Operadic construction of the renormalization grouptheo.inrne.bas.bg/~mitov/configuration/Loday J.-L... · 2017-02-18 · Operadic construction of the renormalization group Jean-Louis

Operadic construction of the renormalizationgroup

Jean-Louis Loday and Nikolay M. Nikolov

February 1, 2012

Abstract First, we give a functorial construction of a group associated to a sym-metric operad. Applied to the endomorphism operad it gives the group of formaldiffeomorphisms. Second, we associate a symmetric operad to any family of deco-rated graphs stable by contraction. In the case of Quantum Field Theory models itgives the renormalization group. As an example we get an operadic interpretationof the group of “diffeographisms” attached to the Connes-Kreimer Hopf algebra.

Contents.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Operadic construction of the group of formal diffeomorphisms . . . . . . . . . . . . . 23. Group associated to a symmetric operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. Feynman diagrams and their combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. The universal contraction operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. Suboperads in R and concrete combinatorial models of QFT . . . . . . . . . . . . . . 137. The group related to the contraction operad and its representation in the group of

formal diffeomorphisms on the space of interactions . . . . . . . . . . . . . . . . . . . . . 178. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Appendix A. A brief introduction to the notion of operad . . . . . . . . . . . . . . . . . . . 24Appendix B. Feynman diagrams in perturbation theory . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1 Introduction

The combinatorics underlying the renormalization of Quantum Field Theory (QFT)is encoded into the Feynman diagrams. The diagram technique is a powerful tool in

Jean-Louis LodayInstitut de Recherche Mathematique Avancee CNRS et Universite de Strasbourg, Zinbiel Instituteof Mathematics

Nikolay M. NikolovINRNE, Bulgarian Academy of Sciences, Tsarigradsko chaussee 72 Blvd., Sofia 1784, Bulgaria

1

2 J.-L. Loday and N.M. Nikolov

perturbative QFT. It was discovered by Connes and Kreimer that the combinatoricsin renormalization can be described by a Hopf algebra structure on the space ofFeynman diagrams since the attached group is the renormalization group. In thispaper our aim is to systematize this procedure by means of symmetric operads. Firstwe show that a family of decorated graphs which is stable for the contraction ofthe internal edges determines a symmetric operad. Second, we show that to anysymmetric operad is attached a (formal) group which takes care of the symmetricgroup action. Combining the two constructions we get the construction of a groupattached to families of diagrams. In the case of QFT we get the renormalizationgroup.

For the notation and terminology on operads we follow [5] for which we referfor details.

2 Operadic construction of the group of formal diffeomorphisms

Let V ≡ RN be a vector space and →x = (x1, . . . ,xN),→y ,→z ∈ V . Consider the formal

power series

→y =→f(→x) = ∞

∑n=1

1n!

N

∑µ1,...,µn =1

→f µ1,...,µn xµ1 · · ·xµn , (1)

→z =→g(→y) = ∞

∑n=1

1n!

N

∑µ1,...,µn =1

→g µ1,...,µn xµ1 · · ·xµn ,

where→f µ1,...,µn = ( fν ;µ1,...,µn)

Nν =1 and →g µ1,...,µn = (gν ;µ1,...,µn)

Nν =1 are the series co-

efficients. Since these series do not have constant terms (i.e., terms with n = 0) it iswell known that their composition

→z =→g(→

f(→x)) = ∞

∑n=1

1n!

N

∑µ1,...,µn =1

→h µ1,...,µn xµ1 · · ·xµn , (2)

can be determined completely algebraically. A less popular fact is the formula for

the coefficients→h µ1,...,µn = (hν ;µ1,...,µn)

Nν =1 of the composition series:

hν ;µ1,...,µn = ∑P∈Part1,...,n

N

∑ρ1,...,ρk =1

gν ;ρ1,...,ρk fρ1;µi1,1 ,...,µi1, j1· · · fρk;µik,1 ,...,µik, jk

, (3)

which, in the case N = 1, is known as the Faa di Bruno formula. Here are thenotations used in Eq. (3):

• the sum is over all partitions

Operadic construction of the renormalization group 3

P =

i1,1, . . . , i1, j1, . . . ,

ik,1, . . . , ik, jk

(4)

of the set 1, . . . ,n;• in particular, k is the cardinality |P| of the partition P and j1, . . . , jk are the

cardinalities of its pieces;• the partitions P are unordered, but we shall introduce a “canonical order”

such that inside each group the elements are in increasing order and thegroups are ordered according to the order of their minimal elements

i`,1 < · · ·< i`, j` , i1,1 < i2,2 < · · ·< ik, jk . (5)

Note that all the coefficients→f µ1,...,µn , →g µ1,...,µn and

→h µ1,...,µn are symmetric in their

indices µ1, . . . ,µn and hence, our convention in Eq. (3) about the order on P isnot essential. However, we shall see that dropping the symmetry condition on thecoefficients still defines an associative product.

Let us try to simplify a little bit Eq. (3) by absorbing some summations: the

coefficients→f µ1,...,µn define a multi-linear map

fn =(→

f µ1,...,µn

): V⊗n→V (6)

and vice versa, every multi-linear map fn : V×n→V defines a system of coefficients→f µ1,...,µn by its matrix elements. Furthermore, the coefficients

→f µ1,...,µn are symmet-

ric in µ1, . . . ,µn iff the map fn is symmetric. Similarly, we set

gn =(→g µ1,...,µn

): V⊗n→V , hn =

(→h µ1,...,µn

): V⊗n→V

(n = 1,2, . . . ). Then Eq. (3) reads

hn = ∑P∈Part1,...,n

gk (

f j1⊗·· ·⊗ f jk

)σP , (7)

where the numbers k, j1, . . . , jk are defined by conventions (4) and (5) together withthe permutation σP ∈ Sn, which is

σP :=(i1,1, . . . , i1, j1 , . . . , ik,1, . . . , ik, jk

).

Thus, the formal power series →y =→f(→x) of formula (1) is encoded by a sequence

f =(

f1, f2, . . . , fn, . . .)∈

∞

∏n=1

Hom(V⊗n,V

)Sn

(Hom(V⊗n,V

)Sn being the subspace of Sn–invariant maps in Hom(V⊗n,V

)). The

multiplication in∞

∏n=1

Hom(V⊗n,V

)Sn ,


h = g • f :=(hn)∞

n=1 ,

that is defined by Eq. (7) is associative. It has a unit, the composition unit:

1 = (idV ,0, . . .)

Furthermore, if we assume that f1 = idV (the identity map of V ), then f has a com-position inverse f−1 =

(1,( f−1)2, . . .

)since for n > 1 we have

0 =(1)

n =(

f−1 • f)

n = ( f−1)n + fn + low order terms ,

which inductively fixes ( f−1)n.The so described group of formal diffeomorphisms is denoted by

FDiff(V ) ∼= idV×∞

∏n=2

Hom(V⊗n,V

)Sn . (8)

Note that the vector space V can be even arbitrary linear vector space: N then willbe the cardinality (possibly, infinite) of the linear basis of V and the series (1) wouldbe neither more nor less formal. We note also that fν ;µ1,...,µn for fixed µ1, . . . ,µn arenonzero only for no more than a finite number of indices ν since they are coordinates

of the vector→f µ1,...,µn . Hence, the correspondence

→f(→x)↔ f defined by (6) remains

valid and the composition (2) is again well defined algebraically.

3 Group associated to a symmetric operad

We now observe that the multiplication (7) has a straightforward generalization in asymmetric operad (see Eq. (9) below). Indeed, it uses two basic structures which areaxiomatized in the operad theory (see Appendix A for a brief introduction). Theseare the composition of multilinear maps and the right action of (or, compositionwith) permutations.

Theorem 1. ([4]) There is a functor together with a subfunctor:Category of

Symmetric operads

→

Category of

Groups

P = P(n)∞

n=1 7→ G(P) = id×∞

∏n=2

P(n)⋃‖P = P(n)∞

n=1 7→ G(P) = id×∞

∏n=2

P(n)Sn ,


where P(n)Sn stands for the subspace of Sn–invariant elements. The multiplicationlaw is given by

(β • α)n = ∑P∈Part1,...,n

γ(βk;α j1 , . . . ,α jk

)σP (9)

for α = (αn)∞n=1 and β = (βn)

∞n=1 and the notations of Eq. (7). On operadic mor-

phisms ϑ : P →P ′ (= ϑn : P(n)→P ′(n)∞n=1) the functor gives

G(ϑ) :=∞

∏n=1

ϑn .

