# Migdal Recursion and Mathematica

I am studying $SU(2)$ lattice field theory, and I am attempting to use migdal recursion for renormalization.

The main equation for Migdal recursion for my case is $$e^{-S_p(U,\lambda a)}=\left[ \sum_r F_r(a)^{\lambda^2}d_r \chi_r (U) \right]^{\lambda^{d-2}}$$ with $d_r$ as the dimension of the representation and $\chi_r$ the trace around a plaquette of a lattice, and $U$ the product of the group matrices assigned to each link in the plaquette. Here $\lambda$ is an expansion factor for the lattice links. We can write $$e^{-S_p(U, a)}=\sum_r F_r(a)d_r \chi_r (U)$$ with $$F_r (a)=\frac{1}{d_r}\int dU\, e^{-S_p(U,a)}\chi^{\ast}_r (U)$$ as the "Fourier" coefficients. The trace can also be written as the sum of the eigenvalues $$\chi_r=\frac{\sin[(2r+1)\theta /2]}{\sin(\theta /2)}$$ Now normalize the coefficients with $e^{-S_p(\mathbb{1},a)}=\sum_r F_r (d_r)^2$ Okay, so far I have

dim[r_]:=2r+1;
chi[r_]:=Sin[dim[r]theta/2]/Sin[theta/2];
F[r_]:=(1/dim[r])NIntegrate[Sin^2[theta/2] act chi[r],{theta,0,4 pi}];


with act defined by

act=Exp[betaf (chi[1/2]-2)+betaa(chi-3)+beta3(chi[3/2]-4)]


Then I define my new normalized coefficients with $$C_r(a)=F_r (a)/e^{-S_p (1,a)}$$ so

norm=Sum[F[r] dim[r]^2,{r,0,9,1/2}];


and

c[r_]:=F[r]/norm;


Now $$e^{-S_p(U,\lambda a)}=\left[ \sum_r \frac{F_{r}^{\lambda^2} d_r \chi_r (U)}{(\sum_r F_r (d_r)^2)^{\lambda^2}} \right]^{\lambda^{d-2}}=\left[ \sum_r \frac{\left( \frac{1}{d_r}\int dU\, e^{-S_p(U,a)}\chi^{\ast}_r (U)\right)^{\lambda^2} d_r \chi_r (U)}{\left(\sum_r \left( \frac{1}{d_r}\int dU\, e^{-S_p(U,a)}\chi^{\ast}_r (U)\right) (d_r)^2\right)^{\lambda^2}} \right]^{\lambda^{d-2}}$$ Is this my recursion relation? I have an initial action to start with, so I can generate subsequent $e^{S}$s. Please ask me to clarify if you think you can help!

Thanks,

• I...don't even know where to begin here, because lattice field theory is so far out of my domain of expertise. My main concern is that no one may answer this question, because SciComp may not have anyone with the right expertise. It seems like a good question. Is this question about the Migdal recursion algorithm itself? I'll leave it here for now. If no one answers it in a week or so, do you have any suggestions about where I should migrate it? Jun 5 '12 at 21:45
• Well, I would like some reassurance that I have made the right substitutions into the generic Migdal formula (eq. 1). I suppose one could even ignore the normalization, just as long as someone with some credit feels I made the appropriate substitutions to set up the recursion correctly. After that I can do the recursion, and my final hope to is find a 4-parameter linear fit to the resulting graph. Also, it could be migrated to Physics if it can't find treatment. Jun 5 '12 at 23:14

Well, I figured I'd close this with what I ended up using. I ended up using the final relation without the normalization $$e^{-S_p(U,\lambda a)}=\left[ \sum_r F_{r}^{\lambda^2} d_r \chi_r (U) \right]^{\lambda^{d-2}}= \left[ \sum_r \left( \frac{1}{d_r}\int dU\, e^{-S_p(U,a)}\chi^{\ast}_r (U)\right)^{\lambda^2} d_r \chi_r (U) \right]^{\lambda^{d-2}}$$ then taking the log of both sides I am able to grab the coefficients from the different order terms in the sum using the orthogonality of the $\chi_r(U)$s. $$\chi_r(WU)\delta_{rs}/d_r =\int dV\chi_r(WV)\chi_{s}^*(UV)$$ That way I iterate as many times as necessary and then integrate again after multiplying through to knock off the coefficients. Now I have been plotting these coefficients against each other and I have a definite trend, and I suspect I can match a fit to the data points, but I need some help with that too, which will be in a follow-up question.