Combinatorial Dyson-Schwinger equations and systems I Feynma. Feynman graphs, rooted trees and combinatorial Dyson-Schwinger equations

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1 Combinatorial and systems I, rooted trees and combinatorial Potsdam November 2013 Combinatorial and systems I Feynma

2 In QFT, one studies the behaviour of particles in a quantum fields. Several types of particles: electrons, photons, bosons, etc. Several types of interactions: an electron can capture/eject a photon, etc. One wants to predict certain physical constants: mass or charge of the electron,etc. Develop the constant in a formal series, indexed by certain combinatorial objects: the. Attach to any Feynman graph a real/complex number: Feynman rules and Renormalization. Combinatorial and systems I Feynma

3 The expansion as a formal series gives formal sums of : the propagators (vertex functions, two-points functions). These formal sums are characterized by a set of equations: the. In order to be "physically meaningful", these functions should be compatible with the extraction/contraction Hopf algebra structure on. This imposes strong constraints on the. Because of a 1-cocycle property, everything can be lifted and studied to the level of decorated rooted trees. Combinatorial and systems I Feynma

4 Feynman definition Combinatorial structures on To a given QFT is attached a family of graphs. 1 A finite number of possible half-edges. 2 A finite number of possible vertices. 3 A finite number of possible external half-edges (external structure). 4 The graph is connected and 1-PI. To each external structure is associated a formal series in the. Combinatorial and systems I Feynma

5 Feynman definition Combinatorial structures on In QED 1 Half-edges: 2 Vertices:. (electron), (photon). 3 External structures:,,. Combinatorial and systems I Feynma

6 Feynman definition Combinatorial structures on Examples in QED,,,,,,,, Combinatorial and systems I Feynma

7 Feynman definition Combinatorial structures on Other examples Φ 3. Quantum Chromodynamics. Combinatorial and systems I Feynma

8 Feynman definition Combinatorial structures on Subgraphs and contraction 1 A subgraph of a Feynman graph Γ is a subset γ of the set of half-edges Γ such that γ and the vertices of Γ with all half edges in γ is itself a Feynman graph. 2 If Γ is a Feynman graph and γ 1,..., γ k are disjoint subgraphs of Γ, Γ/γ 1... γ k is the Feynman graph obtained by replacing γ 1,..., γ k by vertices in Γ. Combinatorial and systems I Feynma

9 Feynman definition Combinatorial structures on Insertion Let Γ 1 and Γ 2 be two. According to the external structure of Γ 1, you can replace a vertex or an edge of Γ 2 by Γ 1 in order to obtain a new Feynman graph. Examples in QED =, =,,. Combinatorial and systems I Feynma

10 Algebras, coalgebras, Hopf algebras Hopf algebra of Let A and B be two vector spaces. The tensor product of A and B is a space A B with a bilinear product : A B A B satisfying a universal property: if f : A B C is bilinear, there exists a unique linear map F : A B C such that F(a b) = f (a, b) for all (a, b) A B. If (e i ) i I is a basis of A and (f j ) j J is a basis of B, then (e i f j ) i I,j J is a basis A B. Combinatorial and systems I Feynma

11 Algebras, coalgebras, Hopf algebras Hopf algebra of The tensor product of vector spaces is associative: (A B) C = A (B C). We shall identify K A, A K and A via the identification of 1 a, a 1 and a. Combinatorial and systems I Feynma

12 Algebras, coalgebras, Hopf algebras Hopf algebra of If A is an associative algebra, its (bilinear) product becomes a linear map m : A A A, sending a b on ab. The associativity is given by the following commuting square: A A A m Id A A Id m A A m A m Combinatorial and systems I Feynma

13 Algebras, coalgebras, Hopf algebras Hopf algebra of If A is unitary, its unit 1 A induces a linear map { K A η : λ λ1 A. The unit axiom becomes: Id η K A η Id A A A K m A Combinatorial and systems I Feynma

