Scalar Product Research Articles

Green's function method for the two-dimensional frustrated spin- 12 Heisenberg magnetic lattice

The magnon Hedin's equations are derived via the Schwinger functional derivative technique, and the resulting self-consistent Green's function method is used to calculate ground state spin patterns and magnetic structure factors for 2-dimensional magnetic systems with frustrated spin-1/2 Heisenberg exchange coupling. Compared to random-phase approximation treatments, the inclusion of a self-energy correction improves the accuracy in the case of scalar product interactions, as shown by comparisons between our method and exact benchmarks in homogeneous and inhomogeneous finite systems. We also find that for cross-product interactions (e.g. antisymmetric exchange), the method does not perform equally well, and an inclusion of higher corrections is in order. Aside from indications for future work, our results clearly indicate that the Green's function method in the form proposed here already shows potential advantages in the description of systems with a large number of atoms as well as long-range interactions.

Simplified models for resonant neutral scalar production with missing transverse energy final states

Additional Higgs bosons appear in many extensions of the Standard Model (SM). While most existing searches for additional Higgs bosons concentrate on final states consisting of SM particles, final states containing beyond the SM (BSM) particles play an important role in many BSM models. In order to facilitate future searches for such final states, we develop a simplified model framework for heavy Higgs boson decays to a massive SM boson as well as one or more invisible particles. Allowing one kind of BSM mediator in each decay chain, we classify the possible decay topologies for each final state, taking into account all different possibilities for the spin of the mediator and the invisible particles. Our comparison of the kinematic distributions for each possible model realization reveals that the distributions corresponding to the different simplified model topologies are only mildly affected by the different spin hypotheses, while there is significant sensitivity for distinguishing between the different decay topologies. As a consequence, we point out that expressing the results of experimental searches in terms of the proposed simplified model topologies will allow one to constrain wide classes of different BSM models. The application of the proposed simplified model framework is explicitly demonstrated for the example of a mono-Higgs search. For each of the simplified models that are proposed in this paper we provide all necessary ingredients for performing Monte-Carlo simulations such that they can readily be applied in experimental analyses.

Wind work at the air-sea interface: a modeling study in anticipation of future space missions

Abstract. Wind work at the air-sea interface is the transfer of kinetic energy between the ocean and the atmosphere and, as such, is an important part of the ocean-atmosphere coupled system. Wind work is defined as the scalar product of ocean wind stress and surface current, with each of these two variables spanning, in this study, a broad range of spatial and temporal scales, from 10 km to more than 3000 km and hours to months. These characteristics emphasize wind work's multiscale nature. In the absence of appropriate global observations, our study makes use of a new global, coupled ocean-atmosphere simulation, with horizontal grid spacing of 2–5 km for the ocean and 7 km for the atmosphere, analyzed for 12 months. We develop a methodology, both in physical and spectral spaces, to diagnose three different components of wind work that force distinct classes of ocean motions, including high-frequency internal gravity waves, such as near-inertial oscillations, low-frequency currents such as those associated with eddies, and seasonally averaged currents, such as zonal tropical and equatorial jets. The total wind work, integrated globally, has a magnitude close to 5 TW, a value that matches recent estimates. Each of the first two components that force high-frequency and low-frequency currents, accounts for ∼ 28 % of the total wind work and the third one that forces seasonally averaged currents, ∼ 44 %. These three components, when integrated globally, weakly vary with seasons but their spatial distribution over the oceans has strong seasonal and latitudinal variations. In addition, the high-frequency component that forces internal gravity waves, is highly sensitive to the collocation in space and time (at scales of a few hours) of wind stresses and ocean currents. Furthermore, the low-frequency wind work component acts to dampen currents with a size smaller than 250 km and strengthen currents with larger sizes. This emphasizes the need to perform a full kinetic budget involving the wind work and nonlinear advection terms as small and larger-scale low-frequency currents interact through these nonlinear terms. The complex interplay of surface wind stresses and currents revealed by the numerical simulation motivates the need for winds and currents satellite missions to directly observe wind work.

Entanglement in joint Lambda bar{Lambda } decay; cont.

