Systematics check: evaluation of type B uncertainties.

The method used in this analysis to evaluate the uncertainties due to the systematic effects is the same as described in [6]. For more details on the systematic studies used here refer to [1]. Just to remind how the systematics checks are considered, let be $\mu_o$ the value obtained with the (nominal) systematic hypothesis $h_o$. The corrected $\mu$ can be expressed as a function of $\mu_o$ and of a shift $g$ due to the systematic effect ${\bf h}$.

$$\mu = \mu_o + g( {\bf h})$$ (8)

The best value of $\mu$ and its variance is, using a Taylor's expansion at the first order:

$$\hat{\mu} = E[\mu] \sim \mu_o + E[\sum_l\frac{\partial g}{\partial h_l}(h_l-h_{ol})] \equiv \mu_o + \sum_l \delta \mu_{l}$$ (9)

$$\sigma^2 = \sigma_o^2 + \sum_l \left (\frac{\partial g}{\partial h_l} \right )^2 \sigma^2_{h_l} \equiv \sigma_o^2 + \sum_l u^2_l$$ (10)

The contributions given by each systematic source are reported in the table 1. From the table 1 we get:

$$N_{PhP} = (232 \pm 17) \cdot 10^{3}$$ (11)

that means

$$\sigma_{e p} = 0.980 \pm 0.073 \; \; \; \mu b$$ (12)