Consider the effective current$\times$current interaction Hamiltonian of weak interaction at the quark level, \begin{equation} \mathcal{H}=\frac{G_F}{\sqrt{2}}[\overline{e}\gamma^\mu(1-\gamma_5)\nu_e][\overline{u}\gamma_\mu(1-\gamma_5) d] \end{equation} 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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.