In the case of EndV we have a natural isomorphism

G(EndV

) ∼= FDiff(V ) . (10)

The most nontrivial part of the above statement is the associativity of the op-eration • (9). It can be proven by straightforward inspection. The existence of aunit and inverse elements follows exactly by the same arguments as for the group offormal diffeomorphisms.

Remark 1. There is a natural group associated with a non–symmetric operad P =Pn

n>1 (see [5, Sect. 5.8.15]). However when this construction is applied to asymmetric operad considered as a non-symmetric it gives a different group.

We will give below some facts about the structure of of the groups related tosymmetric operads. Note that by Eq. (9) we get

(α • β )n = βn +αn + rn(α2, . . . ,αn−1;β2, . . . ,βn−1)

for every α = (αn)∞n=1, β = (βn)

∞n=1 ∈ G(P), where rn is a polynomial function

of its arguments such that

rn(0, . . . ,0;β2, . . . ,βn−1) = 0 = rn(α2, . . . ,αn−1;0, . . . ,0) .

It then follows that

(α−1)n = −αn +qn(α2, . . . ,αn−1) ,

where qn is a polynomial function such that

qn(0, . . . ,0) = 0 .

As a consequence of these formulas we get

Proposition 1. ([4]) Let us set for m > 0

Gm(P) =

α = (αn)∞n=1 ∈ G(P)

∣∣∣α2 = · · ·= αm = 0


(for m = 1, G1(P) := G(P)). Then Gm(P) is a normal subgroup of G(P).

Proof. If α = (αn)∞n=1, β = (βn)

∞n=1 ∈ G(P) and α = (id,0, . . . ,0,αm+1, . . .) then

α−1 = (id,0, . . . ,0,−αm+1,α

′m+1, . . .) ,

β • α = (id,β2, . . . ,βm,β′m+1 = βm+1 +αm+1,β

′m+2, . . .) ,

β • α • β−1 = (id,β2, . . . ,βm,β

′m+1, . . .) • (id,(β−1)2, . . . ,(β

−1)m, . . . . . .)

= (id,0, . . . ,0,β ′′m+1, . . .) ,

so that α−1, β • α • β−1 ∈ Gm(P).

Note thatG(P) = lim

←−G(P)

/Gm(P)

and in the case when the operadic spaces P(n) are finite dimensional the quotientgroups are (finite dimensional) Lie groups. Hence, in the latter case the group G(P)is a pro-Lie group. We use this fact to derive the Lie algebra corresponding to thegroup G(P) together with the exponential map.

Theorem 2. ([4]) The Lie algebra corresponding to the group G(P) is

g(P) = 0 ×∞

∏n=2

P(n)

The Lie bracket on g(P) is built from a pre-Lie bracket

[µ,ν ] = µ ∗ ν−ν ∗ µ

(µ,ν ∈ g(P)), where1

(µ ∗ ν)n = ∑/0 6=J⊆1,...,n

(νk min J µ j

)σPJ

≡ ∑/0 6=J⊆1,...,n

γ(νk; id, . . . , id, µ j

↑min J

, id, . . . , id)σPJ (11)

where j = |J| and the partition PJ is the partitioni∣∣i ∈ 1, . . . ,n\J∪ J.

(Note that the sum in (11) is the subsum in (9) corresponding to partitions P of aform PJ .)

The Lie algebra g(P) is again an inverse limit of finite dimensional Lie algebras

g(P) = lim←−

g(P)/gm(P)

where gm(P) is the ideal

1 i is the ith operadic partial composition


gm(P) =

µ = (µn)∞n=1 ∈ g(P)

∣∣∣µ2 = · · ·= µm = 0.

Note that the quotient group G(P)/Gm(P) and Lie algebra g(P)

/gm(P) are

isomorphic as sets to the setm∏

n=2P(n) and the group and pre–Lie products on this

set are just • (9) and ∗ (11) truncated up to order m.

4 Feynman diagrams and their combinatorics

Feynman diagrams are a powerful tool in perturbation theory. They indicate theterms of perturbative expansions. Furthermore, many manipulation on the corre-sponding formal perturbation series have a combinatorial description by operationson diagrams. For a brief introduction on the appearance on the origin of Feynmandiagrams in QFT we refer the reader to Appendix B.

a) Basic definitions

A Feynman diagram is a finite graph with various decorations.A graph Γ is a set of points, called vertices, with attached flags (or half-edges) to

them. Some pairs of these flags are further joined to become edges connecting thecorresponding vertices. All these structures are contained in the following data: twofinite sets, the set of vertices vert(Γ ) and the set of flags flag(Γ ), and two maps

s : flag(Γ )→ vert(Γ ) , σ : flag(Γ )→ flag(Γ ) (12)

such that σ2 = id. Thus, the map s represents the process of attaching flags to ver-tices, i.e., the flag f is attached to the vertex s( f ). The map σ represents the processof joining flags, i.e., the flag f is joined with the flag σ( f ). In the latter case iff = σ( f ) then we call this flag an external line; such a line is attached to one onlyvertex. If f 6= σ( f ) then the unordered pair f ,σ( f ) form an edge, or an inter-nal line of the graph, which is attached to the vertices s( f ) and s(σ( f )). Whens( f ) = s(σ( f )) but f 6= σ( f ) we have an internal line attached to one and the samevertex. Such an internal line is called a tadpole and it is usually excluded to exist.

To every graph we assign a topological space: its geometric realization. To thisend we assign to each edge a copy of the closed interval [0,1] (without the orienta-tion) and to each vertex a point. Then we glue all of these spaces according to theincidence between the edges and the vertices.

A decorated graph is a graph with some extra data. Forgetting these extra struc-ture we obtain just a graph that is called the body of the decorated graph. We shallconsider graphs with the following decorations:

a) Colors for the vertices and for the flags. They form two sets– a set of colors for the vertices: Colv


– a set of colors for the flags: ColfThen we have maps assigning colors:

cv : vert(Γ )→ Colv , cf : flag(Γ )→ Colf . (13)

b) The second type of decoration we shall consider is an enumeration

ν : vert(Γ )∼= 1, . . . ,n (14)

of the set of vertices.

b) Examples

These are the notion of graph and decorated graph, or also diagram. Here are someexamples to illustrate them.

Example 1. An example of a graph is: vert(Γ ) = 0,1, flag(Γ ) = a,b,c,d,e, f,s(a) = s(b) = s(c) = 0, s(d) = s(e) = s( f ) = 1, σ(a) = a, σ(b) = e, σ(c) = d,σ( f ) = f . The geometric realization is:

Example 2. A decoration for the graph in Example 1 is provided by Colv = •,Colf=

, and coloring maps: cv(0)= cv(1)= •, cf(a)= , cf(b)= , cf(e)=

, cf( f ) = , cf(c) = = cf(d). The result can be drawn as

So, we indicated the colors in this example by shapes, which is common in physics.Also if the colors of two joined flags coincide we indicate this as a color of thecorresponding edge. In the above example we also meet situation of edges of theform and in this case it is also convenient to think of such an edge as anoriented edge . Then we can draw the diagram of this example as

c) Types of graphs and diagrams

A graph is called connected if its geometric realization is a connected space.Another important type of graphs are the so-called one particle irreducible (1PI)

graphs. A graph Γ is called one particle irreducible if it is connected and after cuttingany of its inner edges it remains connected. Here cutting of an inner edge determinedby a pair of flags f 6= σ( f ) means to change the second structure map σ to a newmap σ ′ such that σ ′( f ′) := σ( f ) if f ′ 6= f and f ′ 6= σ( f ), and σ ′( f ′) := f ′ if f ′ = f


or f ′ = σ( f ). We shall impose in addition the requirement that 1PI graphs have notadpoles and have at least two vertices (or equivalently, at least one inner edge).

