Exploring interacting holographic dark energy in a perturbed universe with parameterized post-Friedmann approach

The model of holographic dark energy in which dark energy interacts with dark matter is investigated in this paper. In particular, we consider the interacting holographic dark energy model in the context of a perturbed universe, which was never investigated in the literature. To avoid the large-scale instability problem in the interacting dark energy cosmology, we employ the generalized version of the parameterized post-Friedmann approach to treating the dark energy perturbations in the model. We use the current observational data to constrain the model. Since the cosmological perturbations are considered in the model, we can then employ the redshift-space distortions (RSD) measurements to constrain the model, in addition to the use of the measurements of expansion history, which has never been done in the literature. We find that, for both the cases with \(Q=\beta H\rho _\mathrm{c}\) and with \(Q=\beta H_0\rho _\mathrm{c}\), the interacting holographic dark energy model is more favored by the current data, compared to the holographic dark energy model without interaction. It is also found that, with the help of the RSD data, a positive coupling \(\beta \) can be detected at the \(2.95\sigma \) statistical significance for the case of \(Q=\beta H_0\rho _\mathrm{c}\).     

$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ 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.     

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 \(p>0\).     

Kinetically modified non-minimal inflation with exponential frame function

We consider supersymmetric (SUSY) and non-SUSY models of chaotic inflation based on the \(\phi ^n\) potential with \(n=2\) or 4. We show that the coexistence of an exponential non-minimal coupling to gravity \(f_\mathcal{R}=\mathrm{e}^{c_\mathcal{R}\phi ^{p}}\) with a kinetic mixing of the form \(f_{\mathrm{K}}=c_{\mathrm{K}}f_\mathcal{R}^m\) can accommodate inflationary observables favored by the Planck and Bicep2/Keck Array results for \(p=1\) and 2, \(1\le m\le 15\) and \(2.6\times 10^{-3}\le r_{\mathcal {R}\mathrm{K}}=c_\mathcal{R}/c_{\mathrm{K}}^{p/2}\le 1,\) where the upper limit is not imposed for \(p=1\). Inflation is of hilltop type and it can be attained for subplanckian inflaton values with the corresponding effective theories retaining the perturbative unitarity up to the Planck scale. The supergravity embedding of these models is achieved employing two chiral gauge singlet supefields, a monomial superpotential and several (semi)logarithmic or semi-polynomial Kähler potentials.     

Shift and broadening of emission lines in Nd<Superscript>3+</Superscript>:YAG laser crystal influenced by input energy

Spectroscopic properties of the flashlamp-pumped Nd ³⁺:YAG laser as a function of input energy were studied over the range of 18–75 J. The spectral widths and shifts of quasi-three-level and four-level inter-Stark emissions within the respective intermanifold transitions of \(^{\mathrm {4}}\mathrm {F}_{\mathrm {3/2}}\to ^{\mathrm {4}}{\kern -2.7pt}\mathrm {I}_{\mathrm {9/2}}\) and \(^{\mathrm {4}}\mathrm {F}_{\mathrm {3/2}}\to ^{\mathrm {4}}{\kern -2.7pt}\mathrm {I}_{\mathrm {11/2}}\) were investigated. The emission lines of \(^{\mathrm {4}}\mathrm {F}_{\mathrm {3/2}}\to ^{\mathrm {4}}{\kern -2.7pt}\mathrm {I}_{\mathrm {9/2}}\) shifted towards longer wavelength (red shift) and broadened, while the positions and linewidths of the \(^{\mathrm {4}}\mathrm {F}_{\mathrm {3/2}}\to ^{\mathrm {4}}{\kern -3.5pt}\mathrm {I}_{\mathrm {11/2}}\) transition lines remained constant by increasing the pumping energy. This is attributed to the thermal population as well as one-phonon and multiphonon emission processes in the ground state. This phenomenon degrades the output performance of the lasers.     

Near Critical Preferential Attachment Networks have Small Giant Components

Preferential attachment networks with power law exponent \(\tau >3\) are known to exhibit a phase transition. There is a value \(\rho _{\mathrm{c}}>0\) such that, for small edge densities \(\rho \le \rho _{\mathrm{c}}\) every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities \(\rho >\rho _{\mathrm{c}}\) there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like \(\exp (-c/ \sqrt{\rho -\rho _{\mathrm{c}}})\) for an explicit constant \(c>0\) depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).     

Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale \(M_\mathrm{in}\) below the supersymmetric grand unification scale \(M_\mathrm{GUT}\), as can occur in mirage mediation and other models. In addition to \(M_\mathrm{in}\), such ‘sub-GUT’ models have the 4 parameters of the CMSSM, namely a common gaugino mass \(m_{1/2}\), a common soft supersymmetry-breaking scalar mass \(m_0\), a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs \(\tan \beta \), assuming that the Higgs mixing parameter \(\mu > 0\). We take into account constraints on strongly- and electroweakly-interacting sparticles from \(\sim 36\)/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for \(M_\mathrm{in}\sim 10^5\) to \(10^9 \,\, \mathrm {GeV}\), with \(M_\mathrm{in}\sim M_\mathrm{GUT}\) disfavoured by \(\Delta \chi ^2 \sim 3\) due to the \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with \(m_{\tilde{\chi }^0_{1}} \sim 1 \,\, \mathrm {TeV}\), with annihilation via heavy Higgs bosons H / A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the \((g-2)_\mu \) constraint, finding similar results whether or not it is included.     

Likelihood analysis of the minimal AMSB model

We perform a likelihood analysis of the minimal anomaly-mediated supersymmetry-breaking (mAMSB) model using constraints from cosmology and accelerator experiments. We find that either a wino-like or a Higgsino-like neutralino LSP, \(\tilde{\chi }^0_{1}\), may provide the cold dark matter (DM), both with similar likelihoods. The upper limit on the DM density from Planck and other experiments enforces \(m_{\tilde{\chi }^0_{1}} \lesssim 3 \,\, \mathrm {TeV}\) after the inclusion of Sommerfeld enhancement in its annihilations. If most of the cold DM density is provided by the \(\tilde{\chi }^0_{1}\), the measured value of the Higgs mass favours a limited range of \(\tan \beta \sim 5\) (and also for \(\tan \beta \sim 45\) if \(\mu > 0\)) but the scalar mass \(m_0\) is poorly constrained. In the wino-LSP case, \(m_{3/2}\) is constrained to about \(900\,\, \mathrm {TeV}\) and \(m_{\tilde{\chi }^0_{1}}\) to \(2.9\pm 0.1\,\, \mathrm {TeV}\), whereas in the Higgsino-LSP case \(m_{3/2}\) has just a lower limit \(\gtrsim 650\,\, \mathrm {TeV}\) (\(\gtrsim 480\,\, \mathrm {TeV}\)) and \(m_{\tilde{\chi }^0_{1}}\) is constrained to \(1.12 ~(1.13) \pm 0.02\,\, \mathrm {TeV}\) in the \(\mu >0\) (\(\mu <0\)) scenario. In neither case can the anomalous magnetic moment of the muon, \((g-2)_\mu \), be improved significantly relative to its Standard Model (SM) value, nor do flavour measurements constrain the model significantly, and there are poor prospects for discovering supersymmetric particles at the LHC, though there are some prospects for direct DM detection. On the other hand, if the \(\tilde{\chi }^0_{1}\) contributes only a fraction of the cold DM density, future LHC Open image in new window -based searches for gluinos, squarks and heavier chargino and neutralino states as well as disappearing track searches in the wino-like LSP region will be relevant, and interference effects enable \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) to agree with the data better than in the SM in the case of wino-like DM with \(\mu > 0\).     

Study of the weak annihilation contributions in charmless $$\varvec{B_s\rightarrow VV}$$ decays

In this paper, in order to probe the spectator-scattering and weak annihilation contributions in charmless \(B_s\rightarrow VV\) (where V stands for a light vector meson) decays, we perform the \(\chi ^2\)-analyses for the endpoint parameters within the QCD factorization framework, under the constraints from the measured \(\bar{B}_{s}\rightarrow \) \(\rho ^0\phi \), \(\phi K^{*0}\), \(\phi \phi \) and \(K^{*0}\bar{K}^{*0}\) decays. The fitted results indicate that the endpoint parameters in the factorizable and nonfactorizable annihilation topologies are non-universal, which is also favored by the charmless \(B\rightarrow PP\) and PV (where P stands for a light pseudo-scalar meson) decays observed in previous work. Moreover, the abnormal polarization fractions \(f_{L,\bot }(\bar{B}_{s}\rightarrow K^{*0}\bar{K}^{*0})=(20.1\pm 7.0)\%,(58.4\pm 8.5)\%\) measured by the LHCb collaboration can be reconciled through the weak annihilation corrections. However, the branching ratio of \(\bar{B}_{s}\rightarrow \phi K^{*0}\) decay exhibits a tension between the data and theoretical result, which dominates the contributions to \(\chi _\mathrm{min}^2\) in the fits. Using the fitted endpoint parameters, we update the theoretical results for the charmless \(B_s\rightarrow VV\) decays, which will be further tested by the LHCb and Belle-II experiments in the near future.     

