Global characteristics of the medium produced in ultra-high energy cosmic ray collisions

TL;DR

This paper estimates geometrical and bulk parameters of matter produced in ultra-high energy cosmic ray collisions, suggesting the formation of deconfined quark-gluon matter even in light nuclear interactions at these energies.

Contribution

It provides novel estimations of space-time and particle density parameters for UHECR collisions, indicating potential Bose-Einstein condensation and quark-gluon plasma formation.

Findings

Indicates formation of deconfined quark-gluon matter in light nuclear collisions at UHECR energies

Suggests possible Bose-Einstein condensation of secondary particles

Provides estimations useful for future collider experiments and cosmic ray models

Abstract

Estimations of some geometrical and bulk parameters are presented for the matter produced in various type collisions with ultra-high energy cosmic ray (UHECR) particles. Results for multiplicity density at midrapidity, decoupling time, and energy density are discussed for small and larger collision systems. Based on the analytic functions suggested previously elsewhere, estimations for a wide set of space-time quantities are obtained for emission region created in various particle collisions at energies of UHECR. The space particle densities at freeze-out are derived also and allow the possibility of novel features for secondary particle production like Bose-Einstein condensation at least for nuclear interactions with UHECR particles. The estimations obtained for global and geometrical parameters indicate the creation of deconfined quark-gluon matter with large enough volume and lifetime even in light nuclear collisions at UHECR energies. These quantitative results can be important for both the future collider experiments at center-of-mass energy frontier and the improvement of the phenomenological models for development of the cosmic ray cascades in ultra-high energy domain.

Tables

Table 1. Table 1: Center-of-mass energies (TeV) for collisions with various incoming nuclei.

Particle	$E_p$, TeV
	${10}^5$	$5\times {10}^5$	${10}^6$	$5\times {10}^6$	${10}^7$	$5\times {10}^7$	${10}^8$	$5\times {10}^8$	${10}^9$
${}_{}{}^{1}p_{}^{1+}$	13.70	30.63	43.32	96.86	137.0	306.3	433.2	968.6	1370
${}_{}{}^4\text{He}_{}^{2+}$	9.686	21.66	30.63	68.49	96.86	216.6	306.3	684.9	968.6
${}_{}{}^{14}\text{N}_{}^{7+}$	9.686	21.66	30.63	68.49	96.86	216.6	306.3	684.9	968.6
${}_{}{}^{56}\text{Fe}_{}^{26+}$	9.334	20.87	29.52	66.00	93.34	208.7	295.2	660.0	933.4

Table 2. Table 2: Estimations for global parameters in various collisions at some UHECR energies.

Parameter	$E_p$, TeV	${}_{}{}^{1}p_{}^{1+}$	${}_{}{}^4\text{He}_{}^{2+}$	${}_{}{}^{14}\text{N}_{}^{7+}$	${}_{}{}^{56}\text{Fe}_{}^{26+}$
${\rho }^{\eta }$	${10}^5$	$6.6\pm 0.7$	$46.5\pm 1.7$	$46.5\pm 1.7$	$45.9\pm 1.7$
		–	$13.8\pm 0.6$	–	–
	${10}^7$	$11.1\pm 1.4$	$95\pm 4$	$95\pm 4$	$94\pm 4$
		–	$22.2\pm 1.1$	–	–
	${10}^9$	$18.8\pm 2.6$	$194\pm 9$	$194\pm 9$	$191\pm 9$
		–	$35.6\pm 1.8$	–	–
${\xi }^{-1}{N}_{\text{ch}}$	${10}^5$	$80.5\pm 2.2$	$141\pm 20$	$141\pm 20$	$139\pm 20$
	${10}^7$	$154.7\pm 2.3$	$331\pm 20$	$331\pm 20$	$327\pm 20$
	${10}^9$	$291.6\pm 2.4$	$768\pm 21$	$768\pm 21$	$758\pm 21$
${\rho }_T^E$,	${10}^5$	–	$18\pm 7$	$18\pm 7$	$18\pm 7$
GeV	${10}^7$	–	$45\pm 19$	$45\pm 19$	$45\pm 19$
	${10}^9$	–	$110\pm 50$	$110\pm 50$	$110\pm 50$
${{\xi }^{-1}{\rho }_{\text{Bj}}^E|}_{\tau =1}$,	${10}^5$	–	$1.5\pm 0.6$	$0.63\pm 0.26$	$0.24\pm 0.10$
GeV/fm3	${10}^7$	–	$3.7\pm 1.6$	$1.6\pm 0.7$	$0.62\pm 0.27$
	${10}^9$	–	$9\pm 4$	$4.0\pm 1.9$	$1.6\pm 0.7$
${{\xi }^{-1/4}T|}_{\tau =1}$,	${10}^5$	–	$0.55\pm 0.06$	$0.45\pm 0.05$	$0.35\pm 0.04$
GeV	${10}^7$	–	$0.70\pm 0.08$	$0.57\pm 0.06$	$0.45\pm 0.05$
	${10}^9$	–	$0.88\pm 0.10$	$0.71\pm 0.08$	$0.56\pm 0.07$
${\xi }^{-1/3}{\tau }_{\text{kin}}$,	${10}^5$	–	$13.55\pm 0.17$	$13.55\pm 0.17$	$13.40\pm 0.17$
fm/$c$	${10}^7$	–	$27.7\pm 0.4$	$27.7\pm 0.4$	$27.4\pm 0.4$
	${10}^9$	–	$56.5\pm 0.9$	$56.5\pm 0.9$	$55.8\pm 0.9$