If the body of a decorated graph is connected, then the graph is also called con-nected. Similarly a decorated graph is called 1PI if its body is 1PI.

e) Operations on graphs and diagrams

A subgraph of a graph Γ is a subset J ⊆ vert(Γ ). It determines a graph ΓJ as fol-lows: the set of vertices of ΓJ is vert(ΓJ) := J ⊆ vert(Γ ). The set of flags of ΓJ isflag(ΓJ) := s−1(J) ≡ s−1

(vert(ΓJ)

)and we set the map sJ : flag(ΓJ)→ vert(ΓJ) to

be the restriction of the map s. The map σJ : flag(ΓJ)→ flag(ΓJ) coincides with σ

whenever f and σ( f ) belong to flag(ΓJ): such pairs f ,σ( f ) of different flags arethe inner edges of the subgraph. For the remaining f ∈ flag(ΓJ) we set σJ( f ) = fand they are the outer edges of the subgraph. Note that the outer edges of the graphΓJ are either outer edges of Γ attached to a vertex in J or they are inner edges of Γ

with only one end belonging to J.If the graph Γ is colored then the graph ΓJ determined by a subgraph J has an

induced coloring defined just by the restrictions of the coloring maps cv and cf tovert(ΓJ) and flag(ΓJ), respectively.

If the graph Γ is enumerated, then the graph ΓJ has an induced enumerationprovided by the unique monotonically increasing isomorphism ν(J)∼= 1, . . . , |J|.

Another important operation on graphs is the contraction of a subgraph.For every graph Γ and its subgraph J ⊆ vert(Γ ) we define the contracted graph

Γ /J as follows. We introduce a new vertex vJ , which for the sake of definitenesscan be identified with the set J. Then we set

vert(Γ /J) :=(vert(Γ )\J

)∪vJ ,

flag(Γ /J) :=

f ∈ flag(Γ )∣∣ if s( f ) and s(σ( f )) ∈ J then f = σ( f )

≡ flag(Γ )

∖f ∈ flag(Γ )

∣∣s( f ),s(σ( f )) ∈ J and f 6= σ( f ),

in other words, flag(Γ /J) contains all the flags of flag(Γ ) except those ones thatform the inner edges of the graph ΓJ . The structure maps sΓ /J and σΓ /J are definedas follows:

sΓ /J( f ) := s( f ) if s( f ) /∈ J and sΓ /J( f ) := vJ if s( f ) ∈ J ,

σΓ /J := σ∣∣vert(Γ /J) ,

where the second identity is provided by the fact that flag(Γ /J) is defined as aσ–invariant subset. To summarize, the graph Γ /J is obtained by shrinking all thevertices in J to a single vertex vJ and removing all the internal lines of ΓJ . Note thatif the graph Γ is connected or 1PI, respectively, then so is Γ /J.

If the graph Γ is colored, then for every pair (J,L) consisting of a subset J ⊆vert(Γ ) and an element L ∈ Colv we can define a colored contracted graph Γ /(J,L)constructed as the graph Γ /J endowed with the following coloring maps c′v and c′f:


c′v∣∣vert(Γ )\J := cv

∣∣vert(Γ )\J , c′v(vJ) := L ,

c′f := cf∣∣flag(Γ /J) .

Finally, if we have an enumerated graph Γ , then the contracted graph Γ /J willbe endowed with the enumeration provided by the unique monotonically increasingisomorphism

ν(vert(Γ )\J)∪minν(J) ∼= 1, . . . ,n−|J|+1 .

Note that if the graph Γ has no tadpoles, then the graphs ΓJ and Γ /J have notadpoles for every subgraph J of Γ .

f) Isomorphic diagrams

Let us introduce the notion of an isomorphism of two enumerated diagrams Γ andΓ ′. We shall treat two such diagrams as identical. An isomorphism of graphs Γ ∼=Γ ′

consists of a pair of bijections jv : vert(Γ ) ∼= vert(Γ ′) and j f : flag(Γ ) ∼= flag(Γ ′),which commute with the structure maps s,s′ and σ ,σ ′, respectively. In other words,jv s = s′ j f and j f σ = σ ′ jv. An isomorphism of colored graphs is an isomor-phism of graphs, which in addition satisfies cv = c′v jv and cf = c′f j f (compatibil-ity with the coloring maps). Finally, an isomorphism of enumerated colored graphsis an isomorphism of colored graphs which preserves the enumeration. Let

Dgm(n) := set of all equivalence classes of isomorphicenumerated colored graphs with n vertices. (15)

g) Combinatorial Feynman rules, or, representation of diagrams in a monoid

There is a convenient one-to-one correspondence between the equivalence classes ofisomorphic enumerated colored graphs and the elements (monomials) of a commu-tative monoid. This construction follows on an abstract algebraic (or combinatorial)level the so called “Feynman rules” that assign in QFT to every Feynman diagraman analytic expression. Let

M(n) := the free commutative monoid with a set of generators(1, . . . ,n×Colv

)∪(1, . . . ,n×Colf

)∪(1, . . . ,n×Colf

)×2. (16)

Let us introduce “physical” names and notation for the elements in the above threedisjoint sets. We call the elements of Colf the basic “fields” and denote them byφ , ψ , etc. Then the element (i,φ) ∈ 1, . . . ,n×Colf will be denoted by φ(i) andcalled a “field at the point i”. Next, the elements (i,φ ; j,ψ) ∈

(1, . . . ,n×Colf

)×2

will be denoted by Cφ ,ψ(i, j) and will be called “propagators”. Finally, the elementsL ∈ Colv will be called “interactions” and a pair (i,L) ∈ 1, . . . ,n×Colv will becalled an interaction at the point i and will be denoted by L(i).


Thus, in the above notations the set of generators (16) for the monoid M(n)reads:

L(i)∣∣L ∈ Colv, i = 1, . . . ,n

∪

φ(i)∣∣φ ∈ Colf, i = 1, . . . ,n

∪

Cφ ,ψ(i, j)∣∣φ ,ψ ∈ Colf, i, j = 1 . . . ,n

. (17)

Now, to each enumerated colored graph Γ we assign a monomial in M(n) in thefollowing way. To the vertex ν−1(i) (i.e., to the vertex with number i) we assign L(i)if its color is L ∈ Colv. To each outer edge attached to the vertex ν−1(i) we assignφ(i) if the color of the corresponding flag is φ ∈Colf. To each inner edge connectingthe vertices ν−1(i) and ν−1( j) we assign Cφ ,ψ(i, j) if the colors of the flags attachedto ν−1(i) and ν−1( j) are φ and ψ , respectively. Finally, we multiply all the aboveobtained generators in M(n). The resulting monomial in M(n) is denoted by MΓ .

Example 3. In the case of Example 2 with vertex enumeration ν(0) = 1, ν(1) = 2we have

MΓ = ψ(1)ψ(2)L(1)L(2)CA,A(1,2)Cψ,ψ(1,2) ,

where we denoted now the colors by letters: L := • ∈ Colv and A := , ψ := , ψ

:= .

Proposition 2. The correspondence Γ 7→ MΓ is a bijection Dgm(n) ∼= M(n), i.e.,it is a one-to-one correspondence between the equivalence classes of isomorphicenumerated colored graphs with n vertices and the elements of the monoid M(n).

Proof. It is clear that Γ 7→MΓ maps injectively the equivalence classes of diagramsto elements of M(n). To see that this map is surjective one constructs for everyelement of M(n) a diagram that reproduces this monomial.

5 The universal contraction operad

Recall that Dgm(n) is the set of all equivalence classes of isomorphic enumeratedcolored graphs with n vertices. Let us define

R(n) := HomK

(K(Dgm(n)),K(Colv)

)∼= KDgm(n)×Colv , (18)

where K(I) stands for the vector space over the ground field (ring) K spanned by abasis indexed by I and the existence of the second canonical isomorphism followsin the case when Colv is a finite set, which we shall assume further. This canonicalisomorphism is provided by the decomposition

Q(Γ ) = ∑L∈Colv

q(Γ ,L)L , (19)


where Q ∈ R(n). We shall treat the isomorphism at the second equality in (18) asan identification, R(n) =KDgm(n)×Colv.

We call the elements of R(n) contraction maps. This is motivated by the factthat they can be thought of as prescriptions for contracting subgraphs as we shalldescribe below.

