# Brane world regularization of point particle classical self-energy

Román Linares1<EMAIL_ADDRESS>Hugo A. Morales-Técotl1<EMAIL_ADDRESS>Omar Pedraza2<EMAIL_ADDRESS>Luis O. Pimentel1<EMAIL_ADDRESS>1
Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa,
San Rafael Atlixco 186, C.P. 09340, México D.F., México,  2 Área Académica de
Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo,
Carretera Pachuca-Tulancingo Km. 4.5, C P. 42184, Pachuca, México.

###### Abstract

Physical effects in brane worlds models emerge by the incorporation of field
modes coming from extra dimensions with the usual four dimensional ones. Such
effects can be tested with well established experiments to set bounds on the
parameters of the brane models. In this work we extend a previous result which
gave finite electromagnetic potentials and self energies for a source looking
pointlike to an observer sitting in a 4D Minkowski subspace of a single brane
of a Randall-Sundrum spacetime including compact dimensions, and along which
the source stretches uniformly. We show that a scalar particle produces a
nonsingular static potential, possess a finite self-energy and that
technically its analysis is very similar to the electrostatic case. As for the
latter, we use the deviations from the Coulomb potential to set bounds on the
anti de Sitter radius of the brane model on the basis of two experiments,
namely, one of the Cavendish type and other being the scattering of electrons
by Helium atoms. We found these are less stringent than others previously
obtained using the Lamb shift in Hydrogen.

###### pacs:

11.25.Wx, 11.10Kk, 11.25.Mj,04.25.-g, 03.50.-z

##  I Introduction

Despite the extraordinarily rich accuracy with which the predictions of
electrodynamics have been experimentally tested over the years (see for
instance Goldhaber and Nieto (2010, 1971) and references therein), efforts to
place limits on deviations from its standard formulation continue nowadays.
The nature of the experiments cover a big range of possibilities which include
among others: a) Testing the power in the inverse-square law of Coulomb, b)
Seeking a nonzero value for the rest mass of the photon and c) Considering
more degrees of freedom, allowing mass for the photon while preserving
explicit gauge invariance. It is worth to mention that all these experiments
have probed length scales increasing dramatically over time.

Now, historically, once the Maxwell theory of electromagnetic fields was
established, one of the main concerns in physics was the construction of a
consistent description of electrodynamics and charged particles. The first
serious proposals in this direction were developed by Lorentz Lorentz (1909)
and Abraham Abraham (1903). These proposals and subsequent attempts based on
classical electrodynamics, special relativity and the Lorentz force law led to
the theory known as Classical Electron Theory (CET). Whereas in the Maxwell
theory charges are considered as punctual which produce infinite Lorentz self-
force and infinite electromagnetic self-energies associated with the
singularities of the Liénard-Wiechert potential, in the CET charges are
considered as extended objects that experience a volume-averaged Lorentz
force. Parallel to the development of the CET, quantum mechanics was developed
giving origin to one of the most spectacular theory we have in physics:
Quantum Electrodynamics (QED). This theory is awesome due to the impressive
range of electromagnetic phenomena it covers with spectacular precision. After
the experimental achievements of QED, CET dropped from the list of contenders
for a fundamental theory of electrodynamics interacting with matter. However
despite the great success of QED there are still some features of the theory
that could be waiting for a better explanation, for instance, its property of
renormalizability. It turns out that QED is defined by a perturbative series
that is renormalizable in each order, but it is most likely to be merely
asymptotic in character rather than convergent Dyson (1952), in such a way
that the precision results are obtained only when computations are made to
some order in the expansion series, but without any a priori prescription to
stop the series at some order.

It is thus tempting to investigate theories that avoid singularities. These
are not expected to solve the problem but at least they may contribute to a
better understanding of the singularity issue.

In this context, recently, in a previous work, some of us found that a source
lying on the single brane of a Randall-Sundrum spacetime including compact
dimensions, and which looks pointlike to an observer sitting in usual 3D
space, produces a static potential which is non singular at 3D point position.
Moreover it matches Coulomb’s potential outside a small neighborhood Morales-
Técotl et al. (2007). The presence of the compact dimensions in this setup
serve to localize the gauge field on the brane Dubovsky et al. (2000a, b); Oda
(2000). The aim of this paper is to investigate further some consequences of
the above property to set bounds to the AdS curvature radius $\epsilon$ using
the experimental results from the Cavendish experiment for electromagnetism
and the scattering of electrons by Helium atoms. For the sake of clarity, the
simpler case of a scalar particle is first considered. Remarkably, the
nonsingular character of the potential holds together with the finiteness of
the selfenergy. Indeed, technically, the study of the potentials for both
scalar and electromagnetic is very similar.

Our interest in this work is twofold, on one side it is interesting to explore
how the old problem of divergences acquires a different character in light of
the brane world models, at least classically, and, on the other hand, it is
also interesting from the perspective of the brane-world scenarios Randall and
Sundrum (1999a, b); Antoniadis et al. (1998); Arkani-Hamed et al. (1998);
Antoniadis (1990), which have recently been matter of a copious research,
mainly in high energy physics (see e.g. Allanach et al. (2004); Csaki (2004),
and references therein) and cosmology (see e.g. Maartens (2004); Elizalde
(2006); Maartens and Koyama (2010), and references therein). More recently,
the possibility to obtain information from models with extra dimensions
studying low energy physical phenomena has also been addressed. In particular
we mention the ones that have been performed in the RSII-$p$ setup, such as
the electric charge conservation Dubovsky et al. (2000b), the Casimir effect
between parallel plates Linares et al. (2010); Frank et al. (2008) and the
Hydrogen Lamb shift Morales-Técotl et al. (2007).

The paper is organized as follows. In section II we briefly describe the
RSII-$p$ setup, section III is dedicated to obtain the static potential for a
scalar field whereas in section IV we do the same for the electric case. In
section V.1 we set bounds to the AdS radius $\epsilon$ comparing our
electrostatic results with the experimental values obtained in Cavendish like
experiments of the Coulomb force. Section V.2 is dedicated to the same purpose
but this time we use the experimental results of the scattering process of
electrons by Helium atoms. Finally section VI is devoted to a brief
discussion.

##  II Randall-Sundrum II-$p$ scenarios

The Randall-Sundrum II-$p$ scenarios consist of a ($3+p$)-brane with $p$
compact dimensions and positive tension $\sigma$, embedded in a ($5+p$)
spacetime whose metrics are two patches of anti-de Sitter (AdS5+p) having
curvature radius $\epsilon$. The interest in these models comes from its
property of localizing on the brane: scalar, gauge and gravity fields due to
the gravity produced by the brane itself. This property is valid whenever
there are $p$ extra compact dimensions Dubovsky et al. (2000b); Oda (2000). In
the limiting case $p=0$, the model only localizes scalar and gravity fields.
With this setup and appropriate fine-tuning between the brane tension $\sigma$
and the bulk cosmological constant $\Lambda$, which are related to $\epsilon$
as follows

$\sigma=\frac{2(3+p)}{8\pi\epsilon
G_{5+p}},\quad\Lambda=-\frac{(3+p)(4+p)}{16\pi\epsilon^{2}G_{5+p}}=-\frac{(4+p)\sigma}{4\epsilon},$
(1)

there exists a solution to (5+$p$)D Einstein equations with metric

$ds_{5+p}^{\,2}=e^{-2|y|/\epsilon}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}-\sum_{i=1}^{p}R^{2}_{i}d\theta_{i}^{2}\right]-dy^{2}.$
(2)

Here $\eta_{\mu\nu}$ is the 4D Minkowski tensor, $\theta_{i}\in[0,2\pi]$ are
$p$ compact coordinates, $R_{i}$ are the sizes of compact dimensions,
$G_{5+p}$ is the ($5+p$)D Newton constant. Throughout the paper we will use
the following notation for the 5+$p$ coordinates
$X^{M}\equiv(x^{\mu},\theta_{i},y)$, where $\mu=0,1,2,3$, and $i=1,\dots p$.

