the complex plane among the (complex) spin amplitudes. If the particular reaction has n spin amplitudes, there are n - 1 independent relative phases among them. The overall phase of all amplitudes taken together is not an observable quantity, and can be determined only on the basis of some theoretical assumptions or conventions, for example through interference with other types of interactions (such as the Coulomb force) for which we have a theoretical overall phase. In what follows we will ignore this overall phase and focus our attention only on the n - 1 relative phases among the n amplitudes. The reported feature then consists of these phase differences showing a marked tendency to have values which are multiples of 90 °. In other words, the complex amplitudes show a marked tendency to be, relative to each other, pure real or pure imaginary. This statement can then be investigated as a function of the following variables: (1) Which reaction do we consider? (2) In which optimal formalism do we consider the amplitude-phase differences? While the previous reports all pertained to the planar-transverse system, it is interesting to find out if any regularities appear in other optimal formalisms also. (3) At what energies do we consider the reaction? (4) At what angles do we consider the reaction? (5) The phase difference between which amplitudes do we consider? Previous reports offered data only on the phase differences among all amplitudes together. (6) Is there something to be noted also about the magnitudes of the amplitudes and not only about the phase differences? The aim of this section is to provide detailed information on the above dimensions of the purported amplitude phase pattern. The experimentally determined material is presented in Figs. 1-19. The crucial quantities are reaction amplitudes which were determined by analyzing polarization correlation data or by summing documented partial waves that interpolate cross section and polarization data. These figures show histograms [6] in which the amplitude-phase difference is plotted on the abscissa, and the number of situations in which a given amplitude-phase difference is realized is plotted on the ordinate. The amplitude-phase bins used in these histograms are 5 ° wide. We investigated the effect on this arbitrary choice of the width of the bin on the patterns and found no such effect that would affect our conclusions. In all the histograms presented here events pertaining to various reaction

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energies and reaction angles are aggregated into one histogram. The energies and angles are indicated in the histograms. We also disaggregated the histograms according to energy and angle to the extent that this was possible, without reaching too narrow bins for which the results become statistically meaningless. In general, we did not find anything in this disaggregation that would affect our conclusions. Some remarks on this will follow below. We also generated graphs in which the phase differences between specific amplitudes and specific energies are plotted against the reaction angle. There is of course a very large number of such graphs, which we therefore cannot present here, but they are available on request. They indicate, however, that the phase differences are in general continuous functions of the kinematic variables, something that one would expect in order for the patterns to be considered "real". All histograms show fluctuations from one amplitude-phase-difference bin to the next. These fluctuations are not inconsistent with the magnitudes of the purely statistical fluctuations we expect by taking the square roots of the number of events in a bin. Making the bins twice as wide smooths out these fluctuations to an expected extent. We now turn to the specific aspects of the data listed earlier. 2.1. Which reactions do we consider?

We have tried to investigate all reactions for which sufficient polarization data are available [2-9] so that the amplitudes can be determined reasonably reliably. These included pp and np elastic scattering, pion-nucleon elastic scattering, p i o n - d e u t e r o n elastic scattering, and the reaction pp --, d~ -+. There also may be data on p r o t o n - a l p h a elastic scattering. We have not explored that reaction so far. Also there is now a presumably complete set of data on p r o t o n - d e u t e r o n elastic scattering, but the amplitude analysis of that has not been performed yet. In this latter reaction there are 12 complex amplitudes, and therefore it is possible that the precision by which amplitudes can be determined will be too low for amplitude-phase patterns to be established reliably. In this context it is well to remember that in any amplitude determination, the magnitudes of the complex amplitudes can usually be determined with relatively greater precision and without any discrete ambiguities. Furthermore, such a determination can be carried out from a subset of the experiments, independently of the phases. In contrast, the determination of the phases of the amplitudes can usually be made only with larger uncertainties, and only in conjunction with the determination of the magnitudes. Furthermore, in the determination of the phases discrete ambiguities also enter. Since in the present investigation we are interested in the phase differences and not the magnitudes of the amplitudes, for our purposes high-precision data are necessary. It should also be recalled that in the relationship among amplitudes in different bases both the magnitudes and phases enter. Thus the determination of just the magnitudes of the amplitudes (or of just the phases) in a given basis is not sufficient to obtain the magnitudes (or the phases) in another basis. Thus the

