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Consider now a system which contains three real scalar fields, called φi ( x ) with i = 1, 2, 3. The action we will use is 3 λ 1 1 ∂µ φi ∂µ φi − ∑ m2 φi2 − 2 2 8 i =1 i =1 3 S= d4 x − ∑ 3 ∑ φi2 2 . e. 76) where GF ( x − y) is the single-field scalar propagator? e. 1 to the more complicated system under consideration here). (c) Compute, at order λ, the transition amplitude for the 2 → 2 scattering process in which one particle of type i and one of type j scatter to one particle of type k and one of type l.

5 Cross sections and decay rates In the rest frame of particle two, the incoming flux is simply the number of particles of momentum k1 , per unit area of the beam, per unit time. 64) AT V where we multiplied and divided by |v1 | to arrive at the last expression. So the flux of one particle is |v1 |/V. More generally, when the target is not at rest, we would have to use the relative velocity of the two incoming particles, and the flux is |v1 − v2 |/V. In terms of momenta, this is incoming flux = Third ingredient: normalise with respect to the ingoing flux.

17) This is as simple as it is going to get. 7) we still need to do a Fourier transform, ∞ GF ( Pf , Pi ) = −∞ dX f dXi exp iPi Xi + iPf X f GF ( X f , Xi ) . 18) Of course we expect that the initial and final momenta are not independent, and indeed we will see a Dirac delta function appear in a minute. We thus finally get, for our path integral version of the momentum space propagator, GF ( Pf , Pi ) ∝ = ∞ dT 0 T2 ∞ dT 0 T2 ∞ −∞ dX f dXi exp iPi Xi + iPf X f + ∞ −∞ dX f dXi exp i im2 + ( X f − Xi ) 2 − T 2¯h T 2¯h i i¯h ( X f − Xi + h¯ Pf T )2 − ( Pf )2 T 2¯h T 2 im2 + T .

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