New PDF release: Weak Interaction of Elementary Particles

By L. B. Okun and D. ter Haar (Auth.)

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Hence physical processes take place if the Γ-matrix, which is determined by the relation S = I + iT, differs from zero. In what follows the quantity Mfi of Tfi = (2n)*d*(j>f-Pi)Mfi9 will be called the amplitude of the process. Here pt and pf are the 4-momenta of the initial and final state, while the (5-function ex­ presses the energy-momentum conservation law in its explicit form: ö4(Pr - Ρ,) = δ(ΡΪ ~ Pf) HP} - Ρ0δ(ρ} - tf)d{E, - Ε,). For the sake of brevity, the subscripts / and i will be omitted henceforth.

As is known, the fermion is described by a 4-component spinor u. Let us find out how this spinor is transformed under space inversion. Consider the Dirac equation (p — m) u = 0. This equation can be written in the following way: (Εγ^ — ργ — m) u = 0. Under inversion of coordinates p -> p' = — p, u -> u' — Pu, and we get (Εγ4 + ργ — m) Pu = 0. We ask the question: what form should the operator P have in order that the resulting equation goes over to the initial Dirac equation? For this the operator P should be such that Ρ^ίΕγ^ + ργ - m)P = (Εγ4 - ργ - m), where P1 is the inverse P-matrix: P-'P = 1.

In seconds is 2 x 10- 1 4 cm _ 1Λ -2S 25 77-^ sec. ; — ~ 7 x 10 10 3 x 10 cm/sec The nucleon meson mass ratio Μ\μ is equal to ~ 9 . Hence we obtain 192π3 x 9 5 x 7 x 10" 25 ^ 1Λ 6, x ~ T^YQ Ä 2 x 10" sec. We now carry out an accurate calculation of the decay probabi­ lity with the amplitude For clarity, we divide the calculation into several stages. EXPRESSION FOR \M\2 Since \M\2 = MM*, we must first find an expression for M*. M represents the product of two parentheses, each of which has the form (vOau), where Oa = γΛ(1 + γ5).

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