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precision in the determination of all amplitude p a r a m e t e r s (both magnitudes and phases) in one basis enters when we want to assess the uncertainties of the amplitude p a r a m e t e r s in another basis. This brings us to the analysis of the uncertainties in the histograms. The purely statistical error due to the n u m b e r of events in each bin is only one and the easiest of the components in the overall assessment of the uncertainties in the histograms. A much more difficult one to assess is the uncertainty in the individual amplitude p a r a m e t e r s (and particularly the amplitude phases) which underly each event in the histogram. The uncertainties in the amplitude p a r a m e t e r s are related, in a very complex way, to the uncertainties of the individual experimental observables which serve as the basis for the determination of the amplitudes. In some cases, such as for n u c l e o n - n u c l e o n elastic scattering, the process of going from the experimental data to the amplitude p a r a m e t e r s actually goes through an intermediate process of an energy-dependent phase-shift analysis which has its own sets of uncertainties. The result of all this is that it is not possible to attach a quantitatively firm estimate on the horizontal uncertainties in these histograms. Very rough estimates indicate that the widths of most peaks one sees are not incompatible with the assumption that the widths arise entirely from these uncertainties in the determination of the amplitude phases, but this conclusion must be considered as extremely tentative. The widths, which in some cases are 30 ° to 40 °, could correspond to real dynamics a n d significant deviations away from the multiples of 90 °. This is a possibility that cannot be ruled out by the calculated phases, given the quality of the primary data. Histograms for the above mentioned five reactions are presented in the figures. We see that all such figures exhibit the pattern in which the phase differences tend to be multiples of 90 °. The extent to which the pattern is pronounced varies somewhat, but the pattern is clearly evident in all cases. Given the uncertainties about finite widths and the resulting "slippage" in determining the center of the peaks for particular histograms, the conclusion is weakened somewhat. That is, peaks in some cases may be centered at 330 ° rather than 360 ° (Fig. 1) or 150 ° rather than 180 ° (Figs. 5 and 6). Such caveats aside, however, the overall summary of the large array of phase histograms (for particular frames) for different reactions, at a wide range of energies and scattering angles supports the conclusion. 2.2. In which optimal frames [10] do we consider the amplitude phase differences? Although we noticed the amplitude-phase pattern first in the planar-transverse (PT) system, in subsequent studies we also generated histograms for the phase differences in two other noted optimal frames, the transversity (T) and the helicity (H) systems. Thus the figures offer histograms in all three frames. We see that the pattern is evident in all three frames, to varying degrees. In general, a pattern in one frame would not be expected to be evident in other frames which are related to the first one by linear transformations of the complex amplitudes. Thus the emergence of the pattern gives us additional information on the nature of the pattern.

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pp--->pp Planar-Transverse Phase Differences:

All Energies 10-90 deg All

40

tO O0

30

O O 20

..O

E z

10

0 0

60

120

180

240

300

360

phase differences(degrees) Fig. 1. Histogram at all energies and angles, of all phase differences, for pp elastic scattering in the planar-transverse system.

From the point of a comparison among the three frames, the reader should juxtapose Figs. 1, 2, and 3, then 4, 5, and 6, then 7, 8, and 9, then 10 and 11, and finally 12, 13, and 14. Note that for the reaction ~1 + 0 ~ ~1 + 0 (which is the class PP--->PP Transversity Phase differences: co fl)

lO0-1300 MeV 10-90 (deg) All

l°° t 80 -~

O O ..(3

E z

60'

40'

20'

0 ¸

0

60

120

180

240

300

360

p h a s e differences(degrees) Fig. 2. Histogram at all energies and angles, of all phase differences, for pp elastic scattering in the transversity system.

G.R. Goldstein et aL /Nuclear Physics A578 (1994) 441-470

PP--->PP Helicity

447

100-1300 MeV 10-90 (deg)

Phase Differences:

All

50' O9 I1)

40 1

O 30'

o

..Q

20

E '-I 7

10

0

60

120

180

240

300

360

phase differences(degrees) Fig. 3. Histogram at all energies and angles, of all phase differences, for pp elastic scattering in the helicity system.

pion-nucleon elastic scattering belongs to) the helicity and the planar-transverse systems have the same phase difference between the two spin amplitudes, as tabulated in the appendix, and hence for that reaction two figures suffice.

PN--->PN Planar-Transverse Phase Differences:

100-1300 MeV 10-90 (deg) All

5O tO tO

o o d3

30

20

E Z

10

o o

60

12o

180

240

300

360

phase differences(degrees) Fig. 4. Histogram at all energies and angles, of all phase differences, for np elastic scattering in the planar-transverse system.

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4O

PN--->PN Transversity Phase Differences:

100-1300 MeV 10-90 (deg) All

30

O O

~

20

..Q

E Z

lo

0

o

60

120

18o

240

300

360

p h a s e differences(degrees) Fig. 5. Histogram at all energies and angles, of all phase differences, for np elastic scattering in the transversity system.

2.3. A t what energies do we consider the reaction?

We tried to use all energies at which the data were complete enough to yield an amplitude determination. On the whole, we found the same pattern for a given reaction, at all energies. The clarity of the pattern, however, changed somewhat with energy. In pp elastic scattering, for instance, as Fig. 15 shows, the pattern

pN--->I~N

100-1300 MeV

Helicity

10-90 (deg)

Phase Differences:

All

6o 5O ~9 (,~

4O

O

~

30

..Q E

2o

Z

10 0

0

60

120

180

240

300

360

phase differences(degrees) Fig. 6. Histogram at all energies and angles, of all phase differences, for np elastic scattering in the helicity system.

