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Then the map ι is given by ι : Γ(0) → I. 67) It assigns to each vertex a monomial in the interaction Lagrangian Lint . e. 68) deg ι(v) = #∂0−1 (v) + #∂1−1 (v) , ∀v ∈ Γ(0) . One also requires that there are N external lines labeled by {1, 2, . . , N }. The labelling is by the vertex in {1, 2, . . 69) #∂0−1 (v) + #∂1−1 (v) = 1 , ∀v ∈ {1, 2, . . , N }. 70) (1) Γext = ∪1j=0 ∂j−1 {1, 2, . . , N } ⊂ Γ(1) 3. FEYNMAN DIAGRAMS 44 (1) and its complement Γint ⊂ Γ(1) is the set of internal lines. 71) |Γ| = Γ(1) × [0, 1] ∪∂ (Γ(0) ∪ {1, 2, .

118) ∂Π(−m2 ) = 0. 116) to the self-energy. 113). 113) is not canceled by the mass counterterm alone. 119) Πren (p2 ) = Π(p2 ) − Π(−m2 ) + Π (−m2 ) (p2 − m2 ), 4. DIMENSIONAL REGULARIZATION 56 Figure 15. Graph with three external legs where Π is the derivative of Π with respect to p2 . 118). 120) g 3 (2π)−6 d6 k. 112) gives us the freedom to write the bare coupling in the form g + δg and to introduce the counterterm δg. 112) is fixed by its experimental value. 121) LE = 1 (∂φ)2 (1 − δZ) + 2 m2 − δm2 2 φ2 − g + δg 3 φ .

76) (2π)D δ(p1 + p2 ) (p21 + m2 )−1 . Edges where one pi is paired with one of the k are external. 69) holds. 68) holds. Notice that the two endpoints of an edge may well be the same, as in the tadpole graph of Figure 5. In the φ3 theory this only occurs for tadpole graphs, but in other theories it occurs in more important graphs. For instance, in the case of the φ4 theory this happens in the self-energy graph of Figure 11. Feynman rules: Now that we know how to assign a graph to a pairing we can write down the Feynman rules which give the contribution of the pairing.

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