- #1

LCSphysicist

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- 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?

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

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?