Multiplicity per rapidity in Carruthers and hadron resonance gas approaches

Abstract

The multiplicity per rapidity of the well-identified particles $\pi ^{-}$, $\pi ^{+}$, $k^{-}$, $k^{+}$, $\bar{p}$, p, and $p-\bar{p}$ measured in different high-energy experiments, at energies ranging from 6.3 to 5500 GeV, is successfully compared with the Cosmic Ray Monte Carlo event generator. For these rapidity distributions, we introduce a theoretical approach based on fluctuations and correlations (Carruthers approach) and another one based on statistical thermal assumptions (hadron resonance gas approach). Both approaches are fitted to both sets of results deduced from experiments and simulations. We found that the Carruthers approach reproduces well the full range of multiplicity per rapidity for all produced particles, at the various energies, while the HRG approach fairly describes the results within a narrower rapidity range. While the Carruthers approach seems to match well with the Gaussian normal distribution, ingredients such as flow and interactions should be first incorporated in the HRG approach. We conclude that fluctuations, correlations, interactions, and flow, especially in the final state, assure that the produced particles become isotropically distributed.

Appendix: Mathematical details for Eq. (6)

The Grand canonical partition function reads

$$\begin{aligned} Z(T,V,\mu )=\text{ Tr }\left[ \exp \left( \frac{{\mu }N-H}{\mathtt {T}}\right) \right] , \end{aligned}$$

(27)

where H is Hamiltonian of the system, N is the total number of constituents. In the HRG model, Eq. (27) can be expressed as a sum over all hadron resonances as

$$\begin{aligned} \ln Z(T,V,\mu )=\sum _i{{\ln Z}_i(T,V,\mu )} =\frac{V g_i}{(2{\pi })^3}\int ^{\infty }_0{\pm d^{3}p {\ln } {\left[ 1\pm \exp \left( \frac{{\varepsilon }_i(p)-\mu _{i}}{\mathtt {T}} \right) \right] }}, \end{aligned}$$

(28)

where ± stands for bosons and fermions, respectively, and $\varepsilon _{i}=\left( p^{2}+m_{i}^{2}\right) ^{1/2}$ is the dispersion relation of the ith particle. The total number of particles can be obtained from the partition function as follows.

$$\begin{aligned} N_{i}=T \frac{\partial Z_{i}(T, V)}{\partial \mu _{i}}=\frac{V g_i}{(2{\pi })^3}\int ^{\infty }_0{d^{3}p \left[ \exp \left( \frac{\varepsilon _{i} (p)-\mu _{i}}{\mathtt {T}}\right) \pm 1\right] ^{-1}}, \end{aligned}$$

(29)

The invariant momentum spectrum of partially radiated by a thermal source is given by

$$\begin{aligned} \frac{\mathrm{d}^{3}N_{i}}{\mathrm{d}ym_{T}\mathrm{d}m_{T}\mathrm{d}\phi }=\varepsilon _{i} \frac{V g_i}{(2{\pi })^3}\left[ \exp \left( \frac{\varepsilon _{i} (p)-\mu _{i}}{\mathtt {T}}\right) \pm 1\right] ^{-1}, \end{aligned}$$

(30)

where $m_{T}$ is the transverse mass. This is given by

$$\begin{aligned} m_{T}=\sqrt{m^{2}+p_{T}^{2}}, \end{aligned}$$

(31)

where $p_{T}$ is the transverse momentum. The energy of the ith particle $\varepsilon _{i}$ can be expressed in terms of the rapidity $\left( y\right)$ and $m_{T}$

$$\begin{aligned} \varepsilon =m_{T} \cosh \left( y\right) . \end{aligned}$$

(32)

Then, the rapidity density can be obtained through integral over the full transverse mass $m_{T}$,

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}=\frac{V g_i}{\left( 2{\pi }\right) ^2}\int ^{\infty }_m {dm_{\mathtt {T}} \cosh \left( y\right) m_{\mathtt {T}}^{2} \left[ \exp \left( \frac{m_{\mathtt {T}} \cosh \left( y\right) -\mu _{i}}{T}\right) \pm 1\right] ^{-1}}. \end{aligned}$$

(33)

To make the lower integration limit starts from zero, we define a new variable t

$$\begin{aligned} t = \frac{m_T - m}{T} \cosh \left( y\right) , \end{aligned}$$

(34)

Equation (34) can be arranged as

$$\begin{aligned} m_T = \frac{t T}{\cosh \left( y\right) } + m, \end{aligned}$$

(35)

