\section{Introduction}

One of the key outstanding questions in the modern S-matrix program is understanding the renormalization group (RG) from an on-shell point of view. The Wilsonian picture is fundamentally Euclidean, and referring to ``coarse graining" in position space, or ``integrating out high momentum modes" in momentum space, crucially relies on concepts far removed from the on-shell philosophy. Alternatively, much of the technical content of the RG was earlier realized using the seemingly more complicated ideas of  Bogoliubov--Parasiuk--Hepp--Zimmermann \cite{10.1007/BF02392399,Hepp1966,Zimmermann1969} on ultraviolet (UV) divergences, which have been given a deeper Hopf-algebraic understanding \cite{Connes:1998qv,Connes:1999yr,Connes:2000fe} exhibiting a closer connection to on-shell ideas.  More recently,  analogous RG-like structures were found for infrared (IR) divergences in the context of QCD \cite{Mueller:1979ih,Collins:1980ih,Sen:1981sd,Korchemsky:1988si,Sterman:2002qn,Gardi:2009qi,Agarwal:2021ais} and later soft-collinear effective theory, see, e.g., \cite{Collins:1989gx,Becher:2009qa,StewartLectures,Becher:2014oda,Ma:2019hjq,Beekveldt:2020kzk}.

It is then natural to ask whether these perspectives on UV and IR divergences can be unified into a single picture, one which can connect the emerging geometric underpinnings of scattering amplitudes to the physics of UV divergences and the RG. In this note, we take a step in this direction, by describing a beautiful class of polytopes associated with any Feynman graph, whose facet structure concisely captures and calculates both the leading UV and IR divergences in a unified way. 

Our starting point is the treatment of Feynman integrals in dimensional regularization from a worldline perspective, where instead of integrating over the loop momenta, the only variables are the Schwinger proper times $\alpha=e^\tau$ measuring lengths of each propagator. From this point of view, the limits $\tau \to \mp\infty$ correspond to shrinking and expanding edges, which are naturally associated to the UV and IR singularities respectively. They are thus put on the same footing and the unified analysis of all divergences boils down to examining the asymptotic behavior of the integrand at infinity. Mathematicians have studied this subject, known as \emph{tropical geometry}, for decades, see, e.g., \cite{maclagan2015introduction} for an introduction. We will explore the physical meaning and consequences of this geometry in our setting.

\begin{figure}
    \centering
    \includegraphics[]{figpdfs/fig1-figure0.pdf}
    \caption{\label{fig:scalings}Example scalings of Schwinger parameter responsible for: (a) collinear, (b) soft/collinear, (c) power-law IR, (d) nested soft/collinear, and (e) nested UV singularities in four dimensions. In each case the divergence comes from shrinking/expanding the relevant worldline edge lengths as the product of the variables $\rho,\sigma, \ldots$ to the indicated powers, as they all scale  $ \to \infty$.  Massless and massive particles are denoted with single and doubled lines respectively.}
\end{figure}

As we will see, all of the elaborate structure of UV and IR divergences of a Feynman integral associated with a graph $G$ is captured by the \emph{Feynman polytope} $\mathbf{S}_G$. It can be constructed out of the \emph{Symanzik polytopes} (previously explored in \cite{brown2011multiple,Brown:2015fyf,Schultka:2018nrs,Borinsky:2020rqs,Mizera:2021icv} for IR-safe kinematics) defined fully by the combinatorics of $G$ and the kinematic data of external momenta and particle masses. 

Elements of polyhedral geometry of Feynman integrals previously appeared in sector decomposition \cite{Binoth:2000ps,Binoth:2003ak,Bogner:2007cr,Kaneko:2009qx,Kaneko:2010kj,Borowka:2015mxa,Schlenk:2016epj,Heinrich:2021dbf}, expansion by regions \cite{Pak:2010pt,Ananthanarayan:2018tog,Semenova:2018cwy,Ananthanarayan:2020fhl,Tellander:2021xdz}, and blowups of integration domains \cite{Bloch2006,Brown:2015fyf}.\footnote{Tropical geometry has also recently emerged as a useful tool in studying other, complementary, aspects of scattering amplitudes, see, e.g., \cite{Tourkine:2013rda,Panzer:2019yxl,Cachazo:2019ngv,Drummond:2020kqg,Arkani-Hamed:2020cig,Drummond:2019cxm,Arkani-Hamed:2019mrd,Arkani-Hamed:2019plo,Arkani-Hamed:2020tuz,Chicherin:2020umh,Cachazo:2020wgu,He:2021non}.}
Sector decomposition is a powerful general algorithm for studying singularities of generalized hypergeometric function (see, e.g., \cite{Mastrolia:2018uzb,delaCruz:2019skx,Klausen:2019hrg,Feng:2019bdx,Abreu:2019wzk,Tellander:2021xdz,Mizera:2021icv} for recent progress).  But as such, its application to the study of Feynman diagrams does not make use of any special properties that one might expect would arise for the integrals which arise from physics, meaning one has to laboriously compute $\mathbf{S}_G$ for every $G$ from scratch. One of our central results is that the structure of the integrals handed to us by physics, and the corresponding polyhedral geometry, is indeed very special, allowing us to give an analytic solution to this problem for any scalar Feynman diagram. 

