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By Vincent Rivasseau (Chief Editor)

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Disertori and V. Rivasseau Ann. Henri Poincar´ e where by convention u0 = 1. The Jacobian of this transformation is the determinant of a triangular matrix hence it is given by: n ¯ −1 J = β1 (β1 β2 ) . . (β1 β2 . . βn¯ −2 ) = βin¯ −1−i . 97) i=1 We absorb Λ−n into the term K n¯ since we recall that Λ > m hence that Λ−n = (Λm )−n , and that in Theorem 3 Λm remains in the compact X, hence is bounded away from 0. 95) becomes 2p ||φi ||∞,2 e−(1− |ΓΛ 2p (φ1 , . . n !  −1+ 12 (¯ n−i)− 12 |Ni | βi i=1   ¯ −1 1n 1 ...

The derivative as: ΛΛ0 ∂ ∂Λ Γ2p (φ1 , . . 23), ∂ ΛΛ0 ΛΛ0 0 Γ (φ1 , . . , φ2p ) = T ΓΛΛ 2p (φ1 , . . , φ2p ) + LΓ2p (φ1 , . . , φ2p ). 125) 54 M. Disertori and V. Rivasseau Ann. Henri Poincar´ e 0 The first term T ΓΛΛ 2p (φ1 , . . , φ2p ) is the series obtained when the derivative falls on a tree line propagator (see Figure 23a): ∞ 0 T ΓΛΛ 2p (φ1 , . . n ! n ¯ −1 0≤w1 ≤···≤wn−1 ≤1 q=1 ¯ n ¯ −1 RGki Gk i ∈Dµ q =1 Λ ,wq DΛ0 dwq v (T , Ω) d2 x1 . . d2 xn¯ o−T E,µ Col,Ω λw(v) N δmw(v ) v δζw(v ) v ∂ Λ0 ,wq (¯ xlq , xlq ) D ∂Λ Λ (¯ xlq , xlq ) det M(µ) φ1 (xi1 ) .

The same result holds for ||Gg ||2C . A similar argument can be performed for loop lines with some gradient ap1 plied. Each derivative adds a factor α− 2 in the integral. B: External lines. The only external line essential in spatial integration is the root y1 , then we can choose this point as reference vertex for the whole graph so that it is never interpolated. For the other external lines we take again the easiest formula, the linear one. All gradients generated by moving the external lines in fact apply to the test functions.

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Annales Henri Poincaré - Volume 1 by Vincent Rivasseau (Chief Editor)


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