Estimating experimental uncertainties in K_e4

For our first estimates, we consider only the first order terms in a partial wave expansion of the form factors F, G and H, i.e., we take

This is consistent with the parametrization used by Pais and Treiman, [4] and Rosselet et al. [5] They consider higher order terms, but the coefficients of these terms are found to be consistent with zero by the experiment of Rosselet et al., [5] so we do not consider further terms in our initial estimates.

All the and dependence in the problem is contained in the expression of as a function of these variables and the ; for this reason these two variables are referred to as ``trivial.'' \ appears only in the equations for the in terms of F, G, and H (and the phase space expansion, should we consider higher order terms). This leaves then , g and h with a possible dependence on s_ and s_l . The phase shifts and (which at this order appear only in the combination ) depend only on s_ . For the moment we parametrize , g and h by an expression of the form

where s_ , and y stands for f, g, or h. We take the slope to be the same for , g and h, and no slope in s_l at this stage, again consistent with Ref. [5]. For the dependence of on s_ we will consider average values in a set of 5 bins in s_ , and consider parametrizations of in a later section.

One last important detail remains to be mentioned. The MLM (or asymmetries, for that matter) says nothing about an overall factor in the intensity, as we require the probability density to be normalized to one. Thus we divide out ( for short) from the amplitude, as it is the parameter with the most effect on the integrated intensity. Wherever and appear, they are divided by , so we replace them by new parameters and ( and are unaffected). We then apply the MLM to the set of parameters , , and , and obtain the correlation matrix

The diagonal entries of this matrix are variances of the four parameters, where N is the number of events. The off-diagonal elements represent correlations between the parameters; in this case they are small, but they can be significant, depending on the parametrization used. We do not report the full correlation matrix for each parametrization in this paper, but they are available from our programs if needed for further calculations. They can not be neglected in general if one wants to calculate functions of the parameters we use, and propagate the errors correctly. In the end of this paper, we consider the determination of some highly correlated parameters.

We then extract the error on from the equation

where is the unnormalized probability function inputted to the MLM calculation, and C represents all the constant factors (masses, two's, 's) needed to complete the equation. The relative error is given by in an experiment like KLOE [6] where the statistical error will be dominant. The relative error on the integral (a) is given by the matrix product

Combining this error in quadrature with the statistical error on , we obtain the error we quote for ; combining the error on in quadrature with the errors on and , we obtain the errors we quote for and .

In Table 1 we display the results of this calculation. The central values (our input) are those found by the previous experiment. [5] We have used the program VEGAS [7] to do the necessary integrals in five-dimensional phase space. The normalization of the probability distribution is ensured automatically by the program, and the necessary derivatives also computed numerically. Estimated errors are shown for N=30000 events, the statistics of the previous experiment) and N=300000 events, the anticipated statistics [1] in one ``year'' seconds of running with cm s. All errors in this paper, unless otherwise noted, are statistical errors and can be simply scaled by for different numbers of events. As a general rule, also, the fractional error on is roughly independent of its central value, while the absolute errors of the other parameters remain constant. The last line of Table 1 shows the errors on these parameters found by Rosselet et al.. We do not quote the error on because an error on averaged over the whole of phase space is not very meaningful. The errors shown are independent of the central value of used.

 
Table 1: Central values and estimated errors for , , and

At this point, before going on to further discuss errors in KLOE, it is necessary to say some words about why our estimated errors at ``Rosselet statistics'' are so different from those that Rosselet quotes. The errors given above are purely statistical, but apply to a ``perfect'' detector, i.e., one which covers the whole of phase space with unity efficiency everywhere. This is close to true for KLOE; we will attempt to illustrate this later in this paper, and will describe a more rigorous demonstration in a future paper. However, Rosselet's detector was far from ``perfect.'' In the error we have quoted for , the errors from and the normalization a contribute about equally; the first is about and the second about . Rosselet, however, quotes a relative error K_e4 of , which completely accounts for their large error on . Their fixed target experiment had a overall efficiency for K_e4 , and a highly variable efficiency as well, varying, for example, smoothly from in a very small portion of phase space with large s_l and small s_ , to near zero at large s_ and small s_l . KLOE is in contrast a hermetic detector, operating at a collider running at the resonance, producing self-tagging low momentum pairs. It will have a uniform near- efficiency over all of phase space, minus a few percent of phase space that will be cleanly cut and discarded. [8]

The next step in our analysis was to drop the slope parameter and determine the errors on the parameters in five bins in s_ , chosen so as to have equal numbers of events. Such an analysis with real data would have the advantages of studying the s_ dependence in a more parametrization independent way. If, however, the s_ dependence is correctly given by eq. 10, this method will not determine as accurately, so in general both types of approach are necessary. For our purposes, displaying the error in bins is also important to illustrate the possible accuracies with which may be measured, before we implement a possible parametrization of . In Table 2, we give the estimated errors, taking an average of in each bin as our inputs for y=f, g, and h, with Rosselet values for the and . The errors on are essentially independent of the inputs of its central value. In the last line, for comparison, we display the errors on as measured by Rosselet et al. The improvement is not as drastic as that of was, but is nonetheless a factor of 1.5 to 2. This should be further multiplied by a factor of to per DANE running year. The accuracy on f in bins is even better than we might have expected from the error on multiplied by . This is because the error on gives most of the contribution to the error on the normalization a, and thus a significant contribution to the error on .

 
Table 2: Estimated errors in five bins of 6000 events each.

We have examined in some detail the question of what accuracy can be measured to. We have first of all determined that while appears in , , , , , and , it is only the dependence of that gives us the above accuracy on . This can be seen by replacing in all the , except , by a dummy variable , set equal to the central value of . When we proceed to apply the MLM to the new probability function, we find the same error on as before, within a few percent. If, however, we apply the MLM to the as parameters in their own right (we cannot use the as parameters, because they are functions of the phase space variables) and then take the ratio to determine , we find that the error on increases by . (If we use the asymmetry method to determine the , the error increases another .) We have not taken care to cancel correlated errors in and , but we have checked that the correlated parts of the errors are small relative to the uncorrelated parts. So, this increase appears to be mainly due to the information lost in integrating the over three out of five of the phase space variables before applying the MLM. Equivalently, the better error on can be attributed to applying a more detailed parametrization (therefore more information) from the beginning of the calculation.

Nonetheless, it may be interesting to determine the and their errors as a parameterization independent way to present the data. We have estimated that the combination can be determined to in five bins of 60,000 events each. The other can be determined with absolute errors of one to two times this error.