Constructive Tensor Field Theory

Constructive Tensor Field Theory\star\star$\star$This paper is a contribution
to the Special Issue on Tensor Models, Formalism and Applications. The full
collection is available at
http://www.emis.de/journals/SIGMA/Tensor_Models.html

Vincent RIVASSEAU

V. Rivasseau

Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI,
F-91405 Orsay Cedex, France<EMAIL_ADDRESS>http://www.rivasseau.com/

Received March 23, 2016, in final form August 18, 2016; Published online
August 22, 2016

We provide an up-to-date review of the recent constructive program for field
theories of the vector, matrix and tensor type, focusing not on the models
themselves but on the mathematical tools used.

constructive field theory; renormalization; tensor models

81T08

##  1 Introduction

This paper is the second one of our twin contributions to the “Special Issue
on Tensor Models, Formalism and Applications” of the SIGMA journal. In our
first paper [49] the physical motivations behind the tensor models and their
connection to quantum gravity are discussed. So we advise strongly the reader
not already familiar with tensor models and their relationship to quantum
gravity to read [49] first, as we shall not duplicate the corresponding
material here, but provide below only a minimal introduction to tensor models.
We shall discuss in detail in this paper only the constructive program for
(Euclidean) random tensor models of general rank $d$, which include as special
cases vector (rank 1) and matrix (rank 2) models. For other aspects of the
burgeoning study of random tensor models and their associated field theories
we refer to the other contributions to the special SIGMA issue as well.

Tensor models were introduced as promising candidates for an _ab initio_
quantization of gravity [3, 4, 24, 57]. Indeed they are combinatorial objects
which do not refer to any background space-time, nor even to any background
topology. However such tensors were initially introduced as symmetric in their
indices, a feature which for a long time prevented to investigate them with
rigorous analytic methods. In particular in contrast with the famous ’t Hooft
$1/N$ expansion for random matrix models, there was no way to probe the large
$N$ limit of such symmetric random tensors at rank $d\geq 3$. The modern
reformulation [12, 27, 30, 34] unlocked the theory by considering
_unsymmetrized_ random tensors. These objects have a larger, truly tensorial
symmetry (typically in the complex case a ${\rm U}(N)^{\otimes d}$ symmetry at
rank $d$ instead of the single ${\rm U}(N)$ of symmetric tensors). This larger
symmetry allows to probe their large $N$ limit through $1/N$ expansions of a
new type [10, 11, 26, 28, 31].

Random tensor models can be further divided into fully invariant models, in
which both propagator and interaction are invariant, and tensor field theories
(hereafter TFT) in which the interaction is invariant but the propagator is
not [7]. This propagator can also incorporate a further gauge invariance to
make contact with group field theory [14, 15], in which case one calls them
tensor group field theories (TGFT).

We restrict ourselves in this paper to the relatively simple case of models
with a quartic interaction, for which the typical constructive task is to
prove Borel summability of the free energy or of the connected Schwinger
functions on the “stable side” of the coupling constant. To be non-trivial and
physically interesting, such Borel summability theorems should be either
_uniform in $N$_, the size of the tensor, for tensor models, or _uniform in
the ultraviolet cutoff_ , hence include renormalization, for TFTs or TGFTs.

Our review starts with a section on general mathematical prerequisites:
quartic tensor models, Nevanlinna theorem [58] and the intermediate field
representation (IFR) plus key combinatorial constructive tools, namely the
forest and jungle formulas [2, 13]. The next section presents three specific
constructive methods adapted to tensor models: the loop vertex expansion
(LVE), which is a combination of the intermediate field representation and of
the forest formula first introduced in [46]; the multiscale loop vertex
expansion (MLVE), which is a combination of the intermediate field
representation and of a level-2 jungle formula with a bosonic and a fermionic
part first introduced in [32]; and iterated Cauchy–Schwarz bounds (ICS
bounds), introduced in this context in [43] and [18], for which we provide
here a slightly improved version.

Then in the last section we review the recent Borel summability results on
tensor models in increasing order of difficulty. The quartic melonic tensor
model in any $d$ [29] requires a simple LVE and no ICS bounds. This is also
the case for the melonic ${\rm U}(1)-T^{4}_{3}$ TGFT [38]. The general quartic
tensor model in any $d$ requires a LVE plus ICS bounds [18]. The melonic ${\rm
U}(1)-T^{4}_{4}$ TGFT requires an MLVE but still no ICS bounds [39]. Finally
the melonic $T^{4}_{3}$ TFT requires both MLVE and ICS bounds [19]. We discuss
also briefly the case of matrix models, for which the LVE has to be completed
by an additional topological expansion in order to identify the $1/N$
expansion of the model in a non-perturbative way up to finite genus $g$ [35].

The world of tensor models and field theories is extremely vast, as tensor
interactions encode infinitely many triangulations of any piecewise linear
manifold in any dimension and of a large class of pseudo-manifolds. It also
includes in particular any matrix model, not only invariant ones, but also
non-commutative field theories such as the Grosse–Wulkenhaar model which we
shall discuss briefly too. The constructive program to explore all these
models at the rigorous non-perturbative level has clearly just started.
However a rich harvest of non-perturbative Borel-summability results has
already been obtained for tensor field theories. Such constructive results
essentially have no counterpart yet in the other approaches to quantum
gravity. This is very promising for the tensor track towards quantum gravity
[47]. In our conclusion we indicate what could be the next steps of this
constructive program.

##  2 Mathematical prerequisites

###  2.1 Quartic tensor models

For a general introduction to tensor models we refer to [34]. We simply recall
in this subsection the notations required for the quartic models discussed
thereafter, following roughly [18].

For a complex rectangular random matrix $M$, called a Wishart matrix, the
basic invariant homogeneous of order $2p$ is unique. It is
$\operatorname{Tr}(MM^{\dagger})^{p}$. All other polynomial invariants are
linear combinations of products of these basic bricks. In particular there is
only one111Of course there is another quartic “disconnected” interaction,
namely $[\operatorname{Tr}MM^{\dagger}]^{2}$. We shall not consider further
such “multi-trace” models here. See however [11]. matrix model with a
connected quartic interaction, namely
$\operatorname{Tr}MM^{\dagger}MM^{\dagger}$. The situation is completely
different in the case of tensor models of higher rank $d$, as there are many
more available interactions and in particular many different invariant quartic
interactions.

We consider a Hermitian inner product space $V$ of dimension $N$ and $e_{n}$,
$n=1,\dots,N$ an orthonormal basis in $V$. A covariant tensor of rank $d$ is a
multilinear form ${\bf T}\colon V^{\otimes d}\to\mathbb{C}$. We denote its
components in the canonical dual tensor product basis by

$\displaystyle T_{n_{1}\dots n_{d}}\equiv{\bf
T}(e_{n_{1}},\dots,e_{n_{d}}),\qquad{\bf T}=\sum_{n_{1},\dots
n_{d}}T_{n_{1}\dots n_{d}}e^{n_{1}}\otimes\dots\otimes e^{n_{d}}.$

A priori $T_{n_{1}\dots n_{d}}$ has no symmetry properties, hence its indices
have a well defined position. We call the position of an index its _color_ ,
and we denote ${\cal{D}}$ the set of colors $\\{1,\dots,d\\}$. Subsets
${\cal{C}}\subset{\cal{D}}$ will be called generalized colors; singleton
subsets identify with the ordinary colors.

Using the canonical identification of $V$ with its dual thanks to the Hilbert
scalar product, we can also expand the dual tensor as a conjugated multilinear
map ${\bf T}^{\vee}$ with matrix elements
$\bar{T}_{\bar{n}_{1}\dots\bar{n}_{d}}$ in the canonical dual basis. We write
all the indices in subscript, and we denote the contravariant indices with a
bar. Indices are always understood to be listed in increasing order of their
colors. We denote
$\delta_{n_{{\cal{C}}}\bar{n}_{{\cal{C}}}}=\prod\limits_{c\in{\cal{C}}}\delta_{n_{c}\bar{n}_{c}}$
and $\operatorname{Tr}_{{\cal{C}}}$ the partial trace over the indices
$n_{c}$, $c\in{\cal{C}}$.

Under unitary base change, covariant tensors transform under the tensor
product of $d$ fundamental representations of ${\rm U}(N)$: the group acts
independently on each index of the tensor. For $U^{(1)}\cdots U^{(d)}\in{\rm
U}(N)$,

$\displaystyle{\bf T}\to\big{(}U^{(1)}\otimes\cdots\otimes U^{(d)}\big{)}{\bf
T},\qquad{\bf T}^{\vee}\to{\bf T}^{\vee}\big{(}U^{(1)*}\otimes\cdots\otimes
U^{(d)*}\big{)}.$

In components, it reads

$\displaystyle
T_{a_{{\cal{D}}}}\to\sum_{m_{{\cal{D}}}}U^{(1)}_{a_{1}m_{1}}\cdots
U^{(d)}_{a_{d}m_{d}}\
T_{m_{{\cal{D}}}},\qquad\bar{T}_{\bar{a}_{{\cal{D}}}}\to\sum_{\bar{m}_{{\cal{D}}}}\bar{U}^{(1)}_{\bar{a}_{1}\bar{m}_{1}}\cdots\bar{U}^{(d)}_{\bar{a}_{d}\bar{m}_{d}}\
\bar{T}_{\bar{m}_{{\cal{D}}}}.$

A _tensor invariant_ is a polynomial in the components of the tensor and its
conjugate which is invariant under this action of the external tensor product
of $d$ independent copies of the unitary group ${\rm U}(N)$. Tensor invariants
are linear combinations of products of connected invariants (also called
bubbles or generalized traces). Such connected invariants are built by
contracting indices of the same color of a product of tensor entries into a
connected graph. Hence connected invariants are in one to one correspondence
with $d$-regular edge-colored bipartite connected graphs [34] and can be
enumerated quite precisely [6].

The unique quadratic trace invariant is the (scalar) Hermitian pairing of
${\bf T}^{\vee}$ and ${\bf T}$ which reads

$\displaystyle{\bf T}^{\vee}\cdot_{{\cal{D}}}{\bf
T}=\sum_{n_{{\cal{D}}}\bar{n}_{{\cal{D}}}}\bar{T}_{\bar{n}_{1}\dots\bar{n}_{d}}\delta_{\bar{n}_{{\cal{D}}}n_{{\cal{D}}}}T_{n_{1}\dots
n_{d}},$

In arbitrary rank $d$, the most general quartic connected trace invariants are
associated to non trivial generalized colors, i.e., to subsets
${\cal{C}}\subset{\cal{D}}$, ${\cal{C}}\neq\varnothing$,
${\cal{C}}\neq{\cal{D}}$. The connected quartic invariant $V_{\mathcal{C}}$ is

$\displaystyle V_{{\cal{C}}}({\bf T}^{\vee},{\bf
T})=\operatorname{Tr}_{{\cal{C}}}\big{[}\big{[}{\bf
T}^{\vee}\cdot_{{\cal{D}}\setminus{\cal{C}}}{\bf
T}\big{]}\cdot_{{\cal{C}}}\big{[}{\bf
T}^{\vee}\cdot_{{\cal{D}}\setminus{\cal{C}}}{\bf T}\big{]}\big{]},$

where we denoted $\cdot_{{\cal{C}}}$ the product of operators from
$V^{\otimes{\cal{C}}}$ to $V^{\otimes{\cal{C}}}$. In components this invariant
writes:

$\displaystyle\sum_{n,\bar{n},m,\bar{m}}\big{(}\bar{T}_{\bar{n}}\delta_{\bar{n}_{{\cal{D}}\setminus{\cal{C}}}n_{{\cal{D}}\setminus{\cal{C}}}}T_{n}\big{)}\delta_{n_{{\cal{C}}}\bar{m}_{{\cal{C}}}}\delta_{\bar{n}_{{\cal{C}}}m_{{\cal{C}}}}\big{(}\bar{T}_{\bar{m}}\delta_{\bar{m}_{{\cal{D}}\setminus{\cal{C}}}m_{{\cal{D}}\setminus{\cal{C}}}}T_{m}\big{)}.$

Remark that the invariant for ${\cal{C}}$ is the same as for its complement
${\cal{D}}\setminus{\cal{C}}$. Hence at rank $D$ there are exactly $2^{D-1}-1$
quartic trace invariants. The 7 invariants at rank $d=4$ are shown in Fig. 1.

Figure 1: The quartic invariants of rank 4 [18]: there are four types of
“melonic” ones (one of them shown left, the others being obtained by permuting
the colors) and three types of “necklaces” (one of them shown right, the
others being obtained by color permutations).

A generic quartic tensor model (with standard scaling and interaction set
$\mathcal{Q}$) is then defined by the moments of the (invariant) perturbed
Gaussian measure:

$\displaystyle
d\mu=\left(\prod_{n}N^{D-1}\frac{d\bar{T}_{{n}}dT_{n}}{2\imath\pi}\right)e^{-N^{D-1}\big{(}{\bf
T}^{\vee}\cdot_{{\cal{D}}}{\bf
T}+\lambda\sum\limits_{\mathcal{C}\in\mathcal{Q}}V_{\mathcal{C}}({\bf
T}^{\vee},{\bf T})\big{)}},$ (2.1)

where $\mathcal{Q}$ is some subset of ${\cal{C}}$s. The global scaling factor
$N^{D-1}$ for the action is the unique one such that the model admits a non-
trivial $1/N$ expansion [11, 34]. Of course more general models can be
considered by giving an independent coupling to each interaction, or by
considering dimensions $N_{c}$ depending on the color, or by adding the
disconnected interaction $[{\bf T}^{\vee}\cdot_{{\cal{D}}}{\bf T}]^{2}$, but
we shall not consider them here.

The “melonic” quartic models [18] are obtained by restricting the interactions
to singletons, so that $\mathcal{Q}=\\{\mathcal{C},\,|\mathcal{C}|=1\\}$ in
(2.1). In other words interaction is restricted to the $d$ quartic melonic
invariants, one for each color.

Quartic tensor field theories have the same interactions but a non-symmetric
propagator. The simplest such models are built around the Hilbert space
$V:=L^{2}({\mathbb{Z}})$ and as in usual field theory, the propagator is the
inverse of a Laplacian operator plus a mass term

$\displaystyle
C(n_{\cal{D}},\bar{n}_{\cal{D}})=\frac{\delta_{n_{\cal{D}},\bar{n}_{\cal{D}}}}{\sum\limits_{j=1}^{d}n_{j}^{2}+m^{2}}.$
(2.2)

The melonic quartic TGFT at rank $d$ with propagator (2.2) and a unique
coupling constant $\lambda$ identical for each coupling is nicknamed
$T^{4}_{d}$. We often further simplify the model by putting $m^{2}=1$.

