New algebraic aspects of perturbative and
non-perturbative Quantum Field Theory
Christoph Bergbauer1,4 and Dirk Kreimer2,3
1Freie Universität Berlin, Institut für Mathematik II
Arnimallee 3, 14195 Berlin, Germany
2CNRS at Institut des Hautes Etudes Scientifiques
35 route de Chartres, 91440 Bures-sur-Yvette, France
3Boston University, Center for Mathematical Physics
111 Cummington Street, Boston, MA 02215, USA
4Erwin-Schrödinger-Institut
Boltzmanngasse 9, 1090 Wien, Austria
bergbau@math.fu-berlin.de, kreimer@ihes.fr
April 1, 2007
Abstract
In this expository article we review recent advances in our understand-
ing of the combinatorial and algebraic structure of perturbation theory in
terms of Feynman graphs, and Dyson-Schwinger equations. Starting from
Lie and Hopf algebras of Feynman graphs, perturbative renormalization
is rephrased algebraically. The Hochschild cohomology of these Hopf al-
gebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger
equations. We discuss recent progress in solving simple Dyson-Schwinger
equations in the high energy sector using the algebraic machinery. Finally
there is a short account on a relation to algebraic geometry and number
theory: understanding Feynman integrals as periods of mixed (Tate) mo-
tives.
1 Introduction
As elements of perturbative expansions of Quantum field theories, Feynman
graphs have been playing and still play a key role both for our conceptual
understanding and for state-of-the-art computations in particle physics. This
http://arxiv.org/abs/0704.0232v2
article is concerned with several aspects of Feynman graphs: First, the com-
binatorics of perturbative renormalization give rise to Hopf algebras of rooted
trees and Feynman graphs. These Hopf algebras come with a cohomology the-
ory and structure maps that help understand important physical notions, such
as locality of counterterms, the beta function, certain symmetries, or Dyson-
Schwinger equations from a unified mathematical point of view. This point of
view is about self-similarity and recursion. The atomic (primitive) elements in
this combinatorial approach are divergent graphs without subdivergences. They
must be studied by additional means, be it analytic methods or algebraic geom-
etry and number theory, and this is a significantly more difficult task. However,
the Hopf algebra structure of graphs for renormalization is in this sense a sub-
structure of the Hopf algebra structure underlying the relative cohomology of
graph hypersurfaces needed to understand the number-theoretic properties of
field theory amplitudes [6, 5].
2 Lie and Hopf algebras of Feynman graphs
Given a Feynman graph Γ with several divergent subgraphs, the Bogoliubov
recursion and Zimmermann’s forest formula tell how Γ must be renormalized in
order to obtain a finite conceptual result, using only local counterterms. This
has an analytic (regularization/extension of distributions) and a combinatorial
aspect. The basic combinatorial question of perturbative renormalization is to
find a good model which describes disentanglement of graphs into subdiver-
gent pieces, or dually insertion of divergent pieces one into each other, from the
point of view of renormalized Feynman rules. It has been known now for several
years that commutative Hopf algebras and (dual) Lie algebras provide such a
framework [26, 14, 15] with many ramifications in pure mathematics. From the
physical side, it is important to know that, for example, recovering aspects of
gauge/BRST symmetry [39, 37, 30, 38] and the transition to nonperturbative
equations of motion [12, 28, 29, 36, 3, 35, 32, 34, 4] are conveniently possible in
this framework, as will be discussed in subsequent sections.
In order to introduce these Lie and Hopf algebras, let us now fix a renormaliz-
able quantum field theory (in the sense of perturbation theory), given by a local
Lagrangian. A convenient first example is massless φ3 theory in 6 dimensions.
We look at its perturbative expansion in terms of 1PI Feynman graphs. Each
1PI graph Γ comes with two integers, |Γ| = rankH1(Γ), its number of loops,
and sdd(Γ), its superficial degree of divergence. As usual, vacuum and tadpole
graphs need not be considered, and the only remaining superficial divergent
graphs have exactly two or three external edges, a feature of renormalizability.
Graphs without subdivergences are called primitive. Here are two examples.
Both are superficially divergent as they have three external edges. The first
one has two subdivergences, the second one is primitive. Note that there are
infinitely many primitive graphs with three external edges. In particular, for
every n ∈ N one finds a primitive Γ such that |Γ| = n.
Let now L be the Q-vector space generated by all the superficially divergent
(sdd ≥ 0) 1PI graphs of our theory, graded by the number of loops | · |. There
is an operation on L given by insertion of graphs into each other: Let γ1, γ2 be
two generators of L. Then
γ1 ⋆ γ2 :=
n(γ1, γ2,Γ)
where n(γ1, γ2,Γ) is the number of times that γ1 shows up as a subgraph of Γ
and Γ/γ1 ∼= γ2. Here are two examples:
⋆ = + +
⋆ = 2
This definition is extended bilinearly onto all of L. Note that ⋆ respects the
grading as |γ1 ⋆ γ2| = |γ1| + |γ2|. The operation ⋆ is not in general associative.
Indeed, it is pre-Lie [14, 17]:
(γ1 ⋆ γ2) ⋆ γ3 − γ1 ⋆ (γ2 ⋆ γ3) = (γ1 ⋆ γ3) ⋆ γ2 − γ1 ⋆ (γ3 ⋆ γ3). (1)
To see that (1) holds observe that on both sides nested insertions cancel. What
remains are disjoint insertions of γ2 and γ3 into γ1 which do obviously not
depend on the order of γ2 and γ3. One defines a Lie bracket on L :
[γ1, γ2] := γ1 ⋆ γ2 − γ2 ⋆ γ1.
The Jacobi identity for [·, ·] is satisfied as a consequence of the pre-Lie property
(1) of ⋆. This makes L a graded Lie algebra. The bracket is defined by mutual
insertions of graphs. As usual, U(L), the universal envelopping algebra of L is a
cocommutative Hopf algebra. Its graded dual, in the sense of Milnor-Moore, is
therefore a commutative Hopf algebra H. As an algebra, H is free commutative,
generated by the vector space L and an adjoined unit I. By duality, one expects
the coproduct ofH to disentangle its argument into subdivergent pieces. Indeed,
one finds
∆(Γ) = I⊗ Γ + Γ⊗ I+
γ ⊗ Γ/γ. (2)
The relation γ ( Γ refers to disjoint unions γ of 1PI superficially divergent
subgraphs of Γ.Disjoint unions of graphs are in turn identified with their product
in H. For example,
= I⊗ + ⊗ I+ 2 ⊗ .
The coproduct respects the grading by the loop number, as does the product (by
definition). Therefore H =
n=0 Hn is a graded Hopf algebra. Since H0
it is connected. The counit ǫ vanishes on the subspace
n=1 Hn, called aug-
mentation ideal, and ǫ(I) = 1. As usual, if ∆(x) = I⊗ x+ x⊗ I, the element x
is called primitive. The linear subspace of primitive elements is denoted PrimH.
The interest in H and L arises from the fact that the Bogoliubov recursion
is essentially solved by the antipode of H. In any connected graded bialgebra,
the antipode S is given by
S(x) = −x−
S(x′)x′′, x /∈ H0 (3)
in Sweedler’s notation. Let now V be a C-algebra. The space of linear maps
LQ(H, V ) is equipped with a convolution product (f, g) 7→ f ∗ g = mV (f ⊗
g)∆ where mV is the product in V. Relevant examples for V are suggested by
regularization schemes such as the algebra V = C[[ǫ, ǫ−1] of Laurent series with
finite pole part for dimensional regularization (space-time dimensionD = 6+2ǫ.)
The (unrenormalized) Feynman rules provide then an algebra homomorphism
φ : H → V mapping Feynman graphs to Feynman integrals in 6+2ǫ dimensions.
On V there is a linear endomorphism R (renormalization scheme) defined, for
example minimal subtraction R(ǫn) = 0 if n ≥ 0, R(ǫn) = ǫn if n < 0. If Γ
is primitive, as defined above, then φ(Γ) has only a simple pole in ǫ, hence
(1−R)φ(Γ) is a good renormalized value for Γ. If Γ does have subdivergences,
the situation is more complicated. However, the map S
R : H → V
R(Γ) = −R
φ(Γ)−
′)φ(Γ′′)
provides the counterterm prescribed by the Bogoliubov recursion, and (S
φ)(Γ) yields the renormalized value of Γ. The map S
R is a recursive deforma-
tion of φ ◦ S by R, compare its definition with (3). These are results obtained
by one of the authors in collaboration with Connes [26, 14, 15].
