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Motivated by the need for an absolute polarimeter to determine the beam polarization for the forthcoming RHIC spin program, we study the spin dependence of the proton-proton elastic scattering amplitudes at high energy and small momentum transfer. In particular, we examine experimental evidence for the existence of an asymptotic part of the helicity-flip amplitude φ 5 which is not negligible relative to the largely imaginary average non-flip amplitude φ + = 1 2 (φ 1 + φ 3 ). We discuss theoretical estimates of r 5 = m φ 5 / √ −t Im φ + based upon several approaches: extrapolation of low and medium energy Regge phenomenological results to high energies, models based on a hybrid of perturbative QCD and non-relativistic quark models, and models based on eikonalization techniques. We also apply the rigorous, model-independent methods of analyticity and unitarity. We find the preponderence of evidence at currently available energy indicates that r 5 is small, probably less than 10%. The best available experimental limit comes from Fermilab E704: combined with rather weak theoretical assumptions those data indicate that |r 5 | < 15%. These bounds are important because rigorous methods allow much larger values. Furthermore, in contradiction to a widelyheld prejudice that r 5 decreases with energy, general principles allow it to grow as fast as ln s asymptotically, and some of the models we consider show an even faster growth in the RHIC range. One needs a more precise measurement of r 5 or to bound it to be smaller than 5% in order to use the classical Coulomb-nuclear interference technique for RHIC polarimetry. Our results show how important the measurements of spin dependence at RHIC will be to our understanding of proton structure and scattering dynamics. As part of this study, we demonstrate the surprising result that protonproton elastic scattering is self-analysing, in the sense that all the helicity amplitudes can, in principle, be determined experimentally at small momentum transfer without a knowledge of the magnitude of the beam and target polarization.

We present a study of the tensorial structure of the hadronic matrix elements of the angular momentum operators J. Well known results in the literature are shown to be incorrect, and we have taken pains to derive the correct expressions in three different ways, two involving explicit physical wave packets and the third, totally independent, based upon the rotational properties of the state vectors. Surprisingly it turns out that the results are very sensitive to the type of relativistic spin state used to describe the motion of the particle, i.e., whether a canonical (i.e., boost) state or a helicity state is utilized. We present results for the matrix elements of the angular momentum operators, valid in an arbitrary Lorentz frame, for both helicity states and canonical states. These results are relevant for the construction of angular momentum sum rules, relating the angular momentum of a nucleon to the spin and orbital angular momentum of its constituents. It turns out that it is necessary to distinguish carefully whether the motion of the partons is characterized via canonical or helicity spin states. Fortunately, for the simple parton model interpretation, when the proton moves along OZ, our results for the sum rule based upon the matrix elements of J z agree with the often used sum rule found in the literature. But for the components J x ; J y the results are different and lead to a new and very intuitive sum rule for transverse polarization.We shall return to this question in Secs. VA and V B.Of primary interest are the matrix elements of the angular momentum operators J k or, equivalently, the J ij . Consider the forward matrix element, at t 0, CRITIQUE OF THE ANGULAR MOMENTUM SUM RULES.

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A simple parametrization of the spin density matrix p is given which ensures that p is a positive, Hermitian matrix which satisfies the parity symmetry conditions. There is a minimal number of parameters and they can vary independently; no complicated constraints on their allowed region of variation are required. A similar parametrization is given for the real part of p. Rank conditions, polarized targets, and angular distributions are also discussed.