 Elastic scattering from a quartic scalar contact interaction

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude $|\lambda |$. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is ${\lambda }^2/(128{\pi }^{2}s)$. On a hemisphere representing each event once, the density is ${\lambda }^2/(64{\pi }^{2}s)$. Both conventions give total event cross-section ${\lambda }^2/(32\pi s)$.