Scalar Part Research Articles

Small parameters in infrared quantum chromodynamics

We study the long-distance properties of quantum chromodynamics in an expansion in powers of the three-gluon, four-gluon, and ghost-gluon couplings, but without expanding in the quark-gluon coupling. This is motivated by two observations. First, the gauge sector is well-described by perturbation theory in the context of a phenomenological model with a massive gluon. Second, the quark-gluon coupling is significantly larger than those in the gauge sector at large distances. In order to resum the contributions of the remaining infinite set of QED-like diagrams, we further expand the theory in $1/N_c$, where $N_c$ is the number of colors. At leading order, this double expansion leads to the well-known rainbow approximation for the quark propagator. We take advantage of the systematic expansion to get a renormalization-group improvement of the rainbow resummation. A simple numerical solution of the resulting coupled set of equations reproduces the phenomenology of the spontaneous chiral symmetry breaking: for sufficiently large quark-gluon coupling constant, the constituent quark mass saturates when its valence mass approaches zero. We find very good agreement with lattice data for the scalar part of the propagator and explain why the vectorial part is poorly reproduced.

Inflation and dark matter in the inert doublet model

We discuss inflation and dark matter in the inert doublet model coupled non-minimally to gravity where the inert doublet is the inflaton and the neutral scalar part of the doublet is the dark matter candidate. We calculate the various inflationary parameters like ns, r and Ps and then proceed to the reheating phase where the inflaton decays into the Higgs and other gauge bosons which are non-relativistic owing to high effective masses. These bosons further decay or annihilate to give relativistic fermions which are finally responsible for reheating the universe. At the end of the reheating phase, the inert doublet which was the inflaton enters into thermal equilibrium with the rest of the plasma and its neutral component later freezes out as cold dark matter with a mass of about 2 TeV.

Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

Modified Rodrigues parameters (MRPs) are triplets in {mathbb {R}}^3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

On the Evaluation of the 4-D Reaction Integral for the Scalar Potential in Galerkin’s Method of Moments

Despite the great progress made in the evaluation of the 4-D reaction integrals arising in the method of moments applied to the surface integral equation formulations, there are still some aspects that deserve to be explored further. The focus in this paper lies in the evaluation of the 4-D reaction integral for the scalar potential. We demonstrate that, unless this integral is computed to machine precision in the whole impedance matrix, it is imperative to maintain the trianglewise balance in the evaluation of the scalar potential part. A small imbalance, even at very small error, results in a loss of degrees of freedom (DOFs) for the charge distribution, violating the null-space property of the divergence operator and yielding potentially spurious solutions. The effect, although present, may go unnoticed at the conventional frequency regimes, where the usual mesh sizes can be applied. However, it becomes more pronounced as we approach the quasi-static limit, eventually rendering spurious solutions at the frequencies higher than the classical low-frequency breakdown would.

General Solution of the Inhomogeneous Div-Curl System and Consequences

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $$\mathbb {R}^3$$ . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $$DW=(Df/f)\overline{W}$$ in $$\mathbb {R}^3$$ , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell’s equations with only variable permeability.

Methods for determining the polarisability of the fine structure levels in the ground state of the thulium atom

We have calculated the scalar and tensor parts of the polarisabilities of the fine-structure sublevels J = 7/2 and 5/2 of the thulium atom ground state. The static polarisabilities are in a good agreement with the values known from the literature. We also present experimental techniques for measuring the dynamic scalar polarisabilities of these levels at a wavelength of 532 nm. The measured values agree with our calculations within the errors. The results of this work can be used for designing an optical frequency standard based on laser-cooled thulium atoms.

Holistic quaternion vector convolution filter for RGB-depth video contour detection

A quaternion vector gradient filter is proposed for RGB-depth (RGB-D) video contour detection. First, a holistic quaternion vector system is introduced to synthetically express the color and depth information, by adding the depth to its scalar part. Then, a convolution differential operator for quaternion vector is proposed to highlight edges with both depth and chromatic variations but restrain the gradient of intensity term. In addition, the quaternion vector gradients are adaptively weighted utilizing depth confidence measure and the quadtree decomposition of the coding tree units in the video streaming. Results on the 3-D high-efficiency video coding test sequences and quantitative simulated experiments on Berkeley segmentation datasets both indicate the availability of the proposed gradient-based method on detecting the semantic contour of the RGB-D videos.

