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\title{Gluon distribution in nucleon and its application to analysis of
$ep$ DIS, $pp$
  and $AA$ collisions at high energies and mid-rapidity
 \\ Распределение глюонов в нуклоне и его применение к анализу $ep$
  глубоко неупругого рассеяния (ГНР), $pp$ и $AA$ столкновений при высоких
  энергиях и центральной быстроте.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
\authors{G.I.\,Lykasov$^{a,}$\footnote{E-mail:lykasov@jinr.ru},
  A.I.\,Malakhov$^{a,}$\footnote{E-mail:malakhov@jinr.ru},
  M.A.\,Malyshev$^{b,}$\footnote{E-mail:malyshev@theory.sinp.msu.ru},
  A.V.\,Lipatov$^{a,b,}$\footnote{E-mail:lipatov@theory.sinp.msu.ru},
  A.A.\,Zaitsev$^{a,}$\footnote{E-mail:zaicev@jinr.ru}}
  \setcounter{footnote}{0}
 \authors{Г.И.\,Лыкасов$^{a,}$\footnote{E-mail: lykasov@jinr.ru},
  A.I.\,Малахов$^{a,}$\footnote{E-mail:malakhov@jinr.ru},
  M.A.\,Малышев$^{b,}$\footnote{E-mail: malyshev@theory.sinp.msu.ru},
  A.V.\,Липатов$^{a,b,}$\footnote{E-mail: lipatov@theory.sinp.msu.ru},
  A.A.\,Зайцев$^{a,}$\footnote{E-mail: zaicev@jinr.ru}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\from{$^{a}$\,Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia}  
\from{$^{a}$\,Объединенный институт ядерных исследований, 141980, Дубна, Московская обл., Россия}
\from{$^{b}$\,Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991, Moscow, Russia}
\from{$^{b}$\,Научно-исследовательский институт ядерной физики им. Д.В.\,Скобельцына Московского государственного университета им. М.В.\,Ломоносова, 119991, Москва, Россия}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}

\begin{abstract}
% Russian translation of the abstract
Дается обзор о распределении глюонов по поперечному
импульсу (ПИР) и его применении к анализу $ep$ глубоко неупругого рассеяния
(ГНР) при энергиях HERA, жестких и мягких $pp$ процессах при энергиях (БАК)
и в центральной области быстрот.
\vspace{0.2cm}

A review on the gluon transverse momentum dependent distribution (TMD) and
its application to the analysis of the $ep$ DIS at the HERA energies,
hard and soft $pp$ processes at LHC energies and mid-rapidity is presented.
\end{abstract}

\vspace{1cm}

\noindent
    {\it Keywords:} small-$x$ physics, TMD gluon density, deep inelastic scattering,
    color dipole approach, modified quark-gluon string model
    
    \newpage

    \section{Introduction} \indent

   
    The description of any physical observables, obtained in the collider
    experiments at LHC, is mainly based on different factorization theorems
    in quantum chromodynamics (QCD).
    According to these theorems, hard partonic processes are separated 
    from partonic density functions (PDF). An appropriate QCD evolution
    describing
    the PDF, as a function of the resolution scale $\mu^2$, is given by the
    Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation \cite{DGLAP}.  
    This approach can be successfully applied usually to analyze inclusive
    hard processes, like deep-inelastic lepton-hadron scattering (DIS).      
    The transverse momentum ($k_T$) dependent (TMD or or unintegrated)
    parton densities $f_a(x, {\mathbf k}_T^2, \mu^2)$ (with $a = q$ or $g$)
    satisfy  the the Balitsky-Fadin-Kuraev-Lipatov
    (BFKL) or Catani-Ciafaloni-Fiorani-Marchesini
    (CCFM) evolution equation \cite{CCFM}.
    
