- #1
parton
- 83
- 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.
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.