Solving Higgs Decay Invariant Averaged Amplitude Problem

[dingo_d]

Homework Statement

I have decay of Higgs to fermion and antifermion and I need to find out the invariant, averaged amplitude.

And I wrote down the Feynman diagram, and calculated everything and I came to this part:

\langle|M|^2\rangle=\frac{g_w^2}{4}\frac{m_f^2}{m_w^2}(4p_1\cdot p_2-4m_f^2)

Now to calculate p_1\cdot p_2 I sit in Higgs rest frame, so that the impulses of the fermions are the same (magnitude), but have different sign: \vec{p}_1=-\vec{p}_2.

So what I need to end up with is:

\langle|M|^2\rangle=\frac{g_w^2}{4}\frac{m_f^2}{m_w^2}(2m_h^2-8m_f^2)

But I just cannot get the right result!

p_1\cdot p_2=(E_1+E_2)^2-(\vec{p}_1+\vec{p}_2)^2=(E_1+E_2)^2

If the magnitude of the impulses of the two fermions are the same, and if the masses are the same the energies should also be the same, right? That is, E_1=E_2=E, then the scalar product of two 4-vectors is:

p_1\cdot p_2=4E^2, where E is the energy of the fermion. Since the energy is conserved: E_H=E_1+E_2\Rightarrow E=\frac{E_H}{2}, but since the Higgs is at rest it's mass is equal to it's energy so E=\frac{m_H}{2}, but if I put that back into the scalar product I have:
p_1\cdot p_2=m_H, but that doesn't give me the right answer :\

Where is my mistake?!

 

[dingo_d]

Ok I think I solved it, I was messing up the kinematics. I feel like an idiot -.-"