We have previously investigated joint Lambda bar{Lambda } decay in the reaction e^+ e^- rightarrow gamma Lambda (rightarrow ppi ^-) bar{Lambda }(rightarrow bar{p}pi ^+). The cross-section-distribution functions encountered were relativistically covariant and expressed in terms of scalar products of the four-momentum vectors of the particles involved. In the present, sequel investigation, we work instead in the Lambda bar{Lambda } rest system and with three-momentum scalars. In this configuration our results become directly comparable to those of others, including experiment.

Scalar leptoquarks at the LHC and flavour anomalies: a comparison of pair-production modes at NLO-QCD

We analyse scalar leptoquark pair production at the LHC with predictions including t-channel lepton exchange contributions up to next-to-leading order (NLO) in QCD. In particular, we calculate NLO-QCD predictions for off-diagonal production channels, i.e. channels that involve two different leptoquark eigenstates and are driven solely by diagrams involving Standard Model leptons in the t-channel at leading order, as opposed to diagonal channels where a pair of the same leptoquark eigenstate is produced. We find that reliable theoretical predictions for both channels require NLO accuracy. The relative importance of the off-diagonal modes depends strongly on the considered scenario. In a generic model involving R2 and S3 leptoquarks, at large values of the Yukawa couplings off-diagonal contributions initiated by valence quarks can be up to an order of magnitude higher than the diagonal production. However, we also find that in phenomenologically viable scenarios addressing the flavour anomalies off-diagonal production is generally negligible, with a few exceptions of 10%–30% of the total rate depending on the treatment of the charm density in the proton.

On modified Einstein tensors and two smooth invariants of compact manifolds

Let ( M , g ) (M,g) be a Riemannian n n -manifold, we denote by R i c Ric and S c a l Scal the Ricci and the scalar curvatures of g g . For each real number k > n k>n , the modified Einstein tensors denoted E i n k \mathrm {Ein}_k is defined to be E i n k ≔ S c a l g − k R i c \mathrm {Ein}_k ≔Scal\, g -kRic . Note that the usual Einstein tensor coincides with one half of E i n 2 \mathrm {Ein}_2 and E i n 0 = S c a l . g \mathrm {Ein}_0=Scal.g . It turns out that all these new modified tensors, for 0 > k > n 0>k>n , are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. The positivity of E i n k \mathrm {Ein}_k for some positive k k implies the positivity of all E i n l \mathrm {Ein}_l with 0 ≤ l ≤ k 0\leq l\leq k and so we define a smooth invariant E i n ( M ) \mathbf {Ein}(M) of M M to be the supremum of positive k’s that renders E i n k \mathrm {Ein}_k positive. By definition E i n ( M ) ∈ [ 0 , n ] \mathbf {Ein}(M)\in [0,n] , it is zero if and only if M M has no positive scalar curvature metrics and it is maximal equal to n n if M M possesses an Einstein metric with positive scalar curvature. In some sense, E i n ( M ) \mathbf {Ein}(M) measures how far M M is away from admitting an Einstein metric of positive scalar curvature. In this paper, we prove that E i n ( M ) ≥ 2 \mathbf {Ein}(M)\geq 2 , for any closed simply connected manifold M M of positive scalar curvature and with dimension ≥ 5 \geq 5 . Furthermore, for a compact 2 2 -connected manifold M M with dimension ≥ 6 \geq 6 and of positive scalar curvature, we show that E i n ( M ) ≥ 3 \mathbf {Ein}(M)\geq 3 . We demonstrate as well that the invariant E i n ( M ) \mathbf {Ein} (M) of a manifold M M increases after doing a surgery on M M or by assuming that M M has higher connectivity. We show that the condition E i n ( M ) ≤ n − 2 \mathbf {Ein}(M)\leq n-2 does not imply any restriction on the first fundamental group of M M . We define and prove similar properties for an analogous invariant namely e i n ( M ) \mathbf {ein}(M) . The paper contains several open questions.