    \section{Gluon TMD and and its saturation at low $Q^2$} \indent
    
    The CCFM evolution equation reads:
    \begin{equation}
      \displaystyle f_g(x,{\mathbf k}_T^2,\mu^2) = f_g^{(0)}(x,{\mathbf k}_T^2,\mu_0^2)
      \Delta_s(\mu,\mu_0) + \atop {
        \displaystyle + \int\frac{dz}{z}\int\frac{dq^2}{q^2}
        \Theta(\mu-zq)\Delta_s(\mu,zq) \tilde P_{gg}(z,{\mathbf k}_T^2, q^2)
        f_g\left(\frac{x}{z},{\mathbf k}^{\prime \, 2}_T,q^2\right) },
    \label{eq:CCFM}
    \end{equation}

\noindent
where ${\mathbf k}_T^\prime = {\mathbf q}(1 - z) + {\mathbf k}_T$,
$\tilde P_{gg}(z,{\mathbf k}^2_T,q^2)$ is the CCFM
splitting function \cite{CCFM}, $\Delta_s(\mu,\mu_0)$ is the Sudakov
form factor, see details in \cite{CCFM}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As it is shown in \cite{LLM:2023,LLM2:2023}, the solution of this
equation is very sensitive
to the form of the initial gluon TMD $f_g^{(0)}(x,{\mathbf k}_T^2,\mu_0^2)$
at the starting renormalization scale $\mu_0^2$.
There are a view type of it.
Similar to conventional PDFs,
a construction of the TMD parton distributions in a proton starts from the input
densities,
which are further used as an initial conditions for subsequent non-collinear
QCD evolution.
As it was mentioned above, usually the initial TMD gluon density at some 
starting scale $\mu_0^2$ (which is of order of hadron scale) is taken in the rather
general mathematical form 
with factorized Gauss smearing in transverse momentum ${\mathbf k}_T^2$
(see, for example,\cite{JH2014}):
\begin{gather}
  f_g^{(0)}(x, {\mathbf k}_T^2, \mu_0^2) = a_1 x^{a_2} (1-x)^{a_3} e^{-{\mathbf k}_T^2/q_0^2},
  \label{eq:Jung-Hautmann}
\end{gather}
\noindent
where all the parameters have to be extracted from the experimental data.
Alternatively, a more physically motivated non-factorized expression 
for $f_g^{(0)}(x, {\mathbf k}_T^2, \mu_0^2)$ can be taken 
from the Golec-Biernat-W\"usthoff (GBW) approach \cite{GBW1, GBW2}
based on color dipole picture for deep inelastic scattering (DIS):
Similar to conventional PDFs,
a construction of the TMD parton distributions in a proton starts from the input
densities,
which are further used as an initial conditions for subsequent non-collinear QCD
evolution.
As it was mentioned above, usually the initial TMD gluon density at some 
starting scale $\mu_0^2$ (which is of order of hadron scale) is taken in the
rather general mathematical form 
with factorized Gauss smearing in transverse momentum ${\mathbf k}_T^2$ (see,
for example,\cite{JH2014}):
\begin{gather}
  f_g^{(0)}(x, {\mathbf k}_T^2, \mu_0^2) = a_1 x^{a_2} (1-x)^{a_3} e^{-{\mathbf k}_T^2/q_0^2},
  \label{eq:Jung-Hautmann}
\end{gather}
\noindent
where all the parameters have to be extracted from the experimental data.
Alternatively, a more physically motivated non-factorized expression 
for $f_g^{(0)}(x, {\mathbf k}_T^2, \mu_0^2)$ can be taken 
from the Golec-Biernat-W\"usthoff (GBW) approach \cite{GBW1, GBW2}
based on color dipole picture for deep inelastic scattering (DIS):
\begin{gather}
  f_g^{(0)}(x, {\mathbf k}_T^2, \mu_0^2) = c_g R_0^2(x) {\mathbf k}_T^2 e^{ -R_0^2(x)
    {\mathbf k}_T^2 }, \quad R_0(x) = {1\over Q_0} \left( {x\over x_0} \right)^{\lambda/2},
  \label{eq:GBW}
\end{gather}
\noindent
where $c_g = 3 \sigma_0/(4 \pi^2 \alpha_s)$, $\sigma_0 = 29.12$~mb, $\alpha_s = 0.2$,
$Q_0 = 1$~GeV, $x_0 = 4.1 \cdot 10^{-5}$ and $\lambda = 0.277$.
In this approach, the effect of saturation of $q \bar q$ dipole cross section 
at large distance $r$ between quark and anti-quark in the dipole is taken into
account.
The GBW model has been successfully applied to both inclusive and diffractive
$ep$ scattering at HERA.
However, it meets some difficulties in the accurate description of several hard
LHC processes.
In our previous studies 
we have demonstrated that one could achieve a reasonably well description of 
the latter using the analytical expression (non-factorized with respect to $x$ and 
${\mathbf k}_T^2$) for initial TMD gluon density derived in the modified QGSM 
approach
from the best description of inclusive hadron spectra in the mid-rapidity at low
scales \cite{LLM:2023}.

