Euclidean Region Research Articles

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter ϵ. The results are expressed as one-fold integrals over classical polylogarithms. We provide a set of supplementary files containing our results in machine-readable form, including a proof-of-concept implementation for numerical evaluations of the one-fold integrals valid within a subset of the Euclidean region.

Higher-loop integrated negative geometries in ABJM

In the three-dimensional \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 6 Chern-Simons matter (ABJM) theory, the integrand for the logarithm of the scattering amplitude admits a decomposition in terms of negative geometries, which implies that all the infrared divergences concentrate in the last loop integration. We compute the infrared-finite functions that arise from performing a three-loop integration over the four-loop integrand for the logarithm of the four-point amplitude, for which we use the method of differential equations. Our results provide a direct computation of the four-loop cusp anomalous dimension of the theory, in agreement with the current all-loop integrability-based proposal. We find an apparent simplicity in the leading singularities of the integrated results, provided one works in the frame in which the unintegrated loop variable goes to infinity. Finally, our results suggest an alternating sign pattern for the integrated negative geometries in the Euclidean region.

Integrated approach for facility management of existing buildings using point cloud segmentation

PurposeFacility management gained profound importance due to the increasing complexity of different systems and the cost of operation and maintenance. However, due to the increasing complexity of different systems, facility managers may suffer from a lack of information. The purpose of this paper is to propose a new facility management approach that links segmented assets to the vital data required for managing facilities.Design/methodology/approachAutomatic point cloud segmentation is one of the most crucial processes required for modelling building facilities. In this research, laser scanning is used for point cloud acquisition. The research utilises region growing algorithm, colour-based region-growing algorithm and Euclidean cluster algorithm.FindingsA case study is worked out to test the accuracy of the considered point cloud segmentation algorithms utilising metrics precision, recall and F-score. The results indicate that Euclidean cluster extraction and region growing algorithm revealed high accuracy for segmentation.Originality/valueThe research presents a comparative approach for selecting the most appropriate segmentation approach required for accurate modelling. As such, the segmented assets can be linked easily with the data required for facility management.

Segmentation Clustering Algorithm for Point Cloud Processing of Targets on The Ground

After the ground extraction algorithm, there are multiple objects, including the target in the non-ground point cloud. To effectively extract the features of individual objects and overcome the dependence of the traditional algorithm based on the normal vector and curvature of the region growing point cloud segmentation method on the initial point, the normal vector angle in the neighborhood, and the curvature threshold selection. In this paper, we propose a fast clustering segmentation method for region-growing point clouds based on neighborhood density. The test results show that the point cloud segmentation algorithm proposed in this paper has better performance than the traditional Euclidean clustering point cloud segmentation algorithm and region growing point cloud segmentation algorithm in terms of timeliness and correct segmentation rate. This new segmentation clustering algorithm can quickly implement automatic point cloud clustering segmentation of ground objects.

Critical behavior and stability problem in a scalar field model

As shown in the works [1-3], the asymptotic behavior of the propagator in the Euclidean region of momenta for the model of a complex scalar field φ and a real scalar field χ with the interaction drastically changes depending on the value of the coupling constant. For small values of the coupling, the propagator of the field φ behaves asymptotically as free, while in the strong-coupling region the propagator in the deep Euclidean region tends to be a constant. In this paper, the influence of the vacuum stability problem of this model on this critical behavior is investigated. It is shown that within the framework of the approximations used, the addition of a stabilizing term of type to the Lagrangian leads to a renormalization of the mass and does not change the main effect of changing the ultraviolet behavior of the propagator. PACS number: 11.10.Jj.

