# The gluon condensation in hadron collisions

Wei Zhua111Corresponding author, E-mail<EMAIL_ADDRESS>Qihui Chenb,
Zhiyi Cuia and Jianhong Ruana
aDepartment of Physics, East China Normal University, Shanghai 200241, P.R.
China
bSchool of Physical Science and Technology Southwest Jiaotong University,
Chengdu 610031, P.R. China

###### Abstract

Gluons may converge to a stable state at a critical momentum in hadrons. This
gluon condensation is predicted by a nonlinear QCD evolution equation. We
review the understanding of the gluon condensation and present a clear
physical picture that produces the gluon condensation from the colour glass
condensate. We summarize the applications of the GC effect in the $p-p(A)$
collisions and predict that the $p-Pb$ and $Pb-Pb$ collisions at the LHC are
close to the energy region of the gluon condensation. We warn that for the
next generation of hadron colliders with the increasing of the collision
energy, the extremely strong gamma-rays will be emitted in a narrow space of
the accelerator due to the gluon condensation effect. Such artificial mini
gamma-ray bursts in the laboratory may damage the detectors.

keywords: Gluon condensation; Antishadowing effect; Next generation of LHC

##  1 Introduction

Hadron in infinite momentum frame is consisted of partons (i.e., quarks and
gluons). Gluons distribute mainly in the small $x$ range ($x$ is the Bjorken
variable) and they dominate the high energy hadronic processes. The parton
distribution functions are evolved according to the evolution equations based
on perturbative QCD (pQCD) in the standard model. For example, the gluon
distribution functions satisfy the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) equation in a broad kinematic range, its elemental amplitude is shown
in Fig. 1a, where the correlations among initial partons are neglected [1].
Obviously, these correlations should be considered when the gluon density
becomes large in very small $x$ region, where the wave functions of the
initial partons begin to overlap. Adding the initial gluons to the elemental
amplitude of the DGLAP equation, one can get a series of modified elemental
amplitudes in Figs. 1(b)-1(d), corresponding to the Balitsky-Fadin-Kuraev-
Lipatov (BFKL) [2], Gribov-Levin-Ryskin-Mueller-Qiu -Zhu-Ruan-Shen (GLR-MQ-
ZRS) [3,4] and Zhu, Shen, Ruan (ZSR) [5] equations, respectively. For
convenience, we use the names of authors to mark the different evolution
equations. According to the standard quantum field theory, a complete
evolution equation should include the contributions of all possible Feynman
diagrams for energy-momentum conservation and infrared (IR) safety. Works
[4,5] used the time ordered perturbative theory (TOPT) to derive the evolution
equations in Fig. 1. Interestingly, one of the resulting evolution equation
(ZSR) predicts that the gluons in hadron may converge at a critical momentum
(Fig. 3). This is the gluon condensation (GC).

Figure 1: The corrections of the initial gluons to an elemental amplitude of
the DGLAP equation (a) [1] and they lead to (b) the BFKL equation [2], (c) the
GLR-MQ-ZRS equation [3-4] and (d) the ZSR evolution equation [5],
respectively. The dashed line is a virtual current which probes gluons.

The GC is displayed in the numerical solution of an approximate evolution
equation. A question is that the GC does exactly exist in the nature? or is it
just an approximate solution of the equation, once the approach is improved,
will the GC disappear? This work attempts to answer the above question in the
following two steps.

(1) We briefly review the GC-theory in Sec. 2, where we track the formation
processes of the GC through Figs. 4 and 5. We noticed that the transverse
momenta of the splitting and fusing gluons in hadrons at high energy processes
are randomly changed. It leads to the so-called Lipatov-singularity [2]. This
infrared singularity should be regularized by summing all relating Feynman
diagrams according to the standard quantum field theory, after which, we can
obtain Eq. (2.1), where both the linear and nonlinear parts have the
regularized Lipatov-singularities. The nonlinear part in Eq. (2.1) converts
the weak jumps on the gluon distributions into the strong chaotic
oscillations. The chaotic oscillation generates the strong antishadowing
effect, and the later leads to the GC. Thus, we show a more clear physical
picture of the GC, which is quantitatively described by a delta-like function
with the undetermined GC-critic momentum $(x_{c},k_{c})$.

(2) Then we turned to discuss the applications of the GC effect in Sec. 3. The
GC should induce significant effects in the proton collisions if gluons with
the GC-critic momentum enter the measuring energy region. Unfortunately, the
value of $(x_{c},k_{c})$ can not been entirely determined in the theory since
it relates to the unknown input conditions and high order modifications. We
have not directly observed the GC-effects at the Large Hadron Collider (LHC).
Therefore, the GC effect was used to analyse the astrophysical observations in
[6-9], since the proton energy may be accelerated to cause the GC-effect in
the hadronic collisions. Although the astrophysical phenomena relate to many
complex factors, the GC effect as a general physical effect should be shown in
the astronomical observation phenomenon. Specifically, cosmic gamma-rays can
be generated in $p+p\rightarrow\pi^{0}\rightarrow 2\gamma$. The sharp peak in
the momentum distribution of gluons caused by the GC effect in Fig. 3 can
suddenly increase the cross section in the hadron-hadron collisions, and
results in a typical broken power law in the gamma-ray energy spectra, where
the broken energy $E_{\pi}^{GC}$ is determined by the critical momentum. After
a brief summary we discuss the GC-threshold in the LHC energy range. We find
that the interaction energies in the $p-Pb$ and $Pb-Pb$ collisions in LHC are
close to the GC-energy region. Note that the cross section of pion production
in the hadronic colliders can be increased by several orders of magnitude due
to the GC effect, and about half of the proton kinetic energy converted to the
photons of energy $E_{\gamma}=m_{\pi}/2$ with an extra intensity in a narrow
space at the moment of collision. Therefore, we issue a warning for the next
accelerator plans since such unexpected intense gamma rays in the laboratory
may damage the detectors. Finally, the discussions and summary are given in
Sec. 4, where we explain why the Balitsky-Kovchegov (BK) [10] does not have
the GC solution. We also show the self-consistence among the popular QCD
evolution equations.