As it is suggested in \cite{kt-factorization}, at very low $x$ and $Q^2$
the proton structure function $F_2(x,Q^2)$ determined mainly by the TMD gluon density
$f_g(x,k_T,Q^2)$ can be saturated, i.e., it does not depend on $Q^2$. In
\cite{Mueller:1990}
the partial saturation of the gluon TMD at $Q^2$ less than the saturation
value $Q_S^2$ was suggested, i.e.,
$f_g(x,k_T,Q^2)\leq Q_S^2\simeq Constant\times \ln(1/x)$.
Further developments on the saturation effect were done in
\cite{NZ:1990,NPZ:1994,GBW1,GBW2}.
At asymptotically low $x$, an equivalent description is
  provided by the color dipole model ,\cite{NZ:1990,NPZ:1994,GBW1,GBW2}.
  where gluon saturation effects \cite{Mueller:1990}
(the partial saturation) important at low scales
  (and, of course, at low $Q^2$ in DIS) can be consistently taken into account.
In this approach, the DIS events are
considered as an interaction of the virtual photon decaying into a color dipole
$q{\bar q}$ and a proton.
The saturation of the photon-proton cross section $\sigma^{\gamma^* p}(x, Q^2)$ as
a function of the transverse 
distance $r$ between $q$ and ${\bar q}$ is suggested at large $r$ or small $Q^2$
(see\cite{GBW1, GBW2} for more information).
Within the dipole approach, a satisfactory description of
early HERA data on the proton
  structure function $F_2(x,Q^2)$ and $\sigma^{\gamma^* p}(x, Q^2)$ at low $x$ was
  obtained\cite{GBW1, GBW2}.
  
  The new starting gluon TMD at low  $Q^2$ was obtained recently in
  \cite{LLM:2023}.
  It reads:
\begin{equation}
  f_g(x, {\mathbf k}_T^2) = c_g (1-x)^{b_g} \sum_{n = 1}^3 \left(c_n R_0(x)
  |{\mathbf k}_{T}|\right)^n e^{-R_0(x)|{\mathbf k}_{T}|},
  \label{eg:OurGluon}
\end{equation}
\noindent
where $R_0(x)$ is defined in~(\ref{eq:GBW}) and we kept $x_0 = 4.1 \cdot 10^{-5}$
and $\lambda = 0.22$.
Our best fit for phenomenological parameters
leads to $c_1 = 5$, $c_2 = 3$, $c_3 = 2$ and $Q_0 = 1.233$~GeV.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We treat $b_g$ as a running parameter
at ${\mathbf k}_T^2 \geq Q_0^2$: 
\begin{equation}
  b_g = b_g(0) + {4 C_A \over \beta_0} \ln {\alpha_s(Q_0^2) \over
    \alpha_s({\mathbf k}_T^2)},
  \label{eg:bg}
\end{equation}
\noindent
where $b_g(0) = 5.854$, $C_A = N_c$ and 
$\beta_0 = 11 - 2/3 N_f$ is the first coefficient of the QCD $\beta$-function.
At small ${\mathbf k}_T^2 < Q_0^2$, the fixed value $b_g = b_g(0)$ is used.
%The similar approach has been applied in the investigation of EMC effect from 
%the study of shadowing at low $x$ to antishadowing at
%$x \sim 0.1 - 0.2$\cite{bgForm}.