Table 3. Table 3: Estimations for space-time characteristic of pion source at some UHECR energies.

Particle	$E_p$, TeV	BE parameter
		$R_s$, fm		$R_o$, fm		$R_l$, fm		$V$, fm3
	${10}^5$	$1.83\pm 0.27$	$1.7\pm 0.7$	$1.4\pm 0.6$	$1.4\pm 0.7$	$1.85\pm 0.07$	$1.82\pm 0.17$	$100\pm 30$	$80\pm 60$
${}_{}{}^{1}p_{}^{1+}$	${10}^7$	$2.4\pm 0.4$	$2.1\pm 0.7$	$1.7\pm 0.8$	$1.7\pm 0.9$	$2.18\pm 0.12$	$1.98\pm 0.20$	$190\pm 60$	$130\pm 90$
	${10}^9$	$2.9\pm 0.4$	$2.5\pm 0.9$	$1.9\pm 0.9$	$1.9\pm 1.1$	$2.63\pm 0.19$	$2.14\pm 0.24$	$360\pm 110$	$210\pm 140$
	${10}^5$	$2.0\pm 0.5$	$1.60\pm 0.07$	$1.7\pm 0.5$	$1.72\pm 0.07$	$2.2\pm 0.4$	$2.25\pm 0.09$	$130\pm 70$	$91\pm 9$
${}_{}{}^4\text{He}_{}^{2+}$	${10}^7$	$2.6\pm 1.0$	$1.71\pm 0.07$	$1.7\pm 0.5$	$1.78\pm 0.08$	$2.3\pm 0.4$	$2.50\pm 0.11$	$250\pm 200$	$115\pm 11$
	${10}^9$	$3.6\pm 1.9$	$1.81\pm 0.08$	$1.8\pm 0.6$	$1.84\pm 0.08$	$2.4\pm 0.4$	$2.75\pm 0.12$	$500\pm 500$	$142\pm 14$
	${10}^5$	$3.0\pm 0.8$	$2.43\pm 0.10$	$2.6\pm 0.8$	$2.62\pm 0.11$	$3.3\pm 0.5$	$3.42\pm 0.14$	$460\pm 250$	$320\pm 30$
${}_{}{}^{14}\text{N}_{}^{7+}$	${10}^7$	$4.0\pm 1.6$	$2.60\pm 0.11$	$2.7\pm 0.8$	$2.70\pm 0.12$	$3.5\pm 0.6$	$3.80\pm 0.16$	$900\pm 700$	$400\pm 40$
	${10}^9$	$5.6\pm 2.9$	$2.75\pm 0.12$	$2.7\pm 0.8$	$2.79\pm 0.13$	$3.7\pm 0.6$	$4.17\pm 0.18$	$1800\pm 1900$	$500\pm 50$
	${10}^5$	$4.7\pm 1.2$	$3.86\pm 0.16$	$4.1\pm 1.3$	$4.16\pm 0.18$	$5.2\pm 0.8$	$5.42\pm 0.23$	$1800\pm 1000$	$1270\pm 120$
${}_{}{}^{56}\text{Fe}_{}^{26+}$	${10}^7$	$6.3\pm 2.5$	$4.11\pm 0.18$	$4.2\pm 1.3$	$4.29\pm 0.19$	$5.5\pm 0.9$	$6.01\pm 0.25$	$3500\pm 2800$	$1600\pm 150$
	${10}^9$	$9\pm 5$	$4.36\pm 0.19$	$4.3\pm 1.3$	$4.42\pm 0.20$	$5.8\pm 0.9$	$6.62\pm 0.28$	$7000\pm 7000$	$1900\pm 190$

