# Feynman Amplitudes in Periodically Compactified Spaces - Spin 0

E. Cavalcanti<EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas/MCTI,
22290-180 Rio de Janeiro, RJ, Brazil

###### Abstract

We propose an extension of the Schwinger parametric representation for Feynman
amplitudes in $D$ euclidean dimensions to a scenario where $d$ dimensions are
compactified ($d<D$) through the introduction of periodic boundary conditions
in space. We obtain two valid representations, one useful near the bulk (large
compactification length) and another useful near the dimensional reduction
(small compactification length). Also, to illustrate, we exhibit some Feynman
amplitudes up to three loops in a compactified scalar field theory.

##  I Introduction

One essential task in perturbative quantum field theory is to obtain, for a
given theory, the amplitude of graph $G$ at some arbitrary order. This task is
decades older and its solution is already textbook material, the most known
parametric representations are those of Feynman [1, 2], Schwinger [1, 2, 3]
and Mellin [4, 5] although other efforts, as for example the Complete Mellin
representation [6, 7, 8, 9, 10], were also studied. The employment of these
representations in a general setting employs the Symanzik polynomials, which
can be directly extracted from the topology of the graph by inspecting the
1-trees and 2-trees [2, 3, 11]. That way, one can avoid the otherwise
cumbersome computation of these polynomials and we have an ‘easy’ prescription
to compute higher-order diagrams. This first simple task was essential for the
important developments that follow, for example, proving the renormalizability
of a quantum field theory up to all orders or extracting the asymptotic
behavior of a diagram.

Although this topic, to obtain a parametric representation, is completely
settled for theories in non-compact space-time – let us refer to the euclidean
space-time with $D$ dimensions – it has not yet been established for theories
in compactified dimensions. If the only interest of someone is the proof of
renormalizability, one can justify this lack of understanding concerning
theories in compactified dimensions due to the knowledge that the divergent
behavior in the amplitudes of compactified theories come from the bulk, that
is, the contribution related to the non-compactified space [12, 13]. This
means that there is no need to prove renormalizability again as it comes
directly from the proof in the bulk scenario. However, this does not justify
the absence of this exploration in the literature. Right now, we do not have a
prescription to ‘easily’ compute higher-order diagrams in a scenario with
compactified spaces neither an established asymptotic expansion for this
scenario. The first step, if one is interested in some progress in topics that
depend on higher-order corrections of Feynman diagrams in compactified spaces,
is to establish a useful parametric representation and this is the purpose of
this work.

There are plenty of ways to introduce compactified dimensions, just as there
are many possible choices of boundary conditions. Perhaps the most simple
scenario are periodically compactified theories with just one compactified
dimension, which is exactly the highly explored scenario of field theory at
finite temperature [14, 15, 16, 17, 18], where the inverse temperature
$\beta=1/T$ is introduced as the compactification length of a periodically
compactified dimension throught the Matsubara formalism of imaginary time. In
recent years, there is also a growing interest in field theory with a small
circle compactification [19, 20, 21, 22, 23], which is somewhat equivalent to
finite temperature but allows to impose antiperiodic or even twisted boundary
conditions in the spatial compactification – recall that the boundary
condition in imaginary time is restricted, by the Kubo-Martin-Schwinger (KMS)
condition [14, 15, 16, 17, 18], to follow the bosonic/fermionic nature of the
field, but we are free of it if we deal with spatial compactification. When it
comes to the extension to more compactified dimensions and different boundary
conditions the range of applicability just grows even more: Camir effect [24,
25, 26, 27, 28], superstring theory [29, 30, 31], quantum field theory with an
small extradimension [32, 33, 34, 35, 36, 37, 38, 39, 40], finite volume
considerations QFT models and in particle physics [41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54], and so on. To see more applications we refer to
the works [28, 18, 55, 56] and references therein.

In the last decades, Refs. [18, 55] established the so-called ‘quantum field
theory in toroidal topologies’, a formalism to deal with periodically
compactificatified dimensions. This is born as an extension from the Matsubara
prescription of imaginary time to more dimensions, producing a topology
$\Gamma^{d}_{D}=\mathbb{R}^{D-d}\times\mathbb{S}_{L_{1}}\times\cdots\times\mathbb{S}_{L_{d}}$,
where $\mathbb{S}_{L_{i}}$ represents each of the $d$ compactifications that
can be imposed by periodic/antiperiodic boundary conditions 111One can easily
extend it to quasiperiodic (also called twisted or anyonic) boundary
conditions [57], with $L_{i}$ as the characteristic compactification length.
Within this formulation, many applications were explored both in the field of
particle physics and in the field of condensed matter, as the occurrence of
size-dependent phase transitions [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70]. In recent years, a somewhat similar approach [56] also started to
explore the formulation and applications of periodically compactified QFT.

So far, only few attempts were done [71, 72, 73, 74] in the direction to
establish a parametric representation for Feynman amplitudes in compactified
spaces. Three decades ago Benhamou [71, 72] proposed a parametric
representation for field theory at a finite temperature that recovers the
Symanzik polynomials for the compactified part and that has as the zero-
temperature limit the usual non-compactified parametric representation. One
decade ago there was an attempt [73] to build for scalar field models some
representation for the scenario of periodically compactified dimensions. Also,
a recent work Ref. [74] deals with the parametric representation for fields
with different spins in a compactified space.

In Sec. II we start building the Schwinger parametric representation for a
graph in periodically compactified spaces. We see that there are two possible
paths to follow: 1 - we can consider the scale where the lengths of the
compactified dimensions are very small (Sec. III), meaning that we are close
to a dimensional reduction; 2 - or we can assume that the lengths are very
large (Sec. IV), so we are close to the bulk scenario without
compactifications. Both paths are equivalent and one could transport from one
to another [75, 76], but each one of them is more useful in one regime (near
the dimensional reduction or the bulk), due to quicker convergence. After
this, we illustrate the representation by showing some diagrams in Sec. V with
the useful information to write their parametric representation. In the
conclusions, we indicate some further developments.

##  II Parametric representation of compactified Feynman diagrams

Let us start considering a general scalar scenario [1, 2],

$\widetilde{\mathcal{I}}_{G}=C_{G}\prod_{i=1}^{I}\left[\int\frac{d^{D}K_{i}}{(2\pi)^{D}}\frac{1}{K_{i}^{2}+m_{i}^{2}}\right]\prod_{v=1}^{V}\left[(2\pi)^{D}\delta^{D}\left(P_{v}-\sum_{i}\varepsilon_{vi}K_{i}\right)\right],$
(1)

where $D$ is the number of dimensions (we consider a euclidian space-time),
$C_{G}$ is a factor related to the vertices and the symmetry of the graph,
$K_{i}$ are the internal momenta – that runs from $i=1,\ldots,I$, where $I$ is
the number of internal lines –, $P_{v}$ are the sum of external momenta
inciding over the vertex $v$ – that runs from $v=1,\ldots,V$, where $V$ is the
number of vertices –, and $\varepsilon_{vi}$ is the incidence matrix
($\varepsilon_{vi}=+1$ if the line $i$ starts at the vertex $v$,
$\varepsilon_{vi}=-1$ if the line $i$ ends at the vertex $v$ and
$\varepsilon_{vi}=0$ if the line $i$ is not related to the vertex $v$). To
abbreviate notation, $\prod_{i}$ will always refer to $\prod_{i=1}^{I}$,
unless specified.

We introduce $d$ ($d<D$) periodically compactified dimensions by employing an
extension of the Matsubara formalism of imaginary time [18, 55]. The
prescription is well established to also consider the introduction of a
chemical potential or to employ quasiperiodic/anyonic boundary conditions.
However, for this work, we stick with just the periodic boundary conditions.
We denote by a capital letter the full $D$-dimensional momenta ($K_{i}$) and
by lower case the momenta related to the $D-d$ non-compactified dimensions,
that is

$\displaystyle K_{i}^{2}$ $\displaystyle\rightarrow
k_{i}^{2}+\sum_{\alpha}(\omega_{\alpha}^{k_{i}})^{2},$ (2a)
$\displaystyle\int\frac{d^{D}K_{i}}{(2\pi)^{D}}$
$\displaystyle\rightarrow\int\frac{d^{D-d}k_{i}}{(2\pi)^{D-d}}\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}},$ (2b) $\displaystyle\omega_{\alpha}^{k_{i}}$
$\displaystyle=\frac{2\pi}{L_{\alpha}}n_{\alpha}^{k_{i}}.$ (2c)

Where the index $\alpha$ runs over the $d$ compactified dimensions,
$L_{\alpha}$ is the characteristic length for each of them and
$\omega_{\alpha}^{k_{i}}$ is the $\alpha$-th Matsubara frequency associated
with the momenta $k_{i}$. Notice that $\prod_{\alpha}$ must be undestood as
$\prod_{\alpha=1}^{d}$.

