# Cutkosky’s Theorem for Massive One-Loop Feynman Integrals - Part 1

Maximilian Mühlbauer

###### Abstract.

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of
Feynman integrals in the massive one-loop case up to the involved intersection
index. This is done by applying the techniques to treat singular integrals
developed in [FFLP65]. We write one-loop integrals as an integral of a
holomorphic family of holomorphic forms over a compact cycle. Then we
determine at which points simple pinches occur and explicitly compute a
representative of the corresponding vanishing sphere and the vanishing cell.
This also yields an algorithm to compute the Landau surface of a one-loop
graph without explicitly solving the Landau equations. We also discuss the
bubble, triangle and box graph in detail.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Some Aspects of Projective Space
    2. 2.2 Some Basics on Sheaves and Monodromy
    3. 2.3 Singularities of Integrals
      1. 2.3.1 Leray’s Calculus of Residues
      2. 2.3.2 Simple Pinches
      3. 2.3.3 The Picard-Lefschetz Formula for Simple Pinches
      4. 2.3.4 Variation and Discontinuity Along Products of Loops
  3. 3 Feynman Graphs and Integrals
  4. 4 One-Loop Feynman Graphs
    1. 4.1 Compactification and Stratification
      1. 4.1.1 Remark on the Compactification in [FP66]
    2. 4.2 On the Continuation of the Local Section Defined by $\mathbb{R}\mathbb{P}^{D}$
    3. 4.3 Cutkosky’s Theorem in the One-Loop Case
    4. 4.4 Discontinuity around Points of Higher Codimension
  5. 5 More Advanced Examples
    1. 5.1 The Triangle Graph $C_{3}$
    2. 5.2 The Box Graph $C_{4}$
  6. 6 Conclusion and Outlook

##  1\. Introduction

One of the primary tasks in perturbative quantum field theory is the
computation of Feynman integrals. These are integrals associated to graphs,
essentially given by quadratic functions

(1.1) $Q_{1}(p,m),\ldots,Q_{n}(p,m):\mathbb{C}^{LD}\to\mathbb{C},$

with $n$ the number of edges and $L$ the number of independent cycles of the
underlying graph, which themselves depend (quadratically) on physical
parameters (the external momenta $p$ and masses $m$). The integrals under
consideration are then of the form

(1.2)
$(\prod_{i=1}^{L}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}d^{D}k_{i})\frac{1}{\prod_{i=1}^{n}(Q_{i}(p,m)(k_{1},\ldots,k_{L}))^{\lambda_{i}}}$

and, if they are well-defined at all,111We refer here to the convergence of
the integral in question. It should be remarked that, as it stands, the
expression (1.2) is generally ill-defined, even when it converges: The
integral need not be absolutely convergent so that Fubini’s Theorem does not
apply. Thus one needs to agree on the order of integration to make sense of
(1.2). define functions in the parameters $p$ and $m$. Even in simple cases,
it is unfortunately very difficult (though sometimes possible, see for example
[tHV79] and [ABW13]) to express such an integral in terms of well-understood
mathematical functions. Therefore there is a growing need to understand the
properties of functions defined by Feynman integrals in their own right. To
this end, the benefits of considering a Feynman integral as a function of
general complex momenta instead of just physical Minkowski momenta (living in
$i\mathbb{R}\times\mathbb{R}^{D-1}$ in our setup) have been discovered long
ago. Among many other advantages, this allows us to apply the plethora of
elegant techniques from complex analysis to the problem.
Leaving aside the problem of renormalization (which we conveniently sidestep
in this work by using analytic regulators
$\lambda_{1},\ldots,\lambda_{n}\in\mathbb{C}$), posed by the fact that the
integral (1.2) might be ill-defined for every value of $(p,m)$ even when
integrating over $(\mathbb{R}^{D})^{L}$ instead of
$(i\mathbb{R}\times\mathbb{R}^{D-1})^{L}$, the expression (1.2) does certainly
not make sense for all possible complex values of $(p,m)$. Leaving the masses
$m$ fixed and positive (as we do throughout this text), we can however find an
open neighborhood $U$ in the complex space of external momenta where the
expression is well-defined and yields in fact a holomorphic function on $U$.
This immediately leads to the question along which paths this function can be
analytically continued and what the result of such a continuation is. In other
words we seek to understand the analytic structure of functions defined by
Feynman integrals.
To answer questions of this nature the authors of [FFLP65] developed a
framework, later expanded by Pham in [Pha67] and [Pha11], to deal with
integrals of the form

(1.3) $\int_{\Gamma}\omega(t).$

This is to be understood as follows: Suppose $X$ and $T$ are two complex
analytic manifolds. Denote by $\pi:X\times T\twoheadrightarrow T$ the
canonical projection. We assume there is an analytic subset $S\subset X\times
T$ such that every fiber $S_{t}:=\pi^{-1}(t)\cap S$ (naturally viewed as a
subset of $X$) is a codimension 1 analytic subset of $X$. We take $\omega(t)$
to be a holomorphic $n$-form on $X\backslash S_{t}$ holomorphically dependent
on $t\in T$ (a notion which we define precisely in the preliminaries in
Section 2) and $\Gamma$ to be a compact $n$-cycle in $X\backslash S_{t_{0}}$
for some fixed $t_{0}\in T$. The authors of [FFLP65] where able to show that
in good cases the integral (1.3) defines a holomorphic function in an open
neighborhood of $t_{0}$ which can be analytically continued along any path in
$T$ which does not meet a certain analytic subset $L\subset T$ of codimension
1. This $L$ is called the Landau surface of the integral (1.3). The
discontinuity of such functions along simple loops around points $t\in L$ of
codimension 1 at certain $t^{\prime}\in T\backslash L$ close to $t$ can then
be computed to be

(1.4) $N\int_{\tilde{e}}\omega(t^{\prime})=(2\pi
i)^{m}N\int_{e}\text{Res}^{m}\,\omega(t^{\prime}),$

where $\tilde{e}$ (resp. $e$) is an $n$-cycle in $X\backslash S_{t^{\prime}}$
(resp. ($n-m$)-cycle in $S_{t^{\prime}}$) called the vanishing sphere (resp.
vanishing cycle) which can be computed by means of the local geometry of
$S_{t^{\prime}}$ alone and $\text{Res}^{m}$ is the iterated Leray residue.
Here $N$ is an integer determined by the intersection index of the integration
cycle at $t^{\prime}$ with a homology class in $X$ relative to
$S_{t^{\prime}}$ associated to $\tilde{e}$ and $e$.
We want to apply this program to a class of simple integrals of the form
(1.2), namely integrals associated to one-loop Feynman graphs. The goal is to
prove Cutkosky’s Theorem, first stated in [Cut60], in this case. It describes
the discontinuity of functions defined by Feynman integrals in terms of a
simpler integrals. Suppose we consider the case
$\lambda_{1}=\cdots=\lambda_{n}=1$. Then conjecturally the discontinuity along
certain loops around a problematic point $p$ in Minkowski space evaluated at
points $p^{\prime}$ close to $p$ is given by the formula

(1.5) $(2\pi
i)^{|C|}(\prod_{i=1}^{L}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}d^{D}k_{i})\frac{\prod_{e\in
C}\delta_{+}(Q_{e}(p^{\prime},m)(k))}{\prod_{e\in
E(G)-C}Q_{e}(p^{\prime},m)(k)}$

with $C$ a certain subset of edges. Here the integration over
$\delta_{+}(Q_{e}(p,m)(k))$ means integrating the residue of $Q_{e}(p,m)$
along its positive energy part. Unfortunately this is not yet a mathematically
precise statement and correspondingly there does not seem to be a rigorous
proof of this statement anywhere in the literature. The only attempt the
author is aware of is the paper [BK15] by Bloch and Kreimer which is still
work in progress and has not been published yet.
The plan of this paper is as follows: First we modify Feynman integrals to fit
the form (1.3). The main obstacle is the non-compactness of the integration
domain and the complex ambient space $\mathbb{C}^{LD}$ it lives in. From this
new representation of one-loop Feynman integrals, we derive the well-known
fact that the Landau surface $L$ is given by the Landau equations.222It should
be remarked that although this is common knowledge for physicists, a
mathematical proof does not seem to be available. For an as of now unpublished
attempt in the multiloop case, see [Müh20]. We then proceed to establish that,
outside of a small set of pathological external momenta, the integral is
behaved well enough for the techniques above to apply (more precisely: all
relevant pinches are simple pinches). Therefore the vanishing sphere and cell
are defined and we compute them explicitly. Finally, putting all these
ingredients together, we prove Cutkosky’s Theorem for one-loop graphs up the
the yet undetermined intersection index. The computation of the latter is
postponed to the second part of this work as it involves an array of quite
different techniques than the ones employed here.
The paper has the following structure: In Section 2 we review all the
necessary concepts and techniques needed to state and proof our version of
Cutkosky’s Theorem. This includes remarks on real and complex projective
space, some basics from the theory of sheaves (Subsection 2.2) and the theory
of singular integrals as developed by Pham et. al. (Subsection 2.3). The
theory of Feynman integrals is quickly reviewed in Section 3. The subsequent
Section 4 contains the main part of this paper, leading to a statement and
proof of Cutkosky’s Theorem for one-loop graphs. The proof also yields an
algorithm to compute the Landau surface without solving the Landau equations
explicitly. To the authors knowledge, this algorithm is new. After the results
for general one-loop graphs are established, we look at two advanced examples,
the triangle and the box graph, in detail in Section 5. In the concluding
Section 6 we comment on the relevance of these results and ideas for future
continuation of this work.

##  2\. Preliminaries

Before diving into the details regarding Feynman integrals and Cutkosky’s
Theorem, we introduce some amount of known theory for the convenience of the
reader. In Subsection 2.1, we start with some elementary properties of real
(resp. complex) projective space viewed as a real analytic (resp. complex
analytic) manifold. After that, we recap some basics of the theory of sheaves,
in particular the theory of local systems in Subsection 2.2. These provide a
convenient language to talk about the theory of singular integrals as
initiated in [FFLP65] and developed further by Pham in [Pha67] and [Pha11],
which we review in some detail in Subsection 2.3.

###  2.1. Some Aspects of Projective Space

A considerable amount of this works content relies on various properties of
the real and complex projective spaces $\mathbb{R}\mathbb{P}^{n}$ and
$\mathbb{C}\mathbb{P}^{n}$. Therefore in this subsection we recall some of
their properties that we make use of frequently throughout this work.
Let $\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$. First recall that the
$n$-dimensional projective space $\mathbb{K}\mathbb{P}^{n}$ viewed as a set
can be defined as the quotient of $\mathbb{K}^{n+1}\backslash\\{0\\}$ by the
equivalence relation

(2.1) $x_{1}\sim
x_{2}\quad:\Leftrightarrow\quad\exists\lambda\in\mathbb{K}^{\times}\;:\;x_{1}=\lambda\cdot
x_{2}.$

The equivalence class of any
$(x_{0},\ldots,x_{n})\in\mathbb{K}^{n+1}\backslash\\{0\\}$ with respect to
this equivalence relation is denoted by $[x_{0}:\cdots:x_{n}]$. The quotient
space comes with a natural projection

(2.2)
$\pi_{\mathbb{K}}:\mathbb{K}^{n+1}\backslash\\{0\\}\twoheadrightarrow\mathbb{K}\mathbb{P}^{n},\quad(x_{0},\ldots,x_{n})\mapsto[x_{0}:\cdots:x_{n}].$

As a topological space $\mathbb{K}\mathbb{P}^{n}$ is equipped with the induced
quotient topology. Additionally $\mathbb{R}\mathbb{P}^{n}$ is a real analytic
manifold of dimension $n$ and $\mathbb{C}\mathbb{P}^{n}$ is a complex analytic
manifold of (complex) dimension $n$. For convenience of language, we simply
use the term $\mathbb{K}$-analytic to mean real analytic if
$\mathbb{K}=\mathbb{R}$ and complex analytic if $\mathbb{K}=\mathbb{C}$. Both
$\mathbb{R}\mathbb{P}^{n}$ and $\mathbb{C}\mathbb{P}^{n}$ can be covered by
$n+1$ charts

(2.3)
$\varphi_{i,\mathbb{K}}:U_{i,\mathbb{K}}:=\\{[x]\in\mathbb{K}\mathbb{P}^{n}\;|\;x_{i}\neq
0\\}\overset{\sim}{\to}\mathbb{K}^{n},\quad[x]\mapsto(\frac{x_{0}}{x_{i}},\ldots,\widehat{\frac{x_{i}}{x_{i}}},\ldots,\frac{x_{n}}{x_{i}}),\qquad
i\in\\{0,\ldots,n\\}.$

The transition maps are $\mathbb{K}$-analytic and the projection
$\pi_{\mathbb{K}}$ is $\mathbb{K}$-analytic with respect to the induced
manifold structure. While $\mathbb{C}\mathbb{P}^{n}$ is always orientable (as
it is a complex manifold), the manifold $\mathbb{R}\mathbb{P}^{n}$ is
orientable if and only if $n$ is odd. We denote

(2.4) $H_{\infty,\mathbb{K}}:=\mathbb{K}\mathbb{P}^{n}\backslash
U_{0,\mathbb{K}}$

for $\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$ which we call the hyperplane at
infinity.333Of course our choice to single out the 0th coordinate here is
arbitrary and simply a matter of taste. In principal any hyperplane could be
chosen to be the one “at infinity”. The reason for this terminology is the
fact that $\mathbb{K}^{n}$ embeds into $\mathbb{K}\mathbb{P}^{n}$ via the
inclusion

(2.5) $i:\mathbb{K}^{n}\hookrightarrow\mathbb{K}\mathbb{P}^{n},\quad
z\mapsto[1:z].$

This defines a diffeomorphism (for $\mathbb{K}=\mathbb{R}$) or a biholomorphic
map (for $\mathbb{K}=\mathbb{C}$)
$\mathbb{K}^{n}\overset{\sim}{\to}\mathbb{K}\mathbb{P}^{n}\backslash
H_{\infty}=U_{0,\mathbb{K}}$ with inverse $\varphi_{0,\mathbb{K}}$. The
remaining points in $H_{\infty}$ not in the image of $i$ can be thought of as
additional points added “at infinity”. If it is clear from context if we mean
$\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ we simply write
$\varphi_{i}:=\varphi_{i,\mathbb{K}}$, $H_{\infty}:=H_{\infty,\mathbb{K}}$ and
so on.
It is often useful to work in $\mathbb{K}^{n+1}\backslash\\{0\\}$ instead of
$\mathbb{K}\mathbb{P}^{n}$ and then infer desired results by passing to the
quotient using the projection $\pi_{\mathbb{K}}$. The following useful
proposition is an example of a result which allows such an inference.

######  Proposition 2.1.

Let $f:\mathbb{K}^{n+1}\backslash\\{0\\}\to\mathbb{K}$ be a
$\mathbb{K}$-analytic, homogeneous function such that $df$ is nowhere
vanishing. Then $\pi_{\mathbb{K}}(f^{-1}(0))$ is a $\mathbb{K}$-analytic
submanifold of $\mathbb{K}\mathbb{P}^{n}$.

###### Proof.

The Lie group $\mathbb{K}^{\times}$ acts on
$\mathbb{K}^{n+1}\backslash\\{0\\}$ by sending $x$ to $\lambda x$ for every
$x\in\mathbb{K}^{n+1}\backslash\\{0\\}$ and $\lambda\in\mathbb{K}^{\times}$.
It is not difficult to see that this action is $\mathbb{K}$-analytic, free and
proper. Thus by the Quotient Manifold Theorem, we see that the quotient of
$f^{-1}(0)$ (which is a $\mathbb{K}$-analytic manifold by the Inverse Function
Theorem) by this action is a $\mathbb{K}$-analytic manifold itself. ∎

For our purposes we need the pull-back of the canonical $n$-form $d^{n}z$ on
$\mathbb{C}^{n}$ by $\varphi_{0,\mathbb{C}}$. This is computed in the
following

######  Lemma 2.2.

We have

(2.6)
$\varphi_{0,\mathbb{C}}^{\ast}d^{n}z=\frac{1}{z_{0}^{n+1}}\sum_{i=0}^{n}(-1)^{i}z_{i}dz_{0}\wedge\cdots\wedge\widehat{dz_{i}}\wedge\cdots\wedge
dz_{n}.$

###### Proof.

Obtained by a straightforward calculation. ∎

Note that (2.6) extends only to a meromorphic $n$-form on
$\mathbb{C}\mathbb{P}^{n}$. The pullback of $d^{n}z$ by
$\varphi_{0,\mathbb{C}}$ can not be extended to a holomorphic form: The
manifold $\mathbb{R}\mathbb{P}^{n}$ is compact $n$-cycle in
$\mathbb{C}\mathbb{P}^{n}$, so integrating a holomorphic $n$-form over it
yields a finite result. But since removing the set $H_{\infty}$ of measure
zero from $\mathbb{R}\mathbb{P}^{n}$ and choosing inhomogeneous coordinates
yields
$\int_{\mathbb{R}\mathbb{P}^{n}-H_{\infty}}\varphi_{0,\mathbb{C}}^{\ast}d^{n}z=\int_{\mathbb{R}^{n}}d^{n}z=\infty$,
this can not be correct.

