Matrix Elements Of Operators Research Articles

Bounds on Chaos from the Eigenstate Thermalization Hypothesis.

We show that the known bound on the growth rate of the out-of-time-order four-point correlator in chaotic many-body quantum systems follows directly from the general structure of operator matrix elements in systems that obey the eigenstate thermalization hypothesis. This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.

Electron Capture in Nuclear Structure and Astrophysics

Electron capture on nucleons and nuclei, is one of the most important weak interaction processes in the dynamics and evolution of massive stars. Especially on nuclei of the Fe mass region the role of e− capture is crucial in the phase of stellar core collapse. Furthermore, a realistic treatment of electron capture on heavy nuclei provides signi cant information in the hydrodynamics of core collapse and bounce. In this work, we exploit the advantages of a recently published numerical approach to perform nuclear structure calculations of the electron capture in F e group nuclear isotopes. As a rst concrete example, which is simultaneously o ering a good test of ourmethod,wechoosethereaction56Fe(e−,νe)56Mn* that plays a decisive role in core collapse supernovae. We also improve the previous formalism by constructing compact analytical expressions for the required reduced matrix elements of all basic multipole operators in isospin representation. Such a compact formalism o ers the advantage of performing state-by-state calculations of the transition rates for semi- leptonic nuclear processes through advantageous computer codes written in isospin representation.

Sum rules for CP-violating operators of Weinberg type

We estimate the size of the hadronic matrix elements of CP-violating three-gluon and four-gluon Weinberg operators using sum-rule techniques. In the three-gluon case, we are able to reproduce the expressions given in earlier works, while the four-gluon results obtained in this article are new. Our paper therefore represents the first systematic study of contributions to the electric dipole moment of the neutron due to CP-violating dimension-six and dimension-eight operators. We provide many details on both the derivation of the sum rules as well as the analysis of the uncertainties that plague our final predictions.

The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements

We calculate all contributions ∝TF to the polarized three–loop anomalous dimensions in the M–scheme using massive operator matrix elements and compare to results in the literature. This includes the complete anomalous dimensions γqq(2),PS and γqg(2). We also obtain the complete two–loop polarized anomalous dimensions in an independent calculation. While for most of the anomalous dimensions the usual direct computation methods in Mellin N–space can be applied since all recurrences factorize at first order, this is not the case for γqg(2). Due to the necessity of deeper expansions of the master integrals in the dimensional parameter ε=D−4, we had to use the method of arbitrary high moments to eliminate elliptic contributions in intermediate steps. 4000 moments were generated to determine this anomalous dimension and 2640 moments turned out to be sufficient. As an aside, we also recalculate the contributions ∝TF to the three–loop QCD β–function.

Higher-rank isomonodromic deformations and W-algebras

We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$.

Judd-Ofelt parametrization from emission spectra: The case study of the Eu3+ 5D1 emitting level

Traditional applications of the Judd-Ofelt (JO) theory to the analysis of the Eu3+ optical spectra make use of the emission transitions originating from the 5D0 manifold. In the present paper, we report an alternative method of evaluating the JO intensity parameters from the Eu3+ emission spectra based on the 5D1 → 7F0,1 transitions. The reduced matrix elements of the unit tensor operators are re-calculated for the 5D0,1,2 → 7F0,1,…,6 Eu3+ transitions in the intermediate coupling approximation using the average electrostatic and spin-orbit coupling parameters. The suggested method was tested by analyzing the emission spectra of the Eu3+ doped GdAlO3, LaF3, NaYF4, Y2O3, ZrO2, YNbO4, ZBLA and PIGLZ hosts. It is shown that the developed method is more accurate for the hosts with relatively high 5D1 level population, which emphasizes its high potential and applicability. In addition to the JO analysis, the CIE chromaticity coordinates are calculated for the investigated spectra.

