1. Introduction
The proton, a fundamental building block of the universe, is composed of quarks and gluons. The distribution of gluons has long posed a challenge in particle physics. Quantum chromodynamics (QCD) evolution equations predict that, at high energies, gluon densities become extremely large (See
Appendix A). Under these conditions, nonlinear effects dominate, giving rise to phenomena such as the color glass condensate (CGC) [
1]. However, CGC is not a true physical condensation. Further theoretical advancements introduced the Zhu–Shen–Ruan (ZSR) equation [
2]. This equation indicates that, at high energies, the evolution of gluon distributions within nucleons exhibits chaotic behaviour. Such chaos leads to significant shadowing and antishadowing effects [
2]. Consequently, a large number of gluons accumulate within a narrow phase space defined by a critical momentum
. This phenomenon is known as gluon condensation (GC) [
3]. Although GC is consistent with the frameworks of the Standard Model and nonlinear science, its occurrence was unexpected by the scientific community. As a result, identifying experimental evidence for GC has become our primary research objective.
The number of pions is expected to increase markedly when the dense peak of the gluon distribution in the proton enters the proton–proton (
) interaction region. Interestingly, by incorporating the GC distribution (see
Figure 1) into the general formula for proton collision cross sections, we can unify previously observed anomalies. These include features in the cosmic gamma-ray spectrum on an astronomical scale and in heavy-ion collisions at the Large Hadron Collider (LHC). This unification points to a previously unrecognized internal structure within the proton. We will detail this approach in
Section 2, where we present the relevant formulae used in this work. For further information, please refer to the cited literature.
In many astrophysical processes, protons can be accelerated to extremely high energies. When they collide with other protons or nuclei, they emit gamma rays, which may reveal information about the proton’s interior. The broken power law (BPL) is a simple broken line in double logarithmic coordinates, often observed in high-energy gamma-ray spectra. There are many speculations and uncertainties about its formation [
4]. We will demonstrate that the GC model can provide a simple and unique explanation in
Section 3.
High-energy heavy-ion collisions have long sought the Bose–Einstein condensation (BEC) of pions. A key feature is the large number of pions occupying low-momentum (
) ground states. For instance, Begun and Florkowski used the BEC model to fit LHC (ALICE) data, finding pion coalescence to be 0–2% in central collisions and rising to 19% in peripheral collisions [
5]. This contrasts with the traditional view that the central region is more conducive to BEC formation. Consequently, the existence of BEC pions remains widely debated. In
Section 4, we show that the GC model can produce similar results to Begun and Florkowski without the strict BEC condition.
Thus, on the deepest level of the proton—the very small x gluon distribution—we have uncovered a possible intrinsic connection between the peculiar shape of the cosmic gamma-ray spectrum (at the astronomical scale) and the low momentum pion anomaly found at the LHC (at the particle scale). We will discuss them in the last section.
3. The BPL in Cosmic Gamma Ray Spectra
Let us first focus on the strong GC region (the yellow areas in
Figure 2 and
Figure 3). The
collisions are an extremely common phenomenon in the Universe, and they are the processes of interest to both astronomical observations and accelerator experiments. Following
Figure 2 and
Figure 3 which were done using
origin 10.2.0.188, we need to consider the hadronization of gluon mini-jets. It is a complex problem. Fortunately, GC provides an ingenious solution to bypass the complexity of the hadronization mechanism [
8]. We envisage that when a substantial number of condensed gluons at the threshold
suddenly participate in the
collisions, it inevitably leads to a dramatic increase in the production of secondary pions. Since pions have mass, their yield
is inherently constrained. In principle, the condensed gluons engaging in collisions may simultaneously generate a considerable number of secondary on-mass-shell pions at a given interaction energy; they are capable of saturating all available energy, indicating that nearly all kinetic energies in collisions at the center-of-mass (C.M.) frame are utilized in creating the rest pions. This results in almost no relative momentum for the newly-formed pions, leading to the maximum value of
. While the validity of this saturation approximation will be scrutinized by subsequent observed data, adopting this limit allows us to circumvent the complex hadronization mechanism. Thus, energy conservation and relativistic covariance have
and
where
marks the leading particle and
is the Lorentz factor. Using the empirical relation [
9]
and
, where
is the inelasticity, we obtain the relationships between the pion yield
, the proton energy
, and the pion energy
, which exhibits the typical power law (PL), i.e., they are the straight lines in double logarithmic coordinates (
Figure 4a),
with
, where
and
.
