Matrix Elements Of Operators Research Articles

Reparametrization invariance and partial re-summations of the heavy quark expansion

We extend existing work on reparametrization invariance (RPI) of the heavy-quark expansion (HQE). RPI implies relations between different orders of the HQE. We discuss the total rates of inclusive processes and obtain results which have a manifest RPI and can be expressed through matrix elements of operators and states defined in full QCD. This approach leads to a partial re-summation of higher-order terms in the heavy-quark expansion. It has the advantage that for reparametrization-invariant observables it shows that the number of independent non-perturbative parameters can be reduced.

The two-mass contribution to the three-loop gluonic operator matrix element [formula omitted

We calculate the two-mass QCD contributions to the massive operator matrix element Agg,Q at O(αs3) in analytic form in Mellin N- and z-space, maintaining the complete dependence on the heavy quark mass ratio. These terms are important ingredients for the matching relations of the variable flavor number scheme in the presence of two heavy quark flavors, such as charm and bottom. In Mellin N-space the result is given in the form of nested harmonic, generalized harmonic, cyclotomic and binomial sums, with arguments depending on the mass ratio. The Mellin inversion of these quantities to z-space gives rise to generalized iterated integrals with square root valued letters in the alphabet, depending on the mass ratio as well. Numerical results are presented.

Many-body-localization: strong disorder perturbative approach for the local integrals of motion

For random quantum spin models, the strong disorder perturbative expansion of the local integrals of motion around the real-spin operators is revisited. The emphasis is on the links with other properties of the many-body-localized phase, in particular the memory in the dynamics of the local magnetizations and the statistics of matrix elements of local operators in the eigenstate basis. Finally, this approach is applied to analyze the many-body-localization transition in a toy model studied previously from the point of view of the entanglement entropy.

Scalar, Axial, and Tensor Interactions of Light Nuclei from Lattice QCD.

Complete flavor decompositions of the matrix elements of the scalar, axial, and tensor currents in the proton, deuteron, diproton, and ^{3}He at SU(3)-symmetric values of the quark masses corresponding to a pion mass m_{π}∼806 MeV are determined using lattice quantum chromodynamics. At the physical quark masses, the scalar interactions constrain mean-field models of nuclei and the low-energy interactions of nuclei with potential dark matter candidates. The axial and tensor interactions of nuclei constrain their spin content, integrated transversity, and the quark contributions to their electric dipole moments. External fields are used to directly access the quark-line connected matrix elements of quark bilinear operators, and a combination of stochastic estimation techniques is used to determine the disconnected sea-quark contributions. The calculated matrix elements differ from, and are typically smaller than, naive single-nucleon estimates. Given the particularly large, O(10%), size of nuclear effects in the scalar matrix elements, contributions from correlated multinucleon effects should be quantified in the analysis of dark matter direct-detection experiments using nuclear targets.

K→π matrix elements of the chromomagnetic operator on the lattice

We present the results of the first lattice QCD calculation of the $K \to \pi$ matrix elements of the chromomagnetic operator $O_{CM} = g\, \bar s\, \sigma_{\mu\nu} G_{\mu\nu} d$, which appears in the effective Hamiltonian describing $\Delta S = 1$ transitions in and beyond the Standard Model. Having dimension 5, the chromomagnetic operator is characterized by a rich pattern of mixing with operators of equal and lower dimensionality. The multiplicative renormalization factor as well as the mixing coefficients with the operators of equal dimension have been computed at one loop in perturbation theory. The power divergent coefficients controlling the mixing with operators of lower dimension have been determined non-perturbatively, by imposing suitable subtraction conditions. The numerical simulations have been carried out using the gauge field configurations produced by the European Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing. Our result for the B-parameter of the chromomagnetic operator at the physical pion and kaon point is $B_{CMO}^{K \pi} = 0.273 ~ (70)$, while in the SU(3) chiral limit we obtain $B_{CMO} = 0.072 ~ (22)$. Our findings are significantly smaller than the model-dependent estimate $B_{CMO} \sim 1 - 4$, currently used in phenomenological analyses, and improve the uncertainty on this important phenomenological quantity.

Electronic Transition Dipole Moments in Relativistic Coupled-Cluster Theory: the Finite-Field Method

The determination of the matrix elements of single-electron operators for the transitions between electronic states (primarily, the dipole moments of electron transitions) is one of the critical problems in simulating the atomic and molecular spectra. In this study, the efficiency of a simple version of the finite-field method for calculating these values within the Fock-space coupled-cluster theory has been investigated.

Transition sum rules in the shell model

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the double-commutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous Thomas-Reiche-Kuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$-shell.