In this work we consider two different $(5+p)D$ field theories on RSII-$p$: a
massless scalar field and electrodynamics. They will be subjected to a hybrid
of the two well known consistent compactifications, namely Kaluza-Klein (KK)
Kaluza (1921); Klein (1926a) and warped Randall and Sundrum (1999a). These two
differ among them on whether the compactified manifold is factorizable or not.
The corresponding effective field theories in 4D Minkowski space-time will be
given.

In regard to the KK compactification, it is well known toroidal dimensional
reductions lead to consistent lower dimensional theories which nonetheless can
be questioned in that they do not come with a mechanism to fix the moduli, or
equivalently, the radii of the $p$D torus $T^{p}$ Duff et al. (1986); Green et
al. (1987). Historically, a way out in such cases, has been to conform with
the corresponding phenomenology at low enough energies and set a bound for the
radii (e.g. the use of the classical value of the electron charge required a
radius of the order Planck length in the original KK setting Klein (1926a,
b)). We will adhere to this approach by considering a low energy approximation
so that we truncate the massive KK modes of the compact dimensions but keep
those corresponding to the noncompact dimension just meaning that we assume
the energy scale of the former is much smaller than that of the latter. This
is explicitly performed in the Green’s function in III.1 for the scalar field
and in IV for the gauge field. As for the consistency of the Randall-Sundrum
compactification it has been discussed in Rubakov (2001) (and references
therein).

For completeness we only mention other mechanisms adopted in the literature to
perform a generalized KK compactification. One of them is the so called
Scherk-Schwarz compactification Scherk and Schwarz (1979a, b) or flux
compactifications Grana (2006); Blumenhagen et al. (2007); Douglas and Kachru
(2007). In this mechanism the symmetries of the compactification manifold
and/or the fields are used to produce an effective potential for stabilizing
the size of the extra dimensions. There also exists a quantum proposal by
Candelas and Weinberg Candelas and Weinberg (1984) where the effective
potential for the moduli fields is produced by the Casimir energy of matter
fields or gravity. It remains open to study these possibilities for our
present setup.

A remark regarding the stability of the scenario described by the metric (2)
is here in order. Concerning the world volume of the (3+$p$)-brane,
$M_{4}\otimes T^{p}$, it is clear that the space is stable since it is flat.
On the other hand the stability of the space-time (2), without the $T^{p}$
structure, was studied long ago in Randall and Sundrum (1999a); Garriga and
Tanaka (2000) for static perturbations of the metric and in Sasaki et al.
(2000) for general space-time dependent sources. The stability of other warped
compactifications has also been addressed, for instance in Goldberger and Wise
(1999); Lesgourgues and Sorbo (2004); Maity et al. (2006); Das et al. (2008)
it was discussed the moduli stabilization of the RSI model whereas in Flachi
et al. (2003) it was discussed for more general metrics.

Before ending this section is worth to mention that this setup has been
considered in different low energy physics effects such as the electric charge
conservation Dubovsky et al. (2000b), the Casimir effect between two conductor
hyperplates Linares et al. (2008a, b); Frank et al. (2008); Linares et al.
(2010) and the Liennard-Wiechert potentials and Hydrogen Lamb shift Morales-
Técotl et al. (2007) among others.

##  III Static potential for a scalar field

In this section we compute the potential produced by a static source which is
seen as punctual by an observer living on the usual 3D subspace of the
(3+$p$)-brane. It stretches however along the $p$ compact dimensions thus
forming a $p$-dimensional torus (1).

Figure 1: Schematic view of the charge source for $p$=2. The source is
effectively pointlike from the perspective of an observer sitting in the usual
3d space.

###  III.1 The Green’s function

Let us consider a massless scalar field $\Phi$ described by the action in
(5+$p$)D

$S=\int\,d\,{}^{4}x\,\prod_{i=1}^{p}\,R_{i}d\theta_{i}\,dy\,\sqrt{|g|}\,\left(\frac{1}{2}g^{MN}\partial_{M}\Phi\,\partial_{N}\Phi+\Phi
J_{scalar}\right).$ (3)

The equation of motion for the scalar field is

$\frac{1}{\sqrt{|g|}}\partial_{M}\left(\sqrt{|g|}g^{MN}\partial_{N}\Phi\right)=J_{scalar}$
(4)

where the source is given by

$J_{scalar}=\lambda^{(p)}\delta^{3}\left(\vec{x}-\vec{x}_{0}\right)\delta\left(y-y_{0}\right).$
(5)

Here $\lambda^{(p)}$ is a constant whose dimensions are [charge]/[length]p,
explicitly: $\lambda^{(p)}=\frac{\lambda}{(2\pi)^{p}R_{1}\cdots R_{p}}$, with
$\lambda$ the total charge.

In the background (2), the equation of motion (4) becomes

$e^{2|y|/\epsilon}\left[\Box\Phi-\sum_{i=1}^{p}\frac{1}{R_{i}^{2}}\,\partial_{\theta_{i}}^{\,2}\Phi\right]-\frac{1}{\sqrt{|g|}}\,\partial_{y}\left[\sqrt{|g|}\partial_{y}\Phi\right]=J_{scalar},$
(6)

where $\Box$ stands for the flat 4D D’Alambertian. The corresponding Green’s
equation is

$e^{2|y|/\epsilon}\left[\Box
G-\sum_{i=1}^{p}\frac{1}{R_{i}^{2}}\,\partial_{\theta_{i}}^{\,2}G\right]-\frac{1}{\sqrt{|g|}}\,\partial_{y}\left[\sqrt{|g|}\partial_{y}G\right]=\frac{\delta(y-y^{\prime})\delta^{p}(R_{i}\theta_{i}-R_{i}\theta^{\prime}_{i})\delta^{4}(x-x^{\prime})}{\sqrt{|g|}},$
(7)

where $G$ is the $(5+p)$D Green’s function. This can be expressed in terms of
the eigenfunctions of the differential operators for the different
coordinates. Assuming $\Psi(X^{M})\equiv
e^{ik_{\mu}x^{\mu}}\prod_{i=1}^{p}\Theta_{i}(\theta_{i})\psi(y)$, where the
modes $\Theta_{n}$ and $\psi_{m}$ account for the $\theta$ and $y$ dependence
respectively. These have been discussed previously (see for instance Dubovsky
et al. (2000a); Linares et al. (2010)) and here we only give a summary. The
differential equations governing the $p$ compact modes depending on
$\theta_{i}$ are

$(\partial_{\theta_{i}}^{2}+m_{\theta_{i}}^{2}R_{i}^{2})\Theta_{i}(\theta_{i})=0,\hskip
28.45274pti=1,\cdots,p,$ (8)

whereas for the noncompact modes depending on $y$ one gets

$(\partial_{y}^{2}-\frac{(4+p)}{\epsilon}\mbox{sgn}(y)\partial_{y})+m^{2}e^{2|y|/\epsilon})\psi(y)=0.$
(9)

The $(p+1)$ constants of separation, $m_{\theta_{i}},m$, fulfill the following
dispersion relation

$k^{2}=\sum_{i=1}^{p}m_{\theta_{i}}^{2}+m^{2}\equiv m_{p}^{2}+m^{2}.$ (10)

To account for the compactness of the $p$ dimensions Eq. (8) is solved under
the periodic boundary conditions

$\Theta_{n_{i}}(\theta_{i})=\Theta_{n_{i}}(\theta_{i}+2\pi),$ (11)

and the solutions turn out to be

$\Theta_{n_{i}}(\theta_{i})=\frac{1}{\sqrt{2\pi
R_{i}}}e^{in_{i}\theta_{i}},\,\,\,\mbox{where}\,\,\,n_{i}=m_{\theta_{i}}R_{i}\in\mathbb{Z}.$
(12)

