Six-loop $\varepsilon$ expansion of three-dimensional $\text{U}(n)\times \text{U}(m)$ models

We analyze the Landau-Wilson field theory with $\text{U}(n)\times\text{U}(m)$ symmetry which describes the finite-temperature phase transition in QCD in the limit of vanishing quark masses with $n=m=N_f$ flavors and unbroken anomaly at the critical temperature. The six-loop expansions of the renormalization group functions are calculated within the Minimal Subtraction scheme in $4 - \varepsilon$ dimensions. The $\varepsilon$ series for the upper marginal dimensionality $n^{+}(m,4-\varepsilon)$ -- the key quantity of the theory -- are obtained and resummed by means of different approaches. The numbers found are compared with their counterparts obtained earlier within lower perturbative orders and the pseudo-$\varepsilon$ analysis of massive six-loop three-dimensional expansions. In particular, using an increase in the accuracy of numerical results for $n^{+}(m,3)$ by one order of magnitude, we strengthen the conclusions obtained within previous order in perturbation theory about fairness of the inequality $n^{+}(m,3)>m$. This, in turn, indicates the absence of a stable three-dimensional fixed point for $n=m$, and as a consequence a first-order kind of finite-temperature phase transition in light QCD.