Dispersive analysis of glueball masses

We develop an inverse matrix method to solve for resonance masses from a dispersion relation obeyed by a correlation function. Given the operator product expansion (OPE) of a correlation function in the deep Euclidean region, we obtain the nonperturbative spectral density, which exhibits resonance structures naturally. The value of the gluon condensate in the OPE is fixed by producing the $\rho$ meson mass in the formalism, and then input into the dispersion relations for the scalar, pseudoscalar and tensor glueballs. It is shown that the low-energy limit of the correlation function for the scalar glueball, derived from the spectral density, discriminates the lattice estimate for the triple-gluon condensate from the single-instanton estimate. The spectral densities for the scalar and pseudoscalar glueballs reveal a double-peak structure: the peak located at lower mass implies that the $f_0(500)$ and $f_0(980)$ ($\eta$ ad $\eta'$) mesons contain small amount of gluonium components, and should be included into scalar (pseudoscalar) mixing frameworks. Another peak determines the scalar (pseudoscalar) glueball mass around 1.50 (1.75) GeV with a broad width about 200 MeV, suggesting that the $f_0(1370)$, $f_0(1500)$ and $f_0(1710)$ ($\eta(1760)$) mesons are the glue-rich states. We also predict the topological susceptability $\chi_t^{1/4}=75$-78 MeV, deduced from the correlation function for the pseudoscalar glueball at zero momentum. Our analysis gives no resonance solution for the tensor glueball, which may be attributed to the insufficient nonperturbative condensate information in the currently available OPE.

Lifting heptagon symbols to functions

Seven-point amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.

Landau equation and QCD sum rules for the tetraquark molecular states

The quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionable to apply the Landau equation to study the Feynman diagrams in the QCD sum rules. Furthermore, we carry out the operator product expansion in the deep Euclidean region $p^2\to -\infty$, where the Landau singularities cannot exist. The Landau equation servers as a kinematical equation in the momentum space, and is independent on the factorizable and nonfactorizable properties of the Feynman diagrams in the color space. The meson-meson scattering state and tetraquark molecular state both have four valence quarks, which form two color-neutral clusters, we cannot distinguish the contributions based on the two color-neutral clusters in the factorizable Feynman diagrams. Lucha, Melikhov and Sazdjian assert that the contributions at the order $\mathcal{O}(\alpha_s^k)$ with $k\leq1$ in the operator product expansion, which are factorizable in the color space, are exactly canceled out by the meson-meson scattering states at the hadron side, the tetraquark molecular states begin to receive contributions at the order $\mathcal{O}(\alpha_s^2)$. Such an assertion is questionable, we refute the assertion in details, and choose an axialvector current and a tensor current to examine the outcome of the assertion. After detailed analysis, we observe that the meson-meson scattering states cannot saturate the QCD sum rules, while the tetraquark molecular states can saturate the QCD sum rules. The Landau equation is of no use to study the Feynman diagrams in the QCD sum rules for the tetraquark molecular states, the tetraquark molecular states begin to receive contributions at the order $\mathcal{O}(\alpha_s^0/\alpha_s^1)$ rather than at the order $\mathcal{O}(\alpha_s^2)$.

Reflection positivity and complex analysis of the Yang\u2013Mills theory from a viewpoint of gluon confinement

In order to understand the confining decoupling solution of the Yang–Mills theory in the Landau gauge, we consider the massive Yang–Mills model which is defined by just adding a gluon mass term to the Yang–Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev–Popov ghost term. First of all, we show that massive Yang–Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout–Englert–Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang–Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang–Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov–Stingl form. We try to understand these results in light of the Fradkin–Shenker continuity between confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model.

The heavy fermion contributions to the massive three loop form factors

We compute the non-singlet nh terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals have to be dealt with only in their moment representation, allowing to also consider quantities which obey differential equations, which are not first order factorizable (elliptic and higher), already at this level. To obtain all the associated recursions, up to 8000 moments had to be calculated. A new technique has been applied to solve the associated differential equation systems. Here the decoupling is performed such, that only minimal depth ε–expansions had to be performed for non–first-order factorizing systems, minimizing the calculation of initial values. The pole terms in the dimensional parameter ε can be completely predicted using renormalization group methods, as confirmed by the present results. A series of contributions at O(ε0) have first order factorizable representations. For a smaller number of color–zeta projections this is not the case. All first order factorizing terms can be represented by harmonic polylogarithms. We also obtain analytic results for the non–first-order factorizing terms by Taylor series in a variable x, for which we have calculated at least 2000 expansion coefficients, in an approximation. Based on this representation the form factors can be given in the Euclidean region and in the region q2≈0. Numerical results are presented.

