Dyson-Schwinger systems on rooted trees

Dyson-Schwinger systems on rooted trees

Loïc Foissy

Bertinoro

September 2013

Let I be a set. Rooted trees decorated by I:

q

a

, a ∈ I; qq

ab

, (a, b) ∈ I ² ;

^b

∨ ^q q

_a

^q

^c

=

^c

∨ ^q q

_a

^q

^b

, qq q

a b c

, (a, b, c) ∈ I ³ ; q

∨ ^qq

_a

^q

^d

bc

= ∨ ^qq q

_a

^q

^c

bd

= . . . = ∨ ^qq q

_a

^q

^b

dc

, ∨ ^q q ^q q

a d b c

= ∨ ^q q ^q q

a b d

c

, ∨ ^{q q} _q ^q

a b d c

= ∨ ^{q q} _q ^q

a b c d

, qq qq

a b cd

, (a, b, c, d ) ∈ I ⁴ . Coproduct:

∆( ∨ ^q q ^q q

d c b a

) = ∨ ^q q ^q q

d c b a

⊗ 1 + 1 ⊗ ∨ ^q q ^q q

d c b a

+ qq

ba

⊗ qq

dc

+ q

a

⊗

^b

∨ ^q q

_d

^q

^c

+ q

c

⊗ qq q

db a

+ qq

ba

q

c

⊗ q

d

+ q

a

q

c

⊗ qq

db

.

Study of Dyson-Schwinger systems More realistic Dyson-Schwinger systems

R Dyson-Schwinger systems

Dyson-Schwinger system from QED:

^{= X γ B γ   (1 − (1 + ) 2|γ| (1 ) − 1+2|γ| ) |γ|   ,}

^{= B   (1 (1 + − ) ) 2 2   ,}

^{= B   (1 − (1 + )(1 − ) 2 )   .}

Combinatorial Dyson-Schwinger systems Study of Dyson-Schwinger systems More realistic Dyson-Schwinger systems

Hopf algebra of decorated rooted treesH^I_R Dyson-Schwinger systems

Dyson-Schwinger system from QED:

^{= X n=1 ∞   |γ|=n X B γ     (1 − (1 + ) 2n (1 − ) 1+2n ) n   ,}

^{= B   (1 (1 + − ) ) 2 2   ,}

^{= B   (1 − (1 + )(1 − ) 2 )   .}

Combinatorial Dyson-Schwinger systems Study of Dyson-Schwinger systems More realistic Dyson-Schwinger systems

Hopf algebra of decorated rooted treesH^I_R Dyson-Schwinger systems

Dyson-Schwinger system from QED truncated at order 1:

^{= B   (1 − (1 + ) 2 (1 − ) 3 )   ,}

^{= B   (1 (1 + − ) ) 2 2   ,}

^{= B   (1 − (1 + )(1 − ) 2 )   .}

Lifting to decorated trees:

X ₁ = B ₁

(1 + X ₁ ) ³ (1 − X ₃ ) ² (1 − X ₂ )

, X ₂ = B ₂

(1 + X ₁ ) ² (1 − X ₃ ) ²

, X ₃ = B ₃

(1 + X ₁ ) ² (1 − X ₂ )(1 − X ₃ )

.

X ₁ = q

1

+ 3 qq

11

+ qq

12

+ 2 qq

13

+9 qq q

1 1 1

+ 3 qq q

1 1 2

+ 6 qq q

1 1 3

+ 2 qq q

1 2 1

+ 2 qq q

1 2 3

+ 4 qq q

1 3 1

+ 2 qq q

1 3 2

+ 2 qq q

1 3 3

+3

¹

∨ ^q q

₁

^q

¹

+ 3

¹

∨ ^q q

₁

^q

²

+ 6

¹

∨ ^q q

₁

^q

²

+

²

∨ ^q q

₁

^q

²

+ 2

²

∨ ^q q

₁

^q

³

+ 3

³

∨ ^q q

₁

^q

³

+ . . . X ₂ = q

2

+ 2 qq

21

+ qq

23

+6 qq q

2 1 1

+ 2 qq q

2 1 2

+ 4 qq q

2 1 3

+ 4 qq q

2 3 1

+ 2 qq q

2 3 2

+ 2 qq q

2 3 3

+ ∨ ^q q

2

^q

1

1

+ 4 ∨ ^q q

2

^q

3

1

+ 3 ∨ ^q q

2

^q

3

3

+ . . .

