Showing papers on "Antisymmetric relation published in 2016"
TL;DR: In this paper, the effect of trace and antisymmetric operators on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the O(N) model, in the limit when the spin is much larger than the twist, was investigated.
Abstract: This note is an extension of a recent work on the analytical bootstrapping of O(N) models. An additonal feature of the O(N) model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor (T
μν
) and the ϕ
i
ϕ
i
scalar, we also have other minimal twist operators as the spin-1 current J
μ
and the symmetric-traceless scalar in the case of O(N). We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the O(N) model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the ϵ−expansion are an exact match with our n = 0 case. A plausible holographic setup for the special case when N = 2 is also mentioned which mimics the calculation in the CFT.
67 citations
TL;DR: In this article, an array-based collective surface lattice resonance of both bright and dark, that is symmetric and antisymmetric, was used to control the relative strength of the two surface-lattice modes.
Abstract: We study regular wavelength scale arrays of metallic dimers. By employing dimers made up of two different sized discs, we are able to couple to array-based collective surface lattice resonances of both bright and dark, that is symmetric and antisymmetric, dimer modes and to show that the degree of asymmetry can be used to control the relative strength of the two surface-lattice modes. The collective nature of these excitations can even lead to an antisymmetric surface-lattice resonance that is stronger than the symmetric one; this is in stark contrast to the dark and bright nature of the underlying modes of the individual dimers. We verify these experimental findings, derived from extinction measurements, by comparison with both analytical and numerical modeling.
66 citations
TL;DR: The dual formulation of double field theory at the linearized level was proposed in this article, which is a classically equivalent theory describing the duals of the dilaton, the Kalb-Ramond field and the graviton in a T-duality or O(D, D) covariant way.
Abstract: We present the dual formulation of double field theory at the linearized level. This is a classically equivalent theory describing the duals of the dilaton, the Kalb-Ramond field and the graviton in a T-duality or O(D, D) covariant way. In agreement with previous proposals, the resulting theory encodes fields in mixed Young-tableau representations, combining them into an antisymmetric 4-tensor under O(D, D). In contrast to previous proposals, the theory also requires an antisymmetric 2-tensor and a singlet, which are not all pure gauge. The need for these additional fields is analogous to a similar phenomenon for “exotic” dualizations, and we clarify this by comparing with the dualizations of the component fields. We close with some speculative remarks on the significance of these observations for the full non-linear theory yet to be constructed.
58 citations
TL;DR: In this paper, the same authors performed a global analysis of the laminar flow over a rounded-tip 2 : 1 elliptic cone of 0.86 m length at zero angle of attack and yaw at Mach 7.45 and unit Reynolds number Re′ = 1.015 × 107 m−1.
Abstract: Steady laminar flow over a rounded-tip 2 : 1 elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number 7.45 and unit Reynolds number Re′ = 1.015 × 107 m−1. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major- and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.
58 citations
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TL;DR: In this paper, the authors explore a novel type of transition in certain 6D and 4D quantum field theories, in which the matter content of the theory changes while the gauge group and other parts of the spectrum remain invariant.
Abstract: We explore a novel type of transition in certain 6D and 4D quantum field theories, in which the matter content of the theory changes while the gauge group and other parts of the spectrum remain invariant. Such transitions can occur, for example, for SU(6) and SU(7) gauge groups, where matter fields in a three-index antisymmetric representation and the fundamental representation are exchanged in the transition for matter in the two-index antisymmetric representation. These matter transitions are realized by passing through superconformal theories at the transition point. We explore these transitions in dual F-theory and heterotic descriptions, where a number of novel features arise. For example, in the heterotic description the relevant 6D SU(7) theories are described by bundles on K3 surfaces where the geometry of the K3 is constrained in addition to the bundle structure. On the F-theory side, non-standard representations such as the three-index antisymmetric representation of SU(N) require Weierstrass models that cannot be realized from the standard SU(N) Tate form. We also briefly describe some other situations, with groups such as Sp(3), SO(12), and SU(3), where analogous matter transitions can occur between different representations. For SU(3), in particular, we find a matter transition between adjoint matter and matter in the symmetric representation, giving an explicit Weierstrass model for the F-theory description of the symmetric representation that complements another recent analogous construction.
