Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 9

Let $P=m_p{c}^2$ denote the proton rest energy and $W=220\mathrm{GeV}$ the desired particle's rest energy. For an antiproton of total energy $E_{\bar{p}}$ striking a stationary proton, the incoming squared centre-of-mass energy isFor production of the stated single particle, four-momentum conservation requires $s=W^2$. ThusThe numerical value uses $P=0.938272\mathrm{GeV}$. It is a total incident energy; the corresponding kinetic energy is $E_{\bar{p}}^{\mathrm{fixed}}-P\simeq 25.7902\mathrm{TeV}$. Much of the laboratory energy supplies motion of the outgoing system rather than its invariant mass.

For equal and opposite beam momenta, the laboratory is the centre-of-mass frame. Each beam has the same total energy and their sum is $W$, givingThese are the fixed-target production threshold and collider kinematic comparison for the specified final state. Extra final particles or restrictions from conserved quantum numbers would require a separately specified reaction.

In the parent's rest frame, the identical daughters have equal and opposite momenta, and each has total energy $110\mathrm{GeV}$. With daughter rest energy $Q=90\mathrm{GeV}$, each speed in this frame isTransform to the rest frame of one daughter using the collinear relativistic velocity addition law. The other daughter then has speedAn independent invariant mass check is useful. In the rest frame of one daughter the other's total energy satisfies $W^2=2Q^2+2Q{E}^{\prime }$, so $E^{\prime }=178.8889\mathrm{GeV}$ and the relative Lorentz factor is ${\gamma }_{\mathrm{rel}}=E^{\prime }/Q=161/81$. Hence $w/c=\sqrt{1-{\gamma }_{\mathrm{rel}}^{-2}}$, the same result.

The answer is unchanged if the parent is produced by a fixed-target collision. Its laboratory velocity then differs, but its rest mass and two-body decay kinematics do not. The scalar product of the daughters' four-momenta, equivalently their relative Lorentz factor, depends only on $W$ and $Q$. A common boost cannot change the speed of one measured in the rest frame of the other.^[ref]