Direct fabrication of arbitrary phase masks in optical glass via ultra-short pulsed laser writing of refractive index modifications

We study the possibility to fabricate an arbitrary phase mask in a one-step laser-writing process inside the volume of an optical glass substrate. We derive the phase mask from a Gerchberg–Saxton-type algorithm as an array and create each individual phase shift using a refractive index modification of variable axial length. We realize the variable axial length by superimposing refractive index modifications induced by an ultra-short pulsed laser at different focusing depth. Each single modification is created by applying 1000 pulses with 15 upmuJ pulse energy at 100 kHz to a fixed spot of 25 upmum diameter and the focus is then shifted axially in steps of 10 upmum. With several proof-of-principle examples, we show the feasibility of our method. In particular, we identify the induced refractive index change to about a value of varDelta n = 1.5 cdot 10^{-3}. We also determine our current limitations by calculating the overlap in the form of a scalar product and we discuss possible future improvements.

On factorising twists in AdS3 and AdS2

In this paper we study factorising twists of the massless AdS3 and AdS2 integrable R-matrices, and explore the programme of analysis of form factors which Maillet et al. developed for ordinary spin-chains. We derive the factorising twists from the universal R-matrix of the gl(1|1) Yangian double, and discuss the RTT relations for the two- and three-site monodromy matrix. We show how the twist can be used to compute a simple scalar product. We then implement our construction in the language of free fermions. Finally, we show how to obtain the massless AdS2 quantum R-matrix from the Yangian universal R-matrix, and compute a peculiar factorising twist for this case as well.

Momentum scalar triple product as a measure of chirality in electron ionization dynamics of strongly driven atoms

We formulate a measure that quantifies chirality in single electron ionization triggered in atoms, which are achiral systems. We do so in the context of Ar driven by a different type of optical fields that consists of two noncollinear laser beams giving rise to chirality that varies in space across the focus of the beams. Our computations account for realistic experimental conditions. To define this measure of chirality, we first find the sign of the electron final momentum scalar triple product ${\mathbf{p}}_{x}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{p}}_{y}\ifmmode\times\else\texttimes\fi{}{\mathbf{p}}_{z})$ and multiply it with the probability for an electron to ionize with certain values of the momentum components. Then, we integrate over all values of ${p}_{x}$, ${p}_{y}$, ${p}_{z}$. We show this to be a robust measure of chirality in electron ionization triggered by chiral electric fields.

Calculation of magnetic forces and torques on the Kibble coil

Analytically the force acting on a current-carrying coil in a magnetic field can be calculated in two ways. First, a line integral can be conducted along the coil’s wire, summing up the differential force contributions. Each contribution results from a cross-product of the corresponding differential line segment with the magnetic flux density at that location. Alternatively, the coil’s energy in the field is given as a product of three factors, the number of turns, the current, and the flux threading the coil. The energy can then be obtained by executing a surface integral over the coil’s open surface using the scalar product of the differential surface element with the magnetic flux density as its integrand. The force on the coil is the negative derivative of the energy with respect to the appropriate coordinate. For yoke-based Kibble balances, the latter method is much simpler since most of the flux is contained in the inner yoke of the magnet and can be written as a simple equation. Here, we use this method to provide simple equations and their results for finding the torques and forces that act on a coil in a yoke-based magnet system. We further introduce a straightforward method that allows the calculation of the position and orientation difference between the coil and the magnet from three measurements.

Investigation of the problem of representation of a linear functional in the form of a scoal product

Subject of research: the problem of representing a linear functional in the form of a scalar product in the Hilbert space. For example, this may be the Dirichlet problem for a second-order elliptic equation in variational form.
 Purpose of research: to present a general scheme of the method of iterative extensions for solving elliptic boundary value problems with Dirichlet conditions.
 Methods and objects of research: the object of study in such a problem can be the deformation of the membrane. A continuation of the considered problem in the Hilbert space is given. The extended problem is considered in a subspace of the Hilbert space. The extended problem is studied by the method of iterative extensions in the Euclidean space. An algorithm for implementing the method of iterative extensions is given.
 Main results of research: as a result, it turns out that the general scheme of the method of iterative extensions as applied to the solution of a boundary value problem with Dirichlet conditions for an elliptic equation does not depend on the order of this equation. An example is given to illustrate the conclusion about the efficiency of the method of iterative extensions.

