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Arnold Sommerfeld Center Ludwig–Maximilians–Universität München Prof. Dr. Livia Ferro Summer term 2016 Exercises for Quantum Field Theory (TVI/TMP) Problem set 7, due June 1, 2016 1 Interacting real scalar field theory: φ4 Consider the theory describing an interacting real scalar field with Lagrangian density L[φ] = 1 1 g ∂µ φ ∂ µ φ − m2 φ2 − φ4 . 2 2 4! (1) During the lectures you have seen how the generating functional can be used to derive n-point functions of scalar fields in this theory. Consider a perturbative expansion with parameter g, and focus on the second order contributions (g 2 ). (i) How many vacuum diagrams can you build using two 4-point vertices? Draw them. (ii) How do these diagrams contribute to the 2- and 4-point correlation functions? (iii) Consider the 2-point correlation function. How many second order diagrams can you draw with two external legs? You should find three of them. Compute their symmetry factor. (iv) Consider the 4-point correlation function. How many fully connected second order diagrams can you draw with four external legs? You should find seven of them. Compute their symmetry factor. 2 Interacting complex scalar field theory: (φ∗ φ)2 Consider the theory describing an interacting complex scalar field with Lagrangian density g L[φ, φ∗ ] = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ − (φ∗ φ)2 . 4 (2) This theory has two kinds of sources, J and J ∗ , and so we need a way to tell which is which when we draw the diagrams. Rather than labeling the source blobs with a J or J ∗ , we will indicate which is which by putting an arrow on the attached propagator that points towards the source if it is a J ∗ , and away from the source if it is a J. (i) How do you draw the two point function? (ii) What kind of vertex appears in the diagrams for this theory? Hint: your answer should involve those arrows. (iii) Compute (up to order g) the normalized generating functional R δ exp iSint [−i δJ , −i δJδ ∗ ] exp[− d4 x d4 yJ ∗ (x)∆F (x − y)J(y))] ∗ R Znorm [J, J ] = . δ exp iSint [−i δJ , −i δJδ ∗ ] exp[− d4 x d4 yJ ∗ (x)∆F (x − y)J(y))] |J,J ∗ =0 (3) Draw all the involved diagrams, and verify that there is no vacuum contribution. (iv) Evaluate, up to order g, the 2-point correlation function δ δ hφ∗ (x1 )φ(x2 )i = −i −i ∗ Znorm [J, J ∗ ] . δJ(x1 ) δJ (x2 ) J,J ∗ =0 (4) 3 Interacting real scalar field theory: φ3 Consider the following Lagrangian density for an interacting real scalar field: L= 1 g 1 ∂µ φ ∂ µ φ − m2 φ2 − φ3 . 2 2 3! (5) (i) What is the mass dimension of g in four space-time dimensions? Consider the normalized generating functional: R δ exp iSint [−i δJ ] exp[− 12 d4 x d4 yJ(x)∆F (x − y)J(y))] R Znorm [J] = . δ {exp iSint [−i δJ ] exp[− 21 d4 x d4 yJ(x)∆F (x − y)J(y))]}|J=0 (6) (ii) Evaluate explicitly the numerator up to and including the order g and then give the diagrammatic expression of its various terms. (iii) The 1-point correlator is non-zero. Evaluate it at order g. (iv) Evaluate the 2-point correlator up to and including the order g.