# How are Feynman graphs resummed by the Loop Vertex Expansion?

Vincent Rivasseau, Zhituo Wang
Laboratoire de Physique Théorique, CNRS UMR 8627,
Université Paris XI, F-91405 Orsay Cedex, France
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The purpose of this short letter is to clarify which set of pieces of Feynman
graphs are resummed in a Loop Vertex Expansion, and to formulate a conjecture
on the $\phi^{4}$ theory in non-integer dimension.

LPT-20XX-xx

MSC: 81T08, Pacs numbers: 11.10.Cd, 11.10.Ef
Key words: Feynman graphs, Combinatorics, Loop vertex expansion.

##  1 Introduction

In quantum field theory (hereafter QFT) any connected (i.e. interesting)
quantity is written as a sum of amplitudes for a certain category of connected
graphs

$S=\sum_{G\;\;{\rm connected}}{\cal{A}}_{G}$ (1)

but this formula is not a valid definition of $S$ since usually

$\sum_{G\;\;{\rm connected}}|{\cal{A}}_{G}|=\infty.$ (2)

This phenomenon, known since [1], is basically due to the very large number of
elements at order $n$ in the species [2] of Feynman graphs. Accordingly the
generating functional for the Feynman graphs species, namely the series
$\sum_{n}\frac{\lambda^{n}a_{n}}{n!}$, where $a_{n}$ is the number of Feynman
graphs at order $n$, has zero radius of convergence as power series in
$\lambda$. We call such a species a proliferating species. In zero space-time
dimension, quantum field theory reduces to this generating functional, hence
to graphs counting. In higher dimensions quantum field theory is in fact a
weighted such species, that is Feynman graphs have to be pondered with
weights, called Feynman amplitudes. For an introduction to the structure of
Feynman graphs, see [3]. Nevertheless these Feynman amplitudes tend to behave
as $K^{n}$ at order $n$ (at least in low dimensions). Hence the perturbation
series eg for the $\phi^{4}$ Euclidean Bosonic quantum field theory tends to
behave as $\sum_{n}(-\lambda)^{n}K^{n}n!$ and it has been proved to have zero
radius of convergence in one, two and three dimensions ([4, 5]). Nothing is
yet known for sure in dimension 4 but there are strong reasons to expect also
the renormalized Feynman series to diverge there as well (see [6] and
references therein).

In contrast Cayley’s theorem, which states that the total number of labeled
trees at order $n$ is $n^{n-2}$, implies that the species of trees is not
proliferating. This fact can be related to the local existence theorems for
flows in classical mechanics, since classical perturbation theory is indexed
by trees [7]. These theorems have no quantum counterpart, but constructive
theory can be seen as various recipes to replace the ordinary divergent
Feynman graph expansions by convergent ones, indexed by trees rather than
graphs [8]. It can therefore be considered a bridge between QFT and classical
mechanics, since it repacks the loops which are the fundamental feature of
QFT, and brings the expansion closer to those of classical mechanics.
Historically constructive theory used cumbersome non canonical tools borrowed
from lattice statistical mechanics, such as cluster expansions which did not
respect the rotational invariance of the underlying theory [9, 6]. The Loop
Vertex Expansion [10, 11] is a more canonical way to replace the ordinary
perturbative divergent expansion by a convergent one, which in principle
allows to compute quantities to arbitrary accuracy.

One of us (VR) was recently asked exactly which (pieces of) Feynman graphs are
resummed by this expansion. The answer is contained in the initial papers, but
perhaps not easy to extract. The purpose of this little note is therefore to
explain more explicitly exactly which pieces of which Feynman graphs of
different orders are combined together by the loop vertex expansion to create
a convergent expansion. This reshuffling is fully explicited up to third order
for the simplest of all possible examples, namely the $\phi^{4}_{0}$ quantum
field theory. Finally we also propose a conjecture, which, if true, would
allow to define QFT in non-integer dimensions of space-time.

##  2 Relative Tree Weights in a Graph

A graph may contain many (spanning) forests, and a forest can be extended into
many graphs with loops. So the relationship between graphs and their spanning
forests is not trivial.

The forest formula which we use [13, 14] can be viewed as a tool to associate
to any pair made of a graph $G$ and a spanning forest ${\cal{F}}\subset G$ a
unique rational number or weight $w(G,{\cal{F}})$ between 0 and 1, called the
relative weight of ${\cal{F}}$ in $G$.

