aainstitutetext: School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ, 08540, USAbbinstitutetext: Mathematical Institute, Andrew Wiles
Building, Woodstock Rd, Oxford, UKccinstitutetext: Max-Plank-Institüt fur
Physik, Werner-Heisenberg-Institut, D-80805 München, Germanyddinstitutetext:
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-
Saclay, IUF, Franceeeinstitutetext: LaCIM, Département de Mathématiques,
Université du Québec à Montréal, Montréal, QC, Canada

#  All Loop Scattering for All Multiplicity

N. Arkani-Hamed b H. Frost c G. Salvatori d P-G. Plamondon e H. Thomas
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>pierre-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This is part of a series of papers describing the new curve integral formalism
for scattering amplitudes of the colored scalar tr$\phi^{3}$ theory. We show
that the curve integral manifests a very surprising fact about these
amplitudes: the dependence on the number of particles, $n$, and the loop
order, $L$, is effectively decoupled. We derive the curve integrals at tree-
level for all $n$. We then show that, for higher loop-order, it suffices to
study the curve integrals for $L$-loop tadpole-like amplitudes, which have
just one particle per color trace-factor. By combining these tadpole-like
formulas with the the tree-level result, we find formulas for the all $n$
amplitudes at $L$ loops. We illustrate this result by giving explicit curve
integrals for all the amplitudes in the theory, including the non-planar
amplitudes, through to two loops, for all $n$.

##  1 Introduction and Summary

This is the second in a series of papers laying out a new point of view on
scattering amplitudes—the _curve integral_ formalism—which is based on
combinatorial structures in the space of scattering data. us_qft The curve
integral exposes a number of hidden features of amplitudes. In this paper we
describe a striking example of one such hidden structure. Ordinarily, the
complexity of scattering amplitudes is thought to grow both with increasing
the number of particles $n$, and also with increasing loop order $L$. The case
of amplitudes at both large $n$ and large $L$ is thought to be vastly more
complicated still. In this paper we will see that this expectation is
misleading. First, we find that the tree-level calculation for all $n$ is
simplified by the curve integral formula. Second, we find that the curve
integral naturally decouples the dependence of amplitudes on $n$ and $L$. The
calculation of an $L$-loop amplitude for small $n$ is all that is required to
find a closed formula for all $n$.

The $n$-point tree amplitude in $\mathrm{tr}\phi^{3}$ theory is conventionally
given by a sum of diagrams. There are

$C_{n-2}=\frac{1}{n-1}{2n-4\choose n-2}$ (1)

such diagrams, and this number grows exponentially as $\sim
4^{n}/(\sqrt{\pi}n^{3/2})$ for large $n$. By contrast, the curve integral
formula gives the amplitude as an integral over an $(n-3)$-dimensional space,

${\cal A}_{n}^{\rm tree}=\int\limits_{0}^{\infty}d^{n-3}t\,Z.$ (2)

The integrand, $Z$, is given by computing $n(n-3)/2$ simple _headlight
functions_ , $\alpha_{ij}$, each with no more than $O(n)$ terms:

$\log Z=-\sum_{1\leq
i<j-1}\alpha_{ij}\left((p_{i}+\cdots+p_{j-1})^{2}+m^{2}\right).$ (3)

This simple function is all that is needed to obtain ${\cal A}_{n}^{\rm
tree}$, which is naively a sum over exponentially many Feynman diagrams. We
give explicit formulas for $\alpha_{ij}$ valid at all $n$.

We can give closed formulas for the headlight functions $\alpha_{ij}$ because
they follow from an inductive calculation using two-by-two matrices. These
_curve matrices_ , $M^{\rm tree}_{C}$, have simple polynomial entries. The
curve matrices for the $n$-point amplitude are sufficient to determine the
matrices for the $(n+1)$-point amplitude inductively, by matrix
multiplication. This immediately leads to closed formulas for these matrices
for all $n$.

The simplification of the curve integral at tree level becomes even more
striking at loop level. We first study the very simplest _tadpole-like_ loop
amplitudes, which have just one particle in every color trace-factor. These
_tadpole amplitudes_ are determined by a basic set of $2\times 2$ curve
matrices, $M_{C}^{\rm tadpole}$. The surprise is that, having computed both
the tadpole matrices $M_{C}^{\rm tadpole}$ and the tree matrices $M_{C}^{\rm
tree}$, we find that this is everything needed to compute the curve integral
for amplitudes at _any_ $n$, for fixed $L$. In other words, deriving the curve
integral for an amplitude ${\cal A}_{n,L}$ is only as difficult as finding the
curve integral for an $L$-loop tadpole.

Moreover, the dependence on $n$ and $L$ is completely factorized in the curve
integrals. For any $n$ and $L$, the curve integral amplitude formula takes the
form

${\cal A}_{n,L}=\int d^{E}t\leavevmode\nobreak\ {\cal
K}\left(\frac{\pi^{L}}{\cal
U}\right)^{\frac{D}{2}}\,\times\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}\right),$ (4)

where $E=n+3L-3$. In this formula, the first factor in the integrand,

${\cal K}\left(\frac{\pi^{L}}{\cal U}\right)^{\frac{D}{2}}$ (5)

does not depend on $n$. It can be computed once and for all for a tadpole-like
graph. The dependence on $n$ enters only via the exponential, whose exponent
is a sum of terms which grow only polynomially with $n$.

The simplicity of these factorized formulas is illustrated by Table 1, which
shows the number of curve matrices $M_{C}$ required in order to compute,
first, the $n$-independent prefactor, and, second, the exponent of the
exponential.

For example, this factorization can be seen in the 1-loop planar amplitudes.
In appropriate coordinates, the curve integrals for these amplitudes are

${\cal
A}_{\text{1-loop}}=\int\limits_{-\infty}^{0}dt_{0}\int\limits_{-\infty}^{\infty}dt_{1}\cdots
dt_{n-1}\,\left(\frac{\pi}{t_{0}}\right)^{\frac{D}{2}}\,\exp\left(\frac{1}{t_{0}}{\cal
F}_{0}-{\cal Z}\right).$ (6)

The 1-loop planar tadpole has only two propagators, and defines two associated
curve matrices, $M_{C}^{\rm tadpole}$. Just one of these matrices is required
to compute the $n$-independent pre-factor, which in this case is

$\left(\frac{\pi}{t_{0}}\right)^{\frac{D}{2}}.$ (7)

Whereas, both of the two tadpole curve matrices are needed to compute the
exponent. Given these two tadpole curve matrices, and the tree-level matrices
$M_{C}^{\rm tree}$, both ${\cal F}_{0}$ and ${\cal Z}$ can be computed
directly.

Order | # of Matrices for the prefactor | Total # of Matrices
---|---|---
1-loop Planar | 1 | 2
1-loop Double-trace | 3 | 5
2-loop Planar | 9 | 17
2-loop Genus-one | 13 | 14
2-loop Double-trace | 14 | 23
2-loop Triple-trace | 24 | 33
Table 1: The minimal number of curve matrices needed to derive the curve
integral for all-multiplicity amplitudes (right column), and the number of
matrices needed to derive just the pre-factor (middle column).

Needless to say, this Tree-Loop factorization is not at all manifest in the
conventional diagrammatic approach to perturbation theory. Moreover, this
factorization of the tree- and loop-like parts of the calculation plays a
practical role when computing the amplitudes. In this paper, we illustrate
this by giving concrete, all-$n$ formulas for all amplitudes in the
tr$\phi^{3}$ theory through to two loops: tree-level, 1-loop planar, 1-loop
double-trace, two-loop planar, two-loop triple trace, and the genus-one
single-trace two-loop corrections. We focus on the practical methods for
determining the curve integral formulas, while explaining the Tree-Loop
factorization along the way. The mathematical and conceptual foundations for
the results will be described in greater detail elsewhere.

##  2 Review

In us_qft we gave a formula for amplitudes in a coloured cubic scalar theory
at any order in perturbation theory. Each partial amplitude can be computed,
in $D$ spacetime dimensions, using a _curve integral_ of the form

$\mathcal{A}=\int
d^{D}\ell\int\frac{d^{E}t}{\mathrm{MCG}}\exp\left(-\sum_{C}\alpha_{C}(t)X_{C}\right).$
(8)

with $E=n-3+2L+2g$, if ${\cal A}$ is a partial amplitude with $L$ loops and
genus $g$. This formula resembles a standard Feynman integral with Schwinger
parameters. However, here the Feynman parameters are replaced by piecewise
linear functions, $\alpha_{C}$, of the $E$ integration variables $t_{i}$.
Instead of integrating one Feynman diagram at a time, this formula computes
the full amplitude.

Using the curve integral formula, we can integrate over the loop variables,
$\ell_{a}$, once and for all (rather than separately for each diagram). The
result is (in some $D=2d-\epsilon$ dimension)

$\mathcal{A}=\int\frac{d^{E}t}{\mathrm{MCG}}\left(\frac{\pi^{L}}{\mathcal{U}}\right)^{\frac{D}{2}}\,\exp\left({\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}}\right),$
(9)

where $\mathcal{U},\mathcal{F}_{0},\mathcal{Z}$ are homogeneous polynomials in
the $\alpha_{C}$.

Equations (8) and (9) are formulas for the full amplitude, automatically
summing over all Feynman diagrams. However, counter-intuitively, the starting
point for these formulas is to choose a distinguished Feynman diagram,
$\Gamma$. The Feynman diagrams for the theory are cubic fatgraphs. The main
point in us_qft is that a single given fatgraph, $\Gamma$, contains all the
information we need to compute the sum over _all_ diagrams. The variables
$t_{i}$ are each assigned to each of the $E$ edges of the diagram $\Gamma$.
The propagator factors, $X_{C}$, correspond to paths, $C$, along the diagram
$\Gamma$. And the functions $\alpha_{C}(t)$ are computed directly from the
data of the path $C$, which can be described as series of left and right turns
on $\Gamma$.

To be concrete, consider some curve, $C$, with path

$C=e_{0}Le_{1}Re_{2}\cdots.$ (10)

In other words, $C$ enters via external edge $e_{0}$, takes a left turn, goes
down edge $e_{1}$, takes a right turn, and so on. To find $\alpha_{C}$, we
first replace each turn and edge with a matrix:

$e\mapsto\begin{bmatrix}1&0\\\ 0&y_{e}\end{bmatrix},\leavevmode\nobreak\
L\mapsto\begin{bmatrix}1&0\\\ 1&1\end{bmatrix},\leavevmode\nobreak\
R\mapsto\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}.$ (11)

In this way, each curve defines a matrix, given by the product of these
matrices:

$M_{C}=\begin{bmatrix}1&0\\\ 1&1\end{bmatrix}\begin{bmatrix}1&0\\\
0&y_{1}\end{bmatrix}\begin{bmatrix}1&1\\\
0&1\end{bmatrix}\begin{bmatrix}1&0\\\ 0&y_{2}\end{bmatrix}\cdots.$ (12)

Note that external edges have $y_{e}=1$. Each entry of $M_{C}$ is a polynomial
in the variables $y_{e}$. Given $M_{C}$, the _headlight function_ is

$\alpha_{C}=-\mathrm{Trop}M_{C}^{12}-\mathrm{Trop}M_{C}^{21}+\mathrm{Trop}M_{C}^{11}+\mathrm{Trop}M_{C}^{22},$
(13)

where $\mathrm{Trop}$ denotes the _tropicalization_ of a polynomial.

The distinguished fatgraph, $\Gamma$, also defines a consistent momentum
assignment to every curve, $C$. We again read this directly from the fatgraph.
Momenta, $p_{e}^{\mu}$, are assigned to the edges of $\Gamma$ in the usual
way. If $\Gamma$ has $L$ loops, the momenta are parameterized by a choice of
$L$ loop momentum variables $\ell_{a}^{\mu}$.111This is equivalent to choosing
a labeled basis of the first homology of the graph. Then a curve, $C$, that
enters via edge $e_{1}$, say, is assigned momentum

$P_{C}^{\mu}=p_{e_{1}}^{\mu}+\sum_{\text{right turn}}p_{\text{left}}^{\mu},$
(14)

where, at each right turn, we add the momentum incident on the vertex from the
left.

Given this momentum assignment, each $P_{C}^{\mu}$ can be written as a sum of
a loop-dependent and non-loop-dependent part,

$P_{C}^{\mu}=K_{C}^{\mu}+\sum_{a=1}^{L}\ell_{C}^{a}\ell_{a}^{\mu},$ (15)

for some integers $\ell_{C}^{a}$. In terms of this momentum assignment, the
polynomials $\mathcal{U},\mathcal{F}_{0},\mathcal{Z}$ are given by

$\displaystyle\mathcal{U}=\det\Lambda,\qquad{\cal F}_{0}={\cal
J}^{T}\tilde{\Lambda}{\cal J},\qquad{\cal
Z}=\sum_{C}(K_{C}^{2}+m^{2})\alpha_{C},$ (16)

where the $L\times L$ matrix $\Lambda$ and vector $B$ are given by

$\Lambda^{ab}=\sum_{C}\ell_{C}^{a}\ell_{C}^{b}\alpha_{C},\qquad{\cal
J}^{a,\mu}=\sum_{C}\ell_{C}^{a}K_{C}^{\mu}\alpha_{C}.$ (17)

We write $\tilde{\Lambda}=(\det\Lambda)\,\Lambda^{-1}$ for the adjugate of
$\Lambda$. For practical purposes, it is convenient to leave $\mathcal{U}$ and
$\mathcal{F}_{0}$ expressed in terms of $\Lambda^{ab}$ and ${\cal J}^{a}$.
However, these polynomials also have a beautiful combinatorial definition.
Expanding the determinant, ${\cal U}$ becomes a sum of monomials, each
corresponding to a cut of the diagram $\Gamma$ that reduces it to a tree
graph. ${\cal U}$ is then the sum over all such _maximal cuts_. Moreover, the
monomial terms of $\mathcal{F}_{0}$ correspond to _factorizing_ cuts of
$\Gamma$ into two trees.

The amplitude formulas, (8) and (9), still look a little formal because the
integral requires us to mod out by the Mapping Class Group, $\mathrm{MCG}$.
$\mathrm{MCG}$ is a discrete finitely generated group of symmetries of the
fatgraph, $\Gamma$. Rather than finding a fundamental domain for this group
action (which is tricky in general), we can evaluate the integrals by
introducing an integration kernel. The final formula for the amplitude then
reads,

$\mathcal{A}=\int
d^{E}t\,\mathcal{K}\,\left(\frac{\pi^{L}}{\mathcal{U}}\right)^{D/2}\,\exp\left({\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}}\right),$
(18)

for some kernel $\mathcal{K}(t)$, that we call the Mirzakhani kernel.222${\cal
K}$ is a tropical version of the kernel introduced by Mirzakhani to compute
Weil-Petterson volumes, mirzakhani2007 and which has been applied to physical
calculations in, e.g. Saad:2019lba . General methods for finding a kernel are
given in us_qft . In this article, we will see that the computation of ${\cal
K}$ can be simplified further. Amazingly, both ${\cal U}$ and ${\cal K}$ (and
also the matrix $\Lambda$) are independent of the number of external
particles, $n$. The number of external particles, and their momenta, only
enter the formula through the exponent in the formula, which is linear in the
momentum invariants.

##  3 Tree-level Amplitudes

The curve integral formula for amplitudes is already interesting at tree
level. Naively, a tree-level $n$-point amplitude is a sum over $\sim 4^{n}$
different Feynman diagrams. By contrast, the tree-level curve integral is an
integral over $n-3$ dimensions of an integrand formed by $\sim n^{2}$
headlight functions, $\alpha_{C}$, each of which can be computed by
multiplying $\sim n$ matrices. The resulting formula for the tree level
amplitude is

$\displaystyle{\cal A}_{\mathrm{tree}}(123...n)=\int d^{n-3}t\,Z,$ (19)

where

$\displaystyle-\log Z=\sum_{C}\alpha_{C}X_{C}.$ (20)

This sum is over all $n(n-3)/2$ possible propagators, $C_{ij}$. If
$k_{i}^{\mu}$ are the external momenta, the propagator variables, $X_{C}$, are
given by

$X_{ij}=P_{ij}^{2}+m^{2},$ (21)

where $P_{ij}=k_{i}+\ldots+k_{j-1}$ is the momentum of the $C_{ij}$
propagator. We stress again that computing $Z$ is much easier than enumerating
Feynman diagrams.

In fact, this section finds a simple all-$n$ expression for $Z$. The headlight
functions, $\alpha_{C}$, depend on our choice of distinguished fatgraph,
$\Gamma$. If we choose $\Gamma$ to be the comb graph, as in Figure 1, then we
obtain particularly simple formulas for the $\alpha_{C}$. This leads to the
following compact all-$n$ formula for the amplitude,

$\displaystyle{\cal A}_{\mathrm{tree}}(123...n)=\int
d^{n-3}t\,\exp\left(\sum_{i=1}^{n-3}t_{i}X_{i+1,n}+\sum_{1\leq i\leq
j}^{j=n-3}f_{ij}c_{ij}\right).$ (22)

Here we write $f_{ij}$ for the tropical functions

$\displaystyle
f_{ij}(\mathbf{t})=\max\\{0,t_{i},t_{i}+t_{i+1},...,t_{i}+t_{i+1}+...+t_{j}\\},$
(23)

and $c_{ij}$ are the Mandelstam-like kinematic variables,

$\displaystyle c_{ij}=X_{i,j+3}+X_{i+1,j+2}-X_{i,j+2}-X_{i+1,j+3}.$ (24)

Not only is (22) an all-$n$ formula for the tree amplitude, it also contains
the ingredients we need to compute amplitudes for all-$n$ at higher orders in
perturbation theory. Because of this, we will derive (22) in detail in this
section. The formulas in this section will be used repeatedly in subsequent
calculations in this paper.

###  3.1 Tree-level Headlight Functions

Figure 1: The $n$-point tree-level comb graph.

