Pion condensation is a special state of matter, where pions condense in the same quantum state under certain extreme conditions. This concept was first introduced by Migdal, who discussed the possibility of this phenomenon in nuclear matter in his 1971 paper [1]. With the advancement of experimental techniques and theoretical models, the study of pion condensation has made remarkable progress. Modern studies have focused on the possibility and characterization of pion condensation under extreme conditions (e.g. in neutron star and heavy-ion collisions). It has now evolved into a multidisciplinary cross-study area covering high-density quantum chromodynamics (QCD) theory, numerical simulations, heavy-ion collision experiments and astrophysical observations. The confirmation of pion condensation will not only be an important validation of fundamental physical theories, but also have far-reaching implications for several branches of physics, including particle physics, astrophysics, condensed matter physics and nuclear physics. Unfortunately, there is currently no solid experimental evidence confirming the existence of pion condensation.

In this paper, we demonstrate that once a large number of pions condense in high-energy hadron collisions, the gamma-ray spectra produced by π⁰-decay will take a typical broken power law (BPL) shape, which has already existed in many astronomical observations, but we have not yet recognized it.

High-energy pp collisions produce a large number of secondary hadrons in the central region of rapidity, which are dominated by pions. As observed in high-energy hadron collisions, the particle yield in the central region increases slowly with $\mathrm{ln}\sqrt{S}$, and more of the collision energy $\sqrt{S}$ is converted into the kinetic energy of the thermal motion of the new particles in the center-of-mass (C.M.) system. This scenario favors the formation of quark-gluon plasma (QGP), but it does not meet the high-density and low-temperature conditions required for pion condensation.

Pion condensation is defined as the pion field obtaining a non-zero vacuum expectation value, because a large number of pions accumulate in the ground state. High density and extremely low temperature are two conditions that cause pion condensation. Obviously, in order to condense pions, the temperature in the collision center region must be effectively reduced. We find that this method hides the QCD structure of the proton.

A series of QCD evolution equations studies have found that at high energies, the gluon distribution in the nucleon exhibits chaotic behavior, which induces strong shadowing and antishadowing effects, the latter leading to the aggregation of innumerable gluons in a narrow phase space defined by the critical momentum (x_c, k_c). This is gluon condensation (figure 1) [2–5].

Although the Higgs boson is responsible for producing the current quark mass, recent studies have shown that the hadron mass is mainly derived from the gluon field [6]. Therefore, gluon interactions dominate the multi-production processes in high-energy proton collisions. Once the collision energy exceeds the threshold value corresponding to x_c, a large number of condensed gluons rapidly accumulate in the central region. In this case the pion yield will dramatically increase. Energy conservation requests where A is a constant, T is the absolute temperature in Kelvin and the other half of the C.M. energy $\sqrt{S}$ was taken by the leading particles [7]. One can find that if N_π,b ≫ N_π,a, we have T_b ≪ T_b. Therefore, under the same collision energy, the thermal motion energy of pions must be suppressed, resulting in cooling. It seems that an infinite number of condensed gluons could create an infinite number of pion, however, conservation of energy allows only a maximum number of pion to be created since pion has mass, i.e., almost all kinetic energy is used to produce static particles. This results in the newly formed pions having almost no relative momentum and T_b → 0, which produces a maximum value of N_π. Consequently, a dense and low-temperature physical environment conducive to the formation of pion condensation is established in the central region of rapidity.

The next question is what range the temperature in the collision center region should drop to for pion condensation to occur? Okorokov found Bose–Einstein condensate (BEC) correlations between particles with low relative momentum in high-energy hadronic collisions, although it is not full boson condensation since only a small fraction of low relative momentum pions are correlated [8–10]. This discovery implies that bosons with relatively low thermal motions at high densities can also produce boson correlations at temperatures no lower than the critical BEC temperature. So we assume that the relative momentum of the pion in the condensation state is much smaller than the pion mass. Thus, the condensed pions can be approximated as stationary in the C.M. system, while in Lab., they have the same Lorentz factor γ (figure 2).

