One-loop RG Flow for Adjoint Multiscalar Gauge Theory

We study the one-loop renormalisation of 4d SU(N) Yang-Mills theory with $M$ adjoint representation scalar multiplets. We calculate the coupled one-loop renormalization group flows for this theory by developing an algebraic description, which we find to be characterised by a non-associative algebra of marginal couplings. The 4d one-loop beta function of the gauge coupling $g^2$ vanishes for the case $M=22$, which is intriguing for string theory. There are real fixed flows (fixed points of $\lambda \mathrm{/}{g}^2$) only for $M\ge 406$, rendering one-loop fixed points of the gauge coupling and scalar couplings incompatible.