Therefore, the amplitude for the graph $G$ in compactified dimensions turns to
be

$\widetilde{\mathcal{I}}_{G}=C_{G}\prod_{i}\left[\int\frac{d^{D-d}k_{i}}{(2\pi)^{D-d}}\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}\frac{1}{k_{i}^{2}+\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+m_{i}^{2}}\right]\times\\\
\prod_{v=1}^{V}\left[(2\pi)^{D}\delta^{D-d}\left(p_{v}-\sum_{i}\varepsilon_{vi}k_{i}\right)\prod_{\alpha}\delta\left(\omega_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}\right)\right].$
(3)

To proceed we start dealing with the non-compactified dimensions. At this
point the procediment is closely related to the standard one, but is exhibited
for completeness and clarity. We introduce a integral representation for the
propagator and the conservation deltas, introducing the Schwinger parameters
$u_{i}$ related to each internal line,

$\widetilde{\mathcal{I}}_{G}=C_{G}\prod_{i}\left[\int\frac{d^{D-d}k_{i}}{(2\pi)^{D-d}}\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}\int_{0}^{\infty}du_{i}e^{-u_{i}\left[k_{i}^{2}+\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+m_{i}^{2}\right]}\right]\times\\\
\prod_{v=1}^{V}\left[\int
d^{D-d}y_{v}e^{-iy_{v}\cdot\left(p_{v}-\sum_{i}\varepsilon_{vi}k_{i}\right)}\prod_{\alpha}\left[\int
dz_{v}^{(\alpha)}e^{-iz_{v}^{(\alpha)}\left(\omega_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}\right)}\right]\right],$
(4)

we can interchange the sign of integrals and sums to make evident the integral
over the internal momenta $k_{i}$ and also the sum over the internal
frequencies $\omega_{\alpha}^{k_{i}}$, that is

$\widetilde{\mathcal{I}}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\left[\prod_{v=1}^{V}\int
d^{D-d}y_{v}\right]\left[\prod_{v=1}^{V}\prod_{\alpha}\int
dz_{v}^{(\alpha)}\right]e^{-\sum_{i}u_{i}m_{i}^{2}}e^{-i\sum_{v=1}^{V}\sum_{\alpha}z_{v}^{(\alpha)}\omega_{\alpha}^{p_{v}}}e^{-i\sum_{v=1}^{V}y_{v}\cdot
p_{v}}\times\\\
\prod_{i}\left[\int\frac{d^{D-d}k_{i}}{(2\pi)^{D-d}}e^{-u_{i}k_{i}^{2}}e^{i\sum_{v=1}^{V}y_{v}\cdot\varepsilon_{vi}k_{i}}\right]\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{-u_{i}\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}}e^{i\sum_{v=1}^{V}\sum_{\alpha}z_{v}^{(\alpha)}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}}\right].$
(5)

Notice that the integral over $k_{i}$ is a gaussian and we can complete the
squares and compute the integral to get

$\int\frac{d^{D-d}k_{i}}{(2\pi)^{D-d}}e^{-u_{i}k_{i}^{2}}e^{i\sum_{v=1}^{V}y_{v}\cdot\varepsilon_{vi}k_{i}}=\frac{e^{-\frac{\left(\sum_{v=1}^{V}y_{v}\varepsilon_{vi}\right)^{2}}{4u_{i}}}}{\left(4\pi
u_{i}\right)^{\frac{D-d}{2}}},$ (6)

which substituted back into $\widetilde{\mathcal{I}}_{G}$ produce

$\widetilde{\mathcal{I}}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\left[\prod_{v=1}^{V}\int
d^{D-d}y_{v}\right]\left[\prod_{v=1}^{V}\prod_{\alpha}\int
dz_{v}^{(\alpha)}\right]e^{-\sum_{i}u_{i}m_{i}^{2}}e^{-i\sum_{v=1}^{V}\sum_{\alpha}z_{v}^{(\alpha)}\omega_{\alpha}^{p_{v}}}e^{-i\sum_{v=1}^{V}y_{v}\cdot
p_{v}}\times\\\
\prod_{i}\left[\frac{e^{-\frac{\left(\sum_{v=1}^{V}y_{v}\varepsilon_{vi}\right)^{2}}{4u_{i}}}}{\left(4\pi
u_{i}\right)^{\frac{D-d}{2}}}\right]\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{-u_{i}\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+i\sum_{v=1}^{V}\sum_{\alpha}z_{v}^{(\alpha)}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}}\right].$
(7)

Now we make evident the global conservation required by the delta function. To
do so we employ the change of variables

$\displaystyle y_{v}=\bar{y}_{v}+y_{V},$ $\displaystyle\qquad\forall v\neq V;$
(8a) $\displaystyle y_{V}=\bar{y}_{V};$ (8b) $\displaystyle
z_{v}^{(\alpha)}=\bar{z}_{v}^{(\alpha)}+z_{V}^{(\alpha)},$
$\displaystyle\qquad\forall v\neq V;$ (8c) $\displaystyle
z_{V}^{(\alpha)}=\bar{z}_{V}^{(\alpha)}.$ (8d) With this the sum over all
vertices produce
$\sum_{v=1}^{V}y_{v}\varepsilon_{vi}=\sum_{v=1}^{V-1}\bar{y}_{v}\varepsilon_{vi}+\bar{y}_{V}\sum_{v=1}^{V}\varepsilon_{vi}=\sum_{v=1}^{V-1}\bar{y}_{v}\varepsilon_{vi},$
(8e)

because $\sum_{v=1}^{V}\varepsilon_{vi}=0$ as a property of the incidence
matrix (each internal line starts from one vertex $+1$ and ends at one vertex
$-1$, therefore the sum over all contributions is $0$). Applying this change
of variables,

$\widetilde{\mathcal{I}}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\left[\prod_{v=1}^{V}\int
d^{D-d}\bar{y}_{v}\right]\left[\prod_{v=1}^{V}\prod_{\alpha}\int
d\bar{z}_{v}^{(\alpha)}\right]e^{-\sum_{i}u_{i}m_{i}^{2}}e^{-i\sum_{v=1}^{V-1}\sum_{\alpha}\bar{z}_{v}^{(\alpha)}\omega_{\alpha}^{p_{v}}}e^{-i\sum_{v=1}^{V-1}\bar{y}_{v}\cdot
p_{v}}\\\
e^{-i\sum_{\alpha}\bar{z}_{V}^{(\alpha)}\sum_{v=1}^{V}\omega_{\alpha}^{p_{v}}}e^{-i\bar{y}_{V}\cdot\sum_{v=1}^{V}p_{v}}\prod_{i}\left[\frac{e^{-\frac{\left(\sum_{v=1}^{V-1}\bar{y}_{v}\varepsilon_{vi}\right)^{2}}{4u_{i}}}}{\left(4\pi
u_{i}\right)^{\frac{D-d}{2}}}\right]\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{-u_{i}\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+i\sum_{v=1}^{V-1}\sum_{\alpha}\bar{z}_{v}^{(\alpha)}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}}\right],$
(9)

one can extract the global conservation (for the $v=V$ component),

$\int d^{D-d}\bar{y}_{V}\left[\prod_{\alpha}\int
d\bar{z}_{V}^{(\alpha)}\right]e^{-i\sum_{\alpha}\bar{z}_{V}^{(\alpha)}\sum_{v=1}^{V}\omega_{\alpha}^{p_{v}}}e^{-i\bar{y}_{V}\cdot\sum_{v=1}^{V}p_{v}}=\delta^{D-d}\left(\sum_{v=1}^{V}p_{v}\right)\prod_{\alpha}\delta\left(\sum_{v=1}^{V}\omega_{\alpha}^{p_{v}}\right).$
(10)