Using Equation (
8), we obtain
where
. With this result, we can determine how much proton energy is needed to produce a specific value of
.
Equation (
8) can lead to striking and distinctive features of GC in proton collisions, provided the GC-threshold energy
enters the observable range. For this sake, we examine very high energy cosmic gamma rays. In many astrophysical processes, protons can be accelerated to unprecedentedly high energies, and the gamma-ray spectra released by their collisions with other protons or nuclei may carry special GC information.
Substituting the QED formula of
with Equation (
8) into the gamma-ray spectral energy distribution in the hadronization scenario [
4]
where the spectral index
is the photon loss due to absorption in the medium near the source. The intensity of the proton flux
is taken by power law
for simplicity.
contains the motion factor and the flux dimension; the following analytical solution is obtained after a simple integration. This is a BPL (
Figure 4b)
We refer to this as the GC spectrum, which has been used to explain almost a hundred cases of cosmic gamma-ray spectra, including those from supernova remnants (SNRs), pulsars, active galactic nuclei (AGNs), the Galactic Center, and gamma-ray bursts (GRBs). Of course, GC is not the sole explanation for power-law (PL) features in cosmic gamma-ray spectra. For instance, inverse Compton (IC) scattering also exhibits asymptotic PL behaviour, often described using empirical parametric formulas such as the exponentially cutoff PL:
or the log-parabola:
. Therefore, careful comparison reveals their difference from the GC spectrum can test Equation (
11).
We take the GRB spectra at TeV scale as an example. They are of particular interest because of the rarity of such events and the extreme environments. Therefore, careful comparison reveals that their difference from the GC spectrum can test Equation (
11). This allows protons to be accelerated into the very high energy region to reveal a more complete GC spectrum.
Figure 5 is a collection of examples where the intrinsic spectra are corrected from earth observations to account for extragalactic background light (EBL) absorption of photons travelling through cosmic space.
Figure 5a,b show the fits of GRB190114C with two different lepton schemes [
10]. The solid line is the GC spectrum. It seems difficult to judge which model is better. Note that the IC model requires low-energy (KeV) synchrotron radiation as the source of the initial state, its shape being closely related to the TeV spectrum. The high-energy spectrum predicted by the IC model is governed by the shape of the low-energy spectrum of the synchrotron radiation and does not necessarily exhibit PL asymptotic behaviour. The GC spectrum does not have this constraint. However, a more complete high-energy spectrum of GRB was recently presented for GRB 221009A, as shown in
Figure 5c, which clearly favours the the GC spectrum [
11]. We also note that Foffano et al. present a BPL similar to that of GC using the lepton scheme (
Figure 5d) [
12]; however the low- and medium-energy spectra are obviously drawn according to TeV PL, which has no observational fits, making the results questionable.
Remember that the broken point of the GC-spectra in this example,
GeV, is the signature scale at which GC begins to enter the region of action in heavy nucleus collisions (
or
) [
3]. We will use it below.
4. The BEC of Pions
A primary goal in ultra-relativistic heavy-ion collisions is to investigate new states of matter under extreme conditions, especially during the hadronization phase, where pion condensation remains an intriguing and unresolved puzzle. The collision process can be divided into several stages: initially, the quark–gluon collision phase, where high-energy quarks and gluons interact, producing numerous mini-jets; these mini-jets then gradually form a quark–gluon plasma (QGP) through radiation and rescattering; finally, as the system cools to near-critical temperatures, the QGP undergoes hadronization, resulting in a multitude of hadrons. If the pion chemical potential approaches the pion mass during QGP hadronization, Bose–Einstein statistics predict the macroscopic occupation of pion states with zero momentum , known as the BEC of pions.
Theoretically, the simplest way to identify such pions is by directly observing their transverse momentum distribution. However, this method encounters significant experimental and analytical challenges: (i) Particles with very low momentum are difficult to detect accurately. As momentum nears zero, detector efficiency and precision decrease sharply, making it hard to distinguish condensed pions from ordinary low-energy pions; (ii) The ultra-low momentum region is affected by background contributions from secondary particle scattering, decay processes, and instrumental noise, which can obscure potential condensation signatures. Therefore, statistical models are necessary to analyse the observed data.