##  2 The GC effect in the ZSR equation

The ZSR equation at the cylindrically symmetric approximation reads [5]

$-x\frac{\partial F(x,k_{\perp}^{2})}{\partial x}$
$=\frac{3\alpha_{s}k_{\perp}^{2}}{\pi}\int_{k^{2}_{0}}^{\infty}\frac{dk^{\prime
2}_{\perp}}{k^{\prime 2}_{\perp}}\left\\{\frac{F(x,k^{\prime
2}_{\perp})-F(x,k^{2}_{\perp})}{|k^{\prime
2}_{\perp}-k^{2}_{\perp}|}+\frac{F(x,k^{2}_{\perp})}{\sqrt{k^{4}_{\perp}+4k^{\prime
4}_{\perp}}}\right\\}$ $-\frac{81}{16}\frac{\alpha_{s}^{2}}{\pi
R^{2}_{N}}\int_{k^{2}_{0}}^{\infty}\frac{dk^{\prime 2}_{\perp}}{k^{\prime
2}_{\perp}}\left\\{\frac{k^{2}_{\perp}F^{2}(x,k^{\prime 2}_{\perp})-k^{\prime
2}_{\perp}F^{2}(x,k^{2}_{\perp})}{k^{\prime 2}_{\perp}|{k^{\prime
2}_{\perp}-k^{2}_{\perp}|}}+\frac{F^{2}(x,k^{2}_{\perp})}{\sqrt{k^{4}_{\perp}+4k^{\prime
4}_{\perp}}}\right\\},$ $None$

where $F$ is the unintegrated gluon distribution and the linear part describes
the contributions of the BFKL evolution. We used $F(x/2,k_{\perp}^{2})\simeq
F(x,k_{\perp}^{2})$ near the saturation scale; the value of
$R_{N}=4~{}GeV^{-1}$ is fixed by fitting the available experimental data of
the proton structure functions.

We take two saturated inputs to evolve Eq. (2.1) starting from $x_{0}=4\times
10^{-5}$. They are the Golec-Biernat and Wusthoff (GBW) model [11]

$F_{GBW}(x_{0},k_{\perp}^{2})=\frac{3\sigma_{0}}{4\pi^{2}\overline{\alpha}_{s}}R^{2}_{0}(x)k_{\perp}^{4}\exp(-R^{2}_{0}(x)k_{\perp}^{2}),$
$None$

where $\sigma_{0}=29.12~{}mb$, $R_{0}(x)=1/Q_{s}$, $Q_{s}=1~{}GeV$,
$\overline{\alpha}_{s}=0.2$, and the Kharzeev-Levin (KL) model [12]

$F_{KL}(x_{0},k_{\perp}^{2})=\left\\{\begin{array}[]{ll}f_{0}k_{\perp}^{2}~{}if~{}k_{\perp}^{2}<Q^{2}_{s}\\\
f_{0}Q^{2}_{s}~{}if~{}k_{\perp}^{2}>Q^{2}_{s},\end{array}\right.$ $None$

where $f_{0}=10$. These two inputs are plotted in Fig.2, they describe the so
called colour glass condensate (CGC) [13]. Note that in the calculations we
take $F(x,k_{\perp}^{2})=0$ if $F(x,k_{\perp}^{2})<0$ since
$F(x,k_{\perp}^{2})\geq 0$ according to the definition of the gluon
distribution.

Figure 2: The GBW-, KL- and BFKL-inputs.

Figure 3: The solutions of Eq. (2.1) with the GBW- and KL-inputs, where the
Runge-Kutta method is used.

The solutions of $F(x,k^{2}_{\perp})$ in three-dimensional representation are
given in Fig. 3, they show that gluons in the proton are converged at a
critical momentum $(x_{c}.k^{2}_{c})$. In order to expose its origin, we study
the structure of Eq. (2.1). An important character of the BFKL dynamics is
that the tracks of gluon motion on the transverse momentum $k_{\perp}$-plane
are randomly distributed in the evolution. One can find that the variable
$k^{\prime 2}_{\perp}$ may cross over $k^{2}_{\perp}$ and generates the IR-
divergences both in the linear and nonlinear kernels as shown in Eq. (2.1).
However, any physical processes must be IR-safe. The most used nonlinear
modifications to the BFKL equation take the way to avoid the IR-divergences,
while these IR-divergences are canceled by using the TOPT cutting rule in
works [4,5].

We draw the integrated function $R(x_{0},k^{2}_{\perp},k^{\prime
2}_{\perp})\sim k^{\prime 2}_{\perp}$ of Eq. (2.1) at four different values of
$k^{2}_{T}$ using two inputs (2.2) and (2.3) in Fig. 4. For comparison, a
similar figure is also drawn but evolved by the linear BFKL equation with a
normal BFKL-input [14]

$F_{BFKL}(x_{0},k_{\perp}^{2})=\beta\sqrt{k^{2}_{\perp}}\frac{x_{0}^{-\lambda_{BFKL}}}{\sqrt{\ln(1/x_{0})}}\exp\left(-\frac{\ln^{2}(k^{2}_{\perp}/k^{2}_{s})}{2\lambda"\ln(1/x_{0})}\right),$
$None$

where $\lambda_{BFKL}=12\overline{\alpha}_{s}/(\pi\ln
2),\lambda"=32\overline{\alpha}_{s},\beta=0.01$ and $k^{2}_{s}=1~{}GeV^{2}$.
One can find a common feature of the results: having jumps at $k^{\prime
2}_{\perp}=k^{2}_{\perp}$. It is the direct result of the Lipatov-singularity.
After integral, these jumps generate a series of weak perturbations to the
gluon distribution $F(x,k^{2}_{\perp})$. The above perturbations are
independent in the linear BFKL equation and their effects are negligibly
small. In this case the solutions are almost smooth curves in the
$k_{\perp}^{2}$-dependence. However, the special nonlinear terms in Eq. (2.1)
may occur the coupling among random perturbations and generates chaos. A
standard criterion of chaos is that the system has the positive Lyapunov
exponent, which indicates a strong sensitivity to small changes in the initial
conditions. The Lyapunov exponents of the gluon distributions in Eq. (2.1)
with the inputs Eqs. (2.2) and (2.3) are presented in Fig. 5. The positive
Lyapunov exponent of the solutions of Eq. (2.1) is a strong evidence for the
chaotic solution.

