Theory of second optimization for scan experiment

Abstract

In many high energy experiments, the physics quantities are obtained by measuring the cross sections at a few energy points over an energy region. This was referred to as scan experiment. The optimal design of the scan experiment (how many energy points, what the energies are, and what is the luminosity at each energy point) is of great significance both for scientific research and from economical viewpoint. Two approaches, one has recourse to the sampling technique and the other resorts to the analytical proof, are adopted to figure out the optimized scan scheme for the relevant parameters. The final results indicate that for n parameters scan experiment, n energy points are necessary and sufficient for optimal determination of these n parameters; each optimal position can be acquired by single parameter scan (sampling method), or by analysis of auxiliary function (analytic method); the luminosity allocation among the points can be determined analytically with respect to the relative importance between parameters. By virtue of the second optimization theory established in this paper, it is feasible to accommodate the perfectly optimal scheme for any scan experiment.

Future Physics Programme of BESIII *

“…In this approach it is of utmost importance to determine precisely the beam energy and the beam energy spread. For this purpose, starting from 2007, a high accuracy BEMS, located at the north crossing point (NCP) of BEPCII, was designed, constructed, and finally commissioned at the end of 2010 [188][189][190][191][192]. Two days were spent to perform a scan of the resonance.…”

An optimal scheme for top quark mass measurement near the $\rm{t}\bar{t}$ threshold at future $\rm{e}^{+}{e}^{-}$ colliders

“…It is quite easy to imagine that these 10 points definitely will not contribute equally to the fitted top mass due to their different sensitivity to the variations of the top mass and cross sections. In fact, it has been shown [36] in similar tau lepton mass threshold scan case that the most efficient scheme is to concentrate all luminosity on the single optimal energy point in one free mass parameter fitting(Sometimes the background cross section and the selecting efficiency can also be taken as free parameters to be fitted and this is called the multi-parameter fitting, which we do not consider here). We have checked this result and find that the additional energy points far from the optimal energy point have negligible contribution to minimize the statistical error of the fitted top mass, and this indicates that they are completely unnecessary.…”

Discussion on upper limit of the precision for τ mass measurement

“…By the virtue of optimization theory [23], the relation between the absolute error of m τ (denoted by u) and the total luminosity (denoted by L) is given by the following formula…”

“…where A is a constant depending on the cross section, the derivative of cross section to m τ , and some other quantities [23]. An effective error variation that is called L-profit, is introduced and defined as…”

“…A remark is in order here. During the study of m τ scan, BESIII collaboration have make methodical and meticulous studies on optimization process, to guarantee the effective usage of available luminosity [23,26,27,28,29]. Nevertheless, this effort is not helped with the low R ξ issue.…”