Fit procedure

The main objective of the fit-procedure is to use Eq. 2 to fit the $dE/dx$ distributions in every phase space bin. The full function has 12 parameters (5 amplitudes, 5 positions, width and asymmetry). It is practically impossible to fit all these parameters in every bin. However, some of the parameters are likely to be $p_T$-independent. In the fit, it is assumed that all the relative $dE/dx$ peak positions ($x_i/x_1$) are pt-independent. The electron and deuteron relative positions are fixed to a paremeterisation of the peak positions as function of $\beta\gamma$. This parameterisation was obtained from fits to some of the bins with larger statistics. The parameterisation is as follows:

$$\left\langle\frac{dE}{dx}\right\rangle=A\frac{B}{\beta^2}\left[\ln \frac{\beta^2}{1-\beta^2}-\beta^2-\delta(\beta)\right],$$ (3)

where the first two terms are the original Bethe-Bloch formula. The factors $A$ and $B$ are given in the Bethe-Bloch formula in terms of the ionisation energy of the medium and some fundamental constants, but for our fits we determine them from the data. The third term limits the energy loss to finite values in the logarithmic-rise region. It is calculated as follows:

$$\delta = \left\{ \begin{array}{ll} 0 & \ \ \beta\gamma <... ...(\ln \beta\gamma-b) & \ \ \beta\gamma > a_2 \end{array}\right.$$ (4)

The values of $a_1$ and $a_2$ are calculated by requiring continuity of the term $\delta$. The constanst $b$, $c$ and $d$ are specific to the medium and are tabulated for some gases and solid. In NA49, $d$ is set to 3, and the other two constants are fitted to the data. The full equation is quite complicated, but it is implemented in T49SumGaus::RelRise and T49SumGaus::GetRelRise for easy use. The various constants in the implemented function can be set using T49SumGaus::SetBBPars . In the final version of the fit-macro, the constants are set to T49SumGaus::SetBBPars(1.613702, 10.407406, 2.463701, 0.164279) .

This parametrisation is then used to fix the peak positions of the electrons and deuterons in the fits. The kaon and proton relative positions are determined from a simultaneous fit to all $p_T$ bins at each $p$. The same is done for the asymmetry parameter $\delta$. The remaining parameters are 7 per phase space bin (5 amplitudes, pion peak position and resolution). The deuteron peak amplitude is set to 0 for all bins at negative charge. In addition, the peak amplitudes are constrained to be 0 or larger. For technical reasons, if one of the peak amplitudes ends up at 0 from the free fit, it is fixed to 0 and a refit is done for a better evaluation of the errors and better stability of the simultaneous fit. The fit is carried out by the macro fit_ptb_asym_chisq_deut.C . This macro uses a $\chi^2$-criterium for minimisation. I have experimented with a log-likelihood fit, but did not manage to get those to converge properly.