Mixed Hodge Structures and Renormalization in Physics

This paper is a collaboration between a mathematician and a physicist. It is based on the observation that renormalization of Feynman amplitudes in physics is closely related to the theory of limiting mixed Hodge structures in mathematics. Whereas classical physical renormalization methods involve manipulations with the integrand of a divergent integral, limiting Hodge theory involves moving the chain of integration so the integral becomes convergent and studying the monodromy as the chain varies.

Even methods like minimal subtraction in the context of dimensional or analytic regularization implicitly modify the integrand through the definition of a measure $\int d^{D}k$ via analytic continuation. Still, as a regulator dimensional regularization is close to our approach in so far as it leaves the rational integrand assigned to a graph unchanged. Minimal subtraction as a renormalization scheme differs though from the renormalization schemes which we consider -momentum subtractions essentially- by a finite renormalization. Many of the nice algebro-geometric structures developed below are not transparent in that scheme.

The advantages of the limiting Hodge method are firstly that it is linked to a very central and powerful program in mathematics: the study of Hodge structures and their variations. As a consequence, one gains a number of tools, like weight, Hodge, and monodromy filtrations to study and classify the Feynman amplitudes. Secondly, the method depends on the integration chain, and hence on the graph, but it is in some sense independent of the integrand. For this reason it should adapt naturally e.g. to gauge theories where the numerator of the integrand is complicated.

An important point is to analyse the nature of the poles. Limiting mixed Hodge structures demand that the divergent subintegrals have at worst log poles. This does not imply that we can not apply our approach to perturbative amplitudes which have worse than logarithmic degree of divergence. It only means that we have to correctly isolate the polynomials in masses and external momenta which accompany those divergences such that the corresponding integrands have singularities provided by log-poles. This is essentially automatic from the notion of a residue available by our very methods. As a very pleasant byproduct, we learn that physical renormalization schemes -on-shell subtractions, momentum subtractions, Weinberg’s scheme,- belong to a class of schemes for which this is indeed automatic.

Moreover, for technical reasons, it is convenient to work with projective rather than affine integrals. One of the central physics results in this paper is that the renormalization problem can be reduced to the study of logarithmically divergent, projective integrals. This is again familiar from analytic regulators. The fact that it can be achieved here by leaving the integrand completely intact will hopefully some fine day allow to understand the nature of the periods assigned to renormalized values in quantum field theory.

A remark for Mathematicians: our focus in this paper has been renormalization, which is a problem arising in physics. We suspect, however, that similar methods will apply more generally for example to period integrals whenever the domain of integration is contained in ${\mathbb{R}}^{+n}$ and the integrand is a rational function with polar locus defined by a polynomial with non-negative real coefficients. The toric methods and the monodromy computations should go through in that generality.

Both authors thank Francis Brown, Hélène Esnault and Karen Yeats for helpful discussions. This work was partially supported by NSF grants DMS-0603781 and DMS-0653004. S.B. thanks the IHES for hospitality January-March 2006 and January-March 2008. D.K. thanks Chicago University for hospitality in February 2007.

This paper studies the renormalization problem in the context of parametric representations, with an emphasis on algebro-geometric properties. We will not study the nature of the periods one obtains from renormalizable quantum field theories in an even dimension of space-time. Instead, we provide the combinatorics of renormalization such that a future motivic analysis of renormalized amplitudes is feasible along the lines of [2]. Our result will in particular put renormalization in the framework of a limiting mixed Hodge structure, which hopefully provides a good starting point for an analysis of the periods in renormalized amplitudes. That these amplitudes are provided by numbers which are periods (in the sense of [11]) is an immediate consequence of the properties of parametric representations, and will also emerge naturally below (see Thm.(7.3)).

The main result of this paper is a careful study of the singularities of the first Kirchhoff–Symanzik polynomial, which carries all the short-distance singularities of the theory. The study of this polynomial can proceed via an analysis with the help of projective integrals. Along the way, we will also give useful formulas for parametric representations involving affine integrals, and clarify the role of the second Kirchhoff–Symanzik polynomial for affine and projective integrals.

Our methods are general, but in concrete examples we restrict ourselves to $\phi^{4}_{4}$ theory. Parametric representations are used which result from free-field propagators for propagation in flat space-time. In such circumstances, the advantages of analytic regularizations are also available in our study of parametric representations as we will see. In particular, our use of projective integrals below combines such advantages with the possibility to discuss renormalization on the level of the pairing between integration chains and de Rham classes.

In examples, special emphasis is given to the study of particular renormalization schemes, the momentum scheme (MOM-scheme, Weinberg’s scheme, on-shell subtractions).

Also, we often consider Green functions as functions of a single kinematical scale $q^{2}>0$. Green functions are defined throughout as the scalar coefficient functions (structure functions) for the radiative corrections to tree-level amplitudes $r$. They are to be regarded as scalar quantities of the form $1+{\mathcal{O}}(\hbar)$. Renormalized amplitudes are then, in finite order in perturbation theory, polynomial corrections in $L=\ln q^{2}/\mu^{2}$ ($\mu^{2}>0$) without constant term, providing the quantum corrections to the tree-level amplitudes appearing as monomials in a renormalizable Lagrangian [15]:

(1.1)

$$\phi_{R}(\Gamma)=\sum_{j=1}^{{\textrm{aug}}(\Gamma)}p_{j}(\Gamma)L^{j}.$$

Correspondingly, Green functions become triangular series in two variables

(1.2)

$$G^{r}(\alpha,L)=1+\sum_{j=1}^{\infty}\gamma_{j}^{r}(\alpha)L^{j}=1+\sum_{j=1}^{\infty}c_{j}^{r}(L)\alpha^{j}.$$

The series $\gamma_{j}^{r}(\alpha)$ are related by the renormalization group which leaves only the $\gamma_{1}^{r}(\alpha)$ undetermined, while the polynomials $c_{j}^{r}(L)$ are bounded in degree by $j$. The series $\gamma_{1}^{r}$ fulfill ordinary differential equations driven by the primitive graphs of the theory [16].