Finally tensor group field theories are built around a more general space
$V:=L^{2}(G)$ where $G$ is a Lie group [14, 15] or homogeneous space [40], and
their propagator incorporates a Boulatov projector

$\displaystyle
C(n_{\cal{D}},\bar{n}_{\cal{D}})=\frac{\delta_{n_{\cal{D}},\bar{n}_{\cal{D}}}\delta\Big{(}\sum\limits_{j=1}^{d}n_{j}\Big{)}}{\sum\limits_{j=1}^{d}n^{2}_{j}+m^{2}},$
(2.3)

where the indices $n$ now run over a basis of the dual Fourier space of
$L^{2}(G)$, and $n^{2}_{j}$ is a sloppy notation for the Laplace–Beltrami
operator on $G$ (see [14] for the case $G={\rm SU}(2)$). The melonic quartic
TGFT at rank $d$ with propagator (2.3) and Lie group $G$ will be nicknamed
$G-T^{4}_{d}$. For instance in the case $G={\rm U}(1)$ the Hilbert space $V$
is $L^{2}({\mathbb{Z}})$ as in the previous case. We often consider discrete
tensor indices as momenta, and the equivalent representation with continuous
indices in $G^{d}$ as the corresponding “direct space” representation.

###  2.2 Borel summability

The perturbative expansion in quantum field theory is obtained by performing a
Taylor expansion of the interaction, and then illegally commuting the sum with
the Gaussian integral to obtain a series indexed by Feynman graphs. Let us
return in this section to ordinary quantum field theory, since we want to
discuss the relationship between functional integrals and their perturbative
expansion in the traditional context before turning to tensor models. The
ordinary $\phi^{4}_{d}$ theory in dimension $d$ has propagator kernel

$\displaystyle C(x,y)=\int d^{d}pe^{ip(x-y)}\big{(}p^{2}+m^{2}\big{)}^{-1}$

and its partition function is

$\displaystyle Z(\lambda)=\int d\mu_{C}\,e^{-\lambda\int
d^{d}x\phi^{4}(x)}=\sum_{n=0}^{\infty}\frac{(-\lambda)^{n}}{n!}\sum_{G}A_{G},$
(2.4)

where $G$ runs over labeled Feynman graphs222Labeled Feynman graphs can be
considered as defined by Wick contractions of labeled fields belonging to
labeled vertices. Hence we shall avoid in this brief review the subtle issue
of automorphim groups of graphs and of their corresponding symmetry factors.
with $n$ vertices. The Feynman amplitude of $G$ is

$\displaystyle A_{G}=\int\prod_{v\in V(G)}d^{d}x_{v}\prod_{\ell\in
E(G)}C(x_{v(\ell)},x_{v^{\prime}(\ell)}),$

where the index $v$ runs over the set of $n$ vertices of $G$, the index $\ell$
runs over edges of $G$ and $(v(\ell),v^{\prime}(\ell))$ is the pair of end
vertices of edge $\ell$. Of course such amplitudes may diverge if $d\geq 2$
(see, e.g., [33] for a recent review).

Interesting quantum field theories have many degrees of freedom, in which case
the important physical quantities are intensive rather than extensive
quantities, such as the specific free energy and the connected functions or
cumulants $G^{c}_{2p}$. Therefore the important quantum field theory quantity
is not $Z$ but its logarithm. In particular the connected Green functions are
obtained from the generating functional

$\displaystyle W(J)=\log[Z(\lambda,J)]$

through

$\displaystyle
G^{c}_{2p}(x_{1},\dots,x_{2p})=\frac{\partial^{2p}W(J)}{\partial
J_{1}(x_{1})\cdots\partial J_{2p}(x_{2p}}\Big{|}_{J=0},$

where $Z(\lambda,J)$ is defined by adding a source field $J$ to the partition
function

$\displaystyle Z(\lambda,J)=\int d\mu_{C}\,e^{\int
d^{d}x\phi(x)J(x)-\lambda\int d^{d}x\phi^{4}(x)}.$ (2.5)

Similarly the free energy $p$ also is deduced from the logarithm of $Z$, with
interaction restricted to a finite volume $\Lambda$. One should divide by the
volume cutoff $\Lambda$, and finally remove this cutoff, hence
$p=\lim\limits_{|\Lambda|\to\infty}\frac{1}{|\Lambda|}\log
Z_{\Lambda}(\lambda)$.

The most important property of the perturbative expansion is that it allows
for a quick computation of such physical connected quantities. They are indeed
given by the same sums as (2.4), with the same amplitudes, but _restricted to
connected graphs_. For instance the free energy is given in perturbation
theory by a sum identical to (2.4) but restricted to _connected_ rooted
Feynman vacuum graphs, in which for each graph a particular root vertex has
been fixed to the origin. This rule accounts for the need to break the
translation invariance of the theory and divide by the volume. Most of the
time in this review we shall consider for simplicity only this free energy.

Perturbative series typically diverge. In particular for $d=0$, in which
$A_{G}$ is 1 for any $G$, there are $(4n-1)!!$ labeled Feynman graphs with $n$
vertices, hence the perturbative series for $Z(\lambda)$ is

$\displaystyle\sum_{n=0}^{\infty}\frac{(-\lambda)^{n}}{n!}(4n-1)!!=\sum_{n=0}^{\infty}\frac{(-\lambda)^{n}}{n!}\frac{(4n)!}{2^{2n}(2n)!},$

which has zero radius of convergence. Up to dimension $d\leq 3$ the
_renormalized_ $\phi^{4}$ series can be shown rigorously to also have zero
radius of convergence [17, 36], and it is believed to be also true for $d=4$,
although, to our knowledge, there is no proof of this, and the best result in
this direction remains the quite old reference [16].

In the quantum field theory literature we often read that “since the
perturbation series diverges it can be asymptotic at best”, and that
“perturbation theory cannot capture non-perturbative effects, such as
instantons”. Such statements are both confusing and mathematically wrong.

First, asymptoticity of a series is a very weak notion which has nothing to do
with its _convergence_. _Every_ power series, no matter how convergent or
horribly divergent, is asymptotic to _infinitely many_ smooth functions of a
real-variable. Hence an asymptotic series never encodes the full information
about such a function without additional conditions. The typical example is
the function $f(\lambda)=0$ for $\lambda\leq 0$ and
$f(\lambda)=e^{-\frac{1}{\lambda}}$ for $\lambda>0$ whose asymptotic series at
$\lambda=0$ is $0$, the most convergent of all series. It certainly does not
encode the information to reconstruct $f$.

Second, as we shall clarify below, in the case of a Borel summable series, all
the information contained in the functional integral (including any non-
perturbative effect) is in fact entirely contained in the coefficients of the
perturbative series. It is simply difficult to extract in practice.

To correctly relate a function to its power series, the key is to impose an
analyticity condition, hence to consider functions of a complex variable.
Analyticity is a very strong and rigid condition. For instance the function 0
is the only one asymptotic to the 0 series _in the class of functions analytic
in an open neighborhood of the expansion point_.

Fortunately analyticity is a common feature in quantum field theory. The
partition function (2.4) is usually analytic in a non-empty domain of the
coupling constant. It is certainly analytic for $\Re\lambda>0$ in dimension
$d=0$, or if we include volume and ultra-violet cutoffs, since in this case
the functional integral is uniformly convergent. For $|\lambda|$ sufficiently
small this analyticity extends to $\log Z$, since $Z$ is close to 1 and its
logarithm well-defined; note however that the domain of analyticity for $\log
Z$ typically shrinks as cutoffs are removed.

In the super-renormalizable domain for $\phi^{4}$, namely in dimension 2 or 3,
the point $\lambda=0$ belongs to the boundary of the analyticity domain for
the renormalized free energy [17, 36]. Nevertheless the functional integral
(2.4) can be fully recovered from the perturbative series through Borel
summability [21, 44]. The most useful Borel summability theorem for quantum
field theory is due to Nevanlinna (1919) and was rediscovered by A. Sokal:

######  Theorem 2.1 (Nevanlinna–Sokal [58]).

A function $f(\lambda)$ with $\lambda\in\mathbb{C}$ is said to be Borel
summable in $\lambda$ if

  * •

$f(\lambda)$ is analytic in a disk ${\cal
D}_{R}:=\big{\\{}\lambda\,|\,\Re\big{(}\lambda^{-1}\big{)}>R^{-1}\big{\\}}$
with $R\in\mathbb{R}_{+}$.

  * •

There exists some constants $K$ and $\sigma$ such that $f(\lambda)$ expands at
the origin with uniform Taylor remainder bounds in the disk ${\cal D}_{R}$

$\displaystyle
f(\lambda)=\sum_{k=0}^{n-1}f_{k}\lambda^{k}+R_{n}(\lambda),\qquad|R_{n}(\lambda)|\leq
K\sigma^{n}n!|\lambda|^{n}.$

If $f(\lambda)$ is Borel summable in $\lambda$ then

$\displaystyle B(t)=\sum_{k=0}^{\infty}\frac{1}{k!}f_{k}t^{k}$

is an analytic function for $|t|<\sigma^{-1}$ which admits an analytic
continuation in the strip $\\{z\,|\,|\Im z|<\sigma^{-1}\\}$ such that
$|B(t)|\leq Be^{t/R}$ for some constant $B$ and $f(\lambda)$ is given by the
absolutely convergent integral

$\displaystyle
f(\lambda)=\frac{1}{\lambda}\int_{0}^{\infty}dt\,B(t)e^{-\frac{t}{\lambda}}.$

That is the Taylor expansion of $f(\lambda)$ at the origin is Borel summable,
and $f(\lambda)$ is its Borel sum. The set
$\big{\\{}\lambda\,|\,\Re\big{(}\lambda^{-1}\big{)}>R^{-1},\,R\in\mathbb{R}_{+}\big{\\}}$
is a disk in the complex plane with center at $\frac{R}{2}$ and of radius
$\frac{R}{2}$ (see Fig. 2) as, denoting
$\lambda=\frac{R}{2}+ae^{\imath\gamma}$,

$\displaystyle\Re\big{(}\lambda^{-1}\big{)}>R^{-1}\Leftrightarrow\frac{R^{2}}{4}>a.$

Figure 2: A disk for Nevanlinna’s theorem.

The important remark is that Borel summability provides a uniqueness
criterion: if a power series with coefficients $a_{n}$ is the Taylor expansion
of a Borel summable function $f(\lambda)$ at $\lambda=0$, then $f$ is
_uniquely defined_ by the list of $a_{n}$. This is the case for the partition
function of the $\phi^{4}_{d}$ model with finite cutoffs, and also for the
connected functions without cutoffs after renormalization in the super-
renormalizable domain $d\leq 3$ [21, 44]. It is also the case for the
“infrared” limit of $\phi^{4}_{4}$ with fixed ultraviolet cutoff and at the
critical point, which is a just renormalizable marginal theory asymptotically
free in the infrared direction [22].

When Borel summability holds, the perturbative expansion in fact contains
(although in a quite hidden manner) all information about the functional
integral, _including instanton effects to all orders_ , since a good
definition of these instantons is that they occur as discontinuities between
different branches of analytic continuation of the functional integral [8]. Of
course such effects are very difficult to extract in practice from the
perturbative coefficients, because analytic continuation of a function defined
by its Taylor germ at a point is typically very difficult.

In summary Borel summability is a key step for the construction of a quantum
field theory model, since when it holds, it defines a unique analytic germ
synthesizing Feynman’s two great ideas: the functional integral and the
Feynman graphs. It does not solve all physical issues, such as computing the
long range behavior of the model and its bound states, but it renders such
questions at least mathematically _well-defined_.

###  2.3 Intermediate field representation (IFR)

Any quartic (i.e., four-body) interaction can be decomposed as the gluing of
two more elementary three body-interactions, by splitting the vertex through a
so-called intermediate field. This method (usually called Hubbard–Stratonovic
transformation in condensed matter, and Matthews–Salam representation in high
energy physics) has been extremely fruitful in physics. In the case of the
ordinary $\phi^{4}$ theory defined by (2.5), with a slightly more adapted
normalization for the coupling $\lambda$, it consists in writing

$\displaystyle e^{-\frac{\lambda}{2}\int d^{d}x\phi^{4}(x)}=\int
d\nu(\sigma)e^{i\sqrt{\lambda}\int d^{d}x\phi^{2}(x)\sigma(x)},$

where $d\nu$ is the normalized Gaussian measure with “ultralocal” covariance
kernel $\delta(x-y)$. The functional integral over $\phi$ becomes Gaussian,
hence can be explicitly performed, leading to

$\displaystyle Z(\lambda,J)=\int d\nu(\sigma)\,e^{\langle
J,C^{1/2}R(\sigma)C^{1/2}J\rangle}e^{-\frac{1}{2}\operatorname{Tr}\log(1-i\sqrt{\lambda}C^{1/2}\sigma
C^{1/2})},$

where we used operator notation. $\sigma$ is a local multiplication operator,
diagonal in direct space, $C^{1/2}$ is the square root in operator sense of
the positive operator $C$, hence is not diagonal in direct space333Cyclicity
of the trace means we can also freely replace the operator $C^{1/2}\sigma
C^{1/2}$, by $C\sigma$ or $\sigma C$, leading to slightly shorter formulas,
but this would be unwise as Hermiticity would no longer be visible.. 1 means
the identity operator (with kernel $\delta(x-y)$ in direct space), and
$\operatorname{Tr}K=\int d^{d}xK(x,x)$ simply means integration of the
diagonal part of a kernel $K$. Finally the resolvent $R(\sigma)$ is given by

$\displaystyle R(\sigma)=\big{[}1-i\sqrt{\lambda}C^{1/2}\sigma
C^{1/2}\big{]}^{-1},$

and for any $\sigma$ and any $\lambda$ real positive it is a well defined
operator norm-bounded by 1. More precisely a moment of investigation reveals
that the useful natural domain of definition and analyticity in the $\lambda$
complex plane for such a resolvent is a cardioid, pictured in Fig. 3.

Figure 3: The cardioid domain.

This cardioid is defined, in polar coordinates $\lambda=\rho e^{i\phi}$,
$\rho>0$, $\phi\in]{-}\pi,\pi[$ by the inequality $\rho<[\cos(\phi/2)]^{2}$.
Indeed under this condition we have that $R(\sigma)$ is a well defined
operator norm-bounded by $[\cos(\phi/2)]^{-1}$. In expansion steps of the
resolvents such as

$\displaystyle R(\sigma)=1+i\sqrt{\lambda}C^{1/2}\sigma C^{1/2}R(\sigma)$

the bad factor $[\cos(\phi/2)]^{-1}$ for $\|R\|$ will be always compensated by
a good factor $\sqrt{\rho}<\cos(\phi/2)$ coming from the $\sqrt{\lambda}$
numerator. Remark that any cardioid domain _contains_ a Nevanlinna disk of the
type shown in Fig. 2.