Figure 4: The transverse momentum dependence of the integrated function $R$ in
different evolution equations, where we use the GBW- and KL-inputs in Eq.
(2.1) and the BFKL-input in the BFKL equation. One can find the jump
structure, which will arise the chaotic solutions in Eq. (2.1).

Figure 5: The Lyapunov exponents of the chaotic solutions in Eq. (2.1) using
the GBW and KL inputs. The standard program of the Lyapunov exponents see Ref.
[5].

The critical momentum $k^{2}_{c}$ relates to the strength of the nonlinear
terms of the QCD evolution equation. In order to verify this property, we
multiply the nonlinear terms in Eq. (2.1) by a parameter $A$. Then we
calculate the GC-critical momentum $k_{c}$ using different values of $A$. The
results in Table 1 show that the value of the critical momentum $k_{c}$ in a
certain range is determined by the structure of the evolution equations. Using
this result we can predict the nuclear target dependence of the GC effect in
the $p-A$ or $A-A$ collisions since the nonlinear terms of Eq. (2.1) are
$A$-dependent. $A$ is the mass number of a nucleus.

Table 1: The relations of the GC-critic momentum $k_{c}$ with the nonlinear
corrections.

Figure 6: The Lyapunov exponents of the BK and BFKL equations using the BFKL-
input.

For comparison, we discuss the BK equation [10], which is generally considered
as the typical nonlinear corrections to the BFKL equation at the leading order
approximation. The BK equation can be written in the full momentum space as
[15,16]

$-x\frac{\partial F(x,k_{\perp})}{\partial x}$
$=\frac{\alpha_{s}N_{c}}{2\pi^{2}}\int
d^{2}k_{\perp}^{\prime}\frac{k_{\perp}^{2}}{(k_{\perp}-k_{\perp}^{\prime})^{2}k_{\perp}^{2}}2F(x,k_{\perp}^{\prime})-\frac{\alpha_{s}N_{c}}{2\pi^{2}}F(x,k_{\perp})\int
d^{2}k^{\prime
2}_{\perp}\frac{k_{\perp}^{2}}{(k_{\perp}-k^{\prime}_{\perp})^{2}k_{\perp}^{2}}$
$-\frac{18\alpha_{s}^{2}}{\pi
R^{2}_{N}}\frac{N_{c}^{2}}{N_{c}^{2}-1}\frac{1}{k_{\perp}^{2}}F^{2}(x,k^{2}_{\perp}),$
$None$

or taking the cylindrically symmetric approximation

$-x\frac{\partial F(x,k_{\perp}^{2})}{\partial x}$
$=\frac{3\alpha_{s}k_{\perp}^{2}}{\pi}\int_{k^{\prime
2}\perp}^{\infty}\frac{dk_{\perp}^{\prime 2}}{k^{\prime
2}_{\perp}}\left\\{\frac{F(x,k^{\prime
2}_{\perp})-F(x,k_{\perp}^{2})}{|k^{\prime
2}_{\perp}-k_{\perp}^{2}|}+\frac{F(x,k_{\perp}^{2})}{\sqrt{k_{\perp}^{4}+4k^{\prime
4}_{\perp}}}\right\\}-\frac{81}{4}\frac{\alpha_{s}^{2}}{\pi
R^{2}_{N}}\frac{1}{k_{\perp}^{2}}F^{2}(x,k_{\perp}^{2}).$ $None$

We find that both the BK and BFKL equations don’t have the chaotic solutions
since their Lyapunov exponents are negative (see Fig. 6).

With only the chaotic effect it cannot produce the GC solutions in Eq. (2.1).
The TOPT regularized nonlinear kernel plays its second important role for
converting the chaotic vibration into the strong shadowing and antishadowing
effects, the later eventually forms the GC. Let us illustrate them. In Eq.
(2.1) we have

$\left[\frac{k_{\perp}^{2}F^{2}(x,k^{\prime 2}_{\perp})}{k^{\prime
2}_{\perp}|k^{\prime 2}_{\perp}-k_{\perp}^{2}|}-\frac{k^{\prime
2}_{\perp}F^{2}(x,k_{\perp}^{2})}{k^{\prime 2}_{\perp}|k^{\prime
2}_{\perp}-k_{\perp}^{2}|}\right]_{k^{\prime 2}_{\perp}\sim
k_{\perp}^{2}}\sim\frac{d}{dk^{\prime
2}_{\perp}}\left[\frac{k_{\perp}^{2}}{k^{\prime 2}_{\perp}}F^{2}(x,k^{\prime
2}_{\perp})\right]_{k^{\prime 2}_{\perp}\sim k_{\perp}^{2}}.$ $None$

Once chaos is produced, the fast oscillations of the gluon density will
generate both the negative and positive nonlinear corrections to the increment
$\Delta F(x,k^{2}_{\perp})$ through derivative operations. Usually we call the
negative or positive corrections to the nonlinear evolution equations as the
shadowing or antishadowing effects. The former gradually suppresses the
grownup of the gluon density since $F^{2}$ is also suppressed, while the later
is a positive feedback process and it increases rapidly due to it is
proportional to the increasing $F^{2}$. Let us consider Fig. 7, which presents
the gluon distributions when the nonlinear terms in Eq. (2.1) take positive
absolute values (curve 1), negative absolute values (curve 2) and zero value
(dashed curve), they corresponding to the pure antishadowing effect, the pure
shadowing effect and the linear BFKL evolution, respectively. One can find
that the pure shadowing effect gradually suppresses the increasing gluon
distribution, while the pure antishadowing leads to a fast divergence of the
gluon density.

Figure 7: The net antishadowing effect (curve 1) and net shadowing effect
(curve 2). Dashed curve is a solution of the linear BFKL equation. A pair of
curves 3 and 4 are the solutions of Eq (2.1), where the shadowing and
antishadowing are correlated by momentum conversion and they lead to the GC.

