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Subtracting background effects in the DWT power spectra.

This is a more technical discussion of the problem introduced in Subsection 6.7.2, and the same notation is used. The search for a good background subtraction technique underwent a painfull involution from an ideal, but impossible, to an imperfect, but doable.

Assuming that the interaction signals are a sum of the ``physics'' signal with the background distributed randomly according to $F(x)$, the three distributions - $F(x)$ (non-interactions), $G(x)$ (minimum bias sample of interactions), and $P(x)$ (``valid beam'' ($VB$) sample) - are related in the following way :

$$P(x) = \alpha \int_{x_{min}}^{x_{max}}F(\xi)G(x-\xi)\,d\xi + (1-\alpha)F(x)$$ (102)

Due to finiteness of binning, Eq. B.1 is, practically, a system of linear equations:

$$P_x = \alpha \sum_{\xi =x_{min}}^{x_{max}} F_{\xi}G_{x-\xi} + (1-\alpha) F_x,$$ (103)

where $F$,$G$ and $P$ are normalized histograms and $x$ and $\xi$ - discrete indices of bins. In principle, knowing $P(x)$, $F(x)$ and $\alpha$, one can try to determine $P(x)$ by solving this system of linear equations. Our final goal, however, is correcting the average power spectrum components (PSC).

$$P_x {\mathcal{P}}_{P,x} = \alpha \sum_{\xi_{min}}^{\xi_{max}} F_{... ...hcal{P}}_{F,\xi} + {\mathcal{P}}_{G,x-\xi}) + (1-\alpha)F_x {\mathcal{P}}_{F,x}$$ (104)

This equation (B.3) reflects the fact that the mean of the PSC distribution from the physics run is the weighted sum of the PSC from the minimum bias interactions and from the empty target, and that in the interaction events, the resulting PSC is a sum of PSCs from ``pure'' physics and background. Having solved B.1 for $G(x)$, one should solve B.3 for ${\mathcal{P}}_{G,x}$, whereby the background subtraction problem would be solved.

This is the idealistic approach to background subtraction. It can not be realized in practice because Eq. B.1 - Fredholm's equation of the first kind - is a notorious ill-posed problem, whose solution is, generally, numerically unstable. This means that there is no hope of obtaining a physically meaningful solution to Eq. B.1 in the practical situation with non-zero errorbars. Despite the fact that the solution to Eq. B.1 has been obtained both numerically and using a symbolic processor such as MAPLE, it was of no value because, typically, it was a non-physical oscillating function. Therefore, the following simplfying assumption has been made. The multiplicity binning is coarse enough so that the $RMS$ of the $F(x)$ distribution is only about 1/3 of the bin size. Therefore, approximation of $F(x)$ by a ``$\delta$-function'' can be easily justified:

$$F_{x-\xi} = \delta_{x,\xi}$$ (105)

This simplifies the system of equations B.2 to

$$P_x = \alpha G_x + (1-\alpha)\delta_{x,0}$$ (106)

In other words, the $P$ distribution is obtained from $G$ by increasing the $x=0$ bin only. Equations B.3 also become simpler, so that the solution for ${\mathcal{P}}_{G,x}$ is easy:

$${\mathcal{P}}_{G,x} = \frac{(1-\alpha)}{\alpha}\frac{\delta_{x,0}... ...al{P}}_{P,x} -{\mathcal{P}}_{F,x} ) + {\mathcal{P}}_{P,x} - {\mathcal{P}}_{F,0}$$ (107)

Equation B.6 is already simple enough for all bins but one, namely, the zero bin:

$${\mathcal{P}}_{G,x} = {\mathcal{P}}_{P,x} - {\mathcal{P}}_{F,0}$$ (108)

This result could have been guessed intuitively. For the zero bin, Eq. B.6 is not useful because $G_0$ is not known.