Just like the ordinary representation, the IFR has an associated perturbation
theory, obtained by expanding the exponential of the (non-polynomial)
interaction
$V=\frac{1}{2}\operatorname{Tr}\log\big{(}1-i\sqrt{\lambda}C^{1/2}\sigma
C^{1/2}\big{)}$ as

$\displaystyle e^{-V}=\sum_{n=0}^{\infty}\frac{(-V)^{n}}{n!},$

then commuting (again illegally!) integration with respect to $d\nu$ and
Gaussian integration over $\sigma$, leading to

$\displaystyle
Z(\lambda)=1+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n!}\Big{[}e^{\frac{1}{2}\int
d^{d}x\frac{\delta^{2}}{\delta\sigma(x)^{2}}}V(\sigma)^{n}\Big{]}_{\sigma=0}.$

An intermediated field propagator is represented in Fig. 4 below by a dashed
line, which corresponds to the (ultra-local) covariance of a $\sigma$
intermediate field. The interaction
$V=\frac{1}{2}\operatorname{Tr}\log\big{(}1-i\sqrt{\lambda}C^{1/2}\sigma
C^{1/2}\big{)}$ is non polynomial, but since $\operatorname{Tr}\log 1=0$, it
appears in the perturbative expansion only through its functional derivatives
which we call _loop vertices_. In contrast with the original $\phi^{4}$
vertices, such loop vertices have arbitrary coordination $q\geq 1$, and bear
the initial propagators of the theory on their $q$ _corners_ or arcs.

Several interesting remarks are in order. The IFR exchanges the traditional
role of propagators and vertices. The IFR loop vertices are non-local, as they
contain the propagators of the ordinary theory. The IFR propagators in
contrast are local as they correspond to the vertices of the ordinary theory.
An even deeper remark is that the cyclic character of the trace allows for an
embedding of the graphs in the plane, so that the Feynman graphs of the IFR
should really be considered as _combinatorial maps_.

Figure 4: An intermediate field propagator (dashed line), a loop vertex
(circle, here with 3 corners or arcs and 3 intermediate field half-
propagators) and a ciliated loop vertexwith 4 intermediate field half-
propagators and five arcs/corners.

Finally it is easy to add sources to the picture. Any connected function of
order $2p$ is obtained by the same expansion in which we add to the ordinary
loop vertices $p$ particular _ciliated loop vertices_ [29]. The cilium of the
$j$-th ciliated vertex means a cut with two different external variables
$x_{j}$, $x_{j+1}$ for $j=1,\dots,p$ on both sides of the cilium. Apart from
the cut, ciliated vertices are similar to loop vertices and bear propagators
on each of their corners.

Figure 5: A 2-point graph in the intermediate field theory, here of order 8,
has a single _ciliated_ loop vertex.

The $2p$-point functions of the theory become sums over Feynman graphs of the
intermediate representation with exactly $p$ ciliated vertices, as shown in
Fig. 5. The order of perturbation is simply the number of dashed lines.

###  2.4 The forest formula

A forest formula expands a quantity defined on $n$ points in terms of forests
built on these points. Forest formulas, provided they have a positivity
property are the key combinatorial component of constructive field theory and
were developed systematically in particular by Brydges, Battle and Federbush.
The most beautiful such formula is symmetric under action of the permutation
group on the $n$ points. It was discovered in [13] and developed with
alternative proofs in [2]444Non-symmetric versions appeared earlier in the
constructive literature, but won’t be treated here (see [50] for a recent
reference)..

Consider $n$ points which we identify with the set $V_{n}$ of vertices of the
complete graph $K_{n}$; the set of pairs of such points has $n(n-1)/2$
elements $\ell=(i,j)$ for $1\leq i<j\leq n$ and can be identified with the set
$E_{n}$ of edges $\ell$ of $K_{n}$. The forest formula is often presented as a
Taylor expansion for functions $f$ of $n(n-1)/2$ variables $x_{\ell}$,
$\ell\in E_{n}$ which are smooth, e.g., on an open neighborhood of
$[0,1]^{n(n-1)/2}$. Here we expose a variant borrowed from [33] close to
constructive applications.

Consider the vector space $S_{n}$ of symmetric $n$ by $n$ matrices
$X=\\{X_{ij}\\}$, $i,j=1,\dots,n$. It has dimension $n(n+1)/2$. The set ${\rm
PS}_{n}$ of positive symmetric matrices whose diagonal coefficients are all
equal to 1 and off-diagonal elements are between 0 and 1 is compact and
convex. Symmetric matrices with diagonal elements equal to one and off-
diagonal elements in $[0,1]^{n(n-1)/2}$ do not all belong to ${\rm PS}_{n}$,
for instance the matrix $\begin{pmatrix}1&1&0\cr 1&1&1\cr 0&1&1\end{pmatrix}$
is not positive. Any matrix $X\in{\rm PS}_{n}$ can be parametrized by
$n(n-1)/2$ elements $X_{\ell}$, where $\ell$ runs over the edges of the
complete graph $K_{n}$.

${\rm PS}_{n}$ contains as particularly interesting elements the block
matrices $X^{\Pi}$ for any partition $\Pi$ of $V_{n}$. The block matrix
$X^{\Pi}$ has entries $X_{ij}^{\Pi}=1$ if $i$ and $j$ belong to the same block
of the partition $\Pi$, and 0 otherwise. Two extremal cases are the identity
matrix $\operatorname{Id}$, corresponding to $X^{\rm sing}$, that is to the
maximal partition made of all singletons, and the matrix ${\bf 1}$ with all
entries equal to one, corresponding to $X^{V_{n}}$, that is to the minimal
partition made of a single block.

Let us consider a function $f$ defined and smooth in the interior of ${\rm
PS}_{n}$ with continuous extensions (together with all their derivatives) to
${\rm PS}_{n}$ itself. The forest formula can be expressed as a multi-variate
Taylor formula with integral remainder which expands such a function between
the minimal and maximal block-partition matrices ${\bf 1}$ and
$\operatorname{Id}$. The important point is that the Taylor remainder
integrands stay on the ${\rm PS}_{n}$ convex set. The precise statement is

######  Theorem 2.2 (the forest formula).

$\displaystyle f({{\bf 1}})=\sum_{{\cal{F}}}\int
dw_{\cal{F}}\,\partial_{\cal{F}}f\big{[}X^{\cal{F}}(w_{\cal{F}})\big{]},$

where

  * •

the sum over ${\cal{F}}$ is over forests over $n$ labeled vertices
$i=1,\dots,n$, including the empty forest with no edge. Such forests are
exactly the acyclic edge-subgraphs of the complete graph $K_{n}$,

  * •

$\int dw_{\cal{F}}$ means integration from $0$ to $1$ over one parameter for
each forest edge: $\int
dw_{\cal{F}}\equiv\prod\limits_{\ell\in{\cal{F}}}\int_{0}^{1}dw_{\ell}$. There
is no integration for the empty forest since by convention an empty product is
$1$. A generic integration point $w_{\cal{F}}$ is therefore made of
$|{\cal{F}}|$ parameters $w_{\ell}\in[0,1]$, one for each $\ell\in{\cal{F}}$,

  * •

$\partial_{\cal{F}}=\prod\limits_{\ell\in{\cal{F}}}\partial_{\ell}$ means a
product of first-order partial derivatives with respect to the variables
$X_{\ell}$ corresponding to the edges of ${\cal{F}}$. Again there is no such
derivatives for the empty forest since by convention an empty product is $1$,

  * •

$X^{\cal{F}}(w_{\cal{F}})$ is defined by $X^{\cal{F}}_{ii}(w_{\cal{F}})=1$
$\forall\,i$, and for $i\neq j$, $X^{\cal{F}}_{ij}(w_{\cal{F}})$ is the
infimum of the $w_{\ell}$ parameters for $\ell$ in the unique path
$P^{\cal{F}}_{i\to j}$ from $i$ to $j$ in ${\cal{F}}$, when such a path
exists. If no such path exists, which means that $i$ and $j$ belong to
different connected components with respect to the forest ${\cal{F}}$, then by
definition $X^{\cal{F}}_{ij}(w_{\cal{F}})=0$,

  * •

the symmetric $n$ by $n$ matrix $X^{\cal{F}}(w_{\cal{F}})$ defined in this way
is positive, hence belongs to ${\rm PS}_{n}$, for any value of $w_{\cal{F}}$.

Since $X^{\varnothing}=\operatorname{Id}$, the empty forest term in (2.2) is
$f(\operatorname{Id})$, hence (2.2) indeed interpolates $f$ between ${\bf 1}$
and $\operatorname{Id}$, staying on ${\rm PS}_{n}$ as announced.

###### Proof.

We would like to add a new proof to the seven proofs of [1], hence a new item
among the 1001 proofs which should exist according to that reference. We thank
D. Brydges for suggesting this proof.

We recall first Kruskal’s greedy algorithm [37]. Consider a fixed connected
graph $G$, possibly with self-loops and multiple edges. For any _Hepp sector_
$\sigma$, hence any complete ordering of the edges of $G$, this algorithm
defines a unique particular “leading tree” $T(\sigma)$, which minimizes
$\sum\limits_{\ell\in T}\sigma(\ell)$ over all trees of $G$, where
$\sigma(\ell)$ is the order of $\ell$ in $\sigma$. We call $T(\sigma)$, the
_leading tree_ for $\sigma$. The algorithm simply picks the first edge
$\ell_{1}$ in $\sigma$ (i.e., whose order $\sigma(\ell)$ is minimum) which is
not a self-loop. Then it picks the next edge $\ell_{2}$ in $\sigma$ that does
not add a cycle to the (usually disconnected) graph with vertex set $V$ and
edge set $\ell_{1}$ such that the order $\sigma(\ell_{2})$ is minimal among
edges with this property, and so on. Another way to explain the algorithm is
through a deletion-contraction recursion: following the ordering of the sector
$\sigma$, every edge is either deleted if it is a self-loop or contracted if
it is not. The set of contracted edges is exactly the leading tree
$T(\sigma)$.

Returning to the proof of the forest formula, we first bluntly develop the
function $f$ at first order with integral remainder over _all_ its parameters,
hence over all edges of the complete graph $K_{n}$. The result is simply in
notations compatible with Theorem 2.2

$\displaystyle f({{\bf 1}})=\sum_{S\subset K_{n}}\int
dw_{S}\,\partial_{S}f\big{(}Y^{S}(w_{S})\big{)},$

where $Y^{S}_{\ell}(w_{S})$ is simply $w_{\ell}$ if $\ell\in S$ and 0 if
$\ell\not\in S$.

Now for a given $S$ we decompose the $w_{S}$ integrals according to all Hepp
sectors of $S$

$\displaystyle f({{\bf 1}})=\sum_{S\subset
K_{n}}\sum_{\sigma}\int_{\sigma}dw_{S}\,\partial_{S}f\big{(}Y^{S}(w_{S})\big{)},$

and regroup all terms according to the leading forest ${\cal{F}}(S,\sigma)$
which is made of the leading Kruskal tree in each connected component $S_{j}$
of $S$ for the order on $S_{j}$ induced by $\sigma$. In this way we get

$\displaystyle f({{\bf 1}})=\sum_{\cal{F}}\sum_{S\subset
K_{n},\sigma\atop{\cal{F}}(S,\sigma)={\cal{F}}}\int_{\sigma}dw_{S}\,\partial_{S}f\big{(}Y^{S}(w_{S})\big{)}.$
(2.6)

Now to analyze the result for a fixed ${\cal{F}}$ we remark that any edge
$\ell\in K_{n}-{\cal{F}}$ which does not add a cycle to ${\cal{F}}$ cannot
belong to any $S$ in (2.6), hence the value of $\partial_{S}f$ is always at 0
for such edges. In contrary, any edge $\ell\in K_{n}-{\cal{F}}$ which adds a
cycle to ${\cal{F}}$ can either belong or not belong to $S$ in (2.6).
Regrouping the corresponding two terms, we obtain a value at 0 plus an
integral of a derivative from 0 to the maximal value allowed by the condition
${\cal{F}}(S,\sigma)={\cal{F}}$. But this maximal value is nothing but
$X^{\cal{F}}(w_{\cal{F}})$. In this way we have summed all sectors $\sigma$
inducing a fixed Hepp sector $\tilde{\sigma}$ on ${\cal{F}}$. Finally the
remaining sum over $\tilde{\sigma}$ reconstructs exactly the integration range
from 0 to 1 for all edges of ${\cal{F}}$.

The fact that $X^{\cal{F}}(w_{\cal{F}})$ is positive for any ordering $\sigma$
now stems from the fact that, for any Hepp sector $\tilde{\sigma}$ of
${\cal{F}}$ it is a convex barycentric combination of block matrices

$\displaystyle
X^{\cal{F}}(w_{\cal{F}})=(1-w_{\ell_{1}})\operatorname{Id}+(w_{\ell_{1}}-w_{\ell_{2}})X^{\Pi_{1}}+(w_{\ell_{2}}-w_{\ell_{3}})X^{\Pi_{2}}+\dots+w_{\ell_{k}}X^{\Pi_{k}}.$

Remark however that this barycentric combination depends on $\tilde{\sigma}$.
Hence $X^{{\cal{F}}}$ is in ${\rm PS}_{n}$ for any $w_{\cal{F}}$, as
announced, but for a different reason in each different sector (ordering) of
the parameters $w_{\cal{F}}$. ∎

We give now a useful corollary of this theorem which expands Gaussian
integrals over replicas. Consider indeed a Gaussian measure $d\mu_{C}$ of
covariance $C_{pq}$ on a vector variable $\vec{\tau}$ with $N$ components
$\tau_{p}$. To study approximate factorization properties of the integral of a
product of $n$ functions of the variable $\vec{\tau}$ it is useful to first
rewrite this integral using a replica trick. It means writing the integral
over $n$ identical replicas $\vec{\tau}_{i}$ for $i=1,\dots,n$ with components
$\tau_{p,i}$, with the perfectly well-defined measure with covariance
$[C\otimes{\bf 1}]_{p,i;q,j}=C_{pq}{\bf 1}_{ij}=C_{pq}$:

$\displaystyle I:=\int
d\mu_{C}(\vec{\tau})\prod_{i=1}^{n}f_{i}(\vec{\tau})=\int d\mu_{C\otimes{\bf
1}}(\vec{\tau}_{i})\prod_{i=1}^{n}f_{i}(\vec{\tau}_{i}).$

Applying the forest formula to this Gaussian integral we obtain the following
corollary

######  Corollary 2.3.

$\displaystyle I=\sum_{{\cal{F}}}\int dw_{\cal{F}}\int d\mu_{C\otimes
X^{\cal{F}}(w_{\cal{F}})}(\vec{\tau}_{i})\;\partial^{C}_{\cal{F}}\prod_{i=1}^{n}f_{i}(\vec{\tau}_{i}),$

where $\partial^{C}_{\cal{F}}$ means
$\prod\limits_{\ell=(i,j)\in{\cal{F}}}\Bigl{(}\sum\limits_{p,q}\frac{\partial}{\partial\tau_{p,i}}C_{pq}\frac{\partial}{\partial\tau_{q,j}}\Bigr{)}$.

