Right unitarity triangles,stable CP-violating phases and approximate quark–lepton complementarity

Current experimental data indicate that two unitarity triangles of the CKM quark mixing matrix V   are almost the right triangles with α≈90°$\alpha \approx 90°$. We highlight a very suggestive parametrization of V and show that its CP-violating phase ? is nearly equal to α   (i.e., ?−α≈1.1°$?-\alpha \approx 1.1°$). Both ? and α   are stable against the renormalizaton-group evolution from the electroweak scale MZ$M_Z$ to a superhigh energy scale MX$M_X$ or vice versa, and thus it is impossible to obtain α=90°$\alpha =90°$ at MZ$M_Z$ from ?=90°$?=90°$ at MX$M_X$. We conjecture that there might also exist a maximal CP-violating phase φ≈90°$\phi \approx 90°$ in the MNS lepton mixing matrix U. The approximate quark–lepton complementarity relations, which hold in the standard parametrizations of V and U, can also hold in our particular parametrizations of V and U   simply due to the smallness of |V_ub|$|V_{ub}|$ and |V_e3|$|V_{e3}|$.     

Three PT-symmetric Hamiltonians with completely different spectra

We discuss three Hamiltonians, each with a central-field part H₀$H_0$ and a PT-symmetric perturbation igz$igz$. When H₀$H_0$ is the isotropic Harmonic oscillator the spectrum is real for all g$g$ because H$H$ is isospectral to H₀+g²/2$H_0+g^2/2$. When H₀$H_0$ is the Hydrogen atom then infinitely many eigenvalues are complex for all g$g$. If the potential in H₀$H_0$ is linear in the radial variable r$r$ then the spectrum of H$H$ exhibits real eigenvalues for 0<g<g_c$0<g<g_c$ and a PT phase transition at g_c$g_c$.     

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T$T$, a quantum parameter g$g$, and the ratio p=−_J2/_J1$p=-J_2/J_1$, where _J1>0$J_1>0$ refers to ferromagnetic interactions between first-neighbour sites along the d$d$ directions of a hypercubic lattice, and _J2<0$J_2<0$ is associated with competing antiferromagnetic interactions between second neighbours along m≤d$m\le d$ directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0$g=0$ space, with a Lifshitz point at p=1/4$p=1/4$, for d>2$d>2$, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0$T=0$ phase diagram, there is a critical border, _gc=_gc(p)$g_c=g_c\left(p\right)$ for d≥2$d\ge 2$, with a singularity at the Lifshitz point if d<(m+4)/2$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4$p=1/4$.     

Thermodynamics properties of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H$H$ and transverse magnetic field Ω$\Omega$. Using the effective-field theory (EFT) with correlation in cluster with N=1$N=1$ spin we calculate the thermodynamic properties as a function of temperature with values H$H$ and Ω$\Omega$ fixed. The model consists of ferromagnetic interaction _Jx$J_x$ in the x$x$ direction and antiferromagnetic interaction _Jy$J_y$ in the y$y$ direction, and it is found that for H/_Jy∈[0,2]$H/J_y\in [0,2]$ the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$ (isotropic square lattice).     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Pion radiative weak decays in nonlocal chiral quark models

We analyze the radiative pion decay π⁺→e⁺νeγ${\pi }^{+}\to e^{+}{\nu }_e\gamma$ within nonlocal chiral quark models that include wave function renormalization. In this framework we calculate the vector and axial-vector form factors FV$F_V$ and FA$F_A$ at q²=0$q^2=0$ — where q²$q^2$ is the e⁺νe$e^{+}{\nu }_e$ squared invariant mass — and the slope a   of FV(q²)$F_V(q^2)$ at q²→0$q^2\to 0$. The calculations are carried out considering different nonlocal form factors, in particular those taken from lattice QCD evaluations, showing a reasonable agreement with the corresponding experimental data. The comparison of our results with those obtained in the (local) NJL model and the relation of FV$F_V$ and a   with the form factor in π⁰→γ^?γ${\pi }^0\to {\gamma }^{?}\gamma$ decays are discussed.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state _σi(t)∈{0,1}${\sigma }_i(t)\in \{0,1\}$ of a cell i   does not only depend on the states in its local neighborhood at time t-1$t-1$, but also on the memory of its own past states _σi(t-2),_σi(t-3),…,_σi(t-τ),…${\sigma }_i(t-2),{\sigma }_i(t-3),\dots ,{\sigma }_i(t-\tau ),\dots$ . We assume that the weight of this memory decays proportionally to τ^-α${\tau }^{-\alpha }$, with α?0$\alpha ?0$ (the limit α→∞$\alpha \to \infty$ corresponds to the usual CA). Since the memory function is summable for α>1$\alpha >1$ and nonsummable for 0?α?1$0?\alpha ?1$, we expect pronounced changes of the dynamical behavior near α=1$\alpha =1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance H   of initially close trajectories. We typically expect the asymptotic behavior H(t)∝t^1/(1-q)$H(t)\propto t^{1/(1-q)}$, where q   is the entropic index associated with nonextensive statistical mechanics. In all cases, the function q(α)$q(\alpha )$ exhibits a sensible change at α?1$\alpha ?1$. We focus on the class II rules 61, 99 and 111. For rule 61, q=0$q=0$ for 0?α?_αc?1.3$0?\alpha ?{\alpha }_c?1.3$, and q<0$q<0$ for α>_αc$\alpha >{\alpha }_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size N   indicate that the range of the power-law regime for H(t)$H(t)$ typically diverges ∝N^z$\propto N^z$ with 0?z?1$0?z?1$.     