RichMol: A general variational approach for rovibrational molecular dynamics in external electric fields

In this paper, a general variational approach for computing the rovibrational dynamics of polyatomic molecules in the presence of external electric fields is presented. Highly accurate, full-dimensional variational calculations provide a basis of field-free rovibrational states for evaluating the rovibrational matrix elements of high-rank Cartesian tensor operators and for solving the time-dependent Schrödinger equation. The effect of the external electric field is treated as a multipole moment expansion truncated at the second hyperpolarizability interaction term. Our fully numerical and computationally efficient method has been implemented in a new program, RichMol, which can simulate the effects of multiple external fields of arbitrary strength, polarization, pulse shape, and duration. Illustrative calculations of two-color orientation and rotational excitation with an optical centrifuge of NH3 are discussed.

Renormalization in Large Momentum Effective Theory of Parton Physics

In the large-momentum effective field theory approach to parton physics, the matrix elements of nonlocal operators of quark and gluon fields, linked by straight Wilson lines in a spatial direction, are calculated in lattice quantum chromodynamics as a function of hadron momentum. Using the heavy-quark effective theory formalism, we show a multiplicative renormalization of these operators at all orders in perturbation theory, both in dimensional and lattice regularizations. The result provides a theoretical basis for extracting parton properties through properly renormalized observables in MonteCarlo simulations.

Dynamical Jahn-Teller effect of fullerene anions

The dynamical Jahn-Teller effect of ${\mathrm{C}}_{60}^{n\ensuremath{-}}$ anions ($n=1$--5) is studied using the numerical diagonalization of the linear ${p}^{n}\ensuremath{\bigotimes}8d$ Jahn-Teller Hamiltonian with the currently established coupling parameters. It is found that in all anions the Jahn-Teller effect stabilizes the low-spin states, resulting in the violation of Hund's rule. The energy gain due to the Jahn-Teller dynamics is found to be comparable to the static Jahn-Teller stabilization. The Jahn-Teller dynamics influences the thermodynamic properties via strong variation of the density of vibronic states with energy. Thus the large vibronic entropy in the low-spin states enhances the effective spin gap of ${\mathrm{C}}_{60}^{3\ensuremath{-}}$ quenching the spin crossover. From the calculations of the effective spin gap as a function of the Hund's rule coupling, we found that the latter should amount $40\ifmmode\pm\else\textpm\fi{}5$ meV in order to cope with the violation of Hund's rule and to reproduce the large spin gap. With the obtained numerical solutions, the matrix elements of electronic operators for the low-lying vibronic levels and the vibronic reduction factors are calculated for all anions.

Matrix elements of the proton–neutron symplectic model

The tensor properties of the [Formula: see text] algebra generators are determined in respect to the reduction chain [Formula: see text], which defines a shell-model coupling scheme of the proton–neutron symplectic model (PNSM). They are further used to calculate the matrix elements of the basic [Formula: see text] operators of the PNSM in the space of fully symmetric representations in the [Formula: see text]-coupled basis using a generalized Wigner–Eckart theorem. The obtained results allow further the matrix elements of any physical operator of interest, such as the relevant transition operators or the collective potential, to be calculated. As an illustration, the matrix elements of the basic irreducible tensor terms which appear in the [Formula: see text] decomposition of the long-range full major-shell mixing proton–neutron quadrupole–quadrupole interaction are presented.

Classical limit of position matrix elements for Rydberg atoms

In this work we discuss Bohr’s correspondence principle in a particular physical system, namely the hydrogen atom. It is shown that while diagonal elements of the position operator are equal to zero, off-diagonal elements between two states of different but close energies have a classical limit. The classical limit is reached when both principal quantum numbers, and hence both energies, are large but their difference is small. In this case the matrix elements of the position operator tend to the Fourier component of a classical trajectory, whose parameters are found. The classical limit is proved by finding numerical values of the matrix elements and also by exact treatment of the analytical formulas. This discussion provides a deeper understanding of the correspondence between quantum and classical mechanics and contributes to a better understanding of hydrogen atom wave functions.

Finite continuum quasi distributions from lattice QCD

We present a new approach to extracting continuum quasi distributions from lattice QCD. Quasi distributions are defined by matrix elements of a Wilson-line operator extended in a spatial direction, evaluated between nucleon states at finite momentum. We propose smearing this extended operator with the gradient flow to render the corresponding matrix elements finite in the continuum limit. This procedure provides a nonperturbative method to remove the power-divergence associated with the Wilson line and the resulting matrix elements can be directly matched to light-front distributions via perturbation theory.

Recent progress in applying lattice QCD to kaon physics

Standard lattice calculations in kaon physics are based on the evaluation of matrix elements of local operators between two single-hadron states or a single-hadron state and the vacuum. Recent progress in lattice QCD has gone beyond these standard observables. I will review the status and prospects of lattice kaon physics with an emphasis on non-leptonic K → ππ decay and long-distance processes including K0-K0 mixing and rare kaon decays.

