Combinatorial Dyson-Schwinger equations and systems I
Loïc Foissy
Les Houches May 2014
To a given QFT is attached a family of graphs.
Feynman 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 Feynman graphs.
In QED
1 Half-edges: (electron), (photon).
2 Vertices: .
3 External structures: , , .
Examples in QED
, , , , ,
, , ,
Subgraphs and contraction
1 A subgraph of a Feynman graphΓis a subsetγ of the set of half-edgesΓsuch thatγ and the vertices ofΓwith all half edges inγ is itself a Feynman graph.
2 IfΓis a Feynman graph andγ1, . . . , γk are disjoint
subgraphs ofΓ,Γ/γ1. . . γk is the Feynman graph obtained by contraction ofγ1, . . . , γk.
LetΓ1andΓ2be two Feynman graphs. According to the
external structure ofΓ1, you can replace a vertex or an edge of Γ2byΓ1in order to obtain a new Feynman graph.
Examples in QED
= ,
Construction
LetHFGbe a free commutative algebra generated by the set of Feynman graphs. It is given a coproduct: for all Feynman graph Γ,
∆(Γ) = X
γ1...γk⊆Γ
γ1. . . γk ⊗Γ/γ1. . . γk.
∆( ) = ⊗1+1⊗ + ⊗ .
The Hopf algebraHFGis 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 (HFG)0=K1HFG. What is its dual?
Cartier-Quillen-Milnor-Moore theorem
LetHbe a cocommutative, graded, connected Hopf algebra over a field of characteristic zero. Then it is the enveloping algebra of its primitive elements.
This theorem can be applied to the graded dual ofHFG. Primitive elements ofHFG∗
Basis of primitive elements: for any Feynman graphΓ, fΓ(γ1. . . γk) =]Aut(Γ)δγ1...γk,Γ.
The Lie bracket is given by:
[fΓ1,fΓ2] = X
Γ=Γ1Γ2
fΓ− X
Γ=Γ2Γ1
fΓ.
We define:
fΓ1◦fΓ2 = X
Γ=Γ1Γ2
fΓ.
The product◦is not associative, but satisfies:
f1◦(f2◦f3)−(f1◦f2)◦f3=f2◦(f1◦f3)−(f2◦f1)◦f3.
It is (left) prelie.
In the context of QFT, we shall consider some special infinite sums of Feynman graphs:
Example in QED
=X
n≥1
xn
X
γ∈ (n)
sγγ
.
=−X
n≥1
xn
X
γ∈ (n)
sγγ
.
Example in QED
=−X
n≥1
xn
X
γ∈ (n)
sγγ
.
They live in the completion ofHFG.
How to describe these formal series?
For any primitive Feynman graphγ, one defines the
insertion operatorBγ overHFG. This operator associates to a graphGthe sum (with symmetry coefficients) of the insertions ofGintoγ.
The propagators then satisfy a system of equations involving the insertion operators, called systems of Dyson-Schwinger equations.
Example In QED :
B ( ) = 1
2 +1
2
B ( ) = 1
3 + 1
3 +1
3
In QED:
= X
γ
x|γ|Bγ
1+
1+2|γ|
1+
2|γ|
1+
|γ|
= −xB
1+
2
1+
2
= −xB
1+
2
1+ 1+
Other example (Bergbauer, Kreimer)
X = X
γprimitive Bγ
(1+X)|γ|+1
.
Question
For a given system of Dyson-Schwinger equations(S), is the subalgebra generated by the homogeneous components of(S) a Hopf subalgebra?
Proposition
The operatorsBγsatisfy: for allx ∈HFG,
∆◦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.
The Hopf algebra of rooted treesHR (or Connes-Kreimer Hopf algebra) is the free commutative algebra generated by the set of rooted trees.
q, qq, ∨qqq,qqq
, ∨qqqq, ∨qqq q
, ∨qqqq ,qqqq
,Hq ∨qqq q, ∨qqqq q
, ∨qqq q q
, ∨q∨qq qq , ∨qqq
qq , ∨qqqqq
, ∨qqqq q
,∨q qqqq , qqqqq
, . . . The set of rooted forests is a linear basis ofHR:
1,q,q q, qq,q q q, qq q, ∨qqq,qqq
,q q q, qq q q, qq qq, ∨qqq q,qqq
q, ∨qqqq, ∨qqq q
, ∨qqqq ,qqqq
. . .
