By Kane G.L.
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Given the two initial particles a and b, we can have different particles in the final state. Each of these processes is called a ‘channel‘ and its cross-section is called the ‘partial crosssection’ of that channel. The sum of all the partial cross-sections is the total cross-section. Decays Consider, for example, the three-body decay a! b þ c þ d: again, the final state can be defined with more or fewer details, depending on what is measured. Here the quantity to compute is the decay rate in the measured final state.
The probability of transition per unit time to the final state f of n particles is ! ð n n n 3 Y X X 2 1 d p 4 i 3 M f i ð2πÞ δ Ei À E δ pi À P : ð1:49Þ Γif ¼ 3 2E i¼1 ð2πÞ 2E i i¼1 i¼1 With these expressions, we can calculate the measurable quantities, cross-sections and decay rates, once the matrix elements are known. The Standard Model gives the rules to evaluate all the matrix elements in terms of a set of constants. Even if we do not have the theoretical instruments for such calculations, we shall be able to understand the physical essence of the principal predictions of the model and to study their experimental verification.
The two collide and remain attached in a single body of mass M. The total energy does not vary, but the initial kinetic energy has disappeared. Actually, the rest energy has increased by the same amount. The energy conservation is expressed as 2γ mc2 ¼ Mc2. The mass of the composite body is M > 2m, but charges by just a little. Let us see by how much, as a percentage, for a speed of υ ¼ 300 m/s. This is rather high by macroscopic standards, but small compared to c, β ¼ υ/c ¼ 10À6. Expanding in 2m 1 2 a series, we have M ¼ 2γm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi % 2m 1 þ β .