For S
R to be an algebra homomorphism again, one requires R to be a Rota-
Baxter operator, studied in a more general setting by Ebrahimi-Fard, Guo and
one of the authors in [20, 22, 21]. The Rota-Baxter property is at the algebraic
origin of the Birkhoff decomposition introduced in [15, 16]. In the presence of
mass terms, or gauge symmetries etc. in the Lagrangian, φ, S
R and S
R ⋆φ may
contribute to several form factors in the usual way. This can be resolved by
considering a slight extension of the Hopf algebra containing projections onto
single structure functions, as discussed for example in [15, 32]. For the case of
gauge theories, a precise definition of the coefficients n(γ1, γ2,Γ) is given in [30].
The Hopf algebra H arises from the simple insertion of graphs into each other
in a completely canonical way. Indeed, the pre-Lie product determines the co-
product, and the coproduct determines the antipode. Like this, each quantum
field theory gives rise to such a Hopf algebra H based on its 1PI graphs. It is
no surprise then that there is an even more universal Hopf algebra behind all
of them: The Hopf algebra Hrt of rooted trees [26, 14]. In order to see this,
imagine a purely nested situation of subdivergences like
which can be represented by the rooted tree
To account for each single graph of this kind, the tree’s vertices should actually
be labeled according to which primitive graph they correspond to (plus some
gluing data) which we will suppress for the sake of simplicity. The coproduct
on Hrt – corresponding to the one (2) of H – is
∆(τ) = I⊗ τ + τ ⊗ I+
adm.c
Pc(τ) ⊗Rc(τ)
where the sum runs over all admissible cuts of the tree τ. A cut of τ is a
nonempty subset of its edges which are to be removed. A cut c(τ) is defined
to be admissible, if for each leaf l of τ at most one edge on the path from l to
the root is cut. The product of subtrees which fall down when those edges are
removed is denoted Pc(τ). The part which remains connected with the root is
denoted Rc(τ). Here is an example:
⊗ I+ I⊗
+ 2 • ⊗
+ • • ⊗
✁ ⊗ •.
Compared to Hrt, the advantage of H is however that overlapping divergences
are resolved automatically. To achieve this in Hrt requires some care [27].
3 From Hochschild cohomology to physics
There is a natural cohomology theory on H and Hrt whose non-exact 1-cocycles
play an important ”operadic” role in the sense that they drive the recursion
that define the full 1PI Green’s functions in terms of primitve graphs. In order
to introduce this cohomology theory, let A be any bialgebra. We view A as
a bicomodule over itself with right coaction (id ⊗ ǫ)∆. Then the Hochschild
cohomology of A (with respect to the coalgebra part) is defined as follows [14]:
Linear maps L : A → A⊗n are considered as n-cochains. The operator b, defined
bL := (id⊗ L)∆ +
(−1)i∆iL+ (−1)
n+1L⊗ I (4)
furnishes a codifferential: b2 = 0. Here ∆ denotes the coproduct of A and ∆i
the coproduct applied to the i-th factor in A⊗n. The map L ⊗ I is given by
x 7→ L(x) ⊗ I. Clearly this codifferential encodes only information about the
coalgebra (as opposed to the algebra) part of A. The resulting cohomology is
denoted HH•ǫ (A). For n = 1, the cocycle condition bL = 0 is simply
∆L = (id⊗ L)∆ + L⊗ I (5)
for L a linear endomorphism of A. In the Hopf algebraHrt of rooted trees (where
things are often simpler), a 1-cocycle is quickly found: the grafting operator B+,
defined by
B+(I) = •
B+(τ1 . . . τn) =
τ1 . . . τn
for trees τi
joining all the roots of its argument to a newly created root. Clearly, B+ reminds
of an operad multiplication. It is easily seen that B+ is not exact and therefore
a generator (among others) of HH1ǫ (Hrt). Foissy [23] showed that L 7→ L(I) is an
onto map HH1ǫ(Hrt) → PrimHrt. The higher Hochschild cohomology (n ≥ 2)
of Hrt is known to vanish [23]. The pair (Hrt, B+) is the universal model for all
Hopf algebras of Feynman graphs and their 1-cocycles [14]. Let us now turn to
those 1-cocycles of H. Clearly, every primitve graph γ gives rise to a 1-cocycle
+ defined as the operator which inserts its argument, a product of graphs, into
γ in all possible ways. Here is a simple example:
See [30] for the general definition involving some combinatorics of insertion
places and symmetries.
It is an important consequence of the B
+ satisfying the cocycle condition (5)
R ∗ φ)B+ = (1−R)B̃+(S
R ∗ φ) (6)
where B̃+ is the push-forward of B+ along the Feynman rules φ. In other words,
+ is the integral operator corresponding to the skeleton graph γ. This is the
combinatorial key to the proof of locality of counterterms and finiteness of renor-
malization [13, 28, 2, 3]. Indeed, equation (6) says that after treating all subdi-
vergences, an overall subtraction (1−R) suffices. The only analytic ingredient is
Weinberg’s theorem applied to the primitive graphs. In [2] it is emphasized that
H is actually generated (and determined) by the action of prescribed 1-cocycles
and the multiplication. A version of (6) with decorated trees is available which
describes renormalization in coordinate space [2].
The 1-cocycles B
+ give rise to a number of useful Hopf subalgebras of H. Many
of them are isomorphic. They are studied in [3] on the model of decorated
rooted trees, and we will come back to them in the next section. In [30] one of
the authors showed that in nonabelian gauge theories, the existence of a certain
Hopf subalgebra, generated by 1-cocycles, is closely related to the Slavnov-
Taylor identities for the couplings to hold. In a similar spirit, van Suijlekom
showed that, in QED, Ward-Takahashi identities, and in nonabelian Yang-Mills
theories, the Slavnov-Taylor identities for the couplings generate Hopf ideals I
of H such that the quotients H/I are defined and the Feynman rules factor
through them [37, 38]. The Hopf algebra H for QED had been studied before
in [11, 33, 39].
4 Dyson-Schwinger equations
The ultimate application of the Hochschild 1-cocycles introduced in the previous
section aims at non-perturbative results. Dyson-Schwinger equations, reorga-
nized using the correspondence PrimH → HH1ǫ (H), become recursive equations
inH[[α]], α the coupling constant, with contributions from (degree 1) 1-cocycles.
The Feynman rules connect them to the usual integral kernel representation. We
remain in the massless φ3 theory in 6 dimensions for the moment. Let Γ� be
the full 1PI vertex function,
Γ� = I+
res Γ=�
(normalized such that the tree level contribution equals 1). This is a formal
power series in α with values in H. Here res Γ is the result of collapsing all
internal lines of Γ. The graph res Γ is called the residue of Γ. In a renormalizable
theory, res can be seen as a map from the set of generators of H to the terms
in the Lagrangian. For instance, in the φ3 theory, vertex graphs have residue
�, and self energy graphs have residue −. The number SymΓ denotes the order
of the group of automorphisms of Γ, defined in detail for example in [30, 38].
Similarly, the full inverse propagator Γ− is represented by
Γ− = I−
res Γ=−
. (8)
These series can be reorganized by summing only over primitive graphs, with all
possible insertions into these primitive graphs. In H, the insertions are afforded
by the corresponding Hochschild 1-cocycles. Indeed,
Γ� = I+
γ∈PrimH,res γ=�
α|γ|B
�Q|γ|)
Sym γ
Γ− = I−
γ∈PrimH,res γ=−
α|γ|B
−Q|γ|)
Sym γ
. (9)
The universal invariant charge Q is a monomial in the Γr and their inverses,
where r are residues (terms in the Lagrangian) provided by the theory. In φ3
theory we have Q = (Γ�)2(Γ−)−3. In φ3 theory, the universality of Q (i. e. the
fact that the same Q is good for all Dyson-Schwinger equations of the theory)
comes from a simple topological argument. In nonabelian gauge theories how-
ever, the universality of Q takes care that the solution of the corresponding
system of coupled Dyson-Schwinger equations gives rise to a Hopf subalgebra
and therefore amounts to the Slavnov-Taylor identities for the couplings [30].
The system (9) of coupled Dyson-Schwinger equations has (7,8) as its solution.
Note that in the first equation of (9) an infinite number of cocycles contributes
as there are infinitely many primitive vertex graphs in φ36 theory – the second
equation has only finitely many contributions – here one. Before we describe
how to actually attempt to solve equations of this kind analytically (application
of the Feynman rules φ), we discuss the combinatorial ramifications of this con-
struction in the Hopf algebra. It makes sense to call all (systems of) recursive
equations of the form
X1 = I±
. . .
Xs = I±
combinatorial Dyson-Schwinger equations, and to study their combinatorics.
Here, the Bdn+ are non-exact Hochschild 1-cocycles and the Mn are monomials in
the X1 . . .Xs. In [3] we studied a large class of single (uncoupled) combinatorial
Dyson-Schwinger equations in a decorated version of Hrt as a model for vertex
insertions:
X = I+
αnwnB
where the wn ∈ Q. For example, X = I+αB+(X
2)+α2B+(X
3) is in this class.