Determination of the quark condensate from heavy-light current-current correlators in full lattice QCD

We derive the Operator Product Expansion whose vacuum expectation value gives the time-moments of the pseudoscalar heavy-light current-current correlator up to and including terms in $\alpha_s^2$ multiplying $\langle\overline{\psi}\psi\rangle/M^3$ and terms in $\alpha_s$ multiplying $\langle \alpha_s G^2 \rangle/M^4$, where $M$ is the heavy-quark mass. Using lattice QCD results for heavy-strange correlators obtained for a variety of heavy quark masses on gluon field configurations including $u$, $d$ and $s$ quarks in the sea at three values of the lattice spacing, we are able to show that the contribution of the strange-quark condensate to the time-moments is very substantial. We use our lattice QCD time-moments and the OPE to determine a value for the condensate, fitting the 4th, 6th, 8th and 10th time-moments simultaneously. Our result, $\langle \overline{s}s \rangle^{\overline{\text{MS}}}(2 \text{GeV}) = -(296(11) \,\mathrm{MeV})^3$, agrees well with HPQCD's earlier, more direct, lattice QCD determination~\cite{McNeile:2012xh}. As well as confirming that the $s$ quark condensate is close in value to the light quark condensate, this demonstrates clearly the consistency of the Operator Product Expansion for fully nonperturbative calculations of matrix elements of short-distance operators in lattice QCD.

Interseries transitions between Rydberg excitons in Cu2O

We investigate the infrared optical transitions between excitons of the yellow, green and blue series in the cuprous oxide Cu$_{\mathrm{2}}$O. We show that in many cases the dipole approximation is inadequate and, in particular, that it breaks down in yellow-blue transitions even for moderate principal quantum numbers of $n \approx 10$. The interband matrix elements of the transition operator needed for the evaluation of the excitonic transition strengths are derived from known as well as from fitted band parameters.

Multipole decomposition of the rate of muon-to-electron (μ−⟶e−) conversion in Pb208

The $({\ensuremath{\mu}}^{\ensuremath{-}}\ensuremath{\longrightarrow}{e}^{\ensuremath{-}})$ conversion in $^{208}\mathrm{Pb}$ nucleus is studied. For the elementary process we consider the most general effective Lagrangian responsible for the lepton-flavor violation. For the nuclear structure part, the spectrum of $^{208}\mathrm{Pb}$ is calculated by performing a diagonalization of the $\ensuremath{\delta}$-force interaction in the space of particle-hole pairs for neutrons and protons. We calculate the coherent contribution by direct summation of single-particle excitations. The noncoherent muon-to-electron conversion is calculated by summing over the contributions for all possible intermediate states in $^{208}\mathrm{Pb}$. We perform the multipole decomposition of the transition amplitude and present the nuclear matrix elements of spin-dependent and spin-independent operators appearing in the total rate of the muon to electron conversion process. The results are presented in a way to facilitate their use in model building of the electroweak process.

Diagonal finite volume matrix elements in the sinh-Gordon model

Using the fermionic basis we conjecture exact expressions for diagonal finite volume matrix elements of exponential operators and their descendants in the sinh-Gordon theory. Our expressions sum up the LeClair-Mussardo type infinite series generalized by Pozsgay for excited state expectation values. We checked our formulae against the Liouville three-point functions for small, while against Pozsgay's expansion for large volumes and found complete agreement.

E1 transitions in the extended proton-neutron symplectic model

A central extension of the proton-neutron symplectic model (PNSM) with the semidirect structure $\mathrm{wsp}(12,R)\ensuremath{\equiv}[\mathrm{hw}(6)]\mathrm{sp}(12,R)$ is proposed. This extension, among its generators, contains the electric dipole operator, which allows one to calculate the reduced $E1$ transition strengths. The required matrix elements of the dipole transition operator are given explicitly. Using these matrix elements, I calculate the low-energy $B(E1)$ transition strengths between the states of the ground band and ${K}^{\ensuremath{\pi}}={0}_{1}^{\ensuremath{-}}$ band for $^{154}\mathrm{Sm}$ and $^{238}\mathrm{U}$ and compare them with experiment. The results obtained reproduce well the experimental data for the two nuclei under consideration without the use of an effective charge, which could be considered as a significant achievement of the present approach.