The polytopes begin their lives described as the convex hull of the exponent vectors of Symanzik polynomials. We show that they also have a beautifully simple \emph{facet description}, cut out by linear inequalities. It is this dual definition that makes it possible to understand and compute the leading UV and IR divergences.

An overarching theme in the emerging connection between combinatorial geometry and scattering amplitudes has been the realization that the singularities associated with ``factorization" of amplitudes are reflected in an analogous ``factorization" seen in the facet structure of the geometries. This is also seen in the Symanzik polytopes and ${\bf S}_G$, whose boundary structure transparently encodes the combinatorics of shrinking subgraphs of $G$, thereby precisely encoding the ``Russian doll" hierarchical separation of worldline edge-lengths in $G$ that have the potential to generate divergences. 

The definition of ${\bf S}_G$ as cut out by inequalities further endows each face of $\mathbf{S}_G$ with a number: its UV/IR degree of divergence. The way these faces fit into the polytope dictates how the individual divergences are related to one another. In the UV, this prescription recovers the standard picture of nested and overlapping divergences, in agreement with \cite{Brown:2011pj,Brown:2015fyf}. In the IR, it reveals novel patterns of divergences, including the familiar soft/collinear divergences together with their intricate generalizations. Examples are shown in Fig.~\ref{fig:scalings}.

As an extension of \cite{Arkani-Hamed:2019mrd}, we explain how the leading divergent term of every Feynman integral can be computed in terms of volumes of dual cones. In this formalism, the divergence is viewed as an additional redundancy of the integrand in certain directions at infinity. We give a prescription for computing the coefficient of this divergence as a procedure of ``modding out'' by the enhanced symmetry. We apply this tool to various examples, including the rectangular fishnet diagrams, resulting in a formula for the leading IR divergence at any loop order.

Our goal in this note is to take a geodesic path to the central constructions, and illustrate their utility and power with a few examples. A more leisurely description of these ideas as well as further generalizations and examples will appear in \cite{AHHM}. 

\section{Schwinger Parametrization}
A scalar Feynman integral in parameter space corresponding to a graph $G$ is given by 
\begin{equation}\label{eq:I}
    \mathcal{I}_G = \Gamma(d) \int \frac{\mathrm{d}^\mathrm{E} \alpha}{\text{GL}(1)} \frac{1}{\mathcal{U}^{\mathrm{D}/2-d}\mathcal{F}^{d}},
\end{equation}
where $d = \mathrm{E}-\mathrm{L}\mathrm{D}/2$ is the superficial degree of divergence for a scalar diagram $G$ with $\mathrm{E}$ edges and $\mathrm{L}$ loops in dimension $\mathrm{D}$.  The functions in the integral are the Symanzik polynomials which admit a graphical expansion in terms of spanning trees $T$ and $2$-trees $T'$
\begin{equation}
    \mathcal{U} = \sum_T \prod\limits_{e \notin T}\alpha_e, \quad
    \mathcal{F} = \sum_{T'} p_{T'}^2 \!\!\prod\limits_{e \notin T'}\alpha_e - \mathcal{U} \sum_{e=1}^{\mathrm{E}} m_e^2\alpha_e,
\end{equation}
where $p_{T'}^\mu$ is the total momentum flowing across the $2$-tree, see, e.g., \cite{nakanishi1971graph}. Modding out by overall-scale $\mathrm{GL}(1)$ redundancy means we can set $\alpha_\mathrm{E}=1$. Including Feynman integrals with non-trivial numerators does not pose difficulty and will be considered in \cite{AHHM}.

Below, we will use $\gamma$ to mean a subgraph of $G$, and $G/\gamma$ to denote $G$ with each connected component of $\gamma$ contracted to a vertex. To avoid confusion, we add the subscript ${}_G$ to denote the quantity for a specific diagram $G$, e.g., $\mathrm{L}_\gamma$ denotes the number of loops in $\gamma$. 


\section{\label{sec:general}Tropical Geometry}

Feynman integrals \eqref{eq:I} fall into a general class of integrals whose properties can be understood using tropical geometry \cite{gelfand2009discriminants,Nilsson2013,berkesch2013eulermellin,Arkani-Hamed:2019mrd}. In terms of the variables $\tau_e = \log\alpha_e$, the integrand of \eqref{eq:I} can be approximated at infinity, where the UV/IR divergences come from, by $e^{\mathrm{Trop}}$ with the piecewise-linear function
\begin{equation}\label{tropdef}
\mathrm{Trop} = \tau_G + (d_G -\tfrac{\mathrm{D}}{2})\max(\mathcal{U}_G) - d_G \max (\mathcal{F}_G), 
\end{equation}
where $\tau_\gamma = \sum_{e \in \gamma}\tau_e$ and the maxima are taken over all the monomials in $\mathcal{U}_G$ and $\mathcal{F}_G$. Equivalently, one can capture the same information by the Minkowski sum/difference $\mathbf{S}_G = \mathbf{U}_G \oplus c \mathbf{F}_G$, of the Newton polytopes
\begin{equation}\label{eq:UF}
    \mathbf{U}_G = \mathrm{Newt}(\mathcal{U}_G), \qquad \mathbf{F}_G = \mathrm{Newt}(\mathcal{F}_G),
\end{equation}
which we refer to as \textit{Symanzik polytopes} and study closely in Sec.~\ref{sec:Symanzik}.
To obtain the most refined information about the regions where divergences can possibly come from, it is sufficient to consider $c>0$, see App.~\ref{app:general}.
Every \emph{ray} (boundary of domains of linearity) of $\mathrm{Trop}$ defines the normal to a facet of the Feynman polytope $\mathbf{S}_G$ and hence the two pictures can be used interchangeably.