The numbers $w(G,{\cal{F}})$ are multiplicative over disjoint unions 111And
also over vertex joints of graphs, just as in the universality theorem for the
Tutte polynomial.. Hence it is enough to give the formula for $(G,{\cal{F}})$
only when $G$ is connected and ${\cal{F}}={\cal{T}}$ is a spanning tree in
it222It is enough in fact to compute such weights for 1-particle irreducible
and 1-vertex-irreducible graphs, then multiply them in the appropriate way for
the general case..

The definition of these weights is

######  Definition 2.1.

$w(G,{\cal{T}})=\int_{0}^{1}\prod_{\ell\in{\cal{T}}}dw_{\ell}\prod_{\ell\not\in{\cal{T}}}x^{\cal{T}}_{\ell}(\\{w\\})$
(3)

where $x^{\cal{T}}_{\ell}(\\{w\\})$ is the infimum over the
$w_{\ell^{\prime}}$ parameters over the lines $\ell^{\prime}$ forming the
unique path in ${\cal{T}}$ joining the ends of $\ell$.

######  Lemma 2.1.

The relation

$\sum_{{\cal{F}}\subset G}w(G,{\cal{F}})=1$ (4)

holds for any connected graph $G$.

Proof It is a simple consequence of the forest formula [13, 14] applied to the
lines of the graph $G$.

###  2.1 Examples

For a fixed spanning tree inside a graph, we call loop lines the lines not in
the tree.

Consider the graph $G$ of Figure 1.

Figure 1: A first example

There are $5$ spanning trees inside this graph:

${\cal{T}}_{12}=\\{l_{1},l_{2}\\},\ {\cal{T}}_{13}=\\{l_{1},l_{3}\\},\
{\cal{T}}_{14}=\\{l_{1},l_{4}\\},\ {\cal{T}}_{23}=\\{l_{2},l_{3}\\},\
{\cal{T}}_{24}=\\{l_{2},l_{4}\\}.$

For example, for the tree ${\cal{T}}_{12}=\\{l_{2},l_{4}\\}$, the loop lines
are $l_{1}$ and $l_{3}$.

To take into account the weakening factors $x^{\cal{T}}_{\ell}(\\{w\\})$ of
(4) for each loop line $\ell$, it is convenient to decompose the integration
domain $[0,1]^{|{\cal{T}}|}$ into $|{\cal{T}}|!$ sectors corresponding to
complete orderings of the $w_{\ell}$ parameters for $\ell\in{\cal{T}}$.

Let us compute in this way the relative weights of the five trees of $G$.
First consider the contribution of the tree ${\cal{T}}_{12}$. In this case the
loop lines are $l_{3}$ and $l_{4}$. For each of them we have a factor
$\inf(w_{1}w_{2})$. Hence

$\displaystyle w(G,{\cal{T}}_{12})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}dw_{1}dw_{2}[\inf(w_{1},w_{2})]^{2}$
$\displaystyle=$ $\displaystyle
2\int_{0}^{1}dw_{2}\int_{0}^{w_{2}}dw_{1}w_{1}^{2}=\frac{2}{12}=\frac{1}{6}.$

Next we consider the spanning tree ${\cal{T}}_{13}$. In this case the ”loop
lines“ are $l_{2}$ which connects the vertices $v_{1}$ and $v_{3}$ and $l_{4}$
which connects $v_{2}$ and $v_{3}$. So we have:

$\displaystyle w(G,{\cal{T}}_{13})$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{3}}dw_{1}\int dw_{3}\inf(w_{1}w_{3})w_{3}$
$\displaystyle+$ $\displaystyle\int_{w_{3}<w_{1}}dw_{1}\int
dw_{3}\inf(w_{1}w_{3})w_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dw_{3}\int_{0}^{w_{3}}dw_{1}w_{1}w_{3}+\int_{0}^{1}dw_{1}\int_{0}^{w_{1}}dw_{3}w_{3}^{2}=\frac{1}{8}+\frac{1}{12}=\frac{5}{24}.$

With the same method we find that

$w(G,{\cal{T}}_{14})=w(G,{\cal{T}}_{24})=w(G,{\cal{T}}_{23})=\frac{5}{24},$
(6)

and we have

$\sum_{{\cal{T}}\in G}w(G,{\cal{T}})=\frac{1}{6}+4.\frac{5}{24}=1.$ (7)

Let us treat a second example. Consider the graph $G^{\prime}$ of Fig. 2,
which has 6 edges:

$\\{l_{1},l_{2},l_{3},l_{4},l_{5},l_{6}\\}.$ (8)

To each edge $l_{i}$ we associate a factor $w_{i}$.