H3 and He3 calculations without angular momentum decomposition

Results for the three nucleon (3N) bound state carried out using the "three dimensional" (3D) formalism are presented. In this approach calculations are performed without the use of angular momentum decomposition and instead rely directly on the 3D degrees of freedom of the nucleons. In this paper, for the first time, 3D results for $^{3}$He bound state with the inclusion of the screened Coulomb potential are compared to $^{3}$H calculations. Additionally, using these results, matrix elements of simple current operators related to the description of beta decay of the triton are given. All computations are carried out using the first generation of NNLO two nucleon (2N) and 3N forces from the Bochum - Bonn group.

Evaluation of Matrix Elements Using Diffusion Monte Carlo Wave Functions.

Approaches for using diffusion Monte Carlo (DMC) to evaluate matrix elements involving two vibrational wave functions are explored. In the first part of this study, overlaps between a wave function obtained using DMC and one that can be calculated analytically are evaluated. In this case, the analytical wave function is used as the guiding function for an importance sampled DMC simulation. The accuracy of the calculated overlaps is found to depend strongly on the accuracy of the calculated descendant weights, which are obtained in the DMC simulation. While a single evaluation of the descendant weights is sufficient for obtaining projected probability amplitudes or expectation values of multiplicative operators, averages of multiple independent evaluations of the descendant weights are required to obtain accurate matrix elements. This approach is investigated for one-dimensional model systems as well as H2CO, H2D+, and D2H+. The approach is extended to the evaluation of matrix elements of the dipole moment operator between the ground state and states with one quantum of excitation in one of the OH stretching vibrations in H3O2-. For these calculations, the wave functions for both the ground and excited states are evaluated using DMC. The described methodology opens the possibility of evaluating matrix elements involving two different states, both of which are obtained using DMC.

Eigenstate thermalization hypothesis and out of time order correlators.

The eigenstate thermalization hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length χ_{4}, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.

Lattice QCD determination of neutron-antineutron matrix elements with physical quark masses

Matrix elements of six-quark operators are needed to extract new physics constraints from experimental searches for neutron-antineutron oscillations. This work presents in detail the first lattice quantum chromodynamics calculations of the necessary neutron-antineutron transition matrix elements including calculation methods and discussions of systematic uncertainties. Implications of isospin and chiral symmetry on the matrix elements, power counting in the isospin limit, and renormalization of a chiral basis of six-quark operators are discussed. Calculations are performed with a chiral-symmetric discretization of the quark action and physical light quark masses in order to avoid the need for chiral extrapolation. Non-perturbative renormalization is performed, including a study of lattice cutoff effects. Excited-state effects are studied using two nucleon operators and multiple values of source-sink separation. Results for the dominant matrix elements are found to be significantly larger compared to previous results from the MIT bag model. Future calculations are needed to fully account for systematic uncertainties associated with discretization and finite-volume effects but are not expected to significantly affect this conclusion.

Master formula for ε′/ε beyond the Standard Model

We present for the first time a master formula for ε′/ε, the ratio probing direct CP violation in K→ππ decays, valid in any ultraviolet extension of the Standard Model (BSM). The formula makes use of hadronic matrix elements of BSM operators calculated recently in the Dual QCD approach and the ones of the SM operators from lattice QCD. We emphasize the large impact of several scalar and tensor BSM operators in the context of the emerging ε′/ε anomaly. We have implemented the results in the open source code flavio.

