Tensor Integrals Research Articles

MaRTIn – Massive Recursive Tensor Integration

We present MaRTIn, an extendable all-in-one package for calculating amplitudes up to two loops in an expansion in external momenta or using the method of infrared rearrangement. Renormalisable and non-renormalisable models can be supplied by the user; an implementation of the Standard Model is included in the package. In this manual, we discuss the scope and functionality of the software, and give instructions of its use.

B meson mixing at NNLO: technical aspects

We provide details to several technical aspects which are important for the calculation of next-to-next-to-leading order corrections to the mixing of neutral B mesons. This includes the computation of the master integrals for finite charm and bottom quark masses, projectors for products of up to 22 γ matrices and tensor integrals with up to rank 11.

Tensor reduction of loop integrals

The computational cost associated with reducing tensor integrals to scalar integrals using the Passarino-Veltman method is dominated by the diagonalisation of large systems of equations. These systems of equations are sized according to the number of independent tensor elements that can be constructed using the metric and external momenta. In this article, we present a closed-form solution of this diagonalisation problem in arbitrary tensor integrals. We employ a basis of tensors whose building blocks are the external momentum vectors and a metric tensor transverse to the space of external momenta. The scalar integral coefficients of the basis tensors are obtained by mapping the basis elements to the elements of an orthogonal dual basis. This mapping is succinctly expressed through a formula that resembles the ordering of operators in Wick’s theorem.Finally, we provide examples demonstrating the application of our tensor reduction formula to Feynman diagrams in QCD 2 → 2 scattering processes, specifically up to three loops.

Nontrivial one-loop recursive reduction relation

In [1], we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when addressing certain tensor integrals. Recently, we found a remarkable recursion relation [2, 3], where a tensor integral is reduced to lower-rank integrals and lower terms corresponding to integrals with one or more propagators being canceled. However, the expression of the lower terms is unknown. In this paper, we derive this non-trivial recursion relation for non-degenerate and degenerate cases and provides an explicit expression for the lower terms, thus simplifying and speeding up the reduction process.

A unified formulation of one-loop tensor integrals for finite volume effects

A unified formulation of one-loop tensor integrals is proposed for systematical calculations of finite volume corrections. It is shown that decomposition of the one-loop tensor integrals into a series of tensors accompanied by tensor coefficients is feasible, if a unit space-like four vector nμ, originating from the discretization effects at finite volume, is introduced. A generic formula has been derived for numerical computations of all the involved tensor coefficients. For the vanishing external three-momenta, we also investigate the feasibility of the conventional Passarino-Veltmann reduction of the tensor integrals in a finite volume. Our formulation can be easily used to realize the automation of the calculations of finite volume corrections to any interesting quantities at one-loop level. Besides, it provides finite volume result in a unique and concise form, which is suited for, e.g., carrying out precision determination of physical observable from modern lattice QCD data.

Two-loop amplitude generation in OpenLoops

Numerical tools, such as OpenLoops, provide NLO scattering amplitudes for a very wide range of hard scattering amplitudes in a fully automated way. In order to match the numerical precision of current and future experiments, however, the higher precision of NNLO calculations is essential, and their automation in a similar tool highly desirable.In our approach, D-dimensional amplitudes are decomposed into loop-momentum tensor integrals with coefficients constructed in four dimensions and rational terms. We present a fully generic algorithm for the efficient numerical construction of the tensor coefficients, which constitutes an important building block for an automated NNLO tool.

Two-loop tensor integral coefficients in OpenLoops

We present a new and fully general algorithm for the automated construction of the integrands of two-loop scattering amplitudes. This is achieved through a generalisation of the open-loops method to two loops. The core of the algorithm consists of a numerical recursion, where the various building blocks of two-loop diagrams are connected to each other through process-independent operations that depend only on the Feynman rules of the model at hand. This recursion is implemented in terms of tensor coefficients that encode the polynomial dependence of loop numerators on the two independent loop momenta. The resulting coefficients are ready to be combined with corresponding tensor integrals to form scattering probability densities at two loops. To optimise CPU efficiency we have compared several algorithmic options identifying one that outperforms naive solutions by two orders of magnitude. This new algorithm is implemented in the OpenLoops framework in a fully automated way for two-loop QED and QCD corrections to any Standard Model process. The technical performance is discussed in detail for several 2 → 2 and 2 → 3 processes with up to order 105 two-loop diagrams. We find that the CPU cost scales linearly with the number of two-loop diagrams and is comparable to the cost of corresponding real-virtual ingredients in a NNLO calculation. This new algorithm constitutes a key building block for the construction of an automated generator of scattering amplitudes at two loops.