The relation of the of the cross section of the virtual photon $\gamma^*$
with proton is presented in \cite{GBW1,GBW2} calculated within te two-gluon
exchange approximation at has he following form:
\begin{gather}
  \hat{\sigma}(x,r^2) = \frac{4\pi^2 \alpha_s}{3}\int\frac{d{\mathbf k}_T^2}
      {{\mathbf k}_T^2}\left\{1-J_0(|{\mathbf k}_T| r)\right\}
      f_g(x, {\mathbf k}_T^2),
  \label{def:GBW_dpcrsect}
\end{gather}
\noindent
where $J_0$ is the Bessel function of zero order and
$\alpha_s = 0.2$.
In Fig~\ref{fig1} the cross section $\hat\sigma(x,r^2)$ is presented for te
  starting gluon TMDs LLM-2022 and GBW.
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{sigma-test-llm-1.eps}
%sigma-test-x0-allW.pdf
% {FL-ZEUS.pdf}
\caption{The effective dipole cross section $\hat \sigma(x, r^2)$
    normalized by $\sigma_0$ 
  calculated as a function of $r$ at different values of $x$.
}
\label{fig1}
%fig:sigdip_r
\end{center}
\end{figure}
In Fig.~\ref{fig1} we plot the effective dipole cross section
  $\hat \sigma(x, r^2)$ evaluated 
according to~(\ref{def:GBW_dpcrsect})
as a function of $r$ at different values
of $x$. We find that its saturation dynamics at large $r$ strongly 
depends on $x$ and TMD gluon density in a proton. 
In accordance with~(\ref{def:GBW_crsect}),
the GBW gluon density results in the saturation in the region of $r_s\sim 2/R_0$
(see also discussion\cite{GBW2}). 
The corresponding saturation scale at $x = x_0 = 4.2 \cdot 10^{-5}$ and $Q_0= 1$~GeV 
is $Q_s\simeq 2/r_s\simeq 0.8$~GeV.
The LLM gluon TMD results in approximately the same $Q_s$ at fitted
$x_0 = 1.3 \cdot 10^{-11}$ 
and $Q_0 = 2.2$~GeV, see Fig.~\ref{fig1}.
The effective dipole cross section, $\sigma^{\gamma^* p}(x, Q^2)$ and  
gluon density $f_g(x, {\mathbf k}_T^2)$ should not depend on $Q^2$
at low $Q^2 < Q_s^2$.

With the listed pocket formulas we can now calculate some observables.
The structure function $F_2$ is the sum of the transverse structure
function $F_T$ and longitudinal one $F_L$, i.e.,
\begin{equation}
F_2(x,Q^2)~=~F_T(x,Q^2)~+~F_L(x,Q^2) ~.
\label{def:F2}
\end{equation}
$F_T$ and $F_L$ can be calculated at low
 $x$ neglecting small term $m x^2$ using (\ref{def:GBW1,GBW2}):
\begin{equation}
F_{T,L}(x,Q^2)~=~\frac{Q^2}{4\pi^2\alpha_{em}}\sigma_{T,L}(x,Q^2) ~,
\label{def:FTL}
\end{equation}
In fact, the reduced cross section $\sigma_r$ in the $e~p$ DIS, which is related to
structure functions $F_2$ and $F_L$, is measured; it can be calculated as:
\begin{equation}
  \sigma_r=\frac{Q^4x}{2\pi\alpha^2(1-y^2)}\frac{d^2\sigma}{dxdQ^2}=
  F_2(x,Q^2)~-~f(y)F_L(x,Q^2) ~,
\label{def:sigmar}
\end{equation}
where $d^2\sigma/dxdQ^2$ is the double differential DIS cross section,
$f(y)=y^2/(1+(1-y)^2)$ with inelasticity $y=Q^2/(sx)$ and $s=4E_eE_p$,
$E_e$ and $E_p$ are the
electron and the proton energies respectively.
\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{sigmared-318-1.png}
  \caption{The reduced DIS cross section a function of $x$ at different $Q^2$
    at $\sqrt{s}=318$~GeV. The solid line corresponds to the gluon TMD of type
    LLM \cite{LLM:2023}, the dashed line is the calculation with GBW gluon TMD
    \cite{GBW1,GBW2}.
%The notations are the same as on  Fig.~\ref{sigred1}.
    The data are taken from ZEUS and
    H1~\cite{ZEUS:2000,ZEUS:2001,ZEUS:1998,H1:2009}.
}
\label{sigred3}
\end{center}
\end{figure}
Having determined the parameters we can now easily calculate the structure function
$F_2(x,Q^2)$. We show our results in comparison with low $Q^2$ ZEUS~
\cite{ZEUS:1998} and H1~\cite{H1:2009} data in
Fig.~\ref{F2-res}. A good agreement of LLM results with data is achieved.