X ₃ = q

3

+ 2 qq

31

+ qq

32

+ qq

33

+6 qq q

3 1 1

+ 2 qq q

3 1 2

+ 4 qq q

3 1 3

+ 2 qq q

3 2 1

+ 2 qq q

3 2 3

+ 2 qq q

3 3 1

+ qq q

3 3 2

+ qq q

3 3 3

+

¹

∨ ^q q

₃

^q

¹

+ 2

¹

∨ ^q q

₃

^q

²

+ 2

¹

∨ ^q q

₃

^q

³

+

²

∨ ^q q

₃

^q

²

+

²

∨ ^q q

₃

^q

³

+

³

∨ ^q q

₃

^q

³

+ . . .

Definition

Let f ₁ , . . . , f n ∈ K [[h ₁ , . . . , h n ]] − K . The combinatorial Dyson-Schwinger systems attached to f = (f ₁ , . . . , f _n ) is:

(S) :



 

 

X ₁ = B ⁺ ₁ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B ⁺ _n (f _n (X ₁ , . . . , X _n )),

Such a system has a unique solution (X ₁ , . . . , X _n ) ∈ H \ _R ^{1,...,n} .

The subalgebra generated by the homogeneous components of the X (i)’s is denoted by H _(S) .

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

Hopf.

Graph associated to (S)

Let (S) be associated to (f ₁ , . . . , f _n ). The oriented graph associated to (S) is defined by:

1

The vertices are 1, . . . , n.

2

There is an edge from i to j if, and only if, ∂f _i

∂h _j 6= 0.

Example coming from QED

X ₁ = B ₁

(1 + X ₁ ) ³ (1 − X ₃ ) ² (1 − X ₂ )

, X ₂ = B ₂

(1 + X ₁ ) ² (1 − X ₃ ) ²

, X ₃ = B ₃

(1 + X ₁ ) ² (1 − X ₂ )(1 − X ₃ )

.

Graph:

1 ^//

99 = = = = = = = 2

oo

3 ^yy

OO

^^ ===

=== =

Change of variables

Let (S) be the following system:

(S) :



 

 

X ₁ = B ₁ ⁺ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B _n ⁺ (f _n (X ₁ , . . . , X _n )).

If (S) is Hopf, then for all family (λ ₁ , . . . , λ n ) of non-zero scalars, this system is Hopf:

(S) :



 

 

X ₁ = B ₁ ⁺ (f ₁ (λ ₁ X ₁ , . . . , λ n X n )) .. .

X n = B _n ⁺ (f n (λ ₁ X ₁ , . . . , λ n X n )).

Concatenation

Let (S) and (S ⁰ ) be the following systems:

(S) :



 

 

X ₁ = B ₁ ⁺ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B _n ⁺ (f _n (X ₁ , . . . , X _n )).

(S ⁰ ) :



 

 

X ₁ = B ₁ ⁺ (g ₁ (X ₁ , . . . , X m )) .. .

X m = B _m ⁺ (g m (X ₁ , . . . , X m )).

Concatenation

The following system is Hopf if, and only if, the (S) and (S ⁰ ) are Hopf:



 

 

 

 

 

 

 

 

 

 

X ₁ = B ⁺ ₁ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B ⁺ _n (f _n (X ₁ , . . . , X _n ))

X _n+1 = B ⁺ _n+1 (g ₁ (X _n+1 , . . . , X _n+m )) .. .

X n+m = B ⁺ _n+m (g m (X _n+1 , . . . , X n+m )).

This property leads to the notion of connected (or

indecomposable) system.