\paragraph*{\bf Leading divergences.} Divergences come from cones (faces) generated by $\mathrm{Trop} \geqslant 0$ rays. We focus on logarithmic cases with $\mathrm{Trop} = 0$. Such $m$-dimensional cones (codimension-$m$ faces) give $\log^m$ or $1/\varepsilon^m$ UV/IR singularities. The leading divergence is controlled by cones $C_F$, or \emph{maximally-divergent faces} $F$ for which $m$ is the largest. As a refinement of the standard tropical rules \cite{Arkani-Hamed:2019mrd}, we find
\begin{equation}\label{tropvolume}
\mathcal{I}_G = \Gamma(d_G) \sum_{F} \mathrm{vol}(C_{F})\, \mathcal{J}_{F} + \ldots.
\end{equation}
It involves two ingredients: the volume of the cone $C_{F}$ (bounded by $\mathrm{Trop} \geqslant -1$) gives a numerical factor $\propto 1/\varepsilon^{m}$, while all the kinematic dependence is captured by the modded integral
\begin{equation}
\mathcal{J}_F =  \int \frac{\mathrm{d}^\mathrm{E} \alpha}{\text{GL}(1)^{m+1}}\, \frac{1}{\mathcal{U}_G^{\mathrm{D}/2-d}\mathcal{F}_G^{d}} \bigg|_{C_{F}^\perp}.
\end{equation}
Here we use the tropical approximation for the integrand in the direction $C_F^\perp$ orthogonal to the cone $C_F$, which by definition has an enhanced $\mathrm{GL}(1)^m$ symmetry. So extracting the coefficient of this divergence is precisely modding out by the extra redundancy!
Strictly speaking, the volume and the modded integral are not uniquely defined, but their product is, as illustrated on examples in App.~\ref{app:further-examples}.





\section{\label{sec:Symanzik}Symanzik Polytopes}

The polytopes $\mathbf{U}_G$ and $\mathbf{F}_G$ have been defined as a convex hull of vertices via the Newton polytopes \eqref{eq:UF}.  But the singularities of Feynman integrals are instead tied to their \emph{facet} description, cutting out the polytopes with linear inequalities. 

Now, there is a systematic algorithm for finding the facets of an $n$-dimensional polytope given its vertices: checking whether a collection of vertices is a facet by asking whether the remaining vertices are all on the same side of the plane formed by this set. But clearly, the amount of work to do this grows without bound as the number of vertices and/or the dimension of the polytope become large. 

Thus, what it means to ``understand" infinite families of polytopes, is to be able to analytically describe them in {\it both} the convex hull and facet descriptions. As we now show, the Symanzik polytopes can be ``understood" in this sense, and can be given an extremely simple facet presentation. 


\paragraph*{\bf Inequalities.}
$\mathbf{U}_G$ is the Newton polytope of a homogeneous degree $\mathrm{L}_G$ polynomial and therefore is naturally defined on the hyperplane $a_G = \mathrm{L}_G$, where $\vec a$ is a point in the Newton polytope and $a_\gamma = \sum_{e \in \gamma}a_e$.
On this hyperplane, $\mathbf{U}_G$ is defined by
\begin{equation}
\label{udefn}
    \mathbf{U}_G\!:  \quad a_G = \mathrm{L}_G \quad\mathrm{and}\quad a_\gamma \geqslant \mathrm{L}_\gamma \quad\text{for all } \gamma\text{'s}.
\end{equation}
Analogously, $\mathbf{F}_G$ is cut out by
\begin{align}
\label{fdefn}
    \mathbf{F}_G\!:\quad  a_G = \mathrm{L}_G{+}1  \quad\text{and}\;\, &\begin{cases}
    a_\gamma \geqslant \mathrm{L}_\gamma & \text{if}\quad\mathcal{F}_{G/\gamma} \neq 0,
    \\
    a_\gamma \geqslant \mathrm{L}_\gamma{+}1 &  \text{if}\quad\mathcal{F}_{G/\gamma} = 0,
    \end{cases}\nonumber\\
    & \qquad\qquad\qquad\;\text{for all } \gamma\text{'s}.
\end{align}
Graphs $G/\gamma$ with $\mathcal{F}_{G/\gamma} = 0$ are called \emph{scaleless} and define inequalities crucial for the study of IR divergences. In contrast, for generic kinematics (say, all particles massive), the definition of $\mathbf{F}_G$ only differs from that of $\mathbf{U}_G$ by the choice of hyperplane and indeed is the Minkowski sum $\mathbf{F}_G = \mathbf{U}_G \oplus {\bf\Delta}_{\mathrm{E}-1}$, where ${\bf\Delta}_{\mathrm{E}-1}$ is the $(\mathrm{E}{-}1)$-dimensional simplex.