It turns out [28, 3] that the coefficients cn of X, defined by X =
n=0 α
generate a Hopf subalgebra themselves:
∆(cn) =
Pnk ⊗ ck.
The Pnk are homogeneous polynomials of degree n−k in the cl, l ≤ n. These poly-
nomials have been worked out explicitly in [3]. One notices in particular that
the Pnk are independent of the wn and B
+ , and hence that under mild assump-
tions (on the algebraic independence of the cn) the Hopf subalgebras generated
this way are actually isomorphic. For example, X = I+αB+(X
2) +α2B+(X
and X = I + αB+(X
2) yield isomorphic Hopf subalgebras. This is an aspect
of the fact that truncation of Dyson-Schwinger equations – considering only a
finite instead of an infinite number of contributing cocycles – does make (at
least combinatorial) sense. Indeed, the combinatorics remain invariant. Similar
results hold for Dyson-Schwinger equations in the true Hopf algebra of graphs
H where things are a bit more difficult though as the cocycles there involve
some bookkeeping of insertion places.
The simplest nontrivial Dyson-Schwinger equation one can think of is the linear
X = I+ αB+(X).
Its solution is given by X =
n=0 α
n(B+)
n(I). In this case X is grouplike
and the corresponding Hopf subalgebra of cns is cocommutative [25]. A typical
and important non-linear Dyson-Schwinger equation arises from propagator
insertions:
X = I− αB+(1/X),
for example the massless fermion propagator in Yukawa theory where only the
fermion line obtains radiative corrections (other corrections are ignored). This
problem has been studied and solved by Broadhurst and one of the authors
in [12] and revisited recently by one of the authors and Yeats [35]. As we now
turn to the analytic aspects of Dyson-Schwinger equations, we briefly sketch the
general approach presented in [35] on how to successfully treat the nonlinearity
of Dyson-Schwinger equations. Indeed, the linear Dyson-Schwinger equations
can be solved by a simple scaling ansatz [25]. In any case, let γ be a primitive
graph. The following works for amplitudes which depend on a single scale, so
let us assume a massless situation with only one non-zero external momentum –
how more than one external momentum (vertex insertions) are incorporated by
enlarging the set of primitive elements is sketched in [32]. The grafting operator
+ associated to γ translates to an integral operator under the (renormalized)
Feynman rules
+)(I)(p
2/µ2) =
(Iγ(k, p)− Iγ(k, µ))dk
where Iγ is the integral kernel corresponding to γ, the internal momenta are
denoted by k, the external momentum by p, and µ is the fixed momentum at
which we subtract: R(x) = x|p2=µ2 .
In the following we stick to the special case discussed in [35] where only one
internal edge is allowed to receive corrections. The integral kernel φ(B
+) defines
a Mellin transform
F (ρ) =
Iγ(k, µ)(k
where ki is the momentum of the internal edge of γ at which insertions may
take place (here the fermion line). If there are several insertion sites, obvious
multiple Mellin transforms become necessary. The case of two (propagator) in-
sertion places has been studied, at the same example, in [35].
The function F (ρ) has a simple pole in ρ at 0. We write
F (ρ) =
We denote L = log p2/µ2. Clearly φR(X) = 1 +
n γnL
n. An important result
of [35] is that, even in the difficult nonlinear situation, the anomalous dimension
γ1 is implicitly defined by the residue r and Taylor coefficients fn of the Mellin
transform F. On the other hand, all the γn for n ≥ 2, are recursively defined
in terms of the γi, i < n. This last statement amounts to a renormalization
group argument that is afforded in the Hopf algebra by the scattering formula
of [16]. Curiously, for this argument only a linearized part of the coproduct is
needed. We refer to [35] for the actual algorithm. For a linear Dyson-Schwinger
equation, the situation is considerably simpler as the γn = 0 for n ≥ 2 since X
is grouplike [25].
Let us restate the results for the high energy sector of non-linear Dyson-Schwinger
equations [12, 35]: Primitive graphs γ define Mellin transforms via their integral
kernels B̃
+. The anomalous dimension γ1 is implicitly determined order by order
from the coefficients of those Mellin transforms. All non-leading log coefficients
γn are recursively determined by γ1, thanks to the renormalization group. This
reduces, in principle, the problem to a study of all the primitive graphs and the
intricacies of insertion places.
Finding useful representations of those Mellin transforms – even one-dimensional
ones – of higher loop order skeleton graphs is difficult. However, the two-loop
primitive vertex in massless Yukawa theory has been worked out by Bierenbaum,
Weinzierl and one of the authors in [4], a result that can be applied to other
theories as well. Combined with the algebraic treatment [12, 3, 35] sketched
in the previous paragraphs and new geometric insight on primitive graphs (see
section 5), there is reasonable hope that actual solutions of Dyson-Schwinger
equations will be more accessible in the future.
Using the Dyson-Schwinger analysis, one of the authors and Yeats [34] were
able to deduce a bound for the convergence of superficially divergent ampli-
tudes/structure functions from the (desirable) existence of a bound for the su-
perficially convergent amplitudes.
5 Feynman integrals and periods of mixed (Tate)
Hodge structures
A primitive graph Γ ∈ PrimH defines a real number rΓ, called the residue of
Γ, which is independent of the renormalization scheme. In the case that Γ is
massless and has one external momentum p, the residue rΓ is the coefficient
of log p2/µ2 in φR(Γ) = (1 − R)φ(Γ). It coincides with the coefficient r of the
Mellin transform introduced in the previous section. One may ask what kind of
a number r is, for example if it is rational or algebraic. The origin of this ques-
tion is that the irrational or transcendental numbers that show up for various
Γ strongly suggest a motivic interpretation of the rΓ. Indeed, explicit calcula-
tions [9, 10, 8] display patterns of Riemann zeta and multiple zeta values that
are known to be periods of mixed Tate Hodge structures – here the periods
are provided by the Feynman rules which produce Γ 7→ rΓ. By disproving a
related conjecture of Kontsevich, Belkale and Brosnan [1] have shown that not
all these Feynman motives must be mixed Tate, so one may expect a larger class
of Feynman periods than multiple zeta values. Our detailed understanding of
these phenomena is still far from complete, and only some very first steps have
been made in the last few years. However, techniques developed in recent work
by Bloch, Esnault and one of the authors [7] do permit reasonable insight for
some special cases which we briefly sketch in the following.
Let Γ be a logarithmically divergent massless primitive graph with one ex-
ternal momentum p. It is convenient to work in the ”Schwinger” parametric
representation [24] obtained by the usual trick of replacing propagators
dae−ak
and performing the loop integrations (Gaussian integrals) first which leaves us
with a (divergent) integral over various Schwinger parameters a. It is a classical
exercise [24, 7, 6] to show that in four dimensions, up to some powers of i and
φ(Γ) =
da1 . . . dan
e−QΓ(a,p
2)/ΨΓ(a)
Ψ2Γ(a)
where n is the number of edges of Γ. QΓ and ΨΓ are graph polynomials of Γ,
where ΨΓ, sometimes called Symanzik or Kirchhoff polynomial, is defined as
follows: Let T (Γ) be the set of spanning trees of Γ, i. e. the set of connected
simply connected subgraphs which meet all vertices of Γ. We think of the edges
e of Γ as being numbered from 1 to n. Then
t∈T (Γ)
This is a homogeneous polynomial in the ai of degree |H1(Γ)|. It is easily seen
(scaling behaviour of QΓ and ΨΓ) that rΓ =
∂φR(Γ)
∂ log p2/µ2
is extracted from φ(Γ)
by considering the ai as homogeneous coordinates of P
n−1(R) and evaluating at
p2 = 0 :
σ⊂Pn−1(R)
where σ = {[a1, . . . , an] : all ai can be choosen ≥ 0} and Ω is a volume form
on Pn−1. Let XΓ := {ΨΓ = 0} ⊂ P
n−1. If |H1(Γ)| = 1, the integrand in (10) has
no poles. If |H1(Γ)| > 1, poles will show up on the union ∆ =
γ(Γ,H1(γ) 6=0
of coordinate linear spaces Lγ = {ae = 0 for e edge of γ} – these need to
be separated from the chain of integration by blowing up. The blowups being
understood, the Feynman motive is, by abuse of notation,
Hn−1(Pn−1 −XΓ,∆−∆ ∩XΓ)
with Feynman period given by (10). See [7, 6] for details. Some particularly
accessible examples are the wheel with n spokes graphs
Γn :=
studied extensively in [7]. The corresponding Feynman periods (10) yield ratio-
nal multiples of zeta values [9]
rΓn ∈ ζ(2n− 3)Q.