On m-accretive Schrödinger-type operators with singular potentials on Riemannian manifolds

We consider a Schrödinger-type differential expression _HV=∇^∗∇+V$H_V={\nabla }^{\ast }\nabla +V$, where ∇$\nabla$ is a Hermitian connection on a Hermitian vector bundle E$E$ over a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$ and positive smooth measure dμ$d\mu$, and V$V$ is a locally integrable section of the bundle of endomorphisms of E$E$. We give a sufficient condition for m$m$-accretivity of a realization of _HV$H_V$ in L²(E)$L^2\left(E\right)$.     

Singular unitarity in “quantization commutes with reduction”

Let M$M$ be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G$G$. Let M//G=?⁻¹(0)/G=_M0$M//G={?}^{-1}\left(0\right)/G=M_0$ be the symplectic quotient at value 0 of the moment map ?$?$. The space _M0$M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over _M0$M_0$ and the G$G$-invariant subspace of the quantum Hilbert space over M$M$. In this paper, without any regularity assumption on the quotient _M0$M_0$, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck’s constant of a modified map of the above isomorphism under a “metaplectic correction” of the two quantum Hilbert spaces.     

A new entropy based on a group-theoretical structure

A multi-parametric version of the nonadditive entropy S_q$S_q$ is introduced. This new entropic form, denoted by S_a,b,r$S_{a,b,r}$, possesses many interesting statistical properties, and it reduces to the entropy S_q$S_q$ for b=0$b=0$, a=r:=1−q$a=r:=1-q$ (hence Boltzmann–Gibbs entropy S_BG$S_{BG}$ for b=0$b=0$, a=r→0$a=r\to 0$). The construction of the entropy S_a,b,r$S_{a,b,r}$ is based on a general group-theoretical approach recently proposed by one of us, Tempesta (2016). Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of S_a,b,r$S_{a,b,r}$ with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy S_a,b,r$S_{a,b,r}$ can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles N$N$ of the system, or even stabilizes, by increasing N$N$, to a limiting value.     

Constraints on kinematic models from the latest observational data

Kinematical models are constrained by the latest observational data from geometry-distance measurements, which include 557 type Ia supernovae (SNIa) Union2 data and 15 observational Hubble data. Considering two parameterized deceleration parameter, the values of current deceleration parameter q₀$q_0$, jerk parameter j₀$j_0$ and transition redshift zT$z_T$, are obtained. Furthermore, we show the departures for two parameterized kinematical models from ΛCDM model according to the evolutions of jerk parameter j(z)$j(z)$. Also, it is shown that the constraint on jerk parameter j(z)$j(z)$ is weak by the current geometrical observed data.     

Post-breakthrough scaling in reservoir field simulation

We study the oil displacement and production behavior in an isothermal thin layered reservoir model subjected to water flooding. We use the CMG’s (Computer Modelling Group  ) numerical simulators to solve mass balance equations. The influences of the viscosity ratio (m≡_μoil/_μwater$m\equiv {\mu }_{oil}/{\mu }_{water}$) and the inter-well (injector-producer) distance r$r$ on the oil production rate C(t)$C\left(t\right)$ and the breakthrough time _tbr$t_{br}$ are investigated. Two types of reservoir configuration are used, namely one with random porosities and another with a percolation cluster structure. We observe that the breakthrough time follows a power-law of m$m$ and r$r$, _tbr∝r^αm^β$t_{br}\propto r^{\alpha }{m}^{\beta }$, with α=1.8$\alpha =1.8$ and β=−0.25$\beta =-0.25$ for the random porosity type, and α=1.0$\alpha =1.0$ and β=−0.2$\beta =-0.2$ for the percolation cluster type. Moreover, our results indicate that the oil production rate is a power law of time. In the percolation cluster type of reservoir, we observe that P(t)∝t^γ$P\left(t\right)\propto t^{\gamma }$, with γ=−1.81$\gamma =-1.81$, where P(t)$P\left(t\right)$ is the time derivative of C(t)$C\left(t\right)$. The curves related to different values of m$m$ and r$r$ may be collapsed suggesting a universal behavior for the oil production rate.     

