Spectra calculation

The final step of the analysis is to calculate the spectra. This step mainly involves the co-ordinate transformation from the $p,p_T$-grid in which the fits are done to a $y,p_T$-grid for presenting the data.

The analysis is performed in a grid with bins of 0.05 in $\log{p}$ and 0.1 GeV/$c$ in . The final data are presented on a grid of 0.2 in rapidity and 0.1 GeV/$c$ in . The transformation from the measured bins to the final bins is performed as follows.

The fit results represent $d^2N/d\ensuremath{p_T}dp$. For each bin, the Jacobian prefactor $E p_{z} / p$ is calculated at the center of the bin, using the appropriate mass for each particle. The particle density $y,\ensuremath{p_T}$-space $dN/dpt dy$ then follows:

$$\frac{d^2N}{d\ensuremath{p_T}\, dy}=E\frac{d^2N}{d\ensurema... ...\, dp_z}=E\:\frac{p_z}{p}\frac{d^2N}{d\ensuremath{p_T}\, dp}.$$ (5)

These values, however are still on a rectangular grid in $p,\ensuremath{p_T}$, not $y,\ensuremath{p_T}$. The final results are obtained by interpolating between measured points in $p$ to the values at the chosen grid in $y$ (30 bins from 0 to 6 in $y_{lab}$). The bin size in $\log(p)$ corresponds to $\Delta y \sim 0.1$, so that subsequent points in the interpolation do not share a datapoint. No interpolation in is needed.

The uncertainties on the final datapoints are calculated by interpolating the statistical and systematic uncertainties. In other words: the uncertainties on the $y,\ensuremath{p_T}$ grid points are a weighted average of the uncertainties on the neighboring points from the $p,\ensuremath{p_T}$ grid. This procedure is accurate for systematic uncertainties, but overestimates the statistical uncertainty by up to a factor $\sqrt{2}$, if all points are statistically uncorrelated. Since sometimes two $y,\ensuremath{p_T}$ points share a single point from the original $p,\ensuremath{p_T}$ grid, the present procedure seemed reasonable. In addition, the systematic uncertainty in the end is dominant.

This is also the step where the final phase space cuts are applied and where bins with too few entries are rejected. The macro for this step is make_ypt_syserr.C . In this macro, the acceptance and efficiency corrections are also applied. The output of the macro are fully corrected spectra, including systematic error estimates. The format is simple text-files for convenient plotting.

The final systematic errors are also calculated in this step. For this, the results of four separate fits are used. The largest and smallest yield as obtained from these four fits are used as the systematic error limits. The full procedure of fitting the -spectra and extrapolating is then applied to the nominal result and the upper and lower values to obtain the limits on the rapidity spectrum and the total yield. The underlying assumption is that the systematic variations are strongly correlated over and rapidity.