Note that the above examples show the isolate shadowing and antishadowing
effects. In fact, the shadowing and antishadowing are correlated due to the
local momentum conservation. Although Eq. (2.1) works at the small-$x$ range
and it does not describe the total momentum conservation, the local momentum
conservation is still valid at every QCD vertex. The TOPT cutting rule in the
derivation of Eq. (2.1) emphasises to sum all possible Feynman diagrams
including real and virtual processes at the same order approximation, as they
naturally hold the local momentum conservation [5]. It leads to the
correlation between the negative shadowing effect and the positive
antishadowing effect, since these two effects origin from virtual and real
diagrams, respectively. Thus, a strong antishadowing effect in Eq. (2.1)
$must$ be accompanied by the disappearance of the shadowed gluons due to the
local momentum conservation as shown by a pair of curves 3 and 4 in Fig. 7. A
similar example for the correlation of shadowing and antishadowing appears in
a modified DGLAP equation [4], where the antishadowing effect is weaker than
the shadowing effect since the integral range is restricted and the net
shadowing effect dominates the process. The restriction of the resulting
antishadowing effect has been confirmed by the observed EMC effect [17]. Thus,
we do not need to design a mysterious mechanism to condense the gluons, the
momentum conservation plus the net antishadowing effect in Eq. (2.1) may
realize the GC.

Therefore, the GC is the combined effect of chaos and antishadowing in Eq.
(2.1), which forms a GC-chain: the potential perturbation arises chaotic
oscillations in the BFKL dynamics $\rightarrow$generating the positive
antishadowing corrections$\rightarrow$converting the shadowed gluons to the
gluons with the critical momentum $(x_{c},k_{c})$. Therefore, the gluon
condensation is a natural consequence of random evolution of gluons at very
high energy hadronic processes.

The unintegrated gluon distribution $F(x,k_{\perp}^{2})$ of a bound nucleon in
the nucleus $A$ satisfies the following evolution equation

$-x\frac{\partial F(x,k_{\perp}^{2})}{\partial x}$
$\simeq\frac{3\alpha_{s}k_{\perp}^{2}}{\pi}\int_{k^{2}_{\perp,min}}^{\infty}\frac{dk^{\prime
2}_{\perp}}{k^{\prime 2}_{\perp}}\left\\{\frac{F(x,k^{\prime
2}_{\perp})-F(x,k_{\perp}^{2})}{|k^{\prime
2}_{\perp}-k_{\perp}^{2}|}+\frac{F(x,k_{\perp}^{2})}{\sqrt{k_{\perp}^{4}+4k^{\prime
4}_{\perp}}}\right\\}$ $-\frac{81}{16}\frac{\alpha_{s}^{2}A^{1/3}}{\pi
R^{2}_{N}}\int_{k^{2}_{\perp,min}}^{\infty}\frac{dk^{\prime
2}_{\perp}}{k^{\prime 2}_{\perp}}\left\\{\frac{k_{\perp}^{2}F^{2}(x,k^{\prime
2}_{\perp})-k^{\prime 2}_{\perp}F^{2}(x,k_{\perp}^{2})}{k^{\prime
2}_{\perp}|k^{\prime
2}_{\perp}-k_{\perp}^{2}|}+\frac{F^{2}(x,k_{\perp}^{2})}{\sqrt{k_{\perp}^{4}+4k^{\prime
4}_{\perp}}}\right\\}.$ $None$

We take the K-L model in a nucleus $A$ as the input of Eq. (2.8) at $x_{A}$

$F(x_{A},k_{\perp}^{2})=\left\\{\begin{array}[]{ll}f_{0}k^{2}_{\perp}~{}~{}~{}~{}if~{}k^{2}_{\perp}\leq
Q_{A,s}^{2}\\\ \\\
f_{0}Q^{2}_{A,s}~{}~{}~{}~{}if~{}k^{2}_{\perp}>Q_{A,s}^{2}\\\
\end{array}\right.,$ $None$

where $Q^{2}_{A,s}=Q^{2}_{s}A^{1/3}$, $Q_{s}=1~{}GeV$ is the saturation scale
of a free nucleon [18].

Figure 8: Predicted critic momentum $x_{c}$ (solid curve) in different nuclear
target. The point curve is $x_{A}=1.9\times 10^{-8}A$. The dashed curve is the
possible modified $x_{c}$ due to Eq. (3.16).  Figure 9: Predicted critic
momentum $k^{2}_{c}$ in different nuclear targets.

Generally, the bigger the nucleus, the earlier Eq. (2.8) works. The GC-effect
originates from the chaotic solution of Eq. (2.8). Once a most robust chaos is
created, it will dominate the whole process of the condensation, no matter the
event occurs whether in the longitudinal $(\sim A^{1/3})$ or horizontal $(\sim
A^{2/3})$ area of the nucleus. Therefore, the starting point of evolution in
Eq.(2.8) is proportional to the volume of a nucleus rather than its
longitudinal scale, i.e., we take $x_{A}=x_{p}A$, $x_{p}$ is the starting
point of the Eq. (2.8) in a free nucleon. We use the input Eq. (2.9) and
$x_{p}\equiv 1.9\times 10^{-8}$ to calculate the relation $(x_{c},k_{c})\sim
A$. The results are drawn in Figs. 8 and 9. We will use them to discuss the GC
effect in hadron collisions in the following sections.

##  3 The GC-effect in the planning hadron collides beyond the LHC energies

We abstract some formulas for the applications of the GC effect in high energy
hadron-hadron collisions and the detailed derivations can be found in works
[6-9]. According to QCD, the yield of secondary particles at the high energy
$p-p$ collision relates to the number of excited gluons, which participate in
the multi-interactions. The number of pion will rapidly grow when a lot of
gluons enter the interaction range due to the GC effect. Without concrete
calculations, one can image that this will form an excess phenomenon in the
cosmic-ray spectra. We focus on the central rapidity region, where the multi-
hadrons (mainly pions) are produced due to the excited gluons. This process
contains two steps

$p+p\rightarrow gluons~{}(or~{}gluons~{}and~{}quark-
antiquark~{}pairs)\rightarrow n_{\pi},$ $None$

$n_{\pi}$ is the multiplicity of the secondary pions.

