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

Abstract

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.

General gauge-Yukawa-quartic β-functions at 4-3-2-loop order

“…These endeavors have emphasized the usefulness of parametrizing the unknown β-functions in terms of a basis of tensor structures (TS) constructed from the couplings of the theory in order to attempt to fix the coefficients of the parametrization. By comparing the generic expression to known results in specific theories, the 3-loop β-functions for pure Yukawa theories were almost entirely fixed without the need for any new, arduous loop computations [37,38].…”

On ambiguities and divergences in perturbative renormalization group functions

“…With the community now at the brink of new 3-and 4-loop β-function computations, a more thorough understanding of the ambiguity is required to confidently perform the loop calculations and settle whether the divergences point to some deeper limitation of the regular approach. This discussion also directly points to an ambiguity in recent 3-loop β-function computations in Yukawa theories [14,15] and the 6-loop computation in pure scalar theory [16], both of which treat theories with nontrivial flavor structure.…”

“…Clearly, any such contribution drops from the Green's functions (4.27), and the corresponding parts of N I cannot be determined with that approach. Up to 3-loop order in there is only one such term, in N I given by 15 N…”

“…These corrections contain subdivergences associated with corrections to the coupling of the current with the scalar doublet, hence we also have to compute these corrections up to the 2-loop order 15. It is convenient to parameterize N I with a dummy variableĝI rather than using indices and Kronecker deltas.…”

Four-Loop Gauge and Three-Loop Yukawa Beta-Functions in a General Renormalizable Theory

“…First of all, RG functions in general theories were represented by a linear combination of independent tensor structures (TS) corresponding to contractions of various indices. One can match these "template" expressions to known results in specific models and extract modelindependent coefficients [21,22]. Second, new ideas based on the so-called Weyl Consistency Conditions (WCC) [23][24][25] allow one to find relations between known and unknown TS coefficients, thus, putting constraints on to-be-computed numbers.…”