We discuss the heavy-ion collisions at the Large Hadron Collider (LHC). The GC-threshold energy
GeV in heavy nucleus collisions [
3]. Equation (
9) requires the incident proton to be accelerated to
GeV or
TeV to allow condensed gluons to enter the collision range. Thus, the scenario discussed by Begun and Florkowski,
TeV
collisions, occurs in
Figure 2b and
Figure 3b, which is at
. Considering that the distribution of pions fragmented from gluon mini-jets will become softer, we can expect that partially localised low-momentum pions to be observed at a large rapidity in the LHC (
) collision energy region. This is exactly what is seen by Begun and Florkowski, but it is a GC effect and is not related to BEC.
Let us explain this new discovery.
(i) Why do new pions with the GC effect appear at the edge of rapidity at the LHC? This is a natural consequence of GC kinematics and is well understood and shown in
Figure 2b. The mini-jet that appears at the edge of rapidity consists of a large-
x gluon and a small-
x gluon, so as the collision energy increases, the GC peak at small-
x will appear first at the edge of rapidity. Therefore, the data analysed by Begun and Florkowski falls within the range of
Figure 2b, i.e., the rapidity edge region.
(ii) The
distribution in
Figure 3b shows that the jets indeed distribute the low momentum region. The reason is as follows. The maxima of the gluon transverse momentum distribution in the GC spectrum,
Figure 1, are in a similar region
. Therefore, the main contribution to Equation (
1) comes from
. Thus, we have
, as shown in
Figure 2b. Considering that the momentum of each pion after fragmentation softens, these pions have low momentum.
(iii) The QGP may alter quark–gluon distributions; tracking the evolution of GC in the initial nucleon state requires minimizing QGP interference during hadronization. Fortunately, in the
collisions at
TeV, the GC effect appears in the rapidity edge region. Particles produced in the central rapidity region of heavy ion collisions have weak correlations with particles in the peripheral region. This is due to the following: (a) Mechanism differences. The central region is dominated by QGP collective effects, while the peripheral region is dominated by fragmentation processes, which have different physical origins; (b) Space-time separation. Large rapidity differences lead to the interruption of causal connections, resulting in limited correlation lengths. In fact, the ALICE collaboration measured particle correlations in the central region
and forward region (2.5 < y < 4.0) of
collisions at
TeV [
13]. The results showed that for large
, the correlation function has no modulation in the
direction, with correlation coefficients less than 0.05, indicating negligible correlations. The CMS experiment also reported similar findings [
14]. Therefore, we have omitted the possible QGP influence.
(iv) Why is the distribution of low-momentum pions jumpy? The reason lies in the GC peak. When a large number of condensed gluons enter the collision region, the number of pions increases dramatically. This is not only because more gluons in the proton raise the probability of gluon collisions, but also because gluons are the main component of pion.
(v) Why do Begun and Florkowski find weak isospin symmetry breaking in low momentum pions? A large number of pions with a certain energy accumulate in a narrow space during each collision. Due to the overlap of their wave functions, they may transform into each other during the formation time, i.e., . However, since and the lifetime of ( s) is much shorter than the typical weak decay lifetime of ( s– s), the equilibrium will be broken, and will dominate the secondary processes, allowing us to neglect the contribution of . We have observed this effect in the astronomical events. The above is the case of saturation of pion production in very high energy collisions. The GC phenomenon in the LHC 2.76 TeV region should occur locally in the large rapidity region, where the above isospin asymmetry is much weaker, as seen by Begun and Florkowski.
(vi) BEC is a macroscopic quantum phenomenon. After undergoing rapid expansion and cooling, a heavy-ion collision system may locally enter a “freeze-out” stage characterized by low effective temperature and high particle density. In this regime, the pion chemical potential approaches its mass, leading to a divergence in the particle number distribution in momentum space, with an accumulation particularly near low momentum, especially the zero-momentum state. GC refers to the possible spontaneous clustering of a large number of gluons inside a hadron into a critical momentum state at extremely high energies (very small Bjorken-x). This phenomenon originates from the chaotic dynamics induced by singular structures in nonlinear small-x evolution equations, where strong shadowing/anti-shadowing feedback drives gluons to condensate into the critical momentum mode in the final stage of evolution.