Equations

$$p_N=(Z_1/A_1)p_p,{\sqrt{s_{NN}}|}_{m_N\approx m_p,m_p\ll E_p}\approx \sqrt{(Z_1/A_1)s_{pp}},$$

$$\rho^{\eta}\equiv\xi^{-1}\left.\bigl{(}dN_{\scriptsize{\mbox{ch}}}/d\eta\bigr{)}\right|_{\eta=0}=a_{1}\varepsilon^{a_{2}},~{}~{}~{}\varepsilon\equiv s/s_{0},$$

$${\xi }^{-1}{N}_{\text{mbox}ch}=a_1+a_2{\epsilon }^{a_3},$$

$$\rho^{E}_{\scriptsize{T}}\equiv\xi^{-1}\left.\bigl{(}dE_{\,\scriptsize{T}}/d\eta\bigr{)}\right|_{\eta=0}=(0.46\pm 0.16)\,\varepsilon^{0.200\pm 0.005}.$$

$$\rho^{E}_{\scriptsize{\mbox{Bj}}}(\tau)=\bigl{(}S_{\perp}\tau\bigr{)}^{-1}\times\left.\bigl{(}dE_{\,\scriptsize{T}}/d\eta\bigr{)}\right|_{\eta=0}\approx\bigl{(}\pi R^{2}\tau\bigr{)}^{-1}\times\left.\bigl{(}dE_{\,\scriptsize{T}}/d\eta\bigr{)}\right|_{\eta=0},$$

$$T(\tau)=\bigl{[}30\rho^{E}_{\scriptsize{\mbox{Bj}}}(\tau)/\pi^{2}n_{\scriptsize{\mbox{qg}}}\bigr{]}^{1/4},$$

$$f_{i}(\varepsilon)=a_{1}^{i}\bigl{[}1+a_{2}^{i}(\ln\varepsilon)^{a_{3}^{i}}\bigr{]},~{}~{}~{}i=s,o,l$$

$$V=(2\pi {)}^{3/2}{R}_s^2{R}_l$$

$$\tau_{\scriptsize{\mbox{kin}}}\approx 0.875\bigl{[}\left.\bigl{(}dN_{\scriptsize{\mbox{ch}}}/d\eta\bigr{)}\right|_{\eta=0}\bigr{]}^{1/3}.$$

Full text

I Introduction

The present projects of the future research facilities prove that the accelerator physics in XXI century will be the physics in $\mathcal{O}(100~{}\mbox{TeV})$ domain of the center-of-mass energies. Measurements of interactions of cosmic ray particles with ultra-high initial laboratory energies larger than 0.1–1 EeV with nuclei in the atmosphere allow the new unique possibilities for study of multiparticle production processes at energies (well) above not only the Large Hadron Collider – LHC range but foreseeable-future collider on Earth as well. Collisions at such ultra-high energies can lead to creation of a strongly interacting matter under extreme conditions. Due to the air composition and main components of the ultra-high energy cosmic rays (UHECR) the passage of UHECR particles through atmosphere can be considered as collision of mostly small systems. Among the most challenging problems for collider experiments is the study of the quark-gluon matter created in such collisions. On the other hand, the investigation of main properties of the final-state matter for UHECR particle collisions with atmosphere can be useful for better understanding of the origin and features of UHECR itself. Therefore the estimations of global characteristics of the matter produced in ultra-high energy cosmic ray collisions seems important for both the experiments at present and, possibly, even more for future colliders and the physics of cosmic rays.