Figure 2: Example 2-the eye graph

There are 12 spanning trees:

$\displaystyle\\{l_{1},l_{2},l_{3}\\},\\{l_{1},l_{2},l_{4}\\},\\{l_{1},l_{3},l_{4}\\},\\{l_{2},l_{3},l_{4}\\},\\{l_{1},l_{2},l_{5}\\},\\{l_{1},l_{2},l_{6}\\},$
$\displaystyle\\{l_{3},l_{4},l_{5}\\},\\{l_{3},l_{4},l_{6}\\},\\{l_{1},l_{5},l_{4}\\},\\{l_{1},l_{6},l_{4}\\},\\{l_{3},l_{5},l_{2}\\},\\{l_{3},l_{6},l_{2}\\}.$
(9)

Let us compute the relative weight for each of these spanning trees in
$G^{\prime}$. First of all consider ${\cal{T}}_{123}=\\{l_{1},l_{2},l_{3}\\}$.
The other edges are drawn in dotted lines. See figure(3)

Figure 3: The spanning tree $\\{l_{1},l_{2},l_{3}\\}$

As is easily seen the corresponding loop lines are $l_{4}$, $l_{5}$ and
$l_{6}$. The weakening factor for $l_{5}$ and $l_{6}$ $\inf(w_{1},w_{3})$and
the weakening factor for $l_{4}$ is $\inf(w_{1},w_{2},w_{3})$. Therefore we
have

$\displaystyle
w(G^{\prime},{\cal{T}}_{123})=\int_{0<w_{1}<w_{2}<w_{3}<1}dw_{1}dw_{2}dw_{3}\inf(w_{1},w_{3})^{2}\inf(w_{1},w_{2},w_{3})$
$\displaystyle+$ $\displaystyle\rm{other}\ \rm{permutations}\ \rm{of}\
w_{1},w_{2},w_{3}$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}w_{1}^{3}+\int_{w_{2}<w_{3}<w_{1}}dw_{1}dw_{2}dw_{3}w_{3}^{2}w_{2}$
$\displaystyle+$
$\displaystyle\int_{w_{3}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{3}w_{3}^{3}+\int_{w_{2}<w_{1}<w_{3}}dw_{1}dw_{2}dw_{3}w_{1}^{2}w_{2}$
$\displaystyle+$
$\displaystyle\int_{w_{3}<w_{2}<w_{1}}dw_{1}dw_{2}dw_{3}w_{3}^{3}+\int_{w_{1}<w_{3}<w_{2}}dw_{1}dw_{2}dw_{3}w_{1}^{3}.$

We compute only two of the integrals explicitly as others are obtained by
changing the names of variables.

$\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}\
w_{1}^{3}=\int_{0}^{1}dw_{3}\int_{0}^{w_{3}}dw_{2}\int_{0}^{w_{2}}dw_{1}w_{1}^{3}=\frac{1}{120},$
(10) $\int_{w_{1}<w_{2}<w_{3}}dw_{1}dw_{2}dw_{3}\ w_{3}^{2}\
w_{2}=\frac{1}{60}.$ (11)

So we have

$w(G^{\prime},{\cal{T}}_{123})=\frac{1}{120}\times 4+\frac{1}{60}\times
2=\frac{1}{15}.$ (12)

The relative weights in $G^{\prime}$ of the spanning trees ${\cal{T}}_{124}$,
${\cal{T}}_{134}$ and ${\cal{T}}_{234}$ are the same.

Now we consider the tree $\\{l_{1},l_{2},l_{5}\\}$. (See figure 4).