Analytic result for the nonplanar hexa-box integrals

In this paper, we analytically compute all master integrals for one of the two non-planar integral families for five-particle massless scattering at two loops. We first derive an integral basis of 73 integrals with constant leading singularities. We then construct the system of differential equations satisfied by them, and find that it is in canonical form. The solution space is in agreement with a recent conjecture for the non-planar pentagon alphabet. We fix the boundary constants of the differential equations by exploiting constraints from the absence of unphysical singularities. The solution of the differential equations in the Euclidean region is expressed in terms of iterated integrals. We cross-check the latter against previously known results in the literature, as well as with independent Mellin-Barnes calculations.

On the phase structure of vector-matrix scalar model in four dimensions

The leading-order equations of the 1/N – expansion for a vector-matrix model with interaction gphi _a^*phi _bchi _{ab} in four dimensions are investigated. This investigation shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. For small values of the coupling the phion propagator behaves as free. In the strong-coupling region the asymptotic behavior drastically changes – the propagator in the deep Euclidean region tend to some constant limit. The phion propagator in the coordinate space has a characteristic shell structure. At the critical value of coupling that separates the weak and strong coupling regions, the asymptotic behavior of the phion propagator is a medium among the free behavior and the constant-type behavior in strong-coupling region. The equation for a vertex with zero transfer is also investigated. The asymptotic behavior of the solutions shows the finiteness of the charge renormalization constant. In the strong-coupling region, the solution for the vertex has the same shell structure in coordinate space as the phion propagator. An analogy between the phase transition in this model and the re-arrangement of the physical vacuum in the supercritical external field due to the “fall-on-the-center” phenomenon is discussed.

Fakeons and Lee-Wick models

The “fakeon” is a fake degree of freedom, i.e. a degree of freedom that does not belong to the physical spectrum, but propagates inside the Feynman diagrams. Fakeons can be used to make higher-derivative theories unitary. Moreover, they help us clarify how the Lee-Wick models work. In this paper we study the fakeon models, that is to say the theories that contain fake and physical degrees of freedom. We formulate them by (nonanalytically) Wick rotating their Euclidean versions. We investigate the properties of arbitrary Feynman diagrams and, among other things, prove that the fakeon models are perturbatively unitary to all orders. If standard power counting constraints are fulfilled, the models are also renormalizable. The S matrix is regionwise analytic. The amplitudes can be continued from the Euclidean region to the other regions by means of an unambiguous, but nonanalytic, operation, called average continuation. We compute the average continuation of typical amplitudes in four, three and two dimensions and show that its predictions agree with those of the nonanalytic Wick rotation. By reconciling renormalizability and unitarity in higher-derivative theories, the fakeon models are good candidates to explain quantum gravity.

On the relation between pole and running heavy quark masses beyond the four-loop approximation

The effective charges motivated method is applied to the relation between pole and M̅S̅-scheme heavy quark masses to study high order perturbative QCD corrections in the observable quantities proportional to the running quark masses. The non-calculated five- and six-loop perturbative QCD coefficients are estimated. This approach predicts for these terms the sign-alternating expansion in powers of number of lighter flavors nl, while the analyzed recently infrared renormalon asymptotic expressions do not reproduce the same behavior. We emphasize that coefficients of the quark mass relation contain proportional to π2 effects, which result from analytical continuation from the Euclidean region, where the scales of the running masses and QCD coupling constant are initially fixed, to the Minkowskian region, where the pole masses and the running QCD parameters are determined. For the t-quark the asymptotic nature of the non-resummed PT mass relation does not manifest itself at six-loops, while for the b-quark the minimal PT term appears at the probed by direct calculations four-loop level. The recent infrared renormalon based studies support these conclusions.