Note that the action of the permutation group Sn on Dgm(n) induces an actionon R(n). We shall endow now the so-defined S-module R = R(n)n>1 with astructure of a symmetric operad.

To this end we shall define the partial composition maps:

i : R(n)⊗R( j)→R(n−1+ j) , (20)

i = 1, . . . ,n, j = 1,2, . . . . Let us introduce for every enumerated diagram Γ the sub-sets of vertices J := J(i, j)⊆ vert(Γ ):

J (≡ J(i, j)) :=

ν−1(`)

∣∣`= i+1, . . . , i+ j. (21)

We define for Q′′ ∈ R(n), Q′ ∈ R( j) and Γ that is a representative of an isomor-phism class in Dgm(n−1+ j):

(Q′′ i Q′)(Γ ) = ∑L∈Colv

q′(ΓJ ,L)Q′′(Γ /(J,L)

), (22)

whereQ′(ΓJ) =: ∑

L∈Colvq′(ΓJ ,L)L . (23)

Note that if we set

Q′′(Γ ′′) = ∑L∈Colv

q′′(Γ ′′,L)L ,

Q(Γ ) = (Q′′ i Q′)(Γ ) = ∑L∈Colv

q(Γ ,L)L, (24)

then Eq. (22) reads

q(Γ ,K) = ∑L∈Colv

q′(ΓJ ,L)q′′(Γ /(J,L),K

). (25)

Proposition 3. ([4]) R = R(n)n>1 is a symmetric operad.

The proof is straightforward checking and we omit it.


A general composition in the operad R is illustrated by the following picture:

6 Suboperads in R and concrete combinatorial models ofQuantum Field Theory

In the previous section we have defined a universal operad R on decorated graphs,which can include, at the combinatorial level, any concrete model of Quantum FieldTheory (QFT) provided that we have sufficiently many colors in Colv and Colf.So, the QFT models can be considered as particular suboperads of R. Describingthese suboperads can be quite cumbersome in general and we shall do this in severalsteps. At each step we shall impose certain restrictions on the contraction maps Q ∈R(n). These restrictions include, in particular, requirements that Q should vanish oncertain classes of diagrams that are “not admissible for contraction”.

For instance, excluding tadpoles was a first example of such a restriction on di-agrams. It was “stable with respect to contractions and subdiagrams” and hence,it defined a suboperad in R. More precisely, the statement is that the subspaces inR(n) for every n = 1,2, . . . , which consist of those contraction maps that vanish ondiagrams with tadpoles, form a suboperad.

Let us formulate the argument in a more general principle:

Proposition 4. ([4]) Let Φ = Φ(n)n>1 be a system of subsets Φ(n)⊆ Dgm(n)×Colv for n = 1,2, . . . and let us define

RΦ(n) = KΦ(n) ⊆ KDgm(n)×Colv ≡ R(n) ,

RΦ(n) ≡

Q = ∑qL ∈R(n)∣∣∣q∣∣(Dgm(n)×Colv)\Φ(n)= 0

, (26)


where we use the expansion (19) and embeddings of type KA → KB for A ⊆ B,which are defined by (xa)a∈A 7→ (yb)b∈B such that ya = xa for a ∈ A and yb = 0 forb ∈ B\A.

Then the following conditions are equivalent:

(i) The system RΦ = RΦ(n)n>1 is a suboperad of R.(ii) Each subset Φ(n) is Sn-invariant and the system Φ(n)n>1 has the prop-

erty

(ΓJ ,L) ∈Φ(|J|) and (Γ /(J,L),K) ∈Φ(n−|J|+1)⇒ (Γ ,K) ∈Φ(n) (27)

for every Γ ∈ Dgm(n), J ⊆ vert(Γ ) and K,L ∈ Colv.

Proof. (i)⇒ (ii). The system RΦ(n)n>1 is an S-submodule of R iff every setΦ(n) is Sn–invariant. Let us define for every pair (Γ ,K) ∈ Dgm(n)×Colv an ele-ment Q(Γ ,K) ∈R(n) (n = |vert(Γ )|) by:

Q(Γ ,K)(Γ ) = K and Q(Γ ,K)(Γ′) = 0 for Γ

′ 6= Γ .

Thus, Q(Γ ,K) ∈RΦ(n) iff (Γ ,K) ∈ Φ(n). Then let the left hand side of implication(27) be satisfied so that we have Q(ΓJ ,L) ∈RΦ(|J|) and Q(Γ /(J,L),K) ∈RΦ(n−|J|+1)for J ⊆ vert(Γ ) (21). It follows that

Q(Γ ,K) = Q(ΓJ ,L) i Q(Γ /(J,L),K) ∈ RΦ(n) ,

which in turn implies that the right hand side of implication (27) is satisfied. Fora general J ⊆ vert(Γ ) the property (27) follows by the permutation invariance ofΦ(n).

(ii)⇒ (i). Assume J, Q′, Q′′ and Q = Q′′ i Q′ are given by Eqs. (21), (23) and(24). Let Q′ ∈RΦ( j) and Q′′ ∈RΦ(n). We need to show that then Q ∈RΦ(n−1+j), or equivalently,

q(Γ ,K) = 0 if (Γ ,K) ∈ (Dgm(n−1+ j)×Colv)\Φ(n−1+ j) .

By (27) it follows that if (Γ ,K) ∈ (Dgm(n− 1+ j)×Colv)\Φ(n− 1+ j) then ei-ther (ΓJ ,L) ∈ (Dgm( j)×Colv)\Φ( j) or (Γ /(J,L),K) ∈ (Dgm(n)×Colv)\Φ(n).Hence, according to Eq. (25) it follows that q(Γ ,K) = 0, i.e., Q = Q′′ i Q′ ∈RΦ(n− 1+ j). Thus, we have shown that the system RΦ(n)n>1 is closed withrespect to all partial compositions i.

Corollary 1. The following systems form a suboperad in R:

R1PI(n) :=

Q ∈R(n)∣∣Q(Γ ) = 0 if Γ is not 1PI

.

Let us give another example for a restriction on diagrams that induces a subop-erad. Let us consider a non-empty subset


E ⊂ Colf×2

and call it a set of admissible connections. A colored graph Γ is called E –admissibleif for all flags f ∈ flag(Γ ) such that f 6= σ( f ) we have (cf( f ),cf(σ( f ))) ∈ E . Or inother words, if the pairs of colors of the flags corresponding to the inner edges arecontained in E . As an application of Proposition 4 we get:

Corollary 2. Let E be any symmetric subset in Colf×2 and let RE (n) be the spacethat consists of all contraction maps Q ∈R(n), which vanish on all diagrams thateither are not E –admissible, or have tadpoles. Then

RE (n)

n>1 is a suboperad

of R.

Note that in Corollaries 1 and 2 the sets Φ(n) are of the form

Φ(n) = Dgm′(n)×Colv

for some subsets Dgm′(n)⊆ Dgm(n). In this case condition (27) reads

ΓJ ∈ Dgm′(|J|) and Γ /J ∈ Dgm′(n−|J|+1) ⇒ Γ ∈ Dgm′(n) .

and RΦ is

RΦ(n) =

Q ∈R(n)∣∣Q∣∣Dgm(n)\Dgm′(n)= 0

≡ HomK

(K(Dgm′(n)),KColv

)= KDgm′(n)×Colv .

Example 4. Let us introduce an example of the set E for the case of Quantum Elec-

trodynamics (QED). In this case we use three colors for flags Colf =

Theset of admissible connections is:

E =(

,),(

,)(

,)

.

The diagram of Example 2 was thus E –admissible for QED and as there we can usefor edges single colors, one with no orientation and one with orientation. The non-oriented lines are called “photon lines” and the oriented lines are called “electronlines”.

Our next “selection rule” for contraction maps is by the type of vertices. A vertexis a colored graph with one vertex and no tadpoles. So, it contains only outer edgeswhich are called corolla of the vertex. The number of the external edges of the vertexis called its valency.

Let V ⊆ Dgm(1) be a set of vertices. We call the set V types of vertices in thetheory. Let us define then the system ΦV = ΦV (n)n>1

ΦV (n) =(Γ ,L) ∈ Dgm(n)×Colv

∣∣∣ ∀J ⊆ Γ(if |J|= 1 then ΓJ ∈ V

)and

Γ /(vert(Γ ),L) ∈ V.


