Systematics in the testing of theories through

Systematics in the testing of theories through polarization GARYR. GOLDSTEIN Depflrttnent r.$ Pl?y.ric.s, T~lftsUniversity, Medforrl, MA 02 155, U.S.A. AND

MICHAEL J . MORAVCSIK Department of Physics and lrzstit~~te c.$ Theoretical Scietzc,e, Utliver.sity of Oregon, Eugene, O R 97403, U.S.A.

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Rcccivcd July 4. 1986 The weakest test of a theory for particle rcactions is comparison with an unpolarizcd differential cross section only. The strongest test is through a complete set of complex reaction amplitudes, the attainment of which is a considerable experimental task. It is shown how intermediate stages of tcsting of intermediate stringency can be established. The process is illustrated y. on the spinwise fairly complex reaction of n + p + d

+

Le test le plus faible, pour une theorie de reactions entre particule, est celui qui consiste seulement en une comparaison avec la section differentielk non polarisee. Le test Ic plus fort fait intervenir un ensemble complet d'amplitudes complexes de riaction, ce qui irnplique une tichc cxpir~mentaleconsiderable. On montre comment 11 est possible d'itablir des tests intermidiaires i diffirents niveaux de rigueur Pour illustrer la rnithode, on considtre la reaction n + p + d + y, assez complexe quant au spin. [Traduit par la revue] Can. J. Phys. 65. 445 (1987)

1. Introduction In elementary particle physics, the overwhelming fraction of the experimental data pertain to various particle reactions. In nuclear physics also, a large amount of our information comes from reactions, although, here of course, the structure of bound states (nuclei) also contributes much. In both fields, therefore, dynamical theories need to explain the observed features of reactions as a function of all relevant variables. Because measurements involving unpolarized particles are experimentally simpler, a large portion of the data at our disposal, both in particle and in nuclear physics, is about unpolarized differential cross sections, as functions of the purely kinematic variables of energy, momentum, and angle. From the point of view of testing a theory, such unpolarized cross sections offer an extremely weak criterion, because they are, in most formalisms, merely the sum of the absolute-value squares of the reaction amplitudes. In a multidimensional vector space, e.g., the amplitudes of most reactions, such a sum of magnitude squares is a very feeble piece of information. For this reason, the history of particle physics has offered several examples in the last 4-5 decades of dynamical theories managing to be quite successful in fitting such unpolarized differential cross sections, only to fail drastically as soon as they were confronted with even just a few pieces of polarization data. Such data can be expressed as other types of bilinear products of reaction amplitudes, such as differences of magnitude squares or real or imaginary parts of products of two different complex amplitudes. Such data give qualitatively novel insight into the dynamical mechanism of the reaction, and hence it is no wonder that theoretical guesses, even with adjustable parameters, geared originally to explain unpolarized differential cross sections, find it much more of a challenge to predict these other o b s e ~ a b l e s . Indeed, thanks to this gatekeeper action of polarization data, we continue to be without a theory of strong interactions that has any significant amount of predictive power in agreement with data. In further explorations to finally arrive at a good theory of strong interactions, therefore, the guidance of polarization experiments is cardinal.

