Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

334 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345withG n+1 =〈0|O 0 (x 0 )O 1 (x 1 )O 2 (x 2 ) ···O n (x n )|0〉,where the x j = x j (q) and O j (x j ) are some vertex operators that have to be specified. In thisway, the diversity (6.1) appears just as a manifestation of applying Wick theorem to (6.3) for itsdecomposition in terms of two-points correlation functions. To fix the ideas, think about O 0 (x 0 )as given byO 0 (x 0 ) = Γ + (1)and all remaining others as given by vertex operators involving products of Γ − (y) only, that is:(6.3)(6.4)O j (x j ) ∼ ∏ Γ − (y), j = 1,...,n.Since Γ − (y) and any product of Γ − (y) has vacuum expectation values equal to one,)〈0|Γ − (y)|0〉=1 =〈0|(∏Γ− (y) |0〉,it follows thatn∏〈0| O l (x l )|0〉=1.l=1Then, by using Wick theorem G n+1 reduces to(6.5)(6.6)(6.7)G n+1 = ∏ k〈0|O 0 (x 0 )O k (x k )|0〉,k= 1,...,n.(6.8)Let us build the Green function G n+1 step by step, starting from Eq. (5.7) and using the resultsobtained above:6.1. Leading termsBelow, we give the explicit computation of G n for n = 2, 3, 4, 5.(1) G 1 (q) as 2-point propagatorComparingG 1 (q) =〈0|Γ + (1)Γ − (q)|0〉withG 2 =〈0|O 0 (x 0 )O 1 (x 1 )|0〉we getO 0 (x 0 ) = Γ + (1),O 1 (x 1 ) = Γ − (q).(2) G 2 (q) as a 3-point functionStarting from the expression Eq. (5.7) for G 2 ,G 2 (q) =〈0|Γ + (1)Γ −(2) (q)|0〉,and using the special property established in Appendix A;seeEq.(A.26),(6.9)(6.10)(6.11)(6.12)