Background contributions and fitting methods

sc_mass:fitting

There are two sources of combinatoric backgrounds. The first source is fake candidates combined with additional pions in the event. The second background is from real candidates combined with other pions from the production vertex. (Since the has no measurable lifetime, pions from a real decay are indistinguishable from other pions in the primary vertex.) A third background contribution comes from the which decays via $\lcstar \to \lc \pi^+ \pi^-$ in what is measured to be a nearly pure 3-body decay [64,65]. These pions combined with candidates result in a background in the region $\Delta M < 0.205$ . The shapes of these three backgrounds are shown in sc_mass:sigmac_bgs.

 

Figure 6.5: $\lc \piplus$ backgrounds. a) the background shape from the sidebands. b) the background shape from and $\pi^\pm$ from different events. c) the Monte Carlo generated shape for feed down into the signal region plus a constant background.

[Sidebands]

sc_mass:sideband [Random $\pi$]

sc_mass:ran_pi [feed down]

sc_mass:lcstar sc_mass:sigmac_bgs

Fitting the overall $\lc \pi$ distributions can therefore be quite difficult. We have explored two methods:

1.

Fitting to fixed background shapes with floating normalizations.

2.

Fitting to a free background shape.

In addition, for the and , we can choose to include or ignore the contribution from feed down. This gives us four (two³⁴ in the case of ) different choices of fitting method, which help determine possible fit related systematics.

In fitting the mass peaks, we use a simple Gaussian function. This is not technically correct since we have shown that these states have a measurable width as detailed in sc_width. However, for determining the mass of a state, a Gaussian fit is adequate.

To fit with fixed background shapes, we fit both the distribution from the sidebands, shown in sc_mass:sideband, and the distribution formed by combining a from one event and a pion from the previous event, shown in sc_mass:ran_pi, to a phase space function of the form $N(1 + \alpha(\Delta M-m_\pi)\Delta M^\beta)$. ($\Delta M$ is $M(\lc \pion) - M(\lc)$, the mass difference.) N, $\alpha$, and $\beta$ are allowed to vary freely in fitting each background component. We then fit the signal distribution to

$$N_1(1 + \alpha_1(\Delta M-m_\pi)\Delta M^{\beta_1}) + N_2(1 +... ...ta M^{\beta_2}) + \mathrm{Gaussian} \eqnlabel{sc_mass:fix_bg}$$ (17)

where N₁ and N₂ are allowed to vary freely while $\alpha$ and $\beta$are fixed to the values previously found from the background distributions.

To fit with a ``free'' background, we fit the signal distribution to

$$N(1 + \alpha(\Delta M-m_\pi)\Delta M^{\beta}) + \mathrm{Gaussian} \eqnlabel{sc_mass:free_bg}$$ (18)

and allow N, $\alpha$, $\beta$, and all parameters associated with the Gaussian to vary freely.

In order to determine the contribution from feed down, we make an estimate of the contamination using Monte Carlo. We generate a sample of $\lcstaru \to \lc \pi^+ \pi^-$ decays and apply both the and the and reconstructions. The result of the reconstruction, shown in sc_mass:lcstar, is the contamination from feed down plus random combinatorics. We fit this distribution to the function

$$A_{\text{MC}}(1 - a(\Delta M - b)^2) + C_{\text{BG}} \eqnlabel{sc_mass:lcstar_fit}$$ (19)

where the quadratic term (the contamination) is restricted to the range $\Delta M < 0.205$ . $C_{\text{BG}}$ is a constant representing the random combinatoric background. The value of $A_{\text{MC}}$ represents the amplitude, or height, of the feed down contribution in . In order to estimate this contribution in the data, we normalize to . We assume that the relative reconstruction efficiencies for $\lc \pion^\pm$ and $\lc \piplus \piminus$ are the same in data and so that

$$\frac{A_{\mathrm{data}}}{A_{\mathrm{MC}}} = \frac{Y(\lcstar)_{\mathrm{data}}}{Y(\lcstar)_{\mathrm{MC}}}$$ (20)

where Y is the fitted yield. We then solve for the amplitude of the feed down in the data:

$$A_{\mathrm{data}} = A_{\mathrm{MC}} \times \frac{Y(\lcstar)_{\mathrm{data}}}{Y(\lcstar)_{\mathrm{MC}}} \, .$$ (21)

We use identical cuts on the and soft pion(s) in all four reconstructions (and for and data). To include the contribution into the signal distribution, we fix a and b in sc_mass:lcstar_fit to their fit values, fix A_data to the calculated value, and add this contribution to Equation 6.5 or 6.6. The data and signals used to calculate this contribution, or shape, are shown in sc_mass:lcstar_norm.

 

Figure 6.6: signals from data and used to calculate the feed down contribution. The (which is not fitted) is evident in the data shown on the left. is shown on the right.

sc_mass:lcstar_norm

For the and , we find that all four of these fitting methods give very close agreement for the values of $\Delta M$ as shown by the first four points of sc_mass:method_sys.

For the the two fit values (fixed and free backgrounds) also agree very closely, giving a systematic of only $0.01~\mevcc$.

 

Figure 6.7: and systematic errors from fitting and reconstruction methods. The three plots are respectively, the values of , , and . The first four points are values obtained using fixed background shapes with floating normalizations (`fix'), a free parameter phase space background fit (`free'), and including (`lc') or ignoring the feed down. The fifth and sixth points are obtained by using vectors and not using respectively. The solid and dashed lines are our measurement and statistical errors.

sc_mass:method_sys