 Problem Statement
 In the Peskin & Schroeder textbook, the $\beta$function for the GrossNeveu model is discussed in problem 12.2. After computing it, I have tried checking my results with some solutions found online. My problem is that they all disagree among each other (something quite recurrent for this book actually).
 Relevant Equations

$$\mathcal{L} = \bar{\Psi}_{i}(i\not{\partial} )\Psi_{i}g\sigma \bar{\Psi}_i\Psi_i\frac{1}{2}\sigma^2$$
$$\delta_{Z_{\Psi}}=(ig)^2\int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i\not{k}}{k^2}\frac{i}{(k+p)^2}]=0 $$
$$\delta_{Z_{\sigma}}=N(ig)^2\int \frac{d^{d}k}{(2\pi)^d}\frac{i\not{k}}{k^2} \frac{i(\not{k}+\not{p})}{(k+p)^2} =Ng^2 \int \frac{d^{d}k}{(2\pi)^d}\frac{1}{(k+p)^2} =Ng^2\frac{i}{4\pi\epsilon}+finite$$
$$\delta_{g}=(ig)^3 \int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i(\not{k}+\not{p_1})}{(k+p_1)^2} \frac{i(\not{k}+\not{p_2})}{(k+p_2)^2}] = ig^3 \int \frac{d^{d}k}{(2\pi)^d}\frac{2k^2}{(k+p_1)^2 (k+p_2)^2}+...=2ig^3\frac{i}{4\pi\epsilon} $$
In the Peskin & Schroeder textbook, the ##\beta## function for the GrossNeveu model is discussed in problem 12.2. After computing it, I have tried checking my results with some solutions found online. My problem is that they all disagree among each other (something quite recurrent for this book actually). I have basically followed the original paper from Gross and Neveu:
 We start from the Lagrangian $$\mathcal{L} = \bar{\Psi}_{i}(i\not{\partial} )\Psi_{i}g\sigma \bar{\Psi}_i\Psi_i\frac{1}{2}\sigma^2.$$
 We compute the field strength renormalisation
For ##\Psi: ## $$\delta_{Z_{\Psi}}=(ig)^2\int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i\not{k}}{k^2}\frac{i}{(k+p)^2}]=0 $$
For ##\sigma:## $$\delta_{Z_{\sigma}}=N(ig)^2\int \frac{d^{d}k}{(2\pi)^d}\frac{i\not{k}}{k^2} \frac{i(\not{k}+\not{p})}{(k+p)^2} =Ng^2 \int \frac{d^{d}k}{(2\pi)^d}\frac{1}{(k+p)^2} =Ng^2\frac{i}{4\pi\epsilon}+finite$$
 We compute the coupling renormalisation (though in the GrossNeveu original paper, the authors argue the vertex need not be renormalised. That is a point I don't get)
$$\delta_{g}=(ig)^3 \int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i(\not{k}+\not{p_1})}{(k+p_1)^2} \frac{i(\not{k}+\not{p_2})}{(k+p_2)^2}] = ig^3 \int \frac{d^{d}k}{(2\pi)^d}\frac{2k^2}{(k+p_1)^2 (k+p_2)^2}+...=2ig^3\frac{i}{4\pi\epsilon} $$
 We use relation (12.53)
$$\beta (g) = M\frac{\partial}{\partial M}\left(\delta_g +\frac{1}{2}g\sum_i \delta_{Z_i} \right)=\frac{2g^3}{4\pi}+g\frac{Ng^2}{4\pi}=\frac{g^3}{4\pi}(N+2)$$
It is asymptotically free, but I do fully trust my result. Any comment welcome!
 We start from the Lagrangian $$\mathcal{L} = \bar{\Psi}_{i}(i\not{\partial} )\Psi_{i}g\sigma \bar{\Psi}_i\Psi_i\frac{1}{2}\sigma^2.$$
 We compute the field strength renormalisation
For ##\Psi: ## $$\delta_{Z_{\Psi}}=(ig)^2\int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i\not{k}}{k^2}\frac{i}{(k+p)^2}]=0 $$
For ##\sigma:## $$\delta_{Z_{\sigma}}=N(ig)^2\int \frac{d^{d}k}{(2\pi)^d}\frac{i\not{k}}{k^2} \frac{i(\not{k}+\not{p})}{(k+p)^2} =Ng^2 \int \frac{d^{d}k}{(2\pi)^d}\frac{1}{(k+p)^2} =Ng^2\frac{i}{4\pi\epsilon}+finite$$
 We compute the coupling renormalisation (though in the GrossNeveu original paper, the authors argue the vertex need not be renormalised. That is a point I don't get)
$$\delta_{g}=(ig)^3 \int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i(\not{k}+\not{p_1})}{(k+p_1)^2} \frac{i(\not{k}+\not{p_2})}{(k+p_2)^2}] = ig^3 \int \frac{d^{d}k}{(2\pi)^d}\frac{2k^2}{(k+p_1)^2 (k+p_2)^2}+...=2ig^3\frac{i}{4\pi\epsilon} $$
 We use relation (12.53)
$$\beta (g) = M\frac{\partial}{\partial M}\left(\delta_g +\frac{1}{2}g\sum_i \delta_{Z_i} \right)=\frac{2g^3}{4\pi}+g\frac{Ng^2}{4\pi}=\frac{g^3}{4\pi}(N+2)$$
It is asymptotically free, but I do fully trust my result. Any comment welcome!