The limiting Hodge structure $A(\Gamma)$ which we consider for each Feynman graph $\Gamma$ provides contribution of a graph $\Gamma$ to the coefficients of $\gamma_{1}^{r}$ in the limit. This limit is a period matrix (a column vector here) which has, from top to bottom, the periods provided by a renormalized graph $\Gamma$ as entries. The first entry is the contribution to $\gamma_{1}^{r}$ of a graph with $\textrm{res}(\Gamma)=r$ and the $k$-th is a rational multiple of the contribution to $\gamma_{k}^{r}$. In section 9.1 we determine the rational weights which connect these periods to the coefficients $p_{j}(\Gamma)$ attributed to the renormalization of a graph $\Gamma$.

We include a discussion of the structure of renormalization which comes from an analysis of the second Kirchhoff–Symanzik polynomial. While this polynomial does not provide short-distance singularities in its own right, it leads to integrals of the form

(1.3)

$$\int\omega\ln(f)$$

for a renormalized Feynman amplitude, with $\omega$ a de Rham class determined by the first Kirchhoff–Symanzik polynomial, and $f$ -congruent to one along any remaining exceptional divisor- determined by the second. We do not actually do the monodromy calculation for integrals (1.3) involving a logarithm, but it will be similar to the calculation for (1.5) which we do. A full discussion of the Hodge structure of a Green function seems feasible but will be postponed to future work.

Let ${\mathbb{P}}^{n-1}$ be the projective space of lines in ${\mathbb{C}}^{n}$ which we view as an algebraic variety with homogeneous coordinates $A_{1},\dotsc,A_{n}$. Let $\psi(A_{1},\dotsc,A_{n})$ be a homogeneous polynomial of some degree $d$, and let $X\subset{\mathbb{P}}^{n-1}$ be the hypersurface defined by $\psi=0$. We assume the coefficients of $\psi$ are all real and $\geq 0$. Let $\sigma=\{[a_{1},\dotsc,a_{n}]\ |\ a_{i}\geq 0,\forall i\}$ be the topological $(n-1)$-chain (simplex) in ${\mathbb{P}}^{n-1}$, where $[\ldots]$ refers to homogeneous coordinates. We will also use the notation $\sigma={\mathbb{P}}^{n-1}({\mathbb{R}}^{\geq 0})$. Our assumption about coefficients implies

(1.4)

$$\sigma\cap X=\bigcup_{L\subset X}L({\mathbb{R}}^{\geq 0}),$$

where $L$ runs through all coordinate coordinate linear spaces $L:A_{i_{1}}=\cdots=A_{i_{p}}=0$ contained in $X$ (see (see Fig.1)).

The genesis of the renormalization problem in physics is the need to assign values to integrals

(1.5)

$$\int_{\sigma}\omega$$

where $\omega$ is an algebraic $(n-1)$-form on ${\mathbb{P}}^{n-1}$ with poles along $X$. The problem is an important one for physical applications, and there is an extensive literature (see, for example, [10, 22, 21]) focusing on practical formulae to reinterpret (1.5) in some consistent way as a polynomial in $\log t$. (Here $t$ parametrizes a deformation of the integration chain. As a first approximation, one can think of $\int_{t}^{\infty}\omega$ when $\omega$ has a logarithmic pole at $t=0$.)

A similar problem arises in pure mathematics in the study of degenerating varieties, e.g. a family of elliptic curves degenerating to a rational curve with a node. In the classical setup, one is given a family $f:{\mathcal{X}}\to D$, where $D$ is a disk with parameter $t$. The map $f$ is proper (so the fibres $X_{t}$ are compact). ${\mathcal{X}}$ is assumed to be non-singular, as are the fibres $X_{t},\ t\neq 0$. $X_{0}$ may be singular, though one commonly invokes resolution of singularities to assume $X_{0}\subset{\mathcal{X}}$ is a normal crossing divisor. Choose a basis $\sigma_{1,t},\dotsc,\sigma_{r,t}$ for the homology of the fibre $H_{p}(X_{t},{\mathbb{Q}})$ in some fixed degree $p$. By standard results in differential topology, the fibre space is locally topologically trivial over $D^{*}=D-\{0\}$, and we may choose the classes $\sigma_{i,t}$ to be locally constant. If we fix a smooth fibre $t_{0}\neq 0$, the monodromy transformation $m:H_{p}(X_{t_{0}})\to H_{p}(X_{t_{0}})$ is obtained by winding around $t=0$. An important theorem ([7], III,2) says this transformation is quasi-unipotent, i.e. after possibly introducing a root $t^{\prime}=t^{1/n}$ (which has the effect of replacing $m$ by $m^{n}$), $m-id$ is nilpotent. The matrix

