Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment

It is shown that the deterministic infinite trigonometric products

$$\begin{aligned} \prod _{n\in \mathbb {N}}\left[ 1- p +p\cos \left( \textstyle n^{-s}_{_{}}t\right) \right] =: {\text{ Cl }_{p;s}^{}}(t) \end{aligned}$$

with parameters \( p\in (0,1]\ \& \ s>\frac{1}{2}\), and variable \(t\in \mathbb {R}\), are inverse Fourier transforms of the probability distributions for certain random series \(\Omega _{p}^\zeta (s)\) taking values in the real \(\omega \) line; i.e. the \({\text{ Cl }_{p;s}^{}}(t)\) are characteristic functions of the \(\Omega _{p}^\zeta (s)\). The special case \(p=1=s\) yields the familiar random harmonic series, while in general \(\Omega _{p}^\zeta (s)\) is a “random Riemann-\(\zeta \) function,” a notion which will be explained and illustrated—and connected to the Riemann hypothesis. It will be shown that \(\Omega _{p}^\zeta (s)\) is a very regular random variable, having a probability density function (PDF) on the \(\omega \) line which is a Schwartz function. More precisely, an elementary proof is given that there exists some \(K_{p;s}^{}>0\), and a function \(F_{p;s}^{}(|t|)\) bounded by \(|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})\), and \(C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi \), such that

$$\begin{aligned} \forall \,t\in \mathbb {R}:\quad {\text{ Cl }_{p;s}^{}}(t) = \exp \bigl ({- C_{p;s}^{} \,|t|^{1/s}\bigr )F_{p;s}^{}(|t|)}; \end{aligned}$$

the regularity of \(\Omega _{p}^\zeta (s)\) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that \(\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) \) for some \(C>0\). Graphical evidence suggests that \({\text{ Cl }_{{{1}/{3}};2}^{}}(t)\) is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of \({\text{ Cl }_{{{1}/{3}};2}^{}}\)), and illustrated by random sampling of the Riemann-\(\zeta \) walks, whose branching rules allow the build-up of fractal-like structures.

     

$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ decay in the aligned two-Higgs-doublet model

In the aligned two-Higgs-doublet model, we perform a complete one-loop computation of the short-distance Wilson coefficients \(C_{7,9,10}^{(\prime )}\), which are the most relevant ones for \(b\rightarrow s\ell ^+\ell ^-\) transitions. It is found that, when the model parameter \(\left| \varsigma _{u}\right| \) is much smaller than \(\left| \varsigma _{d}\right| \), the charged scalar contributes mainly to chirality-flipped \(C_{9,10}^\prime \), with the corresponding effects being proportional to \(\left| \varsigma _{d}\right| ^2\). Numerically, the charged-scalar effects fit into two categories: (A) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), corresponding to the (large \(\left| \varsigma _{u}\right| \), small \(\left| \varsigma _{d}\right| \)) region; (B) \(C_7^\mathrm {H^\pm }\) and \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are sizable, but \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left| \varsigma _{u}\right| \), large \(\left| \varsigma _{d}\right| \)) region. Taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_{s}{\gamma }\), as well as the latest model-independent global analysis of \(b\rightarrow s\ell ^+\ell ^-\) data, we obtain the much restricted parameter space of the model. We then study the impact of the allowed model parameters on the angular observables \(P_2\) and \(P_5'\) of \(B^0\rightarrow K^{*0}\mu ^+\mu ^-\) decay, and we find that \(P_5'\) could be increased significantly to be consistent with the experimental data in case B.     

Uncertainties of the $$B\rightarrow D$$B→ D transition form factor from the D-meson leading-twist distribution amplitude

The \(B\rightarrow D\) transition form factor (TFF) \(f^{B\rightarrow D}_+(q^2)\) is determined mainly by the D-meson leading-twist distribution amplitude (DA) , \(\phi _{2;D}\), if the proper chiral current correlation function is adopted within the light-cone QCD sum rules. It is therefore significant to make a comprehensive study of DA \(\phi _{2;D}\) and its impact on \(f^{B\rightarrow D}_+(q^2)\). In this paper, we calculate the moments of \(\phi _{2;D}\) with the QCD sum rules under the framework of the background field theory. New sum rules for the leading-twist DA moments \(\left\langle \xi ^n\right\rangle _D\) up to fourth order and up to dimension-six condensates are presented. At the scale \(\mu = 2 \,\mathrm{GeV}\), the values of the first four moments are: \(\left\langle \xi ^1\right\rangle _D = -0.418^{+0.021}_{-0.022}\), \(\left\langle \xi ^2\right\rangle _D = 0.289^{+0.023}_{-0.022}\), \(\left\langle \xi ^3\right\rangle _D = -0.178 \pm 0.010\) and \(\left\langle \xi ^4\right\rangle _D = 0.142^{+0.013}_{-0.012}\). Basing on the values of \(\left\langle \xi ^n\right\rangle _D(n=1,2,3,4)\), a better model of \(\phi _{2;D}\) is constructed. Applying this model for the TFF \(f^{B\rightarrow D}_+(q^2)\) under the light cone sum rules, we obtain \(f^{B\rightarrow D}_+(0) = 0.673^{+0.038}_{-0.041}\) and \(f^{B\rightarrow D}_+(q^2_{\mathrm{max}}) = 1.117^{+0.051}_{-0.054}\). The uncertainty of \(f^{B\rightarrow D}_+(q^2)\) from \(\phi _{2;D}\) is estimated and we find its impact should be taken into account, especially in low and central energy region. The branching ratio \(\mathcal {B}(B\rightarrow Dl\bar{\nu }_l)\) is calculated, which is consistent with experimental data.     

