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By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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Note that β in [CK01] is the operator associated to the β-function only in the single equation case, otherwise it is simply the anomalous dimension. 12. Suppose Γxk appears in X r with coefficient c and Z ⊗ Γxk consists of all terms in ΔX r with Γ on the right hand side. Then Z = cX r Qk . 13. (Plin ⊗ id)ΔX r = X r ⊗ X r − sj X j ⊗ x∂x X r . j∈R Proof. As in the single equation case every graph appearing on the right hand side of ΔX r also appears in X r and vice versa. Suppose Γxk appears in X r and Z ⊗ Γxk consists of all terms in ΔX with Γ on the right hand side.

13. (Plin ⊗ id)ΔX r = X r ⊗ X r − sj X j ⊗ x∂x X r . j∈R Proof. As in the single equation case every graph appearing on the right hand side of ΔX r also appears in X r and vice versa. Suppose Γxk appears in X r and Z ⊗ Γxk consists of all terms in ΔX with Γ on the right hand side. 12 X r Qk = Z, and Plin (X r Qk ) = Plin X r + Plin Qk = Plin X r + kPlin Q = Plin X r − k sj Plin X j j∈R =X −k r sj X j . j∈R The result follows. 14. γkr = ⎛ ⎞ 1⎝ −sign(sr )γ1r (x)2 + k r |sj |γ1j (x)x∂x γk−1 (x)⎠ .

N the Mellin transform Fγ (ρ1 , . . , ρn ) associated to γ. We are interested in Fγ near the origin. Then, another way to see the analytic Dyson-Schwinger equation as coming from the combinatorial Dyson-Schwinger equation by replacing X r with Gr and γ with Fγ . The factor with exponent ρi indicates the argument for the recursive B+ appearance of the X j which is inserted at the insertion place corresponding to edge i. 7. 5. Broadhurst and Kreimer in [BK01] discuss graphs from massless Yukawa theory where powers of the one loop fermion self energy are inserted into itself.

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