Due to this decomposition, it is usual to define a new amplitude without the
overall conservation,

$\widetilde{\mathcal{I}}_{G}=\delta^{D-d}\left(\sum_{v=1}^{V}p_{v}\right)\prod_{\alpha}\delta\left(\sum_{v=1}^{V}\omega_{\alpha}^{p_{v}}\right)\mathcal{I}_{G},$
(11)

such that,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\left[\prod_{v=1}^{V-1}\int
d^{D-d}\bar{y}_{v}\right]\left[\prod_{v=1}^{V-1}\prod_{\alpha}\int
d\bar{z}_{v}^{(\alpha)}\right]e^{-\sum_{i}u_{i}m_{i}^{2}}e^{-i\sum_{v=1}^{V-1}\sum_{\alpha}\bar{z}_{v}^{(\alpha)}\omega_{\alpha}^{p_{v}}}e^{-i\sum_{v=1}^{V-1}\bar{y}_{v}\cdot
p_{v}}\times\\\
\prod_{i}\left[\frac{e^{-\frac{\left(\sum_{v=1}^{V-1}\bar{y}_{v}\varepsilon_{vi}\right)^{2}}{4u_{i}}}}{\left(4\pi
u_{i}\right)^{\frac{D-d}{2}}}\right]\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{-u_{i}\sum_{\alpha}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+i\sum_{v=1}^{V-1}\sum_{\alpha}\bar{z}_{v}^{(\alpha)}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}}\right].$
(12)

At this point we compute the gaussian integral over the parameter
$\bar{y}_{v}$,

$\left[\prod_{v=1}^{V-1}\int
d^{D-d}\bar{y}_{v}\right]e^{-i\sum_{v=1}^{V-1}\bar{y}_{v}\cdot
p_{v}}e^{-\sum_{i}\frac{\left(\sum_{v=1}^{V-1}\bar{y}_{v}\varepsilon_{vi}\right)^{2}}{4u_{i}}}=\frac{(4\pi)^{\frac{(D-d)}{2}(V-1)}}{\left[\det
d_{G}(u)\right]^{\frac{D-d}{2}}}e^{-\sum_{v_{1},v_{2}=1}^{V-1}p_{v_{1}}p_{v_{2}}\left[d^{-1}_{G}(u)\right]_{v_{1},v_{2}}}.$
(13)

Here, it was defined the symmetric $(V-1)\times(V-1)$ matrix $d_{G}(u)$ as

$\left[d_{G}(u)\right]_{v_{1},v_{2}}=\sum_{i}\frac{\varepsilon_{v_{1}i}\varepsilon_{v_{2}i}}{u_{i}}.$
(14)

This produces, after a bit of organization,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}}\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}\right]\\\
\prod_{\alpha}\left\\{\left[\prod_{v=1}^{V-1}\int
d\bar{z}_{v}^{(\alpha)}\right]e^{-i\sum_{v=1}^{V-1}\bar{z}_{v}^{(\alpha)}\omega_{\alpha}^{p_{v}}}e^{-\sum_{i}u_{i}\left(\omega_{\alpha}^{k_{i}}\right)^{2}+i\sum_{v=1}^{V-1}\sum_{i}\bar{z}_{v}^{(\alpha)}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}}\right\\},$
(15)

or, introducing back the delta functions for the compactified dimensions,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}}\prod_{i}\left[\frac{1}{\prod_{\alpha}L_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{-\sum_{\alpha}u_{i}\left(\omega_{\alpha}^{k_{i}}\right)^{2}}\right]\prod_{v=1}^{V-1}\prod_{\alpha}(2\pi)\delta\left(\omega_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}\right).$
(16)

Where $L=I-V+1$ is the number of loops in the diagram (do not confuse with
$L_{\alpha}$, the length of each compactified dimension), and the polynomials
$U$ and $V$ are the Symanzik polynomials [2, 3], defined here as

$\displaystyle\frac{V(q;u)}{U(u)}$
$\displaystyle=\sum_{v_{1},v_{2}=1}^{V-1}q_{v_{1}}q_{v_{2}}\left[d^{-1}_{G}(u)\right]_{v_{1},v_{2}},$
(17a) $\displaystyle U(u)$ $\displaystyle=\left(\prod_{i}u_{i}\right)\det
d_{G}(u).$ (17b)

These polynomials have the remarkable propertie that they can be obtained
directly from the graph $G$ by inspecting the 1-trees and 2-trees that can be
formed by removing some internal lines. That is,

$\displaystyle U$ $\displaystyle=\sum_{T}\prod_{i\notin T}u_{i}$ (18a)
$\displaystyle V$ $\displaystyle=\sum_{K}Q_{K}^{2}\prod_{i\notin K}u_{i}$
(18b)

meaning that the first polynomial $U$ is the sum over all 1-trees $T$
(connected graphs without loops) and we consider all lines $i$ that are not in
the 1-tree, the removed lines. Also, the second polynomial $V$ is the sum over
all 2-trees $K$ (two separated trees) where we take all parameters $u_{i}$
that do not belong to the 2-tree. $Q_{K}$ is the overall momenta that enters
the 2-tree.

This ends the application of the usual procedure to the non-compactified
dimensions. From now on we deal with the remaining $d$ dimensions. To proceed
we employ the delta functions (in fact Kronecker deltas) to reduce the number
of summations just to the loop summations and then treat the expression.

From this point forward $\prod_{v}$ refers to $\prod_{v=1}^{V-1}$ unless
otherwise specified. As the sum is over the modes, one can extract a factor
from the delta functions

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{L_{\alpha}^{I}}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall
i\end{subarray}}e^{-\sum_{i}u_{i}\left(\omega_{\alpha}^{k_{i}}\right)^{2}}\left(L_{\alpha}\right)^{V-1}\prod_{v}\delta\left(n_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}n_{\alpha}^{k_{i}}\right)\right\\}.$
(19)

Notice that for each $\alpha$-th Matsubara mode we have $I$ summations
(related to the internal lines) and $V-1$ relations between the frequencies
(given by the Kronecker deltas), meaning an overall $L=I+1-V$ free frequencies
to be summed up. This indicates that the notation could be changed to
something like

$\frac{1}{L_{\alpha}^{L}}\prod_{v}\sum_{\begin{subarray}{c}n_{\alpha}^{k_{i}}\in\mathbb{Z}\\\
\forall
i\end{subarray}}e^{-\sum_{i}u_{i}\left(\omega_{\alpha}^{k_{i}}\right)^{2}}\prod_{v}\delta\left(n_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}n_{\alpha}^{k_{i}}\right)=\frac{1}{L_{\alpha}^{L}}e^{-Z[\omega_{\alpha}^{p_{v}}]}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell\in[1,L]\end{subarray}}e^{-Y[\omega_{\alpha}^{k_{\ell}}]},$ (20)

where the index $\ell$ runs over the $L$ independent loops.
$Y[\omega_{\alpha}^{k_{\ell}}]$ is a bilinear form on the $L$ frequencies
related to the loops, $Y$ is a $L\times L$ matrix whose coefficients depend on
$u_{i}$, the external modes $n_{\alpha}^{p_{v}}$ and the incidence matrix.
Also, $Z[\omega_{\alpha}^{p_{v}}]$ is another bilinear form that depends on
the external modes. Let us now proceed to obtain it using the delta functions,
we need to rewrite

$\Delta_{\alpha}=-\sum_{i}u_{i}\left(\omega_{\alpha}^{k_{i}}\right)^{2}$ (21)

using the $V-1$ constraints

$\omega_{\alpha}^{p_{v}}-\sum_{i}\varepsilon_{vi}\omega_{\alpha}^{k_{i}}=0.$
(22)

We choose a prescription where the last lines will be eliminated, and we use
the constraints to define a linear system,