The proof follows directly from rewriting the Gaussian integral as

$\displaystyle\int
d\mu_{C}f(x)=e^{\frac{\partial}{\partial\tau_{i}}C_{ij}\frac{\partial}{\partial\tau_{j}}}f\Big{|}_{\tau=0}.$

This is the corolary used in many quantum field theory applications, such as
cluster expansions or loop vertex expansions in which the forest formula is
used to decouple terms which are coupled through propagators, i.e., through
the covariance of a Gaussian measure.

###  2.5 Jungle formulas

The forest formula allows to test links for connecting nodes in a kind of
minimal and symmetric way. However in many physical situations, and especially
in situations with many scales involved, we would like to introduce a
_hierarchy_ between the links connecting nodes, and we would like to test
first connectedness through the most interesting links, those of the highest
energy/shortest length.

Jungle formulas [2] provide an abstract solution to this question. They are a
simple generalization of forest formulas.

######  Definition 2.4.

Let $m\geq 1$ be an integer. An m-jungle over the set $V_{n}$ of $n$ labeled
vertices is a sequence ${\mathfrak{F}}=({\cal{F}}_{1},\ldots,{\cal{F}}_{m})$
of forests on the complete graph $K_{n}$ built on $V_{n}$ such that
${\cal{F}}_{1}\subset\cdots\subset{\cal{F}}_{m}$.

Keeping the same notations than in the previous subsection we now consider a
function $f(X^{1},\dots,X^{m})$ defined over $m$ copies of the previous space
${\rm PS}_{n}$, hence over $[{\rm PS}_{n}]^{m}$, which is smooth in the
interior with continuous extension of all derivatives to the boundary. We want
to Taylor expand with priority to the first matrix $X^{1}$, then to $X^{2}$
and so on. That is we shall test first how to link the vertices of $V_{n}$
into larger connected components first through a forest spanned by the links
due to Taylor expansion of the off-diagonal part of $X_{1}$ then complete this
forest (if it has still several connected components) by Taylor expanding in
the off diagonal part of the second of the matrix $X_{2}$, but so that the new
links together with the first ones still form a forest on $V_{n}$ and so on.
The result is the $m$-level jungle formula of [2].

######  Theorem 2.5 (the jungle formula).

$\displaystyle f({{\bf 1}})=\sum_{{\mathfrak{F}}}\int
dw_{\mathfrak{F}}\,\partial_{\mathfrak{F}}f\big{[}X^{\mathfrak{F}}(w_{\cal{F}})\big{]},$
(2.7)

where

  * •

the sum over ${\mathfrak{F}}$ is over $m$-jungles over the set $V_{n}$ of $n$
labeled vertices $V_{n}=\\{1,\dots,n\\}$, including the empty jungle
${\mathfrak{F}}=({\cal{F}}_{1},\dots,{\cal{F}}_{m})$ with
${\cal{F}}_{i}=\varnothing$ $\forall\,i$,

  * •

$\int
dw_{\mathfrak{F}}\equiv\prod\limits_{\ell\in{\cal{F}}_{m}}\int_{0}^{1}dw_{\ell}$
means integration from $0$ to $1$ over one parameter for each edge in the
final forest ${\cal{F}}_{m}$ of the jungle,

  * •

$\partial_{\mathfrak{F}}=\prod\limits_{k=1}^{m}\prod\limits_{\ell\in{\cal{F}}_{k}\backslash{\cal{F}}_{k-1}}\partial_{\ell}$
means a product of first-order partial derivatives with respect to the
variables $X^{k}_{\ell}$ corresponding to the edges of
${\cal{F}}_{k}\backslash{\cal{F}}_{k-1}$ $($by convention
${\cal{F}}_{0}=\varnothing)$,

  * •

$X^{\mathfrak{F}}(w_{\mathfrak{F}})$ is the sequence of $m$ matrices
$X^{{\mathfrak{F}},k}(w_{\mathfrak{F}})$, $k=1,\dots,m$, defined by
$X^{{\mathfrak{F}},k}_{ii}(w_{\mathfrak{F}})=1$ $\forall\,i$, and for $i\neq
j$, $X^{{\mathfrak{F}},k}_{ij}(w_{\cal{F}})$ is

    * –

$1$ if $i$ and $j$ are connected by ${\cal{F}}_{k-1}$,

    * –

$0$ if $i$ and $j$ are not connected by ${\cal{F}}_{k}$,

    * –

the infimum of the $w_{\ell}$ parameters for $\ell$ in the intersection of
${\cal{F}}_{k}\backslash{\cal{F}}_{k-1}$ and the unique path of
${\cal{F}}_{k}$ from $i$ to $j$, when $i$ and $j$ are connected by
${\cal{F}}_{k}$ but not by ${\cal{F}}_{k-1}$,

  * •

the symmetric $n$ by $n$ matrices $X^{{\mathfrak{F}},k}(w_{\mathfrak{F}})$
defined in this way are all positive, hence
$X^{{\mathfrak{F}}}(w_{\mathfrak{F}})$ $\in[{\rm PS}_{n}]^{m}$, for any value
of $w_{\mathfrak{F}}$.

The proof is an easy generalization of the proof of the forest formula. Remark
that the two-level jungle formula is a main ingredient in multiscale loop
vertex expansions [32].

##  3 Constructive tools

###  3.1 The loop vertex expansion

The loop vertex expansion (LVE) combines an intermediate field functional
integral representation for QFT quantities with the forest formula and a
replica trick similar to the one of the previous section. It allows the
computation of connected functional QFT integrals such as the free energy or
connected Schwinger functions as convergent sums indexed by spanning trees of
arbitrary size $n$ rather than divergent sums indexed by Feynman graphs.

Initially introduced to analyze _matrix_ models with quartic interactions
[46], the LVE has been extended to analyze random _tensor_ models [18, 19, 29,
43]. In this subsection and the next one we follow [33].

Canonical barycentric weights $w(G,T)$ can then be associated to any pair made
of a connected graph $G$ and a spanning tree $T\subset G$ by considering the
percentage of Hepp sectors in which the tree is leading for Kruskal’s greedy
algorithm:

$\displaystyle w(G,T)=\frac{N(G,T)}{|E|!},$

where $N(G,T)$ is the number of sectors $\sigma$ such that $T(\sigma)=T$.

Obviously these weights are barycentric, which simply means that

$\displaystyle\sum_{T\subset G}w(G,T)=1,$

where the sum runs over all spanning trees of $G$. Moreover we have, in the
notations of Theorem 2.2 the integral representation [53].

######  Lemma 3.1.

$\displaystyle w(G,T)=\int dw_{T}\prod_{\ell\in
G-T}X^{T}_{i(\ell)j(\ell)}(w_{T}).$

###### Proof.

We introduce first parameters $w_{\ell}$ for all the edges in $G-T$, writing

$\displaystyle
X^{T}_{ij}(\\{w\\})=\int_{0}^{1}dw_{\ell}\bigg{[}\prod_{\ell^{\prime}\in
P^{T}_{i\to j}}\chi(w_{\ell}<w_{\ell^{\prime}})\bigg{]},$

where $\chi(\cdots)$ is the characteristic function of the event $\cdots$.
Then we decompose the $w$ integrals according to all possible orderings
$\sigma$:

$\displaystyle\int_{0}^{1}\prod_{\ell\in G}dw_{\ell}\prod_{\ell\not\in
T}\bigg{[}\prod_{\ell^{\prime}\in
P^{T}_{\ell}}\chi(w_{\ell}<w_{\ell^{\prime}})\bigg{]}=\sum_{\sigma}\chi(T(\sigma)=T)\int_{0<w_{\sigma(|E|)}<\cdots<w_{\sigma(1)}<1}\prod_{\ell\in
G}dw_{\ell}.$

Indeed in the domain of integration defined by
$0<w_{\sigma(|E|)}<\cdots<w_{\sigma(1)}<1$ the function
$\prod\limits_{\ell\not\in T}\bigl{[}\prod\limits_{\ell^{\prime}\in
P^{T}_{\ell}}\chi(w_{\ell}<w_{\ell^{\prime}})\bigr{]}$ is 1 or zero depending
whether $T(\sigma)=T$ or not, as this function being 1 is exactly the
condition for Kruskal’s algorithm to pick exactly $T$. Strict inequalities are
easier to use here: of course equal values of $w$ factors have zero measure
anyway. Hence

$\displaystyle\int dw_{T}\prod_{\ell\in
G-T}X^{T}_{i(\ell)j(\ell)}(w_{T})=\frac{N(G,T)}{|E|!}=w(G,T).$ ∎

The LVE expressed any Schwinger function $S$ as a convergent sum over trees of
the _intermediate field representation_ :

$\displaystyle S=\sum_{T}A_{T},\qquad A_{T}=\sum_{G\supset T}w(G,T)A_{G},$

with

$\displaystyle\sum_{T}|A_{T}|<+\infty.$

The usual (divergent) perturbative expansion of $S$ is obtained by the ill
defined commutation of the sums over $T$ and $G$,

$\displaystyle S=\sum_{T}\bigg{(}\sum_{G\supset
T}w(G,T)A_{G}\bigg{)}\text{``$=$''}\sum_{G}\sum_{T\subset
G}w(G,T)A_{G}=\sum_{G}A_{G},$ $\displaystyle\sum_{G}|A_{G}|=\infty.$

We shall limit ourselves here to introduce the LVE in the particularly simple
case of the quartic $N$-vector models, for which the $1/N$ expansion is
governed by rooted plane trees.

More precisely, consider a pair of conjugate vector fields $\\{\phi_{p}\\}$,
$\\{\bar{\phi}_{p}\\}$, $p=1,\dots,N$, with $(\bar{\phi}\cdot\phi)^{2}$
interaction. The corresponding functional integral

$\displaystyle
Z(z,N)=\int\frac{d\bar{\phi}d\phi}{(2i\pi)^{N}}e^{-(\bar{\phi}\cdot\phi)+\frac{z}{2N}(\bar{\phi}\cdot\phi)^{2}}$

is convergent for $\Re z<0$. Note the slightly unusual sign convention for the
interaction term. We rewrite it, using a scalar intermediate field $\sigma$,
as

$\displaystyle Z(z,N)=\int
d\sigma\frac{e^{-\sigma^{2}/2}}{\sqrt{2\pi}}\int\frac{d\bar{\phi}d\phi}{(2i\pi)^{N}}e^{-(\bar{\phi}\cdot\phi)+\sqrt{z/N}(\bar{\phi}\cdot\phi)\sigma}=\int\frac{d\sigma}{\sqrt{2\pi}}e^{-\sigma^{2}/2-N\log(1-\sqrt{z/N}\sigma)}.$

Defining $\tau=\sigma/\sqrt{N}$ one gets

$\displaystyle
Z(z,N)=\int\frac{\sqrt{N}d\tau}{\sqrt{2\pi}}e^{-N[\tau^{2}/2+\log(1-\sqrt{z}\tau)]}.$
(3.1)

The two point function

$\displaystyle
G_{2}(z,N)=\frac{1}{Z(z,N)}\int\frac{d\bar{\phi}d\phi}{(2i\pi)^{N}}\frac{1}{N}\bigg{(}\sum_{p}\bar{\phi}_{p}\phi_{p}\bigg{)}e^{-(\bar{\phi}\cdot\phi)+\frac{z}{2N}(\bar{\phi}\cdot\phi)^{2}},$

can be deduced from the free energy by a Schwinger–Dyson equation

$\displaystyle
0=\frac{1}{Z(z,N)}\int\frac{d\bar{\phi}d\phi}{(2i\pi)^{N}}\frac{1}{N}\sum_{p}\frac{\partial}{\partial\phi_{p}}\big{[}\phi_{p}e^{-(\bar{\phi}\cdot\phi)+\frac{z}{2N}(\bar{\phi}\cdot\phi)^{2}}\big{]},$

which yields

$\displaystyle
G_{2}(z,N)=1+2z\frac{d}{dz}\left(\frac{1}{N}\log\int\frac{\sqrt{N}d\tau}{\sqrt{2\pi}}e^{-N[\tau^{2}/2+\log(1-\sqrt{z}\tau)]}\right).$
(3.2)

A simple saddle point evaluates the integral (3.1) as
$\frac{Ke^{-Nf(\tau_{c})}}{\sqrt{f"(\tau_{c})}}$, where the saddle point of
$f(\tau)=\tau^{2}/2+\log(1-\sqrt{z}\tau)]$ is at $\tau_{c}$ with
$f^{\prime}(\tau_{c})=0$ hence
$\tau_{c}=\frac{1}{{2\sqrt{z}}}[1-\sqrt{1-4z}]$. Also

$\displaystyle\lim_{N\to\infty}\frac{\log Z(z,N)}{N}=-f(\tau_{c}),$

and the two point function in the $N\to\infty$ limit is

$\displaystyle\lim_{N\to\infty}G_{2}(z,N)=1+2z\left(-\partial_{z}f(\tau_{c})-\partial_{\tau}f(\tau_{c})\frac{d\tau_{c}}{dz}\right)=1-2z\frac{-\frac{1}{2\sqrt{z}}\tau_{c}}{1-\sqrt{z}\tau_{c}}$
$\displaystyle\hphantom{\lim_{N\to\infty}G_{2}(z,N)}{}=\frac{1}{2z}\big{[}1-\sqrt{1-4z}\big{]},$
(3.3)

which we recognize as the generating function of the Catalan numbers.

Let us now study Borel summability in $z$ of these quantities uniformly as
$N\to\infty$, using the loop vertex expansion. We start from the intermediate
field representation of the two-point function (3.2) and apply the LVE to get

$\displaystyle G_{2}(z,N)=\sum_{{\cal{T}}}\frac{1}{n!}z^{n}\int
dw_{\cal{T}}\int d\mu_{\cal{T}}\prod_{c\in
C({\cal{T}})}\frac{1}{1-\sqrt{z}\tau_{i(c)}},$ (3.4)

where in (3.4)

  * •

the sum over ${\cal{T}}$ is over rooted plane trees, with one ciliated root
vertex labeled $i=0$ plus $n\geq 0$ ordinary vertices labeled $1,\dots,n$,

  * •

$\int dw_{\cal{T}}$ as in Subsection 2.4 means
$\big{[}\prod\limits_{\ell\in{\cal{T}}}\int_{0}^{1}dw_{\ell}\big{]}$,

  * •

$d\mu_{\cal{T}}$ is the normalized Gaussian measure on the $(n+1)$-dimensional
vector field $\vec{\tau}=(\tau_{i})$, $i=0,1,\dots,n$, running over the
vertices of ${\cal{T}}$, which has covariance
$\frac{X_{ij}^{\cal{T}}(w_{\cal{T}})}{N}$ between vertices $i$ and $j$. Recall
that $X^{\cal{T}}(w_{\cal{T}})$ is defined in Subsection 2.4,

  * •

the product over $c$ runs over the set $C({\cal{T}})$ of the $2n+1$ corners of
the tree, the cilium creating an additional corner on the plane tree, and
$i(c)$ is the index of the vertex to which the corner $c$ belongs.