\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{F2-3.png}
  \caption{The structure function $F_2$ as a function of $x$ at different $Q^2$.
    The notations of the lines are the same as on Fig.~\ref{sigred3}.
    The data are taken from ZEUS~\cite{ZEUS:2000,ZEUS:2001,ZEUS:1998}
    (white circles) and H1~\cite{H1:2009} (black circles).
}
\label{F2-res}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%% insert th inclusive spectra of hard processes %%%%%%%%%%%%%%
\section{Soft pp processes at LHC energies and mid-rapidity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we turn to the production of soft charged hadrons in $pp$
  collisions
at the LHC energies. These processes are very sensitive to the gluon 
density in a proton at low scales $\mu \sim p_T \simeq 1$~GeV.
In the calculations we employ the modified QGSM, see \cite{Kaid1,LAS} and
references therein,
and strictly follow the approach described earlier\cite{LLM:2023}.
So, the inclusive hadron spectrum at low $p_T$
and mid-rapidities splits into two pieces: the quark contribution 
$\rho_q$ and the gluon one $\rho_g$:
\begin{gather}
  \rho(x, p_T)= E {d^3\sigma \over d^3 p} \equiv {1\over \pi}
      {d^3 \sigma \over d^2 p_T dy} = \rho_q(x,p_T) + \rho_g(x,p_T).
\end{gather}
\noindent
First term is calculated within the conventional QGSM
%\cite{SoftQuarkGluonStringModel-1, SoftQuarkGluonStringModel-2}
using only one-Pomeron exchange. It can be done because in the mid-rapidity and
small $x_T=2p_T/\sqrt{s}$ the multi-Pomeron exchanges give negligibly small
contributions \cite{BGLP:2012}.
%\cite{ModifiedSoftQuarkGluonStringModel-2}. 
The second one, $\rho_q(x,p_T)$, is calculated
as a convolution of the "modified" gluon distribution with 
fragmentation function (FF) of gluons into hadrons
$G_{g\rightarrow h}(z, |{\mathbf {\tilde p}}_T|)$ multiplied by the
inelastic $pp$ cross section (see, for example,\cite{LLM:2023}
  for more details). 
  These FFs were calculated at leading (LO) and next-to-leading (NLO)
  orders\cite{Kniehl:1995}
and presented in a factorized form
$G_{g\rightarrow h}(z, |{\mathbf {\tilde p}}_T|) =
G_{g\rightarrow h}(z) I_h^g(|{\mathbf {\tilde p}}_T|)$, where
$I_h^g(|{\mathbf {\tilde p}}_T|)$
could be approximated as:
\begin{gather}
  I_h^g(|{\mathbf {\tilde p}}_T|)=\frac{B_h^g}{2\pi}\exp(-B_h^g
  |{\mathbf {\tilde p}}_T|).
  \label{def:FF_T}
\end{gather}
\noindent
Here $\tilde p_T = p_T - z k_T$ with $p_T$ and $k_T$
being the transverse momenta of the produced hadron $h$ and gluon, respectively.
The FFs of quarks and/or diquarks into hadrons $h$ can be 
expressed in a similar way. 
Below we repeat the calculations\cite{LLM:2023}
with the LLM gluon density and newly fitted values of $x_0$,
$\lambda$ and $Q_0 = 2.2$~GeV.
We find that the best description of the LHC data
%\cite{SoftHadronData-1, SoftHadronData-2, SoftHadronData-3}
collected at
different energies ($\sqrt s = 0.9$, $2.36$, $7$ and $13$~TeV)
is achieved with slope parameters $B_h^g = 4.35$~GeV$^{-1}$,
$B_h^q = B_h^{qq} = 5.42$~GeV$^{-1}$ ($\chi^2/n.d.f.=3.5$).
All other parameters involved into the calculations and listed
in~\cite{LLM:2023}
are unchanged.
Our results are shown in Fig.~\ref{fig4}.
One can see that good agreement with the measurements is
obtained in a wide range of energies. For a comparison we also show results
obtained with the GBW TMD. It results in a different fit of the fragmentation
parameters $B_h^g = 4.75$~GeV$^{-1}$,
$B_h^q = B_h^{qq} = 6.5$~GeV$^{-1}$, which, however, gives a worse description
($\chi^2/n.d.f.=5.9$).