52 citations
TL;DR: The possibility of generating the spin-orbital entanglement by spontaneous electronic ordering caused by electron correlations is theoretically investigated, focusing on the centrosymmetric lattices with local asymmetry at the lattice sites, e.g. zigzag, honeycomb, and diamond structures.
Abstract: The spin-orbit coupling in the absence of spatial inversion symmetry plays an important role in realizing intriguing electronic states in solids, such as topological insulators and unconventional superconductivity. Usually, the inversion symmetry breaking is inherent in the lattice structures, and hence, it is not easy to control these interesting properties by external parameters. We here theoretically investigate the possibility of generating the spin-orbital entanglement by spontaneous electronic ordering caused by electron correlations. In particular, we focus on the centrosymmetric lattices with local asymmetry at the lattice sites, e.g. zigzag, honeycomb, and diamond structures. In such systems, conventional staggered orders, such as charge order and antiferromagnetic order, break the inversion symmetry and activate the antisymmetric spin-orbit coupling, which is hidden in a sublattice-dependent form in the paramagnetic state. Considering a minimal two-orbital model on a honeycomb structure, we scrutinize the explicit form of the antisymmetric spin-orbit coupling for all the possible staggered charge, spin, orbital, and spin-orbital orders. We show that the complete table is useful for understanding of spin-valley-orbital physics, such as spin and valley splitting in the electronic band structure and generalized magnetoelectric responses in not only spin but also orbital and spin-orbital channels, reflecting in peculiar magnetic, elastic, and optical properties in solids.
42 citations
TL;DR: In this paper, a variational Monte Carlo linear method was used for the optimization of the cluster Jastrow antisymmetric geminal power ansatz, achieving a lower-order polynomial cost scaling than would be possible with a naive application of the linear method and greatly improving performance relative to that of the previously employed quasi-Newton approach.
Abstract: We present a novel specialization of the variational Monte Carlo linear method for the optimization of the recently introduced cluster Jastrow antisymmetric geminal power ansatz, achieving a lower-order polynomial cost scaling than would be possible with a naive application of the linear method and greatly improving optimization performance relative to that of the previously employed quasi-Newton approach. We test the methodology on highly multireference triple-bond stretches, achieving accuracy superior to those of the traditional coupled cluster theory and multireference perturbation theory in both the typical example of N2 and the transition-metal-oxide example of [ScO]+.
35 citations
TL;DR: In this article, a double-coil EMAT was proposed to generate symmetric omnidirectional Lamb waves with good rejection of antisymmetric modes with successful symmetric mode generation was confirmed by experimental results that favorably agree with theoretical results.
Abstract: This paper investigates a method to generate only symmetric omnidirectional Lamb waves and proposes specially-tuned double-coil EMATs. While methods to generate omnidirectional Lamb waves have been reported, the symmetric mode is typically accompanied by unwanted antisymmetric modes. The double spiral coil works as a wavelength filter but the relation between the coil geometries and the wavelength of the symmetric mode that minimizes the unwanted antisymmetric mode has not been established. Axisymmetric analysis was performed to find the relation. Excellent rejection of antisymmetric modes with successful symmetric mode generation was confirmed by experimental results that favorably agree with the theoretical results.
28 citations
TL;DR: In this article, the stability of three crossarms branching laterally and symmetrically is investigated and the governing imperfection distribution that should be adopted within finite element analysis to capture the actual load-carrying capacity is also obtained.
27 citations
TL;DR: It is shown that nearly identical efficacies of the symmetric and antisymmetric stretching modes of NH3 in promoting the reaction can be understood in terms of local-mode stretching vibrations of the reactant molecule.
Abstract: Vibrational excitations of reactants sometimes promote reactions more effectively than the same amount of translational energy. Such mode specificity provides insights into the transition-state modulation of reactivity and might be used to control chemical reactions. We report here a state-of-the-art full-dimensional quantum dynamical study of the hydrogen abstraction reaction H + NH3 → H2 + NH2 on an accurate ab initio based global potential energy surface. This reaction serves as an ideal candidate to study the relative efficacies of symmetric and degenerate antisymmetric stretching modes. Strong mode specificity, particularly for the NH3 stretching modes, is demonstrated. It is further shown that nearly identical efficacies of the symmetric and antisymmetric stretching modes of NH3 in promoting the reaction can be understood in terms of local-mode stretching vibrations of the reactant molecule.