(1.6)

$$N:=\log m=-\Big{[}(id-m)+(id-m)^{2}/2+\ldots\Big{]}$$

is thus also nilpotent. This is the mathematical equivalent of locality in physics. It insures that our renormalization of (1.5) will be a polynomial in $\log t$ rather than an infinite series. We take a cohomology class $[\omega_{t}]\in H^{p}(X_{t},{\mathbb{C}})$ which varies algebraically. For example, in a family of elliptic curves $y^{2}=x(x-1)(x-t)$, the holomorphic $1$-form $\omega_{t}=dx/y$ is such a class. Note $\omega_{t}$ is single-valued over all of $D^{*}$. It is not locally constant. The expression

(1.7)

$$\exp\Big{(}-(N\log t)/2\pi i\Big{)}\begin{pmatrix}\int_{\sigma_{1,t}}\omega_{t}\\ \vdots\\ \int_{\sigma_{r,t}}\omega_{t}\end{pmatrix}.$$

is then single-valued and analytic on $D^{*}$. Suppose $\omega_{t}$ chosen such that the entries of the column vector in (1.7) grow at worst like powers of $|\log|t||$ as $|t|\to 0$. A standard result in complex analysis then implies that (1.7) is analytic at $t=0$. We can write this

(1.8)

$$\begin{pmatrix}\int_{\sigma_{1,t}}\omega_{t}\\ \vdots\\ \int_{\sigma_{r,t}}\omega_{t}\end{pmatrix}\sim\exp\Big{(}(N\log t)/2\pi i\Big{)}\begin{pmatrix}a_{1}\\ \vdots\\ a_{r}\end{pmatrix}.$$

Here the $a_{j}$ are constants which are periods of a limiting Hodge structure. The exponential on the right expands as a matrix whose entries are polynomials in $\log t$, and the equivalence relation $\sim$ means that the difference between the two sides is a column vector of (multi-valued) analytic functions vanishing at $t=0$.

We would like to apply this program to the integral (1.5). Let $\Delta:\prod_{1}^{n}A_{j}=0$ be the coordinate divisor in ${\mathbb{P}}^{n-1}$. Note that the chain $\sigma$ has boundary in $\Delta$, so as a first attempt to interpret (1.5) as a period, we might consider the pairing

(1.9)

$$H^{n-1}({\mathbb{P}}^{n-1}-X,\Delta-X\cap\Delta)\times H_{n-1}({\mathbb{P}}^{n-1}-X,\Delta-X\cap\Delta)\to{\mathbb{C}}$$

The form $\omega$ is an algebraic $(n-1)$-form and it vanishes on $\Delta$ for degree reasons, so it does give a class in the relative cohomology group appearing in (1.9) (see the discussion (9.8)-(9.10)). On the other hand, the chain $\sigma$ meets $X$ (1.4), so we do not get a class in homology. Instead we consider a family of coordinate divisors $\Delta_{t}:\prod_{1}^{n}A_{j,t}=0$ with $\Delta_{0}=\Delta$. (For details, see section 6.) For $t=\varepsilon>0$ there is a natural chain $\sigma_{\varepsilon}$ which is what the physicists would call a cutoff. We have $\partial\sigma_{\varepsilon}\subset\Delta_{\varepsilon}$ and $\sigma_{\varepsilon}\cap X=\emptyset$, so $\int_{\sigma_{\varepsilon}}\omega$ is defined. One knows on abstract grounds that the monodromy of

$$H_{n-1}({\mathbb{P}}^{n-1}-X,\Delta_{t}-X\cap\Delta_{t})$$

is quasi-unipotent as above ([7], III,§2). The main mathematical work in this paper will be to compute the monodromy of $\sigma_{\varepsilon}$ in the specific case of Feynman amplitudes in physics. More precisely, $X$ will be a graph hypersurface $X_{\Gamma}$ associated to a graph $\Gamma$ (section 5). We will write down chains $\tau_{\gamma}^{\varepsilon}$, one for each flag of core (one particle irreducible in physics) subgraphs $\gamma=\{\Gamma_{1}\subsetneq\cdots\Gamma_{p(\gamma)}\subsetneq\Gamma\}$, representing linearly independent homology classes in $H_{n-1}({\mathbb{P}}^{n-1}-X,\Delta_{\varepsilon}-X\cap\Delta_{\varepsilon})$. (The combinatorics here is similar to that found in [1], [18].) We will show the monodromy in our case is given by

(1.10)

$$m(\sigma_{\varepsilon})=\sigma_{\varepsilon}+\sum_{\gamma}(-1)^{p(\gamma)}\tau_{\gamma}^{\varepsilon}.$$

We will then exhibit a nilpotent matrix $N$ such that

(1.11)

$$\begin{pmatrix}m(\sigma_{\varepsilon})\\ \vdots\\ m(\tau_{\gamma}^{\varepsilon})\\ \vdots\end{pmatrix}=\exp(N)\begin{pmatrix}\sigma_{\varepsilon}\\ \vdots\\ \tau_{\gamma}^{\varepsilon}\\ \vdots\end{pmatrix}.$$

With this in hand, renormalization is automatic for any physical theory for which $\Gamma$ and its subgraphs are at worst logarithmically divergent after taking out suitable polynomials in masses and momenta. Namely, such a physical theory gives a differential form $\omega_{\Gamma}$ as in (1.5) and we may repeat the above argument:

(1.12)

$$\exp(-(N\log t)/2\pi i)\begin{pmatrix}\int_{\sigma_{t}}\omega\\ \vdots\\ \int_{\tau_{\gamma}^{t}}\omega\\ \vdots\end{pmatrix}$$

is single-valued on the punctured disk. The hypothesis of log divergence at worst for subgraphs of $\Gamma$ will imply that the integrals will grow at worst like a power of $\log$ as $|t|\to 0$,(lemma 9.2). Precisely as in (1.8), one gets the renormalization

(1.13)

$$\int_{\sigma_{t}}\omega_{\Gamma}=\sum_{k=0}^{r}b_{k}(\log t)^{k}+O(t),$$

where $O(t)$ denotes a (multi-valued) analytic function vanishing at $t=0$. The renormalization schemes considered here can be characterized by the condition $b_{0}=0$.

Of course, the requirement that a physical theory have at worst log divergences is a very strong constraint. The difficult computations in section 7 show how general divergences encountered in physics can be reduced to log divergences.

Most of the mathematical work involved concerns the calculation of monodromy for a particular topological chain. It is perhaps worth taking a minute to discuss a toy model. Suppose one wants to calculate $\int_{0}^{\infty}\omega$, where $\omega=\frac{dz}{(z-i)z}$. The integral diverges, so instead we consider $\int_{t}^{\infty}\omega$ as a function of $t=\varepsilon e^{i\theta}$ for $0\leq\theta\leq 2\pi$. If we take the path $[t,\infty]$ to be a great circle, then as $t$ winds around $0$, the path will get tangled in the singularity of $\omega$ at $z=i$. Assuming we do not understand the singularities of our integral far from $0$, this could be a problem. Instead we chose our path to follow the small circle from $\varepsilon e^{i\theta}$ to $\varepsilon$ and then the positive real axis from $\varepsilon$ to $\infty$. The variation of monodromy is the difference in the paths for $\theta=0$ and $\theta=2\pi$. In this case, it is the circle $\{|t|=\varepsilon\}$. If we assume something (at worst superficial log divergence for the given graph and all subgraphs in the given physical theory) about the behavior of $\omega$ near the pole at $0$, then the behavior of our integral for $|t|<<1$ is determined by this monodromy, which is a topological invariant. This is quite different from the usual approach in physics involving complicated algebraic manipulations with $\omega$. A glance at fig.(10) suggests that our toy model is too simple. We have to work with two scales, $\varepsilon<<\eta<<1$. This is because in the more complicated situation, we have to deal with cylinders of small radius $\eta$, but then we have further to slightly deform the boundaries of the cylinder (cf. fig.(12)).

Section 2 is devoted to Hopf algebras of graphs and of trees. These have played a central role in the combinatorics of renormalization. In particular, the insight afforded by passing from graphs to trees is important. Since the combinatorics of core subgraphs is even more complicated than that of divergent subgraphs, it seemed worth going carefully through the construction. Section 3 studies the toric variety we obtain from a graph $\Gamma$ by blowing up certain coordinate linear spaces in the projective space with homogeneous coordinates labeled by the edges of $\Gamma$. The orbits of the torus action are related to flags of core subgraphs of the given graph. In section 4, we use the ${\mathbb{R}}$-structure on our toric variety to construct certain topological chains which will be used to explicit the monodromy. Section 5 recalls the basic properties of the graph polynomial $\psi_{\Gamma}\equiv\psi(\Gamma)$ and the graph hypersurface $X_{\Gamma}:\psi_{\Gamma}=0$. The crucial point is corollary 5.3 which says that the strict transform of $X_{\Gamma}$ on our toric blowup avoids points with coordinates $\geq 0$. Any chain we construct which stays close to the locus of such points necessarily is away from $X_{\Gamma}$ and hence also away from the polar locus of our integrand. Section 6 computes the monodromy of our chain. Section 7 considers how to reduce Feynman amplitude calculations as they arise in physics, including masses and momenta as well as divergences which are worse than logarithmic, to the basic situation where limiting methods can apply. In section 8 we calculate the nilpotent matrix $N$ which is the log of the monodromy transformation, and in section 9 we prove the main renormalization theorem in the log divergent case, to which we have reduced the theory.

In this section we bring together material on graphs and the graph Hopf algebra which will be used in the sequel. We also discuss Hopf algebras related to rooted trees and prove a result (proposition 2.5) relating the Hopf algebra of core graphs to a suitable Hopf algebra of labeled trees. Strictly speaking this is not used in the paper, but it provides the best way we know to understand flags of core subgraphs, and these play a central role in the monodromy computations. Trees labeled by divergent subgraphs have a long history in renormalization theory [12], [13].

A graph $\Gamma$ is determined by giving a finite set $HE(\Gamma)$ of half-edges, together with two further sets $E(\Gamma)$ (edges) and $V(\Gamma)$ (vertices) and surjective maps

(2.1)

$$p_{V}:HE(\Gamma)\to V;\quad p_{E}:HE(\Gamma)\to E.$$

(Note we do not allow isolated vertices.) In combinatorics, one typically assumes all fibres $p_{E}^{-1}(e)$ consist of exactly two half-edges ($e$ an internal edge), while in physics the calculus of path integrals and correlation functions dictates that one admit external edges $e\in E$ with $\#p_{E}^{-1}(e)=1$. If all internal edges of $\Gamma$ are shrunk to $0$, the resulting graph (with no internal edges) is called the residue $\text{res}(\Gamma)$. In certain theories, the vertices are decomposed into different types $V=\amalg V_{i}$, and the valence of the vertices in $V_{i}$, $\#p_{V}^{-1}(v)$, is fixed independent of $v\in V_{i}$.

