Scalar Integrals Research Articles

Sharp upper diameter bounds for compact shrinking Ricci solitons

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

Renormalization Group evolution from on-shell SMEFT

We describe the on-shell method to derive the Renormalization Group (RG) evolution of Wilson coefficients of high dimensional operators at one loop, which is a necessary part in the on-shell construction of the Standard Model Effective Field Theory (SMEFT), and exceptionally efficient based on the amplitude basis in hand. The UV divergence is obtained by firstly calculating the coefficients of scalar bubble integrals by unitary cuts, then subtracting the IR divergence in the massless bubbles, which can be easily read from the collinear factors we obtained for the Standard Model fields. Examples of deriving the anomalous dimensions at dimension six are presented in a pedagogical manner. We also give the results of contributions from the dimension-8 H4D4 operators to the running of V+V−H2 operators, as well as the running of B+B−H2D2n from H4D2n+4 for general n.

STR: A Mathematica package for the method of uniqueness

We present Star–Triangle Relations (STRs), a Mathematica® package designed to solve Feynman diagrams by means of the method of uniqueness in any Euclidean space-time dimension. The method of uniqueness is a powerful technique to solve multi-loop Feynman integrals in theories with conformal symmetry imposing some relations between the powers of propagators and the space-time dimension. In our algorithm, we include both identities for scalar and Yukawa type integrals. The package provides a graphical environment in which it is possible to draw the desired diagram with the mouse input and a set of tools to modify and compute it. Throughout the use of a graphic interface, the package should be easily accessible to users with little or no previous experience on diagrams computation. This manual includes some pedagogical examples of computation of Feynman graphs as the scalar two-loop kite master integral and a fermionic diagram appearing in the computation of the spectrum of the [Formula: see text]-deformed [Formula: see text] SYM in the double scaling limit.

Scalar and vector line integrals: A conceptual analysis and an initial investigation of student understanding

This paper adds to the growing body of research happening in multivariable calculus by examining scalar and vector line integrals. This paper contributes in two ways. First, this paper provides a conceptual analysis for both types of line integrals in terms of how theoretical ways of thinking about definite integrals summarized from the research literature might be applied to understanding line integrals specifically. Second, this paper provides an initial investigation of students’ understandings of line integral expressions, and connects these understanding to the theoretical ways of thinking drawn from the literature. One key finding from the empirical part is that several students appeared to understand individual pieces of the integral expression based on one way of thinking, such as adding up pieces or anti-derivatives, while trying to understand the overall integral expression through a different way of thinking, such as area under a curve.

Hypergeometric expression for a three-loop vacuum integral

Using the corresponding Mellin–Barnes representation, we derive holonomic hypergeometric system of linear partial differential equations (PDEs) satisfied by Feynman integral of a three-loop vacuum with five propagators. Through the multidimensional residue theorem in dimensional regularization, the scalar integral can be written as the summation of multiple hypergeometric functions, whose convergent regions can be obtained by the Horn’s convergent theory. The numerical continuation of the scalar integral from convergent regions to whole kinematic regions can be accomplished with the finite element methods, when the system of PDEs can be treated as the stationary conditions of a functional under the restrictions.

FeynCalc 9.3: New features and improvements

We present FeynCalc 9.3, a new stable version of a powerful and versatile Mathematica package for symbolic quantum field theory (QFT) calculations. Some interesting new features such as highly improved interoperability with other packages, automatic extraction of the ultraviolet divergent parts of 1-loop integrals, support for amplitudes with Majorana fermions and γ-matrices with explicit Dirac indices are explained in detail. Furthermore, we discuss some common problems and misunderstandings that may arise in the daily usage of the package, providing explanations and workarounds. Program summaryProgram Title: FeynCalcCPC Library link to program files:http://dx.doi.org/10.17632/cmpjr5ktmp.2Licensing provisions: GNU GPLv3Programming language: Wolfram Mathematica 8 and higherExternal routines/libraries: FeynArts [1]Journal reference of previous version: Comput. Phys. Commun., 207, 432–444, (2016).Does the new version supersede the previous version?: YesNature of problem: Symbolic calculation of Feynman diagrams in particle physics and suitable standalone expressions in Quantum Field Theory.Solution method: The required algorithms and algebraic identities are implemented in Wolfram Mathematica.Reasons for the new version: Highly improved interoperability with other packages, new routines for Dirac algebra and loop integral evaluation.Summary of revisions:FeynCalc can be loaded with arbitrary packages into the same Mathematica kernel without causing shadowing issues. Algebraic manipulations of Dirac matrices and spinors with explicit Dirac indices are now available. Amplitudes involving Majorana fermions (both written by hand or obtained with FeynArts [1] can be evaluated out-of-the box. The same goes for FeynArts-generated diagrams with 4-fermion operators. The algorithm of [2] is added to extract UV-poles from scalar 1-loop integrals (Passarino–Veltman functions) with an arbitrary number of external legs.Restrictions: In the case of multi-particle (beyond 1→3 and 2→3) or multiloop processes, it is not advisable to perform all algebraic evaluations only with FeynCalc and Mathematica.

