Energy dependent kinetic freeze-out temperature and transverse flow velocity in high energy collisions

Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions and in central nucleus-nucleus collisions over an energy range from a few GeV to above 10 TeV are analyzed by a (two-component) blast-wave model with Boltzmann-Gibbs statistics and with Tsallis statistics respectively. The model results are in similarly well agreement with the experimental data measured by a few productive collaborations who work at the Heavy Ion Synchrotron (SIS), Super Proton Synchrotron (SPS), Relativistic Heavy Ion Collider (RHIC), and Large Hadron Collider (LHC), respectively. The energy dependent kinetic freeze-out temperature and transverse flow velocity are obtained and analyzed. Both the quantities have quick increase from the SIS to SPS, and slight increase or approximate invariability from the top RHIC to LHC. Around the energy bridge from the SPS to RHIC, the considered quantities in proton-proton collisions obtained by the blast-wave model with Boltzmann-Gibbs statistics show more complex energy dependent behavior comparing with the results in other three cases.Comment: 16 pages, 4 figures. The European Physical Journal A, accepted. arXiv admin note: text overlap with arXiv:1805.0334

kkth power residue chains of global fields

In 1974, Vegh proved that if $k$ is a prime and $m$ a positive integer, there is an $m$ term permutation chain of $k$th power residue for infinitely many primes [E.Vegh, $k$th power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that $1,2,2^2,...,2^{m-1}$ is an $m$ term permutation chain of $k$th power residue for infinitely many primes. In this paper, we prove that for any "possible" $m$ term sequence $r_1,r_2,...,r_m$, there are infinitely many primes $p$ making it an $m$ term permutation chain of $k$th power residue modulo $p$, where $k$ is an arbitrary positive integer [See Theorem 1.2]. From our result, we see that Vegh's theorem holds for any positive integer $k$, not only for prime numbers. In fact, we prove our result in more generality where the integer ring $\Z$ is replaced by any $S$-integer ring of global fields (i.e. algebraic number fields or algebraic function fields over finite fields).Comment: 4 page

Group manifold approach to higher spin theory

We consider the group manifold approach to higher spin theory. The deformed local higher spin transformation is realized as the diffeomorphism transformation in the group manifold $\textbf{M}$. With the suitable rheonomy condition and the torsion constraint imposed, the unfolded equation can be obtained from the Bianchi identity, by solving which, fields in $\textbf{M}$ are determined by the multiplet at one point, or equivalently, by $(W^{[a(s-1),b(0)]}_{\mu},H)$ in $AdS_{4}\subset \textbf{M}$. Although the space is extended to $\textbf{M}$ to get the geometrical formulation, the dynamical degrees of freedom are still in $AdS_{4}$. The $4d$ equations of motion for $(W^{[a(s-1),b(0)]}_{\mu},H)$ are obtained by plugging the rheonomy condition into the Bianchi identity. The proper rheonomy condition allowing for the maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss the theory with the global higher spin symmetry, which is in parallel with the WZ model in supersymmetry.Comment: 35 pages,v2: revised version, v3: 38 pages, improved discussion on global HS symmetry, clarifications added in appendix B, journal versio