New level of precision in study of neutron beta-decay. Theoretical basics and experimental backgrounds

The article provides a theoretical substantiation for a significant increase in the level of accuracy in determining the neutron lifetime using an alternative concept of neutron beta decay. Neutrons are distributed among different subsets depending on the sign of the scalar product of the neutron spin vector and momentum of the emitted electron during decay. Therefore, the neutron lifetime can be determined separately for each of the subsets as the inverse frequency of its decay, which is to be averaged over the number of neutrons in this subset. The total neutron lifetime on the total set of neutrons is calculated by averaging the partial (basic) lifetimes, considering their weights. The weights of the basic lifetimes are calculated for two weighing methods, leading, respectively, to the so-called lifetime of "nonpolarized" neutrons, the weighted average lifetime of neutrons and the central lifetime, i.e. the arithmetic mean for the basic lifetimes of neutrons. The dependence between different average neutron lifetimes through the integral asymmetry parameter calculated by using known experimental data leads to simple analytical expressions. The numerical estimate of the weighted average neutron lifetime gives a value that is in good agreement with the results of well-known experiments, which proves the validity of the proposed concept of neutron beta decay. This article includes necessary conditions for a new experiment to bring the neutron lifetime determination to a new level of accuracy.

CP-violation, asymmetries and interferences in toverline{t}phi

In this paper, we use the associated production of top-quark pairs ( toverline{t} ) with a generic scalar boson (ϕ) at the LHC (pp → toverline{t}phi ) to explore the sensitivity of a large set of observables to the sign of the CP mixing angle (α), present in the coupling between the scalar boson and the top quarks. The mass of the scalar boson is set to mϕ = 125 GeV (the Standard Model Higgs boson mass) and its coupling to top-quarks is varied such that α = 0°, 22.5°, 45.0°, 67.5°, 90.0°, 135.0° and 180.0°. Dileptonic final states of the toverline{t}phi system are used (pp → bℓ+νℓ overline{b} ℓ− overline{nu} ℓb overline{b} ), where the scalar boson is expected to decay according to ϕ → boverline{b} . A new method to reconstruct the scalar mass, originally designed for the low mass regime is used, improving the resolution of the Higgs mass by roughly a factor of two. A full phenomenological analysis is performed using Standard Model (SM) background and signal events generated with MadGraph5_aMC@NLO, in turn reconstructed using a kinematical fit. The most sensitive CP-observables are selected to compute Confidence Level (CL) limits as a function of the sign of the top quark Yukawa couplings to the ϕ boson. We also explore the sensitivity to interference terms using differential distributions and angular asymmetries. Given the significant difference between the pure scalar (σ0+) and pure pseudo-scalar (σ0−) production cross section values, it is unlikely the toverline{t}phi channel alone will be sensitive to the sign of the CP-mixing angle or interference terms, even at the end of the LHC. Using the {b}_2^{toverline{t}phi } and {b}_4^{toverline{t}phi } variables, exclusion limits at 95% CL for the CP-even and CP-odd components of the top quark Yukawa couplings are expected to be set to overset{sim }{kappa } ∈ [-0.698,+0.698] and |κ| ∈ [0.878,1.04], respectively, at the end of the High Luminosity phase of the LHC (HL-LHC) by using the dileptonic decay channel alone.

Development of a generalized model for parallel-streaming neural element and structures for scalar product calculation devices

Development of a generalized model for parallel-streaming neural element and structures for scalar product calculation devices

Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields

In this paper we continue our derivation of the correlation functions of open quantum spin 1/2 chains with unparallel magnetic fields on the edges; this time for the more involved case of the XXZ spin 1/2 chains. We develop our study in the framework of the quantum separation of variables, which gives us both the complete spectrum characterization and simple scalar product formulae for separate states, including transfer matrix eigenstates. Here, we leave the boundary magnetic field in the first site of the chain completely arbitrary, and we fix the boundary field in the last site N of the chain to be a specific value along the z-direction. This is a natural first choice for the unparallel boundary magnetic fields. We prove that under these special boundary conditions, on the one side, we have a simple enough complete spectrum description in terms of homogeneous Baxter like TQ-equation. On the other side, we prove a simple enough description of the action of a basis of local operators on transfer matrix eigenstates as linear combinations of separate states. Thanks to these results, we achieve our main goal to derive correlation functions for a set of local operators both for the finite and half-infinite chains, with multiple integral formulae in this last case.