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.

Direct reduction of multiloop multiscale scattering amplitudes

We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix and dimension shift for tensor reduction calculation. During this procedure we further utilize the Feynman parameterization to calculate the coefficients of series representation and obtain the form factors. Conventional methodologies are used only for scalar vacuum bubble integrals to finalize the result in series representation form. Finally, we elaborate our approach by presenting the reduction of a typical two-loop amplitude for W boson production.

Mathematical Aspects of Krätzel Integral and Krätzel Transform

A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored.

Numerical Loop-Tree Duality: contour deformation and subtraction

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.

Towards an efficient method to compute two-loop scalar amplitudes

We report on an ongoing work initiated by Prof. Shimizu, proposing a method to numerically compute two-loop scalar integrals as sums of two-dimensional integrals of generalised one-loop N-point functions analytically computed and integrated over some simple weight functions. The analytic computation of the generalised one-loop N-point functions in a systematic way motivates a novel approach sketched in this talk.

A novel approach to the computation of one-loop three- and four-point functions. III. The infrared divergent case

Abstract This article is the third and last of a series presenting an alternative method for computing the one-loop scalar integrals. It extends the results of the first two articles to the infrared divergent case. This novel method enjoys a couple of interesting features as compared with the methods found in the literature. It directly proceeds in terms of the quantities driving algebraic reduction methods. It yields a simple decision tree based on the vanishing of internal masses and one-pinched kinematic matrices, which avoids a profusion of cases. Lastly, it extends to kinematics more general than the physical, e.g. collider processes, relevant at one loop. This last feature may be useful when considering the application of this method beyond one loop using generalized one-loop integrals as building blocks.

A novel approach to the computation of one-loop three- and four-point functions. II. The complex mass case

Abstract This article is the second of a series of three presenting an alternative method to compute the one-loop scalar integrals, which directly proceeds in terms of the quantities driving the algebraic reduction. The method presented in the first article extends to general complex masses in a systematic way with a few slight adjustments required by the fact that the imaginary parts of these quantities are no longer driven by the Feynman prescription of the propagators but by intricate combinations of imaginary masses, which results in different cases sharing a common structure. As in the case of real masses, it incorporates configurations of kinematics that are more general than those pertaining to physical processes.

Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes

Using a careful choice of infrared (IR) subtraction scheme, we demonstrate cancellation of all terms with transcendental weights 0, 1, 2 from the finite part of the full-color two-loop four-gluon mathcal{N} = 2 supersymmetric QCD amplitude, with Nf massless supersymmetric quarks. This generalizes the previously observed cancellation of weight-2 terms in the superconformal theory, where Nf = 2Nc for gauge group SU(Nc). The subtraction scheme follows naturally both from general IR factorization principles and from an integrand-level analysis of divergences in this amplitude. The divergences are written in terms of scalar triangle integrals whose expressions are known to all orders in the dimensional regulator ϵ = (4 − D)/2. We also present integrated expressions for the full-color two-loop four-point amplitudes with both matter and vectors on external legs in which lower-weight terms also cancel using an appropriate IR scheme. This provides us with values for the two-loop cusp, gluonic, and quark anomalous dimensions in mathcal{N} = 2 supersymmetric QCD, which are cross-checked between the three different amplitudes.

A novel approach to the computation of one-loop three- and four-point functions: I. The real mass case

AbstractThis article is the first of a series of three presenting an alternative method of computing the one-loop scalar integrals. This novel method enjoys a couple of interesting features as compared with the method closely following ’t Hooft and Veltman adopted previously. It directly proceeds in terms of the quantities driving algebraic reduction methods. It applies to the three-point functions and, in a similar way, to the four-point functions. It also extends to complex masses without much complication. Lastly, it extends to kinematics more general than that of the physical, e.g., collider processes relevant at one loop. This last feature may be useful when considering the application of this method beyond one loop using generalized one-loop integrals as building blocks.

