The propagator seagull: general evaluation of a two loop diagram

Abstract

We study a two loop diagram of propagator type with general parameters through the Symmetries of Feynman Integrals (SFI) method. We present the SFI group and equation system, the group invariant in parameter space and a general representation as a line integral over simpler diagrams. We present close form expressions for three sectors, each with three or four energy scales, for any spacetime dimension d as well as the expansion. We determine the singular locus and the diagram's value on it. arXiv:1809.05040v3 [hep-th] 19 May 2019 ∂ 2 O 3 I(M 2 , m 2 t, m 2 4 , p 2 ) = J bubble (1, 1; M 2 , m 2 4 , p 2 )J tad (2; m 2 t) (5.37) with J tad and J bubble defined in (A.1) and (A.2) and given explicitly by J bubble (1, 1; M 2 , m 2 4 , p 2 ) = i 1−d π d/2 (−m 2 4 )

Numerator seagull and extended Symmetries of Feynman Integrals

“…By now, SFI was further developed and numerous diagrams have been analyzed within it [8][9][10][11][12][13][14][15]. Even though these diagrams are relatively basic, the analysis achieved new results including the value of the seagull diagram (to be discussed below) in a 3 mass scale sector and the value of the kite diagram (2-loop 2-leg) throughout its singular locus.…”

“…More precisely, both diagrams have a single irreducible numerator. The vacuum seagull was studied within SFI in [8] and the propagator seagull in [14]. Earlier work on these diagrams includes [25][26][27][28][29][30] and references within [8] and [14].…”

“…The vacuum seagull was studied within SFI in [8] and the propagator seagull in [14]. Earlier work on these diagrams includes [25][26][27][28][29][30] and references within [8] and [14]. The results of [8] were incorporated in a public computer code which relates the observable Standard Model parameters to Lagrangian parameters while accounting for some 2, 3 and even 4-loop contributions [31].…”

“…The sources of the second equation have been simplified using additional relations (see appendix A). The second row introduces a novel equation, beyond the standard SFI system, see [14]; it was generated within xSFI and has coefficients which are higher order polynomials in x.…”

“…For the vacuum seagull the homogeneous solution was determined in [8], and for the propagator seagull it was determined up to a free function g(φ) in [14]. Given the new equation for the propagator seagull in (3.11) the homogeneous solution becomes unique, and is given by…”