where f is the dilution factor that includes the effect of the unpolarized nuclei (see the discussion around Eq.(10).
The uncertainty can therefore be expressed as
which is a valid expansion since these uncertainties are uncorrelated. From , one can obtain the exact expression which can be approximated as , for the usual case of small (implying ). Therefore
It is expected that at CEBAF the beam polarization will be measured with an accuracy of better than 4%, as it has been done at other laboratories. We expect the situation to improve such that we have used 2% in our estimates. The target polarization has been determined to in current designs, but, based on CERN experience, we expect to ultimately do better.
The magnitude of is determined by the requirement that it should allow to discriminate among the various models for , and it has to be consistent with the lower limits imposed by the uncertainties , and . Moreover, it should not represent an unreasonably large number of counts.
Figure 11: A as a function of Q for three models and three angles.
From Figure 12 it can be seen that to distinguish, for example, between the Galster parametrization and the Gari-Krümpelmann (G-K) model in the kinematic region of interest, has to be of the order of at the low points, to on the high momentum transfer side, for a four standard deviation (or better) separation between models. Taking as reference the Galster model, Table 5 illustrates the magnitude of the expected uncertainties in the asymmetry , the experimental asymmetry and the number of counts needed for the desired level of precision. In this table, the values of N were computed using the following additional assumptions: , .
These values of N have been calculated using the expression
We note that the minimum uncertainty in is restricted by the combined uncertainties in , and f which in the present case amount to .
To obtain from the asymmetry, we have to solve the expression for (Equation (5)) for the ratio . Since the different models predict that as increases, this ratio approaches and even exceeds 1 ( at (GeV/c) in the dipole and G-K models), it is inaccurate to neglect the term in the range of the present proposal. The result is that we have a quadratic equation for that can be written as
where , and . The solutions are
By substituting in this expression the asymmetry predicted by a given model, it is seen that the negative root reproduces . Therefore we can write
The purpose of this exercise is to obtain an expression for , based on the usual expansion for the uncertainties
where the uncertainties have been neglected given their very small relative magnitudes.
After the appropriate substitutions are made, we find that
This equation contains the effects of both the uncertainty in as well as the propagation of the uncertainty in the asymmetry.
In Table 6 shows that for the considered earlier, there is a significant effect on . The uncertainty in was taken to be 5%, combining our present knowledge of this quantity at the values of this proposal, with the improved precision for Q 1 (GeV/c) expected from the ongoing measurements of the Basel group is performing at Mainz.
We note that in the estimates given above we have taken care of the nonlinear relationship between the asymmetry and G (see Equation 5). This nonlinearity results from the fact that for small values of G A is proportional to G, while for very large values and large momentum transfer A depends on G . There obviously is a range where A does not at all depend on G! This however does not imply that a measurement of A is not useful; one simply has to analyse the data in a different way. The nominator of equation 5 always depends linearly of G. One therefore can determine G directly from , and not from A= / . This requires the knowledge of the detector efficiency, which can be deduced from a short cross section measurement under kinematics where the cross section is dominated by G. The error bars of figure 13 for the case where the data would correspond to something like Gari-Krumpelman (where at large momentum transfer the blow-up factor becomes of order 2) therefore can significantly be reduced.