By differentiating Eq. (35), we get

$$\begin{aligned} dm_T = \frac{T dt}{\cosh \left( y\right) }, \end{aligned}$$

(36)

By substituting Eqs. (34), (35) and (36) into Eq. (33), we get

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}=\frac{V g_i}{\left( 2{\pi }\right) ^2}\int ^{\infty }_0\frac{\cosh \left( y\right) (\frac{t T}{\cosh \left( y\right) } + m)^2 T \mathrm{d}t}{\cosh \left( y\right) \left[ \exp (t + \frac{m}{T}\cosh \left( y\right) - \frac{\mu _i}{T} ) \pm 1 \right] }, \end{aligned}$$

(37)

Also, Eq. (37) can be rewritten as

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}=\frac{V g_i T^3}{\left( 2{\pi }\right) ^2 \cosh ^2\left( y\right) }\int ^{\infty }_0\frac{ (t + \frac{m \cosh \left( y\right) }{T} )^2 \mathrm{d}t}{ \left[ \exp (t + \frac{m}{T}\cosh \left( y\right) - \frac{\mu _i}{T} ) \pm 1 \right] }. \end{aligned}$$

(38)

Let us define

$$\begin{aligned} c = \frac{m \cosh \left( y\right) - \mu _i}{T}. \end{aligned}$$

(39)

Then, Eq. (38) can be rewritten as

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}=\frac{V g_i T^3}{\left( 2{\pi }\right) ^2 \cosh ^2\left( y\right) }\int ^{\infty }_0\frac{ (t^2 + \frac{2 m \cosh \left( y\right) t}{T} + \frac{m^2 \cosh ^2\left( y\right) }{T^2} ) dt}{ \exp (t + c ) \pm 1}, \end{aligned}$$

(40)

Also, Eq. (40) can be rewritten as

$$\begin{aligned}&\frac{\mathrm{d}N}{\mathrm{d}y}=\frac{V g_i T^3}{\left( 2{\pi }\right) ^2 \cosh ^2\left( y\right) }\nonumber \\&\quad \left\{ \int ^{\infty }_0\frac{t^2 \mathrm{d}t}{\exp (t + c) \pm 1} +\frac{2 m \cosh \left( y\right) }{T}\int ^{\infty }_0\frac{t \mathrm{d}t}{\exp (t + c) \pm 1} +\frac{m^2 \cosh \left( y\right) }{T^2}\int ^{\infty }_0\frac{\mathrm{d}t}{\exp (t + c) \pm 1}\right\} . \end{aligned}$$

(41)

The polylogarithmic function is defined as

$$\begin{aligned} \mp L_is(\mp z) = \frac{1}{\Gamma (s)}\int ^{\infty }_0\frac{t^{s-1} dt}{\frac{\exp (t)}{z} \pm 1}, \end{aligned}$$

(42)

where

$$\begin{aligned} L_{i1}(z)= & {} \log (1 + z) \nonumber \\ \Gamma (s)= & {} (s-1)!, \nonumber \\ L_{in}(z)= & {} \sum _k=1^\infty \frac{z^k}{k^n}. \end{aligned}$$

(43)

By substituting from Eqs. (39), (42) and (43) into Eq. (41), we get

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}= & {} \frac{g_i V}{2\pi ^2}\left\{ \frac{2T^3 L_{i3}}{\cosh ^2\left( y\right) } \left( \mp \exp (-c)\right) \mp \frac{2m T^2}{\cosh \left( y\right) }L_{i2}\left( \mp \exp (-c)\right) \right. \nonumber \\&\pm&\left. m^2 T \log (1 \pm (\mp \exp (-c)))\right\} . \end{aligned}$$

(44)

By substituting Eq. (39) into Eq. (44), the multiplicity per rapidity reads

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}y}= & {} \frac{g_i V}{2\pi ^2}\left\{ \frac{2T^3 L_{i3}}{\cosh ^2\left( y\right) } \left[ \mp \exp \left( -\frac{m \cosh \left( y \right) -\mu _i}{T}\right) \right] \right. \nonumber \\&\mp \left. \frac{2m T^2}{\cosh \left( y\right) }L_{i2}\left[ \mp \exp \left( -\frac{m \cosh \left( y\right) -\mu _i}{T}\right) \right] \right. \nonumber \\&\pm \left. m^2 T \log \left( 1 \pm \left[ \mp \exp \left( -\frac{m\cosh \left( y\right) -\mu _i}{T}\right) \right] \right) \right\} . \end{aligned}$$

(45)