\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{softhadrons-x0=1.3e-11-bfg=4.35-bfq=5.35.pdf}
  \caption{Transverse momentum distributions of soft charged
      hadrons produced in $pp$ collisions at different LHC energies
      in the mid-rapidity region. The notations of the lines are the same as in
      Fig.~\ref{F2-res}. The experimental data are from
      \cite{CMS:2010,ATLAS:2011,ATLAS:2016}.}
\label{fig4}
%fig:sigdip_r
\end{center}
\end{figure}

\section{Production of pions and kaons at moderate energies}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The investigation of strange hadron and pion production in heavy-ion collisions
is a promising tool to search for new physical properties of such processes.
The observation of a sharp peak in the production ratio of $K^+$ 
mesons  to $\pi^+$ mesons in central $PbPb$ and $AuAu$ collisions at
mid-rapidity 
\cite{NA49:2002,NA49:2008} has attracted the attention of both theoreticians
and 
experimentalists, see [3-25]
% \textcolor{red} {[3-25]}
%\cite{Marek:1999}-\cite{NA61/SHINE:2020}
 and references therein. 
 When the initial energy $\sqrt{s_{NN}}$ per nucleon becomes larger than
 30 GeV this ratio falls down.
%{\bf
 According to the suggestion of \cite{Marek:1999,Marek:2014} this peak
 (so-called
  ``horn'') 
  can be due to formation of a quark-gluon plasma (QGP) phase at the
  center-of-mass energy $\sqrt{s}\simeq$ 7 GeV. There are another models
  explaining a reason of this ``horn'', see \cite{LMZ:2022} and referencies
  therein.
  However, the ratio of $K^+$ to $\pi^+$ mesons produced in collisions 
of nuclei lighter than $Pb$ and $Au$, as a function of the initial energy
$\sqrt{s_{NN}}$, in particular, $Be Be$ \cite{NA61_BeBe:2021} 
and $Ar Sc$ \cite{NA61_ArSc_ratio:2021} has no peak,
as revealed by the NA61/SHINE Collaboration
\cite{NA61_BeBe:2021,NA61_ArSc_ratio:2021}. 
The fast increase of this 
ratio, when $\sqrt{s_{NN}}$ grows from the kaon threshold up to 20-30 GeV and
the 
slow increase at larger energies have been observed 
\cite{NA61_BeBe:2021,NA61_ArSc:2021}. 
Moreover, the energy dependence of $K/\pi$ ratios observed 
in $Be Be$ collisions \cite{NA61_BeBe:2021} is similar to the one 
in $p p$ collisions.
 
In this paper we analyze the production of kaons and pions in $Be Be$
collisions 
at mid-rapidity and focus on ratios between their cross-sections as functions of
the initial energy within the same theoretical approach, which was presented 
in \cite{LMZ:2021} for $pp$ collisions. This approach is based on the similarity
of the inclusie hadron spectra produced in $AA$ collisions at mid-rapidity
satisfied
to the conservation lows of four-momenta and quantum nmbers  
\cite{LMZ:2021,c3,c4,c5,c6}.  

The inclusive production of hadron $1$ in the interaction of nucleus $A$ with
nucleus $B$     
\begin{equation}
A  +  B  \rightarrow  1  + \ldots ,                                           
\label{eq:n1}                                                                
\end{equation}                                              
is satisfied by the conservation law of four-momenta in the following form
\cite{c5,c6}
\begin{equation}
{(N_AP_A + N_BP_{B} - p_1)}^2 = 
{(N_Am_0 + N_B m_0 + M)}^2 ,
\label{eq:n2}
\end{equation}
where $N_A$ and $N_B$ are the fractions of the four-momentum transmitted by
nucleus $A$ and nucleus $B$, the forms of $N_A, N_B$  are presented in
\cite{c6}; 
$P_A$, $P_B$, $p_1$ are the four-momenta of nuclei  
$A$,  $B$ and particle $1$, respectively; $m_0$ is the mass of the nucleon;
$M$ is
the mass of the 
particle providing for conservation of the baryon 
number, strangeness and other quantum numbers.
Eq.~\ref{eq:n2} was introduced in \cite{c5,c6} for the production of hadrons in
$AB$ collisions in the kinematics 
forbidden 
for free nucleon-nucleon collisions. In fact, it is valid for initial energies
of colliding nuclei close to the threshold of hadron production.  
It allows us to find the minimal value of $M$,
which provides for the conservation of quantum numbers.   
For $\pi$-mesons $m_1 = m_\pi$  and $M = $0.
For antinuclei $M=m_1$ and for $K^-$-mesons  $M = m_1 = m_K$,
$m_K$ is the mass of the $K$-meson.
For nuclear fragments $M = - m_1$.
For $K^+$-mesons $m_1 = m_K$ and $M = m_\Lambda  - m_0$,
$m_\Lambda$ is the mass of the $\Lambda$-baryon.
Let us note that the isospin effects of the produced hadrons and other nuclear
effects
are out of this approach. Therefore, it is assumed that within the
self-similarity
approach there is no big
difference between the inclusive spectra of $\pi^+$ and $\pi^-$ mesons
produced in $pp$ and $AA$
collisions. However, there is a difference between similar spectra of $K^+$
and $K^-$ mesons, because 
the values of $M$ are different. This is due to the conservation law of
strangeness.

