Calculation of maximum decay length

Question

Consider the following decay:

$$\Upsilon(4S) \rightarrow B^0 + \bar{B}^0.$$

I have to determine the maximum decay length given the lifetime $\tau$ of a $B$-meson (as measured in its rest frame). I am given that $\Upsilon(4S)$ has a certain energy which I call $E_\Upsilon = 12.1 GeV$. The masses are given: $m_\Upsilon = 10.58 GeV$ and $m_{B^0} = 0.527 GeV$.

My attempt:

My reasoning was that one of the $B$-mesons has to have to the largest momentum possible, meaning that we have to restrict the problem to one dimension and we use that one meson has to be stationary. But this would simply give $E_{B^0}= E_\Upsilon - m_{B^0}$. But if I plug this into the equation for momentum conservation, momentum cannot be conserved when we assume one particle is stationary after the decay.

Question:

How can we then find the maximal decay length, since we are dealing with two unknown energies?

Answer 1

This problem is solved in three steps:

1. Calculate the momentum of the $B$ mesons in the decay frame of the $\Upsilon$.

There is one unique solution for the magnitude of momentum of daughters in a two-body decay since they must be equal and opposite. The usual formula is this:

$$p = \frac{\sqrt{(M^2-(m_1 + m_2)^2)(M^2-(m_1 - m_2)^2)}}{2M}$$

2. Boost to the lab frame along the direction of one of the $B$ mesons. Get the Lorentz factor from $\gamma = E_\Upsilon / m_\Upsilon$.

3. Calculate decay length with $\ell = \gamma \beta c \tau$.