###  2.2. Some Basics on Sheaves and Monodromy

The central aim of this work is to understand the multivaluedness of
holomorphic functions defined by Feynman integrals. The theory of sheaves,
particularly the theory of local systems, provides a convenient framework to
formulate and investigate such questions. We start by recalling some of the
basic notions of sheaf theory together with the theorems that are relevant to
us. Everything contained in this subsection is rather elementary and well-
understood. All statements presented here can be found in standard textbooks
on the subject and we recommend [Ten75] or [KS90]. For details on local
systems in particular, the reader is referred to [ZS09]. But since the objects
of investigation in this text are Feynman integrals which are mainly studied
by physicists, the author decided to include this material for the convenience
of the reader.
First recall that a pre-ordered set or poset $(X,\prec)$ is a set $X$ together
with a relation $\prec\subset X\times X$ such that $x\prec x$ as well as
$x\prec y\land y\prec x\;\Rightarrow\;x=y$ and $x\prec y\land y\prec
z\;\Rightarrow\;x\prec z$ for all $x,y,z\in X$ (i.e. $\prec$ is reflexive,
anti-symmetric and transitive). Now recall that to any pre-ordered set
$(X,\prec)$ we can associate a category by taking the objects to be the
elements of $X$ and for any $x,y\in X$ taking the set of morphisms
$\text{Hom}(x,y)$ to contain one element if $x\prec y$ and be empty otherwise.
For any topological space $X$ the open subsets of $X$ form a pre-ordered set
with respect to the subset-relation. The associated category of this pre-
ordered set is denoted by $\text{Ouv}_{X}$ and can be thought of as a
subcategory of Top, the category of topological spaces and continuous maps,
with objects the open subsets $U$ of $X$ and morphisms the natural inclusion
maps. A pre-sheaf on $X$ with values in a category $\mathcal{C}$ is a functor
$\mathcal{F}:\text{Ouv}_{X}^{\text{op}}\to\mathcal{C}$, where
$\mathcal{D}^{\text{op}}$ denotes the opposite category of $\mathcal{D}$ for
any category $\mathcal{D}$. For any two open sets $V\subset U$ with
$i:V\hookrightarrow U$ the natural inclusion and any $s\in\mathcal{F}(U)$, it
is customary to denote $s|_{V}:=\mathcal{F}(i)(s)$ called the restriction of
$s$ to $V$. A morphism of pre-sheaves is just a natural transformation between
the functors. The stalk $\mathcal{F}_{x}$ of $\mathcal{F}$ at $x\in X$ is

(2.7) $\mathcal{F}_{x}:=\lim_{U\ni x}\mathcal{F}(U)$

where the limit is over all open neighborhoods $U$ of $x$. The elements of
$\mathcal{F}_{x}$ are called germs of $\mathcal{F}$ at $x$. Given a section
$s\in\mathcal{F}(U)$, we can thus speak of the germ of $s$ at $x$. A pre-sheaf
$\mathcal{F}$ on $X$ is called a sheaf on $X$ if for every $U\in X$ and every
open cover $\\{U_{i}\\}_{i\in I}$ of $U$ the diagram

(2.8) $\mathcal{F}(U)\overset{\prod_{i\in
I}\mathcal{F}(\tau_{i})}{\to}\prod_{i\in
I}\mathcal{F}(U_{i})\overset{\prod_{i,j\in
I}\mathcal{F}(\tau_{j}^{i})}{\underset{\prod_{i,j\in
I}\mathcal{F}(\tau_{i}^{j})}{\rightrightarrows}}\prod_{i,j\in
I}\mathcal{F}(U_{i}\cap U_{j})$

is an equaliser diagram. Here $\tau_{i}:U_{i}\hookrightarrow U$ and
$\tau^{i}_{j}:U_{i}\cap U_{j}\hookrightarrow U_{i}$ are the natural
inclusions. Recall that this means that $\prod_{i\in I}\mathcal{F}(\tau_{i})$
is injective and that

(2.9) $\text{im}(\prod_{i\in I}\mathcal{F}(\tau_{i}))=\\{x\in\prod_{i\in
I}\mathcal{F}(U_{i})\;|\;\prod_{i,j\in
I}\mathcal{F}(\tau^{i}_{j})(x)=\prod_{i,j\in
I}\mathcal{F}(\tau^{j}_{i})(x)\\}.$

More concretely this means two things: First for any open covering
$\\{U_{i}\\}_{i\in I}$ of an open set $U$, if $s,t\in\mathcal{F}(U)$ satisfy
$s|_{U_{i}}=t|_{U_{i}}$ for all $i\in I$ we have $s=t$ (Locality). Second if
$\\{s_{i}\\}_{i\in I}$ are sections $s_{i}\in\mathcal{F}(U_{i})$ such that
$s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}$ for all $i,j\in I$, there
exists a section $s\in\mathcal{F}(U)$ such that $s|_{U_{i}}=s_{i}$ for all
$i\in I$ (Gluing). For any open set $U\subset X$ the elements of
$\mathcal{F}(U)$ are called the (local) sections over $U$. In particular the
elements of $\mathcal{F}(X)$ are called global sections. A morphism of sheaves
is a morphism of the underlying pre-sheaves.
For every pre-sheaf $\mathcal{F}$ on $X$ we can construct a sheaf
$\mathcal{F}^{+}$ on $X$ called the sheafification of $\mathcal{F}$ together
with a morphism $\theta:\mathcal{F}\to\mathcal{F}^{+}$, which satisfies the
following universal property: Given any sheaf $\mathcal{G}$ on $X$ and any
morphism of sheaves $\phi:\mathcal{F}\to\mathcal{G}$, there exists a unique
morphism $\psi:\mathcal{F}^{+}\to\mathcal{G}$ such that
$\phi=\psi\circ\theta$. One possibility to construct $\mathcal{F}^{+}$ is to
set

(2.10) $\mathcal{F}^{+}(U):=\\{f:U\to\bigcup_{x\in
U}\mathcal{F}_{x}\;|\;\begin{subarray}{c}\forall x\in
X\;:\;f(x)\in\mathcal{F}_{x}\\\ \forall x\in U\;:\;\exists V\text{ ngh. of
}x\;:\;\exists g\in\mathcal{F}(V)\;:\;\forall y\in
V\;:\;g_{y}=s_{y}\end{subarray}\\}$

for every open $U\subset X$. Let $Y$ be another topological space and $f:X\to
Y$ a continuous map. Given a sheaf $\mathcal{F}$ on $Y$, the inverse image
sheaf $f^{-1}\mathcal{F}$ of $\mathcal{F}$ by $f$ is the sheafification of the
pre-sheaf given by

(2.11) $U\mapsto\lim_{V\supset f(U)}\mathcal{F}(V),$

where the limit is taken over all open $V\subset X$ containing $f(U)$. Let $X$
be a topological space and $\mathcal{C}$ a category. Let $\mathcal{F}$ be a
sheaf on $X$ with values in $\mathcal{C}$. Then for any subspace $Y\subset X$
there is a sheaf $\mathcal{F}|_{Y}$ called the restriction of $\mathcal{F}$ to
$Y$ defined by $\mathcal{F}|_{Y}:=i^{-1}\mathcal{F}$ where $i:Y\hookrightarrow
X$ is the natural inclusion map. For any object $A$ in $\mathcal{C}$ there is
a constant pre-sheaf associated to $A$ defined by sending each open set to $A$
and each morphism to $\text{id}_{A}$. The sheafification of this pre-sheaf is
called the constant sheaf associated to $A$. A sheaf $\mathcal{F}$ on $X$ is
called a local system or a locally constant sheaf if for any $x\in X$ there is
an open neighborhood $U\subset X$ of $x$ such that $\mathcal{F}|_{U}$ is a
constant sheaf. The inverse image sheaf of any (locally) constant sheaf is
(locally) constant [ZS09].

######  Proposition 2.3 ([ZS09]).

Any locally constant sheaf on a contractible space is constant.

Local systems on a path-connected space $X$ with fiber $F$ (i.e. a local
system which is locally isomorphic to the constant sheaf with value $F$) are
in a bijective correspondence to homomorphisms from the fundamental group of
$X$ to the automorphism group of $F$. This correspondence can be established
as follows: Let $\mathcal{F}$ be a local system on $X$, let $x\in X$ and let
$\gamma:[0,1]\to X$ be a loop based at $x$. Then the inverse image sheaf
$\gamma^{-1}\mathcal{F}$ is a local system on $[0,1]$ and since $[0,1]$ is
contractible, this is a constant sheaf by Proposition 2.3. Thus

(2.12)
$F\simeq(\gamma^{-1}\mathcal{F})_{0}\simeq\gamma^{-1}\mathcal{F}([0,1])\simeq(\gamma^{-1}\mathcal{F})_{1}\simeq
F$

and we obtain an automorphism of $F$, called the monodromy along $\gamma$, by
composing the above isomorphisms. It can be shown that this automorphism
depends only on the homotopy class $[\gamma]\in\pi_{1}(X,x)$ of $\gamma$ (see
[ZS09]). The other way around suppose we are given a homomorphism
$\rho:\pi_{1}(X,x)\to\text{Aut}(F)$. Let $\tilde{\mathcal{F}}$ be the constant
sheaf with value $F$ on the universal covering space $\tilde{X}$ of $X$. Then
the sections of $\tilde{\mathcal{F}}$ invariant under deck-transformations
form a local system of $X$. It is not difficult to see that these two
operations are inverse to each other.
We also need the notion of a multivalued section:

######  Definition 2.4.

Let $\mathcal{F}$ be a sheaf on a topological space $X$ and $\pi:\tilde{X}\to
X$ the universal covering. A multivalued section of $\mathcal{F}$ is a section
in the (constant) inverse image sheaf $\pi^{-1}\mathcal{F}$.

On well-behaved topological spaces, local sections of local systems can always
be extended to multivalued global sections:

######  Proposition 2.5 ([Pha11]).

Let $X$ be a locally connected topological space and $\mathcal{F}$ a sheaf on
$X$. If $\mathcal{F}$ is locally constant then every local section of
$\mathcal{F}$ can be extended to a multi-valued global section of
$\mathcal{F}$.

A class of sheaves which bares particular importance to us is given in the
following

######  Definition 2.6.

Let $Y$ and $T$ be smooth manifolds and $\pi:Y\to T$ a smooth map. The
homology sheaf in degree $p$ of $Y$ over $T$, denoted by
$\mathcal{F}^{p}_{Y/T}$, is the sheafification of the pre-sheaf

(2.13) $U\mapsto H_{p+\dim T}(Y,\pi^{-1}(T-U)).$

For any section $h$ of $\mathcal{F}_{Y/T}^{p}$ we denote its germ at $t$ by
$h(t)$.

######  Proposition 2.7 ([Pha11]).

Let $Y,T$ and $\pi:Y\to T$ as in Definition 2.6. Then for every $t\in T$ and
every $p\in\mathbb{N}$ there are isomorphisms

(2.14) $H_{p}(\pi^{-1}(t))\simeq H_{p+\dim
T}(Y,\pi^{-1}(T-\\{t\\}))\simeq\lim_{U\ni t}H_{p+\dim
T}(Y,\pi^{-1}(T-U))=(\mathcal{F}_{Y/T}^{p})_{t},$

where the limit is taken over all open neighborhoods of $t$.

When we consider germs of sections of $\mathcal{F}^{p}_{Y/T}$, we usually
apply the isomorphism from Proposition 2.7 above implicitly without mention as
long as no confusion can arise.
The sheaves from Definition 2.6 play an important role for the analytical
continuation of functions defined by integrals in our setup. In fact the
extension of local sections to larger domains in these sheaves correspond
directly to an analytic continuation as we will see in the next Subsection.

###  2.3. Singularities of Integrals

We want to understand Feynman integrals, revisited in some detail in the
following Section 3, as holomorphic functions in the external momenta (and
possibly the masses of the virtual particles). To do so, we employ the
framework from [FFLP65] which deals with the analytic properties of functions
defined by integrals in a sufficiently general manner. We begin by revisiting
the basic ideas of this framework. Let $Y$ and $T$ be two complex analytic
manifolds of dimension $n+m$ and $m$ respectively and let $\pi:Y\to T$ be a
smooth submersion. We denote the fibers by $Y_{t}:=\pi^{-1}(t)$ for all $t\in
T$. By the Implicit Function Theorem, there exists for every $y\in Y$ a
coordinate neighborhood $U\subset Y$ of $y$ with coordinates

(2.15)
$\varphi:=(x_{1},\ldots,x_{n},t_{1},\ldots,t_{m})\;:\;U\to\mathbb{C}^{n+m}$

such that there is a chart $(V,\psi)$ of $T$ around $\pi(y)$ with
$\psi\circ\pi=(t_{1},\ldots,t_{m})$. In particular, the fibers $Y_{t}$ are
smooth manifolds for all $t\in T$. This can be used to define families of
differential forms on the fibers $Y_{t}$ which depend holomorphically on $t$.
We use the common multi-index notation for wedge products: For any

(2.16) $I=\\{i_{1},\ldots,i_{p}\\}\subset\\{1,\ldots,n\\}$

with $i_{1}<\cdots<i_{p}$ we write

(2.17) $dx^{I}:=dx_{i_{1}}\wedge\cdots\wedge dx_{i_{p}}.$

The following definition and notation is adapted from [Pha11]:

######  Definition 2.8.

We say that a differential $p$-form $\omega$ on $Y$ is a holomorphic $p$-form
relative to $T$ if it can be expressed in local coordinates (2.15) as above as

(2.18) $\omega=\sum_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=p\end{subarray}}f_{I}(x,t)dx^{I},$

with holomorphic functions $f_{I}$. We denote the space of all holomorphic
$p$-forms relative to $T$ by $\Omega^{p}(Y/T)$. The germ of $\omega$ at $t\in
T$ is denoted by $\omega(t)$.

It makes sense to define a codifferential which acts only on the $x$-part of a
differential form relative to $T$. Thus we define linear maps

(2.19) $d_{Y/T,p}:\Omega^{p}(Y/T)\to\Omega^{p+1}(Y/T)$

for all $p\in\mathbb{N}$ whose action on $\omega\in\Omega^{p}(Y/T)$ in local
coordinates reads

(2.20) $d_{Y/T,p}\;\omega=\sum_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=p\end{subarray}}\sum_{i=1}^{n}\frac{\partial f_{I}}{\partial
x_{i}}(x,t)dx_{i}\wedge dx^{I}.$

As usual we simply write $d_{Y/T}$ for the corresponding endomorphism of
$\bigoplus_{p\in\mathbb{N}}\Omega^{p}(Y/T)$. It is easy to check that
$d_{Y/T}\circ d_{Y/T}=0$ and thus

(2.21)
$0\to\Omega^{0}(Y/T)\overset{d_{Y/T,0}}{\to}\Omega^{1}(Y/T)\overset{d_{Y/T,1}}{\to}\cdots\overset{d_{Y/T,p-1}}{\to}\Omega^{p}(Y/T)\overset{d_{Y/T,p}}{\to}\cdots$

is a cochain complex. Correspondingly, just as in the regular de Rham complex
of differential forms, we say that $\omega$ is a closed differential $p$-form
relative to $T$ if $d_{Y/T}\omega=0$.
Note that if $h$ is a section of $\mathcal{F}_{Y/T}^{p}$ then according to
Proposition 2.7 we have

(2.22) $h(t)\in\lim_{U\ni t}H_{p+\dim T}(Y,\pi^{-1}(T-U))\simeq H_{p}(Y_{t}).$

Thus if $\omega\in\Omega^{p}(Y/T)$, it makes sense to integrate $\omega(t)$
over $h(t)$. The next proposition shows why this is a good idea.

######  Proposition 2.9 ([Pha11]).

Let $\omega\in\Omega^{p}(Y/T)$ and $h\in\mathcal{F}^{p}_{Y/T}(T)$. If $\omega$
is closed then

(2.23) $T\to\mathbb{C},\qquad t\mapsto\int_{h(t)}\omega(t)$

defines a holomorphic function on $T$.

This proposition is the basis for our investigation of holomorphic functions
defined by integrals. In practice however, the situation is scarcely as nice
as in Proposition 2.9. Usually we are not provided with a global section $h$,
but only with a local section. More specifically, often times (in particular
for one-loop Feynman integrals) the situation is as follows: The manifold $Y$
has a product structure minus some analytic subset of problematic points. I.e.
there is a compact complex analytic manifold $X$ of dimension $n$ and an
analytic subset $S\subset X\times T$ such that $Y=(X\times T)\backslash S$. In
this case the map $\pi:Y\twoheadrightarrow T$ is taken to be the canonical
projection. We denote the fiber of $S$ at $t$ by $S_{t}:=Y_{t}\cap S$ for all
$t\in T$. Suppose we are only provided with a fixed $p$-cycle $\Gamma$ in the
fiber $Y_{t_{0}}$ for some given $t_{0}\in T$. This is already sufficient to
define at least a local section:

######  Lemma 2.10.

There is a $\dim T$-simplex $\sigma$ in $T$ such that
$t_{0}\in\overset{\circ}{\sigma}$ and $\Gamma\times\sigma$ defines an element
in $\mathcal{F}_{Y/T}^{p}(\overset{\circ}{\sigma})$ (where
$\overset{\circ}{\sigma}$ denotes the interior of $\sigma$).

###### Proof.

Since $Y$ is a $T_{4}$-space (every metric space is a $T_{4}$-space [vQu00])
and $\Gamma$ as well as $S_{t_{0}}$ are closed sets, there exists open
neighborhoods $V_{1}\subset X$ of $\Gamma$ and $V_{2}\subset X$ of $S_{t_{0}}$
such that $V_{1}\cap V_{2}=\emptyset$. Due to continuity and the compactness
of the fibers there exists an open neighborhood $U\subset T$ of $t_{0}$ such
that $S_{t}\subset V_{2}$ for all $t\in U$. Then $\Gamma\cap S_{t}\subset
V_{1}\cap V_{2}=\emptyset$ for all $t\in U$. Thus $\Gamma\times U$ is a subset
of $Y$. We may assume that there exists a $\dim T$-simplex $\sigma\subset U$
in $T$. Hence, viewing $\Gamma\times\sigma$ as a $(p+\dim T)$-cycle in
$X\times T$, we compute

(2.24) $\partial(\Gamma\times\sigma)=\underbrace{\partial\Gamma}_{=0}\times
U+(-1)^{p}\cdot\Gamma\times\partial\sigma.$

Now $\Gamma\times\partial\sigma=0$ in $H_{p+\dim
T}(Y,\pi^{-1}(T-\overset{\circ}{\sigma}))$ (this is already true on the level
of chains) since it is contained within
$\pi^{-1}(T-\overset{\circ}{\sigma})=X\times(T-\overset{\circ}{\sigma})$. Thus
we indeed have $\partial(\Gamma\times\sigma)=0$ as claimed. We conclude that
$\Gamma\times\sigma$ defines a local section in the homology sheaf of degree
$p$ relative to $T$. ∎

The situation described above applies for example to one-loop Feynman
integrals (defined further below, see for example equation (4.7)), where
$t_{0}$ corresponds to a Euclidean (i.e. real) configuration of momenta. Now
the question is if this local section can be extended to a (possibly multi-
valued) global section, generally relative to a slightly smaller base space
$T^{\ast}\subset T$. A central result by Pham in this direction is the
following

######  Proposition 2.11 ([Pha11]).

Let $T^{\ast}\subset T$ be an open subset. If
$\pi|_{\pi^{-1}(T^{\ast})}:\pi^{-1}(T^{\ast})\to T^{\ast}$ defines a locally
trivial $C^{\infty}$-fibration then $\int_{h(t_{0})}\omega(t_{0})$ defines a
multi-valued holomorphic function on $T^{\ast}$.

The idea for the proof of Proposition 2.11 is rather simple: By Lemma 2.10 the
initial integration domain $\Gamma$ defines a local section of
$\mathcal{F}^{p}_{Y/T^{\ast}}$. If
$\pi|_{\pi^{-1}(T^{\ast})}:\pi^{-1}(T^{\ast})\to T^{\ast}$ defines a locally
trivial $C^{\infty}$-fibration then the sheaf $\mathcal{F}^{p}_{Y/T^{\ast}}$
is locally constant. Hence every local section can be extended to a multi-
valued global section according to Proposition 2.5.
As it stands, this is a little bit to abstract for our purposes. Much more
useful to us is a more detailed discussion of the situation where $Y$ has a
product structure and $\pi$ is the canonical projection as described above.
For this purpose we recall the following notions:

######  Definition 2.12 ([FFLP65]).

A fiber bundle of pairs is a tuple $(E,S,B,\pi,F)$ consisting of topological
spaces $E$, $B$, $F$, a subset $S\subset E$ and a continuous surjection
$\pi:E\twoheadrightarrow B$ such that $(E,B,\pi,F)$ is a fiber bundle which
has local trivializations at every point $b\in B$ whose restriction to $S$ are
local trivializations of $(S,B,\pi|_{S},\pi^{-1}(\\{\text{pt.}\\})\cap S)$
(where pt. denotes any point in $B$).
Furthermore we shall say that $(E,S,B,\pi,F)$ is smooth as a fiber bundle of
pairs if the following conditions are satisfied:

  * •

$E$, $B$ and $F$ are smooth manifolds.

  * •

$\pi$ is a smooth map.

  * •

For any local trivialization $g$, the inverse $g^{-1}$ is differentiable with
respect to $t$.

  * •

For any local trivialization $g$, the inverse $g^{-1}$ lifts smooth vector
fields on $B$ to locally Lipschitzian vector fields on $E$.

To simplify the notation, we denote a smooth fiber bundle of pairs
$(E,S,B,\pi,F)$ by $\pi:(E,S)\to B$.

######  Definition 2.13 ([Hi76]).

Let $V$ and $X$ be a smooth manifold. An isotopy from $V$ to $X$ is a
continuous map $\sigma:V\times[0,1]\to X$ such that $\sigma(\cdot,t)$ is an
embedding for all $t\in[0,1]$. If $V\subset X$ and $\sigma(\cdot,0)$ is the
natural inclusion, we say that $\sigma$ is an isotopy of $V$ in $X$. If $V=X$,
$\sigma(\cdot,0)=\text{id}_{X}$ and $\sigma(\cdot,t)$ is a diffeomorphism for
all $t\in[0,1]$, we say that $\sigma$ is an ambient isotopy.