There are several possible sources of low-momentum pions, each contributing differently to the correlation function. For example:
Resonance decay. Long-lived resonances produce a large number of low-
pions (“halo”) in the final state, which mainly broadens the peak widths [
15,
16].
Coherent hydrodynamic flow. A combination of thermal emission and collective flow leads to an enhanced abundance of low-
particles. The spectrum at
appears nearly exponential [
17,
18].
HBT enhancement due to bosonic symmetry. This results in a single-scale broad peak without
-spikes [
19,
20].
BEC emission. Inverse particle cascades or coherent fields generate a “coalescence peak” as
[
5,
21].
GC. As shown in
Figure 2 and
Figure 3, the GC contribution to the correlation function can be approximately represented as a columnar distribution in the large-rapidity region at LHC energies. We plan to investigate this in our future work.
5. Discussion
Based on the above results, we give the following predictions. The heavy-ion collision energy at the LHC has been upgraded from
TeV to
TeV, corresponding to the transition shown in
Figure 2c and
Figure 3c, where low-momentum pions exceeding
may be detected in the forward rapidity side region. The
collisions at
TeV seem to be a stronger effect; however, it is only half as strong as a
collision of the same energy. The GC thresholds are
TeV and 20 TeV for the O–O collisions at
TeV and
collisions at
TeV [
3], respectively. Thus, we do not discuss them in this work. Future upgrades, such as the HE-LHC for
collisions, where the C.M. energy may exceed
TeV, and FCC-hh or SppC up to
TeV, may allow us to observe how the GC transitions from weak to strong and to unravel its mysteries. As shown in
Figure 2 and
Figure 3, with an energy of
TeV or more, the GC effect will become very strong and dominate the whole rapidity region. We warn that strong gamma rays similar to artificial miniature gamma-ray bursts can be accidentally generated, which may damage the detector.
Although GC and BEC in heavy-ion collisions have yet to be confirmed by direct experiments, comparing their differences is insightful. BEC conditions necessitate that the chemical potential approaches (but does not exceed) the pion mass, . The BEC phase transition is characterized by a significant number of near-zero-momentum pions produced in hadron collisions, with a phase space density surpassing the BEC critical threshold, . Here, represents the pion number density and is the thermal de Broglie wavelength. Additionally, condensed pions should display unique correlations. Although these BEC conditions might arise during QGP hadronization, their experimental verification is highly challenging.
In contrast, the GC model predicts low-momentum pions that are independent of the BEC mechanism, requiring comparatively relaxed conditions. However, it requires that the proton’s energy exceeds the GC threshold. In ultra-high energy cosmic ray events, the unique BPL spectrum of GC can be compared with other models to determine if low-momentum pions dominate. Whether or not the GC effect also satisfies the BEC condition remains a separate question. One possibility is that GC produces enough zero-momentum pions to contribute to BEC.
Conclusions. In this work, we present, for the first time, a unified explanation for anomalies observed in the cosmic gamma-ray spectrum at astrophysical scales and low-momentum pion clustering at the particle scale in heavy-ion collisions at the LHC, attributing both phenomena to GC. Specifically, (1) by employing energy conservation and relativistic covariance, we circumvent the complexities of the hadronization process and derive an analytical solution for gamma-ray spectra that accurately reproduces the BPL features observed in nearly one hundred astrophysical sources, including supernova remnants and active galactic nuclei; and (2) we demonstrate that the low-momentum pion clustering observed by the ALICE collaboration, previously interpreted as BEC, can instead be understood as a manifestation of GC effects. Importantly, the formation of GC does not require the stringent conditions necessary for BEC, and it can be tested in future high-energy collider experiments such as the HE-LHC and FCC-hh. We also caution about the potential generation of artificial gamma-ray bursts in ultra-high-energy collisions, emphasizing the importance of detector protection. This series of findings indicates that GC constitutes a novel and under-recognized structural phenomenon within the Standard Model framework, opening new avenues in particle physics, astrophysics, and nonlinear dynamics. Not only does this deepen our understanding of the proton’s internal structure, but it also provides a powerful tool for exploring matter behaviour under extreme conditions in the Universe.