We will typically work with labeled graphs which are triples $(\Gamma,A,\phi:A\cong E(\Gamma))$. We refer to $A$ as the set of edges.

A graph is a topological space with Betti numbers $|\Gamma|=h_{1}(\Gamma)=\dim H_{1}(\Gamma,{\mathbb{Q}})$ and $h_{0}(\Gamma)$. We say $\Gamma$ is connected if $h_{0}=1$. Sometimes $h_{1}$ is referred to as the loop number.

A subgraph $\gamma\subset\Gamma$ is determined (for us) by a subset $E(\gamma)\subset E(\Gamma)$. We write $\Gamma/\!/\gamma$ for the quotient graph obtained by contracting all edges of $\gamma$ to points. If $\gamma$ is not connected, $\Gamma/\!/\gamma$ is different from the naive quotient $\Gamma/\gamma$. If $\gamma=\Gamma$, we take $\Gamma/\!/\Gamma=\emptyset$ to be the empty set. It will be convenient when we discuss Hopf algebras below to have the empty set as a graph.

Also, for $\gamma=e$ a single edge, we have the contraction $\Gamma/\!/e=\Gamma/e$. In this case we also consider the cut graph $\Gamma-e$ obtained by removing $e$ and also any remaining isolated vertex.

A graph $\Gamma$ is said to be core ($1PI$ in physics terminology) if for any edge $e$ we have $|\Gamma-e|<|\Gamma|$.

A cycle $\gamma\subset\Gamma$ is a core subgraph such that $|\gamma|=1$. If $\Gamma$ is core, it can be written as a union of cycles (see e.g. the proof of lemma 7.4 in [2]).

Let ${\mathcal{P}}$ be a class of graphs. We assume $\emptyset\in{\mathcal{P}}$ and that $\Gamma\in{\mathcal{P}}$ and $\Gamma^{\prime}\cong\Gamma$ implies $\Gamma^{\prime}\in{\mathcal{P}}$. We say ${\mathcal{P}}$ is closed under extension if given $\gamma\subset\Gamma$ we have

(2.3)

$$\gamma,\Gamma\in{\mathcal{P}}\Leftrightarrow\gamma,\Gamma/\!/\gamma\in{\mathcal{P}}.$$

Easy examples of such classes of graphs are ${\mathcal{P}}=\text{core graphs}$, and ${\mathcal{P}}=\text{log divergent graphs}$, where $\Gamma$ is log divergent (in $\phi^{4}_{4}$ theory) if it is core and if further $\#E(\Gamma_{i})=2|\Gamma_{i}|$ for every connected component $\Gamma_{i}\subset\Gamma$. (Both examples are closed under extension by virtue of the identity $|\gamma|+|\Gamma/\!/\gamma|=|\Gamma|$.) Examples which arise in physical theories are more subtle. Verification of (2.3) requires an analysis of which graphs can arise from a given Lagrangian. To verify ${\mathcal{P}}=\{\Gamma\ |\ \textrm{sdd}(\Gamma)\geq 0\}$ satisfies (2.3) one must consider self-energy graphs and the role of vertices of kinetic type as discussed above.

In particular, in massless $\phi_{4}^{4}$ theory divergent graphs are closed under extension, and so is the class of graphs for which $4|\Gamma|-2|\Gamma^{[1]}|+2|\Gamma^{[0],\textrm{kin}}|+2|\Gamma^{[0],\textrm{mass}}|\geq 0$. Note that this may contain superficially convergent graphs if there are sufficiently many two-point vertices of mass type. It pays to include them in the class of graphs to be considered, which enables one to discuss the effect of mass in the renormalization group flow.

Associated to a class ${\mathcal{P}}$ which is closed under extension as above, we define a (commutative, but not cocommutative) Hopf algebra $H=H_{\mathcal{P}}$ as follows. As a vector space, $H$ is freely spanned by isomorphism classes of graphs in ${\mathcal{P}}$. (A number of variants are possible. One may work with oriented graphs, for example. In this case, the theory of graph homology yields a (graded commutative) differential graded Hopf algebra. One may also rigidify by working with disjoint unions of subgraphs of a given labeled graph.) $H$ becomes a commutative algebra with $1=[\emptyset]$ and product given by disjoint union. Define a comultiplication $\Delta:H\to H\otimes H$:

(2.4)

$$\Delta(\Gamma)=\sum_{\begin{subarray}{c}\gamma\subset\Gamma\\ \gamma\in{\mathcal{P}}\end{subarray}}\gamma\otimes\Gamma/\!/\gamma.$$

One checks that (2.3) implies that (2.4) is coassociative. Since $H$ is graded by loop numbers and each $H_{n}$ is finite dimensional, the theory of Hopf algebras guarantees the existence of an antipode, so $H$ is a Hopf algebra.

