Understanding Scattering Process in QFT Integral

In summary, the conversation discusses an issue with evaluating an integral involving a delta function in the context of studying scattering processes in QFT. The issue arises from applying a formula involving the delta function and its derivatives, which results in a denominator becoming zero. There is also a question about the use of a constant in the formula and the calculation of variables involving energy and momentum.
  • #1
LCSphysicist
646
162
Homework Statement
All the problem is printed below
Relevant Equations
.
I have been studying scattering process in QFT, but i am stuck now because i can't understand how this integral was evaluated:
$$\int dp\space \frac{1}{\sqrt{p^2+c²}}\frac{1}{\sqrt{p^2+k²}}\space p² \space d\Omega \space \delta(E_{cm}-E_{1}-E_{2})$$$

Where Ecm = c + k, E1 is the factor in the denominator involving c and E2 the factor in the denominator involving k.

Now, $$$\delta(\sqrt{p^2+c²} +\sqrt{p^2+k²} - (k+c))$$.

$$\sqrt{p^2+c²} +\sqrt{p^2+k²} - (k+c) = 0$$ has solution for p=0, so shouldn't the integral becomes

$$\int dp\space \frac{1}{\sqrt{p^2+c²}}\frac{1}{\sqrt{p^2+k²}}\space p² \space d\Omega \space \frac{\delta(p-0)}{(p/E_{1}+p/E_{2})_{p=0}}$$

I just applied the fact that $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$

Why is this wrong? And also, how to evaluate the integral so?
 
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  • #2
May we take
[tex]\int d\Omega = 4\pi[/tex]
in your formula ?

[tex]f(p)=\sqrt{p^2+c^2}-c-\sqrt{p^2+k^2}+k[/tex]
How about calculating f'(p) to use your last formula with ##p_i## satisfying
[tex]f(p_i)=\sqrt{p_i^2+c^2}-c-\sqrt{p_i^2+k^2}+k=0[/tex]?
 
Last edited:
  • #3
anuttarasammyak said:
May we take
[tex]\int d\Omega = 4\pi[/tex]
in your formula ?

[tex]f(p)=\sqrt{p^2+c^2}-c-\sqrt{p^2+k^2}+k[/tex]
How about calculating f'(p) to use your last formula with ##p_i## satisfying
[tex]f(p_i)=\sqrt{p_i^2+c^2}-c-\sqrt{p_i^2+k^2}+k=0[/tex]?
"[tex]f(p)=\sqrt{p^2+c^2}-c-\sqrt{p^2+k^2}+k[/tex]
How about calculating f'(p) to use your last formula with ##p_i## satisfying
[tex]f(p_i)=\sqrt{p_i^2+c^2}-c-\sqrt{p_i^2+k^2}+k=0[/tex]?" i did it in the last step. But as you can see, it would makes the denominator zero.
"May we take
[tex]\int d\Omega = 4\pi[/tex]
in your formula ?
"
As far as i know we can do it, so the real main problem is in fact calculating the variables involving the E and the P.
 
  • #4
Let me clear some points. I assume In CM [tex]p_1=-p_2=p[/tex].
[tex]E_{CM}=\sqrt{p^2+k^2}+\sqrt{p^2+c^2}[/tex]
with
[tex]E_1+E_2=\sqrt{p_1^2+k}+\sqrt{p_2^2+c^2}[/tex]
Is it right?
 

Related to Understanding Scattering Process in QFT Integral

1. What is QFT and how does it relate to scattering processes?

QFT, or quantum field theory, is a theoretical framework that combines quantum mechanics and special relativity to describe the behavior of particles in terms of fields. In this framework, scattering processes refer to the interactions between particles, which can be studied using mathematical tools such as Feynman diagrams and QFT integrals.

2. What is the significance of understanding scattering processes in QFT?

Understanding scattering processes in QFT is crucial for studying the behavior of particles at a fundamental level and making predictions about their interactions. This knowledge is essential for a wide range of fields, including particle physics, cosmology, and condensed matter physics.

3. How do QFT integrals help in understanding scattering processes?

QFT integrals are mathematical tools used to calculate the probability of different scattering processes occurring. They involve summing over all possible paths or interactions between particles, taking into account their quantum properties. This allows us to make predictions about the likelihood of different outcomes and test the validity of theories.

4. What are some challenges in understanding scattering processes in QFT?

One of the main challenges in understanding scattering processes in QFT is the complexity of the calculations involved. QFT integrals can be difficult to solve analytically, and often require advanced mathematical techniques and numerical simulations. Additionally, the behavior of particles at high energies or in extreme conditions, such as in the early universe, can be difficult to model accurately.

5. How does the study of scattering processes in QFT contribute to our understanding of the universe?

Studying scattering processes in QFT allows us to gain insights into the fundamental building blocks of the universe and the laws that govern their behavior. It also helps us to test and refine our theories of particle interactions and the structure of space-time. This knowledge is essential for understanding the origins and evolution of the universe, as well as for developing new technologies based on quantum phenomena.

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