A Marginal Dimension of a Weakly Diluted Quenched m-Vector Model

We calculate a marginal order parameter dimension $m_c$ which in a weakly diluted quenched $m$-vector model controls the crossover from a universality class of a ``pure’’ model ($m>m_c$) to a new universality class ($m<m_c$). Exploiting the Harris criterion and the field-theoretical renormalization group approach allows us to obtain $m_c$ as a five-loop $ϵ$-expansion as well as a six-loop pseudo-$ϵ$ expansion. In order to estimate the numerical value of $m_c$ we process the series by precisely adjusted Pad'e–Borel–Leroy resummation procedures. Our final result $m_c=1.912\pm 0.004<2$ stems from the longer and more reliable pseudo-$ϵ$ expansion, suggesting that a weak quenched disorder does not change the values of $xy$-model critical exponents as it follows from the experiments on critical properties of ${\mathrm{H}\mathrm{e}}^4$ in porous media.