%% `Hektor': On the antipode of Connes-Kreimer's Hopf algebra of graphs
%%  2 January 2003 (HF + JMGB)


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\title{The uses of Connes and Kreimer's algebraic formulation\\
of renormalization theory}

\author{
H\'ector Figueroa\dag\ and
Jos\'e M. Gracia-Bond\'{\i}a\ddag
\\[1pc]
\dag\,Department of Mathematics, Universidad de Costa Rica,\\
2060 San Pedro, Costa Rica
\\[1pc]
\ddag\,Department of Physics, Universidad de Costa Rica,\\
2060 San Pedro, Costa Rica\\
and\\
Department of Theoretical Physics, Universidad de Zaragoza,\\
50009 Zaragoza, Spain}

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%% Abbreviations:

\newcommand{\Cc}{\mathcal{C}}      %% a chain
\newcommand{\D}{\mathcal{D}}       %% another chain
\newcommand{\Dl}{\Delta}           %% coproduct
\newcommand{\dl}{\delta}           %% depth grading
\newcommand{\eps}{\varepsilon}     %% abbreviation for \varepsilon
\newcommand{\F}{\mathcal{F}}       %% a forest
\newcommand{\FF}{\mathbb{F}}       %% real-or-complex field
\newcommand{\Ga}{\Gamma}           %% abbreviation for \Gamma
\newcommand{\ga}{\gamma}           %% abbreviation for \gamma
\renewcommand{\H}{\mathcal{H}}     %% a bialgebra
\DeclareMathOperator{\Hom}{Hom}    %% homomorphism space
\newcommand{\id}{\mathrm{id}}      %% identity operation
\renewcommand{\L}{\mathcal{L}}     %% a set of lines
\newcommand{\la}{\lambda}          %% abbreviation for \lambda
\newcommand{\longto}{\mathop{\longrightarrow}\limits} %% arrow+rider
\newcommand{\Om}{\Omega}           %% abbreviation for \Omega
\newcommand{\ox}{\otimes}          %% tensor product
\newcommand{\oxyox}{\otimes\cdots\otimes} %% repeated tensor product
\newcommand{\row}[3]{{#1}_{#2},\dots,{#1}_{#3}} %% list:  a_1,...,a_n
\newcommand{\sepword}[1]{\qquad\mbox{#1}\qquad} %% well-spaced words
\newcommand{\set}[1]{\{\,#1\,\}}   %% set notation
\newcommand{\sg}{\sigma}           %% abbreviation for \sigma
\newcommand{\Th}{\Theta}           %% abbreviation for  \Theta
\newcommand{\tsum}{\mathop{\textstyle\sum}\nolimits} %% small sum-opr
\newcommand{\V}{\mathcal{V}}       %% a set of vertices

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\begin{document}

\maketitle

\begin{abstract}
We show how, modulo the distinction between the antipode and the
``twisted'' or ``renormalized'' antipode, Connes and Kreimer's
algebraic paradigm trivializes the proofs of equivalence of the
(corrected) Dyson--Salam, Bogoliubov--Parasiuk--Hepp and Zimmermann
procedures for renormalizing Feynman amplitudes. We discuss the
outlook for a parallel simplification of computations in quantum field
theory, stemming from the same algebraic approach.
\end{abstract}

\medskip

\noindent \textit{Keywords}:
Feynman diagrams, renormalization, graded Hopf algebras, antipodes.

\noindent PACS numbers: 11.10.Gh, 02.20.Uw


\section{Introduction}

The present authors have dealt in a Hopf algebraic context with the
relation between the Dyson--Salam, Bogoliubov--Parasiuk and Zimmermann
renormalization schemes in~\cite{Ananke}. This we did using the
algebra of rooted trees $H_R$~\cite{ConnesKrTrees} as a proxy for the
complexities of the combinatorics of Feynman graphs in renormalization.

The point of~\cite{Ananke} was that the differences of the diverse
schemes could largely be tracked down to avatars of the convolution
operation in spaces of homomorphisms of the Hopf algebra. This was
illustrated by the apparently simple-minded, but tremendously
effective, computation of antipode images in $H_R$ by means of the
convolution geometric series.

In this paper we return on the subject, now in terms of Hopf algebras
of the Feynman graphs themselves.

It must be acknowledged that the deeply conceptual approach by Connes
and Kreimer till now has failed to impress many physicists who
practice renormalization theory. For that approach to become
mainstream, it should be shown to simplify both \textit{proofs} and
\textit{calculations} in perturbative renormalization theory.

The proof given by Zimmermann~\cite{Zimmermann} of the equivalence of
his forest formula and Bogo\-liubov and Parasiuk's scheme has a
reputation of difficulty. The Dyson--Salam scheme has been thoroughly
analyzed and purged of difficulties in~\cite{Manoukian}. But the proof
of equivalence between the forest and the (corrected) Dyson--Salam
formulae given in~\cite{Manoukian} is anything but simple.

In contrast, here we show that the equivalence between all the
aforementioned schemes boils down to downright basic facts in Hopf
algebra theory. So basic in fact, that we can only conclude that the
combinatorics of perturbative renormalization finds its definitive
expression in Hopf algebraic terms. Not only the equivalence proofs
result from uniqueness of the Hopf antipode, but we are able to show
that the Dyson--Salam scheme corresponds \textit{identically} to the
convolution geometric series. All our arguments are elementary and
short, much more so, by the way, than in~\cite{Ananke}: in the
previous paper, we did not quite see the forest for the trees.

At the end of the paper we briefly review the perspectives for the
Connes--Kreimer method to simplify other theorems and
\textit{computations} in renormalization, by means of reduction to the
case of elements that are primitive (i.e., without subdivergences).
Here the situation is more mixed; still, the hope remains that
sizeable simplifications can be gleaned from the algebraic approach.


\section{Bialgebras of graphs}

The basics of graded bialgebra theory are recalled in the Appendix.
{}From now on, we assume the reader is familiar with them.

Bialgebras of Feynman graphs, encoding the combinatorics of
renormalization, were introduced by Connes and Kreimer
in~\cite{ConnesKrRHI}. The precise definition we use in this paper was
first given in~\cite{Etoile}. To fix ideas, and whenever an example is
given, we think of the (massless) $\varphi^4_4$ scalar model.
Nevertheless, the constructions hold in any given quantum field
theory, such as the $\varphi^3_6$ model considered
in~\cite{ConnesKrRHI}.

We recall that a \textit{graph} or diagram $\Ga$ of the theory is
specified by a set $\V(\Ga)$ of \textit{vertices} and a set $\L(\Ga)$
of \textit{lines} (propagators) among them; \textit{external} lines
are attached to only one vertex each, \textit{internal} lines to two.
Diagrams with no external lines will not be taken into account ---and
in $\varphi^4_4$ theory only graphs with an even number of external
lines are to be found. Also tadpole diagrams, in which a line connects
a vertex to itself, are excluded in this paper.

