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Fermion-scalar interaction

  1. Apr 12, 2010 #1
    What is the most general reasonable form of the Lorentz invariant interaction term between a fermion field [tex]\psi[/tex] and a scalar field [tex]\phi[/tex]?

    A common choice for the interaction is something like [tex]\psi^{\dagger}A\psi\phi[/tex], with [tex]A[/tex] being a Lorentz invariant matrix (like [tex]\gamma^{5}[/tex]). However, I don't see why an interaction couldn't include terms with arbitrary number of derivatives of fields, as long as it's Lorentz invariant (e.g. [tex]\partial_{\mu}\psi^{\dagger}A\partial^{\mu}\psi\phi[/tex] or [tex]\psi^{\dagger}A\partial_{\mu}\psi\partial^{\mu}\phi[/tex]).

    Is there any physical reason for discarding such interaction terms with derivative coupling, or maybe we simply don't need them to describe real world interactions?
  2. jcsd
  3. Apr 12, 2010 #2
    Yes, there is a reason. Their coefficients have dimensions of [tex](mass)^{-n}[/tex] (with [tex]n \geq 1[/tex]). We usually work with effective theories, which are presumed to reduce to some more accurate theory at some very high energy scale. When a coupling constant has units of some power of mass, that mass will typically be around the energy scale at which the more accurate theory becomes important. So for experiments at energy scales much smaller than that scale, these couplings can be considered to be very small. Also, when a coupling has dimensions of inverse mass, it usually causes the theory to be nonrenormalizable. This is not a problem if the coupling is very small, because by the time you move the cutoff to a scale where it becomes a problem, the more accurate theory takes over. But if the coupling constant is large (compared to the scale you're working at), it usually is a problem.
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