For an isotopy $\sigma:V\times[0,1]\to X$, we denote its track by

(2.25) $\hat{\sigma}:V\times[0,1]\to
X\times[0,1],\quad(x,t)\mapsto(\sigma(x,t),t)$

and its support by

(2.26) $\text{Supp}\,\sigma:=\text{cl}\\{x\in X\;|\;\exists
t\in[0,1]\,:\,\sigma(x,t)\neq\sigma(x,0)\\}.$

Under reasonable circumstances, isotopies in $X$ can be extended to ambient
isotopies of $X$.

######  Theorem 2.14 ([Hi76]).

Let $X$ be a manifold, $A\subset X$ a compact subset and $U\subset X$ an open
neighborhood of $A$. If $\sigma:U\times[0,1]\to X$ is an isotopy of $U$ in $X$
such that $\hat{\sigma}(U\times[0,1])$ is open in $X\times I$ then there
exists an ambient isotopy $\tilde{\sigma}$ of $X$ with compact support such
that $\tilde{\sigma}$ agrees with $\sigma$ in a neighborhood of
$A\times[0,1]$.

Suppose again that we are in the situation above where $Y=(X\times
T)\backslash S$ with an analytic subset $S\subset X\times T$ and suppose that
$\pi:(X\times T,S)\to T$ is a (locally trivial) smooth fiber bundle of pairs,
i.e. $\pi:X\to T$ is a fiber bundle which has local trivializations which also
locally trivialize $\pi|_{S}:S\to T$. Then in particular $\pi:Y\to T$ defines
a locally trivial $C^{\infty}$-fibration and Proposition 2.11 applies. This
can be thought of quite intuitively: If $\gamma:[0,1]\to T$ is any path from
$t_{0}\in T$ to $t_{1}\in T$, then the pull-back bundle of pairs of
$\pi:(X\times T,S)\to T$ by $\gamma$ has a contractible base space and is thus
globally trivial by Proposition 2.3. A trivialization of this pull-back bundle
of pairs is an ambient isotopy

(2.27) $\sigma:X\times[0,1]\to X$

such that $\sigma(\Gamma,s)\cap S_{\gamma(s)}=\emptyset$ for all $s\in[0,1]$.
In the context of our problem this can be viewed as a continuous deformation
of the integration cycle $\Gamma$ away from the points were the differential
form is singular. Phrased differently, the ambient isotopy $\sigma$ allows us
to extend the local section defined by $\Gamma$ along $\gamma$:

######  Proposition 2.15.

Let $\gamma:[0,1]\to T$ be a path, $\Gamma\subset X\backslash S(\gamma(0))$ a
$p$-cycle and $\sigma:X\times[0,1]\to X$ an ambient isotopy such that

(2.28) $\sigma(\Gamma,s)\cap S(\gamma(s))=\emptyset$

for all $s\in[0,1]$. Then there is an open neighborhood $U$ of $\gamma([0,1])$
such that $\Gamma$ defines a section of $\mathcal{F}_{Y/T}(U)$.

###### Proof.

We know from Proposition 2.10 that for every $s\in[0,1]$ there exists a $\dim
T$-simplex $\alpha_{s}$ in $T$ such that
$\gamma(s)\in\overset{\circ}{\alpha}_{s}$ and such that
$\sigma(\Gamma,s)\times\alpha_{s}$ defines an element in
$\mathcal{F}_{Y/T}^{p}(\overset{\circ}{\alpha}_{s})$. We want to show that
these can be glued together to yield a local section of
$\mathcal{F}_{Y/T}^{p}$ on
$U:=\bigcup_{s\in[0,1]}\overset{\circ}{\alpha}_{s}$. By the sheaf axioms it
suffices to show that for any $s_{1},s_{2}\in[0,1]$ such that
$\overset{\circ}{\alpha}_{s_{1}}\cap\overset{\circ}{\alpha}_{s_{2}}\neq\emptyset$,
the two sets $\sigma(\Gamma,s_{1})\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})$
and $\sigma(\Gamma,s_{2})\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})$ define the
same element in
$\mathcal{F}_{Y/T}^{p}(\overset{\circ}{\alpha}_{s_{1}}\cap\overset{\circ}{\alpha}_{s_{2}})$.
We can assume $s_{1}<s_{2}$ without loss of generality. Now let

(2.29)
$\tau:=\sigma^{\ast}(\Gamma\times[s_{1},s_{2}])\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})$

where $\sigma^{\ast}$ denotes the map on the level of cycles induced by
$\sigma$. Then the boundary of $\tau$ is

(2.30)
$\partial\tau=(\partial\sigma^{\ast}(\Gamma\times[s_{1},s_{2}]))\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})+(-1)^{p+1}\sigma^{\ast}(\Gamma\times[s_{1},s_{2}])\times\partial(\alpha_{s_{1}}\cap\alpha_{s_{2}}).$

The second term
$\sigma^{\ast}(\Gamma\times[s_{1},s_{2}])\times\partial(\alpha_{s_{1}}\cap\alpha_{s_{2}})$
is 0 in
$\mathcal{F}_{Y/T}(\overset{\circ}{\alpha}_{s_{1}}\cap\overset{\circ}{\alpha}_{s_{2}})$
and we compute the first factor of the first term to be

(2.31)
$\begin{split}\partial\sigma^{\ast}(\Gamma\times[s_{1},s_{2}])&=\sigma^{\ast}(\partial(\Gamma\times[s_{1},s_{2}]))=\sigma^{\ast}(\underbrace{\partial\Gamma}_{=0}\times[s_{1},s_{2}]+(-1)^{p}(\Gamma\times
s_{2}-\Gamma\times s_{1}))\\\
&=(-1)^{p}(\sigma^{\ast}(\Gamma,s_{2})-\sigma^{\ast}(\Gamma,s_{1})).\end{split}$

Thus we conclude that

(2.32)
$\sigma(\Gamma,s_{2})\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})-\sigma(\Gamma,s_{1})\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})=(\sigma(\Gamma,s_{2})-\sigma(\Gamma,s_{1}))\times(\alpha_{s_{1}}\cap\alpha_{s_{2}})$

is the boundary of $(-1)^{p}\cdot\tau$ which completes the proof. ∎

######  Corollary 2.16.

In the same situation as in Proposition 2.15, the integral
$\int_{\Gamma}\omega(\gamma(0))$ defines a holomorphic function on some open
neighborhood of $\gamma(0)$ which can be analytically continued along $\gamma$
by

(2.33) $\int_{\sigma(\Gamma,s)}\omega(\gamma(s)).$

###### Proof.

Follows directly from Proposition 2.9 and Proposition 2.15. ∎

Due to this result, we make the following definition:

######  Definition 2.17.

Let $\gamma:[0,1]\to T$ be a path, $\Gamma\subset X$ and
$\sigma:X\times[0,1]\to X$ an ambient isotopy from $\Gamma$ to
$\sigma(\Gamma,1)$. We say that $\sigma$ is adapted to $\gamma$ if

(2.34) $\sigma(\Gamma,s)\cap S(\gamma(s))=\emptyset$

for all $s\in[0,1]$.

With this definition in place, Corollary 2.16 can be rephrased as

######  Proposition 2.18.

Let $\gamma:[0,1]\to T$ be a path from $t_{0}\in T$ to some $t\in T$ and
$\sigma:X\times[0,1]\to X$ an ambient isotopy adapted to $\gamma$. Then the
integral can be analytically continued along $\gamma$ via

(2.35) $\int_{\sigma(\Gamma,s)}\omega(\gamma(s)).$

We need some criteria to determine if an appropriate ambient isotopy for such
an analytic continuation exists. Above we discussed how it is sufficient for
$\pi:(X\times T,S)\to T$ to be a fiber bundle of pairs. A special case that is
of particular interest to us is the case where the fibers $S_{t}$ are finite
unions of complex analytic manifolds in general position. Recall that for
smooth submanifolds $S_{1},\ldots,S_{k}\subset X$ of a smooth manifold $X$, we
say that $S_{1},\ldots,S_{k}$ are in general position if the normal vectors at
every intersection point are linearly independent. More concretely, for every
$x\in\bigcup_{i=1}^{k}S_{i}$ set

(2.36) $I_{x}:=\\{i\in\\{1,\ldots,k\\}\;|\;x\in S_{i}\\}$

and let $s_{1,x},\ldots,s_{k,x}:X\to\mathbb{R}$ be local equations for
$S_{1},\ldots,S_{k}$ around $x\in\bigcup_{i=1}^{k}S_{i}$. Then
$S_{1},\ldots,S_{k}$ are in general position if and only if for every
$x\in\bigcup_{i=1}^{k}S_{i}$ there are coordinates $\phi$ around $x$ such that
the gradients of $\\{s_{i,x}\circ\varphi^{-1}\\}_{i\in I_{x}}$ at $\phi(x)$
are linearly independent. We have the following result on the analytic
continuation in this case.

######  Proposition 2.19 ([FFLP65]).

Suppose that for all $t\in T$ the fiber $S_{t}=\bigcup_{i=1}^{k}(S_{i})_{t}$
is the union of finitely many complex analytic manifolds $(S_{i})_{t}$
depending smoothly on $t$. If the $(S_{1})_{t},\ldots,(S_{k})_{t}$ are in
general position for all $t\in T$ then $\pi:(X\times T,S)\to T$ is a smooth
fiber bundle of pairs. In particular $\int_{\Gamma}\omega(t_{0})$ can be
analytically continued along any path in $T$. Furthermore the analytic
continuation to $t\in T$ can be written as

(2.37) $\int_{\Gamma^{\prime}}\omega(t)$

for an appropriate cycle $\Gamma^{\prime}$.

For our purposes, this is unfortunately not enough. It turns out that after
compactification, the relevant $(S_{1})_{t},\ldots,(S_{k})_{t}$ in the case of
Feynman integrals are generally not in general position at any $t\in T$. A
more refined criterion can be obtained by equipping $S$ with a Whitney
stratification and we introduce the necessary theory here.

######  Definition 2.20 ([GM88]).

Let $X$ be a topological space and $(\mathcal{S},\prec)$ a partially ordered
set. A $\mathcal{S}$-decomposition of $X$ is a locally finite collection of
disjoint locally closed sets $A_{i}\subset X$ such that the following hold:

  1. (1)

$X=\bigcup_{i\in\mathcal{S}}A_{i}$.

  2. (2)

For all $i,j\in\mathcal{S}$ we have

(2.38)
$A_{i}\cap\bar{A}_{j}\neq\emptyset\;\Leftrightarrow\;A_{i}\subset\bar{A}_{j}\;\Leftrightarrow\;i\prec
j.$

The $A_{i}$ are called the pieces, their connected components the strata of
the $\mathcal{S}$-decomposition.

To apply techniques from differential topology, it is useful to require that
the pieces $A_{i}$ fit together nicely.

######  Definition 2.21 ([GM88]).

Let $M$ be a smooth manifold and $X,Y\subset M$ two smooth submanifolds. We
say that the pair $(X,Y)$ satisfies Whitney’s condition B if the following
holds: Let $y\in Y$ and suppose $x_{n}$ is a sequence in $X$ converging to $y$
and $y_{n}$ is a sequence in $Y$ converging to $y$. Then if the sequence of
secants $\widebar{x_{n}y_{n}}$ converges to some line $l$ and the sequence of
tangent planes $T_{x_{n}}X$ converges to some plane $T$,444This is to be
understood as convergence in the relevant Graßmannians. we have $l\subset T$.

A few remarks are in order: First note that if $X$ and $Y$ are submanifolds of
$M$ such that $X\cap Y=\emptyset$, the above condition is vacuous. If
$Y\subset\bar{X}$ and $X\cup Y$ is a smooth manifold, it is also evident that
Whitney’s condition B is satisfied for the pair $(X,Y)$ (simply check the
condition in coordinates on $X\cup Y$).

######  Definition 2.22 ([GM88]).

Let $Y$ be a smooth manifold, $S\subset Y$ a closed subset and $\mathcal{S}$ a
partially ordered set. A Whitney stratification of $S$ is a
$\mathcal{S}$-decomposition such that all pairs of strata satisfy Whitney’s
condition B. We call the set $S$ together with a Whitney stratification of $S$
a stratified set.

Suppose that $S\subset Y$ is a closed subset which decomposes into a finite
union of smooth manifolds $S_{1},\ldots,S_{n}\subset Y$ in general position.
Then there is a natural Whitney stratification of $S$ with strata given by the
connected components of

(2.39) $S_{i}-\bigcup_{i<j}S_{i}\cap S_{j},\quad S_{i}\cap
S_{j}-\bigcup_{i<j<k}S_{i}\cap S_{j}\cap S_{k},\quad\cdots$

The pieces are the union of all strata with equal codimension (i.e. equally
many manifolds involved in the intersection) and the underlying partial order
is given by the dimension of the pieces.

######  Definition 2.23 ([Pha11]).

Let $Y$ be a smooth manifold with a Whitney stratification. We say that $Y$ is
a stratified bundle if there exists a Whitney stratified set $X$ such that $Y$
is locally homeomorphic to $X\times T$ by homomorphisms which map every
stratum in $Y$ to the product of a stratum in $X$ with $T$.

Now we can formulate the following criterion:

######  Theorem 2.24 (Thom’s Isotopy Theorem, [Pha11]).

Let $Y$ and $T$ be two differentiable manifolds with $T$ connected and let
$\pi:Y\to T$ be a proper differential map. Suppose that $Y$ is a stratified
set and the restriction of $\pi$ to each stratum is a submersion. Then
$\pi:Y\to T$ is a stratified bundle.

Suppose $S\subset Y$ is a closed subset equipped with a Whitney
stratification. Then the stratification of $S$ together with the connected
components of $Y-S$ yield a Whitney stratification of $Y$ [Pha11]. If this
makes $\pi:Y\to T$ a stratified bundle then in particular $\pi:(Y,S)\to T$ is
a fiber bundle of pairs.
Supposing again that we are in the situation where $Y=(X\times T)\backslash S$
as above, we can equip $S$ with a Whitney stratification (every analytic set
admits such a stratification, see [Wh65]). Denote by $\\{A_{i}\\}_{i\in I}$
the collection of strata and for every $i\in I$ denote by $cA_{i}\subset
A_{i}$ the set of all points at which the restriction of $\pi$ to $A_{i}$
fails to be a submersion. Then we can conclude that the integral of interest
defines a (multivalued) holomorphic function outside of the following set:

######  Definition 2.25.

In the same situation as in Proposition 2.19, we call the set

(2.40) $L:=\pi(\bigcup_{i\in I}cA_{i})$

the Landau surface of the integral $\int_{\Gamma}\omega(t_{0})$.

It should be remarked that it can be shown that the Landau surface is an
analytic set [Pha11] (a consequence of Remmert’s Proper Mapping Theorem).
Thus, if we assume $T$ to be connected, $L$ is either all of $T$ or
$T\backslash L$ is a non-empty open subset. We conclude that
$\int_{\Gamma}\omega(t_{0})$ defines a holomorphic function on
$T^{\ast}:=T\backslash L$.
The fact that a given local section may only be extended to a global section
in the multivalued sense immediately gives rise to the question of the
discontinuity of the function defined by a given integral. This can be easily
expressed in terms of the monodromy of the section $h$ in question. Let
$t_{0}\in T^{\ast}$ and let $\gamma:[0,1]\to T^{\ast}$ be a loop based at
$t_{0}$. Then continuing the germ $h(t_{0})$ along $\gamma$ yields a class
$\gamma^{\ast}h(t_{0})\in H_{\dim_{\mathbb{R}}X}(Y_{t_{0}})$ (where
$\dim_{\mathbb{R}}X$ is the dimension of $X$ viewed as a real manifold) which
can be shown to only depend on the homotopy class of $\gamma$ [Pha11]. Then
analytically continuing the integral along $\gamma$ yields

(2.41) $\int_{\gamma^{\ast}h(t_{0})}\omega(t_{0}).$

We define the variation of $h(t_{0})$ along $\gamma$ by

(2.42)
$\text{Var}_{[\gamma]}h(t_{0})=\gamma^{\ast}h(t_{0})-h(t_{0})=(\gamma^{\ast}-1)h(t_{0})$

and the discontinuity of $\int_{h(t_{0})}\omega(t_{0})$ along $\gamma$ by

(2.43)
$\text{Disc}_{[\gamma]}\int_{h(t_{0})}\omega(t_{0})=\int_{\text{Var}_{[\gamma]h(t_{0})}}\omega(t_{0}).$

The variation can be understood as homomorphism

(2.44)
$\text{Var}:\pi_{1}(T^{\ast},t_{0})\to\text{Aut}(H_{\bullet}(Y_{t_{0}})).$

The task is now to compute $\text{Var}_{[\gamma]h(t_{0})}$. This is generally
a difficult problem. A rather simple case which can be applied in particular
to one-loop Feynman integrals however is extensively studied in [Pha11]. These
are the so called simple pinches we discuss further below.
The idea to understand these simple pinches is to localize the problem such
that the variation yields a homology class which has a representative with
support contained in an arbitrarily small neighborhood of a pinch point. We
can not expect that the problem can be localized like this for any arbitrary
loop. But there is a class of loops whose homotopy classes span the
fundamental group of $T\backslash L$ if $T$ is simply connected which allow
for such a localization:
Let $u\in L$ be a point of codimension 1. Then there are coordinates
$(t_{1},\ldots,t_{m})$ of $T$ defined in a neighborhood $U\subset T$ of $u$
such that $L$ locally looks like the set $\\{t_{1}=0\\}$. Let $\theta:[0,1]\to
T\backslash L$ be a path from some basepoint $u_{0}\in T\backslash L$ to some
$u_{1}\in U-L$. Let $\omega:[0,1]\to U-L$ be a loop based at $u_{1}$ that
traces a circle in the coordinate $t_{1}$ with $t_{2},\ldots,t_{m}$ fixed.
Then $\gamma:=\theta\omega\theta^{-1}$ is a loop in $T\backslash L$ based at
$u_{0}$. A loop constructed like this is called a simple loop. In an abuse of
language, we shall call elements of the fundamental group which can be
represented by a simple loop also simple loops. In the cases we study here, it
is always possible to restrict our attention to simple loops without
sacrificing any amount of generality due to the following

######  Proposition 2.26 ([Pha11]).

The fundamental group $\pi_{1}(T\backslash L,u_{0})$ of $T\backslash L$ is
spanned by simple loops if and only if $T$ is simply connected.

####  2.3.1. Leray’s Calculus of Residues

In the preceding subsubsection, we have seen how to compute the discontinuity
of a function defined by an integral of the form (1.3) by extending a local
section of $\mathcal{F}_{Y/T}^{n}$ to a multivalued global section. To obtain
Cutkosky’s Theorem in Section 4, we also need to employ the multi-variant
version of the Residue Theorem, going back Leray [Ler59], as already eluded to
in the introduction.
First recall the regular Residue Theorem from complex analysis: Let
$U\subset\mathbb{C}$ be a simply connected open set and let $a\in U$ be a
point. Then the theorem states that for a holomorphic function
$f:U\backslash\\{a\\}\to\mathbb{C}$ and a positively oriented simple closed
curve $\gamma:[0,1]\to U\backslash\\{a\\}$ we have

(2.45) $\oint_{\gamma}f(z)dz=2\pi i\cdot\text{Res}(f,a),$

where $\text{Res}(f,a)$ is the residue of $f$ at $a$.555Of course there are
more general formulations of the Residue Theorem (e.g. for a finite list of
points $(a_{1},\ldots,a_{n})$). But this version suffices here for the
illustration of the concept. To generalize this to multiple dimensions, we
essentially have to answer two questions: What is the multi-dimensional analog
for the curve $\gamma$ along which we integrate and what is the multi-
dimensional analog of the residue?
First we attend to the first question. We follow [Pha11]. Recall that for a
smooth manifold $M$ and a smooth submanifold $S\subset M$ a tubular
neighborhood of $S$ in $M$ is a vector bundle $\pi:E\to S$ together with a
smooth map $J:E\to M$ such that:

  * •

If $i:S\hookrightarrow M$ is the natural embedding and $0_{E}$ is the zero
section we have $J\circ 0_{E}=i$.