Due to the simple topology of the Γn, the geometry of the pairs (XΓn ,∆Γn) are
well understood and the corresponding motives have been worked out explicitly
[7]. The methods used are however nontrivial and not immediately applicable
to more general situations.
When confronted with non-primitive graphs, i. e. graphs with subdivergences,
there are more than one period to consider. In the Schwinger parameter picture,
subdivergences arise when poles appear along exceptional divisors as pieces of
∆ are blown up. This situation can be understood using limiting mixed Hodge
structures [6], see also [31, 36] for a toy model approach to the combinatorics
involved. In [6] it is also shown how the Hopf algebra H of graphs lifts to the
category of motives. For the motivic role of solutions of Dyson-Schwinger equa-
tions we refer to work in progress. Finally we mention that there is related work
by Connes and Marcolli [18, 19] who attack the problem via Riemann-Hilbert
correspondences and motivic Galois theory.
Acknowledgements. We thank Spencer Bloch and Karen Yeats for discus-
sion on the subject of this review. The first named author (C. B.) thanks the
organizers of the ICMP 2006 and the IHES for general support. His research is
supported by the Deutsche Forschungsgemeinschaft. The IHES, Boston Univer-
sity and the Erwin-Schrödinger-Institute are gratefully acknowledged for their
kind hospitality. At the time of writing this article, C. B. is visiting the ESI as
a Junior Research Fellow.
References
[1] P. Belkale and P. Brosnan. Matroids, motives, and a conjecture of Kontse-
vich. Duke Math. J. 116, (1):147–188, 2003. math.AG/0012198.
[2] C. Bergbauer and D. Kreimer. The Hopf algebra of rooted trees in Epstein–
Glaser renormalization. Ann. Henri Poincare, 6:343–367, 2004. hep-
th/0403207.
[3] C. Bergbauer and D. Kreimer. Hopf algebras in renormalization theory: Lo-
cality and Dyson-Schwinger equations from Hochschild cohomology. IRMA
Lect. Math. Theor. Phys., 10:133–164, 2006. hep-th/0506190.
[4] I. Bierenbaum, D. Kreimer, and S. Weinzierl. The next-to-ladder approxi-
mation for Dyson-Schwinger equations. Phys. Lett. , B646:129–133, 2007.
hep-th/0612180.
[5] S. Bloch. Mixed Hodge structures and motives in physics. Talk at the
conference on Motives and Algebraic Cycles, Fields Institute, March 2007.
[6] S. Bloch. Motives Associated to Graphs. Takagi Lectures, Kyoto, November
2006. Available online at http://www.math.uchicago.edu/ bloch/.
[7] S. Bloch, H. Esnault, and D. Kreimer. On Motives Associated
to Graph Polynomials. Commun. Math. Phys., 267:181–225, 2006.
math.AG/0510011.
[8] D. Broadhurst, J. Gracey, and D. Kreimer. Beyond the triangle and
uniqueness relations: Non-zeta counterterms at large N from positive knots.
Z. Phys., C75:559–574, 1997. hep-th/9607174.
[9] D. Broadhurst and D. Kreimer. Knots and numbers in Φ4 theory to 7 loops
and beyond. Int. J. Mod. Phys., C6:519–524, 1995. hep-ph/9504352.
[10] D. Broadhurst and D. Kreimer. Association of multiple zeta values with
positive knots via feynman diagrams up to 9 loops. Phys. Lett. , B393:403–
412, 1997. hep-th/9609128.
[11] D. Broadhurst and D. Kreimer. Renormalization automated by Hopf alge-
bra. J. Symb. Comput., 27:581, 1999. hep-th/9810087.
[12] D. Broadhurst and D. Kreimer. Exact solutions of Dyson-Schwinger
equations for iterated one-loop integrals and propagator-coupling duality.
Nucl. Phys. , B600:403–422, 2001. hep-th/0012146.
[13] J. Collins. Renormalization. Cambridge Monographs on Mathematical
Physics. Cambridge University Press, 1984.
[14] A. Connes and D. Kreimer. Hopf algebras, renormalization and non-
commutative geometry. Commun. Math. Phys., 199:203–242, 1998. hep-
th/9808042.
[15] A. Connes and D. Kreimer. Renormalization in quantum field theory and
the Riemann– Hilbert problem I: The Hopf algebra structure of graphs
and the main theorem. Comm. Math. Phys., 210:249–273, 2000. hep-
th/9912092.
[16] A. Connes and D. Kreimer. Renormalization in quantum field theory and
the Riemann-Hilbert problem II: The beta-function, diffeomorphisms and
the renormalization group. Commun. Math. Phys., 216:215–241, 2001. hep-
th/0003188.
[17] A. Connes and D. Kreimer. Insertion and elimination: The doubly infinite
Lie algebra of Feynman graphs. Ann. Henri Poincaré, 3:411–433, 2002.
hep-th/0201157.
[18] A. Connes and M. Marcolli. From Physics to Number Theory via Noncom-
mutative Geometry, Part II: Renormalization, the Riemann-Hilbert cor-
respondence, and motivic Galois theory. In Frontiers in Number Theory,
Physics, and Geometry II, pages 617–713. Springer, 2006. hep-th/0411114.
[19] A. Connes and M. Marcolli. Quantum fields and motives. J. Geom. Phys. ,
56(1):55–85, 2006. hep-th/0504085.
[20] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Integrable renormalization I:
The ladder case. J. Math. Phys., 45:3758–3769, 2004. hep-th/0402095.
[21] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Spitzer’s identity and the
algebraic Birkhoff decomposition in pQFT. J. Phys., A37:11037–11052,
2004. hep-th/0407082.
[22] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Integrable renormalization II:
The general case. Ann. Henri Poincaré, 6:369–395, 2005. hep-th/0403118.
[23] L. Foissy. Les algèbres de Hopf des arbres enracinés I-II. Bull. Sci. Math.,
126:193–239 and 249–288, 2002.
[24] C. Itzykson and J. -B. Zuber. Quantum field theory. McGraw–Hill, 1980.
[25] D. Kreimer. Étude for linear Dyson-Schwinger equations. IHES Preprint
P/06/23.
[26] D. Kreimer. On the Hopf algebra structure of perturbative quantum field
theories. Adv. Theor. Math. Phys., 2:303–334, 1998. q-alg/9707029.
[27] D. Kreimer. On overlapping divergences. Commun. Math. Phys., 204:669,
1999. hep-th/9810022.
[28] D. Kreimer. Factorization in quantum field theory: An exercise in Hopf
algebras and local singularities. 2003. Contributed to Les Houches School of
Physics: Frontiers in Number Theory, Physics and Geometry, Les Houches,
France, 9-21 Mar 2003. hep-th/0306020.
[29] D. Kreimer. What is the trouble with Dyson-Schwinger equations?
Nucl. Phys. Proc. Suppl., 135:238–242, 2004. hep-th/0407016.
[30] D. Kreimer. Anatomy of a gauge theory. Annals Phys. , 321:2757–2781,
2006.
[31] D. Kreimer. The residues of quantum field theory: Numbers we should
know. In C. Consani and M. Marcolli, editors, Noncommutative Geometry
and Number Theory (Bonn, 2003), pages 187–204. Vieweg, 2006. hep-
th/0404090.
[32] D. Kreimer. Dyson Schwinger equations: From Hopf algebras to number
theory. In I. Binder and D. Kreimer, editors, Universality and Renor-
malization, volume 50 of Fields Inst. Comm., pages 225–248. AMS, 2007.
hep-th/0609004.
[33] D. Kreimer and R. Delbourgo. Using the Hopf algebra structure of QFT
in calculations. Phys. Rev. , D60:105025, 1999. hep-th/9903249.
[34] D. Kreimer and K. Yeats. Recursion and growth estimates in renormalizable
quantum field theory. hep-th/0612179.
[35] D. Kreimer and K. Yeats. An etude in non-linear Dyson-Schwinger equa-
tions. Nucl. Phys. Proc. Suppl. , 160:116–121, 2006. hep-th/0605096.
[36] I. Mencattini and D. Kreimer. The structure of the Ladder Insertion-
Elimination Lie algebra. Commun. Math. Phys., 259:413–432, 2005. math-
ph/0408053.
[37] W. van Suijlekom. The Hopf algebra of Feynman graphs in QED.
Lett. Math. Phys. , 77:265–281, 2006. hep-th/0602126.
[38] W. van Suijlekom. Renormalization of gauge fields: A Hopf algebra ap-
proach. 2006. hep-th/0610137.
