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The mass spectrum of different meson particles is generated using an effective Lagrangian of the extended linear-sigma model (eLSM) for scalar and pseudoscalar meson fields and quark flavors, up, down, strange, and charm. Analytical formulas for the masses of scalar, pseudoscalar, vector, and axialvector meson states are derived assuming global chiral symmetry. The various eLSM parameters are analytically deduced and numerically computed. This enables accurate estimations of the masses of sixteen noncharmed and thirteen charmed meson states at vanishing temperature. The comparison of these results to a recent compilation of the particle data group (PDG) allows us to draw the conclusion that the masses of sixteen noncharmed and thirteen charmed meson states calculated in the eLSM are in good agreement with the PDG. This shows that the eLSM, with its configurations and parameters, is an effective theoretical framework for determining the mass spectra of various noncharmed and charmed meson states, particularly at vanishing temperature.

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

In the QCD-like effective model, the Polyakov linear-sigma model, the isospin sigma field (σ¯3=fK±−fK0) and the third generator of the matrix of the explicit symmetry breaking [h3=ma02fK±−fK0] are estimated in terms of the decay constants of the neutral (fK0) and charged Kaon (fK±) and the mass of a0 meson. Both quantities σ¯3 and h3 are then evaluated, at finite baryon (μB), isospin chemical potential (μI), and temperature (T). Thereby, the dependence of the critical temperature on isospin chemical potential could be mapped out in the (T−μI) phase diagram In the QCD-like effective model, the Polyakov linear-sigma model, the isospin sigma field (σ¯3=fK±−fK0) and the third generator of the matrix of the explicit symmetry breaking [h3=ma02fK±−fK0] are estimated in terms of the decay constants of the neutral (fK0) and charged Kaon (fK±) and the mass of a0 meson. Both quantities σ¯3 and h3 are then evaluated, at finite baryon (μB), isospin chemical potential (μI), and temperature (T). Thereby, the dependence of the critical temperature on isospin chemical potential could be mapped out in the (T−μI) phase diagram. The in-medium modifications of pseudoscalars (Jpc=0−+), scalars (Jpc=0++), vectors (Jpc=1−−), and axial-vectors (Jpc=1++) meson states are then analyzed in thermal and dense medium. We conclude that the QCD phase diagram (T−μI) is qualitatively similar to the (T−μB) phase diagram. We also conclude that both temperature and isospin chemical potential enhance the in-medium modifications of the meson states a0, σ, η′, π, f0, κ, η, K, ρ, ω, κ*, ϕ, a1, f1, K*, and f1*. Regarding their chemical potential, at high temperatures the various meson states likely dissolve into colored partonic phase. In this limit, the meson masses form a universal bundle. Thus, we conclude that the increase in the chemical potential similar to temperature derives the colorless confined meson states into the colored deconfined parton phase.

At finite isospin chemical potential μI, the tension between measured decays and partial branching ratios of neutral and charged bosons as functions of dimuon mass squared and the Standard Model (SM) isospin asymmetry can be analyzed in nonperturbative QCD-effective models, for instance, the Polyakov linear sigma-model. With almost first-principle derivation of the explicit isospin symmetry breaking, namely, σ¯3=fK±−fK0 the isospin sigma field, and h3=ma02fK±−fK0 the third generator of the matrix of the explicit symmetry breaking H=Taha. fK± and fK0 are decay constants of K± and K0, respectively. ma0 is the mass of a0 meson. Accordingly, the QCD phase structure could be extended to finite μI. With the thermal and density dependence of a0, fK±, and fK0, σ¯3 and h3 are accordingly expressed in dependence on the temperatures and the chemical potentials. We find that the resulting critical chiral temperatures Tχ decrease with the increase in μB and/or μI. We conclude that the (Tχ−μI) boundary has almost the same structure as that of the (Tχ−μB) plane.

Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise accurate measurement of the first fundamental form ds2=gμνdxμdxν, the author derived a quantum-induced revision of the fundamental tensor. To this end, the four-dimensional Riemann manifold is extended to the eight-dimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime; the minimum measurable length could be imposed ad hoc on the Finsler structure. The present script introduces an approach to quantize the fundamental tensor and first fundamental form. Based on gravitized quantum mechanics, the resulting relativistic generalized uncertainty principle (RGUP) is directly imposed on the Finsler structure, F(x^0μ,p^0ν), which is obviously homogeneous to one degree in p^0μ. The momentum of a test particle with mass m¯=m/mp with mp is the Planck mass. This unambiguously results in the quantized first fundamental form ds˜2=[1+(1+2βp^0ρp^0ρ)m¯2(|x¨|/A)2]gμνdx^μdx^ν, where x¨ is the proper spacelike four-acceleration, A is the maximal proper acceleration, and β is the RGUP parameter. We conclude that an additional source of curvature associated with the mass m¯, whose test particle is accelerated at |x¨|, apparently emerges. Thereby, quantizations of the fundamental tensor and first fundamental form are feasible.

We analyzed the transverse momentum spectra (pT) reported by the NA61/SHINE and NA49 experiments in inelastic proton–proton (pp) and central Lead–Lead (Pb−Pb), Argon–Scandium (Ar−Sc), and Beryllium–Beryllium (Be−Be) collisions with the Blast-wave model with Boltzmann–Gibbs (BWBG) statistics. The BGBW model was in good agreement with the experimental data. We were able to extract the transverse flow velocity (βT), the kinetic freeze-out temperature (T0), and the kinetic freeze-out volume (V) from the pT spectra using the BGBW model. Furthermore, we also obtained the initial temperature (Ti) and the mean transverse momentum () by the alternative method. We observed that T0 increases with increasing collision energy and collision cross-section, representing the colliding system’s size. The transverse flow velocity was observed to remain invariant with increasing collision energy, while it showed a random change with different collision cross-sections. In the same way, the kinetic freeze-out volume and mean transverse momentum increased with an increase in collision energy or collision cross-section. The same behavior was also seen in the freeze-out temperature, which increased with increasing collision cross-sections. At chemical freeze-out, we also determined both the chemical potential and temperature and compared these with the hadron resonance gas model (HRG) and different experimental data. We report that there is an excellent agreement with the HRG model and various experiments, which reveals the ability of the fit function to manifest features of the chemical freeze-out.

It is conjectured that in cosmological applications the particle current is not modified but finite heat or energy flow. Therefore, comoving Eckart frame is a suitable choice, as it merely ceases the charge and particle diffusion and conserves charges and particles. The cosmic evolution of viscous hadron and parton epochs in casual and non-casual Eckart frame is analyzed. By proposing equations of state deduced from recent lattice QCD simulations including pressure p, energy density ρ, and temperature T, the Friedmann equations are solved. We introduce expressions for the temporal evolution of the Hubble parameter H˙, the cosmic energy density ρ˙, and the share η˙ and the bulk viscous coefficient ζ˙. We also suggest how the bulk viscous pressure Π could be related to H. We conclude that the relativistic theory of fluids, the Eckart frame, and the finite viscous coefficients play essential roles in the cosmic evolution, especially in the hadron and parton epochs.

Based on recent perturbative and non-perturbative lattice calculations with almost quark flavors and the thermal contributions from photons, neutrinos, leptons, electroweak particles, and scalar Higgs bosons, various thermodynamic quantities, at vanishing net-baryon densities, such as pressure, energy density, bulk viscosity, relaxation time, and temperature have been calculated up to the TeV-scale, i.e., covering hadron, QGP, and electroweak (EW) phases in the early Universe. This remarkable progress motivated the present study to determine the possible influence of the bulk viscosity in the early Universe and to understand how this would vary from epoch to epoch. We have taken into consideration first- (Eckart) and second-order (Israel–Stewart) theories for the relativistic cosmic fluid and integrated viscous equations of state in Friedmann equations. Nonlinear nonhomogeneous differential equations are obtained as analytical solutions. For Israel–Stewart, the differential equations are very sophisticated to be solved. They are outlined here as road-maps for future studies. For Eckart theory, the only possible solution is the functionality, H(a(t)), where H(t) is the Hubble parameter and a(t) is the scale factor, but none of them so far could to be directly expressed in terms of either proper or cosmic time t. For Eckart-type viscous background, especially at finite cosmological constant, non-singular H(t) and a(t) are obtained, where H(t) diverges for QCD/EW and asymptotic EoS. For non-viscous background, the dependence of H(a(t)) is monotonic. The same conclusion can be drawn for an ideal EoS. We also conclude that the rate of decreasing H(a(t)) with increasing a(t) varies from epoch to epoch, at vanishing and finite cosmological constant. These results obviously help in improving our understanding of the nucleosynthesis and the cosmological large-scale structure.