On non-consensus motions of dynamical linear multiagent systems

The first-principle density functional theory (DFT) calculations were employed to investigate the electronic structures, magnetic properties and half-metallicity of \(\text {Ti}_{2}\text {IrZ}\) (Z \(=\) B, Al, Ga, and In) Heusler alloys with \(\text {AlCu}_{2}\text {Mn}\)- and \(\text {CuHg}_{2}\text {Ti}\)-type structures within local density approximation and generalised gradient approximation for the exchange correlation potential. It was found that \(\text {CuHg}_{2}\text {Ti}\)-type structure in ferromagnetic state was energetically more favourable than \(\text {AlCu}_{2}\text {Mn}\)-type structure in all compounds except \(\text {Ti}_{2}\text {IrB}\) which was stable in \(\text {AlCu}_{2}\text {Mn}\)-type structure in non-magnetic state. \(\text {Ti}_{2}\text {IrZ}\) (Z \(=\) B, Al, Ga, and In) alloys in \(\text {CuHg}_{2}\text {Ti}\)-type structure were half-metallic ferromagnets at their equilibrium lattice constants. Half-metallic band gaps were respectively equal to 0.87, 0.79, 0.75, and 0.73 eV for \(\text {Ti}_{2}\text {IrB}\), \(\text {Ti}_{2}\text {IrAl}\), \(\text {Ti}_{2}\text {IrGa}\), and \(\text {Ti}_{2}\text {IrIn}\). The origin of half-metallicity was discussed for \(\text {Ti}_{2}\text {IrGa}\) using the energy band structure. The total magnetic moments of \(\text {Ti}_{2}\text {IrZ}\) (Z \(=\) B, Al, Ga, and In) compounds in \(\text {CuHg}_{2}\text {Ti}\)-type structure were obtained as \(2\mu _{\mathrm{B}}\) per formula unit, which were in agreement with Slater–Pauling rule (\(M_{\mathrm{tot}} =Z_{\mathrm{tot}}-\)18). All the four compounds were half-metals in a wide range of lattice constants indicating that they may be suitable and promising materials for future spintronic applications.     

Aharonov and Bohm versus Welsh eigenvalues

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the \(\delta \) type and a fixed strength \(\beta \) supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux \(\alpha \in [0,\frac{1}{2}]\) in the center. It is shown that if \(\beta \ne 0\), there is a critical value \(\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})\) such that the discrete spectrum has an accumulation point when \(\alpha <\alpha _{\mathrm {crit}}\), while for \(\alpha \ge \alpha _{\mathrm {crit}}\) the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed \(\alpha \in (0,\frac{1}{2})\) and \(|\beta |\) small enough.     

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 \(\alpha \).     

Rapid communication: $$K_{S}^{0}$$ Production from beryllium target using 120 $$\hbox {GeV}/\hbox {c}$$GeV/c protons beam interactions at the MIPP experiment

We have measured the cross-section for the \(K_{S}^{0}\) production from beryllium target using 120 \(\hbox {GeV}/\hbox {c}\) protons beam interactions at the main injector particle production (MIPP) experiment at Fermilab. The data were collected with target having a thickness of 0.94% of the nuclear interaction length. The \(K_{S}^{0}\) inclusive differential cross-section in bins of momenta is presented covering momentum range from \(0.4\,\hbox {GeV}/\hbox {c}\) to \(30\,\hbox {GeV}/\hbox {c}\). The measured inclusive \(K_{S}^{0}\) production cross-section amounts to \(39.54\pm 1.46\delta _{\mathrm {stat}}\pm 6.97\delta _{\mathrm {syst}}\) mb and the value is compared with the prediction of FLUKA hadron production model.     

Induced-gravity GUT-scale Higgs inflation in supergravity

Models of induced-gravity inflation are formulated within Supergravity employing as inflaton the Higgs field which leads to a spontaneous breaking of a \(U(1)_{B-L}\) symmetry at \(M_\mathrm{GUT}=2\cdot 10^{16}~{\mathrm{GeV}}\). We use a renormalizable superpotential, fixed by a U(1) R symmetry, and Kähler potentials which exhibit a quadratic non-minimal coupling to gravity with or without an independent kinetic mixing in the inflaton sector. In both cases we find inflationary solutions of Starobinsky type whereas in the latter case, others (more marginal) which resemble those of linear inflation arise too. In all cases the inflaton mass is predicted to be of the order of \(10^{13}~{\mathrm{GeV}}\). Extending the superpotential of the model with suitable terms, we show how the MSSM \(\mu \) parameter can be generated. Also, non-thermal leptogenesis can be successfully realized, provided that the gravitino is heavier than about \(10~{\mathrm{TeV}}\).     