It follows that ΦV satisfies condition (ii) of Proposition 4 and hence,

RV := RΦV,

is a suboperad of R.Thus, a physical theory can be defined as intersection of the operads

RE ,V := R1PI∩RE ∩RV . (28)

In the next section we shall consider the main examples of physical theories.

Remark 2. If Φii∈I is a collection of systems Φi = Φi(n)n>1 each satisfyingcondition (ii) of Proposition 4 then⋂

i∈I

RΦi = RΦ where Φ = Φ(n)n>1 with Φ(n) =⋂i∈I

Φi(n) ,

and Φ also satisfies condition (ii) of Proposition 4.

Example 5. (Massless ϕn theory.) It is known in physics that the theory of ϕn,where ϕ is a real scalar field, is a renormalizable theory in space-time dimensionD if n

(D2 −1

)6D. So, for D = 2 all ϕn-theories are renormalizable. For D > 2, ϕn

is a renormalizable theory for n 6 2DD−2 . In particular, for D = 3 we have n 6 6; for

D = 4, n 6 4; for D = 5,6, n 6 3. For D > 6 there is no renormalizable ϕn-theorywith n > 2. Some of the above renormalizable theories belong to a more restrictiveclass of theories called “super–renormalizable”. In these theories only a finite num-ber of diagrams need a renormalization. The renormalizable ϕn–theories that are notsuper renormalizable are those ones for which we have equality n

(D2 −1

)= D, i.e.,

ϕ6 in D = 3, ϕ4 in D = 4 and ϕ3 in D = 6. In the latter theories the renormalizationis related to an operad of type (28) and we shall describe these cases below.

In a renormalizable (but not super-renormalizable) massless ϕn theory we haveonly one color for vertices and lines. So, the set E of admissible connections is theonly possible Colf×Colf. The set V of types of vertices in ϕn-theory contains atwo-valent vertex and a one (n− 2k)-valent vertex for all k with 2 < n− 2k 6 n.In particular, for ϕ4 in D = 4 and ϕ3 in D = 6 there are two types of vertices: atwo-valent one and an n-valent one with n = 4 and 3, respectively. For ϕ6 in D = 3there are three types of vertices with valences 2, 4 and 6 respectively.

The next two examples of renormalizable theories are in four dimensional space–time.

Example 6. (QED completed.) The pair (E ,V ) for QED is

E =

V =


In particular, we have two colors for the vertices and three colors for the flags (cf.Example 5).

One can consider also massless QED, where we have only one color for thevertices and again three colors for the flags. So, the additional color for the verticesabove corresponds to the mass of the electron.

Example 7. (Quantum Chromodynamics.) The pair (E ,V ) for Quantum Chromo-dynamics (QCD) is

E =

V =

In particular, we have one color for the vertices and three colors for the flags.

7 The group related to the contraction operad and itsrepresentation in the group of formal diffeomorphisms on thespace of interactions

Having defined a symmetric operad R for each particular QFT model we have au-tomatically a group G(R) associated to it. This group is precisely the operadic con-struction of the renormalization group.

7.1 Notions of renormalization group

There are several widespread notions of renormalization group in physics and theydo not lead to equal objects although they are closely related to each other. We shallreview below some of them. For recent related works we refer the reader to [7], [1].

In renormalization theory a physical quantity U (an observable for instance, ora correlation function in QFT) is derived as a function U = U(κ1, . . . ,κN ;ε) (≡U(

→κ ;ε)) of various parameters including:• physical constants κ1, . . . ,κN . In QFT these are called coupling constants.• An additional subsidiary parameter ε > 0 called a regularization parameter.

It makes meaningful the value of U(κ1, . . . ,κN ;ε) that is usually ill-definedfor ε → 0. The latter limit corresponds exactly to the actual physical valueof U and the purpose of the renormalization is to understand how to do it.• There might be further variables but we consider them as a “part” of U (so

that U is then valued in some vector or function space).Furthermore, in perturbation theory, one has defined U only as a formal power seriesin the coupling constants


U(→κ ;ε) =

∞

∑n=0

1n!

N

∑i1,...,in =1

Ui1,...,in(ε)κi1 · · ·κin , (29)

with coefficients Ui1,...,in(ε) that are functions in ε > 0. The renormalization issuenow is to find such a change of the physical parameters:

→κ′ =

→K(

→κ ;ε) , κ

′i =

∞

∑n=1

1n!

N

∑i1,...,in =1

→K i;i1,...,in(ε)κi1 · · ·κin , (30)

again as a formal power series, so that after the substitution2

U ren(→κ ;ε) := U

(→K(

→κ ;ε);ε

)=

∞

∑n=0

1n!

N

∑i1,...,in =1

U reni1,...,in(ε)κi1 · · ·κin , (31)

the resulting coefficients U reni1,...,in(ε) would have a finite limit for ε → 0. We set the

final renormalized physical quantity U ren to be

U ren(→κ ) := lim

ε→0U ren(

→κ ;ε) . (32)

The existence of such a formal diffeomorphism→κ ′ =

→K(

→κ ;ε) (30) for a given in

advance series U(→κ ;ε) (29) so that the limit (32) exists is far from being a trivial

statement. This phenomena is called renormalizability of U . The physical interpre-tation of this procedure is that we pass by the change (30) to a new set of couplingconstants called “renormalized couplings” so that the initial “bare couplings” be-come infinite (meaningless) for ε → 0.

Still, the above renormalization procedure has a built in ambiguity. Namely, if

we have one solution→K(

→κ ;ε) (30) of this problem then any composition

→K 1(

→κ ;ε) =

→K(→X(

→κ );ε

)with a formal diffeomorphism

→X(

→κ ) will also be a solution. Thus, the group of for-

mal diffeomorphisms of the couplings→κ appears naturally as acting on the renor-

malization schemes. This is the first notion of a renormalization group. It is simplythe group of formal diffeomorphism.

We see that the above concept of renormalization is rather general. It leads alsoto the most primary concept of a renormalization group and so, it should be re-lated to any other such notion. More precisely, any other notion of a renormalizationgroup should have a representation (a homomorphism) in the group of formal diffeo-morphisms of the coupling constants. In this case we speak about “renormalization

2 in terms of formal power series; note that the series→K(

→κ ;ε) starts from n = 1 but for U(

→κ ;ε) we

do not have such a restriction


group action”, i.e., it is an action of the corresponding group by formal diffeomor-phisms of the couplings.

We pass now to a second notion of the renormalization group that is specificfor QFT and it is finer than the above one. In QFT there are additional technicalfeatures of the renormlization procedure. Namely, each of the terms Ui1,...,in(ε) inseries (29) is additionally expanded in a finite sum labeled by a Feynman graph withn vertices. The renormalization adds to every diagram contribution a counter-termtogether with recursively determined counter-terms for subdiagrams. Without goingmore into the details we will only mention that the ambiguity in the renormaliza-tion in QFT is described exactly by contraction maps introduced in Sect. 5. So, weobtain now a finer notion of renormalization group that is formed by sequences ofcontraction maps. One further shows that the composition in this group is exactlygiven by the rule following from the operadic structure on contraction maps. Thelatter is shown in [6, Sect. 2.6] in a more general context of renormalization than thegraph-combinatorial one.

Thus, from this second perspective the renormalization group appears exactly asa group related to the contraction operad on Feynman diagrams. Then, as explainedabove, there should be related a “renormalization group action”, i.e., a homomor-phism from this group to the group of formal diffeomorphisms of the couplings.The existence and the derivation of this homomorphism follow also from the gen-eral renormalization theory and are not a part of the present work. However, ourresult is that the resulting homomorphism corresponds to an operadic morphism viathe functor established in Theorem 1. Let us summarize all this:

There is an operadic morphism, Ξ : RE ,V → End RV , from the contraction op-erad to the operad End RV over the vector space spanned by the set of type ofvertices V . The latter set indexes the set of coupling constants in the QFT modelthat is determined by the combinatorial data (E ,V ). The induced map between therelated groups

G(Ξ) : G(RE ,V

)→G

(End RV

) ∼= FDiff(RV)

(33)

coincides with the renormalization group action determined from the renormaliza-tion theory.