$\omega_{\alpha}^{p_{v}}-\sum_{\ell}\varepsilon_{v\ell}\omega_{\alpha}^{k_{\ell}}=\sum_{v_{0}=1}^{V-1}\varepsilon_{v,L+v_{0}}\omega_{\alpha}^{k_{L+v_{0}}},$
(23)

that determine the frequencies to be eliminated
($\omega_{\alpha}^{k_{L+v_{0}}}$) as a function of the $L$ frequencies to be
kept ($\omega_{\alpha}^{k_{\ell}}$). In matricial notation, this means
trivally that

$Q=\bar{\epsilon}\overline{W}\Rightarrow\overline{W}=\bar{\epsilon}^{-1}Q,$
(24a) with
$\displaystyle(Q)_{v}=\omega_{\alpha}^{p_{v}}-\sum_{\ell}\varepsilon_{v\ell}\omega_{\alpha}^{k_{\ell}},$
(24b) $\displaystyle(\bar{\epsilon})_{v,v_{0}}=\varepsilon_{v,L+v_{0}},$ (24c)
$\displaystyle(\overline{W})_{v_{0}}=\omega_{\alpha}^{k_{L+v_{0}}}.$ (24d)

Of course, for this to work, we need the $(V-1)\times(V-1)$ matrix
$\bar{\epsilon}$ to be invertible. This is not guaranteed for all choices of
labeling of lines and vertices when building the incidence matrix
$\varepsilon_{v,i}$. But we can always reorganize the labels to guarantee a
choice where $\bar{\epsilon}$ is invertible. There are a few remarks here.
First, here we always choose to eliminate the last internal lines, this choice
does not affect the results, is just a matter of aesthetics (we want to keep
the same labels for the number of loops, so the $L$ frequencies related to the
$L$ loops are exactly the first $L$ internal lines). Second, different choices
of $\bar{\epsilon}$ will not affect the appearance of the Symanzik polynomials
$U$ and $V$, as one should expect for consistency. However, the matrix $Y$
will depend on this choice. Anyway, one can show that through a suitable
rearrangement of the modes ($n_{\alpha}^{k_{\ell}}$) one can relate a matrix
$Y$ obtained by some $\bar{\epsilon}$ to another matrix $Y$ obtained by
another choice of $\bar{\epsilon}$. Therefore, we do not lose generality by
specifying some prescriptions to obtain $\bar{\epsilon}$.

If we substitute back Eq. (24a) into Eq. (21),

$\Delta_{\alpha}=-\sum_{\ell}u_{\ell}\left(\omega_{\alpha}^{k_{\ell}}\right)^{2}-\sum_{v_{0}=1}^{V-1}u_{L+j}\left(\omega_{\alpha}^{k_{L+j}}\right)^{2}=-\sum_{\ell}u_{\ell}\left(\omega_{\alpha}^{k_{\ell}}\right)^{2}-\sum_{v_{0},v_{1},v_{2}=1}^{V-1}u_{L+j}\left[(\bar{\epsilon}^{-1})_{v_{0},v_{1}}Q_{v_{1}}\right]\left[(\bar{\epsilon}^{-1})_{v_{0},v_{2}}Q_{v_{2}}\right],$

and then apply the expression for $Q_{v}$, Eq. (24b), we obtain after some
algebraic manipulations that, in matricial notation,

$\Delta_{\alpha}=-(W_{\alpha}-M^{-1}B_{\alpha})^{t}M(W_{\alpha}-M^{-1}B_{\alpha})+(-F_{\alpha}+B_{\alpha}^{t}M^{-1}B_{\alpha}),$
(25a) where $W_{\alpha},M,B_{\alpha}$ and $F_{\alpha}$ are:
$\displaystyle(W_{\alpha})_{\ell}$ $\displaystyle=\omega_{\alpha}^{k_{\ell}},$
(25b) $\displaystyle(M)_{\ell_{1},\ell_{2}}$
$\displaystyle=u_{\ell_{1}}\delta_{\ell_{1},\ell_{2}}+\sum_{v_{0},v_{1},v_{2}}u_{L+v_{0}}\left[(\bar{\epsilon}^{-1})_{v_{0},v_{1}}\varepsilon_{v_{1},\ell_{1}}\right]\left[(\bar{\epsilon}^{-1})_{v_{0},v_{2}}\varepsilon_{v_{2},\ell_{2}}\right],$
(25c) $\displaystyle(B_{\alpha})_{\ell}$
$\displaystyle=\sum_{v_{0},v_{1},v_{2}}u_{L+v_{0}}\left[(\bar{\epsilon}^{-1})_{v_{0},v_{1}}\varepsilon_{v_{1},\ell}\right]\left[(\bar{\epsilon}^{-1})_{v_{0},v_{2}}\omega_{\alpha}^{P_{v_{2}}}\right],$
(25d) $\displaystyle F_{\alpha}$
$\displaystyle=\sum_{v_{0},v_{1},v_{2}}u_{L+v_{0}}\left[(\bar{\epsilon}^{-1})_{v_{0},v_{1}}\omega_{\alpha}^{P_{v_{1}}}\right]\left[(\bar{\epsilon}^{-1})_{v_{0},v_{2}}\omega_{\alpha}^{P_{v_{2}}}\right].$
(25e) Here, $M$ is a $L\times L$ matrix. For convience, we can also define a
tilde notation to indicate that the factor $2\pi/L_{\alpha}$ is extracted,
$\displaystyle(\widetilde{W}_{\alpha})_{\ell}$
$\displaystyle=n_{\alpha}^{k_{\ell}},$ (25f)
$\displaystyle(\widetilde{B}_{\alpha})_{\ell}$
$\displaystyle=\sum_{v_{0},v_{1},v_{2}}u_{L+v_{0}}\left[(\bar{\epsilon}^{-1})_{v_{0},v_{1}}\varepsilon_{v_{1},\ell}\right]\left[(\bar{\epsilon}^{-1})_{v_{0},v_{2}}n_{\alpha}^{P_{v_{2}}}\right].$
(25g)

At this point we can get back to the original expression, Eq. (19), and
rewrite the amplitude related to some graph $G$ as,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{L_{\alpha}^{L}}e^{-F_{\alpha}+B_{\alpha}^{t}M^{-1}B_{\alpha}}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell\in[1,L]\end{subarray}}e^{-\frac{4\pi^{2}}{L_{\alpha}^{2}}(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})^{t}M(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})}\right\\}.$
(26)

With the previous procedure (integrate over the momenta and then integrate
over the parameters related to the delta function) we found the standard
definition of the Symanzik polynomials, see Eq. (17), in terms of the
incidence matrix. Using the alternative procedure (apply first the delta
function and reduce the number of integrals) the Symanzik polynomials can be
expressed as 222One can check it repeating the procedure for the non-compact
dimensions. This is discussed in Ref. [77].

$\displaystyle U(u)$ $\displaystyle=\det M,$ (27a)
$\displaystyle\frac{V(\omega_{\alpha}^{p},u)}{U(u)}$
$\displaystyle=F_{\alpha}-B_{\alpha}^{t}M^{-1}B_{\alpha}.$ (27b)

With $M$, $B$ and $F$ defined as in Eq. (25). Therefore, one can simplify the
representation to

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(P)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{L_{\alpha}^{L}}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell\in[1,L]\end{subarray}}e^{-\frac{4\pi^{2}}{L_{\alpha}^{2}}(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})^{t}M(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})}\right\\},$
(28)

remember that $P_{v}=(p_{v},\omega_{\alpha}^{p_{v}})$ is the full external
momenta, meaning that $V(P)=V(p)+\sum_{\alpha}V(\omega_{\alpha}^{p},u)$.

From this point forward we will consider two different scenarios: the regime
near dimensional reduction ($L_{\alpha}\rightarrow 0$) and the regime near
bulk ($L_{\alpha}\rightarrow\infty$). Both of them are equivalent
analytically, in the sense that it is possible to transport from one to
another, but each of them is more suitable for a different length scale. At
first, we show the behavior when the compactification lengths are small (near
a dimensional reduction), in this scenario the expression is exactly the one
from Eq. (26) but requires some treatment, this is done in Sec. III. Then, in
Sec. IV, we show the behavior when the compactification lengths are large
(near the bulk), for this we employ a Jacobi theta identification that
modifies the summation in Eq. (26). The difference between both procedures is
that, although both are indeed valid for all compactified sizes, each one
converges faster on the specified limit. That said, although it can be done in
principle it is not indicated to employ the representation from Sec. III to
study the bulk limit neither the representation from Sec. IV to study the
dimensional reduction limit.

