Combinatorial Dyson-Schwinger equations and systems I

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Combinatorial Dyson-Schwinger equations and systems I

Loïc Foissy

Bertinoro September 2013

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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 Feynman graphs.

Attach to any Feynman graph a real/complex number:

Feynman rules and Renormalization.

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The expansion as a formal series gives formal sums of Feynman graphs: the propagators (vertex functions, two-points functions).

These formal sums are characterized by a set of equations: the Dyson-Schwinger equations.

In order to be "physically meaningful", these functions should be compatible with the extraction/contraction Hopf algebra structure on Feynman graphs. This imposes strong constraints on the Dyson-Schwinger equations.

Because of a 1-cocycle property, everything can be lifted and studied to the level of decorated rooted trees.

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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.

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In QED

1 Half-edges:

(electron),

^(photon).

2 Vertices:

^.

3 External structures:

^,

^.

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Examples in QED

^,

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Other examples Φ³.

Quantum Chromodynamics.

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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, . . . , γ_k are disjoint

subgraphs ofΓ,Γ/γ₁. . . γ_k is the Feynman graph obtained by replacingγ1, . . . , γk by vertices inΓ.

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Insertion

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

⁼⁼

^,

^.

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LetAandBbe two vector spaces.

The tensor product ofAandBis a spaceA⊗Bwith a bilinear product⊗:A×B−→A⊗Bsatisfying a universal property: iff :A×B−→Cis bilinear, there exists a unique linear mapF :A⊗B−→Csuch thatF(a⊗b) =f(a,b)for all(a,b)∈A×B.

If(e_i)i∈I is a basis ofAand(f_j)j∈J is a basis ofB, then (e_i⊗f_j)i∈I,j∈J is a basisA⊗B.

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The tensor product of vector spaces is associative:

(A⊗B)⊗C=A⊗(B⊗C).

We shall identifyK ⊗A,A⊗K andAvia the identification of 1⊗a,a⊗1 anda.

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IfAis an associative algebra, its (bilinear) product becomes a linear mapm:A⊗A−→A, sendinga⊗bonab. The

associativity is given by the following commuting square:

A⊗A⊗A^m⊗Id ^//

Id⊗m

A⊗A

m

A⊗A _m ^//A

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IfAis unitary, its unit 1_Ainduces a linear map η:

K −→ A λ −→ λ1_A.

The unit axiom becomes:

K ⊗A ^η⊗Id ^//

%%J

JJ JJ JJ JJ

J A⊗A

m

A⊗K

ooId⊗η

yytttttttttt

A

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Dualizing these diagrams, we obtain the coalgebra axioms Coalgebra

A coalgebra is a vector spaceCwith a map∆ :C −→C⊗C such that:

C ^∆ ^//

∆

C⊗C

Id⊗∆

C⊗C

∆⊗Id//C⊗C⊗C

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Coalgebra

There exists a mapε:C−→K, called the counit, such that:

K ⊗C

%%K

KK KK KK KK

K^oo^ε⊗Id C⊗C ^Id^⊗ε^//C⊗K

yyssssssssss

C

∆

OO

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IfAis an algebra, thenA⊗Ais an algebra, with:

(a₁⊗b₁).(a₂⊗b₂) = (a₁.a₂)⊗(b₁.b₂).

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.

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Examples

IfGis a group,KGis a Hopf algebra, with∆(x) =x ⊗x for allx ∈G.

Ifgis a Lie algebra, its enveloping algebra is a Hopf algebra, with∆(x) =x ⊗1+1⊗x for allx ∈g.

IfH is a finite-dimensional Hopf algebra, then its dual is also a Hopf algebra.

IfHis a graded Hopf algebra, then its graded dual is also a Hopf algebra.

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Construction

LetH_FGbe 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. . . γ_k ⊗Γ/γ₁. . . γ_k.

∆(

^{) =}

^⊗1^+1⊗

⁺

^⊗

^.

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The Hopf algebraH_FGis 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=K1_H_FG. 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.

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This theorem can be applied to the graded dual ofH_FG. Primitive elements ofH_FG^∗

Basis of primitive elements: for any Feynman graphΓ, f_Γ(γ₁. . . γ_k) =]Aut(Γ)δ_γ₁_...γ_k_,Γ.