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).     

Constraining the electric charges of some astronomical bodies in Reissner–Nordström spacetimes and generic <Emphasis Type="Italic">r</Emphasis><Superscript>−2</Superscript>-type power-law potentials from orbital motions

We put independent model dynamical constraints on the net electric charge Q of some astronomical and astrophysical objects by assuming that their exterior spacetimes are described by the Reissner-Nordström, metric, which induces an additional potential \({U_{\rm RN} \propto Q^2 r^{-2}}\). From the current bounds \({\Delta \dot \varpi}\) on any anomalies in the secular perihelion rate \({\dot \varpi}\) of Mercury and the Earth–mercury ranging Δρ, we have \({\left|Q_{\odot}\right| \lesssim 1-0.4 \times 10^{18}\ {\rm C}}\). Such constraints are ~60–200 times tighter than those recently inferred in literature. For the Earth, the perigee precession of the Moon, determined with the Lunar Laser Ranging technique, and the intersatellite ranging Δρ for the GRACE mission yield \({\left|Q_{\oplus} \right| \lesssim 5-0.4 \times 10^{14}\ {\rm C}}\). The periastron rate of the double pulsar PSR J0737-3039A/B system allows to infer \({\left|Q_{\rm NS} \right| \lesssim 5\times 10^{19}\ {\rm C}}\). According to the perinigricon precession of the main sequence S2 star in Sgr A^*, the electric charge carried by the compact object hosted in the Galactic Center may be as large as \({\left|Q_{\bullet} \right| \lesssim 4\times 10^{27} \ {\rm C}}\). Our results extend to other hypothetical power-law interactions inducing extra-potentials \({U_{\rm pert} = \Psi r^{-2}}\) as well. It turns out that the terrestrial GRACE mission yields the tightest constraint on the parameter \({\Psi}\), assumed as a universal constant, amounting to \({|\Psi| \lesssim 5\times 10^{9}\ {\rm m^4\ s^{-2}}}\).     

The decays of $$\bar{B}^0$$, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of \(\bar{B}^0_s\), \(\bar{B}^0\) and \(B^-\) into \(\eta _c\) plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of \(\bar{B}\) meson into \(\eta _c\) and a \(q\bar{q}\) pair. After hadronization of this \(q\bar{q}\) component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the \(\bar{B}^0\) and \(\bar{B}^0_s\) decays into \(\eta _c\) and \(\rho ^0\), \(K^*\) are evaluated and compared to the \(\eta _c\) and \(\phi \) production. The calculation is based on the postulation that the scalar mesons \(f_0(500)\), \(f_0(980)\) and \(a_0(980)\) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the \(\pi \pi \), \(K \bar{K}\) and \(\pi \eta \) invariant mass distributions for the decays of \(\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0_s \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0 \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta \), \(B^- \rightarrow \eta _c K^0 K^-\) and \(B^- \rightarrow \eta _c \pi ^- \eta \) are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, \(f_0(500), f_0(980)\) and \(a_0(980)\).     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Coupled channels dynamics in the generation of the $$\Omega (2012)$$ resonance

We look into the newly observed \(\Omega (2012)\) state from the molecular perspective in which the resonance is generated from the \(\bar{K} \Xi ^*\), \(\eta \Omega \) and \(\bar{K} \Xi \) channels. We find that this picture provides a natural explanation of the properties of the \(\Omega (2012)\) state. We stress that the molecular nature of the resonance is revealed with a large coupling of the \(\Omega (2012)\) to the \(\bar{K} \Xi ^*\) channel, that can be observed in the \(\Omega (2012) \rightarrow \bar{K} \pi \Xi \) decay which is incorporated automatically in our chiral unitary approach via the use of the spectral function of \(\Xi ^*\) in the evaluation of the \(\bar{K} \Xi ^*\) loop function.     