##  III Near dimensional reduction representation

As far as the author knows, although the topic of dimensional reduction is
well-known in the subject of finite temperature field theory (a dimensional
reduction occurs in the very high-temperature limit) the only attempt to
understand the behavior of a Feynman amplitude with many periodically
compactified spaces near a dimensional reduction comes from [75, 76]. The
present context, where we produce a parametric representation for a scalar
field theory in a periodically compactified space, allows a clear and easier
evaluation of the dimensional reduction.

At first, let us make clear that we consider the dimensional reduction in the
sense of Fisher [78]. That is, one does not say that all length parameters are
zero ($L_{\alpha}=0$), but rather that the length is small enough so that its
contribution can be mostly ignored. So we can take the limit where
$L_{\alpha}\rightarrow 0$ and keeps track only of the dominant contribution.

Taking the amplitude $I_{G}$ as in Eq. (26), we see that the dominant
contribution from the summation comes from a combination of modes
$n_{\alpha}^{(\ell)}$ that minimize the exponential suppression. All other
internal modes are responsible for subdominant contributions in this limit.
The dominant contribution is

$\mathcal{I}_{G}\sim
C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left[\frac{e^{-\frac{4\pi^{2}}{L_{\alpha}^{2}}\min
G_{\alpha}}}{L_{\alpha}^{L}}\right].$ (29)

Where $G_{\alpha}$ depends on the parameters $u_{i}$ and the external modes
$n_{\alpha}^{p_{v}}$ as

$G_{\alpha}=\widetilde{F}_{\alpha}-\widetilde{B}_{\alpha}^{t}M^{-1}\widetilde{B}_{\alpha}+(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})^{t}M(\widetilde{W}_{\alpha}-M^{-1}\widetilde{B}_{\alpha})$
(30)

and the minimization is with respect to the internal modes. That is, one
chooses a set $\widetilde{W}_{\alpha}$ such that $G_{\alpha}$ has its minimum
value. If $M^{-1}\widetilde{B}_{\alpha}=0$ this is trivially
$\widetilde{W}_{\alpha}=0$. The component $L_{\alpha}^{L}$ can be easily
absorbed by the definition of the coupling constant (that is inside the
constant $C_{G}$), but $G_{\alpha}$ is not easily absorbed. This component
introduces a new contribution to the Symanzik polynomials. That is, in the
limit of dimensional reduction one do not simply get the same amplitude in
reduced dimensions, but there is some surviving information from the original
dimensions.

There is another perspective, where one considers a completely static mode
approach. In this perspective all modes, internal and external, are null, and
this eliminates the contribution $G_{\alpha}$ producing simply

$\mathcal{I}_{G}\Big{|}_{\\{n_{\alpha}^{k_{\ell}},n_{\alpha}^{p_{v}}\\}=0}=\frac{C_{G}}{\prod_{\alpha}L_{\alpha}^{L}}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(p)}{U}}}{U^{\frac{D-d}{2}}},$
(31)

notice we can absorb the remaining factor related with the size in the
constant $C_{G}$ associated with the vertices. This is exactly the usual
parametric representation of Feynman diagram but only considering the $D-d$
non-compactified and large dimensions.

Sometimes it might look that both procedures are equivalent. The difference
between both becomes more evident if we consider a scenario where the graph
$G$ is a subdiagram to be evaluated in the ultraviolet regime, in this
scenario the external momenta are large and there is no justification to use a
zero-mode approximation for the external frequencies. This simple remark can
be responsible to produce large differences in behavior.

The remaining terms, subdominant contributions, to the amplitude in the small
length regime are obtained in straightforward way. Just as the dominant
contribution is the minimum with respect to the internal modes ($G_{\alpha}$),
the next leading order is a small deviation around this minimum. To make
notation clear we apply a shift in the internal modes, that is

$n_{\alpha}^{(\ell)}\rightarrow n_{\alpha}^{(\ell)}+j_{\alpha}^{\ell},$ (32)

where $j_{\alpha}^{\ell}\in\mathbb{Z}$, this is chosen such that the minimum
now lie on $n_{\alpha}^{(\ell)}=0$. Let us write this shift as

$\widetilde{W}_{\alpha}\rightarrow\widetilde{W}_{\alpha}-\widetilde{Y}_{\alpha}=\widetilde{W}_{\alpha}+M^{-1}\widetilde{B}_{\alpha}-\widetilde{Z}_{\alpha},$
(33)

so that $G_{\alpha}$ can be rewritten as

$G_{\alpha}=\widetilde{F}_{\alpha}-\widetilde{B}_{\alpha}^{t}M^{-1}\widetilde{B}_{\alpha}+(\widetilde{W}_{\alpha}-\widetilde{Z}_{\alpha})^{t}M(\widetilde{W}_{\alpha}-\widetilde{Z}_{\alpha}).$
(34)

We must choose a vector $\widetilde{Y}_{\alpha}\in\mathbb{Z}^{L}$ such that
the components of the resultant vector $\widetilde{Z}_{\alpha}$ all lies in
the interval
$(\widetilde{Z}_{\alpha})_{\ell}\in\left[-\frac{1}{2},\frac{1}{2}\right]$.
This guarantees that the minimum value of $G_{\alpha}$ occurs at
$\widetilde{W}_{\alpha}=0$. The simple choice is that $\widetilde{Y}_{\alpha}$
is the nearest integer to the real value $M^{-1}\widetilde{B}_{\alpha}$, which
we denote by $\lfloor M^{-1}\widetilde{B}_{\alpha}\rceil$. Therefore, the
vector $\widetilde{Z}_{\alpha}$ is just

$\widetilde{Z}_{\alpha}=M^{-1}\widetilde{B}_{\alpha}-\lfloor
M^{-1}\widetilde{B}_{\alpha}\rceil.$ (35)

With this simple change of notation the order of dominance is easier to write.
The dominant contribution, as already stated, occur at
$\sum_{\ell}|n_{\alpha}^{(\ell)}|=0$, meaning that
$n_{\alpha}^{(\ell)}=0,\,\forall\ell$. The first correction occurs at
$\sum_{\ell}|n_{\alpha}^{(\ell)}|=1$ and so on. This produces the simple
expression,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(P)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{L_{\alpha}^{L}}e^{-Z_{\alpha}^{t}MZ_{\alpha}}\right\\}\\\
+C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(P)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{L_{\alpha}^{L}}\sum^{{}^{\prime}}_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell\in[1,L]\end{subarray}}e^{-\frac{4\pi^{2}}{L_{\alpha}^{2}}(\widetilde{W}_{\alpha}-\widetilde{Z}_{\alpha})^{t}M(\widetilde{W}_{\alpha}-\widetilde{Z}_{\alpha})}\right\\}.$
(36)

Where $\sum^{\prime}$ denotes that the zero mode is already removed.

In the next section, we deal with the scenario near the bulk (large
compactification length) and rewrite the parametrization using a new
distribution of sectors that change the $u$ parameters to $t$ parameters. Near
the bulk, this reparametrization has the advantage of letting us see the
appearance of the modified Bessel function of the second kind, characteristic
of periodically compactified problems. However, when discussing the scenario
near dimensional reduction this new parametrization does not contribute a lot,
so we let the explanation of it for the next section. For completeness,
however, we show below how the amplitude looks like after this
reparametrization.