[39] D. Volovich, I. and Prokhorenko. Renormalizations in quantum electro-
dynamics, and Hopf algebras. Tr. Mat. Inst. Steklova, 245(Izbr. Vopr. p-
adich. Mat. Fiz. i Anal. ):288–295, 2004.
	Introduction
	Lie and Hopf algebras of Feynman graphs
	From Hochschild cohomology to physics
	Dyson-Schwinger equations
	Feynman integrals and periods of mixed (Tate) Hodge structures

  In this expository article we review recent advances in our understanding of
the combinatorial and algebraic structure of perturbation theory in terms of
Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf
algebras of Feynman graphs, perturbative renormalization is rephrased
algebraically. The Hochschild cohomology of these Hopf algebras leads the way
to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent
progress in solving simple Dyson-Schwinger equations in the high energy sector
using the algebraic machinery. Finally there is a short account on a relation
to algebraic geometry and number theory: understanding Feynman integrals as
periods of mixed (Tate) motives.

Introduction
As elements of perturbative expansions of Quantum field theories, Feynman
graphs have been playing and still play a key role both for our conceptual
understanding and for state-of-the-art computations in particle physics. This
http://arxiv.org/abs/0704.0232v2
article is concerned with several aspects of Feynman graphs: First, the com-
binatorics of perturbative renormalization give rise to Hopf algebras of rooted
trees and Feynman graphs. These Hopf algebras come with a cohomology the-
ory and structure maps that help understand important physical notions, such
as locality of counterterms, the beta function, certain symmetries, or Dyson-
Schwinger equations from a unified mathematical point of view. This point of
view is about self-similarity and recursion. The atomic (primitive) elements in
this combinatorial approach are divergent graphs without subdivergences. They
must be studied by additional means, be it analytic methods or algebraic geom-
etry and number theory, and this is a significantly more difficult task. However,
the Hopf algebra structure of graphs for renormalization is in this sense a sub-
structure of the Hopf algebra structure underlying the relative cohomology of
graph hypersurfaces needed to understand the number-theoretic properties of
field theory amplitudes [6, 5].
2 Lie and Hopf algebras of Feynman graphs
Given a Feynman graph Γ with several divergent subgraphs, the Bogoliubov
recursion and Zimmermann’s forest formula tell how Γ must be renormalized in
order to obtain a finite conceptual result, using only local counterterms. This
has an analytic (regularization/extension of distributions) and a combinatorial
aspect. The basic combinatorial question of perturbative renormalization is to
find a good model which describes disentanglement of graphs into subdiver-
gent pieces, or dually insertion of divergent pieces one into each other, from the
point of view of renormalized Feynman rules. It has been known now for several
years that commutative Hopf algebras and (dual) Lie algebras provide such a
framework [26, 14, 15] with many ramifications in pure mathematics. From the
physical side, it is important to know that, for example, recovering aspects of
gauge/BRST symmetry [39, 37, 30, 38] and the transition to nonperturbative
equations of motion [12, 28, 29, 36, 3, 35, 32, 34, 4] are conveniently possible in
this framework, as will be discussed in subsequent sections.
In order to introduce these Lie and Hopf algebras, let us now fix a renormaliz-
able quantum field theory (in the sense of perturbation theory), given by a local
Lagrangian. A convenient first example is massless φ3 theory in 6 dimensions.
We look at its perturbative expansion in terms of 1PI Feynman graphs. Each
1PI graph Γ comes with two integers, |Γ| = rankH1(Γ), its number of loops,
and sdd(Γ), its superficial degree of divergence. As usual, vacuum and tadpole
graphs need not be considered, and the only remaining superficial divergent
graphs have exactly two or three external edges, a feature of renormalizability.
Graphs without subdivergences are called primitive. Here are two examples.
Both are superficially divergent as they have three external edges. The first
one has two subdivergences, the second one is primitive. Note that there are
infinitely many primitive graphs with three external edges. In particular, for
every n ∈ N one finds a primitive Γ such that |Γ| = n.
Let now L be the Q-vector space generated by all the superficially divergent
(sdd ≥ 0) 1PI graphs of our theory, graded by the number of loops | · |. There
is an operation on L given by insertion of graphs into each other: Let γ1, γ2 be
two generators of L. Then
γ1 ⋆ γ2 :=
n(γ1, γ2,Γ)
where n(γ1, γ2,Γ) is the number of times that γ1 shows up as a subgraph of Γ
and Γ/γ1 ∼= γ2. Here are two examples:
⋆ = + +
⋆ = 2
This definition is extended bilinearly onto all of L. Note that ⋆ respects the
grading as |γ1 ⋆ γ2| = |γ1| + |γ2|. The operation ⋆ is not in general associative.
Indeed, it is pre-Lie [14, 17]:
(γ1 ⋆ γ2) ⋆ γ3 − γ1 ⋆ (γ2 ⋆ γ3) = (γ1 ⋆ γ3) ⋆ γ2 − γ1 ⋆ (γ3 ⋆ γ3). (1)
To see that (1) holds observe that on both sides nested insertions cancel. What
remains are disjoint insertions of γ2 and γ3 into γ1 which do obviously not
depend on the order of γ2 and γ3. One defines a Lie bracket on L :
[γ1, γ2] := γ1 ⋆ γ2 − γ2 ⋆ γ1.
The Jacobi identity for [·, ·] is satisfied as a consequence of the pre-Lie property
(1) of ⋆. This makes L a graded Lie algebra. The bracket is defined by mutual
insertions of graphs. As usual, U(L), the universal envelopping algebra of L is a
cocommutative Hopf algebra. Its graded dual, in the sense of Milnor-Moore, is
therefore a commutative Hopf algebra H. As an algebra, H is free commutative,
generated by the vector space L and an adjoined unit I. By duality, one expects
the coproduct ofH to disentangle its argument into subdivergent pieces. Indeed,
one finds
∆(Γ) = I⊗ Γ + Γ⊗ I+
γ ⊗ Γ/γ. (2)
The relation γ ( Γ refers to disjoint unions γ of 1PI superficially divergent
subgraphs of Γ.Disjoint unions of graphs are in turn identified with their product
in H. For example,
= I⊗ + ⊗ I+ 2 ⊗ .
The coproduct respects the grading by the loop number, as does the product (by
definition). Therefore H =
n=0 Hn is a graded Hopf algebra. Since H0
it is connected. The counit ǫ vanishes on the subspace
n=1 Hn, called aug-
mentation ideal, and ǫ(I) = 1. As usual, if ∆(x) = I⊗ x+ x⊗ I, the element x
is called primitive. The linear subspace of primitive elements is denoted PrimH.
The interest in H and L arises from the fact that the Bogoliubov recursion
is essentially solved by the antipode of H. In any connected graded bialgebra,
the antipode S is given by
S(x) = −x−
S(x′)x′′, x /∈ H0 (3)
in Sweedler’s notation. Let now V be a C-algebra. The space of linear maps
LQ(H, V ) is equipped with a convolution product (f, g) 7→ f ∗ g = mV (f ⊗
g)∆ where mV is the product in V. Relevant examples for V are suggested by
regularization schemes such as the algebra V = C[[ǫ, ǫ−1] of Laurent series with
finite pole part for dimensional regularization (space-time dimensionD = 6+2ǫ.)
The (unrenormalized) Feynman rules provide then an algebra homomorphism
φ : H → V mapping Feynman graphs to Feynman integrals in 6+2ǫ dimensions.
On V there is a linear endomorphism R (renormalization scheme) defined, for
example minimal subtraction R(ǫn) = 0 if n ≥ 0, R(ǫn) = ǫn if n < 0. If Γ
is primitive, as defined above, then φ(Γ) has only a simple pole in ǫ, hence
(1−R)φ(Γ) is a good renormalized value for Γ. If Γ does have subdivergences,
the situation is more complicated. However, the map S
R : H → V
R(Γ) = −R
φ(Γ)−
′)φ(Γ′′)
provides the counterterm prescribed by the Bogoliubov recursion, and (S
φ)(Γ) yields the renormalized value of Γ. The map S
R is a recursive deforma-
tion of φ ◦ S by R, compare its definition with (3). These are results obtained
by one of the authors in collaboration with Connes [26, 14, 15].
For S
R to be an algebra homomorphism again, one requires R to be a Rota-
Baxter operator, studied in a more general setting by Ebrahimi-Fard, Guo and
one of the authors in [20, 22, 21]. The Rota-Baxter property is at the algebraic
origin of the Birkhoff decomposition introduced in [15, 16]. In the presence of
mass terms, or gauge symmetries etc. in the Lagrangian, φ, S
R and S
R ⋆φ may
contribute to several form factors in the usual way. This can be resolved by
considering a slight extension of the Hopf algebra containing projections onto
single structure functions, as discussed for example in [15, 32]. For the case of
gauge theories, a precise definition of the coefficients n(γ1, γ2,Γ) is given in [30].