Given a graph $\Ga$, a \textit{subdiagram} $\ga$ of $\Ga$ is specified
by a subset of at least two elements of $\V(\Ga)$ and a subset of the
lines that join these vertices in $\Ga$. By exception, the empty
subset $\emptyset$ will be admitted as a subdiagram of $\Ga$. As well
as $\Ga$ itself. Clearly, the external lines for a subdiagram $\ga$
include not only a subset of original incident lines, but some
internal lines of $\Ga$ not included in $\ga$. The connected pieces of
$\Ga$ are the maximal connected subdiagrams. A diagram is
\textit{proper} (or 1PI) when the number of its connected pieces would
not increase on the removal of a single internal line; otherwise it is
called \textit{improper}. An improper graph is the union of proper
components plus subdiagrams containing a single line.

A \textit{subgraph} of a proper graph is a subdiagram that contains
all the elements of $\L(\Ga)$ joining its vertices in the whole graph;
as such, it is determined solely by the vertices. When a subdiagram
contains several connected pieces, each one of them being a subgraph,
we still call it a subgraph. A subgraph of an improper graph $\Ga$,
distinct from $\Ga$ itself, is a proper subdiagram each of whose
components is a subgraph with respect to the proper components of
$\Ga$.

We write $\ga \subseteq \Ga$ if and only if $\ga$ is a subgraph of
$\Ga$ as defined (not just a subdiagram): this is the really important
concept for us. For renormalization in configuration
space~\cite{Carme}, it is more convenient to deal with subgraphs than
with more general subdiagrams. Zimmermann showed long ago that only
subtractions corresponding to subgraphs need be used~\cite{ZimVW}, and
this dispenses us from dealing with subdiagrams that are not
subgraphs.

Two subgraphs $\ga_1,\ga_2$ of $\Ga$ are said to be
\textit{nonoverlapping} when $\ga_1 \cap \ga_2 = \emptyset$ or
$\ga_1 \subseteq \ga_2$ or $\ga_2 \subseteq \ga_1$; otherwise they are
overlapping. Given $\ga \subseteq \Ga$, the quotient graph or cograph
$\Ga/\ga$ (reduced graph in Zimmermann's parlance) is defined by
shrinking $\ga$ in $\Ga$ to a point, that is to say, $\ga$ (bereft of
its external lines) is considered as a vertex of $\Ga$, and all the
lines in $\Ga$ not belonging to $\ga$ belong to $\Ga/\ga$. This is
modified in the obvious way when $\ga$ represents a propagator
correction. The graphs $\Ga$ and $\Ga/\ga$ have the same external
structure. A nonempty $\Ga/\ga$ will be proper iff $\Ga$ is proper
---the situation considered in~\cite{ConnesKrRHI}.

Now, the bialgebra $\H$ is defined as the polynomial algebra generated
by the empty set $\emptyset$ and the connected Feynman graphs that are
(superficially) divergent and/or have (superficially) divergent
subgraphs (renormalization parts in Zimmermann's parlance), with set
union as the product operation (hence $\emptyset$ is the unit element
$1\in\H$). The counit is given by $\eps(\Ga) := 0$ on any generator,
except $\eps(\emptyset) = 1$.

The really telling operation is the coproduct $\Dl: \H \to \H \ox \H$;
as it is to be a homomorphism of the algebra structure, we need only
define it on connected diagrams. By definition, the coproduct of $\Ga$
is given by
\begin{equation}
\Dl \Ga\, := \Ga \ox 1 + 1 \ox \Ga
+ \sum_{\emptyset\varsubsetneq\ga\varsubsetneq\Ga} \ga \ox \Ga/\ga\,
= \sum_{\emptyset \subseteq \ga \subseteq \Ga} \ga \ox \Ga/\ga.
\label{eq:coprod} %% (1)
\end{equation}
The sum is over all divergent, proper, not necessarily connected
subgraphs of $\Ga$, such that \textit{each piece} is divergent,
including (and then with the possible exception of, as $\Ga$ need not
be divergent nor proper) the empty set and $\Ga$ itself. We put
$\Ga/\Ga = 1$. When appropriate, the sum runs also over different
types of local counterterms associated to $\ga$~\cite{ConnesKrRHI};
this is not needed in our example model.

It is natural to exclude the appearance of tadpole parts in $\Ga/\ga$,
and this we do hereafter. This tadpole-free condition was not used or
remarked in~\cite{Etoile}, nor in~\cite{GudrunIII}, which employs the
same definition. We show in Figure~\ref{fg:tadpole-corr} how his
situation can happen. The cograph corresponding to the ``bikini''
subgraph in the upper part of the graph in
Figure~\ref{fg:tadpole-corr} is a tadpole correction and can be
outlawed from the coproduct.

\begin{figure}[ht]
\begin{center}
\vspace{2pc}
\parbox{5pc}{
\begin{picture}(50,18)
\put(-20,5){$\Delta$}
\put(-10,5){$\Biggl($}
\put(25,10){\circle{40}}
\put(25,-10){\circle{20}}
\qbezier(25,30)(35,10)(50,20)
\qbezier(25,30)(15,10)(0,20)
\put(52,5){$\Biggr)$}
\end{picture}
}\qquad
would contain the term
\qquad
\parbox{7pc}{
\begin{picture}(80,10)
\put(0,5){\line(-1,2){5}}
\put(0,5){\line(-1,-2){5}}
\put(10,5){\circle{20}}
\put(30.2,5){\circle{20}}
\put(40.2,5){\line(1,2){5}}
\put(40.2,5){\line(1,-2){5}}
\put(52,2){$\otimes$}
\put(85,5){\circle{30}}
\put(85,-10){\circle{15}}
\put(85,20){\line(3,2){10}}
\put(85,20){\line(-3,2){10}}
\put(82,18){$\bullet$}
\end{picture}
}
\end{center}
\vspace{1pc}
\caption{Cograph which is a tadpole part}
\label{fg:tadpole-corr}
\vspace{1pc}
\end{figure}

For the proof of the bialgebra properties of $\H$, we refer
to~\cite{Polaris}; for graphical examples of coproducts,
see~\cite{Etoile}.

Actually $\H$ is a connected, graded bialgebra. Obvious grading
operators are available: if $\#(\Ga)$ denotes the number of vertices
in $\Ga$ (i.e., the coupling order), then we define the degree of a
generator (connected element) $\Ga$ as $\nu(\Ga) := \#(\Ga) - 1$; the
degree of a product is the sum of the degrees of the factors. This
grading is compatible with the coproduct, and clearly scalars are the
only degree~0 elements. Other gradings are by the number $I(\Ga)$ of
internal lines in $\Ga$ and by loop number
$\ell(\Ga) := I(\Ga) - \nu(\Ga)$. For the $\varphi^4_4$ model,
$\ell(\Ga) = \nu(\Ga) + 1$ for two-point graphs and
$\ell(\Ga) = \nu(\Ga)$ for four-point graphs. A more relevant grading
will emerge in the next section.

