Theory of a Continuous H$_{c2}$ Normal-to-Superconducting Transition

I study the $H_{c2}$ transition within the Ginzburg-Landau model, with $m$-component order parameter $\psi_i$. I find a renormalized fixed point free energy, exact in $m\rightarrow\infty$ limit, suggestive of a $2$nd-order transition in contrast to a general belief of a $1$st-order transition. The thermal fluctuations for $H\neq 0$ force one to consider an infinite set of marginally relevant operators for $d<d_{uc}=6$. I find $d_{lc}=4$, predicting that the ODLRO does not survive thermal fluctuations in $d=2,3$. The result is a solution to a critical fixed point that was found to be inaccessible within $\epsilon=6-d$-expansion, previously considered in E.Brezin, D.R.Nelson, A.Thiaville, Phys.Rev.B {\bf 31}, 7124 (1985), and was interpreted as a $1$st-order transition.