Abstract
We compute the two-loop beta -function of scalar and spinorial quantum electrodynamics as well as pure Yang–Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using implicit regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisation constants are fully defined as basic divergent integrals to arbitrary loop order. Moreover, we confirm the hypothesis that momentum routing invariance in the loops of Feynman diagrams implemented via setting well-defined surface terms to zero deliver non-abelian gauge invariant amplitudes within IREG just as it has been proven for abelian theories.
Highlights
For typical collider observables, due to the high multiplicity of jets, the real radiation is subjected to intricate phase-space constraints and calculations are often performed numerically
No analog of -scalar field needed in schemes such as dimensional reduction (DRED) and four-dimensional helicity (FDH) [17,45,46,47,48] to assure unitarity and proper cancellation of divergences is introduced in implicit regularisation (IREG) as a matter of principle as it operates in the physical dimension
To extract any deviation between theory and experimental data in the standard model (SM) as well as test beyond the SM (BSM) theories, precision observables demand at least N 2 L O and N 3 L O approximations
Summary
Ultraviolet (UV) and infrared (IR) divergences are ubiquitous beyond leading order in S-matrix calculations and must be judiciously removed in order to automated computation codes for the evaluation of Feynman amplitudes. Renormalisation group (RG) logs are resummed via RG evolution equations whereas Sudakov logs originate from IR and collinear singularities and may be resummed through exponentiation of IR and collinear poles Such resummations at and beyond next-to-leading-log (N L L) assure the validity of the perturbative series, leading to non-perturbative contributions to high energy cross sections. No analog of -scalar field needed in schemes such as DRED and four-dimensional helicity (FDH) [17,45,46,47,48] to assure unitarity and proper cancellation of divergences is introduced in IREG as a matter of principle as it operates in the physical dimension.
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