Measurement Method

sc_width:method

The usual method for determining the width of a resonance is to fit to a function convoluted with a fixed-$\sigma$ Gaussian function which represents the experimental resolution. Because the width of the states is comparable to our experimental resolution and the statistics are limited, this method has certain instabilities as shown in sc_width:validity.

In order to measure the natural widths of narrow states, we must accurately know the experimental resolution, . To find the central value of $\Gamma$ we assume , the estimate of the experimental resolution from the is the true experimental resolution. Deviations from are discussed in the next section.

Once we have determined the experimental resolution, we fit the invariant mass distributions to the function

$$N(1 + \alpha(\Delta M-m_\pi)\Delta M^{\beta}) + \mbox{convoluted \bw} \eqnlabel{sc_width:free_bg}$$ (25)

where N, $\alpha$, and $\beta$ are allowed to vary freely. The signal shape is a convoluted with a Gaussian function with $\sigma = \ensuremath{\sigma_{\text{MC}}}\xspace$. The parameters are also allowed to vary. The resulting fits to the data are shown in sc_width:sigc_data, as are the fits to the used to extract the experimental resolution.

 

Figure: mass differences for data and . The top plots are for the data, the bottom . The left plots show the and the right plots show the . The events are and generated with $\Gamma(\sigc) = 0$.

sc_width:sigc_data