Estimating force using interaction vertices

[spaghetti3451]

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.

 

[nrqed]

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.

What would you take as the coupling constant between the fermion and the Higgs?