II Observables and approximating functions

Although the potential sources of UHECR are still quite far from understanding, it is reasonable to suggest the same acceleration mechanism for protons and heavier components of the UHECR which affects the charged component of the nuclei. Some electromagnetic fields can be such a mechanism but not the shock waves from explosion processes. Therefore within the nucleus $(A_{1},Z_{1})$ incident on a particle at rest $(A_{2},Z_{2})$ with nucleon numbers $A_{1}$, $A_{2}$ and charges $Z_{1}e$, $Z_{2}e$ in conditions with electromagnetic field set for protons of laboratory momentum $p_{p}$, the collision will be characterized by the momentum per nucleon of the incoming nucleus in laboratory reference system and the center-of-mass energy per nucleon-nucleon pair

[TABLE]

where $s_{NN/pp}=2m_{N/p}(E_{N/p}+m_{N/p})$ is standard Mandelstam invariant variable, $E_{N/p}$ and $m_{N/p}$ is the energy in the laboratory frame and mass of nucleon/proton PDG-PRD-98-030001-2018 .

In the present study the global parameters are estimated with the help of the extrapolation technique. The corresponding extrapolations use the parameterizations that describe the existing experimental data in dependence on $s$111Below the index “NN / pp” will be omitted for this parameter for brevity if the statement is applicable for both the $p+p$ and nucleus-nucleus collisions.. It should be noted that analytic functions given below were obtained on the basis of the Standard Model (SM) without invoking any hypothesis concerning contributions from physics beyond it. The justification of the approach for ultra-high energy domain can be found elsewhere Okorokov-PAN-82-134-2019 .

The energy dependence of the pseudorapidity ($\eta$) density of secondary charged particles produced at $\eta=0$ per nucleon-nucleon pair can be approximated in particular by the universal power expression

[TABLE]

where $s_{0}=1$ GeV2, $\xi\equiv 0.5\langle N_{\scriptsize{\mbox{part}}}\rangle$, $\langle N_{\scriptsize{\mbox{part}}}\rangle$ is the average number of participants and $\langle N_{\scriptsize{\mbox{part}}}\rangle=2$ for $p+p$ collisions, free parameters are $a_{1}=0.75\pm 0.06$, $a_{2}=0.114\pm 0.003$ for nonsingle diffractive $p+p$ collisions at $\sqrt{\smash[b]{s_{pp}}}>20$ GeV Okorokov-arXiv-1606.08665-2016 , $a_{1}=1.94\pm 0.08$, $a_{2}=0.103\pm 0.002$ for $p(d)+A$ interactions EPJC-79-307-2019 , and $a_{1}=2.70\pm 0.07$, $a_{2}=0.155\pm 0.004$ for 0–5% most central $A+A$ collisions PRL-116-222302-2016 ; PLB-790-35-2019 . The total multiplicity of the charged particles produced in nuclear collisions depends on $s$ and can be approximated by the power function

[TABLE]

where $a_{1}=-7.36\pm 0.16$ $(-6.72\pm 1.44)$, $a_{2}=6.97\pm 0.12$ $(5.42\pm 1.11)$, $a_{3}=0.133\pm 0.001$ $(0.180\pm 0.020)$ for $p+p$ (central $A+A$) collisions PRD-93-054046-2016 .

Taking into account the experimental results for the central heavy ion collisions PRC-71-034908-2005 ; PRL-109-152303-2012 one can deduce the following approximation for the energy dependence of the transverse energy ($E_{\,\scriptsize{T}}$) density at $\eta=0$ normalized by participant pairs

[TABLE]

The Bjorken energy density and temperature of the matter created in the collisions of UHECR particles in atmosphere depend on the time duration ($\tau$) since the collision moment. The corresponding estimations for most central collisions are following:

[TABLE]

[TABLE]

where $S_{\perp}$ is the nuclei transverse overlap area, the radius for incoming nucleus is estimated as the radius of spherically-symmetric object $\forall\,A>1:R=r_{0}A^{1/3}$ with $r_{0}=(1.25\pm 0.05)$ fm book and for $p+p$ collisions $R=(0.875\pm 0.006)$ fm Okorokov-IJMPA-33-1850077-2018 . Here the Stefan–Boltzmann relation is used for derivation of the $T(\tau)$ and $n_{\scriptsize{\mbox{qg}}}$ is the number of degrees of freedom of the quark-gluon matter with $N_{f}=3$ active quark flavors within the present work.