In \cite{c5,c6} the parameter of self-similarity is introduced in the
following 
form
\begin{equation} 
\Pi=\min \frac{1}{2} \left[ (u_A N_A + u_B N_B)^2\right]^{1/2}  ,
\label{eq:n3} 
\end{equation}                                            
where $u_A$ and $u_B$ are the four-velocities of nuclei $A$ and $B$,
respectively. 
The minimization over 
$N$ presented in Eq.~(\ref{eq:n3}) allows us to find the parameter $\Pi$. This
parameter introduced in \cite{c5} was obtained as the analytical form in
\cite{c6} for nucleus-nucleus
collisions in the mid-rapidity region, however, it can
also be applied successfully for the analysis of 
pion production in $pp$ collisions. 

The inclusive spectrum of particle 1 produced in the $AB$ collision
can be parametrized as a general universal function dependent on the 
self-similarity parameter $\Pi$, as it was shown in \cite{Baldin_AA:1996}.
\begin{equation}
E d^3 \sigma_{AB}/d^3p~=~A_A^{\alpha(N_A)}\cdot A_B^{\alpha(N_{B})}\cdot F(\Pi)
%\exp(-\Pi/C_2),
\label{eq:n4} 
\end{equation}
where $\alpha(N_A)=1/3 + N_A/3$, $\alpha(N_B)=1/3 + N_B/3$ 
and function $F(\Pi)$ is the inclusive spectrum of hadron production in
the $A B$ collision 
\cite{LM:2018}.

For symmetric colliding nuclei $N_A=N_B=N$ the function $\Pi$ is found from 
the minimization of Eq.~\ref{eq:n3} by solving the equation. This assumption 
has been suggested in \cite{c5,c6} 
\begin{eqnarray}
\frac{d\Pi}{d N}=0
\label{def:minimPi} 
\end{eqnarray}  
The exact solution of Eq.~\ref{def:minimPi} at $y=0$,
as 
\begin{eqnarray}
N=\frac{\Pi}{\mbox{cosh}(Y)}\equiv\frac{2m_0\Pi}{\sqrt{s}}, 
\label{def:N} 
\end{eqnarray}  
was obtained in \cite{c6,c5}, for details see, also \cite{LM:2018}.
In Eq.~\ref{def:N} $Y$ is rapidity of colliding nuclei.  
Therefore, $\alpha(N)=1/3 + 2m_0\Pi/(3\sqrt{s})$.
For symmetric nuclei Eq.~\ref{eq:n4} is presented in the following form
\begin{equation}
E d^3 \sigma_{AA}/d^3p~=~A^{2\alpha(N)}\cdot F(\Pi)
\label{eq:AA_sp} 
\end{equation} 
\begin{eqnarray}
F(\Pi)=\bigg[ A_q \mbox{exp}\Big(-\frac{\Pi}{C_q}\Big) + \\
\nonumber
A_g\sqrt{p_T}\phi_1(s) \mbox{exp}\Big(-\frac{\Pi}{C_g}\Big)\bigg] \sigma_{tot}
\label{def:F} 
\end{eqnarray}
where 
\begin{eqnarray}
\Pi(s,m_{1T},y)~=~\left\{\frac{m_{1T}}{2m_0\delta_h}+
\frac{M}{\sqrt{s}\delta_h}\right\}\mbox{cosh}(y)G,
\label{eq:n10}
\end{eqnarray}
\begin{eqnarray}
G~=~\left\{1+\sqrt{1+\frac{M^2-m_1^2}
{(m_{1T}+2Mm_0/\sqrt{s})^2\mbox{cosh}^2(y)}\delta_h}\right\}~.
\label{def:G}.
\end{eqnarray}
Here 
$\phi_1(s)~=~1-\sigma_{nd}(s)/\sigma_{tot}(s)$, see \cite{LM:2018},\\
$\delta_h=\left(1 - \frac{s_{th}^h}{s} \right)$;
$s_{th}^{\pi}\simeq 4m_0^2$; 
$s_{th}^{K^+}=\left(m_0 + m_K + m_\Lambda \right)^2$;
$s_{th}^{K^-}=(2m_0+2m_K)^2$;
$M = m_\Lambda - m_0; m_\Lambda = $ 1.115 GeV; 
$m_K = $ 0.494 GeV; $s_0 = $ 1 GeV; $m_0 = 0.938$ GeV;
$p_{1T}$ and $m_{1T}$ are the transverse momentum and transverse mass
of the produced hadron $1$; 
$\sigma_{nd} = (\sigma_{tot} - \sigma_{el} - \sigma_{SD})$ is the 
non-difrractive cross-section;
$\sigma_{tot},\sigma_{SD}$ and $\sigma_{el}$ are the total
cross-section, the single difrractive cross-section and the elastic
cross-section of $pp$ collisions, respectively.