  * •

There exist $U\subset E$ and $V\subset M$ with $0_{E}[S]\subset U$ and
$S\subset V$ such that $J|_{U}:U\overset{\sim}{\to}V$ is a diffeomorphism.

We are mainly interested in $V$ and in an abuse of language call this a
tubular neighborhood of $S$ as well. Note that there is an associated
retraction $\mu:V\to S$ by applying $(J|_{U})^{-1}$, retracting to the zero
section $0_{E}$ and going back to $M$ via $J|_{U}$. Now suppose $X$ is a
complex analytic manifold and $S\subset X$ a closed analytic submanifold of
codimension 1. Fix a closed tubular neighborhood $V\subset X$ of $S$. The
associated retraction $\mu:V\to S$ induces a disk bundle structure over $S$.
Then if $\sigma$ is a simplex in $S$, its preimage $\mu^{-1}(\sigma)$ is
homeomorphic to $\sigma\times D$, where $D$ is the unit disk. On the level of
chains we obtain

(2.46) $\mu^{\ast}\sigma=D\otimes\sigma.$

The boundary of $\mu^{\ast}\sigma$ is thus

(2.47) $\partial\mu^{\ast}\sigma=\partial
D\otimes\sigma+D\otimes\partial\sigma.$

Using this result, we define a homomorphism $\delta_{\mu}:C_{p}(S)\to
C_{p+1}(X-S)$ by setting $\delta_{\mu}(\sigma):=\partial D\otimes\sigma$ for
each simplex $\sigma$ and extending by linearity. Note that $\delta_{\mu}$
anti-commutes with the boundary $\partial$:

(2.48) $\partial\delta_{\mu}\sigma=\partial(\partial D\otimes\sigma)=-\partial
D\otimes\partial\sigma=-\delta_{\mu}\partial\sigma$

Hence $\delta_{\mu}$ descends to a homomorphism

(2.49) $\delta_{\ast}:H_{p}(S)\to H_{p+1}(X\backslash S)$

which we call the Leray coboundary. It can be shown that this map does not
depend on the choice of $\mu$ which is why we dropped the subscript. Note that
this construction gives a simple closed curve $\gamma$ if we consider the one-
dimensional case where $S$ is just a point.
Now we turn to the second question. To answer it, let us consider a closed
differential $p$-form $\omega$ on $X\backslash S$. For all $x\in S$, let
$s_{x}$ be a local equation for $S$ near $x$. If for all $x\in S$ the form
$s_{x}\cdot\omega$ can be extended to a differential $p$-form on a
neighborhood of $x$, we say that $\omega$ has a polar singularity of order $1$
along $S$.

######  Proposition 2.27 ([Pha11]).

If $\omega$ has a polar singularity of order 1 along $S$ then for every $x\in
X$ there exist differential forms $\psi_{x}$ and $\theta_{x}$ defined on a
neighborhood of $x$ such that

(2.50) $\omega=\frac{ds_{x}}{s_{x}}\wedge\psi_{x}+\theta_{x}.$

Furthermore $\psi_{x}|_{S}$ is closed and depends only on $\omega$. We call
$\psi_{x}|_{S}$ the residue of $\omega$.

In the situation of Proposition 2.27, we denote

(2.51)
$\text{res}[\omega]:=\left.\frac{s_{x}\omega}{ds_{x}}\right|_{S}:=\psi_{x}|_{S}\quad\in\quad\Omega^{p-1}(S).$

The $(p-1)$-form $\text{res}[\omega]$ is called the residue form of $\omega$.
Now if $\omega$ is any closed form on $X\backslash S$, then $\omega$ is
cohomologous to a closed form $\tilde{\omega}$ on $X\backslash S$ with a
simple pole along $S$ [Pha11]. Thus it makes sense to define a homology class

(2.52) $\text{Res}[\omega]:=[\text{res}[\tilde{\omega}]]\quad\in\quad
H^{p-1}(S)$

and it can be shown that it only depends on the homology class of
$\tilde{\omega}$ in $X\backslash S$ [Pha11].

######  Theorem 2.28 (Residue Theorem, [Pha11]).

Let $\gamma$ be a $(p-1)$-cycle in $S$ and $\omega$ a closed differential
$p$-form on $X\backslash S$. Then the following identity holds:

(2.53) $\int_{\delta\gamma}\omega=2\pi i\int_{\gamma}\mathrm{Res}[\omega]$

We also have to deal with situations in which $S$ is not a submanifold but a
union of $N$ closed submanifolds $S_{1},\ldots,S_{N}\subset X$ in general
position. To achieve this, we can iterate the construction above. For each
$k\in\\{1,\ldots,N\\}$ the manifold $S_{1}\cap\cdots\cap S_{k}$ is a closed
submanifold in which $S_{k+1},\ldots,S_{N}$ intersect in general position.
Thus we obtain a sequences of maps

(2.54) $H^{p}(X\backslash(S_{1}\cup\cdots\cup
S_{N}))\overset{\text{Res}_{1}}{\to}H^{p-1}(S_{1}-S_{2}\cup\cdots\cup
S_{N})\overset{\text{Res}_{2}}{\to}H^{p-2}(S_{1}\cap S_{2}-S_{3}\cup\cdots
S_{N})\overset{\text{Res}_{3}}{\to}\cdots$

and

(2.55) $H_{p}(X\backslash(S_{1}\cup\cdots\cup
S_{N}))\overset{\delta_{1}}{\leftarrow}H_{p-1}(S_{1}-S_{2}\cup\cdots\cup
S_{N})\overset{\delta_{2}}{\leftarrow}H_{p-2}(S_{1}\cap S_{2}-S_{3}\cup\cdots
S_{N})\overset{\delta_{3}}{\leftarrow}\cdots$

This allows us to define the composite maps

(2.56)
$\text{Res}^{m}:=\text{Res}_{m}\circ\cdots\circ\text{Res}_{1}\quad\text{and}\quad\delta^{m}:=\delta_{1}\circ\cdots\circ\delta_{m}.$

Similarly to the case of one manifold $S$, we obtain the following

######  Theorem 2.29 (Iterated Residue Theorem, [Pha11]).

Let $\gamma$ be a $(p-m)$-cycle in $S$ and $\omega$ a closed differential
$p$-form on $X\backslash S$. Then the following identity holds:

(2.57) $\int_{\delta^{m}\gamma}\omega=(2\pi
i)^{m}\int_{\gamma}\mathrm{Res}^{m}[\omega]$

The (iterated) Leray residue has the following nice property that we employ
later on:

######  Proposition 2.30.

The Leray residue commutes with pullbacks, i.e. for any differential map
$f:Y\to X$ between two smooth manifolds $X$ and $Y$, any submanifold $S\subset
X$ of codimension 1 and any differential form $\omega$ on $X\backslash S$ we
have

(2.58) $f^{\ast}\mathrm{Res}^{m}[\omega]=\mathrm{Res}^{m}[f^{\ast}\omega].$

###### Proof.

It suffices to show the case $m=1$ since the case for general $m$ follows by
induction. We can assume without loss of generality that $\omega$ has a polar
singularity of order 1 along $S$ (otherwise we replace $\omega$ by a
cohomologous form with this property). First note that $f^{\ast}\omega$ is a
closed differential form on $Y\backslash f^{-1}(S)$. According to Proposition
2.27 we can write $\omega=\frac{ds_{x}}{s_{x}}\wedge\psi_{x}+\theta_{x}$. We
compute

(2.59)
$f^{\ast}\omega=f^{\ast}(\frac{ds_{x}}{s_{x}}\wedge\psi_{x}+\theta_{x})=f^{\ast}\frac{ds_{x}}{s_{x}}\wedge
f^{\ast}\psi_{x}+f^{\ast}\theta_{x}=\frac{d(s_{x}\circ f)}{s_{x}\circ f}\wedge
f^{\ast}\psi_{x}+f^{\ast}\theta_{x}.$

Now $s_{x}\circ f$ is a local equation for $f^{-1}(S)$ around any point in
$f^{-1}(x)$. Thus we conclude

(2.60)
$\mathrm{Res}[f^{\ast}\omega]=[(f^{\ast}\psi_{x})|_{S}]=f^{\ast}[\psi_{x}|_{S}]=f^{\ast}\mathrm{Res}[\omega]$

as claimed. ∎

As mentioned in the introduction, Cutkosky’s Theorem involves the use of
$\delta$-functions (which are of course not functions in the conventional
sense). These are usually defined in the language of functional analysis. But
in the context of singular integrals we can define the $\delta$-function and
its derivatives evaluated at a test function by integrating the residue of a
differential form induced by the given test function.

######  Definition 2.31 ([Pha11]).

For any test function $f$ set $\omega:=fdx_{1}\wedge\cdots dx_{n}$. Then we
define

(2.61)
$\delta^{(a)}(S)[f]:=\int_{S}\delta^{(a)}(f):=\int_{S}\left.\frac{d^{\alpha}\omega}{ds^{\alpha+1}}\right|_{S}.$

####  2.3.2. Simple Pinches

Now we want to compute the variation of homology classes for a rather simple
situation in which the singular points of the differential form is a finite
union of manifolds which are in general position except at isolated points.
Again, we follow [Pha11].

######  Definition 2.32.

Let $Y$ and $T$ be two complex analytic manifolds, $\pi:Y\to T$ a smooth map
and $S_{1},\ldots,S_{N}\subset Y$ complex analytic submanifolds of codimension
1. Denote by $(S_{i})_{t}:=\pi^{-1}(t)\cap S_{i}$ the fiber of $S_{i}$ over
$t$ for all $i\in\\{1,\ldots,N\\}$ and $t\in T$. We say that the system
$S_{1},\ldots,S_{N}$ has a simple pinch at $y\in Y$ if there is a coordinate
chart $(\varphi=(x_{1},\ldots,x_{n},t_{1},\ldots,t_{m}),U)$ in a neighborhood
$U\subset Y$ of $y$ such that there are local equations

(2.62) $s_{1}=x_{1},\quad\ldots\quad s_{N-1}=x_{N-1}$

and

(2.63) $s_{N}=t_{1}-x_{1}-\cdots-x_{N-1}-x_{N}^{2}-\cdots-x_{n}^{2}.$

for $S_{1},\ldots,S_{N}$ around $y$.

In should be remarked that the case $N=n+1$ is a little different from the
remaining cases $N\leq n$. The first one is called a linear pinch while the
remaining cases are called quadratic pinches.
Suppose $S_{1},\ldots,S_{N}$ has a simple pinch at $y\in Y$ and let
$(\varphi,U)$ be the coordinate chart from Definition 2.32. Then $t:=\pi(y)$
must necessarily be a point of codimension 1 in the Landau surface. Suppose we
fix $t\in T$ such that $t_{1}$ is real and positive and denote
$U_{t}:=Y_{t}\cap U$. Then

(2.64)
$U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t}\simeq\\{(0,\ldots,0,x_{N},\ldots,x_{n})\in\varphi(U)\;|\;x_{N}^{2}+\cdots+x_{n}^{2}=t_{1}\\}\simeq
S^{n-N}_{\mathbb{C}},$

where $S^{k}_{\mathbb{C}}$ denotes the complex unit sphere of dimension
$k\in\mathbb{N}$.666Equation (2.64) also holds in the linear case $N=n+1$ if
we agree on the convention that the -1 dimensional sphere is empty. It is
well-known that the complex $k$-sphere deformation retracts to the real
$k$-sphere $S^{k}$. Thus in case $n>N$ we obtain

(2.65) $H_{k}(U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})\simeq
H_{k}(S^{n-N})\simeq\begin{cases}\mathbb{Z}&\text{if }k=0,n-N\\\
0&\text{otherwise}\end{cases}$

for the homology groups of $U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t}$ and
the generator of $H_{n-N}(U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})$ is
represented by a real $(n-N)$-sphere contained within
$U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t}$. On the level of cycles, this
is called the vanishing sphere denoted by

(2.66)
$e:=\\{(x_{1},\ldots,x_{n})\in\mathbb{C}^{n}\;|\;x_{1},\ldots,x_{n}\in\mathbb{R},\;s_{1}(x,t)=\cdots=s_{N}(x,t)=0\\}.$

It is the iterated boundary of the vanishing cell

(2.67)
$\mathbf{e}:=\\{(x_{1},\ldots,x_{n})\in\mathbb{C}^{n}\;|\;x_{1},\ldots,x_{n}\in\mathbb{R},\;s_{1}(x,t),\ldots,s_{N}(x,t)\geq
0\\}.$

This means that $e=(\partial_{1}\circ\cdots\circ\partial_{N})(\mathbf{e})$
where $\partial_{i}$ is the operator taking the boundary within $(S_{i})_{t}$
for all $i\in\\{1,\ldots,N\\}$. Furthermore we define the vanishing cycle

(2.68) $\tilde{e}:=(\delta_{1}\circ\cdots\circ\delta_{N})(e)$

by taking the iterated Leray coboundary of the vanishing sphere. These define
homology classes

(2.69) $[e]\in H_{n-N}(U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t}),$ (2.70)
$[\mathbf{e}]\in H_{n}(U_{t},(S_{1})_{t}\cup\cdots\cup(S_{N})_{t}),$ (2.71)
$[\tilde{e}]\in H_{n}(U_{t}-(S_{1})_{t}\cup\cdots\cup(S_{N})_{t}).$

These three classes are the protagonists in the calculation of the variation
in the situation of a simple pinch. Due to the following proposition, it
suffices in principle to know one of the three.

######  Proposition 2.33 ([FFLP65]).

In the situation above, the maps $\partial_{i}$ and $\delta_{i}$ are
isomorphisms on the level of homology. In particular $[\mathbf{e}]$ generates
the relative homology group
$H_{n}(U_{t},(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})$ and $[\tilde{e}]$
generates the homology group
$H_{n}(U_{t}-(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})$ of the complement.

In fact we can use this generating property as the definition of the three
classes in question.777Of course this is a slight abuse of language since
above, we defined the three objects in question as cycles. However, this does
not cause any problems since all computations in this work depend only on the
homology classes.

######  Definition 2.34.

The vanishing cell $[\mathbf{e}]$, vanishing sphere $[e]$ and vanishing cycle
$[\tilde{e}]$ is a generator of
$H_{n-N}(U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})$,
$H_{n}(U_{t},(S_{1})_{t}\cup\cdots\cup(S_{N})_{t})$ and
$H_{n}(U_{t}-(S_{1})_{t}\cup\cdots\cup(S_{N})_{t})$ respectively.

Note that this definition does only determine $[\mathbf{e}]$, $[e]$ and
$[\tilde{e}]$ up to orientation ($\mathbb{Z}$ has two generators: 1 and -1).
In what follows it will become evident that this choice of orientation does
not matter as all calculation in which these three classes appear are
independent of the chosen orientation.
There are two marginal cases that we also need to cover: In case $n=N$, we are
dealing with the 0-sphere which consists of two points. Thus

(2.72) $H_{k}(U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t})\simeq
H_{k}(S^{n-N})\simeq\begin{cases}\mathbb{Z}^{2}&\text{if }k=0\\\
0&\text{otherwise}\end{cases}$

and the 0th homology group is generated by the two points of the 0-sphere. In
case linear case $n=N+1$, the intersection
$U_{t}\cap(S_{1})_{t}\cap\cdots\cap(S_{N})_{t}$ is empty and the vanishing
sphere does not exist. In accordance with [Pha11], we agree on the convention
that all associated vanishing classes are 0 in this case.

####  2.3.3. The Picard-Lefschetz Formula for Simple Pinches

We have seen that an element $[\gamma]\in\pi_{1}(T\backslash L,t_{0})$ gives
rise to an automorphism $[\gamma]_{\ast}$ of $H_{\bullet}(Y_{t_{0}}\backslash
S_{t_{0}})$. In the situation of a simple pinch, the problem can be localized
in the sense that the variation $([\gamma]_{\ast}-\text{id})[h]$ of a class
$[h]\in H_{n}(Y_{t_{0}}\backslash S_{t_{0}})$ can be represented by a cycle
with support entirely contained within $U$, the domain of the coordinate chart
$(\varphi,U)$ appearing in the definition of a simple pinch. Since the
homology group
$H_{n}((U)_{t_{0}}-(S_{1})_{t_{0}}\cap\cdots\cap(S_{N})_{t_{0}})$ is spanned
by a single element (the vanishing cycle), it is clear that the variation must
thus be some integer multiple $N\cdot[\tilde{e}]$ of the vanishing
cycle.888This also holds in the marginal case $N=n$ (see [Pha11]) but the
argument is different. The discontinuity of the integral (2.23) corresponding
to $[\gamma]$ is thus given by

(2.73) $\int_{N\tilde{e}}\omega=N\cdot\int_{\tilde{e}}\omega$

and the remaining task is to determine the integer $N$. This can be done by a
Picard-Lefschetz type formula introduced in [FFLP65]. To understand its
content, we first need the notion of an intersection index. There are quite
general definitions of this concept.999For example in [Pha11], the
intersection index of two currents is defined. But for our purposes it
suffices to consider the intersection index of two closed oriented manifolds
with- or without boundary intersecting transversally and with dimensions
adding up to the dimension of the whole space.

######  Proposition 2.35 ([Pha11]).

Let $M$ be a smooth manifold and $S_{1},S_{2}\subset M$ two orientable
submanifolds (with or without boundary) such that

(2.74) $S_{1}\cap\partial S_{2}=S_{2}\cap\partial S_{1}=\emptyset.$

If $S_{1}$ and $S_{2}$ intersect transversally, then $S_{1}\cap S_{2}$
consists of a finite number of isolated points and the intersection index
$\langle S_{2},S_{1}\rangle$ of $S_{1}$ and $S_{2}$ is given by

(2.75) $\langle S_{2},S_{1}\rangle=N_{+}-N_{-}\quad\in\quad\mathbb{Z},$

where $N_{+}$ (resp. $N_{-}$) is the number of points in $S_{1}\cap S_{2}$ at
which the orientations of $S_{1}$ and $S_{2}$ match (resp. do not match).

We can express the integer $N$ from (2.73) as the intersection index of the
vanishing cell with the integration cycle as follows:

######  Theorem 2.36 (Picard-Lefschetz Formula, [Pha11]).

The integer $N$ in equation (2.73) is given by

(2.76) $N=(-1)^{\frac{(n+1)(n+2)}{2}}\cdot\langle\mathbf{e}|h\rangle$

where $h$ is the integration cycle at $t$.

In this paper, we do not explicitly compute the intersection index relevant
for one-loop Feynman integrals since it involves a very different array of
techniques, mostly from homological algebra. Therefore this step is postponed
to the second part of this work.