25 citations
TL;DR: In this article, the authors presented the thickness-extensional vibration of a rectangular piezoelectric thin film bulk acoustic wave filter with two pairs of electrodes symmetrically deposited on the center of the zinc oxide film.
Abstract: This paper presents the thickness-extensional vibration of a rectangular piezoelectric thin film bulk acoustic wave filter with two pairs of electrodes symmetrically deposited on the center of the zinc oxide film. The two-dimensional scalar differential equations which were first derived to describe in-plane vibration distribution by Tiersten and Stevens are employed. The Ritz method with trigonometric functions as basis functions is used based on a variational formulation developed in our previous paper. Free vibration resonant frequencies and corresponding modes are obtained. The modes may separate into symmetric and antisymmetric ones for such a structurally symmetric filter. Trapped modes with vibrations mainly under the driving electrodes are exhibited. The six corner-type regions of the filter neglected by Tiersten and Stevens for an approximation are taken into account in our analysis. Results show that their approximation can lead to an inaccuracy on the order of dozens of ppm for the fundamental mode, which is quite significant in filter operation and application.
TL;DR: In this paper, a short overview of the results of the microscopic theory of the Dzyaloshinskii-Moriya coupling and other related exchange-relativistic effects such as exchange anisotropy, antisymmetric magnetoelectric coupling, and electron-nuclear supertransferred hyperfine interactions in strongly correlated 3D compounds is presented.
TL;DR: In this article, an equation of motion formalism for excited states in variational Monte Carlo is derived, and a pilot implementation for the Jastrow-modified antisymmetric geminal power is tested.
Abstract: An equation of motion formalism for excited states in variational Monte Carlo is derived, and a pilot implementation for the Jastrow-modified antisymmetric geminal power is tested. In single excitations across a range of small molecules, this combination is shown to be intermediate in accuracy between configuration interaction singles and equation of motion coupled cluster with singles and doubles. For double excitations, energy errors are found to be similar to those for coupled cluster.
TL;DR: The results show that the symmetric e-ph coupling plays the role of destabilizing the polaron as a result of temperature induced phonons that, via the asymmetric coupling, affects the charge distribution of thePolaron.
Abstract: The influence of the interplay between symmetric and antisymmetric inter-molecular electron-phonon (e-ph) coupling mechanisms on the polaron stability and mobility in organic semiconductors has been theoretically investigated at a molecular level. A semi-empirical Holstein-Peierls model is used which in addition to the symmetric and antisymmetric inter-molecular e-ph interactions also includes an antisymmetric intra-molecular e-ph coupling. Our results show that the symmetric e-ph coupling plays the role of destabilizing the polaron as a result of temperature induced phonons that, via the symmetric coupling, affects the charge distribution of the polaron. Considering this kind of coupling, the parameter space for which the polaron is dynamically stable is strongly temperature-dependent. For the combination of symmetric e-ph coupling strength and temperature, which results in a stable polaron, the velocity of the polaron, and therefore also the charge carrier mobility, is not affected by the symmetric e-ph coupling strength.
TL;DR: In this paper, the authors derived global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions.
Abstract: We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Riviere in the two-dimensional setting.
TL;DR: In this paper, the spontaneous violation of Lorentz symmetry by the vacuum condensation of an antisymmetric two-tensor was considered, and the coset construction for nonlinear realization of spacetime symmetries was employed to build the most general low-energy effective action for the Goldstone modes interacting with photons.
Abstract: This paper considers the spontaneous violation of Lorentz symmetry by the vacuum condensation of an antisymmetric two-tensor. The coset construction for nonlinear realization of spacetime symmetries is employed to build the most general low-energy effective action for the Goldstone modes interacting with photons. We analyze the model within the context of the Standard-Model extension and noncommutative QED. Experimental bounds for some parameters of the model are discussed, and we readdress the subtle issues of stability and causality in Lorentz-noninvariant scenarios. To set a sensible low-energy effective model, in addition to the two photon polarizations only one Goldstone mode must be dynamical, and the enhancement of the stability by accounting for interaction terms points to a protection against observational Lorentz violation.
TL;DR: The spin-orbit coupling in the absence of spatial inversion symmetry plays an important role in realizing intriguing electronic states in solids, such as topological insulators and unconventional superconductivity as discussed by the authors.