\paragraph*{\bf Boundary structure.}

Saturating the above inequalities defines faces of $\mathbf{U}_G$ and $\mathbf{F}_G$ with the following factorization properties:
\begin{equation}
\partial\mathbf{U}_G \supset \mathbf{U}_\gamma \times \mathbf{U}_{G/\gamma} \quad\text{on}\quad a_\gamma = \mathrm{L}_\gamma
\end{equation}
and
\begin{align}\label{eq:del-F}
    \partial\mathbf{F}_G \supset
    \begin{cases}
    \mathbf{U}_\gamma \times \mathbf{F}_{G/\gamma} & \text{on}\quad a_\gamma = \mathrm{L}_\gamma,\\
    \mathbf{F}_\gamma \times \mathbf{U}_{G/\gamma} & 
    \text{on}\quad a_\gamma = \mathrm{L}_\gamma{+}1.
    \end{cases}
\end{align}
It follows that $\mathbf{S}_G$ factorizes in the same way as $\mathbf{F}_G$. Furthermore, we claim that \emph{facets} (codimension-$1$ faces) of $\mathbf{S}_G$ are those faces of \eqref{eq:del-F} for which every instance of $\mathbf{U}_{\gamma'}$ in the factorization above has $\gamma'$ 1VI (recall that a diagram is called 1VI if it cannot be disconnected by removal of a single vertex) and, in the $\mathcal{F}_{G/\gamma} = 0$ case, no subgraph $\tilde \gamma \subset \gamma$ already has $\mathcal{F}_{G/\tilde \gamma} = 0$ and $\mathrm{L}_{\tilde\gamma} = \mathrm{L}_\gamma$.



\paragraph*{\bf Compatibility of facets.} Feynman integrals motivate us to study the properties of $\mathbf{S}_G$.  In particular, it is natural to ask if two facets labelled by $\gamma_1$ and $\gamma_2$ are compatible, i.e., they meet.  The following identity
\begin{align}
    a_{\gamma_1}+a_{\gamma_2} = a_{\gamma_1 \cup \gamma_2}+a_{\gamma_1 \cap \gamma_2}
\end{align}
furnishes a necessary condition for compatibility:
\begin{multline}
\label{compatibility}
     (1+c)(\mathrm{L}_{\gamma_1}+\mathrm{L}_{\gamma_2}) +(c)_{\gamma_1}+(c)_{\gamma_2} \geqslant\\
     (1+c)(\mathrm{L}_{\gamma_1\cup\gamma_2}+\mathrm{L}_{\gamma_1\cap\gamma_2}) +(c)_{\gamma_1\cup\gamma_2}+(c)_{\gamma_1\cap\gamma_2}.
\end{multline}
The $c$'s in the parentheses are only included when the $\mathbf{F}_G$ facet is of $a_\gamma \geqslant \mathrm{L}_\gamma{+}1$ type.

Note that the above condition means Feynman polytopes are in general \emph{not} generalized permutohedra, in contrast with \cite{Brown:2015fyf,Schultka:2018nrs,Borinsky:2020rqs}.


\section{Tropicalization}

Recall that rays in the Trop space are dual to facets of the corresponding polytope.
Given the explicit form of the $\text{Trop}$ function \eqref{tropdef}, one can simply plug in the ray corresponding to a given inequality and get the associated value of $\text{Trop}$ along that ray.  However, equipped with the explicit \textit{constants} in the inequalities for $\mathbf{U}_G$ and $\mathbf{F}_G$, we already know the value of Trop along the facet rays: they are given by the constant in the inequality.

\paragraph*{\bf Degrees of divergence.} The two polytopes are defined by intersecting the same inequalities with different subspaces.  Because of projective invariance, we can uniformly shift the points on $\mathbf{U}_G$ and $\mathbf{F}_G$, which has no effect on the $\mathrm{Trop}$ function, with
\begin{align}
    \label{hyperplaneshifts}
    \vec a \to  \vec a- \tfrac{\mathrm{L}_G}{\mathrm{E}_G}\vec 1_G, \hspace{7mm}  \vec a \to  \vec a-\tfrac{\mathrm{L}_G+1}{\mathrm{E}_G}\vec 1_G,
\end{align}
respectively,
where $\vec 1_G$ is a vector with all entries equal to $1$. At the level of the integrand, this is equivalent to distributing the overall $\alpha_1\alpha_2\cdots \alpha_\mathrm{E}$ between the $\mathcal{U}_G$ and $\mathcal{F}_G$.  Note that no change of integration variables is implied here, but the inequalities in \eqref{udefn} and \eqref{fdefn} are shifted.
The resulting inequalities for $\mathbf{U}_G$ are:
\begin{align}
    \label{shiftedineqsU}
    a_\gamma  \geqslant \frac{\mathrm{L}_\gamma \mathrm{E}_G-\mathrm{L}_G \mathrm{E}_\gamma}{\mathrm{E}_G}
\end{align}
and for $\mathbf{F}_G$:
\begin{align}
\label{shiftedineqsF}
a_\gamma \geqslant
   \begin{cases}
   \frac{\mathrm{L}_\gamma \mathrm{E}_G-(\mathrm{L}_G+1)\mathrm{E}_\gamma}{\mathrm{E}_G}& \text{if}\quad\mathcal{F}_{G/\gamma} \neq 0,\\
    \frac{(\mathrm{L}_\gamma+1)\mathrm{E}_G-(\mathrm{L}_G+1)\mathrm{E}_\gamma}{\mathrm{E}_G}&\text{if}\quad\mathcal{F}_{G/\gamma} = 0,
   \end{cases}
\end{align}
for the two types of $\mathbf{F}_G$ facets.  The value of Trop along a ray is given by substituting the saturation of that ray's inequality in the maxima. Since our conventions identify the value of Trop with the constant in the outward-pointing normal inequality, we substitute the constants in \eqref{shiftedineqsU} and \eqref{shiftedineqsF} with a minus sign, see App.~\ref{app:general}. This allows us to compute the value of Trop along any extremal ray.  The two types of faces are distinguished by their inequality in $\mathbf{F}_G$ and give two cases:
\begin{align}\label{eq:Trop-gamma}
    \text{Trop} = \begin{cases}
    -d_\gamma & \text{if}\quad \mathcal{F}_{G/\gamma} \neq 0, \\
    d_{G/\gamma} & \text{if}\quad \mathcal{F}_{G/\gamma} = 0.
    \end{cases}
\end{align}
The first case is familiar and associated with superficially UV divergent subgraphs when $d_\gamma \leqslant 0$.  We will see that the second case is associated with IR divergences when $d_{G/\gamma} \geqslant 0$ and $G/\gamma$ is scaleless.

