International Journal of Modern Physics A

In addition to the diagonalization of a normal matrix by a unitary similarity transformation, there are two other types of diagonalization procedures that sometimes arise in quantum theory applications — the singular value decomposition and the Autonne–Takagi factorization. In this pedagogical review, each of these diagonalization procedures is performed for the most general $2\times 2$ matrices for which the corresponding diagonalization is possible, and explicit analytical results are provided in each of the three cases.

The explanation of the small neutrino mass can be depicted using some handsome models like type-I and inverse seesaw where the Standard Model gauge singlet heavy right-handed neutrinos are deployed. The common thing in these two models is a lepton number violating parameter, however, its order of magnitude creates a striking difference between them making the nature of the right-handed heavy neutrinos a major play factor. In the type-I seesaw a large lepton number violating parameter involves the heavy right-handed neutrinos in the form of Majorana fermions while a small lepton number violating parameter being involved in the inverse seesaw demands the pseudo-Dirac nature of the heavy right-handed neutrinos. Such heavy neutrinos are accommodated in these models through the sizable mixings with the Standard Model light neutrinos. In this paper we consider the purely inverse seesaw scenario to study the pair production of the pseudo-Dirac heavy neutrinos followed by their various multilepton decay modes through the leading branching fraction at the leading order and next-to-leading order QCD at the LHC with a center-of-mass energy of 13 TeV and a luminosity of 3000 fb${}^{-1}$. We also consider a prospective 100 TeV hadron collider with luminosities of 3000 fb${}^{-1}$ and 30,000 fb${}^{-1}$, respectively to study the process. Using anomalous multilepton search performed by the CMS at the 8 TeV with 19.5 fb${}^{-1}$ luminosity we show prospective search reaches of the mixing angles for the three lepton and four lepton events at the 13 TeV LHC and 100 TeV hadron collider.

This work aims to investigate the wormhole solutions in the background of $f(R,\phi ,X)$ theory of gravity, where $R$ is Ricci scalar, $\phi$ is scalar potential, and $X$ is the kinetic term. We consider spherically symmetric static space–time for exploring the wormhole geometry with anisotropic fluid. For our current analysis, we consider a particular equation of state parameter to study the behavior of traceless fluid and examine the physical behavior of energy density and pressure components. Furthermore, we also choose a particular shape function and explore the energy conditions. It can be noticed that energy conditions are violated for both shape functions. The violation of energy conditions indicates the existence of exotic matter and wormhole. Therefore, it can be concluded that our results are stable and realistic. The interesting feature of this work is to show two- and three-dimensional plotting for the analysis of wormhole geometry.

The problem of determining the main parameters of light nuclei from precision atomic spectroscopy is considered. Within the framework of the quasipotential method in quantum electrodynamics, the energy interval $(2S-1S)$ in muonic ions of lithium, beryllium and boron is calculated. Corrections of orders ${\alpha }^3÷{\alpha }^6$, which are determined by relativistic effects, effects of vacuum polarization, nuclear structure and recoil, as well as combined corrections, including the above, are taken into account. Nuclear structure effects are expressed in terms of the nuclear charge radius in the case of one-photon interaction and the electromagnetic form factors of nuclei in the case of two-photon interaction. The obtained numerical values for the $(2S-1S)$ interval can be used for comparison with future experimental data and for a more accurate determination of the nucleus charge radii.

We analyze the recent results of next-to-next-to-leading (NNLO) singlet BFKL eigenvalue in $N=4$ SYM written in terms of harmonic sums. The nested harmonic sums building known NNLO BFKL eigenvalue for specific values of the conformal spin have poles at negative integers. We sort the harmonic sums according to the complexity with respect to their weight and depth and use their pole decomposition in terms of the reflection identities to find the most complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the conformal spin. The obtained result is compatible with the Bethe–Salpeter approach to the BFKL evolution.

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are ${\mathrm{\Delta }}_1$-dual each other.

The adaptive perturbation method decomposes a Hamiltonian by the diagonal elements and nondiagonal elements of the Fock state. The diagonal elements of the Fock state are solvable but can contain the information about coupling constants. We study the harmonic oscillator with the interacting potential, ${\lambda }_1{x}^4/6+{\lambda }_2{x}^6/120$, where ${\lambda }_1$ and ${\lambda }_2$ are coupling constants, and $x$ is the position operator. In this study, each perturbed term has an exact solution. We demonstrate the accurate study of the spectrum and $〈x^2〉$ up to the next leading-order correction. In particular, we study a similar problem of Higgs field from the inverted mass term to demonstrate the possible nontrivial application of particle physics.

We have analyzed the Barrow holographic dark energy (BHDE) in the framework of flat FLRW universe by considering the various estimations of Barrow exponent △. Here, we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein–Hawking. To understand the recent accelerated expansion of the universe, consider the Hubble horizon as the IR cutoff. The cosmological parameters, especially the density parameter $({\mathrm{\Omega }}_D)$, the equation of the state parameter $({\omega }_D)$, energy density $({\rho }_D)$ and the deceleration parameter $(q)$ are studied in this paper and found the satisfactory behaviors. Moreover we additionally focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_m(z)$ to discriminant BHDE model from the $\mathrm{\Lambda }$CDM model. Here we determined and plotted the trajectories of evolution for statefinder $(r,s)$, $(r,q)$ and $O_m(z)$ diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent △, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime and exhibits the phantom-divide line during the cosmological evolution.