Theoretical and physical insights of acoustic radiation forces beyond Gor’kov potential

Acoustic radiation forces exerted on small objects by standing waves can be simply described by the Gor’kov potential that provides a basis for designs of acoustofluidic devices for trapping and separation of particles. However, when considering forces exerted on large objects by general sound beams, theoretical and physical insights into the forces must rely on momentum conservation. Nevertheless, the relative complexity of analysis of the force starting from stress tensor or spatial integral of acoustic radiation pressure creates a barrier on the insights in the frame of momentum conservation. Despite the early availability of momentum conservation frame for acoustic radiation pressure on interface between two media, the insights into acoustic radiation forces on large objects beyond Gor’kov potential are still limited. These insights are now enhanced by our recent theory of momentum conservation frame for three-dimensional acoustic radiations forces. The theory has simplified analytical results for acoustic radiation forces into compact and physically meaningful formulas. These formulas provide an understanding for forces and torques exerted by sound beams on both large objects and interface between two media in various scenarios, which will be summarized and reviewed.

Distinction of Asymmetric Load From Eccentricity or Synchronous Electric Generator Winding Fault Using Search Coil Measurements

Forces acting on the stator teeth of the 400 kVA, a four-pole synchronous generator was analyzed for asymmetric electric load conditions. The measurement of the flux density was performed with the measuring coils installed on the stator teeth and calculations were done by the finite-element method. The measured data were processed to determine the radial forces of the stator teeth through the Maxwell stress tensor integration algorithm. The processed data were used for the detection of asymmetric load condition. The methodology for the distinction of the asymmetric load from the eccentricity or winding fault is presented. The parameters of the measuring coil sensors and their adequate position can be determined with finite-element calculation before performing sensor installation on the machine.

Superintegrable systems with spin and second-order tensor and pseudo-tensor integrals of motion

We investigate a quantum non-relativistic system describing the interaction of two particles with spin and spin 0, respectively. Assuming that the Hamiltonian is rotationally invariant and parity conserving we identify all such systems which allow additional tensor and pseudo-tensor integrals of motion that are second-order matrix polynomials in the momenta. Previously we found all the scalar, pseudo-scalar, vector and axial vector integrals of motion. No non-obvious tensor integrals exist. However, nontrivial pseudo-tensor integrals do exist. Together with our earlier results we give a complete list of such superintegrable Hamiltonian systems allowing second-order integrals of motion.

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.

On the evaluation of two-loop electroweak box diagrams for e+e\u2212 \u2192 HZ production

Precision studies of the Higgs boson at future e+e− colliders can help to shed light on fundamental questions related to electroweak symmetry breaking, baryogenesis, the hierarchy problem, and dark matter. The main production process, e+e−→ HZ, will need to be controlled with sub-percent precision, which requires the inclusion of next-to-next-to-leading order (NNLO) electroweak corrections. The most challenging class of diagrams are planar and non-planar double-box topologies with multiple massive propagators in the loops. This article proposes a technique for computing these diagrams numerically, by transforming one of the sub-loops through the use of Feynman parameters and a dispersion relation, while standard one-loop formulae can be used for the other sub-loop. This approach can be extended to deal with tensor integrals. The resulting numerical integrals can be evaluated in minutes on a single CPU core, to achieve about 0.1% relative precision.

Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

Stress-Energy in Liouville Conformal Field Theory

We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere {mathbb {S}}^2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on {mathbb {S}}^2. In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In Kupiainen et al. (Commun Math Phys 371:1005–1069, 2019) the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in Oikarinen (Ann Henri Poincaré 20(7):2377–2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.