In this, the extreme opposite to just having unpolarized differential cross sections is to perform so many polarization experiments that from their set, all complex reaction amplitudes can be determined unambiguously. This is the most an experimentalist can accomplish, and once having arrived at this ultimate goal of phenomenology, it is then the task of theorists to predict these complex amplitudes as functions of the purely kinematic parameters. To reach such a refined stage of having determined all the complex amplitudes is, however, often an impressive experimental challenge. We have examples where this has been accomplished, such as for elastic proton-proton scattering in a wide range of energies and angles up to 6 GeVqc-', for pion-nucleon elastic scattering at various energies, for proton-deuteron scattering at some energies, etc. For many other reactions, however, such a complete knowledge of all complex reaction amplitudes is not likely to be attained in the near future. It is, therefore, very important to ask if there are intermediate steps of information seeking, between the minimal state of knowing only the unpolarized differential cross section, on the one hand, and knowing all complex amplitudes, on the other. The present paper shows how, with very little theoretical effort, a number of such intermediate stages can be established, and how the experimental requirements to measure the appropriate polarization quantities for those stages can also be very well defined. In fact, various choices are available, suited to the particular experimental techniques at our disposal. The class of such intermediate stages explored in this paper involves various linear combinations of magnitude squares of amplitudes. There may be other intermediate stages also involving other types of bilinear products of the amplitudes, although considerable searching for them so far has remained unsuccessful. One might interject that measuring any polarization quantity constitutes a determination of some combination of amplitudes and hence adds information to what was obtained from unpolarized differential cross sections. This is, of course, true; but if the particular polarization quantities measured merely

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CAN. J . PHYS. V O L , 6 5 . 1987

define a complicated set of constraints among all the ampltiude parameters, their practical usefulness in testing the theory is limited. What one would want is a situation in which a subset of all amplitude parameters is completely determined by a subset of polarization measurements, leaving the oomplementary subset completely undetermined. In that case, the comparison with theoretical predictions is more stringent, and deviations can be interpreted much more easily. This is, therefore, the situation we want to investigate. The approach discussed in this paper may be juxtaposed with the approach more commonly used in the past, in which the choice of polarization experiments was determined primarily by the type of technology that happened to be on hand and looked easiest. For the same reasons, the comparison of theory with experiment has been carried out on the observable level rather than on the amplitude level. This latter approach, while having some advantages in terms of short-term convenience, suffers from a number of shortcomings, some of which are as follows. (a) Different observables that in terms of experimental techniques do not appear to have anything in common may, nevertheless, provide exactly or nearly exactly the same kind of information about the reaction amplitudes and hence about the theory to be tested. Therefore, performing both of these experiments is superfluous. To establish this, however, one needs to perform the amplitude decomposition of these observables. For example, the amplitudes of elastic proton-proton scattering at 6 Gev-L.-I are such that the two observables D L , and Dss provide virtually the same information once we know (from other experiments) that the transversity anlplitudes 6 and E are small. However, the two experiments may look very different to the experinlentalist. (b) The theoretical expressions for observables (as contrasted with an~plitudes)are usually much more complicated; they involve bilinear products of amplitudes. All theories yield, as primary quantities, reaction amplitudes. The purpose of the comparison of experiment with theory is not only to ascertain if agreement is present but also to track down the causes of any disagreement in order to be able to modify the theory (if possible) to bring about better agreement. For this purpose, comparing measurements of observables with theoretical predictions for observables is far less definitive and revealing than comparing measurements with theory on the level of amplitude parameters. (c) In the planning of experimental programs, often several options of approxinlately equal technical difficulty are available. The choice among them, therefore, needs to be made on the basis of which set promises to provide new information having the greatest amount of novelty and decisiveness with respect to the predictions of the theory. This decision can be made more easily and more reliably on the amplitude level. (d) In numerous instances, there is no reliable theory to be tested for the reaction under investigation, or there are several competing theories. In such situations, it is particularly important to ascertain the features of these theories (in terms of their predictions for reaction amplitudes) in which they differ or fail. This can be done much more easily and unambiguously on the amplitude level than on an observable level. (e) In decisions concerning the developnlent of new techniques for performing polarization experiments, it is important to know what type of new technique would provide what degree of new, novel, and decisive information. This can be determined, to a large extent even in the absence of specific

