Electron and positron collision producing a b0 meson pair

[physconomics]

Homework Statement

Electrons and positrons collide head-on with beam energies 9GeV and E respectively, producing B0 and anti-B0 mesons at the threshold energy. The B0 mesons undergo decay with a mean proper lifetime of 1.5 × 10−12 s. Calculate the mean distance that the B0 mesons travel before decay, as observed in the rest frame of the laboratory. (Neglect rest-mass energies of electron and positron)

Relevant Equations

E^2 = p^2c^2 + m^2c^4
E = ymc^2
P = ymu
conservation of energy and momentum

Conservation of Energy: 9GeV + E = 5.3GeV + 5.3GeV
Therefore E = 1.6GeV for the threshold energy.

How would I find the velocity of B0 mesons so that I can calculate their mean distance?
Then it would just be distance = velocity of b0 * mean proper lifetime
Right?

 

[PeroK]

Conservation of Energy: 9GeV + E = 5.3GeV + 5.3GeV
Therefore E = 1.6GeV for the threshold energy.

Are you sure about that? What happened to conservation of momentum?

PS If the meson are created with their rest energy, then their speed is zero and they wouldn't go anywhere.

 

[physconomics]

Are you sure about that? What happened to conservation of momentum?

PS If the meson are created with their rest energy, then their speed is zero and they wouldn't go anywhere.

I'm confused doesn't the question say they're created at the threshold energy?

 

[PeroK]

I'm confused doesn't the question say they're created at the threshold energy?

Yes, but the threshold energy is only the rest mass energy in the centre of momentum (COM) frame. In any other frame it is higher.

If the question said that the electron and positron had the same energy, then the lab frame would be the COM frame and each would have an energy of $5.3GeV$.

If the positron has an energy of $1.6GeV$, then the system has significant momentum in the lab frame, hence must have residual KE in the lab frame (conservation of momentum).

 

[PeroK]

PS I guess the question may be ambiguous as by "threshold" energy it means the "minimum/threshold" energy, given that the electron has an energy of $9GeV$.

Nevertheless, there clearly is no solution at a total energy of $10.6GeV$ - given the electron has $9 GeV$. And, even if there were a solution with no residual momentum/energy, the mesons would be at rest and would travel no distance.

Hint: don't worry about calculating the threshold energy. Just trust the energy-momentum conservation equations.

 

[physconomics]

PS I guess the question may be ambiguous as by "threshold" energy it means the "minimum/threshold" energy, given that the electron has an energy of $9GeV$.

Nevertheless, there clearly is no solution at a total energy of $10.6GeV$ - given the electron has $9 GeV$. And, even if there were a solution with no residual momentum/energy, the mesons would be at rest and would travel no distance.

Hint: don't worry about calculating the threshold energy. Just trust the energy-momentum conservation equations.

Ah okay, I see, thank you! I've used four vectors and then the invariant to get E = 3.12GeV. I think I've got the second part too, using conservation of energy and time dilation. Thank you! :)

 

[PeroK]

Ah okay, I see, thank you! I've used four vectors and then the invariant to get E = 3.12GeV. I think I've got the second part too, using conservation of energy and time dilation. Thank you! :)

What answer did you get?

 

[physconomics]

What answer did you get?

I got 2.49x10^(-4)m

 

[PeroK]

I got 2.49x10^(-4)m

Yes, that looks correct!