Take $\Gamma$ to be the comb graph with end points labelled $1$ and $n$, and
with the ordering $1234...n$. For each $i,j$, there is a unique curve,
$C_{ij}$, that connects external line $i$ to external line $j$. The paths of
these curves are

$\displaystyle C_{ij}$ $\displaystyle=Re_{i-1}Le_{i}Le_{i+1}...Le_{j-2}R,$
(25) $\displaystyle C_{1j}$ $\displaystyle=Le_{1}Le_{2}...Le_{j-2}R,$ (26)
$\displaystyle C_{in}$ $\displaystyle=Re_{i-1}Le_{i}Le_{i+1}...Le_{n-3}L.$
(27)

where $1<i<j-1$ and $j<n$. Using the replacements (11), we find that these
curves have curve matrices

$\displaystyle M_{ij}=\begin{bmatrix}F_{i-1,j-3}&F_{i-1,j-2}\\\
y_{i-1}F_{i,j-3}&y_{i-1}F_{i,j-2}\end{bmatrix},$ (28) $\displaystyle
M_{1j}=\begin{bmatrix}1&1\\\ F_{1,j-3}&F_{1,j-2}\end{bmatrix},\qquad
M_{in}=\begin{bmatrix}F_{i-1,n-3}&y_{i-1,n-3}\\\
y_{i-1}F_{i,n-3}&y_{i-1,n-3}\end{bmatrix},$ (29)

where $y_{i,j}=y_{i}y_{i+1}\ldots y_{j}$ and the $F_{ij}$ are the
polynomials333The polynomials $F_{ij}(y)$ are the _F-polynomials_ associated
to this A-type cluster algebra.

$\displaystyle F_{ij}=1+y_{i}+y_{i}y_{i+1}+...+y_{i}y_{i+1}...y_{j},$ (30)

These matrices will be used throughout our calculations in this paper.

Write $f_{ij}=\text{Trop}F_{ij}$ for the tropicalization of the F-polynomials,
(30). It follows from (28) that the headlight functions are

$\displaystyle\alpha_{ij}(\mathbf{t})$
$\displaystyle=f_{i-1,j-3}+f_{i,j-2}-f_{i-1,j-2}-f_{i,j-3}$ (31)
$\displaystyle\alpha_{1j}(\mathbf{t})$ $\displaystyle=f_{1,j-2}-f_{1,j-3},$
(32) $\displaystyle\alpha_{in}(\mathbf{t})$
$\displaystyle=-t_{i-1}+f_{i-1,n-3}-f_{i,n-3},$ (33)

with the tropical functions $f_{ij}$ given by

$\displaystyle
f_{ij}(\mathbf{t})=\max\\{0,t_{i},t_{i}+t_{i+1},...,t_{i}+t_{i+1}+...+t_{j}\\}.$
(34)

###  3.2 The All-multiplicity Amplitude

To obtain the amplitude, we first consider the assignment of momenta to
curves. Using the momentum assignment rule, (14), the momentum of the curve
$C_{ij}$ is (with $i<j-1$)

$\displaystyle P_{ij}$ $\displaystyle=p_{i}+p_{i+1}+...+p_{j-1},$ (35)

where $p_{a}^{\mu}$ are the external momenta. Because $\Gamma$ is planar, we
can alternatively assign these momenta using _dual momentum variables_ ,
$z_{a}^{\mu}$, so that

$P_{ij}^{\mu}=z_{j}^{\mu}-z_{i}^{\mu},$ (36)

where, say, $z_{i}=p_{1}+\cdots+p_{i-1}$. In either case, the propagator
factors are

$\displaystyle
X_{ij}=m^{2}+\left(\sum_{a=i}^{j-1}p_{a}\right)^{2},\qquad\text{with}\leavevmode\nobreak\
X_{ii+1}=0,\leavevmode\nobreak\ X_{1n}=0.$ (37)

With these momentum assignments understood, the amplitude is

$\displaystyle{\cal A}_{n}=\int d^{n-3}t\,Z,$ (38)

where

$\displaystyle-\log Z=\sum_{1\leq i<j-1}^{n}\alpha_{ij}X_{ij}.$ (39)

The $\alpha_{ij}(t)$ are given by (31–33).

An alternative form of $\log Z$ is obtained by expanding the $\alpha_{ij}(t)$.
This gives

$\displaystyle-\log Z=-\sum_{i=1}^{n-3}t_{i}X_{i+1,n}+\sum_{1\leq
i<j-1}^{j=n}(f_{i,j-2}+f_{i-1,j-3}-f_{i-1,j-2}-f_{i,j-3})X_{ij},$ (40)

where, in the second sum, we set $f_{kl}\equiv 0$ if $k=0$, or $l=n-2$. The
second sum can be rewritten to find

$\displaystyle\log Z=\sum_{i=1}^{n-3}t_{i}X_{i+1,n}+\sum_{1\leq i\leq
j}^{j=n-3}f_{ij}c_{ij},$ (41)

where the Mandelstam-like variables $c_{ij}$ are

$c_{ij}=X_{i,j+3}+X_{i+1,j+2}-X_{ij+2}-X_{i+1,j+3}.$ (42)

Evaluating these using (37), we find

$\displaystyle c_{ij}=\left\\{\begin{matrix}s_{i,j+3}-m^{2}&j=i\\\
s_{1n-3}-m^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(i,j)=(1,n-3)\\\ s_{i,j+3}&\text{otherwise},\end{matrix}\right.$ (43)

where the $s_{ij}=(k_{i}+k_{j})^{2}$ are the Mandelstam variables.

###  3.3 Grafting Trees

Throughout must of this paper, we will join comb tree graphs to other, higher-
loop fatgraphs. By convention, we will take a $n+1$-point comb graph,
$\Gamma$, and join its $(n+1)$st leg to an external leg of some other graph,
$\Gamma^{\prime}$. The result is a larger graph $\Gamma\sqcup\Gamma^{\prime}$.
Consider a curve, $C$, that enters the graph through external line $i$ of
$\Gamma$, and then proceeds into $\Gamma^{\prime}$. It has a path

$C=Re_{i-1}Le_{i}L\cdots Le_{n-1}D,$ (44)

for some path $D$. Now that $e_{n-1}$ is an internal edge of the larger graph
$\Gamma\sqcup\Gamma^{\prime}$, it is assigned its own variable, $y_{n-1}$. The
curve matrix $M_{C}$ then factors into a product

$M_{C}=M_{i,n+1}\begin{bmatrix}1&0\\\ 0&y_{n-1}\end{bmatrix}M_{D}.$ (45)

Given this, it is convenient to define the matrix

$W_{i}=M_{i,n+1}\begin{bmatrix}1&0\\\ 0&y_{n-1}\end{bmatrix},$ (46)

so that $M_{C}=W_{i}M_{D}$. For $i>1$ we have

$W_{i}=\begin{bmatrix}F_{i-1,n-2}&y_{i-1,n-1}\\\
y_{i-1}F_{i,n-2}&y_{i-1,n-1}\end{bmatrix},$ (47)

and for $i=1$ we have

$W_{1}=\begin{bmatrix}1&0\\\ F_{1,n-2}&y_{1,n-1}\end{bmatrix}.$ (48)

When multiple tree graphs $\Gamma^{1},\ldots,\Gamma^{h}$ are joined to
different trace-factors of a graph $\Gamma$, it is helpful to record the
trace-factor by adding an additional index and writing $W_{i}^{A}$ (for
$A=1,\ldots,h$) for the tree-level matrix starting from $i$ in $\Gamma^{A}$.

##  4 The Planar 1-loop Amplitudes

Curve integrals allow us to compute the 1-loop planar amplitudes for all
multiplicity in several different ways. In this section, we illustrate the
curve integral method by deriving an all-$n$ formula using the _wheel_
diagram. Later, in Section 7, we revisit this amplitude with new tools, that
allow us to disentangle the loop-like and tree-like parts of this calculation.

###  4.1 1-Loop Headlight Functions

Figure 2: An $n$-point 1-loop planar graph.

Take $\Gamma$ to be the 1-loop _wheel diagram_ , in Figure 2, with $n$
external legs labeled $1,2,...,n$, cyclically. This fatgraph has finitely many
curves. There are $n(n+1)/2$ curves connecting external legs, that we call
$C_{ij}$ with $1\leq i\leq n$ and $i<j$ (cyclically). There are also $n$
spiraling curves $C_{i0}$ that start at an external leg, $i$, and end in a
spiral around $\Gamma$’s internal boundary.

Examining $\Gamma$, in Figure 2, the curves $C_{ij}$ ($i\leq j$) and $C_{i0}$
have paths

$\displaystyle C_{ij}=Re_{i}Le_{i+1}L\cdots Le_{j-1}R,$ (49) $\displaystyle
C_{i0}=R(e_{i}Le_{i+1}\cdots Le_{i-1}R)^{\infty}.$ (50)

Substituting the matrices in equation (11), these paths determine the curve
matrices of these curves:

$\displaystyle M_{ij}=\begin{bmatrix}F_{i,j-1}&F_{ij}\\\
y_{i}F_{i+1,j-1}&y_{i}F_{i+1,j}\end{bmatrix},\qquad
M_{i0}=\frac{1}{1-y_{1n}}\begin{bmatrix}F_{i,i-2}&0\\\
y_{i}F_{i+1,i-1}&0\end{bmatrix},$ (51)

where

$\displaystyle{F}_{ij}=1+y_{i}+y_{i}y_{i+1}+\ldots+y_{i}y_{i+1}...y_{j}.$ (52)

These curve matrices define headlight functions. Write

$\displaystyle\mathrm{Trop}{F}_{ij}={f}_{ij}=\max(0,t_{i},t_{i}+t_{i+1},...,t_{i}+t_{i+1}+...+t_{j}),$
(53)

for the tropicalization of the $F$-polynomials, where the indices are
understood cyclically (mod $n$). Then (in the region $\sum_{i=1}^{n}t_{i}<0$)

$\displaystyle\alpha_{{ij}}$
$\displaystyle=f_{i,j-1}+f_{i+1,j}-f_{ij}-f_{i+1,j-1}$ (54)
$\displaystyle\alpha_{i0}$ $\displaystyle=-t_{i}-f_{i+1,i-1}+f_{i,i-2}.$ (55)

Note that the $\alpha_{i0}$ satisfy the identity

$\displaystyle\sum_{i=1}^{n}\alpha_{i0}=-\sum_{i=1}^{n}t_{i}.$ (56)

This identity will be useful for simplifying the formula for the amplitude.

###  4.2 The Loop Integrand

Since $\Gamma$ is a planar diagram, it is convenient to use _dual momentum
variables_ to record momentum assignments. With dual variables $x_{i}^{\mu}$
($i=0,1,\ldots,n$), the propagator factors are

$X_{ij}=(x_{j}-x_{i})^{2}+m^{2}.$ (57)

If $p_{i}^{\mu}$ ($i=1,\ldots,n$) are the external momenta, take

$\displaystyle x_{i}^{\mu}=p_{1}^{\mu}+\ldots+p_{i-1}^{\mu}.$ (58)

It is convenient to use $x_{0}^{\mu}$ as the loop momentum variable.

With these momentum assignments, the integrand (pre-loop integration) is given
by the curve integral

$\displaystyle{\cal I}_{\mathrm{1-loop}}=\int\limits_{\sum t_{i}\leq
0}d^{n}t\,Z,$ (59)

where

$-\log
Z=\sum_{i<j}^{\text{cyclic}}X_{ij}\alpha_{ij}+\sum_{i=1}^{n}X_{i0}\alpha_{i0}.$
(60)

A more explicit formula for $Z$ follows from substituting the expressions for
the headlight functions. This gives

$\displaystyle\log
Z=\sum_{i=1}^{n}t_{i}X_{i,0}+\sum_{i=1}^{n}f_{i,i-2}(X_{i-1,0}-X_{i,0})+\sum_{i\leq
j}^{\text{cyclic}}f_{ij}c_{ij},$ (61)

where $c_{ij}=X_{ij}+X_{i-1,j+1}-X_{i,j+1}-X_{i-1,j}$.

###  4.3 The All-multiplicity Amplitude

The integrand, ${\cal I}$, depends on the loop momentum variable
$x_{0}^{\mu}$. But the integral over $x_{0}^{\mu}$ is Gaussian. Performing
this integration in $D=2d-\epsilon$ dimensions gives the curve integral
formula for the amplitude,

$\displaystyle{\cal A}_{\text{1-loop}}=\int\limits_{\sum_{i}t_{i}\leq
0}d^{n}\mathbf{t}\left(\frac{\pi}{\mathcal{U}}\right)^{\frac{D}{2}}\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right),$
(62)

where $\mathcal{U},\mathcal{F}_{0},\mathcal{Z}$ are the _surface Symanzik
polynomials_ for $\Gamma$. In terms of the headlight functions,

$\mathcal{U}=\sum_{i=1}^{n}\alpha_{i0},\qquad\mathcal{F}_{0}=\sum_{i,j=1}^{n}\alpha_{i0}\alpha_{j0}z_{i}\cdot
z_{j},$ (63)
$\mathcal{Z}=\sum_{i<j-1}^{\text{cyclic}}\alpha_{ij}X_{ij}+\sum_{i=1}^{n}\alpha_{i0}(z_{i}^{2}+m^{2}).$
(64)

We find even more explicit formulas for ${\cal U},{\cal F}_{0}$ by
substituting the formulas for the headlight functions. This gives:

$\displaystyle\mathcal{U}=-\sum_{i=1}^{n}t_{i},\qquad\mathcal{F}_{0}=\sum_{i,j=1}^{n}t_{i}t_{j}z_{i}\cdot
z_{j}+f_{i,i-2}f_{j,j-2}\,p_{i}\cdot p_{j}.$ (65)

##  5 The Telescopic Property

The computations of the headlight functions, $\alpha_{C}$, are not very
sensitive to the number of external particles, $n$. This is because the
transfer matrices, $M_{C}$, are computed multiplicatively. This is related to
a fundamental _telescopic property_ of the headlight functions, which we
explain in this section. The telescopic property leads to simplifications of
the curve integral formulas.

Consider a path, $C$, that has not yet reached an external line of some
fatgraph, $\Gamma$. The path has an associated curve matrix,

$M_{C}=\begin{bmatrix}a&b\\\ c&d\end{bmatrix},$ (66)

for some $a,b,c,d$, which are polynomials for the $y$-variables. The
associated headlight function for this matrix is

$\alpha_{C}=\text{Trop}(a)+\text{Trop}(d)-\text{Trop}(b)-\text{Trop}(c).$ (67)

Suppose that $C$ ends just before traversing an edge, $e_{*}$, with variable
$y_{*}$. Then $C$ can be extended in one of two ways, as in Figure 3. It can
traverse $e_{*}$ and turn left, to get a longer path: $Ce_{*}L$. Or it can
turn right instead: $Ce_{*}R$. The associated transfer matrices are

$M_{Cy_{*}L}=\begin{bmatrix}a+by_{*}&by_{*}\\\
c+dy_{*}&dy_{*}\end{bmatrix},\qquad M_{Cy_{*}R}=\begin{bmatrix}a&a+by_{*}\\\
c&c+dy_{*}\end{bmatrix}.$ (68)

If we stop here to compute the new headlight functions of these paths, we find

$\displaystyle\alpha_{Ce_{*}L}$
$\displaystyle=\text{Trop}(b)+\text{Trop}(c+dy_{*})-\text{Trop}(d)-\text{Trop}(a+by_{*})$
(69) $\displaystyle\alpha_{Ce_{*}R}$
$\displaystyle=\text{Trop}(a+by_{*})+\text{Trop}(c)-\text{Trop}(a)-\text{Trop}(c+dy_{*}).$
(70)

But note that the sum of these two functions is equal to $\alpha_{C}$,
corresponding to the matrix $M_{C}$:

$\alpha_{Ce_{*}L}+\alpha_{Ce_{*}R}=\alpha_{C}.$ (71)

In other words, the sum $\alpha_{Ce_{*}L}+\alpha_{Ce_{*}R}$ can be computed
without finding $M_{Cy_{*}L}$ and $M_{Cy_{*}R}$ separately. It is only
necessary to compute the matrix $M_{C}$.

Figure 3: An incomplete curve can be extended in one of two ways: to the left,
or to the right.

This relation can be telescopically nested to compute larger sums of headlight
functions. Suppose $\Gamma$ can be decomposed into a join of two graphs,
$\Gamma=\Gamma^{1}\sqcup\Gamma^{2}$, where $\Gamma^{1}$ is a tree graph. And
suppose that $C$ is a path in $\Gamma$ with an endpoint at the join of the two
graphs. We can sum over all ways to extend $C$ to an external line of the tree
graph $\Gamma^{1}$. If $\Gamma^{1}$ has $k$ external lines (excluding the
join), we obtain $k$ curves $C_{1},...,C_{k}$. Then applying (71) gives

$\sum_{i=1}^{k}\alpha_{C_{k}}=\alpha_{C}.$ (72)

Similarly, if $C$ is a path in $\Gamma$ with _both_ of its endpoints at the
join of the two graphs, we can sum over all ways to extend $C$ to a complete
path. Suppose $C_{ij}$ is the curve obtained by extending $C$ to have
endpoints $i$ and $j$ in $\Gamma_{0}$ (with, say, $i\leq j$). Then

$\sum_{\begin{subarray}{c}i=1\\\ i\leq
j\end{subarray}}^{k}\alpha_{C_{ij}}=\alpha_{C}.$ (73)

In other words, the telescopic property dramatically simplifies any sum over
headlight functions $\alpha_{C}$ for a class of curves, $C$, that differ only
by their endpoint on a tree subgraph.

##  6 Decoupling Trees and Loops

One drastic consequence of the telescopic property is that the curve integral
integrands _factorize_ into the product of a loop-like factor, which is
independent of $n$, and a second $n$-dependent factor which contains the
kinematics.

To make this factorization manifest, we define a class of _Tree-Loop graphs_.
We say that an $L$-loop fatgraph is a _tadpole graph_ if it has exactly one
particle per trace-factor. This generalises the usual notion of a tadpole
(i.e. a graph with just one external leg). A Tree-Loop graph is then formed by
joining tree graphs to the legs of a tadpole graph. For example, Figure 4 is a
Tree-Loop graph formed by joining a 1-loop tadpole and an $(n+1)$-point tree
graph.