dynamical models, by the analysis of the observables in terms of reaction amplitudes, which is the contribution of the present paper. In any specific situation, it is not difficult to explore, on paper, both of these approaches and then assess the "cost-effectiveness" of both to decide on which experimental program to embark. The present paper offers a choice between two options instead of forcing the experimenter to follow a single course that may not be as useful in the interplay of theory and experiment as one would wish. Finally, it may be noted that the distinction between the two approaches is not merely in the difficulty of acquiring a given kind of information; hence, it cannot be circumvented by sirnply having access to large computers. The two approaches, in general, yield different kinds of information, especially in the intermediate stages when not enough types of polarization experiments have been performed to determine all reaction amplitudes completely. In recent years, we have come to a sufficiently detailed understanding of the general polarization structure of reactions with particles of arbitrary spins that it is indeed easy to describe a class of polarization experiments that give information on the individual magnitudes of the complex reaction amplitudes. We do this and then illustrate the results on a spinwise relatively sophisticated reaction, n + p + d + y, which has been the subject of experimental work and will continue to be so. The method given here, however, can be immediately transferred to any other reaction. Section 2, which discusses the general structure, is somewhat abstract. The less involved and more practical reader, therefore, may want to turn directly to Sect. 3, which deals with the application to the abovementioned reaction, and to Sect. 4, which summarizes the results and converts them into an operational procedure. That section also gives a numerical example.

2. T h e general structure As indicated, we will explore the set of experiments that can give partial information about the amplitude parameters from a partial set of experiments by measuring the magnitude squares of the amplitudes. For our discussion, we will use the optimal formalism (for the relevant features of optimal polarization formalism, see ref. 1.) of polarization phenomena for two principal reasons. (a) In optimal formalism, the relationship between primary observables and the magnitude squares of the amplitudes is given by one-by-one matrices; hence, the structure of the matrix relating the secondary observables and the magnitude squares is completely dynamics independent and depends only on how we choose to define the secondary observables. (b) Using optimal formalism, we can specify extremely sirnply and quickly which magnitude squares go with which observables and how, so that a measurement of a set of observables can be immediately translated into information on particular reaction amplitudes. We will see these two points illustrated in our discussion. In the general structure of the optimal formalism for a reaction with particles with arbitrary spins, the absolute-value squares of an~plitudesare contained in the submatrices denoted by I,. For a reaction A,

+ A?+

B,

+ Bz +...+

B,,

where the spins of the particles are s , , s?, s;,

.ri, . . . , s,:, for the

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GOLDSTEIN A N D MORAVCSIK

case when only Lorentz invariance restricts the reaction matrix and no additional symmetries hold, there are [2. I]

x = (2sl + 1) (2sz + 1)(2si + 1 ) . . . (2s,{ + I )

such IM-type submatrices, each of them one-by-one. Their general form is I l l (11 ( 2 ) (?I. ( 1 ) ( I 1 Ill (111 [2.21 ~ ( u , [,L , , u,?u,?, 5 p 1 C p 1 3 . . . tb,,tp,,)

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=

~D(U;,',

t:,',

. . . ,tL:')l'

where the left-hand side is a primary observable in which the particle Ai(i = 1,2) is characterized by the diagonal spin indices u:,) u:,), the particle B,(j = 1,2, . . . , n) is characterized by the diagonal spin indices tCtgI1,and ~ ( u : , ' , urj; tg,', . . . , t;:,)) is the reaction amplitude with those particular spinprojection values. In the above expression, a,= 1 , 2 , . . . , 2 s i l a n d p , = 1 , 2, . . . , 2 s i + I . There are two important observations to be made. First, these observables, which are related to the absolute-value squares of the amplitudes, have only diagonal spin arguments, which means that they only involve polarizations in the direction of the quantization axes of the particles. Second, the quantization axes at this point are still completeiy arbitrary. This means that we have a multiply infinite set of choices for a set of amplitudes, the magnitudes of which we are to determine from our subset of experiments. Of course, once we fix the quantization direction of each particle, the corresponding experiments that determine the magnitudes are also fixed. They are the various spin o b s e ~ a b l e swith indices only in these quantization directions. Depending, therefore, on the experimental facilities we have and, more specifically, on the direction in which each particle can be polarized in the easiest way, we can choose the most convenient optimal frame to use. There is, therefore, the opportunity for the experimentalists to determine in what system of amplitudes the program of determination of the magnitudes of the amplitudes should be carried out. This completes the analysis when we use primary observables and when only Lorentz invariance is imposed on the reaction matrix. We now turn to the same situation with respect , are to symmetries but using the secondary o b s e ~ a b l e s which the ones directly used in experiments. One of these is the unpolarized state of the particle, that is, the sum over all diagonal elements:

