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Introduction and definitions

The total energy $E$ and a three-dimensional vector of momentum $\vec{p}$ of a particle, emitted in an interaction, along with that particle's internal quantum numbers (mass $m$, charge, spin, strangeness, baryon number, etc), carry important information about the dynamics of the interaction. The simplest way to analyze this information is by using statistical distributions of particles with respect to various kinematical variables, derived from its Minkowski energy-momentum four-vector $(E,\vec{p})$. Among such variables are

• transverse momentum $p_T = \sqrt{p_x^2+p_y^2}$,
• transverse energy $m_T = \sqrt{p_T^2+m^2}$,
• transverse kinetic energy $k_T = m_T - m$,
• rapidity $y = 1/2 \ln ((E + p_z)/(E - p_z))$,
• pseudorapidity $\eta = -\ln (\tan (\theta/2))$, where $\theta$ is zenith angle.

It is a popular practice to use $m$, $p_T$, $y$ and azimuthal angle $\phi$ instead of ( $E, \vec{p}$), because in the former set of variables three ( $m, p_T, \phi$) are Lorentz-invariant with respect to translations along the $z$ axis, whereas $y$ is Lorentz-transformed by a simple addition of a number, so that $\Delta y$ is Lorentz-invariant.

In particle physics, one uses the notion of a differential cross-section of particle production

$$\frac{E\,d^3\sigma}{\,dp^3}$$ (3)

With our preferred set of kinematical variables, one notices that

$$\frac{E\,d^3\sigma}{\,dp^3} = \frac{E\,d^3\sigma}{\,d\phi p_T\,dp... ...\,d^3\sigma}{\,d\phi p_T\,dp_T\,dy} = \frac{\,d^3\sigma}{\,d\phi m_T\,dm_T\,dy}$$ (4)

Whatever reference is chosen to measure $\phi$, in practice one will see a set of interaction events averaged over all possible azimuthal orientations of the colliding system, unless one's experiment is more sensitive to some of them than to others. For experiments which do not distinguish azimuthal orientations (like NA44), nature performs a Monte-Carlo integration of Eq. 4.1 over $\phi$, and one is left with

$$\frac{\,d^2\sigma}{m_T\,dm_T\,dy}.$$ (5)

It may be more direct to talk about

$$\frac{\,d^2N}{m_T\,dm_T\,dy},$$ (6)

$N$ being the number of particles of given identity emitted in an interaction event. What one measures however is

$$\frac{\,d^2n}{m_T\,dm_T\,dy} = A(y,m_T) \frac{\,d^2N}{m_T\,dm_T\,dy},$$ (7)

where $A(y,m_T)$ is an acceptance function. $A(y,m_T) < 1$ due to experimental inefficiencies. Technically, $A(y,m_T)$ is represented in the analysis by a TURTLE[40]-based Monte Carlo simulation procedure and takes into account the effects of focusing and analyzing optics of the spectrometer as well as tracking efficiency and decays:

$$MC_{output} = MC_{input} A(y,m_T)$$ (8)

The output from the single particle analysis will be presented in form of 1D distributions, or integrals of Eq. 4.4:

$$\frac{\,dN}{\,dy} = \int^\infty_m \frac{\,d^{2}N}{\,dy \,dm_T} \,... ...,dN}{m_T\,dm_T} = \int^{y_{max}}_{y_{min}} \frac{\,d^{2}N}{\,dy m_T\,dm_T} \,dy$$ (9)