Figure 4: A Tree-Loop fatgraph for the planar 1-loop amplitudes.

Take some $L$-loop tadpole fatgraph, $\Gamma_{\ell}$, with $h$ external legs
(one for each trace-factor). Then, for some comb tree graphs $\Gamma^{A}$
($A=1,\ldots,h$) we can form a Tree-Loop graph

$\Gamma=\Gamma_{\ell}\sqcup\bigsqcup_{A=1}^{h}\Gamma_{A},$ (74)

where $\Gamma^{A}$ is joined to leg $A$ of $\Gamma_{\ell}$. Now consider the
curves on this Tree-Loop graph, $\Gamma$. A curve that connects particle $i$
on trace-factor $A$ to particle $j$ on trace-factor $B$ can be decomposed into
three parts:

$C=C_{i}^{A}C_{AB}C_{j}^{B},$ (75)

where $C_{i}^{A}$ and $C_{j}^{B}$ are paths in the tree graphs, and $C_{AB}$
is a path in the tadpole, $\Gamma_{\ell}$. The curve matrix for $C$ is then
the product

$M_{C}=W_{i}^{A}M_{C_{AB}}\left(W_{j}^{B}\right)^{T},$ (76)

where the $W_{i}^{A}$ are the tree-level matrices we have already computed for
all $n$ in Section 3.3. In other words, to find the curve matrices for
$\Gamma$, it suffices to compute only the the matrices $M_{C_{AB}}$ for the
curves on the tadpole graph, $\Gamma_{\ell}$.

Note that the momentum assignments also become easier to compute. For the
curve $C=C_{i}^{A}C_{AB}C_{j}^{B}$, the momentum can be written as a sum

$P_{C}={P}_{C_{AB}}+z_{j}^{B}-z_{i}^{A},$ (77)

where $P_{C_{AB}}$ is the momentum of $C_{AB}$ on the tadpole graph,
$\Gamma_{\ell}$. Here, $z_{i}^{A}$ is the momentum

$z_{i}^{A}=p_{i+1}+\cdots+p_{n_{A}}$ (78)

formed by summing momenta on trace-factor $A$. $n_{A}$ is the number of
particles incident on trace-factor $A$, and the particles are ordered
$(12\cdots n_{A})$ by convention. So the momenta of curves on $\Gamma$ follow
from the momentum assignments for the tadpole graph, $\Gamma_{\ell}$.

The upshot is that $n$-point amplitudes at $L$ loops with $h$ trace-factors,
can be computed by studying an $h$-point $L$-loop fatgraph.

##  7 Planar One-Loop Amplitudes Revisited

In Section 4 we computed the planar one-loop amplitudes. We now revisit these
amplitudes, by deriving them from a study of the 1-loop planar tadpole.

The 1-loop planar tadpole graph, $\Gamma_{\ell}$, is shown in Figure 5. The
tadpole has two curves: a single loop, $P$, and a spiral, $S$. We discard the
clockwise spiral.444In general, only one helicity of spirals enter into the
formulas for amplitudes. As explained in us_qft , including both spirals leads
to overcounting of Feynman diagrams. The paths of these curves are

$P=RxR,\qquad\text{and}\qquad S=R(xL)^{\infty}.$ (79)

So the two curve matrices are

$\displaystyle P=\begin{bmatrix}1&1+x\\\
0&x\end{bmatrix},\qquad\text{and}\qquad S=\begin{bmatrix}\frac{1}{1-x}&0\\\
\frac{x}{1-x}&0\end{bmatrix}.$ (80)

The headlight functions are

$\alpha_{P}=t_{x}-\max(0,t_{x}),\qquad\alpha_{S}=-t_{x}.$ (81)

Note that $P$ has zero momentum, whereas $S$ has momentum $x_{0}^{\mu}$, if
$x_{0}^{\mu}$ is the loop momentum variable running through the one edge of
the graph.

As an aside, consider evaluating the tadpole amplitude. The curve integral
formula for the loop integrand gives

${\cal I}_{\rm
tadpole}=\int\limits_{-\infty}^{0}dt_{x}\,\exp\left({-\alpha_{P}m^{2}-\alpha_{S}(m^{2}+\ell^{2})}\right)=\int\limits_{-\infty}^{0}dt_{x}\,e^{t_{x}\ell^{2}}.$
(82)

After integrating out the loop momentum, the curve integral for the amplitude
is

${\cal A}_{\rm
tadpole}=\int\limits_{-\infty}^{0}dt_{x}\,\left(\frac{\pi}{t_{x}}\right)^{\frac{D}{2}},$
(83)

as expected.

Figure 5: The Planar One-loop tadpole (left), and its two curves: $S$ and $P$.

To compute the $n$-point amplitude, join a comb tree graph $\Gamma^{1}$ to the
one leg of the tadpole. The result is the _Tree-Loop_ graph, $\Gamma$, in
Figure 4. The curves on $\Gamma$ can be written in Tree-Loop factorised form.
In addition to the curves on $\Gamma^{1}$, $C_{ij}$ (with $i<j$), that do not
intersect the tadpole, the other curves are (with $i\leq j$)

$\displaystyle C_{ji}=W_{i}^{1}\,P\,\left({W}_{j}^{1}\right)^{T},\qquad
C_{i0}=W_{i}^{1}S,$ (84)

where $W_{i}^{1}$ is the curve that connects edge $i$ on the tree, to the edge
that joins $\Gamma^{1}$ to the the tadpole. $(\leavevmode\nobreak\ )^{T}$
denotes word reversal. Note that $M_{C^{T}}$ is given by the matrix transpose,
$M_{C}^{T}$. The curve matrices for these paths are given in Section 3.3.

Using (80), the curve matrix for $C_{i0}$, for example, is

$M_{i0}=W_{i}^{1}S=\begin{bmatrix}1&0\\\
F_{1,n-2}&y_{1,n-1}\end{bmatrix}\begin{bmatrix}\frac{1}{1-x}&0\\\
\frac{x}{1-x}&0\end{bmatrix}=\frac{1}{1-x}\begin{bmatrix}1&0\\\
F_{1,n-2}+xy_{1,n-1}&0\end{bmatrix},$ (85)

so that the headlight function for this curve is

$\alpha_{i0}=-\max(0\leavevmode\nobreak\ ,\leavevmode\nobreak\
t_{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \ldots\leavevmode\nobreak\
,\leavevmode\nobreak\ t_{1}+\cdots+t_{n-2}\leavevmode\nobreak\
,\leavevmode\nobreak\ t_{1}+\cdots+t_{n-1}+t_{x}),$ (86)

where $t_{x}$ is the tropical variable for $x$. The other headlight functions
can be similarly computed.

Substituting the headlight functions into our curve integral formula gives a
new formula for the planar one-loop amplitudes. In fact, an interesting
simplification arises. Recall that ${\cal U}$ is given by

${\cal U}=\sum_{i=1}^{n}\alpha_{i0}.$ (87)

Applying the telescopic property, this simplifies to

${\cal U}=\alpha_{S}=-t_{x}.$ (88)

The amplitude then takes the form

${\cal A}_{\text{1-loop}}=\int\limits_{t_{x}<0}d^{n}{\bf
t}\,\left(\frac{\pi}{-t_{x}}\right)^{\frac{D}{2}}\,\exp\left(\frac{{\cal
F}_{0}}{-t_{x}}-{\cal Z}\right),$ (89)

where ${\bf t}=(t_{1},\ldots,t_{n-1},t_{x})$. In this integral, the first
factor,

$\left(\frac{\pi}{-t_{x}}\right)^{\frac{D}{2}},$ (90)

is independent of $n$, and is precisely what is obtained when we evaluate the
(divergent) tadpole amplitude.

This is a first sign that Tree-Loop factorization simplifies the final
amplitude formulas. As we will see in the following sections, Tree-Loop
factorization becomes even more useful at higher orders in the perturbation
series.

##  8 The Non-planar One-loop Amplitudes

As a more complicated example, consider the one-loop double-trace amplitudes,
for any number of particles, $n=n_{1}+n_{2}$. The all-multiplicity case
follows from an analysis of the 1-loop double-trace tadpole graph. The final
formula we obtain for the amplitude takes the form (with $E=n_{1}+n_{2}$)

${\cal A}_{n_{1},n_{2}}=\int d^{E}t\,{\cal K}\,\left(\frac{\pi}{\cal
U}\right)^{\frac{D}{2}}\,\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}\right).$ (91)

Here, ${\cal U}$ and ${\cal K}$ have simple formulas that follow from studying
the tadpole graph. Whereas ${\cal F}_{0}$ and ${\cal Z}$ have all-$n$ formulas
involving the kinematics, that we derive by adding trees to the tadpole
fatgraph. The number of terms in ${\cal F}_{0}$ and ${\cal Z}$ grows
polynomially with $n$.

###  8.1 The Tadpole

Figure 6: The non-planar one-loop tadpole-like graph (left), and examples of
curves on the graph (right).

The tadpole fatgraph, $\Gamma_{\ell}$, is given in Figure 6. It has two trace-
factors, each with one particle, labelled $1$ and $2$. There are two curves on
$\Gamma_{\ell}$ that begin and end at the same point, and carry zero momentum:

$C_{11}=Le_{1}Le_{2}L,\qquad C_{22}=Le_{1}Le_{2}L.$ (92)

There are also curves that connect $1$ and $2$. For example, reading off from
the fatgraph, the first few of these curves are

$C_{12}^{-1}=Re_{2}L,\qquad C_{12}^{0}=Le_{1}R,\qquad
C_{12}^{1}=Le_{1}(Le_{2}Re_{1})R.$ (93)

These three curves differ from each other by Dehn twists around the closed
path

$\Delta=Le_{2}Re_{1}.$ (94)

Dehn twists around $\Delta$ generate the MCG for this diagram.

Let $p_{1}^{\mu}=-p^{\mu}$ and $p_{2}^{\mu}=p^{\mu}$ be the momentum assigned
to the external legs. Then the momentum associated to $\Delta$ is

$P_{\Delta}=-p^{\mu}.$ (95)

Assign a momentum routing to the internal edges of the fatgraph ($e_{1}$ and
$e_{2}$) by setting, say, $p_{e_{2}}=\ell-p$, and $p_{e_{1}}=\ell-2p$. Then
the three curves have momenta

$P_{12}^{-1}=\ell-p,\qquad P_{12}^{0}=\ell,\qquad P_{12}^{1}=\ell+p.$ (96)

###  8.2 Aside on the Role of the Mirzakhani Kernel

$\Gamma_{\ell}$ is sufficiently simple that it is a feasible exercise to write
down paths for _all_ curves on the fatgraph. The complete set of curves can be
labelled as

$C_{11},\qquad C_{22},\qquad C_{12}^{w},$ (97)

for $w\in\mathbb{Z}$. Here, $C_{12}^{w+1}$ is obtained from $C_{12}^{w}$ by
inserting a twist around the closed curve $\Delta$. The curves
$C^{-1}_{12},C^{0}_{12},C^{1}_{12}$ are as above. Moreover, the momentum
assignments in (96) generalise to

$P_{12}^{m}=\ell+mp,\qquad P_{11}=0,\qquad P_{22}=0.$ (98)

With these momentum assignments, the tadpole amplitude can be given by the
curve integral

${\cal
A}_{\text{tadpole}}=\int\frac{d^{2}t}{\mathrm{MCG}}\,\left(\frac{\pi}{\cal
U}\right)^{\frac{D}{2}}\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal Z}\right),$
(99)

where it is still necessary to quotient by the action of MCG. In this formula,
the surface Symanzik polynomials are formally given as infinite sums:

$\displaystyle{\cal U}=\sum_{w=-\infty}^{\infty}\alpha_{12}^{w},\qquad{\cal
F}_{0}={\cal J}^{\mu}{\cal J}_{\mu},$ (100) $\displaystyle{\cal
Z}=m^{2}(\alpha_{11}+\alpha_{22})+\sum_{w=-\infty}^{\infty}(m^{2}+w^{2}p^{2})\alpha_{12}^{w},$
(101)

where

${\cal J}^{\mu}=p^{\mu}\sum_{w=-\infty}^{\infty}\,w\,\alpha_{12}^{w}.$ (102)

In practice, these infinite expressions are not useful for computations. This
emphasizes the important role played by the _Mirzakhani kernel_ , ${\cal K}$.
The integral becomes

${\cal A}_{\rm tadpole}=\int{d^{2}t}\,{\cal K}\,\left(\frac{\pi}{\cal
U}\right)^{\frac{D}{2}}\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal Z}\right).$
(103)

In the region where ${\cal K}\neq 0$, most headlight functions, $\alpha_{C}$,
are vanishing. In general, only finitely many headlight functions contribute
to this formula. So ${\cal U},{\cal F}_{0}$ and ${\cal Z}$ are all given
instead by _finite sums_. And it is only necessary to find the paths for a
finite number of curves in order to use the formula.

###  8.3 MCG and Mirzakhani Kernel

The MCG is generated by Dehn twists around the closed curve $\Delta$. Write
$\gamma_{\Delta}$ for the action of this Dehn twist on curves. $\Delta$ does
not intersect $C_{11}$ or $C_{22}$, so these curves are MCG invariant. Whereas
$\Delta$ intersects any curve, $C_{12}$, connecting $1$ and $2$. So these
curves are acted on by $\gamma_{\Delta}$.

The curves $C_{12}$ form a single coset under the action of $\gamma_{\Delta}$.
Take $C_{12}^{0}$ as a coset representative. Then a Mirzakhani kernel is given
by

${\cal K}=\frac{\alpha_{12}^{0}}{\rho},$ (104)

where $\rho$ is the sum

$\rho=\sum_{C_{12}}\alpha_{C_{12}}$ (105)

over all curves connecting $1$ and $2$.

However, here a simplification arises. Draw $C_{12}^{0}$ on the fatgraph
$\Gamma_{\ell}$. There are only _four_ curves that do not intersect
$C_{12}^{0}$. These are

$C_{11},\leavevmode\nobreak\ \leavevmode\nobreak\ C_{22},\leavevmode\nobreak\
\leavevmode\nobreak\ C_{12}^{-1},\leavevmode\nobreak\ \leavevmode\nobreak\
C_{12}^{1},$ (106)

whose paths are given above (see (92) and (93)). So the Mirzakhani kernel is
in fact

${\cal
K}=\frac{\alpha_{12}^{0}}{\alpha_{12}^{-1}+\alpha_{12}^{0}+\alpha_{12}^{1}}.$
(107)

Moreover, the surface Symanzik polynomials for the tadpole, which were given
as formal expressions in Section 8.2, also simplify in the region where ${\cal
K}$ is non-vanishing. They become (if $p^{\mu}$ is on-shell, say)

$\displaystyle{\cal
U}=\alpha_{12}^{-1}+\alpha_{12}^{0}+\alpha_{12}^{1},\qquad{\cal
F}_{0}=p^{2}(\alpha_{12}^{1}-\alpha_{12}^{-1})^{2}$ (108) $\displaystyle{\cal
Z}=m^{2}(\alpha_{11}+\alpha_{22}+\alpha_{12}^{0}).$ (109)

In summary, the pre-factor appearing in the tadpole amplitude formula,

${\cal K}\,\left(\frac{\pi}{\cal U}\right)^{\frac{D}{2}},$ (110)

can be computed using the paths for 3 curves. While the whole integrand for
the integral can be computed using the paths for a set, ${\cal S}$ of 5
curves. The curve matrices for these five curves is given in Appendix A.1.

As a final remark, using the Appendix, the headlight functions for the five
curves are

$\displaystyle\alpha_{12}^{-1}=\max(0,t_{2})-t_{2},\qquad\alpha_{12}^{0}=\max(0,t_{1}),\leavevmode\nobreak\
$ (111)
$\displaystyle\alpha_{12}^{1}=\max(0,2t_{1},2t_{1}+t_{2})-2\max(0,t_{1}),\leavevmode\nobreak\
$ (112)
$\displaystyle\alpha_{11}=\alpha_{22}=t_{1}+t_{2}-\max(0,t_{1},t_{1}+t_{2}).$
(113)

Restricting to the region $t_{1}>0$, where ${\cal K}\neq 0$, they simplify to
the headlight functions become

$\displaystyle\alpha_{12}^{-1}=\max(0,t_{2})-t_{2},\qquad\alpha_{12}^{0}=t_{1},$
(114)
$\displaystyle\alpha_{12}^{1}=\max(0,t_{2}),\qquad\alpha_{11}=t_{2}-\max(0,t_{2}).$
(115)

So, in fact, ${\cal K}$ and ${\cal U}$ take especially simple forms in
coordinates:

${\cal K}=\frac{t_{1}}{t_{1}+|t_{2}|},\qquad{\cal U}=t_{1}+|t_{2}|.$ (116)

We will see below that these formulas continue to hold for the all-
multiplicity formula.

###  8.4 The Tree-Loop Fatgraph

Figure 7: The Tree-Loop graph for the 1-loop non-planar amplitude.

To compute the $n$-point amplitude, consider joining the two ends of the
tadpole $\Gamma_{\ell}$ to two comb tree graphs, $\Gamma^{1}$ and
$\Gamma^{2}$, with $n_{1}$ and $n_{2}$ particles respectively. The result is
the Tree-Loop graph for the problem, $\Gamma$, as in Figure 7. In this
section, we see how to find the curves on $\Gamma$ and their momenta
explicitly.

