Related papers: On vacuum integration
In this note we present the important details regarding the massless vacuum integrations which are not outlined in the literature. In particular, it has been shown how the delta-function represents either UV-regime or IR-regime. In the case…
We present the second part of a paper series devoted to the study of the multi-loop effective potential evolution in $\varphi^4$-theory using the conformal symmetry. In this paper, we demonstrate that the conformal symmetry can still be…
A useful connection between two-loop massive vacuum integrals and one-loop off-shell triangle diagrams with massless internal particles is established for arbitrary values of the space-time dimension n.
We derive analytic results for scalar massless bosonic vacuum sum-integrals at two loops. Building upon a recent factorization proof of massive two-loop vacuum integrals, we are able to solve the corresponding Matsubara sums and map the…
We obtain finite parts (as well as $\epsilon$-pole parts) of massive three-loop vacuum diagrams with three-point and/or four-point interaction vertices by reducing them to tetrahedron diagrams with both massive and massless lines, whose…
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of…
We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using…
We begin a series of two papers that is devoted to the study of the multi-loop effective potential evolution in $\varphi^4$-theory using the conformal symmetry. In the first part, we introduce and describe in detail the vacuum…
We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using…
The scalar two-loop self-energy master diagram is studied in the case of arbitrary masses. Analytical results in terms of Lauricella- and Appell-functions are presented for the imaginary part. By using the dispersion relation a…
We present analytical results for all master integrals for massless three-point functions, with one off-shell leg, at four loops. Our solutions were obtained using differential equations and direct integration techniques. We review the…
Loop amplitudes are conveniently expressed in terms of master integrals whose coefficients carry the process dependent information. Similarly before integration, the loop integrands may be expressed as a linear combination of propagator…
The approach to the constructing explicit solutions of the recurrence relations for multi-loop integrals are suggested. The resulting formulas demonstrate a high efficiency, at least for 3-loop vacuum integrals case. They also produce a new…
In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive…
Three-loop vacuum integrals are an important building block for the calculation of a wide range of three-loop corrections. Until now, only results for integrals with one and two independent mass scales are known, but in the electroweak…
We report about recent progress in computing four-loop massive correlators. The expansion of these correlators in the external momentum leads to vacuum integrals. The calculation of these vacuum integrals can be used to determine Taylor…
We present compact, fully analytical expressions for singular parts of a class of three-loop diagrams which cannot be factorized into lower-loop integrals. As a result of the calculations we obtain the analytical expression for the…
The implementation of an algorithm for three-loop massive vacuum integrals, based on the explicit solution of the recurrence relations, in REDUCE and FORM is described.
In the recent years there has been an enormous development in the evaluation of higher order quantum corrections. An essential ingredient in the practical calculations is provided by vacuum diagrams, i.e. integrals without external momenta.…
We provide an update on a long-term project that aims at evaluating massive vacuum integrals at the five-loop frontier, with high precision and in various space-time dimensions. A number of applications are sketched, mainly concerning the…