## A W boson decays at rest to an up quark and a bottom quark. THe W has a mass of 80.4 GeV and the mass of the up quark can be neglected. FInd the values of the energies and momenta of the two quarks, working in natural units.

In natural units the energy of particle is defined by E^{2} = p^{2} + m^{2} and then as always energy and momentum must must be conserved so we have the following equations:

(1) E_{w} = E_{u} + E_{b}

(2) p_{w} = p_{u} + p_{b}

(3) E_{w}^{2} = p_{w}^{2} + m_{w}^{2}

(4) E_{u}^{2} = p_{u}^{2} + m_{u}^{2}

(5) E_{b}^{2} = p_{b}^{2} + m_{b}^{2}

The W boson is at rest and so p_{w} = 0. Subbing that into equation 2 we find that |p_{u}|= |p_{b}| = p. For momentum to be conserved the up and bottom quarks have the same magnitude of momentum but in opposite directions. Also we are assuming that m_{u} = 0. So we can rewrite equations 3, 4 and 5 as:

(3) E_{w}^{2} = m_{w}^{2} = 80.4^{2}

(4) E_{u}^{2} = p^{2}

(5) E_{b}^{2} = p^{2} + m_{b}^{2} = p^{2} + 4.5^{2}

So now we can sub these results into equation 1 and solve:

E_{w} = E_{u} + E_{b}