Lurking in the background there is a character (i.e., multiplicative)
map $f$ (the ``Feynman rule'') of $\H$ into an algebra $V$ of Feynman
amplitudes: for instance, in dimensional regularization the character
takes values in a ring of Laurent series (with finite order poles) in
the regularization parameter. In physics, the Feynman rules are
essentially fixed by the interpretation of the theory, and thus one
tends to identify $\Ga$ with $f(\Ga)$.


\section{The importance of convolution}

Given a unital algebra $(A,m,u)$ and a counital coalgebra
$(C,\Dl,\eps)$ over $\FF$, the \textit{convolution} of two elements
$f,g$ of the vector space of $\FF$-linear maps $\Hom(C,A)$ is defined
as the map $f * g \in \Hom(C,A)$ given by the composition
$$
C \longto^\Dl C \ox C \longto^{f\ox g} A \ox A \longto^m A.
$$
In other words, $f * g = m (f \ox g) \Dl$. This product
turns $\Hom(C,A)$ into a unital algebra, where the unit is the map
$u \eps$, as is easily checked. In particular, linear endomorphisms of
a bialgebra can be convolved.

A bialgebra $H$ in which the identity map $\id_H$ is invertible under
convolution is called a \textit{Hopf algebra}, and its convolution
inverse $S$ is called the coinverse or \textit{antipode}; that is to
say, $\id_H * S = S * \id_H = u \eps$. The antipode is clearly unique.
It is known to be of order two for commutative bialgebras. Also, in a
commutative bialgebra $S$ is a homomorphism: $S(ab) = S(a)S(b)$.

In particular, if $\Dl(a) = \sum_j a'_j \ox  a''_j$, then
\begin{equation}
\eps(a) 1_H = u \eps(a) = m(S \ox \id)\Dl(a) = \tsum_j S(a'_j) a''_j,
\label{eq:antipode}% (2)
\end{equation}
and likewise $\eps(a) 1_H = \sum_j a'_j S(a''_j)$. Since any left
inverse under convolution automatically equals any right inverse
provided both exist, any map $S$ satisfying \eqref{eq:antipode} is the
antipode.

The main outcome of equation~\eqref{eq:gr-coprod} in the Appendix for
connected graded bialgebras is that these \textit{are always Hopf}.
Indeed, as is done in~\cite{Ananke}, one can try to compute the
antipode $S\colon H \to H$ by exploiting its very definition as the
convolution inverse of the identity in $H$, via a geometric series:
\begin{equation}
S := (\id)^{*-1} = (u \eps -(u \eps -\id))^{*-1}
      = u \eps + (u \eps -\id) + (u \eps -\id)^{*2}
      + \cdots
\label{eq:geom-series}% (3)
\end{equation}


\begin{prop}% Proposition 1.
Let $H$ be a connected, graded bialgebra, then the geometric series
expansion of $S(a)$ has at most $n+1$ terms when $a \in H^{(n)}$.
\end{prop}

\begin{proof}
If $a \in H^{(0)}$ the claim holds since $(u \eps - \id)1 = 0$.
Assume that the claim holds for the elements of $H^{(k)}$ when
$k \leq n-1$, and let $a \in H^{(n)}$; then by~\eqref{eq:gr-coprod}
\begin{align*}
(u \eps -\id)^{*(n+1)}(a)
&= (u \eps -\id) * (u \eps -\id)^{*n}(a)
\\
&= m [(u \eps -\id) \ox (u \eps -\id)^{*n}] \Dl(a)
\\
&= m [(u \eps -\id) \ox (u \eps -\id)^{*n}]
(a \ox 1 + 1 \ox a + \Dl'a).
\end{align*}
The first two terms vanish because $(u \eps - \id)1 = 0$. By the
induction hypothesis each of the summands of the third term are also
zero.
\end{proof}

As a corollary, connected graded bialgebras are always Hopf, with
antipode indeed given by the geometric series~\eqref{eq:geom-series}.
One of the advantages of this formulation is that we obtain fully
explicit formulae for $S$ from the coproduct.


\begin{prop}% Proposition 2.
If $a \in H^{(n)}$, $\Dl'(a) = \sum_{j_1} a'_{j_1} \ox a''_{j_1}$,
$\Dl'(a'_{j_1}) = \sum_{j_2} a'_{j_1j_2} \ox a''_{j_1j_2}$, and in
general $\Dl'(a'_{\row j1k})
= \sum_{j_{k+1}} a'_{\row j1{k+1}} \ox a''_{\row j1{k+1}}$, then for
$1 \leq k\leq n -1$,
\begin{equation}
(u \eps -\id)^{*k+1}(a) = (-1)^{k+1} \sum_{\row j1k}
a'_{\row j1k} \, a''_{\row j1k} \cdots a''_{j_1 j_2} \, a''_{j_1}.
\label{eq:many-conv}% (4)
\end{equation}
\end{prop}

\begin{proof}
To abbreviate we use the notation $\sg := u\eps - \id$. Then
$\sg(a) = -a$ if $a \in H^{(n)}$ with $n \geq 1$, because then
$\eps(a) = 0$. Moreover,
\begin{align*}
\sg^{*2}(a) &= m (\sg \ox \sg) (a \ox 1 + 1 \ox a + \Dl'a)
\\
&= \sum_{j_1} \sg(a'_{j_1}) \, \sg(a''_{j_1})
   = \sum_{j_1} a'_{j_1} \, a''_{j_1},
\end{align*}
so the statement holds for $k = 1$. If the statement holds for $k-1$,
then
\begin{align*}
\sg^{*k+1}(a) &= m (\sg^{*k} \ox \sg) (a \ox 1 + 1 \ox a + \Dl'a)
\\
&= \sum_{j_1} \sg^{*k}(a'_{j_1}) \, \sg(a''_{j_1})
   = - \sum_{j_1} \sg^{*k}(a'_{j_1}) \, a''_{j_1}
\\
&=  (-1)^{k+1} \sum_{{\row j1k}}
a'_{\row j1k} \, a''_{\row j1k} \cdots a''_{j_1 j_2} \, a''_{j_1},
\end{align*}
since, by the induction hypothesis,
$\sg^{*k}(a'_{j_1}) = (-1)^k \sum_{{\row j2k}}
a'_{j_1,\row j2k} \, a''_{j_1,\row j2k} \cdots a''_{j_1,j_2}$.
\end{proof}


If $\Dl'_i$ denotes the map $H^{\ox i} \to H^{\ox i+1}$ where $\Dl'$
is applied on the first tensor factor only, and $m_i$ the map
$H^{\ox(i+1)} \to H^{\ox i}$, where $m$ is applied on the first two
tensor factors only, then we can rewrite \eqref{eq:many-conv} as
\begin{equation}
(u \eps -\id)^{*k+1} = (-1)^{k+1} m\,m_2 \cdots m_k\,
\Dl'_k \cdots \Dl'_2 \Dl'.
\label{eq:many-conv-bis}% (5)
\end{equation}
By splitting the powers in the form $\sg^{*(k+1)} = \sg * \sg^k$, one
can obtain a twin formula of \eqref{eq:many-conv}, on which the
coproduct is applied successively on the last tensor factor, instead
of on the first factor; this formula was given in~\cite{Ananke}.