∆(t) = X cadmissible cut
Pc(t)⊗Rc(t).
cutc ∨qqq q
q
∨q qq
q
∨q q q
q
∨q qq
q
∨q qq
q
∨q q q
q
∨q qq
q
∨q q q
total Admissible ? yes yes yes yes no yes yes no yes
Wc(t) ∨qqq q
qq qq q q∨q q qqqq q q qq qq q q qq q q q q q q ∨qqq
q Rc(t) ∨qqq
q
qq ∨qqq qqq × q qq × 1
Pc(t) 1 qq q q × qq q q q × ∨qqq q
The grafting operator ofHR is the mapB:HR −→HR, associating to a forestt1. . .tnthe tree obtained by grafting t1, . . . ,tn on a common root. For example:
B(qq q) = ∨qqq q
.
Proposition For allx ∈HR:
∆◦B(x) =B(x)⊗1+ (Id⊗B)◦∆(x).
SoBis a 1-cocycle ofHR.
Universal property
LetAbe a commutative Hopf algebra and letL:A−→Abe a 1-cocycle ofA. Then there exists a unique Hopf algebra morphismφ:HR−→Awithφ◦B =L◦φ.
This will be generalized to the case of several 1-cocycles with the help of decorated rooted trees.
HR is graded by the number of vertices andBis homogeneous of degree 1.
LetY =Bγ(f(Y))be a Dyson-Schwinger equation in a suitable Hopf algebra of Feynman graphsHFG, such that
|γ|=1.
There exists a Hopf algebra morphismφ:HR −→HFG, such thatφ◦B =Bγ◦φ. This morphism is homogeneous of degree 0.
LetX be the solution ofX =B(f(X)). Thenφ(X) =Y and for alln≥1,φ(X(n)) =Y(n).
Consequently, if the subalgebra generated by theX(n)’s is Hopf, so is the subalgebra generated by theY(n)’s.
Letf(h)∈C[[h]].
The combinatorial Dyson-Schwinger equations associated tof(h)is:
X =B(f(X)), whereX lives in the completion ofHR. This equation has a unique solutionX =P
X(n), with:
X(1) = p0q, X(n+1) =
n
X X
a +...+a =n
pkB(X(a1). . .X(ak)),
X(1) = p0q, X(2) = p0p1qq, X(3) = p0p21qqq
+p02p2 ∨qqq, X(4) = p0p31qqqq
+p02p1p2 ∨qqqq
+2p20p1p2 ∨qqq q
+p03p3 ∨qqqq.
Examples
Iff(h) =1+h:
X = q+ qq + qqq + qqqq
+ qqqqq +· · ·
Iff(h) = (1−h)−1:
X = q+ qq + ∨qqq + qqq
+ ∨qqqq +2 ∨qqq q
+ ∨qqqq + qqqq q
q q q qqq q
q q q
q ∨qqqq qq q
q q
∨qqq ∨qqq
q ∨q qqq qqq
Letf(h)∈C[[h]]. The homogeneous components of the unique solution of the combinatorial Dyson-Schwinger equation
associated tof(h)generate a subalgebra ofHRdenoted byHf. Hf is not always a Hopf subalgebra
For example, forf(h) =1+h+h2+2h3+· · ·, then:
X = q+ qq + ∨qqq + qqq
+2 ∨qqqq +2 ∨qqq q
+ ∨qqqq + qqqq
+· · ·
So:
∆(X(4)) = X(4)⊗1+1⊗X(4) + (10X(1)2+3X(2))⊗X(2) +(X(1)3+2X(1)X(2) +X(3))⊗X(1)
+X(1)⊗(8 ∨qqq +5qqq ).
Iff(0) =0, the unique solution ofX =B(f(X))is 0. From now, up to a normalization we shall assume thatf(0) =1.