We simplify the process pp → X to pp → pp + N_π. According to figure 2 we write energy conservation where ${\tilde{m}}_{p}$ marks the leading particle and γ_i is the Lorentz factor. Note that the leading particle carries about half of the total energy. On the other hand, the square of relativistic invariant total energy $S={({p}_{1}+{p}_{2})}^{2}$ in the two systems is

The proportion of the total collision energy occupied by the leading particles and the central particles is independent of the selection of the reference frame. Using the following empirical relation [7], we remove the physical quantities about the leading particles in equations (2) and (3) and k ≃ 1/2 is the inelasticity. Thus, we have the relations among N_π, E_π and N_p where $a\equiv 0.5\mathrm{ln}(2{m}_{p}/{\rm{GeV}})-\mathrm{ln}({m}_{\pi }/{\rm{GeV}})+\mathrm{ln}k$ and $b\,\equiv \mathrm{ln}(2{m}_{p}/{\rm{GeV}})-2\mathrm{ln}({m}_{\pi }/{\rm{GeV}})+\mathrm{ln}k$. They are straight lines in double logarithmic coordinates, called the power law (PL) (figure 3(a)). This is the result of the condensation of a large number of pions in the pp collision (see figure 2), therefore, we regard it as a replantation of pion condensation.

We can also measure the gamma-ray spectrum from the condensed π⁰-decay to receive the pion condensation signal. The gamma-ray spectral energy distribution is [12]where the spectral index β_γ is the photon loss due to absorption in the medium near the source. In accelerator laboratory, the intensity of the proton flux N_p is known, and the power law is taken here for simplicity. C_γ contains the motion factor and the flux dimension.

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., π⁺π⁻⇌2π⁰. However, since ${m}_{{\pi }^{+}}+{m}_{{\pi }^{-}}\gt 2{m}_{{\pi }^{0}}$ and the lifetime of π⁰ (10⁻¹⁶ s) is much shorter than the typical weak decay lifetime of π^± (10⁻⁶ s - 10⁻⁸ s), the equilibrium will be broken, and π⁰ will dominate the secondary processes, allowing us to neglect the contribution of π^±. This result is consistent with the fact: that there is no significant enhancement of the neutrino signal accompanying gamma-ray spectrum, which is a decay product of π^±. Substituting the standard formula of π⁰ → 2γ with equation (6) into equation (7), the following analytical solution is obtained after a simple integration, which is a BPL (figure 3(b)),We call it the condensation-spectrum.

Unfortunately, all hadron collider experiments in the laboratory have not detected the above condensation signal. Considering a series of works [13–20] concerning the applications of the condensation-spectrum (equation (8)), we find that the parameters ${E}_{\pi }^{GC}$ and ${E}_{\pi }^{cut}$ are target-A dependent in pA collisions. Specifically, ${E}_{\pi }^{GC}\epsilon [100\,GeV,20\,\,TeV]$ from heavy nuclei to proton, and it corresponds to $\sqrt{S}\gg 1{0}^{6}\,GeV$ [5]. It means that the pion condensation threshold is beyond the maximum energy available at the accelerator. Therefore, we turn our attention to cosmic rays because their energy may exceed the gluon condensation threshold, where we found that the condensation spectrum equation (8) can widely interpret very high energy (VHE) gamma-ray spectra in active galactic nuclei, gamma-ray bursts, pulsars, galactic center excess, supernova remnants, even electron/positron and proton/nuclei spectra, where more than 70 astronomical events are recorded [13–20]. Now let us add a few more typical examples:

1. PKS 0625-354 [21, 22]

This is bright in the x-ray and gamma-ray bands, displaying very high-energy radiation, and is therefore observed and studied by a wide variety of astronomical equipment. Red-shift z ∼ 0.054. The variability observed in PKS 0625-354 can be explained by the hadronic model, since changes in the conditions inside the jet (e.g., magnetic field strength, particle density, or acceleration efficiency) lead to rapid changes in the emission. Moreover, the estimation of the radiated power favors the hadronic scenario because of the small proton radiation losses and the higher acceleration efficiency.

