Estimating force using interaction vertices

• spaghetti3451
In summary, the fermion-fermion interaction due to the exchange of the Higgs field can be determined using the Lagrangian given, using the Higgs expectation value of ##\nu \approx 246\ \text{GeV}## and the interaction vertices. The strength of this interaction compared to the electromagnetic force can be calculated for both small and large distances compared to the inverse Higgs mass ##1/m_{H} \approx (125\ \text{GeV})^{-1}##, with a parametric answer being sufficient. The same comparison can be made for the top quark, which carries the same electric charge. However, the Higgs mass is independent of ##\nu## and does not factor into the
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?

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.

failexam said:

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?

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?

1. What is the concept of estimating force using interaction vertices?

The concept of estimating force using interaction vertices is based on the fundamental understanding that forces arise from the interactions between particles. In physics, an interaction vertex is a point where particles interact and exchange energy, resulting in a force between them. By analyzing these interaction vertices, scientists can estimate the strength and direction of the force between particles.

2. How do scientists determine the strength of a force using interaction vertices?

Scientists use mathematical models and calculations to analyze the characteristics of interaction vertices, such as the type of particles involved and the energy exchanged. By applying known laws and principles of physics, they can determine the strength of the force between the particles at the interaction vertex.

3. Can estimating force using interaction vertices be applied to all types of forces?

Yes, estimating force using interaction vertices is a universal concept that can be applied to all types of forces, including gravitational, electromagnetic, and nuclear forces. In fact, this approach has been crucial in understanding the behavior of particles in the subatomic world and in developing theories such as quantum mechanics.

4. How accurate is estimating force using interaction vertices?

Estimating force using interaction vertices is a highly accurate method, as it is based on well-established laws and principles of physics. However, the accuracy of the estimated force depends on the precision of the measurements and the complexity of the interactions between particles.

5. What are the practical applications of estimating force using interaction vertices?

Estimating force using interaction vertices has various practical applications in different fields of science. For example, it is essential in understanding the behavior of subatomic particles in particle accelerators and in developing new technologies such as nuclear power plants. It is also used in fields such as astrophysics, where it helps in studying the interactions between celestial bodies and their forces.

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