In the subsequent subsections we will construct the morphism Ξ : RE ,V →End RV . We shall continue our considerations on a general ground field (ring) Kbut the above application uses the case K= R.

7.2 Bosons and fermions

We introduce a subdivision of the set of fields, i.e. the set Colf of flags’ colors,into two disjoint subsets called bosons and fermions. According to this we assign(Z/2Z)–parities to the set of generators (17) of the monoid M(n). For a bosonic φ


the element φ(i) is even and for fermionic φ , φ(i) is odd. The parity of the propaga-tor Cφ ,ψ(i, j) is the sum of the parities of the coupled fields φ and ψ . Usually bosonsare coupled only to bosons and fermions - to fermions, so that the propagators arethen always even. Finally, the interactions L(i) are even as well.

Recall that we introduced in Sect. 4 g a canonical isomorphism Dgm(n)∼=M(n)between the set Dgm(n) of all classes of isomorphic enumerated colored diagramswith n vertices and the elements in the free monoid M(n) generated by the set (17).Let us introduce the linear envelope of the monoid M(n):

M (n) := K(M(n)) ∼= K(Dgm(n)) , (34)

which is thus an algebra.3 In the more general case of presence of fermions weredefine the algebra structure on M (n) (34) and set

M (n) := the graded commutative algebra generated by the set (17). (35)

Note that in all the constructions up to now the division of the fields (i.e., the setColf) into bosons and fermions is inessential.

7.3 The Wick generating operator of diagrams

Let us assume first that we have a theory only with bosons so that the algebras M (n)are commutative.

Let us have n vertices I1, . . . , In ∈V and consider them as one enumerated coloredgraph that is completely disconnected (i.e., it has no inner lines). The monomial inM(n) corresponding to this diagram is thus I1(1) · · · In(n)≡ I1⊗·· ·⊗ In, where thenumber in bracket “( j)” indicates the number assigned to the corresponding vertex.Denote

WickEn (I1, . . . , In) := ∑all possible ways of connecting the vertices

I1(1), . . . , In(n) into E –admissible enumerated coloredgraphs with no tadpoles

= I1(1) · · · In(n)+ · · · , (36)

where E ⊆ Colf×2 is a set of admissible connections as defined in Sect. 6. Thisdefines us a multilinear map

WickEn :(KV)×n→M (n) .

3 However, we remark that the algebra structure induced by the monoid structure of M(n) is quitedifferent from the algebra structure on the space of diagrams that is usually used in the Connes–Kreimer approach.


Proposition 5. ([4]) Under the isomorphism Dgm(n) ∼= M(n) (Proposition 2) thefollowing equation holds

WickEn (I1, . . . , In)

=

[∏

16 i< j6nexp

(∑

(φ ,ψ)∈E

Cφ ,ψ(i, j)∂ 2

∂φ(i)∂ψ( j)

)]I1(1) · · · In(n) . (37)

Proof. Let us denote here for short Ii(i) by Ii with a slight abuse of the notation. Wefirst note that the sum of all possible ways of connecting the vertices Ii and I j withk lines Cφ ,ψ(i, j) gives

1k!

(Cφ ,ψ(i, j)

∂ 2

∂φ(i)∂ψ( j)

)k

IiI j .

Indeed, let Ii = I′i φ(i)mi and I j = I′jψ( j)m j , where the monomials I′i and I′j do not

contain φ(i) or ψ( j). In other words, assume that Ii contains mi attached edges ofthe color φ and I j contains m j attached edges of the color ψ . Then connecting Iiand I j with k lines Cφ ,ψ(i, j) gives I′jφ(i)

mi−kI′jψ( j)m j−k and there are k!(mi

k

)(mik

)possible ways for doing this. But

1k!

(Cφ ,ψ(i, j)

∂ 2

∂φ(i)∂ψ( j)

)k

IiI j = k!(

mi

k

)(mi

k

)I′jφ(i)

mi−k I′jψ( j)m j−k .

On the other hand,

∑kφ ,i,ψ, j > 0

for all i < j andfor all (φ ,ψ) ∈ E

∏16 i< j6n

(1

kφ ,i,ψ, j!

(Cφ ,ψ(i, j)

∂ 2

∂φ(i)∂ψ( j)

)kφ ,i,ψ, j)

= ∏16 i< j6n

exp

(∑

(φ ,ψ)∈E

Cφ ,ψ(i, j)∂ 2

∂φ(i)∂ψ( j)

),

and the summand in the left hand side corresponding to kφ ,i,ψ, ji< j is the operatorthat produces the sum of all possible connections of the vertices with kφ ,i,ψ, j inneredges Cφ ,ψ(i, j).

In the presence of fermions Eq. (37) continues to generate the terms in the righthand side of Eq. (36) but with some signs that depend on the order of writing of theremaining generators of M (n). The derivatives ∂

∂φ(i) are understood as left Grass-man derivatives for odd φ(i).


7.4 Construction of operadic morphism Ξ : RE ,V → End KV

The operadic morphism Ξ : RE ,V → End KV consists of a sequence of linear maps

Ξn : RE ,V (n)→ End KV (n) ≡ Hom((KV)⊗n

,KV). (38)

The ansatz for Ξn is

Ξn(Q)(I1⊗·· ·⊗ In) = Q(

WickEn (I1, . . . , In)

)∈ KV , (39)

where Q ∈RE ,V (n)⊆R(n) is generally given by Eq. (19) and Q is then set to be

Q(Γ ) = ∑L∈Colv

q(Γ ,L)[Γ/(vert(Γ ),L)

], Q : K(Dgm(n))→KDgm(1) , (40)

i.e., Q(Γ ) contracts the diagram Γ to a sum of single vertices according to the colorprescription of Q : K(Dgm(n))→K(Colv).

Let us explain by words the meaning of Eq. (38). The value of Ξn(Q)(I1⊗·· ·⊗In) is a sum of single vertices obtained by making first a sum over all possibleways of connecting the vertices I1(1), . . . , In(n) into enumerated diagrams; then wecontract each of the terms in the latter sum to a sum of single vertices via Q. Shortlyspeaking, Ξn(Q)(I1⊗ ·· · ⊗ In) is the Q–contraction of all possible connections ofI1, . . . , In into diagrams.

Proposition 6. ([4]) Equation (40) determines an operadic morphism.

Proof. We use the notations of Eqs. (20) – (25) of Sect. 5. We should prove that

Ξn−1+ j(Q′′ i Q′

)= Ξn(Q′′)i Ξ j(Q′) .

The latter means to check that

Ξn−1+ j(Q′′ i Q′

)(I1⊗·· ·⊗ In−1+ j

)= Ξn(Q′′)

(I1⊗·· ·⊗ Ii−1⊗Ξ j(Q′)(Ii⊗·· ·⊗ Ii+ j)⊗ Ii+ j+1⊗·· ·⊗ In−1+ j

).

By Eq. (39) we rewrite the latter equation as

(Q′′ i Q′)(

WickEn−1+ j

(I1⊗·· ·⊗ In−1+ j

))= Q′′

(WickE

n(I1⊗·· ·⊗ Ii−1⊗ Q′

(WickE

j(Ii⊗·· ·⊗ Ii+ j

))⊗ Ii+ j+1⊗·· ·⊗ In−1+ j

)).

Let us explain by words as above the meaning of the latter equation. In the lefthand side we start with vertices I1(1), . . . , In(n) and make a sum over all possibleways of connecting them into enumerated diagrams. To each term Γ of this sum we


apply (Q′′ i Q′)

that contracts first the subdiagram ΓJ (J = i+ 1, . . . , i+ j) to (asum of) single vertices via Q′ and then Q′′ contracts further the resulting contracteddiagrams to a final sum of single vertices. In the right hand side we first make allpossible connections to diagrams in the subset of vertices Ia for a ∈ J and contractthem to a sum of single vertices I′ via Q′. Then we make all possible connectionsamong the vertices

I1(1), . . . , Ii(i), I′(i+1), Ii+ j+1(i+2), . . . , In−1+ j(n)

and contract them to single vertices via Q′′. It is a combinatorial statement that bothprocedures coincide.

