Examples of invariant mass distributions and signal distributions after the extraction procedure described below are shown in Figures 1, 2, 3, 4 for Σ∗+ and Σ∗- respectively.
Figure 1: Σ∗+ invariant mass distribution.
Figure 2: Σ∗- invariant mass distribution.
Figure 3: Σ∗+ signal after background subtraction.
Figure 4: Σ∗+ signal after background subtraction.
Invariant mass distribution
Figure 5 shows, for the Σ∗+ and for all pt bins, the invariant mass distribution (in black) for the Λ π candidate pairs. Event mixing (EM) was used to calculate the combinatorial background. Λ ’s from a good i-th event have been paired with π ’s from up to five subsequent good events, with characteristics similar to the i-th event (tracks multiplicity difference not greater than 10 and vertex Z difference not greater than 1 cm). The event-mixing background is shown in red in Figure 5.
Figure 5: Invariant mass distribution and event-mixed background for the Λπ candidate pairs in the 11 pt bins for Σ∗+.
Normalization of the EM distribution
The EM distribution contains by deﬁnition more entries than the same-event invariant mass distribution and needs therefore to be normalized to it. Problems arise since the EM distribution has a different shape with respect to the same-event invariant mass distribution not only in the signal region but also in an extended region on both sides of the signal peak. Only in the rightmost part of the invariant mass window both distributions have the same shape, providing a criterium (a region) were the EM distribution can be normalized to the invariant mass distribution. This criterium holds for the first nine pt bins but fails for the last two ones due to the limited invariant mass window. For the latter the very last part of the invariant mass window was used as normalization region. This is seen in Figure 5 and in table 3 which summarizes the normalization region for each pt bin and the normalization constants. Not surprisingly, the normalization constants are compatible with 5 (the number of mixed events) for all the pt bins but the last two ones. For each pt bin different normalization regions have been tried out to check the eﬀect on the final results.
Table 1: Event mixing normalization
Figure 6 shows the invariant mass distribution after the subtraction of the normalized EM background. The distribution is fitted with a constant straight line in the EM normalization region as a cross check. Table 4 shows the results of the straight line fit. The different shape of the invariant mass distribution and the EM background around the signal region produce a residual background after the EM subtraction. The residual background takes contributions from two different effects:
- correlated Λ - π pairs from the kinematics of the event. The extend of this residual background is pt dependent and is reproduced in the Monte Carlo. The four lower pt bins are mostly unaffected by this contribution.
- correlated Λ - π pairs from the Λ∗→ Λππ channel. This contribution affects all pt bins with comparable intensity.
Figure 6: Invariant mass distribution after the subtraction of the normalized EM background. The distribution is ﬁtted with a constant straight line in the EM normalization region as a cross check. Table 4 shows the results of the ﬁt.
Table 2: Constant straight line ﬁt values in the EM normalization region.
Fitting the residual background from Monte Carlo
Figure 7 shows the invariant mass distribution for Monte Carlo (MC) simulated data after the EM-background subtraction (EM-background also from MC data) and after the further subtraction of the MC-true signal. The MC residual background is fitted with a polynomial of third order in the region from 1.26 GeV/c2 (just left of the signal region) and the lower edge of the normalization region. The χ2 probability is as low as 4% for the 3000-4000 pt bin but not less than 20% for all the other pt bins. A second order polynomial could not reproduce well the shape of the residual background in all the region considered, and higher orders are producing an overﬁt of the data. Slight changes in the fitting regions have been tried out to check the effect on the ﬁnal results. The fitting polynomial from the MC residual background in Figure 7 is scaled (one parameter) to fit the residual background from real data in Figure 6. The fit is performed in the region from 1.46 (just right of the signal region) and the lower edge of the normalization region. Figure 8 shows the polynomial fit to the residual background. The polynomial is extrapolated in the signal region providing a model-guided way to describe the background below the signal. Note that, as already noticed, the residual background is almost absent in the lower pt bins.
Figure 7: Residual background from Monte Carlo data. The distribution is ﬁtted with a third order polynomial.
Figure 8: The polynomial from the fit of the residual background in MC data (Figure 7) is scaled to fit the EM-background-subtracted invariant mass distribution from Figure 6. Only one parameter is used.
Contamination from the Λ (1520)
Figure 9 shows the invariant mass after the residual background subtraction. The MC-true invariant mass distribution for Λπ pairs coming from the Λ∗→ Λππ decay has been calculated for events from the Λ(1520) dedicated enhanced-signal Monte Carlo simulation and ﬁtted with a Gaussian (see Figure 10). The mean and the σ of the Gaussian have then been used to constrain the Gaussian peak (mean and σ ) in the combined fit (Gaussian plus Breit-Wigner) of the distribution in Figure 9. Table 3 summarizes the results of the combined fit. Both the Gaussian means and the widths are in agreement within errors with the MC corresponding values. The Gaussian parameters (mean and width) have also been kept ﬁxed and the difference accounted for as systematic uncertainty.
Figure 9: The invariant mass distribution after the subtraction of the residual background. The peak from the Λ (1520) is contaminating the distribution in the lefthand side of the signal.
Figure 10: MC-true invariant mass distribution for Λπ pairs coming from the Λ∗→ Λππ decay. The peaks are ﬁtted with a Gaussian.
Table 3: Results of the combined ﬁt (Gaussian for the Λ∗-peak, BW for the signal). Statistical errors only.
Masses and widths from the Breit-Wigner function in the combined fit are presented in Figures 11, 12, 13 and 14. Values for the Σ∗− are also shown in the figures. Finally, the signal functions have been integrated to extract the raw yields. The statistical errors in the signal distributions have been propagated through the integrals to provide the statistical uncertainties on the raw yields. Table 4 summarizes the raw yields for the Σ∗+, together with their relative statistical uncertainties.
Figure 11: Masses as function of pt for the Σ∗+.
Figure 12: Masses as function of pt for the Σ∗-.
Figure 13: Widths as function of pt for the Σ∗+.
Figure 14: Widths as function of pt for the Σ∗-.
Table 4: Raw yields for the Σ∗+, together with their relative statistical uncertainties.
In order to extract the absolute yields, the raw yields (NRAW) were corrected for the decay branching ratio (0.88) and for losses due to in-ﬂight decays, geometrical acceptance, and detector efficiency (Ncor = NRAW/BR, where BR indicates the decay branching ratio). The global efficiencies Σ∗ were determined from Monte Carlo simulations with PYTHIA 6.4 (tune Perugia0) event generator and with a GEANT3-based simulation of the ALICE detector response.
Table 5: Efficiency corrections as a function of pt for all four Σ∗ species. The errors are statistical.