The Lie bracket is given by:

[f_Γ₁,f_Γ₂] = X

Γ=Γ1Γ2

f_Γ− X

Γ=Γ2Γ1

f_Γ.

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We define:

f_Γ₁◦f_Γ₂ = X

Γ=Γ1Γ2

f_Γ.

The product◦is not associative, but satisfies:

f₁◦(f₂◦f₃)−(f₁◦f₂)◦f₃=f₂◦(f₁◦f₃)−(f₂◦f₁)◦f₃.

It is (left) prelie.

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In the context of QFT, we shall consider some special infinite sums of Feynman graphs:

Propagators in QED

⁼^X^n≥1^xⁿ^^^^^^γ∈

^X ⁽ⁿ⁾^s^γ^γ^^^^^^.

⁼⁻^X^n≥1^xⁿ^^^^^^γ∈

^X ⁽ⁿ⁾^s^γ^γ^^^^^^.

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Propagators in QED

⁼⁻^X^n≥1^xⁿ^^^^^^γ∈

^X ⁽ⁿ⁾^s^γ^γ^^^^^^.

They live in the completion ofH_FG.

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How to describe the propagators?

For any primitive Feynman graphγ, one defines the

insertion operatorB_γ overH_FG. 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.

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Example In QED :

B_B

⁽

⁾⁾ ⁼⁼ ¹²¹³

⁺⁺¹³¹²

⁺¹³

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Results

In QED:

⁼ ^X^γ ^x^|γ|^B^γ^^^¹⁺

¹⁺

^|γ|¹⁺^1+2|γ|

^2|γ|^^^

⁼ ^−xB

^^^¹¹⁺⁺

²²^^^

⁼ ^−xB

^^^¹⁺

¹⁺

² ^^^

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Other example (Bergbauer, Kreimer)

X = X

γprimitive B_γ

(1+X)^|γ|+1

.

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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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Proposition

The operatorsB_γsatisfy: for allx ∈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.

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Cartier-Quillen cohomology

letC be a coalgebra and let(B, δ_G, δ_D)be aC-bicomodule.

D_n=L(B,C^⊗n).

For alll∈D_n: b_n(L) =

n

X

i=1

(−1)ⁱ(Id^⊗(i−1)⊗∆⊗Id^⊗(n−i))◦L

+(Id⊗L)◦δ_G+ (−1)ⁿ⁺¹(L⊗Id)◦δD.

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A particular case

We takeB=C,δ_G(b) = ∆(b)andδ_D(b) =b⊗1. A 1-cocycle ofCis a linear mapL:C−→C, such that for allb∈C:

(Id⊗L)◦∆(b)−∆◦L(b) +b⊗1=0.

SoB_γ is a 1-cocycle ofH_FGfor all primitive Feynman graph.

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The Hopf algebra of rooted treesH_R (or Connes-Kreimer Hopf algebra) is the free commutative algebra generated by the set of rooted trees.

q, qq, ∨^qq^q,qqq

, ∨^qqq^q, ∨^qq^q q

, ∨^q^q_q^q ,qqqq

,_H^q ∨^qq^q ^q, ∨^qqq^q q

, ∨^qq^q q q

, ∨^q∨q^{q q}^q , ∨^qq^q

qq , ∨^q^q_q^q^q

, ∨^q^q_q^q q

,∨^{q q}qqq , qqqqq

, . . .

The set of rooted forests is a linear basis ofH_R: 1,q,q q, qq,q q q, qq q, ∨^qq^q,qqq

,q q q, qq q q, qq qq, ∨^qq^q _q,qqq

q, ∨^qqq^q, ∨^qq^q q

, ∨^q^q_q^q ,qqqq

. . .

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∆(t) = X cadmissible cut

P^c(t)⊗R^c(t).

cutc ∨^qq^q 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

W^c(t) ∨^qq^q q

qq qq q q∨^q ^q _qq^q_q _{q q qq} _{qq q q} qq q q q q q q ∨^qq^q

q R^c(t) ∨^qq^q

q

qq ∨^qq^q _qq^q × q qq × 1

P^c(t) 1 qq q q × qq q q q × ∨^qq^q q

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The grafting operator ofH_R is the mapB:H_R −→H_R, associating to a forestt₁. . .tnthe tree obtained by grafting t₁, . . . ,t_n on a common root. For example:

B(qq q) = ∨^qq^q q

.