\smallskip

While all the foregoing is happily elementary, the following important
fact must be registered: a new grading on $H$ has been obtained,
defined simply by declaring the degree of a generator $a$ as $k$ when
$\sg^{*k}(a) \neq 0$ and $\sg^{*k+1}(a) = 0$. It is easily seen that
this indeed defines a grading $\dl$; the degree of a product is the
sum of the degrees of the factors.

For a connected diagram $\Ga \in \H$ this grading coincides with
the maximal length of a chain of subdivergences inside $\Ga$ (see
below), and in this context we call it \textit{depth}. We can
combine $\dl$ with the gradings $\#$ or $\ell$ to obtain a
bidegree: note that we have proved $\dl(\Ga) \leq\ \#(\Ga)$, for
any $\Ga$.

This grading by depth is the same $k$-primitivity grading already
pondered in the seminal paper~\cite{KreimerOriginal} and studied in
the context of $H_R$ by Broadhurst and Kreimer~\cite{BK}. Here the
concept is even more pertinent, as the correlation between loop number
and depth in field theory is weaker than the correlation between the
number of tree vertices and $k$-primitivity in $H_R$: for instance,
it is well known that in the $\varphi^4_4$ model there are three
5-loop diagrams which are (1-)primitive~\cite{GreenBook}.

We write $\H^{(k)}$ for the space of elements of primitivity
degree~$k$.

\smallskip

Anticipating the following discussion, note that there are other ways
to show that a connected graded bialgebra is a Hopf algebra. One can
take advantage of the equation $m (S\ox\id) \Dl(a) = 0$ whenever
$a \in H^{(n)}$ for $n \ge 1$, to introduce in the context the
Bogoliubov recursive formula:
\begin{equation}
S_B(a) := - a - \sum_{j} S_B(a'_j) a''_j,
\label{eq:Bogol-recur}% (6)
\end{equation}
if $\Dl'a = \sum_j a'_j \ox  a''_j$.


\begin{prop}% Proposition 3.
\label{pr:Bogol-antp}
If $H$ is a connected, graded bialgebra, then $S(a) = S_B(a)$.
\end{prop}

\begin{proof}
The statement holds, by a direct check, if $a \in H^{(1)}$. Assume
that $S(b) = S_B(b)$ whenever $b \in H^{(k)}$ with $k \leq n$, and let
$a \in H^{(n+1)}$. Then
\begin{align*}
S(a) &= \sg(a) + \sum_{i=1}^n \sg^{*i} * \sg(a)
    = -a + m \biggl( \sum_{i=1}^n \sg^{*i} \ox \sg \biggr) \Dl(a)
\\
&= -a + m \sum_{i=1}^n \sg^{*i} \ox \sg (a \ox 1 + 1 \ox a + \Dl'a)
\\
&= -a + \sum_j \sum_{i=1}^n \sg^{*i} (a'_j) \, \sg(a''_j)
  = -a - \sum_j \sum_{i=1}^n \sg^{*i} (a'_j) \, a''_j
\\
&=  -a - \sum_j S_B(a'_j) \,a''_j = S_B(a),
\end{align*}
where the penultimate equality uses the inductive hypothesis.
\end{proof}


Taking into account that we can also write
$S(a) = \sg(a) + \sum_{i=1}^n \sg * \sg^{*i}(a)$, it follows that the
twin formula
$$
S'_B(a) := - a - \sum_{j} a'_j S'_B(a''_j),
$$
also provides an expression for the antipode.

We record~\eqref{eq:Bogol-recur} in the language of the bialgebra $\H$
of graphs
\begin{equation}
S(\Ga) := - \Ga -
\sum_{\emptyset \varsubsetneq \ga \varsubsetneq \Ga} S(\ga) \, \Ga/\ga.
\label{eq:Bogol-recurgraph}% (7)
\end{equation}
For a primitive diagram, $S(\Ga) = -\Ga$.

\smallskip

Now it is time to reveal our strategy. Perhaps the main path-breaking
insight of~\cite{KreimerOriginal} and subsequent papers by Kreimer and
coworkers is the introduction of the ``twisted antipode''. Let us
usher in the other personages of this drama. There is a linear map
$T\colon V \to V$, which effects the subtraction of ultraviolet
divergencies in each renormalization scheme.

The twisted (or ``renormalized'') antipode $S_{T,f}$ is a map
$\H \to V$ defined by $S_{T,f}(\emptyset) = 1$; $S_{T,f} = T\circ
f\circ S$ for primitive diagrams, and then recursively:
$$
S_{T,f} \Ga = -[T\circ f]\Ga - T\biggl[
\sum_{\emptyset\varsubsetneq\ga\varsubsetneq\Ga}
S_{T,f}(\ga) \, f(\Ga/\ga) \biggr].
$$
In other words, $S_{T,f}$ is the map that produces the counterterms in
perturbative field theory. The Hopf algebra approach works most
effectively because in many cases $S_{T,f}$ is multiplicative; for
that, it is not necessary for $T$ to be an endomorphism of the algebra
of amplitudes $V$, but the following weaker
condition~\cite{ConnesKrRHI,Calypso,KreimerChen} is sufficient:
$$
T(hg) = T(T(h)g) + T(hT(g)) - T(h)T(g).
$$
This condition endows $\H$ with the structure of a Rota--Baxter algebra
(see~\cite{Dirkoflate} and references therein); it is fulfilled in the
BPHZ formalism and the dimensional regularization scheme with minimal
subtraction, for which the present paradigm is most cleanly
formulated~\cite{CKII}.

Finally, the renormalization map $R_{T,f}$ is given by
$$
R_{T,f} := S_{T,f} * f.
$$
%%% in this order!
In view of a previous remark, $R_{T,f}$ is also a homomorphism;
compatibility with the coproduct operation is given by its very
definition as a convolution.

(In Epstein--Glaser renormalization, things are a bit more
complicated, since $S_{T,f}$ is not properly defined, and $R_{T,f}$
involves a map between two different spaces of Feynman amplitudes;
still, the homomorphism condition for $R_{T,f}$ can by enforced, and
$R_{T,f}$ is compatible with the Hopf algebra structure in a suitable
sense~\cite{Flora}.)

In what follows, we shall assume that $S_{T,f}$ has been defined to be
a homomorphism, and we concentrate on the computation of $S$.
According to the dictum, Hopf algebras simplify combinatorics by
reducing it to algebra. Connes and Kreimer's algebraic approach to the
renormalization schemes separates neatly their combinatorics from the
analytical procedures and renders the first an essentially trivial
application of Hopf algebra.

\smallskip


Now, for the combinatorial aspect in renormalization theory, there are
on the market mainly the recursive formula by Bogoliubov, Zimmermann's
forest formula and the corrected Dyson--Salam formula. The last one is
most natural in the context of the primitivity grading. They just
amount to different ways to compute the antipode. It must be already
evident that the recursive formula by Bogoliubov corresponds to the
definition of $S_B$. Now, in order to prove the equivalence of two
combinatorial schemes, it is enough to prove that both yield the
antipode, either directly, as in the proof of Proposition~3, or by
using the uniqueness of the antipode. This we systematically proceed
to do in the sequel. That the coming proofs are all short, utterly
simple, or decidedly trivial, is our main point and asset.