Theorem
Letf(h)∈C[[h]], withf(0) =1. The following assertions are equivalent:
1 Hf is a Hopf subalgebra ofHR.
2 There exists(α, β)∈C2such that(1−αβh)f0(h) =αf(h).
3 There exists(α, β)∈C2such thatf(h) =1 ifα=0 or f(h) =eαhifβ =0 orf(h) = (1−αβh)−β1 ifαβ6=0.
1=⇒2. We putf(h) =1+p1h+p2h2+· · ·. ThenX(1) = q. Let us write:
∆(X(n+1)) =X(n+1)⊗1+1⊗X(n+1) +X(1)⊗Y(n) +. . . .
1 By definition of the coproduct,Y(n)is obtained by cutting a leaf in all possible ways inX(n+1). So it is a linear span of trees of degreen.
2 AsHf is a Hopf subalgebra,Y(n)belongs toHf. Hence, there exists a scalarλnsuch thatY(n) =λnXn.
lemma Let us write:
X =X
t
att.
For any rooted treet:
λ|t|at =X
t0
n(t,t0)at0,
wheren(t,t0)is the number of leaves oft0 such that the cut of this leaf givest.
We here assume thatf is not constant. We can prove that p16=0.
Fort the ladder(B)n(1), we obtain:
pn−11 λn=2(n−1)pn−21 p2+pn1. Hence:
λn =2p2
p1(n−1) +p1. We putα=p1andβ=2p2
p21 −1, then:
λn=α(1+ (n−1)(1+β)).
Fort the corollaB(qn−1), we obtain:
λnpn−1=npn+ (n−1)pn−1p1. Hence:
α(1+ (n−1)β)pn−1=npn. Summing:
(1−αβh)f0(h) =αf(h).
X(1) = q, X(2) = αqq, X(3) = α2
(1+β)
2 ∨qqq + qqq ,
X(4) = α3 (1+2β)(1+β)
6 ∨qqqq + (1+β) ∨qqq q
+ (1+β) 2
q
∨qqq + qqqq !
,
X(5) = α4
(1+3β)(1+2β)(1+β)
24 Hq ∨qqq q+(1+2β)(1+β) 2 ∨qqqq
q
+(1+β)2 2 ∨q∨qqqq
+ (1+β) ∨qqq qq
+(1+2β)(1+β) 6
q
∨q qqq
+(1+β)2 ∨qqq q q
+ (1+β) ∨qqqq q
+(1+β)2 ∨q qqqq + qqqqq
.
Particular cases
If(α, β) = (1,−1),f =1+handX(n) = (B)n(1)for alln.
If(α, β) = (1,1),f = (1−h)−1and:
X(n) = X
|t|=n
]{embeddings oftin the plane}t.
Si(α, β) = (1,0),f =ehand:
X(n) = X
|t|=n
1
]{symmetries oft}t.
(Left) prelie algebra
A prelie algebragis a vector space with a linear product◦such that for allx,y,z∈g:
x◦(y ◦z)−(x◦y)◦z=y◦(x ◦z)−(y◦x)◦z.
Associated Lie bracket
If◦is a prelie product ong, its antisymmetrization is a Lie bracket.
Primitive elements of the dual ofHR For any rooted treet let us define:
ft :
HR −→ C F −→ stδF,t.
The family(ft)is a basis of the primitive elements ofHR∗. The Lie bracket is given by:
[ft1,ft2] = X
t0=t1t2
ft0− X
t0=t2t1
ft0.
q q q
We define:
ft1◦ft2 = X
t0=t1t2
ft0.
This product is prelie.
Theorem (Chapoton-Livernet)
As a prelie algebra,Prim(HR∗)is freely generated byfq.
Faà di Bruno prelie algebra
gFdB has a basis(ei)i≥1, and the prelie product is defined by:
ei◦ej = (j+λ)ei+j. For alli,j,k ≥1:
ei◦(ej◦ek)−(ei◦ej)◦ek =k(k+λ)ei+j+k.
Theorem
Ifβ 6=−1 andα=1,
∆(X) =X⊗1+
∞
X
j=1
(1+λX)1+λj ⊗X(j),
withλ= −1 1+β. Ifβ =−1 andα=1,
∆(X) =1⊗X+X⊗1+X⊗X.
Corollary
Ilα6=0, the prelie algebra of the primitive elements of the dual of the Hopf algebra generated by theX(i)’s has a basis(ei)i≥1.
Ifβ 6=−1,ei◦ej = (λ+j)ei+j (Faà di Bruno case).