+

2,. t 1

if we consider an initial particle, and 2s; + I

if we consider a final-state particle. There are then another 2 s or 2s' other combinations of the primary arguments to form linearly independent secondary observables. For a spin-112 particle, there is only one of these, usually taken to be ( + + ) - (--) = A. For a spin-1 particle, they are usually taken to be ( + + ) - (--) = A and (++) - 2(00) ( - - ) -- A. In general, we denote these linear combinations of the primary arguments by

+

i = 1, 2, for the initial particles, and

2,; I I

v.61

g:;

(dLJJ,I , . . . , v!:,l,,;

I

I)

E

z'

v:!)

PI=

t(.1)( 1 ) ],

I

pl

tpl

j = 1, 2, . . . ,rz, for the final particles.

We can, for the sake of sirnplicity of notation, also include the unpolarized state in the above notation, indexing it by k = 1 and m = I. In that case, each k, goes over 1 , 2 , . . . ,2si + I ; each inJ goes over 1 , 2 , . . . ,2sl + 1, and we have [2.7]

A; = c{jl(l,I , . . . , I )

and

For these secondary observables, the original set of one-byone submatrices combine into one large matrix of size x-by-x (with x given by [2.1 I). In this matrix, the coefficient in the row of the observable, [2.91

~(c:.:',c':'; g:,:,', . . . , g:::,:)

and the column of the magnitude square,

Because each of the c""s and g""s are constructed so that they are linearly independent of each other for the same i (or j ) , and because the large matrix is an outer product of the small and g"', the large matrix also has a matrices for each nonzero determinant, and so the 1, matrix of the secondary observables completely determines the magnitude squares of the amplitudes. If we now turn to the case of primary observables with not only Lorentz invariance but also parity conservation, then in imposing the constraints of parity conservation, we have to distinguish among different cases, depending on the quantization directions chosen for the particles. As has been concluded in earlier papers, the simplest and therefore most advisable choice in the case of parity conservation is the pure transversity frame in which the quantization direction of each particle is perpendicular to the reaction plane. This plane is uniquely defined for four-particle reactions, or for coplanar reactions with any number of particles. For a reaction with more than four particles that is non-coplanar, parity conservation does not reduce the number of reaction amplitudes and does not set up relationships among them. In that case, therefore, the situation remains as it was in the pure Lorentzinvariant case. If, however, the reaction is coplanar (with four or more particles), parity conservation halves the number of reaction amplitudes (for reactions involving fermions only or a mixture of fermions and boson!) and approximately halves it for pure boson reactions. This is the situation we need to discuss, and the pure transversity frame is the most natural one to use. In it, half of the reaction amplitudes simply vanishes, while the other half remains untouched. Correspondingly, half of the primary observables in 1 , also vanish. We then have simply a set of one-by-one submatrices numbering half as many as in the pure Lorentz-invariance case. Thus, mathematically, the situation is virtually unchanged, and so the conclusions we make for the pure Lorentz-invariant case remain in effect. With parity conservation, the only other choice (beside the transversity one) for a particle's quantization direction is some-

cLiJ

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CAN. 1. PHYS. VOL. 65. 1987

TABLE1 . The I, submatrix of the relationship between obscwables and bilinear products of reaction amplitudes for

the reaction n + p -t d + y in the optimal formalism with only Lorentz invariance imposed. The column headings show the indices of the amplitudes, the magnitude squares of which appear in the table, and the row headings give the observables, the notation for which is given in the text after 12.41. Ln the table itself, + means + 1 , and - means - 1