To match the modes across the brane along the non-compact dimension, equation
(9) is solved with the following boundary conditions

$\psi(y=0^{+})=\psi(y=0^{-})\qquad{\rm
and}\qquad\partial_{y}\psi(y=0^{+})=\partial_{y}\psi(y=0^{-}).$ (13)

In this case the solutions include a massless zero mode localized on the brane

$\psi_{0}(y)=\sqrt{\frac{2+p}{2\epsilon}}$ (14)

which satisfies the normalization condition
$2\int_{0}^{\infty}dye^{-(2+p)|y|/\epsilon}\psi_{0}^{2}(y)=1$, as well as
massive modes given by

$\psi_{m}(y)=e^{\gamma
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}\left[a_{m}J_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right)+b_{m}N_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right)\right],$
(15)

where $J_{\gamma}$ and $N_{\gamma}$ are the Bessel and Neumann functions,
respectively. In this expression

$\gamma\equiv\frac{4+p}{2},$ (16)

and the coefficients $a_{m}$ y $b_{m}$ are

$a_{m}=-\frac{A_{m}}{\sqrt{1+A_{m}^{2}}},\quad
b_{m}=\frac{1}{\sqrt{1+A_{m}^{2}}},\quad
A_{m}=\frac{N_{\gamma-1}\left(m\epsilon\right)}{J_{\gamma-1}\left(m\epsilon\right)}.$
(17)

Notice that in this case, the localization of the massive modes on the brane
is better for increasing $p$ , since the modes are modulated exponentially by
a factor of $e^{-p|y|/(2\epsilon)}$. The normalization condition for the
massive modes is
$\int_{-\infty}^{\infty}dye^{-(p+2)|y|/\epsilon}\psi_{m}(y)\psi_{m^{\prime}}(y)=\delta(m-m^{\prime})$.

With the eigenfunctions at hand it is straightforward to use them to write
down the Green’s function. It takes the form

$G(x,\theta_{i},y;x^{\prime},\theta^{\prime}_{i},y^{\prime})=\prod_{i=1}^{p}\sum_{\\{n\\}}\frac{e^{in_{i}\theta_{i}}e^{-in_{i}\theta_{i}^{\prime}}}{2\pi
R_{i}}\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}\left[\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{k^{2}-m_{p}^{2}}+\int_{0}^{\infty}dm\frac{\psi_{m}(y)\psi_{m}(y^{\prime})}{k^{2}-m^{2}-m_{p}^{2}}\right],$
(18)

where $\\{n\\}$ denotes
$\\{n_{1},n_{2},\dots,n_{p}\,|\,n_{1}\in\mathbb{Z},\dots,n_{p}\in\mathbb{Z}\\}$.

At this point it is convenient to introduce an approximation that will allow
us to obtain analytic expressions of the potential. We have massive modes from
both compact dimensions (12) and the noncompact one (15). Since we are
interested in the low energy regime we assume $m_{\theta_{i}}\ll
m\ll\epsilon^{-1}$. Hence we will set $n_{1}=\dots=n_{p}=0$, and, as for the
noncompact modes (15) we use

$\psi_{m}(y)\approx-e^{\gamma
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}J_{\gamma}\left(m\epsilon\,e^{y/\epsilon}\right).$
(19)

In such low energy regime and upon integrating over the $p$ compact extra
dimensions, we end up with an effective 5D Green’s function

$G_{5D}(x,y;x^{\prime},y^{\prime})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}\left[\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{k^{2}}+\int_{0}^{\infty}dm\frac{\psi_{m}(y)\psi_{m}(y^{\prime})}{k^{2}-m^{2}}\right],$
(20)

where the massless mode is given by (14) and the massive modes by (19).
Although we have only taken the zero modes of the compact extra dimensions,
notice that their imprints remain in the 5D Green’s function through (16) and
(19).

###  III.2 Static potential

Now we are in position to compute the static potential. In this case the
useful Green’s function is

$\displaystyle G(\vec{x},y;\vec{x^{\prime}},y^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt^{\prime}G(\vec{x},t=0,y;\vec{x^{\prime}},t^{\prime},y^{\prime})$
(21) $\displaystyle=$
$\displaystyle\frac{\psi_{0}(y)\psi_{0}(y^{\prime})}{4\pi
r}+\int_{0}^{\infty}dm\psi_{m}(y)\psi_{m}(y^{\prime})\frac{e^{-mr}}{4\pi r}$

where $r=|\vec{x}-\vec{x}^{\prime}|$. As usual the potential is obtained upon
integrating the Green’s function times the source, Eq. (5), and we are
interested in its form at the brane, i.e. $y=0$, namely,

$\displaystyle\varphi(r,y=0)$ $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}dy^{\prime}G(\vec{x},y=0;\vec{x^{\prime}},y^{\prime})\,J_{scalar}(\vec{x^{\prime}},y^{\prime};\vec{x_{0}},y_{0})$
(22) $\displaystyle=$
$\displaystyle\frac{\lambda^{(p)}\psi_{0}(0)\psi_{0}(y_{0})}{4\pi
r}+\lambda^{(p)}\int_{0}^{\infty}dm\psi_{m}(0)\psi_{m}(y_{0})\frac{e^{-mr}}{4\pi
r},$

where now $r=|\vec{x}-\vec{x}_{0}|$, and (19) takes the asymptotic value

$\psi_{m}(0)\approx\frac{1}{\Gamma(\gamma-1)}\sqrt{\frac{m\epsilon}{2}}\left(\frac{m\epsilon}{2}\right)^{\gamma-2}.$
(23)

Finally, the potential becomes

$\varphi(r)=\frac{\lambda^{(p)}}{4\pi
r}\left(\frac{2+p}{2\epsilon}\right)-\lambda^{(p)}\int_{0}^{\infty}dm\frac{1}{\Gamma(\gamma-1)}\left(\frac{m\epsilon}{2}\right)^{\gamma-1}e^{\gamma
y_{0}/\epsilon}J_{\gamma}\left({m}{\epsilon}e^{y_{0}/\epsilon}\right)\frac{e^{-mr}}{4\pi
r}.$ (24)

Next we further assume the source to be located at the brane, i.e. $y_{0}=0$.
The explicit form of (24) now depends on whether the number of extra compact
dimensions, $p$, is odd or even, and so we discuss each case separately.

###  III.3 Odd number of extra compact dimensions

In the case that $p$ takes odd values, $\gamma$ takes semi-integer values and
is useful to use the relation

$m^{l+1/2}J_{l+1/2}\left(m\epsilon\right)=(-1)^{l}\sqrt{\frac{2}{\pi}}\epsilon^{l+1/2}\left(\frac{d}{\epsilon
d\epsilon}\right)^{l}\frac{\sin(m\epsilon)}{\epsilon},$ (25)

in the integrand of (24). Upon evaluation of the integral we get

$\varphi(r)=\frac{\lambda^{(p)}}{4\pi
r}\frac{2+p}{2\epsilon}-\frac{(-1)^{\gamma-\frac{1}{2}}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-3/2}\sqrt{\pi}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-\frac{1}{2}}\left(\frac{\pi}{2\epsilon}-\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right).$
(26)

Let us notice the first term of this expression is divergent at $r=0$. However
such a term cancels out with the first term within parenthesis for every odd
$p$. This leads to the form of the effective potential

$\varphi(r)=\frac{(-1)^{\gamma-\frac{1}{2}}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-3/2}\sqrt{\pi}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-\frac{1}{2}}\left(\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right).$
(27)

