combinatorial dyson-schwinger equations

Feynman graphs and Dyson-Schwinger equationsReformulation with trees

ResultsCombinatorial Dyson-Schwinger systems

Equations with several 1-cocycles

Combinatorial Dyson-Schwinger equations

Loïc Foissy

BerlinFebruary 2012

Loïc Foissy Combinatorial Dyson-Schwinger equations




Feynman graphsDyson-Schwinger equationsHopf algebras axioms

Feynman graphs1 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 theFeynman graphs, called the propagator.






Examples in QED

�,�,�,�,�,

�,�

,�

,�Loïc Foissy Combinatorial Dyson-Schwinger equations





Propagators in QED

� =∑n≥1

xn

∑γ∈� (n)

sγγ

.

�= −

∑n≥1

xn

∑γ∈�

(n)

sγγ

.

�= −

∑n≥1

xn

∑γ∈�

(n)

sγγ

.






How to describe the propagators?

The algebra HDF is the free commutative algebra generated bythe Feynman graphs of a given QFT.

For any primitive Feynman graph γ, one defines the insertionoperator Bγ over HDF . This operators associates to a graph Gthe sum (with symmetry coefficients) of the insertions of G intoγ.

The propagators then satisfy a system of equations involvingthe insertion operators, called systems of Dyson-Schwingerequations.






ExampleIn QED :

B

�(�

) =12�+

12�

B

�(�) =

13�+

13�+

13�






Then:

� =∑γ∈Γ

x |γ|Bγ

(

1 +�)1+2|γ|

(1 +�

)|γ| (1 +�

)2|γ|

�= −xB

�

(

1 +�)2

(1 +�

)2

�= −xB

�

(

1 +�)2

(1 +�

)(1 +�

)






Let A be a vector space. The tensor square of A is a spaceA⊗ A with a bilinear product ⊗ : A× A −→ A⊗ A with auniversal 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) productbecomes a linear map m : A⊗ A −→ A, sending ei ⊗ ejover ei .ej . The associativity is given by the followingcommuting square:

A⊗ A⊗ Am⊗Id //

Id⊗m��

A⊗ A

m��

A⊗ A m// A






Dualizing the associativity axiom, we obtain thecoassociativity axiom: a coalgebra is a vector space C witha map ∆ : C −→ C ⊗ C such that:

C∆ //

∆��

C ⊗ C

Id⊗∆��

C ⊗ C∆⊗Id

// C ⊗ C ⊗ C

A Hopf algebra is both an algebra and a coalgebra, withthe compatibility:

∆(xy) = ∆(x)∆(y).

(And a technical condition of existence of an antipode).






Examples

If G is a group, KG is a Hopf algebra, with ∆(x) = x ⊗ x forall x ∈ G.If g is a Lie algebra, its enveloping algebra is a Hopfalgebra, with ∆(x) = x ⊗ 1 + 1⊗ x for all x ∈ g.The algebra of Feynman graphs HDF is a graded Hopfalgebra. For example:

∆(�) =�⊗1+1⊗�+�

⊗�.






QuestionFor a given system of Dyson-Schwinger equations (S), is thesubalgebra generated by the homogeneous components of (S)a Hopf subalgebra?





IntroductionHopf algebra of rooted treesCombinatorial Dyson-Schwinger equations

PropositionThe operators Bγ satisfy: for all x ∈ HDF ,

∆ ◦ Bγ(x) = Bγ(x)⊗ 1 + (Id ⊗ Bγ) ◦∆(x).

This relation allows to lift any system of Dyson-Schwingerequation to the Hopf algebra of decorated rooted trees.

We first treat the case of a single equation with a single1-cocycle.






The Hopf algebra of rooted trees HR (or Connes-Kreimer Hopfalgebra) is the free commutative algebra generated by the setof rooted trees. The set of rooted forests is a linear basis of HR:

1, q , q q , qq , q q q , qq q , q∨qq, qqq , q q q , qq q q , qq qq , q∨qq q , qqq q , q∨qq q

, q∨qqq,

q∨qq q , qqqq . . .






