Integral Analysis: Equality of Lorentz Structures

[ppg]

Hi, I am a newcomer, and i have a question about an integral shown as follows,

$$\int d^{3}p_{2} \frac{1}{2E_{2}} d^{3}p_{3} \frac{1}{2E_{3}} \delta^{4}(p-p_{2}-p_{3}) p_{3}^{\alpha} p_{3}^{\beta} p_{2}^{\mu}$$

Is this integral equal to (just considering Lorentz structures)

$$A(p^{2})g^{\alpha\beta}p^{\mu}+B(p^{2})g^{\beta\mu}p^{\alpha}+B(p^{2})g^{\mu\alpha}p^{\beta}$$

where the coefficients of the last two terms should be the same because of the symmetry of the integral variables $$p_{3}^{\alpha} p_{3}^{\beta}$$

Thank u for ur help!

 

[CompuChip]

It's been a while since I did such HEP integrals, so I can't give you an answer off the top of my head. But I want to give you a tip, since you have posted the formulas in valid LaTeX code anyway: if you wrap them in tex-tags (put [ tex] at the beginning and [/ tex] at the end, and take out the spaces in both they will be nicely rendered by LaTeX. For inline (i.e. $ ... $) you can replace tex by itex in both tags. That will make your post a lot more readable, probably giving you faster responses as well.

Quote my message to see the codes used:
$$\int d^{3}p_{2} \frac{1}{2E_{2}} d^{3}p_{3} \frac{1}{2E_{3}} \delta^{4}(p-p_{2}-p_{3}) p_{3}^{\alpha} p_{3}^{\beta} p_{2}^{\mu}$$
[...] integral variables $p_{3}^{\alpha} p_{3}^{\beta}$.

 

[sir-pinski]

Integral analysis is a powerful tool for solving complex mathematical problems, and it can also be applied to physics problems, especially in the study of Lorentz structures. The integral shown in your question involves a delta function and Lorentz structures, and it is important to understand the equality of these structures in order to properly solve the integral.

To answer your question, yes, the integral is indeed equal to the expression you provided. This can be seen by considering the Lorentz structures in the integral. The first term, A(p^{2})g^{\alpha\beta}p^{\mu}, represents the Lorentz structure with two indices raised and one index lowered, while the second and third terms, B(p^{2})g^{\beta\mu}p^{\alpha} and B(p^{2})g^{\mu\alpha}p^{\beta}, represent the Lorentz structures with one index raised and one index lowered.

The symmetry of the integral variables, p_{3}^{\alpha} p_{3}^{\beta}, ensures that the coefficients of the last two terms are the same, as you correctly pointed out. This is because the integral is invariant under Lorentz transformations, and the Lorentz structures must also be invariant. Therefore, the coefficients of the last two terms must be the same in order for the integral to be equal to the expression you provided.

In conclusion, the equality of Lorentz structures is an important concept in integral analysis, and it is crucial to understand it in order to solve problems involving Lorentz structures. I hope this explanation has helped clarify your understanding. Best of luck with your future studies!