The fit function

The basic peak shape is assumed to be a sum of asymmetric Gaussian with widths $\sigma_{i,l}$ depending on the particle type (index $i$) and track length (number of points $l$):

$$\sigma_{i,l} = \sigma_0 \left(\frac{x_i}{x_1}\right)^{0.625}/\sqrt{l}.$$ (1)

The basic parameter setting the width is $\sigma_0$ which is fitted for all bins. The dependence on path-length is assumed to be $1/\sqrt{l}$. The exact dependence on $l$ is not a real concern, since the number-of-points distribution is quite strongly peaked in each phase space bin. The dependence of the width on the $dE/dx$ peak position $x_{i}$ is parameterised as a power law with power 0.625. This number was extracted from simultaneous fits to $m^2$ from TOF and $dE/dx$ from the TPC, as illustrated in Figs 1 [2]. Fig 2 shows the result of the two-dimensional fits for the higher beam energies (40, 80 and 158 GeV). The points clearly cluster at values of $\alpha$ slightly above 0.5, but significantly below unity. The value 0.625 was takena s a certal value and the sensitivity to this parameter was investigated and found to be small (a few per cent at maximum, for 40 and above).

Figure 1: (upper left) Distribution of the $m^2$ measurement from TOF and the $dE/dx$ measurement in the MTPCs for positive tracks close to mid-rapidity with transverse momenta close to 0.5 GeV. (lower left) Projection of the two-dimensional histogram on the $dE/dx$ axis. (upper right) Projection of the two-dimensional histogram ion the $m^2$ axis. In each projected histogram, a projection of the two-dimensional fit is also shown. Indicated are the pion, kaon and proton peaks.

Figure 2: Value of the scaling parameter $\alpha$ for the width of the $dE/dx$ peaks as a function of the position, as determined in different bins of total momentum and $p_t$. The different panels show results at the three different beam energies. The dashed lines are at $0.625 \pm 0.125$.

The Gaussian peaks are allowed to be asymmetric to reflect the remainder of the tail of the Landau-distribution which is still present in our truncated mean measure of $dE/dx$ . The total formula for fitting the $dE/dx$ distribution is then

$$\sum_{i=d,p,K\pi,e} { A_i \frac{1}{\sum_l{n_l}} \sum_l \fr... ...c{1}{2}\left(\frac{x-x_i}{(1\pm\delta)\sigma_{i,l}}\right)^2},$$ (2)

where $A_i$ are the amplitudes (yields) of each peak. The $n_l$ are numbers of tracks with a certain length, so the second sum is simply the weighted average of the line-shape from the different track-lengths in the sample.

The fit function is evaluated in the T49SumGaus class.