Terms Of Matrix Elements Research Articles

Development of the automatic procedures for spinor matrix element calculation with massive particles

We present a new version the procedure for calculation helicity amplitudes with massive particles. First, we introduce a massive 4-momentum into higher dimensional space, where it can be treated as massless. Secondly, we apply the symmetry properties of gauge theories and get quite simple matrix elements in terms of the field strength.

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Unpolarized quarks

Traditionally, lattice QCD computations of generalized parton distributions (GPDs) have been carried out in a symmetric frame, where the transferred momentum is symmetrically distributed between the incoming and outgoing hadrons. However, such frames are inconvenient since they require a separate calculation for each value of the momentum transfer, increasing significantly the computational cost. In this work, by focusing on the quasi-distribution approach, we lay the foundation for faster and more effective lattice QCD calculations of GPDs exploiting asymmetric frames, with freedom in the transferred momentum distribution. An important ingredient of our approach is the Lorentz covariant parameterization of the matrix elements in terms of Lorentz-invariant amplitudes, which allows one to relate matrix elements in different frames. We also use this amplitude approach to propose a new definition of quasi-GPDs that is frame-independent and, more importantly, may lead to smaller power corrections in the matching relations to the light-cone GPDs. We demonstrate the efficacy of the formalism through numerical calculations using one ensemble of $N_f$=2+1+1 twisted mass fermions with a clover improvement. The value of the light-quark masses lead to a pion mass of about 260 MeV. Concentrating on the proton, and limiting ourselves to a vanishing longitudinal momentum transfer to the target, we extract the invariant amplitudes from matrix element calculations in both the symmetric and asymmetric frame, and obtain results for the twist-2 light-cone GPDs for unpolarized quarks, that is, $H$ and $E$.

Prompt hadroproduction of C-even quarkonia in the light-front $k_T$ -factorization approach

We present a new approach for the prompt production of quarkonia which is based on the k_TkT-factorization method. The production of even C-parity quarkonia proceeds via the fusion of two (off-shell) gluons. Especially in the kinematics of the LHCb experiment these processes are thus expected to be a sensitive probe of the small-xx gluon distribution. We calculate the relevant off-shell matrix elements in terms of the light-front wave functions of the quarkonium states. We present our results for scalar and pseudoscalar charmonia and discuss photon transition form factors as well as cross sections for prompt hadroproduction. We compare our results for the \eta_cηc to recent LHCb data.

General Analysis of the Reaction e^+ + e^- → N + Ñ + π^0

The general analysis of the reaction , in the case of longitudinally polarized electron beam, has been performed in the one-photon-nnihilation approximation, accounting for the polarization states of the final nucleon. This analysis is useful for the description of the continuum (non-resonant) and resonant (with different possible vector mesons or excited baryons in the intermediate virtual states of the Feynman diagrams) contributions. The conservation of the hadron electromagnetic currents and P-invariance of the hadron electromagnetic interaction were used to express the matrix element in terms of the six complex independent invariant amplitudes. The general structure of the hadronic tensor for the case of unpolarized final hadrons and polarized nucleon has been derived. The spin-independent part of the hadronic tensor is determined by five structure functions and the spin-dependent one by 13 structure functions. The transversal, longitudinal and normal components of the nucleon polarization four-vector are expressed by means of the four-vectors of the particle momenta. The five independent invariant variables which describe the reaction have been introduced. The limits of the changing of these variables have been considered. The kinematical double invariant variables regions are given in the figure. The kinematics, suitable to study the invariant mass distributions, is investigated.

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969 Zh. Eksp. Teor. Fiz. 56 1375–90) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in a different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system ‘charged particle + electromagnetic radiation’, induced by a (strong) coherent field or by a black-body radiation component (with a Planck–Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh–Jeans limit, however, in the Wien limit it completely fails.

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

Abstract We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in an different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system “charged particle + electromagnetic radiation”, induced by a (strong) coherent field or by a black-body radiation component (with a Planck-Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it completely fails.

Simulating the dynamics of electronic observables via reduced-dimensionality generalized quantum master equations.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.