G.R. Goldstein et aL / Nuclear Physics A578 (1994) 441-470 PI-D--->PI-D Planar-Transverse Phase Differences:

449

117-324 MeV 10-170 (deg) All

50

o9 O9

40-

0 ,.i...-

30"

.1~

2o-

Z

lo-

0

E

0

0

60

-

120

~ 180

240

300

360

Phase difference(degrees) Fig. 7. Histogram at all energies and angles, of all phase differences, for ~-d elastic scattering in the planar-transverse system.

became more pronounced if we disregarded the lowest energies in the energy range we considered. For the reaction pp ~ d~ -+, the pattern was clearer at 300 M e V and at 700-800 M e V than in between the two energies (see Fig. 16). The pattern persisted up to the highest energies we could utilize, which was p i o n nucleon scattering data at 45 GeV. We find it remarkable that a pattern of strong interaction exists which extends over such a large energy range.

PI-D--->pi-D Transversity Phase Differences:

117-324 MeV 10-170 (deg) All

40

O9 O9

30

O O

20

..Q

E Z

0 0

60

120

180

240

300

360

Phase difference (degrees) Fig. 8. Histogram at all energies and angles, of all phase differences, for ~'d elastic scattering in the transversity system.

G.R. Goldstein et aL /Nuclear PhysicsA578 (1994) 441-470

450

PI-D--->PI-D Helicity Phase Differences:

117-324 MeV 10-170 (deg) All

100

~•

80t

O

o

$

..Q

60

4o.

E Z

20.

00

60

120

180

240

300

360

Phase difference(degrees) Fig. 9. Histogram at all energies and angles, of all phase differences, for ~-d elastic scattering in the helicity system.

2.4. A t what angles do we consider the reaction?

A s w i t h e n e r g i e s , w e t r i e d to u s e all a n g l e s at w h i c h d a t a w e r e a v a i l a b l e . U n f o r t u n a t e l y at h i g h e r e n e r g i e s t h e c r o s s s e c t i o n s fall o f f s h a r p l y w i t h i n c r e a s i n g

PI+P--->PI+P Energies:All & 45 GeV/c Planar Transverse 10-170 (deg.) PhaseDifferences: All 80.

t/)

60'

"6 ~ ,IQ

40.

E z 20.

O. o

60

120

180

240

300

360

Phase defferences (degrees) Fig. 10. Histogram at all energies and angles, of all phase differences, for 7rN elastic scattering in the planar-transverse (or helicity) system.

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G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

PI+P--->PI+P Transversity Phase Differences:

100-1200 MeV 10-170 (deg) All

20

O o

10

.x3

E Z 0¸ 0

60

120

180

240

300

360

Phase differences(degrees) Fig. 11. Histogram at all energies and angles, of all phase differences, for ~'N elastic scattering in the transversity system.

angles so that at those energies only relatively small angles appear in the data. For lower energies, up to a GeV or so, the entire angular range is available. On the whole, we found the pattern appearing in the whole angular ranges at our disposal. There is some tendency for pp elastic scattering for the pattern to be

330-800 Mev 10-170 (deg) All

PP--->PI-D Planar-Transverse Phase Differences: 60

50-

O

40-

O 30-

..(3

E "1

20-

Z 10-

00

60

120

180

240

300

Phase differences(degrees)

360

Fig. 12. H i s t o g r a m at all e n e r g i e s a n d angles, of all p h a s e differences, for pp ~ d~ -+ s c a t t e r i n g in the

planar-transverse system.

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

452

PP--->pi-D Transversity Phase Differences:

330-800 MeV 10-170 (deg) All

200

09

o O

1O0

..(3

E Z

0"

0

60

120

180

240

300

360

Phase Differences(degrees) Fig. 13. Histogram at all energies and angles, of all phase differences, for pp ~ d ~ transversity system.

scattering in the

more pronounced if we eliminate some of the smallest scattering angles (say, below 30 ° or 40°). This, together with the sharpening of the pattern as we go to higher energies (mentioned in the previous subsection) might suggest that the pattern is characteristic particularly of the small-range interaction.

80

tJ)

PP--->PI-D Helicity Phase differences:

330-800 MeV 10-170 (deg) All

60

o 0

40

.Q

E -I

Z

20

o o

60

Phase

12o

18o

240

300

360

Differences(degrees)

Fig. 14. Histogram at all energies and angles, of all phase differences, for pp --* d ~ * scattering in the helicity system.

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

pp--->pp Planar-Transverse Phase differences:

453

600-1300 MeV 10-90 (deg) All

40

CD 1/1

30

O O

~

20

E Z

10

0 0

60

120

180

240

300

360

Phase differences(degrees) Fig. 15. Histogram for energies above 600 MeV, at all angles, of all phase differences, for pp elastic scattering in the planar-transverse system.

W e also e x p l o r e d if t h e p a t t e r n m a y b e d u e to s o m e special r e l a t i o n s h i p s at special a n g l e s (0 ° a n d 90 °) w h i c h a m p l i t u d e s have to satisfy i n d e p e n d e n t l y o f dynamics. T h i s was d o n e by g e n e r a t i n g p a t t e r n s in w h i c h t h e special angles a n d

pp-->pi-d PLanar-Transverse

Phase Differences:

330& 700-800 Mev 10-170(deg) All

40-

t#) O t#)

30.

d "6 20. .Q

E Z 10'

o

o

60

lZO

180

240

300

360

Phase differences(degrees) Fig. 16. Histogram at energies of 300 and 700-800 MeV and all angles, of all phase differences, for pp ~ d T r + in the planar-transverse system.

