phspacenmassless - Phase space for N massless particles k...

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Phase space for N massless particles P = Z Y k dp + k d 2 p k (2 π ) 3 2 p + k (2 π ) 4 δ X k p + k - E 2 ! δ X k p k ! δ X k p 2 k 2 p + k - E 2 ! (1) = Z Y k dp + k d 2 p k (2 π ) 3 2 p + k (2 π ) 4 δ X k p + k - E 2 ! Z dx + d 2 x (2 π ) 3 exp i x · X k p k - ix + X k p 2 k 2 p + k - E 2 !! = Z Y k dp + k d 2 p k (2 π ) 3 2 p + k (2 π ) 4 δ X k p + k - E 2 ! Z dx + d 2 x (2 π ) 3 exp i E 2 2 x + x 2 - ix + X k p 2 k 2 p + k - E 2 !! = Z Y k dp + k d 2 p k (2 π ) 3 2 p + k (2 π ) 4 δ X k p + k - E 2 ! Z dx + (2 π ) 3 2 2 iπx + E exp - ix + X k p 2 k 2 p + k - E 2 !! = Z Y k dp + k (2 π ) 3 (2 π ) 4 δ X k p + k - E 2 ! 1 πE ∂E Z d 2 p k 2 p + k δ X k p 2 k 2 p + k - E 2 ! = Z Y k dp + k (2 π ) 3 (2 π ) 4 δ X k p + k - E 2 ! 1 πE ∂E Z d 2 N Q δ ± Q 2 - E 2 ² = ± E 2 ² N - 1 Z Y k dx k (2 π ) 3 (2 π ) 4 δ X k x k - 1 ! 1 πE ∂E ± E 2 ² N - 1 Z d 2 N Q δ ( Q 2 - 1 ) = N - 1 2 N - 2 E 2 N - 4 (2 π ) 3 N - 3 Z 0 Y k dx k δ X k x k - 1 ! Z d 2 N Q δ ( Q 2 - 1 ) (2) The integral over Q is just proportional to the volume of a 2 N - 1 dimensional sphere: Z d 2 N Q δ ( Q 2 - 1 ) = π N Γ( N ) (3) and a short calculation shows that Z 0 Y k dx k δ X k x k - 1 ! = 1 Γ( N ) . (4) The same method works for any even transverse dimension
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