Inclusive production of heavy quarkonia in pNRQCD

We develop a formalism for computing inclusive production cross sections of heavy quarkonia based on the nonrelativistic QCD and the potential nonrelativistic QCD effective field theories. Our formalism applies to strongly coupled quarkonia, which include excited charmonium and bottomonium states. Analogously to heavy quarkonium decay processes, we express nonrelativistic QCD long-distance matrix elements in terms of quarkonium wavefunctions at the origin and universal gluonic correlators. Our expressions for the long-distance matrix elements are valid up to corrections of order 1/{N}_c^2 . These expressions enhance the predictive power of the nonrelativistic effective field theory approach to inclusive production processes by reducing the number of nonperturbative unknowns, and make possible first-principle determinations of long-distance matrix elements once the gluonic correlators are known. Based on this formalism, we compute the production cross sections of P-wave charmonia and bottomonia at the LHC, and find good agreement with measurements.

The Matrix Elements of Area Operator in (2+1) Euclidean Loop Quantum Gravity

In this article we discuss the matrix elements of area operator in (2+1) Euclidean Loop Quantum Gravity. The Euclidean signature is chosen because it has the same group as (3+1) Lorentzian Loop Quantum Gravity that is SU (2). The techniques used follows what has been done on the volume operator in (3+1) Lorentzian Loop Quantum Gravity. Although both are referred to as volume operator, they both have significantly different forms, so the initial steps of the discussion will be different. The result show that we can express the matrix elements in terms of standard recoupling scheme basis.

Model-independent framework for determining finite-volume effects of spatially nonlocal operators

We present a model-independent framework to determine finite-volume corrections of matrix elements of spatially-separated current-current operators. We define these matrix elements in terms of Compton-like amplitudes, i.e. amplitudes coupling single-particle states via two current insertions. We show that the infrared behavior of these matrix elements is dominated by the single-particle pole, which is approximated by the elastic form factors of the lowest-lying hadron. Therefore, given lattice data on the relevant elastic form factors, the finite-volume effects can be estimated non-perturbatively and without recourse to effective field theories. For illustration purposes, we investigate the implications of the proposed formalism for a class of scalar theories in two and four dimensions.

Comparison of Microscopic Interacting Boson Model and Quasiparticle Random Phase Approximation 0νββ Decay Nuclear Matrix Elements

The fundamental nature of the neutrino is presently a subject of great interest. A way to access the absolute mass scale and the fundamental nature of the neutrino is to utilize the atomic nuclei through their rare decays, the neutrinoless double beta (0νββ) decay in particular. The experimentally measurable observable is the half-life of the decay, which can be factorized to consist of phase space factor, axial vector coupling constant, nuclear matrix element, and function containing physics beyond the standard model. Thus reliable description of nuclear matrix element is of crucial importance in order to extract information governed by the function containing physics beyond the standard model, neutrino mass parameter in particular. Comparison of double beta decay nuclear matrix elements obtained using microscopic interacting boson model (IBM-2) and quasiparticle random phase approximation (QRPA) has revealed close correspondence, even though the assumptions in these two models are rather different. The origin of this compatibility is not yet clear, and thorough investigation of decomposed matrix elements in terms of different contributions arising from induced currents and the finite nucleon size is expected to contribute to more accurate values for the double beta decay nuclear matrix elements. Such comparison is performed using detailed calculations on both models and obtained results are then discussed together with recent experimental results.

A simple molecular orbital picture of RIXS distilled from many-body damped response theory.

Ab initio calculations of resonant inelastic x-ray scattering (RIXS) often rely on damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as the sum over all electronic states of the system [sum-over-states (SOS) expressions]. By invoking resonance arguments, this expression can be reduced to a few terms, an approximation commonly exploited for the interpretation of computed cross sections. We present an alternative approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from many-body calculations using the damped response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix elements between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states such that the RIXS moments can be evaluated using complex response one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary components of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS process into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para-nitroaniline, and 4-amino-4'-nitrostilbene. These examples highlight the significance of both the near-resonance and off-resonance channels.