8 Outlook

We make here a connection with the Connes–Kreimer Hopf algebra of “formal dif-feographisms” ([2]), which in details will appear in a forthcoming work.

The first step towards the comparison with the Connes–Kreimer approach is tostudy the dual (commutative) Hopf algebra to the Lie algebra associated with a sym-metric operad. In fact, it can be associated directly to a symmetric co-operad. Whenthis construction is applied to the contraction operads on diagrams we obtain a Hopfalgebra that is very close to the Connes–Kreimer Hopf algebra. However, there isan important difference. On a technical level, in our approach a subdiagram is al-ways contracted to a vertex, while in the Connes–Kreimer theory some subdiagramsthat have two external lines can be contracted also to an edge with no intermediatevertex.

The origin for this difference comes from physics. The Connes–Kreimer Hopfalgebra incorporates an additional step in the renormalization called a “field renor-malization”. Let us briefly explain this. Our set of vertices V corresponds to allthe monomials in the Lagrangian of a given QFT model. Some of these verticesof valence two correspond to quadratic terms in the Lagrangian, which are called“kinetic terms” since they basically determine the propagators. For this reason inphysics there are no physical parameters related to these terms: we always normal-ize them with some standard normalization coefficients like

12(∂φ) · (∂φ) , ψ(γ ·∂ )ψ ,

for a scalar and a spinor field, respectively (γ · ∂ being the Dirac operator). On theother hand, as a result of the renormalization the coefficients in front of these kineticterms are changed (renormalized). Then we absorb this change by a redefinitionof the field strengths. For instance, in the above examples we pass to new fieldsφ ′ = Zφ φ , ψ ′ = Zψ ψ and ψ

′ = Zψ ψ so that the kinetic terms are changed by Z2φ

andZψ Zψ , respectively, in such a way that compensate the renormalization change.


In some more details, recall that in our combinatorial language the fields are theelements of the set Colf. The vertices of the theory form the set V and it can beconsidered as a subset in the monoid M(1) and hence in the algebra M (1) (cf.Eq. (34)). By the construction, every monomial in M (1) is generated by Colv and

Colf. For example, a monomial I ∈ V ⊂M (1) has a general form I = Lk∏

i=1φk for

some L ∈ Colv and φi ∈ Colf. The multiplicative group (K∗)Colf acts on KColf viamultiplication:

φ 7→ Zφ φ , (Zφ )φ∈Colf ∈ (K∗)Colf .

This induces an action of (K∗)Colf on KV by

I (= Lk

∏i=1

φk) 7→(

k

∏i=1

Zφk

)I .

By this additional action we can compensate the action of FDiff(KV ) in some direc-tions in KV that form a subset Vkin of V , which is in one-to-one correspondence withColf, Vkin ∼= Colf. This reduces the renormalization group action to FDiff(KV \Vkin).

Acknowledgements We thank Dorothea Bahns, Kurusch Ebrahimi-Fard, Alessandra Frabetti,Klaus Fredenhagen and Raymond Stora for fruitful discussions. The work was partially supportedby the French-Bulgarian Project Rila under the contract Egide-Rila N112. N.N. thanks the CourantResearch Center ”Higher order structures in mathematics” (Gottingen) and the II. Institute forTheoretical Physics at the University of Hamburg for support and hospitality.

Appendix A. A brief introduction to the notion of operad

We shall introduce here a minimal setup for understanding the assumptions of Theo-rem 1. This is the abstract structure on the sequence of spaces

Hom

(V⊗n,V

)∞

n=1,which allows us to define the multiplication in Eq. (7) on a more abstract level. Thelatter formula contains two structures: the first one is the composition gk

(f j1⊗

·· ·⊗ f jk

)and the second one is the composition with permutations from the right.

These data together with the distinguished element id = idV ∈ Hom(V⊗1,V

)deter-

mine a symmetric operad, or just an operad (see the classical definition in [5, Sect.5.3]).

We shall consider the definitions in parallel with the main example of operads,the endomorphism operad EndV provided by the sequence

Hom

(V⊗n,V

)∞

n=1.

A symmetric operad P consists of the following data:• a sequence of (vector) spaces

P(n)

∞

n=1, each P(n) being a right Sn–module; such a sequence is also called an S-module.

• a collection of (multi–linear) maps


P(k)×P( j1)×·· ·×P( jk)γ( j1,..., jk)−→ P( j1 + · · · + jk)(

νk, µ(1)j1, . . . , µ

(k)jk

)7−→ γ(νk; µ

(1)j1, . . . ,µ

(k)jk)

(41)

called operadic compositions. Here, with a slight abuse in the notation weomitted the labels j1, . . . , jk of γ since we indicate them also as subscriptsin the arguments µ

(1)j1, . . . , µ

(k)jk

(then, the upper indices are inserted to dis-tinguish further the µ’s if necessary).• The operadic unit that is an element id ∈P1.

In the example EndV the above data are respectively:• EndV (n) := Hom(V⊗n,V ) and the right Sn–action is fn 7→ ( fn)

σn := fn σn(where σn ∈ Sn acts on V⊗n on the left via σn(u1⊗ ·· · ⊗ un) = u

σ−1n (1)⊗

·· ·⊗uσ−1n (n)).

• the operadic compositions (41) are:

EndV (k)×EndV ( j1)×·· ·×EndV ( jk)γ( j1,..., jk)−→ EndV ( j1 + · · · + jk)(

gk , f (1)j1, . . . , f (k)jk

)7−→ gk

(f (1)j1⊗·· ·⊗ f (k)jk

) ;(42)

• the operadic unit is idV ∈ EndV (1) = Hom(V,V ).

Before giving the axioms for (symmetric) operads let us draw the operadic com-

position pictorially by assigning to an element µn ∈P(n) a “black box”

eradic composition pictorially by assign to an element

box” withn inputs on the top and one output on the bottom. The operadic composition (41)is then drawn by connecting inputs with outputs and encircling the result in a newbox:

γ7→

Now we proceed with the conditions (axioms) on the data defining symmetricoperads:

• Associativity of the operadic compositions states that the two ways of per-forming the operadic compositions coincide:


=

In more details, the above law reads

γ( j1, . . . , jk)(idP(n)⊗ γ(i1,1, . . . , i1, j1)⊗·· ·⊗ γ(ik,1, . . . , ik, jk)

)= γ(i1,1, . . . , ik, jk)

(γ( j1, . . . , jk)⊗ idP(i1,1)⊗·· ·⊗ idP(ik, jk )

), (43)

where the left hand side acts on the space

P(n)⊗(P( j1)⊗P(i1,1)⊗·· ·⊗P(i1, j1)

)⊗·· ·

⊗(P( jk)⊗P(ik,1)⊗·· ·⊗P(ik, jk)

)(n = j1 + · · ·+ jk), while the right hand side acts on(P(n)⊗P( j1)⊗·· ·⊗P( jk)

)⊗P(i1,1)⊗·· ·⊗P(i1, j1)⊗ . . .

⊗P(ik,1)⊗·· ·⊗P(ik, jk) ,

so the equality in Eq. (43) is understood after the obvious reordering of thearguments.

• The operadic unit id is a two–side unit for the operadic compositions:

γ(id; µn)(≡ γ(n)(id; µn)) = µn ,

γ(µn; id⊗n)(≡ γ(1, . . . ,1)(µn; id, . . . , id)) = µn .

Thus, the first step in the generalization of Eq. (7) is to replace gk (

f j1⊗·· ·⊗f jk

)by γ(νk; µ j1 , . . . µ jk) (≡ γ( j1, . . . , jk)

(νk; µ j1 , . . . µ jk

)).

The second structure used in Eq. (7) which replaces the composition with σP isthe sequence of right actions Sn→ End

(P(n)

)op (“op” is for the reverting the orderin the multiplication). We shall write the action of σn ∈ Sn of µn ∈P(n) by (µn)

σn

and thus,(µn)

σn σ ′n =((µn)

σn)σ ′n .