Investigation of frustrated and dimerized classical Heisenberg chains

We have considered the 1D dimerized frustrated antiferromagnetic (ferromagnetic) Heisenberg model with arbitrary spin S$S$. The exact classical magnetic phase diagram at zero temperature is determined using the LK cluster method. The cluster method results show that the classical ground-state phase diagram of the model is very rich, including first-order and second-order phase transitions. In the absence of dimerization, a second-order phase transition occurs between antiferromagnetic (ferromagnetic) and spiral phases at the critical frustration _αc=±0.25${\alpha }_c=\pm 0.25$, a well-known result. In the vicinity of the critical points _αc${\alpha }_c$, the exact classical critical exponent of the spiral order parameter is found to be 1/2$1/2$. In the case of a dimerized chain (δ≠0$\delta \ne 0$), the spiral order shows stability and exists in some part of the ground-state phase diagram. We have found two first-order phase boundaries separating antiferromagnetic (uud and duu) phases from the spiral phase.     

What if supersymmetry breaking appears below the GUT scale?

We consider the possibility that the soft supersymmetry-breaking parameters m_1/2$m_{1/2}$ and m₀$m_0$ of the MSSM are universal at some scale Min$M_{\mathrm{in}}$ below the supersymmetric grand unification scale MGUT$M_{\mathrm{GUT}}$, as might occur in scenarios where either the primordial supersymmetry-breaking mechanism or its communication to the observable sector involve a dynamical scale below MGUT$M_{\mathrm{GUT}}$. We analyze the (m_1/2,m₀)$(m_{1/2},m_0)$ planes of such sub-GUT CMSSM models, noting the dependences of phenomenological, experimental and cosmological constraints on Min$M_{\mathrm{in}}$. In particular, we find that the coannihilation, focus-point and rapid-annihilation funnel regions of the GUT-scale CMSSM approach and merge when Min∼¹⁰¹² GeV$M_{\mathrm{in}}\sim {10}^{12}\text{GeV}$. We discuss sparticle spectra and the possible sensitivity of LHC measurements to the value of Min$M_{\mathrm{in}}$.     

Chaotic inflation,radiative corrections and precision cosmology

We employ chaotic (?²${?}^2$ and ?⁴${?}^4$) inflation to illustrate the important role radiative corrections can play during the inflationary phase. Yukawa interactions of ?  , in particular, lead to corrections of the form −κ?⁴ln(?/μ)$-\kappa {?}^4\ln (?/\mu )$, where κ>0$\kappa >0$ and μ   is a renormalization scale. For instance, ?⁴${?}^4$ chaotic inflation with radiative corrections looks compatible with the most recent WMAP (5 year) analysis, in sharp contrast to the tree level case. We obtain the 95% confidence limits 2.4×¹⁰⁻¹⁴?κ?5.7×¹⁰⁻¹⁴$2.4\times {10}^{\mathrm{-14}}?\kappa ?5.7\times {10}^{\mathrm{-14}}$, 0.931?ns?0.958$0.931?n_s?0.958$ and 0.038?r?0.205$0.038?r?0.205$, where ns$n_s$ and r   respectively denote the scalar spectral index and scalar to tensor ratio. The limits for ?²${?}^2$ inflation are κ?7.7×¹⁰⁻¹⁵$\kappa ?7.7\times {10}^{\mathrm{-15}}$, 0.929?ns?0.966$0.929?n_s?0.966$ and 0.023?r?0.135$0.023?r?0.135$. The next round of precision experiments should provide a more stringent test of realistic chaotic ?²${?}^2$ and ?⁴${?}^4$ inflation.     

Fixed points of the SRG evolution and the on-shell limit of the nuclear force

We study the infrared limit of the similarity renormalization group (SRG) using a simple toy model for the nuclear force aiming to investigate the fixed points of the SRG evolution with both the Wilson and the Wegner generators. We show how a fully diagonal interaction at the similarity cutoff λ→0$\lambda \to 0$ may be obtained from the eigenvalues of the Hamiltonian and quantify the diagonalness by means of operator norms. While the fixed points for both generators are equivalent when no bound-states are allowed by the interaction, the differences arising from the presence of the Deuteron bound-state can be disentangled very clearly by analyzing the evolved interactions in the infrared limit λ→0$\lambda \to 0$ on a finite momentum grid. Another issue we investigate is the location on the diagonal of the Hamiltonian in momentum-space where the SRG evolution places the Deuteron bound-state eigenvalue once it reaches the fixed point. This finite momentum grid setup provides an alternative derivation of the celebrated trace identities, as a by product. The different effects due to either the Wilson or the Wegner generators on the binding energies of A=2,3,4$A=2,3,4$ systems are investigated and related to the occurrence of a Tjon-line which emerges as the minimum of an avoided crossing between E_α=4E_t−3E_d$E_{\alpha }=4E_t-3E_d$ and E_α=2E_t$E_{\alpha }=2E_t$. All infrared features of the flow equations are illustrated using the toy model for the two-nucleon S$S$-waves.