From Special Relativity to Feynman Diagrams.pdf

2 π 4 δ 4 p out p in t f i t i f n 2 π 8 δ 4 p

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( 2 π ) 4 δ 4 ( P out P in ) T f i T i f + n ( 2 π ) 8 δ 4 ( P out P n 4 ( P in P n ) T f n T in = 0 , (12.100) where we have used the property of {| n } of being a complete set of states, n δ f n δ in = δ f i . Writing δ 4 ( P out P n 4 ( P in P n ) = δ 4 ( P out P in 4 ( P out P n ) , we can deduce from ( 12.100 ) the following equation:
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12.3 Dynamics of Interaction Processes 459 T f i T i f = n i ( 2 π ) 4 δ 4 ( P out P n ) T f n T in , (12.101) where all the kinematical quantities are subject to the constraint P in = P out . Since T ab is proportional to the (small) coupling constant λ , to lowest order in λ, T is hermitian: T f i = T i f . Suppose now that the kinematical constraints only allow elastic processes. This means that δ 4 ( P in P n ) T ni is different from zero only for states | n describing two (free) particles of rest masses m 1 , m 2 . In the CM frame the final state is totally defined by the scattering angle θ between q and p . The value θ = 0, in particular, corresponds to the forward scattering in which the initial and final states coincide | ψ in = | ψ out (i.e. i = f and p = q ). In this case ( 12.101 ) reads: 2 Im ( T ii ) = n ( 2 π ) 4 δ 4 ( P out P n ) | T in | 2 . (12.102) Observe now that we can replace the sum over the intermediate states n , by the integral over the momenta q 1 , q 2 of the corresponding two particles and the sum pol .( n ) over their polarizations: n pol .( n ) d 3 q i ( 2 π ) 3 V i . (12.103) Equation ( 12.102 ) will then read: 2 Im ( T ii ) = pol .( n ) | T in | 2 d ( 2 ) . (12.104) Note that the left hand side is proportional to the total cross-section of the elastic scattering σ t ( in ) d σ( in ; n ) = cV 1 V 2 f ( v 1 , v 2 ) pol .( n ) | T in | 2 d ( 2 ) , (12.105) where we have used ( 12.75 ). Equation ( 12.104 ) can now be written in the following form: 2 Im ( T ii ) = f ( v 1 , v 2 ) cV 1 V 2 σ t . (12.106) We have now performed the replacement V 1 /( 2 E ) yet. This is done by writing the left hand side in terms of T ii , defined in ( 12.78 ): T ii = 1 V 1 V 2 4 E 1 E 2 T ii . (12.107)
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460 12 Fields in Interaction The normalization volumes cancel and we end up with: Im ( T ii ) = 2 c 2 ( p 1 · p 2 ) 2 m 2 1 m 2 2 c 4 σ t = 2 cE | p | σ t , (12.108) where E is the total energy in the CM frame and we have used the comment in footnote10. Equation ( 12.108 ) directly descends from the unitarity property of S and relates the imaginary part of the forward scattering amplitude to the total cross section of the process. It describes the content of the optical theorem. 12.3.5 Natural Units Our analysis would simplify considerably if we could get rid of all the factors , c occurring in our formulas. This can be done, for instance, by choosing length (or mass) as the only fundamental quantity and by defining in terms of it the units of mass (or length) and time so that = c = 1 . Let us denote by [ c ] and [ ] the new dimensional quantities corresponding to the units where c = 1 and = 1 . Then we may write 1 kg = c × ( 1 m ) 1 [ c 1 ] , 1 s = c × 1 m [ c 1 ] .
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