Calculating Divergent Amplitude in Phi-4 Theory

• Daniel_C
In summary, the conversation discusses an integral in scalar field theory and the calculation of a divergent amplitude. The integral contains terms such as $\lambda^2$ and is divergent in the $\ln{\Lambda^2}$ term. Renormalization will be used to tackle this divergence. The speaker requests help with the evaluation of the integral and mentions the availability of QFT textbooks for reference.

Daniel_C

Homework Statement
Calculate the divergent amplitude of this diagram in phi-4 theory.
Relevant Equations
Feynman rules in momentum space for phi-4 theory.
For the diagram

In scalar field theory, I have obtained an integral which looks like

$$\int_{0}^{\Lambda} \frac{d^4 q}{(2\pi)^4} \frac{i}{q^2 - m^2 + i\varepsilon} \frac{i}{(p - q)^2 - m^2 + i\varepsilon}$$

I am required to calculate this and obtain the divergent amplitude

$$i\mathcal{M} = ia\lambda^2[\ln{\Lambda^2} - \ln{(p_1 + p_2)^2} ]$$

The terms like $\lambda^2$ come from outside the integral, they arise due to the vertices in the diagram. I'm only really interested in how to actually go through and do this integral.

The integral is divergent in the $\ln{\Lambda^2}$ term, but we are going to tackle renormalization soon.

I'd appreciate it if someone could provide working out for the integral so that I have an example for future integrals to come.

Do you at least have a textbook or some more specific questions? The evaluation of this integral should be in any QFT textbook (and there are some nice ones available for free online), so I don't think it would be very useful for me to work everything out here on a forum post. But if you have a more specific question about the evaluation I'd be happy to help.

1. What is the Phi-4 theory and how does it relate to calculating divergent amplitude?

The Phi-4 theory is a mathematical model used in theoretical physics to describe the behavior of quantum fields. It is often used in the study of particle physics. Calculating divergent amplitude in Phi-4 theory involves using mathematical techniques to analyze the behavior of the theory and determine the magnitude of divergent amplitudes.

2. Why is calculating divergent amplitude important in Phi-4 theory?

Calculating divergent amplitude is important in Phi-4 theory because it helps physicists understand the behavior of quantum fields and make predictions about the behavior of particles. Divergent amplitudes can also provide insight into the underlying structure of the theory and help guide the development of new theories.

3. What are the methods used to calculate divergent amplitude in Phi-4 theory?

There are several methods used to calculate divergent amplitude in Phi-4 theory, including perturbative expansions, dimensional regularization, and renormalization. These methods involve manipulating mathematical equations and applying various techniques to obtain meaningful results.

4. How does the calculation of divergent amplitude in Phi-4 theory contribute to the understanding of the universe?

The calculation of divergent amplitude in Phi-4 theory is a crucial step in the development and refinement of theories that describe the behavior of particles and fields at the quantum level. This understanding is essential for accurately predicting and explaining phenomena in the universe, such as the behavior of particles in high-energy collisions.

5. What are some current challenges in calculating divergent amplitude in Phi-4 theory?

One of the main challenges in calculating divergent amplitude in Phi-4 theory is the occurrence of infinities in the mathematical equations, which require careful handling through techniques such as renormalization. Additionally, as the theory becomes more complex, the calculations can become increasingly difficult and time-consuming. This highlights the need for continued research and development in this area of theoretical physics.