Abstract
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}} 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.
Highlights
JHEP12(2020)012 determined in [61] and cross-checked in [62] using Weyl consistency conditions [20, 63, 64]
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 minimal subtraction scheme (MS) scheme
Quartic four-loop β functions and field anomalous dimensions have been obtained in [20] using the same techniques
Summary
Any perturbatively renormalisable QFT in four dimension can be embedded in the template Lagrangian [9,10,11]. All Yukawa couplings, scalar quartic and cubic interactions as well as fermion and scalar masses can be embedded into the respective tensor structures Yiaj , λabcd, habc, mij and m2ab. These quantities are chosen to be symmetric in all their scalar or fermionic indices, e.g. λabcd = λcabd. In combination with the general ansatz (2.1), all momentum integrals and spinor summations can be resolved, and the RGEs (2.4) can be expressed in terms of contracted generalised couplings This provides a template to conveniently obtain RGEs for any renormalisable QFT by using the embedding into (2.1), without the need of loop calculations. This comes at the cost of having to conduct an involved computation for the general theory (2.1) once
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