- #1

spaghetti3451

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## Homework Statement

Suppose that a hypothetical fermion ##\psi## interacts with the Higgs field ##h## obey the Lagrangian

$$\bar{\psi}(i\gamma^{\mu}{\partial_{\mu}}-y\nu)\psi-y\bar{\psi}h\psi-y_{\mu}\bar{\psi}_{\mu}h\psi_{\mu} + \frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2} - \frac{\lambda}{6}\nu h^{3} -\frac{\lambda}{24}h^{4}$$

Use the Higgs expectation value ##\nu \approx 246\ \text{GeV}## and the interaction vertices to deduce the force between two fermions, due to the exchange of the Higgs ##h##. How strong is it compared to the electromagnetic force? (Consider both the cases of distances small or large compared to the inverse Higgs mass ##1/m_{H} \approx (125\ \text{GeV})^{-1}##.) Here a parametric answer is sufficient, ignore factors of ##2##.

How would the same comparison go for the top quark (##m_{t} \sim 170\ \text{GeV}##), which carries the same electric charge?

## Homework Equations

## The Attempt at a Solution

The interaction vertex coupling the Higgs to the fermion is ##-igy##. But the Higgs mass is independent of ##\nu##, so the Higgs Feynman propagator does not have a factor of ##\nu##. I was wondering how ##\nu## figures into the scattering amplitude for the scattering of two fermions.