The Dual QCD @ Work: 2018

The Dual QCD (DQCD) framework, based on the ideas of ’t Hooft and Witten, and developed by Bill Bardeen, Jean-Marc Gérard and myself in the 1980s is not QCD, a theory of quarks and gluons, but a successful low energy approximation of it when applied to K → ππ decays and K¯0 - K0 mixing. After years of silence, starting with 2014, this framework has been further developed in order to improve the SM prediction for the ratio ε'/ε, the ΔI = 1/2 rule and B^K. Most importantly, this year it has been used for the calculation of all K → ππ hadronic matrix elements of BSM operators which opened the road for the general study of ε'/ε in the context of the SM effective theory (SMEFT). This talk summarizes briefly the past successes of this framework and discusses recent developments which lead to a master formula for ε'/ε valid in any extension of the SM. This formula should facilitate the search for new physics responsible for the ε'/ε anomaly hinted by 2015 results from lattice QCD and DQCD.

HQE parameters from unquenched lattice data on pseudoscalar and vector heavy-light meson masses

We present a new lattice determination of some of the parameters appearing both in the Operator Product Expansion (OPE) analysis of the inclusive semileptonic B-meson decays and in the Heavy Quark Expansion (HQE) of the pseudoscalar (PS) and vector (V) heavy-light meson masses. We perform a lattice QCD (LQCD) computation of PS and V heavy-light meson masses for heavy-quark masses mh in the range from [see formula in PDF] to ≃ [see formula in PDF]. We employed the Nf = 2 + 1 + 1 gauge configurations of the European Twisted Mass Collaboration (ETMC) at three values of the lattice spacing a ≃ (0.062,0.082,0.089) fm with pion masses in the range Mπ ≃ (210 - 450) MeV. The heavy-quark mass is simulated directly on the lattice up to ≃ [see formula in PDF]. The interpolation to the physical [see formula in PDF] is performed using the ETMC ratio method and adopting the kinetic mass scheme. We obtain [see formula in PDF] (1 GeV) = 4.61(20) GeV (m̅b(m̅b) = 4.26(18) GeV in the MS scheme). The lattice data are analyzed in terms of the HQE and the matrix elements of dimension-4 and dimension-5 operators are extracted with good precision, namely: [see formula in PDF] and [see formula in PDF]. The data also allow for an estimate of the dimension-6 operator matrix elements.

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.

The two-mass contribution to the three-loop pure singlet operator matrix element

We present the two-mass QCD contributions to the pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the structure function F2(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals that include square root letters in the alphabet, depending also on the mass ratio through the main argument. Numerical results are presented.

The Kubo-Greenwood formula as a result of the random phase approximation for the electrons of the metal

For calculation of short laser pulse absorption in metal the imaginary part of permittivity, which is simply related to the conductivity, is required. Currently to find the static and dynamic conductivity the Kubo-Greenwood formula is most commonly used. It describes the electromagnetic energy absorption in the one-electron approach. In the present study, this formula is derived directly from the expression for the permittivity expression in the random phase approximation, which in fact is equivalent to the method of the mean field. The detailed analysis of the role of electron-electron interaction in the calculation of the matrix elements of the velocity operator is given. It is shown that in the one-electron random phase approximation the single-particle conductive electron wave functions in the field of fixed ions should be used. The possibility of considering the exchange and correlation effects by means of an amendment to a local function field is discussed.

Nucleon transverse momentum-dependent parton distributions in lattice QCD: Renormalization patterns and discretization effects

Lattice QCD calculations of transverse momentum-dependent parton distribution functions (TMDs) in nucleons are presented, based on the evaluation of nucleon matrix elements of quark bilocal operators with a staple-shaped gauge connection. Both time-reversal odd effects, namely, the generalized Sivers and Boer-Mulders transverse momentum shifts, as well as time-reversal even effects, namely, the generalized transversity and one of the generalized worm-gear shifts are studied. Results are obtained on two different $n_f = 2+1$ flavor ensembles with approximately matching pion masses but very different discretization schemes: domain-wall fermions (DWF) with lattice spacing $a=0.084$ fm and pion mass 297 MeV, and Wilson-clover fermions with $a=0.114$ fm and pion mass 317 MeV. Comparison of the results on the two ensembles yields insight into the length scales at which lattice discretization errors are small, and into the extent to which the renormalization pattern obeyed by the continuum QCD TMD operator continues to apply in the lattice formulation. For the studied TMD observables, the results are found to be consistent between the two ensembles at sufficiently large separation of the quark fields within the operator, whereas deviations are observed in the local limit and in the case of a straight link gauge connection, which is relevant to the studies of parton distribution functions. Furthermore, the lattice estimates of the generalized Sivers shift obtained here are confronted with, and are seen to tend towards, a phenomenological estimate extracted from experimental data.