It is now obvious why (3.3) is true; since the covariance of the $\tau$ fields
vanishes as $N\to\infty$ the limit of $G_{2}(z,N)$ is obtained by putting
every $\tau_{i(c)}$ factor to 0 in every corner resolvent, in which case we
exactly get a weight $z^{n({\cal{T}})}$ for each rooted plane tree, hence we
recover the Catalan generating function (the $1/n!$ is canceled by relabeling
of the vertices).

We can now use (3.4) to prove analyticity and Borel summability of the free
energy and correlation functions of the model in the variables $z$ and $1/N$
in the cardioid domain of Fig. 3 (see [23] for an early reference to Borel
summability of the $1/N$ expansion of vector models).

Let us set $z=|z|e^{i\pi+i\phi}$ for $|\phi|<\pi$. We have
$\sqrt{z}=i\sqrt{|z|}e^{i\phi/2}$. Each resolvent
$\frac{1}{1-\sqrt{z}\tau_{i(c)}}$ is bounded in norm by $[\cos(\phi/2)]^{-1}$,
hence using the fact that there are $2n+1$ such resolvents, we obtain
analyticity of representation (3.4) for $4|z|<[\cos(\phi/2)]^{2}$, the
cardioid domain of Fig. 3.

But in fact we can extend the analyticity domain into the extended cardioid
domain of Fig. 6, a domain introduced for quartic vector models in [9]. Indeed
using the parametric representation of resolvents

$\displaystyle\frac{1}{1-\sqrt{z}\tau}=\int_{0}^{\infty}d\alpha_{c}e^{-\alpha_{c}(1-\sqrt{z}\tau)}$

we can explicitly integrate over the measure $d\mu_{{\cal{T}}}$ and get the
integral representation

$\displaystyle G_{2}(z,N)=\sum_{{\cal{T}}}\frac{1}{n!}z^{n}\int
dw_{\cal{T}}\left[\prod_{c\in{\cal{T}}}\int_{0}^{\infty}d\alpha_{c}e^{-\alpha_{c}}\right]e^{\frac{z}{2N}\sum\limits_{ij}\big{(}\sum\limits_{c\in
i}\alpha_{c}\big{)}X^{\cal{T}}_{ij}(w_{\cal{T}})\big{(}\sum\limits_{c\in
j}\alpha_{c}\big{)}}.$

The formula above can be further simplified. Putting $\beta_{i}=\sum_{c\in
i}\alpha_{c}$ we have

$\displaystyle
G_{2}(z,N)=\sum_{{\cal{T}}}\frac{1}{n!}z^{n}\left[\prod_{i=0}^{n}\int_{0}^{\infty}d\beta_{i}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}e^{-\beta_{i}}\right]\int
dw_{\cal{T}}e^{\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}},$
(3.5)

where $d_{i}$ is the degree of $i$, hence the number of corners of $i$.

Figure 6: The extended cardioid domain of vector models. The crosshatching
represents the analytic continuation above the negative axis cut; there is a
symmetric analytic continuation below the cut.

Setting $z=|z|e^{i\pi+i\phi}$ and $\beta=|\beta|e^{i\psi}$ we have, for
$-\pi/2\leq\phi+2\psi\leq\pi/2$ and $-\pi/2<\psi<\pi/2$, both
$\cos(\phi+2\psi)\geq 0$ and $\cos\psi>0$, hence

$\displaystyle\left|\left[\prod_{i=0}^{n}\int_{0}^{e^{i\psi}\infty}d\beta_{i}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}e^{-\beta_{i}}\right]e^{\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}}\right|$
$\displaystyle\qquad{}\leq\prod_{i=0}^{n}\int_{0}^{\infty}d|\beta_{i}|\frac{|\beta_{i}|^{d_{i}-1}}{(d_{i}-1)!}e^{-|\beta_{i}|\cos\psi}\leq(\cos\psi)^{-\sum\limits_{i=0}^{n}d_{i}}=(\cos\psi)^{-2n-1}.$

Therefore $G_{2}(z,N)$ is analytic in the extended cardioid $C=C_{+}\cup
C_{-}$, where $C_{+}$ is the union of the quarter-disk $0\leq\phi<\pi/2$,
$4|z|<1$ and of the domain $\pi/2\leq\phi<3\pi/2$,
$4|z|<[\cos(\phi/2-\pi/4)]^{2}$. $C_{-}$ is the complex conjugate domain.

To prove that this convergent analytic function is the Borel sum of its
perturbative series at any fixed $N$ also requires uniform Taylor estimates of
the type $K^{p}p!|z|^{p}$ for the Taylor remainder at order $p$ in at least a
disk tangent to the imaginary axis (Nevanlinna’s criterion). They follow from
Taylor expanding the exponential of the $\beta$ quadratic form with an
integral remainder:

$\displaystyle
e^{\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}}=\sum_{q=0}^{p-1}\frac{z^{q}}{q!(2N)^{q}}\bigg{[}\sum_{ij}\beta_{i}x^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}\bigg{]}^{q}$
$\displaystyle\hphantom{e^{\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}}=}{}+\int_{0}^{1}dt\frac{(1-t)^{p-1}}{(p-1)!}\frac{z^{p}}{(2N)^{p}}\bigg{[}\sum_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}\bigg{]}^{p}e^{t\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}}.$

The sum over $q$, i.e., the $p$ first terms, are exactly the perturbative
expansion up to order $p$ hence support a $K^{p}p!|z|^{p}$ bound. The Taylor
remainder term for any tree ${\cal{T}}$ in the disk $-\pi/2\leq\phi\leq\pi/2$,
where we can take $\psi=0$ can be bounded as

$\displaystyle\left|\left[\prod_{i=0}^{n}\int_{0}^{\infty}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}d\beta_{i}e^{-\beta_{i}}\right]\int_{0}^{1}dt\frac{(1-t)^{p}}{p!}\frac{z^{p}}{(2N)^{p}}\right.$
$\displaystyle\left.\qquad\quad{}\times\int
dw_{\cal{T}}\bigg{[}\sum_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}\bigg{]}^{p}e^{t\frac{z}{2N}\sum\limits_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}}\right|$
$\displaystyle\qquad{}\leq\frac{|z|^{p}}{p!}\left[\prod_{i=0}^{n}\int_{0}^{\infty}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}d\beta_{i}e^{-\beta_{i}}\right]\left[\sum_{i=0}^{n}\beta_{i}\right]^{2p}$
$\displaystyle\qquad{}=\frac{|z|^{p}}{p!}\left(\prod_{i=0}^{n}\frac{1}{(d_{i}-1)!}\right)\int_{0}^{\infty}d\beta
e^{-\beta}\beta^{2p}\int_{\beta_{1}+\dots+\beta_{n}=\beta}\prod_{i=0}^{n}\beta_{i}^{d_{i}-1}d\beta_{i}$
$\displaystyle\qquad{}=\frac{|z|^{p}}{p!}\left(\prod_{i=0}^{n}\frac{1}{(d_{i}-1)!}\right)(2p+2n+1)!\int_{u_{1}+\dots+u_{n}=1}\prod_{i=0}^{n}u_{i}^{d_{i}-1}du_{i}\leq
4^{n}K^{p}p!|z|^{p}.$

These Taylor estimates for a single rooted plane tree can be summed over all
rooted plane trees (using the $|z|^{n}$ factor in (3.5)) in the half-disk
defined by $16|z|<1$ and $-\pi/2\leq\arg z\leq\pi/2$ (shown in red on Fig. 6).
Hence in this half-disk (which is uniform in $N$) we obtain the desired Taylor
estimates, which is more than enough to check that the expansions (3.4) and
(3.5) represent indeed for all $N$ the unique Borel sum of the perturbative
series.

Interesting functions are the real and imaginary parts along the real axis
$0\leq z<1/8$ which are

$\displaystyle G^{\rm
mean}_{2}(z,N)=\frac{G_{2}(z,N)_{+}+G_{2}(z,N)_{-}}{2},\qquad G^{\rm
cut}_{2}(z,N)=\frac{G_{2}(z,N)_{+}-G_{2}(z,N)_{-}}{2i},$

where $G_{+}$ is analytically continued to $\phi=+\pi$ and $G_{-}$ is
analytically continued to $\phi=-\pi$. Taking $\psi=-\pi/4$ in the first case
and $\psi=+\pi/4$ is the second case, one obtains explicitly convergent
integral representations for these quantities, namely

$\displaystyle G^{\rm
mean}_{2}(z,N)=\sum_{{\cal{T}}}\frac{1}{n!}z^{n}\left[\prod_{i\in
V({\cal{T}})}\int_{0}^{\infty}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}d\beta_{i}e^{-\frac{\sqrt{2}}{2}\beta_{i}}\right]$
$\displaystyle\hphantom{G^{\rm mean}_{2}(z,N)=}{}\times\int
dw_{\cal{T}}\cos\left((2n+1)\frac{\pi}{4}+\frac{\sqrt{2}}{2}\sum_{i}\beta_{i}+\frac{z}{2N}\sum_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}\right),$
$\displaystyle G^{\rm cut}_{2}(z,N)=\sum_{{\cal{T}}}\frac{1}{n!}z^{n}\int
dw_{\cal{T}}\left[\prod_{i\in
V({\cal{T}})}\int_{0}^{\infty}\frac{\beta_{i}^{d_{i}-1}}{(d_{i}-1)!}d\beta_{i}e^{-\frac{\sqrt{2}}{2}\beta_{i}}\right]$
$\displaystyle\hphantom{G^{\rm cut}_{2}(z,N)=}{}\times\int
dw_{\cal{T}}\sin\left((2n+1)\frac{\pi}{4}+\frac{\sqrt{2}}{2}\sum_{i}\beta_{i}+\frac{z}{2N}\sum_{ij}\beta_{i}X^{\cal{T}}_{ij}(w_{\cal{T}})\beta_{j}\right).$

where the factors $(2n+1)\frac{\pi}{4}$ come from the rotation of the $\beta$
integrals, using $\sum d_{i}=2n+1$. These convergent integrals extend half-way
to the Catalan singularity $z_{\text{Catalan}}=1/4$. Indeed bounding the
cosine or sinus function by 1 we obtain convergence, but loosing a factor
$(\sqrt{2})^{\sum d_{i}}=2^{n}\sqrt{2}$.

One can still check easily that the limit for $N\to\infty$ of the mean
integral for positive $z$ is the Catalan function. Indeed the cosine function
simplifies in that case. Rotating the $\beta$ integrals back in position we
obtain again the factor 1 for each rooted plane tree.

The extended cardioid is an analyticity domain in $z$ which holds for any
$N\geq 1$. In other words it is common to all $N$-vector models, including the
particular $N=1$ scalar case, However as $N\to\infty$ we could hope for larger
and larger domains of analyticity which approach the $z=1/4$ singularity when
$N\to\infty$; but we do not know, even in this simple vector model case, how
to prove this.

In the case of quartic large $N$ matrix [46] and large $N$ tensor models [18,
19, 29, 43], the LVE also provides analyticity in cardioid-like domains.

The constructive treatment of renormalizable models requires a multiscale
analysis, hence a multiscale version of the loop vertex expansion (MLVE).
Following [32], we sketch now how this expansion works in the case of a super-
renormalizable toy model which is a slight modification of the vector model
above.

###  3.2 Multiscale loop vertex expansion

Consider the same pair of conjugate vector fields $\\{\phi_{p}\\}$,
$\\{\bar{\phi}_{p}\\}$, $p=1,\dots,N$, with the same
$\frac{\lambda^{2}}{2}(\bar{\phi}\cdot\phi)^{2}$ bare interaction as in the
previous section, but with a different Gaussian measure
$d\mu(\bar{\phi},\phi)$ which breaks the ${\rm U}(N)$ invariance of the
theory. It has diagonal covariance (or propagator) which decreases as the
inverse power of the field index:

$\displaystyle
d\eta(\bar{\phi},\phi)=\left(\prod_{p=1}^{N}p\frac{d\bar{\phi}_{p}d\phi_{p}}{2\pi\imath}\right)e^{-\sum_{p=1}^{N}p\bar{\phi}_{p}\phi_{p}},\qquad\int
d\eta(\bar{\phi},\phi)\bar{\phi}_{p}\phi_{q}=\frac{\delta_{pq}}{p}.$

This propagator renders the perturbative amplitudes of the model finite in the
$N\to\infty$ limit, except for a mild divergence of self-loops which yields a
logarithmically divergent sum
$L_{N}=\sum\limits_{p=1}^{N}\frac{1}{p}\simeq\log N$. These divergences are
easily renormalized by using a vector-Wick-ordered $\phi^{4}$ interaction,
namely $\frac{1}{2}[\lambda(\bar{\phi}\cdot\phi-L_{N})]^{2}$. Remark that this
interaction (contrary to the $\phi^{4}_{2}$ case) remains positive for
$\lambda$ real at all values of $(\bar{\phi},\phi)$. The renormalized
partition function of the model is

$\displaystyle Z(\lambda,N)=\int
d\eta(\bar{\phi},\phi)\,e^{-\frac{\lambda^{2}}{2}(\bar{\phi}\cdot\phi-
L_{N})^{2}}.$

The intermediate field representation decomposes the quartic interaction using
an intermediate scalar field $\sigma$:

$\displaystyle e^{-\frac{\lambda^{2}}{2}(\bar{\phi}\cdot\phi-L_{N})^{2}}=\int
d\nu(\sigma)\,e^{\imath\lambda\sigma(\bar{\phi}\cdot\phi-L_{N})},$

where $d\nu(\sigma)=\frac{1}{\sqrt{2\pi}}e^{-\frac{\sigma^{2}}{2}}$ is the
standard Gaussian measure with covariance 1. Integrating over the initial
fields $(\bar{\phi}_{p},\phi_{p})$ leads to

$\displaystyle Z(\lambda,N)=\int
d\nu(\sigma)\,\prod_{p=1}^{N}\frac{1}{1-\imath\frac{\lambda\sigma}{p}}e^{-\imath\frac{\lambda\sigma}{p}}=\int
d\nu(\sigma)\,e^{-\sum\limits_{p=1}^{N}\log_{2}\bigl{(}1-\imath\frac{\lambda\sigma}{p}\bigr{)}},$

where $\log_{2}(1-x)\equiv x+\log(1-x)=O(x^{2})$.

Applying the ordinary LVE of the previous section to this functional integral
would express $\log Z(\lambda,N)$ as a sum over trees, but there is no simple
way to remove the logarithmic divergence of all leaves of the tree without
generating many intermediate fields in numerators which, when integrated
through the Gaussian measure, would create an apparent divergence of the
series. The MLVE is designed to solve this problem.

