# Are these assumptions valid?

## Homework Statement

I'm not going to write the full question as its not neccessary, but this is the gist of it.

Electrons are fired at protons and a search is made for a neutral particle created in the impact. Both the electron and proton still exist after the impact, along with this new particle. What is the maximum mass that can be produced. Use any reasonable approximations, but justify your arguments.

Basically it strikes me that the maximum mass of this new particle would be in the case that both the electron and proton are stationary after the collision. But is this a physically possible scenario??

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## Homework Statement

I'm not going to write the full question as its not neccessary, but this is the gist of it.

Electrons are fired at protons and a search is made for a neutral particle created in the impact. Both the electron and proton still exist after the impact, along with this new particle. What is the maximum mass that can be produced. Use any reasonable approximations, but justify your arguments.

Basically it strikes me that the maximum mass of this new particle would be in the case that both the electron and proton are stationary after the collision. But is this a physically possible scenario??
Yes, stationary in the center of mass frame.

Just take the kinetic energy of the initially moving electron and set that energy to equal mc^2 after the collision and you have the mass of the new particle.

In this case is it the case that:

$$(\Sigma E)^2 - (\Sigma p)^2 = (m_e + m_p + m_Q)^2$$

I'm using the non SI units that allow c to equal one.

so $$(\Sigma E)^2$$ = Ee = $${p_e}^2+{m_e}^2$$, $$(\Sigma P) = {p_e}^2$$

Therefore:
$${m_e}^2 = {m_e}^2 + {m_p}^2 + {m_Q}^2 + 2{m_e}{m_p} + 2{m_p}{m_Q} + 2{m_e}{m_Q}$$

From here I seem a little confused. I can't seem to rearrange to solve for mQ (which I just realised I haven't explicitly defined as my name for the new particle). Any help would be appreciated, but I will be banging on with it in the mean time.

Thanks

I think I have some working wrong here - please ignore. I will be going to see lecturers and/or peers tomorrow about this problem.

You just write down conservation of four momentum:

p1 + p2 = p1' + p2' + pq

Then you reason as follows. If the mass of the new particle is maximal, then you are at threshold. In the center of mass frame the total momentum is zero, and then all the 3-momenta on the right hand side are zero. You can then extract the mass of the q-particle using conservation of energy. So, all you need to know is the energy of the electron + proton before the collision in the center of mass frame, but that's just the invariant mass. If E is the total energy in tthe COM frame then (1 = electron, 2 = proton, q = neutral particle):

E^2 = (p1 + p2)^2 = p1^2 + p2^2 + 2 p1 dot p2 =

m1^2 + m2^2 + 2 E_e m2 (if proton was initially at rest) =

m1^2 + m2^2 + 2 (Eekin + m1) m2 =

(m1 + m2 )^2 + 2 Eekin m2

So, the energy is:

E = sqrt[(m1 + m2 )^2 + 2 Eekin m2]

The total energy in the COM frame after the collision is:

m1 + m2 + mq, because all the particles are at rest. Therefore:

mq = sqrt[(m1 + m2 )^2 + 2 Eekin m2] - m1 - m2