It turns out, and this is perhaps the most illuminating result, that
the Dyson--Salam scheme corresponds identically (i.e., without need of
further cancellations) to the geometric series formula.


\section{Convolution and the Dyson--Salam formula}

The present framework applies to proper and improper graphs. For
brevity, in what follows we concentrate on proper graphs.


\begin{defn}% Definition 1.
A \textit{chain} $\Cc$ of a proper, connected graph $\Ga$ is a sequence
$\emptyset \varsubsetneq \ga_1 \varsubsetneq \ga_2 \varsubsetneq\cdots
\varsubsetneq \ga_k \varsubsetneq \Ga$ of proper, divergent,
\textit{not necessarily connected} subgraphs of $\Ga$. We denote by
$C(\Ga)$ the set of chains of $\Ga$. The \textit{length} of a chain
$\Cc$ is the number $l(\Cc) = k + 1 =: |\Cc| + 1$, and we write
$\Om(\Cc) := \ga_1\,(\ga_2/\ga_1)\dots(\ga_{k-1}/\ga_k)\,(\Ga/\ga_k)$.

With this notation we can define the antipode as follows:
\begin{equation}
S_{DS}(\Ga) := \sum_{\Cc \in C(\Ga)} (-1)^{l(\Cc)} \Om(\Cc).
\label{eq:ds-antip}% (8)
\end{equation}
\end{defn}


This definition corresponds, on the one hand, to the correct version of
the Dyson--Salam formula for renormalization. On the other hand,
formula~\eqref{eq:ds-antip} is totally analogous to the explicit
expression for the antipode given by Schmitt for his incidence Hopf
algebras~\cite{Schmitt}.


\begin{prop}% Proposition 4.
\label{pr:DS-antp}
$S_{DS}$ so defined is an antipode for $\H$.
\end{prop}

\begin{proof}
We prove that $S_{DS}$ is an inverse, under convolution, of $\id$. By
definition,
\begin{align*}
S_{DS} * \id (\Ga)
&= \sum_{\ga \subseteq \Ga} S_{DS}(\ga) \, \Ga/\ga
= S_{DS}(\Ga) + \sum_{\emptyset \subseteq  \ga \varsubsetneq \Ga}
S_{DS}(\ga) \, \Ga/\ga
\\
&= S_{DS}(\Ga) + \sum_{\emptyset \subseteq  \ga \varsubsetneq \Ga}
\sum_{\D \in C(\ga)}  (-1)^{l(\D)} \Om(\D) \, \Ga/\ga.
\end{align*}

Now, if $\D \in C(\ga)$, say $\D = \{\row{\ga}1k\}$, then
$\Cc = \{\row{\ga}1k,\ga\} \in C(\Ga)$. Moreover,
\begin{equation}
\Om(\Cc) = \Om(\D) \, \Ga/\ga,  \sepword{and}  l(\Cc) = l(\D) + 1.
\label{eq:chain-length}% (9)
\end{equation}
On the other hand, given a chain $\Cc = \{\row{\ga}1n\} \in C(\Ga)$,
then $\D = \{\row{\ga}1{n-1}\} \in C(\ga_n)$, and
\eqref{eq:chain-length} holds. Therefore
$$
S_{DS} * \id (\Ga)
= S_{DS}(\Ga) - \sum_{\Cc \in C(\Ga)} (-1)^{l(\Cc)} \Om(\Cc)
= 0 = u \eps(\Ga);
$$
in other words, $S_{DS}$ is a left inverse for $\id$, and therefore
it is an antipode.
\end{proof}

As a corollary $S = S_B = S_{DS}$. Nevertheless, it is more
instructive to check that $S = S_{DS}$ \textit{identically}, as
follows.


\begin{prop}% Proposition 5.
\label{pr:DS-antpbis}
$S_{DS}$ coincides with $S$ without cancellations.
\end{prop}

\begin{proof}
First, given a proper, connected graph $\Ga$, we rewrite
$$
S_{DS}(\Ga) := \sum_k (-1)^{k+1} \sum_{\Cc \in C_k(\Ga)} \Om(\Cc),
$$
where $C_k(\Ga)$ denote the set of chains of length $k+1$. Thus, it is
enough to prove that
$$
(-1)^{k+1} \sum_{\Cc \in C_k(\Ga)} \Om(\Cc) =
(u \eps -\id)^{*(k+1)}(\Ga) = \sg^{*(k+1)}(\Ga).
$$
To prove this first we notice that
\begin{equation}
\sum_{\emptyset\varsubsetneq\ga_1 \varsubsetneq\ga_2
\cdots\varsubsetneq\ga_k}
\ga_1 \ox \ga_2/\ga_1 \oxyox \ga_k/\ga_{k-1} \ox \Ga/\ga_k
= \Dl'_k \cdots \Dl'_2 \Dl'(\Ga).
\label{eq:foldup}% (10)
\end{equation}
Indeed, by definition of the coproduct the statement is true for
$k = 1$. Moreover, if \eqref{eq:foldup} holds for $k-1$, then
\begin{align*}
\sum_{\emptyset\varsubsetneq\ga_1 \varsubsetneq\ga_2
\cdots\varsubsetneq\ga_k \varsubsetneq\Ga}
&\ga_1 \ox \ga_2/\ga_1 \oxyox \ga_k/\ga_{k-1} \ox \Ga/\ga_k
\\
&= \sum_{\emptyset \varsubsetneq \ga_2 \varsubsetneq \ga_3
\cdots \varsubsetneq \ga_k  \varsubsetneq \Ga}
\Dl'_k(\ga_2 \ox \ga_3/\ga_2 \oxyox \ga_k/\ga_{k-1} \ox
\Ga/\ga_k)
\\
&= \Dl'_k \biggl( \sum_{\emptyset\varsubsetneq\ga_1 \varsubsetneq\ga_2
\cdots\varsubsetneq \ga_{k-1} \varsubsetneq \Ga}
\ga_1 \ox \ga_2/\ga_1 \oxyox \ga_{k-1}/\ga_{k-2} \ox\Ga/\ga_{k-1}
\biggr)
\\
&= \Dl'_k \Dl'_{k-1} \cdots \Dl'_2 \Dl'(\Ga).
\end{align*}
Thus, by \eqref{eq:many-conv-bis}
\begin{align*}
(-1)^{k+1} \sum_{\Cc \in C_k(\Ga)} \Om(\Cc)
&= (-1)^{k+1}
\sum_{\emptyset\varsubsetneq\ga_1 \varsubsetneq\ga_2
\cdots\varsubsetneq\ga_k \varsubsetneq\Ga}
\ga_1 \, \bigl(\ga_2/\ga_1\bigr) \cdots
\bigl( \ga_k/\ga_{k-1} \bigr) \, \bigl(\Ga/\ga_k \bigr)
\\
&= (-1)^{k+1} m\,m_2 \cdots m_k\,\Dl'_k \cdots \Dl'_2 \Dl' (\Ga)
= \sg^{*(k+1)}(\Ga).
\qquad\qquad\qed
\end{align*}
\hideqed
\end{proof}

The proof shows how the chains are generated from the coproduct.

