2nd Born term calculation (potential scattering)

[CyBeaR]

2nd Born term calculation problem (potential scattering)

Homework Statement

I'm trying to get a 2nd Born term for scattering amplitude in momentum representation (potential scattering).
I take potential of the form:
$$V \left( r \right) = \gamma r^{-1} \exp \left( \frac{-r}{R} \right) \; , R >0$$
and it's form in momentum representation is:
$$\left\langle \mathbf{p'} \vert V \vert \mathbf{p} \right\rangle = \frac{\gamma}{2 \pi^2} \frac{1}{R^{-2}+ \left(\mathbf{p'}-\mathbf{p}\right)^2}$$

Homework Equations

2nd Born term is of the form:
$$\left\langle \mathbf{p'} \vert VGV \vert \mathbf{p} \right\rangle = \int d^3 q \frac{\left\langle \mathbf{p'} \vert V \vert \mathbf{q} \right\rangle \left\langle \mathbf{q} \vert V \vert \mathbf{p} \right\rangle }{k^2 - q^2 + i \epsilon }$$
where $$k$$ is the length of the momenta $$\mathbf{p'}$$ and $$\mathbf{p}$$ and $$0< k \leq 1$$
After applying Feynman method:
$$\frac{1}{ab} = \frac{1}{2} \int_{-1}^1 d\alpha \left( a \frac{1+\alpha}{2}+b \frac{1-\alpha }{2} \right)^{-2}$$
with:
$$a=R^{-2}+ \left(\mathbf{p'}-\mathbf{q}\right)^2 , \; b=R^{-2}+ \left(\mathbf{q}-\mathbf{p}\right)^2$$
to:
$$\int d^3 \mathbf{q}\frac{1}{a b \left( k^2 - q^2 + i \epsilon \right)}$$
i'm getting two integrals to evaluate. One (easly done by using partial fractions):
$$-2 i \pi^2 k \int_0^1 d\alpha (f^2 - k^2 \tau^2 \alpha^2)^{-1}=-\frac{i \pi^2}{\tau f} ln\left|\frac{f+k\tau}{f-k\tau}\right|$$
and 2nd:
$$- 2 \pi^2 R^{-2} \int_0^1 d\alpha (4 R^{-2}+\tau^2 - \tau^2 \alpha^2)^{-1/2} (f^2 - k^2 \tau^2 \alpha^2)^{-1}$$
which is the problematic one.
Variable info:
-> $$\tau = 2 k sin\left( \frac{\theta}{2} \right)$$
-> $$\theta$$ is scattering angle and $$0\leq \theta \leq \pi$$
-> $$f^2 = R^{-4} + 4R^{-2}k^2 + k^2 \tau^2$$

The Attempt at a Solution

I've tried substitution like:
$$\sqrt{4R^{-2}+\tau^2 -\tau^2 \alpha^2}=\alpha \lambda -\sqrt{4R^{-2}+\tau^2}$$
or
$$\lambda = \frac{4R^{-2}+\tau^2-\tau^2 \alpha^2}{\sqrt{4R^{-2}+\tau^2}-\tau \alpha}$$
- where $$\lambda$$ is new integration variable in both cases - but that leads to cumbersome results.

Maybe someone can explain to me how to calculate second integral or share some tricks which can be applied to such integrals.
Any advice will be greatly appreciated.
Cheers!

 

[mighty2000]

Hello there,

Thank you for sharing your problem with us. I understand the importance of finding solutions to complex calculations like this one. After reading through your forum post, I have a few suggestions that may help you in your calculation.

Firstly, have you considered using numerical methods to solve the integral? This may be a faster and more accurate approach compared to analytical methods, especially for complex integrals like the one you are dealing with. There are many numerical integration techniques available, such as the trapezoidal rule or Simpson's rule, that may be useful in this case.

Another suggestion would be to try and simplify the integral before trying to evaluate it. This can be done by using trigonometric identities or by making appropriate substitutions. For example, you could try substituting \tau and f^2 in terms of \alpha to make the integral more manageable.

Lastly, have you checked for any existing methods or techniques that may be applicable to your integral? It is always helpful to see if other scientists have faced a similar problem and have found a solution or approach that may be useful to you.

I hope these suggestions help you in finding a solution to your problem. Keep up the good work and don't hesitate to seek help or collaborate with other scientists to find solutions to complex problems. Good luck!