$\mathcal{I}_{G}=\frac{1}{(\prod_{\alpha}L_{\alpha})^{L}}C_{G}\left[\prod_{i=1}^{I-1}\int_{0}^{1}dt_{i}t_{i}^{I-1-i}\right]\frac{\Gamma\left(I-\frac{(D-d)L}{2}\right)}{(4\pi)^{\frac{(D-d)}{2}L}{\overline{U}}^{\frac{(D-d)}{2}}}\frac{1}{\left[\mathcal{M}^{2}(t_{i},m_{i})+\frac{\overline{V}(P;t)}{\overline{U}(t)}+{\overline{Z}}_{\alpha}^{t}\overline{M}\overline{Z}_{\alpha}\right]^{I-\frac{(D-d)L}{2}}}\\\
+\frac{1}{(\prod_{\alpha}L_{\alpha})^{L}}C_{G}\left[\prod_{i=1}^{I-1}\int_{0}^{1}dt_{i}t_{i}^{I-1-i}\right]\frac{\Gamma\left(I-\frac{(D-d)L}{2}\right)}{(4\pi)^{\frac{(D-d)}{2}L}{\overline{U}}^{\frac{(D-d)}{2}}}\times\\\
\sum^{{}^{\prime}}_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell,\forall\alpha\end{subarray}}\frac{1}{\left[\mathcal{M}^{2}(t_{i},m_{i})+\frac{\overline{V}(P;t)}{\overline{U}(t)}+(\overline{W}_{\alpha}-{\overline{Z}}_{\alpha})^{t}\overline{M}(\overline{W}_{\alpha}-\overline{Z}_{\alpha})\right]^{I-\frac{(D-d)L}{2}}}$
(37)

The remaining sum is a zeta-like summation, in the scenario of very small
compactification lengths, it can be truncated at low terms (as discussed
before, each new contribution is subdominant).

##  IV Near bulk representation

Starting from Eq. (26) we apply the Jacobi theta identity,

$\sum_{\mathbf{a}\in\mathbb{Z}^{n}}e^{-\pi tY[\mathbf{a}+\mathbf{g}]}e^{2\pi
i\mathbf{h}\cdot\mathbf{a}}.=e^{-2\pi
i\mathbf{h}\cdot\mathbf{g}}t^{-\frac{n}{2}}|Y|^{-\frac{1}{2}}\sum_{\mathbf{a}\in\mathbb{Z}^{n}}e^{-\frac{\pi}{t}Y^{-1}[\mathbf{a-h}]}e^{2\pi
i\mathbf{a}\cdot\mathbf{g}},$ (38)

with $t=4\pi/L_{\alpha}^{2}$, $n=L$, the matrix $Y=M$,
$(\textbf{g})_{n}=-(M^{-1}\widetilde{B}_{\alpha})_{\ell}$,
$(\textbf{a})_{n}=(\widetilde{W}_{\alpha})_{\ell}=n_{\alpha}^{k_{\ell}}$. So
that we obtain

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{(D-d)}{2}L}}\frac{e^{-\frac{V(P)}{U}}}{U^{\frac{D-d}{2}}}\prod_{\alpha}\left\\{\frac{1}{(4\pi)^{\frac{L}{2}}\left[\det
M\right]^{\frac{1}{2}}}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell\in[1,L]\end{subarray}}e^{-\frac{L_{\alpha}^{2}}{4}\widetilde{W}_{\alpha}^{t}M^{-1}\widetilde{W}_{\alpha}}e^{2\pi
i\widetilde{W}_{\alpha}^{t}M^{-1}\widetilde{B}_{\alpha}}\right\\}.$ (39)

Recall the definition of the Symanzik polynomials $U$ and $V$ as in Eq. (27),
the amplitude for a graph $G$ when $d$ dimensions are compactified can then be
written as

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{D}{2}L}}\frac{e^{-\frac{V(P;u)}{U(u)}}}{U(u)^{\frac{D}{2}}}\times\\\
\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell,\forall\alpha\end{subarray}}e^{-\sum_{\alpha}\frac{L_{\alpha}^{2}}{4U(u)}\sum_{\ell_{1},\ell_{2}}n_{\alpha}^{k_{\ell_{1}}}n_{\alpha}^{k_{\ell_{2}}}A_{\ell_{1},\ell_{2}}(u)}e^{\frac{2\pi
i}{U(u)}\sum_{\alpha,\ell_{1},\ell_{2}}n_{\alpha}^{k_{\ell_{1}}}A_{\ell_{1},\ell_{2}}(u)(\widetilde{B}_{\alpha}(u))_{\ell_{2}}},$
(40)

or, in compact notation,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{D}{2}L}}\frac{e^{-\frac{V(P;u)}{U(u)}}}{U(u)^{\frac{D}{2}}}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell,\forall\alpha\end{subarray}}e^{-\sum_{\alpha}\frac{L_{\alpha}^{2}}{4U(u)}\textbf{n}_{\alpha}^{t}A(u)\textbf{n}_{\alpha}}e^{\frac{2\pi
i}{U(u)}\sum_{\alpha}\textbf{n}_{\alpha}^{t}A(u)\widetilde{B}_{\alpha}(u)}.$
(41)

here $\textbf{n}_{\alpha}$ is vector with components
$(\textbf{n}_{\alpha})_{\ell}=n_{\alpha}^{k_{\ell}}$ and the matrix $A$ is
$A=M^{-1}\det M$.

Based on all we have done so far, the prescription one must follow is:

  1. 1.

From the graph extract the 1-trees and produce $U(u)$ as usual, see Eq. (18);

  2. 2.

From the graph extract the 2-trees and produce $V(u)$ as usual, see Eq. (18);

  3. 3.

From the incidence matrix $\varepsilon_{v,i}$ extract a $(V-1)\times(V-1)$
matrix $\bar{\epsilon}$ that is invertible. By convention, take the last
internal lines;

  4. 4.

Using $\bar{\epsilon}$, $\varepsilon_{v,i}$ and $\omega_{\alpha}^{p_{v}}$
compute the $M$, Eq. (25c), and $\widetilde{B}$, Eq. (25g). The matrix $A$ is
defined as $A=M^{-1}\det M$.

Unfortunately, up to this point, we do not have a prescription that allows us
to recover the matrix $A$ or the vector $\widetilde{B}$ directly from the
topology of diagram, like $U$ and $V$ that are extracted by drawing the
1-trees and 2-trees related to the graph, see Eq. (18).

The Symanzyk polynomials $U$ and $V$ are solely related to the graph, however,
we also need $A$ and $B$, which are determined from our prescription of
$\bar{\epsilon}$. One might inquire whether they are different depending on
the choice of prescription and indeed they are. Unlike $U$ and $V$, which do
not feel our choice of $\bar{\epsilon}$, the factors $A$ and $B$ depend on it.
However, we can transform the summation modes in such a way to transport from
one representation to another, meaning that they are all equivalent.

One can easily notice from Eq. (40) that the zeroth mode now represents the
bulk scenario (if we take $L_{\alpha}\rightarrow\infty$ we get an exponential
supression, so the only survivor is $n_{\alpha}^{(\ell)}=0$). This will
produce the standard representation,

$\mathcal{I}_{G}=C_{G}\left[\prod_{i}\int_{0}^{\infty}du_{i}\right]\frac{e^{-\sum_{i}u_{i}m_{i}^{2}}}{(4\pi)^{\frac{D}{2}L}}\frac{e^{-\frac{V(P;u)}{U(u)}}}{U(u)^{\frac{D}{2}}},$

as should be expected. This shows the consistency of the representation, that
recovers the expected scenario when the compactification is removed. As far as
we know, this simple and expected connection between the parametric
representation in the compactified scenario $(D,d)$ and the non-compactified
scenario $(D)$ has never been explicitly stated.

In the following, we change the parametrization and proceed to show that the
usual sum over the Bessel-K functions appears naturally. This is expected when
dealing with Feynman amplitudes in periodically compactified spaces as shown
in previous works [18, 55, 28] restricted to contributions of Feynman graphs
of low orders.

