The variable that we use for the fits is ...

u = a + b ln(x) ... (1)

with x=dEdx expressed in units of keV/cm.

The inverse of (1) is ...

x = exp [ (u-a) / b ] ... (2)

We have chosen as a practical matter for general display,

a= 1.5, b=2.8

As noted by other collaborators in STAR, distributions in reconstructed dEdx are "Gaussian+tail" in shape. It can be seen that a transformation like (1) makes variable u Gaussian-like. The variable u is linearly related to the Z-variable (see Manuel Calderón de la Barca Sánchez's thesis for a discussion of the Z-variable), so they both convey the same statistical information, and therefore are equivalent for extracting yields.

Z= [ln(x/xo)] / s_xo

xo = dEdx(for mass assumption mo). Example: mo=mpion

s_xo= standard deviation of xo. (~ resolution).

If one measures a standard distribution s_u for variable u, from (1) it follows that, approximately,

s_x = s_u * x / b ... (3)

where x is expressed in terms of the variable u used in the fits by eq. (2).

The procedure we have followed to obtain yields involves

1) Convert x to u

2) Obtain spectra of u for a given y, dy and dmt. We have chosen dy=0.2, dmt=0.025 GeV/c^2, for mt=0 to 2.5. (40 bins, dmt= 0.025 GeV/c^2 from mt=0 to 1; 10 bins dmt= 0.150 GeV/c^2 from mt=1 to 2.5)

3) (Since mt^2 = (pt^2 + m^2), the corresponding dpt are smaller than dmt by mt/pt.)

4) First pass. Fit distribution of variable u to 4 Gaussians. This is a essencially free centroid, free width for each Gaussian.

5) Extract centroids for IDENTIFIED species (in selected mt bins). Extract also the corresponding measured bg for the identified specie.

6) Convert extracted u to x and s_x using eqs. (2) and (3). Extract pion resolution near the minimum ionizing region. Fix this resolution for all species near minimun ionizing region (in particular, for electrons.)

7) Step 6) produces a first pass values for x covering a signifcant region of bg (see results), in fact, roughly from 0.2 to 1000.

8) First pass calibration. Plot x vs. bg and fit each plot to Aihong's function (the most general case; we have restricted some parameters in some of the analysis done):


m1 * (1.+1./ t^ 2)^ m2 * {abs[ln(m3* t^ 2)]^m4 + m5 * (1.+1./ t^ 2)^m6 } - m7 ...............(4)

The following parameters give good fit for au130. We have used them as general initial conditions for the fits:

m1=1.2403; m2=0.31426; m3=12.428; m4=0.41779; m5=1.6385; m6=0.72059; m7=2.3503

9) Second pass. Use the calibration to set centroids and fixed widths of Gaussians, allowing the centroids to vary by 2%.

10) Repeat steps 4)-8) to obtain a better, very close to final, calibration.

11) Third pass. Use latest calibration to fix centroids.

Examine closely fits in IDENTIFIED regions, making sure fits are good.

12) DONE. The yields are obtained from the fits to 4 Gaussian with fixed centroids and widths for all histograms corresponding to each mt bin.

13) One notices that, since the TPC energy loss resolution, s_x / x, is approximately constant (say 10%), s_x has a bg - dependence similar to that of x. So a fit to a function (4) would be appropiate to calibrate s_x. This procedure could be used to establish the bg- dependence of the width in the low bg region where dEdx changes rapidally as a function of pt (kinematic widening of dEdx.)