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How to extract the one-dimensional distributions

Acceptance of the NA44 spectrometer is discussed in section  3.7 and presented on Fig. 3.3. The $1/m_T\,dN/\,dm_T$ spectra fall with $m_T$ and their shapes within the NA44 acceptance can be justifiably fit [93] by a single $\exp(-m_T/T)$. The $T$ will be referred to as a temperature parameter or a slope parameter. In determining $\,dN/\,dy$ and $1/m_T\,dN/\,dm_T$ for kaons and pions we use spectrometer settings, or portions thereof, with $\Delta y = 0.2-0.6$. The slope parameters do not depend appreciably on $y$ within this small range. Then the double differential multiplicity can be factorized as

$$\frac{\,d^{2}N}{\,dy\,dm_T} = B(y) T(m_T)$$ (10)

In parallel to the equations (4.7), for the observable particle counts one can define

$$\frac{\,dn}{\,dy} = \int^\infty_m \frac{\,d^{2}n}{\,dy\,dm_T} \,dm_T$$ (11)

$$\frac{\,dn}{\,dm_T} = \int^{y_{max}}_{y_{min}} \frac{\,d^{2}n}{\,dy\,dm_T} \,dy$$ (12)

(Technically speaking, they are projections of 2D histograms.) Integration of Eq.  4.5 assuming that Eq.  4.8 holds yields

$$\frac{\,dn}{\,dy} = B(y) \int^\infty_m T(m_T) A(y,m_T) \,dm_T$$ (13)

If Eq. 4.8 is true, $\,dN/\,dy$ from Eq. 4.7 can be rewritten as

$$\frac{\,dN}{\,dy} = B(y) \int^\infty_m T(m_T) \,dm_T$$ (14)

Eliminating $B(y)$, we conclude the ``theoretical justification'' for what is known as a 1D acceptance correction:

$$\frac{\,dN}{\,dy} = \frac{\,dn}{\,dy} \frac{ \int^\infty_m T(m_T) \,dm_T}{\int^\infty_m T(m_T) A(y,m_T) \,dm_T}$$ (15)

Similarly, for the $m_T$ spectrum we get

$$\frac{\,dN}{\,dm_T} = \frac{\,dn}{\,dm_T} \frac{\int^{y_{max}}_{y_{min}}B(y)\,dy} {\int^{y_{max}}_{y_{min}}B(y)A(y,m_T)\,dy}$$ (16)

Similar equations can be obtained for $p_T$ or $k_T$. In principle, one needs to know $B(y)$ and $T(m_T)$ in order to determine the true shape of $m_T$ spectrum and $\,dN/\,dy$. In practice, for pions $B(y)$ was taken to be a Gaussian with $\sigma=1.4$ centered at mid-rapidity; for kaons - a Gaussian with $\sigma=1.1$. For the acceptance correction in $y$, I used

$$1/m_T \,dN/\,dm_T = \sum_{i=1}^8 w_i \exp(-m_T/T_i)$$ (17)

Here $w_i$ are 8 variable parameters, $T_i$ are fixed ``temperatures''. For pions, all 8 parameters were used (see Table 4.1); for kaons, good fits could be obtained with only one $T$ parameter (see Table 5.1). The original idea of fitting the pion $mT-m$ distributions with the multi-temperature formula  4.15 to test sufficiency of a simple statistical description with a single temperature and a chemical potential, by transforming the $mT-m$ spectrum into a ``temperature spectrum''. The pion spectra seem to be dominated by a temperature component around 200 MeV.

Table 4.1: Parameters of the multi-temperature fits to the transverse kinetic energy distributions of pions. See Section 4.1.2, Eq. 4.15. The fitting range is $0 < k_T <1.5 GeV$

$T_i$, GeV	$w_i$ ($\pi^+$)	$w_i$ ($\pi^-$)
0.06	$(0.43 \pm 0.12)\times 10^{-1}$	$(0.51 \pm 0.17)\times 10^{-1}$
0.10143	$(0.25 \pm 0.11)\times 10^{-1}$	$(0.76 \pm 0.24)\times 10^{-1}$
0.14286	$(0.1 \pm 0.2)\times 10^{-2}$	$(0. \pm 0.19)\times 10^{-2}$
0.18429	$(0.185 \pm 0.06)\times 10^{-1}$	$(0.6 \pm 0.7)\times 10^{-2}$
0.22571	$(0.53 \pm 0.06)\times 10^{-1}$	$(0.76 \pm 0.04)\times 10^{-1}$
0.26714	$(0.48 \pm 0.2)\times 10^{-2}$	$(0.4 \pm 0.6)\times 10^{-3}$
0.30857	$(0.1 \pm 0.2)\times 10^{-3}$	$(0. \pm 0.4)\times 10^{-3}$
0.350	$(0.2 \pm 0.4)\times 10^{-3}$	$(0. \pm 0.2)\times 10^{-3}$
$\chi ^2/NDF$	88./66	87./66