Matrix Elements Of Operators Research Articles

Factorization at subleading power and endpoint-divergent convolutions in h \u2192 \u03b3\u03b3 decay

It is by now well known that, at subleading power in scale ratios, factorization theorems for high-energy cross sections and decay amplitudes contain endpoint-divergent convolution integrals. The presence of endpoint divergences hints at a violation of simple scale separation. At the technical level, they indicate an unexpected failure of dimensional regularization and the overline{mathrm{MS}} subtraction scheme. In this paper we start a detailed discussion of factorization at subleading power within the framework of soft-collinear effective theory. As a concrete example, we factorize the decay amplitude for the radiative Higgs-boson decay h → γγ mediated by a b-quark loop, for which endpoint-divergent convolution integrals require both dimensional and rapidity regulators. We derive a factorization theorem for the decay amplitude in terms of bare Wilson coefficients and operator matrix elements. We show that endpoint divergences caused by rapidity divergences cancel to all orders of perturbation theory, while endpoint divergences that are regularized dimensionally can be removed by rearranging the terms in the factorization theorem. We use our result to resum the leading double-logarithmic corrections of order {alpha}_s^n{ln}^{2n+2}left(-{M}_h^2/{m}_b^2right) to the decay amplitude to all orders of perturbation theory.

Bilinear quark operators in the RI/SMOM scheme at three loops

We consider the renormalization of the matrix elements of the bilinear quark operators ψ¯ψ, ψ¯γμψ, and ψ¯σμνψ at next-to-next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the MS‾ scheme and the regularization invariant symmetric momentum subtraction (RI/SMOM) scheme. The obtained results can be used to reduce the errors in determinations of quark masses from lattice QCD simulations. The results are given in Landau gauge.

Wavefunctions and energy eigenvalues of charge carriers in zigzag carbon nanotubes and in armchair nanoribbons in the vicinity of Dirac point under the influence of longitudinal electric field

Abstract Finite length graphene nanoribbon with armchair edges impacted by the external electrical field longitudinal to the edges was investigated, and approximate solutions for the electrons energy and wavefunctions in the neighborhood of the Dirac point were obtained. Matrix elements for the position operators are also calculated. Based on the four-point unit cell for elongated zigzag carbon nanotubes and staggered armchair-edge graphene nanoribbons under the influence of longitudinal electric field, the eigenvalues and eigenfunctions are obtained in vicinity of the Dirac point. It is shown that for the not-perturbed Hamilton operator H 0 the eigenvalues of armchair nanoribbon and zigzag nanotube coincide if their lengths are the same, while their spectrum is equidistant. Similarly to carbyne and zigzag edge nanotubes, for armchair edge nanoribbons based on the four-point unit phase cell in the vicinity of Dirac points, the wavefunction is expressed in the form of two-component Dirac function. It is shown that in the case of semiconducting N armchair-edge nanoribbons, where N + 1 = 3 p + δ , δ = ± 1 , N > > 1 , electronic properties can be described using the two uncoupled Majorana equations of 1 + 1 dimension for the particle with non-zero mass. Solutions of these equations are charge-neutral Majorana fields φ ± ( δ ) . These results are of practical interest in designing the supercomputers and experiments in this area. The possibility of generation of terahertz radiation by the Cherenkov mechanism and by analogy with cascade lasers is also discussed.

Renormalization and matching for the Collins-Soper kernel from lattice QCD

The Collins-Soper kernel, which governs the energy evolution of transverse- momentum dependent parton distribution functions (TMDPDFs), is required to accurately predict Drell-Yan like processes at small transverse momentum, and is a key ingredient for extracting TMDPDFs from experiment. Earlier we proposed a method to calculate this kernel from ratios of the so-called quasi-TMDPDFs determined with lattice QCD, which are defined as hadronic matrix elements of staple-shaped Euclidean Wilson line operators. Here we provide the one-loop renormalization of these operators in a regularization-independent momentum subtraction (RI′/MOM) scheme, as well as the conversion factor from the RI′/MOM-renormalized quasi-TMDPDF to the overline{mathrm{MS}} scheme. We also propose a procedure for calculating the Collins-Soper kernel directly from position space correlators, which simplifies the lattice determination.

Lattice QCD Matrix Elements for the B_{s}^{0}-B[over ¯]_{s}^{0} Width Difference beyond Leading Order.

