# Applying Wick's theorem in second-order products

Consider the effective current$$\times$$current interaction Hamiltonian of weak interaction at the quark level, $$$$\mathcal{H}=\frac{G_F}{\sqrt{2}}[\overline{e}\gamma^\mu(1-\gamma_5)\nu_e][\overline{u}\gamma_\mu(1-\gamma_5) d]$$$$ where $$e$$-electron, $$\nu_e$$- electron-neutrino, $$u$$- up-quark, $$d$$- down-quark fields. When we consider double beta decay we have to consider second-order S-matrix of the time-ordered product for $$\mathcal{H}$$, \begin{align} \mathcal{S}^{2}-1&=\frac{(-i)^2}{2!}\int d^4x_1 d^4x_2 \mathcal{T}\bigg(\mathcal{H}(x_1)\mathcal{H}(x_2)\bigg)\\ &=\frac{(-i)^2}{2!}\int d^4x_1 d^4x_2 \mathcal{T}\bigg([\overline{e}(x_1)\gamma^\mu(1-\gamma_5)\nu_e(x_1)][\overline{u}(x_1)\gamma_\mu(1-\gamma_5) d(x_1)]\\ &\qquad\qquad\qquad\qquad\qquad \times [\overline{e}(x_2)\gamma^\nu(1-\gamma_5)\nu_e(x_2)][\overline{u}(x_2)\gamma_\nu(1-\gamma_5) d(x_2)]\bigg) \end{align} where $$\mathcal{T}$$ is the time-ordered operator. Can we apply the Wick's theorem for the operator products only for the lepton part, leaving the time-ordered symbol intact for the quarks as follows (considering only the first term of the Wick's expansion), \begin{align} \mathcal{S}^{2}-1&=\frac{(-i)^2}{2!}\int d^4x_1 d^4x_2 \mathcal{N}\bigg([\overline{e}(x_1)\gamma^\mu(1-\gamma_5)\nu_e(x_1)][\overline{e}(x_2)\gamma^\nu(1-\gamma_5)\nu_e(x_2)]\bigg)\\ &\qquad\qquad\qquad\qquad \times \mathcal{T}\bigg([\overline{u}(x_1)\gamma_\mu(1-\gamma_5) d(x_1)][\overline{u}(x_2)\gamma_\nu(1-\gamma_5) d(x_2)]\bigg) \end{align} where $$\mathcal{N}$$ is normal-ordered operator. On a related note, if the above is allowed then what is the justification that lepton fields can be moved across the quark fields to yield factorized lepton$$\times$$lepton and quark$$\times$$quark type operator products?