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n + + + + - - p + + - - + + - - + d + + + + + + + + O n p d y

+ + + + - - + - - + + 0 0 0 0 0 + - + - + - + - + + - + A A A A + + + + + + + + + + + + + + A A A A + - + - + - + - + - + - + A A A A + + - - + + - - + + - - + + A A A A + - - + + - - + + - - + + - A A A A + + + + - - - - + + + + - A A A A + - + - - + - + + + - - +

+ + + + + + - + - + - + + - + + - + + + + + - + - - + + - - + + + + + A A A A + + + + + + + A A A A + - + - + - + A A A A + + - + + A A A A + - - + + A A A A + + + + A A A A + - + - - + + A A A A + + A A A A + - + - + +

A A A A A A A A

A A A A A A A A

A A A A A A A A

A A A A A A A A

+ + + + + + +

-

-

-

-

-

-

-

-

-

-

+ + + + + +

+ - - - - + + - - - - - - + - + + + + + + - + - + - + + - + + - - + + - - - - - + - +

-

-

-

+ + + + + -

-

-

-

+ + + + + + + + +

+ + + +

-

-

- 0 0 + + + + - + - +

+ -

+ + + -

where in the reaction plane, thus putting the particle in what is called a planar frame. There is an infinite number of planar frames, because the quantization direction can lie anywhere in the reaction plane. For each particle we can choose a different quantization direction (unless further symmetries, such a s timereversal invariance or identical-particle constraints force a correlation among the quantization directions of the particles). S o , we have a nlultiply infinite set of hybrid frames in which some particles are in the transversity frame and others are in some planar frame. For a particle in the planar frame, the constraints of parity conservation d o not cause some amplitudes to vanish and others to remain untouched; instead, amplitudes become either pairwise equal to each other o r pairwise equal to the negative of another. If, in this case, we want to construct the l,, submatrix for secondary observables, and want to write it in the irreducible form containing only linearly independent amplitudes, the resulting matrix is somewhat more complicated, also containing coefficients other than + 1 o r - 1: than it was for the case of a pure transversity frame. Nevertheless, the large matrix is still an outer product of small matrices that are constructed to have nonzero determinants, it still has a nonzero determinant, and hence again, the lM observables uniquely determine the magnitude squares. The above discussion is rather abstract and generalized, and it is probably not easily comprehensible to those not extensively acclimatized to dealing with the structure of the optimal formalism. For that reason, and also for practical reasons, w e

turn in the next section to a specific reaction and investigate the information that can b e obtained from the l Msubmatrix.

+

+

3. Example: the reaction n p 4 d y For this reaction x = 24; the photon counts (from the point of view of spin states) as a spin- 1/ 2 particle, as d o all massless particles with nonzero spins. In this case, n = 2 , and so the lM-typeone-by-one submatrices pertaining to the primary obs e r v a b l e ~have the form f3.11

L(uznu:", u E p ~ E pti, ; tidrti, ti.,) =

I D ( d n , uEn; ti,, tYpT)I2

with a,, a,, and p, = 1, 2 and pd = 1, 2 , 3. Thus we have 2 x 2 x 2 X 3 = 24 such one-by-one submatrices of the 1, type' W e simplify the notation and use for the spin arguments the following correspondence Of symbols: [3.2]

u:"=+,-

uEp =

+,-

SYp,

=

+,ti, = +,o,-

For the secondary observables, we use the ones mentioned in the previous subsection for spin-1/2 and spin-1 particles. T h e relationship between the secondary observables and the magnitude squares is then given by Table 1. This holds for the most general case, when only Lorentz invariance holds. In our reaction, parity conservation also holds, and hence w e now simplify Table 1 by imposing these additional constraints.