Stokes polytopes and intersection theory

Intersection numbers of Stokes polytopes living in complex projective space are computed using the techniques employed to find the inverse string Kawai-Lewellen-Tye (KLT) matrix elements in terms of intersection numbers of associahedra. To do this requires an appropriate convex realization of Stokes polytopes in ${\mathbb{CP}}^{n}$ loaded with suitable generalizations of the Koba-Nielsen factor. The procedure is carried out explicitly for the lower point cases, and the prescription for the generic higher point cases is laid out as well. The intersection numbers are identified as scattering amplitudes corresponding to a theory the coupling constants of which are determined entirely in terms of the combinatorial weights of the Stokes polytopes. A parameter ${\ensuremath{\alpha}}^{\ensuremath{'}}$ having units of length is used to define the intersection numbers in a manner that yields the amplitudes of ${\ensuremath{\phi}}^{4}$ theory to leading order when the limit of vanishing ${\ensuremath{\alpha}}^{\ensuremath{'}}$ limit is taken. Most importantly, we contrast and compare this method of understanding quartic vertices with previous string-theoretic attempts to obtain quartic interaction amplitudes.

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on Z + \mathbb {Z}_+ to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.

Erratum: Scalar and vector form factors of D→π(K)ℓν decays with Nf=2+1+1 twisted fermions [Phys. Rev. D 96 , 054514 (2017)

We present a lattice determination of the vector and scalar form factors of the $D \to \pi(K) \ell \nu$ 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 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 we determine the vector and scalar form factors in the whole kinematical region from $q^2 = 0$ up to $q^2_{max} = (M_D - M_{\pi(K)})^2$ 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 $f_+^{D \pi(K)}(q^2)$ and $f_0^{D \pi(K)}(q^2)$ for several selected values of $q^2$, is provided and also the corresponding covariance matrix is available. At zero 4-momentum transfer we get: $f_+^{D \to \pi}(0) = 0.612 ~ (35)$ and $f_+^{D \to K}(0) = 0.765 ~ (31)$. Using the experimental averages for $|V_{cd}| f_+^{D \to \pi}(0)$ and $|V_{cs}| f_+^{D \to K}(0)$, 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: $|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 0.949 ~ (78)$.

Electric dipole moment of Hg199 atom from P , CP -odd electron-nucleon interaction

We calculate the effect of the P, CP-odd electron-nucleon interaction on the electric dipole moment of the $^{199}$Hg atom by evaluating the nuclear spin matrix elements in terms of the nuclear shell model. It is found that the neutron spin matrix element of the $^{199}$Hg nucleus is $\langle \Psi |\, \sigma_{nz} | \Psi \rangle \approx -0.4$ with a dominant configuration of $p_{1/2}$ orbital neutron. We also derive constraints on the CP phases of Higgs-doublet models, supersymmetric models, and leptoquark models from the latest experimental limit $|d_{\rm Hg}| < 7.4 \times 10^{-30}e$ cm.

Donor-acceptor symmetric and antisymmetric tunneling matrix elements: a pathway model investigation of protein electron transfer.

In protein electron transfer reaction rate calculations, the electronic Hamiltonian is apportioned into donor-acceptor (D-A) and protein bridge subspaces, and a two-state system is defined for the D-A subspace. Löwdin partitioning is used to perform the two-state reductions necessary to compute the tunneling matrix element between D and A sites. Here, a method of performing donor and acceptor state analysis for a non-orthogonal basis set in both the weak and strong electronic coupling regimes is developed. The electron tunneling current and coupling are obtained in terms of D-A symmetric and antisymmetric interatomic tunneling elements, and are then used to compare pathway models. These interatomic tunneling elements are both proportional to the Green's function elements of the isolated protein bridge. To facilitate a perturbative treatment of antisymmetric interatomic tunneling currents, we found a well-known expression for the D-A tunneling matrix element in terms of transformed Green's function matrix elements of the isolated protein bridge. Also, the relationship of the tunneling matrix element to BO pathways is discussed using the symmetric interatomic coupling. Finally, the definition of the average interatomic and atomic pathway coupling allows us obtain the quantum interference between interatomic tunneling pathways.

