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Beam Parameter Fitting

Theoretically, the luminosity spectrum is determined by a set of beam parameters: the horizontal, vertical and longitudinal beam sizes (tex2html_wrap_inline1527) and the number of particles in a bunch(N). In the empirical function, two independent-valuables appeared as tex2html_wrap_inline1531 and tex2html_wrap_inline1533. They can be calculated from the measured spectrum, since the measured one is the luminosity spectrum convoluted by the cross section of the Bhabha scattering process, which is called the weighted luminosity spectrum. In this analysis, one of them is obtained by assuming a fixed value of the other, and vice versa, i.e. one parameter fitting in a likelihood method. The beam energy was set to be 250 GeV and the relevant parameters are listed in Table 14.1, which are called ``nominal" beam parameters.

For this study, two kinds of Bhabha events were generated by the method described in the previous section. They are called an `experimental-data sample' (e-sample) and a `likelihood-function sample' (l-sample). The Monte Carlo statistics of e-samples corresponds to 10fbtex2html_wrap_inline1313 and 1fbtex2html_wrap_inline1313 for the angular regions of tex2html_wrap_inline1487 and tex2html_wrap_inline1541, respectively. The polar angles of electrons and positrons in the Bhabha events were smeared with a Gaussian distribution of 1mrad resolution; then, the weighted luminosity-spectrum was calculated as a function of tex2html_wrap_inline1429 from their acollinearity angles. A number of e-samples were generated with several sets of B and tex2html_wrap_inline1533. To determine a likelihood function, a l-sample was generated with 10 times more statistics than the e-samples with the nominal beam parameters. The weighted luminosity spectrum was obtained in the same way as the e-sample. Dividing the spectrum into 50 bins for 450tex2html_wrap_inline1549500GeV, the normalized likelihood function tex2html_wrap_inline1551 was defined by
where tex2html_wrap_inline1553 is the bin-number corresponding to tex2html_wrap_inline1429, tex2html_wrap_inline1557 is the number of events in the tex2html_wrap_inline1553'th bin, and tex2html_wrap_inline1561 is the total event number of the l-sample. For the e-sample, the log-likelihood was calculated by
where tex2html_wrap_inline1563 is the total event number of the e-sample and tex2html_wrap_inline1565 is the likelihood of the j'th event in the e-sample.

Figure 14.23: Log-likelihood distributions of the e-samples as a function of (a) tex2html_wrap_inline1569 and (b) tex2html_wrap_inline1533, where the parameter was varied for tex2html_wrap_inline142710% relative to the nominal value, while the other was fixed to the nominal value and vice versa.

Figures 14.23(a) and (b) show the log-likelihood distributions as a function of B and tex2html_wrap_inline1533, respectively. From these figures, it can be seen that the beam parameter of B or tex2html_wrap_inline1533 can be determined with an accuracy of several % by large (tex2html_wrap_inline1487) and small(tex2html_wrap_inline1541) angle Bhabha events for an integrated luminosity of 10fbtex2html_wrap_inline1313 and 1fbtex2html_wrap_inline1313, respectively.

As described in the previous section, since the energy scan should be made for the integrated luminosity of a few fbtex2html_wrap_inline1313 each at a number of CM energies around toponium resonance states[2], a 1 mrad angular resolution must be necessary in the small angular region (tex2html_wrap_inline1541). In these cases, the integrated luminosity of the peak in the luminosity spectrum must be precisely determined, although the peak may be smeared by the beam energy spread. It is seen that the present study has demonstrated the feasibility toward this end.

next up previous contents
Next: References Up: Luminosity Spectrum Previous: Measurement of the Luminosity

Toshiaki Tauchi
Thu, May 29, 1997 04:47:48 PM