Two-Loop Rational Terms for Spontaneously Broken Theories

Abstract

Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the underlying symmetry open the door to the efficient determination of rational counterterms for the full Standard Model at two loops. The renormalisation-scheme dependence is analysed in detail, and we show that rational counterterms need to be determined only once and for all in a generic renormalisation scheme for the symmetric phase and, a posteriori, they can be easily adapted to a wide range of physical renormalisation schemes for the spontaneously broken phase. As a first application we determine the full set of $\mathcal{O}({\alpha }_{\mathrm{s}}^2)$ rational counterterms of UV origin for the full Standard Model, i.e. for all superficially UV-divergent two-loop vertex functions involving combinations of gluons, quarks, electroweak vector bosons and scalar bosons.

arXiv commentv2: acknowledgements added. v3: Minor typos fixed and clarifying remarks added. Version accepted by JHEP

References in corpus (38)

• New features of FORM
• OpenLoops 2
• The renormalization of the axial anomaly in dimensional regularization
• Automation of one-loop QCD corrections
• Group theory factors for Feynman diagrams
• Complete Corrections of O(αα_s) to the Decay of the Z Boson into Bottom Quarks
• Full one-loop amplitudes from tree amplitudes
• Application of the Background-Field Method to the electroweak Standard Model
• The Multithreaded version of FORM
• On the Rational Terms of the one-loop amplitudes
• Automated one-loop calculations: a proof of concept
• Recola2: REcursive Computation of One-Loop Amplitudes 2
• MATAD: a program package for the computation of MAssive TADpoles
• Electroweak Radiative Corrections for Collider Physics
• Facts of life with gamma(5)
• Feynman Rules for the Rational Part of the QCD 1-loop amplitudes
• Direct Extraction Of One Loop Rational Terms
• Renormalization of vacuum expectation values in spontaneously broken gauge theories
• Techniques for QCD calculations by numerical integration
• Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity
• NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 1. quark-antiquark annihilation
• From loops to trees by-passing Feynman's theorem
• Algebraic evaluation of rational polynomials in one-loop amplitudes
• Numerical NLO QCD calculations
• Feynman rules for the rational part of the Electroweak 1-loop amplitudes
• A four-dimensional approach to quantum field theories
• Two-Loop Renormalization in the Standard Model Part I: Prolegomena
• Removing infrared divergences from two-loop integrals
• Numerical Loop-Tree Duality: contour deformation and subtraction
• The rational parts of one-loop QCD amplitudes I: The general formalism
• Systematic Implementation of Implicit Regularization for Multi-Loop Feynman Diagrams
• Factorization and Regularization by Dimensional Reduction
• Two-Loop Renormalization in the Standard Model Part II: Renormalization Procedures and Computational Techniques
• Two-loop off-shell QCD amplitudes in FDR
• Primary Feynman rules to calculate the epsilon-dimensional integrand of any 1-loop amplitude
• The Four Dimensional Helicity Scheme Beyond One Loop
• Two-Loop Rational Terms in Yang-Mills Theories
• Rational Terms of UV Origin at Two Loops