Figure 4: The spanning tree $\\{l_{1},l_{2},l_{5}\\}$

To the loop line $l_{3}$ is associated a weakening factor $\inf(w_{1},w_{5})$.
To the loop line $l_{4}$ is associated a weakening factor $\inf(w_{2},w_{5})$.
To the loop line $l_{6}$ is associated a weakening factor $w_{5}$. So we have

$\displaystyle
w(G^{\prime},{\cal{T}}_{125})=\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}\inf(w_{1},w_{5})\inf(w_{2},w_{5})w_{5}$
$\displaystyle+$ $\displaystyle\rm{other}\ \rm{permutations}\ \rm{of}\
w_{1},w_{2},w_{5}$ $\displaystyle=$
$\displaystyle\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}+\int_{w_{5}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{3}$
$\displaystyle+$
$\displaystyle\int_{w_{2}<w_{5}<w_{1}}dw_{1}dw_{2}dw_{5}w_{5}^{2}w_{2}+\int_{w_{2}<w_{1}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}$
$\displaystyle+$
$\displaystyle\int_{w_{1}<w_{5}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{2}w_{1}+\int_{w_{5}<w_{2}<w_{1}}dw_{1}dw_{2}dw_{5}w_{5}^{3}.$

We have

$\int_{w_{1}<w_{2}<w_{5}}dw_{1}dw_{2}dw_{5}w_{1}w_{2}w_{5}=\frac{1}{48},$ (14)
$\int_{w_{5}<w_{1}<w_{2}}dw_{1}dw_{2}dw_{5}w_{5}^{3}=\frac{1}{120},$ (15)
$\int_{w_{2}<w_{5}<w_{1}}dw_{1}dw_{2}dw_{5}w_{2}w^{2}_{5}=\frac{1}{60}.$ (16)

Similarly we get

$w(G^{\prime},{\cal{T}}_{125})=\frac{1}{120}\times 2+\frac{1}{60}\times
2+\frac{1}{48}\times 2=\frac{11}{120}.$ (17)

By the same method we find that this is also the relative weight of trees
${\cal{T}}_{126},{\cal{T}}_{345},{\cal{T}}_{346},{\cal{T}}_{125},{\cal{T}}_{145},{\cal{T}}_{146},{\cal{T}}_{235}$
and ${\cal{T}}_{236}$.

We can check again that

$\sum_{{\cal{T}}\in
G^{\prime}}w(G^{\prime},{\cal{T}})=4.\frac{1}{15}+8.\frac{11}{120}=1.$ (18)

##  3 Resumming Feynman Graphs

###  3.1 Naive Repacking

Consider the expansion (1) of a connected quantity $S$. The most naive way to
reorder Feynman perturbation theory according to trees rather than graphs is
to insert for each graph the relation (4)

$S=\sum_{G}A_{G}=\sum_{G}\sum_{{\cal{T}}\subset G}w(G,{\cal{T}}){\cal{A}}_{G}$
(19)

and exchange the order of the sums over $G$ and ${\cal{T}}$. Hence it writes

$S=\sum_{\cal{T}}{\cal{A}}_{\cal{T}},\quad{\cal{A}}_{\cal{T}}=\sum_{G\supset{\cal{T}}}w(G,{\cal{T}}){\cal{A}}_{G}.$
(20)

This rearranges the Feynman expansion according to trees, but each tree has
the same number of vertices as the initial graph. Hence it reshuffles the
various terms of a given, fixed order of perturbation theory. Remark that if
the initial graphs have say degree 4 at each vertex, only trees with degree
less than or equal to 4 occur in the rearranged tree expansion.

For Fermionic theories this is typically sufficient and one has for small
enough coupling

$\sum_{{\cal{T}}}|{\cal{A}}_{\cal{T}}|<\infty$ (21)

because Fermionic graphs mostly compensate each other at a fixed order by
Pauli’s principle; mathematically this is because these graphs form a
determinant and the size of a determinant is much less than what its
permutation expansion suggests. This is well known [15, 16, 17].

But this repacking fails for Bosonic theories, because the only compensations
there occur between graphs of different orders. Hence if we were to perform
this naive reshuffling, eg on the $\phi^{4}_{0}$ theory we would still have

$\sum_{T}|{\cal{A}}_{T}|=\infty.$ (22)

##  4 The Loop Vertex Expansion

The loop vertex expansion overcomes this difficulty by exchanging the role of
vertices and propagators before applying the forest formula. The corresponding
regrouping is completely different and each tree resums an infinite number of
pieces of the previous graphs. It relies on a technical tool (which physicists
call the intermediate field representation) which decomposes any interaction
of degree higher than three in terms of simpler three-body interactions. It is
particularly natural for 4-body interactions but can be generalized to higher
interactions as well [18].

