Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Combinatorial Dyson-Schwinger equations Loïc Foissy
Berlin February 2012
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Feynman graphs 1
A finite number of possible edges.
2
A finite number of possible vertices.
3
A finite number of possible external edges (external structure).
4
Conditions of connectivity.
To each external structure is associated a formal series in the Feynman graphs, called the propagator.
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Examples in QED
,
,
,
,
,
Loïc Foissy
,
,
,
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Propagators in QED
=
X n≥1
xn
X n≥1
γ∈
xn
X n≥1
(n)
xn
Loïc Foissy
sγ γ .
X
γ∈
=−
sγ γ .
X
=−
(n)
sγ γ .
X
γ∈
(n)
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
How to describe the propagators? The algebra HDF is the free commutative algebra generated by the Feynman graphs of a given QFT. For any primitive Feynman graph γ, one defines the insertion operator Bγ over HDF . This operators 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 Dyson-Schwinger equations.
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Example In QED :
B
B
(
(
) =
1 2
) =
1 3
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+
+
1 2
1 3
+
1 3
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Then:
=
X γ∈Γ
x |γ| Bγ
= −xB
|γ|
1+
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2|γ|
1+
2
1+
2
1+
= −xB
1+2|γ|
1+
1+
2
1+
1+
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Let A be a vector space. The tensor square of A is a space A ⊗ A with a bilinear product ⊗ : A × A −→ A ⊗ A with a universal property. If (ei )i∈I is a basis of A, then (ei ⊗ ej )i,j∈I is a basis A ⊗ A. If A is an associative algebra, its (bilinear) product becomes a linear map m : A ⊗ A −→ A, sending ei ⊗ ej over ei .ej . The associativity is given by the following commuting square: m⊗Id
A⊗A⊗A Id⊗m
A⊗A
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/A⊗A
m
m
/A
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Dualizing the associativity axiom, we obtain the coassociativity axiom: a coalgebra is a vector space C with a map ∆ : C −→ C ⊗ C such that: C ∆
C⊗C
/C⊗C
∆
Id⊗∆
/C⊗C⊗C
∆⊗Id
A Hopf algebra is both an algebra and a coalgebra, with the compatibility: ∆(xy ) = ∆(x)∆(y ). (And a technical condition of existence of an antipode). Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
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 ⊗ 1 + 1 ⊗ x for all x ∈ g.
The algebra of Feynman graphs HDF is a graded Hopf algebra. For example:
∆(
⊗1+1⊗
)=
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+
⊗
Combinatorial Dyson-Schwinger equations
.
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Feynman graphs Dyson-Schwinger equations Hopf algebras axioms
Question For a given system of Dyson-Schwinger equations (S), is the subalgebra generated by the homogeneous components of (S) a Hopf subalgebra?
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
Proposition The operators Bγ satisfy: for all x ∈ HDF , ∆ ◦ Bγ (x) = Bγ (x) ⊗ 1 + (Id ⊗ Bγ ) ◦ ∆(x). This relation allows to lift any system of Dyson-Schwinger equation to the Hopf algebra of decorated rooted trees. We first treat the case of a single equation with a single 1-cocycle.
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
The Hopf algebra of rooted trees HR (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 HR : q q q q q qq qq q q q ∨q q q q q q q q qq 1, q , q q , q , q q q , q q , ∨q , q , q q q , q q q , q q , ∨q q , q q , ∨q , ∨q , q ,
Loïc Foissy
Combinatorial Dyson-Schwinger equations
qq q q ...
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
The coproduct is given by admissible cuts: X ∆(t) = P c (t) ⊗ R c (t). c admissible cut q q q
q qq
∨q ∨q cutc Admissible ? yes yes W c (t) R c (t) P c (t)
q q q
∨q
q q q ∨q
1
q qq
∨q yes
q qq
∨q yes
qq qq
qq q ∨q
qq
∨q
q qq q q qq
qq
q
q
qq
∨q no
q qq
∨q ∨q yes yes
q qq
q q q
∨q no
q q q
total yes
q q qq
qq q q
qq q q
qqqq
∨q
×
q
qq
×
1
×
qq q
qq
×
∨q
q q q
q q q
q q q q qq q q qq qq q q q q q ∆( ∨q ) = ∨q ⊗1+1⊗ ∨q + q ⊗ q + q ⊗ ∨q + q ⊗ q + q q ⊗ q + q q ⊗ q . Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
The grafting operator of HR is the map B + : HR −→ HR , associating to a forest t1 . . . tn the tree obtained by grafting t1 , . . . , tn on a common root. For example: q qq qq q B ( ) = ∨q . +
Proposition For all x ∈ HR : ∆ ◦ B + (x) = B + (x) ⊗ 1 + (Id ⊗ B + ) ◦ ∆(x).
