Feynman one-loop integral ##I_{21}##

In summary, the conversation discusses the general formula for I, and the final result of the conversation is that $$I_{2,1}=\frac{1}{\epsilon}\bigg(2\Delta I_{20}+\frac{\Delta}{(4\pi)^2}\bigg)+O(\epsilon^0)$$
  • #1
RicardoMP
49
2
Homework Statement
I need to determine Feynman one-loop integrals to work out some Feynman diagrams, in particular ##I_{2,1}##.
Relevant Equations
$$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$
Starting from the general formula:
$$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$
I arrived to the following:
$$I_{2,1}=\frac{\Delta}{(4\pi)^2}\frac{(2-\frac{\epsilon}{2})}{(\epsilon-1)}[\frac{2}{\epsilon}-\gamma+ln(\frac{4\pi M^2}{\Delta})-\gamma\frac{\epsilon}{2}ln(\frac{4\pi M^2}{\Delta})+O(\epsilon)]$$
The term ##\frac{1}{\epsilon-1}## is giving me some trouble so I expanded it and, after removing terms proportional to ##\epsilon##, finally got:

$$I_{2,1}=-2\Delta I_{20}-\frac{\Delta}{(4\pi)^2}$$

Can someone confirm if this is the correct result?
 
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  • #2
Edit: I think I got the right result. After some more work I got to :$$I_{2,1}=\frac{1}{\epsilon}\bigg(2\Delta I_{20}+\frac{\Delta}{(4\pi)^2}\bigg)+O(\epsilon^0)$$
 

Related to Feynman one-loop integral ##I_{21}##

What is Feynman one-loop integral ##I_{21}##?

Feynman one-loop integral ##I_{21}## is a mathematical expression used in quantum field theory to calculate corrections to particle interactions at the one-loop level. It involves integrating over a loop of virtual particles and is an essential part of many calculations in theoretical physics.

How is Feynman one-loop integral ##I_{21}## calculated?

Feynman one-loop integral ##I_{21}## is calculated using Feynman diagrams, which represent the possible ways in which a particle interaction can occur. The integral is then evaluated using techniques from calculus and complex analysis.

What is the significance of Feynman one-loop integral ##I_{21}## in physics?

Feynman one-loop integral ##I_{21}## plays a crucial role in understanding the behavior of particles at the quantum level. It allows physicists to make precise predictions about the outcomes of particle interactions and has been used to successfully explain many experimental results.

What are the challenges associated with calculating Feynman one-loop integral ##I_{21}##?

Calculating Feynman one-loop integral ##I_{21}## can be a complex and time-consuming process, as it involves performing multiple integrations and dealing with infinities that arise in quantum field theory. As a result, sophisticated mathematical techniques and computer programs are often used to assist with the calculations.

How does Feynman one-loop integral ##I_{21}## relate to other concepts in physics?

Feynman one-loop integral ##I_{21}## is closely related to other concepts in theoretical physics, such as renormalization and the calculation of scattering amplitudes. It is also connected to the broader field of quantum field theory, which seeks to understand the fundamental interactions between particles at the quantum level.

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