Deriving The Vertex Feynman Rule For Interacting Scalar Fields

Introduction

In the realm of Quantum Field Theory (QFT), Feynman diagrams serve as a powerful tool for calculating scattering amplitudes and decay rates. The vertex Feynman rule, in particular, plays a crucial role in determining the interactions between particles. In this article, we will derive the vertex Feynman rule for interacting scalar fields, which will enable us to analyze the decay of a scalar field into two other scalar fields.

The Interaction Lagrangian

The interaction Lagrangian between two real scalar fields, $\phi$ and $\chi$, is given by:

$$\mathcal{L}_\text{int}=-g\chi\phi^2,$$

where $g$ is the coupling constant. This Lagrangian describes the interaction between the two scalar fields, with the coupling constant $g$ determining the strength of the interaction.

Deriving the Vertex Feynman Rule

To derive the vertex Feynman rule, we need to consider the process of a scalar field $\phi$ decaying into two other scalar fields, $\chi$ and $\phi$. The Feynman diagram for this process is shown below:

  +---------------+
  |  φ  |  χ  |  φ  |
  +---------------+
  |  (p)  |  (k)  |  (q)  |
  +---------------+

In this diagram, the scalar field $\phi$ is represented by a solid line, while the scalar field $\chi$ is represented by a dashed line. The momenta of the particles are denoted by $p$, $k$, and $q$.

Calculating the Vertex Feynman Rule

To calculate the vertex Feynman rule, we need to evaluate the following expression:

$$\mathcal{M}=\int\frac{d^4k}{(2\pi)^4}\frac{i}{k^2-m^2}\frac{-ig}{\sqrt{2}}\bar{\chi}(k)\phi(p)\phi(q).$$

This expression represents the amplitude for the process of a scalar field $\phi$ decaying into two other scalar fields, $\chi$ and $\phi$. The integral over the momentum $k$ is performed using the Feynman parameterization.

Feynman Parameterization

The Feynman parameterization is a technique used to evaluate the integral over the momentum $k$. It involves introducing a new variable, $x$, such that:

$$k^2=(1-x)p^2+xq^2.$$

This allows us to rewrite the integral as:

$$\mathcal{M}=\int_0^1dx\int\frac{d^4p}{(2\pi)^4}\frac{i}{(1-x)p^2+xq^2-m^2}\frac{-ig}{\sqrt{2}}\bar{\chi}(k)\phi(p)\phi(q).$$

Evaluating the Integral

To evaluate the integral, we need to perform the following steps:

1. Evaluate the integral over the momentum $p$.
2. Perform the integration over the Feynman parameter $x$.

Evaluating the Integral over the Momentum p

The integral over the momentum $p$ can be evaluated using the following expression:

$$\int\frac{d^4p}{(2\pi)^4}\frac{i}{(1-x)p^2+xq^2-m^2}\phi(p)\phi(q).$$

This integral can be evaluated using the Feynman parameterization, which involves introducing a new variable, $y$, such that:

$$p^2=(1-y)q^2+yk^2.$$

This allows us to rewrite the integral as:

$$\int\frac{d^4p}{(2\pi)^4}\frac{i}{(1-x)p^2+xq^2-m^2}\phi(p)\phi(q)=\int_0^1dy\int\frac{d^4k}{(2\pi)^4}\frac{i}{(1-y)q^2+yk^2-m^2}\phi(p)\phi(q).$$

Performing the Integration over the Feynman Parameter x

The integration over the Feynman parameter $x$ can be performed using the following expression:

$$\int_0^1dx\frac{i}{(1-x)p^2+xq^2-m^2}.$$

This integral can be evaluated using the following result:

$$\int_0^1dx\frac{i}{(1-x)p^2+xq^2-m^2}=\frac{i}{p^2-m^2}\ln\left(\frac{p^2-m^2}{q^2-m^2}\right).$$