G.R. Goldstein et al. /Nuclear Physics A578 (1994) 441-470

454

pp-->pi+d Transversity: Phase differences:

All Energies 10-80 (deg) All

1O0 -

80

0

~1

6O

¢'~

40

o

E z

0

0

60

120

180

240

300

360

Phase differences (degrees) Fig. 17. Histogram at all energies and at angles between 10° and 80°, of all phase differences, for pp ~ d~ + scattering in the transversity system. their vicinities were excluded. No significant change in the pattern was observed as a result of this exclusion. An illustration of this is given in Fig. 17.

2.5. The phase difference between which amplitudes do we consider? In the histograms in Figs. 1-17 the phase differences among all amplitudes are plotted. There are two questions that emerge in connection with that agglomeration. First, in order to treat all amplitudes on an equal footing, we plotted all phase differences among amplitudes at a given reaction energy and reaction angle. For example, for pp elastic scattering, there are five complex amplitudes, and hence 10 amplitude-phase differences. We plotted all of them, even though only four of the set represent a linearly independent set. Does this introduce a bias that may artificially produce the pattern we see? As one can easily convince oneself, the answer is not only "no", but in fact one can see that plotting all phase differences instead of only four independent ones tends to deemphasize the pattern in which the phase differences are multiples of 90 °. Thus, in this sense, all the histograms presented here somewhat deemphasize the pattern, which would be sharper if we had plotted only a set of independent phase differences. The second question that arises is whether the pattern of phase differences between specific two amplitudes is consistent, that is, whether that is the same for the aggregation of the kinematic variables appearing in Figs. 1-17. The answer is

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455

very much in the affirmative, at least in nucleon-nucleon elastic scattering and in pion-deuteron elastic scattering, where we could generate statistically singificant histograms for individual phase differences. For pion-nucleon elastic scattering there are two complex amplitudes and hence this issue does not arise. For pp ~ d~"+ the disaggregation into individual phase differences yielded histograms with too low statistics to offer meaningful results. The documentation for this decomposition into individual amplitude-phase differences is provided in Figs. 18 and 19. We see from these that a consistent overall picture can be derived from these histograms about the relative phases of the amplitudes independently of reaction energy and angle. For pp elastic scatterpp-..>pp Planar-Transverse Phase Differences:

100-1300 MeV 10-90 (deg) a-b

12 ,

10-

0

8

0

~

6

E

4

Z

2"

0

60

120

180

240

300

360

Phase differences(degrees) pp.-->pp Planar-Transverse Phase Differences: 8

o9

600-1300 MeV 10-90 (deg) a-c

,

6

O O

$

'

E ~

z

2-

0

' 0

60

~ 120

180

240

300

360

P h a s e differences(degrees) Fig. 18. (a)-(j) Histograms at all energies and angles, of phase differences between specific amplitudes, for pp elastic scattering in the p l a n a r - t r a n s v e r s e system.

G.R. Goldstein et aL /Nuclear Physics A578 (1994) 441-470

456

ing the amplitudes a, c, d, and e are generally collinear (with even their signs usually the same over the whole kinematic range) and b is at 90 ° from the others. For p i o n - d e u t e r o n elastic scattering all amplitudes are collinear, and their signs are also, on the whole, consistent over the kinematic range. Again there are uncertainties in individual plots that can weaken these general conclusions. A critical viewer could conclude that in Figs. 18a, b, c the large-angle peak is actually at 330 ° rather than 360 ° and the middle peak in Figs. 18d, e, f and 19c is at 150 ° rather than 180 ° . This could reflect some other tendencies altogether, that may have to do with rather conventional behavior of amplitudes. While this latter interpretation cannot be excluded by the histograms, again, we think that there is an overall trend that is summarized by phase accumulations at multiples of 90 ° . PP--->PP

600-1300 M e V 10-90 (deg)

Planar-Transverse Phase Differences:

a-d

20

o0 (/)

0 10 ¸

c~

E "-I

z

0

60

120

180

240

300

360

Phase differences(degrees) PP--->PP

600-1300 M e V 10-90 (deg)

Planar-transverse Phase Differences:

o0

a-e

8

0 6

o ..Q

4

E Z

2

60

120

180

240

300

Phase differences(degrees) Fig. 18.

(continued).

360

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

457

This completes the summary of the phenomenological features of the pattern we see. As a postscript, one might ask whether one can discern any pattern in the magnitudes of the amplitudes also. In one of our previous papers we indicated such a magnitude pattern for pp elastic scattering. Our investigation of the other reactions, however, does not appear to indicate any magnitude pattern. This question will be subject to further study. On the basis of the above phenomenological conclusions, we now give, in the next section, a possible and tentative interpretation of the existence of the pattern we claim to have established above.