Updated constraints on the Georgi-Machacek model and its electroweak phase transition and associated gravitational waves

With theoretical constraints such as perturbative unitarity and vacuum stability conditions and updated experimental data of Higgs measurements and direct searches for exotic scalars at the LHC, we perform an updated scan of the allowed parameter space of the Georgi-Machacek (GM) model. With the refined global fit, we examine the allowed parameter space for inducing strong first-order electroweak phase transitions (EWPTs) and find only the one-step phase transition is phenomenologically viable. Based upon the result, we study the associated gravitational wave (GW) signals and find most of which can be detected by several proposed experiments. We also make predictions on processes that may serve as promising probes to the GM model in the near future at the LHC, including the di-Higgs productions and several exotic scalar production channels.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

A unified structure-based constitutive model for both tension and shear of passive myocardium

This paper presents a hyperelastic model for replicating the passive response of ventricular myocardium and pericardium using fourth-order structural tensors. Models that can accurately predict mechanical response of cardiac samples are useful in understanding diseased tissues or postulating potential biomaterials for the replacement of damaged tissue in surgical applications. An orthotropic model is postulated here and is demonstrated to perform well in recreating experimental results in a variety of planar biaxial testing protocols and six modes of simple shear. The work outlines the fourth-order structural tensor (FOST) approach to representations of hyperelasticity for prescribing individual microstructural components (or interaction between them) that contribute to the overall elastic response of a biological tissue, through selection of specific FOSTs and constructing fourth-order material tensors possessing the constitutive properties and directional biases of that constituent. A fourth-order scalar product is then performed between this and an appropriately chosen fourth-order finite strain measure that returns a physically meaningful contribution to the strain energy density function for the overall tissue. A structure-based model for ventricular myocardium and pericardium is presented in this work using this approach. In the case of human myocardium, a consistent set of seven material constants simultaneously replicates five planar biaxial tension protocols and six modes of simple shear response.

Tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions

The demand for precision predictions in the field of high energy physics has dramatically increased over recent years. Experiments conducted at the LHC, as well as precision measurements at the intensity frontier such as Belle II require equally precise theoretical predictions to make full use of the acquired data. To match the experimental precision, second-, third- and, for certain quantities, even higher-order calculations in perturbative quantum field theory are required.To facilitate such calculations, computer software automating as many steps as possible is required. Yet, each calculation poses different challenges and thus, a high level of configurability is required. In this context we present tapir: a tool for identification, manipulation and minimization of Feynman integral families. It is designed to integrate in toolchains based on the computer algebra system FORM, the use of which is common practice in the field. tapir can be used to reduce the complexity of multi-loop problems with cut-filters, topology mapping, partial fraction decomposition and alike. Program summaryProgram Title:tapirCPC Library link to program files:https://doi.org/10.17632/ptc9t46xyn.1Developer's repository link:https://gitlab.com/tapir-devs/tapirLicensing provisions: GPLv3Programming language:python 3, C++Nature of problem: Multi-loop computations require the automatization of a large number of different tasks related to Feynman integral topologies. Among them are the identification and minimization of integral topologies, partial fraction decomposition of topologies in the case of linearly dependent propagators as well as mapping scalar products of loop momenta to scalar functions.Solution method: The minimization of topologies is performed by comparison of their respective Nickel indices [1], even further minimization utilizes Pak's algorithm [2]. To efficiently map scalar products of loop momenta to scalar functions FORM [3] code is generated.Additional comments including restrictions and unusual features: Minimization based on Pak's algorithm slows down for many lines and scales. A coarser minimization using the Nickel indices, however, is still possible.

Lehre der Trigonometrie anhand realistischer Aufgaben im Online-Unterricht

The aim of our study was to explore the effects of the active use of realistic exercises in the field of trigonometry. We taught a group of 14 pupils, who were in grade 11. The most of them told us they did not plan mathematics-related studies in the future. We included realistic exercises into our teaching plan, which covered the fields of scalar product, as well as the sine and cosine theorems. Our teaching experiment was done within the framework of online teaching. Effects on the motivation, performance and results of the students were taken into consideration. We also attempted to examine the effects of online teaching on motivation and whether the use of realistic exercises is worthwhile in an online classroom environment. Performance of the students showed a tendency of improvement when they were dealing with the material through realistic exercises even despite the teaching happened online.
 Subject Classification: 97C70, 97D40, 97G60