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Sample text

Perhaps the cleanest argument showing that this is true for any amplitude can be given using path integrals, see Sec. 12 in Ref. 10 and also Refs. 11 and 14. One can show that given a Lagrangian containing a higher dimensional operator with a part proportional to the equations of motion = −(c1 + L = LD≤4 + c F (ϕ) δLD≤4 , δϕ all correlation functions of the form ϕ(x1 ) . . ϕ(xn )F (ϕ(y)) (49) δLD≤4 δϕ(y) ∂2 η ∂2 c2 (a) vanish. η c2 (b) Fig. 6. Diagrams with one non-derivative quartic interaction and one quartic interaction containing ∂ 2 .

Two important, and very tightly constrained experimentally, universal operators are OS = H † σ i HAiµν B µν , OT = H † Dµ H 2 (54) , i (55) Aiµν where σ are the Pauli matrices, meanwhile Bµν and are the U (1)Y and SU (2)L field strengths, respectively. The operator OS introduces kinetic mixing between Bµ and A3µ when the vev is substituted for H. The second operator, OT , violates the custodial symmetry. The custodial symmetry guarantees the treelevel relation between the W and Z masses, MW = MZ cos θw , where θw is the weak mixing angle.

48). Diagram (b) is associated with the 3! · 3 factor, where 3! comes from the permutations of lines without the derivative and 3 comes from placing the derivative on either of the 3 external lines. Combined with 3! in Eq. (50), we get 3! · 3 + 3! = 4! to reproduce the coefficient of the ϕ4 term in Eq. (48). 6. Summary We have constructed several effective theories so far. It is a good moment to pause and review the observations we made. To construct an EFT one needs to identify the light fields and their symmetries, and needs to establish a power counting scheme.

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