Predicting the B_{s}^{0}-B[over ¯]_{s}^{0} width difference ΔΓ_{s} relies on the heavy quark expansion and on hadronic matrix elements of ΔB=2 operators. We present the first lattice QCD results for matrix elements of the dimension-7 operators R_{2,3} and linear combinations R[over ˜]_{2,3} using nonrelativistic QCD for the bottom quark and a highly improved staggered quark (HISQ) action for the strange quark. Computations use MILC Collaboration ensembles of gauge field configurations with 2+1+1 flavors of sea quarks with the HISQ discretization, including lattices with physically light up or down quark masses. We discuss features unique to calculating matrix elements of these operators and analyze uncertainties from series truncation, discretization, and quark mass dependence. Finally we report the first standard model determination of ΔΓ_{s} using lattice QCD results for all hadronic matrix elements through O(1/m_{b}). The main result of our calculations yields the 1/m_{b} contribution ΔΓ_{1/m_{b}}=-0.022(10) ps^{-1}. Adding this to the leading order contribution, the standard model prediction is ΔΓ_{s}=0.092(14) ps^{-1}.

Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy.

The multiconfiguration time-dependent Hartree (MCTDH) method is a powerful method for solving the time-dependent Schrödinger equation in quantum molecular dynamics. It is, however, hampered by the so-called curse of dimensionality which results in exponential scaling with respect to the number of degrees of freedom in the system and, thus, limits its applicability to small- and medium-sized molecules. To avoid this scaling, we derive equations of motion for a series of truncated MCTDH methods using a many-mode second-quantization formulation where the configuration space is restricted based on mode-combination levels as also done in the vibrational configuration interaction and vibrational coupled cluster methods for solving the time-independent Schrödinger equation. The full MCTDH wave function is invariant with respect to the choice of constraint (or gauge) operators, but restricting the configuration space removes this invariance. We, thus, analyze the remaining redundancies and derive equations for variationally optimizing the non-redundant matrix elements of the constraint operators. As an alternative, we also present a constraint that keeps the density matrices block diagonal during the propagation and the two choices are compared. Example calculations are performed on formyl fluoride and a series of high-dimensional Henon-Heiles potentials. The results show that the MCTDH[n] methods can be applied to large systems and that an optimal choice of constraint operators is key to obtaining the correct physical behavior of the wave function.

Shell model calculations in the A ~ 100 mass region

The properties of nuclei with 39≤Ζ≤47 and Ν=51-52 are investigated in large scale shell-model calculations. The doubly closed nucleus 100Sn is selected as the reference state and the nuclei under examination are described in terms of proton holes and one up two neutrons outside the inert core. The proton holes are distributed in a model space consisting of the orbitals g9/2, p1/2, p3/2 while f5/2 is sometimes also considerd. Similary the model space for the neutron particles in­ cludes the orbitals g7/2, d5/2, d3/2, s1/2 and in certain cases h11/2. The effective two-body interaction and the matrix elements of the effective operators were deter­ mined by introducing second-order corrections to the Sussex matrix elements. The single proton holes as well as the single-neutron energies were treated as parameters which were determined by least-squares fit to the observed levels of 39≤Ζ≤47, Ν=50 and Ν=51 respectively. The results of the calculation were found to be in satisfactory agreement with experimental data and this enable us to make predictions about the properties of some exotic nuclei in the vicinity of 100Sn.

Nonleptonic Decays of Doubly Charmed Baryons

In this lecture, we provide a basic introduction into the topic of charmed baryons and their nonleptonic two-body decays. Some features of the baryon weak decays on the quark level are discussed in detail in the framework of effective field theory. The calculation of the matrix elements of the four-quark operators arising in the effective theory proceeds by using the covariant constituent quark model. The model allows one to evaluate not only the factorizing tree-level diagrams but also more complicated diagrams with the internal W–exchange. The technique required for such calculation is discussed in some detail. Finally, the numerical results are presented, and comparison of the contributions coming from the W–exchange diagrams with those from the tree-level are carefully performed.

Shell model calculations in the A=90-98 mass region. A study of the N=51. nuclei

The properties of nuclei with 39 < Ζ < 47 and N=51 are investigated in large scale shell-model calculations. The doubly closed nucleus 100Sn is selected as the reference state and the nuclei under examination are described in terms of proton holes and a single neutron outside the inert core. The proton holes are distributed in a model space consisting of the orbitals g9/2, p1/2, p3/2 while f5/2 is sometimes also considerd. Similary the model space for the single neutron includes the orbitals g7/2, d5/2, d3/2, s1/2 and in certain cases h11/2. The effective two-body interaction and the matrix elements of the effective operators were determined by introducing second-order correctios to the Sussex matrix elements. The single proton holes as well as the single-neutron energies were treated as parameters which were determined by least-squares fit to the observed levels of 39 < Ζ < 47, N=50 and Ν=51 respectively. The results of the calculation were found to be in satisfactory agreement with experimental data and this enable us to make predictions about the properties of some exotic nuclei in the vicinity of 100Sn.

