# Mistake in Peskin & Schroeder, Renormalization of Linear Sigma Model?

In section 11.4 of Peskin & Schroeder's "Introduction to Quantum Field Theory", the authors calculate the effective potential of the linear sigma model to one-loop order:

\begin{align*} V_{\text{eff}}(\phi_0)&=-\frac 1 2 \mu^2\phi_0^2 + \frac \lambda 6 \phi_0^4 \\ &\hspace{1cm}-\underset{\text{divergent term}}{\underbrace{\frac{\Gamma(-d/2)}{2(4\pi)^{d/2}}\left[(N-1)(\lambda\phi_o^2-\mu^2)^{d/2}+(3\lambda\phi_0^2-\mu^2)^{d/2}\right]}}\\ &\hspace{1cm}+\frac 1 2 \delta_{\mu}\phi_0^2 + \frac 1 6 \delta_{\lambda}\phi_0^4 \tag{11.84} \end{align*}

where $$d$$ is the spacetime dimension and $$\delta_{\mu,\lambda}$$ are counterterms, and $$\phi_0^2\equiv \sum_i \phi_{i,0}^2$$ is the minimum energy field configuration (constant in spacetime). This expression is divergent for $$d=0,2,4$$, since the gamma-function has a pole at those values. The authors say the following:

Thus, Eq. (11.84) becomes a finite expression in the limit $$d\rightarrow 2$$ if we set $$\delta_{\mu}=-\lambda (N+2) \frac{\Gamma \left(1-\frac d 2 \right)} {(4\pi)} + \text{finite.}$$

But I do not think this is true. When I evaluate the divergent term in Eq. (11.84) above in the limit $$d\rightarrow 2$$ and keep the divergent terms unevaluated, I get:

\begin{align} \frac{\Gamma(-d/2)}{2(4\pi)}\left[(N-1)(\lambda\phi_o^2-\mu^2)+(3\lambda\phi_0^2-\mu^2)\right]&=\frac{\Gamma(1-d/2)}{(-\frac d 2 )2(4\pi)}\left[(N+2)\lambda\phi_o^2- N\mu^2\right]\\ &=-\frac{\Gamma(1-d/2)}{(4\pi)}[\underset{\color{green}{\text{[A]}}}{\underbrace{(N+2)\lambda\phi_o^2}}- \underset{\color{red}{\text{[B]}}}{\underbrace{N\mu^2}}] \end{align}

[Question] You can see that $$\color{green}{\text{[A]}}$$ is the term the author cites/cancels, but what about term $$\color{red}{\text{[B]}}$$? That still contributes to the overall divergence.

[2$$^{\text{nd}}$$ Question] Note also that if we simply cancel that "additional" divergence with, say, $$\delta_{\mu}$$, then the counterterm will actually depend on $$\phi_0^2$$. Is this an issue?

And by the way, nowhere in that section does the author take the limit/case $$\mu^2\rightarrow 0$$.

Since the term proportional to $$\mu^2$$ is constant in the fields, it is irrelevant in the effective potential because it is a rigid constant shift of the potential itself.