Extension

Let (S) be the following system:

(S) :



 

 

X ₁ = B ₁ ⁺ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B _n ⁺ (f _n (X ₁ , . . . , X _n )).

Then (S ⁰ ) is an extension of (S):

(S ⁰ ) :



 

 



 

 



X ₁ = B ₁ ⁺ (f ₁ (X ₁ , . . . , X _n )) .. .

X _n = B _n ⁺ (f _n (X ₁ , . . . , X _n ))

X _n+1 = B _n+1 ⁺ (1 + a ₁ X ₁ ).

Iterated extensions

(S) :



 

 

 

 



 

 

 

 



X ₁ = B ₁

(1 − βX ₁ ) ⁻

^β1

, X ₂ = B ₂ (1 + X ₁ ),

X ₃ = B ₃ (1 + X ₁ ),

X ₄ = B ₄ (1 + 2X ₂ − X ₃ ),

X ₅ = B ₅ (1 + X ₄ ).

Dilatation

(S ⁰ ) is a dilatation of (S):

(S) :

( X ₁ = B ₁ ⁺ (f(X ₁ , X ₂ )), X ₂ = B ₂ ⁺ (g(X ₁ , X ₂ )),

(S ⁰ ) :



 

 

 

 



 

 

 

 



X ₁ = B ₁ ⁺ (f (X ₁ + X ₂ + X ₃ , X ₄ + X 5 )),

X ₂ = B ₂ ⁺ (f (X ₁ + X ₂ + X ₃ , X ₄ + X 5 )),

X ₃ = B ₃ ⁺ (f (X ₁ + X ₂ + X ₃ , X ₄ + X ₅ )),

X ₄ = B ₄ ⁺ (g(X ₁ + X ₂ + X ₃ , X ₄ + X ₅ )),

X ₅ = B ₅ ⁺ (g(X ₁ + X ₂ + X ₃ , X ₄ + X ₅ )).

Fundamental systems

Let β ₁ , . . . , β _k ∈ K . The following system is an example of a fundamental system:



 

 

 

 

 



 

 

 

 

 



X _i = B _i



(1 − β _i X _i )

k

Y

j=1

(1 − β _j X _j ) ⁻

1+βj βj

n

Y

j=k +1

(1 − X _j ) ⁻¹



 if i ≤ k ,

X _i = B _i



(1 − X _i )

k

Y

j=1

(1 − β _j X _j ) ⁻

1+βj βj

n

Y

j=k+1

(1 − X _j ) ⁻¹





if i > k .

Cyclic systems

The following systems are cyclic: if n ≥ 2,



 

 



 

 



X ₁ = B ⁺ ₁ (1 + X ₂ ), X ₂ = B ⁺ ₂ (1 + X ₃ ),

.. .

X _n = B ⁺ _n (1 + X ₁ ).

Graph on a cyclic system: an oriented cycle.

Theorem

Let (S) be an SDSE. If it is Hopf, then, for all i, j ∈ I, for all n ≥ 1, there exists a scalar λ ^(i,j) _n such that for all tree t ⁰ , which root is decorated by i:

X

t

n _j (t, t ⁰ )a _t = λ ^(i,j) _|t

0

| a _t

⁰

,

where n _j (t, t ⁰ ) is the number of leaves ` of t decorated by j such

that the cut of ` gives t ⁰ .

We shall denote by a ⁽ⁱ⁾ _j the coefficient of h _j in f _i and by a ⁽ⁱ⁾ _j,k the coefficient of h _j h _k in f _i .

Lemma

∂f _i

∂h _j 6= 0 if, and only if, a ⁽ⁱ⁾ _j 6= 0.

Theorem

Let us assume that (S) is Hopf. Let us fix i.

1

For all path i = i ₁ → i ₂ → . . . → i _k in the graph of (S)

λ ^(i,j) _k = a ⁽ⁱ _j

^k

⁾ +

k−1

X

p=1

(1 + δ _j,i

_p+1

) a ⁽ⁱ _j,i

^p

⁾

p+1

a _i ⁽ⁱ

^p

⁾

p+1

.

