International Journal of Modern Physics A

For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)=\frac{1}{g^2}\widehat{V}(gr)$ the perturbation theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wave function $y$. It is shown that PT in powers of $g$ developed in RB equation leads to Taylor expansion of $y$ at small $r$ while being developed in GB equation leads to a new form of semiclassical expansion at large $(gr)$: it coincides with loop expansion in path integral formalism. In complementary way PT for large $g$ developed in RB equation leads to an expansion of $y$ at large $r$ and developed in GB equation leads to an expansion at small $(gr)$. Interpolating all four expansions for $y$ leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at $r\in [0,\infty )$ for any coupling constant $g\ge 0$ and dimension $D>0$. As a concrete application, the low-lying states of the cubic anharmonic oscillator $V=r^2+g{r}^3$ are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is $\lesssim 10^{-4}$ for $r\in [0,\infty )$ for coupling constant $g\ge 0$ and dimension $D=1,2,\dots .$ In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for $g\ge 0$ and $D=1,2,\dots .$

Quantum chromodynamics is a fundamental non-Abelian gauge theory of strong interactions. The physical quantum chromodynamics vacuum state is a linear superposition of the $n$-vacua states with different topological numbers. Because of the configuration of the gauge fields, the tunneling events can induce the local parity-odd domains. Those interactions that occur in these domains can be affected by these effects. Considering the hadron (nucleon) system, we introduce the parity-odd parton distribution functions in order to describe the parity-odd structures inside the hadron in this paper. We obtain 8 parity-odd parton distribution functions at leading twist for spin-1/2 hadrons and present their properties. By introducing the parity-odd quark–quark correlator, we find the parity-odd effects vanish from the macroscopic point of view. In this paper, we consider the high energy semi-inclusive deeply inelastic scattering process to investigate parity-odd effects by calculating the spin asymmetries.

The Schrödinger spectrum of a hydrogen atom, modeled as a two-body system consisting of a point electron and a point proton, changes when the usual Coulomb interaction between point particles is replaced with an interaction which results from a modification of Maxwell’s law of the electromagnetic vacuum. Empirical spectral data thereby impose bounds on the theoretical parameters involved in such modified vacuum laws. In the present paper the vacuum law proposed, in the 1940s, by Bopp, Landé–Thomas, and Podolsky (BLTP) is scrutinized in such a manner. The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature — the Bopp length ${\varkappa }^{-1}$ —, to the effect that the electrostatic pair interaction deviates significantly from Coulomb’s law only for distances much shorter than ${\varkappa }^{-1}$. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, $E_{\ell ,n}(\varkappa )$, for all $\ell \in \{0,1,2,\dots \}$ and $n>\ell$. The energy levels $E_{0,1}(\varkappa )$, $E_{0,2}(\varkappa )$, and $E_{1,2}(\varkappa )$ are also computed numerically and plotted versus ${\varkappa }^{-1}$. It is found that the BLTP theory predicts a nonrelativistic correction to the splitting of the Lyman-$\alpha$ line in addition to its well-known relativistic fine-structure splitting. Under the assumption that this splitting does not go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-$\alpha$ line suggest that ${\varkappa }^{-1}$ must be smaller than $\approx 10^{-18}\mathrm{\text{m}}$. Finite proton size effects are found not to modify this conclusion. As a consequence, the electrostatic field energy of an elementary point charge, although finite in BLTP electrodynamics, is much larger than the empirical rest mass ($\times c^2$) of an electron. If, as assumed in all “renormalized theories” of the electron, the empirical rest mass of a physical electron is the sum of its bare rest mass and its electrostatic field energy, then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.

We discuss a recently found family of exact and analytic, finite and accelerating, $(1+1)$-dimensional solutions of perfect fluid relativistic hydrodynamics to describe the pseudorapidity densities and longitudinal HBT-radii and to estimate the lifetime parameter and the initial energy density of the expanding fireball in Au+Au collisions at RHIC with $\sqrt{s_{NN}}=130$ GeV and 200 GeV colliding energies. From these exact solutions of relativistic hydrodynamics, we derive a simple and powerful formula to describe the pseudorapidity density distributions in high-energy proton–proton and heavy-ion collisions, and derive the scaling of the longitudinal HBT radius parameter as a function of the pseudorapidity density. We improve upon several oversimplifications in Bjorken’s famous initial energy density estimate, and apply our results to estimate the initial energy densities of high-energy reactions with data-driven pseudorapidity distributions. When compared to similar estimates at the LHC energies, our results indicate a surprising and nonmonotonic dependence of the initial energy density on the energy of heavy-ion collisions.

A fully canonical quantization of electromagnetic field in the presence of a bi-anisotropic absorbing magneto-dielectric cylindrical shell is provided. The mode expansions of the dynamical quantum fields, contained in the theory, is achieved and the ladder operators of the system are introduced. Using the Frobenius’s series technique, the Maxwell’s equations in the presence of the bi-anisotropic absorbing magneto-dielectric cylindrical shell are solved and the space–time dependence of the quantized electromagnetic field is obtained. Applying the conservation principle of the angular momentum, the net quantum vacuum torque exerted on the bi-anisotropic absorbing magneto-dielectric cylindrical shell is calculated. The net quantum vacuum torque exerted on the cylindrical shell is calculated in the vacuum state and the thermal state of the system. The quantum vacuum torque on the cylindrical shell identically vanishes when the bi-anisotropic absorbing magneto-dielectric cylindrical shell is converted to an isotropic one.

We calculate the $U$(1) axial-anomaly in two and four dimensions using a modified path integral method coupled to a Pauli–Villars regulator field in the noncommutative QED. Pauli–Villars regularization method provides us with unambiguous way to connect the modified path integral formalism with perturbative axial Ward identities at each step of calculations.

In this paper, we assign the newly reported state $X(3842)$ to be a $D$-wave $\bar{c}c$ meson, and study its mass and decay constant with QCD sum rules by considering the contributions of vacuum condensates up to dimension-6 in the operator product expansion. The predicted mass $M_{X(3842)}=\left(\right.3.844_{-0.0823}^{+0.0675}\pm 0.020\left)\right.$ GeV is in agreement with the experimental data $M_{X(3842)}=(3842.71\pm 0.16\pm 0.12)$ MeV from the LHCb collaboration. This result supports assigning $X(3842)$ as the $1^3{D}_3$ charmonium meson. In this case, its predicted strong decay width with the ${}^3{P}_0$ decay model is compatible with the experimental data.