Let us deeply understand the hadronization processes in the QCD-view. Based on
the Dyson-Schwinger (DS) equation, the recent research by Roberts, et al.,
shows that the effective quark mass in hadron might be dynamically generated
due to the gluon nonlinearities [19]. In this mechanism the nonperturbative
gluon propagator is constructed by a lot of partons (i.e., gluons and sea
quarks, note that the later is originated from gluons) in the hadron infinite
momentum frame. A massless quark has got a small current quark mass through
the Higgs mechanism and it further acquires the effective mass through the
nonperturbative gluon propagator in the QCD self-energy diagram (Fig. 10).
These ”dressed” quarks are called as the constituent quarks. Now we compare
the gluon distributions of the gluon condensate and the glasma. At the same
collision energy $\sqrt{s}$, the gluon condensate has abundant gluons to
greatly generate the number of the constituent quarks and increase the number
$n_{\pi}$, i.e., $n_{\pi,GC}\gg n_{\pi,Glasma}$. On the other hand, we have
$1/2\sqrt{s}\geq n_{\pi}m_{\pi}$ due to the restriction of the energy
conservation. As a limit, we image that almost all available energy of the
collision at the center-of-mass system is used to create pions.

Figure 10: The self-energy diagram of obtaining effective constituent quark
mass. Works [19] use it to explain the formation of pion mass. Empty circle is
the DS vertex and black spots indicate the nonperturbative propagators, which
contain the multi-parton components.

Combining the relativistic invariant and energy conservation, one straightly
writes [6,7]

$(2m_{p}^{2}+2E_{p}m_{p})^{1/2}=E^{*}_{p1}+E^{*}_{p2}+n_{\pi}m_{\pi},$ $None$
$E_{p}+m_{p}=m_{p}\gamma_{1}+m_{p}\gamma_{2}+n_{\pi}m_{\pi}\gamma,$ $None$

where $E^{*}_{pi}$ is the energy of leading proton at the center-of-mass
system, $\gamma_{i}$ is the corresponding Lorentz factor. We can easily get
the solutions of $n_{\pi}(E_{p},E_{\pi})$ in the $p-p$ collision

$\ln n_{\pi}=0.5\ln E_{p}+a,~{}~{}\ln n_{\pi}=\ln E_{\pi}+b,$ $None$

where $E_{\pi}\in[E_{\pi}^{GC},E_{\pi}^{max}]$. The parameters

$a\equiv 0.5\ln(2m_{p})-\ln m_{\pi}+\ln 1/2,$ $None$

and

$b\equiv\ln(2m_{p})-2\ln m_{\pi}+\ln 1/2.$ $None$

These equations give the one-to-one relation among $n_{\pi}$, $E_{p}$ and
$E_{\pi}^{GC}$. Equation (3.4) gives the following special relation

$E_{p}=e^{-2(a-b)}E_{\pi}^{2}=\frac{2m_{p}}{m_{\pi}^{2}}E_{\pi}^{2},$ $None$

The upper limit $E_{\pi}^{max}=14(E_{\pi}^{GC})^{2}$ (in the GeV-unit) if the
charged particles can be accelerated continuously. However the actual
$E_{\pi}$ cannot reach such high limit because the restriction of acceleration
mechanism. Therefore, we use a smaller cut-energy $E_{\pi}^{cut}$ to replace
$E_{\pi}^{max}$. $E_{\pi}^{cut}$ relates to the accelerator mechanism and
$E_{\pi}^{cut}\gg E_{\pi}^{GC}$. According to the numerical simulation,
$E_{\pi}^{cut}$ is required at least one order of magnitude greater than
$E_{\pi}^{GC}$, i.e., $E_{\pi}^{cut}\geq 10E_{\pi}^{GC}$. $E_{\pi}^{GC}$ and
$(x_{c},k_{c})$ have the following relation [6,7]

$E_{\pi}^{GC}=\exp\left(0.5\ln\frac{k^{2}_{c}}{2m_{p}x_{c}}+a-b\right)=\frac{m_{\pi}}{2m_{p}}\frac{k_{c}}{\sqrt{x_{c}}}.$
$None$

Using this equation, one can predict the GC-threshold in Fig. 11. We find that
the distribution of $E_{\pi}^{GC}$ can be roughly divided into three ranges:
(i) $E_{\pi}^{GC}\approx 100~{}GeV$ for intermediate and heavy nuclei
$(A>100)$, (ii) $E_{\pi}^{GC}$ quickly increases for light nuclei ($A<20$) and
(iii) the area between them $(20<A<100)$.

Figure 11: The predicted GC-threshold $E_{\pi}^{GC}$ in the $p-A$ (or $A-A$)
collisions (solid curve), which are obtained by Eq. (3.8) and the data of
Figs. 8 and 9. The dashed curve is the possible modified GC-threshold
$E_{\pi}^{GC}$ due to the corrections of Eq. (3.8).

The contributions of proton and nucleus in the $p-A$ collisions dominate the
rapidity distribution on the two sides of rapidity space, respectively. Note
that the GC effect begins work when the gluons with $x_{A}$ takes the first to
participate the $A^{\prime}-A$ collisions if $A>A^{\prime}$, and enhances the
total cross section of the collisions, therefore, we have
$E_{\pi}^{GC}(p-A)\simeq E_{\pi}^{GC}(A-A)$, although the GC-signal strength
of the former is half weaker than the latter.

Table 2: The predicted center-of-mass energy $\sqrt{s_{pA}^{cut}}$ appearing
the GC effect and the relating quantities in different hadronic collisions (in
the GeV-unit).