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
Universal property Let A be a commutative Hopf algebra and let L : A −→ A such that for all a ∈ A: ∆ ◦ L(a) = L(a) ⊗ 1 + (Id ⊗ L) ◦ ∆(a). Then there exists a unique morphism Hopf algebra morphism φ : HR −→ A with φ ◦ B + = L ◦ φ.
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
Let f (h) ∈ K [[h]]. The combinatorial Dyson-Schwinger equations associated to f (h) is: X = B + (f (X )), where X lives in the completion of HR . P This equation has a unique solution X = xn , with: q x1 = pn0 , X X pk B + (xa1 . . . xak ), xn+1 = k =1 a1 +...+ak =n
where f (h) = p0 + p1 h + p2 h2 + . . . Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
Examples If f (h) = 1 + h: q X = q+ q +
q q q qq + qq +
qq q qq
+ ···
If f (h) = (1 − h)−1 :
X
q qq qq q q qq q qq q ∨q = q + q + ∨q + q + ∨q + 2 ∨q + q + q q q q qq q q q ∨ q q qq q q q q q q q q +H∨ + 3 ∨q + ∨q + 2 ∨q + 2 ∨q +
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q qq q
q q q qqq q q ∨q q ∨qq ∨q +2 q + q +
Combinatorial Dyson-Schwinger equations
qq qq q
+ ···
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Introduction Hopf algebra of rooted trees Combinatorial Dyson-Schwinger equations
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 HR denoted by Hf . Hf is not always a Hopf subalgebra For example, for f (h) = 1 + h + h2 + 2h3 + · · · , then: q qq q qq qq qq q ∨q q q X = q + q + ∨q + q + 2 ∨q + 2 ∨q + q +
qq qq + · · ·
So: ∆(x4 ) = x4 ⊗ 1 + 1 ⊗ x4 + (10x12 + 3x2 ) ⊗ x2
q q
q
q +(x13 + 2x1 x2 + x3 ) ⊗ x1 + x1 ⊗ (8 ∨q + 5 q ).
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
Theorem Let f (h) ∈ K [[h]], with f (0) = 1. The following assertions are equivalent: 1
Hf is a Hopf subalgebra of HR .
2
There exists (α, β) ∈ K 2 such that (1 − αβh)f 0 (h) = αf (h).
3
There exists (α, β) ∈ K 2 such that f (h) = 1 if α = 0 or − β1
f (h) = eαh if β = 0 or f (h) = (1 − αβh)
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if αβ 6= 0.
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
Idea of the proof. 1 =⇒ 2. We put f (h) = 1 + p1 h + p2 h2 + · · · . If Hf is a Hopf subalgebra, for all n ≥ 1 there exists a scalar αn such that: (Z q ⊗ Id) ◦ ∆(xn+1 ) = αn xn . Considering the coefficient of (B + )n (1), we obtain: p1n−1 αn = 2(n − 1)p1n−1 p2 + p1n . Considering the coefficient of B + ( q n−1 ), we obtain: αn pn = (n + 1)pn+1 + npn p1 . p2 We put α = p1 and β = 2 − 1, then: p1 (1 − αβh)f 0 (h) = αf (h). Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
2. =⇒ 1. By the description of X : x1 =
q,
q
x2 = α q , qq 2 (1 + β) ∨ q + x3 = α 2 x4
x5
qq q ,
q q q qq (1 + 2β)(1 + β) qq q (1 + β) ∨q 3 ∨q + (1 + β) ∨q + q + = α 6 2 q (1+3β)(1+2β)(1+β) q q q q (1+2β)(1+β) q q q H∨ q + ∨q 24 2 q qq q q q q qq (1+β)2 ∨q q (1+2β)(1+β) ∨q q ∨ q ∨ q + + (1 + β) 4 + . 2 6 = α q q qq q qq ∨ q qq q q q q ∨ q q (1+β) (1+β) + 2 ∨q + (1 + β) q + 2 q + q Loïc Foissy
Combinatorial Dyson-Schwinger equations
qq ! qq ,
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
Particular cases If (α, β) = (1, −1), xn = (B + )n (1) for all n (ladder of degree n). If (α, β) = (1, 1), X xn = ]{embeddings of t in the plane}t. |t|=n
Si (α, β) = (1, 0), xn =
X |t|=n
1 t. ]{symmetries of t}
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
Hence, we have a family of Hopf subalgebras H(α,β) of HR indexed by (α, β). If α = 0, H(α,β) = K [ q ]. If α 6= 0, by the Cartier-Quillen-Milnor-Moore theorem, ∗ H(α,β) is an enveloping algebra. Its Lie algebra has a basis (Zi )i≥1 and for all i, j: [Zi , Zj ] = (β + 1)(j − i)Zi+j .