We fix an integer $M>1$ and define the $j$-th _slice_ , as made of the indices
$p\in I_{j}\equiv[M^{j-1},M^{j}-1]$. The ultraviolet cutoff $N$ is chosen as
$N=M^{j_{\max}}-1$, with $j_{\max}$ an integer. We can also fix an infrared
cutoff $j_{\min}$. Hence there are $j_{\max}-j_{\min}$ slices in the theory,
and the ultraviolet limit corresponds to the limit $j_{\max}\to\infty$. The
intermediate field representation writes:

$\displaystyle Z(\lambda,N)=\int
d\nu(\sigma)\prod_{j=j_{\min}}^{j_{\max}}e^{-V_{j}},\qquad V_{j}=\sum_{p\in
I_{j}}\log_{2}\left(1-\imath\frac{\lambda\sigma}{p}\right).$

The factorization of the interaction over the set of slices
${\mathcal{S}}=[j_{\min},\dots,j_{\max}]$ can be encoded into an integral over
Grassmann numbers. Indeed,

$\displaystyle a=\int d\bar{\chi}d\chi\,e^{-\bar{\chi}a\chi}=\int
d\mu(\bar{\chi},\chi)\,e^{-\bar{\chi}(a-1)\chi},$

where $d\mu(\bar{\chi},\chi)=d\bar{\chi}d\chi\,e^{-\bar{\chi}\chi}$ is the
standard normalized Grassmann Gaussian measure with covariance 1. Hence,
denoting $W_{j}(\sigma)=e^{-V_{j}}-1$,

$\displaystyle Z(\lambda,N)=\int
d\nu(\sigma)\left(\prod_{j=j_{\min}}^{j_{\max}}d\mu(\bar{\chi}_{j},\chi_{j})\right)e^{-\sum\limits_{j=j_{\min}}^{j_{\max}}\bar{\chi}_{j}W_{j}(\sigma)\chi_{j}}.$

We now rewrite the partition function as

$\displaystyle Z(\lambda,N)=\int
d\nu_{\mathcal{S}}\,e^{-W},\qquad\\!d\nu_{{\mathcal{S}}}=d\nu_{{\bf
1}_{\mathcal{S}}}(\\{\sigma_{j}\\})\,d\mu_{\mathbb{I}_{\mathcal{S}}}(\\{\bar{\chi}_{j},\chi_{j}\\}),\qquad\\!W=\sum_{j=j_{\min}}^{j_{\max}}\bar{\chi}_{j}W_{j}(\sigma_{j})\chi_{j}.$

This is the starting point for the MLVE. The first step is to expand to
infinity the exponential of the interaction

$\displaystyle Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\int
d\nu_{{\mathcal{S}}}\,(-W)^{n}=\sum_{n=0}^{\infty}\frac{1}{n!}\int
d\nu_{{\mathcal{S}},V}\,\prod_{a=1}^{n}(-W_{a}),$

where the $a$-th vertex is

$\displaystyle W_{a}=\sum_{j=j_{\min}}^{j_{\max}}W_{a,j},\qquad
W_{a,j}=\bar{\chi}^{a}_{j}W_{j}(\sigma^{a}_{j})\chi^{a}_{j},$

and has now its own (replicated) bosonic and fermionic fields
$\sigma_{j}^{a}$, $\bar{\chi}^{a}_{j}$, $\chi^{a}_{j}$ and the replica measure
is completely degenerate:

$\displaystyle d\nu_{{\mathcal{S}},V}=d\nu_{{\bf 1}_{\mathcal{S}}\otimes{\bf
1}_{V}}(\\{\sigma^{a}_{j}\\})\,d\mu_{\mathbb{I}_{\mathcal{S}}\otimes{\bf
1}_{V}}(\\{\bar{\chi}^{a}_{j},\chi^{a}_{j}\\}).$

The obstacle to factorize this integral over vertices lies now in the bosonic
and fermionic degenerate blocks ${\bf 1}_{V}$. In order to deal with these
couplings we apply the $m=2$ jungle formula (2.7) with priority to the bosonic
links. It means that in the measure $d\nu$ one introduces first the matrix
$x_{V}$ with coupling parameters $x_{ab}=x_{ba}$, $x_{aa}=1$ between the
vertex bosonic replicas

$\displaystyle Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\int d\nu_{{\bf
1}_{\mathcal{S}}\otimes
x_{V}}(\\{\sigma^{a}_{j}\\})\,d\mu_{\mathbb{I}_{\mathcal{S}}\otimes{\bf
1}_{V}}(\\{\bar{\chi}^{a}_{j},\chi^{a}_{j}\\})\prod_{a=1}^{n}\left(-\sum_{j=j_{\min}}^{j_{\max}}W_{a,j}\right)\Big{|}_{x_{ab}=1}$

and apply the forest formula. We denote ${\cal{F}}_{B}$ a bosonic forest with
$n$ vertices labelled $\\{1,\dots,n\\}$, $\ell_{B}$ a generic edge of the
forest and $a(\ell_{B}),b(\ell_{B})$ the end vertices of $\ell_{B}$. The
result of the first forest formula is

$\displaystyle
Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{{\cal{F}}_{B}}\int
dw_{{\cal{F}}_{B}}\int d\nu_{{\bf 1}_{\mathcal{S}}\otimes
X(w_{\ell_{B}})}(\\{\sigma^{a}_{j}\\})\,d\mu_{\mathbb{I}_{\mathcal{S}}\otimes{\bf
1}_{V}}(\\{\bar{\chi}^{a}_{j},\chi^{a}_{j}\\})$
$\displaystyle\hphantom{Z(\lambda,N)=}{}\times\partial_{{\cal{F}}_{B}}\prod_{a=1}^{n}\left(-\sum_{j=j_{\min}}^{j_{\max}}W_{a,j}\right),$

where

$\displaystyle\int
dw_{{\cal{F}}_{B}}=\prod_{\ell_{B}\in{\cal{F}}_{B}}\int_{0}^{1}dw_{\ell_{B}},\qquad\partial_{{\cal{F}}_{B}}=\prod_{\ell_{B}\in{\cal{F}}_{B}}\left(\sum_{j,k=j_{\min}}^{j_{\max}}\frac{\partial}{\partial\sigma^{a(\ell_{B})}_{j}}\frac{\partial}{\partial\sigma^{b(\ell_{B})}_{k}}\right),$

and $X_{ab}(w_{\ell_{B}})$ is the infimum over the parameters $w_{\ell_{B}}$
in the unique path in the forest ${\cal{F}}_{B}$ connecting $a$ and $b$, and
the infimum is set to $1$ if $a=b$ and to zero if $a$ and $b$ are not
connected by the forest.

The forest ${\cal{F}}_{B}$ partitions the set of vertices into blocks
${\cal{B}}$ corresponding to its trees. Remark that the blocks can be
singletons (corresponding to the trees with no edges in ${\cal{F}}_{B}$). We
denote $a\in{\cal{B}}$ if the vertex $a$ belongs to a bosonic block
${\cal{B}}$. A vertex belongs to a unique bosonic block. Contracting every
bosonic block to an “effective vertex” we obtain a graph which we denote
$\\{n\\}/{\cal{F}}_{B}$. We introduce decoupling parameters
$y_{{\cal{B}}{\cal{B}}^{\prime}}=y_{{\cal{B}}^{\prime}{\cal{B}}}$ for the
fermionic fields $\chi^{{\cal{B}}}_{j}$ corresponding to the blocks of
${\cal{F}}_{B}$ (i.e., for the effective vertices of $\\{n\\}/{\cal{F}}_{B}$).
Applying (a second time) the forest formula, this time for the $y$’s, is
exactly the level 2 jungle formula as it leads to a set of fermionic edges
${\cal{L}}_{F}$ forming a forest in $\\{n\\}/{\cal{F}}_{B}$ (hence connecting
bosonic blocks). We denote $L_{F}$ a generic fermionic edge connecting blocks
and ${\cal{B}}(L_{F}),{\cal{B}}^{\prime}(L_{F})$ the end blocks of the
fermionic edge $L_{F}$. We obtain

$\displaystyle
Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{{\cal{F}}_{B}}\sum_{{\cal{L}}_{F}}\int
dw_{{\cal{F}}_{B}}\int dw_{{\cal{L}}_{F}}\int d\nu_{{\bf
1}_{\mathcal{S}}\otimes X(w_{\ell_{B}})}(\\{\sigma^{a}_{j}\\})$
$\displaystyle\hphantom{Z(\lambda,N)=}{}\times
d\mu_{\mathbb{I}_{\mathcal{S}}\otimes
Y(w_{L_{F}})}(\\{\bar{\chi}^{\cal{B}}_{j},\chi^{\cal{B}}_{j}\\})\partial_{{\cal{F}}_{B}}\partial_{{\cal{L}}_{F}}\prod_{{\cal{B}}}\prod_{a\in{\cal{B}}}\left(-\sum_{j=j_{\min}}^{j_{\max}}\bar{\chi}^{{\cal{B}}}_{j}W_{j}(\sigma^{a}_{j})\chi^{{\cal{B}}}_{j}\right),$
(3.6)

where

$\displaystyle\int
dw_{{\cal{L}}_{F}}=\prod_{L_{F}\in{\cal{L}}_{F}}\int_{0}^{1}dw_{L_{F}},$
$\displaystyle\partial_{{\cal{L}}_{F}}=\prod_{L_{F}\in{\cal{L}}_{F}}\left(\sum_{j=j_{\min}}^{j_{\max}}\left(\frac{\partial}{\partial\bar{\chi}_{j}^{{\cal{B}}(L_{F})}}\frac{\partial}{\partial\chi_{j}^{{\cal{B}}^{\prime}(L_{F})}}+\frac{\partial}{\partial\bar{\chi}_{j}^{{\cal{B}}^{\prime}(L_{F})}}\frac{\partial}{\partial\chi_{j}^{{\cal{B}}(L_{F})}}\right)\right),$

and $Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})$ is the infimum over
$w_{\ell_{F}}$ in the unique path in ${\cal{L}}_{F}$ connecting ${\cal{B}}$
and ${\cal{B}}^{\prime}$, this infimum being set to $1$ if
${\cal{B}}={\cal{B}}^{\prime}$ and to zero if ${\cal{B}}$ and
${\cal{B}}^{\prime}$ are not connected by ${\cal{L}}_{F}$. Note that the
fermionic edges are oriented. Expanding the sums over $j$ in the last line of
equation (3.6) we obtain a sum over slice assignments $J=\\{j_{a}\\}$ to the
vertices $a$, where $j_{a}\in[j_{\min},j_{\max}]$. Taking into account that
$\partial_{\sigma^{a}_{j}}W(\sigma^{a}_{j_{a}})=\delta_{jj_{a}}\partial_{\sigma^{a}_{j_{a}}}W(\sigma^{a}_{j_{a}})$
we obtain:

$\displaystyle
Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{{\cal{F}}_{B}}\sum_{{\cal{L}}_{F}}\sum_{J}\int
dw_{{\cal{F}}_{B}}\int dw_{{\cal{L}}_{F}}$
$\displaystyle\hphantom{Z(\lambda,N)=}{}\times\int d\nu_{{\bf
1}_{\mathcal{S}}\otimes
X(w_{\ell_{B}})}(\\{\sigma^{a}_{j}\\})d\mu_{\mathbb{I}_{\mathcal{S}}\otimes
Y(w_{L_{F}})}\big{(}\big{\\{}\bar{\chi}^{\cal{B}}_{j},\chi^{\cal{B}}_{j}\big{\\}}\big{)}$
$\displaystyle\hphantom{Z(\lambda,N)=}{}\times\partial_{{\cal{F}}_{B}}\partial_{{\cal{L}}_{F}}\prod_{{\cal{B}}}\prod_{a\in{\cal{B}}}\big{(}{-}\bar{\chi}^{{\cal{B}}}_{j_{a}}W_{j_{a}}(\sigma^{a}_{j_{a}})\chi^{{\cal{B}}}_{j_{a}}\big{)}.$

In order to compute the derivatives in $\partial_{{\cal{L}}_{F}}$ with respect
to the block fermionic fields $\chi^{{\cal{B}}}_{j}$ and
$\bar{\chi}^{{\cal{B}}}_{j}$ we note that such a derivative acts only on
$\prod\limits_{a\in{\cal{B}}}\bigl{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{)}$
and, furthermore,

$\displaystyle\frac{\partial}{\partial\bar{\chi}_{j}^{{\cal{B}}}}\prod_{a\in{\cal{B}}}\bigl{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{)}=\left(\sum_{a^{\prime}\in{\cal{B}}}\delta_{jj_{a^{\prime}}}\frac{\partial}{\partial\bar{\chi}_{j_{a^{\prime}}}^{{\cal{B}}}}\right)\prod_{a\in{\cal{B}}}\big{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\big{)},$
$\displaystyle\frac{\partial}{\partial\chi_{j}^{{\cal{B}}}}\prod_{a\in{\cal{B}}}\bigl{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{)}=\left(\sum_{a^{\prime}\in{\cal{B}}}\delta_{jj_{a^{\prime}}}\frac{\partial}{\partial\chi_{j_{a^{\prime}}}^{{\cal{B}}}}\right)\prod_{a\in{\cal{B}}}\big{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\big{)}.$

It follows that the Grassmann Gaussian integral is

$\displaystyle\Bigg{[}e^{\sum\limits_{{\cal{B}},{\cal{B}}^{\prime}}Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})\sum\limits_{a\in{\cal{B}},b\in{\cal{B}}^{\prime}}\delta_{j_{a}j_{b}}\frac{\partial}{\partial\bar{\chi}_{j_{a}}^{{\cal{B}}}}\frac{\partial}{\partial\chi_{j_{b}}^{{\cal{B}}^{\prime}}}}$
$\displaystyle\qquad{}\times\prod_{L_{F}\in{\cal{L}}_{F}}\left(\sum_{a\in{\cal{B}}(L_{F}),b\in{\cal{B}}^{\prime}(L_{F})}\delta_{j_{a}j_{b}}\left(\frac{\partial}{\partial\bar{\chi}_{j_{a}}^{{\cal{B}}(L_{F})}}\frac{\partial}{\partial\chi_{j_{b}}^{{\cal{B}}^{\prime}(L_{F})}}+\frac{\partial}{\partial\bar{\chi}_{j_{b}}^{{\cal{B}}^{\prime}(L_{F})}}\frac{\partial}{\partial\chi_{j_{a}}^{{\cal{B}}(L_{F})}}\right)\right)$
$\displaystyle\qquad{}\times\prod_{{\cal{B}}}\prod_{a\in{\cal{B}}}\bigl{(}\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{)}\Bigg{]}_{\chi^{{\cal{B}}}_{j},\bar{\chi}^{{\cal{B}}}_{j}=0}.$

The sums over $a\in{\cal{B}}(\ell_{F})$ and $b\in{\cal{B}}^{\prime}(\ell_{F})$
yield a sum over all the possible ways to hook the edge
$L_{F}\in{\cal{L}}_{F}$ to vertices in its end blocks. Each term represents a
detailed fermionic edge $\ell_{F}$ in the original graph (having the same
$w_{\ell_{F}}=w_{L_{F}}$ parameter). The sum over ${\cal{L}}_{F}$ becomes
therefore a sum over detailed fermionic forests ${\cal{F}}_{F}$ in the
original graph (in which the bosonic blocks are not contracted)and we obtain