All two-loop scalar self-energies and tadpoles in general renormalisable field theories

We calculate the complete tadpoles and self-energies at the two-loop order for scalars in general renormalisable theories, a crucial component for calculating two-loop electroweak corrections to Higgs-boson masses or for any scalar beyond the Standard Model. We renormalise the amplitudes using mass-independent renormalisation schemes, based on both dimensional regularisation and dimensional reduction. The results are presented here in Feynman gauge, with expressions for all 121 self-energy and 25 tadpole diagrams given in terms of scalar and tensor integrals with the complete set of rules to reduce them to a minimal basis of scalar integrals for any physical kinematic configuration. In addition, we simplify the results to a set of only 16 tadpole and 58 self-energy topologies using relations in order to substitute the ghost and Goldstone-boson couplings that we derive. To facilitate their application, we also provide our results in electronic form as a new code TLDR. We test our results by applying them to the Standard Model and compare with analytic expressions in the literature.

Reduction of Feynman integrals in the parametric representation

In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum space. Tensor integrals can directly be parametrized without performing tensor reductions. The integrands of parametric integrals are functions of Lorentz scalars, instead of four momenta. The complexity of a calculation is determined by the number of propagators that are present rather than the number of all the linearly independent propagators. Furthermore, the symmetries of Feynman integrals under permutations of indices are transparent in the parametric representation. Since all the indices of the propagators are nonnegative, an algorithm to solve those identities can easily be developed, which can be used for automatic calculations.

Tensor invariants and integration of differential equations

The connection between tensor invariants of systems of differential equations and explicit integration of them is discussed. A general result on the integrability of dynamical systems admitting a complete set of integral invariants in the sense of Cartan is proved. The existence of an invariant -form is related to the representability of the dynamical system in Hamiltonian form (with a symplectic structure which may be degenerate). This general idea is illustrated using an example of linear systems of differential equations. A general concept of flags of tensor invariants is introduced. General relations between the Kovalevskaya exponents of quasi-homogeneous systems of differential equations and flags of quasi-homogeneous tensor invariants having a certain structure are established. Results of a general nature are applied, in particular, to show that the general solution of the equations of rotation for a rigid body is branching in the Goryachev–Chaplygin case. Bibliography: 50 titles.

Tensor invariants and integration of differential equations

Обсуждается связь тензорных инвариантов систем дифференциальных уравнений с проблемой их точного интегрирования. Доказана общая теорема об интегрируемости динамических систем, допускающих полный набор интегральных инвариантов по Картану. Наличие инвариантной $1$-формы связано с возможностью представления динамической системы в гамильтоновой форме (возможно, с вырожденной симплектической структурой). Эта общая идея продемонстрирована на примере линейных систем дифференциальных уравнений. Введено общее понятие флагов тензорных инвариантов. Установлены общие соотношения между показателями Ковалевской квазиоднородных систем дифференциальных уравнений и флагами квазиоднородных тензорных инвариантов известной структуры. Результаты общего характера применены, в частности, для доказательства ветвления общего решения уравнений вращения твердого тела в случае Горячева-Чаплыгина. Библиография: 50 названий.

Compressibility of arterial wall – Direct measurement and predictions of compressible constitutive models

Volumetric compressibility and Poisson's ratios of fibrous soft tissues are analyzed in this paper on the basis of constitutive models and experimental data. The paper extends the previous work of Skacel and Bursa (J Mech Behav Biomed Mater, 54, pp. 316–327, 2016), dealing with incompressible behaviour of constitutive models, to the area of compressibility. Both recent approaches to structure-based constitutive modelling of anisotropic fibrous biomaterials (based on either generalized structure tensor or angular integration) are analyzed, including their compressibility-related aspects. New experimental data related to compressibility of porcine arterial layer are presented and compared with the theoretical predictions of analyzed constitutive models. The paper points out the drawbacks of recent models with distributed fibres orientation since none of the analyzed constitutive models seems to be capable to predict the experimentally observed Poisson's ratios and volume change satisfactory.

Application of the Feynman-tree theorem together with BCFW recursion relations

Recently, it has been shown that on-shell scattering amplitudes can be constructed by the Feynman-tree theorem combined with the BCFW recursion relations. Since the BCFW relations are restricted to tree diagrams, the preceding application of the Feynman-tree theorem is essential. In this way, amplitudes can be constructed by on-shell and gauge-invariant tree amplitudes. Here, we want to apply this method to the electron–photon vertex correction. We present all the single, double, and triple phase-space tensor integrals explicitly and show that the sum of amplitudes coincides with the result of the conventional calculation of a virtual loop correction.