Abstract: The spin-orbit coupling in the absence of spatial inversion symmetry plays an important role in realizing intriguing electronic states in solids, such as topological insulators and unconventional superconductivity. Usually, the inversion symmetry breaking is inherent in the lattice structures, and hence, it is not easy to control these interesting properties by external parameters. We here theoretically investigate the possibility of generating the spin-orbital entanglement by spontaneous electronic ordering caused by electron correlations. In particular, we focus on the centrosymmetric lattices with local asymmetry at the lattice sites, e.g., zig-zag, honeycomb, and diamond structures. In such systems, conventional staggered orders, such as charge order and antiferromagnetic order, break the inversion symmetry and activate the antisymmetric spin-orbit coupling, which is hidden in a sublattice-dependent form in the paramagnetic state. Considering a minimal two-orbital model on a honeycomb lattice, we scrutinize the explicit form of the antisymmetric spin-orbit coupling for all the possible staggered charge, spin, orbital, and spin-orbital orders. We show that the complete table is useful for understanding of spin-valley-orbital physics, such as spin and valley splitting in the electronic band structure and generalized magnetoelectric responses in not only spin but also orbital and spin-orbital channels, reflecting in peculiar magnetic, elastic, and optical properties in solids.
TL;DR: In this article, a review of the co-adjoint orbit formulation of finite dimensional quantum mechanics is presented, and the notion of quantum Fisher information index and metric is interpreted. And the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation is discussed.
Abstract: In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.
TL;DR: It is demonstrated that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric formof PME favours full conversion of antimatter into matter.
Abstract: When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter-this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case and the opposite in the time-antisymmetric case). We demonstrate that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric form of PME favours full conversion of antimatter into matter. At this stage, it is impossible to make an experimentally justified choice in favour of the symmetric or antisymmetric versions of thermodynamics since we have no experience of thermodynamic properties of macroscopic objects made of antimatter, but experiments of this kind may become possible in the future.
TL;DR: In this article, a free vibration analysis of orthotropic beams with local and non-local formulation using the high-order theory including the Poisson effect is presented, taking into account the transverse shear effects introducing a new displacement shape function and a parabolic distribution of the transversal shear strains through the thickness of the beam.
Abstract: Free vibration analysis of orthotropic beams with local and nonlocal formulation using the high-order theory including the Poisson effect is presented in this paper. The theory takes into account the transverse shear effects introducing a new displacement shape function and a parabolic distribution of the transverse shear strains through the thickness of the beam. Hence it is unnecessary to use shear correction factors. The governing equations are derived from the principle of virtual displacements. The couplings among the axial, torsion, and bending deformations are investigated in the one-dimensional beam model. The free vibration solutions are finally presented for the nonlocal higher-order beam/column models. The influence of the various geometrical and material parameters, thickness ratio, and number of symmetric and antisymmetric layers of the laminate material has been investigated to find the natural frequencies. The numerical results obtained in the present study for several examples are ...
TL;DR: In this article, the problem of computing the topological string partition function for 5-brane web diagrams with an O7$^-$-plane has been studied and compared.
Abstract: We consider the computation of the topological string partition function for 5-brane web diagrams with an O7$^-$-plane. Since upon quantum resolution of the orientifold plane these diagrams become non-toric web diagrams without the orientifold we are able to apply the topological vertex to obtain the Nekrasov partition function of the corresponding 5d theory. We apply this procedure to the case of 5d $SU(N)$ theories with one hypermultiplet in the antisymmetric representation and to the case of 5d pure $USp(2N)$ theories. For these cases we discuss the dictionary between parameters and moduli of the 5d gauge theory and lengths of 5-branes in the web diagram and moreover we perform comparison of the results obtained via application of the topological vertex and the one obtained via localisation techniques, finding in all instances we consider perfect agreement.
DOI•
08 Jul 2016TL;DR: This paper shows that a large number of zeros in the simulated transformed system matrix is obtained when orthogonal transforms applied to the spatial domain have only symmetric and antisymmetric basis functions, and proposes a new method to obtain better transforms that retain the required symmetry properties.