The coproduct is given by admissible cuts:

∆(t) =∑

c admissible cutPc(t)⊗ Rc(t).

cutc q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqqtotal

Admissible ? yes yes yes yes no yes yes no yes

W c(t) q∨qqq qq qq q q∨qq qqq q q q qq qq q q qq q q q q q q q∨qqqRc(t) q∨qqq qq q∨qq qqq × q qq × 1

Pc(t) 1 qq q q × qq q q q × q∨qqq∆( q∨qqq

) = q∨qqq⊗1+1⊗ q∨qqq

+ qq ⊗ qq + q⊗ q∨qq+ q⊗ qqq + qq q⊗ q + q q⊗ qq .






The grafting operator of HR is the map B+ : HR −→ HR,associating to a forest t1 . . . tn the tree obtained by graftingt1, . . . , tn on a common root. For example:

B+( qq q) = q∨qqq.

PropositionFor all x ∈ HR:

∆ ◦ B+(x) = B+(x)⊗ 1 + (Id ⊗ B+) ◦∆(x).






Universal propertyLet A be a commutative Hopf algebra and let L : A −→ A suchthat for all a ∈ A:

∆ ◦ L(a) = L(a)⊗ 1 + (Id ⊗ L) ◦∆(a).

Then there exists a unique morphism Hopf algebra morphismφ : HR −→ A with φ ◦ B+ = L ◦ φ.






Let f (h) ∈ K [[h]]. The combinatorial Dyson-Schwingerequations associated to f (h) is:

X = B+(f (X )),

where X lives in the completion of HR.

This equation has a unique solution X =∑

xn, with:x1 = p0 q ,

xn+1 =n∑

k=1

∑a1+...+ak=n

pkB+(xa1 . . . xak ),

where f (h) = p0 + p1h + p2h2 + . . .






Examples

If f (h) = 1 + h:

X = q + qq + qqq + qqqq + qqqqq + · · ·

If f (h) = (1− h)−1:

X = q + qq + q∨qq+ qqq + q∨qq q

+ 2 q∨qqq+

q∨qq q + qqqq+ q∨qq

�Hq q+ 3 q∨qq qq

+ q∨qq qq+ 2 q∨qq∨qq

+ 2 q∨qqqq+

q∨qq qq + 2 q∨qq qq + qqq∨q q

+ qqqqq + · · ·






Let f (h) ∈ K [[h]]. The homogeneous components of the uniquesolution of the combinatorial Dyson-Schwinger equationassociated 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:

X = q + qq + q∨qq+ qqq + 2 q∨qq q

+ 2 q∨qqq+

q∨qq q + qqqq + · · ·

So:

∆(x4) = x4 ⊗ 1 + 1⊗ x4 + (10x21 + 3x2)⊗ x2

+(x31 + 2x1x2 + x3)⊗ x1 + x1 ⊗ (8 q∨qq

+ 5 qqq).Loïc Foissy Combinatorial Dyson-Schwinger equations




When is Hf a Hopf subalgebra?What are these Hopf algebras?

TheoremLet f (h) ∈ K [[h]], with f (0) = 1. The following assertions areequivalent:

1 Hf is a Hopf subalgebra of HR.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) = eαh if β = 0 or f (h) = (1− αβh)−1β if αβ 6= 0.






Idea of the proof.

1 =⇒ 2. We put f (h) = 1 + p1h + p2h2 + · · · . If Hf is a Hopfsubalgebra, for all n ≥ 1 there exists a scalar αn such that:

(Z q ⊗ Id) ◦∆(xn+1) = αnxn.

Considering the coefficient of (B+)n(1), we obtain:

pn−11 αn = 2(n − 1)pn−1

1 p2 + pn1 .

Considering the coefficient of B+( qn−1), we obtain:

αnpn = (n + 1)pn+1 + npnp1.

We put α = p1 and β = 2p2

p1− 1, then:

(1− αβh)f ′(h) = αf (h).