\section{Generalized Weinberg's theorem}
The above geometric picture makes Weinberg's theorem \cite{PhysRev.118.838} obvious: if all subdiagrams $\gamma \subset G$ have $d_{\gamma}>0$ (with appropriately defined $d_\gamma$ for non-scalar diagrams), the diagram $G$ is UV convergent.
With \eqref{eq:Trop-gamma}, we are now equipped to state its IR counterpart: if all subdiagrams $\gamma \subset G$ for which $G/\gamma$ is scaleless have $d_{G/\gamma} < 0$, the diagram $G$ is IR finite. Together, the two theorems give conditions for any diagram to be UV/IR finite, cf. \cite{Lowenstein:1975rg}.


\section{\label{sec:UV} Ultraviolet}
\begin{figure}[!t]
	\centering
	\includegraphics[]{figpdfs/fig71-figure0.pdf}
	\quad
    \raisebox{-.15\height}{\includegraphics[]{figpdfs/fig72-figure0.pdf}}
	\caption{\label{eyeofsauron}Two UV-divergent Feynman diagrams $G_L$ and $G_R$ considered in Sec.~\ref{sec:UV}.
	}
\end{figure}


We now provide some illustrative examples.  As a UV example, we consider the diagram $G_L$ in Fig.~\ref{eyeofsauron} (left). It contributes a leading logarithmic divergence in $\mathrm{D} = 4-2\varepsilon$ and therefore has a two-dimensional divergent Trop = 0 space (recall that one divergence comes from the overall $\Gamma(d_{G_L}) = \tfrac{1}{3\varepsilon} + \dots$).  In order to calculate the leading divergence, we first need to identify the divergent rays spanning the Trop = 0 space.  These correspond to the superficially divergent subgraphs, of which there are three: $\gamma_{1256}$, $\gamma_{3456}$, $\gamma_{56}$ where the subscripts denote the edges of the subgraph.  Next, we need to know the compatibility of these divergent rays in $\mathbf{S}_{G_L}$.  This is given by the compatibility criterion (\ref{compatibility}) and demonstrates that $\gamma_{56}$ is compatible with each of the other two subgraphs, but $\gamma_{1256}$ and $\gamma_{3456}$ are not compatible with each other.  This means that we have two two-dimensional cones and they share a ray.  We must compute the contribution to the leading divergence from each two-dimensional cone separately and add them.

We may now leverage eq.~(\ref{tropvolume}) to obtain
\begin{align}
    \label{eyedivergence}
    \mathcal{I}_{G_L} = \Gamma(d_{G_L}) \frac{1}{d_{\gamma_{56}}}\left(\frac{1}{d_{\gamma_{1256}}}+\frac{1}{d_{\gamma_{3456}}}\right)+\ldots,
\end{align}
where the finite integral is accounted for as it is merely a product of bubble integrals which equal one. Since $d_{\gamma_{1256}} = d_{\gamma_{3456}} = 2\varepsilon$ and $d_{\gamma_{56}} = \varepsilon$, the leading UV divergence contributes $\mathcal{I}_{G_L} = \frac{1}{3\varepsilon^3} + \ldots$.  The structure of the rays in this example captures the familiar RG intuition of sequentially shrinking loops.  One simplicial cone is generated by rays $r_{1256}$ and $r_{56}$ and captures the shrinking of the two corresponding subgraphs, one nested in the other.  Any ray in this cone corresponds to a direction of scaling integration variables where the bubble $\gamma_{56}$ shrinks at least as fast as the collective $\gamma_{1256}$, capturing the physics of shrinking successive UV subgraphs to effective vertices.

\begin{figure}[!t]
    \centering
    \includegraphics[]{figpdfs/fig81-figure0.pdf}
	\includegraphics[]{figpdfs/fig82-figure0.pdf}
    \caption{Divergent rays of the diagrams from Fig.~\ref{eyeofsauron}. Left: Two two-dimensional cones which share a single ray $r_{56}$, reflecting the fact that there is an overlapping divergence. Right: Triangulation of the three-dimensional cone into two simplicial cones separated by the red plane.  In this triangulation each simplex has the interpretation of nested shrinking subgraphs with ordering implied by the rays.}
    \label{eyeinbubblerays}
\end{figure}