14 Algebras, coalgebras, Hopf algebras Hopf algebra of Dualizing these diagrams, we obtain the coalgebra axioms Coalgebra A coalgebra is a vector space C with a map : C C C such that: C C C C C Id Id C C C Combinatorial and systems I Feynma

15 Algebras, coalgebras, Hopf algebras Hopf algebra of Coalgebra There exists a map ε : C K, called the counit, such that: ε Id K C C C Id ε C K C Combinatorial and systems I Feynma

16 Algebras, coalgebras, Hopf algebras Hopf algebra of If A is an algebra, then A A is an algebra, with: (a 1 b 1 ).(a 2 b 2 ) = (a 1.a 2 ) (b 1.b 2 ). Bialgebra and Hopf algebra A bialgebra is both an algebra and a coalgebra, such that the coproduct and the counit are algebra morphisms. A Hopf algebra is a bialgebra with a technical condition of existence of an antipode. Combinatorial and systems I Feynma

17 Algebras, coalgebras, Hopf algebras Hopf algebra of Examples If G is a group, KG is a Hopf algebra, with (x) = x x for all x G. If g is a Lie algebra, its enveloping algebra is a Hopf algebra, with (x) = x x for all x g. If H is a finite-dimensional Hopf algebra, then its dual is also a Hopf algebra. If H is a graded Hopf algebra, then its graded dual is also a Hopf algebra. Combinatorial and systems I Feynma

18 Algebras, coalgebras, Hopf algebras Hopf algebra of Construction Let H FG be a free commutative algebra generated by the set of. It is given a coproduct: for all Feynman graph Γ, (Γ) = γ 1... γ k Γ/γ 1... γ k. γ 1...γ k Γ ( ) = Combinatorial and systems I Feynma

19 Algebras, coalgebras, Hopf algebras Hopf algebra of The Hopf algebra H FG is graded by the number of loops: Γ = E(Γ) V (Γ) + 1. Because of the 1-PI condition, it is connected, that is to say (H FG ) 0 = K 1 HFG. What is its dual? Cartier-Quillen-Milnor-Moore theorem Let H be a cocommutative, graded, connected Hopf algebra over a field of characteristic zero. Then it is the enveloping algebra of its primitive elements. Combinatorial and systems I Feynma

20 Algebras, coalgebras, Hopf algebras Hopf algebra of This theorem can be applied to the graded dual of H FG. Primitive elements of H FG Basis of primitive elements: for any Feynman graph Γ, The Lie bracket is given by: f Γ (γ 1... γ k ) = Aut(Γ)δ γ1...γ k,γ. [f Γ1, f Γ2 ] = Γ=Γ 1 Γ 2 f Γ Γ=Γ 2 Γ 1 f Γ. Combinatorial and systems I Feynma

21 Algebras, coalgebras, Hopf algebras Hopf algebra of We define: f Γ1 f Γ2 = Γ=Γ 1 Γ 2 f Γ. The product is not associative, but satisfies: f 1 (f 2 f 3 ) (f 1 f 2 ) f 3 = f 2 (f 1 f 3 ) (f 2 f 1 ) f 3. It is (left) prelie. Combinatorial and systems I Feynma

22 Insertion operators Examples of In the context of QFT, we shall consider some special infinite sums of : Propagators in QED = n 1 = n 1 x n x n γ (n) γ (n) s γ γ. s γ γ. Combinatorial and systems I Feynma

23 Insertion operators Examples of Propagators in QED = n 1 x n They live in the completion of H FG. s γ γ. γ (n) Combinatorial and systems I Feynma

24 Insertion operators Examples of How to describe the propagators? For any primitive Feynman graph γ, one defines the insertion operator B γ over H FG. This operator associates to a graph G the sum (with symmetry coefficients) of the insertions of G into γ. The propagators then satisfy a system of equations involving the insertion operators, called systems of. Combinatorial and systems I Feynma