As an example, let us work out the case in which we have only one compact
extra dimension, ie $p=1\,\Rightarrow\,\gamma=5/2$. From (27) we obtain

$\varphi(r)=\frac{2q_{s}^{(1)}}{3\pi\epsilon}\left[3\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}+\frac{5}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)}+2\frac{\frac{r^{2}}{\epsilon^{2}}}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{2}}\right],$
(28)

where $q_{s}^{(1)}=\frac{3\lambda^{(1)}}{8\pi\epsilon}$. The finite value of
the potential at the 3D point position of the source takes the value

$\lim_{r\rightarrow 0}\varphi(r)=\frac{16q_{s}^{(1)}}{3\pi\epsilon},$ (29)

evidently regularized by the existence of $\epsilon$ and $R$. Using (28) we
can compute the effective self-energy of the point charge, as determined by a
3D observer

$E_{self}^{(p=1)}:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\nabla\varphi\right)^{2}=\frac{85\left(q_{s}^{(1)}\right)^{2}}{9\epsilon}.$
(30)

###  III.4 Even number of extra compact dimensions

In the case that $p$ takes even values $\gamma$ is integer and we can use the
relation

$m^{l}J_{l}(m\epsilon)=(-1)^{l}\epsilon^{l}\left(\frac{d}{\epsilon
d\epsilon}\right)^{l-1}\left(-\frac{J_{1}(m\epsilon)m}{\epsilon}\right),$ (31)

in (24), to obtain 111It is worth mentioning here an alternative approach to
get the same results for both odd an even $p$. It amounts to using the
completeness relation of the noncompact modes. In such a case the coefficient
in front of $1/r$ is proportional to the square of a Dirac delta and by using
either dimensional regularization or distribution operations the coefficient
vanishes thus obtaining the same result, ie that the potential is finite at
the 3D position of the source Morales-Técotl et al. (2007).

$\varphi(r)=\frac{(-1)^{\gamma+1}\lambda^{(p)}\epsilon^{2\gamma-1}}{2^{\gamma-1}\Gamma(\gamma-1)}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\gamma-1}\left(\frac{r}{\epsilon^{2}\sqrt{r^{2}+\epsilon^{2}}}\right).$
(32)

As an example, let us consider the lowest even value for $p$:
$p=2\,\Rightarrow\gamma=3$. From (32) we obtain

$\varphi(r)=\frac{q_{s}^{(2)}}{8\epsilon}\left[\frac{8}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{4}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}+\frac{3}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{5}{2}}}\right],$
(33)

where $q_{s}^{(2)}=\frac{\lambda^{(2)}}{2\pi\epsilon}$. In this case the
finite value of the potential at the 3D position of the source is

$\lim_{r\rightarrow 0}\varphi(r)=\frac{15q_{s}^{(2)}}{8\epsilon},$ (34)

whereas the source self-energy is given by

$E_{self}^{(p=2)}=\frac{51975\pi^{2}\left(q_{s}^{(2)}\right)^{2}}{65536\epsilon}.$
(35)

##  IV Electrostatic potential

The procedure to compute this potential is similar to the one we used in the
scalar case. In the spirit of avoiding repetition, we describe briefly the
computation giving special emphasis to the aspects that are different with
respect to the scalar case. A previous discussion of the photon Green’s
function analysis in the RSII-$p$ scenario can be found in Dubovsky et al.
(2000b). We begin by considering the (5+$p$)D action

$S=\int\,d\,{}^{4}x\,\prod_{i=1}^{p}\,R_{i}d\theta_{i}\,dy\,\sqrt{|g|}\,\left(\frac{1}{4}g^{MN}g^{PQ}F_{MP}F_{NQ}+A_{M}J^{N}_{gauge}\right),$
(36)

leading to the equation of motion

$\displaystyle\frac{1}{\sqrt{|g|}}\partial_{M}\bigg{(}\sqrt{|g|}\,\,g^{MP}g^{NQ}F_{PQ}\bigg{)}=-j^{N}_{gauge}.$
(37)

We consider a static source along the brane, uniformly distributed along the
$p$ extra compact dimensions, namely

$\displaystyle\sqrt{|g|}j_{gauge}^{N}=\rho^{(p)}\delta^{N}_{0}\delta^{3}(\vec{x}-\vec{x}_{0})\delta(y-y_{0}),$
(38)

where $\rho^{(p)}$ is the charge density. Now we write down the equation of
motion for the gauge field in the background (2). In order to do this, it is
convenient to fix the gauge $A^{y}=0$ and $A^{\theta_{i}}=0$, which is
consistent with the value $J^{\theta_{i}}=0$ for the components of the current
density in the directions of the compact extra dimensions. Thus Eq. (37)
becomes

${\cal
O}\hat{A}^{\sigma}-e^{-p|y|/\epsilon}\partial^{\sigma}\partial_{\mu}\hat{A}^{\mu}=-R^{-p}e^{p|y|/\epsilon}\sqrt{g}\,\,j^{\sigma},$
(39)

where we assume equal size compact dimensions, $R_{i}=R,i=1,\dots,p$, and the
differential operator $\cal O$ is defined as

${\cal
O}:=e^{-(p+2)|y|/\epsilon}\left(-\partial^{2}_{y}+\frac{p+2}{\epsilon}\,sgn(y)\partial_{y}+e^{2|y|/\epsilon}\Box\right),$
(40)

and $\hat{A}^{\nu}=\eta^{\nu\mu}A_{\mu}$. Inspection of equation (39) reveals
the term $\partial_{\mu}\hat{A}^{\mu}$ is pure gauge on the brane, so we drop
it from now on Dubovsky et al. (2000b).

To solve (39) let us notice that the differential operator (40) is invariant
under the change $y\to-y$, so the solutions will inherit such symmetry. This
is important since we are looking for the potential on the brane. We shall
adopt again the Green’s function method. As in the scalar case, the necessary
tools are the eigenfunctions and eigenvalues of the differential equation.

The eigenfunctions and eigenvalues for the $p$ compact modes are the same as
those for the scalar field, Eq. (12). As for the noncompact modes depending
upon $y$ and subject to the boundary conditions (13) they fulfill again a
Bessel equation and have the following form

$\phi_{0}=\sqrt{\frac{p}{2\epsilon}},\quad\quad\phi_{m}(y)=e^{\nu
y/\epsilon}\sqrt{\frac{m\epsilon}{2}}\left[a_{m}J_{\nu}\left({m\epsilon}\,e^{y/\epsilon}\right)+b_{m}N_{\nu}\left({m\epsilon}\,e^{y/\epsilon}\right)\right],$
(41)

where

$\nu\equiv\frac{p+2}{2},$ (42)

and the constants $a_{m}$, $b_{m}$ are defined as in (17), with $\gamma$
replaced by $\nu$. The modes are normalized in the form
$\int_{-\infty}^{\infty}dy\,e^{-p|y|/\epsilon}\phi_{0}^{2}=1$ and
$\int_{-\infty}^{\infty}dye^{-p|y|/\epsilon}\phi_{m}(m\epsilon)\phi_{m^{\prime}}(m^{\prime}\epsilon)=\delta(m-m^{\prime})\,$.
Formally the Green’s function, its low energy approximation and the static
potential on the brane are obtained from (18), (20) and (24), replacing the
scalar modes $\psi$ by the gauge modes $\phi$ as well as the factor $\gamma$
by $\nu$. The electrostatic Green’s function on the brane takes the form

$G_{gauge}(\vec{x}-\vec{x}^{\prime},y=0,y^{\prime})=\frac{p}{2\epsilon}\frac{1}{4\pi
r}-\frac{1}{4\pi r}\frac{e^{\nu
y^{\prime}/\epsilon}}{\Gamma(\nu-1)}\left(\frac{\epsilon}{2}\right)^{\nu-1}\int_{0}^{\infty}dm\,m^{\nu-1}J_{\nu}\left(my^{\prime}\right)e^{-mr}\,.$
(43)

Since we are interested in the potential for a source located on the brane, we
have to evaluate the above expression in the limit $y^{\prime}\rightarrow 0$.
As in the scalar case this limit is different depending on whether $p$ is
either even or odd. They are given explicitly below.