\paragraph*{\bf Leading divergences and the RG.}
In general, the maximal cones Trop = 0 cones need not be simplicial. A given cone's contribution then need not be interpreted in the familiar nested shrinking picture consistent with our familiar RG intuitions.  Nonetheless, there is a triangulation of the cone that does this.  Consider the diagram $G_R$ from Fig.~\ref{eyeofsauron} (right), whose rays are depicted in Fig.~\ref{eyeinbubblerays} (right).  The triangulation divides the cone into two simplices, with rays that again imply an ordering of scaling rate.  That is, any ray in the simplex generated by $r_{345678}, r_{3478}, r_{78}$ has components $\gamma_{78}$ shrink the fastest, followed by $\gamma_{34}$ and then $\gamma_{56}$.  The nested shrinking is therefore one natural way of organizing the calculation of the leading the divergence, though not uniquely so.  Using \eqref{tropvolume}, we can verify that the two triangulations produce the same leading contribution:
\begin{align}
    &\frac{1}{d_{\gamma_{345678}}d_{\gamma_{78}}}\left(\frac{1}{d_{\gamma_{3478}}}+\frac{1}{d_{\gamma_{5678}}}\right)\nonumber\\ =&\,\frac{1}{d_{\gamma_{3478}}d_{\gamma_{5678}}}\left(\frac{1}{d_{\gamma_{78}}}+\frac{1}{d_{\gamma_{345678}}}\right) = \frac{1}{3\varepsilon^3},
\end{align}
and hence the leading contribution in this case is $\mathcal{I}_{G_R} = \Gamma(d_{G_R}) \frac{1}{3\varepsilon^3} + \ldots = \frac{1}{12\varepsilon^4}+\dots$.
Similarly, the phenomenon of overlapping divergences for the diagram $G_L$ is illustrated in Fig.~\ref{eyeinbubblerays} (left), where the ray $r_{56}$ is shared between two two-dimensional cones.

The fact that leading UV divergences are organized in a ``Russian doll" structure of shrinking nested subgraphs, reflects the Wilsonian way of organizing the generation of logarithmic divergences scale-by-scale. This is a simple but important first step in understanding the origin of the renormalization group. Of course the full structure of the RG goes beyond this basic fact about leading divergences of individual diagrams, and organizes the UV divergences into leading, subleading etc. logarithms captured by running coupling constants. We will discuss a geometric perspective on this understanding in \cite{AHHM}. 





\section{\label{sec:IR} Infrared}

\begin{figure}[!t]
    \centering
    \includegraphics[scale=.7]{figpdfs/fig2-figure0.pdf}
    \caption{Left: Collinear divergence illustrated for the box diagram $G_B$ with one massless corner (single line) and the remaining ones massive (doubled line). A subgraph $\gamma_{14}$ corresponding to the graph's unique divergent ray in $\mathrm{D} = 4-2\varepsilon$.  Right: The resulting $G_B/\gamma_{14}$ is scaleless.}
    \label{IRcorner}
\end{figure}

Combinatorics of leading divergences is much richer in the IR.
The rays for which $\text{Trop} = -d_{G/\gamma}$ correspond to scaleless $G/\gamma$'s.  We are particularly interested in $G/\gamma$'s for which  $d_{G/\gamma} = 0$, as they correspond to logarithmic IR divergences.
Such $G/\gamma$'s can then be blown up into a graph $G$ by inserting a $\gamma$ into it. By studying the possible structures of $G/\gamma$'s, we can understand the ways in which IR divergences can appear.

All scaleless graphs must have massless internal edges.  There are then two ways to make scaleless graphs: momentum conservation or masslessness.  Either the only spanning $2$-trees isolate a single massless external momentum or they isolate all external momenta and vanish by momentum conservation.  In order to have a divergence, the sign of Trop tells us that  we want $d_{G/\gamma}$ to be as convergent as possible; the more superficially convergent $d_{G/\gamma}$, the more divergent the contribution to the Feynman integral $\mathcal{I}_G$.  But we also need $G/\gamma$ to be scaleless. There is a basic tension between being scaleless and being superficially convergent.  We can make a graph arbitrarily superficially convergent by adding external lines, i.e., bivalent vertices in the amputated graph, but we cannot have more than two such vertices with external kinematics and be scaleless. 

\paragraph*{\bf In which dimensions do IR divergences exist?}
This discussion allows us to bound the dimensions in which IR divergences appear.  In particular, we can ask: what is the highest dimension in which a marginal IR divergence may exist?  For $n > 3$, we will maximize $\mathrm{E}-\mathrm{L} \mathrm{D}/2$ when we have a bubble filled with trivalent vertices.  In this case, with $2v$ trivalent vertices, we have 
\begin{align}
    d_{G/\gamma} = \frac{(4-\mathrm{D}) + v(6-\mathrm{D})}{2}.
\end{align}
We see that as $v\to \infty$ then $\mathrm{D} \to 6$ becomes marginal, therefore one will not find IR divergences in six or higher dimensions.
Since adding numerators can only soften IR divergences, these bounds apply to a general theory.
Fig.~\ref{IRdimensions} catalogues the leading examples of marginal scaleless graphs in the relevant dimensions.