\smallskip

As an application, we obtain a nonrecursive formula for
$S_{T,f}$: if $\Cc = \{\row{\ga}1k\}$ is a chain in $C(\Ga)$, write
$$
\Om_{T,f}(\Cc) := T\biggl(T\biggl[ \cdots T\Bigl(T \Bigl[
T\bigl(f(\ga_1)\bigr) \, f(\ga_1/\ga_2)\Bigr] f(\ga_2/\ga_3)\Bigr)
\cdots f(\ga_{k-1}/\ga_k) \biggr] f(\Ga/\ga_k)\biggr).
$$
Let us use the temporary notation
$$
\tilde S_{T,f}\Ga :=
\sum_{\Cc \in C(\Ga)} (-1)^{l(\Cc)}\Om_{T,f}(\Cc)
= \sum_{k=0}^{\dl(\Ga)}
    (-1)^{k+1} \sum_{\Cc \in C_k(\Ga)} \Om_{T,f}(\Cc).
$$


\begin{prop}% Proposition 6.
\label{pr:twisted-DS}
$S_{T,f} = \tilde S_{T,f}$.
\end{prop}

\begin{proof}
We shall proceed by induction on the bidegree (no other method seems
available here). A simple check gives the statement for
$\Ga \in \H^{(1)}$. Assume the claim is true for graphs in $\H^{(l)}$
with $l \leq k$, and let $\Ga \in \H^{(k+1)}$, then
\begin{align*}
S_{T,f}\Ga
&= -[T\circ f]\Ga - T\Biggl[
\sum_{\emptyset\varsubsetneq\ga\varsubsetneq\Ga}
S_{T,f}(\ga) f(\Ga/\ga) \Biggr]
\\
&= -[T\circ f]\Ga - T\Biggl[
\sum_{\emptyset\varsubsetneq\ga\varsubsetneq\Ga} \sum_{\D \in C(\ga)}
(-1)^{l(\D)}\Om_{T,f}(\D)  f(\Ga/\ga) \Biggr],
\end{align*}
since each $\ga \in \H^{(l)}$ for some $l \le k$. Now, if $\D \in
C(\ga)$, then $\Cc = \D \cup \{\ga\} \in C(\Ga)$,
$$
l(\Cc) = l(\D) + 1,  \sepword{and}
\Om_{T,f}(\Cc) = T\bigl[\Om_{T,f}(\D) \, f(\Ga/\ga)\bigr].
$$
Conversely, if $\Cc = \{\row{\ga}1n\} \in C(\Ga)$ is not the trivial
chain $\{\emptyset\}$, then $\D = \{\row{\ga}1{n-1}\}$ is a chain in
$C(\ga_n)$, and
$$
(-1)^{l(\Cc)}\Om_{T,f}(\Cc)
= - T \bigl[ (-1)^{l(\D)}\Om_{T,f}(\D) f(\Ga/\ga_n)\bigr].
$$
Therefore
$$
S_{T,f} = \sum_{\Cc \in C(\Ga)} (-1)^{l(\Cc)}\Om_{T,f}(\Cc)
= \tilde S_{T,f}.
\eqno\qed
$$
\hideqed
\end{proof}

\smallskip

The morals of the story so far are: first, there is nothing in
Bogoliubov's procedure that will not be valid in \textit{any}
connected, graded bialgebra; second, Schmitt's formula for his
incidence Hopf algebras coincides identically with the geometric
series formulae~\eqref{eq:geom-series} and~\eqref{eq:many-conv};
third, the latter in the field theory context gives rise to the
Dyson--Salam formula. We turn our attention now to Zimmermann's forest
formula.



\section{Zimmermann's forest formula}

\begin{defn}% Definition 2.
A (normal) \textit{forest} $\F$ of a proper, connected graph $\Ga$ is
a set of proper, divergent and connected subdiagrams, none of them
equal to $\Ga$, such that any pair of elements are nonoverlapping.
Again we include the forest $\{\emptyset\}$ as a special case.
$F(\Ga)$ denotes the set of forests of $\Ga$. The \textit{density} of
a forest $\F$ is the number $d(\F) = |\F| + 1$, where $|\F|$ is the
number of elements of $\F$. Given $\ga \in \F$ we say that $\ga'$ is a
\textit{predecessor} of $\ga$ in $\F$ if $\ga'\varsubsetneq \ga$ and
there is no element $\ga''$ in $\F$ such that
$\ga'\varsubsetneq \ga'' \varsubsetneq \ga$. Let
$$
\Th(\F) := \prod_{\ga \in \F\cup\{\Ga\}}  \ga / \tilde\ga,
$$
where $\tilde\ga$ denote the disjoint union of all predecessors of
$\ga$. When $\ga$ is minimal, $\tilde\ga = \emptyset$, and
$\ga / \tilde\ga = \ga$.
\end{defn}


Notice that if a forest $\F$ is a chain, then $\Th(\F) =\Om(\F)$, and
conversely if a chain $\Cc$ is a forest, $\Om(\Cc) = \Th(\Cc)$.
Obviously not every forest is a chain; but also not every chain is a
forest, because product subgraphs can occur in chains and cannot in
forests. There are diagrams like the one in the $\varphi^4_4$ model
pictured in Figure~\ref{fg:diagrama-prieto}, for which the sets of
chains and forests coincide; but in general there are fewer forests
than chains.

\begin{figure}[ht]
\begin{center}
\vspace{1pc}
\begin{picture}(0,50)
\put(0,0){\line(3,1){72}}
\put(0,0){\line(-3,1){72}}
\put(0,0){\line(1,2){20}}
\put(0,0){\line(-1,2){20}}
\put(20,40){\line(2,-1){52}}
\put(-20,40){\line(-2,-1){52}}
\qbezier(-20,40)(0,60)(20,40)
\qbezier(-20,40)(0,20)(20,40)
\end{picture}
\end{center}
\caption{Diagram $\Ga$ without extra cancellations in $S_Z(\Ga)$ with
respect to $S_{DS}(\Ga)$}
\label{fg:diagrama-prieto}
\vspace{1pc}
\end{figure}

Zimmermann's version for the antipode is defined by
$$
S_Z(\Ga) := \sum_{\F \in F(\Ga)} (-1)^{d(\F)} \Th(\F).
$$

\begin{prop}% Proposition 7.
\label{pr:Zimm-antp}
$S_Z$ provides another formula for the antipode of $\H$.
\end{prop}

\begin{proof}
Once more, the idea is to prove that $S_Z$ is an inverse, under
convolution, of $\id$. By definition
\begin{align*}
S_Z * \id (\Ga)
&= \sum_{\ga \subseteq \Ga} S_Z(\ga) \, \Ga/\ga
= S_Z(\Ga) + \sum_{\emptyset \subseteq \ga \varsubsetneq \Ga}
S_Z(\ga) \, \Ga/\ga
\\
&= S_Z(\Ga) + \sum_{\emptyset \subseteq \ga \varsubsetneq \Ga} \Bigl(
\prod_{i=1}^{n_\ga} S_Z(\la^\ga_i) \Bigr)\, \Ga/\ga
\\
&= S_Z(\Ga) + \sum_{\emptyset \subseteq \ga \varsubsetneq \Ga}
\prod_{i=1}^{n_\ga} \Bigl( \sum_{\F_i \in F(\la^\ga_i)}
(-1)^{d(\F_i)} \Th(\F_i) \Bigr) \, \Ga/\ga,
\end{align*}
where $n_\ga$ is the number of connected components of $\ga$, and
$\la^\ga_i$, $i= 1,\dots,n_\ga$ are the connected components of $\ga$.