25 Insertion operators Examples of Example In QED : B ( ) = B ( ) = Combinatorial and systems I Feynma

26 Insertion operators Examples of In QED: = γ = xb = xb 1+2 γ (1 + ) x γ B γ ) γ ) 2 γ (1 + (1 + 2 (1 + ) ) 2 (1 + 2 (1 + ) ) ) (1 + (1 + Combinatorial and systems I Feynma

27 Insertion operators Examples of Other example (Bergbauer, Kreimer) X = B γ ((1 + X) γ +1). γ primitive Combinatorial and systems I Feynma

28 Insertion operators Examples of Question For a given system of (S), is the subalgebra generated by the homogeneous components of (S) a Hopf subalgebra? Combinatorial and systems I Feynma

29 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Proposition The operators B γ satisfy: for all x H FG, B γ (x) = B γ (x) 1 + (Id B γ ) (x). This relation allows to lift any system of Dyson-Schwinger equation to the Hopf algebra of decorated rooted trees. Combinatorial and systems I Feynma

30 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Cartier-Quillen cohomology let C be a coalgebra and let (B, δ G, δ D ) be a C-bicomodule. D n = L(B, C n ). For all l D n : b n (L) = n ( 1) i (Id (i 1) Id (n i) ) L i=1 +(Id L) δ G + ( 1) n+1 (L Id) δ D. Combinatorial and systems I Feynma

31 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? A particular case We take B = C, δ G (b) = (b) and δ D (b) = b 1. A 1-cocycle of C is a linear map L : C C, such that for all b C: (Id L) (b) L(b) + b 1 = 0. So B γ is a 1-cocycle of H FG for all primitive Feynman graph. Combinatorial and systems I Feynma

32 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? The Hopf algebra of rooted trees H R (or Connes-Kreimer Hopf algebra) is the free commutative algebra generated by the set of rooted trees.,,,,,,,,,,,,, The set of rooted forests is a linear basis of H R : 1,,,,,,,,,,,,,,,,,,,...,... Combinatorial and systems I Feynma

33 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? The coproduct is given by admissible cuts: (t) = P c (t) R c (t). c admissible cut cutc total Admissible? yes yes yes yes no yes yes no yes W c (t) R c (t) 1 P c (t) 1 ( ) = Combinatorial and systems I Feynma

34 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? The grafting operator of H R is the map B : H R H R, associating to a forest t 1... t n the tree obtained by grafting t 1,..., t n on a common root. For example: B( ) =. Proposition For all x H R : B(x) = B(x) 1 + (Id B) (x). So B is a 1-cocycle of H R. Combinatorial and systems I Feynma

35 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Universal property Let A be a commutative Hopf algebra and let L : A A be a 1-cocycle of A. Then there exists a unique Hopf algebra morphism φ : H R A with φ B = L φ. This will be generalized to the case of several 1-cocycles with the help of decorated rooted trees. Combinatorial and systems I Feynma

36 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? H R is graded by the number of vertices and B is homogeneous of degree 1. Let Y = B γ (f (Y )) be a Dyson-Schwinger equation in a suitable Hopf algebra of H FG, such that γ = 1. There exists a Hopf algebra morphism φ : H R H FG, such that φ B = B γ φ. This morphism is homogeneous of degree 0. Let X be the solution of X = B(f (X)). Then φ(x) = Y and for all n 1, φ(x(n)) = Y (n). Consequently, if the subalgebra generated by the X(n) s is Hopf, so is the subalgebra generated by the Y (n) s. Combinatorial and systems I Feynma

37 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Definition Let f (h) K [[h]]. The combinatorial associated to f (h) is: X = B(f (X)), where X lives in the completion of H R. This equation has a unique solution X = X(n), with: X(1) = p 0, n X(n + 1) = p k B(X(a 1 )... X(a k )), k=1 a a k =n where f (h) = p 0 + p 1 h + p 2 h Combinatorial and systems I Feynma