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

The envelope travelling wave solutions to the Gerdjikov–Ivanov model

Valence universal multireference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of three-body transformed Hamiltonian (\(\widetilde{{H}}_3\)) on ionisation potentials through full connected triple excitations \(S_{3}^{(1,0) }\). \(\widetilde{H}_3 \) is constructed using CCSDT1-A model of Bartlett et al for the ground-state calculation. Contribution of transformed Hamiltonian through full connected triples \(\overline{\widetilde{H}_3 S_3^{\left( {1,0} \right) }}\) involves huge amount of computational operations that is time-consuming. Investigation on \(\hbox {Cl}_{2}\) and \(\hbox {F}_{2}\) molecules using cc-pVDZ and cc-pVTZ basis sets shows that the above effect varies from 0.001 eV to around 0.5 eV, suggesting that inclusion of \(\overline{\widetilde{H} _3 S_3^{\left( {1,0} \right) } }\) is essential for highly accurate calculations.     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

Antihydrogen formation via antiproton scattering on positronium between the $e^{-} +\bar {H}(n = 2)$ and e−+H̄(n=3)$e^{-}+\bar {H}(n = 3)$ thresholds

The charge exchange reaction \(\bar {\mathrm {p}} + \text {Ps} \rightarrow \mathrm {e}^{-} + \bar {\mathrm {H}} \), of interest for the future experiments (GBAR, AEGIS, ATRAP, ...) aiming to produce antihydrogen atoms, is investigated in the energy range between the \(\mathrm {e}^{-}+\bar {\mathrm {H}}(n = 2)\) and \(\mathrm {e}^{-}+\bar {\mathrm {H}}(n = 3)\) thresholds. An ab-initio method based on the solution of the Faddeev-Merkuriev equations is used. Special focus is put on the impact of the Feshbach resonances and the Gailitis-Damburg oscillations, appearing in the vicinity of the \(\bar {\mathrm {p}} +\text {Ps}(n = 2)\) threshold, on the \(\bar {\mathrm {H}}\) production cross section.     

A Note on Quantum Coherence

In this paper, we discuss the coherence of the reduced state in system H _A ?H _B under taking different quantum operations acting on subsystem H _B . Firstly, we show that for a pure bipartite state, the coherence of the final subsystem H _A under the sum of two orthonormal rank 1 projections acting on H _B is less than or equal to the sum of the coherence of the state after two orthonormal projections acting on H _B , respectively. Secondly, we obtain that the coherence of reduced state in subsystem H _A under random unitary channel \({\Phi }(\rho )={\sum }_{s}\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B , is equal to the coherence of the state after each operation \({\Phi }_{s}(\rho )=\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B for every s. In addition, for general quantum operation \({\Phi }(\rho )={\sum }_{s}F_{s}\rho F_{s}^{\ast }\) on H _B , we get the relation

$$ C\left (\left ((I\otimes {\Phi })\rho ^{AB}\right )^{A}\right )\leq \sum \limits _{s}C\left (\left ((I\otimes {\Phi }_{s})\rho ^{AB}\right )^{A}\right ). $$

     

Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma -1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}-1}{\frac{2}{\gamma }-1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1-\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.     

Equilibration in the Kac Model Using the GTW Metric $$d_2$$

We use the Fourier based Gabetta–Toscani–Wennberg metric \(d_2\) to study the rate of convergence to equilibrium for the Kac model in 1 dimension. We take the initial velocity distribution of the particles to be a Borel probability measure \(\mu \) on \(\mathbb {R}^n\) that is symmetric in all its variables, has mean \(\vec {0}\) and finite second moment. Let \(\mu _t(dv)\) denote the Kac-evolved distribution at time t, and let \(R_\mu \) be the angular average of \(\mu \). We give an upper bound to \(d_2(\mu _t, R_\mu )\) of the form \(\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,\) where \(\lambda _1 = \frac{n+2}{2(n-1)}\) is the gap of the Kac model in \(L^2\) and B depends only on the second moment of \(\mu \). We also construct a family of Schwartz probability densities \(\{f_0^{(n)}: \mathbb {R}^n\rightarrow \mathbb {R}\}\) with finite second moments that shows practically no decrease in \(d_2(f_0(t), R_{f_0})\) for time at least \(\frac{1}{2\lambda }\) with \(\lambda \) the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).     

Isospin analysis of charmless <Emphasis Type="Italic">B</Emphasis>-meson decays

We discuss the determination of the CKM angle \(\alpha \) using the non-leptonic two-body decays \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) using the latest data available. We illustrate the methods used in each case and extract the corresponding value of \(\alpha \). Combining all these elements, we obtain the determination \(\alpha _\mathrm{dir}={({86.2}_{-4.0}^{+4.4} \cup {178.4}_{-5.1}^{+3.9})}^{\circ }\). We assess the uncertainties associated to the breakdown of the isospin hypothesis and the choice of the statistical framework in detail. We also determine the hadronic amplitudes (tree and penguin) describing the QCD dynamics involved in these decays, briefly comparing our results with theoretical expectations. For each observable of interest in the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) systems, we perform an indirect determination based on the constraints from all the other observables available and we discuss the compatibility between indirect and direct determinations. Finally, we review the impact of future improved measurements on the determination of \(\alpha \).     

