Special Rel Colliding Particles Problem

[cpfoxhunt]

Homework Statement

This should be quite a simple problem, I'm tying myself in knots with it though regardless. Anyway,

An electron of energy 9.0 GeV and a positron of energy E collide head on to produce a B meson and an anti B meson (B nought mesons), each with a mass of 5.3 GeV/c^2 . What is the minimum positron energy required to produce the B Meson pair? (You may neglect the rest mass energies of the electron and the positron).

Homework Equations

Invarience of the interval? Lorentx transforms for energy and momentum?

The Attempt at a Solution

Obviously not a linear subtraction (I wish). In the CM (ZM/COM) frame, it seems to me that the electron and the positron have equal energies, E, where E= 5.3GeV

Their momenta are equal and opposite, and the value for the invarient of the whole system is 4*(5.3 GeV)^2

gamme = g
If I then use E' = g(E - vp) and take p to be zero as the unprimed frame is the cm frame, I can work out the velocity - but then I get stuck, and I'm a bit dubious about this wole last step. (The idea would then be to transform the total energy by the same amount and subtract the 9 from it)

Any help would be greatly appreciated, as would any quicker (non 4 vector based please because this is first year undergrad stuff), methods.

Thanks
Cpfoxhunt

 

[Dick]

The minimum energy positron produces a B anti B pair at rest in their relative center of mass system. You can consider this as a single particle of mass 2*M_B. That means using 4-vectors, (E_e+E_p)^2-(p_e+p_p)^2=(2*M_B)^2. You also know E_e^2-p_e^2=E_p^2-p_p^2=M_e^2. I'm setting c=1 and don't be afraid of 4-vectors. They are your friend. That's basically two equations in two unknowns.

 

[cpfoxhunt]

That's 4 vectors? I'd call that the invarient quantity first for the system and then for the individual particules. I'm a abit confused though - is that enough information to eliminate all the unknowns?

And just for completeness, are there any other simple ways of doing the problem along the lines of the method I was originally trying to do?

(Thinking about it I can see that you have three equations, three unknowns and some nice cancelling in your method, thanks a lot)

 

[Dick]

I'd call it 4 vectors. Notice when I wrote p_e+p_p I meant in a vector sense. p_e and p_p are pointing in opposite directions. I don't think it's simpler to start fiddling with explicit lorentz transforms.