The condition on these actions is that• all the γ’s are equivariant. There are two equivariance laws. The first one is

simpler:

γ( j1, . . . , jk)(νk;(µ(1)j1)σ (1)

, . . . ,(µ(k)jk)σ (k)

)

= γ( j1, . . . , jk)(νk; µ(1)j1, . . . ,(µ

(k)jk))σ (1)×···×σ (k)

, (44)


and the second one is

γ( j1, . . . , jk)((νk)σ ; µ

(1)j1, . . . ,µ

(k)jk))

= γ( jσ−1(1), . . . , jσ−1(k))(νk; µ(σ−1(1))jσ−1(1)

, . . . ,µ(σ−1(k))jσ−1(k)

)(σ) j1 ,..., jk , (45)

where (σ) j1,..., jk for σ ∈ Sk is a permutation belonging to S j1+ ···+ jk thatpermutes the blocks

(1, . . . , j1), ( j1 +1, . . . , j1 + j2), . . . .

More precisely,

(σ) j1,..., jk =(. . . , j1 + · · ·+ jσ−1(`)−1 +1, . . . , j1 + · · ·+ jσ−1(`), . . .

),

where ` = 1, . . . ,k. We note that the assignment σ 7→ (σ) j1,..., jk : Sk →S j1+ ···+ jk is not a group homomorphism but satisfies a certain chain rule:

(σσ′) j1,..., jk = (σ) j

σ ′−1(1),..., jσ ′−1(k)(σ ′) j1,..., jk .

The operads make a category. An operadic morphism ϑ : P →P ′ is a se-quence of linear maps ϑ =

ϑn : P(n)→P ′(n)

∞

n=1 such that they preserve allthe structures above:

ϑ j1+ ···+ jk γ( j1, . . . , jk) = γ( j1, . . . , jk)(ϑk×ϑ j1×·· ·×ϑ jk

), (46)

ϑ1(id) = id′ , (47)ϑn((µn)

σn)= ϑn

(µn)σn . (48)

In the case of a representation of P , that is a morphism a morphism ϑ : P →EndV , the space V is called an algebra over the operad P . In this case if µn ∈P(n) then µn := ϑ(µn) ∈ Hom(V⊗,V ), i.e., µn becomes an n-ary operation on Vand Eqs. (46)–(46) read

γ( j1, . . . , jk)(νk; µ

(1)j1, . . . ,µ

(k)jk

)= νk

(µ(1)j1×·· ·× µ

(k)jk

), (49)

id = idV , (50)

(µn)σn = µn σn . (51)

Hence, all the identities between the elements of P(n) under the γ’s and the per-mutations are transformed into relations between operations. This is the reason whythey are called “algebras” over the operad P . We see thus that an operad can betaught as a notion of type of algebras.

In particular, there is a symmetric operad Lie such that any operad morphismLie→ EndV endows V with a structure of a Lie algebra. The operadic spaces Lie(n)


are generated by two elements id ∈ Lie(1) and λ ∈ Lie(2) under the γ’s and thepermutation action. They are subject to the relations

λ τ2 = −λ ,

γ(1,2)(λ ; id,λ )+ γ(1,2)(λ ; id,λ )τ3 + γ(1,2)(λ ; id,λ )τ23 = 0 ,

where τn is the cyclic permutation in Sn. The first identity above corresponds to theanti-symmetry of the Lie bracket and the second one is the Jacobi identity. Under arepresentation Lie→ EndV the element λ is mapped to an actual Lie bracket on V .

Even more simple are the operads Ass and Com corresponding to associativealgebras and commutative associative algebras, respectively. We refer the reader to[5, Sect. 5] for further details.

Appendix B. Feynman diagrams in perturbation theory

We demonstrate in this section how the Feynman diagrams appear from the point ofview of probability theory. This interpretation is not accidental for QFT as it is wellknown that QFT has a statistical interpretation via the famous Feynman integralsover trajectories.

Let fi be a finite collection of Gaussian random variables over a measure spaceΩ with a probability measure µ0. Let us consider a perurbation of the measure µ0 ofa form 1

Z exp(−I)dµ0(ω), where Z =∫

Ω

exp(−I)dµ0(ω) is a normalization constant.

The function I in the exponent is considered as a “small” polynomial in the randomvariables fi,

I = ∑v∈V

κv ∑j1,..., j|v|

v j1,..., j|v| f j1(ω) · · · f j|v|(ω) ,

where the coefficients κv (indexed by a set V 3 v) are the parameters of the pertur-bation called “coupling constants”. The moments of the random variables fi arecentral objects of interest and they are called in physics also “correlation functions”.For the perturbed measure these moments are⟨

fi1 · · · fim⟩

(52)

=1Z

∫Ω

fi1(ω) · · · fim(ω) exp(− ∑

v∈Vκv ∑

j1,..., j|v|

v j1,..., j|v| f j1(ω) · · · f j|v|(ω))

dµ0(ω) .

Expanding formally the exponent (see Remark 3 below) in a power series of thecoupling constants we obtain

Operadic construction of the renormalization group 29⟨fi1 · · · fim

⟩=

1Z

∞

∑n=0

1n! ∑

v1,...,vn∈V

κv1 · · ·κvn (53)

× ∑j1,1,..., jn,|vn |

v j1,1,..., j1,|v1 |· · ·v j1,1,..., j1,|vn |

⟨fi1 · · · fim f j1,1 · · · f j1,|v1|

· · · f jn,1 · · · f jn,|vn|

⟩0,

where 〈F〉0 =∫

Ω

F(ω)dµ0(ω) is the mean value with respect to µ0. Now the mo-

ments of fi with respect to µ0, which appear in the right–hand side of Eq. (53) areeasily computed by the Wick theorem that is a characteristic property of a Gaussianfamily of random variables. It states that every moment of such a family is equalto the sum of products of 1– and 2– moments over all partitioning of the enteringvariables. Assuming that all 1–moments of fi are zero we obtain then

⟨fi1 · · · fim

⟩=

1Z

∞

∑n=0

1n! ∑

v1,...,vn∈V

κv1 · · ·κvn (54)

× ∑j1,1,..., jn,|vn |

∑all possible

pairing

,

where we denote by a pairing the correlation 〈 fi f j〉0. Now the terms in theright–hand side of Eq. (53) has a straightforward graphical interpretation:

which is a graph with vertices labeled by v ∈ V and edges labeled by a pairing .Thus, we can write

⟨fi1 · · · fim

⟩=

1Z

∞

∑n=0

1n! ∑

v1,...,vn∈V

κv1 · · ·κvn ∑Γn

∆Γn(i1, . . . , im) , (55)

where the sum is over all possible graphs Γn with made of the vertices v1, . . . ,vn and∆Γn(i1, . . . , im) is the expression of the corresponding term in Eq. (54). We can writeit as

∆Γn(i1, . . . , im) = ∑j’s

(n

∏k=1

v jk,1,..., jk,|vk |

)(∏

eG(se, te)

)(56)

where e runs over all edges of the graph Γn with ends labeled by fse and fte so thatG(se, te) = 〈 fse fte〉0. The normalization coefficient 1

Z in (55) is further canceled bythe sum of all ‘vacuum” diagrams, i.e., diagrams with no external lines, and we get

⟨fi1 · · · fim

⟩=

∞

∑n=0

1n! ∑

v1,...,vn∈V

κv1 · · ·κvn ∑connected Γn

∆Γn(i1, . . . , im) . (57)


Remark 3. As we mentioned the expansion (53) is formal. It is easy to realize thatthis expansion is divergent and the left hand side is non-analytic in the couplingconstant around their 0 value. However, one can prove that the obtained series isBorel summable.

The correspondence that produces Eq. (56) is called “Feynman rules” and itscombinatorial version is introduced in Sect. 4 g.

References

1. R. Brunetti, M. Duetsch, K. Fredenhagen, Adv. Theor. Math. Phys. 13 (2009) 1–562. A. Connes, D. Kreimer, Comm. Math. Phys. 199 (1998) 203–2423. A. Frabetti, J. Algebra 319 (2008), no. 1, 3774134. J.-L. Loday, N.M. Nikolov, Renormalization from the operadic point of view, in preparation5. J.-L. Loday, B. Vallette, Algebraic Operads, (2011), submitted.6. N.M. Nikolov, arXiv:0903.01877. R. Stora, Causalite et Groupes de Renormalisation Perturbatifs, in T. Boudjedaa, A. Makhlouf

(Eds.), Theorie Quantique des Champs Methode et Applications, (2007) p.67

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