Computing one-loop corrections to effective vertices with two scales in the EFT for Multi-Regge processes in QCD

The computation of one-loop corrections to Reggeon-Particle-Particle effective vertices with two scales of virtuality is considered in the framework of gauge-invariant effective field theory for Multi-Regge processes in QCD. Rapidity divergences arising in loop integrals are regulated by “tilted Wilson lines” prescription. General analysis of rapidity divergences at one loop is given and necessary scalar integrals with one and two scales of virtuality are computed. Two examples of effective vertices at one loop are considered: the effective vertex of interaction of (space-like) virtual photon with one Reggeized and one Yang-Mills quark and the effective vertex of Reggeized gluon to Yang-Mills gluon transition with an insertion of the operator tr[GμνGμν] carrying the (space-like) off-shell momentum. All terms ∼r±ϵ in the rapidity-regulator variable r cancel between diagrams and only log⁡r-divergence is left. It is checked on several examples, that obtained results indeed allow one to reproduce Regge limit of one-loop QCD scattering amplitudes.

Integrability of black hole orbits in maximal supergravity

We study the dynamics of a pair of extremal (half-BPS) black holes in mathcal{N} = 8 supergravity, as a potentially solvable model of gravitational dynamics. As a diagnosis of hidden symmetries, we ask whether the perihelion of the orbits precesses over time. We consider black hole charge vectors with arbitrary misalignment. First, we use scattering amplitude methods to compute the leading post-Newtonian correction for general mass ratios. This computation is greatly simplified by introducing a suitable on-shell superspace. Second, we study the probe limit to all orders in velocity and Newton’s constant through a ten-dimensional brane setup. In all cases we find no precession. We relate this to the absence of scalar triangle integrals.

Three-loop vacuum integral with four-propagators using hypergeometry * * Supported partly by the National Natural Science Foundation of China (NNSFC) (11535002, 11705045) and the youth top-notch talent support program of the Hebei Province

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams. In this study, we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators. The result can be described in terms of generalized hypergeometric functions of triple variables. Based on the triple hypergeometric functions, we establish the systems of homogeneous linear partial differential equations (PDEs) satisfied by the scalar integral of three-loop vacuum diagram with four propagators. The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.

Design and implementation of quasi-optical components for the upgrade of the TCV EC-system

The EC-system of the TCV tokamak is progressively being upgraded with the addition of two MW-class dual-frequency gyrotrons (84 and 126 GHz/2s/1MW). In order to connect the two gyrotrons to the existing low field side and top launchers, new waveguide routing from gyrotron hall to TCV tokamak was designed and dedicated Matching Optics Units (MOU) have been developed. The internal optics of the system have been determined aiming at optimal coupling to the HE11 waveguides. The laws of quasi-optics were used to find quadratic surfaces to shape an incoming Gaussian beam representative of the gyrotron output into a beam matching the proper field distribution at the waveguide entrance and with HE11 content compatible with the system requirements. A solution with one flat (movable) mirror and two shaping mirrors was found and characterized with the physical optics code GRASP. The resulting field distribution is then truncated and projected onto the HE11 component to evaluate the design solution (coupled power at the waveguide entrance >98.4% and HE11 content >96.8% for both frequencies). The model and the results of this analysis will be presented and compared to a model based on the Rayleigh–Sommerfeld scalar diffraction integral. GRASP was also used to evaluate preliminary misalignment effects in terms of coupled power to the waveguide and the MOU design moved to the manufacturing phase. In parallel to the gyrotron integration and to extend the level of flexibility of the TCV EC-system, a modular closed divertor chamber is developed, requiring the X3 top-launcher to be redesigned. Preliminary antenna conceptual design studies including new curvature to cope with the requirements of modularity and flexibility will be presented.

Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension d has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions F12, vertices need additionally Kampé de Fériet-Appell functions F1, and box integrals also Lauricella-Saran functions FS. In this study, alternative recursive Mellin-Barnes representations are used for the representation of n-point functions in terms of (n−1)-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions F1 and Saran functions FS at arbitrary kinematical arguments.