Dynamic equations for an isotropic spherical shell using the power series method and surface differential operators

Dynamic equations for an isotropic spherical shell are derived by using a series expansion technique. The displacement field is split into a scalar (radial) part and a vector (tangential) part. Surface differential operators are introduced to decrease the length of all equations. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using the expansions of the displacement components, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the four of lowest order and to express higher order expansion functions in terms of those of lowest orders. Applying the boundary conditions on the surfaces of the spherical shell and eliminating all but the four lowest order expansion functions give the shell equations as a power series in the shell thickness. After lengthy manipulations, the final four shell equations are obtained in a relatively compact form which are given to second order in shell thickness explicitly. The eigenfrequencies are compared to exact three-dimensional theory with excellent agreement and to membrane theory.

Mass Function of Galaxy Clusters in Relativistic Inhomogeneous Cosmology

The current model ($\Lambda$CDM) with the underlying FLRW metric relies on the assumption of local isotropy, hence homogeneity of the Universe. Difficulties arise when one attempts to justify this model as an average description of the Universe from first principles of general relativity, since in general, the Einstein tensor built from the averaged metric is not equal to the averaged stress--energy tensor. In this context, the discrepancy between these quantities is called cosmological backreaction and has been the subject of scientific debate among cosmologists and relativists for more than $20$ years. Here we present one of the methods to tackle this problem, i.e. averaging the scalar parts of the Einstein equations, together with its application, the mass function of galaxy clusters.

Anisotropy of some FCC transition metals

The norm of elastic constant tensor and the norms of the irreducible parts of the elastic constants of FCC elemental transition metals Nickel, Copper, Silver, Gold, Platinum and Rhodium are calculated. The relation of the scalar parts norm and the other parts norms and the anisotropy of these elements are presented. The norm ratios are used to study anisotropy of these elements

Visualizing Quaternion Multiplication

Quaternion rotation is a powerful tool for rotating vectors in 3-D; as a result, it has been used in various engineering fields, such as navigations, robotics, and computer graphics. However, understanding it geometrically remains challenging, because it requires visualizing 4-D spaces, which makes exploiting its physical meaning intractable. In this paper, we provide a new geometric interpretation of quaternion multiplication using a movable 3-D space model, which is useful for describing quaternion algebra in a visual way. By interpreting the axis for the scalar part of quaternion as a 1-D translation axis of 3-D vector space, we visualize quaternion multiplication and describe it as a combined effect of translation, scaling, and rotation of a 3-D vector space. We then present how quaternion rotation formulas and the derivative of quaternions can be formulated and described under the proposed approach.

Phenomenological description of spin effects in electromagnetic and strong interactions of quarks

Phenomenological description of interactions of relativistic quarks by the Dirac equation with the Cornell potential is given. The general form of the initial equation containing the vector and scalar parts of the Cornell potential is used at the arbitrary connection between these parts. The Hamiltonian in the Foldy-Wouthuysen representation is derived in the general form with allowance for the electromagnetic interactions. Unlike precedent investigations, it is relativistic and exact for terms of the zeroth and first powers in the Planck constant and also for such terms of the second power which describe contact interactions. General quantum mechanical equations of motion for the momentum and the spin are derived and the classical limit of the Hamiltonian and the equations of motion are found for the first time. A connection between the angular velocity of the quark spin precession and the force acting on it is determined. The energy of the spin-orbit interaction is rather high (of the order of 100 MeV). The terms describing the spin-orbit and contact interactions have opposite signs for the scalar and the vector parts of the Cornell potential. The evolution of the quark helicity and the spin-spin interaction of the quarks are also calculated.

Cosmological memory effect

The ``memory effect'' is the permanent change in the relative separation of test particles resulting from the passage of gravitational radiation. We investigate the memory effect for a general, spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmology by considering the radiation associated with emission events involving particle-like sources. We find that if the resulting perturbation is decomposed into scalar, vector, and tensor parts, only the tensor part contributes to memory. Furthermore, the tensor contribution to memory depends only on the cosmological scale factor at the source and observation events, not on the detailed expansion history of the universe. In particular, for sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of ($1+z$).