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
When is Hf a Hopf subalgebra? What are these Hopf algebras?
Theorem If α 6= 0 and β = −1, H(α,β) is isomorphic to the Hopf algebra of symmetric functions. If α 6= 0 and β 6= −1, H(α,β) is isomorphic to the Faà di Bruno Hopf algebra. In other words, H(α,β) is the coordinate ring of the group of formal diffeomorphisms of the line that are tangent to the identity: G = {f (h) = h + a1 h2 + . . . | a1 , a2 , . . . ∈ K }, ◦ .
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
In the case of Dyson-Schwinger systems, we have to use trees with decorated vertices. The combinatorial Dyson-Schwinger systems have the form: + X1 = B1 (f1 (X1 , . . . , Xn )) .. (S) : . Xn = Bn+ (fn (X1 , . . . , Xn )), where f1 , . . . , fn ∈ K [[h1 , . . . , hn ]] − K , with constant terms equal to 1. Such a system has a unique solution (X1 , . . . , Xn ) ∈ H\ {1,...,n} . The subalgebra generated by the homogeneous components of the Xi ’s is denoted by H(S) . If this subalgebra is Hopf, we shall say that the system is Hopf. Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Description of two families of Dyson-Schwinger systems: 1 2
Fundamental systems, Cyclic systems.
Four operations on Dyson-Schwinger systems: 1 2 3 4
Change of variables, Concatenation, Dilatation, Extension.
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Main theorem Let (S) be Hopf combinatorial Dyson-Schwinger system. Then (S) is obtained from the concatenation of fundamental or cyclic systems with the help of a change of variables, a dilatation and a finite number of extensions.
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Fundamental systems Let β1 , . . . , βk ∈ K . The following system is an example of a fundamental system: 1+βj n k Y Y − (1 − Xj )−1 Xi = Bi (1 − βi Xi ) (1 − βj Xj ) βj j=k +1 j=1 if i≤ k , 1+βj k n Y Y − Xi = Bi (1 − Xi ) (1 − βj Xj ) βj (1 − Xj )−1 j=1 j=k +1 if i > k .
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Cyclic systems The following systems are cyclic: if n ≥ 2, X1 = B1+ (1 + X2 ), X2 = B2+ (1 + X3 ), .. . Xn = Bn+ (1 + X1 ).
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Change of variables Let (S) be the following system: + X1 = B1 (f1 (X1 , . . . , Xn )) .. (S) : . Xn = Bn+ (fn (X1 , . . . , Xn )). If (S) is Hopf, then for all family (λ1 , . . . , λn ) of non-zero scalars, this system is Hopf: + X1 = B1 (f1 (λ1 X1 , . . . , λn Xn )) .. (S) : . Xn = Bn+ (fn (λ1 X1 , . . . , λn Xn )). Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Concatenation Let (S) and (S 0 ) be the following systems: + X1 = B1 (f1 (X1 , . . . , Xn )) .. (S) : . Xn = Bn+ (fn (X1 , . . . , Xn )). + X1 = B1 (g1 (X1 , . . . , Xm )) .. (S 0 ) : . + Xm = Bm (gm (X1 , . . . , Xm )).
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
Concatenation The following system is Hopf if, and only if, the (S) and (S 0 ) are Hopf: X1 = B1+ (f1 (X1 , . . . , Xn )) .. . Xn = Bn+ (fn (X1 , . . . , Xn )) + Xn+1 = Bn+1 (g1 (Xn+1 , . . . , Xn+m )) . .. + Xn+m = Bn+m (gm (Xn+1 , . . . , Xn+m )).