Now, if $\F_i \in F(\la^\ga_i)$, then $\F := \bigcup_{i=1}^{n_\ga}
(\F_i \cup \{\la^\ga_i\})$ is a forest of $\Ga$. Moreover
$$
|\F| = \sum_{i=1}^{n_\ga} |\F_i| + n_\ga = \sum_{i=1}^{n_\ga} d(\F_i),
$$
so
\begin{equation}
d(\F) = \sum_{i=1}^{n_\ga} d(\F_i) + 1.
\label{eq:density}%  (11)
\end{equation}
On the other hand, in $\F$, $\Ga/\tilde\Ga = \Ga/\ga$, hence
\begin{equation}
\Th(\F) =  \Bigl(\prod_{i=1}^{n_\ga} \Th(\F_i) \Bigr) \Ga/\tilde\Ga
= \Bigl(\prod_{i=1}^{n_\ga} \Th(\F_i) \Bigr) \, \Ga/\ga.
\label{eq:big-theta}% (12)
\end{equation}
Conversely, if $\F \in F(\Ga)$, and if $\row{\ga}1k$ are the maximal
elements of $\F$, then the sets
$\F_i:= \set{\ga \in\F : \ga \varsubsetneq \ga_i}$ constitute a forest
of $\ga_i$. Since all the elements of a forest are connected diagrams,
$\row{\ga}1k$ are the connected components of
$\ga = \prod_{i=1}^k \ga_i$ and clearly \eqref{eq:density} and
\eqref{eq:big-theta} hold. Therefore,
$$
S_Z * \id (\Ga)
= S_Z(\Ga) - \sum_{\F \in F(\Ga)} (-1)^{d(\F)} \Th(\F)
= 0 = u \eps(\Ga).
$$
Thus, $S_Z$ is a left inverse for $\id$, and therefore is an antipode.
\end{proof}

The proofs of Propositions~\ref{pr:DS-antp} and~\ref{pr:Zimm-antp} are
parallel; the difference lies in minor combinatorial details. Even so,
it is clear that Zimmermann's formula (although more sensitive in the
practice to the details of the renormalization method) is, from the
combinatorial viewpoint, altogether subtler than Bogoliubov's or Dyson
and Salam's. It is more economical in that all the cancellations
implicit in the convolution formula~\eqref{eq:geom-series} are taken
into account and suppressed; this we already made clear in the context
of the algebra of rooted trees~\cite{Ananke}. Thus, the ``commerce''
between quantum field theory and Hopf algebra theory has not been
one-way: Zimmermann's formula is advantageously applicable to a large
class of bialgebras~\cite{KreimerOriginal}.

The reader will have no difficulty in writing the nonrecursive forest
formula for the twisted coinverse $S_{T,f}$.



\section{The bidegree and computations in quantum field theory}

Let us indicate first that Kreimer has announced~\cite{Dirkoflate} a
new proof of finiteness of the renormalized graphs and Green
functions, based on a cohomological reinterpretation of the basic
coproduct equation~\eqref{eq:coprod}.


Whether the Connes--Kreimer algebraic paradigm will become useful to
simplify \textit{computations} ---in distinction to ``merely''
proofs--- in realistic field theories, seems to hinge to a large
extent on the practical usefulness of the depth bigrading. The fact is
that simplifications in sums of Feynman diagrams do occur, and they
usually involve, beyond trivialization of the topology, reduction in
depth. For instance, in the $\varphi^4_4$ model we have (for the
corresponding amplitudes in configuration space):

\begin{figure}[h]
\begin{center}
\vspace{2pc}
\parbox{2pc}{
\begin{picture}(0,0)
\put(-10,-15){\line(0,1){30}}
\put(-10,-15){\line(2,3){20}}
\put(10,-15){\line(0,1){30}}
\put(10,-15){\line(1,0){8}}
\put(-10,15){\line(-1,0){8}}
\qbezier(-10,-15)(0,-25)(10,-15)
\qbezier(-10,-15)(0,-5)(10,-15)
\qbezier(-10,15)(0,5)(10,15)
\qbezier(-10,15)(0,25)(10,15)
\end{picture}
}
+
\qquad\quad
\parbox{2pc}{
\begin{picture}(0,0)
\put(0,-20){\line(1,1){20}}
\put(0,-20){\line(-1,1){20}}
\put(-26,0){\line(1,0){52}}
\qbezier(0,0)(10,-10)(0,-20)
\qbezier(0,0)(-10,-10)(0,-20)
\qbezier(-20,0)(0,20)(20,0)
\end{picture}
}\qquad
=
\hspace{4.5em}
\parbox{4pc}{
\begin{picture}(0,0)
\put(30,0){\line(1,0){6}}
\put(-30,0){\line(-1,0){6}}
\qbezier(-30,0)(-20,10)(-10,0)
\qbezier(-30,0)(-20,-10)(-10,0)
\qbezier(-10,0)(0,10)(10,0)
\qbezier(-10,0)(0,-10)(10,0)
\qbezier(10,0)(20,10)(30,0)
\qbezier(10,0)(20,-10)(30,0)
\qbezier(-30,0)(0,40)(30,0)
\end{picture}
}
+ \ $\displaystyle\frac{1}{2}$
\hspace{2.5em}
\parbox{3pc}{
\begin{picture}(0,0)
\put(20,0){\line(1,0){6}}
\put(-20,0){\line(-1,0){6}}
\qbezier(-20,0)(-10,10)(0,0)
\qbezier(-20,0)(-10,-10)(0,0)
\qbezier(0,0)(10,10)(20,0)
\qbezier(0,0)(10,-10)(20,0)
\qbezier(-20,0)(0,30)(20,0)
\end{picture}
}
\end{center}
\vspace{1pc}
\caption{$\zeta(3)$ coefficients vanish in the sum of two graphs with
the same symmetry factor}
\label{fg:magic-sum}
\vspace{1pc}
\end{figure}

A naive hope in that respect, to wit, that every diagram be eventually
expressed in terms of primitive elements (so renormalization proceeds
``at a stroke'') is quickly dashed. A Hopf algebra $H$ is
\textit{primitively generated} when the smallest subalgebra of $H$
containing all its primitive elements is $H$ itself. The structure
theorem for commutative connected graded algebras~\cite{BlueBook}
makes it plain that Hopf algebras of Feynman graphs are far from being
primitively generated; neither the Hopf algebra of rooted trees nor
its noncommutative geometry subalgebra
$H_{\mathrm{CM}}$~\cite{ConnesMHopf} is primitively generated.