It is worth noting that the existence of marginal IR divergences in a given dimension implies the existence of power IR divergences in lower dimensions.
In particular, scalar theories such as $\phi^3$ exhibit power IR divergences in $\mathrm{D}=4$ \cite{AHHM}, see Fig.~\ref{fig:scalings}c, giving an additional raison d'être for numerators in gauge theories: softening IR divergences to logarithmic.

\begin{figure}[!t]
    \centering
    \includegraphics[]{figpdfs/fig9-figure0.pdf}
    \caption{Leading $G/\gamma$ graphs which are scaleless and marginal in the relevant dimensions.  We write $\mathrm{D}\to 6$ as marginality is only attained in the limit of an infinite number of rungs.}
    \label{IRdimensions}
\end{figure}

\paragraph*{\bf Soft/collinear divergences.}

The standard collinear divergence comes from a ray $r_{14}$ corresponding to shrinking $\gamma_{14}$ (or expanding $\gamma_{23}$), as illustrated in Fig.~\ref{fig:scalings}a. It is singular because $G_B/\gamma_{14}$ is scaleless, see Fig.~\ref{IRcorner}. 
The cone is therefore a single ray $r_{14}$, which according to \eqref{tropvolume} gives, in the conventions of Fig.~\ref{IRcorner}
\begin{align}\!
\vcenter{\hbox{\includegraphics[width=7mm]{figpdfs/inline1-figure0.pdf}}}\!\!&=\!  -\frac{\Gamma(d_{G_B}\!)}{d_{G_{\!B}\!/\gamma_{14}}}\!\! \int\!\!\! \frac{\mathrm{d}^4 \alpha/\text{GL(1)}^2}{(s\alpha_1\alpha_3{+}t\alpha_2\alpha_4{+}p_1^2\alpha_1\alpha_2{+}p_3^2\alpha_3\alpha_4)^2} \!+\!\dots\nonumber\\
&= -\frac{1}{\varepsilon}  \int_{0}^{\infty}\!\! \frac{\mathrm{d} \alpha_1 \mathrm{d} \alpha_2 }{(s \alpha_1+t\alpha_2+p_1^2\alpha_1\alpha_2+p_3^2)^2} + \dots,\nonumber\\
&= -\frac{1}{\varepsilon}  \frac{\log\left(\frac{st}{p_1^2 p_3^2}  \right)}{s t-p_1^2p_3^2 } +\dots,
\label{eq:one-corner-box}
\end{align}
where we used the two $\mathrm{GL}(1)$'s to fix $\alpha_3 = \alpha_4 = 1$ and $d_{G_B} = 2{+}\varepsilon$ with $d_{G_B/\gamma_{14}} = \varepsilon$,
see App.~\ref{app:further-examples} for more details.

The soft/collinear divergence is illustrated in Fig.~\ref{fig:scalings}b (see also \cite{YelleshpurSrikant:2019khx}) and comes from a two-dimensional cone generated by two IR rays, say $r_{14}$ and $r_{34}$, sharing an edge. Its leading divergence is
\begin{align}
\vcenter{\hbox{\includegraphics[width=7mm]{figpdfs/inline2-figure0.pdf}}} &= \frac{\Gamma(d_{G_B})}{d_{G_B\!/\gamma_{14}}\, d_{G_B\!/\gamma_{34}}} \int  \frac{\mathrm{d}^4 \alpha/\text{GL(1)}^3}{(s\alpha_1\alpha_3+t\alpha_2\alpha_4)^2} + \dots \nonumber\\
&= \frac{1}{\varepsilon^2 st} +\dots,
\end{align}
where we used the enhanced $\mathrm{GL}(1)^3$ to fix $\alpha_2=\alpha_3=\alpha_4=1$. Generalizations to nested IR divergences can be quite intricate, see, e.g., Fig.~\ref{fig:scalings}d.







\paragraph*{\bf Fishnet diagrams.} 

For scalar integrals in $\mathrm{D}=4$, it is natural to study the generalization to \emph{fishnet} diagrams, i.e., planar arrays of $M \times N$ squares \cite{Gurdogan:2015csr,Chicherin:2017frs,Loebbert:2020tje}, see Fig.~\ref{fishnetexample} for $M=N=4$ with two massless corners.
As discussed in Sec.~\ref{sec:general},
calculation of the divergent contribution from a graph is performed by identifying the rays generating divergent cones and applying \eqref{tropvolume}.  In the case of IR-divergent fishnets, the combinatorics of these rays is rich, but admits a simple description.  It suffices to study the case with two diagonally-opposed massless corners, as the combinatorics of IR rays associated with neighboring corners is trivial, i.e., the geometry is a product.  This means that for any assignment of the external masses, we can treat the diagonals independently and multiply the result.  A single leading-dimensional cone of the Trop = 0 space is in correspondence with a staircase walk (dashed lines) separating the fishnet into two Young tableau-shaped sets of edges (blue and red) as in Fig.~\ref{fishnetexample}. 
Then the compatible rays $\gamma$ are given by the complements of 
all the Young \emph{subdiagrams} based at the top-left (blue) or bottom-right corner (red), including the dashed bordering edges.  For example, the blue shading with the dashed lines is the $\gamma$ corresponding to the tableaux given by the red shading.