####  IV.0.1 $p$ odd

In this case the potential gets the form

$A^{0}(r)=\frac{\sigma^{(5+p)}}{4\pi
R^{p}r}\sqrt{\frac{2}{\pi}}\frac{(-1)^{\nu}\epsilon^{2\nu-1}}{\Gamma(\nu-1)(2)^{\nu-1}}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\nu-\frac{1}{2}}\left[\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\epsilon}\right]\,,$
(44)

where $r=|\vec{x}-\vec{x}_{0}|$. As an example notice that for one extra
compact dimension $p=1$ one gets

$A^{0}(r)=\frac{2e}{\epsilon\pi}\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}\right),\quad
e=\frac{e^{(6)}}{2R\epsilon^{2}}\,,$ (45)

which reduces to the Coulomb potential for $r\gg\epsilon$ and is finite at the
3D source position

$\lim_{r\rightarrow 0}A^{0}(r)=\frac{4e}{\pi\epsilon}.$ (46)

The self-energy in this case is

$E_{self}^{(p=1)}:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\left(\nabla
A^{0}\right)^{2}=\frac{5e^{2}}{32\pi^{3}\epsilon}.$ (47)

####  IV.0.2 $p$ even

Now $\nu$ is an integer and

$A^{0}(r)=\frac{(-1)^{\nu}\sigma^{(5+p)}\epsilon^{2\nu-1}}{2^{\nu-1}\Gamma(\nu-1)R^{p}}\frac{1}{4\pi
r}\left(\frac{d}{\epsilon
d\epsilon}\right)^{\nu-1}\left(\frac{r}{\epsilon^{2}\sqrt{r^{2}+\epsilon^{2}}}\right)\,.$
(48)

Notice that for $p=2$,

$A^{0}(r)=\frac{e}{\epsilon}\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}\right),\quad
e=\frac{e^{(7)}}{R^{2}\epsilon}\,,$ (49)

which becomes the Coulomb potential for $r\gg\epsilon$ and its finite at the
3D source position:

$\lim_{r\rightarrow 0}A^{0}(r)=\frac{3e}{2\epsilon}.$ (50)

The source self-energy is now

$E_{self}^{(p=2)}=\frac{315e^{2}}{16384\pi\epsilon}.$ (51)

The static potentials for $p=1$, $p=2$ and Coulomb’s are compared in Fig. (2).
Remarkably as we have mentioned, the electrostatic potentials corrected by the
extra dimensions are finite at the 3D position of the charge.

It is interesting and natural to explore possible consequences of the modified
electrostatic potentials we just obtained using known experiments like the
Cavendish and scattering ones. We do so in the following section.

Figure 2: Electrostatic potential of the point particle for the standard 4D
Coulomb case and $p=1,2$.

##  V Phenomenology of the electrostatic potential

###  V.1 Cavendish experiment

From the different results obtained to verify the accuracy of the
electrostatic force, we have chosen the ones obtained by Plimpton and Lawton
Plimpton and Lawton (1936) and more recent modifications (see Tu et al.
(2005); Tu and Luo (2004) for a review of the different experiments). The
reason is that this belongs to a series of experiments in which the main idea
was to test the accuracy of Coulomb’s force between charged particles using
similar techniques as the one used by Cavendish to test the gravitational
force (see for instance Goldhaber and Nieto (2010) for a recent review on the
different perspectives and experiments performed to test different aspects of
electrodynamics). In the modern version of Cavendish experiment we have a
modified electromagnetic potential for a charge $Q$

$V=V_{C}+\delta V,$

where $\delta V$ is the modification to the Coulomb potential. The idea behind
the concentric charged spheres experiments is that only for the Coulomb
potential the interior of a charged sphere is field free and therefore the
potential there is a constant. Then the potential difference between a charged
outer sphere and the uncharged inner sphere is zero only if the potential is
that of Coulomb. Any deviation from this would imply a nonvanishing potential
difference between the spheres that can be measured.

The potential of a sphere with a charge $Q$ and radius $c$ at a distance $r$
from the center is

$U(Q,r,c)=\frac{Q}{2cr}\left[f(r+c)-f(|r-c|)\right],$

where the function $f$ is given by

$f(r)=\int_{0}^{r}{ds\;sV(s,Q=1)},$

It is easy to verify that for $V=\alpha/r$, $U(Q,r<c,c)=const.$, that is the
potential is constant in the interior.

In the simplest version of the Cavendish experiment one has an outer sphere of
radius $b$, charged to a certain voltage, and then measures the relative
voltage difference to the uncharged inner sphere of radius $a<b$,

$\gamma_{ab}=\left|\frac{{\mathcal{V}}_{b}-{\mathcal{V}}_{a}}{{\mathcal{V}}_{b}}\right|\;\;=\;\;\left|\frac{U(Q,b,b)-U(Q,a,b)}{U(Q,b,b)}\right|.$
(52)

Plimpton and Lawton found that $|\gamma_{ab}|\leq 3\times 10^{-10}$ with
$a=0.696$ m, $b=0.762$m. Here we will calculate $|\gamma_{ab}|$ for the two
potentials corresponding to $p=1$ and $p=2$, namely

$\displaystyle v_{1}$ $\displaystyle=$
$\displaystyle\frac{2Q}{\epsilon\pi}\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan(\frac{r}{\epsilon})}{\frac{r}{\epsilon}}\right),$
$\displaystyle v_{2}$ $\displaystyle=$
$\displaystyle\frac{Q}{\epsilon}\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2(1+\frac{r^{2}}{\epsilon^{2}})^{\frac{3}{2}}}\right).$
(53)

The results, to first order in $\epsilon$ are

$|\gamma_{1ab}|=\frac{\epsilon}{\pi
b},\;\;|\gamma_{2ab}|=\frac{\epsilon}{4b}.$ (54)

Taking into account the experimental bound of Plimpton and Lawton this means
that $\epsilon\leq 7.18\times 10^{-10}m$ or $\epsilon\leq 9.14\times
10^{-10}m$ for $p=1$ and $p=2$, respectively.

The more recent version of the Cavendish experiment employs four concentric
spheres of radii a, b, c, d in increasing order. The Outer sphere has a charge
$Q$ and the next one -$Q$. Then the potential at radius r is given by

$U(Q,r,c,d)=\frac{Q}{2dr}\left[f(r+d)-f(|r-d|)\right]-\frac{Q}{2cr}\left[f(r+c)-f(|r-c|)\right],$
(55)

The experiment sets a bound for the ratio of the potential differences between
the two uncharged spheres and the two outer spheres

$\gamma_{abcd}=\left|\frac{{\mathcal{V}}_{b}-{\mathcal{V}}_{a}}{{\mathcal{V}}_{c}-{\mathcal{V}}_{d}}\right|=\left|\frac{U(Q,b,c,d)-U(Q,a,c,d)}{U(Q,c,c,d)-Q(Q,d,c,d)}\right|.$
(56)

Williams et al. Williams et al. (1971) found that $|\gamma_{abcd}|\leq 2\times
10^{-16}$ with $a=0.60$ m, $b=0.94$m, $c=0.947$m and $d=1.27$m. Here we will
calculate $|\gamma_{abcd}|$ for the two potentials corresponding to $p=1$ and
$p=2$, and using the experimental limits to constrain $\epsilon$. A
straightforward calculation gives, to leading order in $\epsilon$