In fact, only elements for which the coproduct is invariant under the
flip map $a \ox b \mapsto b \ox a$ can be primitively generated. In
any Hopf algebra associated to a field theory there exist ``ladder''
subalgebras of diagrams with only completely nested subgraphs, and
these subalgebras are primitively generated. However, this is of scant
practical use, as then the recursive methods~\cite{Isabella} carry off
the award for computational simplicity.

According to the structure theorem, commutative Hopf algebras can be
decomposed as algebras as a tensor product
$$
H = S(P(H)) \ox S(W_H),
$$
where $S(P(H))$ denotes the polynomial algebra generated by all the
primitive elements in $H$, and $S(W_H)$ the polynomial algebra on a
(nonunique) suitable subspace $W_H$ of $H$. To get a handle on (a
representative for) $W_\H$ for bialgebras of Feynman graphs is on the
order of the day.

The depth grading for the algebra $H_R$ of rooted trees has been
investigated, beyond~\cite{BK}, in the
paper~\cite{Chryssomalakosetal}. The strategy suggested
in~\cite{Chryssomalakosetal} looks feasible in bialgebras of graphs.
By use of the dual algebra, so-called normal coordinate elements
(appropriate sums of products of graphs) can be found, for which the
antipode (although not the twisted antipode in general) is
\textit{diagonal}.

Let $Z_\ga$ be the dual element of a graph $\ga$. Any graph $\Ga$ has
an associated normal element $\psi_\Ga$, and any graph can be
decomposed into a sum of products of normal elements. Let a sequence
of graphs $J = (\ga_1,\dots,\ga_k)$ be given; we say that $\Ga$ is
compatible with $J$ if
\begin{equation}
\left< Z_{\ga_1} \oxyox Z_{\ga_k},\Dl^{k-1}\Ga \right> \neq 0.
\label{eq:compatibility}%  (13)
\end{equation}
The normal decomposition is as follows:
$$
\Ga = \psi_\Ga + \sum_{k\geq2}
\frac{\left< Z_{\ga_1} \oxyox Z_{\ga_k},\Dl^{k-1}\Ga \right>}{k!}
\,\psi_{\ga_1}\dots\psi_{\ga_k},
$$
where we sum in practice over a finite number of compatible sequences.

Ladder normal coordinate elements are primitive, and for non-ladder
ones the indications are that the complexity of their renormalization
is substantially lessened with respect to that of the graphs
themselves; they are instrumental in the description of $W_\H$.

Also, Kreimer has introduced~\cite{Dirkoflate} a ``shuffle'' product
of diagrams, based on a variant of~\eqref{eq:compatibility}, that
seems to hold promise of eventual factorization of perturbative field
theory into primitive elements. In this respect, as in other
tantalizing subjects springing from the Connes--Kreimer
paradigm~\cite{Dirkbook}, we are barely starting to scratch the
surface.


\section*{Acknowledgments}

We thank Joseph C. V\'arilly for illuminating discussions and
\TeX{}nical help. We are grateful for the hospitality of the
Departamento de F\'{\i}sica Te\'orica of the Universidad de Zaragoza.
Support from the Vicerrector\'{\i}a de Investigaci\'on of the
Universidad de Costa Rica is acknowledged.


\appendix

\section{Appendix: Graded bialgebras}

A bialgebra H is a vector space over a field $\FF$ (here taken to be of
characteristic~0) equipped with two structures: an algebra structure
and a coalgebra structure, related by some compatibility conditions.
The algebra structure is described by two maps: the product
$m\colon H \ox H \to H$, and the unit map $u\colon \FF \to H$. The
conditions imposed on these maps are:
\begin{enumerate}
\item
Associativity: $m(m \ox \id) = m(\id \ox m) : H \ox H \ox H \to H$;
\item
Unity: $m(u \ox \id) = m(\id \ox u) = \id : H \to H$.
\end{enumerate}


A coalgebra is obtained is obtained by reversing arrows in the
defining maps for an algebra; it is, therefore, also described by two
maps: the coproduct $\Dl\colon H \to H \ox H$, and the counit
$\eps\colon H \to \FF$. The requirements are:
\begin{enumerate}
\addtocounter{enumi}{2}
\item
Coassociativity:
$(\Dl \ox \id)\Dl = (\id \ox \Dl)\Dl : H \to H \ox H \ox H$;
\item
Counity: $(\eps \ox \id) \Dl = (\id \ox \eps) \Dl = \id : H \to H$.
\end{enumerate}

Finally, to obtain a bialgebra one stipulates
\begin{enumerate}
\addtocounter{enumi}{4}
\item
Compatibility: $\Dl$ and $\eps$ are unital algebra homomorphisms.
\end{enumerate}

This requirement turns out to be equivalent to asking that $m$ and
$u$ be coalgebra morphisms.


\begin{defn}% Definition 3.
A bialgebra $H = \bigoplus^\infty_{n=0} H^{(n)}$ graded as a vector
space is called a \textit{graded bialgebra} when the grading is
compatible with both the algebra and the coalgebra structures:
$$
H^{(n)}H^{(m)} \subseteq H^{(n+m)}  \sepword{and}
\Dl(H^{(n)}) \subseteq \bigoplus_{p+q = n} H^{(p)} \ox H^{(q)}.
$$
It is called \textit{connected} when the first piece consists of
scalars only: $H^{(0)} = \FF\, 1$.
\end{defn}


A most useful property of connected graded bialgebras is that when
$a \in H^{(n)}$, the coproduct can be written as
\begin{subequations}
\label{eq:gr-coprod}
\begin{equation}
\Dl a = a \ox 1 + 1 \ox a + \sum_j a'_j \ox  a''_j,
\label{eq:gr-coprod-full}% (14a)
\end{equation}
where the elements $a'_j$ and $a''_j$ all have degree between 1 and
$n-1$. The proof is easy and found in many
places~\cite{Bourbaki,Montgomery,Polaris,Calypso,Kastler}. To simplify
the notation we define
\begin{equation}
\Dl'a := \Dl a - a \ox 1 - 1 \ox a = \sum_j a'_j \ox  a''_j.
\label{eq:gr-coprod-trunc}% (14b)
\end{equation}
\end{subequations}
Coassociativity of $\Dl'$ is easily obtained from the coassociativity
of $\Dl$. An element $a \in H$ is called (1-)primitive when
$\Dl'a = 0$.

Applying $\eps \ox \id$ to \eqref{eq:gr-coprod-full} gives
$a = (\eps \ox \id)(\Dl a) = \eps(a)1 + a + \sum_j \eps(a'_j)\,a''_j$;
therefore, if $a \in H^{(n)}$ with $n \geq 1$, the connectedness
condition forces $\eps(a) = 0$.



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\end{document}