####  2.3.4. Variation and Discontinuity Along Products of Loops

So far, we only discussed how to compute the variation and discontinuity for
simple pinches along simple loops around points of codimension 1. To fully
understand the discontinuities of one-loop Feynman integrals, it is useful to
be able to relate the variation along products of loops to the variation along
the individual factors. For the case of simple pinches, we have the following:

######  Lemma 2.37.

Let $[\gamma_{1}],[\gamma_{2}]\in\pi_{1}(T\backslash L,t_{0})$ be two simple
loops. Then

(2.77)
$\mathrm{Var}_{[\gamma_{1}]\cdot[\gamma_{2}]}=\mathrm{Var}_{[\gamma_{1}]}+\mathrm{Var}_{[\gamma_{2}]}+\mathrm{Var}_{[\gamma_{2}]}\mathrm{Var}_{[\gamma_{1}]}$

and

(2.78)
$\mathrm{Disc}_{[\gamma_{1}]\cdot[\gamma_{2}]}=\mathrm{Disc}_{[\gamma_{1}]}+\mathrm{Disc}_{[\gamma_{2}]}+\mathrm{Disc}_{[\gamma_{2}]}\mathrm{Disc}_{[\gamma_{1}]}.$

###### Proof.

Let $h\in H_{n}(Y_{t_{0}})$. Denote
$\text{Var}_{[\gamma_{i}]}h=:N_{i}\tilde{e}_{i}$ for $i=1,2$. We compute

(2.79)
$\begin{split}\text{Var}_{[\gamma_{1}]\cdot[\gamma_{2}]}h&=\gamma_{2}^{\ast}\gamma_{1}^{\ast}h-h\\\
&=\gamma_{2}^{\ast}(h+N_{1}[\tilde{e}_{1}])-h\\\
&=N_{2}[\tilde{e}_{2}]+N_{1}\gamma_{2}^{\ast}[\tilde{e}_{1}]\\\
&=N_{1}[\tilde{e}_{1}]+N_{2}[\tilde{e}_{2}]+(\gamma_{2}^{\ast}-1)(N_{1}\tilde{e}_{1})\\\
&=\text{Var}_{[\gamma_{1}]}h+\text{Var}_{[\gamma_{2}]}h+\text{Var}_{[\gamma_{2}]}\text{Var}_{[\gamma_{1}]}h.\end{split}$

The corresponding formula for the discontinuity follows immediately. ∎

We can easily generalize this to arbitrary finite products of loops:

######  Proposition 2.38.

Let $[\gamma_{1}],\ldots,[\gamma_{k}]\in\pi_{1}(T^{\ast},t_{0})$ be simple
loops. Then

(2.80) $\text{Var}_{[\gamma_{1}]\cdots[\gamma_{k}]}=\sum_{i=1}^{k}\sum_{1\leq
j_{1}<\cdots<j_{i}\leq
k}\text{Var}_{[\gamma_{j_{i}}]}\cdots\text{Var}_{[\gamma_{j_{1}}]}$

and

(2.81) $\text{Disc}_{[\gamma_{1}]\cdots[\gamma_{k}]}=\sum_{i=1}^{k}\sum_{1\leq
j_{1}<\cdots<j_{i}\leq
k}\text{Disc}_{[\gamma_{j_{i}}]}\cdots\text{Disc}_{[\gamma_{j_{1}}]}.$

###### Proof.

We proceed by induction on $k$. For $k=1$ the formulas are trivial. For the
induction step we compute

(2.82)
$\begin{split}\text{Var}_{[\gamma_{1}]\cdots[\gamma_{k+1}]}&=\text{Var}_{[\gamma_{1}]\cdots[\gamma_{k}]}+\text{Var}_{[\gamma_{k+1}]}+\text{Var}_{[\gamma_{k+1}]}\text{Var}_{[\gamma_{1}]\cdots[\gamma_{k}]}\\\
&=\sum_{i=1}^{k}\sum_{1\leq j_{1}<\cdots<j_{i}\leq
k}\text{Var}_{[\gamma_{j_{i}}]}\cdots\text{Var}_{[\gamma_{j_{1}}]}+\text{Var}_{[\gamma_{k+1}]}\\\
&+\text{Var}_{[\gamma_{k+1}]}\sum_{i=1}^{k}\sum_{1\leq j_{1}<\cdots<j_{i}\leq
k}\text{Var}_{[\gamma_{j_{i}}]}\cdots\text{Var}_{[\gamma_{j_{1}}]}\\\
&=\sum_{1\leq j_{1}\leq
k+1}\text{Var}_{[\gamma_{j_{1}}]}+\sum_{i=2}^{k+1}\sum_{1\leq
j_{1}<\ldots<j_{i}\leq
k+1}\text{Var}_{[\gamma_{j_{i}}]}\cdots\text{Var}_{[\gamma_{j_{1}}]}\\\
&=\sum_{i=1}^{k+1}\sum_{1\leq j_{1}<\cdots<j_{i}\leq
k+1}\text{Var}_{[\gamma_{j_{i}}]}\cdots\text{Var}_{[\gamma_{j_{1}}]},\end{split}$

where we used Lemma 2.37 in the first step and the induction hypothesis in the
second step. Again the formula for the discontinuity follows immediately. ∎

With Proposition 2.38, we can compute the discontinuity along any loop around
points where codimension 1 parts of the Landau surface intersect in general
position.

##  3\. Feynman Graphs and Integrals

Now we turn to the study of Feynman integrals as holomorphic functions in the
external momenta. To ensure that there are no ambiguities regarding the
notation, conventions and terminology employed in this work and to revisit the
basic notions of the field for the non-expert reader, we quickly recall the
definition of Feynman graphs and integrals. For a more detailed exposition,
the reader is referred to [Wei95] or [We06]. A graph always means a finite
multi-graph in this text. For a graph $G$ we denote its underlying set of
vertices by $V(G)$ and its underlying (multi-)set of edges by $E(G)$. The
first Betti number of $G$, which is the maximal number of independent cycles,
is denoted by $h_{1}(G)$. It is customary to say that $G$ has $h_{1}(G)$
loops. For us a Feynman integral means a graph together with some additional
information specifying at which vertices how many particles are in- or
outgoing. More specifically we make the following

######  Definition 3.1.

A Feynman graph $(G,\phi)$ is a graph $G$ together with a map
$\phi:V(G)\to\mathbb{N}$ called the external structure.

This is not the most general definition of Feynman graphs. In many physical
theories, for example quantum electro- or chromodynamics, one wants to
distinguish different types of edges (see Figure 1 for two examples). The
essential features of the analytic structure of the parameter-dependent
integrals associated to such a graph are however already captured in our
setting. So we do not go into detail regarding these more general Feynman
graphs and integrals.
A Feynman graph (or Feynman diagram as they are sometimes called) is a
representation of a collection of possible ways elementary particles can
interact: The external structure represents in- and outgoing particles. It
assigns each vertex a number of external momenta. It is common to draw a
pictorial representation of a Feynman graph $(G,\phi)$ by drawing the
underlying graph $G$ as usual and then attaching $\phi(v)$ lines not connected
to a second vertex to each vertex $v\in V(G)$.

Figure 1. Two Feynman graphs in quantum electrodynamics, both contributing to
the probability amplitude of electron-positron scattering.

Each Feynman graph $G$ is assigned a Feynman integral by applying the Feynman
rules to it, which contributes to the probability amplitude of a given
process. There are several equivalent descriptions of this procedure. Here we
focus on the so called momentum space representation which is obtained from a
Feynman graph $(G,\phi)$ in the following manner: First we equip the graph $G$
with an arbitrary orientation and denote by $\mathcal{E}\in
M(|V(G)|\times|E(G)|;\mathbb{Z})$ the corresponding incidence matrix. For each
edge $e\in E(G)$ we write down a factor of
$\frac{1}{(k_{e}^{2}+m_{e}^{2})^{\lambda_{e}}}$ (called the propagator), where
$k_{e}\in\mathbb{C}^{D}$ is called the internal momentum and
$m_{e}\in\mathbb{R}_{\geq 0}$ the mass associated to the edge $e$. The
$\lambda_{e}$ are generally complex numbers with positive real part called
analytic regulators. For each vertex $v\in V(G)$ we write down a constant
factor (which we ignore in this text since it does not influence the analytic
structure) and assign $\phi(v)$ external momenta
$p_{v,1},\ldots,p_{v,\phi(v)}\in\mathbb{C}^{D}$ to it. We denote the total
external momentum at $v$ by $p_{v}:=\sum_{i=1}^{\phi(v)}p_{v,i}$. Then we
enforce momentum conservation at every vertex $v\in V(G)$, i.e. we insist on
the internal and external momenta satisfying $\sum_{e\in
E(G)}\mathcal{E}_{v,e}k_{e}+p_{v}=0$ for every $v\in V(G)$.101010Note that
this implies that all external momenta are counted as incoming in our
convention. Usually one wants to factor out the overall momentum conservation,
i.e. the condition $\sum_{v\in V(G)}p_{v}=0$, as it does not depend on the
internal momenta $k_{e}$. To do this, one may fix some vertex $v_{0}\in V(G)$
and drop the momentum conservation at $v_{0}$. The Feynman integral $I(G)$
corresponding to a Feynman graph $G$ thus reads

(3.1) $I(G)(p):=\int_{\mathbb{R}^{|E(G)|D}}d^{|E(G)|D}k\prod_{e\in
E(G)}\frac{1}{(k_{e}^{2}+m_{e}^{2})^{\lambda_{e}}}\prod_{v\in
V(G)\backslash\\{v_{0}\\}}\delta(\sum_{e\in
E(G)}\mathcal{E}_{v,e}k_{e}+p_{v}).$

Integrating out the $\delta$s leads to a linear system of equations for the
$k_{e}$ and it can be shown that all but $h_{1}(G)$ of the internal momenta
can be eliminated by solving this system. The remaining internal momenta are
then integrated over and the result reads

(3.2) $I(G)=\int_{\mathbb{R}^{h_{1}(G)D}}\frac{d^{h_{1}(G)D}k}{\prod_{e\in
E(G)}((K_{e}(k)+P_{e}(p))^{2}+m_{e}^{2})^{\lambda_{e}}},$

where $K_{e}:\mathbb{C}^{h_{1}(G)D}\to\mathbb{C}^{D}$ and
$P_{e}:\mathbb{C}^{(\sum_{v\in V(G)}\phi(v))D}\to\mathbb{C}^{D}$ are linear
maps for all $e\in E(G)$. The resulting propagators and thus the maps $K_{e}$
and $P_{e}$ are not uniquely determined, but the result of the integration (if
it is well-defined at all) is independent of the remaining freedom of choice.
In physics, one is almost always concerned with Minkowski momenta to adhere to
the principals of special relativity. This means in the physics literature the
internal momenta are considered to be real, but the momentum-squares are
typically defined as $k^{2}:=-k_{0}^{2}+k_{1}^{2}+\cdots+k_{D-1}^{2}$.111111It
is customary to start indexing the components of momenta with 0 instead of 1.
It should also be mentioned that we employ a different sign convention than
most particle physicists (it appears that the author’s convention is in fact
met with open hatred, see for example the footnote on page 2 in [Tan10]), who
like to set $k^{2}=k_{0}^{2}-k_{1}^{2}-\cdots-k_{D-1}^{2}$. In this case the
propagator needs to be $(k_{e}^{2}-m_{e}^{2})^{-\lambda_{e}}$ instead.
Unfortunately this immediately leads to problems since the integration domain
now includes the poles at $k_{e}^{2}+m_{e}^{2}=0$. The usual ploy to avoid
this is to introduce a small complex shift in the propagator by replacing
$(k_{e}^{2}+m_{e}^{2})^{-\lambda_{e}}$ with
$(k_{e}^{2}+m_{e}^{2}-i\epsilon)^{\lambda_{e}}$ for some $0<\epsilon\ll 1$.
Then the integration is carried out and the limit $\epsilon\to 0^{+}$ is taken
at the very end of the calculation. This is called the
$i\epsilon$-prescription. We shall see that this is not satisfactory for our
purposes and we take a different route: In our setting it is necessary to
consider complex internal momenta. The Minkowski momenta can then be
identified with those momenta that have purely imaginary 0th component and all
remaining components real. We however start with an entirely real integration
domain $\mathbb{R}^{D}$ (in the physics literature this is known as a
Euclidean Feynman integral) instead of choosing to integrate over all
Minkowski momenta $i\mathbb{R}\times\mathbb{R}^{D-1}$, which never meets the
zero locus of $k_{e}^{2}+m_{e}^{2}$ as long as $m_{e}^{2}>0$. This serves as
the starting point for an analytic continuation. In Section 2 we saw how an
analytic continuation requires us to continuously deform the integration
domain as we move along a path in the space of external momenta. In particular
we show in Section 4 how this is done explicitly for one-loop Feynman
integrals in the case where we want to continue from Euclidean external
momenta to Minkowski external momenta and explain how this is in agreement
with the $i\epsilon$-prescription. For now, we simply define our Feynman
integrals as Euclidean integrals.

######  Definition 3.2.

Let $G$ be a Feynman graph. The corresponding Feynman integral in momentum
space representation in $D\in\mathbb{N}$ dimensions is

(3.3) $I(G)(p):=\int_{\mathbb{R}^{h_{1}(G)D}}d^{h_{1}(G)D}k\prod_{e\in
E(G)}\frac{1}{((K_{e}(k)+P_{e}(p))^{2}+m_{e}^{2})^{\lambda_{e}}},$

where $K_{e}$ and $P_{e}$ are the linear maps obtained as described above.

As mentioned above, we have omitted some constant factors in this definition
as they do not play a role in the analytical structure. These factors are only
needed to compare numerical values obtained from Feynman integrals with the
experiment. Note that it suffices to understand the bridge-less graphs $G$, or
one-particle-reducible (1PI) graphs as they are called in the physics
literature, to understand all Feynman integrals, as integrals corresponding to
graphs with bridges factorize into 1PI contributions.
Up to this point, we have not stated what complex manifold we would like to
(or even can) consider a Feynman integral to be a function on. In this work,
we focus on the dependence on the external momenta. The masses are regarded as
fixed and positive (the massless case works differently and we postpone the
discussion of this case to future research). Thus an obvious choice would be
$(\mathbb{C}^{D})^{\sum_{v\in V(G)}\phi(v)}$. But this can be simplified.
First of all the external momenta are restricted to a hyperplane by overall
momentum conservation. In particular we can express one of the momenta as
minus the sum of all the others, so that $(\mathbb{C}^{D})^{\sum_{v\in
V(G)}\phi(v)-1}$ would be a sufficient space to work with.¸ Furthermore it is
well-known in physics that a Feynman integral is Lorentz-invariant, i.e.
applying the same Lorentz transformation to all external momenta does not
change the value of the integral. In our setup, this statement takes a
slightly different form: Since we need to work with arbitrary complex momenta
and not just Minkowski momenta, our transformation group is different. But the
general idea stays the same: The integral should be invariant under all linear
transformations of the external momenta leaving the products $p_{i}p_{j}$
unchanged. This group is the Lie group $O(D,\mathbb{C})$ of complex orthogonal
$D\times D$-matrices. We define an action of $O(D,\mathbb{C})$ on
$(\mathbb{C}^{D})^{n}$ by

(3.4) $g\cdot(x_{1},\ldots,x_{n})=(g\cdot x_{1},\ldots,g\cdot x_{n})$

for all $g\in O(D,\mathbb{C})$ and all $n\in\mathbb{N}^{\ast}$, where the
multiplication on the right is just regular matrix-vector multiplication. The
invariance of Feynman integrals under $O(D,\mathbb{C})$ for the one-loop case
is proven in the next section.

##  4\. One-Loop Feynman Graphs

We now begin our investigation of one-loop Feynman graphs, i.e. Feynman graphs
$G$ with $h_{1}(G)=1$. As mentioned in Section 3, it suffices to consider the
1-particle-irreducible graphs to understand their analytic structure. The 1PI
graphs with one loop are the cycle graphs $C_{n}$ (viewed as Feynman graphs)
shown in the following figure:

$p_{n}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$$p_{3}$$k+\sum_{i=1}^{3}p_{i}$$p_{4}$$k+\sum_{i=1}^{4}p_{i}$$p_{5}$

On the level of (multi-)sets this means

(4.1) $V(C_{n}):=\\{1,\ldots,n\\},\quad
E(C_{n}):=\\{\\{1,2\\},\\{2,3\\},\ldots,\\{n-1,n\\},\\{1,n\\}\\}.$

Note that we assigned exactly one external momentum to each vertex, i.e. the
external structure $\phi$ of the graphs under consideration is simply given by
$\phi(v)=1$ for all $v\in V(C_{n})$.121212For physicists this means we
consider Feynman graphs in $\phi^{3}$-theory, where $\phi$ stands for a scalar
field and the exponent gives the power with which this field occurs in the
Lagrangian. The later is also the allowed valency for the vertices of graphs
appearing in a perturbative expansion. We could consider more general external
structures, but this bears no relevance to our discussion: If there is more
than one line attached to a vertex, the integral depends only on the sum of
all external momenta going into that vertex. In the coordinates we chose, the
general one-loop Feynman integral in momentum space representation in $D$
dimensions from Definition 3.2 reads

(4.2)
$I(C_{n})(p)=\int_{\mathbb{R}^{D}}\frac{d^{D}k}{\prod_{i=1}^{n}((k+\sum_{j=1}^{i-1}p_{j})^{2}+m_{i}^{2})^{\lambda_{i}}}=\int_{\mathbb{R}^{D}}\frac{d^{D}k}{\prod_{i=1}^{n}((k+P^{(i)}(p))^{2}+m_{i}^{2})^{\lambda_{i}}}.$

To ease notation, we set $P^{(i)}(p):=\sum_{j=1}^{i-1}p_{j}$ for all
$i\in\\{1,\ldots,n\\}$. Recall that we agreed to fix all masses $m_{i}$ to be
real and positive. Of course the integral (4.2) is not well-defined in
general. But for $p\in(\mathbb{R}^{D})^{n-1}$ the integral 4.2 converges
absolutely if and only if $2\text{Re}(\lambda)>D$. This is an application of
Weinberg’s famous Power Counting Theorem [Wei60] in its simplest form (where
no subdivergencies need to be considered since all proper subgraphs of $C_{n}$
are forests and hence trivially “converge” as there is no integration to be
performed).
The Feynman graph $C_{2}$ will accompany us as a running example throughout
this section to illustrate all the ideas as they occur:

######  Example 4.1.

Consider the Feynman graph $C_{2}$:

$C_{2}\quad=\quad$ $p_{1}$$k+p_{1}$$k$$p_{2}$

In terms of (multi-)sets, this means $C_{2}=(G,\varphi)$ with

(4.3) $G=(\\{1,2\\},\\{\\{1,2\\},\\{1,2\\}\\}).$

and $\varphi(v)=1$ for $v=1,2$. The corresponding Feynman integral in $D$
dimensions reads

(4.4)
$\int_{\mathbb{R}^{D}}\frac{d^{D}k}{(k^{2}+m_{1}^{2})^{\lambda_{1}}((k+p_{1})^{2}+m_{2}^{2})^{\lambda_{2}}}.$

It does not depend on $p_{2}$ and in fact momentum conservation demands
$p_{2}=-p_{1}$.

