Past exam of the symmetry and relativity course of the University of Oxford / 2015 / 3 / c / Solution

Choose the kaon momentum along $+x$ and the first pion along $+y$. Momentum conservation givesBoth pions have mass $m_{\pi }$. Apply the on-shell condition to the second pion, using $E_2=E-E_1$:Substitute $E^2-p^2{c}^2=M^2{c}^4$ and $E_1^2-p_1^2{c}^2=m_{\pi }^2{c}^4$. The equal daughter masses cancel, leavingWith the supplied rest energies and $E=600$ MeV,The second pion travels below the parent axis, atReflection in the $x$ axis gives the equivalent opposite-sign configuration. The condition $E_1\ge m_{\pi }{c}^2$ requires $E\le M^2{c}^2/(2m_{\pi })\simeq 886.84$ MeV for this perpendicular first-pion geometry. The given $600$ MeV lies within that range; the geometry cannot be extrapolated to an arbitrarily boosted kaon.