New series representations for the two-loop massive sunset diagram
- PDF / 1,634,183 Bytes
- 18 Pages / 595.276 x 790.866 pts Page_size
- 6 Downloads / 189 Views
Regular Article - Theoretical Physics
New series representations for the two-loop massive sunset diagram B. Ananthanarayan1, Samuel Friot2,3 , Shayan Ghosh4,a 1
Centre for High Energy Physics, Indian Institute of Science, Bangalore, Karnataka 560012, India Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France 3 Univ Lyon, Univ Claude Bernard Lyon1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622 Villeurbanne, France 4 Helmholtz-Institut für Strahlen- und Kernphysik & Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
2
Received: 29 November 2019 / Accepted: 9 June 2020 © The Author(s) 2020
Abstract We derive new convergent series representations for the two-loop sunset diagram with three different propagator masses m 1 , m 2 and m 3 and external momentum p by techniques of analytic continuation on a well-known triple (3) series that corresponds to the Lauricella FC function. The convergence regions of the new series contain regions of interest to physical problems. These include some ranges of masses and squared external momentum values which make them useful from Chiral Perturbation Theory to some regions of the parameter space of the Minimal Supersymmetric Standard Model. The analytic continuation results presented for the Lauricella series could be used in other settings as well. 1 Introduction The sunset integral is among the simplest of two-loop integrals that appear in the perturbation expansion of quantities in various quantum field theories, including the Standard Model (SM). In the convention of the classic work [1], the general massive sunset diagram of Fig. 1 is defined as T ( p 2 , m 21 , m 22 , m 23 , α, β, γ ) dnr dnq 1 ≡ 2 4 n−4 i π (2π μ) (2π μ)n−4 1 α β γ 2 2 q2 − m1 r 2 − m2 (q + r − p)2 − m 23
(1)
where the external momentum p 2 = s can take any value, n = 4 − 2, being the dimensional regularization parameter, and m 1 , m 2 , m 3 are the masses of the three distinct particles in the propagators. The most general sunset integral, therefore, can have up to four independent mass scales. a e-mail:
[email protected] (corresponding author)
0123456789().: V,-vol
The sunset integral with general powers of the propagators (α, β, γ ) can be reduced or obtained from integration by parts into a linear combination of no more than four master integrals (MI) [2] (also see [3] for a useful discussion). These MI are the sunset integral with the following propagator powers: (α, β, γ ) = (1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2). It may also be noted that all tensor and vector integrals, defined with momentum factors and appropriate Lorentz indices inside the integrals, as well as arbitrary derivatives with respect to the momentum and / or the masses can also be reduced to or obtained from the same set of MI. Thus the study of the MI is of great importance in the context of the sunset integrals. Important special cases for renormalization theory include the fact that at p 2 = 0 the residues in dimensional regularization can be evaluated in closed form, see
Data Loading...