This property leads to the notion of connected (or indecomposable) system. Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
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Extension Let (S) be the following system: + X1 = B1 (f1 (X1 , . . . , Xn )) .. (S) : . Xn = Bn+ (fn (X1 , . . . , Xn )). Then (S 0 ) is an extension of (S): X1 = B1+ (f1 (X1 , . . . , Xn )) .. . 0 (S ) : X = Bn+ (fn (X1 , . . . , Xn )) n + Xn+1 = Bn+1 (1 + a1 X1 ). Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Iterated extensions (S) :
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
−1 X1 = B1 (1 − βX1 ) β , X2 = B2 (1 + X1 ), X3 = B3 (1 + X1 ), X4 = B4 (1 + 2X2 − X3 ), X5 = B5 (1 + X4 ).
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
Dilatation (S 0 ) is a dilatation of (S): ( (S) :
X1 X 2 X3 (S 0 ) : X4 X 5
Formulation with trees Obtained results Examples Operations on Dyson-Schwinger systems
X1 = B1+ (f (X1 , X2 )), X2 = B2+ (g(X1 , X2 )),
= B1+ (f (X1 + X2 + X3 , X4 + X5 )), = B2+ (f (X1 + X2 + X3 , X4 + X5 )), = B3+ (f (X1 + X2 + X3 , X4 + X5 )), = B4+ (g(X1 + X2 + X3 , X4 + X5 )), = B5+ (g(X1 + X2 + X3 , X4 + X5 )). Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
A single equation Systems
We now consider equations of the form: X Bj f (j) (X ) , (E) : X = j∈J
where: J is a set. For all j, Bj is a 1-cocyle of a certain degree. For all j, f (j) is a formal series such that f (j) (0) = 1.
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A single equation Systems
Lemma Let us assume that (E) is Hopf. If Bi and Bj have the same degree, then f (i) = f (j) have the same degree. This allows to assume that J ⊆ N∗ .
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Combinatorial Dyson-Schwinger equations
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A single equation Systems
Main theorem Under these hypothesis, if there is at most one constant f (j) , there exists λ, µ ∈ K such that: X +1 − λj µ B (1 − µX ) if µ 6= 0, j j∈J X (E) : X = Bj ejλX if µ = 0. j∈J
Example For λ = 1 and µ = −1, the following equation gives a Hopf subalgebra: X X = Bn (1 + X )n+1 . n≥1 Loïc Foissy
Combinatorial Dyson-Schwinger equations
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A single equation Systems
Main theorem If there are at least two constant f (j) , there exists α ∈ K , and m ∈ N such that: X X Bj (1). (E) : X = Bj (1 + αX ) + j∈J∩mN
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j∈J\mN
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
A single equation Systems
We now consider systems of the form : X + X1 = B1,i (f1,i (X1 , . . . , Xn )) i∈J 1 .. (S) : . X + X = Bn,i (fn,i (X1 , . . . , Xn )), n i∈Jn
where for all k , i, Bk ,i is a 1-cocycle of degree i.
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A single equation Systems
Theorem We assume that 1 ∈ Jk for all k . Then (S) is entirely determined by f1,1 , . . . , fn,1 .
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
A single equation Systems
Fundamental system Xi Xi
=
X
1+βj k n Y Y q − Bi,q (1 − βi Xi ) (1 − βj Xj ) βj (1 − Xj )−q
q∈Ji
j=1
j=k +1
if i ≤ k , 1+β k n X Y − β jq Y = Bi,q (1 − Xi ) (1 − βj Xj ) j (1 − Xj )−q q∈Ji
j=1
j=k +1
if i > k .
Loïc Foissy
Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
A single equation Systems
For example, we choose n = 3, k = 2, β1 = −1/3 β2 = 1, J1 = N∗ , J2 = J3 = {1}. After a change of variables h1 −→ 3h1 , we obtain: X (1 + X1 )1+2k X1 = B1,k , (1 − X2 )2k (1 − X3 )k k ≥1 (1 + X1 )2 (S) : X2 = B2 , (1 − X2 )(1 − X3 ) 2 X = B (1 + X1 ) . 3 3 (1 − X2 )
This is the example of the introduction, with X1 = X2 = −
, X3 = −
Loïc Foissy
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Combinatorial Dyson-Schwinger equations
Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles
A single equation Systems
Cyclic systems X X1 = B1,j 1 + X1+j , j∈I1 . . (S) : . X X = B 1 + X n,j n n+j . j∈I1
Loïc Foissy
Combinatorial Dyson-Schwinger equations