If ${\mathcal{P}}^{\prime}\subset{\mathcal{P}}$ with Hopf algebras $H^{\prime},H$ ( e.g. take ${\mathcal{P}}$ to be core graphs, and ${\mathcal{P}}^{\prime}\subset{\mathcal{P}}$ divergent core graphs) then the map $H\twoheadrightarrow H^{\prime}$ obtained by sending $\Gamma\mapsto 0$ if $\Gamma\not\in{\mathcal{P}}^{\prime}$ is a homomorphism of Hopf algebras. For example, the divergent Hopf algebra carries the information needed for renormalization [13], while the core Hopf algebra $H_{\mathcal{C}}$ determines the monodromy. In terms of groupschemes, one has ${\rm Spec\,}(H_{\text{log. div.}})\hookrightarrow{\rm Spec\,}(H_{{\mathcal{C}}})$ is a closed subgroupscheme, and renormalization can be viewed as a morphism from the affine line with coordinate $L$ to ${\rm Spec\,}(H_{\text{log. div.}})$. Already here we use that for divergent graphs with $\textrm{sdd}(\Gamma)>0$, we can evaluate them as polynomials in masses and external momenta with coefficients determined from log divergent graphs, see below.

Let $\Gamma_{i},i=1,2$ be core graphs (a similar discussion will be valid for other classes of graphs) and let $v_{i}\in\Gamma_{i}$ be vertices. Let $\Gamma=\Gamma_{1}\cup\Gamma_{2}$ where the two graphs are joined by identifying $v_{1}\sim v_{2}$. Then $\Gamma$ is core (cf. proposition 3.2). Further, core subgraphs $\Gamma^{\prime}\subset\Gamma$ all arise as the image of $\Gamma_{1}^{\prime}\amalg\Gamma_{2}^{\prime}\to\Gamma$ for $\Gamma_{i}^{\prime}\subset\Gamma_{i}$ core. Thus

(2.5)

$$\Delta(\Gamma)=\sum\Gamma^{\prime}\otimes\Gamma/\!/\Gamma^{\prime}=\Big{(}\sum\Gamma_{1}^{\prime}\otimes\Gamma_{1}/\!/\Gamma_{1}^{\prime}\Big{)}\Big{(}\sum\Gamma_{2}^{\prime}\otimes\Gamma_{2}/\!/\Gamma_{2}^{\prime}\Big{)}+\\ \sum(\Gamma^{\prime}-\Gamma_{1}^{\prime}\cdot\Gamma_{2}^{\prime})\otimes(\Gamma_{1}/\!/\Gamma_{1}^{\prime}\cdot\Gamma_{2}/\!/\Gamma_{2}^{\prime})+\sum\Gamma^{\prime}\otimes\Big{(}\Gamma/\!/\Gamma^{\prime}-\Gamma_{1}/\!/\Gamma_{1}^{\prime}\cdot\Gamma_{2}/\!/\Gamma_{2}^{\prime}\Big{)}.$$

It follows that the vector space $I\subset H_{\mathcal{C}}$ spanned by elements $\Gamma-\Gamma_{1}\cdot\Gamma_{2}$ as above satisfies $\Delta(I)\subset I\otimes H_{\mathcal{C}}+H_{\mathcal{C}}\otimes I$. Since $I$ is an ideal, we see that $\overline{H}_{\mathcal{C}}:=H_{\mathcal{C}}/I$ is a commutative Hopf algebra. Roughly speaking, $\overline{H}_{\mathcal{C}}$ is obtained from $H_{\mathcal{C}}$ by identifying one vertex reducible graphs with products of the component pieces.

Generalization to theories with more vertex and edge types are straightforward.

Fig.(4) gives the wheel with three spokes. This graph, which in $\phi^{4}$ theory (external edges to be added such that each vertex is four-valent) has a residue $6\zeta(3)$ for conceptual reasons [2], has a coproduct (we omit edge labels and identify terms which are identical under this omission, which gives the indicated multiplicities)

(2.6)		$\displaystyle\Delta\left(\;\raisebox{-11.38109pt}{\epsfbox{wsix.eps}}\;\right)=\;\raisebox{-11.38109pt}{\epsfbox{wsix.eps}}\;$	$\displaystyle\otimes$	$\displaystyle\mathbb{I}$
	$\displaystyle+\mathbb{I}$	$\displaystyle\otimes$
	$\displaystyle+4\;\raisebox{-11.38109pt}{\epsfbox{wthr.eps}}\;$	$\displaystyle\otimes$
	$\displaystyle+3\;\raisebox{-11.38109pt}{\epsfbox{wfour.eps}}\;$	$\displaystyle\otimes$
	$\displaystyle 6\;\raisebox{-11.38109pt}{\epsfbox{wfive.eps}}\;$	$\displaystyle\otimes$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{wtad.eps}}\;.$

For example, the three possible labelings for the four-edge cycle in the third line are $4523$, $5631$ and $6412$. While the graph has a non-trivial coproduct in the core Hopf algebra, it is a primitive element in the renormalization Hopf algebra. It is tempting to hope that the core coproduct relates to the Hodge structure underlying the period which appears in the residue of this graph.

