Positive Sigma ( \(\Sigma^{+}\)) baryons can be created along with positive \(k\) mesons \(k^{+}\) in high-energy collisions of protons with protons, in the reactions \(p+p ightarrow \Sigma^{+}+k^{+}+n\), where \(n\) is a neutron.

(a) Find the minimum (i.e., threshold) energy required for the incident proton if the target proton is at rest in the lab. The masses of the particles (in units \(\mathrm{MeV} / \mathrm{c}^{2}\) ) are \(p: 938.3 ; \Sigma: 1189.4 ; k^{+}: 493.7\), and \(n: 939.6\).

(b) Find the minimum energy of each proton required in a \((p, p)\) collider experiment, in which the total momentum is zero.

(c) Find the velocity of the CM (i.e., zero-momentum) frame in this experiment, relative to the lab frame in which one initial proton is at rest, expressed as \(V_{\mathrm{CM}} / c\).