Scheme invariants in ϕ 4 theory in four dimensions

For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the $$ \overline{\mathrm{MS}} $$ MS ¯ scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and from those derive β functions for scalar masses and cubic interactions. As an example, the results are applied to compute all renormalisation group equations in U(n) × U(n) scalar theories.

“…In summary, we obtain the general result for the purely scalar part of any three-and four-loop scalar field anomalous dimension, in accordance with [61,62]…”

Section: Discussionsupporting

confidence: 85%

Section: Jhep12(2020)012 3 Three-loop Scalar Contributionsmentioning

confidence: 99%

Section: Jhep12(2020)012 3 Three-loop Scalar Contributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Results for β-functions and anomalous dimensions in general scalar fermion theories are presented to three loops. Various constraints on the individual coefficients for each diagram following from supersymmetry are analysed. The results are used to discuss potential fixed points in the ε-expansion for scalar fermion theories, with arbitrary numbers of scalar fields, and where there are just two scalar couplings and one Yukawa coupling. For different examples the fixed points follow a similar pattern as the numbers of fermions is varied. For diagrams with subdivergences there are extensive consistency constraints arising from the existence of a perturbative a-function and these are analysed in detail. Further arbitrary scheme variations which preserve the form of β functions and anomalous dimensions in terms of 1PI diagrams are also discussed. The existence of linear and quadratic scheme invariants is demonstrated and the consistency condition are shown to be expressible in terms of these invariants.

Towards general scalar-Yukawa renormalisation group equations at three-loop order

Steudtner

2021

For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the $$ \overline{\mathrm{MS}} $$ MS ¯ scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.