Anatomy of varepsilon '/varepsilon beyond the standard model

We present for the first time a model-independent anatomy of the ratio varepsilon '/varepsilon in the context of the Delta S = 1 effective theory with operators invariant under QCD and QED and in the context of the standard model effective field theory (SMEFT) with the operators invariant under the full SM gauge group. Our goal is to identify the new physics scenarios that are probed by this ratio and which could help to explain a possible deviation from the SM that is hinted by the data. To this end we derive a master formula for varepsilon '/varepsilon , which can be applied to any theory beyond the standard model (BSM) in which the Wilson coefficients of all contributing operators have been calculated at the electroweak scale. The relevant hadronic matrix elements of BSM operators are from the Dual QCD approach and the SM ones from lattice QCD. Within SMEFT, the constraints from K^0 and D^0 mixing as well as electric dipole moments limit significantly potential new physics contributions to varepsilon '/varepsilon . Correlations of varepsilon '/varepsilon with Krightarrow pi nu {bar{nu }} decays are briefly discussed. Building on our EFT analysis and the model-independent constraints, we discuss implications of a possible deviation from the SM in varepsilon '/varepsilon for model building, highlighting the role of the new scalar and tensor matrix elements in models with scalar mediators.

The O(α2) initial state QED corrections to e+e− annihilation to a neutral vector boson revisited

We calculate the non-singlet, the pure singlet contribution, and their interference term, at O(α2) due to electron-pair initial state radiation to e+e− annihilation into a neutral vector boson in a direct analytic computation without any approximation. The correction is represented in terms of iterated incomplete elliptic integrals. Performing the limit s≫me2 we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in m2/s. In this way, we also confirm the validity of the factorization of massive partons in the Drell–Yan process. We also add non-logarithmic terms at O(α2) which have not been considered in [1]. The corrections are of central importance for precision analyses in e+e− annihilation into γ⁎/Z⁎ at high luminosity.

Multiplicative Renormalizability of Operators defining Quasiparton Distributions.

Extracting parton distribution functions (PDFs) from lattice QCD calculation of quasi-PDFs has been actively pursued in recent years. We extend our proof of the multiplicative renormalizability of the quasiquark operators of Ishikawa etal. [Phys Rev. D 96, 094019 (2017)] to quasigluon operators, and demonstrated that quasigluon operators could be multiplicatively renormalized to all orders in perturbation theory, without mixing with other operators. We find that using a gauge-invariant UV regulator is essential for achieving this proof. With the multiplicative renormalizability of both quasiquark and quasigluon operators, and QCD collinear factorization of hadronic matrix elements of there operators into PDFs, extracting PDFs from lattice QCD calculated hadronic matrix elements of quasiparton operators could have a solid theoretical foundation.

BSM hadronic matrix elements for \u03b5\u2032/\u03b5 and K \u2192 \u03c0\u03c0 decays in the Dual QCD approach

We calculate for the first time all four-quark hadronic matrix elements of local operators possibly contributing to K → ππ decays and in particular to the ratio ε′/ε beyond the Standard Model (BSM). To this end we use the Dual QCD (DQCD) approach. In addition to 7 new mirror operators obtained from the SM ones by flipping the chirality, we count 13 BSM four-quark operators of a given chirality linearly independent of each other and of the aforesaid 14 operators for which hadronic matrix elements are already known. We present results in two bases for all these operators, one termed DQCD basis useful for the calculation of the hadronic matrix elements in the DQCD approach and the other called SD basis suited to the short distance renormalization group evolution above the 1 GeV scale. We demonstrate that the pattern of long distance evolution (meson evolution) matches the one of short distance evolution (quark-gluon evolution), a property which to our knowledge cannot be presently achieved in any other analytical framework. The highlights of our paper are chirally enhanced matrix elements of tensor-tensor and scalar-scalar BSM operators. They could thereby explain the emerging ε′/ε anomaly which is strongly indicated within DQCD with some support from lattice QCD. On the other hand we do not expect the BSM operators to be relevant for the ΔI = 1/2 rule.