As we have known, the GC effect causes a big excess in the cross section of
hadron collisions at center-of-mass energy $\sqrt{s^{GC}}$. In Tab. 2 we give
these thresholds using Eqs. (3.7), (3.8) and the modified curve is shown in
Fig. 11. For example, we take $E_{\pi}^{GC}(p-p)=20$ TeV, thus,
$E_{pp}^{GC}=100\times(2\times 10^{4})^{2}$ GeV$=4\times 10^{10}$ GeV and
$\sqrt{s_{pp}^{GC}}=\sqrt{2m_{p}E_{pp}^{GC}}=3\times 10^{2}$ TeV. To observe
the GC effect in the $p-p$ collision, one needs $E_{\pi}^{cut}\geq
10E_{\pi}^{GC}$, which requires $\sqrt{s_{pp}^{cut}}\geq 3\times 10^{3}$ TeV.
The maximum energies of the $p-p$ collision at the LHC is $\sqrt{s_{pp}}=13$
TeV $\ll 3\times 10^{3}$ TeV. Obviously, we can not record any GC-signals from
the $p-p$ collision in the recent LHC.

Now we consider the the $p-Pb$ and $Pb-Pb$ collisions at the LHC, their
maximum energies are 8.16 TeV and 5.02 TeV, respectively. Referring to Fig.
11, we take $E_{\pi}^{GC}(Pb-Pb)=100$ GeV for them. The results are presented
in Tab. 2. The energy region where the GC-signal appears is
$\sqrt{s_{pPb}^{cut}}>10$ TeV. Although the energies of these collisions in
the LHC are already close to the GC-thresholds, it is still necessary to
further increase the collision energy.

We emphasize that the GC effect may efficiently convert the kinetic energy of
the parent protons into a large number of secondary particles. In fact, the
multiplicity of mini gluon-jets in the hadronic collisions with the GC effect
is about $10^{3}-10^{4}$ times larger than that without the GC model [5].
Besides, there is a highest gluon-pion conversion rate in the GC model. Thus,
about half of the kinetic energy of protons are converted to the large numbers
of photons with energy $E_{\gamma}=m_{\pi}/2$ in the center of mass (CM) frame
in the $Pb-Pb$ collision. Such monochrome gamma-rays have extra high strength
in a narrow space, and may damage the detectors in the laboratory.

We noticed that the Auger collaboration indirectly used the cosmic ray data at
the top of the atmosphere and found that $pA$ cross section at $\sqrt{s}\sim
100~{}TeV$ is the normal value $\sim 567~{}mb$ with no big increment [21].
According to Fig. 12 we guess that the GC-threshold
$E_{\pi}^{GC}(p-atm)>1~{}TeV$ for light nuclei at the top of atmosphere. which
implies $\sqrt{s_{p-atm}^{GC}}>10~{}TeV$, and
$\sqrt{s_{p-atm}^{cut}}>100~{}TeV$. This result means that the observed energy
by Auger collaboration at $\sqrt{s}=100~{}TeV$ is close to the lower-limit of
the GC-effect range if the above mentioned indirect estimations are correct.

The more direct method of probing the GC effect in the laboratory is the
Electron Ion Collider (EIC) [22] and the Large Hadron electron Collider (LHeC)
[23]. The designed collision energy of upcoming EIC(US) is
$\sqrt{s_{eA}}\simeq 140~{}GeV$, which is in the CGC range
($x>10^{-5},Q^{2}>1~{}GeV^{2})$. According to our estimation in Fig.8, the
range of the GC effect is $(x_{c}\simeq 10^{-6},k^{2}_{c}\simeq 5~{}GeV^{2})$,
which exceeds the range of the EIC(US). However, the center-of-mass energy of
the proposed Large Hadron electron Collider (LHeC) $\sqrt{s_{eA}}\sim 1$ TeV
will be able to cover a very low $x$-range: $x\sim 10^{-6}$ at
$Q^{2}>1GeV^{2}$ in $eA$ collisions, where we may record the GC-signal.

Since the huge number of gluons are condensed in a critical momentum
$(x_{c},k_{c})$, it should greatly increase the hadron cross section and
release strong gamma-rays. We warn that further increase of the hadron
collision energies in the next LHC plans may lead to unexpected intense gamma-
rays in the accelerator, they look like the artificial mini GRBs and may
damage the detectors

##  4 Discussions

(1) The evolution of the BFKL dynamics may become nonperturbative near the
singular range mentioned above. Fortunately, both the lattice simulations and
the nonperturbative dynamics of the QCD show that the effective strong
coupling constant is restricted by $\alpha_{s}/\pi\leq B$ ($B$ is a constant)
[24].

(2) The equation (2.1) is based on the leading order approximation, where the
higher order corrections are neglected. An important question is: will the
chaos effects disappear in the evolution equation after considering higher
order corrections? Works [5] have discussed this problem and we abstract some
conclusions as follows. The GC effect origins from the singular nonlinear
evolution kernel and local momentum conservation, they are general structure
at the leading $\ln(1/x)$-resummation. The multi-singular structure from
higher order corrections should be cancelled by the contributions of the
virtual Feynman diagrams at a same order level. The resulting nonlinear
evolution equation is still keeping the conditions of existing GC. Although
the higher order QCD corrections may change the value of $k_{c}$, however, the
simple form of the GC solution contains only a few parameters $(x_{c},k_{c})$
and they can be determined by the experimental data.

(3) The BK equation is generally considered as a typical nonlinear correction
to the BFKL equation at the $LL(1/x)$ approximation and it is the most widely
used small $x$ evolution in the community. Why the BK equation doesn’t have
the chaotic solution? We try to answer this question. The BK equation is
usually written by using the scattering amplitude $N(x_{\perp},x)$ in the
transverse coordinator space

$-x\frac{\partial N(x_{\perp},x)}{\partial x}$
$=\frac{\alpha_{s}N_{c}}{2\pi^{2}}\int
d^{2}x^{\prime}_{\perp}\frac{x_{\perp}^{2}}{x_{\perp}^{2}(x_{\perp}-x^{\prime}_{\perp})^{2}}\left[N(x^{\prime}_{\perp},x)+N(x_{\perp}-x^{\prime}_{\perp},x)-N(x_{\perp},x)\right.$
$-N(x^{\prime}_{\perp},x)N(x_{\perp}-x^{\prime}_{\perp},x)].$ $None$

