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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 :

[itex]m_1\vec{u}_1+m_2\vec{u}_2 = m_1\vec{v}_1+m'_2\vec{v}_2[/itex]

and

[itex]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[/itex]

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

[itex]\frac{(m_1\vec{u}_1+m_2\vec{u}_2)^2}{(m_1+m_2)(m_1+m'_2)}[/itex]

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.

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# Dark matter scattering

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