Evaluation of the general three-loop vacuum Feynman integral

Abstract

We discuss the systematic evaluation of 3-loop vacuum integrals with arbitrary masses. Using integration by parts, the general integral of this type can be reduced algebraically to a few basis integrals. We define a set of modified finite basis integrals that are particularly convenient for expressing renormalized quantities. The basis integrals can be computed numerically by solving coupled first-order differential equations, using as boundary conditions the analytically known special cases that depend on only one mass scale. We provide the results necessary to carry this out, and introduce an implementation in the form of a public software package called 3VIL (3-loop Vacuum Integral Library), which efficiently computes the numerical values of the basis integrals for any specified masses. 3VIL is written in C, and can be linked from C, C++, or FORTRAN code. Contents

Standard model parameters in the tadpole-free pure MS¯ scheme

“…where the lists of 2-loop and 1-loop basis integrals required are: with the 2-loop vacuum integral function I(x, y, z) as defined as in previous papers e.g. [26,125,126], and the coefficients C…”

“…with n-loop order contributions ∆ n that are free of spurious imaginary parts and infrared divergences and do not depend at all on the Goldstone boson squared mass. (The 1/λ in this equation is the source of the tadpole effects noted above if one chooses to expand in terms of v tree rather than v.) The full 3-loop contributions were given in [14] in terms of 2-loop and 3-loop basis integrals that can be efficiently evaluated numerically using the computer code 3VIL [26], ‡ and the 4-loop contribution was obtained at leading order in QCD in [15]. However, a numerical illustration of these effects was deferred.…”

Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation

“…At the three-loop level, this statement was confirmed in refs. [65][66][67]. In the framework of integration-by-parts (IBP) relations [68,69], there is a so-called {dim} relation (see the discussion in ref.…”

Matching relations for decoupling in the standard model at two loops and beyond

“…Decoupling of lighter fermions in the QCD+QED effective theory 15 IV. Numerical results 18 V. Outlook 22 References 22 † The Landau gauge Standard Model effective potential and its minimization condition are presently known to full 2-loop [8, 9] and 3-loop [10, 11] orders, and the 4-loop part only at leading order in QCD [12].These results make use of Goldstone boson resummation [13,14], and employ 3-loop vacuum integral basis functions defined and evaluated by [15]; for an alternative evaluation method see [16]. In particular, refs.…”

“…These results make use of Goldstone boson resummation [13,14], and employ 3-loop vacuum integral basis functions defined and evaluated by [15]; for an alternative evaluation method see [16]. In particular, refs.…”