Terms Of Matrix Elements Research Articles

D → π and D → K semileptonic form factors with Nf = 2 + 1 + 1 twisted mass fermions

We present a lattice determination of the vector and scalar form factors of the D → π(K)lv semileptonic decays, which are relevant for the extraction of the CKM matrix elements |Vcd| and |Vcs| from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 +1 flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. The matrix elements of both vector and scalar currents are determined for a plenty of kinematical conditions in which parent and child mesons are either moving or at rest. Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data and included in the decomposition of the current matrix elements in terms of additional form factors. After the extrapolations to the physical pion mass and to the continuum limit the vector and scalar form factors are determined in the whole kinematical region from q2 = 0 up to [see formula in PDF] accessible in the experiments, obtaining a good overall agreement with experiments, except in the region at high values of q2 where some deviations are visible.

Forces in Molecules. New Quantum Relations

New quantum-mechanical expressions are obtained for forces between the particles in molecules. A new interpretation of quantum transitions in molecular isomeric structures is proposed in terms of matrix elements of force operators.

Scalar and vector form factors of D→π(K)ℓν decays with Nf=2+1+1 twisted fermions

We present a lattice determination of the vector and scalar form factors of the $D\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\ell}\ensuremath{\nu}$ and $D\ensuremath{\rightarrow}K\ensuremath{\ell}\ensuremath{\nu}$ semileptonic decays, which are relevant for the extraction of the CKM matrix elements $|{V}_{cd}|$ and $|{V}_{cs}|$ from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with ${N}_{f}=2+1+1$ flavors of dynamical quarks, at three different values of the lattice spacing ($a\ensuremath{\simeq}0.062,0.082,0.089\text{ }\text{ }\mathrm{fm}$) and with pion masses as small as 210 MeV. Quark momenta are injected on the lattice using nonperiodic boundary conditions. The matrix elements of both vector and scalar currents are determined for plenty of kinematical conditions in which parent and child mesons are either moving or at rest. Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data and included in the decomposition of the current matrix elements in terms of additional form factors. After the extrapolations to the physical pion mass and to the continuum limit, we determine the vector and scalar form factors in the whole kinematical region from ${q}^{2}=0$ up to ${q}_{\mathrm{max}}^{2}=({M}_{D}\ensuremath{-}{M}_{\ensuremath{\pi}(K)}{)}^{2}$ accessible in the experiments, obtaining a good overall agreement with experiments, except in the region at high values of ${q}^{2}$ where some deviations are visible. A set of synthetic data points, representing our results for ${f}_{+}^{D\ensuremath{\pi}(K)}({q}^{2})$ and ${f}_{0}^{D\ensuremath{\pi}(K)}({q}^{2})$ for several selected values of ${q}^{2}$, is provided and also the corresponding covariance matrix is available. At zero four-momentum transfer, we get ${f}_{+}^{D\ensuremath{\rightarrow}\ensuremath{\pi}}(0)=0.612(35)$ and ${f}_{+}^{D\ensuremath{\rightarrow}K}(0)=0.765(31)$. Using the experimental averages for $|{V}_{cd}|{f}_{+}^{D\ensuremath{\rightarrow}\ensuremath{\pi}}(0)$ and $|{V}_{cs}|{f}_{+}^{D\ensuremath{\rightarrow}K}(0)$, we extract $|{V}_{cd}|=0.2330(137)$ and $|{V}_{cs}|=0.945(38)$, respectively. The second row of the CKM matrix is found to be in agreement with unitarity within the current uncertainties: $|{V}_{cd}{|}^{2}+|{V}_{cs}{|}^{2}+|{V}_{cb}{|}^{2}=0.949(78)$.

An approach to establish the connection between configuration and su(n + 1) algebraic spaces in molecular physics: application to ammonia

ABSTRACTAn approach aimed to connect configuration and algebraic spaces is discussed. This approach emerges as the need to translate a vibrational description in configuration space to an algebraic representation based on unitary dynamical algebras, where a straightforward connection does not exist e.g. the vibron model case. Our method is based on the mapping of the algebraic to configuration states, a premise that allows arbitrary operators in configuration to be expanded in terms of generators of the dynamical algebra. The coefficients are determined through a minimisation procedure and given in terms of matrix elements defined in configuration space. We apply the general formalism to the Morse potential, representing anharmonic vibrations in a molecule, as a benchmark case where a dynamical symmetry exits, and to the symmetric double-Morse potential, representing vibrations that can tunnel through a potential barrier, as an example in which a dynamical symmetry is not present. We discuss how the tunnelling effect in the double Morse can be described very simply in the su(2) algebraic representation, taking the ammonia inversion vibrational spectrum as an example.