This quite universal and powerful trick is linked to various deep physical and
mathematical tools, such as the color 1/N expansion and the Matthews-Salam and
Hubbard-Stratonovich methods in physics and the Kaufmann bracket of a knot and
many similar ideas in mathematics.

It is easy to describe the intermediate field method in terms of functional
integrals, as it is a simple generalization of the formula

$e^{-\lambda\phi^{4}/2}=\int
e^{-\sigma^{2}/2}e^{i\sqrt{\lambda}\sigma\phi^{2}}d\sigma.$ (23)

In this section we introduce the graphical procedure equivalent to this
formula.

In the case of a $\phi^{4}$ graph $G$ each vertex has exactly four half-lines
hence there are exactly three ways to pair these half-lines into two pairs.
Hence each fully labeled (vacuum) graph of order $n$ (with labels on vertices
and half-lines), which has $2n$ lines can be decomposed exactly into $3^{n}$
labeled graphs $G^{\prime}$ with degree 3 and two different types of lines

\- the $2n$ old ordinary lines

\- $n$ new dotted lines which indicate the pairing chosen at each vertex (see
Figure 5).

Figure 5: The extension and collapse for the order 1 graph

Such graphs $G^{\prime}$ are called the 3-body extensions of $G$ and we write
$G^{\prime}{\ \rm ext\ }G$ when $G^{\prime}$ is an extension of $G$. Let us
introduce for each such extension $G^{\prime}$ an amplitude
$A_{G^{\prime}}=3^{-n}A_{G}$ so that

$A_{G}=\sum_{G^{\prime}{\ \rm ext\ }G}A_{G^{\prime}}$ (24)

when $G^{\prime}$ is an extension of $G$.

Now the ordinary lines of any extension $G^{\prime}$ of any $G$ must form
cycles. These cycles are joined by dotted lines.

######  Definition 4.1.

We define the collapse $\bar{G}^{\prime}$ of such a $G^{\prime}$ graph as the
graph obtained by contracting each cycle to a ”bold” vertex (see Figure 5). We
write $\bar{G}^{\prime}{\ \rm coll\ }G^{\prime}$ if $\bar{G}^{\prime}$ is the
collapse of $G^{\prime}$, and define the amplitude of the collapsed graph
$\bar{G}^{\prime}$ as equal to that of $G^{\prime}$.

Remark that collapsed graphs, made of bold vertices and dotted lines, can have
now arbitrary degree at each vertex. Remark also that several different
extensions of a graph $G$ can have different collapsed graphs, see Figure 5.

Now the loop vertex expansion rewrites

$S=\sum_{G}A_{G}=\sum_{G^{\prime}{\ \rm ext\
}G}A_{G^{\prime}}=\sum_{\bar{G}^{\prime}{\ \rm coll\ }G^{\prime}{\ \rm ext\
}G}A_{\bar{G}^{\prime}}.$ (25)

Now we perform the tree repacking according to the graphs $\bar{G}^{\prime}$
with the $n$ dotted lines and not with respect to $G$. This is a completely
different repacking:

$A_{\bar{G}^{\prime}}=\sum_{\bar{\cal{T}}\in\bar{G}^{\prime}}w(\bar{G}^{\prime},\bar{\cal{T}})A_{\bar{G}^{\prime}},$
(26)

so that

$S=\sum_{G^{\prime}{\ \rm ext\
}G}A_{\bar{G}^{\prime}}=\sum_{\bar{\cal{T}}\in\bar{G}^{\prime}}A_{\bar{\cal{T}}},$
(27)
$A_{\bar{\cal{T}}}=\sum_{\bar{G}^{\prime}\supset\bar{\cal{T}}}w(\bar{G}^{\prime},\bar{\cal{T}})A_{\bar{G}^{\prime}}.$
(28)

The ”miracle” is that

######  Theorem 4.1.

For $\lambda$ small

$\sum_{\bar{\cal{T}}}|A_{\bar{\cal{T}}}|<\infty$ (29)

the result being the Borel sum of the initial perturbative series [12].