2. =⇒ 1. By the description of X :

x1 = q ,x2 = α qq ,x3 = α2

((1 + β)

2q∨qq

+ qqq) ,

x4 = α3

((1 + 2β)(1 + β)

6q∨qq q

+ (1 + β) q∨qqq+

(1 + β)

2q∨qq q + qqqq ) ,

x5 = α4

(1+3β)(1+2β)(1+β)24 q∨qq

�Hq q+ (1+2β)(1+β)

2 q∨qq qq+ (1+β)2

2 q∨qq∨qq+ (1 + β) q∨qqqq

+ (1+2β)(1+β)6

q∨qq qq

+ (1+β)2 q∨qq qq

+ (1 + β)q∨qq qq + (1+β)

2 qqq∨q q

+ qqqqq

.






Particular casesIf (α, β) = (1,−1), xn = (B+)n(1) for all n (ladder of degreen).If (α, β) = (1, 1),

xn =∑|t |=n

]{embeddings of t in the plane}t .

Si (α, β) = (1, 0),

xn =∑|t |=n

1]{symmetries of t}

t .






Hence, we have a family of Hopf subalgebras H(α,β) of HRindexed 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 .






TheoremIf α 6= 0 and β = −1, H(α,β) is isomorphic to the Hopfalgebra of symmetric functions.

If α 6= 0 and β 6= −1, H(α,β) is isomorphic to the Faà diBruno Hopf algebra. In other words, H(α,β) is thecoordinate ring of the group of formal diffeomorphisms ofthe line that are tangent to the identity:

G =({f (h) = h + a1h2 + . . . | a1, a2, . . . ∈ K}, ◦

).





Formulation with treesObtained resultsExamplesOperations on Dyson-Schwinger systems

In the case of Dyson-Schwinger systems, we have to use treeswith decorated vertices. The combinatorial Dyson-Schwingersystems have the form:

(S) :

X1 = B+

1 (f1(X1, . . . , Xn))...

Xn = B+n (fn(X1, . . . , Xn)),

where f1, . . . , fn ∈ K [[h1, . . . , hn]]− K , with constant terms equalto 1. Such a system has a unique solution(X1, . . . , Xn) ∈ H{1,...,n}. The subalgebra generated by thehomogeneous components of the Xi ’s is denoted by H(S).

If this subalgebra is Hopf, we shall say that the system is Hopf.






Description of two families of Dyson-Schwinger systems:1 Fundamental systems,2 Cyclic systems.

Four operations on Dyson-Schwinger systems:1 Change of variables,2 Concatenation,3 Dilatation,4 Extension.






Main theoremLet (S) be Hopf combinatorial Dyson-Schwinger system. Then(S) is obtained from the concatenation of fundamental or cyclicsystems with the help of a change of variables, a dilatation anda finite number of extensions.






Fundamental systemsLet β1, . . . , βk ∈ K . The following system is an example of afundamental system:

Xi = Bi

(1− βiXi)k∏

j=1

(1− βjXj)−

1+βjβj

n∏j=k+1

(1− Xj)−1

if i ≤ k ,

Xi = Bi

(1− Xi)k∏

j=1

(1− βjXj)−

1+βjβj

n∏j=k+1

(1− Xj)−1

if i > k .






Cyclic systemsThe following systems are cyclic: if n ≥ 2,

X1 = B+1 (1 + X2),

X2 = B+2 (1 + X3),

...Xn = B+

n (1 + X1).






Change of variables

Let (S) be the following system:

(S) :

X1 = B+

1 (f1(X1, . . . , Xn))...

Xn = B+n (fn(X1, . . . , Xn)).

If (S) is Hopf, then for all family (λ1, . . . , λn) of non-zero scalars,this system is Hopf:

(S) :

X1 = B+

1 (f1(λ1X1, . . . , λnXn))...

Xn = B+n (fn(λ1X1, . . . , λnXn)).






ConcatenationLet (S) and (S′) be the following systems:

(S) :

X1 = B+

1 (f1(X1, . . . , Xn))...

Xn = B+n (fn(X1, . . . , Xn)).

(S′) :

X1 = B+

1 (g1(X1, . . . , Xm))...

Xm = B+m(gm(X1, . . . , Xm)).