Diquark mass differences from unquenched lattice QCD**Supported by National Science Foundation of China (11575197, 10835002, 11405178, 11335001), joint funds of NSFC (U1232109), MG and ZL are partially supported by the Youth Innovation Promotion Association of CAS (2015013, 2011013), YC and ZL acknowledge support of NSFC and DFG (CRC110)

We calculate diquark correlation functions in the Landau gauge on the lattice using overlap valence quarks and 2+1-flavor domain wall fermion configurations. Quark masses are extracted from the scalar part of quark propagators in the Landau gauge. The scalar diquark quark mass difference and axial vector scalar diquark mass difference are obtained for diquarks composed of two light quarks and of a strange and a light quark. The light sea quark mass dependence of the results is examined. Two lattice spacings are used to check the discretization effects. The coarse and fine lattices are of sizes 243 × 64 and 323 × 64 with inverse spacings 1/a = 1.75(4) GeV and 2.33(5) GeV, respectively.

Impact of warped extra dimensions on the dipole coefficients in b → sγ transitions

We calculate the electro- and chromomagnetic dipole coefficients $C_{7\gamma,8g}$ and $\tilde C_{7\gamma,8g}$ in the context of the minimal Randall-Sundrum (RS) model with a Higgs sector localized on the IR brane using the five-dimensional (5D) approach, where the coefficients are expressed in terms of integrals over 5D propagators. Since we keep the full dependence on the Yukawa matrices, the integral expressions are formally valid to all orders in $v^2/M_{\rm KK}^2$. In addition we relate our results to the expressions obtained in the Kaluza-Klein (KK) decomposed theory and show the consistency in both pictures analytically and numerically, which presents a non-trivial cross-check. In Feynman-'t Hooft gauge, the dominant corrections from virtual KK modes arise from the scalar parts of the $W^\pm$-boson penguin diagrams, including the contributions from the scalar component of the 5D gauge-boson field and from the charged Goldstone bosons in the Higgs sector. The size of the KK corrections depends on the parameter $y_\ast$, which sets the upper bound for the anarchic 5D Yukawa matrices. We find that for $y_\ast\gtrsim1$ the dominant KK corrections are proportional to $y_\ast^2$. We discuss the phenomenological implications of our results for the branching ratio ${\rm Br}(\bar B\to X_s\gamma)$, the time-dependent CP asymmetry $S_{K^*\gamma}$, the direct CP asymmetry $A_{\rm CP}^{b\to s\gamma}$ and the CP asymmetry difference $\Delta A_{\rm CP}^{b\to s\gamma}$. We can derive a lower bound on the first KK gluon resonance of $3.8\,$TeV for $y_\ast=3$, requiring that at least $10\%$ of the RS parameter space covers the experimental $2\sigma$ error margins. We further discuss the branching ratio ${\rm Br}(\bar B\to X_sl^+l^-)$ and compare our predictions for $C_{7\gamma,9,10}$ and $\tilde C_{7\gamma,9,10}$ with phenomenological results derived from model-independent analyses.

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman–Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman–Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson’s model.

Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering ``degenerate'' Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of scalar-tensor theories. The quartic Horndeski Lagrangian and its extension beyond Horndeski belong to these degenerate cases. We also identify new families of scalar-tensor theories with the property that they are degenerate despite the nondegeneracy of the purely scalar part of their Lagrangian.

On the role of dynamical quark mass generation in chiral symmetry breaking in QCD

The phenomenon of dynamical quark mass generation is studied in QCD within the framework of a gauge invariant formalism. An exact relationship is established between the equation satisfied by the scalar part of the two-point gauge invariant quark Green's function and the quark-antiquark bound state equation in the chiral limit. A possible nontrivial solution of the former yields a massless pseudoscalar solution of the bound state equation with vanishing total momentum. The result is also corroborated by the corresponding Ward-Takahashi identity. The problem is explicitly solved in two-dimensional QCD in the large-Nc limit.

Klein–Gordon and Dirac Equations with Thermodynamic Quantities

We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein-Gordon, and Dirac equations. Our analyze includes two main subsections: ($i$) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts ($ii$) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong ($A \gg 1$). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution to the sum in partition function. We give the analytical results for high temperatures.

CMB anisotropies generated by a stochastic background of primordial magnetic fields with non-zero helicity

We consider the impact of a stochastic background of primordial magnetic fields with non-vanishing helicity on CMB anisotropies in temperature and polarization. We compute the exact expressions for the scalar, vector and tensor part of the energy-momentum tensor including the helical contribution, by assuming a power-law dependence for the spectra and a comoving cutoff which mimics the damping due to viscosity. We also compute the parity-odd correlator between the helical and non-helical contribution which generate the TB and EB cross-correlation in the CMB pattern. We finally show the impact of including the helical term on the power spectra of CMB anisotropies up to multipoles with ℓ ∼ \U0001d4aa(103).