The proof of the theorem will not be recalled here (see [10, 11, 12]) but it
relies on the positivity property of the $x^{\cal{T}}_{\ell}(\\{w\\})$
symmetric matrix, and the representation of each $A_{\bar{\cal{T}}}$ amplitude
as an integral over a corresponding normalized Gaussian measure of a product
of resolvents bounded by 1. This convergence would not be true if we had
chosen naive $w({\cal{T}},G)$ barycentric weights such as 1/5 for each of the
five trees of the graph in Figure 1.

This method is valid for any $\phi^{4}$ model in any dimension with cutoffs
[11]. It is not limited to $\phi^{4}$ but works eg for any stable interaction
at the cost of introducing more intermediate particles until three body
elementary interactions are reached [18]. It also reproduces correctly the
large $N$ behaviour of $\phi^{4}$ matrix models, which was the key property
for which this expansion was found [10].

##  5 Examples

In this section we give the extension and collapse of the Feynman graphs for
$Z$ and $\log Z$ for the $\phi^{4}_{0}$ model up to order 3. We also recover
the combinatorics of those graphs through the ordinary functional integral
formula for the loop vertex expansion formula of [12].

The extension and collapse at order 1 was shown in Figure (6). In this case
the tree structure is easy. We find only the trivial ”empty” tree with one
vertex and no edge and the ”almost trivial” tree with two vertices and a
single edge. The weight for these trees is 1.

Figure 6: The extension and collapse for order 1 graph, with combinatoric
weight shown below, and the list of corresponding trees.

At second order we find one disconnected Feynman graph and two connected ones.
Only the connected ones survive in the expansion of $\log Z$.

Figure 7: The extension and collapse for order 2 graph and the number of
graphs.

The corresponding graphs and tree structures are shown in Figure (7) and
Figure(8). Using the loop vertex expansion formula we begin to see that graphs
that come from different order of the expansion of $\lambda$ are associated to
the same trees by the loop vertex expansion. Indeed we recover contributions
for the trivial and almost trivial trees of the previous figure. But we find
also a new contribution belonging to a tree with two edges.

Figure 8: The connected graphs and the tree structure from the Loop vertex
expansion. Figure 9: The order 3 vacuum graph and the number of graphs. Figure
10: The extension and collapse for order 3 graph. Figure 11: The graph
structure and combinatorics from the loop vertex expansion at order 3. The
symbols like 1122 means we have 4 loop vertices V, two of them have one
$\sigma$ field each and two of them have two $\sigma$ fields each, as we could
read directly from this figure. Figure 12: The tree structure of order 3
graphs.

At order three the computation becomes a bit more involved but the process is
clear. We could start from the ordinary Feynman graphs and get the graphs of
loop vertex expansion by extension and collapse. This is shown in Figure (10).
The number under each collapsed graph means the number of the corresponding
graphs, as in the previous case. The tree structure is shown in Figure(12). In
this figure the weight factor $w$ means always $w(G,{\cal{T}})$. We could also
get the graphs and combinatorics by using directly the loop vertex expansion,
namely we integrate the $\phi$ fields and consider only the Wick contractions
of the $\sigma$ fields. This is shown in the appendix and Figure (11). In this
process we expand both $\exp V$ and the vertex
$V=\mbox{tr}\log(1+2i\sqrt{2\lambda}\sigma)$. The interactions terms are then
the loop vertices $V$ with various attached $\sigma$ fields. This is shown on
the left hand of Figure (11). For example, the symbol $123$ means we consider
the $V^{3}/3!$ term in $\exp V$. We expand one of the $V$ to order
$\lambda^{1/2}$, namely with one $\sigma$ field attached, one to the order
$\lambda$, namely with 2 $\sigma$ fields attached and the third one to
$\lambda^{3/2}$, namely with 3 $\sigma$ fields. Then we contract the sigma
fields with respect to the Gaussian measure, obtaining all the contracted
graphs. The total number of $123$ graphs could be read directly from this
Gaussian integration. To get the combinatoric factor of each graph we need to
compute the relative weights of these graphs. This is shown in the following
example:

######  Example 5.1.

Figure 13: The example of ’123’ contractions.