The linear part corresponds to the BFKL evolution equation, while the
nonlinear evolution kernel is regularized by the connecting amplitude
$N(x^{\prime}_{\perp},x)N(x_{\perp}-x^{\prime}_{\perp},x)$ in the dipole
splitting model to avoid the Lipatov-singularity. In order to illustrate the
physical meaning of this regularization method, we use

$N(x,k_{\perp})=\int\frac{d^{2}x_{\perp}}{2\pi}\exp(-ik_{\perp}\cdot
x_{\perp})\frac{N(x_{\perp},x)}{x_{\perp}^{2}},$ $None$

to rewrite the BK equation in the transverse momentum space

$-x\frac{\partial N(x,k_{\perp}^{2})}{\partial x}$
$=\frac{3\alpha_{s}}{\pi}\int_{k^{\prime 2}\perp}^{\infty}\frac{dk^{\prime
2}_{\perp}}{k^{\prime 2}_{\perp}}\left\\{\frac{k^{\prime
2}_{\perp}N(x,k^{\prime
2}_{\perp})-k^{2}_{\perp}N(x,k_{\perp}^{2})}{|k^{\prime
2}_{\perp}-k_{\perp}^{2}|}+\frac{k^{2}_{\perp}N(x,k_{\perp}^{2})}{\sqrt{k_{\perp}^{4}+4k^{\prime
4}_{\perp}}}\right\\}-\frac{3\alpha_{s}}{\pi}N^{2}(x,k_{\perp}^{2}).$ $None$

If we define

$N(x,k_{\perp})\equiv\frac{27\alpha_{s}}{16k_{\perp}^{2}R^{2}_{N}}F(x,k_{\perp}),$
$None$

one can obtain Eq. (2.6) for the unintegrated gluon distribution
$F(x,k_{\perp})$.

Let us to expose a relation between the ZSR and BK equations. The Lipatov-
singularity in the nonlinear evolution kernel is arisen by the crossed
diagrams of two different amplitudes in Fig. 1d and they imply the random
evolution in the transverse momentum space. A simple, but unreasonable way for
avoiding the singularity is to remove these crossed diagrams. In this case,
one can find that Fig. 1d reduces to Fig. 1c. The latter is the contributions
of the gluon fusion to the DGLAP evolution equation at the
$DLL$-approximation, i.e.,

$Q^{2}\frac{\partial G(x,Q^{2})}{\partial
Q^{2}}=-\frac{36\alpha_{s}^{2}}{8Q^{2}R^{2}_{N}}\frac{N_{c}^{2}}{N_{c}^{2}-1}\int\frac{dx^{\prime}}{x^{\prime}}G^{2}(x^{\prime},Q^{2}),$
$None$

where the transverse momenta of gluons are strongly ordered [3,4]. Note that
the relation between gluon distribution $G(x,Q^{2})$ and unintegrated gluon
distribution $F(x,k^{2}_{\perp})$ is

$G(x,Q^{2})\equiv\int^{Q^{2}}_{k^{2}_{\perp,min}}\frac{dk^{2}_{\perp}}{k^{2}_{\perp}}F(x,k^{2}_{\perp}).$
$None$

We use $F(x,k^{2}_{\perp})$ to replace $G(x,Q^{2})$. From Eq.(5.5) we have

$\Delta
G(x,Q^{2})=-18\alpha_{s}^{2}\frac{N^{2}_{c}}{N_{c}^{2}-1}\int^{Q^{2}}_{Q^{2}_{min}}\frac{dk^{2}_{\perp}}{k^{2}_{\perp}}\int\frac{dx^{\prime}}{x^{\prime}}F^{(2)}(x^{\prime},k^{2}_{\perp}),$
$None$

or

$\Delta F(x,k^{2}_{\perp})=\left.Q^{2}\frac{\partial\Delta
G(x,Q^{2})}{\partial
Q^{2}}\right|_{Q^{2}=k^{2}_{\perp}}=-\frac{18\alpha_{s}^{2}}{\pi
k^{2}_{\perp}R^{2}_{N}}\frac{N^{2}_{c}}{N^{2}_{c}-1}\int\frac{dx^{\prime}}{x^{\prime}}F^{2}(x^{\prime},k^{2}_{\perp}),$
$None$

and

$-x\frac{\partial F(x,k^{2}_{\perp})}{\partial x}=-\frac{18\alpha_{s}^{2}}{\pi
k^{2}_{\perp}R^{2}_{N}}\frac{N^{2}_{c}}{N^{2}_{c}-1}F^{2}(x,k^{2}_{\perp}),$
$None$

Where we define

$F^{(2)}(x,k^{2}_{\perp})\equiv\frac{1}{\pi^{2}R_{N}^{2}}F^{2}(x,k^{2}_{\perp}).$
$None$

Combining Eq. (4.9) with the linear BFKL equation, we can obtain the BK
equation (2.6).

This example exposes a fact: the nonlinear part of the BK equation neglects
the contributions of the random evolution of gluons on the transverse space.
Therefore, the BK equation losses the GC-source since it avoids the Lipatov-
singularity in the nonlinear kernel by using a dipole model. Thus, the BK
equation and its general form [13] have not the GC solution.

Interestingly, we show the following evolution of the QCD evolution equations
with increasing gluon densities:

$The~{}DGLAP~{}equation~{}(|a|^{2})~{}[Violation~{}of~{}unitarity]$
$\rightarrow~{}The~{}BFKL~{}equation~{}(|b|^{2})~{}[Violation~{}of~{}unitarity]$
$\rightarrow~{}The~{}GLR-MQ-
ZRS~{}equation~{}(|a|^{2}+|c|^{2})~{}[Shadowing~{}and~{}antishadowing]$
$\rightarrow~{}The~{}BKequation~{}(|a|^{2}+|c|^{2})~{}[Saturation]$
$\rightarrow~{}The~{}ZSR~{}equation~{}(|b|^{2}+|d|^{2})~{}[Gluon~{}condensation],$

where the brackets mark the amplitude structure using Fig. 1, and square
brackets contain the predicted effects. This picture reflects the self-
consistence of the evolution equations including Eq. (2.1).