600-1300 MeV 10-90 (deg)

PP--->PP

Planar-Transverse Phase differences:

b-c

6

(/)

5

o)

O

4

O 3

..Q

E

2

Z 1-

ot

100

0

200

300

400

P h a s e differences(degreed)

PP--->PP

600-1300 MeV 10-90 (deg)

Planar-Transverse Phase Differences

b-d

10 -'r"

03 f~

o O

E Z

0

60

120

180

240

300

P h a s e differences(degrees) Fig. 18. (continued).

360

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G.R. Goldstein et a L / Nuclear Physics A578 (1994) 441-470

3. An interpretation of the amplitude-phase pattern The appearance of peaks in the histograms for relative phases among spin amplitudes in various frames suggests that some coherent spin-dependent dynamical process is contributing to the scattering. By coherence here we mean a dynamical process that persists over many energies and angles and maintains a definite phase relationship among spin amplitudes over the full range of kinematic variables. The definite phase results in constructive or destructive contributions to

PP--->PP Planar-Transverse Phase Differences:

600-1300 MeV 10-90 (deg) b-e

12

if) o3 09 ¢0

O

10

8

O 6

..(3

E

4

Z 2

0 0

60

120

180

240

300

360

P h a s e differences(degrees)

PP--->PP Planar-Transverse Phase Differences:

600-1300 MeV 10-90 (deg) c-d

8 ,

09 O3 09

O o ..(3

E Z

0-f 0

60

120

180

240

300

P h a s e differences(degrees) Fig. 18. (continued).

360

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G.R. Goldstein et al. /Nuclear Physics A578 (1994) 441-470

amplitudes when summed over the kinematics. Alternatively, the mechanism of coherence leads to alignment of the spin amplitudes in the complex plane. While there are many single processes that have such a property for small ranges of energy a n d / o r angle, e.g. resonance formation or single partial wave dominance, the effect seen in the histograms is one that persists when large ranges

PP--->PP Planar-transverse Phase Differences:

10

600-1300 MeV 10-90 (deg) c-e

8-!

l

i

o 6o

O L_

4"

E 2"

Z

0

. 60

0

~ 120

180

240

300

360

Phase differences(degrees) pp-->pp Planar-Transverse Phase Differences:

(D

600-1300 MeV 10-90 deg a-c

6"

o o

..Q

E "3 Z

2.

0.d' 0

60

120

180

240

300

Phase differences(degrees) F i g . 18. ( c o n t i n u e d ) .

360

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

460

~+d--->n+d Planar-Transverse: Phase Differences:

All Energies 10-170 deg a-b

30

O

20 ¸

O

E Z

0 0

1 O0

200

300

400

Phase Differences (degrees)

~+d--->=+d Planar-Transverse: Phase Differences:

All Energies 10-170 deg a-c

10

tar)

8

t~

O

6

O

$ ..Q

E

4

Z

0 0

. 1 O0

~ 200

300

400

Phase Differences (degrees) Fig. 19. (a)-(f) Histograms at all energies and angles, of phase differences between specific amplitudes for ~ d elastic scattering in the planar-transverse system.

G.R. Goldstein et aL / Nuclear Physics A578 (1994) 441-470

~+d--->~+d Planar-Transverse: Phase Differences:

461

All Energies 10-170 deg a-d

10

Or) if)

0 0

E

4

Z

i

0

100

200

300

400

Phase Differences (degrees)

~+d--->=+d Planar-Transverse: Phase Differences:

All Energies 10-170 deg b-c

30

U') O0

0

20

0

$ ..Q

E

10

Z

0

100

200

300

Phase Differences (degrees) Fig. 19. (continued).

400

462

G.R. Goldstein et al. /Nuclear Physics A578 (1994) 441-470

of kinematic variables are accumulated. A mechanism like single-particle exchange in a crossed channel would give rise to such a coherence, since the spin couplings will all have the same phase (up to a sign) in a planar frame and the energy and scattering angle dependence is smooth in the direct channel. ~+d--->~+d Planar-Transverse: Phase Differences:

All Energies 10-170 deg b-d

20

ffl

o 0 ..0

E -3

Z

100

200

300

400

Phase Differences (degrees) ~+d--->~+d Planar-Transverse: Phase Differences:

All Energies 10-170 deg c-d

15

O9 (/) 0

10

0

E -3

5"

Z

O'-'r 0

100

200

300

Phase Differences (degrees) Fig. 19. (continued).