Double beta decay without invoking closure

The nuclear matrix elements of PU operators entering in the 0v ßß-decay of 76Ge-->76Se have been calculated explicitly in the context of QRPA within the model space 0f7/2 - 0h11/2. T h e validity of the closure approximation has been tested and seems to be quite satisfactory for those matrix elements which are not usually suppressed. Our results indicate that they are dominated by multipoles other than 0+ and 1+ and that the matrix elements are comparable to those of shell model calculations.

The three-loop single mass polarized pure singlet operator matrix element

We calculate the massive polarized three-loop pure singlet operator matrix element AQq(3),PS in the single mass case in the Larin scheme. This operator matrix element contributes to the massive polarized three-loop Wilson coefficient HQq(3),PS in deep-inelastic scattering and constitutes a three-loop transition matrix element in the variable flavor number scheme. We provide analytic results in Mellin N and in x space and study the behaviour of this operator matrix element in the region of small and large values of the Bjorken variable x.

Screening properties of quark-gluon plasma obtained from distribution and correlation functions of the constituent quasiparticle model

Based on the constituent quasiparticle model of quark-gluon plasma (QGP), the matrix elements of the density operator and the Wigner function in the color phase space are presented in form of color path integrals over Wiener and SU(3) group Haar measures. Monte Carlo calculations of quark and gluon momentum distributions and spatial pair distribution functions have been carried out for the strongly coupled QGP plasma in thermal equilibrium at zero baryon chemical potential. The Debye screening mass and the running coupling constant have been obtained from the spatial pair distribution function and are in agreement with the available lattice QCD data. At densities related to the average interparticle distance more than 0.4 fm the gluon bound states in the form of glueballs have been found. Comparison with the Maxwell Boltzmann distribution shows significant influence of interparticle interaction on high energy asymptotics of the momentum distribution functions, resulting in appearance of quantum tails. The new color pair correlation function has been introduced, and related new color screening mass has been discussed.

Explicit form for the kernel operator matrix elements in eigenfunction basis of harmonic oscillator

In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems.Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.

Emergence of conformal symmetry in quantum spin chains: Antiperiodic boundary conditions and supersymmetry

Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and anti-periodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local operators and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we explore the emergence of superconformal symmetry in the quantum spin chain.

On Complex Gamma-Function Integrals

It was observed recently that relations between matrix elements of certain operators in the ${\rm SL}(2,\mathbb R)$ spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with ${\rm SL}(2,\mathbb C)$ symmetry group and ${\rm L}_2(\mathbb C)$ as a local Hilbert space give rise to a new type of $\Gamma$-function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals.

The three-loop polarized pure singlet operator matrix element with two different masses

We present the two-mass QCD contributions to the polarized pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the polarized structure function g1(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the polarized heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals. These integrals depend on the mass ratio through the main argument, and the alphabet includes square–root valued letters.

Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group

For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

Calculation of Transition Probabilities in Quantum Mechanics with a Nonnegative Distribution Function in the Maple Computer Algebra System

In the Maple computer algebra system, an algorithm is implemented for symbolic and numerical computations for finding the transition probabilities for hydrogen-like atoms in quantum mechanics with a nonnegative quantum distribution function (QDF). Quantum mechanics with a nonnegative QDF is equivalent to the standard theory of quantum measurements. However, the presence in it of a probabilistic quantum theory in the phase space gives additional possibilities for calculating and interpreting the results of quantum measurements. The methods of computer algebra seem to be necessary for the relevant calculations. The calculation of the matrix elements of operators is necessary for determining the energy levels, oscillator strengths, and radiation transition parameters for atoms and ions with an open shell. Transition probabilities are calculated and compared with experimental data. They are calculated using the Galerkin method with the Sturm functions of the hydrogen atom as coordinate functions. The verification of the model showed good agreement between the calculated and experimentally measured transition probabilities.

Entanglement and matrix elements of observables in interacting integrable systems.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

Update on splitting of lifetimes of c and b hyperons within the heavy quark expansion and decays ΞQ→ΛQπ

It is argued that the results of recent precision measurements by LHCb of the lifetimes of charmed and bottom hyperons are very well consistent with the description within heavy quark expansion and allow to accurately determine matrix elements of light-flavor nonsinglet four-quark operators over the hyperons and to accurately reproduce the difference of lifemes of $\Lambda_b$ and $\Xi_b^-$. When combined with the recent LHCb results on the decay $\Xi_b^- \to \Lambda_b \pi^-$ this leads to prediction of a lower bound on the rate of the decays $\Xi_c \to \Lambda_c \pi$.