\begin{figure}[!t]
    \centering
    \scalebox{0.5}{
    \includegraphics[]{figpdfs/fig31-figure0.pdf}}
    \hspace{1.25cm}\scalebox{1.15}{
   \includegraphics[]{figpdfs/fig32-figure0.pdf}}
    \caption{Left: Half of a top-dimensional contribution to a four-point fishnet diagram with opposite massless corners.  The IR subgraphs (red and blue plus respective sub-tableaux) are complementary to those appearing in the inequalities, i.e., these are the edges becoming long/soft.  We additionally label for the red portion one of the standard tableaux corresponding to a cell in the triangulation. Right: Scaling parameters $\rho_i \to \infty$ of the lower-right edges for a square labeled by integer $i$ in the standard tableaux.}
    \label{fishnetexample}
\end{figure}

Since such cones are in general not simplicial, in order to leverage \eqref{tropvolume} one needs a triangulation.  For concreteness, consider the red subdiagram in Fig.~\ref{fishnetexample}.  One triangulation is given by cells corresponding to the \emph{standard} tableaux, see Fig.~\ref{fishnetexample}. Scaling of the bottom and right edges of every red box labeled with integer $i$ is given by $\rho_i \rho_{i+1} \cdots \rho_6$, where each $\rho_i$ parameterizes scaling along a ray generating the simplicial cell.  Each cell clearly contributes the volume $\frac{1}{6!}$, so the non-trivial combinatorics is in the counting of the standard tableaux, given by the hook-length formula.  We claim that summing over the volume contributions from all the cells yields
\begin{equation}
    \label{catalan}
    \frac{2^{MN}}{(MN)!} C_{M,N} = \!\!\sum\limits_{\lambda_{i} > \lambda_{i+1}}\frac{f^{(\lambda_1, \dots, \lambda_M)}f^{(\bar \lambda_M, \dots, \bar \lambda_1)}}{(\lambda_1{+}\dots{+} \lambda_M)!(\bar\lambda_1{+}\dots{+} \bar\lambda_M)!}  ,
\end{equation}
where $\bar \lambda_i = N-\lambda_i$ and $f^{(\lambda_1, \dots, \lambda_M)}$ counts the number of the standard Tableaux with shape $\lambda = (\lambda_1, \dots, \lambda_M)$, and $C_{M, N}$ is the multi-dimensional Catalan number \cite{MultiCatalan} with standard definition:
\begin{equation}
    C_{M,N} = (M N)!\frac{\text{sf}(M-1)\text{sf}(N-1)}{\text{sf}(M+N-1)}
\end{equation}
with the superfactorial $\text{sf}(N) = 0!1!2!\cdots N!$.

For the fishnet with one massless corner, there is no sum over partitions and we merely have to count the number of standard tableaux for the $M \times N$ rectangle, which is given by $C_{M, N}$:
\begin{equation}
\label{3massfish}
\raisebox{-2em}{\includegraphics[]{figpdfs/fig3massfish-figure0.pdf}}
\;=\; \frac{F_{M,N}(s, t, p_i^2)}{(MN)!} \frac{C_{M,N}}{\varepsilon^{MN}} + \ldots,
\end{equation}
where $F_{M,N}(s, t, p_i^2)$ is a non-trivial function of kinematics computed by the $\text{GL}(1)^{MN}$-modded integral in the way described above, see \eqref{eq:one-corner-box} for a special case. And in the case of two diagonally opposed massless edges, we employ \eqref{catalan} yielding:
\begin{equation}
\label{2massfish}
\raisebox{-2em}{\includegraphics[]{figpdfs/fig2massfish-figure0.pdf}}
\;=\; \frac{2^{MN}}{(MN)!} \frac{C_{M, N}}{\varepsilon^{MN} s^N t^M} + \ldots,
\end{equation}
The leading contribution to the fully massless fishnet is obtained by squaring the kinematic-independent part of \eqref{2massfish} and keeping the same kinematic dependence.\footnote{Similarly, the contribution with two neighboring massless corners is obtained by squaring the kinematic-independent part of \eqref{3massfish} and doing the appropriate modded integral giving $1/(s^N t^M)$. The contribution with the three massless corners is obtained by multiplying kinematic-independent parts of \eqref{3massfish} and \eqref{2massfish} and again obtaining $1/(s^N t^M)$ from the modded integral.} See \cite{AHHM} for details and \cite{Basso:2021omx,Olivucci:2021pss} for alternative perspectives.



\section{Outlook}

In this work we introduced polytopes capturing the structure of UV/IR divergences of Feynman integrals. Recently, Symanzik polytopes also played a role in the study of kinematic singularities, known as anomalous thresholds \cite{Mizera:2021icv}, and it would be interesting to relate the two approaches. Other future direction include a systematic approach to extracting subleading divergences, inclusion of kinematic numerators, as well as summing over multiple Feynman diagrams under the same integral, further pursued in \cite{AHHM}.




\begin{acknowledgments}
We thank Francis Brown for introducing us to Symanzik polytopes and for sharing his unpublished notes \cite{BrownIHES}. We also thank Nick Early, Hofie Hannesdottir, Song He, Thomas Lam, Erik Panzer,  Giulio Salvatori, and Simon Telen for stimulating discussions. This work is supported by the grant DE-SC0009988 from the U.S. Department of Energy.
S.M. gratefully acknowledges the funding provided by Frank and Peggy Taplin. 
\end{acknowledgments}


