Due diligence first. The first term of the first equation you wrote is missing a sign, as you should have easily checked yourself, from the coefficient of $\ln ~ \pi$. That is,  Write down $W(q)$ right above the equation you start with, around your minimum  *q* =2, i.e. $q\equiv 2(1+\epsilon)$, so that $p= 2(1-\epsilon) +O(\epsilon^2)$, dismissing any higher orders of $\epsilon$. 

The relevant term in the derivative w.r.t. $2\epsilon$ then is $\pi^{-n2\epsilon/4} N$, so you catch the paper's typo.