Ifβ =−1,ei◦ej =ei+j (symmetric case).
In QFT, generally Dyson-Schwinger equations involve several 1-cocycles, for example [Bergbauer-Kreimer]:
X =
∞
X
n=1
Bn((1+X)n+1),
whereBnis the insertion operator into a primitive Feynman graph withnloops.
LetIbe a set. Set of rooted trees decorated byI:
qa,a∈I; qqab,(a,b)∈I2; b∨qqaqc =c∨qqaqb,qqq
ab c
,(a,b,c)∈I3;
q
∨qqaqd
c
b = ∨qqqaqc
d
b =. . .= ∨qqqaqb
c d , ∨qqq
q
a d b c
= ∨qqq q
a b d
c
, ∨qqqq
a b d c
= ∨qqqq
a b c d
, qqqq
ab c d
,(a,b,c,d)∈I4. The Connes-Kreimer construction is extended to obtain the
Hopf algebraHRI.
∆( ∨qqq q
d c b a
) = ∨qqq q
d c b a
⊗1+1⊗ ∨qqq q
d c b a
+ qqba ⊗ qqdc + qa ⊗ ∨qqdq c b
+q ⊗ qqq
b a
+ qqa q ⊗ q + q q ⊗ qqb.
For alld ∈I, there is a grafting operatorBd :HRI −→HRI . For example, ifa,b,c,d ∈I:
Ba(qqbc qd) = ∨qqq q
a d b c
.
Proposition
For alla∈I,x ∈HRI:
∆◦Ba(x) =Ba(x)⊗1+ (Id⊗Ba)◦∆(x).
Universal property
LetAbe a commutative Hopf algebra and for alla∈I, let La:A−→Asuch that for allx ∈A:
∆◦La(x) =La(x)⊗1+ (Id⊗La)◦∆(x).
Then there exists a unique Hopf algebra morphism φ:HRI −→Awithφ◦Ba=La◦φfor alla∈A.
Definitions
LetIbe a graded set and letfi(h)∈C[[h]]for alli ∈I.
The combinatorial Dyson-Schwinger equations associated to(fi(h))i∈I is:
X =X
i∈I
Bi(fi(X)),
whereX lives in the completion ofHRI . This equation has a unique solutionX =P
X(n).
The subalgebra ofHRI generated by theX(n)’s is denoted byH(f).
We shall say that the equation is Hopf ifH(f)is a Hopf subalgebra.
Lemma
Let us assume that the equation associated to(f)is Hopf. If fi(0) =0, thenfi =0.
We now assume thatfi(0) =1 for alli ∈I.
Lemma
Let us assume that the equation associated to(f)is Hopf. If i,j∈Ihave the same degree, thenfi =fj.
Grouping 1-cocycles by degrees, we now assume thatI⊆N∗.
Let us choosei ∈I. We restrict our solution toi, that is to say we delete any tree with a decoration which is not equal toi.
The obtained elementX0 is solution of:
X0 =Bi(fi(X0)),
and this equation is Hopf. By the study of equations with only one 1-cocycle:
Lemma
For alli ∈I, there existsαi, βi ∈Csuch that :
fi =
( eαihifβi =0, (1−αiβih)−1/βi ifβi 6=0.
One of the following assertions holds:
1 there existsλ, µ∈Csuch that, if we put:
Q(h) = (
(1−µh)−λµ ifµ6=0, eλh ifµ=0,
then:
(E) :x =X
i∈I
Bj
(1−µx)Q(x)i .
2 There existsm≥0 andα∈C− {0}such that:
Theorem
For allλ, µ∈C, the algebra generated by the components of the solution of the Dyson-Schwinger equation of the first type is a Hopf subalgebra.
Corollary
Ifµ6=−1 andλ=1+µ,
∆(X) =X ⊗1+
∞
X
j=1
(1+λ0X)1+λj0 ⊗X(j),
withλ0= −1 1+µ.
we assume that 1∈I.
Theorem
X =X
i∈I m|j
Bi(1+αX) +X
i∈I m/|i
Bi(1),
withα∈C− {0}. The dual ofH(f) is the enveloping algebra of a pre-Lie algebrag, such that:
ghas a basis(fi)i≥1. For alli,j ≥1:
0 ifm/| j,