In particular, λ ^(i,j) ₁ = a ⁽ⁱ⁾ _j .

2

For all p ₁ , · · · , p _n ∈ N :

a ⁽ⁱ⁾ _(p

1

,··· ,p

_j

+1,···,p

n

) = 1

p _j + 1 λ ^(i,j) _p

1

+···+p

_n

+1 − X

l∈I

p _l a ^(l) _j

! a ⁽ⁱ⁾ _(p

1

,··· ,p

n

) .

Lemma

Let (S) be a Hopf SDSE. In the graph associated to (S):

i ^//

j

l k

= ⇒ i ^//

j

l ^// k

or i ^//

< < < < < < < < j

l k

.

Let us assume that a ⁽ⁱ⁾ _k = 0. As a ⁽ⁱ⁾ _j 6= 0, j 6= k . As a ⁽ⁱ⁾ _k = 0, a

^j

∨ ^q q

_i

^q

^k

= a ⁽ⁱ⁾ _j,k = 0.

Then:

λ ^(i,k) ₂ a ⁽ⁱ⁾ _j = λ ^(i,k) ₂ a _qq

_i^j

= a qq q

ij

k

+ a

q

∨ ^q

_i

^q

^k

j

= a ⁽ⁱ⁾ _j a ^(j) _k + 0;

Hence:

(i,k)

a ^(j) 6= 0.

Moreover, As a ⁽ⁱ⁾ _l 6= 0, l 6= k . Then:

a ⁽ⁱ⁾ _l λ ^(i,k ₂ ⁾ = λ ^(i,k) ₂ a _qq

_i^l

= a qq q

i l

k

+ a

q

∨ ^q

_i

^q

^k

l

= a ⁽ⁱ⁾ _l a ^(l) _k + 0.

so:

λ ^(i,k) ₂ = a ^(l) _k . Hence:

a ^(l) _k = a ^(j) _k 6= 0.

1

A first special case is given by i = k : i ^{oo //}

j

l

= ⇒ ;; i ^{oo //}

j

l

or i _OO ^{oo //}

j

l

.

2

A second special case is given by i = l:

i ^//

;; j

k

= ⇒ _;; i ^//

<

< <

< <

< <

< j

k

.

Proposition

Let (S) be a Hopf Dyson-Schwinger system with the following graph:

1 ^{oo //} 2 .

Up to a change of variables, two cases can occur:

1

(S) :

X ₁ = B ₁ (1 + X ₂ ), X ₂ = B ₂ (1 + X ₁ ).

2

(S) :

X ₁ = B ₁ ((1 − X ₂ ) ⁻¹ ),

X ₂ = B ₂ ((1 − X ₁ ) ⁻¹ ).

We put:

f ₁ (h ₂ ) =

∞

X

i=0

a _i h ⁱ ₂ , f ₂ (h ₁ ) =

∞

X

i=0

b _i h ⁱ ₁ .

Up to a change of variables, assume that a ₁ = b ₁ = 1. Then:

λ ^(1,1) ₃ = λ ^(1,1) ₃ a qq q

12

1

= 2a ∨ ^q _{q q} ^q

1 2 1

1

= 2b ₂ .

On the other hand:

2a ₂ b ₂ = λ ^(1,1) ₃ a q

∨ ^q

₁

^q

²

2

= a

q

∨ ^{q q} q

1 2 2

1

= 2a ₂ .

So 2a ₂ b ₂ = 2a ₂ and a ₂ = 0 or b ₂ = 1. Similarly, b ₂ = 0 or a ₂ = 1. Finally:

a = b = 0 or 1.

In the first case, f ₁ (h ₂ ) = 1 + h ₂ and f ₂ (h ₁ ) = 1 + h ₁ . In the second case, consider the path 1 → 2 → 1 → . . . of length n.

If n = 2k is even:

λ ^(1,2) _n = 2 + 2(k − 1) = 2k = n.

