Dimensional Crossover in Anisotropic Percolation on $${\mathbb {Z}}^{d+s}$$

We consider bond percolation on \({\mathbb {Z}}^d\times {\mathbb {Z}}^s\) where edges of \({\mathbb {Z}}^d\) are open with probability \(p<p_c({\mathbb {Z}}^d)\) and edges of \({\mathbb {Z}}^s\) are open with probability q, independently of all others. We obtain bounds for the critical curve in (p, q), with p close to the critical threshold \(p_c({\mathbb {Z}}^d)\). The results are related to the so-called dimensional crossover from \({\mathbb {Z}}^d\) to \({\mathbb {Z}}^{d+s}\).     

Universality for 1d Random Band Matrices: Sigma-Model Approximation

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of \(W\times W\) random Gaussian blocks (parametrized by \(j,k \in \Lambda =[1,n]^d\cap \mathbb {Z}^d\)) with a fixed entry’s variance \(J_{jk}=\delta _{j,k}W^{-1}+\beta \Delta _{j,k}W^{-2}\), \(\beta >0\) in each block. Taking the limit \(W\rightarrow \infty \) with fixed n and \(\beta \), we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit \(\beta , n\rightarrow \infty \), we prove that in the dimension \(d=1\) the behaviour of the sigma-model approximation in the bulk of the spectrum, as \(\beta \gg n\), is determined by the classical Wigner–Dyson statistics.     

A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs

We consider oriented long-range percolation on a graph with vertex set \({\mathbb {Z}}^d \times {\mathbb {Z}}_+\) and directed edges of the form \(\langle (x,t), (x+y,t+1)\rangle \), for x, y in \({\mathbb {Z}}^d\) and \(t \in {\mathbb {Z}}_+\). Any edge of this form is open with probability \(p_y\), independently for all edges. Under the assumption that the values \(p_y\) do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on \({\mathbb {Z}}^d\).     

Discrete One-Dimensional Coverage Process on a Renewal Process

Consider the following coverage model on \(\mathbb {N}\), for each site \(i \in \mathbb {N}\) associate a pair \((\xi _i, R_i)\) where \((\xi _i)_{i \ge 0}\) is a 1-dimensional undelayed discrete renewal point process and \((R_i)_{i \ge 0}\) is an i.i.d. sequence of \(\mathbb {N}\)-valued random variables. At each site where \(\xi _i=1\) start an interval of length \(R_i\). Coverage occurs if every site of \(\mathbb {N}\) is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.     

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations \({\varGamma }^+ \times {\varGamma }^-\) over \({\mathbb {R}}^d\). Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator \(L^+(\gamma ^-) + \frac{1}{\varepsilon }L^-\), \(\varepsilon > 0\). Here \(L^-\) describes the environment process on \({\varGamma }^-\) and \(L^+(\gamma ^-)\) describes the system process on \({\varGamma }^+\), where \(\gamma ^-\) indicates that the corresponding birth-and-death rates depend on another locally finite configuration \(\gamma ^- \in {\varGamma }^-\). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states \(\mu _t^{\varepsilon }\) on \({\varGamma }^+ \times {\varGamma }^-\). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let \(\mu _{\mathrm {inv}}\) be the invariant measure for the environment process on \({\varGamma }^-\). In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of \(\mu _t^{\varepsilon }\) onto \({\varGamma }^+\) converges weakly to an evolution of states on \({\varGamma }^+\) associated with the averaged Markov birth-and-death operator \({\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)\).     

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields \(\psi \) and \(\phi \) defined on a countable set Z. For instance, \(Z=\mathbb {Z}^d\) or \(Z=\mathbb {Z}^d\times I\), where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of \(\psi \) and \(\phi \) enjoy a certain decay property, then all joint cumulants between \(\psi \) and \(\phi \) are \(\ell _2\)-summable in the precise sense described in the text. The decay assumption for the cumulants of \(\psi \) and \(\phi \) is a restricted \( \ell _1\) summability condition called \(\ell _1\)-clustering property. One immediate application of the results is given by a stochastic process \(\psi _t(x)\) whose state is \(\ell _1\)-clustering at any time t: then the above estimates can be applied with \(\psi =\psi _t\) and \(\phi =\psi _0\) and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any \(\ell _1\)-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green–Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants     

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.     