Polarization amplitudes in $\tau^{-} \rightarrow \nu_{\tau} VP$τ-→ντVP decay beyond the Standard Model

We use a recent formalism of the weak hadronic reactions $\tau^- \to \nu_{\tau} M_1 M_2$ to study the helicity amplitudes in $\tau^- \to \nu_{\tau} V P$ decay. This recent formalism maps the transition matrix elements at the quark level into hadronic matrix elements, and finally writes the weak matrix elements in terms of easy analytical formulas evaluated by means of an elaborate angular momentum algebra. The formalism provides directly the amplitudes for the different spin third components of the vector mesons involved. We extend the formalism to a general case, with the operator $\gamma^\mu -\alpha\gamma^\mu \gamma_5$, that can account for different models beyond the standard model and study in detail the $\tau^- \to \nu_{\tau} K^{*0} K^{-} $ reaction for the different helicities of the $K^{*0}$. The results are shown in terms of the $\alpha$ parameter that differs for each model. We find that $\frac{d \Gamma}{d M_{\rm inv}^{(K^{*0} K^{-} )}}$ is very different for the different components $M=\pm 1, 0$ and in particular the magnitude $\frac{d \Gamma}{d M_{\rm inv}^{(K^{*0} K^{-} )}}|_{M=+1} -\frac{d \Gamma}{d M_{\rm inv}^{(K^{*0} K^{-} )}}|_{M=-1} $ is very sensitive to the $\alpha$ parameter, which makes the investigation of this magnitude a most welcome initiative to test different models beyond the standard model.

Helicity amplitudes in B rightarrow D^{*} bar{nu } l decay

We use a recent formalism of the weak hadronic reactions that maps the transition matrix elements at the quark level into hadronic matrix elements, evaluated with an elaborate angular momentum algebra that allows finally to write the weak matrix elements in terms of easy analytical formulas. In particular they appear explicitly for the different spin third components of the vector mesons involved. We extend the formalism to a general case, with the operator gamma ^mu -alpha gamma ^mu gamma _5, that can accommodate different models beyond the Standard Model and study in detail the B rightarrow D^{*} bar{nu } l reaction for the different helicities of the D^*. The results are shown for each amplitude in terms of the alpha parameter that is different for each model. We show that frac{d varGamma }{d M_mathrm{inv}^{(nu l)}} is very different for the different components M=pm ,1, 0 and in particular the magnitude frac{d varGamma }{d M_mathrm{inv}^{(nu l)}}|_{M=-1} -frac{d varGamma }{d M_mathrm{inv}^{(nu l)}}|_{M=+1} is very sensitive to the alpha parameter, which suggest to use this magnitude to test different models beyond the standard model. We show that our formalism implies the heavy quark limit and compare our results with calculations that include higher order corrections in heavy quark effective theory. We find very similar results for both approaches in normalized distributions, which are practically identical at the end point of M_mathrm{inv}^{(nu l)}= m_B- m_{D^*}.

Tensor form factor of D→π(K)ℓν and D→π(K)ℓℓ decays with Nf=2+1+1 twisted-mass fermions

We present the first lattice Nf=2+1+1 determination of the tensor form factor $f_T^{D \pi(K)}(q^2)$ corresponding to the semileptonic and rare $D \to \pi(K)$ decays as a function of the squared 4-momentum transfer $q^2$. Together with our recent determination of the vector and scalar form factors we complete the set of hadronic matrix elements regulating the semileptonic and rare $D \to \pi(K)$ transitions within and beyond the Standard Model, when a non-zero tensor coupling is possible. Our analysis is based on the gauge configurations produced by ETMC with Nf=2+1+1 flavors of dynamical quarks, which include in the sea, besides two light mass-degenerate quarks, also the strange and charm quarks with masses close to their physical values. We simulated at three different values of the lattice spacing and with pion masses as small as 220 MeV. The matrix elements of the tensor current are determined for plenty of kinematical conditions in which parent and child mesons are either moving or at rest. As in the case of the vector and scalar form factors, Lorentz symmetry breaking due to hypercubic effects is clearly observed also in the data for the tensor form factor 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 and infinite volume limits we determine the tensor form factor in the whole kinematical region accessible in the experiments. A set of synthetic data points, representing our results for $f_T^{D \pi(K)}(q^2)$ for several selected values of $q^2$, is provided and the corresponding covariance matrix is also available. At zero four-momentum transfer we get $f_T^{D \pi}(0) = 0.506 (79)$ and $f_T^{D K}(0) = 0.687 (54)$, which correspond to $f_T^{D \pi}(0)/f_+^{D \pi}(0) = 0.827 (114)$ and $f_T^{D K}(0)/f_+^{D K}(0)= 0.898 (50)$.