If n = 2k + 1 is odd:

λ ^(1,2) _n = 1 + 2k = n.

So:

λ ^(1,2) _n = n for all n ≥ 1.

Hence, for all n ≥ 1, a _n+1 = a _n and finally f ₁ (h ₂ ) = (1 − h ₂ ) ⁻¹ .

Similarly, f ₂ (h ₁ ) = (1 − h ₁ ) ⁻¹ .

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.

If (S) is a Hopf, the dual of H _(S) is the enveloping algebra of a prelie algebra g _(S) .

Description of g _(S)

It has a basis (e _i (p)) _{1≤i≤n,p≥1} . The prelie product is given by:

e _i (p) ◦ e _j (q) = λ ^(j,i) _q e _j (p + q).

As a consequence, g _i = Vect (e _i (p), p ≥ 1) is a prelie subalgebra. In the fundamental case, there are three possibilities:

1

i ≤ k , with β i = −1. Then e _i (p) ◦ e _i (q) = e _i (p + q): g _i is an associative, commutative algebra.

2

i > k . Then e _i (p) ◦ e _i (q) = 0: g _i is a trivial prelie algebra.

3

i ≤ k and β i 6= −1. Then b _j 6= 0, and g _i is a Faà di Bruno prelie algebra with parameter given by:

λ _i = −β _i

1 + β _i .

Proposition

Let (S) be a fundamental SDSE. If k < n or if there exists i ≤ k , such that β _i 6= −1, then the Lie algebra g _(S) can be

decomposed in a semi-direct product:

g _(S) = (M ₁ ⊕ . . . ⊕ M _k ) o g ₀ ,

where:

g ₀ is a Lie subalgebra of g _(S) , isomorphic to the Faà di Bruno Lie algebra, with basis (f _n ⁰ ) n≥1 such that for all m, n ≥ 1:

[f _m ⁰ , f _n ⁰ ] = (n − m)f _n+m ⁰ .

For all 1 ≤ i ≤ k, M _i is an abelian Lie subalgebra of g _(S) ,

with basis (f _n ⁱ ) n≥1 .

Proposition

For all 1 ≤ i ≤ k , M _i is a left g ₀ -module in the following way:

f _m ⁰ .f _n ⁱ = nf _m+n ⁱ .

Proposition

Let (S) be a cyclic SDSE, possibly with dilatations and

extensions. The prelie g _(S) admits a basis (e _i (k )) 1≤i≤n,k≥1 such that:

e _i (k )◦e _j (l) =

e _j (k + l) if there exists a path from j to i of length l, 0 otherwise.

This prelie product is associative.

We now consider systems of the form :

(S) :



 

 

 

 

 

 

X ₁ = X

i∈J

₁

B _1,i ⁺ (f _1,i (X ₁ , . . . , X _n ))

.. . X n = X

i∈J

_n

B _n,i ⁺ (f _n,i (X ₁ , . . . , X n )),

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

Theorem

We assume that 1 ∈ J _k for all k. Then (S) is entirely

determined by f _1,1 , . . . , f _n,1 .

Fundamental system



 

 

 

 

 



 

 

 

 

 



X _i = X

q∈J

_i

B _i,q



(1 − β i X _i )

k

Y

j=1

(1 − β j X _j ) ⁻

1+βj

βj

q n

Y

j=k+1

(1 − X _j ) ^−q



 if i ≤ k,

X _i = X

q∈J

i

B _i,q



(1 − X _i )

k

Y

j=1

(1 − β _j X _j ) ⁻

1+βj

βj

q n

Y

j=k+1

(1 − X _j ) ^−q





if i > k.

For example, we choose n = 3, k = 2, β ₁ = −1/3 β ₂ = 1, J ₁ = N ^∗ , J ₂ = J ₃ = {1}. After a change of variables h ₁ −→ 3h ₁ , we obtain:

(S) :



 

 

 

 

 

 

 

 

X ₁ = X

k≥1

B _1,k

(1 + X ₁ ) ^1+2k (1 − X ₂ ) ^2k (1 − X ₃ ) ^k

,

X ₂ = B ₂

(1 + X ₁ ) ² (1 − X ₂ )(1 − X ₃ )

, X ₃ = B ₃

(1 + X ₁ ) ² (1 − X ₂ )

.