Asymptotic Behavior and Critical Coupling in Scalar Yukawa Model

The solution of the equation for the pion propagator in the leading order of the [Formula: see text] – expansion for a vector-matrix model with interaction [Formula: see text] in four dimensions shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant.

Note on lattice regularization and equal-time correlators for parton distribution functions

We show that a recent interesting idea to circumvent the difficulties with the continuation of parton distribution functions to the Euclidean region, that consists in looking at equal time correlators between proton states of infinite momentum, encounters some problems related to the power divergent mixing pattern of DIS operators, when implemented within the lattice regularization.

The solution of euclidean norm trust region SQP subproblems via second-order cone programs: an overview and elementary introduction

It is well known that convex sequential quadratic programming (SQP) subproblems with an Euclidean norm trust region constraint can be reduced to second-order cone programs for which the theory of Euclidean Jordan algebras leads to efficient interior-point algorithms. Here, a brief and self-contained outline of the principles of such an implementation is given and the application to SQP subproblems as well as to cubic regularization problems is discussed. All identities relevant for the implementation are derived from scratch and are compared to interior-point methods for linear programs (LPs). Sparsity of the data of the SQP subproblem can be maintained essentially in the same way as for interior-point methods for LPs. The presentation is intended as an introduction for students and for colleagues who may have heard about Jordan algebras but did not yet find the time to get involved with them. A simple Matlab implementation is made available and the discussion of implementational aspects addresses a scaling property that is critical for SQP subproblems.

Deuteron Electromagnetic Form Factors in AdS/QCD

We extend a soft-wall AdS/QCD approach to a description of deuteron properties. Our framework is based an effective action formulated in terms of AdS fields, which are holographically equivalent to the deuteron and photon fields. This action produces the equation of motion for the deuteron wave function and the \(Q^2\)-dependent electromagnetic current, which are then used to calculate the deuteron electromagnetic form factors and structure functions in the Euclidean region. We show that the predicted deuteron quantities are expressed through a universal function, which is defined by a single scale parameter \(\kappa \) and which has the correct \(1/Q^{10}\) power scaling at large \(Q^2\).

Primordial scalar power spectrum from the Euclidean big bounce

In effective models of loop quantum cosmology, the holonomy corrections are associated with deformations of space-time symmetries. The most evident manifestation of the deformations is the emergence of an Euclidean phase accompanying the non-singular bouncing dynamics of the scale factor. In this article, we compute the power spectrum of scalar perturbations generated in this model, with a massive scalar field as the matter content. Instantaneous and adiabatic vacuum-type initial conditions for scalar perturbations are imposed in the contracting phase. The evolution through the Euclidean region is calculated based on the extrapolation of the time direction pointed by the vectors normal to the Cauchy hypersurface in the Lorentzian domains. The obtained power spectrum is characterized by a suppression in the IR regime and oscillations in the intermediate energy range. Furthermore, the speculative extension of the analysis in the UV reveals a specific rise of the power.

Hermitian vs <i>PT</i>-Symmetric Scalar Yukawa Model

A comparative analysis of a model of complex scalar field φ and real scalar field χ with interaction for the real and purely imaginary values of coupling g in perturbative and non-perturbative regions is provided. In contrast to the usual Hermitian version (real g), which is asymptotically free and energetically unstable, the non-Hermitian PT-symmetric theory (imaginary g) is energetically stable and not asymptotically free. The non-perturbative approach based on Schwinger-Dyson equations reveals new interesting feature of the non-Hermitian model. While in the Hermitian version of theory the phion propagator has the non-physical non-isolated singularity in the Euclidean region of momenta, the non-Hermitian theory substantially free of this drawback, as the singularity moves in the pseudo-Euclidean region.