New series representations for the two-loop massive sunset diagram

Abstract

We derive new convergent series representations for the two-loop sunset diagram with three different propagator masses $$m_1,\, m_2$$ m 1 , m 2 and $$m_3$$ m 3 and external momentum p by techniques of analytic continuation on a well-known triple series that corresponds to the Lauricella $$F_C^{(3)}$$ F C ( 3 ) 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.

Massive One-loop Conformal Feynman Integrals and Quadratic Transformations of Multiple Hypergeometric Series

“…In fact, even better than this, the convergence regions of all the series representations that we have obtained as analytic continuations of the result of region B are wider than those analytically continuing the result of region A (although the former do not include the latter in general). Therefore, the set of analytic continuations of region B's result considerably reduces, in the present case of the massive one-loop conformal 3-point integral, the domain of the conformal variables space which does not belong to any of the convergence regions of the set of series representations associated with region A's result (this particular "unreachable" domain, that we have studied and called white region in some previous works [10,11,12], is inherent to the series representations of many MB integrals).…”

“…Let us close here this parenthesis and come back to the explicit computation of the series representations associated with the MB integral in Eq. (10).…”

“…A comparison of the MB representations in Eqs. (10) and (17) shows that our method to find the series representations of Eq. ( 17) will give the same set of conic hulls S than the one associated with Eq.(20).…”

On the evaluation of the Appell $F_2$ double hypergeometric function

“…It is easy to conclude from [13] that the analytic continuations of the F C triple series, derived from the Mellin-Barnes representation of the F C function, give access to a restricted region of its three variables space. Therefore, in order to obtain analytic expressions for the sunset outside this region, some transformations of the F C Lauricella series have been obtained in [14], using an alternative method. This method, which uses quadratic transformations of the Gauss 2 F 1 hypergeometric series as intermediate steps in the derivation of new series representations for F 4 [15] (and, as a by product, for F C ) can be seen as an extension of a classical work of Olsson [16] which focused on the question of the analytic contination of the Appell F 1 series and of its F D multivariable generalisation, using linear transformations of 2 F 1 .…”

“…This method, which uses quadratic transformations of the Gauss 2 F 1 hypergeometric series as intermediate steps in the derivation of new series representations for F 4 [15] (and, as a by product, for F C ) can be seen as an extension of a classical work of Olsson [16] which focused on the question of the analytic contination of the Appell F 1 series and of its F D multivariable generalisation, using linear transformations of 2 F 1 . The approach of [14,15] can however not give the full answer to the problem of finding series representations that can be used to evaluate the F C function for all values of its variables.…”