The smooth energy dependences of the Bose–Einstein (BE) correlation parameters allow the study of the geometry and space-time extent of the emission region of secondary particles produced in UHECR collisions. The linear scales (radii) of the homogeneity region for the 3D Gaussian source of the charged pion pairs with low relative momentum can be parameterized by the universal function Okorokov-AHEP-2015-790646-2015

[TABLE]

with the appropriate set of parameters for each direction $i$ of the Pratt–Bertsch coordinate system. The volume of the homogeneity region is calculated as Okorokov-AHEP-2015-790646-2015

[TABLE]

and time duration since the collision moment until kinetic freeze-out stage called also BE decoupling time and characterizing the total duration of the longitudinal expansion of final-state matter is PLB-696-328-2011

[TABLE]

III Results of extrapolations

The energy range for protons in laboratory reference system considered in the present paper is $E_{p}=10^{17}$–$10^{21}$ eV. This range includes the energy domain corresponded to the Greisen–Zatsepin–Kuzmin (GZK) limit Greisen-PRL-16-748-1966 and somewhat expands it, taking into account, on the one hand, both possible uncertainties of theoretical estimations for the limit values for UHECR and experimental results, namely, measurements of several events with $E_{p}>10^{20}$ eV and the absence of UHECR particle flux attenuation up to $E_{p}\sim 10^{20.5}$ eV Okorokov-PAN-81-508-2018 and, on the other hand, the energies corresponding to the nominal value $\sqrt{\smash[b]{s_{pp}}}=14$ TeV of the commissioned LHC as well as to the parameters for the main international projects high energy LHC – HE-LHC ($\sqrt{\smash[b]{s_{pp}}}=27$ TeV) and Future Circular Collider – FCC ($\sqrt{\smash[b]{s_{pp}}}=100$ TeV). Therefore the estimations below can be useful for both the UHECR physics and the collider experiments.

Here the following set of nuclei \mathcal{G}_{Y}\equiv\bigl{\{}\mathcal{G}_{Y}^{i}\bigr{\}}_{i=1}^{4}=\bigl{\{}{{}^{1}p^{1+},{}^{4}\mbox{He}^{2+},{}^{14}\mbox{N}^{7+},{}^{56}\mbox{Fe}^{26+}}\bigr{\}} is considered. The nuclei correspond to the four groups of elements which are the main components of cosmic rays with studied energies PPNP-63-293-2009 . It should be noted that the free parameter values in (2)–(5) for $A+A$ have been obtained for heavy ion collisions and usually for the most central bin. Consequently, the estimations derived within the present work are for most central collisions. In any case the applicability of just the aforementioned analytic relations for light nuclei requires the additional justification and careful verification. Therefore the future estimations for light nucleus-nucleus collisions can be considered as preliminary with taking into account this feature.

Table 1 shows the $\sqrt{\smash[b]{s_{NN}}}$ values for the laboratory energies of the incoming nuclei from the set $\mathcal{G}_{Y}$ corresponding to some fixed (”nominal”) values of proton energy $E_{p}$. The UHECR with highest energies under consideration allow the study of the final-state matter created in PeV domain for $\sqrt{\smash[b]{s_{NN}}}$ which is far above any further accelerator facilities projected now. As stressed above, the present analysis supposes the absence of the noticeable contributions of a new physics up to the $\sqrt{\smash[b]{s_{NN}}}\sim 1$ PeV.