The Hopf algebra H arises from the simple insertion of graphs into each other
in a completely canonical way. Indeed, the pre-Lie product determines the co-
product, and the coproduct determines the antipode. Like this, each quantum
field theory gives rise to such a Hopf algebra H based on its 1PI graphs. It is
no surprise then that there is an even more universal Hopf algebra behind all
of them: The Hopf algebra Hrt of rooted trees [26, 14]. In order to see this,
imagine a purely nested situation of subdivergences like
which can be represented by the rooted tree
To account for each single graph of this kind, the tree’s vertices should actually
be labeled according to which primitive graph they correspond to (plus some
gluing data) which we will suppress for the sake of simplicity. The coproduct
on Hrt – corresponding to the one (2) of H – is
∆(τ) = I⊗ τ + τ ⊗ I+
adm.c
Pc(τ) ⊗Rc(τ)
where the sum runs over all admissible cuts of the tree τ. A cut of τ is a
nonempty subset of its edges which are to be removed. A cut c(τ) is defined
to be admissible, if for each leaf l of τ at most one edge on the path from l to
the root is cut. The product of subtrees which fall down when those edges are
removed is denoted Pc(τ). The part which remains connected with the root is
denoted Rc(τ). Here is an example:
⊗ I+ I⊗
+ 2 • ⊗
+ • • ⊗
✁ ⊗ •.
Compared to Hrt, the advantage of H is however that overlapping divergences
are resolved automatically. To achieve this in Hrt requires some care [27].
3 From Hochschild cohomology to physics
There is a natural cohomology theory on H and Hrt whose non-exact 1-cocycles
play an important ”operadic” role in the sense that they drive the recursion
that define the full 1PI Green’s functions in terms of primitve graphs. In order
to introduce this cohomology theory, let A be any bialgebra. We view A as
a bicomodule over itself with right coaction (id ⊗ ǫ)∆. Then the Hochschild
cohomology of A (with respect to the coalgebra part) is defined as follows [14]:
Linear maps L : A → A⊗n are considered as n-cochains. The operator b, defined
bL := (id⊗ L)∆ +
(−1)i∆iL+ (−1)
n+1L⊗ I (4)
furnishes a codifferential: b2 = 0. Here ∆ denotes the coproduct of A and ∆i
the coproduct applied to the i-th factor in A⊗n. The map L ⊗ I is given by
x 7→ L(x) ⊗ I. Clearly this codifferential encodes only information about the
coalgebra (as opposed to the algebra) part of A. The resulting cohomology is
denoted HH•ǫ (A). For n = 1, the cocycle condition bL = 0 is simply
∆L = (id⊗ L)∆ + L⊗ I (5)
for L a linear endomorphism of A. In the Hopf algebraHrt of rooted trees (where
things are often simpler), a 1-cocycle is quickly found: the grafting operator B+,
defined by
B+(I) = •
B+(τ1 . . . τn) =
τ1 . . . τn
for trees τi
joining all the roots of its argument to a newly created root. Clearly, B+ reminds
of an operad multiplication. It is easily seen that B+ is not exact and therefore
a generator (among others) of HH1ǫ (Hrt). Foissy [23] showed that L 7→ L(I) is an
onto map HH1ǫ(Hrt) → PrimHrt. The higher Hochschild cohomology (n ≥ 2)
of Hrt is known to vanish [23]. The pair (Hrt, B+) is the universal model for all
Hopf algebras of Feynman graphs and their 1-cocycles [14]. Let us now turn to
those 1-cocycles of H. Clearly, every primitve graph γ gives rise to a 1-cocycle
+ defined as the operator which inserts its argument, a product of graphs, into
γ in all possible ways. Here is a simple example:
See [30] for the general definition involving some combinatorics of insertion
places and symmetries.
It is an important consequence of the B
+ satisfying the cocycle condition (5)
R ∗ φ)B+ = (1−R)B̃+(S
R ∗ φ) (6)
where B̃+ is the push-forward of B+ along the Feynman rules φ. In other words,
+ is the integral operator corresponding to the skeleton graph γ. This is the
combinatorial key to the proof of locality of counterterms and finiteness of renor-
malization [13, 28, 2, 3]. Indeed, equation (6) says that after treating all subdi-
vergences, an overall subtraction (1−R) suffices. The only analytic ingredient is
Weinberg’s theorem applied to the primitive graphs. In [2] it is emphasized that
H is actually generated (and determined) by the action of prescribed 1-cocycles
and the multiplication. A version of (6) with decorated trees is available which
describes renormalization in coordinate space [2].
The 1-cocycles B
+ give rise to a number of useful Hopf subalgebras of H. Many
of them are isomorphic. They are studied in [3] on the model of decorated
rooted trees, and we will come back to them in the next section. In [30] one of
the authors showed that in nonabelian gauge theories, the existence of a certain
Hopf subalgebra, generated by 1-cocycles, is closely related to the Slavnov-
Taylor identities for the couplings to hold. In a similar spirit, van Suijlekom
showed that, in QED, Ward-Takahashi identities, and in nonabelian Yang-Mills
theories, the Slavnov-Taylor identities for the couplings generate Hopf ideals I
of H such that the quotients H/I are defined and the Feynman rules factor
through them [37, 38]. The Hopf algebra H for QED had been studied before
in [11, 33, 39].
4 Dyson-Schwinger equations
The ultimate application of the Hochschild 1-cocycles introduced in the previous
section aims at non-perturbative results. Dyson-Schwinger equations, reorga-
nized using the correspondence PrimH → HH1ǫ (H), become recursive equations
inH[[α]], α the coupling constant, with contributions from (degree 1) 1-cocycles.
The Feynman rules connect them to the usual integral kernel representation. We
remain in the massless φ3 theory in 6 dimensions for the moment. Let Γ� be
the full 1PI vertex function,
Γ� = I+
res Γ=�
(normalized such that the tree level contribution equals 1). This is a formal
power series in α with values in H. Here res Γ is the result of collapsing all
internal lines of Γ. The graph res Γ is called the residue of Γ. In a renormalizable
theory, res can be seen as a map from the set of generators of H to the terms
in the Lagrangian. For instance, in the φ3 theory, vertex graphs have residue
�, and self energy graphs have residue −. The number SymΓ denotes the order
of the group of automorphisms of Γ, defined in detail for example in [30, 38].
Similarly, the full inverse propagator Γ− is represented by
Γ− = I−
res Γ=−
. (8)
These series can be reorganized by summing only over primitive graphs, with all
possible insertions into these primitive graphs. In H, the insertions are afforded
by the corresponding Hochschild 1-cocycles. Indeed,
Γ� = I+
γ∈PrimH,res γ=�
α|γ|B
�Q|γ|)
Sym γ
Γ− = I−
γ∈PrimH,res γ=−
α|γ|B
−Q|γ|)
Sym γ
. (9)
The universal invariant charge Q is a monomial in the Γr and their inverses,
where r are residues (terms in the Lagrangian) provided by the theory. In φ3
theory we have Q = (Γ�)2(Γ−)−3. In φ3 theory, the universality of Q (i. e. the
fact that the same Q is good for all Dyson-Schwinger equations of the theory)
comes from a simple topological argument. In nonabelian gauge theories how-
ever, the universality of Q takes care that the solution of the corresponding
system of coupled Dyson-Schwinger equations gives rise to a Hopf subalgebra
and therefore amounts to the Slavnov-Taylor identities for the couplings [30].
The system (9) of coupled Dyson-Schwinger equations has (7,8) as its solution.
Note that in the first equation of (9) an infinite number of cocycles contributes
as there are infinitely many primitive vertex graphs in φ36 theory – the second
equation has only finitely many contributions – here one. Before we describe
how to actually attempt to solve equations of this kind analytically (application
of the Feynman rules φ), we discuss the combinatorial ramifications of this con-
struction in the Hopf algebra. It makes sense to call all (systems of) recursive
equations of the form
X1 = I±
. . .
Xs = I±
combinatorial Dyson-Schwinger equations, and to study their combinatorics.
Here, the Bdn+ are non-exact Hochschild 1-cocycles and the Mn are monomials in
the X1 . . .Xs. In [3] we studied a large class of single (uncoupled) combinatorial
Dyson-Schwinger equations in a decorated version of Hrt as a model for vertex
insertions:
X = I+
αnwnB
where the wn ∈ Q. For example, X = I+αB+(X
2)+α2B+(X
3) is in this class.
It turns out [28, 3] that the coefficients cn of X, defined by X =
n=0 α
generate a Hopf subalgebra themselves:
∆(cn) =
Pnk ⊗ ck.