400

G.R. Goldstein et al. / Nuclear PhysicsA578 (1994) 441-470

463

O f course, what is seen in the histograms is a set of coherent peaks on top of a seemingly randomly distributed uniform background. W h a t is the meaning of the background, which varies in relative overall size depending on the reaction and frame? Calling this background the incoherent contribution suggests an interpretation - p h e n o m e n a that vary in phase rapidly over energy and angle. An example of this would be a set of many resonances of different mass, width, and spin. Each resonance contributes to all spin amplitudes with a definite phase at each kinematic point. Combining many resonances that overlap somewhat in energy, however, produces total phases that vary over the kinematic range. What emerges from this separation of coherent and incoherent dynamical mechanisms is a two-component picture of the kinematically aggregated spin amplitudes. An example of this is just the combination of a summation of simple resonances plus an exchange picture as proposed above. A somewhat more conservative interpretation could be advocated at this point. Recall that zero scattering in a particular spin channel corresponds to 0 ° phase shift and hence a real amplitude for that channel. If most channels have little scattering, their amplitudes will be coherent and resonances will produce large deviations that vary rapidly in phase as a function of energy. So the incoherent component will remain a resonant phenomenon, but no coherent mechanism would be needed - a much more modest supposition. Of course the assumption of small amplitudes getting large weighting because they are relatively real is indistinguishable from other assumptions about coherent p h e n o m e n a on the basis of the amplitude phases alone. But other data on the magnitudes suggest that many are quite large, especially the helicity non-flip amplitudes in the elastic processes. Exchanges, more complex unitarization effects, diffraction, or absorption are all possibilities for the coherent peaks. It may be that the different dynamical mechanisms that are already expected to operate in different regions of energy actually produce similar phase relations. Then no new p h e n o m e n a are being revealed by the phase patterns. This more conservative point of view remains to be tested over the large range of data utilized herein. It must be admitted that the two-component picture being explored here is hardly unique, but it is meant to be a plausible mechanism over all the energies explored, given what is known about dynamics already. A two-component scheme like this has the danger of "double counting" in the intermediate-energy region, in the duality sense of the late 1960's. But we are concerned here with an interpretation of the full complex spin amplitudes, not only their imaginary parts. Those amplitudes are evaluated at lower energies as well and at larger angles than those duality arguments considered. We will make contact with those duality arguments nevertheless, in the discussion below. To test this exemplar of a two-component picture, we construct a specific model for 7rN elastic-scattering amplitudes that is not far from what is generally believed. We take the well-established I = 3 1 and ~3 ~-N resonances and their p a r a m e t e r s from the Particle D a t a G r o u p compilation [11], up to a resonance mass of 2640 MeV. For the coherent part of the contribution we take real elementary-particle exchanges, i.e. Born terms - an f meson (with I t = 0) that contributes only to the

464

G.R. Goldstein et al. /Nuclear Physics A578 (1994) 441-470 PHASE

. . . . . .

200

HISTOGRAM

I

--

. . . .

I'

'4

~±p e l a s t i c Two

Component

PL^8 ---- 5 0 0

tO

Mode 3000

l~eV/c

I,

,

150

o

I00

z 50

, I,,,, 0

i00

200 PHASE

300

ANGLE

Fig. 20. ~" + p elastic-scattering phase histogram from the two-component model for PLAB = 500 to 3000 M c V / c and all angles in planar-transverse or helicity system.

helicity or p l a n a r - t r a n s v e r s e non-flip amplitude and a P' meson (also with I t = 0), at the p mass, that contributes to the flip amplitude. These exchanges are not m e a n t to be the only terms nor are they m e a n t to describe reality precisely. Rather they are chosen to represent the most important contributions to the process as revealed by the histograms. The results of aggregating the energies and angles as predicted by the simple model is shown in Fig. 20. Laboratory m o m e n t a from 500 to 3000 MeV, in 100 M e V steps, along with center-of-mass angles from 0 ° to 180 °, in 5 ° steps, are accumulated with each kinematic point considered as an "event". As expected, there is a "randomly fluctuating" background and a large peak at 180 °. Consider the background in more detail first. The peak disappears when the exchange couplings are set to zero leaving only the fluctuating background, as Fig. 21 shows. The appearance of the latter would be quite variable, depending on the particular choice of m o m e n t a range and resonance parameters. If only near-forward scattering angles are chosen, the pattern develops considerable peaking in the region of 220 °, as shown in Fig. 22. This is in agreement with the old notion of duality; the imaginary parts of the amplitudes due to resonances alone (in the near-forward direction and at fixed t) oscillate about the particle-exchange contributions as the energy is varied monotonically, giving rise to a definite phase

465

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

HISTOGRAM

PHASE 120

,

,

,

'

I

'

'

'

i

. . . .

t

'

'-I

~±p elastic TWO Component Model

i00 i

A Pt^8 -- 500 to 3000 MeVtc

~q

t IHelicity System

80

~/~ o

60

z

40

/ 1Exchanges ExcIuded

:

2O

o

,,

0

,

,

l

i00

,

,

,

I

,

200 PHASE ANGLE

,

,

,

I

300

Fig. 21. The same as Fig. 20 with exchangesexcluded.

relation. When all angles are included, in Fig. 21, the tendency to peak disappears. The resonances contribute a rather stochastic set of phase values, as anticipated. Turning on the exchange contribution gives a sharp peak at phase angle 180 °, whose height is dependent on the coupling constants of the exchanges. One example is the histogram shown in Fig. 20, for which the two exchange couplings are set equal to + 5 and - 5 in some appropriate units. Compare this to the histogram for the helicity amplitudes that was determined by the data, Fig. 10, over the lower-energy range. The same features are clear, corroborating this two-component picture, as we expected. Can this interpretation be universal? The ~'N case is as simple as can be, having only two amplitudes. Consider the NN elastic-scattering case. Not only are there more amplitudes, but the role of resonances is minimal, if dibaryons exist at all. However, it is known that many different exchanges contribute in the t- and u-channels to the five amplitudes. In the NN problem, identical particle constraints require both channels to receive the same exchanges. Note that t-channel one-particle exchanges fall off at least linearly with t (and correspondingly for u-channel terms), if not exponentially as in Regge parameterization. Hence at large angles, near 90 °, these exchanges would be swamped by resonances in the s-channel if they existed. Even if that were the case, what we have seen, in the ~-N case, is that