The estimations for global characteristics described above are presented in Table 2 for three ”nominal” values of $E_{p}$. The quantities $N_{\scriptsize{\mbox{ch}}}$, $\rho^{E}_{\scriptsize{\mbox{Bj}}}$, $T$ and $\tau_{\scriptsize{\mbox{kin}}}$ depend on $\langle N_{\scriptsize{\mbox{part}}}\rangle$. The energy dependence of the last parameter provides additional uncertainty for the global characteristics under consideration especially in UHECR energy domain and for light nuclei because of (very) limited data. In order to avoid this source of dispersion the appropriate scale factors are added for some global characteristics in Table 2. For the laboratory frame realized for UHECR passage through the atmosphere the $\rho^{\eta}$ does not depend on $A$ and $Z$ of the target nucleus and is the same for any asymmetric $p+\mbox{A}$ collisions for given $E_{p}$. Therefore the values for $p+\mbox{He}$ interactions are only shown on the second line for $\rho^{\eta}$. The collisions of all considered types, $p+p$, $p+A$, and $A+A$, are characterized by the large values of $\rho^{\eta}$ and scaled $N_{\scriptsize{\mbox{ch}}}$ for all nucleus from the set $\mathcal{G}_{Y}$. For instance, the multiplicity density (2) in $p+p$ interactions at $E_{p}=10^{19}$ eV is already equal to the value of this parameter in most central $\mbox{Pb}+\mbox{Pb}$ collisions at $\sqrt{\smash[b]{s_{NN}}}=5.02$ TeV PRL-116-222302-2016 . The values of $\rho^{E}_{\scriptsize{T}}$ and scaled parameter (5) at $\tau=1$ fm/$c$ are extremely high for the medium created in the final state of collisions of any nuclei from the $\mathcal{G}_{Y}$ already at $E_{p}=10^{17}$ eV. The Bjorken energy density with taking into account the scale factor is well above the estimation for the critical value of energy density $\rho^{E}_{c}=(0.34\pm 0.16)$ GeV/fm3 IJMPE-24-1530007-2015 for transition from the hadronic phase to the quark-gluon one for any nuclei and energies under study (Table 2). The values of $\left.\xi^{-1}\rho^{E}_{\scriptsize{\mbox{Bj}}}\right|_{\tau=1}$ in small system collisions exceed significantly the value of the parameter in $\mbox{Pb}+\mbox{Pb}$ collisions at $\sqrt{\smash[b]{s_{NN}}}=2.76$ TeV which is $\approx 0.07$ GeV/fm3 PRL-109-152303-2012 . Thus, one can expect the values for energy densities $\rho^{E}_{\scriptsize{T}}$ and $\rho^{E}_{\scriptsize{\mbox{Bj}}}$ correspond to the creation of the quark-gluon deconfined phase state in the collisions of the particles from the set $\mathcal{G}_{Y}$ at the energies under consideration. The matter created in the symmetric collisions of UHECR particles \bigl{\{}\mathcal{G}_{Y}^{i}\bigr{\}}_{i=2}^{4} is characterized by the temperature larger significantly than the critical one $T_{c}=(156.5\pm 1.5)$ MeV NPA-982-211-2019 . One can note the temperature can achieve the value which is of about 1 GeV in light nucleus-nucleus collisions at the highest center-of-mass energies corresponding to the $E_{p}=10^{21}$ eV. In this case one can suggest that the non-perturbative effects in medium will be weaker than those for the matter studied in the collider experiments. Therefore the UHECR collisions for the $E_{p}$ range under study create the quark-gluon matter which reaches the thermodynamic equilibrium and lives the sufficiently long time. The last statement is confirmed by the values of scaled $\tau_{\scriptsize{\mbox{kin}}}$ (Table 2). Furthermore the creation of the quark-gluon matter is already expected in $\mbox{He}+\mbox{He}$ collisions at $\sqrt{\smash[b]{s_{NN}}}$ corresponding to the low-boundary $E_{p}=10^{17}$ eV, i.e. the LHC domain.