For reasons that become apparent further below, we exclude a certain subset of
momenta from our parameter space:

######  Definition 4.2.

For all $n\in\mathbb{N}^{\ast}$ we define

(4.5)
$T_{n}:=\\{p\in(\mathbb{C}^{D})^{n-1}\;|\;\det\begin{pmatrix}p_{1}p_{1}&p_{1}p_{2}&\cdots&p_{1}p_{n-1}\\\
p_{2}p_{1}&p_{2}p_{2}&\cdots&p_{2}p_{n-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
p_{n-1}p_{1}&p_{n-1}p_{2}&\cdots&p_{n-1}p_{n-1}\end{pmatrix}\neq 0\\}.$

The configurations of external momenta in $(\mathbb{C}^{D})^{n-1}\backslash
T_{n}$ are colinear and behave rather differently. Examples have shown that
for these momentum configurations, Feynman integrals exhibit poles instead of
essential singularities and hence there is no associated monodromy. A
discussion of these points is beyond the scope of this paper and postponed to
future research. Note that $p\in T_{n}$ implies that the momenta
$p_{1},\ldots,p_{n-1}$ are linearly independent (over $\mathbb{C}$) and in
particular $T_{n}\neq\emptyset$ if and only if $D\geq n-1$. In the course of
this section we will find that for momenta in $T_{n}$ only simple pinches
occur which can be analyzed by the techniques from Subsection 2.3.

###  4.1. Compactification and Stratification

In the form (4.2), we can not yet apply the techniques from Subsection 2.3 to
$I(C_{n})$. We first have to compactify the integration cycle as well as its
ambient space. In the one-loop case this can be done without substantial
problems. For details on the problems that occur when working with multiple
loops, see [Müh20]. There are various ways to achieve a compactification but
we stick to the arguably simplest one for the purpose of this work. Further
below we show that in the cases we are interested in, genuine pinches which
trap the integration cycle appear only at points $k\in\mathbb{C}^{D}$ at
“finite distance”, i.e. outside of the set of additional points the
compactification introduces. So the chosen compactification is not
particularly important for our purposes.
There is one rather obvious way to achieve the desired compactification in the
case of odd $D$. We view the integration domain $\mathbb{R}^{D}$ as being
embedded in the complex analytic manifold $\mathbb{C}^{D}$. The ambient space
$\mathbb{C}^{D}$ can in turn be viewed as being embedded in the compact
complex analytic manifold $\mathbb{C}\mathbb{P}^{D}$. Applying the pull-back
of the inverse of the natural inclusion

(4.6) $i:\mathbb{C}^{D}\hookrightarrow\mathbb{C}\mathbb{P}^{D},\quad
z\mapsto[1:z]$

restricted to its image (i.e. viewed as a biholomorphic map
$\mathbb{C}^{D}\overset{\sim}{\to}i(\mathbb{C}^{D})=\mathbb{C}\mathbb{P}^{D}\backslash
H_{\infty}$) to the integral (4.2), we obtain

(4.7)
$I(C_{n})(p)=\int_{\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2})^{\lambda_{i}}}=:\int_{\mathbb{R}\mathbb{P}^{D}}\omega_{n,D}(p),$

where

(4.8) $\Omega_{D}:=u\cdot d^{D}k-\sum_{i=0}^{D-1}(-1)^{i}k_{i}\cdot du\wedge
dk_{0}\wedge\cdots\wedge\widehat{dk_{i}}\wedge\cdots\wedge dk_{D-1}$

is the differential form from Lemma 2.2 and

(4.9)
$\pi:\mathbb{C}^{D+1}\backslash\\{0\\}\twoheadrightarrow\mathbb{C}\mathbb{P}^{D},\quad(u,z)\mapsto[u:z]$

is the projection from (2.2). Here we denote the additional (homogeneous)
coordinate introduced by the inclusion into complex projective space by $u$
instead of $k_{0}$ since, as mentioned above, it is customary in the physics
literature to index the components of $k$ from 0 to $D-1$ instead of from 1 to
$D$. This notation also helps to render the conceptual difference between the
coordinate $u$ and the coordinates given by the $D$ components of $k$ more
visible. Note also that we replaced the integration domain $i(\mathbb{R}^{D})$
by its closure $\widebar{i(\mathbb{R}^{D})}=\mathbb{R}\mathbb{P}^{D}$. This
does not affect the value of the integral as
$\mathbb{R}\mathbb{P}^{D}-i(\mathbb{R}^{D})=H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}$
has Lebesgue measure 0.
The same idea works for even $D$ with a slight modification: In this case
$\mathbb{R}\mathbb{P}^{D}$ is not orientable, so the integral (4.7) does not
make any sense. However, we can still apply the program from [Pha11] by
lifting to the oriented double cover $S^{D}$ of $\mathbb{R}\mathbb{P}^{D}$.
The geometric part of the analysis, which does not require the integration
cycle to be oriented, can be performed on the level of
$\mathbb{R}\mathbb{P}^{D}$ embedded in the compact space
$\mathbb{C}\mathbb{P}^{D}$ while the actual integration can be performed on
the double cover. The details of this can be found in the second part of this
work.
Now we need to establish some notation. We write

(4.10)
$Q_{i}(p):\mathbb{C}^{D+1}\to\mathbb{C},\quad(u,k)\mapsto(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}\qquad\forall
i\in\\{1,\ldots,n\\}$

for all $p\in(\mathbb{C}^{D})^{n-1}$. For all $i\in\\{1,\ldots,n\\}$ we set

(4.11) $S_{i}:=\\{([u:k],p)\in\mathbb{C}\mathbb{P}^{D}\times
T_{n}\;|\;(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}=0\\},$

and the corresponding fiber (with respect to the obvious projection) at $p\in
T_{n}$ by

(4.12)
$S_{i}(p):=\\{[u:k]\in\mathbb{C}\mathbb{P}^{D}\;|\;(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}=0\\}.$

Furthermore we set $S:=\bigcup_{i=1}^{n}S_{i}$ and
$S(p):=\bigcup_{i=1}^{n}S_{i}(p)$ for all $p\in T_{n}$. Note that $S_{i}(p)$
(resp. $S_{i}$) is a complex analytic closed submanifold of
$\mathbb{C}\mathbb{P}^{D}$ (resp. $\mathbb{C}\mathbb{P}^{D}\times T_{n}$) of
(complex) codimension 1 for all $p\in T_{n}$ and all $i\in\\{1,\ldots,n\\}$ by
Proposition 2.1. Indeed, the gradient of the defining equation
$Q_{i}(p)(u,k)=0$ in $\mathbb{C}^{D+1}\backslash\\{0\\}$ vanishes nowhere:

(4.13) $\frac{\partial
Q_{i}(p)}{\partial(u,k)}(u,k)=2\begin{pmatrix}k+uP^{(i)}(p)\\\
kP^{(i)}+u((P^{(i)}(p))^{2}+m_{i}^{2})\end{pmatrix}\overset{!}{=}0$

implies $k=-uP^{(i)}(p)$ by the first $D$ equations and thus
$u^{2}m_{i}^{2}=0$ by the last equation. Since $m_{i}^{2}\neq 0$ this means
$u=0$ which in turn implies $k=0$, a contradiction. Thus the zero locus $S(p)$
is the union of a finite number of closed complex analytic submanifolds for
all $p\in T_{n}$. The integration domain $\mathbb{R}\mathbb{P}^{D}$ is now a
compact $D$-cycle in $\mathbb{C}\mathbb{P}^{D}\backslash S(p)$ for any
$p\in(\mathbb{R}^{D})^{n-1}$. Indeed for
$(u,k)\in\mathbb{R}^{D+1}\backslash\\{0\\}$ and $p\in(\mathbb{R}^{D})^{n-1}$,
the equation

(4.14) $Q_{i}(p)(u,k)=(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}\overset{!}{=}0$

implies $u=0$ and thus $k=0$, again a contradiction, for all
$i\in\\{1,\ldots,n\\}$. The integrand of (4.7) is a holomorphic $D$-form on
$\mathbb{C}\mathbb{P}^{D}\backslash S(p)$ for all $p\in T_{n}$ which depends
holomorphically on $p$, i.e.

(4.15) $\omega_{n,D}\in\Omega^{D}(((\mathbb{C}\mathbb{P}^{D}\times
T_{n})\backslash S)/T_{n}).$

Therefore we can conclude that $I(C_{n})$ defines a holomorphic function
outside of its Landau surface. Now we would like to compute the Landau surface
of $I(C_{n})$ as in Definition 2.25. Further below we introduce a
stratification on $S$ which allows us to see that the Landau surface is given
precisely by all $p\in T_{n}$ such that the finite parts of the $S_{i}(p)$ are
not in general position and effectively ignore what happens at infinity. This
is the case if and only if there is an index set $I\subset\\{1,\ldots,n\\}$
and a point $k\in\bigcap_{i\in I}(S_{i}(p)-H_{\infty})$ (where we implicitly
use inhomogeneous coordinates) such that the normal vectors of the $S_{i}(p)$
with $i\in I$ at $k$ are linearly dependent. Thus the Landau surface consists
of all points $p\in T_{n}$ for which a solution

(4.16)
$(\alpha,k,p)\in(\mathbb{C}^{n}\backslash\\{0\\})\times\mathbb{C}^{D}\times
T_{n}$

with

(4.17) $\alpha_{i}=0\quad\text{or}\quad Q_{i}(p)(1,k)=0\qquad\forall
i\in\\{1,\ldots,n\\}$

and

(4.18) $\sum_{i=1}^{n}\alpha_{i}\frac{\partial Q_{i}(p)}{\partial
k}(1,k)=0\qquad\Leftrightarrow\qquad\sum_{i=1}^{n}\alpha_{i}(k+P^{(i)}(p))=0$

exists. These equations are the famous Landau equations. As a remark, it
should be mentioned that $\sum_{i=1}^{n}\alpha_{i}=0$ implies
$\sum_{i=1}^{n}\alpha_{i}P^{(i)}(p)=0$. This means that the momenta
$p_{1},\ldots,p_{n-1}$ are linearly dependent so that $p\notin T_{n}$. Hence
when solving the Landau equations, we can always assume
$\sum_{i=1}^{n}\alpha_{i}=1$ without loss of generality by dividing $\alpha$
by $\sum_{i=1}^{n}\alpha_{i}\neq 0$. We denote the set of all $(\alpha,k,p)$
satisfying these equations by

(4.19)
$B_{n}:=\\{(\alpha,k,p)\in(\mathbb{C}^{n}\backslash\\{0\\})\times\mathbb{C}^{D}\times
T_{n}\;|\;(\alpha,k,p)\text{ satisfies }\eqref{eq:landau_equations_1}\text{
and }\eqref{eq:landau_equations_2}\\}.$

Let

(4.20) $\pi:(\mathbb{C}^{n}\backslash\\{0\\})\times\mathbb{C}^{D}\times
T_{n}\twoheadrightarrow T_{n},\quad(\alpha,k,p)\mapsto p$

be the canonical projection and for now define the Landau surface of $C_{n}$
to be $L_{n}:=\pi(B_{n})$. Note that $L_{n}$ is a closed set. In a moment we
shall see that $L_{n}$ agrees with the Landau surface of $I(C_{n})$ as in
Definition 2.25, justifying this terminology.
Note that if we were to consider the case $u=0$ as well, the equations to test
for linear dependence of the normal vectors become

(4.21)
$k^{2}=0,\qquad\sum_{i=1}^{n}a_{i}k=0,\qquad\sum_{i=1}^{n}\alpha_{i}kP^{(i)}(p)=0.$

We remark that (4.21) can only have a solution if
$\sum_{i=1}^{n}\alpha_{i}=0$. Nevertheless, these equations generally have a
solution for every $p\in T_{n}$ which poses a problem. To circumvent this
issue, we introduce a Whitney stratification of $S$ such that some strata lie
entirely in the finite part of $\mathbb{C}\mathbb{P}^{D}$ and the remaining
ones entirely in $H_{\infty}$. Then we show that the restriction of the
canonical projection to the strata at infinity is a submersion.

######  Example 4.3.

Again considering the simplest example $C_{2}$, the corresponding Landau
equations read

(4.22) $\alpha_{1}=0\quad\text{or}\quad k^{2}+m_{1}^{2}=0,$ (4.23)
$\alpha_{2}=0\quad\text{or}\quad(k+p_{1})^{2}+m_{2}^{2}=0,$ (4.24)
$\alpha_{1}k+\alpha_{2}(k+p_{1})=0.$

First assume that $\alpha_{1}=0$. Then $\alpha_{2}\neq 0$ and by equation
(4.24) we have $k+p_{1}=0$. But by (4.23) this implies $m_{2}=0$ which we
explicitly excluded. A similar argument applies to the case $\alpha_{2}=0$ so
that a solution $(\alpha,k,p)$ to the Landau equations must satisfy
$\alpha_{1},\alpha_{2}\neq 0$. Now assume $\alpha_{1}+\alpha_{2}=0$. Then
(4.24) implies $p_{1}=0$ (so $p\notin T_{2}$ which we already knew from the
discussion above) and from (4.22) and (4.23) we get $m_{1}^{2}=m_{2}^{2}$.
Since we assumed the masses to be real and positive, this means $m_{1}=m_{2}$.
If $\alpha_{1}+\alpha_{2}\neq 0$ on the other hand, we can assume
$\alpha_{1}+\alpha_{2}=1$ without loss of generality by dividing by
$\alpha_{1}+\alpha_{2}$ if necessary. Then from (4.24) we obtain
$k=-\alpha_{2}p_{1}$. Plugging this into (4.22), we obtain
$\alpha_{2}^{2}=-\frac{m_{1}^{2}}{p_{1}^{2}}$ (note that $p_{1}^{2}=0$ is not
possible due to $m_{1}\neq 0$) and thus we get

(4.25)
$(1\pm\sqrt{-\frac{m_{1}^{2}}{p_{1}^{2}}})^{2}p_{1}^{2}+m_{2}^{2}=0\qquad\Leftrightarrow\qquad-
p_{1}^{2}=(m_{1}\pm m_{2})^{2}.$

from (4.23). Note that equation (4.25) also covers the case
$\alpha_{1}+\alpha_{2}=0$. In conclusion the Landau surface of $C_{2}$ is
given by

(4.26) $L_{2}=\\{p_{1}\in T_{2}\;|\;-p_{1}^{2}=(m_{1}\pm m_{2})^{2}\\}$

which can be conveniently written as the zero locus of

(4.27) $T_{2}\to\mathbb{C},\quad
p_{1}\mapsto\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})=(-p_{1}^{2}-(m_{1}+m_{2})^{2})(-p_{1}^{2}-(m_{1}-m_{2})^{2})$

with $\lambda:\mathbb{C}^{3}\to\mathbb{C}$ being the Källén-function defined
by

(4.28) $\lambda(a,b,c):=a^{2}+b^{2}+c^{2}-2(ab+bc+ca)$

for all $a,b,c\in\mathbb{C}$.

Let us take a moment to reflect on the combinatorial structure of the Landau
surfaces associated to one-loop graphs. Let $n\geq 2$ and suppose $I\subset
E(C_{n})$. Now we consider solutions $(\alpha,k,p)\in B_{n}$ to the Landau
equations (4.17) and (4.18) with $\alpha_{i}\neq 0$ for all $i\in I$ and
$\alpha_{i}=0$ if $i\notin I$. We denote the set of such solutions by
$B_{n,I}\subset B_{n}$. Note that such solutions are in bijective
correspondence with solutions $(\alpha^{\prime},k^{\prime},p^{\prime})$ to the
Landau equations of $C_{n}/(E(G)\backslash I)$, the graph with all edges not
in $I$ contracted to a point. Therefore the Landau surface $L_{n}$ consists of
solutions with $\alpha_{i}\neq 0$ for all $i\in\\{1,\ldots,n\\}$ and parts
which are the Landau surfaces of smaller graphs with some external momenta
replaced by the appropriate sums of momenta obtained from shrinking the edges
corresponding to vanishing $\alpha_{i}$. Motivated by this observation we
define

(4.29) $L_{n,I}:=\pi(B_{n,I})$

for all $I\subset E(C_{n})$, again with

(4.30) $\pi:(\mathbb{C}^{n}\backslash\\{0\\})\times\mathbb{C}^{D}\times
T_{n}\twoheadrightarrow T_{n},\qquad(\alpha,k,p)\mapsto p$

the canonical projection. Clearly we have $L_{n}=\bigcup_{I\subset
E(C_{n})}L_{n,I}$.131313Remark: We can also form the sets
$\tilde{L}_{n,I}:=\bigcup_{J\supset I}L_{n,J}$. Then the collection of sets
$\tilde{L}_{n,I}$ is partially ordered by inclusion and in fact constitutes a
finite simplicial poset. The associated simplicial complex consists of a
single simplex. Gluing together all simplices obtained from a graph $C_{n}$ by
permuting the masses $m_{1},\ldots,m_{n}$ along their common faces yields the
complex of holocolored one-loop graphs as investigated in [BM19]. Furthermore
$L_{n,I}=\emptyset$ whenever $|I|\leq 1$. This decomposition of the Landau
surface is useful to organize calculations but also nice from a theoretical
point of view due to the following two propositions. Their proof will be
postponed to Subsection 4.3 where we have a more convenient description of
$L_{n}$ at our disposal.

######  Proposition 4.4.

Let $n\geq 2$ and $I\subset E(C_{n})$ with $|I|\geq 2$. Then $L_{n,I}$ is a
complex analytic submanifold of codimension 1 in $T_{n}$.

######  Proposition 4.5.

Let $I_{1}\subsetneq\cdots\subsetneq I_{k}\subset E(C_{n})$. Then the
$L_{n,I_{1}},\ldots,L_{n,I_{k}}$ intersect in general position.

##### Stratification

Let $I=\\{i_{1},\ldots,i_{m}\\}\subset\\{1,\ldots,n\\}$ with $i_{1}<\cdots
i_{m}$ and let us consider the defining equations for $S_{i_{1}}\cap\cdots\cap
S_{i_{m}}$. The intersection of these $m$ manifolds is given by all
$([u:k],p)\in\mathbb{C}\mathbb{P}^{D}\times(T_{n}\backslash L_{n})$ such that
the following $m$ equations hold simultaneously:

(4.31) $(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}=0\qquad\forall i\in I$

Now plugging the first equation (the one labeled by $i_{1}$) into the
remaining $m-1$ equations yields the following equivalent set of equations:

(4.32) $(k+uP^{(i_{1})}(p))^{2}+u^{2}m_{i_{1}}^{2}=0$

and

(4.33)
$2u(P^{(i)}(p)-P^{(i_{1})}(p))(k+uP^{(i)}(p))+u^{2}((P^{(i)}(p)-P^{(i_{1})}(p))^{2}-m_{i_{1}}^{2}+m_{i}^{2})=0\qquad\forall
i\in I\backslash\\{i_{1}\\}.$

For notational convenience, we define the holomorphic function

(4.34) $\begin{split}f_{i,j}:(\mathbb{C}^{D+1}\backslash\\{0\\})\times
T_{n}&\to\mathbb{C}\\\
((u,k),p)\;\mapsto\;2(P^{(j)}(p)-P^{(i)}(p))(k+uP^{(j)}(p))&+u((P^{(j)}(p)-P^{(i)}(p))^{2}-m_{i}^{2}+m_{j}^{2})\end{split}$

for all $i,j\in\\{1,\ldots,n\\}$ so that the last $m-1$ equations read

(4.35) $u\cdot f_{i_{1},i}((u,k),p)=0\qquad\forall i\in
I\backslash\\{i_{1}\\}.$

Note that

(4.36) $f_{i,j}((0,k),p)=2(P^{(j)}(p)-P^{(i)}(p))k$

and that

(4.37) $P^{(j)}(p)-P^{(i)}(p)=\sum_{l=i}^{j-1}p_{l}$

for all $i,j\in\\{1,\ldots,n\\}$ with $j\geq i$. For all
$l\in\\{1,\ldots,n-1\\}$ we also compute

(4.38)
$\partial_{p_{l}}[(k+uP^{(i_{1})}(p))^{2}+u^{2}m_{i_{1}}^{2}]|_{u=0}=\begin{cases}2k&\text{if
}1\leq l\leq i_{1}-1\\\ 0&\text{otherwise}\end{cases}$

and

(4.39) $\partial_{p_{l}}f_{i,j}((u,k),p)|_{u=0}=\begin{cases}2k&\text{if
}i\leq l\leq j-1\\\ 0&\text{otherwise}\end{cases}\qquad\forall
i,j\in\\{1,\ldots,n\\}\text{ with }j\geq i.$

Now we define

(4.40)
$B_{I,\text{fin}}:=\\{([u:k],p)\in\mathbb{C}\mathbb{P}^{D}\times(T_{n}\backslash
L_{n})\;|\;(k+uP^{(i_{1})}(p))^{2}+u^{2}m_{i_{1}}^{2}=0,\;\forall i\in
I\backslash\\{i_{1}\\}\,:\,f_{i_{1},i}((u,k),p)=0\\}$

and

(4.41) $B_{I,\infty}:=\\{([u:k],p)\in B_{I,\text{fin}}\;|\;u=0\\}.$

######  Lemma 4.6.

The sets $B_{I,\text{fin}}$ (resp. $B_{I,\infty}$) define codimension $|I|$
(resp. codimension $|I|+1$) complex analytic submanifolds of
$\mathbb{C}\mathbb{P}^{D}\times(T_{n}\backslash L_{n})$.