We introduce the Hopf algebra of decorated non-planar rooted trees $H_{\mathcal{T}}$ using non-empty finite sets as decorations (decorations will be sets of edge labels of Feynman graphs below) to label the vertices of the rooted tree Hopf algebra $H_{\mathcal{T}}(\emptyset)$. Products in $H_{\mathcal{T}}$ are disjoint unions of trees (forests). We write the coproduct as

(2.7)

$$\Delta(T)=T\otimes\mathbb{I}+\mathbb{I}\otimes T+\sum_{\textrm{admissible cuts $C$}}P^{C}(T)\otimes R^{C}(T).$$

Edges are oriented away from the root and a vertex which has no outgoing edge we call a foot. An admissible cut is a subset of edges of a tree such that no path from the root to any vertex of $T$ traverses more than one element of that subset. Such a cut $C$ separates $T$ into at least $2$ components. The component containing the root is denoted $R^{C}(T)$, and the product of the other components is $P^{C}(T)$.

A ladder is a tree without side branching. Decorated ladders generate a sub-Hopf algebra $L_{{\mathcal{T}}}\subset H_{{\mathcal{T}}}$. A general element in $L_{{\mathcal{T}}}$ is a sum of bamboo forests, that is disjoint unions of ladders. Decorated ladders have an associative shuffle product

(2.8)

$$L_{1}\star L_{2}:=\sum_{k\in\text{shuffle}(\ell_{1},\ell_{2})}L(k)$$

where $\ell_{i}$ denotes the ordered set of decorations for $L_{i}$ and $\text{shuffle}(\ell_{1},\ell_{2})$ is the set of all ordered sets obtained by shuffling together $\ell_{1}$ and $\ell_{2}$.

For any decoration $\ell$, one has an operator [3]

(2.12)

$$B_{+}^{\ell}:H_{\mathcal{T}}\to H_{\mathcal{T}}$$

which carries any forest to the tree obtained by connecting a single root vertex with decoration $\ell$ to all the roots of the forest. This operator is a Hochschild 1-cocycle, i.e.

(2.13)

$$\Delta B_{+}^{\ell}=B_{+}^{\ell}\otimes\mathbb{I}+(\text{id}\otimes B_{+}^{\ell})\Delta.$$

Let ${\mathcal{J}}\subset H_{\mathcal{T}}$ be the smallest ideal containing the ideal ${\mathcal{K}}$ as in lemma 2.1 and stable under all the operators $B_{+}^{\ell}$. Generators of ${\mathcal{J}}$ as an abelian group are obtained by starting with elements of ${\mathcal{K}}$ and successively applying $B_{+}^{\ell}$ for various $\ell$ and multiplying by elements of $H_{\mathcal{T}}$. It follows from (2.13) that $\Delta{\mathcal{J}}\subset{\mathcal{J}}\otimes H_{\mathcal{T}}+H_{\mathcal{T}}\otimes{\mathcal{J}}$. Define

(2.14)

$$\overline{H}_{\mathcal{T}}:=H_{\mathcal{T}}/{\mathcal{J}}.$$

A flag in a core graph $\Gamma$ is a chain

(2.15)

$$f:=\emptyset\subsetneq\Gamma_{1}\subsetneq\cdots\subsetneq\Gamma_{n}=\Gamma$$

of core subgraphs. Write $F(\Gamma)$ for the collection of all maximal flags of $\Gamma$. One checks easily that for a maximal flag, $n=|\Gamma|$. Let us consider an example.

(2.16)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.17)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{cdef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.18)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.19)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.20)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcde.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.21)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcdf.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.22)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{cdef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.23)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{cdef.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.24)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcde.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.25)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcdf.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.26)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcde.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$
(2.27)			$\displaystyle\subsetneq$	$\displaystyle\;\raisebox{-11.38109pt}{\epsfbox{abcdf.eps}}\;\subsetneq\;\raisebox{-11.38109pt}{\epsfbox{abcdef.eps}}\;,$

are the twelve flags for the graph given in Fig.(5). We omitted the edge labels in the above flags. Note that only the first two , (2.16,2.17) are relevant for the renormalization Hopf algebra to be introduced below.

To the flag $f$ we associate the ladder $L(f)$ with $n$ vertices decorated by $\Gamma_{i}-\Gamma_{i-1}$. (More precisely, the foot is decorated by $\Gamma_{1}$ and the root by $\Gamma-\Gamma_{n-1}$.). Define

(2.28)

$$\rho_{L}:H_{\mathcal{C}}\to L_{\mathcal{T}};\quad\rho_{L}(\Gamma):=\sum_{f\in F(\Gamma)}L(f)$$

Here the set of labels $D$ will be the set of subsets of graph labels.

In fact, the tree structure associated to a maximal flag $f$ of $\Gamma$ is rather more intricate than just a ladder. Though we do not use this tree structure in the sequel, we present the construction in some detail to help in understanding the difference between the core and renormalization Hopf algebra.

We want to associate a forest $T(f)$ to the flag $f$, and we proceed by induction on $n=|\Gamma|$. We can write $\Gamma=\bigcup\Gamma^{(j)}$ in such a way that all the $\Gamma^{(j)}$ are core and one vertex irreducible, and such that $|\Gamma|=\sum|\Gamma^{(j)}|$. This decomposition is unique. If it is nontrivial, we define $T(f)=\prod T(f^{(j)})$ where $f^{(j)}$ is the induced flag from $f$ on $\Gamma^{(j)}$. We now may assume $\Gamma$ is one vertex irreducible. If the $\Gamma_{i}$ in our flag are all one vertex irreducible, we take $T(f)=L(f)$ to be a ladder as above. Otherwise, let $m<n$ be maximal such that $\Gamma_{m}\subsetneq\Gamma$ is one vertex reducible. By induction, we have a forest $T(f|\Gamma_{m})$. To define $T(f)$, we glue the foot of the ladder with decorations $\Gamma_{m+1}-\Gamma_{m},\dotsc,\Gamma-\Gamma_{n-1}$ to all the roots of $T(f|\Gamma_{m})$. (For an example, see figs.(6) and (7).)