Entanglement Involved in Time Evolution of Two-Mode Squeezed State in Single-Mode Diffusion Channel

We derive the evolution law of an initial two-mode squeezed vacuum state \( \text {sech}^{2}\lambda e^{a^{\dag }b^{\dagger }\tanh \lambda }\left \vert 00\right \rangle \left \langle 00\right \vert e^{ab\tanh \lambda }\) (a pure state) passing through an a-mode diffusion channel described by the master equation

$$\frac{d\rho \left( t\right) }{dt}=-\kappa \left[ a^{\dagger}a\rho \left( t\right) -a^{\dagger}\rho \left( t\right) a-a\rho \left( t\right) a^{\dagger}+\rho \left( t\right) aa^{\dagger}\right] , $$

since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from this channel is a two-mode mixed state. Performing partial trace over the b-mode of ρ(t) yields a new chaotic field, \(\rho _{a}\left (t\right ) =\frac {\text {sech}^{2}\lambda }{1+\kappa t \text {sech}^{2}\lambda }:\exp \left [ \frac {- \text {sech}^{2}\lambda }{1+\kappa t\text {sech}^{2}\lambda }a^{\dagger }a \right ] :,\) which exhibits higher temperature and more photon numbers, showing the diffusion effect. Besides, measuring a-mode of ρ(t) to find n photons will result in the collapse of the two-mode system to a new Laguerre polynomial-weighted chaotic state in b-mode, which also exhibits entanglement.

     

Strong decay of $$\Lambda _c(2940)$$ as a 2 P state in the $$\Lambda _c$$Λc family

Considering the mass, parity and \(D^0 p\) decay mode, we tentatively assign the \(\Lambda _c(2940)\) as the \(P-\)wave states with one radial excitation. Then, via studying the strong decay behavior of the \(\Lambda _c(2940)\) within the \(^3P_0\) model, we obtain that the total decay widths of the \(\Lambda _{c1}(\frac{1}{2}^-,2P)\) and \(\Lambda _{c1}(\frac{3}{2}^-,2P)\) states are 16.27 and 25.39 MeV, respectively. Compared with the experimental total width \(27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}~\mathrm {MeV}\) measured by LHCb Collaboration, both assignments are allowed, and the \(J^P=\frac{3}{2}^-\) assignment is more favorable. Other \(\lambda \)-mode \(\Sigma _c(2P)\) states are also investigated, which are most likely to be narrow states and have good potential to be observed in future experiments.     

Looking for new physics in leptonic and semileptonic decays of <Emphasis Type="Italic">B</Emphasis>-meson

We probe possible new physics (NP) effects beyond the standard model (SM) in the decays \({\overline B ^0} \to \pi \tau \overline \upsilon ,{\overline B ^0} \to \rho \tau \overline \upsilon ,and{\overline B ^0} \to \tau \overline \upsilon \), based on an effective Hamiltonian including non-SM operators. Experimental constraints on different NP scenarios are provided by recent measurements of the ratios \({{R\left( {{D^{\left( * \right)}}} \right) \equiv B\left( {{{\overline B }^0} \to {D^{\left( * \right)}}\tau \overline \upsilon } \right)} \mathord{\left/ {\vphantom {{R\left( {{D^{\left( * \right)}}} \right) \equiv B\left( {{{\overline B }^0} \to {D^{\left( * \right)}}\tau \overline \upsilon } \right)} {B\left( {{{\overline B }^0} \to {D^{\left( * \right)}}\mu \overline \upsilon } \right)}}} \right. \kern-\nulldelimiterspace} {B\left( {{{\overline B }^0} \to {D^{\left( * \right)}}\mu \overline \upsilon } \right)}}\), as well as the branching \(B\left( {{B^ - } \to \tau \overline \upsilon } \right)\). The corresponding hadronic form factors and leptonic decay constants are calculated in the covariant confined quark model developed by us.     

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces \({\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}\). These coordinate algebras are quadratic ones associated with an \(\mathcal {R}\)-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces \({\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}\). Among these, particularly well behaved ones have deformation parameter \(\mathbf{u} \in {\mathbb {S}}^2\). Quotients include seven spheres \({\mathbb {S}}^{7}_\mathbf{u}\) as well as noncommutative quaternionic tori \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3\). There is invariance for an action of \({{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)\) on the torus \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}\) in parallel with the action of \(\mathrm{U}(1) \times \mathrm{U}(1)\) on a ‘complex’ noncommutative torus \({\mathbb {T}}^2_\theta \) which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.