Electrostatic limit of the T-matrix for electromagnetic scattering: Exact results for spheroidal particles

The T-matrix, often obtained with Waterman’s extended boundary condition method (EBCM), is a widely-used tool for fast calculations of electromagnetic scattering by particles. Here we investigate the quasistatic or long-wavelength limit of this approach, where it reduces to an electrostatics problem. We first present a fully electrostatic version of the EBCM/T-matrix method (dubbed ES-EBCM). Explicit expressions are then given to quantitatively express the long-wavelength limit of the EBCM matrix elements in terms of those of the ES-EBCM formalism. From this connection we deduce a number of symmetry properties of the ES-EBCM matrices. We then investigate the matrix elements of the ES-EBCM formalism in the special case of prolate spheroids. Using the general electrostatic solution in spheroidal coordinates, we derive fully analytic expressions (in the form of finite sums) for all matrix elements. Those can be used for example for studies of the convergence of the T-matrix formalism. We illustrate this by discussing the validity of the Rayleigh hypothesis, where analytical expressions highlight clearly the link with analytical continuation of series.

Nonresonant two-photon transitions in length and velocity gauges

We reexamine the invariance of two-photon transition matrix elements and corresponding two-photon Rabi frequencies under the "gauge" transformation from the length to the velocity gauge. It is shown that gauge invariance, in the most general sense, only holds at exact resonance, for both one-color as well as two-color absorption. The arguments leading to this conclusion are supported by analytic calculations which express the matrix elements in terms of hypergeometric functions, and ramified by a "master identity" which is fulfilled by off-diagonal matrix elements of the Schroedinger propagator under a the transformation from the velocity to the length gauge. The study of the gauge dependence of atomic processes highlights subtle connections between the concept of asymptotic states, the gauge transformation of the wave function, and infinitesimal damping parameters for perturbations and interaction Hamiltonians that switch off the terms in the infinite past and future [of the form exp(-epsilon * | t |)]. We include a pertinent discussion.

Effects of modal dispersion on few-photon–qubit scattering in one-dimensional waveguides

We study one- and two-photon scattering from a qubit embedded in a one-dimensional waveguide in the presence of modal dispersion. We use a resolvent based analysis and utilize techniques borrowed from the Lee model studies. Modal dispersion leads to atom-photon bound states which necessitate the use of multichannel scattering theory. We present multichannel scattering matrix elements in terms of the solution of a Fredholm integral equation of the second kind. Through the use of the Lippmann-Schwinger equation, we derive an infinite series of Feynman diagrams that represent the solution to the integral equation. We use the Feynman diagrams as vertex correction terms to come up with closed form formulas that successfully predict the trapping rate of a photon in the atom-photon bound state. We verify our formalism through Krylov-subspace based numerical studies with pulsed excitations. Our results provide the tools to calculate the complex correlations between scattered photons in a dispersive environment.

Lattice QCD study of the Boer-Mulders effect in a pion

The three-dimensional momenta of quarks inside a hadron are encoded in transverse momentum-dependent parton distribution functions (TMDs). This work presents an exploratory lattice QCD study of a TMD observable in the pion describing the Boer-Mulders effect, which is related to polarized quark transverse momentum in an unpolarized hadron. Particular emphasis is placed on the behavior as a function of a Collins-Soper evolution parameter quantifying the relative rapidity of the struck quark and the initial hadron, e.g., in a semi-inclusive deep inelastic scattering (SIDIS) process. The lattice calculation, performed at the pion mass m_pi = 518 MeV, utilizes a definition of TMDs via hadronic matrix elements of a quark bilocal operator with a staple-shaped gauge connection; in this context, the evolution parameter is related to the staple direction. By parametrizing the aforementioned matrix elements in terms of invariant amplitudes, the problem can be cast in a Lorentz frame suited for the lattice calculation. In contrast to an earlier nucleon study, due to the lower mass of the pion, the calculated data enable quantitative statements about the physically interesting limit of large relative rapidity. In passing, the similarity between the Boer-Mulders effects extracted in the pion and the nucleon is noted.

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra suq(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.

Lattice QCD calculations of transverse momentum-dependent parton distributions (TMDs)

An ongoing program of evaluating TMD observables within Lattice QCD is reviewed, summarizing recent progress with respect to several challenges faced by such calculations. These lattice calculations are based on a definition of TMDs through hadronic matrix elements of quark bilocal operators containing staple-shaped gauge connections. A parametrization of the matrix elements in terms of invariant amplitudes serves to cast them in the Lorentz frame preferred for a lattice calculation. Data on the naively T-odd Sivers and Boer-Mulders effects as well as the transversity TMD are presented.

