• scalar chromodynamics
• beta functions
• large n

The main goal of this work is to write a program able to compute explicitly invariant expressions for the 2-loop MS-bar beta-functions in a gauge scalar field theory with a generic quartic potential.
Despite tensor formulae in terms of the gauge generators having been known for decades [MV85], reducing them to an easily evaluable form is not trivial.
The difficulty is in handling the contributions to the scalar potential generated already at 1-loop by the gauge interactions.

After reviewing previous results, in chapter 3 we rederive the 1-loop formulae and describe our program for finding explicit invariants.
Then we specialize to the case of three dimensional theories with only one local and one global symmetry, that is scalar chromodynamics with n flavours.
In condensed matter one such model with an emergent gauge field has recently been proposed to describe spin density wave fluctuations in cuprates [SSST19].

Large n methods, which we briefly review in chapter 4, predict the existence of a stable charged RG fixed point, that is with nonzero gauge coupling [VNK84].
Numerical experiments on a lattice do indeed find that the phase transition becomes continuous when the number of flavours is large enough [BFPV21].
In chapter 5 we find the epsilon-expansion estimate of the minimum number of flavours for which this happens.

Independently of this work, two other implementations have been published recently [Tho21] [SS21], which do not however find explicit invariants except for a few matter representations.
Ours is at least an order of magnitude faster on all models tested, however it only supports massless fermions without Yukawa interactions.
A more complete version of the program is in preparation and its source code will be published separately.