# SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their head whether the value of SU(2)'s critical point on the phase diagram has any volume dependence? I have tried to find out in the literature but so far I only found two papers (http://arxiv.org/abs/hep-lat/0404015, http://arxiv.org/abs/hep-lat/9509091) on SU(2) critical phenomena and I haven't been able to sort out that fact yet. Any help would be appreciated.

EDIT:

sorry, I mean in the $\beta_{\frac{1}{2}}$ and $\beta_{1}$ ("beta fundamental and beta adjoint") plane, with a Wilson action $\beta_{1/2}(\chi_{1/2}-2)+\ldots$ keeping only the first three terms in the expansion (up to 3/2). $j$ is the representation. I am doing Migdal-Kadanoff recursion and plotting the different $\beta$s against each other (this is the flow of the $\beta$s throughout the recursion). There is a first order phase transition line in the $\beta_{1/2}$vs $\beta_1$ plane that ends at a critical point, I would like to know if that line (and the critical point) moves around depending on the volume.

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This is pretty cryptic. You mean SU(2) pure gauge theory on the lattice (Wilson action?) with $\beta=1/g^2$ (or $2N_c/g^2$?). What is the index j? What do you mean by flow? (as a fct of $N_\tau$?) – Thomas Oct 17 '12 at 15:27
I tried to make it better. Sorry 'bout that. – kηives Oct 17 '12 at 21:31
This might not be referring to gauge theory--- it might be a Wess-Zumino model. What dimension are you in, and what's the action? – Ron Maimon Oct 18 '12 at 4:33
@RonMaimon it's pure gauge, and in 4D. The initial action before recursion is $S_{plaq}=-[\beta_{1/2}(\chi_{1/2}-2)+\beta_{1}(\chi_{1}-3)+\beta_{3/2}(\chi_{3/‌​2}-4)]$ – kηives Oct 18 '12 at 13:52