One-dimensional backreacting holographic <Emphasis Type="Italic">p</Emphasis>-wave superconductors

We analytically and numerically study the properties of one-dimensional holographic p-wave superconductors in the presence of backreaction. We employ the Sturm–Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the \(\hbox {AdS}_{{3}}\)/\(\hbox {CFT}_{{2}}\) correspondence and determine the relation between the critical temperature \(T_{c}\) and the chemical potential \(\mu \) for different values of the mass m of a charged spin-1 field \(\rho _{\mu }\) and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find that increasing the backreaction and the mass parameter causes the greater values for \({T_{c}}/{\mu }\). Thus, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent \( \beta \) is 1 / 2, which is the same as in the mean field theory. Moreover, both methods confirm the existence of a second order phase transition.     

Revisiting radiative decays of $$1^{+-}$$ heavy quarkonia in the covariant light-front approach

We revisit the calculation of the width for the radiative decay of a \(1^{+-}\) heavy \(Q \bar{Q}\) meson via the channel \(1^{+-} \rightarrow 0^{-+} +\gamma \) in the covariant light-front quark model. We carry out the reduction of the light-front amplitude in the non-relativistic limit, explicitly computing the leading and next-to-leading order relativistic corrections. This shows the consistency of the light-front approach with the non-relativistic formula for this electric dipole transition. Furthermore, the theoretical uncertainty in the predicted width is studied as a function of the inputs for the heavy-quark mass and wave function structure parameter. We analyze the specific decays \(h_{\mathrm{c}}(1P) \rightarrow \eta _{\mathrm{c}}(1S) + \gamma \) and \(h_{\mathrm{b}}(1P) \rightarrow \eta _{\mathrm{b}}(1S) + \gamma \). We compare our results with experimental data and with other theoretical predictions from calculations based on non-relativistic models and their extensions to include relativistic effects, finding reasonable agreement.     

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 \(\Omega =2^{-4},...,2^{7}\).     

Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

We present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schrödinger equation starting from an interacting N-particle system of Bosons. The interaction potential we consider is given by \(W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)\) for some spherically symmetric and compactly supported potential \(W \in L^\infty ({\mathbb {R}}^2, {\mathbb {R}})\). The class of initial wave functions is chosen such that the variance in energy is small. Furthermore, we assume that the Hamiltonian \( H_{W_\beta , t}=-\sum _{j=1}^N \Delta _j+\sum _{1\le j< k\le N} W_\beta (x_j-x_k) +\sum _{j=1}^N A_t(x_j)\) fulfills stability of second kind, that is \( H_{W_\beta , t} \ge -\,CN\). We then prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in either Sobolev trace norm, if \(\Vert A_t\Vert _p < \infty \) for some \(p>2\), or in trace norm, for more general external potentials. For trapping potentials of the form \(A(x)=C |x|^s\; , C>0\), the condition \( H_{W_\beta , t} \ge -\,CN\) can be fulfilled for a certain class of interactions \(W_\beta \), for all \(0< \beta < \frac{s+1}{s+2}\), see Lewin et al. (Proc Am Math Soc 145:2441–2454, 2017).     

The effective neutrino mass of neutrinoless double-beta decays: how possible to fall into a well

The neutrinoless double-beta (\(0\nu 2\beta \)) decay is currently the only feasible process in particle and nuclear physics to probe whether massive neutrinos are the Majorana fermions. If they are of a Majorana nature and have a normal mass ordering, the effective neutrino mass term \(\langle m\rangle ^{}_{ee}\) of a \(0\nu 2\beta \) decay may suffer significant cancellations among its three components and thus sink into a decline, resulting in a “well” in the three-dimensional graph of \(|\langle m\rangle ^{}_{ee}|\) against the smallest neutrino mass \(m^{}_1\) and the relevant Majorana phase \(\rho \). We present a new and complete analytical understanding of the fine issues inside such a well, and identify a novel threshold of \(|\langle m\rangle ^{}_{ee}|\) in terms of the neutrino masses and flavor mixing angles: \(|\langle m\rangle ^{}_{ee}|^{}_* = m^{}_3 \sin ^2\theta ^{}_{13}\) in connection with \(\tan \theta ^{}_{12} = \sqrt{m^{}_1/m^{}_2}\) and \(\rho =\pi \). This threshold point, which links the local minimum and maximum of \(|\langle m\rangle ^{}_{ee}|\), can be used to signify observability or sensitivity of the future \(0\nu 2\beta \)-decay experiments. Given current neutrino oscillation data, the possibility of \(|\langle m\rangle ^{}_{ee}| < |\langle m\rangle ^{}_{ee}|^{}_*\) is found to be very small.