The curves on $\Gamma$ can be obtained from the curves on $\Gamma_{\ell}$. For
example, the curve $C_{12}^{0}$ can be extended to a path

$C_{ij}^{0}=W_{i}^{1}C_{12}^{0}\left(W_{j}^{2}\right)^{T},$ (117)

that connects particle $i$ on trace-factor $1$ and particle $j$ on trace-
factor $2$. Here, $W_{i}^{A}$ is the path through the tree $\Gamma_{A}$ from
particle $i$ to the edge joining the tree to $\Gamma_{\ell}$, and
$(\leavevmode\nobreak\ )^{T}$ denotes path-reversal. Similarly, a curve that
begins and ends on a single trace-factor, $A$, (with $A=1$ or $2$) has a path
of the form (with $i<j$)

$C_{ij}=W_{ij}^{A},\qquad\text{or}\qquad
C_{ji}=W_{i}^{A}C_{AA}\left(W_{j}^{A}\right)^{T},$ (118)

where $W_{ij}^{A}$ is the curve connecting particles $i,j$ in $\Gamma^{A}$
that does not pass through $\Gamma_{\ell}$.

The momenta of curves on $\Gamma$ also follow from the momentum assignment to
curves on the tadpole, $\Gamma_{\ell}$. Let $p_{i}^{1}$ and $p_{j}^{2}$
($i=1,\ldots,n_{1}$, $j=1,\ldots,n_{2}$) be the external momenta. By momentum
conservation,

$\sum_{i=1}^{n_{1}}p_{i}^{1}=p^{\mu},\qquad\sum_{j=1}^{n_{2}}p_{j}^{2}=-p^{\mu}.$
(119)

Then the momentum of $C_{ij}^{0}$, for example, is given by

$P_{ij}^{0}=\ell+z_{j}^{2}-z_{i}^{1},$ (120)

where recall that the curve $P_{12}^{0}$ on the tadpole has momentum $\ell$.
Here,

$z_{i}^{A}=p_{i+1}^{A}+\cdots+p_{n_{A}}^{A}$ (121)

are the dual-like variables on each trace-factor. (The general rule was given
as equation (77).) Similarly, the curves in (118) have momenta (with $A=1$ or
$2$)

$P_{ij}^{A}=p^{A}_{i+1}+\cdots+p^{A}_{j},\qquad
P_{ji}^{A}=z_{j}^{A}-z_{i}^{A}.$ (122)

###  8.5 The All-multiplicity Amplitude

We now use the Tree-Loop graph, $\Gamma$, to find the curve integral for the
all-multiplicity amplitude. We find that the end result is not that much more
complicated than the tadpole case. This is true in general, for reasons
explained further in Section 9.

The MCG action on $\Gamma$ is induced by the MCG on the tadpole,
$\Gamma_{\ell}$. It is generated by the Dehn twist $\gamma_{\Delta}$, which
acts non-trivially only on the curves $C_{ij}^{w}$,

$C_{ij}^{w}=C_{i}^{1}C_{12}^{w}C_{j}^{2},$ (123)

that connect trace-factor 1 and trace-factor 2. Here, $C_{12}^{w}$ are the
curves connecting 1 and 2 on the tadpole. $\gamma_{\Delta}$ does not alter the
tree-parts of the curve. So this set of curves is decomposed into cosets: one
coset for every pair $i,j$ of particles. Take $C_{ij}^{0}$ as coset
representatives. Then a Mirzakhani kernel is

${\cal K}=\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\frac{\alpha_{ij}^{0}}{\rho},$
(124)

where $\rho$ is the sum

$\rho=\sum_{C_{ij}^{w}}\alpha_{ij}^{w}$ (125)

over all such curves.

However, the telescopic property dramatically simplifies this expression. In
particular,

$\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}{\alpha_{ij}^{0}}=\alpha^{0},$ (126)

where $\alpha^{0}$ is the headlight function of the curve $C_{12}^{0}$ on the
tadpole graph. Likewise, $\rho$ simplifies to

$\rho=\alpha^{-1}+\alpha^{0}+\alpha^{1},$ (127)

where we recall (from Section 8.3) that these are the only terms that
contribute in the region $\alpha^{0}\neq 0$. So the all-multiplicity
Mirzakhani kernel is

${\cal K}=\frac{\alpha^{0}}{\alpha^{-1}+\alpha^{0}+\alpha^{1}},$ (128)

which is identical to the ${\cal K}$ found for the tadpole.

In the region $\alpha^{0}\neq 0$, $\alpha_{C}$ vanishes unless $C$ is a curve
that does not intersect the curve $C_{12}^{0}$ on the fatgraph. In other
words, the only curves we need to consider are those constructed from the
special set of five curves

$C_{11},\leavevmode\nobreak\ \leavevmode\nobreak\ C_{22},\leavevmode\nobreak\
\leavevmode\nobreak\ C_{12}^{-1},\leavevmode\nobreak\ \leavevmode\nobreak\
C_{12}^{0}\leavevmode\nobreak\ \leavevmode\nobreak\ C_{12}^{1},$ (129)

considered in Section 8.3. ${\cal U}$ is a sum over all curves that carry a
loop momentum. These are the curves joining trace-factor 1 and trace-factor 2.
So

${\cal
U}=\sum_{w=-1}^{1}\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}\alpha_{ij}^{w}=\alpha^{-1}+\alpha^{0}+\alpha^{1},$
(130)

which has again been simplified using the telescopic relation. Note that this
${\cal U}$ is identical to the one obtained for the tadpole case

Using the coordinate expressions for ${\cal K}$ and ${\cal U}$, equation
(116), the all-multiplicity amplitude is then given by

${\cal
A}_{n_{1},n_{2}}=\int\limits_{t_{1}>0}d^{E}t\,{\frac{t_{1}}{t_{1}+|t_{2}|}}\,\left(\frac{\pi}{t_{1}+|t_{2}|}\right)^{\frac{D}{2}}\,\exp\left(\frac{{\cal
F}_{0}}{t_{1}+|t_{2}|}-{\cal Z}\right).$ (131)

The remaining surface Symanzik polynomials are given by finite sums of curves.
In particular, ${\cal F}_{0}={\cal J}^{\mu}{\cal J}_{\mu}$ where

${\cal
J}^{\mu}=\sum_{w=-1}^{1}\sum_{i,j}\alpha_{ij}^{m}(z_{j}^{2}-z_{i}^{1}+wp).$
(132)

And ${\cal Z}$ is a sum over the finite set of all curves that do not
intersect $C_{12}^{0}$:

${\cal Z}=\sum_{A=1,2}\sum_{i\neq
j}\alpha_{ij}^{A}\left(\left(P_{ij}^{A}\right)^{2}+m^{2}\right)+\sum_{w=-1}^{1}\sum_{i,j}\alpha_{ij}^{w}\left((z_{j}^{2}-z_{i}^{1}+wp)^{2}+m^{2}\right).$
(133)

###  8.6 Comment on Evaluating the Curve Integral

The curve integral formula for the amplitude, (131), has two parts. There is
the pre-factor

${\cal K}\,\left(\frac{\pi}{\cal U}\right)^{\frac{D}{2}},$ (134)

and the exponential part,

$\int d^{E-2}t\,\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal Z}\right),$ (135)

which contains the kinematics. To evaluate the pre-factor, it was only
necessary to know the curve matrices for 3 curves on the tadpole graph. (These
matrices are given in appendix A.1.)

To evaluate the kinematic part, it was necessary to use the set of five curves
on the tadpole graph that are non-intersecting with $C_{12}^{0}$:

$C_{11},\leavevmode\nobreak\ \leavevmode\nobreak\ C_{22},\leavevmode\nobreak\
\leavevmode\nobreak\ C_{12}^{-1},\leavevmode\nobreak\ \leavevmode\nobreak\
C_{12}^{0},\leavevmode\nobreak\ \leavevmode\nobreak\ C_{12}^{1}.$ (136)

Out of these five curves, we can build the relevant set of curves on $\Gamma$:

$C_{ij}^{1},\leavevmode\nobreak\ C_{kl}^{2},\leavevmode\nobreak\
C_{ik}^{-1},\leavevmode\nobreak\ C_{ik}^{0},\leavevmode\nobreak\ C_{ik}^{1},$
(137)

for $i,j=1,\ldots,n_{1}$, $k,l=1,\ldots,n_{2}$ and $i\leq j$, $k\leq l$. There
are

$3n_{1}n_{2}+n_{1}(n_{1}-2)+n_{2}(n_{2}-2)$ (138)

such curves. The curve matrices for all these curves follow from taking
products of the five tadpole curve matrices (for the curves (136)), as well as
the tree matrices

$W_{ij}^{1},\leavevmode\nobreak\ W_{kl}^{2},\leavevmode\nobreak\
W_{i}^{1},\leavevmode\nobreak\ W_{k}^{2},$ (139)

which were computed in Section 3.3. There are

$\frac{1}{2}n_{1}(n_{1}-1)+\frac{1}{2}n_{2}(n_{2}-1)$ (140)

such matrices. So both the number of terms in the exponent of (135), and the
number of matrices needed to compute them, grow polynomially in $n_{1},n_{2}$.

##  9 Prefactors for All-multiplicity

For the 1-loop double-trace amplitudes studied in Section 8, both the
Mirzakhani kernel, ${\cal K}$, and the surface Symanzik polynomial, ${\cal
U}$, did not depend on the number of particles. In this section, we explain
that this _always_ happens, at all orders in perturbation theory, when using a
Tree-Loop graph.

###  9.1 Mirzakhani Kernels

Figure 8: Turning a fatgraph into a surface. Figure 9: The two types of cosets
that arise when forming the Mirzakhani kernel: curves that lower the genus
(left), and curves that decrease the number of trace-factors (right).

Consider any Tree-Loop graph, $\Gamma$, with $h$ trace-factors labelled
$A=1,2,\ldots,h$. It can be a useful book-keeping device when constructing the
Mirzakhani kernel to also draw the surface, $S$, obtained by fattening
$\Gamma$. Examples of surfaces obtained from fatgraph are in Figure 8. A path
$C$ on $\Gamma$ lifts to a curve on $S$, and one can draw $C$ on $S$ without
knowing its exact path.

There are two types of curves that arise when constructing the Mirzakhani
kernel, as shown in Figure 9. Starting from a fixed trace-factor, $A$, we
consider the set of curves, ${\cal S}_{A}$, that contain: all non-separating
curves555A non-separating curve is one that does not divide the surface into
two regions. that return to trace-factor $A$; and all curves that connecting
trace-factor $A$ to a second trace-factor $B$ ($B\neq A$).

The first step in constructing the Mirzakhani kernel is to consider the coset
decomposition of the set of curves ${\cal S}_{A}$ under the $\mathrm{MCG}$
action. The set can be written as the disjoint union

${\cal S}_{A}={\cal S}_{AA}\sqcup\bigsqcup_{B\neq A}{\cal S}_{AB},$ (141)

where ${\cal S}_{AB}$ are the curves connecting distinct trace-factors, $A$
and $B$, and ${\cal S}_{AA}$ are the set of non-separating curves that begin
and end on trace-factor $A$.

Any curve in the set ${\cal S}_{AB}$ can be written in the factorized form

$C=W_{i}^{A}C_{AB}\left(W_{j}^{B}\right)^{T},$ (142)

for some external lines $i$ and $j$, and some curve $C_{AB}$ connecting the
two trace-factors on the tadpole sub-graph, $\Gamma_{\ell}$. The set ${\cal
S}_{AB}$ decomposes into cosets, corresponding to each possible choice of
start and end points, $i$ and $j$. Fix some choice of curve $C^{0}_{AB}$
connecting the two trace-factors on the tadpole sub-graph, $\Gamma_{\ell}$.
Then the curves

$C_{ij}^{0}=W_{i}^{A}C_{AB}^{0}\left(W_{j}^{B}\right)^{T},$ (143)

for each $i$ and $j$ can be taken as coset representatives. In building a
Mirzakhani kernel, we sum over each of these cosets, to obtain the following
terms:

${\cal
K}_{AB}=\sum_{i=1}^{n_{A}}\sum_{j=1}^{n_{B}}\frac{\alpha_{ij}^{0}}{\rho}.$
(144)

where $n_{A}$ and $n_{B}$ are the number of particles on each trace-factor,
and

$\displaystyle\rho=\sum_{C\in{\cal S}}\alpha_{C}.$ (145)

The telescopic property gives a large simplification of the terms ${\cal
K}_{AB}$. The property implies that

$\sum_{i,j}\alpha_{C_{ij}^{0}}=\alpha_{C^{0}_{AB}}.$ (146)

So the terms in ${\cal K}_{AB}$ simplify to become

${\cal K}_{AB}=\frac{\alpha_{AB}^{0}}{\rho}.$ (147)

Moreover, the telescopic property also simplifies the denominator, $\rho$,
which can now be written as

$\rho=\sum_{B=1}^{h}\sum_{C_{AB}}\alpha_{C_{AB}},$ (148)

where the second sum is over curves $C_{AB}$ on the tadpole graph that connect
$A$ and $B$. It follows that ${\cal K}_{AB}$ only depends on tadpole graph
curves.

The analysis of the set ${\cal S}_{AA}$ of non-separating curves is similar.
After applying the telescopic property, the sum over cosets of this set gives

${\cal K}_{AA}=\frac{\alpha_{AA}^{0}}{\rho},$ (149)

for some choice of non-separating curve $C_{AA}^{0}$ on the tadpole graph. The
Mirzakhani kernel for the set ${\cal S}$ is therefore

${\cal K}=\sum_{B=1}^{h}\frac{\alpha_{AB}^{0}}{\rho},$ (150)

and this is precisely a Mirzakhani kernel for the tadpole graph,
$\Gamma_{\ell}$.

Inductively applying this result, to obtain a Mirzakhani kernel for the full
$\mathrm{MCG}$, we can conclude that: _A Mirzakhani kernel for a Tree-Loop
graph $\Gamma$ is given by the Mirzakhani kernel ${\cal K}$ of its tadpole
sub-graph, $\Gamma_{\ell}$._

###  9.2 The Surface Symanzik Polynomial

The surface Symanzik function ${\cal U}$ for a Tree-Loop fatgraph $\Gamma$
also does not depend on $n$. This is because the loop momentum $\ell_{C}$
carried by a curve, $C=C_{i}^{A}C_{AB}C_{j}^{B}$, is just the loop momentum
carried by the tadpole part of its path, $C_{AB}$. Similarly, the loop
momentum $\ell_{C}$ carried by $C=C_{i}^{A}C_{AA}C_{j}^{A}$ is the loop
momentum carried by the subpath $C_{AA}$. The entries of the $\Lambda$ matrix
can therefore be written as

$\Lambda^{ab}=\sum_{A,B=1}^{h}\sum_{C_{AB}}\sum_{i,j}\,\ell_{C_{AB}}^{a}\ell_{C_{AB}}^{b}\alpha_{C_{iABj}},$
(151)

where $C_{iABj}=C_{i}^{A}C_{AB}C_{j}^{B}$. By the telescopic property,

$\sum_{i,j}\,\alpha_{C_{iABj}}=\alpha_{C_{AB}}.$ (152)

So it follows that the matrix entry $\Lambda^{ab}$ for the Tree-Loop graph
$\Gamma$ is equal to the matrix entry for the tadpole subgraph
$\Gamma_{\ell}$:

$\Lambda^{ab}=\sum_{A,B=1}^{h}\sum_{C_{AB}}\ell_{C_{AB}}^{a}\ell_{C_{AB}}^{b}\alpha_{C_{AB}}=\sum_{\begin{subarray}{c}C\\\
{\rm on}\leavevmode\nobreak\
\Gamma_{\ell}\end{subarray}}\ell_{C}^{a}\ell_{C}^{b}\alpha_{C}.$ (153)

We can conclude that: the surface Symanzik function for a Tree-Loop graph is
equal to the surface Symanzik ${\cal U}$ of its tadpole sub-graph,
$\Gamma_{\ell}$.

##  10 Summary of Tree-Loop factorization

In this section we summarize how Tree-Loop fatgraphs, and the general results
in Sections 6 and 9, can be used in practice to compute amplitudes at
arbitrary multiplicity. In the subsequent sections, we then apply this method
to all 2-loop amplitudes.

Let $\Gamma$ be any Tree-Loop graph with $h$ trace-factors and $L$ loops. Let
$\Gamma_{\ell}$ be its tadpole subgraph. As a first step, we can compute a
Mirzakhani kernel, ${\cal K}$, for the tadpole graph by analysing the MCG
action on $\Gamma_{\ell}$.

Given this ${\cal K}$, recall that ${\cal K}$ restricts the domain of
integration. We can restrict our attention to the finite set of curves, ${\cal
S}$, on $\Gamma_{\ell}$, that have non-vanishing headlight functions on this
domain. Using this set of curves, we can compute the surface Symanzik
polynomial

${\cal U}=\det\Lambda,$ (154)

where $\Lambda$ is the $L\times L$ $\Lambda$-matrix of the tadpole graph. This
is computed by working out the momentum assignments, $P_{C}$, to the curves
$C\in{\cal S}$. Each momentum takes the form

$P_{C}^{\mu}=K_{C}^{\mu}+\sum_{a=1}^{L}\ell_{C}^{a}\ell_{a}^{\mu},$ (155)

for some choice of loop momentum variables $\ell_{a}^{\mu}$. Then the
$\Lambda$-matrix is

$\Lambda^{ab}=\sum_{C\in{\cal S}}\ell_{C}^{a}\ell_{C}^{b}\alpha_{C},$ (156)

which is always a finite sum.

After these preliminaries, we have almost all the data needed to write down
the curve integral for all-multiplicity. The final result takes the form

$\displaystyle{\cal A}_{\Gamma}=\int
d\textbf{t}\mathcal{K}\left(\frac{\pi^{L}}{\mathcal{U}}\right)^{D/2}\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right),$
(157)

where ${\cal K}$ and ${\cal U}$ have already been computed.