$\gamma_{1abcd}=\frac{cd\left(\frac{4(c-d)(c+d)\left(-2a^{2}+c^{2}+d^{2}\right)}{\left(a^{2}-c^{2}\right)^{2}\left(a^{2}-d^{2}\right)^{2}}+\frac{4\left(2b^{2}(c-d)(c+d)-c^{4}+d^{4}\right)}{\left(b^{2}-c^{2}\right)^{2}\left(b^{2}-d^{2}\right)^{2}}\right)}{3\pi(c-d)}\epsilon^{3}+O\left(\epsilon^{4}\right),$
(57)
$\gamma_{2abcd}=\frac{cd\left(\frac{\frac{\frac{1}{(a+c)^{3}}+\frac{1}{(a-c)^{3}}}{c}+\frac{\frac{1}{(d-a)^{3}}-\frac{1}{(a+d)^{3}}}{d}}{a}+\frac{\frac{\frac{1}{(c-b)^{3}}-\frac{1}{(b+c)^{3}}}{c}+\frac{\frac{1}{(b+d)^{3}}+\frac{1}{(b-d)^{3}}}{d}}{b}\right)}{16(c-d)}\epsilon^{4}+O\left(\epsilon^{5}\right).$
(58)

Taking into account the experimental value obtained by Williams et al.
($|\gamma_{abcd}|\leq 2\times 10^{-16}$ ) the corresponding bounds for
$\epsilon$ are $\epsilon\leq 2.80\times 10^{-7}m$ or $\epsilon\leq 4.02\times
10^{-6}m$ for $p=1$ and $p=2$, respectively. In this case the two sphere
experiment gives a tighter constraint on $\epsilon$. The reason for this may
be the peculiarities of the modification of the Coulomb potentials that in our
case contains positive powers of $r$.

###  V.2 Scattering by Helium atoms

We shall study the collision of a particle of charge $ze$ and mass $m$ with an
atom of atomic number $Z$. Notice that an exact formulation of this problem
requires the use of a many-body Hamiltonian which describes all the particles
of the system, however we shall make the assumption that the complicated
interaction of the incident particle with the constituents of the atom can be
accounted for by an effective electrostatic potential $V(r)$ in which the
incident particle travels.

It is physically reasonable that the electrostatic potential in which the
incident particle travels is well approximated by

$V(\vec{r})=ze\left[Zev_{1,2}(\vec{r})+e\int\rho(\vec{r}^{\prime})v_{1,2}\left(|\vec{r}-\vec{r}^{\prime}|\right)d^{3}\vec{r}^{\prime}\right],$
(59)

were $\vec{r}$ is the position vector of the incident particle and
$v_{1,2}(\vec{r})$ are given by (45) and (49). The first term is due to the
field of the nucleus and the second term is the potential of the atomic
electrons, described in terms of an effective electron density $\rho$. It is
worth mentioning that in this description we are neglecting all effects of
symmetry and spin. For neutral atoms, the density satisfies

$\int\rho(\vec{r})d^{3}\vec{r}=Z.$ (60)

When the incident particle carries sufficiently high energy, the scattering
amplitudes can be easily evaluated by the Born approximation

$f(\theta)=-\frac{m}{2\pi\hbar^{2}}\int
e^{i\vec{q}\cdot\vec{r}}V(\vec{r})d^{3}\vec{r},$ (61)

where $\vec{q}=\vec{k}_{0}-\vec{k}$, and $\vec{k}_{0}$ and $\vec{k}$ are the
initial and final momentum, respectively. Since the scattering is elastic,
$|\vec{k}|=|\vec{k}_{0}|=k$. Thus introducing Eq. (59) in (61) and making the
following change of variable $\vec{R}=\vec{r}-\vec{r}^{\prime}$ we have

$\displaystyle f_{1,2}(\theta)$ $\displaystyle=$
$\displaystyle-\frac{me^{2}}{2\pi\hbar^{2}}\left[zZ\int
e^{i\vec{q}\cdot\vec{r}}v_{1,2}(\vec{r})d^{3}\vec{r}-z\int
e^{i\vec{q}\cdot\vec{R}}v_{1,2}\left(\vec{R}\right)d^{3}\vec{R}\int\rho(\vec{r}^{\prime})e^{i\vec{q}\cdot\vec{r}^{\prime}}d^{3}\vec{r}^{\prime}\right],$
(62) $\displaystyle=$
$\displaystyle-\frac{me^{2}z}{2\pi\hbar^{2}}\left[Z-F(\vec{q})\right]\int
e^{i\vec{q}\cdot\vec{r}}v_{1,2}(\vec{r})d^{3}\vec{r},$

$F(\vec{q})$ is called the form factor of the atom. We defined $F(\vec{q})$ as

$F(\vec{q})=\int\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^{3}\vec{r}.$ (63)

When the potential is spherically symmetric, the angular integration can be
performed to give

$f_{1,2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\frac{\sin(qr)}{qr}v_{1,2}(r)r^{2}dr,$
(64)

with $q=|\vec{q}|=2k\sin(\theta/2)$ and $r=|\vec{r}|$. The evaluation of this
integral depends on the form that $v_{1,2}(r)$ takes. We first calculate
$f_{1}(\theta)$

$f_{1}(\theta)=-\frac{4me^{2}z}{\pi\hbar^{2}}\frac{1}{\epsilon
q}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\sin(qr)\left(\frac{1}{1+\frac{r^{2}}{\epsilon^{2}}}+\frac{\arctan\left(\frac{r}{\epsilon}\right)}{\frac{r}{\epsilon}}\right)rdr,$
(65)

using the following relations (see Gradshteyn and Ryzhik (1980))

$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{1+\frac{r^{2}}{\epsilon^{2}}}rdr$
$\displaystyle=$ $\displaystyle\frac{\pi}{2}\epsilon^{2}e^{-q\epsilon},$ (66)
$\displaystyle\int_{0}^{\infty}\sin(qr)\frac{\arctan\left(\frac{r}{\epsilon}\right)}{r}rdr$
$\displaystyle=$ $\displaystyle\frac{\pi}{2}\frac{e^{-q\epsilon}}{q},$ (67)

$f_{1}(\theta)$ can be written as

$f_{1}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\left[\frac{1}{q^{2}}+\frac{\epsilon}{q}\right]e^{-q\epsilon}.$
(68)

Considering the form of $v_{2}(r)$, $f_{2}(\theta)$ can be expressed as

$f_{2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\frac{1}{\epsilon
q}\left[Z-F(\vec{q})\right]\int_{0}^{\infty}\sin(qr)\left(\frac{1}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}+\frac{1}{2\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}\right)rdr.$
(69)

Now let us consider the integrals

$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{\sqrt{1+\frac{r^{2}}{\epsilon^{2}}}}rdr$
$\displaystyle=$ $\displaystyle\epsilon^{2}K_{1}(q\epsilon),$ (70)
$\displaystyle\int_{0}^{\infty}\frac{\sin(qr)}{\left(1+\frac{r^{2}}{\epsilon^{2}}\right)^{\frac{3}{2}}}rdr$
$\displaystyle=$ $\displaystyle\epsilon^{3}qK_{0}\left(q\epsilon\right),$ (71)

where $K_{0}(x)$ and $K_{1}(x)$ are Bessel functions of zeroth and first
order, respectively. Thus $f_{2}(\theta)$ takes the form

$f_{2}(\theta)=-\frac{2me^{2}z}{\hbar^{2}}\left[Z-F(\vec{q})\right]\left[\frac{\epsilon}{q}K_{1}(q\epsilon)+\frac{\epsilon^{2}}{2}K_{0}(q\epsilon)\right].$
(72)

For Helium we can calculate the electron density as

$\rho(r)=Z\left(\frac{b^{3}}{\pi a_{0}^{3}}\right)e^{\frac{-2br}{a_{0}}},$
(73)

with $b$ being the effective charge and having the value 1.69 for Helium while
$a_{0}$ is the Bohr radius. The form factor becomes