38 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? X(1) = p 0, X(2) = p 0 p 1, X(3) = p 0 p1 2 + p0 2 p 2, X(4) = p 0 p1 3 + p0 2 p 1p 2 + 2p0 2 p 1p 2 + p0 3 p 3. Combinatorial and systems I Feynma

39 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Examples If f (h) = 1 + h: X = If f (h) = (1 h) 1 : X = Combinatorial and systems I Feynma

40 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Let f (h) K [[h]]. The homogeneous components of the unique solution of the combinatorial Dyson-Schwinger equation associated to f (h) generate a subalgebra of H R denoted by H f. H f is not always a Hopf subalgebra For example, for f (h) = 1 + h + h 2 + 2h 3 +, then: So: X = (X(4)) = X(4) X(4) + (10X(1) 2 + 3X(2)) X(2) +(X(1) 3 + 2X(1)X(2) + X(3)) X(1) +X(1) (8 + 5 ). Combinatorial and systems I Feynma

41 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? If f (0) = 0, the unique solution of X = B(f (X)) is 0. From now, up to a normalization we shall assume that f (0) = 1. Theorem Let f (h) K [[h]], with f (0) = 1. The following assertions are equivalent: 1 H f is a Hopf subalgebra of H R. 2 There exists (α, β) K 2 such that (1 αβh)f (h) = αf (h). 3 There exists (α, β) K 2 such that f (h) = 1 if α = 0 or f (h) = e αh if β = 0 or f (h) = (1 αβh) 1 β if αβ 0. Combinatorial and systems I Feynma

42 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? 1 = 2. We put f (h) = 1 + p 1 h + p 2 h 2 +. Then X(1) =. Let us write: (X(n + 1)) = X(n + 1) X(n + 1) + X(1) Y (n) By definition of the coproduct, Y (n) is obtained by cutting a leaf in all possible ways in X(n + 1). So it is a linear span of trees of degree n. 2 As H f is a Hopf subalgebra, Y (n) belongs to H f. Hence, there exists a scalar λ n such that Y (n) = λ n X n. Combinatorial and systems I Feynma

43 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? lemma Let us write: X = t a t t. For any rooted tree t: λ t a t = t n(t, t )a t, where n(t, t ) is the number of leaves of t such that the cut of this leaf gives t. Combinatorial and systems I Feynma

44 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? We here assume that f is not constant. We can prove that p 1 0. For t the ladder (B) n (1), we obtain: p n 1 1 λ n = 2(n 1)p n 2 1 p 2 + p n 1. Hence: λ n = 2 p 2 p 1 (n 1) + p 1. We put α = p 1 and β = 2 p 2 p1 2 1, then: λ n = α(1 + (n 1)(1 + β)). Combinatorial and systems I Feynma

45 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? For t the corolla B( n 1 ), we obtain: λ n p n 1 = np n + (n 1)p n 1 p 1. Hence: Summing: α(1 + (n 1)β)p n 1 = np n. (1 αβh)f (h) = αf (h). Combinatorial and systems I Feynma

46 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? X(1) =, X(2) = α, X(3) = α 2 ( (1 + β) 2 + ), ( X(4) = α 3 (1 + 2β)(1 + β) 6 X(5) = α 4 (1+3β)(1+2β)(1+β) 24 + (1+β)2 2 + (1+β) 2 + (1 + β) (1 + β) (1 + β) + (1+2β)(1+β) + (1 + β) 2 + (1+2β)(1+β) 6 + (1+β) 2 + ) +,. Combinatorial and systems I Feynma

47 Hopf algebra of rooted trees Combinatorial When is H f a Hopf subalgebra? Particular cases If (α, β) = (1, 1), f = 1 + h and X(n) = (B) n (1) for all n. If (α, β) = (1, 1), f = (1 h) 1 and: X(n) = {embeddings of t in the plane}t. t =n Si (α, β) = (1, 0), f = e h and: X(n) = t =n 1 {symmetries of t} t. Combinatorial and systems I Feynma