$\displaystyle
Z(\lambda,N)=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{{\cal{J}}}\sum_{J}\int
dw_{\cal{J}}\int
d\nu_{{\cal{J}}}\partial_{\cal{J}}\prod_{{\cal{B}}}\prod_{a\in{\cal{B}}}\bigl{(}W_{j_{a}}(\sigma^{a}_{j_{a}})\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{)},$

where

  * •

the sum over $J$ means
$\sum\limits_{j_{1}=j_{\min}}^{j_{\max}}\cdots\sum\limits_{j_{n}=j_{\min}}^{j_{\max}}$,

  * •

the sum over ${\cal{J}}$ runs over all two level jungles, hence over all
ordered pairs ${\cal{J}}=({\cal{F}}_{B},{\cal{F}}_{F})$ of two (each possibly
empty) disjoint forests on $V$, such that
$\bar{\cal{J}}={\cal{F}}_{B}\cup{\cal{F}}_{F}$ is still a forest on $V$. The
forests ${\cal{F}}_{B}$ and ${\cal{F}}_{F}$ are the bosonic and fermionic
components of ${\cal{J}}$. The edges of ${\cal{J}}$ are partitioned into
bosonic edges $\ell_{B}$ and fermionic edges $\ell_{F}$,

  * •

$\int dw_{\cal{J}}$ means integration from 0 to 1 over parameters $w_{\ell}$,
one for each edge $\ell\in\bar{\cal{J}}$. $\int
dw_{\cal{J}}=\prod\limits_{\ell\in\bar{\cal{J}}}\int_{0}^{1}dw_{\ell}$. A
generic integration point $w_{\cal{J}}$ is therefore made of $|\bar{\cal{J}}|$
parameters $w_{\ell}\in[0,1]$, one for each $\ell\in\bar{\cal{J}}$,

  * •

$\displaystyle\partial_{\cal{J}}=\prod_{\genfrac{}{}{0.0pt}{}{\ell_{B}\in{\cal{F}}_{B}}{\ell_{B}=(c,d)}}\left(\frac{\partial}{\partial\sigma^{c}_{j_{c}}}\frac{\partial}{\partial\sigma^{d}_{j_{d}}}\right)\prod_{\genfrac{}{}{0.0pt}{}{\ell_{F}\in{\cal{F}}_{F}}{\ell_{F}=(a,b)}}\delta_{j_{a}j_{b}}\left(\frac{\partial}{\partial\bar{\chi}^{{\cal{B}}(a)}_{j_{a}}}\frac{\partial}{\partial\chi^{{\cal{B}}(b)}_{j_{b}}}+\frac{\partial}{\partial\bar{\chi}^{{\cal{B}}(b)}_{j_{b}}}\frac{\partial}{\partial\chi^{{\cal{B}}(a)}_{j_{a}}}\right),$

where ${\cal{B}}(a)$ denotes the bosonic blocks to which $a$ belongs,

  * •

the measure $d\nu_{{\cal{J}}}$ has covariance $X(w_{\ell_{B}})\otimes{\bf
1}_{\mathcal{S}}$ on bosonic variables and
$Y(w_{\ell_{F}})\otimes\mathbb{I}_{\mathcal{S}}$ on fermionic variables, hence
$\int d\nu_{{\cal{J}}}f$ is the value at $\sigma=\bar{\chi}=\chi=0$ of

$\displaystyle
e^{\frac{1}{2}\sum\limits_{a,b=1}^{n}X_{ab}(w_{\ell_{B}})\frac{\partial}{\partial\sigma^{a}_{j_{a}}}\frac{\partial}{\partial\sigma^{b}_{j_{b}}}+\sum\limits_{{\cal{B}},{\cal{B}}^{\prime}}Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})\sum\limits_{a\in{\cal{B}},b\in{\cal{B}}^{\prime}}\delta_{j_{a}j_{b}}\frac{\partial}{\partial\bar{\chi}_{j_{a}}^{{\cal{B}}}}\frac{\partial}{\partial\chi_{j_{b}}^{{\cal{B}}^{\prime}}}}f,$

  * •

$X_{ab}(w_{\ell_{B}})$ is the infimum of the $w_{\ell_{B}}$ parameters for all
the bosonic edges $\ell_{B}$ in the unique path $P^{{\cal{F}}_{B}}_{a\to b}$
from $a$ to $b$ in ${\cal{F}}_{B}$. The infimum is set to zero if such a path
does not exists and to $1$ if $a=b$,

  * •

$Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})$ is the infimum of the
$w_{\ell_{F}}$ parameters for all the fermionic edges $\ell_{F}$ in any of the
paths $P^{{\cal{F}}_{B}\cup{\cal{F}}_{F}}_{a\to b}$ from some vertex
$a\in{\cal{B}}$ to some vertex $b\in{\cal{B}}^{\prime}$. The infimum is set to
$0$ if there are no such paths, and to $1$ if such paths exist but do not
contain any fermionic edges.

Remember that the symmetric $n$ by $n$ matrix $X_{ab}(w_{\ell_{B}})$ is
positive for any value of $w_{\cal{J}}$, hence the Gaussian measure
$d\nu_{{\cal{J}}}$ is well-defined. The matrix
$Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})$ is also positive, with all
elements between 0 and 1. Since the slice assignments, the fields, the measure
and the integrand are now factorized over the connected components of
$\bar{\cal{J}}$, the logarithm of $Z$ is exactly the same sum but restricted
to the two-levels spanning trees:

$\displaystyle\log
Z(\lambda,N)=\sum_{n=1}^{\infty}\frac{1}{n!}\sum_{{\cal{J}}\,{\rm
tree}}\sum_{J}\int dw_{\cal{J}}\int
d\nu_{{\cal{J}}}\partial_{\cal{J}}\prod_{{\cal{B}}}\prod_{a\in{\cal{B}}}\bigl{[}W_{j_{a}}(\sigma^{a}_{j_{a}})\chi^{{\cal{B}}}_{j_{a}}\bar{\chi}^{{\cal{B}}}_{j_{a}}\bigr{]},$
(3.7)

where the sum is the same but conditioned on
$\bar{\cal{J}}={\cal{F}}_{B}\cup{\cal{F}}_{F}$ being a _spanning tree_ on
$V=[1,\dots,n]$. In [32], it is proven in detail that

######  Theorem 3.2.

The series (3.7) is absolutely convergent for $\lambda\in[-1,1]$ uniformly in
$j_{\max}$.

######  Theorem 3.3.

The series (3.7) is absolutely convergent for $\lambda\in\mathbb{C}$,
$\lambda=|\lambda|e^{\imath\gamma}$ in the domain $|\lambda|^{2}<(\cos
2\gamma)$ uniformly in $j_{\max}$.

We sketch below the proof of Theorem 3.2, referring the reader to [32] for
details. By Cayley’s theorem the number of two level trees over $n\geq 1$
vertices is exactly $2^{n-1}n^{n-2}$.

The Grassmann Gaussian integral evaluates to

$\displaystyle\bigg{(}\prod_{{\cal{B}}}\prod_{\genfrac{}{}{0.0pt}{}{a,b\in{\cal{B}}}{a\neq
b}}(1-\delta_{j_{a}j_{b}})\bigg{)}\bigg{(}\prod_{\genfrac{}{}{0.0pt}{}{\ell_{F}\in{\cal{F}}_{F}}{\ell_{F}=(a,b)}}\delta_{j_{a}j_{b}}\bigg{)}\Bigl{(}{\bf
Y}^{\hat{b}_{1}\dots\hat{b}_{k}}_{\hat{a}_{1}\dots\hat{a}_{k}}+{\bf
Y}^{\hat{a}_{1}\dots\hat{b}_{k}}_{\hat{b}_{1}\dots\hat{a}_{k}}+\dots+{\bf
Y}_{\hat{b}_{1}\dots\hat{b}_{k}}^{\hat{a}_{1}\dots\hat{a}_{k}}\Bigr{)},$

where the sum runs over the $2^{k}$ ways to exchange an $a_{i}$ and a $b_{i}$.
Each ${\bf Y}^{\hat{a}_{1}\dots\hat{b}_{k}}_{\hat{b}_{1}\dots\hat{a}_{k}}$
factor is a determinant of a matrix made of
$Y_{{\cal{B}}{\cal{B}}^{\prime}}(w_{\ell_{F}})$ interpolating factors (see
[32] for the precise definition). Its absolute value is therefore bounded by 1
thanks to Hadamard’s inequality, because the corresponding matrix is positive
with diagonal entries equal to 1.

The bosonic integral is a bit more cumbersome, as one should first evaluate
the effect of the bosonic derivatives on the exponential vertex kernels
$W_{j}$ through the Faà di Bruno formula, whose combinatoric is easy to
control. It leads to a sum over similar exponential kernels but multiplied by
some polynomials.

To bound the remaining bosonic functional integral one first separates the
exponential kernels from the polynomials by some Cauchy–Schwarz estimate with
respect to the bosonic Gaussian measure. The exponential terms being positive,
the corresponding piece is bounded by 1. The polynomial piece is then
explicitly evaluated. This generates a dangerous product of local factorials
of the number of fields in the bosonic blocks, but allows also a good factor
$M^{-j}$ from the propagator of scale $j$ for each occupied bosonic scale $j$.

But here comes the key point. The Grassmann Gaussian integrals ensure that the
occupied scales in any bosonic block of the first forest formula are all
_distinct_. Therefore the good factor collected from the propagator easily
beats the local factorials. The worst case is indeed when the $p$ occupied
scales in the block are lowest, in which case
$\prod\limits_{j=1}^{p}M^{-j}=M^{-p(p+1)/2}$ which easily beats $p!$.

For just renormalizable theories it is not so easy to beat the dangerous
factors by the decay of the propagators, and the constructive expansion must
proceed even more carefully, essentially expanding the functional integral in
each scale in a much more detailed way.

###  3.3 Iterated Cauchy–Schwarz bounds

A Cauchy Schwarz bound requires to decompose an expression as a scalar product
between two halves, $I=\langle S_{1}.S_{2}\rangle\implies|I|\leq\langle
S_{1}.S_{1}\rangle^{1/2}\langle S_{2}.S_{2}\rangle^{1/2}$. A key problem in
the LVE is to bound the resolvents still present in the tree amplitudes by
using their uniform bound in norm (in a cardioid domain of the coupling
constant). The idea of ICS bounds is to get rid of resolvents inductively, by
applying many times this inequality. Typically at any inductive step a
resolvent is sandwiched in the scalar product between $S_{1}$ and $S_{2}$ and
disappears in the right hand side of the bound (thanks to the bound on its
norm), resulting in tree amplitudes with at least one resolvent less.

ICS bounds for a general quartic tensor model were introduced in [18], but we
describe here a new slightly different version. More precisely in [18] a tree
amplitude of the LVE is decomposed in two halves by identifying a pair of
opposite resolvents. However in this process the two halves may not have the
same number of vertices, and the bound obtained can involve new trees of
different orders. Here we shall describe how to work at fixed order by pairing
a resolvent with its opposite corner in the tree, so that the two halves $A$
and $B$ remain of the same order.

Consider a tensor model with quartic interactions for generalized colors
${\cal{C}}\in{\cal{D}}$. Let $N$ be the range for each color index. The
perturbative amplitudes at order $n$, are simply $\lambda^{n}N^{F-(D-1)n}$,
where $F$ is the number of _faces_ of the graph.

Among the graphs of the IFR are in particular the trees. A planar colored tree
of order $n$, is a tree joining $n+1$ loop vertices, in which each edge bears
a generalized color ${\cal{C}}$, corresponding to the subset ${\cal{C}}$ of
indices of the quartic invariant for ${\cal{C}}$. Such a tree has therefore
exactly $2n$ _corners_ , and an amplitude $N^{F-(d-1)n}$.

However in an LVE or MLVE a typical term is indexed by such planar colored
trees but in which some corners of the tree can bear resolvents operators $R$
instead of ordinary propagators $1$. The goal of the ICBS bounds is to prove
that in a cardioid domain of the coupling constant, the amplitude for any such
tree is bounded (uniformly in $\sigma$) by the supremum over all trees at the
same order $n$ of the same amplitude but of the ordinary perturbative type,
that is without any resolvent.

A resolvent-dressed tree is a pair $(T,A,A^{\dagger})$ made of a tree $T$ and
a subset $S$ of the corners of $T$. Its amplitude is

$\displaystyle A_{T}^{A,A^{\dagger}}=\operatorname{Tr}\prod_{a\in
A}C^{1/2}RC^{1/2}\prod_{a\in
A^{\dagger}}C^{1/2}R^{\dagger}C^{1/2}\prod_{a\not\in A\cup
A^{\dagger}}C\prod_{e}\delta_{{\cal{C}}_{e}},$

where the product over $e$ runs over all edges of the tree, the edge factors
are $\delta$ functions

$\displaystyle\delta_{{\cal{C}}_{e}}=\prod_{c\in{\cal{C}}_{e}}\delta_{i_{c},j_{c}}\delta_{i^{\prime}_{c}j^{\prime}_{c}},$

where the operator $\delta_{i_{c},j_{c}}\delta_{i^{\prime}_{c}j^{\prime}_{c}}$
joins the color indices $i_{c}$ and $j_{c}$ of the four corners touched by
edge $e$, and finally the trace means that indices have to be summed over all
faces of the tree.

We assume that in a cardioid domain of the complex plane for the coupling
constant a uniform $N$ and $\sigma$-independent bound for the resolvent holds:

$\displaystyle\|R\|\leq K.$

The same independent bound automatically follows for the matrix coefficients
of the resolvent

$\displaystyle|R_{n,\bar{n}}|\leq K\qquad\forall\,n,\,\bar{n},$

since any coefficient of a matrix is bounded in absolute value by its norm.

Iterated Cauchy–Schwarz bounds then allow to bound the absolute value of any
resolvent-dressed tree amplitude by the supremum over similar tree amplitudes
but without any resolvent for trees $T^{\prime}$ of the same order than $T$:

######  Theorem 3.4.

Under the hypotheses above, any resolvent-dressed tree $A_{T}^{S}$ of order
$n$ obeys the bound

$\displaystyle\big{|}A_{T}^{A,A^{\dagger}}\big{|}\leq
K^{2n}\sup_{T^{\prime}}A_{T^{\prime}}^{\varnothing}.$

###### Proof.

Consider a dressed tree $(T,A,A^{\dagger})$ of order $n$ with
$p=|A|+|A^{\dagger}|$ resolvents. If $p>0$ we select an arbitrary corner
$\gamma$ containing a resolvent $R$ of $T$ (if the resolvent is of the
conjugate type $R^{\dagger}$ the reasoning is identical). Turning around the
tree, there is a unique corner $\tilde{\gamma}$ opposite to $\gamma$, that is
such that the path from $\gamma$ to $\tilde{\gamma}$ has $n-1$ corners whether
we turn around the tree clockwise or counterclockwise. If $p=0$, we select any
corner and its opposite pair.

