Recent zbMATH articles in MSC 30E20https://www.zbmath.org/atom/cc/30E202022-05-16T20:40:13.078697ZWerkzeugRegularization of a class of summary equationshttps://www.zbmath.org/1483.300702022-05-16T20:40:13.078697Z"Garif'yanov, F. N."https://www.zbmath.org/authors/?q=ai:garifyanov.farkhat-nurgayazovich"Strezhneva, E. V."https://www.zbmath.org/authors/?q=ai:strezhneva.elena-vasilevnaSummary: Let \(D\) be an arbitrary quadrangle with boundary \(\Gamma \). We consider a four-element linear summary equation. The solution is sought in the class of functions which are holomorphic outside \(D\) and vanish at infinity. The boundary values satisfy the Hölder condition on any compact set which does not contain the vertices. At the vertices, singularities at most of logarithmic order are allowed. The coefficients of the equation are holomorphic in \(D\) and their boundary values satisfy the Hölder condition on \(\Gamma \). The free term satisfies the same conditions. The solution is sought in the form of the Cauchy type integral over \(\Gamma\) with unknown density. To regularize the obtained functional equation, we use the Carleman problem. Previously, a Carleman shift is introduced on \(\Gamma \); it transfers each side to itself and reverses orientation; the midpoints of the sides are fixed under the shift. We indicate some applications of this summary equation to the problem of moments for entire functions of exponential type.The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domainhttps://www.zbmath.org/1483.300712022-05-16T20:40:13.078697Z"Gençtürk, İlker"https://www.zbmath.org/authors/?q=ai:gencturk.ilkerSummary: In this study, by using some integral representations formulas, we study solvability conditions and explicit solution of the Dirichlet-Neumann problem, an example for a combined boundary value problem, for the Bitsadze equation in a ring domain.A novel approach to the computation of one-loop three- and four-point functions. I: The real mass casehttps://www.zbmath.org/1483.811122022-05-16T20:40:13.078697Z"Guillet, J. Ph"https://www.zbmath.org/authors/?q=ai:guillet.j-ph"Pilon, E."https://www.zbmath.org/authors/?q=ai:pilon.eric"Shimizu, Y."https://www.zbmath.org/authors/?q=ai:shimizu.yuji|shimizu.yasuhiro|shimizu.yasushi|shimizu.yuuko|shimizu.yoshiaki|shimizu.yuya|shimizu.yusuke|shimizu.yoshifumi-r|shimizu.yuichi|shimizu.yukiko-s|shimizu.yuma|shimizu.yoshinori|shimizu.yuuki|shimizu.yasutaka|shimizu.yoshimasa|shimizu.yoshimitsu|shimizu.yosuke|shimizu.yasuyuki|shimizu.yoshiyuki|shimizu.youichiro|shimizu.yuki"Zidi, M. S."https://www.zbmath.org/authors/?q=ai:zidi.m-sThe article presents a novel method for the computation of certain correlation functions in perturbative quantum field theory, namely the three and four point functions evaluated at one loop. It is the first part of a planned series of three articles.
The computation of Feynman diagrams is the most important theoretical tool in the search for new phenomena in high energy physics. At the high precision frontier theoretical computations are matched with experimental results. The continuous increase of experimental accuracy demands an equal development of theoretical methods. While numerous computations can be done numerically, there are structural limitations in computing power. Hence the need of analytical results, even partial.
The series of articles intends to present a method to evaluate certain generalized one-loop type N-point Feynman-type integrals, which enter as building blocks of more complicated amplitudes. The present paper focuses on three and four point functions, where the integrals have the form of a certain rational function integrated over a certain simplex-like domain. In particular the authors present their methods using as a proof of concept the one loop amplitudes in a scalar theory. The method efficiently trivializes the integrals over Feynman parameters as boundary terms, using a Stokes type identity, and obtains all the necessary analytic continuations in a systematic fashion. The paper consider the case with real internal masses, postponing the cases of complex or zero masses to two subsequent articles.
This method is illustrated in detail during the course of the paper, with technical proofs and results explained in 6 appendices. The paper is very clearly written and easy to read, and contains several interesting manipulations of Feynman integrals. The results and the methods explained are likely to prove quite useful in the evaluation of higher loop amplitudes. The paper is mostly aimed to expert in the field and assumes that the reader is already familiar with the computations of loop amplitudes.
Reviewer: Michele Cirafici (Trieste)