In summary, we present our new understanding of the GC effect in a nonlinear
QCD evolution equation. Through the numerical solutions of this evolution
equation, we find that the random evolution of gluons in the traverse momentum
space generates the Lipatov-singularity both in the linear and nonlinear
terms. After regularization of these singularities according to the standard
quantum field theory, the equation emerges a special nonlinear structure,
which causes the chaotic oscillations and then generates the strong net
antishadowing effect. Due to the restriction of the local momentum
conservation, the antishadowing effect gathers the shadowed gluons to a state
at a critical momentum, and forms the gluon condensate. We estimate the
parameters in the GC solution, which predict that the hadron collisions $p-Pb$
and $Pb-Pb$ in the LHC are already close to the energy region of the GC
effect. Since the huge number of gluons are condensed in a critical momentum,
it should greatly increase the hadron cross section and release strong gamma-
rays. We warn that further increase of the hadron collision energies in the
next LHC plans may lead to unexpectedly intense gamma-rays in the accelerator,
they look like the artificial mini GRBs and may damage the detectors.

Acknowledgments: This work is supported by the National Natural Science of
China (No.11851303). Q.H. Chen acknowledges support from the National Natural
Science of China (No.12147208).

## References

  * [1] V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438;
Yu.L. Dokshitzer, Sov. Phys. JETP. 46 (1977) 641;
G. Altarelli, G. Parisi, Nucl. Phys. B 126 (1977) 298.

  * [2] L. N, Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338;
V. S. Fadin, E.A. Kuraev, L. N. Lipatov, Phys. Lett. B60 (1975) 50;
E. A. Kuraev, L.N. Lipatov, V. S. Fadin, Sov. Phys. JETP. 44 (1976) 443;
E. A. Kuraev, L.N. Lipatov, V. S. Fadin, Sov. Phys. JETP. 45 (1977) 199;
I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822.

  * [3] L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. 100 (1983) 1;
A.H. Mueller, J.W. Qiu, Nucl. Phys. B 268 (1986) 427.

  * [4] W. Zhu, Nucl. Phys. B 551(1999) 245;
W. Zhu, J.H. Ruan, Nucl. Phys. B 559 (1999) 378;
W. Zhu, Z.Q. Shen, HEP. $\&$ NP. 29 (2005) 109.

  * [5] W. Zhu, Z.Q. Shen, J.H. Ruan, Chin. Phys. Lett. 25 (2008) 3605;
W. Zhu, Z.Q. Shen, J.H. Ruan, Nucl. Phys. B 911 (2016) 1;
W. Zhu, J.S. Lan, Nucl. Phys. B916 (2017) 647.

  * [6] W. Zhu, J.S. Lan, J.H. Ruan, Int. J. Mod. Physics. E 27 (2018) 1850073.

  * [7] F. Feng, J.H. Ruan, F. Wang, W. Zhu, Astrophys. J. 868 (2018) 2;
W. Zhu, P. Liu, J.H. Ruan, L. Feng, F. Wang, Astrophys. J. 896 (2020) 106;
W. Zhu, Z.C. Zheng, P. Liu, L.H. Wan, J.H. Ruan, F. Wang, JCAP. 01 (2021) 038;
J.H. Ruan, Z.C. Zheng, W. Zhu, JCAP. 08 (2021) 065.

  * [8] W. Zhu, P. Liu, J.H. Ruan, F. Wang, Astrophys. J. 889 (2020) 127.
  * [9] W. Zhu, P. Liu, J.H. Ruan, R.Q. Wang, F. Wang, JCAP, 09 (2020) 011.
  * [10] I. Balitsky, Nucl. Phys., B 463 (1996) 99;
Yu. Kovchegov, Phys. Rev. D 60 (1999) 034008;
Yu. Kovchegov, Phys. Rev. D 61 (2000) 074018.

  * [11] K. Golec-Biernat, M. Wüsthoff, Phys. Rev. D 59 (1998) 014017;
K. Golec-Biernat, M. Wüsthoff, Phys. Rev. D 59 (1998) 014017.

  * [12] D. Kharzeev, E. Levin, Phys. Lett. B 523 (2001) 79.
  * [13] L.D. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 2233;
L.D. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 3352;
L.D. McLerran, R. Venugopalan, Phys. Rev. D 50 (1994) 2225.

  * [14] A.J. Askew, J. Kwiecinki, A.D. Martin, P.J. sutton, Phys. Rev. D 49 (1994) 4402.
  * [15] C. Marquet, G. Soyez, Nucl. Phys. A 760 (2005) 208.
  * [16] J.H. Ruan, Z.Q. Shen, J.F. Yang, W. Zhu, Nucl. Phys. B 760 (2007) 128.
  * [17] M. Aaneodo, Phys. Rep. 240 (1994) 301.
  * [18] Y.V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, (2012) Cambridge university press.
  * [19] V. Mathieu, PoS FACESQCD2010, 002 (2010) arXiv:hep=ph/1102.3875;
L. Chang, C. Mezrag, H. Moutarde, C.D. Roberts, J. Rodrguez-Quintero, P.C.
Tandy, Phys. Lett. B 737 (2014) 23;
C. D. Roberts, Insights into the Origin of Mass, Summary of a plenary
presentation at ”INPC 2019 - 27th International Nuclear Physics Conference”,
Glasgow, UK, 2019, arXiv:hep-ph/1909.12832;
J. R. Quintero, L. Chang, K. Raya, C. D. Roberts, INPC 2019 proceedings,
arXiv:hep-ph/1909.13802.

  * [20] F.A. Aharonian, Very high energy cosmic gamma radiation: a crucial window on the extreme universe, World Scientific, 2004.
  * [21] R. U. Abbasi, M. Abe, T. Abu-Zayyad, Phys. Rev. D 92 (2015) 032007.
  * [22] R. Abdul Khalek, et al., Science Requirements and Detector Concepts for the Electron-Ion Collider, (EIC Yellow Report), (2021) BNL-220990-2021-FORE, JLAB-PHY-21-3198, LA-UR-21-20953.
  * [23] P. Agostini1, et al., (LHeC and FCC-he Study Group) arXiv:hep-ph/2007.14491.
  * [24] C.D. Roberts, arXiv:hep-ph/1909.12832.