Exchange Effects in the Radiative Capture Reactions3H(α, γ)7Li and3He(α, γ)7Be

The mirror 3 H(α, γ) 7 Li and 3 He(α, γ) 7 Be reactions have been considered using the algebraic versions of the resonating group model and of the orthogonality conditions model. Exchange effects in interaction of the colliding nuclei and influence of the corresponding exchange terms in matrix elements of the interaction potential on calculated astrophysical S-factors for the reactions have been studied. At the present time one of the most important aims of nuclear reaction theory is to build so called ab initio approaches to description of nuclear reactions, which use realistic NN-potentials and exactly account for all exchange effects caused by the antisymmetry of a wave function of nuclear system. Such approaches are in continuous progress but, nevertheless, turn out to be very complicated procedures, which are, in fact, unrealizable for sufficiently heavy colliding nuclei. Though, there are alternative approaches suitable for heavier systems. They are based on the resonating group model (RGM) [1] or its algebraic version (AVRGM) [2] with effective NN-potentials and, as above mentioned approaches, rigorously allow for the exchange effects. However, complexities rapidly grow even in such approaches at increasing system mass. Thus, a development of reliable approximations within the existing microscopic approaches is rather topical problem of nuclear reaction theory. Such approximations will allow to simplify the calculations significantly and even to make ones realizable for sufficiently complicated systems.

Transition matrices and orbitals from reduced density matrix theory.

In this contribution, we report two different methodologies for characterizing the electronic structure reorganization occurring when a chromophore undergoes an electronic transition. For the first method, we start by setting the theoretical background necessary to the reinterpretation through simple tensor analysis of (i) the transition density matrix and (ii) the natural transition orbitals in the scope of reduced density matrix theory. This novel interpretation is made more clear thanks to a short compendium of the one-particle reduced density matrix theory in a Fock space. The formalism is further applied to two different classes of excited states calculation methods, both requiring a single-determinant reference, that express an excited state as a hole-particle mono-excited configurations expansion, to which particle-hole correlation is coupled (time-dependent Hartree-Fock/time-dependent density functional theory) or not (configuration interaction single/Tamm-Dancoff approximation). For the second methodology presented in this paper, we introduce a novel and complementary concept related to electronic transitions with the canonical transition density matrix and the canonical transition orbitals. Their expression actually reflects the electronic cloud polarisation in the orbital space with a decomposition based on the actual contribution of one-particle excitations from occupied canonical orbitals to virtual ones. This approach validates our novel interpretation of the transition density matrix elements in terms of the Euclidean norm of elementary transition vectors in a linear tensor space. A proper use of these new concepts leads to the conclusion that despite the different principles underlying their construction, they provide two equivalent excited states topological analyses. This connexion is evidenced through simple illustrations of (in)organic dyes electronic transitions analysis.

Lattice QCD Studies of Transverse Momentum-Dependent Parton Distribution Functions

Transverse momentum-dependent parton distributions (TMDs) relevant for semi-inclusive deep inelastic scattering and the Drell–Yan process can be defined in terms of matrix elements of a quark bilocal operator containing a staple-shaped gauge link. Such a definition opens the possibility of evaluating TMDs within lattice QCD. By parametrizing the aforementioned matrix elements in terms of invariant amplitudes, the problem can be cast in a Lorentz frame suited for the lattice calculation. Results for selected TMD observables are presented, including a particular focus on their dependence on a Collins–Soper-type evolution parameter, which quantifies proximity of the staple-shaped gauge links to the light cone.

SO(4) algebraic approach to the three-body bound state problem in two dimensions

We use the permutation symmetric hyperspherical three-body variables to cast the non-relativistic three-body Schrödinger equation in two dimensions into a set of (possibly decoupled) differential equations that define an eigenvalue problem for the hyper-radial wave function depending on an SO(4) hyper-angular matrix element. We express this hyper-angular matrix element in terms of SO(3) group Clebsch-Gordan coefficients and use the latter's properties to derive selection rules for potentials with different dynamical/permutation symmetries. Three-body potentials acting on three identical particles may have different dynamical symmetries, in order of increasing symmetry, as follows: (1) S3 ⊗ OL(2), the permutation times rotational symmetry, that holds in sums of pairwise potentials, (2) O(2) ⊗ OL(2), the so-called “kinematic rotations” or “democracy symmetry” times rotational symmetry, that holds in area-dependent potentials, and (3) O(4) dynamical hyper-angular symmetry, that holds in hyper-radial three-body potentials. We show how the different residual dynamical symmetries of the non-relativistic three-body Hamiltonian lead to different degeneracies of certain states within O(4) multiplets.