GOLDSTEIN A N D MORAVCSIK

TABLE2. The same as Tablc I cxccpt that parity conservation is also imposed. 'This table holds in the hybrid optimal frame in which n, p, and d arc in the transversity frame and y is in the helicity frame

-

+ + + +

A A A A

+ +

-

-

-

-

-

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+ + + - + + + + +

+ + - + t l + + + + O 0 0 O n p d y y + + + + + + + + A A A A + + + + + + + + A A A A + - + - + - + A A A A + + - - + + + - + - + + A A A A A A A A + + + + A A A A + + A A A A + + - A A A A + - - + n + + - p + - + - +

-

A A A A

A A A A

A A A A

-

-

-

-

-

-

+ + -

+ + +

+ -

-

+ + + + - 2 - 2 - 2 - 2 + + + + T + - + - - 2 + 2 - 2 + 2 + - + - K Z + + - - -2 -2 + 2 + 2 - - D2 + - - + -2 + 2 + 2 -2 + - - + H Z

NOTE:AI'I indices are N (norni;~lto the reaction plane), tr is the unpolarized differential cross section; A,, is the proton asymmetry: A,, is the neutron asynllnetry: A,,,, is the neutron-proton spin correlation: P is the deuteron vector polarization; K l is the proton-deuteron spin transfer ( d vcctor); D l is the neutron-deuteron spin transfer (d tcnsor); H I is the n - p - d triple spin correlation (d vcctor); T is the deuteron tensor polarization; K , is the proton-deuteron spin transfer (d tensor); D 2 is the neutron-deuteron spin transfcr (d tcnsor); and H I is the n-p-d triple spin correlation ( d tensor).

We use a hybrid optimal frame in which the n, p , and d are in the transversity frame but the photon is in the helicity frame, because for massless particles, the helicity frame is the simplest. We see that the advantage of this hybrid frame is that we can determine all magnitude squares without resorting to polarized photons. The constraints of parity conservation then involve every second amplitude in the heading of Table 1 becoming equal to the negative or positive value of the one just before it. Whether it is plus or minus in these pairwise relationships does not matter for the matrix, because all amplitudes enter as squares. The 1, submatrix for the secondary observables for Lorentz invariance and parity conservation therefore becomes a 12-by-12 matrix, and the additional 12 observables in Table 1 vanish. The ones that vanish are those with the photon argument of A, as we would expect, because all the other arguments are the diagonal ones in the transversity frame; parity conservation makes all observables in which only one particle has a polarization in the reaction plane vanish. The resulting 12 observables and 12 magnitude squares are given in Table 2. Here we can see explicitly that the determinant is nonzero, because the matrix has the form of an outer product between the two matrices

and

neither of which has a zero determinant.

449

Table 2 now shows us immediately how, by measuring an increasingly larger number of the 12 observables, we gradually close in on the determination of the 12 magnitude squares; and we can also see immediately what type of information we receive depending on what kind of subset of experiments we perform. W e now expand on this point in detail. In particular. there are two different aspects to discuss. The first question pertains to the amount of information to be gained as we perform an increasingly larger number of experiments from Table 2. This is easy to answer. Because the experimental observables in that table are all linearly independent of each other, while we perform any 1 , 2 , . . . , m ,. . . , 12 experiments out of the 12, we thereby determine 1 , 2 , . . . , rn, . . . , 12 linear coinbinations of the twelve magnitude squares. For example, performing the first two experiments gives two such linear combinations; one contains the six magnitude squares in which the index of the proton is +; the other, those six in which the index is - . It is not necessarily true, of course, that these linear combinations contain an equal number of magnitude squares. For example, performing the first and the ninth gives two combinations; one containing four magnitude squares; the other, eight. Neither is it necessary that these linear combinations be nonoverlapping. For example, performing the first and the fifth measurements gives two combinations of eight. Nevertheless, the table gives a systematic way of tracking down what set of experiments gives information on which linear combinations of magnitude squares; and the more experinlents we perform, the smaller the combinations become and hence the better we can zero in on any one particular magnitude square that may interest us. This brings us to the second aspect; namely, if one can tell which magnitude squares will appear in the combinations measured by a set of observables. It is not difficult to answer this question in some special situations. Consider, for example, in Table 2 all four experiments (the first four in the table) in which we sum over the polarization states of d; i.e., we d o not measure the polarization of the deuteron. It is easy to see that such a set of four experiments determines four sums of magnitude squares given in the table; namely, that of the first, fifth, and ninth; that of the second, sixth, and tenth; that of the third, seventh, and eleventh; and that of the fourth, eighth, and tenth. In other words, we get sums of magnitude squares that have the same indices for the first two particles but differ in (and cover the range of) the index of the third particle, the one that is always unpolarized in our experiments. This is not too surprising, especially a posteriori. W e can convert the result into the following rule of thumb. When we consider the subset of experiments in which one particle remains unpolarized (the same one in all the experiments of the subset), that subset of experiments determines magnitude squares in the same way as if the unpolarized particle were a spin7zero particle. The difference is that instead of using the magnitude square of one single amplitude containing that spinIzero particle, we have to use a sum of magnitude squares with indices identical to those for the other particles but differing in (and covering the range of) the index of the unpolarized particle. Note that this is not a general rule for dealing with all polarization quantities in a reaction. It holds, in this form, only for the lMtype of submatrix in which we have magnitude squares only. The situation in other submatrices is more complicated. The above special case, where each observable leaves the same particle unpolarized, probably occurs frequently in laboratory situations because the polarization of each particle is a