A relation between the positive and negative spectra of elliptic operators

We study the spectral properties of pairs of operators \(-\Delta \pm V\) and show that if their negative spectra are discrete, then their essential spectra fill the positive semi-axis. Analogous statements are proved for more general operators of the form \(m(i\nabla )\pm V\) as well as for operators on the lattice \(\mathbb {Z}^d\).     

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).     

Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function \(l:(0,\infty ) \rightarrow [0,\infty )\) to create the random field \(\Psi (y)=\sum _{x\in \eta }l(|x-y|),\) where \(\eta \) is a homogeneous Poisson process in \({\mathbb {R}}^d.\) The field \(\Psi \) is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets \(\Psi _{\ge h}(y)\) containing the points \(y\in {\mathbb {R}}^d\) such that \(\Psi (y)\ge h.\) In the case where l has unbounded support, we give, for any \(d\ge 2,\) a necessary and sufficient condition on l for \(\Psi _{\ge h}(y)\) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is \(\Psi \) almost surely. Moreover, in this case and for \(d=2,\) we prove uniqueness of the infinite component of \(\Psi _{\ge h}\) when such exists, and we also show that the so-called percolation function is continuous below the critical value \(h_c\).     

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.     

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 \(a_0(980)\).     

Two Types of Maximally Entangled Bases and Their Mutually Unbiased Property in ℂd⊗ℂd′

We first construct a new maximally entangled basis in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\ (k\in Z^{+})\) which is diffrent from the one in Tao et al. (Quantum Inf. Process. 14, 2291 (2015)), then we generalize such maximally entangled basis into arbitrary bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}\). We also study the mutual unbiased property of the two types of maximally entangled bases in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\). In particular, explicit examples in \(\mathbb {C}^{2} \otimes \mathbb {C}^{4}\), \(\mathbb {C}^{2} \otimes \mathbb {C}^{8}\) and \(\mathbb {C}^{3} \otimes \mathbb {C}^{3}\) are presented.     

Modularity of logarithmic parafermion vertex algebras

The parafermionic cosets \(\mathsf {C}_{k} = {\text {Com}} ( \mathsf {H} , \mathsf {L}_{k}(\mathfrak {sl}_{2}) )\) are studied for negative admissible levels k, as are certain infinite-order simple current extensions \(\mathsf {B}_{k}\) of \(\mathsf {C}_{k}\). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to \(\mathsf {C}_{k}\), irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-modules are obtained from those of \(\mathsf {L}_{k}(\mathfrak {sl}_{2})\). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible \(\mathsf {B}_{k}\)-modules. The irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the \(\mathsf {B}_{k}\) are \(C_2\)-cofinite vertex operator algebras.     

CP violations in predictive neutrino mass structures

We study the CP-violation effects from two types of neutrino mass matrices with (i) \((M_\nu )_{ee}=0\), and (ii) \((M_\nu )_{ee}=(M_\nu )_{e\mu }=0\), which can be realized by the high-dimensional lepton number violating operators \(\bar{\ell }_R^c\gamma ^\mu L_L (D_\mu \Phi )\Phi ^2\) and \(\bar{\ell }_R^c l_R (D_\mu {\Phi })^2\Phi ^2\), respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range \(0.002\,\mathrm{eV}\lesssim m_0\lesssim 0.007\,\mathrm{eV}\). Furthermore, for a given value of \(m_0\), there are two solutions for the two Majorana phases \(\alpha _{21}\) and \(\alpha _{31}\), whereas the Dirac phase \(\delta \) is arbitrary. For (ii), the parameters of \(m_0\), \(\delta \), \(\alpha _{21}\), and \(\alpha _{31}\) can be completely determined. We calculate the CP-violating asymmetries in neutrino–antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.     