The remaining two functions, ${\cal F}_{0}$ and ${\cal Z}$ are given by sums
over the finitely many curves on $\Gamma$ that can be produced by extending
the curves ${\cal S}$ to the full graph. In particular,

${\cal F}_{0}={\cal J}_{\mu}\tilde{\Lambda}{\cal J}^{\mu},$ (158)

where $\tilde{\Lambda}=\det(\Lambda)\Lambda^{-1}$ denotes the adjugate of the
$\Lambda$-matrix, and ${\cal J}^{\mu}$ is an $L$-vector given by

${\cal J}^{a,\mu}=\sum_{C_{AB}\in{\cal
S}}\sum_{C=W_{i}^{A}C_{AB}W^{B}_{j}}\,\ell_{C}^{a}\,\alpha_{C}\,\left(K_{C_{AB}}^{\mu}+z_{j}^{B}-z_{i}^{A}\right).$
(159)

The sum is over all curves $C=W_{i}^{A}C_{AB}W^{B}_{j}$ that can be arrived at
by extending a curve $C_{AB}$ in the set ${\cal S}$ to a curve on the full
graph $\Gamma$. Likewise, ${\cal Z}$ is given by a similar sum

${\cal Z}=\sum_{C_{AB}\in{\cal
S}}\sum_{C=W_{i}^{A}C_{AB}W^{B}_{j}}\,\left(K_{C_{AB}}^{\mu}+z_{j}^{B}-z_{i}^{A}\right)^{2}\alpha_{C}+\sum_{\begin{subarray}{c}C\\\
{\rm tree}\end{subarray}}K_{C}^{2}\alpha_{C}.$ (160)

We also include in this sum curves $C=W_{ij}^{A}$ which are entirely in the
tree-part of the tree-loop graph. If we have already computed the momentum
assignments, $P_{C}$, for the tadpole graph curves in ${\cal S}$, these sums
are easy to evaluate. In particular, the headlight function $\alpha_{C}$, for
a curve $C=W_{i}^{A}C_{AB}W^{B}_{j}$, is given by computing the curve matrix

$M_{C}=W_{i}^{A}\,M_{AB}\,\left(W^{B}_{j}\right)^{T}.$ (161)

The $W_{i}^{A}$ matrices have been computed for all-multiplicity in Section
3.3.

In summary, to find the all-multiplicity formula for the amplitude, it
suffices to first study the tadpole graph. For the tadpole graph, one first
derives a kernel, $\mathcal{K}$, and the corresponding finite set of curves,
$\mathcal{S}$, that are compatible with ${\cal K}$. For each of these curves,
$C\in\mathcal{S}$, one computes the corresponding curve matrix, $M_{C}$, and
momentum, $P_{C}=\ell_{C}+K_{C}$. These data depend only on the tadpole
subgraph $\Gamma_{\ell}$, and they suffice to compute the all-multiplicity
formula. The dependence on the particle data in $\mathcal{A}$ enters only
through the vector ${\cal J}^{\mu}$ and the function $\mathcal{Z}$. But the
computation of these functions explained above does not depend on the topology
of the graph $\Gamma_{\ell}$.

##  11 The Planar 2-Loop Amplitudes

Figure 10: The 2-loop planar tadpole.

This section uses the Tree-Loop method described in Section 10 to compute the
planar 2-loop amplitudes. Most of the computation is concerned with the planar
2-loop tadpole, and then the all-multiplicity result is derived at the end.

###  11.1 MCG and Mirzakhani Kernel

Figure 11: The closed curve, $\Delta$, on the 2-loop planar tadpole graph.
Figure 12: The surface picture obtained by fattening the tadpole graph, with
two MCG cosets: curves $C_{0a}$ that end in a spiral around $a$ (left) and
curves $C_{0b}$ that end in a spiral around $b$ (right).

A 2-loop planar tadpole fatgraph, $\Gamma_{\ell}$, is given in Figure 10. The
graph is planar, with two closed loops we label $a$ and $b$. There is one
nontrivial closed curve on $\Gamma_{\ell}$, $\Delta$, shown in Figure 11. This
closed curve has (anti-clockwise) path

$\Delta=zRyRzRwRxRwL.$ (162)

Dehn twists around $\Delta$, $\gamma_{\Delta}$, generate the MCG of the graph.

To find a Mirzakhani kernel, consider how Dehn twists around $\Delta$ act on
the set of curves with one endpoint on the trace-factor. (To produce the
Mirzakhani kernel, we ignore curves that factorize $\Gamma_{\ell}$ into two
graphs.) Using the surface diagram, Figure 12, obtained by fattening
$\Gamma_{\ell}$ to a surface, it is easy to identify that there are two cosets
under MCG: the curves $C_{0a}$ that end in a spiral around $a$, and the curves
$C_{0b}$ that end in a spiral around $b$.

Exploring possible paths on the fatgraph, $\Gamma_{\ell}$, we identify two
possible coset representatives,

$\displaystyle C_{0a}^{0}=LwR(xL)^{\infty},\qquad\text{and}\qquad
C_{0b}^{0}=RzR(yL)^{\infty}.$ (163)

These are the shortest paths connecting $0$ to a spiral around $a$ or $b$.
Other paths can be obtained by acting with Dehn twists $\gamma_{\Delta}$.
Acting once with $\gamma_{\Delta}$ gives paths

$\displaystyle C_{0a}^{1}=RzRyRzRwR(xL)^{\infty},\qquad C_{0b}^{1}=R\Delta
zR(yL)^{\infty}.$ (164)

Whereas acting with $\gamma_{\Delta}$ (or twisting ‘clockwise’ around
$\Delta$) gives paths

$\displaystyle C_{0a}^{-1}=LwLxLwLzLyLzRwR(xL)^{\infty},\qquad
C_{0b}^{-1}=LwLxLwLzR(yL)^{\infty}.$ (165)

The Mirzakhani kernel is then

${\cal K}=\frac{\alpha_{0a}^{0}}{\rho}+\frac{\alpha_{0b}^{0}}{\rho},$ (166)

where $\rho$ is a sum over all curves of the form $C_{0a}$ or $C_{0b}$ that
are compatible with either $C_{0a}^{0}$ or $C_{0b}^{0}$. We write

$\rho=\sum_{C_{0a}\in{\cal S}}\alpha_{C_{0a}}+\sum_{C_{0b}\in{\cal
S}}\alpha_{C_{0b}},$ (167)

where ${\cal S}$ be the set of _all_ curves on $\Gamma_{\ell}$ which are
compatible with either $C_{0a}^{0}$ or $C_{0b}^{0}$.

We find that there are 17 curves in ${\cal S}$. All of these curves and their
paths are listed in Appendix A.2.

###  11.2 Tadpole Curves and Momenta

The tadpole fatgraph is planar, so the momentum assignments can be solved
using dual momentum variables: $x_{0}^{\mu},x_{a}^{\mu},x_{b}^{\mu}$. Curves
that begin at the external line, $0$, and end in a spiral, $a$ or $b$, have
propagators

$\displaystyle X_{0a}=(x_{a}-x_{0})^{2}+m^{2},\qquad
X_{0b}=(x_{b}-x_{0})^{2}+m^{2}.$ (168)

Likewise curves that begin spiraling around $a$ and end spiraling around $b$
have propagator

$X_{ab}=(x_{a}-x_{b})^{2}+m^{2}.$ (169)

Finally, any curve that begins and ends at $0$ has zero momentum. It is
convenient to set $x_{0}=0$ and use $x_{a}$ and $x_{b}$ as loop momentum
variables.

The $\Lambda$-matrix is constructed from the headlight functions $\alpha_{C}$
for curves $C$ that carry loop momentum. But from this momentum assignment, it
follows that the only curves that carry loop momentum, $x_{a}$ and $x_{b}$,
are the curves $C_{0a},C_{0b}$, that have one spiral, and any curve, $C_{ab}$,
that spirals around $a$ and spirals around $b$. In fact, there is just one
curve $C_{ab}$. In the set of curves ${\cal S}$, listed in Appendix A.2, we
see that of a total of 17 curves, only 9 of them carry loop momentum and
appear in $\Lambda$.

Using (168) and (169), the $\Lambda$-matrix is

$\Lambda=\begin{bmatrix}\alpha_{ab}+\alpha_{0a}&-\alpha_{ab}\\\
-\alpha_{ab}&\alpha_{ab}+\alpha_{0b}\end{bmatrix}.$ (170)

Here, $\alpha_{0a},\alpha_{0b}$ is a convenient notation for the sums

$\alpha_{0a}=\sum_{C_{0a}\in{\cal
S}}\alpha_{C_{0a}},\qquad\alpha_{0b}=\sum_{C_{0b}\in{\cal S}}\alpha_{C_{0b}}.$
(171)

The surface Symanzik polynomial, ${\cal U}$, is given by ${\cal
U}=\det\Lambda$. Expanding the determinant as a sum of monomials gives

${\cal
U}=\alpha_{ab}\alpha_{0a}+\alpha_{ab}\alpha_{0b}+\alpha_{0a}\alpha_{0b}.$
(172)

In summary, ${\cal K}$ and ${\cal U}$ can be computed from the headlight
functions for a set of 9 curves, whose paths are listed in Appendix A.2. In
addition to these, ${\cal S}$ has an additional 8 curves which do not carry
loop momentum. These 8 curves do not contribute to computing ${\cal K}$,
${\cal U}$, ${\cal F}_{0}$, but are needed in what follows to compute when
computing ${\cal Z}$.

###  11.3 The Tree-Loop Fatgraph

Figure 13: The Tree-Loop fatgraph for the 2-loop planar amplitudes.

To find the all-$n$ amplitude, attach a tree graph $\Gamma^{0}$ to the
external leg of the tadpole, to obtain the Tree-Loop graph, $\Gamma$, in
Figure 13.

Since $\Gamma$ is still planar, the momenta of curves can still be assigned
using dual momentum variables, $x_{i}^{\mu}$ (for $i=1,\ldots,n,a,b$). If the
external momenta are $p_{i}^{\mu}$, we can take

$x_{i}^{\mu}=p_{1}^{\mu}+\cdots+p_{i-1}^{\mu}.$ (173)

The propagator of a curve that connects line $i$ to line $j$ is then

$X_{ij}=(x_{j}-x_{i})^{2}+m^{2}.$ (174)

The propagators for a curve that begins at line $i$ and ends in a spiral ($a$
or $b$) are

$X_{ia}=(x_{a}-x_{i})^{2}+m^{2},\qquad X_{ib}=(x_{b}-x_{i})^{2}+m^{2}.$ (175)

We again take $x_{a},x_{b}$ as loop momentum variables.

To compute ${\cal F}_{0}$ and ${\cal Z}$, we need to enumerate the curves on
$\Gamma$. But these are obtained directly from the curves on the tadpole
graph, $\Gamma_{\ell}$, by adding tree paths in all possible ways. All the
work was in enumerating the curves on $\Gamma_{\ell}$. We find that the
complete list of loop-carrying curves on $\Gamma$ is

$C_{ia}=W_{i}C_{0a},\qquad C_{ib}=W_{i}C_{0b},\qquad C_{ab}$ (176)

for all curves $C_{0a},C_{0b}\in{\cal S}$ on the tadpole fatgraph. In
addition, we have curves that do not carry loop momentum, (with $i\leq j$)

$\displaystyle C_{ij}=W_{i}C_{00}\,W_{j}^{T},\qquad C_{ij}=W_{ij},\qquad
C_{ji}=W_{i}C_{B}\,W_{j}^{T},$ (177)

for all tadpole curves $C_{00}\in{\cal S}$, and where $C_{B}$ is the boundary
curve on the tadpole fatgraph.

The paths for all the tadpole curves in ${\cal S}$ are given in Appendix A.2
and can be used to find the curve matrices $M_{C}$ for all of these curves by
taking matrix products.

###  11.4 The All-multiplicity Amplitude

The resulting curve integral for the amplitude is (with $E=n+2$)

${\cal A}_{n}=\int d^{E}t\,{\cal K}\,\left(\frac{\pi^{2}}{\cal
U}\right)^{\frac{D}{2}}\,\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}\right),$ (178)

where ${\cal U},{\cal K}$ are given by the tadpole calculation, equations
(172) and (166). It remains to find ${\cal F}_{0}$ and ${\cal Z}$.

The formulas for ${\cal F}_{0}$ and ${\cal Z}$ then follow from the momentum
assignments to these curves. ${\cal F}_{0}$ is given by

$\mathcal{F}_{0}={\cal J}^{T}\tilde{\Lambda}{\cal J},$ (179)

where the vector $\mathcal{J}$ follows from the momentum assignments, (175),
and is given by

${\cal J}^{\mu}=\sum_{i=1}^{n}z_{i}^{\mu}\begin{bmatrix}\sum_{C_{ia}\in{\cal
S}}\alpha_{C_{ia}}\\\ \sum_{C_{ib}\in{\cal S}}\alpha_{C_{ib}}\end{bmatrix}.$
(180)

$\tilde{\Lambda}$ is the matrix adjugate of the matrix, $\Lambda$, computed
for the tadpole, (170). The adjugate is

$\tilde{\Lambda}=\begin{bmatrix}\alpha_{ab}+\alpha_{0b}&\alpha_{ab}\\\
\alpha_{ab}&\alpha_{ab}+\sum_{k}\alpha_{0a}^{k}\end{bmatrix}.$ (181)

Finally, ${\cal Z}$ is a sum over all curves on $\Gamma$ that can be obtained
by from the set ${\cal S}$,

${\cal Z}=\sum_{C}\alpha_{C}(K_{C}^{2}+m^{2}),$ (182)

where $K_{C}^{\mu}$ is the part of the momentum of $C$ that does not depend on
$z_{a}^{\mu},z_{b}^{\mu}$. Explicitly, this is

${\cal
Z}=\sum_{i<j}^{n}X_{ij}\left(\alpha_{ij}+\alpha_{ji}+\sum_{C_{00}\in{\cal
S}}\alpha_{W_{i}C_{00}W_{j}^{T}}\right)\\\
+\sum_{i=1}^{n}(z_{i}^{2}+m^{2})\left(\sum_{C_{ia}\in{\cal
S}}\alpha_{C_{ia}}+\sum_{C_{ib}\in{\cal S}}\alpha_{C_{ib}}\right).$ (183)

##  12 The 2-Loop Triple-trace Amplitudes

In the previous example, the $\mathrm{MCG}$ had only one generator, just as in
the 1-loop double-trace amplitudes. The triple-trace amplitudes are first
example with a more interesting $\mathrm{MCG}$. The $\mathrm{MCG}$ in this
case is still abelian, but has three generators.

###  12.1 MCG and Mirzakhani Kernel

Figure 14: A 2-loop triple-trace tadpole graph.

A triple-trace 2-loop tadpole fatgraph, $\Gamma_{\ell}$, is given in Figure
14. There are three nontrivial closed curves on $\Gamma_{\ell}$, shown in
Figure 15. These closed curves have the paths:

$\displaystyle\Delta_{0}=wRy_{2}Ry_{2}RwRzRx_{2}Rx_{2}RzL,$ (184)
$\displaystyle\Delta_{1}=x_{1}Rx_{2}L,\qquad\Delta_{2}=y_{1}Ly_{2}R.$ (185)

Each of these closed curves surrounds one of the three trace-factors. Dehn
twists around each of these $\Delta_{i}$, $\gamma_{\Delta_{i}}$, generate the
$\mathrm{MCG}$. The $\Delta_{i}$ curves in Figure 15 are non-intersecting. It
follows that the Dehn twists around the $\Delta_{i}$ commute, and the
$\mathrm{MCG}$ is isomorphic to $\mathbb{Z}^{3}$.

To find a Mirzakhani kernel, consider one trace-factor at a time. A curve,
$C_{AB}$, connecting trace-factors $A$ and $B$, intersects $\Delta_{A}$ and
$\Delta_{B}$, and is acted on by Dehn twists around these two curves. Consider
the curves with one endpoint on trace-factor 0. From the surface diagram,
Figure 16, these curves decompose into two cosets: the curves $C_{01}$
connecting 0 and 1, and the curves $C_{02}$ connecting 0 and 2.

Exploring paths on the fatgraph, $\Gamma_{\ell}$, we identify two possible
coset representatives:

$\displaystyle C_{01}^{00}=LzLx_{2}R,\qquad C_{02}^{00}=RwRy_{2}L.$ (186)

These are the shortest paths in the two cosets. Other paths in these cosets
can be obtained by acting with the Dehn twists:
$\gamma_{\Delta_{0}},\gamma_{\Delta_{1}},\gamma_{\Delta_{2}}$. Write

$\rho=\sum\alpha_{C_{01}}+\sum\alpha_{C_{02}}$ (187)

for the sum over curves with a single endpoint on trace-factor 0.

The stabilizer of $C_{01}^{00}$ is generated by $\gamma_{\Delta_{2}}$. To mod
out by the stabilizer, consider the set of curves connecting 1 and 2. From the
surface diagram, Figure 16, it can be seen that these curves form two cosets.
Looking for curves compatible with $C_{01}^{00}$, we find

$\displaystyle C_{12}^{01}=Rx_{1}LzLwLy_{1}R,\qquad
C_{12}^{11}=Lx_{2}RzLwLy_{1}R,$ (188)

which can be taken as coset representatives. Note that $C_{12}^{10}$ is
obtained from $C_{12}^{00}$ by acting with $\gamma_{\Delta_{1}}$.

Likewise, the stabilizer of $C_{02}^{00}$ is generated by
$\gamma_{\Delta_{1}}$. To mod out by this stabilizer, again consider the
curves connecting 1 and 2. From the surface diagram, Figure 16, it can be seen
that these curves form two cosets. Looking for curves compatible with
$C_{02}^{00}$, we find

$\displaystyle C_{12}^{00}=Rx_{1}LzLwRy_{2}L,\qquad
C_{12}^{01}=Lx_{1}LzLwLy_{1}R,$ (189)

which can be taken as coset representatives.

A Mirzakhani is then

${\cal
K}=\frac{\alpha_{01}^{00}\alpha_{12}^{01}}{\rho\rho^{\prime}}+\frac{\alpha_{01}^{00}\alpha_{12}^{11}}{\rho\rho^{\prime}}+\frac{\alpha_{02}^{00}\alpha_{12}^{00}}{\rho\rho^{\prime}}+\frac{\alpha_{02}^{00}\alpha_{12}^{01}}{\rho\rho^{\prime}},$
(190)

where $\rho^{\prime}$ is a sum

$\rho^{\prime}=\sum\alpha_{C_{12}}$ (191)

over curves connecting 1 and 2. Only finitely many curves appear in $\rho$ and
$\rho^{\prime}$. Namely, only curves compatible with one of the pairs of
curves appearing in the numerators of ${\cal K}$.

