If a $\overline{B^0}$ meson decays into $D^+$, $e^-$ and $\overline{\nu}_e$ we have a matrix element $$\langle{D^+ \, e^- \, \overline{\nu}_e |\, (\overline{c_L} \gamma^{\mu} b_L) \, (\overline{e_L} \gamma_{\mu} \nu_{eL} )| \overline{B} }\rangle = \langle{D^+ | \, \overline{c_L} \gamma^{\mu} b_L \,| \overline{B} }\rangle \langle{ e^- \, \overline{\nu}_e |\, \overline{e_L} \, \gamma_{\mu} \nu_{eL}|0 }\rangle .$$ I know that the first matrix element is difficult to compute since there are QCD- effects which cannot be treated perturbatively. My questions now are if $|0\rangle$ is the free vacuum state or the interacting vacuum state and how can this matrix element be computed with Feynman rules since we have an electron and antineutrino which transform into vacuum under consideration of the weak current $\overline{e_L} \, \gamma_{\mu} \nu_{eL} $.
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$\begingroup$ The leptonic factor matrix element vacuum is the "empty" free one: the operator creates an antineutrino and an electron. What is your point? $\endgroup$– Cosmas ZachosCommented May 19 at 19:59
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One performs these calculations using interaction picture fields and also rewrites the interacting theory vacuum in terms of the free theory vacuum, $|0\rangle$, which is the one in your equation.
The leptonic matrix element can be evaluated straightforwardly, the result is $\langle e^- \bar{\nu}_e | \bar{e} \gamma_\mu (1-\gamma_5) \nu_e | 0\rangle = \bar{u}_e(p_e) \gamma_\mu (1-\gamma_5) v_{\nu_e}(p_{\nu_e})$. This leptonic matrix element is the same as in other decays, see e.g. neutron decay.
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$\begingroup$ Thank you. Then I can write the electron and neutrinos in terms of annihilation and creation operators. And how can I rewrite the interacting theory vacuum state in terms of the free theory in this explicit example? And do you have a source with neutron decay ? $\endgroup$ Commented May 20 at 10:45