Complete set of observables for photoproduction of two pseudoscalars on a nucleon

The problem of determining completely the spin amplitudes of photoproduction of two pseudoscalar mesons on a nucleon from observables is studied. The procedure of reconstruction of the scattering matrix elements from a complete set of observables is based on the expressions of all observables as quadratic Hermitian forms in the reaction matrix elements which are derived explicitly. Their inversion allows one to find explicit solutions for the reaction matrix elements in terms of observables. Two methods for finding a complete set of observables are presented. In particular, one set was found that does not contain a triple polarization observable.

Balitsky-JIMWLK evolution equation at NLO

Wilson line operators are infinite gauge factors ordered along the straight lines of the fast moving particles. Scattering amplitudes of proton-Nucleus or Nucleus-Nucleus collisions at high-energy are written in terms of matrix elements of these operators and the energy dependence of such amplitudes is obtained by the evolution equation with respect to the rapidity parameter: the Balitsky-JIMWLK evolution equation. A brief description of the derivation of the Balitsky-JIMWLK evolution equation at leading order and next- to-leading order will be presented.

Branching ratios and directCPasymmetries inD→PPdecays

We propose a theoretical framework for analyzing two-body nonleptonic $D$ meson decays, based on the factorization of short-distance (long-distance) dynamics into Wilson coefficients (hadronic matrix elements of four-fermion operators). The parametrization of hadronic matrix elements in terms of several nonperturbative quantities is demonstrated for the $D\to PP$ decays, $P$ denoting a pseudoscalar meson. We consider the evolution of Wilson coefficients with energy release in individual decay modes, and the Glauber strong phase associated with the pion in nonfactorizable annihilation amplitudes, that is attributed to the unique role of the pion as a Nambu-Goldstone boson and a quark-anti-quark bound state simultaneously. The above inputs improve the global fit to the branching ratios involving the $\eta'$ meson, and resolves the long-standing puzzle from the $D^0\to\pi^+\pi^-$ and $D^0\to K^+K^-$ branching ratios, respectively. Combining short-distance dynamics associated with penguin operators and the hadronic parameters determined from the global fit to branching ratios, we predict direct CP asymmetries, to which the quark loops and the scalar penguin annihilation give dominant contributions. In particular, we predict $\Delta A_{\rm CP}\equiv A_{\rm CP}(K^+K^-)-A_{\rm CP}(\pi^+\pi^-)=-1.00\times 10^{-3}$, lower than the LHCb and CDF data.

Effective-potential model for high-LRydberg atoms

A description of fine-structure patterns in nonpenetrating high-$L$ Rydberg atoms and ions is derived in a perturbative model in which the energy denominators occurring in the second-order perturbation theory are evaluated using the adiabatic expansion. The patterns of Rydberg energies that result are dominated by the expectation value of an effective potential containing a range of tensor orders and increasing negative powers of the Rydberg electron's radial coordinate. The coefficients of each term in the effective potential are expressed in terms of matrix elements and energies of the free ion at the core of the Rydberg system. Smaller corrections to these patterns due to application of the effective potential in second order and due to relativistic and spin contributions are also described. The effective potential provides a framework for extracting ion properties from measurements of high-$L$ fine-structure patterns.

Electronic band structures and optical properties of type-II superlattice photodetectors with interfacial effect

The electronic band structures and optical properties of type-II superlattice (T2SL) photodetectors in the mid-infrared (IR) range are investigated. We formulate a rigorous band structure model using the 8-band k · p method to include the conduction and valence band mixing. After solving the 8 × 8 Hamiltonian and deriving explicitly the new momentum matrix elements in terms of envelope functions, optical transition rates are obtained through the Fermi's golden rule under various doping and injection conditions. Optical measurements on T2SL photodetectors are compared with our model and show good agreement. Our modeling results of quantum structures connect directly to the device-level design and simulation. The predicted doping effect is readily applicable to the optimization of photodetectors. We further include interfacial (IF) layers to study the significance of their effect. Optical properties of T2SLs are expected to have a large tunable range by controlling the thickness and material composition of the IF layers. Our model provides an efficient tool for the designs of novel photodetectors.

TRANSVERSE MOMENTUM-DEPENDENT PARTON DISTRIBUTIONS FROM LATTICE QCD

Starting from a definition of transverse momentum-dependent parton distributions for semi-inclusive deep inelastic scattering and the Drell-Yan process, given in terms of matrix elements of a quark bilocal operator containing a staple-shaped Wilson connection, a scheme to determine such observables in lattice QCD is developed and explored. Parametrizing the aforementioned matrix elements in terms of invariant amplitudes permits a simple transformation of the problem to a Lorentz frame suited for the lattice calculation. Results for the Sivers and Boer-Mulders transverse momentum shifts are presented, focusing in particular on their dependence on the staple extent and the Collins-Soper evolution parameter.