Extending $\mathcal{P}\mathcal{T}$ Symmetry from Heisenberg Algebra to E2 Algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=−iu, [u,v]=0. We can construct the Hamiltonian H=J ²+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT\mathcal{P}\mathcal{T}-symmetric and non-Hermitian Hamiltonian H=J ²+igu, where again g is real. As in the case of PT\mathcal{P}\mathcal{T}-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT\mathcal{P}\mathcal{T}-symmetric Hamiltonian, a region of unbroken PT\mathcal{P}\mathcal{T} symmetry in which all the eigenvalues are real and a region of broken PT\mathcal{P}\mathcal{T} symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.     

The MV formalism for $${\mathrm{IBL}}_\infty $$- and $${\mathrm{BV}}_\infty $$BV∞-algebras

We develop a new formalism for the quantum master equation \(\Delta e^{S/\hbar } = 0\) and the category of \(\mathrm{IBL}_\infty \)-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of \(\mathrm{IBL}_\infty \)-algebras and \(\mathtt{L}_\infty \)-algebras and, on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.     

To Quantify the Difference of η-Inner Products in P T $\mathcal {P}\mathcal {T}$ -Symmetric Theory

International Journal of Theoretical Physics - In this paper, we consider a typical continuous two dimensional $\mathcal {P}\mathcal {T}$ -symmetric Hamiltonian and propose two different approaches...     

$\mathcal{PT}$-Symmetric Quantum Electrodynamics—$\mathcal{PT}$QED

The construction of PT\mathcal{PT}-symmetric quantum electrodynamics is reviewed. In particular, the massless version of the theory in 1+1 dimensions (the Schwinger model) is solved. Difficulties with unitarity of the S-matrix are discussed.     

Conformal classical Yang–Baxter equation,S-equation and $${\mathcal {O}}$$O-operators

Letters in Mathematical Physics - Conformal classical Yang–Baxter equation and S-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In...     

Exact solutions of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by improved $${\textbf{tan}} \left( {\frac{{\boldsymbol{\phi}} \left( {\boldsymbol{\xi}} \right)}{{\textbf{2}}}} \right)$$-expansion method

This paper carries out exact solutions of the perturbed nonlinear Schödinger’s equation withy Kerr law nonlinearity by using the improved tan\(\left( {\frac{\phi \left( \xi \right)}{2}} \right)\)-expansion method. The exact solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be easier and faster by means of symbolic computational system and can be applied to the other nonlinear evolution equations in mathematical physics.     

Vitali $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ Anatol’evich Zverev (On his 80th birthday)

Chronicle

Vitali $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ Anatol’evich Zverev (On his 80th birthday)     

Time-Dependent Vacuum Energy Induced by {\mathcal{D}}–Particle Recoil

We consider cosmology in the framework of a material reference system of particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as 1/t ² for large times t. If this energy density is dominant, the Universe expands with a scale factor R(t)t ². We show that this possibility is compatible with recent observational constraints from high–redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology.     

A complete $\mathcal{O}(\alpha_{\mathrm{S}}^{2})$ calculation of the signal–background interference for the Higgs diphoton decay channel

We analyze a noncommutative model of BTZ spacetime based on deformation of the standard symplectic structure of phase space, i.e., a modification of the standard commutation relations among coordinates and momenta in phase space. We find a BTZ-like solution that is nonperturbative in the non-trivial noncommutative structure. It is shown that the use of deformed commutation relations in the modified non-canonical phase space eliminates the horizons of the standard metric.     

Rogue waves in the nonlocal $${mathcal {PT}}$$ PT -symmetric nonlinear Schrödinger equation

Letters in Mathematical Physics - Rogue waves in the nonlocal $${mathcal {PT}}$$ -symmetric nonlinear Schrödinger (NLS) equation are studied by Darboux transformation. Three types of rogue...     

In memory of Vladimir Mikha $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}$$ lovich Kudryashov (December 11, 1930–February 26, 2003)

Chronicle

In memory of Vladimir Mikha $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ lovich Kudryashov (December 11, 1930–February 26, 2003)     

In memory of Alekse $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ Sergeevich Nikiforov (March 28, 1927–October 1, 2002)

Chronicle

In memory of Alekse $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ Sergeevich Nikiforov (March 28, 1927–October 1, 2002)     

Third-order spectral branch points in Krein space related setups: $$\mathcal{P}\mathcal{T}$$ -symmetric matrix toy model,MHD α 2-dynamo and extended Squire equation

The spectra of self-adjoint operators in Krein spaces are known to possess real sectors as well as sectors of pair-wise complex conjugate eigenvalues. Transitions from one spectral sector to the other are a rather generic feature and they usually occur at exceptional points of square root branching type. For certain parameter configurations two or more such exceptional points may happen to coalesce and to form a higher-order branch point. We study the coalescence of two square root branch points semi-analytically for a -symmetric 4 × 4 matrix toy model and illustrate its occurrence numerically in the spectrum of the 2 × 2 operator matrix of the magneto-hydrodynamic α ²-dynamo and an extended version of the hydrodynamic Squire equation. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Backbending Phenomena in Even–Even $${}^{\boldsymbol{162{-}172}}$$ Hf Isotopes

Physics of Atomic Nuclei - This study probes the backbending phenomena in even–even $${}^{162{-}172}$$ Hf isotopes. Experimental ground-state rotational energies up to $$J^{\pi}=36^{+}$$...     