Figure 15: The three nontrivial closed curves,
$\Delta_{0},\Delta_{1},\Delta_{2}$, on the 2-loop triple-trace tadpole graph.

Let ${\cal S}$ be the set of _all_ curves on $\Gamma_{\ell}$ which are
compatible with one of the pairs of curves appearing in the numerators of
${\cal K}$.

We find that there are 33 curves in ${\cal S}$. 24 of these are curves that
carry loop momentum, and so appear in ${\cal U}$, ${\cal K}$ and ${\cal
F}_{0}$. Whereas 9 of these are curves do not carry loop momentum and only
contribute to ${\cal Z}$. All 33 of these curves and their paths are listed in
Appendix A.4.

Figure 16: The surface obtained by fattening the tadpole fatgraph, and the
four MCG-inequivalent cuts of the surface that give the Mirzakhani kernel.

###  12.2 Tadpole Curves and Momenta

The momentum assignment rule shows that the closed curves have momenta

$P_{\Delta_{0}}=p_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
P_{\Delta_{1}}=p_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
P_{\Delta_{2}}=p_{2}.$ (192)

A momentum routing can be chosen such that

$P_{01}^{00}=\ell_{1},\qquad P_{02}^{00}=\ell_{2},\qquad
P_{12}^{00}=\ell_{1}+\ell_{2},$ (193)

for some loop momentum variables $\ell_{1},\ell_{2}$. Acting with the
$\mathrm{MCG}$, we obtain all other curves. A Dehn twist by $\Delta_{i}$
acting on $C_{ij}$ adds $\pm p_{i}$ to its momentum. So:

$\displaystyle P_{01}^{ab}$ $\displaystyle=\ell_{1}+ap_{0}+bp_{1},$ (194)
$\displaystyle P_{02}^{ab}$ $\displaystyle=\ell_{2}+ap_{0}+bp_{2},$ (195)
$\displaystyle P_{12}^{ab}$ $\displaystyle=\ell_{1}+\ell_{2}+ap_{1}+bp_{2}.$
(196)

Finally, the curves that begin and end at the same trace-factor are homologous
to the boundaries of the fatgraph. This means that the momentum of such a
curve $C_{AA}$ ($A=0,1,2$) is a linear combination of the boundary momenta:
$p_{0},p_{1},p_{2}$.

The $\Lambda$-matrix is constructed from $\alpha_{C}$ for curves $C$ that
carry loop momentum. The only curves carrying loop momenta are those curves
connecting two distinct trace-factors. Given (193), the $\Lambda$-matrix is

$\Lambda=\begin{bmatrix}\alpha_{01}+\alpha_{12}&\alpha_{12}\\\
\alpha_{12}&\alpha_{02}+\alpha_{12}\end{bmatrix},$ (197)

where

$\alpha_{AB}=\sum_{C_{AB}\in{\cal S}}\alpha_{C_{AB}}$ (198)

is the sum over those curves $C_{AB}\in{\cal S}$, connecting trace-factors $A$
and $B$, that are compatible with the Mirzakhani kernel, ${\cal K}$. The
surface Symanzik polynomial ${\cal U}$ is given by

${\cal
U}=\det\Lambda=\alpha_{01}\alpha_{02}+\alpha_{01}\alpha_{12}+\alpha_{02}\alpha_{12}.$
(199)

###  12.3 The All-multiplicity Amplitudes

To find the all-multiplicity amplitudes, attach tree comb graphs
$\Gamma^{0},\Gamma^{1},\Gamma^{2}$ to $\Gamma_{\ell}$, to obtain the Tree-Loop
graph, $\Gamma$. Let the number of external particles on each trace-factor be
$n_{0},n_{1},n_{2}$, respectively, with $n=n_{0}+n_{1}+n_{2}$. From a curve
$C_{AB}$ on $\Gamma_{\ell}$, we obtain curves of the form

$C_{iABj}=W_{i}^{A}C_{AB}\left(W_{j}^{B}\right)^{T}$ (200)

connecting particle $i$ on trace-factor $A$ to particle $j$ on trace-factor
$B$. The momentum assigned to such a curve is (see Section 10)

$P_{C_{iABj}}=P_{C_{AB}}+z_{j}^{B}-z_{i}^{A}.$ (201)

And the headlight function $\alpha_{C}$ is obtained by multiplying the matrix
$M_{C_{AB}}$ with the tree matrices $W_{i}^{A}$ and
$\left(W_{j}^{B}\right)^{T}$.

The curve integral for the amplitude is then (with $E=n$)

${\cal A}_{n_{1},n_{2},n_{3}}=\int d^{E}t\,{\cal K}\,\left(\frac{\pi^{2}}{\cal
U}\right)^{\frac{D}{2}}\exp\left({\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}}\right).$ (202)

Here, ${\cal K}$ and ${\cal U}$ are given as above by the tadpole calculation
(equations (190) and (199)).

The formulas for ${\cal F}_{0}$ and ${\cal Z}$ follow from the momentum
assignments. For a curve $C_{iABj}$ connecting trace-factors $A$ and $B$,
write

$P_{C}=K_{C}+\ell_{C_{AB}},$ (203)

where $\ell_{C_{AB}}$ is the part depending on the loop momentum variables.
From (193),

$\ell_{C_{01}}=\ell_{1},\qquad\ell_{C_{02}}=\ell_{2},\qquad\ell_{C_{12}}=\ell_{1}+\ell_{2}.$
(204)

It follows that J is

${\cal J}=\left[\begin{array}[]{c}\sum_{\begin{subarray}{c}C_{01}\in{\cal
S}\\\
C=W_{i}^{0}C_{01}\left(W_{j}^{1}\right)^{T}\end{subarray}}K_{C}\,\alpha_{C}+\sum_{\begin{subarray}{c}C_{12}\in{\cal
S}\\\
C=W_{i}^{1}C_{12}\left(W_{j}^{2}\right)^{T}\end{subarray}}K_{C}\,\alpha_{C}\\\
\sum_{\begin{subarray}{c}C_{02}\in{\cal S}\\\
C=W_{i}^{0}C_{02}\left(W_{j}^{2}\right)^{T}\end{subarray}}K_{C}\,\alpha_{C}+\sum_{\begin{subarray}{c}C_{12}\in{\cal
S}\\\
C=W_{i}^{1}C_{12}\left(W_{j}^{2}\right)^{T}\end{subarray}}K_{C}\,\alpha_{C}.\end{array}\right],$
(205)

Then ${\cal F}_{0}$ is, as usual ${\cal F}_{0}={\cal
J}^{T}\tilde{\Lambda}\cal{J}$, where the adjugate matrix of $\Lambda$ is

$\tilde{\Lambda}=\begin{bmatrix}\alpha_{02}+\alpha_{12}&-\alpha_{12}\\\
-\alpha_{12}&\alpha_{01}+\alpha_{12}\end{bmatrix}.$ (206)

Finally,

${\cal Z}=\sum_{C_{AB}\in{\cal S}}\sum_{C}((K_{C})^{2}+m^{2})\alpha_{C},$
(207)

where the second sum is over curves
$C=W_{i}^{A}C_{AB}\left(W_{j}^{B}\right)^{T}$ formed by extending $C_{AB}$ to
a curve on $\Gamma$.

As in the previous examples, evaluating the curve integral for ${\cal
A}_{n_{1},n_{2},n_{3}}$ is simplified by using the form of the Mirzakhani
kernel ${\cal K}$, which has four terms. So the amplitude is a sum of four
curve integrals. For example, the monomial $\alpha_{01}^{00}\alpha_{02}^{00}$
appears in ${\cal K}$. Restricting to the region
$\alpha_{01}^{00}\alpha_{02}^{00}$, the curve integral simplifies. Cutting
$\Gamma_{\ell}$ along $C_{01}^{00},C_{02}^{00}$ gives a 7-point tree graph,
which has 14 possible curves. So this contribution to ${\cal
A}_{n_{1},n_{2},n_{3}}$ can be computed using 14 matrices, valid for all
multiplicities.

##  13 The 2-Loop Amplitudes at Genus One

Figure 17: The 2-loop genus-one tadpole fatgraph.

The first non-planar contribution to the single trace amplitude at two loops
is given by genus one diagrams. These give a first example of a non-abelian
Mapping Class Group, which is isomorphic in this case to ${\rm
SL}_{2}\mathbb{Z}\times\mathbb{Z}$.

###  13.1 MCG and Mirzakhani Kernel

Figure 18: The three closed curves, $\Delta_{0},\Delta_{1},\Delta_{2}$, on the
2-loop genus-one tadpole graph. Dehn twists around these curves generate a
$\mathbb{Z}\times\text{SL}_{2}\mathbb{Z}$ Mapping Class Group.

The genus-one tadpole fatgraph, $\Gamma_{\ell}$, is given in Figure 17. There
are three non-trivial closed curves on $\Gamma_{\ell}$. These closed curves
are shown in Figure 18 and they have paths:

$\displaystyle\Delta_{A}=y_{1}Ry_{2}Ly_{3}R,\qquad\Delta_{B}=y_{1}Ry_{2}Ry_{4}L,$
(208)
$\displaystyle\Delta_{0}=y_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}Ly_{1}Ry_{4}Ry_{3}R.$
(209)

$\Delta_{0}$ is the closed curve surrounding the trace-factor of the fatgraph.
$\Delta_{A}$ and $\Delta_{B}$ are homologous to the A- and B- cycles of the
torus (if $\Gamma_{\ell}$ is drawn embedded on the torus).

Dehn twists $\gamma_{\Delta}$ around these closed curves generate
$\mathrm{MCG}$. $\Delta_{0}$ does not intersect either of $\Delta_{A}$ or
$\Delta_{B}$, so $\gamma_{\Delta_{0}}$ commutes with the other two Dehn
twists. Whereas $\Delta_{A}$ and $\Delta_{B}$ intersect, so that Dehn twists
around these closed curves do _not_ commute. Since $\Delta_{A}$ and
$\Delta_{B}$ are the A- and B- cycles of the torus, these Dehn twists generate
the MCG of the torus, which gives an ${\rm SL}_{2}\mathbb{Z}$ subgroup of the
full $\mathrm{MCG}$. The $\mathrm{MCG}$ is therefore isomorphic to
$\mathbb{Z}\times{\rm SL}_{2}\mathbb{Z}$.

All open curves on $\Gamma_{\ell}$ begin and end on the external line, $0$.
Using the surface diagram, Figure 19, one can see that these curves decompose
into two cosets. Consider a curve $C^{0}$ that wraps around the torus. A new
curve, $C^{1/2}$, can be obtained from $C^{0}$ by moving _one_ endpoint around
$\Delta_{0}$, leaving the other endpoint fixed. $C^{1/2}$ cannot be obtained
from $C^{0}$ by acting with Dehn twists. However, all other curves on the
surface can be obtained from these two curves by acting with
$\mathrm{MCG}$.666In terms of the homology $H_{1}(S,\partial S)$ of the
surface relative to the boundary, the closed curves
$\Delta_{0},\Delta_{A},\Delta_{B}$ represent classes that generate the
homology. A curve $C^{0}$ represents a class that does not contain
$[\Delta_{0}]$ when expanded in this basis. A curve $C^{1/2}$ represents a
class that does contain $[\Delta_{0}]$.

Exploring paths on the fatgraph, we find short paths

$C^{0/1;0}=Ly_{2}Ry_{4}Ly_{1}L,\qquad
C^{0/1;1/2}=Ly_{2}Ly_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R$ (210)

which can be taken as coset representatives for the two $\mathrm{MCG}$ cosets.
$C^{0/1;1/2}$ is obtained from $C^{0/1;0}$ by shifting its second endpoint
around $\Delta_{0}$, and so is not in the MCG orbit of $C^{0/1;0}$. Note that
both curves intersect $\Delta_{A}$, but _not_ $\Delta_{B}$. So the stabilizer
of both curves is the $\mathbb{Z}$ subgroup generated by
$\gamma_{\Delta_{B}}$. Write

$\rho=\sum\alpha_{C}$ (211)

for the sum over all curves $C$ that are compatible with either $C^{0/1;0}$ or
$C^{0/1;1/2}$. Then

${\cal K}_{A}=\frac{\alpha^{0/1;0}}{\rho}+\frac{\alpha^{0/1;1/2}}{\rho}$ (212)

is a Mirzakhani kernel that partially mods out by $\mathrm{MCG}$, but does not
mod out by the stabilizer generated by $\gamma_{\Delta_{B}}$.

Cutting the surface along $C^{0/1;0}$ (or $C^{0/1;1/2}$) gives an annulus with
2 marked points on one boundary, and 1 marked point on the other. $\Delta_{B}$
is the closed loop on the annulus. So the curves compatible with $C^{0/1;0}$
(or with $C^{0/1;1/2}$) decompose into two cosets under the $\mathbb{Z}$
subgroup.

To mod out by the stabilizer, we need to find curves that intersect
$\Delta_{B}$ and are compatible with either $C^{0/1;0}$ or $C^{0/1;1/2}$.
Consider curves compatible with $C^{0;0}$. We find, for example, the paths

$C^{1/0;0}=Ry_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ly_{1}L,\qquad
C^{1/0;1/2}=Ly_{2}Ly_{3}Ry_{1}L.$ (213)

These are both compatible with $C^{0/1;0}$, intersect $\Delta_{B}$, but are
not related to each other by $\gamma_{\Delta_{B}}$. So we can take $C^{1/0;0}$
and $C^{1/0;1/2}$ as our two coset representatives.

Finally, looking for curves compatible with $C^{0/1;1/2}$ we find, for
example,

$C^{1/0;1/2}=Ly_{2}Ly_{3}Ry_{1}L,\qquad
C^{1/0;1}=Ly_{2}Ly_{3}Ly_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R.$ (214)

These both intersect $\Delta_{A}$, but are not related to each other by
$\gamma_{\Delta_{A}}$. So we can take $C^{1/0;1/2}$ and $C^{1/0;1}$ as two
coset representatives for the curves compatible with $C^{0/1;1/2}$.

Write

$\rho_{A}=\sum\alpha_{C}$ (215)

for the sum over all curves, $C$, which are compatible with $C^{0/1;0}$ or
$C^{0/1;1/2}$, and which intersect $\Delta_{A}$. Then a final Mirzakhani
kernel for the full MCG is

${\cal
K}=\frac{\alpha^{0/1;0}}{\rho}\frac{\alpha^{1/0;0}+\alpha^{1/0;1/2}}{\rho_{A}}+\frac{\alpha^{0/1;1/2}}{\rho}\frac{\alpha^{1/0;1/2}+\alpha^{1/0;1}}{\rho_{A}}.$
(216)

The four terms in ${\cal K}$ correspond to the four MCG-inequivalent cuts
shown in Figure 19.

Let ${\cal S}$ be the set of all curves which are compatible with one of the
four pairs of curves appearing in the numerators of ${\cal K}$. There are 14
curves in ${\cal S}$. Of these, 13 curves carry loop momentum, and contribute
to the computation of ${\cal K}$ and ${\cal U}$. Only 1 curve in the set does
not carry loop momentum: the curve $C_{B}$ that follows the boundary of
$\Gamma_{\ell}$, and carries zero momentum:

$C_{B}=Ly_{2}Ly_{3}Ly_{4}Ly_{2}Ly_{1}Ly_{3}Ly_{4}Ly_{1}L.$ (217)

The paths for all 14 of the curves in ${\cal S}$ are listed in Appendix A.5.

Figure 19: The four MCG-inequivalent pairs of curves that cut the torus to a
disk. Each pair of curves corresponds to a distinct term in the Mirzakhani
kernel, ${\cal K}$.

###  13.2 Momentum Assignments

By momentum conservation, the external momentum entering the tadpole graph is
zero. Assign a momentum routing to the graph by assigning edge 3 the loop
momentum variable $\ell_{1}^{\mu}$, and edge 4 the loop momentum variable
$\ell_{2}^{\mu}$. Then the momenta of the closed curves are

$P_{\Delta_{0}}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
P_{\Delta_{A}}=\ell_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
P_{\Delta_{B}}=\ell_{2}.$ (218)

Moreover this momentum routing induces a momentum assignment to all open
curves. For example, using the momentum assignment rule,

$P_{C^{0/1;0}}=P_{C^{0/1;1/2}}=\ell_{2},\qquad
P_{C^{1/0;0}}=P_{C^{1/0;1/2}}=P_{C^{1/0;1}}=\ell_{1}.$ (219)

Another way to understand this momentum assignment is that, on the surface
(Figure 19), the curves $C^{0/1;w}$ above are homologous to $\Delta_{B}$
(ignoring the boundary trace-factor), and so has the same momentum as
$\Delta_{B}$. Likewise, the curves $C^{1/0;w}$ are homologous to
$\Delta_{A}$.777As explained in us_qft , the momenta assigned to curves,
$P_{C}^{\mu}$, satisfy the same additive relations as the classes of curves in
homology.

Now consider any curve $C$ in the set ${\cal S}$. The momentum assigned to $C$
is necessarily a linear combination of $\ell_{1}$ and $\ell_{2}$:

$P_{C}=p\ell_{1}+q\ell_{2},$ (220)

for some integers $p$ and $q$. It is convenient to write

$C=C^{p/q;w}$ (221)

for any such curve $C$ with momentum $P_{C}=p\ell_{1}+q\ell_{2}$. Here, $w$ is
an index that distinguishes distinct curves with the same momentum.888Here and
in the appendix, we have adopted a particularly careful choice of indexes $w$
such that the Dehn twist $\gamma_{\Delta_{0}}$ acts on curves as
$C^{p/q;w}\mapsto C^{p/q;w+1}$. But any choice of indexes will suffice for the
computations.