Evaluating the Final Expression

The final expression for the vertex Feynman rule can be evaluated using the following result:

$$\mathcal{M}=\int\frac{d^4k}{(2\pi)^4}\frac{i}{k^2-m^2}\frac{-ig}{\sqrt{2}}\bar{\chi}(k)\phi(p)\phi(q)=\frac{-ig}{\sqrt{2}}\frac{i}{p^2-m^2}\ln\left(\frac{p^2-m^2}{q^2-m^2}\right)\bar{\chi}(k)\phi(p)\phi(q).$$

Conclusion

In this article, we have derived the vertex Feynman rule for interacting scalar fields. The vertex Feynman rule is a crucial component of Feynman diagrams, which are used to calculate scattering amplitudes and decay rates in Quantum Field Theory. The derivation of the vertex Feynman rule involves evaluating the integral over the momentum $k$ and performing the integration over the Feynman parameter $x$. The final expression for the vertex Feynman rule can be used to analyze the decay of a scalar field into two other scalar fields.

References

• Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Addison-Wesley.
• Itzykson, C., & Zuber, J. B. (1980). Quantum field theory. McGraw-Hill.
• Weinberg, S. (1995). The quantum theory of fields. Cambridge University Press.
  Q&A: Deriving the Vertex Feynman Rule for Interacting Scalar Fields ====================================================================

Introduction

In our previous article, we derived the vertex Feynman rule for interacting scalar fields. In this article, we will address some common questions and concerns that readers may have regarding the derivation of the vertex Feynman rule.

Q: What is the significance of the vertex Feynman rule?

A: The vertex Feynman rule is a crucial component of Feynman diagrams, which are used to calculate scattering amplitudes and decay rates in Quantum Field Theory. The vertex Feynman rule describes the interaction between particles, and it is used to determine the probability of a particular process occurring.

Q: What is the difference between the vertex Feynman rule and the propagator?

A: The vertex Feynman rule and the propagator are two distinct components of Feynman diagrams. The propagator describes the probability of a particle propagating from one point to another, while the vertex Feynman rule describes the interaction between particles.

Q: How is the vertex Feynman rule used in Feynman diagrams?

A: The vertex Feynman rule is used to calculate the amplitude of a particular process in a Feynman diagram. The amplitude is a complex number that represents the probability of the process occurring. The vertex Feynman rule is used to determine the amplitude by integrating over the momentum of the particles involved in the process.

Q: What is the Feynman parameterization, and how is it used in the derivation of the vertex Feynman rule?

A: The Feynman parameterization is a technique used to evaluate the integral over the momentum of the particles involved in a process. It involves introducing a new variable, x, such that the momentum of the particles can be expressed in terms of x. The Feynman parameterization is used in the derivation of the vertex Feynman rule to simplify the integral over the momentum.

Q: What is the significance of the coupling constant in the vertex Feynman rule?

A: The coupling constant is a parameter that determines the strength of the interaction between particles. In the vertex Feynman rule, the coupling constant is used to determine the amplitude of the process. A larger coupling constant indicates a stronger interaction between particles.

Q: How is the vertex Feynman rule used in particle physics?

A: The vertex Feynman rule is used in particle physics to calculate the probability of various processes, such as particle decays and scattering. The vertex Feynman rule is used to determine the amplitude of these processes, which is a complex number that represents the probability of the process occurring.

Q: What are some common applications of the vertex Feynman rule in particle physics?

A: Some common applications of the vertex Feynman rule in particle physics include:

• Calculating the probability of particle decays, such as the decay of a Higgs boson into two photons.
• Calculating the probability of scattering processes, such as the scattering of electrons off protons.
• Determining the properties of particles, such as their mass and spin.

Conclusion

In this article, we have addressed some common questions and concerns regarding the derivation of the vertex Feynman rule for interacting scalar fields. The vertex Feynman rule is a crucial component of Feynman diagrams, and it is used to calculate the probability of various processes in particle physics.

References

• Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Addison-Wesley.
• Itzykson, C., & Zuber, J. B. (1980). Quantum field theory. McGraw-Hill.
• Weinberg, S. (1995). The quantum theory of fields. Cambridge University Press.