Having selected opposite corners we can divide the tree $T$ into two halves
$S_{1}$ and $S_{2}$, made of the clockwise and counterclockwise paths form
$\gamma$ to $\tilde{\gamma}$. Every edge from $S_{1}$ to $S_{2}$ is a scalar
product because of the form of the $\delta_{e}$ factors. Hence in a certain
tensor product Hilbert space of many elements we can write
$A_{T}^{A,A^{\dagger}}$ as

$\displaystyle\big{|}A_{T}^{A,A^{\dagger}}\big{|}=|\langle
A_{S_{1}}RA_{S_{2}}\rangle|\leq K\langle
A_{S_{1}}A_{S_{1}}^{\dagger}\rangle^{1/2}\langle
A_{S_{2}}A_{S_{2}}^{\dagger}\rangle^{1/2}.$

In this way we have bounded the initial tree with $p$ resolvents by the
arithmetic mean of two trees with total number of resolvents at most $2(p-1)$.
Iterating this bound for all trees still containing resolvents, we obtain,
after $r$ iterations

$\displaystyle\big{|}A_{T}^{A,A^{\dagger}}\big{|}\leq\prod
K^{k_{r}}\prod_{j}|A_{S_{j}}|^{2^{-r}}.$

It is not true that in such a process after a finite number of steps all
resolvents disappear (counterexamples are easily built). Nevertheless they
_rarefy_ at each step, and this is enough to complete the proof of Theorem
3.4. Let us define $p_{r}$ the total number of resolvents contained in all the
trees $S_{j}$ and $m_{r}$ the number of trees containing at least one
resolvent. Since any tree is of order $n$ it can contain at most $2n$
resolvents, hence $m_{r}\geq p_{r}/(2n)$. Furthermore $p_{r+1}\leq
2p_{r}-2m_{r}$ Therefore the sequence $q_{r}=2^{-r}p_{r}$ satisfies
$q_{r+1}\leq q_{r}(1-\frac{1}{2n})$ hence tends to 0 as $r\to\infty$.

Now a rough relatively trivial bound to evaluate a tree with $p$ resolvents
consists in paying a full additional factor $\prod_{f}N^{2q_{f}}$ for each
face $f$ meeting $q_{f}$ resolvents along the face. Since a resolvent is along
at most $d$ different faces we certainly find that for any $r$

$\displaystyle\big{|}A_{T}^{A,A^{\dagger}}\big{|}\leq[f(N)]^{p}\sup_{T^{\prime}}A_{T^{\prime}}^{\varnothing}$

for a certain function $f(N)$. Hence by the induction

$\displaystyle\big{|}A_{T}^{A,A^{\dagger}}\big{|}\leq[f(N)]^{q_{r}}\sup_{T^{\prime}}A_{T^{\prime}}^{\varnothing},$

and letting $r\to\infty$ completes the proof of Theorem 3.4, since $q_{r}\to
0$ as $r\to\infty$. ∎

##  4 Results

We now review the various tensor models for which (uniform) Borel summability
has been proved. Rather than summarizing or paraphrasing the original papers,
we focus on explaining the reader which of the previous techniques are
required for which model and why.

###  4.1 Melonic quartic models

This is the simplest tensor model, restricted to a quartic melonic
interactions, hence the set $\mathcal{Q}$ of generalized colors in (2.1) is
restricted to _singletons_. This model is fully treated in [29] via a single
loop vertex expansion and does not require any ICS bounds. The LVE graphs in
which all resolvents $R$ are replaced by one are exactly the tree graphs of
the IFR of the model, and they coincide exactly with the melonic graphs of the
initial representation of the tensor model. Hence the LVE automatically
computes the leading order of quartic melonic tensor models as the $\sigma=0$,
$R(\sigma)=1$ approximation of the LVE. Sub leading orders are of course
obtained by further expanding the resolvent factors of the LVE.

Why is it that the melonic quartic model does not require the technique of ICS
bounds to prove Borel summability? This is essentially because in this case
the propagators of the IFR correspond to insertions of the type
$1\otimes\cdots\otimes\sigma_{c}\otimes\cdots\otimes 1$ which all _commute_.

As a result one can use the parametric representation of the resolvent

$\displaystyle\big{(}1+i\sqrt{\lambda}\sigma\big{)}=\int_{0}^{\infty}e^{-\alpha(1+i\sqrt{\lambda}\sigma)}d\alpha$
(4.1)

and regroup all identical colors. The bound now factorizes over the
_subforests of the same color_ in the tree and this allows for a constructive
bound of the same order than the leading melonic graphs of the model.

###  4.2 General quartic models

For these models the interactions are no longer restricted to the melonic
case. Intermediate fields are matrices corresponding to generalized colors
${\cal{C}}$ which are not necessarily singletons. In particular they no longer
commute when the subsets ${\cal{C}}$ and ${\cal{C}}^{\prime}$ have a non-
trivial intersection ${\cal{C}}\cap{\cal{C}}^{\prime}\neq\varnothing$. This is
why the parametric representation (4.1) is not enough to get rid of resolvents
in the loop vertex expansion, and ICS bounds have to be used [18].

###  4.3 $\boldsymbol{{\rm U}(1)-T^{4}_{3}}$ TGFT

This field theory is built on the ${\rm U}(1)$ group, for tensors of rank 3,
with quartic melonic interactions and a Boulatov-type projector, hence the
propagator, in momentum space is

$\displaystyle
C(n)=\frac{\delta(n_{1}+n_{2}+n_{3})}{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+1}$

for $n=(n_{1},n_{2},n_{3})\in{\mathbb{Z}}^{3}$.

The delta projector reduces by 1 the effective dimension of the model, hence
this model has the same power counting than a rank 2 non-commutative field
theory without any Boulatov projector, hence is fully convergent (no
ultraviolet divergencies). Therefore it can be built by a single loop vertex
expansion [38].

###  4.4 $\boldsymbol{{\rm U}(1)-T^{4}_{4}}$ TGFT

This field theory [39] is built on the ${\rm U}(1)$ group, for tensors of rank
4, with quartic melonic interactions and a Boulatov-type projector, hence the
propagator, in momentum space is

$\displaystyle
C(n)=\frac{\delta(n_{1}+n_{2}+n_{3}+n_{4})}{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}+1}$

for $n=(n_{1},n_{2},n_{3},n_{4})\in{\mathbb{Z}}^{4}$.

The delta projector reduces by 1 the effective dimension of the model, hence
this model has the same power counting than a rank 3 quartic tensor model
without any Boulatov projector, hence requires a rather mild renormalization.

Therefore it requires a multi-loop vertex expansion. However it does not
require any iterated Cauchy–Schwarz bounds, since insertions of different
colors commute! This is surprising, given that a priori they do not commute
with the $C$ propagator, whose denominators mixes colors. However it is a
peculiarity and simplifying feature that the Boulatov projector, when combined
with quartic melonic interactions leads to a global momentum conservation rule
along all the propagators of a given loop vertex.

Indeed consider a $\sigma$ insertion of color $c$ in a loop vertex. A priori
the $n_{c}$ index may change into $m_{c}$. However the propagators $\delta$
functions immediately before and after the insertion enforce that
$n_{c}=-\sum\limits_{c^{\prime}\neq c}n_{c^{\prime}}$. Therefore since the
$n_{c^{\prime}}$ are conserved through the $\sigma_{c}$ insertions, we obtain
that $n_{c}=m_{c}$. In fact the random intermediate field matrices
$\sigma_{c}$ reduce to their diagonal part in momentum space, and the
intermediate fields can therefore be considered as vectors rather than
matrices [41].

Now this conservation rule along any loop vertex ensures the commutation of
all intermediate fields along the loop vertex, hence the argument with the
parametric representation of [29] can be adapted to this situation and ICS
bounds are not necessary.

###  4.5 $\boldsymbol{T^{4}_{3}}$ TFT

This super-renormalizable field theory is built on the ${\rm U}(1)$ group, for
tensors of rank 3, with quartic melonic interactions and an ordinary
Laplacian-based projector which in momentum space is

$\displaystyle C(n)=\frac{1}{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+1}$

for $n=(n_{1},n_{2},n_{3})\in{\mathbb{Z}}^{3}$.

The model has a power counting almost similar to the one of the ordinary
$\phi^{4}_{2}$ theory. It has for each color $c$ two vacuum divergent graphs,
one linearly divergent and one logarithmically divergent. It has also a single
logarithmically divergent two-point graph (the “melonic tadpole”), again with
a single vertex, which requires a mass renormalization (see Fig. 7).

Figure 7: From left to right, the divergent tadpole, the convergent tadpole
and the two vacuum divergent graphs.

Therefore the proof of Borel summability for this model requires both a multi-
loop vertex expansion and ICS bounds [19]. Indeed insertions of intermediate
fields of different colors along a loop vertex no longer need to commute,
since they do not commute with the propagators along a given loop vertex.
Indeed these propagators may have now different momenta running through them,
since there is no longer the conservation rule of the quartic melonic models
with Boulatov projectors.

An interesting aspect of the renormalization of this model is that it is
performed directly in the IFR representation. Hence the initial
$\operatorname{Tr}\log(1+i\sqrt{\lambda}\Sigma)$ interaction is replaced by a
subtracted interaction $\operatorname{Tr}\log_{2}(1+i\sqrt{\lambda}\Sigma)$,
where $\log_{2}(1+x):=-x+\log(1+x)$. In addition to the ICS bounds to get rid
of resolvents in the trees created by the bosonic forest formula, there is
also a non-trivial argument to bound the non perturbative part left in the
exponential by the MLVE. This non trivial “non-perturbative” bound can be
traced to the divergence of the vacuum energy graphs, and can be considered a
kind of tensor analog in the IFR representation of the famous Nelson bound for
$\phi^{4}_{2}$, which was the birth act of constructive field theory [45].

###  4.6 Quartic matrix model

The paper [35] deals with a Gaussian complex random matrix model with a
quartic perturbation. Its partition function is defined by the integral over
complex $N\times N$ matrices $M$:

$\displaystyle Z(\lambda,N)=\int
dMe^{-\operatorname{Tr}(MM^{\dagger}-\frac{\lambda}{2N}MM^{\dagger}MM^{\dagger})}.$

The model can be shown Borel summable with a single loop vertex expansion but
the trees of that expansion form only a part of the planar sector of the
theory which is the leading order at large $N$. To extract this leading sector
is possible through an additional expansion called the topological expansion
in [35]. Starting form the LVE trees, one can perform a first expansion step
on the resolvents, testing the presence of an additional field $\sigma$. This
field is then contracted with the Gaussian measure, and the process is
continued until a first non-planar Wick contraction appear. The part of the
expansion for planar graphs does not diverge since the number of planar graphs
at order $n$ is bounded by $({\rm const})^{n}$.

Continuing this additional topological expansion until at most $p+1$
independent crossing, it is possible to extract the complete terms of the
$1/N$ expansion up to order $p$ and to prove that the remainder is still Borel
summable, with a uniform bound in $N^{-(p+1)}$.

A similar expansion could be used for tensor models such as those of [11],
which mix a melonic sector and a planar sector behavior.

##  5 Conclusion: open questions

Super-renormalizable and just renormalizable tensor field theories form a rich
world [5]. A key physical issue is to move towards realistic quantum gravity
models, and to better connect the “tensor field theory” approach to other
approaches to quantum gravity. In particular understanding which tensor models
admit a realistic emergent four dimensional (Euclidean) space time with a real
time analytic continuation seems to us the key problem. However we shall not
discuss this issue here, addressing the reader to our twin review [49] and
focus here only on a brief discussion of the next technical steps of the
constructive tensor program.

The next obvious step in this program is to build super-renormalizable
theories with power counting of the same level than ordinary $\phi^{4}_{3}$,
namely the $T^{4}_{4}$ model with propagator $(p^{2}+1)^{-1}$. This is well
under way and it seems that it can be treated like $T^{4}_{3}$ by just the
same 2-jungle formula of the MLVE and ICS bounds; but the list of divergent
graphs is much longer (there are 12 primitively divergent graphs) and the
$\operatorname{Tr}\log$ interaction should therefore be pushed further,
resulting in particular in much more complicated non perturbative estimates
compared to $T^{4}_{3}$ [51].

The model ${\rm U}(1)-T^{4}_{5}$ with Boulatov projectors has a comparable
finite amount of divergent graphs. It should be easier than $T^{4}_{4}$, since
intermediate fields can be represented by diagonal matrices in the momentum
basis.

The next, more difficult step is the construction of a just renormalizable
model such as the $T^{4}_{5}$ model or the ${\rm U}(1)-T^{4}_{6}$ model with
Boulatov projector [41, 55, 56]. Both models are asymptotically free [5, 48,
54], hence the construction should be possible but it is not clear whether it
can be done with the same tools or if a more advanced expansion (such as a
multiscale loop vertex expansion with higher-order jungle formulas) will be
required.

Another important direction is to treat models with higher-order interactions,
for instance the just renormalizable models with six-order melonic
interactions introduced in [7] and [14]. Here the starting point might be the
intermediate field representation introduced in [52] and improved in [42].
Beware however that even in zero dimension we do not know yet how to perform a
loop vertex expansion in this representation [42].

We would like to complete this review by indicating two other challenging
research directions for matrix models and non-commutative field theory

  * •

The first direction is to extend the analyticity domain in the coupling
constant of quartic models beyond the cardioid characteristically obtained
through the LVE. In the case of vector models we already know that this is
possible and that the optimal angular domain of analyticity extends to 3$\pi$
(see Fig. 6 and the last section of [33]). In matrix or tensor models nothing
of this type is known.

This is a key issue for many reasons, in particular to understand the
instanton cuts which are responsible for the usual non-perturbative effects of
$\phi^{4}$ theory, and how they should disappear in the limit $N\to\infty$,
unveiling a new singularity at finite distance on the negative coupling
constant which is the one responsible for the single and double scaling limits
of quantum gravity in two dimensions, hence for the emergence of continuous
surfaces [20].

  * •

Construct non-perturbatively the full non-planar sector of the
Grosse–Wulkenhaar model [25]. Although in a certain limit $\theta\to\infty$ of
infinite non-commutativity, this sector disappears, again it would be very
interesting to understand rigorously how to construct the full theory at
finite $\theta$. It would be extremely interesting, as the theory is
asymptotically safe, to check that this property persists at the non-
perturbative level.

### Acknowledgements

We thank A. Abdesselam, T. Delepouve, R. Gurau, V. Lahoche, L. Lionni, J.
Magnen, K. Noui, M. Smerlak, A. Tanasa, F. Vignes-Tourneret and ZhiTuo Wang
for contributing to various aspects of this constructive program.

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