466

G.R. Goldstein et al. /Nuclear Physics A578 (1994) 441-470 PHASE HISTOGRAM

40

--1

I

I

I

n±p e l a s t i c _Two

30

Component

PLAB =

1 to

Helioity p~ o

Model

3 GeV/c

System

Center-of-Mass

Angles

20

k

< 35 °

D Z i0

0

100

200 PHASE

300

ANDL~,

Fig. 22. The same as Fig. 21 with PLAB = 1 to 3 G e V / c and only angles less than 35 ° in the center of mass.

the " c o h e r e n c e " of the phases still prevails when forming our aggregated histograms - the resonances' phases wander about "incoherently". Work on constructing an actual two-component model for the five NN amplitudes is now underway, but the interpretation is still speculative. As for the 90 ° and 270 ° peaks, if absorption or diffractive rescattering is taken into account, factors of i will be introduced into otherwise real exchange amplitudes. Those are not just overall factors, but are additional terms in some eikonal expansion or other iterative scheme that incorporates the existence of an imaginary diffractive contribution to all the non-flip amplitudes in the helicity frame. Along with real exchanges, such rescattering is a beginning of an iteration scheme for obtaining the p e a k structure. The background-type contribution must come from many other tand u-channel exchanges and very inelastic dibaryon poles.

4. Some concluding remarks Finally we want to offer some more speculative and tentative remarks concerning the significance of the phase pattern and its interpretation. The interpretation, as mentioned, does not hypothesize completely new mechanisms for strong interactions of particles, but utilizes primarily the mechanism that

G.R. Goldsteinet al. / Nuclear PhysicsA578 (1994) 441-470

467

has been perhaps the only element firmly established and experimentally tested in the theory of strong interactions over the years, namely the original Yukawa idea that forces are generated by particle exchange. The problem in the past with this mechanism has been that although the validity of it has been confirmed in kinematic situations when the pure one-particle exchange could be expected to be the main contribution to dynamics, in more general situations there was no calculational procedure to evaluate additions and corrections to such one-particle exchanges. We think that it is possible that in the point of view presented in sect. 3, such a calculational procedure based on successive approximations (i.e. essentially on perturbation theory) might be possible. The one-particle exchanges yield, when aggregated over kinematic variables as explained in this paper, a good first approximation. Such contributions are clearly not the whole story, since such expressions are not unitary. It might be then possible to unitarize such one-particle-exchange contributions by some (e.g. geometrical) unitarization procedure. While such a process is not unambiguous, the differences among the various methods of unitarization tend to be small. These small corrections to the unitarized one-particle-exchange contributions (which cause the striking patterns we exhibited) would then be calculated by some perturbation calculations. This speculative procedure is different from our previous ways of looking at strong interactions in which the "expansion" was done in such a way that the series did not appear to converge. We suggest an alternative way of classifying the various contributions in which such a convergent expansion may be possible, thus allowing us a quantitative calculation of strong-interaction reactions. Whether this hope for an alternative expansion of dynamical mechanisms can be converted into reality will have to await further work. Its urgency will be enhanced if the pattern strongly suggested by the histograms presented here is confirmed by future amplitude analyses and more precise experiments.

Acknowledgement The work was supported by grants from the US Department of Energy, to which we are particularly grateful. We (F.A. and G.R.G.) are also especially appreciative of the support and hospitality of R. Hwa and N. Deshpande at the Institute of Theoretical Science, University of Oregon, during many visits when most of this work was being done.

Appendix The relations among planar-transverse, helicity, and transversity frame amplitudes In this appendix we use the following notation for the amplitudes for AB ~ CD: D r ( a , c; d, b) where F is the frame - h for helicity, Pt for planar-transversity, and

468

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

t for transversity. The arguments of D F are the projections of the spins of particles C, A, D, B, respectively, in the indicated frames. (1) ~-N --+ ~-N: Det(+, +) = - D h ( + , - ) , n e t ( + , - - ) = D h ( + , +). (2) pd ~ pd: (a) Planar-transverse frame to helicity frame: DPt(+I, + a ) = ~ -1[ D h ( + a , --1) -- Dh(0, 0) + Dh( + l, +1)], 1 n e t ( + l , --1)=~-[Dh(+l, +l)+nh(0,0)+nh(+l,

--1)],

nPt( + 1, 0) = --Dh( + 1, 0), Det(0,

0) =

Dh( + 1, -- 1) -- Dh( + 1, + 1).

(b) Transversity to helicity: Dr( + 1, + 1)

= ~lDhtO~ , 0 ) ,

Dt(+l,-1)=½[Dh(+l,

+ l) -- Dh( + l, --1) -- Dh(0, 0)] ,

nt(-1, -1)=½[nh(+l,

+ l) -- Dh( + l, --1) + nh(0, 0)]

- iv/2 n h ( + 1, 0),

Dt(0, 0)

=

Dr'( + 1, + 1) + Dh( + 1, -- 1).

