Migdal Recursion and Mathematica

Question

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[1]-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,

Answer 1

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.

Thanks anyways,