We consider the $123$ case for example. This is shown more explicitly in
Figure(13). We use $a,b,\cdots,f$ to label the $\sigma$ fields attached to the
vertices. After the Wick contractions we get three different graphs $A$, $B$
and $C$. The number of possibilities to get $A$ is $3$, the number to get $B$
is $2\times 3=6$ and the number to get $C$ is also 6. So the relative weight
for graph $A$ is $3/(3+6+6)=1/5$ and the relative weights for $B$ and $C$ are
both $6/(3+6+6)=2/5$. As we could read directly from the loop vertex formula
that the total number of $123$ contraction graphs is $960$, we get finally the
combinatoric factor of graph $A$ to be $960\times 1/5=192$, and the
corresponding factors for graphs $B$ and $C$ are $960\times 2/5=384$. This
result agrees with the one coming from the Feynman graph computation.

From these examples we find that the structure of loop vertex expansion is
totally different from that of Feynman graph calculus. At each order of the
loop vertex expansion we combine terms in different orders of $\lambda$.

##  6 Non-integer Dimension

Let us now consider, eg for $0<D\leq 2$ the Feynman amplitudes for the
$\phi^{4}_{D}$ theory. They are given by the following convergent parametric
representation

$A_{D,G}=\int_{0}^{\infty}d\alpha\frac{e^{-m^{2}\sum_{\ell}\alpha_{\ell}}}{U_{G}^{D/2}}$
(30)

where $m$ is the mass and $U_{G}$ is the Kirchoff-Symanzik polynomial for $G$

$U_{G}=\sum_{{\cal{T}}\in G}\prod_{\ell\not\in{\cal{T}}}\alpha_{\ell}.$ (31)

All the previous decompositions working at the level of graphs, they are
independent of the space-time dimension. We can therefore repack the series of
Feynman amplitudes in non integer dimension to get the $D$ dimensional tree
amplitude:

$A_{D,\bar{\cal{T}}}=\sum_{G\supset\bar{\cal{T}}}w(\bar{\cal{T}},G)A_{D,G}$
(32)

We know that for $D=0$ and $D=1$ the loop vertex expansion is convergent.
Therefore it is tempting to conjecture , for instance at least for $D$ real
and $0\leq D<2$ (that is when no ultraviolet divergences require
renormalization)

######  Conjecture 6.1.

For $\lambda$ small

$\sum_{\bar{\cal{T}}}|A_{D,\bar{\cal{T}}}|<\infty$ (33)

the result being the Borel sum of the initial perturbative series.

Progress on this conjecture would be extremely interesting as it would allow
to bridge quantum field theories in various dimensions of space time, and ie
perhaps justify the Wilson-Fisher $4-\epsilon$ expansion that allows good
numerical approximate computations of critical indices in 3 dimensions.

We know however that when renormalization is needed, ie for $D\geq 2$, this
approach has to be completed with the introduction of the correct
counterterms. Presumably in this case the tree expansion should be adapted to
select optimal trees with respect to renormalization group scales. This is
work in progress.

An other possible approach to quantum field theory in non integer dimension,
also based on the forest formula but more radical, is proposed in [19].

##  7 Conclusion

The lessons we may draw from the Loop Vertex Expansion are

  * •

Interactions should be decomposed into three body elementary interactions. The
corresponding fields might be more fundamental than the initial ones.

  * •

Tree formulas solve the constructive problem ie resum perturbation theory at
the cost of loosing locality of the new vertices.

It may be also interesting to further understand why trees are so central both
in the parametric formulas (30) for single Feynman amplitudes and in the non-
perturbative treatment of the theory. The answer might imply a complete
refoundation of quantum field theory around the notion of trees, rather than
Feynman graphs or even functional integrals [19].

##  8 Appendix

In this Appendix we compute the weight of collapsed Feynman graphs using the
Loop Vertex Expansion.