Abstract: In this paper, we study the compression of the magnetic particle imaging system matrix for imaging setups in which a field free point is moved along a Lissajous trajectory. We show that a large number of zeros in the simulated transformed system matrix is obtained when orthogonal transforms applied to the spatial domain have only symmetric and antisymmetric basis functions. For measured system matrices, this property only holds approximately, because of noise induced by the scanner hardware. The required symmetry properties are naturally fulfilled by some standard orthogonal transforms such as the type-two discrete cosine transformand the discrete Chebychev transform. However, these transforms are not yet optimal for compressing system matrices, and we propose a new method to obtain better transforms that retain the required symmetry properties.
TL;DR: In this article, the authors studied the protection of subradiant states by the symmetry of the atomic distributions in the Dicke limit, in which collective Lamb shifts cannot be neglected.
Abstract: We study the protection of subradiant states by the symmetry of the atomic distributions in the Dicke limit, in which collective Lamb shifts cannot be neglected. We find that antisymmetric states are subradiant states for distributions with reflection symmetry. Continuous symmetry can also be used to achieve subradiance. This study is relevant to the problem of robust quantum memory with long storage time and fast readout.
TL;DR: The diagonal terms in the added mass matrix for a two-dimensional surface-piercing structure, which satisfies a geometric condition known as the John condition, are proven to be non-negative as mentioned in this paper.
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TL;DR: An equation of motion formalism for excited states in variational Monte Carlo is derived, and a pilot implementation for the Jastrow-modified antisymmetric geminal power is tested, and energy errors are found to be similar to those for coupled cluster.
Abstract: An equation of motion formalism for excited states in variational Monte Carlo is derived and a pilot implementation for the Jastrow-modified antisymmetric geminal power is tested. In single excitations across a range of small molecules, this combination is shown to be intermediate in accuracy between configuration interaction singles and equation of motion coupled cluster with singles and doubles. For double excitations, energy errors are found to be similar to those for coupled cluster.
TL;DR: In this paper, a vector field can then be decomposed into longitudinal and transverse components, and those transverse component written as positive-and negative-helicity components.
Abstract: The autocorrelation between two members of a galaxy population is symmetric under the interchange of the two galaxies being correlated. The cross-correlation between two different types of galaxies, separated by a vector $\mathbf{r}$, is not necessarily the same as that for a pair separated by $\ensuremath{-}\mathbf{r}$. Local anisotropies in the two-point cross-correlation function may thus indicate a specific direction which when mapped as a function of position trace out a vector field. This vector field can then be decomposed into longitudinal and transverse components, and those transverse components written as positive- and negative-helicity components. A locally asymmetric cross-correlation of the longitudinal type arises naturally in halo clustering, even with Gaussian initial conditions, and could be enhanced with local-type non-Gaussianity. Early-Universe scenarios that introduce a vector field may also give rise to such effects. These antisymmetric cross-correlations also provide a new possibility to seek a preferred cosmic direction correlated with the hemispherical power asymmetry in the cosmic microwave background and to seek a preferred location associated with the cosmic microwave background cold spot. New ways to seek cosmic parity breaking are also possible.
TL;DR: A generalized beamforming approach for composite structures with nonuniform cross section that eliminates systematic damage localization errors of conventional algorithms.
TL;DR: In this article, the vacuum expectation values of 1/6 supersymmetric Wilson loops in higher dimensional representations of the gauge group in ABJM theory were computed for the m-symmetric and m-antisymmetric representations by exploiting standard matrix model techniques.
Abstract: We compute the vacuum expectation values of 1/6 supersymmetric Wilson loops in higher dimensional representations of the gauge group in ABJM theory. We present results for the m-symmetric and m-antisymmetric representations by exploiting standard matrix model techniques. At leading order, in the saddle point approximation, our expressions reproduce holographic results from both D6 and D2 branes corresponding to the antisymmetric and symmetric representations, respectively. We also compute 1/N corrections to the leading saddle point results.
TL;DR: In this article, the quantum equivalence of free massive antisymmetric tensor fields of the second and third rank to the free Proca theory and massive scalar fields with minimal coupling to gravity was confirmed using explicit covariant calculations of nonlocal form factors based on the heart-kernel technique.
Abstract: In a recent paper, Buchbinder, Kirillova, and Pletnev presented formal arguments concerning the quantum equivalence of free massive antisymmetric tensor fields of the second and third rank to the free Proca theory and massive scalar field with minimal coupling to gravity, respectively. We confirm this result using explicit covariant calculations of nonlocal form factors based on the heart-kernel technique and discuss the discontinuity of quantum contributions in the massless limit.