###  IV.1 Change of parametrization

We employ a new parametrization that sectorize the $u_{i}$-parameters such
that there is one parameter ($s$) that runs over the positive real axis and
all others ($t_{i}$) just lie in the interval $[0,1]$. This is done by the
sectorization

$\displaystyle u_{1}$ $\displaystyle=st_{1}\ldots t_{I-1},$ $\displaystyle
u_{2}$ $\displaystyle=st_{1}\ldots t_{I-2}(1-t_{I-1}),$ $\displaystyle u_{3}$
$\displaystyle=st_{1}\ldots t_{I-3}(1-t_{I-2}),$ $\displaystyle\vdots$
$\displaystyle=\vdots$ $\displaystyle u_{I-1}$
$\displaystyle=st_{1}(1-t_{2}),$ $\displaystyle u_{I}$
$\displaystyle=s(1-t_{1}).$

This change of variables produce
$\prod_{i=1}^{I}du_{i}=s^{I-1}ds\prod_{i=1}^{I-1}t_{i}^{I-1-i}dt_{i}$. And, as
all $u_{i}$ have a factor $s$, this will factor out in a way that allows us to
define

$\displaystyle U(u_{i})$ $\displaystyle=s^{L}\overline{U}(t_{i}),$ (42a)
$\displaystyle V(P;u_{i})$ $\displaystyle=s^{L+1}\overline{V}(P;t_{i}),$ (42b)
$\displaystyle A(u_{i})$ $\displaystyle=s^{L-1}\overline{A}(t_{i}),$ (42c)
$\displaystyle\widetilde{B}(u_{i})$
$\displaystyle=s\overline{\widetilde{B}}(t_{i}).$ (42d)

Notice that one must determine the new expressions by direct inspection.

With this notation, the amplitude related to the diagram becames

$\mathcal{I}_{G}=C_{G}\int_{0}^{\infty}dss^{I-1}\left[\prod_{i=1}^{I-1}\int_{0}^{1}dt_{i}t_{i}^{I-1-i}\right]\frac{e^{-s\mathcal{M}^{2}(t_{i},m_{i})}}{(4\pi)^{\frac{D}{2}L}}\frac{e^{-s\frac{\overline{V}(P;t)}{\overline{U}(t)}}}{s^{\frac{DL}{2}}{\overline{U}}^{\frac{D}{2}}}\sum_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell,\forall\alpha\end{subarray}}e^{-\frac{1}{s}\sum_{\alpha}\frac{L_{\alpha}^{2}}{4\overline{U}(t)}\textbf{n}_{\alpha}^{t}\overline{A}(t)\textbf{n}_{\alpha}}e^{\frac{2\pi
i}{\overline{U}(t)}\sum_{\alpha}\textbf{n}_{\alpha}^{t}\overline{A}(t)\overline{\widetilde{B}}_{\alpha}},$
(43)

with

$\mathcal{M}^{2}[t_{i},m_{i}^{2}]=\sum_{i}\frac{u_{i}}{s}m_{i}^{2}.$ (44)

When $m_{i}=m$ we get simply $\mathcal{M}^{2}=m^{2}$.

By inspection, we have two kinds of integrals regarding the parameter $s$.
When all modes are zero ($n_{\alpha}^{(\ell)}=0$) we have a gamma function,
when any of them is different from zero we get the integral that defines the
modified Bessel function of the second kind (Bessel-K), that is

$\mathcal{I}_{G}=C_{G}\left[\prod_{i=1}^{I-1}\int_{0}^{1}dt_{i}t_{i}^{I-1-i}\right]\frac{\Gamma\left(I-\frac{DL}{2}\right)}{(4\pi)^{\frac{D}{2}L}{\overline{U}}^{\frac{D}{2}}}\frac{1}{\left[\mathcal{M}^{2}(t_{i},m_{i})+\frac{\overline{V}(P;t)}{\overline{U}(t)}\right]^{I-\frac{DL}{2}}}\\\
+\frac{2C_{G}}{(4\pi)^{\frac{D}{2}L}}\left[\prod_{i=1}^{I-1}\int_{0}^{1}dt_{i}t_{i}^{I-1-i}\right]\sum^{{}^{\prime}}_{\begin{subarray}{c}n_{\alpha}^{(\ell)}\in\mathbb{Z}\\\
\forall\ell,\forall\alpha\end{subarray}}e^{\frac{2\pi
i}{\overline{U}(t)}\sum_{\alpha}\textbf{n}_{\alpha}^{t}\overline{A}(t)\overline{\widetilde{B}}_{\alpha}}\left(\frac{\sum_{\alpha}L_{\alpha}^{2}\textbf{n}_{\alpha}^{t}\overline{A}(t)\textbf{n}_{\alpha}}{4\overline{U}(t)\left[\mathcal{M}^{2}(t_{i},m_{i})+\frac{\overline{V}(P;t)}{\overline{U}(t)}\right]}\right)^{\frac{I-DL/2}{2}}\times\\\
K_{I-\frac{DL}{2}}\left(\sqrt{\frac{\sum_{\alpha}L_{\alpha}^{2}\textbf{n}_{\alpha}^{t}\overline{A}(t)\textbf{n}_{\alpha}}{\overline{U}(t)}}\sqrt{\mathcal{M}^{2}(t_{i},m_{i})+\frac{\overline{V}(P;t)}{\overline{U}(t)}}\right).$
(45)

Where $\sum^{{}^{\prime}}$ represents that we do not sum at the point where
all modes are zero.

This completes the analysis. Once again we remark that this expression is
suitable for large $L_{\alpha}$. This is much more evident at this point. If
we try to extract information from small $L_{\alpha}$ we approach the limit
$\lim_{z\rightarrow 0}K_{\nu}(z)$, where the Bessel-K function diverge and one
must be very careful with the treatment, as discussed in Ref. [75, 76]. Also,
as the length parameter, $L_{\alpha}$ is reduced each term of the sum gets
bigger and the convergence of the summations takes longer. This is the
justification why we also exhibit the scenario near the dimensional reduction
in Sec. III.

In the following section, we show the needed terms $U,V,A$ and $B$ for some
Feynman graphs.

##  V Some examples in $\phi^{3}$

In this section, we assume that our model is the simple toy model $g\phi^{3}$
and exhibit the relevant information to write the amplitudes of certain
diagrams both in the near dimensional reduction representation, Sec. III, and
the near bulk representation, Sec. IV, for systems living in periodically
compactified topologies. The required information, for each diagram, to show
the two Symanzik polynomials ($U$ and $V$) and some complementary information
as the matrix $M$ (see Eq. (25c)), $A$ (recall that $A=M^{-1}\det M=UM^{-1}$)
and the vector $B$ (see Eq. (25d)). In what follows one can obtain the
Symanzik polynomials by any of the three equivalent expressions Eq. (17), Eq.
(18) ou Eq. (27).

As a first example we consider the fish diagram, Eqs. (46), with external
momenta $Q$. Due to the existence of only one loop, the matrix is reduced to a
number. We define for convinience that $u_{12}=u_{1}+u_{2}$, and we follow
this notation in all other examples.

Figure 1: Fish diagram

$\displaystyle U$ $\displaystyle=u_{12}=s$ (46a) $\displaystyle V/Q^{2}$
$\displaystyle=u_{1}u_{2}=s^{2}\left\\{t(1-t)\right\\}$ (46b) $\displaystyle
M$ $\displaystyle=u_{12}=s$ (46c) $\displaystyle A$ $\displaystyle=1$ (46d)
$\displaystyle B$
$\displaystyle=u_{2}\omega^{p}_{\alpha}=s(1-t)\omega^{p}_{\alpha}$ (46e)

In this simple scenario, the expression near bulk is

$\mathcal{I}_{G}=C_{G}\int_{0}^{1}dt\frac{\Gamma\left(2-\frac{D}{2}\right)}{(4\pi)^{\frac{D}{2}}}\frac{1}{\left[\mathcal{M}^{2}(t_{i},m_{i})+P^{2}t(1-t)\right]^{2-\frac{D}{2}}}+\frac{2C_{G}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{1}dt\sum^{{}^{\prime}}_{\begin{subarray}{c}n_{\alpha}\in\mathbb{Z}\\\
\forall\alpha\end{subarray}}e^{2\pi
i\sum_{\alpha}(1-t)n_{\alpha}n_{\alpha}^{p}}\times\\\
\left(\frac{\sum_{\alpha}L_{\alpha}^{2}n_{\alpha}^{2}}{4\left[\mathcal{M}^{2}(t_{i},m_{i})+P^{2}t(1-t)\right]}\right)^{\frac{2-D/2}{2}}K_{2-\frac{D}{2}}\left(\sqrt{\sum_{\alpha}L_{\alpha}^{2}n_{\alpha}^{2}}\sqrt{\mathcal{M}^{2}(t_{i},m_{i})+P^{2}t(1-t)}\right).$
(47)

The first component is the well-known bulk contribution and the second
component is the finite-size contribution. We can get a bit further and study,
for example, the asymptotic behavior of this amplitude with respect to the
external momenta $P$. This falls outside the scope of this work and we shall
return to this point in a near future.