As a consequence of lemma 2.4 we may partition the flags $F(\Gamma)$ associated to a core $\Gamma$ as follows. Given $f\in F(\Gamma)$, Let $\Gamma_{m}\subset\Gamma$ be maximal in the flag $f$ such that $\Gamma_{m}$ is $1$-vertex reducible. The flag $f$ induces a flag $f_{m}$ on $\Gamma_{m}$, and we know that it is a shuffle of flags $f_{m}^{(j)}$ on $\Gamma_{m}^{(j)}$ where $\Gamma_{m}=\bigcup\Gamma_{m}^{(j)}$ as in the lemma. We say two flags are equivalent, $f\sim f^{\prime}$, if $f$ and $f^{\prime}$ agree at $\Gamma_{m}$ and above, and if they simply correspond to two different shuffles of the flags $f_{m}^{(j)}$. We now have

(2.33)

$$T(f)\equiv\sum_{f^{\prime}\sim f}L(f^{\prime})\mod{\mathcal{J}}.$$

Indeed, $T(f)$ is obtained by successive $B_{+}^{\ell}$ operations applied to the forest $T(f|\Gamma_{m})$. The latter, by remark 2.2, coincides with the righthand side of (2.33). We conclude

Here $\rho_{T}(\Gamma)$ is the sum $T(f)$ over equivalence classes of flags $f$ as above. We will barely use $H_{\mathcal{T}}$ in the following, and introduced it for completeness and the benefit of the reader used to it.

In a similar manner, one may define homomorphisms

(2.35)

$$\rho_{\mathcal{R}}:H_{\mathcal{R}}\to\overline{H}_{\mathcal{T}}$$

for any one of the renormalization Hopf algebras obtained by imposing restrictions on external leg structure. For a graph $\Gamma$, let, as before, the residue of $\Gamma$, $\textrm{res}(\Gamma)$, be the graph with no loops obtained by shrinking all its internal edges to a point. What remains are the external half edges connected to that point (cf. section 2.1). Note that ”doubling” an edge by putting a two-point vertex in it does not change the residue.

In $\phi_{4}^{4}$ theory for example, graphs have $2m$ external legs, with $m\geq 0$. For a renormalizable theory, there is a finite set of external leg structures $\mathcal{R}$ such that we obtain a renormalization Hopf algebra for that set.

For example, for massive $\phi_{4}^{4}$ theory, there are three such structures: the four-point vertex, and two two-point vertices, of kinetic type and mass type.

Let us now consider flags associated to core graphs. Such chains $\cdots\Gamma_{i}\subsetneq\Gamma_{i+1}\subsetneq\cdots\subsetneq\Gamma$ correspond to decorated ladders, and the coproduct on the level of such ladders is a sum over all possibilities to cut an edge in such a ladder, splitting the chain

(2.36)

$$[\cdots\subsetneq\Gamma_{i}]\otimes[\Gamma_{i+1}/\!/\Gamma_{i}\subsetneq\cdots\subsetneq\Gamma/\!/\Gamma_{i}].$$

So let us call such an admissible cut renormalization-admissible, if all core graphs $\Gamma_{i}$, $\Gamma/\!/\Gamma_{i}$ obtained by the cut have residues in ${\mathcal{R}}$.

The set of renormalization-admissible cuts is a subset of the admissible cuts of a core graph, and the coproduct respects this. Hence the renormalization Hopf algebra $H_{\mathcal{R}}$ is a quotient Hopf algebra of the core Hopf algebra.

If we enlarge the set ${\mathcal{R}}$ to include other local field operators appearing for example in an operator product expansion we get quotient Hopf algebras between the core and the renormalization Hopf algebra.

External edges are usually labeled by data which characterize the amplitude under consideration. Let $\sigma$ be such data. For graphs $\Gamma$ with a given residue ${\textrm{res}}(\Gamma)$, there is a finite set $\tau\in\{\sigma\}_{{\textrm{res}}(\Gamma)}$ of possible data $\tau$. A choice of such data determines a labeling of the corresponding vertex to which a subgraph shrinks. Let $\Gamma/\!/\gamma_{\tau}$ be that co-graph with the corresponding vertex labeling.

One gets a Hopf algebra structure on pairs $(\Gamma,\sigma)$ by using the renormalization coproduct $\Delta(\Gamma)=\Gamma^{\prime}\otimes\Gamma^{\prime\prime}$ by setting $\Delta(\Gamma,\sigma)=\sum_{\tau\in\{\sigma\}_{\textrm{res}(\Gamma^{\prime})}}(\Gamma^{\prime},\tau)\otimes(\Gamma^{\prime\prime}_{\tau},\sigma)$. We regard the decomposition into external leg structures as a partition of unity and write

(2.37)

$$\sum_{\tau\in\{\sigma\}_{\textrm{res}(\Gamma)}}(\Gamma,\tau)=(\Gamma,\mathbb{I}).$$

In our applications we only need this for (sub)graphs $\gamma$ with $|\textrm{res}(\gamma)|=2$, and the use of these notions will become clear in the applications below.