Dark Matter Scattering: Solving for v1 with m'2

[Machoire]

Hi,

I recently read the following article http://arxiv.org/pdf/0805.2895v4.pdf which deals with dark matter scattering and I have difficulties in deriving formula (1).

The problem is the following, we have a nucleon (mass m1, velocity u1) scattering from a dark matter particle (mass m2, velocity u2) into an outgoing nucleon (mass m1, velocity v1) and a secondary dark matter particle (masse m'2, velocity v2, with m'2 = m2 - Δm). And we want to know the outgoing velocity of the nucleon v1 as a function of u1 and u2. Both u1 and u2 are assumed to be small compared to c (no relativistic effects).

I started with momentum and energy conservation :

$m_1\vec{u}_1+m_2\vec{u}_2 = m_1\vec{v}_1+m'_2\vec{v}_2$

and

$m_1\vec{u}^2_1+m_2\vec{u}^2_2 = m_1\vec{v}^2_1+m'_2\vec{v}^2_2 - 2\Delta m c^2$

But then I'm stuck because I get a formula like equation (1) of the article but in which the following term is missing

$\frac{(m_1\vec{u}_1+m_2\vec{u}_2)^2}{(m_1+m_2)(m_1+m'_2)}$

and I do not know where it comes from. It looks like it is the product of the center of mass velocity before and after scattering.

Did I forget something in the momentum/energy conservation equations ? More generally, how to deal with this kind of process where total mass is not conserved so that center of mass velocity is discontinuous ?

Thanks in advance for your help.

 

[AndyW]

Hi,

Thank you for bringing up this interesting question. The missing term in your equation is indeed related to the center of mass velocity before and after scattering. To understand this, let's look at the derivation of equation (1) in the article.

The equation is derived using the concept of the center of mass frame. In this frame, the total momentum of the system is zero. Using this, we can rewrite the momentum conservation equation as:

m_1\vec{u}_1+m_2\vec{u}_2 = (m_1+m_2)\vec{V}_CM

where \vec{V}_CM is the center of mass velocity. Now, using the fact that the total energy of the system is conserved, we can rewrite the energy conservation equation as:

\frac{1}{2}m_1\vec{u}^2_1+\frac{1}{2}m_2\vec{u}^2_2 = \frac{1}{2}(m_1+m_2)\vec{V}^2_CM - \Delta m c^2

Solving for \vec{V}_CM, we get:

\vec{V}_CM = \frac{m_1\vec{u}_1+m_2\vec{u}_2}{m_1+m_2} + \frac{\Delta m c^2}{(m_1+m_2)^2}\frac{m_1\vec{u}_1+m_2\vec{u}_2}{m_1+m'_2}

This is the missing term in your equation. It represents the contribution of the center of mass velocity to the outgoing velocity of the nucleon, and it is necessary to account for the non-conservation of total mass in this scattering process.

I hope this helps to clarify the issue. Feel free to ask any further questions. Good luck with your research!