For the $\phi^{4}_{0}$ model we have:

$Z=\frac{1}{\sqrt{2\pi}}\int d\phi
e^{-\frac{1}{2}\phi^{2}-\lambda\phi^{4}}=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}-\frac{1}{2}\log(1+2i\sqrt{2\lambda}\sigma)}.$ (34)

We define

$V=\frac{1}{2}\log(1+2i\sqrt{2\lambda}\sigma).$ (35)

In what follows we compute the vacuum graphs up to order $3$ in $\lambda$. We
expand $Z$ into powers of $V$:

$Z=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[1-V+\frac{1}{2!}V^{2}-\frac{1}{3!}V^{3}+\frac{1}{4!}V^{4}-\frac{1}{5!}V^{5}+\frac{1}{6!}V^{6}],$
(36)

and we have

$\displaystyle\log(1+2i\sqrt{2\lambda}\sigma)$ $\displaystyle=$ $\displaystyle
2\sqrt{2\lambda}i\sigma+4\lambda\sigma^{2}-\frac{16\sqrt{2}i}{3}\lambda^{3/2}\sigma^{3}-16\lambda^{2}\sigma^{4}$
(37) $\displaystyle+$
$\displaystyle\frac{128\sqrt{2}i}{5}\lambda^{5/2}\sigma^{5}+\frac{256}{3}\lambda^{3}\sigma^{6}.$

The first term

$\frac{1}{\sqrt{2\pi}}\int d\sigma e^{-\frac{1}{2}\sigma^{2}}1=1$ (38)

is trivial.

The order $V$ terms give:

$\displaystyle-\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V=\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[-2\lambda\sigma^{2}+8\lambda^{2}\sigma^{4}-\frac{128}{3}\lambda^{3}\sigma^{6}]$
(39) $\displaystyle=$ $\displaystyle-2\lambda+24\lambda^{2}-640\lambda^{3}.$

The $V^{2}$ terms give:

$\displaystyle\frac{1}{2!}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V^{2}=\frac{1}{2!}(\frac{1}{2})^{2}\frac{1}{\sqrt{2\pi}}\int
d\sigma
e^{-\frac{1}{2}\sigma^{2}}[-8\lambda\sigma^{2}+16\lambda^{2}\sigma^{4}$ (40)
$\displaystyle-$ $\displaystyle\frac{64\times
8}{9}\lambda^{3}\sigma^{6}+\frac{128}{3}\lambda^{2}\sigma^{4}-\frac{128\times
8}{5}\lambda^{3}\sigma^{6}-128\lambda^{3}\sigma^{6}]$ $\displaystyle=$
$\displaystyle-\lambda+22\lambda^{2}-\frac{320}{3}\lambda^{3}-624\lambda^{3}.$

The $V^{3}$ terms give:

$\displaystyle-\frac{1}{3!}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}V^{3}=-\frac{1}{3!}(\frac{1}{2})^{3}\frac{1}{\sqrt{2\pi}}\int
d\sigma
e^{-\frac{1}{2}\sigma^{2}}[64\lambda^{3}\sigma^{6}-96\lambda^{2}\sigma^{4}$
(41) $\displaystyle+$ $\displaystyle
384\lambda^{3}\sigma^{6}+512\lambda^{3}\sigma^{6}]$ $\displaystyle=$
$\displaystyle 6\lambda^{2}-300\lambda^{3}.$

The $V^{4}$ terms give:

$\displaystyle\frac{1}{4!}(\frac{1}{2})^{4}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}[64\lambda^{2}\sigma^{4}-\frac{2048}{3}\lambda^{3}\sigma^{6}-768\lambda^{3}\sigma^{6}]$
(42) $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{2}-\frac{80}{3}\lambda^{3}-30\lambda^{3}.$

The $V^{5}$ terms give:

$\displaystyle-\frac{1}{5!}(\frac{1}{2})^{5}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}\ 1280\lambda^{3}\sigma^{6}=-5\lambda^{3}.$ (43)

The $V^{6}$ term gives:

$\displaystyle-\frac{1}{6!}(\frac{1}{2})^{6}\frac{1}{\sqrt{2\pi}}\int d\sigma
e^{-\frac{1}{2}\sigma^{2}}\ 512\lambda^{3}\sigma^{6}=-\frac{1}{6}\lambda^{3}.$
(44)

So up to $3$rd order in $\lambda$ we recover

$Z=-3\lambda+\frac{105}{2}\lambda^{2}-\frac{10395}{6}\lambda^{3}=-4!!\lambda+\frac{8!!}{2!}\lambda^{2}-\frac{12!!}{3!}\lambda^{3},$
(45)

which of course coincide with the number of ordinary Wick contractions derived
by the regular $\lambda\phi^{4}$ Feynman expansion.

Acknowledgments We thank H. Knörrer for asking the question which lead to
writing this paper.

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