With regard to mass corrections there are two 1PI (one-particle irreducible)
diagrams at two loops (see Fig. 2 and Fig. 3). For first diagram, Fig. 2, the
functions allows to reproduce both the near dimensional reduction and near
bulk scenario are exhibited in Eqs. (48).

Figure 2: 1PI diagrams at two loops

$\displaystyle U$
$\displaystyle=u_{5}u_{1234}+u_{14}u_{23}=s^{2}\left\\{t_{1}\left[1-t_{1}+t_{1}t_{2}\left(1-t_{3}t_{4}-t_{2}(1-t_{3}t_{4})^{2}\right)\right]\right\\}$
(48a) $\displaystyle V/Q^{2}$
$\displaystyle=u_{5}u_{12}u_{34}+u_{12}u_{3}u_{4}+u_{1}u_{2}u_{34}$
$\displaystyle=s^{3}\left\\{t_{1}t_{2}^{2}t_{3}\left[1-t_{1}-t_{2}t_{3}+t_{1}t_{2}t_{3}+t_{1}t_{4}(1-t_{2})(1-t_{3}t_{4})+t_{1}^{2}(1-t_{4})(1-t_{3})(1-t_{2}(1-t_{3}t_{4}))\right]\right\\}$
(48b) $\displaystyle M$ $\displaystyle=\begin{pmatrix}u_{145}&-u_{5}\\\
-u_{5}&u_{235}\end{pmatrix}=s\begin{pmatrix}1-t_{2}+t_{1}t_{2}t_{3}t_{4}&-1+t_{1}\\\
-1+t_{1}&1-t_{1}+t_{2}(1-t_{3}t_{4})\end{pmatrix}$ (48c) $\displaystyle A$
$\displaystyle=\begin{pmatrix}u_{235}&u_{5}\\\
u_{5}&u_{145}\end{pmatrix}=s\begin{pmatrix}1-t_{1}+t_{2}(1-t_{3}t_{4})&1-t_{1}\\\
1-t_{1}&1-t_{2}+t_{1}t_{2}t_{3}t_{4}\end{pmatrix}$ (48d) $\displaystyle B$
$\displaystyle=\omega^{p}_{\alpha}\begin{pmatrix}u_{4}\\\
u_{3}\end{pmatrix}=st_{1}\omega^{p}_{\alpha}\begin{pmatrix}1-t_{2}\\\
t_{2}(1-t_{3})\end{pmatrix}$ (48e)

Figure 3: 1PI diagrams at two loops

For the second 1PI diagram, see Fig. 3, the two scenarios are reproduced if we
use the functions in Eqs (49).

$\displaystyle U$
$\displaystyle=u_{5}u_{2}+u_{134}u_{25}=s^{2}t_{1}\left\\{\left[1-t_{2}t_{3}(1-t_{4})\right]\left[1-t_{1}+t_{1}t_{2}t_{3}(1-t_{4})\right]+t_{2}t_{3}(1-t_{1})(1-t_{4})\right\\}$
(49a) $\displaystyle V/Q^{2}$
$\displaystyle=u_{4}u_{2}u_{5}+u_{4}u_{13}u_{25}$
$\displaystyle=s^{3}t_{1}^{2}t_{2}\left\\{t_{3}(1-t_{1})(1-t_{2})(1-t_{4})+(1-t_{2})(1-t_{3}+t_{3}t_{4})(1-t_{1}+t_{1}t_{2}t_{3}(1-t_{4}))\right\\}$
(49b) $\displaystyle M$ $\displaystyle=\begin{pmatrix}u_{1345}&-u_{5}\\\
-u_{5}&u_{25}\end{pmatrix}=s\begin{pmatrix}1-t_{1}t_{2}t_{3}+t_{1}t_{2}t_{3}t_{4}&-1+t_{1}\\\
-1+t_{1}&1-t_{1}+t_{1}t_{2}t_{3}(1-t_{4})\end{pmatrix}$ (49c) $\displaystyle
A$ $\displaystyle=\begin{pmatrix}u_{25}&u_{5}\\\
u_{5}&u_{1345}\end{pmatrix}=s\begin{pmatrix}1-t_{1}+t_{1}t_{2}t_{3}(1-t_{4})&1-t_{1}\\\
1-t_{1}&1-t_{1}t_{2}t_{3}+t_{1}t_{2}t_{3}t_{4}\end{pmatrix}$ (49d)
$\displaystyle B$ $\displaystyle=\omega^{p}_{\alpha}\begin{pmatrix}u_{4}\\\
0\end{pmatrix}=st_{1}\omega^{p}_{\alpha}\begin{pmatrix}1-t_{2}\\\
0\end{pmatrix}$ (49e)

At three loops we have a large number of 1PI diagrams that contribute to the
correction of the full propagator. One such example is shown in Fig. 4. One
can use directly the matrices $A$ and the vector $B$ to build the
representation near the bulk, see Eq. (45), and use $M$ to build the
representtion near the dimensional reduction, see Eq. (37).

Figure 4: Example with three loops

$\displaystyle U=u_{34}u_{56}u_{1278}+u_{34}u_{18}u_{27}+u_{3}u_{4}u_{1278}$
(50a) $\displaystyle V/Q^{2}=u_{12}u_{34}u_{56}u_{78}+u_{1}u_{2}u_{34}u_{78}$
$\displaystyle+u_{12}u_{3}u_{4}u_{78}+u_{12}u_{34}u_{7}u_{8}$ (50b)
$\displaystyle M=\begin{pmatrix}u_{14568}&-u_{456}&u_{4}\\\
-u_{456}&u_{24567}&-u_{4}\\\ u_{4}&-u_{4}&u_{34}\end{pmatrix}$ (50c)
$\displaystyle A=\begin{pmatrix}\begin{matrix}u_{2567}u_{34}\\\
+u_{3}u_{4}\end{matrix}&\begin{matrix}u_{3}u_{4}\\\
+u_{34}u_{56}\end{matrix}&-u_{4}u_{27}\\\ \begin{matrix}u_{3}u_{4}\\\
+u_{34}u_{56}\end{matrix}&\begin{matrix}u_{1568}u_{34}\\\
+u_{3}u_{4}\end{matrix}&u_{4}u_{18}\\\
-u_{4}u_{27}&u_{4}u_{18}&\begin{matrix}u_{456}u_{1278}\\\
+u_{18}u_{27}\end{matrix}\end{pmatrix}$ (50d) $\displaystyle
B=\omega^{p}_{\alpha}\begin{pmatrix}u_{8}\\\ u_{7}\\\ 0\end{pmatrix}$ (50e)

##  VI Conclusion

We manage to show two useful parametric representations for Feynman diagrams
in periodically compactified spaces. One representation is more suitable near
the bulk and recovers the expected standard Schwinger parametric
representation for non-compactified theories at the bulk. The other
representation is more suitable near the dimensional reduction and recovers
that only a completely static mode approximation can generate a dimensional
reduction. Both expressions allow one to obtain the behavior with respect to
the compactification length and shall be helpful for those interested in
higher-order Feynman diagrams in compactified spaces to study a myriad of
phenomena as depicted by works in quantum field theory with compactifications
[14, 18, 55, 25, 26, 27, 28, 56].

Of course, we still do not get here the full picture and there are many
remaining tasks to be done: 1 - There is a need to extend the present work for
fields with non-zero spins; 2 - It might be useful to consider scenarios where
space is compactified by a different prescription other than a periodic one
(Dirichlet, Neumann, Robin, …); 3 - It is of utmost importance to extract
asymptotic information from the parametric representations so we can easily
understand the behavior of a higher-order graph in compactified space. Each of
these topics are the subject of future works.

###### Acknowledgements.

The author thanks the Brazilian agency Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) for financial support.

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