ConcatenationThe following system is Hopf if, and only if, the (S) and (S′) areHopf:

X1 = B+1 (f1(X1, . . . , Xn))

...Xn = B+

n (fn(X1, . . . , Xn))

Xn+1 = B+n+1(g1(Xn+1, . . . , Xn+m))

...Xn+m = B+

n+m(gm(Xn+1, . . . , Xn+m)).

This property leads to the notion of connected (orindecomposable) system.






ExtensionLet (S) be the following system:

(S) :

X1 = B+

1 (f1(X1, . . . , Xn))...

Xn = B+n (fn(X1, . . . , Xn)).

Then (S′) is an extension of (S):

(S′) :

X1 = B+

1 (f1(X1, . . . , Xn))...

Xn = B+n (fn(X1, . . . , Xn))

Xn+1 = B+n+1(1 + a1X1).






Iterated extensions

(S) :

X1 = B1

((1− βX1)

− 1β

),

X2 = B2(1 + X1),

X3 = B3(1 + X1),

X4 = B4(1 + 2X2 − X3),

X5 = B5(1 + X4).






Dilatation(S′) is a dilatation of (S):

(S) :

{X1 = B+

1 (f (X1, X2)),

X2 = B+2 (g(X1, X2)),

(S′) :

X1 = B+1 (f (X1 + X2 + X3, X4 + X5)),

X2 = B+2 (f (X1 + X2 + X3, X4 + X5)),

X3 = B+3 (f (X1 + X2 + X3, X4 + X5)),

X4 = B+4 (g(X1 + X2 + X3, X4 + X5)),

X5 = B+5 (g(X1 + X2 + X3, X4 + X5)).





A single equationSystems

We now consider equations of the form:

(E) : X =∑j∈J

Bj

(f (j)(X )

),

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.






LemmaLet us assume that (E) is Hopf. If Bi and Bj have the samedegree, then f (i) = f (j) have the same degree.

This allows to assume that J ⊆ N∗.






Main theorem

Under these hypothesis, if there is at most one constant f (j),there exists λ, µ ∈ K such that:

(E) : X =

∑j∈J

Bj

((1− µX )−

λjµ

+1)

if µ 6= 0,∑j∈J

Bj

(ejλX

)if µ = 0.

ExampleFor λ = 1 and µ = −1, the following equation gives a Hopfsubalgebra:

X =∑n≥1

Bn

((1 + X )n+1

).






Main theorem

If there are at least two constant f (j), there exists α ∈ K , andm ∈ N such that:

(E) : X =∑

j∈J∩mNBj(1 + αX ) +

∑j∈J\mN

Bj(1).






We now consider systems of the form :

(S) :

X1 =∑i∈J1

B+1,i(f1,i(X1, . . . , Xn))

...Xn =

∑i∈Jn

B+n,i(fn,i(X1, . . . , Xn)),

where for all k , i , Bk ,i is a 1-cocycle of degree i .






TheoremWe assume that 1 ∈ Jk for all k . Then (S) is entirelydetermined by f1,1, . . . , fn,1.






Fundamental system

Xi =∑q∈Ji

Bi,q

(1− βiXi)k∏

j=1

(1− βjXj)−

1+βjβj

qn∏

j=k+1

(1− Xj)−q

if i ≤ k ,

Xi =∑q∈Ji

Bi,q

(1− Xi)k∏

j=1

(1− βjXj)−

1+βjβj

qn∏

j=k+1

(1− Xj)−q

if i > k .






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:

(S) :

X1 =∑k≥1

B1,k

((1 + X1)

1+2k

(1− X2)2k (1− X3)k

),

X2 = B2

((1 + X1)

2

(1− X2)(1− X3)

),

X3 = B3

((1 + X1)

2

(1− X2)

).

This is the example of the introduction, with X1 =� ,

X2 = −�

, X3 = −�

.






Cyclic systems

(S) :

X1 =∑j∈I1

B1,j

(1 + X1+j

),

...Xn =

∑j∈I1

Bn,j

(1 + Xn+j

).


combinatorial dyson-schwinger equations

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