The Pnk are homogeneous polynomials of degree n−k in the cl, l ≤ n. These poly-
nomials have been worked out explicitly in [3]. One notices in particular that
the Pnk are independent of the wn and B
+ , and hence that under mild assump-
tions (on the algebraic independence of the cn) the Hopf subalgebras generated
this way are actually isomorphic. For example, X = I+αB+(X
2) +α2B+(X
and X = I + αB+(X
2) yield isomorphic Hopf subalgebras. This is an aspect
of the fact that truncation of Dyson-Schwinger equations – considering only a
finite instead of an infinite number of contributing cocycles – does make (at
least combinatorial) sense. Indeed, the combinatorics remain invariant. Similar
results hold for Dyson-Schwinger equations in the true Hopf algebra of graphs
H where things are a bit more difficult though as the cocycles there involve
some bookkeeping of insertion places.
The simplest nontrivial Dyson-Schwinger equation one can think of is the linear
X = I+ αB+(X).
Its solution is given by X =
n=0 α
n(B+)
n(I). In this case X is grouplike
and the corresponding Hopf subalgebra of cns is cocommutative [25]. A typical
and important non-linear Dyson-Schwinger equation arises from propagator
insertions:
X = I− αB+(1/X),
for example the massless fermion propagator in Yukawa theory where only the
fermion line obtains radiative corrections (other corrections are ignored). This
problem has been studied and solved by Broadhurst and one of the authors
in [12] and revisited recently by one of the authors and Yeats [35]. As we now
turn to the analytic aspects of Dyson-Schwinger equations, we briefly sketch the
general approach presented in [35] on how to successfully treat the nonlinearity
of Dyson-Schwinger equations. Indeed, the linear Dyson-Schwinger equations
can be solved by a simple scaling ansatz [25]. In any case, let γ be a primitive
graph. The following works for amplitudes which depend on a single scale, so
let us assume a massless situation with only one non-zero external momentum –
how more than one external momentum (vertex insertions) are incorporated by
enlarging the set of primitive elements is sketched in [32]. The grafting operator
+ associated to γ translates to an integral operator under the (renormalized)
Feynman rules
+)(I)(p
2/µ2) =
(Iγ(k, p)− Iγ(k, µ))dk
where Iγ is the integral kernel corresponding to γ, the internal momenta are
denoted by k, the external momentum by p, and µ is the fixed momentum at
which we subtract: R(x) = x|p2=µ2 .
In the following we stick to the special case discussed in [35] where only one
internal edge is allowed to receive corrections. The integral kernel φ(B
+) defines
a Mellin transform
F (ρ) =
Iγ(k, µ)(k
where ki is the momentum of the internal edge of γ at which insertions may
take place (here the fermion line). If there are several insertion sites, obvious
multiple Mellin transforms become necessary. The case of two (propagator) in-
sertion places has been studied, at the same example, in [35].
The function F (ρ) has a simple pole in ρ at 0. We write
F (ρ) =
We denote L = log p2/µ2. Clearly φR(X) = 1 +
n γnL
n. An important result
of [35] is that, even in the difficult nonlinear situation, the anomalous dimension
γ1 is implicitly defined by the residue r and Taylor coefficients fn of the Mellin
transform F. On the other hand, all the γn for n ≥ 2, are recursively defined
in terms of the γi, i < n. This last statement amounts to a renormalization
group argument that is afforded in the Hopf algebra by the scattering formula
of [16]. Curiously, for this argument only a linearized part of the coproduct is
needed. We refer to [35] for the actual algorithm. For a linear Dyson-Schwinger
equation, the situation is considerably simpler as the γn = 0 for n ≥ 2 since X
is grouplike [25].
Let us restate the results for the high energy sector of non-linear Dyson-Schwinger
equations [12, 35]: Primitive graphs γ define Mellin transforms via their integral
kernels B̃
+. The anomalous dimension γ1 is implicitly determined order by order
from the coefficients of those Mellin transforms. All non-leading log coefficients
γn are recursively determined by γ1, thanks to the renormalization group. This
reduces, in principle, the problem to a study of all the primitive graphs and the
intricacies of insertion places.
Finding useful representations of those Mellin transforms – even one-dimensional
ones – of higher loop order skeleton graphs is difficult. However, the two-loop
primitive vertex in massless Yukawa theory has been worked out by Bierenbaum,
Weinzierl and one of the authors in [4], a result that can be applied to other
theories as well. Combined with the algebraic treatment [12, 3, 35] sketched
in the previous paragraphs and new geometric insight on primitive graphs (see
section 5), there is reasonable hope that actual solutions of Dyson-Schwinger
equations will be more accessible in the future.
Using the Dyson-Schwinger analysis, one of the authors and Yeats [34] were
able to deduce a bound for the convergence of superficially divergent ampli-
tudes/structure functions from the (desirable) existence of a bound for the su-
perficially convergent amplitudes.
5 Feynman integrals and periods of mixed (Tate)
Hodge structures
A primitive graph Γ ∈ PrimH defines a real number rΓ, called the residue of
Γ, which is independent of the renormalization scheme. In the case that Γ is
massless and has one external momentum p, the residue rΓ is the coefficient
of log p2/µ2 in φR(Γ) = (1 − R)φ(Γ). It coincides with the coefficient r of the
Mellin transform introduced in the previous section. One may ask what kind of
a number r is, for example if it is rational or algebraic. The origin of this ques-
tion is that the irrational or transcendental numbers that show up for various
Γ strongly suggest a motivic interpretation of the rΓ. Indeed, explicit calcula-
tions [9, 10, 8] display patterns of Riemann zeta and multiple zeta values that
are known to be periods of mixed Tate Hodge structures – here the periods
are provided by the Feynman rules which produce Γ 7→ rΓ. By disproving a
related conjecture of Kontsevich, Belkale and Brosnan [1] have shown that not
all these Feynman motives must be mixed Tate, so one may expect a larger class
of Feynman periods than multiple zeta values. Our detailed understanding of
these phenomena is still far from complete, and only some very first steps have
been made in the last few years. However, techniques developed in recent work
by Bloch, Esnault and one of the authors [7] do permit reasonable insight for
some special cases which we briefly sketch in the following.
Let Γ be a logarithmically divergent massless primitive graph with one ex-
ternal momentum p. It is convenient to work in the ”Schwinger” parametric
representation [24] obtained by the usual trick of replacing propagators
dae−ak
and performing the loop integrations (Gaussian integrals) first which leaves us
with a (divergent) integral over various Schwinger parameters a. It is a classical
exercise [24, 7, 6] to show that in four dimensions, up to some powers of i and
φ(Γ) =
da1 . . . dan
e−QΓ(a,p
2)/ΨΓ(a)
Ψ2Γ(a)
where n is the number of edges of Γ. QΓ and ΨΓ are graph polynomials of Γ,
where ΨΓ, sometimes called Symanzik or Kirchhoff polynomial, is defined as
follows: Let T (Γ) be the set of spanning trees of Γ, i. e. the set of connected
simply connected subgraphs which meet all vertices of Γ. We think of the edges
e of Γ as being numbered from 1 to n. Then
t∈T (Γ)
This is a homogeneous polynomial in the ai of degree |H1(Γ)|. It is easily seen
(scaling behaviour of QΓ and ΨΓ) that rΓ =
∂φR(Γ)
∂ log p2/µ2
is extracted from φ(Γ)
by considering the ai as homogeneous coordinates of P
n−1(R) and evaluating at
p2 = 0 :
σ⊂Pn−1(R)
where σ = {[a1, . . . , an] : all ai can be choosen ≥ 0} and Ω is a volume form
on Pn−1. Let XΓ := {ΨΓ = 0} ⊂ P
n−1. If |H1(Γ)| = 1, the integrand in (10) has
no poles. If |H1(Γ)| > 1, poles will show up on the union ∆ =
γ(Γ,H1(γ) 6=0
of coordinate linear spaces Lγ = {ae = 0 for e edge of γ} – these need to
be separated from the chain of integration by blowing up. The blowups being
understood, the Feynman motive is, by abuse of notation,
Hn−1(Pn−1 −XΓ,∆−∆ ∩XΓ)
with Feynman period given by (10). See [7, 6] for details. Some particularly
accessible examples are the wheel with n spokes graphs
Γn :=
studied extensively in [7]. The corresponding Feynman periods (10) yield ratio-
nal multiples of zeta values [9]
rΓn ∈ ζ(2n− 3)Q.
Due to the simple topology of the Γn, the geometry of the pairs (XΓn ,∆Γn) are
well understood and the corresponding motives have been worked out explicitly
[7]. The methods used are however nontrivial and not immediately applicable
to more general situations.