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CAN. J . PHYS. VOL. 65. 1987

different kind of technology. Nevertheless, we must also investigate the situation when the subset has no such regular pattern and consists of an arbitrary selection of obsewables out of the entire list. In this case, the situation is more complicated. In this general context, the following question is of interest: Is it possible to select a subset of experiments in I, so that this subset determines a subset of magnitude squares? It is not difficult to see that unfortunately, the answer in general is "no." To show this, first note that the structure of the 1, matrix of the secondary observables is that of a nested combination of the small matrices of secondary observables for each particle. This matrix for a spin- 112 particle (with secondary observables A and A) is

while for a spin-l particle it is given by [3.4]. Then the structure of the overall I M matrix for the secondary observables is

where MI is given by [3.3] and is of the form

where M , is given by [3.5]. This nested structure assures that unless for at least one of the participating particles the matrix for its secondary obsewables contains a subset of q obsewables containing only q magnitude squares, there will be no such subset for the overall IM matrix of secondary observables either. Such a subset, however, does not exist for any individual particle, and so it does not exist for the whole submatrix either. Because the secondary observables are selected in such a way that they are orthogonal to each other in the space of the magnitude squares, the inverse of the overall 1, matrix for the secondary observables (which gives the magnitude squares in terms of the secondary observables) is the transpose of the overall I M matrix; hence, it does not contain those blocks of zeros that are needed to produce a subset of magnitude squares determinable from a subset of observables. Thus, there is, in general, no subset of obsewables that can determine a subset of magnitude squares. To determine individual magnitude squares, we must perform the entire list of experiments in the submatrix IM. In the above argument we use the fact that the secondary observables are defined so that they are orthogonal to each other. What if we abandon that requirement? An investigation of the actual cases for spins 112, I, 312, and 2 indicates that resorting to such nonorthogonal observables does not help. The situation is constrained by the requirements that in each secondary observable (except in the unpolarized differential cross section), the sum of the coefficients of the magnitude squares must be zero; in fact, the secondary observables must be invariant under a rotation of the quantization axis by 180". These requirements are important experimentally because they assure a great reduction in, or an elimination of, systematic errors in the measurement of the secondary observables.

