oh4 - Quantum Field Theory Example Sheet 4 Dr David Tong November 2007 1 A real scalar feld with \u03c6 4 interaction has the Lagrangian L = 1 2 \u2202 \u03bc

# oh4 - Quantum Field Theory Example Sheet 4 Dr David Tong...

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Quantum Field Theory: Example Sheet 4 Dr David Tong, November 2007 1. A real scalar field with φ 4 interaction has the Lagrangian L = 1 2 μ φ∂ μ φ 1 2 m 2 φ 2 λ 4! φ 4 (1) Use Dyson’s formula and Wick’s theorem to show that the leading order contribution to 3-particle 3-particle scattering includes the amplitude p 3 2 p p 1 p 1 2 p p 3 / / / = ( ) 2 i ( p 1 + p 2 + p 3 ) 2 m 2 (2) Check that this result is consistent with the Feynman rules for the theory. What other diagrams also contribute to this process? 2. Examine ( 0 | S | 0 ) to order λ 2 in φ 4 theory. Identify the different diagrams with the different contributions arising from an application of Wick’s theorem. Confirm that to order λ 2 , the combinatoric factors work out so that the the vacuum to vacuum amplitude is given by the exponential of the sum of distinct vacuum bubble types, ( 0 | S | 0 ) = exp ( + + + ... ) (3) 3. Consider the Lagrangian for 3 scalar fields φ i , i = 1 , 2 , 3, given by L = 3 summationdisplay i =1 1 2 ( μ φ i )( μ φ i ) 1 2 m 2 ( 3 summationdisplay i =1 φ 2 i ) λ 8 ( 3 summationdisplay i =1 φ 2 i ) 2 (4) Show that the Feynman propagator for the free field theory (i.e. λ = 0) is of the form ( 0 | i ( x ) φ j ( y ) | 0 ) = δ ij D F ( x y ) (5) where D F ( x y ) is the usual scalar propagator. Write down the Feynman rules of the theory. Compute the amplitude for the scattering φ i φ j φ k φ l to lowest order in λ .