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.

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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












Examples in QED

,

,

,

,

,

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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

Propagators in QED






 =

X n≥1

  xn  

X n≥1

γ∈

  xn  

X n≥1

(n)

  xn  

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   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.

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Combinatorial Dyson-Schwinger equations

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Example In QED :

B

B

(

(

) =

1 2

) =

1 3

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+

+

1 2

1 3

+

1 3

Combinatorial Dyson-Schwinger equations

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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

  

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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

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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

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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

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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.

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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

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 ,

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Combinatorial Dyson-Schwinger equations

qq q q ...

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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).

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Combinatorial Dyson-Schwinger equations

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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.

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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

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 ).

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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

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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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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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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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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

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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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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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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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Cyclic systems The following systems are cyclic: if n ≥ 2,  X1 = B1+ (1 + X2 ),      X2 = B2+ (1 + X3 ), ..   .    Xn = Bn+ (1 + X1 ).

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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

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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 )).

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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

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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

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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 ).

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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

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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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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

Feynman graphs and Dyson-Schwinger equations Reformulation with trees Results Combinatorial Dyson-Schwinger systems Equations with several 1-cocycles

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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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

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 = −

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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