Comments on all-loop constraints for scattering amplitudes and Feynman integrals

We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from $$ \overline{Q} $$ Q ¯ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

Generalization of the Optical Theorem to an Arbitrary Multipole Excitation of a Particle near a Transparent Substrate

In the present paper, the generalization of the optical theorem to the case of a penetrable particle deposited near a transparent substrate that is excited by a multipole of an arbitrary order and polarization has been derived. In the derivation we employ classic Maxwell’s theory, Gauss’s theorem, and use a special representation for the multipole excitation. It has been shown that the extinction cross-section can be evaluated by the calculation of some specific derivatives from the scattered field at the position of the multipole location, in addition to some finite integrals which account for the multipole polarization and the presence of the substrate. Finally, the present paper considers some specific examples for the excitation of a particle by an electric quadrupole.

Certain Finite Integrals Related to the Products of Special Functions

The aim of this paper is to establish a theorem associated with the product of the Aleph-function, the multivariable Aleph-function, and the general class of polynomials. The results of this theorem are unified in nature and provide a very large number of analogous results (new or known) involving simpler special functions and polynomials (of one or several variables) as special cases. The derived results lead to significant applications in physics and engineering sciences.

Two-loop helicity amplitudes for gg → ZZ with full top-quark mass effects

We calculate the two-loop QCD corrections to gg → ZZ involving a closed top-quark loop. We present a new method to systematically construct linear combinations of Feynman integrals with a convergent parametric representation, where we also allow for irreducible numerators, higher powers of propagators, dimensionally shifted integrals, and subsector integrals. The amplitude is expressed in terms of such finite integrals by employing syzygies derived with linear algebra and finite field techniques. Evaluating the amplitude using numerical integration, we find agreement with previous expansions in asymptotic limits and provide ab initio results also for intermediate partonic energies and non-central scattering at higher energies.

Local behavior of homeomorphisms with a finite mean over spheres

It is well known that the modulus method is one of the most powerful tools for studying mappings. Distortion estimates of the modulus of paths families are established in many known classes, in particular, the modulus does not change under conformal mappings, is finitely distorted under qu\-a\-si\-con\-for\-mal mappings, at the same time, its behavior under mappings with finite distortion depends on the dilatation coefficient. One common case is the study of mappings for which this coefficient is integrable in the domain. In the context of our research, this case has been studied in detail in our previous publications and its consideration has mostly been completed. In particular, we obtained results on the local, boundary, and global behavior of homeomorphisms, the inverse of which satisfy the weight Poletsky inequality, provided that the corresponding majorant is integrable. In contrast, the focus in this paper is on mappings for which a similar inequality may contain non integrable weights. Study of the situation of non integrable majorants, in turn, is associated with the specific behavior of the weight modulus of the annulus, which is achieved on a certain function and up to constant is equal to $(n-1)$-degree of the Lehto integral. To the same extent, these results are also related to finding the extremal in the weight modulus of the ring. The basic theorem contains the result about equicontinuity of homeomorphisms with the inverse Poletsky inequality, when the corresponding weight has finite integrals on some set of spheres, and the set of corresponding radii of these spheres must have a positive Lebesgue measure. According to Fubini's theorem, the mentioned result summarizes the corresponding statement for any integrable majorants and is fundamental in the sense that it is easy to give examples of non integrable functions with finite integrals by spheres. In addition, since conformal and quasiconformal mappings satisfy the Poletsky inequality with a constant majorant in the forward and inverse directions, the basic theorem may be considered as a generalization of previously known statements in these classes. Note that the main result does not contain any geometric constraints on the definition and image domains of the mappings, in particular, the definition domain is assumed to be arbitrary, and the image domain is supposed to be only a bounded domain in Euclidean $n$-dimensional space. The proof of the main theorem is given by the contradiction, namely, we assume that the statement about equicontinuity of the corresponding family of mappings is incorrect, and we obtain a contradiction to this assumption due to upper and lower estimates of the modulus of families of paths.

Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Dual-frequency offset transreflector antennas

The paper presents the results of computer simulation of the developed dual‑frequency antennas with a common radiating aperture, operating in the millimeter and centimeter wavelength range. The operation of antennas in the Kaand W‑bands is based on the well‑known dual reflector polarization rotating antennas constructing principles. The twistreflector of the considered antennas is combined with the radiation surface of the centimeter wavelength range (X‑band) waveguide‑slot array. Transreflector is made by offset scheme. For the Kaand X‑bandsantenna, the transreflector is a paraboloid of rotation part. In the case of the Wand X‑band antenna, the transreflector has a flat Fresnel zoned antenna structure. Computer simulation is based on the method of finite integrals, which provides a reliable result at an appropriately chosen sampling step. The calculated characteristics confirm the operability of the considered antenna options.