Proposition For allx ∈H_R:

∆◦B(x) =B(x)⊗1+ (Id⊗B)◦∆(x).

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Universal property

LetAbe a commutative Hopf algebra and letL:A−→Abe a 1-cocycle ofA. Then there exists a unique Hopf algebra morphismφ:H_R−→Awithφ◦B =L◦φ.

This will be generalized to the case of several 1-cocycles with the help of decorated rooted trees.

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H_R 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 graphsH_FG, such that

|γ|=1.

There exists a Hopf algebra morphismφ:H_R −→H_FG, such thatφ◦B =B_γ◦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

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Definition

Letf(h)∈K[[h]].

The combinatorial Dyson-Schwinger equations associated tof(h)is:

X =B(f(X)), whereX lives in the completion ofH_R. This equation has a unique solutionX =P

X(n), with:







X(1) = p₀q, X(n+1) =

n

X

k=1

X

a1+...+ak=n

p_kB(X(a₁). . .X(a_k)),

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X(1) = p₀q, X(2) = p₀p₁qq, X(3) = p₀p²₁qqq

+p₀²p₂ ∨^qq^q, X(4) = p₀p³₁qqqq

+p₀²p₁p₂ ∨^q^q_q^q

+2p²₀p₁p₂ ∨^qq^q q

+p₀³p₃ ∨^qqq^q.

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Examples

Iff(h) =1+h:

X = q+ qq + qqq + qqqq

+ qqqqq +· · ·

Iff(h) = (1−h)⁻¹:

X = q+ qq + ∨^qq^q + qqq

+ ∨^qqq^q +2 ∨^qq^q q

+ ∨^q^q_q^q + qqqq q

q q q qqq q qq qq ∨^qq^qq _q^q _q

q q

∨^q^q^q ∨^q_q^q

q ∨^{q q}qq _qq^q

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solution of the combinatorial Dyson-Schwinger equation associated tof(h)generate a subalgebra ofH_Rdenoted byH_f. H_f is not always a Hopf subalgebra

For example, forf(h) =1+h+h²+2h³+· · ·, then:

X = q+ qq + ∨^qq^q + qqq

+2 ∨^qqq^q +2 ∨^qq^q q

+ ∨^q^q_q^q + 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)

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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)∈K[[h]], withf(0) =1. The following assertions are equivalent:

1 H_f is a Hopf subalgebra ofH_R.

2 There exists(α, β)∈K²such that(1−αβh)f⁰(h) =αf(h).

3 There exists(α, β)∈K²such thatf(h) =1 ifα=0 or f(h) =e^αhifβ =0 orf(h) = (1−αβh)⁻^β¹ ifαβ6=0.

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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 AsH_f is a Hopf subalgebra,Y(n)belongs toH_f. Hence, there exists a scalarλnsuch thatY(n) =λnXn.

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lemma Let us write:

X =X

t

a_tt.

For any rooted treet:

λ_|t|a_t =X

t⁰

n(t,t⁰)a_t⁰,

wheren(t,t⁰)is the number of leaves oft⁰ such that the cut of this leaf givest.

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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:

α(1+ (n−1)(1

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Fort the corollaB(qⁿ⁻¹), we obtain:

λnp_n−1=np_n+ (n−1)p_n−1p₁. Hence:

α(1+ (n−1)β)pn−1=npn. Summing:

(1−αβh)f⁰(h) =αf(h).

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X(1) = q, X(2) = αqq, X(3) = α²

(1+β)

2 ∨^q^q^q + qqq ,

X(4) = α³ (1+2β)(1+β)

6 ∨^qq^q^q + (1+β) ∨^qq^q q

+ (1+β) 2

q

∨^q_q^q + qqqq !

,

X(5) = α⁴





(1+3β)(1+2β)(1+β)

24 _H^q ∨^qq^q ^q+(1+2β)(1+β) 2 ∨^qqq^q

q

+^(1+β)₂ ² ∨^q∨^qq^q^q

+ (1+β) ∨^qq^q qq

+(1+2β)(1+β) 6

q

∨^q qqq





 .

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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 =e^hand:

X(n) = X

|t|=n

1

]{symmetries oft}t.