The 3–3–1 Model with RH Neutrinos and Associated ZH Production at High-Energy ${\bf \textit{e}}^{+}{\bf \textit{e}}^{-}$ Collider

In the context of SU(3) _C ?? SU(3) _L ?? U(1) _X (3–3–1) model with right-handed neutrinos, we study the Higgsstrahlung process e ^?+? e ^???→ZH and calculate the cross section of this process at leading order. Our numerical results showed that the production cross sections for this process can be significantly large as \(M_{Z'}\approx \sqrt{s}\). With reasonable values of the Z′ mass M _Z′, Z′ exchange can generate large corrections to the cross sections of this process, which might be detected in the future high-energy linear e ^?+? e ^??? collider experiments.     

A $$bb{\bar{b}}{\bar{b}}$$ di-bottomonium at the LHC?

We study the case of a di-bottomonium \(bb{\bar{b}}{\bar{b}}\) particle at the LHC. Its mass and decay width are obtained within a new model of diquark–antidiquark interactions in tetraquarks. The same model solves several open problems on the phenomenology of the experimentally better known X, Z states. We show that the \(bb\bar{b}{\bar{b}}\) tetraquark is expected approximately 100 MeV below threshold, and compare to a recent analysis by LHCb seeking it in the \(\Upsilon \mu \mu \) final state.     

The structures and properties of FeSin/FeSi\hbox{$_{\mathsf{n}}^{+}$}+n/FeSi\hbox{$_{\mathsf{n}}^{-}$}−n (n = 1 ~ 8) clusters

The geometry, stability, and electronic properties of iron-doped silicon clusters FeSi _n /FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n (n = 1 ~ 8) have been systematically investigated using the density functional theory (DFT) approach at the B3LYP/6-311+G* level. Our results show that the ground state structures of FeSi _n /FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n change from planar to three-dimensional for n > 3. Bipyramidal structures, or their face-capped isomers, are favored for the larger clusters. For neutral FeSi _n clusters, their ground state structures are the trigonal, tetragonal, capped tetragonal, capped pentagonal, and combined tetragonal bipyramids for n = 4 ~ 8, respectively. The lowest-energy structures of the anionic FeSi\hbox{$_{n}^{-}$}?n clusters essentially retain similar frameworks to their neutral counterparts, while those of the cationic FeSi\hbox{$_{n}^{+}$}+n clusters are significantly deformed; this is confirmed by their calculated ionization potential and electronic affinity values. For most of the stable structures, the spin electronic configurations are s = 1 or 2 for neutral FeSi _n , s = 3/2 or 5/2 for ionic FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n. The average binding energy values generally increase with increasing cluster size, indicating the clusters can continue to gain energy during the growth process. Fragmentation and second-order energy peaks (maxima) are found at n = 2, 5, and 7 for FeSi _n /FeSi\hbox{$_{n}^{-}$}?n, n = 4 and 6 for FeSi\hbox{$_{n}^{+}$}+n, suggesting that these clusters possess higher relative stability. Furthermore, the HOMO-LUMO gap values show that anionic FeSi\hbox{$_{n}^{-}$}?n have greater chemical reactivity than cationic FeSi\hbox{$_{n}^{+}$}+n and neutral FeSi _n , except when n = 7.     

$${\mathcal{N}=2}$$ Superconformal Algebra and the Entropy of Calabi–Yau Manifolds

We use the representation theory of \({\mathcal{N}=2}\) superconformal algebra to study the elliptic genera of Calabi–Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKähler (D ? 3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi–Yau 3-fold has a vanishing entropy. At D > 3, using our previous results on hyperKähler manifolds, we find \({S_{CY_D}\sim 2\pi \sqrt{\frac{(D-3)^2}{2(D-1)}n}}\). When D is even, we find the behavior of CY entropy behaving as \({S_{CY_D}\sim 2 \pi\sqrt{\frac{D-1}{2}n}}\). These agree with Cardy’s formula at large D.     

Consequences of R-parity violating interactions for anomalies in $${\bar{B}}\rightarrow D^{(*)} \tau {\bar{\nu }}$$ and $$b\rightarrow s \mu ^+\mu ^-$$b→sμ+μ-

We investigate the possibility of explaining the enhancement in semileptonic decays of \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\), the anomalies induced by \(b\rightarrow s\mu ^+\mu ^-\) in \({\bar{B}}\rightarrow (K, K^*, \phi )\mu ^+\mu ^-\) and violation of lepton universality in \(R_K = \mathrm{Br}({\bar{B}}\rightarrow K \mu ^+\mu ^-)/\mathrm{Br}({\bar{B}}\rightarrow K e^+e^-)\) within the framework of R-parity violating MSSM. The exchange of down type right-handed squark coupled to quarks and leptons yields interactions which are similar to leptoquark induced interactions that have been proposed to explain the \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\) by tree level interactions and \(b\rightarrow s \mu ^+\mu ^-\) anomalies by loop induced interactions, simultaneously. However, the Yukawa couplings in such theories have severe constraints from other rare processes in B and D decays. Although this interaction can provide a viable solution to the \(R(D^{(*)})\) anomaly, we show that with the severe constraint from \({\bar{B}} \rightarrow K \nu {\bar{\nu }}\), it is impossible to solve the anomalies in the \(b\rightarrow s \mu ^+\mu ^-\) process simultaneously.