$F(q)=\frac{Z}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}.$ (74)

The differential scattering cross section for elastic processes thus become

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)^{(p=1)}$ $\displaystyle=$
$\displaystyle\left(\frac{2zZ}{a_{0}q^{2}}\right)^{2}\left[1-\frac{1}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}\right]^{2}\left[1+q\epsilon\right]^{2}e^{-2q\epsilon},$
(75) $\displaystyle\left(\frac{d\sigma}{d\Omega}\right)^{(p=2)}$
$\displaystyle=$
$\displaystyle\left(\frac{2zZ}{a_{0}q^{2}}\right)^{2}\left[1-\frac{1}{\left(1+\frac{a_{0}^{2}q^{2}}{4b^{2}}\right)^{2}}\right]^{2}\left[q\epsilon
K_{1}(q\epsilon)+\frac{q^{2}\epsilon^{2}}{2}K_{0}(q\epsilon)\right]^{2}.$ (76)

For incident electrons, we set $z=-1$ and $Z=2$ for the Helium atom. To
complete the analysis we compare the theoretical results with the
corresponding experimental ones. This comparison is made explicit in Figures
(3) and (4). For both $p=1,2$ a best agreement is attained when $\epsilon\sim
10^{-10}m$.

Figure 3:  Comparison of experimental differential cross section Jansen et al.
(1976) with that corresponding to one compact dimension, Eq. (75). Figure 4:
Comparison of experimental differential cross section Jansen et al. (1976)
with that corresponding to two compact dimensions, Eq. (76).

##  VI Discussion

The ever increasing accuracy with which electrodynamics has been tested
naturally lends itself to consider it as a probe to set bounds for possible
deviations coming from the existence of extra dimensions. Amongst different
models the so called Randall-Sundrum ones including a single 3-brane and $p$
extra compact dimensions (RSII-$p$) have provided simple scenarios that yield
effects well under control. Take for example the Casimir force Linares et al.
(2010); Frank et al. (2008): In a nutshell the field modes corresponding to
the extra dimensions add up to modify the usual Casimir force expression and
the deviations are assumed to be bounded by the uncertainties in the
experimental data. This in turn sets bounds for the parameters of the brane
model.

In this work we have explored the static potential produced by a scalar and a
charged sources, respectively, in RSII-$p$. These sources are effectively
pointlike from the perspective of an observer sitting in the usual 3D space.
However they stretch uniformly along the $p$ compact dimensions thus having
the structure of a $T^{p}$ torus. Remarkably the effective potentials turn out
to be non-singular at the position in 3D space. At first one may think this is
related to the fact the sources stretch along the extra dimensions, similarly
as in models of charged spherical shells Rohrlich (2007). This is not the case
as a more careful look reveals: the potential produced by either a charged
ring or a torus is not finite at the source itself Andrews (2006); Kondratev
et al. (2008); Bannikova et al. (2010). The RSII-p scenario thus allows to
regularize the 3D potentials and selfenergies. Indeed the combined limit
having AdS radius and compact size going to zero yields the usual standard
divergent result.

We have determined the potentials in the low energy regime in terms of light
modes; this entails approximating the continuous modes given in terms of
Bessel functions by their small argument form whereas for the compact modes we
keep the zero mode only. Within this approximation a delicate balance occurs
between part of the massive sector contribution to the potential and the zero
mode. Since the zero mode is responsible for the usual singular $1/r$ term,
the potential characteristic of massless fields, the balance just described
regularizes such a divergence. Moreover the remaining effective potential
becomes the usual $1/r$ within a few times $\epsilon$ away from $r=0$ and
provides finite selfenergies as determined from the usual 3D formulae.

To probe the effective potentials we proposed to consider two types of
experiments. First we adopted the long known Cavendish experiment with two and
four conducting spheres that is used to test the form of the Coulomb force. To
be consistent with know experimental results for the case of two spheres a
value of $\epsilon\sim 10^{-10}m$ is required. The four spheres setting
however turns out to produce a milder bound $\epsilon\sim 10^{-7}m$, probably
due to the positive powers of the correcting terms of the effective potentials
when developing around $1/r$. The second possibility we studied to test our
effective potentials was to consider electrons scattered off by Helium. A
comparison of the differential cross section modified by the RSII-p scenario
with the curve fitting experimental data indicates consistency with a value of
$\epsilon\sim 10^{-11}m$. In a previous work Morales-Técotl et al. (2007) we
used the Lamb shift to set a bound of $\epsilon\sim 10^{-14}m$ for $p=1$, and
$\epsilon\sim 10^{-13}m$, for $p=2$, which clearly are stronger than the ones
obtained in the present work.

The fact that for both the scalar and electromagnetic case the potentials
become well behaved leads naturally to the question of whether the same
results holds for the gravitational case. This is work under study and will be
reported elsewhere. Indeed, historically, finiteness of the potentials have
led in the past to the idea that gravity regulates the self-energy of the
charged point particle Arnowitt et al. (1962) as well as nonlinear field
equations to achieve the finiteness of the electric field Born and Infeld
(1934).

In the low energy regime we have focused on in this work there are some other
possible directions which can be pursued. These include a reanalysis of the
radiation reaction problem in both electromagnetic an gravitational cases as
well as the complete understanding of the regularization of the potentials and
in particular its relation to the topology of the sources together with their
dimensionality.

###### Acknowledgements.

HAMT acknowledges partial support from grant CONACyT-NSF Strong backreaction
effects in quantum cosmology.