However, the traditional hadronic scenario cannot fit the observed VHE spectrum. Next there was a general preference for the lepton scenario. The PKS 0625-354 broad-band spectral energy distribution shows two humps that can be interpreted as synchrotron-self-Compton (SSC) emission products. However, a careful review of the lepton program is not satisfactory. The VHE spectra cannot be described by a single-region SSC scheme. After modification of multiple sources in the low-energy region, different models, the multi-zone SSC scheme and the lepton-hadron SSC scheme can give similar results, suggesting that the correlation between the two peaks is not necessarily inevitable, and that this is a key point in determining whether or not SSC is the radiation mechanism. Figure 4 is the condensation-spectrum of PKS 0625-354 using equation (8). It can be found a specific spectrum with sharp broken power law at 200 GeV as shown by the condensation spectrum. The data is corrected for EBL attenuation. In order to test the PL of Φ_γ predicted by pion condensation, figure 3(b) was not multiplied by ${E}_{\gamma }^{2}$ and we use Chi-square per degree of freedom (χ²/d. o. f. ) to assess the fitting goodness.

2. MGRO J1908+06 [23, 24]

This is an ultra-high-energy gamma-ray source located in the galactic plane. It is notable for emitting ultra-high energies in excess of 200 TeV. J1908+06 has a corresponding Fermi-LAT GeV counterpart. The morphology and spectral properties of this counterpart suggest a common origin with the TeV emission, reinforcing the view that the high-energy processes of the source are complex and multifaceted. Figure 5 is an explanation of the condensation-spectrum.

3. 1ES 2344+514 [25, 26]

BL Lacertae 1ES 2344+514 is a member of the blazar category. It consists of extremely complex physical processes with extreme variations over time. Since its discovery in 1995, the high-energy peaked blazar 1ES 2344+514 has been observed in the multiwavelength by several other telescopes. Figure 6 shows the GC spectrum of 1ES 2344+514. One can find a clear signal of pion condensation.

4. J1356-645 [27, 28]

Since the pp collisions are a common phenomenon in the Universe, the pion condensation signals can occur in a variety of environments. J1356-645 is a known gamma-ray pulsar. Reference [28] reanalyzes the GeV gamma-ray emission in the direction of HESS J1356-645 with more than 13 years of Fermi Large Area Telescope (LAT) data. They find that the spectrum in the energy range in [1GeV, 1TeV] can be described by a PL. It can be explained by the condensation-spectrum (figure 7).

We review the history of BEC. Bose in 1924 proposed a theory about the statistical distribution of photons, and Einstein found that Bose’s method was not only applicable to photons, but also to other particles with integer spins (i.e. bosons). He further proposed that at very low temperatures, bosons could condense into the same quantum state to form a completely new state of matter. However, it was not until 1995 that Cornell and Wieman used laser cooling and magnetic trapping techniques to cool sodium atoms to temperatures close to absolute zero, and succeeded in observing a large number of sodium atoms undergoing the buildup predicted by BEC. The phenomenon of BEC has become an important way to study superfluidity, superconductors, and other quantum phenomena.

Therefore, cooling dense and unstable pions is a key challenge in realizing pion condensation. In this work, we point out that the gluons condensed in protons, which was predicted by QCD , can effectively reduce the temperature in the central region of ultra-high energy proton collisions. Thus creating a favorable environment for pion condensation. Although the existing hadron colliders do not have enough energy to produce pion condensation, the prevalence of ultra-high-energy proton collisions in the Universe is not difficult to create a large number of examples of pion condensation, which can be confirmed by the special shape of the gamma-ray spectrum. We hope this will open a new window for the pion condensation research.

In summary, we show that once a large number of pions condense in high-energy hadron collisions, the gamma energy spectra produced by the pion multiplicity and the π⁰-decay show a special PL, the latter of which has already been documented in many astronomical observations, but we have not recognized it. Further analysis supports that the above pion condensation may arise from the gluon condensation in proton. In the pp collisions, a large number of condensed gluons in protons pour into the central region, dramatically increasing the number of pions, which is cooled due to the restriction of energy conservation. In the limit, almost all of the available collision energy is used to create pions, and the central region briefly forms a dense cryogenic region with the potential to form pion condensation. This result not only unravels the mystery of pion condensation, which has been hidden for half a century, but also adds a paradigm for bridging the QCD structure of the proton with cosmic phenomena.