###### Proof.

By the Implicit Function Theorem, it suffices to check that the gradients of
the defining functions of $B_{I,\text{fin}}$ and $B_{I,\infty}$ are linearly
independent. For all points $([u:k],p)\in B_{I,\text{fin}}$ with $u\neq 0$,
this follows immediately from $p\notin L_{n}$ and the definition of $L_{n}$.
According to equations (4.38) and (4.39), the derivatives for
$B_{I,\text{fin}}$ with respect to $p$ at $([0:k],p)\in B_{I,\text{fin}}$
(divided by 2) read

(4.42) $\begin{pmatrix}k\\\ \vdots\\\ k\\\ 0\\\ \vdots\\\ 0\\\ 0\\\ \vdots\\\
0\\\ 0\\\ \vdots\\\ 0\\\ 0\\\ \vdots\end{pmatrix},\quad\begin{pmatrix}0\\\
\vdots\\\ 0\\\ k\\\ \vdots\\\ k\\\ 0\\\ \vdots\\\ 0\\\ 0\\\ \vdots\\\ 0\\\
0\\\ \vdots\end{pmatrix},\quad\begin{pmatrix}0\\\ \vdots\\\ 0\\\ k\\\
\vdots\\\ k\\\ k\\\ \vdots\\\ k\\\ 0\\\ \vdots\\\ 0\\\ 0\\\
\vdots\end{pmatrix},\quad\begin{pmatrix}0\\\ \vdots\\\ 0\\\ k\\\ \vdots\\\
k\\\ k\\\ \vdots\\\ k\\\ k\\\ \vdots\\\ k\\\ 0\\\
\vdots\end{pmatrix},\quad\ldots$

Since $k\neq 0$, these gradients are clearly linearly independent. For
$B_{I,\infty}$, the argument is similar. The only difference is that
$B_{I,\infty}$ has the additional defining equation $u=0$ whose gradient is
clearly linearly independent of the remaining vectors. ∎

Now, using our manifolds $B_{I,\text{fin}}$ and $B_{I,\infty}$, we introduce a
Whitney stratification on $S$ which allows us to distinguish between the
finite and infinite part of $S$. To this end we first define the sets

(4.43)
$\mathcal{T}:=\\{(I,x)\;|\;I\subset\\{1,\ldots,n\\},\;I\neq\emptyset,\;x\in\\{\text{fin},\infty\\}\\}$

and

(4.44) $\mathcal{S}:=\\{1,\ldots,n\\}\times\\{\text{fin},\infty\\}.$

We equip $\mathcal{S}$ with a partial order $\prec$ defined by

(4.45)
$\forall(m_{1},x_{1}),(m_{2},x_{2})\in\mathcal{S}\,:\,(m_{1},x_{1})\prec(m_{2},x_{2})\quad:\Leftrightarrow\quad
m_{2}\leq m_{1}\land(x_{1},x_{2})\neq(\text{fin},\infty).$

For any non-empty $I=\\{i_{1},\ldots,i_{m}\\}$ with $i_{1}<\cdots<i_{m}$ we
now define

(4.46) $A_{I,\text{fin}}:=B_{I,\text{fin}}-\bigcup_{J\supsetneq
I}B_{J,\text{fin}}-H_{\infty}\times(T_{n}\backslash L_{n}),$

as well as

(4.47) $A_{I,\infty}:=B_{I,\infty}-\bigcup_{J\supsetneq I}B_{J,\infty}.$

Note that $A_{I,\text{fin}}\cap A_{J,\infty}=\emptyset$ for all non-empty
$I,J\subset\\{1,\ldots,n\\}$. Also note that

(4.48) $A_{I,\text{fin}}=\\{([u:k],p)\in
B_{I,\text{fin}}\;|\;\begin{subarray}{c}u\neq 0,\;\forall
i<i_{1}\,:\,(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}\neq 0,\\\ \forall i\notin
I,\,i>i_{1}\,:\,f_{i_{1},i}((u,k),p)\neq 0\end{subarray}\\}$

and

(4.49) $A_{I,\infty}=\\{([u:k],p)\in
B_{I,\infty}\;|\;\begin{subarray}{c}\forall
i<i_{1}\,:\,(k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}\neq 0,\\\ \forall i\notin
I,\,i>i_{1}\,:\,f_{i_{1},i}((u,k),p)\neq 0\end{subarray}\\}.$

In what follows, we show that the $A_{I,x}$ (where $(I,x)\in\mathcal{T}$) can
be used as strata for a Whitney stratification of $S$.

######  Lemma 4.7.

For all non-empty $(I,x)\in\mathcal{T}$ we have

(4.50) $\bar{A}_{I,x}=B_{I,x}.$

###### Proof.

Let $(I,x)\in\mathcal{T}$. Clearly, $B_{I,x}$ is a closed set. Thus it
suffices to show that $B_{I,x}\subset\bar{A}_{I,x}$. So let $([u:k],p)\in
B_{I,x}$ and let $U\subset B_{I,x}$ be an open neighborhood of $([u:k],p)$. We
may assume without loss of generality that $U$ contains a coordinate
neighborhood $V$ of $([u:k],p)$. By equations (4.48) and (4.49), the set
$A_{I,x}$ is given by all points in $B_{I,x}$ at which a finite set of
holomorphic functions does not vanish. By using coordinates, we can view these
as holomorphic functions on $V$. Denote these functions by
$f_{1},\ldots,f_{m}:V\to\mathbb{C}$. Then
$\prod_{i=1}^{m}f_{i}:V\to\mathbb{C}$ is a holomorphic function on $V$ which
is not identically zero. Hence it can not vanish on all of $V$. But this means
$V$ (and thus also $U$) contains a point
$([u^{\prime}:k^{\prime}],p^{\prime})\in U$ such that
$f_{i}((u^{\prime},k^{\prime}),p)\neq 0$ for all $i\in\\{1,\ldots,m\\}$. This
means $([u^{\prime}:k^{\prime}],p^{\prime})\in A_{I,x}$ and we conclude that
$([u:k],p)$ lies in the closure $\bar{A}_{I,x}$ of $A_{I,x}$. ∎

Now we combine the $A_{I,x}$ to form the pieces of a stratification. For all
$(m,x)\in\mathcal{S}$ we set

(4.51) $A_{m,x}:=\bigcup_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=m\end{subarray}}A_{I,x}.$

######  Proposition 4.8.

The $A_{m,x}$ form an $\mathcal{S}$-decomposition of $S$.

###### Proof.

Clearly the collection of the $A_{m,x}$ is locally finite (as there are only
finitely many pieces) and every $A_{m,x}$ is a locally closed set. Furthermore
it is not difficult to see that

(4.52)
$\bigcup_{(I,1)\in\mathcal{T}}A_{I,x}=S-H_{\infty}\times(T_{n}\backslash
L_{n})\qquad\text{and}\qquad\bigcup_{(I,\infty)\in\mathcal{T}}A_{I,x}=S\cap(H_{\infty}\times(T_{n}\backslash
L_{n}))$

so that

(4.53)
$\bigcup_{(m,x)\in\mathcal{S}}A_{m,x}=\bigcup_{(I,x)\in\mathcal{T}}A_{I,x}=S.$

The frontier condition is also easy to verify: According to Lemma 4.7 we have

(4.54) $\bar{A}_{m,x}=\bigcup_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=m\end{subarray}}\bar{A}_{I,x}=\bigcup_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=m\end{subarray}}B_{I,x}$

for all $(m,x)\in\mathcal{S}$. Since $B_{I,\infty}\subset H_{\infty}$ and
$A_{I,\text{fin}}\cap H_{\infty}=\emptyset$ for all non-empty
$I\subset\\{1,\ldots,n\\}$, we immediately see that
$A_{m_{1},x_{1}}\cap\bar{A}_{m_{2},x_{2}}=\emptyset$ for all
$(m_{1},x_{1}),(m_{2},x_{2})\in\mathcal{S}$ whenever $x_{1}=\text{fin}$ and
$x_{2}=\infty$. Furthermore note that

(4.55) $A_{I,\text{fin}}\cap B_{J,\text{fin}}=A_{I,\infty}\cap
B_{J,\text{fin}}=A_{I,\infty}\cap B_{J,\infty}=\emptyset$

for all non-empty subsets $I,J\subset\\{1,\ldots,n\\}$ such that $J$ is not a
subset of $I$. In case $J\subset I$ it is also easy to see that

(4.56) $A_{I,\text{fin}}\subset B_{J,\text{fin}},\quad A_{I,\infty}\subset
B_{J,\text{fin}},\quad A_{I,\infty}\subset B_{J,\infty}.$

Now applying equation (4.54) readily yields the desired frontier condition. ∎

######  Proposition 4.9.

$A_{I,x}$ is a smooth manifold for every $(I,x)\in\mathcal{T}$ and $A_{m,x}$
is a smooth manifold for every $(m,x)\in\mathcal{S}$.

###### Proof.

By Lemma 4.6, we know that the $B_{I,x}$ are smooth (even complex analytic)
manifolds. By construction, each $A_{I,x}$ is an open subset of $B_{I,x}$
(with respect to the topology on $B_{I,x}$) and thus a smooth manifold itself.
Since the $A_{I,x}$ are disjoint, we see immediately that the $A_{m,x}$ are
smooth manifolds as well (as they are the disjoint union of smooth manifolds).
∎

######  Proposition 4.10.

The decomposition above is a Whitney stratification of $S$.

###### Proof.

By Proposition 4.9 we know that each $A_{I,x}$ is a smooth manifold. It
remains to check Whitney’s condition B for any pair of strata. All pairs
$(A_{I,1},A_{J,1})$ satisfy condition B since the $A_{I,1}$ form the canonical
stratification (see equation (2.39)) of the manifolds

(4.57) $S_{1}-(H_{\infty}\times(T_{n}\backslash
L_{n})),\;\ldots,\;S_{n}-(H_{\infty}\times(T_{n}\backslash L_{n}))$

which are in general position. For all pairs $(A_{I,\infty},A_{J,\infty})$ and
$(A_{I,1},A_{J,\infty})$ with $J\subset I$ it is also easy to see that
condition B holds: Locally these pairs look like

(4.58) $\\{x\in\mathbb{C}^{a}\;|\;x_{1}\neq
0\\}\quad\text{and}\quad\\{x\in\mathbb{C}^{a}\;|\;x_{1}=\cdots=x_{b}=0\\}$

for appropriate $a,b\in\mathbb{N}$ with $a\geq b$. In these coordinates, the
condition can be readily verified. ∎

With our Whitney stratification in place, we can now deduce that $I(C_{n})$
defines a holomorphic function outside its Landau surface $L_{n}$ by showing
that the restriction of the projection to any stratum $A_{I,\infty}$ at
infinity is always a submersion. Thus, informally speaking, we can ignore the
non-general position of the $S_{i}$ at infinity and can focus the discussion
on finite internal momenta.

######  Theorem 4.11.

The one-loop Feynman integral $I(C_{n})$ defines a holomorphic function on
$T_{n}\backslash L_{n}$.

###### Proof.

By Proposition 4.10, the (connected components of the) $A_{I,x}$ constitute
the strata of a Whitney stratification of $S$. Thus we can apply Thom’s
Isotopy Theorem 2.24. It is clear that $L_{n}$ is the set where the projection
restricted to the strata of the form $A_{I,1}$ is not a submersion. Hence it
suffices to show that $\pi|_{A_{I,\infty}}$ is a submersion for all non-empty
$I\subset\\{1,\ldots,n\\}$. So let
$I=\\{i_{1},\ldots,i_{m}\\}\subset\\{1,\ldots,n\\}$ be non-empty with
$i_{1}<\cdots<i_{m}$. The tangent space at $([u:k],p)\in A_{I,\infty}$ can be
identified with the orthogonal complement of the vectors

(4.59) $\begin{pmatrix}k\\\ 0\\\ \vdots\\\
0\end{pmatrix},\quad\begin{pmatrix}P^{(i_{2})}(p)-P^{(i_{1})}(p)\\\
(\partial_{p_{1}}P^{(i_{2})}(p)-\partial_{p_{1}}P^{(i_{1})}(p))\cdot k\\\
\vdots\\\
(\partial_{p_{n-1}}P^{(i_{2})}(p)-\partial_{p_{n-1}}P^{(i_{1})}(p))\cdot
k\end{pmatrix},\quad\cdots\quad,\quad\begin{pmatrix}P^{(i_{m})}(p)-P^{(i_{1})}(p)\\\
(\partial_{p_{1}}P^{(i_{m})}(p)-\partial_{p_{1}}P^{(i_{1})}(p))\cdot k\\\
\vdots\\\
(\partial_{p_{n-1}}P^{(i_{m})}(p)-\partial_{p_{n-1}}P^{(i_{1})}(p))\cdot k\\\
\end{pmatrix}$

which is given by the solution to the homogeneous linear system of equations
given by the matrix

(4.60)
$\setcounter{MaxMatrixCols}{18}\begin{pmatrix}k^{T}&0&\cdots&0&0&\cdots&0&0&\cdots&0&0&\cdots&0&\cdots&0&0&\cdots&0\\\
(\sum_{l=i_{1}}^{i_{2}-1}p_{l})^{T}&0&\cdots&0&k^{T}&\cdots&k^{T}&0&\cdots&0&0&\cdots&0&\cdots&0&0&\cdots&0\\\
(\sum_{l=i_{1}}^{i_{3}-1}p_{l})^{T}&0&\cdots&0&k^{T}&\cdots&k^{T}&k^{T}&\cdots&k^{T}&0&\cdots&0&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
(\sum_{l=i_{1}}^{i_{m}-1}p_{l})^{T}&0&\cdots&0&k^{T}&\cdots&k^{T}&k^{T}&\cdots&k^{T}&k^{T}&\cdots&k^{T}&\cdots&k^{T}&0&\cdots&0\end{pmatrix}.$

The differential of the projection restricted to $A_{I,\infty}$ takes a vector
$\begin{pmatrix}x\\\ p^{\prime}\end{pmatrix}$ in this tangent space to
$p^{\prime}$. So we need to check if for every $p^{\prime}\in T_{n}\backslash
L_{n}$ there is a solution $x\in\mathbb{C}^{D}$ to the inhomogeneous system of
linear equations

(4.61) $\begin{pmatrix}k^{T}\\\ (\sum_{l=i_{1}}^{i_{2}-1}p_{l})^{T}\\\
\vdots\\\ (\sum_{l=i_{1}}^{i_{m}-1}p_{l})^{T}\end{pmatrix}\cdot
x=\begin{pmatrix}0\\\ k^{T}\cdot\sum_{l=i_{1}}^{i_{2}-1}p_{l}^{\prime}\\\
\vdots\\\ k^{T}\cdot\sum_{l=i_{1}}^{i_{m}-1}p_{l}^{\prime}\end{pmatrix}.$

We distinguish two cases: If $m\leq D$ we can consider the smaller system

(4.62) $\begin{pmatrix}(\sum_{l=i_{1}}^{i_{2}-1}p_{l})^{T}\\\ \vdots\\\
(\sum_{l=i_{1}}^{i_{m}-1}p_{l})^{T}\end{pmatrix}\cdot
x=\begin{pmatrix}k^{T}\cdot\sum_{l=i_{1}}^{i_{2}-1}p_{l}^{\prime}\\\ \vdots\\\
k^{T}\cdot\sum_{l=i_{1}}^{i_{m}-1}p_{l}^{\prime}\end{pmatrix}.$

Since $p\in T_{n}$, the $\sum_{l=i_{1}}^{i_{j}-1}p_{l}$ must be linearly
independent and hence the matrix on the right has the full rank $m-1$. The
solution space of (4.62) is thus an affine space of dimension $D-(m-1)\geq 1$.
In this space there clearly exists an $x$ such that $k^{T}x=0$ so that (4.61)
can indeed be solved. If $m=D+1$ on the other hand (note that we can not have
$m>D+1$ since $m\geq n$ and thus $n>D+1$ which implies $p\notin T_{n}$) we
show that the matrix on the left-hand side of (4.61) has rank $m$ which
immediately implies the existence of a solution. First note that in this case
we have $I=\\{1,\ldots,n\\}$ which means
$\sum_{l=i_{1}}^{i_{j}-1}p_{l}=\sum_{l=1}^{j-1}p_{l}$ for all
$j\in\\{1,\ldots,n\\}$. It suffices to show that $k$ is linearly independent
of the $\sum_{l=1}^{j-1}p_{l}$ which is equivalent to $k$ being linearly
independent of the $p_{j}$. To see this, let us suppose the opposite is true.
Then there exist $\lambda_{1},\ldots,\lambda_{n-1}\in\mathbb{C}$ (not all
equal to zero) such that $k=\sum_{i=1}^{n-1}\lambda_{i}p_{i}$. But
$([0:k],p)\in A_{I,\infty}$ requires

(4.63)
$(P^{(j)}(p)-P^{(1)}(p))k=\sum_{i=1}^{n-1}\sum_{l=1}^{j-1}\lambda_{i}p_{i}p_{l}\overset{!}{=}0\qquad\forall
j\in\\{2,\ldots,n\\}.$

But this implies $p\notin T_{n}$, a contradiction. In conclusion
$\pi|_{A_{I,\infty}}$ is indeed a submersion. ∎

With our compactification in place we can show the invariance of the
holomorphic function defined by $I(C_{n})$ with respect to the action (3.4) of
$O(D,\mathbb{C})$. To this end we consider the following extension of the
action (3.4) to $\mathbb{C}\mathbb{P}^{D}$:

(4.64)
$\tilde{\sigma}:O(D,\mathbb{C})\times\mathbb{C}\mathbb{P}^{D}\to\mathbb{C}\mathbb{P}^{D},\quad(g,[u:z])\mapsto
g\cdot[u:z]:=[u:g\cdot z].$

Note that this is well-defined since if $(u^{\prime},z^{\prime})=(\lambda
u,\lambda z)$ for some $\lambda\in\mathbb{C}^{\times}$, we have

(4.65) $g\cdot[u^{\prime}:z^{\prime}]=[\lambda u:g\cdot(\lambda z)]=[\lambda
u:\lambda(g\cdot z)]=g\cdot[u:z]$

for all $g\in O(D,\mathbb{C})$. Furthermore this action is clearly continuous.
Note also that the Landau surface $L_{n}$ is invariant under our action on
$T_{n}$: If $g\in O(D,\mathbb{C})$ and $(\alpha,k,g\cdot p)\in B_{n}$ is a
solution to the Landau equations, then $(\alpha,g^{-1}\cdot k,p)$ is also a
solution. Hence $p\in L_{n}\;\Leftrightarrow\;g\cdot p\in L_{n}$ for all $g\in
O(D,\mathbb{C})$. In fact this even shows $L_{n,I}=g\cdot L_{n,I}$ for all
$I\subset\\{1,\ldots,n\\}$ and $g\in O(D,\mathbb{C})$. To show the invariance
of our integral under $O(D,\mathbb{C})$ we need some information on how the
differential forms in question transforms under this group. The following
lemma establishes the very simple transformation behavior of $\omega_{n,D}(p)$
under $O(D,\mathbb{C})$:

######  Lemma 4.12.

For all $g\in O(D,\mathbb{C})$ we have

(4.66) $(\tilde{\sigma}(g,\cdot))^{\ast}\omega_{n,D}(g\cdot
p)=\det(g)\cdot\omega_{n,D}(p).$

###### Proof.

Obtained from a straightforward calculation. ∎

######  Proposition 4.13.

Let $n\geq 2$. The multivalued holomorphic function $I(C_{n}):T_{n}\backslash
L_{n}\to\mathbb{C}$ is invariant under the action (3.4) of $O(D,\mathbb{C})$,
i.e. for any $p\in T_{n}\backslash L_{n}$ and any $g\in O(D,\mathbb{C})$ we
have

(4.67) $I(C_{n})(g\cdot p)=I(C_{n})(p)$

when evaluating the left and right hand side on the same branch.