\section{Summary}
Suggested gluon TMD at low $Q^2$ is applied successfully to the analysis
of the $ep$ DIS, $pp$ and $AA$ collisions at high and midle energies and
mid-rapidity. 
Note that the form for $F(\pi)$ in Eq.~\ref{def:F} was suggeted first in
\cite{BGLP:2012} for best description of charge hadrons produced in $pp$
collisions
at LC energies and low transverse momenta in the mid-rapidity region assuming
the creation of nonpertrbatie gluons at low $Q^2$ and their transverse momenta
$k_T$. In the latest our papers \cite{LLM:2023,LLM2:2023} these gluon distribtions
were related to the saturation effect in the $ep$ DIS mentioned in Sect.2.

\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{Ratio_pp_KPi_proposal_2023.png}
%    softhadrons-x0=1.3e-11-bfg=4.35-bfq=5.35.pdf}
  \caption{Left:the ratio of cross sections $\sigma_{K^+}/\sigma_{\pi^+}$.
    in $pp$ collisions as a function of $\sqrt{s}$.
  Right:the similar ratio for $\sigma_{K^-}/\sigma_{\pi^-}$.}
\label{fig5}
%fig:sigdip_r
\end{center}
\end{figure}


\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{Ratio_BeBe_proposal_2023.png}
%    softhadrons-x0=1.3e-11-bfg=4.35-bfq=5.35.pdf}
  \caption{Left:the ratio of cross sections $\sigma_{K^+}/\sigma_{\pi^+}$
  in $BeBe$ collisions as a function of $\sqrt{s}$.
  Right:the similar ratio for $\sigma_{K^-}/\sigma_{\pi^-}$.}
\label{fig6}
%fig:sigdip_r
\end{center}
\end{figure}

\begin{figure}
\begin{center}
  \includegraphics[width=15cm]{Kaons_ArSc_NA61_Baldin.png}
%{Ratio_BeBe_proposal_2023.png}
%    softhadrons-x0=1.3e-11-bfg=4.35-bfq=5.35.pdf}
  \caption{Left: inclusive $p_T$-spectra of $K^+$-mesons produced in $ArSc$
    collisions at mid-rapidity ($y\approx$0) and different initial energies $\sqrt{s_{NN}}$.
    Right: the similar $p_T$-spectra but for $K^-$-mesons.
  }
\label{fig7}
%fig:sigdip_r
\end{center}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{\bf


{\sl Acknowledgements}. We thank S.P.~Baranov, 
H.~Jung, S.~Schmidt  and S.~Taheri Monfared for their interest, important
comments
and remarks. 
Establishing new TMD and fragentation functions parameters as well as soft
hadron
production simulations and the dipole cross sections calculations were supported
by the 
Russian Science Foundation under grant~22-22-00387. The reduced cross section
and
structure function calculation and corresponding TMD refitting was performed
with
support of Russian Science Foundation under grant~22-22-00119.


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