4. Summary and numerical example We have seen that the I M submatrix for any reaction offers a well-defined and easy instrument for systematically obtaining

an increasingly more constraining body of information about the reaction amplitudes. The I M submatrix gives information only about the magnitudes of amplitudes; but depending on the quantization directions chosen in defining the particular optimal frame in which we consider the I , submatrix, we get information on the magnitudes of many different sets of amplitudes (which are linear combinations of each other). We have also seen that a subset of experiments in the I, submatrix provides a well-defined set of linear combinations of magnitude squares, but unfortunately not a subset of individual magnitude squares (unless we measure all the observables in the I M submatrix). When the subset of observables we select from the entire I M submatrix of secondary obsewables contains only experiments in which the same particle is always unpolarized, the linear combinations of magnitude squares that we can determine contain sums of magnitude squares with the indices of the other particles fixed, but the index of the abovementioned unpolarized particle covers all possible values. On the basis of these results, the following procedure is recommended in applications. (i) Determine, on the basis of experimental constraints, which is the easiest direction in which to polarize the particles in the reaction to be studied. (ii) From the theory to be tested by the experiment, calculate the optimal reaction-amplitude magnitudes, in the frame in which the quantization axes are the polarization directions chosen in (i). (iii) Measure as many polarization obsewables as possible with the indices corresponding to the polarization directions chosen in (i). (iv) From the set of observables measured in (iii), determine as many linear combinations of magnitude squares as possible, and compare them with the predictions obtained in (ii). If the experimental possibilities for (i) are broad, one can reverse the order of (i) and (ii), and determine the polarization direction on the basis of the question: In which system does the theory offer the firmest prediction? For example, the theory, in a particular frame, may make a prediction symptomatic for that theory only, for one or several of the amplitudes. But for other amplitudes, the prediction may follow from general considerations. Hence, the prediction may not be characteristic of this particular theory. In another frame, such a separation of symptomatic and general predictions may not be possible because the two may mix for all amplitudes. In that case, it is particularly advantageous to select a particular frame in which the separation is pronounced. Then, the experimental setup can accommodate this choice by selecting the quantization direction of the chosen frame to be the polarization direction. A hypothetical but concrete numerical example might help to illustrate the procedure. Let us assume that for n + p -+ d + y, we have a theory that, in the hybrid frame in which Table 2 is given, makes a prediction (at a certain energy and angle) for the magnitudes of the twelve amplitudes as follows:

GOLDSTEIN A N D MORAVCSIK

45 1

ments appear in I,, in which the polarization of all four particles needs to be determined. Indeed, we never have to deal with polarized photons. The photon is on an unequal footing in this respect because we chose its quantization direction to be the helicity direction, and thus different from the quantization directions of the other three particles. This is an experimental advantage.

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and it is claimed that n? and a~ are particularly sensitive to the specific features of the theory to be tested. We can then see from Table 2 that the set of experiments

14'21

(A,A;A,A),

(A,A;A,A),

(A,A;A,A),

A , A; A

(A,A; A,A),

(A, A; A, A)

,

can provide la212+ In,]'. In fact, because (if the theoretical predictions can be approximately trusted) we have

we do not even need (A,A; A,A) and (A, A; A, A); even the first four observables in [4.2] provide a test to a level of about 5%. To separate la2(' from In4(', that is, to determine the two magnitude squares separately, we know from our discussion that we would have to measure all twelve observables in I,. Note, by the way, that in the frame we chose, no experi-

Acknowledgements We are indebted to Norman Davison of the Triuniversity Meson Facility (TRIUMF) for sharing with us his interest in the reaction n + p -+ d + y and thereby stimulating this investigation. One of us (MJM) is grateful to TRIUMF and, in particular, to Erich Vogt and Harold Fearing for the hospitality provided during a two-week visit, which opened the opportunity for interaction with Norman Davison. This work was in part supported by the U.S. Department of Energy. 1. G. R. GOLDSTEIN and M. J. MORAVCSIK. Ann. Phys. (N.Y.), 98, 128 (1976); 126, 176 (1980); Nucl. Instrum. Methods Phys. Res. Sect. A, A227, 108 (1984); M. J . Moravcsik. Polarization as a probe of high energy physics. In Proceedings of the Tenth Hawaii Topical Conference on High Energy Physics. 1985.