Determinations of $$|V_{cb}|$$ and $$|V_{ub}|$$|Vub| from baryonic $$\Lambda _b$$Λb decays

We present the first attempt to extract \(|V_{cb}|\) from the \(\Lambda _b\rightarrow \Lambda _c^+\ell \bar{\nu }_\ell \) decay without relying on \(|V_{ub}|\) inputs from the B meson decays. Meanwhile, the hadronic \(\Lambda _b\rightarrow \Lambda _c M_{(c)}\) decays with \(M=(\pi ^-,K^-)\) and \(M_c=(D^-,D^-_s)\) measured with high precisions are involved in the extraction. Explicitly, we find that \(|V_{cb}|=(44.6\pm 3.2)\times 10^{-3}\), agreeing with the value of \((42.11\pm 0.74)\times 10^{-3}\) from the inclusive \(B\rightarrow X_c\ell \bar{\nu }_\ell \) decays. Furthermore, based on the most recent ratio of \(|V_{ub}|/|V_{cb}|\) from the exclusive modes, we obtain \(|V_{ub}|=(4.3\pm 0.4)\times 10^{-3}\), which is close to the value of \((4.49\pm 0.24)\times 10^{-3}\) from the inclusive \(B\rightarrow X_u\ell \bar{\nu }_\ell \) decays. We conclude that our determinations of \(|V_{cb}|\) and \(|V_{ub}|\) favor the corresponding inclusive extractions in the B decays.     

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 \(\Omega =2^{-4},...,2^{7}\).     

Monojet signatures from heavy colored particles: future collider sensitivities and theoretical uncertainties

In models with colored particle \(\mathcal {Q}\) that can decay into a dark matter candidate X, the relevant collider process \(pp\rightarrow \mathcal {Q}\bar{\mathcal {Q}}\rightarrow X\bar{X}\,+\,\)jets gives rise to events with significant transverse momentum imbalance. When the masses of \(\mathcal {Q}\) and X are very close, the relevant signature becomes monojet-like, and Large Hadron Collider (LHC) search limits become much less constraining. In this paper, we study the current and anticipated experimental sensitivity to such particles at the High-Luminosity LHC at \(\sqrt{s}=14\) TeV with \(\mathcal {L}=3\) ab\(^{-1}\) of data and the proposed High-Energy LHC at \(\sqrt{s}=27\) TeV with \(\mathcal {L}=15\) ab\(^{-1}\) of data. We estimate the reach for various Lorentz and QCD color representations of \(\mathcal {Q}\). Identifying the nature of \(\mathcal {Q}\) is very important to understanding the physics behind the monojet signature. Therefore, we also study the dependence of the observables built from the \(pp\rightarrow \mathcal {Q}\bar{\mathcal {Q}} + j \) process on \(\mathcal {Q}\) itself. Using the state-of-the-art Monte Carlo suites MadGraph5_aMC@NLO+Pythia8 and Sherpa, we find that when these observables are calculated at NLO in QCD with parton shower matching and multijet merging, the residual theoretical uncertainties are comparable to differences observed when varying the quantum numbers of \(\mathcal {Q}\) itself. We find, however, that the precision achievable with NNLO calculations, where available, can resolve this dilemma.     

Two Higgs doublet models and $$b\rightarrow s$$ exclusive decays

We computed the leading order Wilson coefficients relevant to all the exclusive \(b\rightarrow s\ell ^+\ell ^-\) decays in the framework of the two Higgs doublet model (2HDM) with a softly broken \(\mathbb {Z}_2\) symmetry by including the \(\mathcal {O}(m_b)\) corrections. We elucidate the issue of appropriate matching between the full and the effective theory when dealing with the (pseudo-)scalar operators for which keeping the external momenta different from zero is necessary. We then make a phenomenological analysis by using the measured \({\mathcal {B}}(B_s\rightarrow \mu ^+\mu ^-)\) and \({\mathcal {B}}(B\rightarrow K \mu ^+\mu ^-)_{\mathrm {high}-q^2}\), for which the hadronic uncertainties are well controlled, and we discuss their impact on various types of 2HDM. A brief discussion of the decays with \(\tau \)-leptons in the final state is provided too.