This is the example of the introduction, with X ₁ = ^,

X ₂ = − ^{, X 3 = − .}

X ₁ = q

(1,1)

+ 3 qq

(1,(1,1)1)

+ qq

(1,2 1)

+ qq

(1,3 1)

+ q

(1,2)

+ 9 qq q

(1,1) (1,1) (1,1)

+3 qq q

(1,1) (1,1) 2

+ 6 qq q

(1,1) (1,1) 3

+ 2 qq q

(1,1) 2 (1,1)

+ 2 qq q

(1,1) 2 3

+ 4 qq q

(1,1) 3 (1,1)

+ 2 qq q

(1,1) 3 2

+ 2 qq q

(1,1) 3 3

+3

^(1,1)

∨ ^q q

_(1,

^q

^(1,₁₎¹⁾

+ 3

^(1,1)

∨ ^q q

_(1,

^q

²₁₎

+ 6

^(1,1)

∨ ^q q

_(1,

^q

²₁₎

+

²

∨ ^q q

_(1,

^q

²₁₎

+ 2

²

∨ ^q q

_(1,

^q

³₁₎

+3

³

∨ ^q q

_(1,

^q

³₁₎

+ 3 qq

(1,(1,1)2)

+ 5 qq

(1,(1,2)1)

+ 2 qq

(1,2 2)

+ 4 qq

(1,3 2)

+ q

(1,3)

+ . . . X ₂ = q

2

+ 2 qq

2(1,1)

+ qq

23

+6 qq q

2 (1,1) (1,1)

+ 2 qq q

2 (1,1) 2

+ 4 qq q

2 (1,1) 3

+ 4 qq q

2 3 (1,1)

+ 2 qq q

2 3 2

+ 2 qq q

2 3 3

+

^(1,1)

∨ ^q q

₂

^q

^(1,1)

+ 4

^(1,1)

∨ ^q q

₂

^q

³

+ 3

³

∨ ^q q

₂

^q

³

+ 2 qq

2(1,2)

+ . . . X ₃ = q

3

+ 2 qq

3(1,1)

+ qq

32

+ qq

33

+ 6 qq q

3 (1,1) (1,1)

+ 2 qq q

3 (1,1) 2

+4 qq q

3(1,1) 3

+ 2 qq q

32 (1,1)

+ 2 qq q

32 3

+ 2 qq q

33 (1,1)

+ qq q

33 2

+ qq q

33 3

+

^(1,1)

∨ ^q q

₃

^q

^(1,1)

+2

^(1,1)

∨ ^q q

₃

^q

²

+ 2

^(1,1)

∨ ^q q

₃

^q

³

+

²

∨ ^q q

₃

^q

²

+

²

∨ ^q q

₃

^q

³

+

³

∨ ^q q

₃

^q

³

+ 2 qq

3(1,2)

+ . . .

Cyclic systems

(S) :



 

 

 

 

 

 

X ₁ = X

j∈I

₁

B _1,j

1 + X _1+j ,

.. . X _n = X

j∈I

₁

B _n,j

1 + X _n+j

.

n = 3:

X ₁ = q

(1,1)

+ q

(1,2)

+ qq

(1,(2,1)1)

+ q

(1,3)

+ qq

(1,(3,2)1)

+ qq q

(1,1) (2,1) (3,1)

, X ₂ = q

(2,1)

+ q

(2,2)

+ qq

(2,(3,1)1)

+ q

(2,3)

+ qq

(2,(1,2)1)

+ qq q

(2,1) (3,1) (1,1)

, X ₃ = q

(3,1)

+ q

(3,2)

+ qq

(3,(1,1)1)

+ q

(3,3)

+ qq

(3,(2,2)1)

+ qq q

(3,1) (1,1) (2,1)

.