Action of $\phi^4$ theory in lattice field theory

Question

Euclidian action of $\phi^4$ thory is given by: \begin{equation} \int d^Dx L_E=\int d^Dx\left ( \frac{1}{2}(\partial_\mu \phi)^2 +\frac{m_0 ^2}{2}\phi^2+\frac{g_0}{4!} \phi^4\right), \end{equation} Now I want to write it in lattice field theory but I am not getting it in same form as given in notes, I am expected to get

\begin{equation} S_E[\phi]=\sum_x \left\{\phi(x)^2 +\lambda[\phi(x)^2-1]^2\right\}-\beta\sum_{<xy>}\phi(x)\phi(y) \end{equation}

where, $g_0=4!\frac{\lambda a^{D-4}}{\beta^2}$ and $m_0^2a^2=(1-2\lambda)\frac{2}{\beta} -2D$.

For transformation I am using: $\partial_\mu\phi(x)\rightarrow \frac{\phi(x+\hat\mu a)-\phi(x)}{a}$ and $\int d^D x\rightarrow a^D \sum_x $ , and $\phi^2 \rightarrow \frac{a^{D-2}}{\beta}\phi^2$ , and shifting lagrangian $L_E\rightarrow L_E+\lambda$

Answer 1

In short:

1. use discrete traslational invariance (on a periodic lattice) in the lattice derivative term, i.e. $\sum_x \phi^2(x+\mu) = \sum_x \phi^2(x)$; this way you get the neirest-heighbours sum and a redefinition of the mass term;
2. rescale the fields $\phi \to \sqrt{\beta}\varphi$
3. redefine the mass term (add-subtract a $\varphi^2$ term) and add an irrelevant constant to complete the square, i.e. $\varphi^4 \to (\varphi^2-1)^2$
4. impose (we have a complete unknown, $\beta$) that $\varphi^2$ coeff. is $1$ and $(\varphi^2-1)^2$ coeff = $\lambda$, and you get the quoted relation between bare parameters $(m_0, g) \to (\beta, \lambda)$