(3) pp --) rr+d: (a) Planar-transverse frame to helicity frame: Det(+l; +, + ) = ½ [ n h ( + l ;

+, + ) + D h ( - - 1 ; +, + ) - - v ~ - D h ( 0 ; - - , +)],

DPt(0; +, + ) = - - ½ V / 2 [ n h ( + l ; +, --) + D h ( + l ; --, +)], DPt(-1; +, + ) = ½[Dh(+l; +, + ) + n h ( - - 1 ; +, + ) + ~ - D h ( o ; - - , +)], D P t ( 0 ; - , +)=½Vf2[Dh(+l; +, +)--Dh(--1; +, +)], Det(+l;-,

+)= ½[Dh(+l;-,

nPt(+l; +,-)=

+)-Dh(+l;

+,-)-

--½[Dh(+l;--, +)--nh(+l;

V'2 Dh(0; +, +)],

+,--)

+V~ nh(0; +, +)]. (b) Transversity frame to helicity frame: D t ( + l ; + , - - ) = --½[Dh(+l; +, +)--Dh(--1; +, +) + V~ Dh(0; --, +)] - - l~t' [ n h ( + l ;

+, --)

+ D h ( + l ; - - , +)--v~-Dh(0; +, +)],

G.R. Goldstein et al. / Nuclear PhysicsA578 (1994) 441-470

Dt(+l;-,

469

+ ) = --½v~Dh(0;--, +) -- ½i[Dh(+l; + , - - ) +Dh(+l; --, +)],

Dt(0; + , + ) = ½f2-[Dh(+l; +, +)+Dh(--1; +, +)] +½iv~[Dh(+I;-, +)-Dh(+l; +,-)], nt(o;-,-)

= - i v ~ - [ n h ( + l ; +, + ) + n h ( - - 1 ; +, +)] + l i v ~ [ n h ( + l ; - - , + ) - - n h ( + l ; +,--)],

Dt(-1; +, - ) = --½[Dh(+I; +, +)+Dh(--1; +, +) +v~Dh(0;--, +)] -- ½i[Dh(+l; + , - - ) + D h ( + l ; - , + ) + ~-Dh(0; +, +)], D r ( - 1 ; - , + ) = '[Dh(+I; + +)--Dh(--1; + +)+l/~-Dh(0; --½i[Dh(+l; + , - - ) + D h ( + l ; - - ,

, +)]

+)

-1/2 Dh(0; +, +)]. (4) pp ~ pp: (a) Planar-transverse frame to helicity frame: DVt(+, +; +, + ) = a + z , DVt(+, +; - , - ) = c + z , DPt(+,-; +,-) =d+z, DVt(+,-;-,

+)=e-z,

DPt(+, +; +, - ) = -b, where a,b,c,d,e are helicity amplitudes corresponding to a = D h ( + , +; +, +), b = D h ( + , +; + , - ) , c =Dh(+, +; , ), d=Dh(+,-; +,-), e = D h ( + , - ; - , +), z = - l ( a + c + d - e ) . (b) Transversity flame to helicity frame: Dt(+, +; +, + ) = ¼ v ~ ( a + c + d

- e + 4ib),

Dt(,,,

)=¼v~(a+c+d

- e - 4ib),

)=¼v~(a+c-d

+e),

Dt(+, +;

,

D t ( + , - ; + , - ) = ¼v~-(-a + c -

d-e),

Dt(+,-;-,

-e).

+)=¼v~(a-c-d

470

G.R. Goldstein et al. / Nuclear Physics A578 (1994) 441-470

References [1] [2] [3] [4] [5] [6] [7] [8]

G.R. Goldstein and M.J. Moravcsik, Phys. Lett. B 102 (1981) 189 M.J. Moravcsik, F. Arash and G.R. Goldstein, Phys. Rev. D 31 (1985) 2360 F. Arash, M.J. Moravcsik and G.R. Goldstein, Intern. J. Mod. Phys. A 2 (1987) 739 F. Arash, M.J. Moravcsik and G.R. Goldstein, Phys. Rev. D 31 (1985) 667 G.R. Goldstein and M.J. Moravcsik, Phys. Lett. B 199 (1987) 563 F. Arash, M.J. Moravcsik and G.R. Goldstein, Mod. Phys. Lett. A 4 (1989) 529 N. Ghahramany, G.R. Goldstein and M.J. Moravcsik, Phys. Rev. D 29 (1983) 1086 F. Arash, H. Garcilazo, G.R. Goldstein and M.J. Moravcsik, The planar-transverse phase pattern in pion-deuteron scattering, Mod. Phys. Lett. A (1990), in press [9] F. Arash, M.J. Moravcsik, G.R. Goldstein and D.V. Bugg, Phys. Rev. Lett. 62 (1989) 517 [10] G.R. Goldstein and M.J. Moravcsik, Ann. of Phys. 98 (1976) 128; 126 (1980) 176; 142 (1982) 219 [11] Particle Data Group, Phys. Lett. B 204 (1988)