###### Proof.

Now let $p\in T_{n}\backslash L_{n}$ and $g\in O(D,\mathbb{C})$. Let us first
suppose that $g$ lies in the connected component of $O(D,\mathbb{C})$
containing the identity $e$. In particular this means $\det(g)=1$. Then if
$\gamma_{g}:[0,1]\to O(D,\mathbb{C})$ is a path from $e$ to $g$ we can define
a path

(4.68) $\gamma:[0,1]\to T_{n}\backslash L_{n},\quad s\mapsto\gamma_{g}(s)\cdot
p.$

from $p$ to $g\cdot p$. Note that this is also well-defined as the Landau
equations (4.17) and (4.18) are invariant under our action of
$O(D,\mathbb{C})$ as mentioned above. We know that we can write

(4.69) $I(C_{n})(p)=\int_{\Gamma}\omega_{n,D}(p)$

for an appropriate $D$-cycle $\Gamma\subset\mathbb{C}\mathbb{P}^{D}\backslash
S(p)$. Defining

(4.70)
$\sigma:\mathbb{C}\mathbb{P}^{D}\times[0,1]\to\mathbb{C}\mathbb{P}^{D},\quad([z],s)\mapsto\gamma_{g}(s)\cdot[z],$

we have $\sigma(\Gamma,s)\cap S(\gamma(s))=\emptyset$ for all $s\in[0,1]$.
Indeed if there was $[u:k]\in\sigma(\Gamma,s)\cap S(\gamma(s))$, there would
exist $i\in\\{1,\ldots,n\\}$ such that

(4.71)
$(k+uP^{(i)}(\gamma(s)))^{2}+u^{2}m_{i}^{2}=((\gamma_{g}(s))^{-1}k+uP^{(i)}(p))^{2}+u^{2}m_{i}^{2}=0.$

Since $\sigma([u:(\gamma_{g}(s))^{-1}k],s)=[u:k]$, this means

(4.72) $[u:(\gamma(s))^{-1}k]\in\sigma(\Gamma,0)\cap S(\gamma(0))=\Gamma\cap
S(p)$

in contradiction $\Gamma\cap S(p)=\emptyset$. Thus by Corollary 2.16 we can
analytically continue $I(C_{n})$ from $p$ to $g\cdot p$ along $\gamma$ with

(4.73)
$\begin{split}I(C_{n})(\gamma(s))&=\int_{\sigma(\Gamma,s)}\omega_{n,D}(\gamma(s))=\int_{\gamma_{g}(s)\cdot\Gamma}\omega_{n,D}(\gamma_{g}(s)\cdot
p)\\\ &=\int_{\Gamma}(\sigma(\cdot,s))^{\ast}\omega_{n,D}(\gamma_{g}(s)\cdot
p)=\int_{\Gamma}\omega_{n,D}(p)=I(C_{n})(p)\end{split}$

for all $s\in[0,1]$.
Now if $g$ is not in the connected component containing $e$, then $g$ can be
expressed as a product $g=r\cdot g^{\prime}$ of an element $g^{\prime}$ in the
connected component containing $e$ and a reflection

(4.74) $r=\begin{pmatrix}-1&0&\cdots&0\\\ 0&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\end{pmatrix}.$

Thus it suffices to check the claim for $g=r$. Using $\det(r)=-1$, we obtain

(4.75) $\begin{split}I(C_{n})(r\cdot
p_{\text{in}})&=\int_{\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((k+uP^{(i)}(r\cdot
p_{\text{in}}))^{2}+u^{2}m_{i}^{2})^{\lambda_{i}}}\\\
&=-\int_{r\cdot\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((r\cdot
k+uP^{(i)}(r\cdot p_{\text{in}}))^{2}+u^{2}m_{i}^{2})^{\lambda_{i}}}\\\
&=\int_{\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((k+uP^{(i)}(p_{\text{in}}))^{2}+u^{2}m_{i}^{2})^{\lambda_{i}}}=I(C_{n})(p_{\text{in}})\end{split}$

for all $p_{\text{in}}\in(\mathbb{R}^{D})^{n-1}$ (where $I(C_{n})$ is
evaluated on the principal branch). Thus for any path $\gamma:[0,1]\to
T_{n}\backslash L_{n}$ from $p_{\text{in}}$ to $p$, the analytic continuation
along the path $\gamma^{\prime}:[0,1]\to T_{n}\backslash L_{n}$ from $r\cdot
p_{\text{in}}$ to $r\cdot p$ defined by $\gamma^{\prime}(s)=r\cdot\gamma(s)$
for all $s\in[0,1]$ yields the same result. ∎

####  4.1.1. Remark on the Compactification in [FP66]

The authors of [FP66], a paper which was never published but reprinted in
[HT66], employ a different compactification: They consider specifically the
box graph $C_{4}$ which they compactify as follows: The integration domain
$\mathbb{R}^{D}$ can be thought of as part of a $D$-sphere in
$\mathbb{R}^{D+1}$ using the stereographic projection. More precisely we
consider the diffeomorphism

(4.76)
$f:S^{D}(-1,0,\ldots,0)\backslash\\{0\\}\overset{\sim}{\to}\mathbb{R}^{D},\quad(x_{0},\ldots,x_{D})\mapsto(\frac{x_{1}}{x_{0}},\ldots,\frac{x_{D}}{x_{0}}).$

Here $S^{D}(-1,0\ldots,0)$ is the real $D$-sphere with radius 1 and center
$(-1,0,\ldots,0)$. Applying the pullback $f^{\ast}$ to the Feynman integral
(4.2), one obtains

(4.77)
$I(C_{n})=\int_{S^{D}(-1,0,\ldots,0)}\frac{x_{0}^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((k+x_{0}\cdot
P^{(i)}(p))^{2}+x_{0}^{2}m_{i}^{2})^{\lambda_{i}}}$

after adding the set $\\{0\\}$ of Lebesgue measure 0 to the integration
domain. While this compactifies the integration domain, the natural choice for
the complex ambient space, the complex sphere

(4.78)
$S_{\mathbb{C}}^{D}:=\\{(x_{0},k)\in\mathbb{C}^{D+1}\;|\;(x_{0}+1)^{2}+k^{2}=1\\}$

around $(-1,0,\ldots,0)$ with radius 1, is not compact. Thus the authors view
$S_{\mathbb{C}}^{D}$ as embedded in $\mathbb{C}\mathbb{P}^{D+1}$ in a second
step. I.e. they apply the pullback of the inverse of the inclusion

(4.79) $i:\mathbb{C}^{D}\hookrightarrow\mathbb{C}\mathbb{P}^{D+1},\qquad
z\mapsto[1:z]$

restricted to its image like we do in the first step. The integral they obtain
after applying all transformations reads

(4.80)
$I(C_{n})=\int_{S_{\mathbb{P}}^{D}}\frac{x_{0}^{2\lambda-D-1}\cdot\Omega_{D}}{\prod_{i=1}^{n}((k+x_{0}\cdot
P^{(i)}(p))^{2}+x_{0}^{2}m_{i}^{2})^{\lambda_{i}}},$

where

(4.81)
$\begin{split}S_{\mathbb{P}}^{D}:&=\\{[u:x_{0}:k]\in\mathbb{C}\mathbb{P}^{D+1}\;|\;(x_{0}+u)^{2}+k^{2}=u^{2}\\}\\\
&=\\{[u:x_{0}:k]\in\mathbb{C}\mathbb{P}^{D+1}\;|\;x_{0}^{2}+k^{2}=-2ux_{0}\\}.\end{split}$

This compactification comes with significant restrictions with respect to the
dimension: The form of equation 4.80 suggests that the differential to be
integrated might be singular at points $[u:x_{0}:k]\in S_{\mathbb{P}}^{D}$
with $x_{0}=0$ and $k=0$ which would be an issue. The term
$x_{0}^{2\lambda-D-1}$ could potentially cancel the singular nominator at such
points but if it actually does depends on $D$ and $\lambda$. A careful
computation of the relevant differential form in charts shows that this
cancellation happens if and only if $D\leq\text{Re}\,\lambda$. Particularly in
the case $\lambda_{1}=\cdots=\lambda_{n}=1$ one is usually interested in this
is a problem: In this case we have $\lambda=n$ and for $p\in T_{n}$ we need
$D\geq n-1$. Hence this compactification can only work if $D=n$ or $D=n-1$.
For example the box graph $C_{4}$ in 4 dimensions as investigated in [FP66]
can be compactified in this manner while the triangle graph $C_{3}$ in 4
dimensions can not.

###  4.2. On the Continuation of the Local Section Defined by
$\mathbb{R}\mathbb{P}^{D}$

Before discussing the discontinuity of $I(C_{n})$, we first establish some
results on the continuation of the local section defined by the initial
integration cycle $\mathbb{R}\mathbb{P}^{D}$.
The next lemma allows us to show that if there is an ambient isotopy realizing
a continuation, we can always choose it in a way that fixes the integration
cycle at infinity.

######  Lemma 4.14.

Let $K\subset T_{n}$ be a compact set. Then there exists an open neighborhood
$U_{\infty}\subset\mathbb{C}\mathbb{P}^{D}$ of
$H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}$ such that

(4.82) $U_{\infty}\cap S(p)=\emptyset\qquad\forall p\in K.$

###### Proof.

First note that for any $p\in T_{n}$ we have

(4.83) $Q_{i}(p)(0,k)=k^{2}\neq 0\qquad\forall k\in\mathbb{R}^{D},\;\forall
i\in\\{1,\ldots,n\\}.$

Thus $H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}\cap S(p)=\emptyset$. Now let
$d:\mathbb{C}\mathbb{P}^{D}\times\mathbb{C}\mathbb{P}^{D}\to\mathbb{R}_{\geq
0}$ be a metric on $\mathbb{C}\mathbb{P}^{D}$ inducing its standard topology.
Then

(4.84) $d(H_{\infty}\cap\mathbb{R}\mathbb{P}^{D},S(p))>0$

for all $p\in T_{n}$. Now $d(H_{\infty}\cap\mathbb{R}\mathbb{P}^{D},S(\cdot))$
can be viewed as a continuous function on $T_{n}$ which necessarily takes on a
minimum $d_{\text{min}}>0$ on the compact set $K$. Thus if we denote by
$D([z],d_{\text{min}})\subset\mathbb{C}\mathbb{P}^{D}$ the open disc of radius
$d_{\text{min}}$ around $[z]$, we have $D([z],d_{\text{min}})\cap
S(p)=\emptyset$ for all $[z]\in H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}$ and
$p\in K$. Consequently the open set

(4.85) $U_{\infty}:=\bigcup_{[z]\in
H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}}D([z],d_{\text{min}})$

fulfills the requirement of the lemma. ∎

######  Corollary 4.15.

Let $\gamma:[0,1]\to T_{n}$ be a path with
$\gamma(0)\in(\mathbb{R}^{D})^{n-1}$. Suppose there exists an ambient isotopy
$\sigma$ of $\mathbb{C}\mathbb{P}^{D}$ such that $\sigma(\cdot,s)$ takes
$S(\gamma(0))$ to $S(\gamma(s))$ for all $s\in[0,1]$. Then there exists an
ambient isotopy $\sigma^{\prime}$ of $\mathbb{C}\mathbb{P}^{D}$ such that
$\sigma^{\prime}(\mathbb{R}\mathbb{P}^{D},s)\cap S(\gamma_{p}(s))=\emptyset$
and

(4.86)
$\sigma^{\prime}(H_{\infty}\cap\mathbb{R}\mathbb{P}^{D},s)=H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}$

for all $s\in[0,1]$.

###### Proof.

Let $U_{\infty}$ be an open neighborhood of $H_{\infty}$ as in Lemma 4.14.
Furthermore let $U$ be an open neighborhood of $S(\gamma(0))$ disjoint from
$U_{\infty}$. Then $\sigma$ restricts to an isotopy of $U\cup U_{\infty}$ such
that the track $\hat{\sigma}((U\cup U_{\infty})\times[0,1])$ is open. We can
modify this ambient isotopy by letting it be the identity on $U_{\infty}$. Now
by Theorem 2.14 there exists an ambient isotopy $\sigma^{\prime}$ of
$\mathbb{C}\mathbb{P}^{D}$ which agrees with $\sigma$ on the compact set
$S(\gamma(0))\cup(H_{\infty}\cap\mathbb{R}\mathbb{P}^{D})$. In particular
$\sigma^{\prime}(H_{\infty}\cap\mathbb{R}\mathbb{P}^{D},s)=H_{\infty}\cap\mathbb{R}\mathbb{P}^{D}$
for all $s\in[0,1]$. ∎

Now we discuss a class of path along which we can always analytically
continue. For any $I\subset\\{0,\ldots,D-1\\}$ consider the continuous matrix
valued function $s\mapsto M_{I}(s)\in M(D\times D,\mathbb{C})$ defined by

(4.87) $(M_{I}(s))_{ij}:=\begin{cases}e^{i\frac{\pi}{2}s}&\text{if }i=j\in
I\\\ 1&\text{if }i=j\notin I\\\ 0&\text{otherwise}.\end{cases}$

Note that we again start our indexing at $0$ instead of 1 and that
$M_{I}(0)=\text{id}_{\mathbb{C}^{D}}$. Then for any
$p\in(\mathbb{C}^{D})^{n-1}$ we can define a path

(4.88) $\gamma_{I,p}:[0,1]\mapsto(\mathbb{C}^{D})^{n-1},\quad
s\mapsto(M_{I}(s)p_{1},\ldots,M_{I}(s)p_{n-1}).$

Furthermore we define the continuous map

(4.89)
$\sigma_{I,p}:\mathbb{C}\mathbb{P}^{D}\times[0,1]\to\mathbb{C}\mathbb{P}^{D},\quad([u:k],s)\mapsto[u:M_{I}(s)k].$

######  Proposition 4.16.

Let $p\in(\mathbb{R}^{D})^{n-1}$ and $I\subset\\{0,\ldots,D-1\\}$. Then for
every $s_{1}\in(0,1)$ the ambient isotopy
$\sigma_{I,p}|_{\mathbb{C}\mathbb{P}^{D}\times[0,s_{1}]}$ is adapted to
$\gamma_{I,p}|_{[0,s_{1}]}$. In particular the integral $I(C_{n})$ can be
analytically continued from $p$ to $\gamma_{I,p}(s_{1})$ along
$\gamma_{I,p}|_{[0,s_{1}]}$.

###### Proof.

According to Proposition 2.18, it suffices to show

(4.90) $\sigma_{I,p}(\mathbb{R}\mathbb{P}^{D},s)\cap
S(\gamma_{I,p})=\emptyset\qquad\forall s\in[0,s_{1}].$

So let $s\in[0,s_{1}]$. We write $I=\\{i_{1},\ldots,i_{|I|}\\}$ and
$J:=\\{0,\ldots,D-1\\}\backslash I=\\{j_{1},\ldots,j_{|J|}\\}$ with
$i_{1}<\cdots<i_{|I|}$ and $j_{1}<\ldots<j_{|J|}$. Furthermore we denote

(4.91) $k_{I}:=(k_{i_{1}},\ldots,k_{i_{|I|}})\quad\text{and}\quad
k_{J}:=(k_{j_{1}},\ldots,k_{j_{|J|}})$

as well as

(4.92)
$P_{I}^{(l)}(p^{\prime}):=(P_{i_{1}}^{(l)}(p^{\prime}),\ldots,P_{i_{|I|}}^{(l)}(p^{\prime}))\quad\text{and}\quad
P_{J}^{(l)}(p^{\prime}):=(P_{j_{1}}^{(l)}(p^{\prime}),\ldots,P_{j_{|J|}}^{(l)}(p^{\prime}))$

for all $l\in\\{1,\ldots,n\\}$ and all $p^{\prime}\in T_{n}$. Now let
$[u:k]\in\mathbb{R}\mathbb{P}^{D}$. Then $(u,M_{I}(s)k)$ is a representative
of $\sigma_{I,p}([u:k],s)$ and for any $l\in\\{1,\ldots,n\\}$ we have

(4.93)
$\begin{split}Q_{l}(\gamma_{I,p}(s))(u,M_{I}(s)k)&=(M_{I}(s)k+uM_{I}(s)P^{(l)}(p))^{2}+u^{2}m_{l}^{2}\\\
&=e^{i\pi
s}(k_{I}+uP_{I}^{(l)}(p))^{2}+(k_{J}+uP_{J}^{(l)}(p))^{2}+u^{2}m_{l}^{2}.\end{split}$

Now either $k_{I}+uP_{I}^{(l)}(p)=0$ and we have

(4.94)
$Q_{l}(\gamma_{I,p}(s))(u,M_{I}(s)k)=(k_{J}+uP_{J}^{(l)}(p))^{2}+u^{2}m_{l}^{2}>0$

or $k_{I}+uP_{I}^{(l)}(p)\neq 0$ in which case
$Q_{l}(\gamma_{I,p}(s))(u,M_{I}(s)k)$ has non-zero imaginary part. In either
case we conclude $Q_{l}(\gamma_{I,p}(s))(u,M_{I}(s)k)\neq 0$. ∎

###  4.3. Cutkosky’s Theorem in the One-Loop Case

Now we are in a position to compute the discontinuity of the general one-loop
integral (4.7) around simple loops up to the intersection index. This requires
us to compute the vanishing sphere.
Let us first consider the geometry of the zero locus of $Q_{j}(p)(1,\cdot)$ at
an arbitrary point $p\in(\mathbb{C}^{D})^{n-1}$. It is given by all
$k\in\mathbb{C}^{D}$ such that

(4.95)
$(k+P^{(j)}(p))^{2}+m_{j}^{2}=0\qquad\Leftrightarrow\qquad(ik+iP^{(j)}(p))^{2}=m_{j}^{2}.$

This describes a complex $(D-1)$-sphere with radius $m_{j}$. It contains the
real $(D-1)$-sphere

(4.96) $\\{k\in\mathbb{C}^{D}\;|\;k+P^{(j)}(p)\in
i\mathbb{R}^{D},\;(ik+iP^{(j)}(p))^{2}=m_{j}^{2}\\}$

to which it deformation retracts. To compute the vanishing sphere we want to
understand the homology of the finite part of $S_{1}(p)\cap\cdots\cap