Meaning of approximation $\cos(\phi)=\frac{\phi^2}{2}\mathrm{e}^{-\frac{1}{2}\langle{\phi^2}\rangle}$ in a field theory? In Appendix E.1 (linking to a pdf) of Giamarchi's book "Quantum physics in one dimension", when deriving renormalization group equations (irrelevant to this question at all), formula (E.18)
is used to bring a potential $\cos{\phi}$ to a quadratic form $\phi^2\mathrm{e}^{-\frac{1}{2}\langle{\phi^2}\rangle}$, which is just (E.19), where $\langle\rangle$ means averaging over the noninteracting fast-mode part in the action.
I cannot understand the explanation prior to (E.18) and the referenced paper (a free pdf, 2nd paragraph below eq. (9)). They imply a vague (maybe only for me) relation to eliminating the infinity background or so. Also, I guess there's a minus sign missing in (E.19)?
A snapshot of that part is as follows

 A: I found it easy in the end. Simply because $\langle \phi^2 \rangle$ is unbounded, one has to expand the normally ordered operator instead of the original one containing infinite background. Here, the implicit assumption is that we do the average over a harmonic Hamiltonian, enabling the use of the Debye-Waller formula in the 2nd equality underneath $$\langle \cos\phi \rangle = \frac{\langle\mathrm{e}^{\mathrm{i}\phi}\rangle + \langle\mathrm{e}^{-\mathrm{i}\phi}\rangle}{2} = \mathrm{e}^{-\frac{1}{2}\langle \phi^2 \rangle} = 1-\frac{1}{2}\langle \phi^2 \rangle + \frac{1}{2!}\frac{1}{4}\langle \phi^2 \rangle^2 - \frac{1}{3!}\frac{1}{8}\langle \phi^2 \rangle^3 + \cdots$$
And we also have the straightforward $$ :\cos\phi: = 1-\frac{1}{2!}:\phi^2: + \frac{1}{4!}:\phi^4: + \cdots$$
On the other hand, using Wick's theorem, we have $$ \cos\phi = 1 - \frac{1}{2!} \phi^2 + \frac{1}{4!} \phi^4 = 1 - \frac{1}{2!} (:\phi^2:+\langle\phi^2\rangle) + \frac{1}{4!} (:\phi^4:+6\langle\phi^2\rangle:\phi^2:+3\langle\phi^2\rangle^2) + \cdots$$
And it is easy to verify that $\cos\phi=:\cos\phi: \langle \cos\phi \rangle$. One then expand $:\cos\phi:$.
However, I am still wondering where the missing minus sign has gone....???
