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Can a neutrino act as a virtual particle between two electrons to mediate electron-electron fermonic interaction analogous to how a photon acts as a virtual particle between two electrons to mediate a bosonic electron-electron interaction?

What would the Lagrangian look like for an electron-electron interaction mediated by neutrinos?

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Neutrinos are fermions like electrons. There can be no $ee\nu$ vertex -- this is not possible as it would violate conservation of angular momentum. –  suresh Aug 2 at 1:48
    
@suresh I apologize for my generalization. I should have said two neutrinos instead, which I think would conserve angular momentum. –  linuxfreebird Aug 2 at 13:07
    
Thanks for the clarification. Then, Kyle's answer will do as the loop diagram involving two $e\nu W$ vertices will work. In other words, such a process does NOT need new terms in the SM lagrangian. If you want effective Lagrangians, it will look like a four-fermi interaction after integrating out the W-bosons with the coupling constant proportional to $G_F^2$ ($G_F$ is the Fermi constant) –  suresh Aug 4 at 3:43

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I'm a bit rusty on my qed, but I'll give this a shot. The simplest case would be described by a diagram similar to:

enter image description here

But the $e^--e^--\nu_e$ vertex doesn't exist (also note that I can't draw the required arrow on the neutrino) - the vertices of the standard model (with the exception of vertices involving the Higgs and neutrino oscillations) are:

enter image description here

With these, the closest interaction to what you describe that I can see is:

enter image description here

There are virtual neutrinos as you specified, but also virtual $W$ bosons. If you rotate that diagram 90 degrees, there's an $e^--e^+$ scattering mediated by virtual neutrinos and $W$ bosons, but again, not quite what you asked for.

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If anyone less rusty than me could give that a quick once over and leave a comment if there are any glaring errors, would be appreciated. Would be nice just to know roughly how much I retained from when I tried learning particle physics. –  Kyle Aug 1 at 23:50
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I think it looks good - but why do you say that the neutrinos in your bottom diagram are not virtual? As internal lines, they seem to be more or less by definition virtual. –  ACuriousMind Aug 1 at 23:53
    
@Kyle I think your saying that a W boson is required for the interaction. Thanks for the answer. What would the Lagrangian look like for two electrons? –  linuxfreebird Aug 1 at 23:59
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@linuxfreebird: Lagrangian are not things accompanying specific interactions, they are the general principle from which the interactions arise. The "Lagrangian for 2 electrons" is the same as the Lagrangian for any other particle of the Standard Model, it is (a subset of) the Standard Model Lagrangian. –  ACuriousMind Aug 2 at 0:03
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This is correct. It's called a "box diagram." It'll be a tiny effect, though. In an upcoming $e$-$e$ weak interaction experiment, the expected asymmetry due to interference between electromagnetic repulsion and $Z^0$ exchange is predicted to be about 35 parts per billion (that is, flip a coin a billion times, get 35 more tails than heads). If we handwave that the $e$-$W$ coupling is as strong as the $e$-$Z$ coupling, this interaction would contribute an asymmetry around $(35\times10^{-9})^2 \approx 10^{-15}$. Not experimentally accessible. –  rob Aug 2 at 2:10

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