The $\Lambda$-matrix follows from the momentum assignments, (220). It is given
by

$\Lambda=\left[\begin{array}[]{cc}\sum\limits_{\begin{subarray}{c}C=C^{p/q;w}\\\
C\in{\cal
S}\end{subarray}}\,p^{2}\,\alpha_{C}&\sum\limits_{\begin{subarray}{c}C=C^{p/q;w}\\\
C\in{\cal S}\end{subarray}}\,pq\,\alpha_{C}\\\
\sum\limits_{\begin{subarray}{c}C=C^{p/q;w}\\\ C\in{\cal
S}\end{subarray}}\,pq\,\alpha_{C}&\sum\limits_{\begin{subarray}{c}C=C^{p/q;w}\\\
C\in{\cal S}\end{subarray}}\,q^{2}\,\alpha_{C}\end{array}\right],$ (222)

The surface Symanzik polynomial ${\cal U}$ is then the determinant, ${\cal
U}=\det\Lambda$.

As an aside, note that this determinant can be expanded as the sum over pairs
$C,D$ of curves in ${\cal S}$:

$\det\Lambda=\sum_{\begin{subarray}{c}C=C^{p/q;w}\\\
D=C^{r/s;w^{\prime}}\end{subarray}}\left|\begin{matrix}p&r\\\
q&s\end{matrix}\right|^{2}\alpha_{C}\,\alpha_{D}.$ (223)

It can be checked that the determinant factors appearing in this sum, $(ps-
qr)^{2}$, are equal to $1$ if and only if the two curves $C$ and $D$ cut
$\Gamma_{\ell}$ to a tree. Otherwise the determinant factor is zero. So ${\cal
U}$ is precisely a sum over maximal cuts of $\Gamma_{\ell}$ that can be formed
from the set ${\cal S}$:

${\cal U}=\sum_{\begin{subarray}{c}C,D\in{\cal S}\\\ \text{max.
cut}\end{subarray}}\alpha_{C}\alpha_{D}.$ (224)

###  13.3 The All-multiplicity Amplitudes

To find the all-mulitplicity genus-one amplitudes, attach a tree comb graph,
$\Gamma^{0}$, to the external leg of $\Gamma_{\ell}$, to obtain a Tree-Loop
graph, $\Gamma$. Let $\Gamma$ have $n$ external legs. From any curve
$C^{p/q;w}\in{\cal S}$ on $\Gamma_{\ell}$, we can obtain a curve

$C_{ij}^{p/q;w}=W_{i}C^{p/q;w}\left(W_{j}\right)^{T},$ (225)

connecting particles $i$ and $j$, and such a curve has momentum (see Section
10)

$P_{ij}^{p/q;w}=P_{C^{p/q;w}}+z_{j}-z_{i}.$ (226)

The headlight functions for these curves are obtained by matrix multiplication
from the $n$ tree-level matrices $W_{i}$ ($i=1,\ldots,n$) and the 13 matrices
$M_{C}$ for the curves $C^{p/q;w}$ in ${\cal S}$.

Note that there are also a set of curves that do not carry loop momentum and
have tree-like propagators. These are (for $i<j-1$)

$C_{ij}=W_{ij},$ (227)

with momentum $P_{ij}=p_{i}+\cdots+p_{j-1}$, and also (for $i\leq j$)

$C_{ji}=W_{i}C_{B}\left(W_{j}\right)^{T},$ (228)

with momentum $P_{ji}=p_{i+1}+\cdots+p_{j}$.

The curve integral for the amplitude is then (with $E=n+2$)

${\cal A}_{n}=\int d^{E}t\,{\cal K}\,\left(\frac{\pi^{2}}{\cal
U}\right)^{\frac{D}{2}}\exp\left({\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}}\right).$ (229)

Here, ${\cal K}$ (equation (216)) and ${\cal U}$ (equation (222)) are computed
using the 13 curves $C^{p/q;w}$ in ${\cal S}$.

The formulas for ${\cal F}_{0}$ and ${\cal Z}$ follow from the momentum
assignments. The vector ${\cal J}^{\mu}$ is

${\cal
J}^{\mu}=\left[\begin{array}[]{c}\sum\limits_{\begin{subarray}{c}C^{p/q;w}\in{\cal
S}\\\ C=W_{i}C^{p/q;w}W_{j}^{T}\end{subarray}}\,p\,\alpha_{C}\,z_{ij}^{\mu}\\\
\sum\limits_{\begin{subarray}{c}C^{p/q;w}\in{\cal S}\\\
C=W_{i}C^{p/q;w}W_{j}^{T}\end{subarray}}\,q\,\alpha_{C}\,z_{ij}^{\mu}\end{array}\right],$
(230)

where $z_{ij}^{\mu}=z_{j}^{\mu}-z_{i}^{\mu}$. Then ${\cal F}_{0}={\cal
J}^{T}\tilde{\Lambda}{\cal J}$, where $\tilde{\Lambda}$ is the the adjugate of
the $\Lambda$-matrix (equation (222)). Finally,

${\cal Z}=\sum_{\begin{subarray}{c}C\in{\cal S}\\\
D=W_{i}CW_{j}^{T}\end{subarray}}\left(z_{ij}^{2}+m^{2}\right)\alpha_{D}+\sum_{\begin{subarray}{c}i=1\\\
i<j-1\end{subarray}}^{j=n}\left(z_{ji}^{2}+m^{2}\right)\alpha_{C_{ij}}.$ (231)

##  14 The 2-Loop Double-trace Amplitudes

Figure 20: A tadpole graph for the 2-loop double-trace amplitudes.

To complete our list of 2-loop amplitude formulas, we give here a curve
integral formula for the 2-loop double-trace amplitudes. The analysis is very
similar to the computations in the foregoing sections.

###  14.1 MCG and Mirzakhani

Figure 21: The surface obtained by fattening the double-trace 2-loop fatgraph
is a sphere with one puncture and two trace-factors. There are five MCG-
inequivalent cuts of this surface to a disk, which appear in the Mirzakhani
kernel, ${\cal K}$. Note that the first two (and the last two) cuts are MCG-
inequivalent due to the different order in which the curves are incident at
the trace-factors.

The double-trace tadpole fatgraph, $\Gamma_{\ell}$, is given in Figure 20. The
$\mathrm{MCG}$ is generated by Dehn twists around the closed curves,
$\Delta_{1}$ and $\Delta_{2}$, that surround trace-factor 1 and trace-factor 2
of $\Gamma_{\ell}$, respectively. $\Delta_{1}$ and $\Delta_{2}$ are non-
intersecting, and so these two Dehn twists commute, and the $\mathrm{MCG}$ is
isomorphic to the free abelian group $\mathbb{Z}^{2}$.

Using the surface diagram, Figure 21, one can identify the MCG cosets. In
fact, there are five MCG-inequivalent cuts of the surface to a disk (shown in
the figure). Exploring paths on the tadpole fatgraph, $\Gamma_{\ell}$, we find
that the following five pairs of curves can be taken as representatives for
the five cuts:

$\displaystyle(C_{1,0}^{0},C_{1,2}^{1,0}),\leavevmode\nobreak\
\leavevmode\nobreak\ (C_{1,0}^{0},C_{1,2}^{0,0}),\leavevmode\nobreak\
\leavevmode\nobreak\ (C_{1,0}^{0},C_{2,0}^{0}),\leavevmode\nobreak\
\leavevmode\nobreak\ (C_{2,0}^{0},C_{1,2}^{0,0}),\leavevmode\nobreak\
\leavevmode\nobreak\ (C_{2,0}^{0},C_{1,2}^{0,1}),$ (232)

where the explicit paths for the 5 curves appearing here are given in Figure
22. The paths for these curves are also listed in Appendix A.3. We have chosen
to label curves as $C_{AB}^{w}$, where $A$ and $B$ ($A,B=0,1,2$) are trace-
factors that the curve begins and ends on. $A=0$ is the trace-factor with no
particles. $w$ is some decoration to distinguish distinct curves with the same
endpoints.

Let ${\cal S}$ be the finite set of curves compatible with at least one of the
five cuts in equation (232). All curves in this set and their paths are listed
in Appendix A.3. There are 23 curves in total, of which 14 curves connect
distinct trace-factors. Consider the sum, ${\cal U}$, over all possible cuts
that can be formed from the set of curves ${\cal S}$:

${\cal
U}=\alpha_{01}\alpha_{12}+\alpha_{01}\alpha_{02}+\alpha_{02}\alpha_{12},$
(233)

where $\alpha_{AB}$ is a sum over curves in ${\cal S}$ connecting trace-
factors $A$ and $B$,

$\alpha_{AB}=\sum\limits_{C_{AB}\in{\cal S}}\alpha_{C_{AB}}.$ (234)

Then a Mirzakhani kernel is given by

$\displaystyle\mathcal{K}=\frac{\alpha_{10}^{0}\alpha_{12}^{10}+\alpha_{10}^{0}\alpha_{12}^{00}+\alpha_{10}^{0}\alpha_{20}^{0}+\alpha_{20}^{0}\alpha_{12}^{00}+\alpha_{20}^{0}\alpha_{12}^{01}}{\mathcal{U}}.$
(235)

Figure 22: The five curves on $\Gamma_{\ell}$ which we use to construct all
five MCG-inequivalent cuts.

###  14.2 The All-multiplicity Amplitudes

To assign momenta to the curves in the set ${\cal S}$, choose a routing of
momenta on $\Gamma_{\ell}$ (Figure 20) such that edge $x_{1}$ has momentum
$\ell_{1}^{\mu}$ and edge $x_{2}$ has momentum $\ell_{2}^{\mu}$, for some loop
momentum variables, $\ell_{1},\ell_{2}$. Let $p_{1}=p$ and $p_{2}=-p$ be the
momenta of the two trace-factors. Then the momentum assignment rule gives, for
example,

$P_{C_{01}^{0}}=\ell_{1},\qquad P_{C_{02}^{0}}=\ell_{2}.$ (236)

The only curves that carry a loop momentum are those connecting distinct
trace-factors. In Appendix A.3, these curves are labelled such that their
momenta are given by

$\displaystyle P_{C_{12}^{w_{1},w_{2}}}$
$\displaystyle=\ell_{1}-\ell_{2}+(w_{1}-w_{2}-1)p,$ (237) $\displaystyle
P_{C_{01}^{w}}$ $\displaystyle=\ell_{1}+wp,$ (238) $\displaystyle
P_{C_{02}^{w}}$ $\displaystyle=\ell_{2}-wp.$ (239)

In the notation of the summary, Section 10, these curves have momentum

$P_{C}^{\mu}=K_{C}^{\mu}+\ell_{C}^{1}\ell_{1}^{\mu}+\ell_{C}^{2}\ell_{1}^{\mu},$
(240)

where $\ell_{C_{12}}=(1,-1)$, $\ell_{C_{01}}=(1,0)$ and $\ell_{C_{02}}=(0,1)$.

The all-multiplicity amplitude is then given by

${\cal A}=\int d^{E}t\,{\cal K}\left(\frac{\pi^{2}}{\cal
U}\right)^{\frac{D}{2}}\,\exp\left(\frac{{\cal F}_{0}}{\cal U}-{\cal
Z}\right),$ (241)

where ${\cal K}$ is given by (235), ${\cal U}$ is given by (233). ${\cal
F}_{0}$ and ${\cal Z}$ are given by the formulas in Section 10, with the
momenta assignments as above.

##  15 Outlook

This paper opens a systematic investigation of all-loop scattering amplitudes
in the tr$\phi^{3}$ theory at all $n$. We have seen that certain especially
simple fatgraph representatives for the curve integral formalism manifest a
remarkable decoupling of $n$ and $L$, and we illustrated this at all $n$ for
all amplitudes in the theory through to two loops.

In practice, the computations described in this paper are best done using a
computer. The curve integral formalism is based on simple combinatorial rules,
which are very amenable to implement in code. This will play an important role
in applications of these formulas beyond two-loops. One possible application
of these computer methods could lie in the numerical evaluation of amplitudes.
Tropical methods have recently proved useful for the evaluation of individual
Feynman integrals. Borinsky_2023 ; feyntrop

The key result that underlies all the formulas in this paper is the existence
of an explicit positive parametrization for Teichmüller spaces, which is most
closely related to the point of view of fockgoncharov . This parametrization
is given by the definition of the so-called $u$-variables as rational
functions in positive $y$-variables. us_qft . However, to arrive at formulas
for amplitudes, we have to take the tropicalization of this
parametrization.999The headlight functions, $\alpha_{C}(t)$, are defined as
the tropicalization of the $u$-variables, $u_{C}(y)$. Tropical moduli spaces
have a rich topological structure, which is an active area of
research.101010Tropical moduli spaces are central to recent calculations of
previously unknown Betti numbers in the cohomology of ${\cal M}_{g,n}$, and in
calculations of the cohomology of the Kontsevich graph complex, relevant to
certain problems in number theory. Brown:2021umn ; borinsky2020euler ;
andersen2021kontsevich It remains to be seen what importance these topological
facts about tropical moduli spaces have for the new formulas in this paper,
which compute amplitudes as integrals over tropical moduli spaces.

We close by discussing in greater detail some particularly interesting physics
questions to explore using the results in this paper.

###  15.1 Large $n$ limits

Perhaps the most striking fact about the curve integral formulas is that an
integral over a $\sim n$ dimensional space, built out of $\sim n^{2}$ simple
_headlight functions_ , each with no more than $\sim n$ terms, magically
produces the amplitude: which is naively the sum over $\sim 4^{n}$ Feynman
diagrams. This suggests that the curve integral can be used to define a
sensible large $n$ limit of amplitudes. Large $n$ limits have been studied for
certain special cases, Voloshin:1992mz ; Douglas:1989ve and processes with
multi-state emission of many particles may be important in phenomenomology.

Clearly, the external kinematics should be chosen carefully for the large $n$
limit to be well-defined. For instance, we can pick a smooth curve $C$ in some
$D$ dimensional space, and at any $n$, define our canonical momentum polygons
by picking a collection of ordered points on $C$; the density of points
increases as $n\to\infty$, and one should expect the final amplitude to just
be a function of $C$ itself. Another option is to think of the $X_{ij}$ as
evaluating some smooth function $X(u,v)$ on the “kinematic spacetime” on the
discrete lattice associated with the $n$ point problem. We can begin with the
simplest situation where $X(u,v)=X_{0}=X_{ij}$ is just a constant. In this
case, working for simplicity at tree-level, the sum over all diagrams is
simply the total number of diagrams weighted by $1/X_{0}^{n-3}$. Since the
total number of diagrams scales as $4^{n}$, we have that at large $n$ the
amplitude scales as ${\cal M}_{n}\sim(4/X_{0})^{n}$. Thus ${\rm
lim}_{n\to\infty}(1/n){\rm log}{\cal M}_{n}\to\log(4/X_{0})$ has a good large
$n$ behavior, reminiscent of the large $N$ behavior in the ’t Hooft limit,
where the action is enhanced by an overall power of $N$. It is then natural to
understand ${\rm lim}_{n\to\infty}(1/n){\rm log}{\cal M}_{n}(X)$ for smooth
perturbations $X(u,v)=X_{0}+\delta X(u,v)$. In both these scenarios for taking
the large $n$ limit of kinematics, we expect that curve integral
representation will also have a “smooth” limit; with the integral of a
piecewise linear action on an $(n-3)$ dimensional space asymptoting to a sort
one-dimensional path integral with an action given by an integral over
tropical functionals, and one can hope that critical points of this unusual
“tropical action” will determine the leading behavior of the amplitude in this
limit.

###  15.2 Large $L$ limits

In this paper we have focused on understanding the large $n$ limit of
amplitudes. This was made possible by choosing a particularly simple reference
triangulation. This suggests a similar strategy for understanding large $L$
limits as well. For instance in the planar limit, we can work with an
especially simple fat graph for a tadpole at $L$-loops, consisting of a string
of $L$ bubbles. The enumeration of all the relevant curves that survive on the
support on the Mirzakhani kernel, together with the computation of the
corresponding matrices, can be carried out recursively in a straightforward
way. It would be interesting to examine the behavior of the amplitudes at
large $L$ in the ’t Hooft limit.

In a similar vein, we can simply count all planar vacuum diagrams; this is
tantamount to simply setting all the $X$ variables to one, with no loop
integrations. Diagram counting is the same thing as studying the tr$\phi^{3}$
_matrix model_. This subject is usually studied using the techniques of random
matrix theory, with the the famous and rich connections to two dimensional
quantum gravity and string theory revealed in double-scaling limits.
Gross:1989ni ; Gross:1989vs ; Saad:2019lba These methods are however tailored
to matrix integrals and do not straightforwardly generalize to field theory
and non-trivial loop integrals. With a judicious choice of representative fat
graphs for general vacuum diagrams, the curve integral formalism yields a
different way of counting diagrams. This should not recover the results from
the matrix model literature, but suggests a path to generalize those results
to higher dimensions and to compute subleading terms in $1/L$.

###  15.3 Renormalization

One of the most important aspects of fundamental physics that has not yet been
properly understood from the on-shell amplitude perspective is the
exponentiation of UV divergences and the Wilsonian renormalization group. The
reason appears obvious: the Wilsonian viewpoint is maximally off-shell,
integrating out models of local quantum fields. Moreover, already when
studying individual Feynman integrals, it can be intricate to identify and
remove those divergences which need to be canceled. In this vein, tropical
methods have recently proved useful. arkanimizerahillman ;
hillman2023subtraction

The curve integral formalism offers a fresh perspective on this physics and is
expected to manifest renormalization and renormalizability in a new way. Here
again, the UV divergences depend on $n$ and $L$ in a way which suggests
treating $n$ and $L$ independently, involving separate physics. The
logarithmic UV divergences at $n$ points should be associated with specific
$n$-dependent factors, reflecting the fact they can be reabsorbed into a
redefinition of coupling constants. Whereas, the $L$-dependence must reflect
the summation of leading and subleading UV log-divergences associated with the
RG. It is plausible that the exponentiation of UV divergences is naturally
reflected in the recursive structure of the fan itself.