The space-time extents of the emission region are evaluated for $p+p$ collisions with the help of the results obtained for secondary pions in Okorokov-AHEP-2016-5972709-2016 . The experimental results for BE correlations for other nuclei from the set $\mathcal{G}_{Y}$ are absent. Thus the estimations for BE parameters in symmetric nuclear collisions for the subset \bigl{\{}\mathcal{G}_{Y}^{i}\bigr{\}}_{i=2}^{4} are calculated based on the 3D analysis of the available experimental data for the $\sqrt{\smash[b]{s_{NN}}}$ dependence of the scaled BE radii of the charged-pion emission region created in various nuclear interactions Okorokov-AHEP-2015-790646-2015 . Table 3 summarizes the estimations for the space-time extents of the source for secondary particle (pions) at some UHECR energies. For given parameter and $E_{p}$ value the first-column values are based on the results of approximations of the energy dependence of femtoscopic radii by the general expression (7), while the second columns show the estimations deduced with the help of the results for the specific case of the fit function (7) at $\forall\,i=s,o,l:a_{3}^{i}=1.0$ (fixed). The $p+p$ interactions produce the quasi-spherical source with equal radii within large errors, while the cylindrical shape of the emission region is clearly seen for nucleus-nucleus collisions with $R_{l}>R_{s}=R_{o}$ within uncertainties. Furthermore the excess of $R_{l}$ over the transverse-plane radii increases with collision energy. As seen in Table 3, the final-state matter in collisions of UHECR particles with atmosphere occupies a noticeable volume at freeze-out even for lightest system interactions ($p+p$, $\mbox{He}+\mbox{He}$) at low boundary energy under consideration. The source radii in the symmetric ${}^{14}\mbox{N}^{7+}$ collisions are comparable in order of magnitude with the space-time extents of the emission region in $\mbox{Cu}+\mbox{Cu}$ collisions at RHIC energies $\sqrt{\smash[b]{s_{NN}}}=$62.4–200 GeV Okorokov-AHEP-2015-790646-2015 , especially for longitudinal axis. The similar relations are valid for the radii in $\mbox{Fe}+\mbox{Fe}$ collisions for ultra-high energy cosmic rays and heavy-ion ($\mbox{Au}+\mbox{Au}$) collisions at RHIC energies $\sqrt{\smash[b]{s_{NN}}}=$62.4–200 GeV Okorokov-AHEP-2015-790646-2015 . Thus, the growth of space extents of the emission region with collision energy expected in the case of the general view of (7) provides the radius values for symmetric ${}^{56}\mbox{Fe}^{26+}$ collisions similar to those in heavy-ion interactions at $\sqrt{\smash[b]{s_{NN}}}\simeq 0.1$ TeV already at low boundary $E_{p}=10^{17}$ eV of the energy domain studied. Therefore the estimations for the emission region geometry shown in Table 3 especially for the general case of (7) at energies close to the GZK limit $E_{p}\gtrsim 10^{19}$ eV prove the collisions of some UHECR light nuclei with air as the source of bulk of the strongly interacting medium as large as in the modern collider experiments with heavy-ion beams at $\sqrt{\smash[b]{s_{NN}}}\simeq 0.1$ TeV.

Global characteristics of the medium created in ultra-high energy small system collisions studied here for \bigl{\{}\mathcal{G}_{Y}^{i}\bigr{\}}_{i=1}^{4} and approach developed elsewhere Okorokov-AHEP-2016-5972709-2016 allow the investigation of lasing behavior for secondary pions in the UHECR energy domain. The results of the calculations with help of the power function (3) at some fixed energies shown by symbols in Fig. 1 are added by the smooth dependencies of the charged particle density on $\sqrt{\smash[b]{s_{NN}}}$ derived with hybrid approximation of scaled $N_{\scriptsize{\mbox{ch}}}$ PRD-93-054046-2016 for the more complete picture. The hypothesis of the Bose–Einstein condensation corresponding to the lasing feature for pion production seems unfavorable in $p+p$ collisions even at highest $E_{p}=10^{21}$ eV (Fig. 1). It cannot be possible to derive the estimations for light nucleus collisions due to the unknown $\langle N_{\scriptsize{\mbox{part}}}\rangle$. However, pion laser regime is supported in Fig. 2 shown for heavy-ion collisions at ultra-high energies for completeness.

IV Conclusions

Summarizing the foregoing, one can draw the following conclusions.

The medium produced in the collisions of the UHECR particles with air is characterized by high energy density at midrapidity and temperatures well above the critical ones for the creation of the quark-gluon matter already at $\sqrt{\smash[b]{s_{NN}}}$ corresponding to the $E_{p}=10^{17}$ eV. BE decoupling time is about 10 fm/$c$ on order of value even in helium nucleus collisions. Therefore for the first time the quantitative analysis of the wide set of global and geometric characteristics strongly indicates that the long-lived medium in quark-gluon phase can be produced in light nuclei collisions at UHECR energies. The particle source created in small system collisions at $E_{p}\gtrsim 10^{19}$ eV is characterized by the large space-time extents which support the hypothesis of the creation of blobs of the quark-gluon matter under extreme conditions in UHECR interactions.

Future experimental and theoretical study for collisions of light nuclei at ultra-high energies are important for the verification of the estimations obtained within the paper.

Acknowledgments

This work was supported in part within the Program for Improving the Competitive Ability of National Research Nuclear University MEPhI (Contract no. 02.a03.21.0005 of August 27, 2013).