When confronted with non-primitive graphs, i. e. graphs with subdivergences,
there are more than one period to consider. In the Schwinger parameter picture,
subdivergences arise when poles appear along exceptional divisors as pieces of
∆ are blown up. This situation can be understood using limiting mixed Hodge
structures [6], see also [31, 36] for a toy model approach to the combinatorics
involved. In [6] it is also shown how the Hopf algebra H of graphs lifts to the
category of motives. For the motivic role of solutions of Dyson-Schwinger equa-
tions we refer to work in progress. Finally we mention that there is related work
by Connes and Marcolli [18, 19] who attack the problem via Riemann-Hilbert
correspondences and motivic Galois theory.
Acknowledgements. We thank Spencer Bloch and Karen Yeats for discus-
sion on the subject of this review. The first named author (C. B.) thanks the
organizers of the ICMP 2006 and the IHES for general support. His research is
supported by the Deutsche Forschungsgemeinschaft. The IHES, Boston Univer-
sity and the Erwin-Schrödinger-Institute are gratefully acknowledged for their
kind hospitality. At the time of writing this article, C. B. is visiting the ESI as
a Junior Research Fellow.
References
[1] P. Belkale and P. Brosnan. Matroids, motives, and a conjecture of Kontse-
vich. Duke Math. J. 116, (1):147–188, 2003. math.AG/0012198.
[2] C. Bergbauer and D. Kreimer. The Hopf algebra of rooted trees in Epstein–
Glaser renormalization. Ann. Henri Poincare, 6:343–367, 2004. hep-
th/0403207.
[3] C. Bergbauer and D. Kreimer. Hopf algebras in renormalization theory: Lo-
cality and Dyson-Schwinger equations from Hochschild cohomology. IRMA
Lect. Math. Theor. Phys., 10:133–164, 2006. hep-th/0506190.
[4] I. Bierenbaum, D. Kreimer, and S. Weinzierl. The next-to-ladder approxi-
mation for Dyson-Schwinger equations. Phys. Lett. , B646:129–133, 2007.
hep-th/0612180.
[5] S. Bloch. Mixed Hodge structures and motives in physics. Talk at the
conference on Motives and Algebraic Cycles, Fields Institute, March 2007.
[6] S. Bloch. Motives Associated to Graphs. Takagi Lectures, Kyoto, November
2006. Available online at http://www.math.uchicago.edu/ bloch/.
[7] S. Bloch, H. Esnault, and D. Kreimer. On Motives Associated
to Graph Polynomials. Commun. Math. Phys., 267:181–225, 2006.
math.AG/0510011.
[8] D. Broadhurst, J. Gracey, and D. Kreimer. Beyond the triangle and
uniqueness relations: Non-zeta counterterms at large N from positive knots.
Z. Phys., C75:559–574, 1997. hep-th/9607174.
[9] D. Broadhurst and D. Kreimer. Knots and numbers in Φ4 theory to 7 loops
and beyond. Int. J. Mod. Phys., C6:519–524, 1995. hep-ph/9504352.
[10] D. Broadhurst and D. Kreimer. Association of multiple zeta values with
positive knots via feynman diagrams up to 9 loops. Phys. Lett. , B393:403–
412, 1997. hep-th/9609128.
[11] D. Broadhurst and D. Kreimer. Renormalization automated by Hopf alge-
bra. J. Symb. Comput., 27:581, 1999. hep-th/9810087.
[12] D. Broadhurst and D. Kreimer. Exact solutions of Dyson-Schwinger
equations for iterated one-loop integrals and propagator-coupling duality.
Nucl. Phys. , B600:403–422, 2001. hep-th/0012146.
[13] J. Collins. Renormalization. Cambridge Monographs on Mathematical
Physics. Cambridge University Press, 1984.
[14] A. Connes and D. Kreimer. Hopf algebras, renormalization and non-
commutative geometry. Commun. Math. Phys., 199:203–242, 1998. hep-
th/9808042.
[15] A. Connes and D. Kreimer. Renormalization in quantum field theory and
the Riemann– Hilbert problem I: The Hopf algebra structure of graphs
and the main theorem. Comm. Math. Phys., 210:249–273, 2000. hep-
th/9912092.
[16] A. Connes and D. Kreimer. Renormalization in quantum field theory and
the Riemann-Hilbert problem II: The beta-function, diffeomorphisms and
the renormalization group. Commun. Math. Phys., 216:215–241, 2001. hep-
th/0003188.
[17] A. Connes and D. Kreimer. Insertion and elimination: The doubly infinite
Lie algebra of Feynman graphs. Ann. Henri Poincaré, 3:411–433, 2002.
hep-th/0201157.
[18] A. Connes and M. Marcolli. From Physics to Number Theory via Noncom-
mutative Geometry, Part II: Renormalization, the Riemann-Hilbert cor-
respondence, and motivic Galois theory. In Frontiers in Number Theory,
Physics, and Geometry II, pages 617–713. Springer, 2006. hep-th/0411114.
[19] A. Connes and M. Marcolli. Quantum fields and motives. J. Geom. Phys. ,
56(1):55–85, 2006. hep-th/0504085.
[20] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Integrable renormalization I:
The ladder case. J. Math. Phys., 45:3758–3769, 2004. hep-th/0402095.
[21] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Spitzer’s identity and the
algebraic Birkhoff decomposition in pQFT. J. Phys., A37:11037–11052,
2004. hep-th/0407082.
[22] K. Ebrahimi-Fard, L. Guo, and D. Kreimer. Integrable renormalization II:
The general case. Ann. Henri Poincaré, 6:369–395, 2005. hep-th/0403118.
[23] L. Foissy. Les algèbres de Hopf des arbres enracinés I-II. Bull. Sci. Math.,
126:193–239 and 249–288, 2002.
[24] C. Itzykson and J. -B. Zuber. Quantum field theory. McGraw–Hill, 1980.
[25] D. Kreimer. Étude for linear Dyson-Schwinger equations. IHES Preprint
P/06/23.
[26] D. Kreimer. On the Hopf algebra structure of perturbative quantum field
theories. Adv. Theor. Math. Phys., 2:303–334, 1998. q-alg/9707029.
[27] D. Kreimer. On overlapping divergences. Commun. Math. Phys., 204:669,
1999. hep-th/9810022.
[28] D. Kreimer. Factorization in quantum field theory: An exercise in Hopf
algebras and local singularities. 2003. Contributed to Les Houches School of
Physics: Frontiers in Number Theory, Physics and Geometry, Les Houches,
France, 9-21 Mar 2003. hep-th/0306020.
[29] D. Kreimer. What is the trouble with Dyson-Schwinger equations?
Nucl. Phys. Proc. Suppl., 135:238–242, 2004. hep-th/0407016.
[30] D. Kreimer. Anatomy of a gauge theory. Annals Phys. , 321:2757–2781,
2006.
[31] D. Kreimer. The residues of quantum field theory: Numbers we should
know. In C. Consani and M. Marcolli, editors, Noncommutative Geometry
and Number Theory (Bonn, 2003), pages 187–204. Vieweg, 2006. hep-
th/0404090.
[32] D. Kreimer. Dyson Schwinger equations: From Hopf algebras to number
theory. In I. Binder and D. Kreimer, editors, Universality and Renor-
malization, volume 50 of Fields Inst. Comm., pages 225–248. AMS, 2007.
hep-th/0609004.
[33] D. Kreimer and R. Delbourgo. Using the Hopf algebra structure of QFT
in calculations. Phys. Rev. , D60:105025, 1999. hep-th/9903249.
[34] D. Kreimer and K. Yeats. Recursion and growth estimates in renormalizable
quantum field theory. hep-th/0612179.
[35] D. Kreimer and K. Yeats. An etude in non-linear Dyson-Schwinger equa-
tions. Nucl. Phys. Proc. Suppl. , 160:116–121, 2006. hep-th/0605096.
[36] I. Mencattini and D. Kreimer. The structure of the Ladder Insertion-
Elimination Lie algebra. Commun. Math. Phys., 259:413–432, 2005. math-
ph/0408053.
[37] W. van Suijlekom. The Hopf algebra of Feynman graphs in QED.
Lett. Math. Phys. , 77:265–281, 2006. hep-th/0602126.
[38] W. van Suijlekom. Renormalization of gauge fields: A Hopf algebra ap-
proach. 2006. hep-th/0610137.
[39] D. Volovich, I. and Prokhorenko. Renormalizations in quantum electro-
dynamics, and Hopf algebras. Tr. Mat. Inst. Steklova, 245(Izbr. Vopr. p-
adich. Mat. Fiz. i Anal. ):288–295, 2004.
	Introduction
	Lie and Hopf algebras of Feynman graphs
	From Hochschild cohomology to physics
	Dyson-Schwinger equations
	Feynman integrals and periods of mixed (Tate) Hodge structures