A hint of how renormalization is encoded in the curve integral picture can be
seen for the 1-loop planar amplitudes discussed in Sections 4 and 7. In
$D=2d-\epsilon$ dimensions, these amplitudes have $1/\epsilon$ UV divergences.
The divergences are conventionally removed by redefinition of the propagators
or vertices in the Lagrangian, with the exact correction required depending on
the dimension. But the divergences can also be removed directly inside the
curve integral, which has the form (after loop integration)

${\cal A}=\int\limits_{\sum t_{i}\leq 0}d^{n}t\,F,$ (242)

for some $F(t)$. Every cone in the integration region of this curve integral
corresponds to some fatgraph $\sigma$. The integral over a cone will give a UV
divergence if (and only if) $\sigma$ contains a divergent subgraph, $\gamma$.
By Euler’s relation, if $\gamma$ has loop order $L_{\gamma}$ and $n_{\gamma}$
external lines, then the superficial divergence of $\gamma$ is

$d_{\gamma}=n_{\gamma}+3(L_{\gamma}-1)-\frac{1}{2}LD,$ (243)

and it gives a divergence if $d_{\gamma}<0$. We can remove these divergences
from the curve integral by defining tropical functions, $\mathcal{R}_{D}(y)$,
such that

$\mathcal{R}_{D}|_{\sigma}=\left\\{\begin{matrix}0&\qquad\text{if a subgraph
of $\sigma$ has $d_{\gamma}<0$}\\\ 1&\qquad\text{else}\end{matrix}\right.$
(244)

Then define a _regularized amplitude_ by

${\cal A}^{\text{reg}}_{\text{1-loop}}=\int\limits_{\sum t_{i}\leq
0}d^{n}t\,{\cal R}_{D}\,F.$ (245)

The exact form of ${\cal R}_{D}$ depends on the dimension, $D$.

For instance, in $D=2+\epsilon$ dimensions, the divergent subgraphs are 1-loop
tadpoles, which have $d_{\gamma}=-\epsilon/2<0$. In this case, one option for
${\cal R}_{D}$ turns out to be

$\mathcal{R}_{2}=\max\left(0,1-\sum_{i}g_{ii}^{a}\partial_{t_{a}}\alpha_{C_{ii}}(t)\right),$
(246)

where the sum is over all _tadpole propagator_ curves, $C_{ii}$, and ${\bf
g}_{ii}$ is the $g$-vector of $C_{ii}$. Similar functions can be defined for
higher dimensions.

It will be fascinating to find generalizations of this to all loops, such that
UV divergences are governed by tropical functions on the Feynman fan, to
better understand renormalization and renormalizability from an on-shell
perspective.

## Acknowledgments

NAH is supported by the DOE under grant DE-SC0009988; further support was made
possible by the Carl B. Feinberg cross-disciplinary program in innovation at
the IAS. PGP is supported by ANR grant CHARMS (ANR-19-CE40-0017) and by the
Institut Universitaire de France (IUF). HF is supported by Merton College,
Oxford, and received additional support from ERC grant GALOP (ID: 724638). GS
is supported by the European Union’s Horizon 2020 research and innovation
programs Novel structures in scattering amplitudes (No. 725110) of Johannes
Henn. HT is supported by NSERC Discovery Grant RGPIN-2022-03960 and the Canada
Research Chairs program, grant number CRC-2021-00120.

##  Appendix A Curve Glossary

For reference, and as a proof of principle, we record here all the curve paths
needed to compute every amplitude up to 2-loops. In practice, these lists are
most easily obtained by computer.

###  A.1 1-loop Non-planar 2-point

The Mirzakhani kernel in Section 8 takes the form

${\cal K}=\frac{\alpha_{12}^{0}}{\rho}.$ (247)

So the set of curves, ${\cal S}$, contributing to the curve integral for the
amplitude are those which are compatible with $C_{12}^{0}$. There are five
such curves on the tadpole graph.

One way to enumerate these curves is to cut the tadpole graph along
$C_{12}^{0}$, to produce a 5-point tree graph. The five curves we need are
then the five curves on three 5-point tree graph.

Of the 5 curves in ${\cal S}$, 3 of them carry loop momentum and are given by

$\displaystyle C_{12}^{-1}$ $\displaystyle=1Re_{2}L2,$ (248) $\displaystyle
C_{12}^{0}$ $\displaystyle=1Le_{1}R2,$ (249) $\displaystyle C_{12}^{1}$
$\displaystyle=1Le_{1}Le_{2}Re_{1}R2.$ (250)

In addition to these, there are the two curves that follow the boundaries of
the fatgraph, and which contribute to ${\cal Z}$. These boundary curves have
paths

$\displaystyle C_{11}$ $\displaystyle=1Le_{1}Le_{2}L1,$ (251) $\displaystyle
C_{22}$ $\displaystyle=2Le_{1}Le_{2}L2.$ (252)

Note that it is straightforward to use these paths to obtain the curve
matrices for these five curves by matrix multiplication. One finds

$\displaystyle M_{11}=M_{22}$ $\displaystyle=\begin{bmatrix}1&0\\\
1+y_{1}+y_{1}y_{2}&y_{1}y_{2}\end{bmatrix}$ (253) $\displaystyle M^{-1}_{12}$
$\displaystyle=\begin{bmatrix}1+y_{2}&y_{2}\\\ y_{2}&y_{2}\end{bmatrix}$ (254)
$\displaystyle M^{0}_{12}$ $\displaystyle=\begin{bmatrix}1&1\\\
1&1+y_{1}\end{bmatrix}$ (255) $\displaystyle M^{1}_{12}$
$\displaystyle=\begin{bmatrix}1&1+y_{1}\\\
1+y_{1}&1+2y_{1}+y_{1}^{2}+y_{1}^{2}y_{2}\end{bmatrix}$ (256)

For the rest of this Appendix, we record only the paths of the curves.

###  A.2 2-loop Planar Tadpole

The following is the finite set $\mathcal{S}$ of 16 non-boundary curves
compatible with the Mirzakhani kernel ${\cal K}$ in Section 11:

$\displaystyle 0LwR(xL)^{\infty}$ $\displaystyle 0RzR(yL)^{\infty}$
$\displaystyle 0LwLxLwR0$ $\displaystyle 0RzLyLzL0$ $\displaystyle 0RzRyRzL0$
$\displaystyle(Rx)^{\infty}LwLzR(yL)^{\infty}$
$\displaystyle(Ry)^{\infty}LzRwR(xL)^{\infty}$ $\displaystyle
0LwLxLwLzR(yL)^{\infty}$ $\displaystyle 0RzRyRzRwR(xL)^{\infty}$
$\displaystyle(Ry)^{\infty}LzRwRxRwR0$
$\displaystyle(Rx)^{\infty}LwLzLyLzRwR(xL)^{\infty}$
$\displaystyle(Ry)^{\infty}LzRwLxLwLzR(yL)^{\infty}$ $\displaystyle
0LwLxLwLzLyLzRwR(xL)^{\infty}$ $\displaystyle 0RzRyRzRwRxRwLzR(yL)^{\infty}$
$\displaystyle 0LwLxLwLzLyLzRwRxRwR0$ $\displaystyle 0RzRyRzRwLxLwLzLyLzL0$

The momentum of a curve $C_{AB}$ ($A,B=0,a,b$) connecting trace-factors $A$
and $B$ is given in dual variables by $x_{B}^{\mu}-x_{A}^{\mu}$. $x_{a}^{\mu}$
and $x_{b}^{\mu}$ are the loop momentum variables. So it follows that, of
these 16 curves, 9 of them carry loop momentum and so contribute to
$\mathcal{K}$, $\mathcal{U}$ and ${\cal F}_{0}$. The remaining 7 contribute
only to ${\cal Z}$.

In addition, the tadpole graph has one boundary curve, $C_{00}$, which
contributes to $\mathcal{Z}$, and has the path

$\displaystyle C_{00}$ $\displaystyle=0RzRyRzRwRxRwR0$

###  A.3 2-loop Double Trace

The following is the finite set $\mathcal{S}$ of 21 non-boundary curves
compatible with the Mirzakhani kernel ${\cal K}$ in Section 14:

$\displaystyle C_{12}^{01}$ $\displaystyle=Ly_{2}Rx_{1}L$ $\displaystyle
C_{12}^{10}$ $\displaystyle=Ly_{1}Rx_{2}L$ $\displaystyle C_{12}^{00}$
$\displaystyle=Rx_{2}RzLx_{1}L$ $\displaystyle C_{12}^{11}$
$\displaystyle=Ly_{1}LzRy_{2}R$ $\displaystyle C_{11}^{0}$
$\displaystyle=Ly_{1}Rx_{2}Ry_{2}Rx_{1}L$ $\displaystyle C_{22}^{0}$
$\displaystyle=Ly_{2}Rx_{1}Ry_{1}Rx_{2}L$ $\displaystyle C_{10}^{0}$
$\displaystyle=R(x_{1}Ly_{2}Lx_{2}Ly_{1})^{\infty}$ $\displaystyle C_{20}^{0}$
$\displaystyle=R(x_{2}Ly_{1}Lx_{1}Ly_{2})^{\infty}$ $\displaystyle
C_{11}^{-1/2}$ $\displaystyle=Ly_{1}Rx_{2}Ry_{2}LzRy_{1}R$ $\displaystyle
C_{11}^{1/2}$ $\displaystyle=Rx_{1}Ly_{2}Lx_{2}RzLx_{1}L$ $\displaystyle
C_{22}^{-1/2}$ $\displaystyle=Ly_{2}Rx_{1}Ry_{1}LzRy_{2}R$ $\displaystyle
C_{22}^{1/2}$ $\displaystyle=Rx_{2}Ly_{1}Lx_{1}RzLx_{2}L$ $\displaystyle
C_{12}^{1-1}$ $\displaystyle=Ly_{1}Rx_{2}Ry_{2}LzLx_{2}L$ $\displaystyle
C_{12}^{-11}$ $\displaystyle=Ly_{2}Rx_{1}Ry_{1}LzLx_{1}L$ $\displaystyle
C_{12}^{-10}$ $\displaystyle=Rx_{1}RzRy_{1}Lx_{1}RzLx_{2}L$ $\displaystyle
C_{12}^{0-1}$ $\displaystyle=Rx_{2}RzRy_{2}Lx_{2}RzLx_{1}L$ $\displaystyle
C_{10}^{-1}$ $\displaystyle=Ly_{1}LzR(y_{2}Lx_{2}Ly_{1}Lx_{1})^{\infty}$
$\displaystyle C_{10}^{1}$
$\displaystyle=Rx_{1}RzR(y_{1}Lx_{1}Ly_{2}Lx_{2})^{\infty}$ $\displaystyle
C_{20}^{-1}$ $\displaystyle=Ly_{2}LzR(y_{1}Lx_{1}Ly_{2}Lx_{2})^{\infty}$
$\displaystyle C_{20}^{1}$
$\displaystyle=Rx_{2}RzR(y_{2}Lx_{2}Ly_{1}Lx_{1})^{\infty}$ $\displaystyle
C_{00}$
$\displaystyle=(x_{2}Ry_{2}Rx_{1}Ry_{1})^{\infty}LzR(y_{2}Lx_{2}Ly_{1}Lx_{1})^{\infty}$

The only curves that carry loop momentum are those curves $C_{AB}$ connecting
distinct trace-factors, $A$ and $B$.

It follows that, of these 21 curves, 14 of them carry loop momentum and so
contribute to $\mathcal{K}$, $\mathcal{U}$ and ${\cal F}_{0}$. The remaining 7
contribute only to ${\cal Z}$.

In addition, the tadpole graph has two boundary curves, which contribute to
$\mathcal{Z}$, and have paths

$\displaystyle C_{11}$ $\displaystyle=1Rx_{1}RzRy_{1}R1$ $\displaystyle
C_{22}$ $\displaystyle=2Rx_{2}RzRy_{2}R2$

###  A.4 2-loop Triple Trace

The following is the finite set $\mathcal{S}$ of 30 non-boundary curves
compatible with the Mirzakhani kernel ${\cal K}$ in Section 12:

$\displaystyle 0LzLx_{2}R1$ $\displaystyle 1Rx_{1}LzR0$ $\displaystyle
2Ly_{1}RwL0$ $\displaystyle 2Ry_{2}LwL0$ $\displaystyle
0LzLx_{2}Lx_{1}Rx_{2}R1$ $\displaystyle 1Rx_{1}LzLwLy_{1}R2$ $\displaystyle
2Ly_{1}RwRzLx_{2}R1$ $\displaystyle 2Ry_{2}LwRzLx_{2}R1$ $\displaystyle
2Ry_{2}LwRzRx_{1}L1$ $\displaystyle 2Ry_{2}Ry_{1}Ly_{2}LwL0$ $\displaystyle
0LzLx_{2}Lx_{1}LzLwLy_{1}R2$ $\displaystyle 0LzLx_{2}Lx_{1}LzLwRy_{2}L2$
$\displaystyle 1Lx_{2}RzLwLy_{1}Ly_{2}LwL0$ $\displaystyle
1Lx_{2}RzLwLy_{1}Ly_{2}Ry_{1}R2$ $\displaystyle 1Rx_{1}LzLwLy_{1}Ly_{2}LwL0$
$\displaystyle 1Rx_{1}LzLwLy_{1}Ly_{2}Ry_{1}R2$ $\displaystyle
1Rx_{1}Rx_{2}Lx_{1}LzLwLy_{1}R2$ $\displaystyle
1Rx_{1}Rx_{2}Lx_{1}LzLwRy_{2}L2$ $\displaystyle
0LzLx_{2}Lx_{1}LzLwLy_{1}Ly_{2}Ry_{1}R2$ $\displaystyle
1Lx_{2}RzLwLy_{1}Ly_{2}LwRzLx_{2}R1$ $\displaystyle
1Rx_{1}LzLwLy_{1}Ly_{2}LwRzLx_{2}R1$ $\displaystyle
1Rx_{1}LzLwLy_{1}Ly_{2}LwRzRx_{1}L1$ $\displaystyle
1Rx_{1}Rx_{2}Lx_{1}LzLwLy_{1}Ly_{2}LwL0$ $\displaystyle
2Ly_{1}RwRzLx_{2}Lx_{1}LzLwLy_{1}R2$ $\displaystyle
2Ry_{2}LwRzLx_{2}Lx_{1}LzLwLy_{1}R2$ $\displaystyle
2Ry_{2}LwRzLx_{2}Lx_{1}LzLwRy_{2}L2$ $\displaystyle
0LzLx_{2}Lx_{1}LzLwLy_{1}Ly_{2}LwRzLx_{2}R1$ $\displaystyle
0LzLx_{2}Lx_{1}LzLwLy_{1}Ly_{2}LwRzRx_{1}L1$ $\displaystyle
2Ly_{1}RwRzLx_{2}Lx_{1}LzLwLy_{1}Ly_{2}LwL0$ $\displaystyle
2Ry_{2}LwRzLx_{2}Lx_{1}LzLwLy_{1}Ly_{2}LwL0$

The only curves that carry loop momentum are those curves $C_{AB}$ connecting
distinct trace-factors, $A$ and $B$.

It follows that, of these 30 curves, 24 of them carry loop momentum and so
contribute to $\mathcal{K}$, $\mathcal{U}$ and ${\cal F}_{0}$. The remaining 6
contribute only to ${\cal Z}$.

In addition, the tadpole graph has three boundary curves, which contribute to
$\mathcal{Z}$, and have paths

$\displaystyle C_{11}$ $\displaystyle=1Rx_{1}Rx_{2}R1$ $\displaystyle C_{22}$
$\displaystyle=2Ry_{2}Ry_{1}R2$ $\displaystyle C_{00}$
$\displaystyle=0RwRy_{2}Ry_{1}RwRzRx_{1}Rx_{2}RzR0$

###  A.5 2-loop Genus One Tadpole

The following is the finite set $\mathcal{S}$ of 12 non-boundary curves
compatible with the Mirzakhani kernel ${\cal K}$ in Section 13:

$\displaystyle C^{0/1;0}$ $\displaystyle=0Ly_{2}Ry_{4}Ly_{1}L0$ $\displaystyle
C^{1/0;1/2}$ $\displaystyle=0Ly_{2}Ly_{3}Ry_{1}L0$ $\displaystyle C^{1/-1;0}$
$\displaystyle=0Ry_{1}Ry_{4}Ry_{3}Ry_{1}L0$ $\displaystyle C^{1/-1;1/2}$
$\displaystyle=0Ly_{2}Ly_{3}Ly_{4}Ly_{2}R0$ $\displaystyle C^{1/1;1/2}$
$\displaystyle=0Ly_{2}Ly_{3}Ry_{1}Ry_{2}Ry_{4}Ly_{1}L0$ $\displaystyle
C^{0/1;1/2}$ $\displaystyle=0Ly_{2}Ly_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0$
$\displaystyle C^{1/0;0}$
$\displaystyle=0Ry_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ly_{1}L0$ $\displaystyle
C^{0/1;-1/2}$
$\displaystyle=0Ry_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ly_{4}Ly_{1}L0$
$\displaystyle C^{1/0;1}$
$\displaystyle=0Ly_{2}Ly_{3}Ly_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0$
$\displaystyle C^{1/1;0}$
$\displaystyle=0Ry_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ly_{1}Ry_{2}Ry_{4}Ly_{1}L0$
$\displaystyle C^{1/1;1}$
$\displaystyle=0Ly_{2}Ly_{3}Ry_{1}Ry_{2}Ly_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0$
$\displaystyle C^{1/-1;1}$
$\displaystyle=0Ly_{2}Ly_{3}Ly_{4}Ly_{2}Ly_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0$
$\displaystyle C^{0/1;1}$
$\displaystyle=0Ly_{2}Ly_{3}Ly_{4}Ly_{2}Ly_{1}Ly_{3}Ly_{4}Ly_{1}Ry_{2}Ly_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0.$

All 12 of these curves carry loop momentum and so contribute to $\mathcal{K}$,
$\mathcal{U}$ and ${\cal F}_{0}$.

In addition, the tadpole graph has one boundary curve, which contributes to
$\mathcal{Z}$, and has path

$\displaystyle C_{00}$
$\displaystyle=0Ry_{1}Ry_{4}Ry_{3}Ry_{1}Ry_{2}Ry_{4}Ry_{3}Ry_{2}R0.$

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