# Special Rel Colliding Particles Problem

1. Dec 29, 2007

### cpfoxhunt

1. The problem statement, all variables and given/known data
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).

2. Relevant equations
Invarience of the interval? Lorentx transforms for energy and momentum?

3. 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

2. Dec 29, 2007

### 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.

3. Dec 30, 2007

### 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)

4. Dec 30, 2007

### 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.