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Average fermion current

  1. Jan 24, 2013 #1
    Hi!

    I have a little problem.

    Consider a 4-fermion interaction (neglecting constant factors) of the form [itex] \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} [/itex] .
    I want to average this interaction over a background consisting of fermions (so it corresponds to the situation where fermions propagate in a background consisting of fermions).

    To this purpose, the left-handed current [itex] \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} [/itex] is approximated by the average value [itex] \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle [/itex]

    There is the following relation for the averaged value produced by this interaction:
    [tex]
    \begin{align}
    \begin{split}
    \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \to & \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} + \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \rangle
    \\
    & \quad + \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \rangle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} + \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} \rangle
    \\
    & \quad - \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \rangle - \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \rangle \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} \rangle.
    \end{split}
    \end{align}
    [/tex]

    But I don't really understand how to obtain this relation. The first two terms look reasonable, but I don't understand the remaining ones.

    Ok, the 3rd and 4th terms might correspond to an "exchange term", where fermion c and d are interchanged, because in the case where c is equal to d, we cannot distinguish between the propagating and the background fermion (on the other hand, if c is not equal to d, this exchange term should be 0). But why is there no additional minus sign, because of Fermi-Dirac statistics?

    Maybe it has something to do with a Fierz transformation, where the minus sign cancels out, i.e.,
    [tex] \overline{\psi}_{a \mathrm{L}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} = \overline{\psi}_{a \mathrm{L}} \gamma^{\lambda} \psi_{d \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}}.[/tex]

    And how could the last two terms be interpreted?

    I hope somebody could help me understanding the relation above.
     
  2. jcsd
  3. Jan 26, 2013 #2
    can you provide a reference for that averaging procedure.How is it defined.Also it is not Fierz reshuffling otherwise you will also get pseudoscalar and pseudovector part(apart from identity)
     
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