## References

  * Goldhaber and Nieto (2010) A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 82, 939 (2010), eprint 0809.1003.
  * Goldhaber and Nieto (1971) A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 43, 277 (1971).
  * Lorentz (1909) H. A. Lorentz, Teubner, Leipzig (1909), (orig. Leyden (1895)).
  * Abraham (1903) M. Abraham, Ann. Phys. 10, 105 (1903).
  * Dyson (1952) F. J. Dyson, Phys. Rev. 85, 631 (1952).
  * Morales-Técotl et al. (2007) H. A. Morales-Técotl, O. Pedraza, and L. O. Pimentel, Gen. Rel. Grav. 39, 1185 (2007), eprint physics/0611241.
  * Dubovsky et al. (2000a) S. L. Dubovsky, V. A. Rubakov, and P. G. Tinyakov, Phys. Rev. D62, 105011 (2000a), eprint hep-th/0006046.
  * Dubovsky et al. (2000b) S. L. Dubovsky, V. A. Rubakov, and P. G. Tinyakov, JHEP 08, 041 (2000b), eprint hep-ph/0007179.
  * Oda (2000) I. Oda, Phys. Lett. B496, 113 (2000), eprint hep-th/0006203.
  * Randall and Sundrum (1999a) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999a), eprint hep-th/9906064.
  * Randall and Sundrum (1999b) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999b), eprint hep-ph/9905221.
  * Antoniadis et al. (1998) I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B436, 257 (1998), eprint hep-ph/9804398.
  * Arkani-Hamed et al. (1998) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B429, 263 (1998), eprint hep-ph/9803315.
  * Antoniadis (1990) I. Antoniadis, Phys. Lett. B246, 377 (1990).
  * Allanach et al. (2004) B. C. Allanach et al. (Beyond the Standard Model Working Group) (2004), eprint hep-ph/0402295.
  * Csaki (2004) C. Csaki (2004), eprint hep-ph/0404096.
  * Maartens (2004) R. Maartens, Living Rev. Rel. 7, 7 (2004), eprint gr-qc/0312059.
  * Elizalde (2006) E. Elizalde, J. Phys. A39, 6299 (2006), eprint hep-th/0607185.
  * Maartens and Koyama (2010) R. Maartens and K. Koyama (2010), eprint 1004.3962.
  * Linares et al. (2010) R. Linares, H. A. Morales-Técotl, and O. Pedraza, Phys. Rev. D81, 126013 (2010), eprint 1003.4286.
  * Frank et al. (2008) M. Frank, N. Saad, and I. Turan, Phys. Rev. D78, 055014 (2008), eprint 0807.0443.
  * Kaluza (1921) T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921, 966 (1921).
  * Klein (1926a) O. Klein, Z. Phys. 37, 895 (1926a).
  * Duff et al. (1986) M. J. Duff, B. E. W. Nilsson, and C. N. Pope, Phys. Rept. 130, 1 (1986).
  * Green et al. (1987) M. B. Green, J. Schwarz, and E. Witten (1987).
  * Klein (1926b) O. Klein, Nature 118, 516 (1926b).
  * Rubakov (2001) V. A. Rubakov, Phys. Usp. 44, 871 (2001), eprint hep-ph/0104152.
  * Scherk and Schwarz (1979a) J. Scherk and J. H. Schwarz, Phys.Lett. B82, 60 (1979a).
  * Scherk and Schwarz (1979b) J. Scherk and J. H. Schwarz, Nucl.Phys. B153, 61 (1979b).
  * Grana (2006) M. Grana, Phys.Rept. 423, 91 (2006), eprint hep-th/0509003.
  * Blumenhagen et al. (2007) R. Blumenhagen, B. Kors, D. Lust, and S. Stieberger, Phys.Rept. 445, 1 (2007), eprint hep-th/0610327.
  * Douglas and Kachru (2007) M. R. Douglas and S. Kachru, Rev.Mod.Phys. 79, 733 (2007), eprint hep-th/0610102.
  * Candelas and Weinberg (1984) P. Candelas and S. Weinberg, Nucl.Phys. B237, 397 (1984).
  * Garriga and Tanaka (2000) J. Garriga and T. Tanaka, Phys.Rev.Lett. 84, 2778 (2000), eprint hep-th/9911055.
  * Sasaki et al. (2000) M. Sasaki, T. Shiromizu, and K.-i. Maeda, Phys.Rev. D62, 024008 (2000), eprint hep-th/9912233.
  * Goldberger and Wise (1999) W. D. Goldberger and M. B. Wise, Phys.Rev.Lett. 83, 4922 (1999), eprint hep-ph/9907447.
  * Lesgourgues and Sorbo (2004) J. Lesgourgues and L. Sorbo, Phys.Rev. D69, 084010 (2004), eprint hep-th/0310007.
  * Maity et al. (2006) D. Maity, S. SenGupta, and S. Sur, Phys.Lett. B643, 348 (2006), eprint hep-th/0604195.
  * Das et al. (2008) S. Das, A. Dey, and S. SenGupta, Europhys.Lett. 83, 51002 (2008), eprint 0704.3119.
  * Flachi et al. (2003) A. Flachi, J. Garriga, O. Pujolas, and T. Tanaka, JHEP 0308, 053 (2003), eprint hep-th/0302017.
  * Linares et al. (2008a) R. Linares, H. A. Morales-Técotl, and O. Pedraza, Phys. Rev. D77, 066012 (2008a), eprint arXiv:0712.3963 [hep-ph].
  * Linares et al. (2008b) R. Linares, H. A. Morales-Técotl, and O. Pedraza, Phys. Rev. D78, 066013 (2008b), eprint 0804.2042.
  * Plimpton and Lawton (1936) S. J. Plimpton and W. E. Lawton, Phys. Rev 50, 1066 (1936).
  * Tu et al. (2005) L.-C. Tu, J. Luo, and G. T. Gillies, Rept. Prog. Phys. 68, 77 (2005).
  * Tu and Luo (2004) L.-C. Tu and J. Luo, Metrologia 41, S136 (2004).
  * Williams et al. (1971) E. R. Williams, J. E. Faller, and H. A. Hill, Phys. Rev. Lett. 26, 721 (1971).
  * Gradshteyn and Ryzhik (1980) I. Gradshteyn and I. Ryzhik, _Table of Integrals, Series and Products_ (Academic Press Inc., 1980), ISBN 0122947606\.
  * Jansen et al. (1976) R. H. J. Jansen, F. J. de Heer, H. J. Luyken, B. van Wingerden, and H. J. Blaauw, Journal of Physics B: Atomic and Molecular Physics 9, 185 (1976).
  * Rohrlich (2007) F. Rohrlich, _Classical Charged Particles_ (World Scientific, 2007), ISBN 9812700048\.
  * Andrews (2006) M. Andrews, Journal of Electrostatics 64, 664 (2006).
  * Kondratev et al. (2008) B. P. Kondratev, A. S. Dubrovskii, N. G. Trubitsyna, and E. S. Mukhametshina, Technical Physics 53, 948 (2008).
  * Bannikova et al. (2010) E. Y. Bannikova, V. G. Vakulik, and V. M. Shulga (2010), eprint 1009.4324.
  * Arnowitt et al. (1962) R. L. Arnowitt, S. Deser, and C. W. Misner (1962), eprint gr-qc/0405109.
  * Born and Infeld (1934) M. Born and L. Infeld, Proc. Roy. Soc. Lond. A144, 425 (1934).


# A new simple family of non-periodic tilings with square tiles

Nikolay Vereshchagin
Moscow State University, HSE University, Yandex This paper was prepared within
the framework of the HSE University Basic Research Program.

###### Abstract

We define a new family of non-periodic tilings with square tiles that is
mutually locally derivable with some family of tilings with isosceles right
triangles. Both families are defined by simple local rules, and the proof of
their non-periodicity is as simple as that of the non-periodicity of
Robinson’s tilings.

Keywords: non-periodic tilings, Domino problem, aperiodic tile sets,
substitution tilings

##  1 Introduction

The study of non-periodic tilings of the plane by square tiles is motivated by
the Domino problem: to construct an algorithm that, given local rules to
attach tiles, finds out whether it is possible to tile the entire plane
according to these rules. The non-existence of such an algorithm was proved by
Berger in [2]. The main step in the proof is the construction of local rules
such that any tiling according to these rules is non-periodic. The
construction of such a family was subsequently simplified by Robinson [5]. An
even simpler construction using the same idea is given in [3].

In this paper, we propose a new such family. The local rules for these tilings
and the proof of non-periodicity are approximately as easy as those in [5] and
[3]. The novel thing is that our tilings are based on tilings by isosceles
right triangles. In fact, we first construct a family of non-periodic tilings
with triangles for which the local rules ensure that the triangles can be
grouped into square tiles. Then we rewrite the local rules for triangle
tilings as local rules to attach resulting square tiles. Similarly, tilings by
isosceles triangles with angles divisible by $36^{\circ}$ yield Penrose
tilings P3 by rhombuses, which is called the Robinson decomposition of Penrose
tilings. To use conventional terminology, our square tilings are mutually
locally derivable to triangular tilings.

In the next section, we define our family of tilings with square tiles, and in
Section 3 we define tilings with triangles and provide proofs. The proof of
the non-periodicity of tilings by triangles (Theorem 3), and hence of the non-
periodicity of the family of tilings by square tiles (Theorem 1), uses a well-
known technique based on self-similarity. Namely, we consider the following
composition scheme: two triangles with a common leg are combined into one
triangle. Applying this scheme to any tiling satisfying local rules we obtain
a new tiling that again satisfies the same rules. It is well known that any
family of tilings with such a property can only consist of non-periodic
tilings. Moreover, we prove that all tilings in our family are substitution
tilings (Theorem 4). This means that any its finite fragment occurs in some
supertile. The latter are defined as tilings that can be obtained from single
tiles by several decompositions.

##  2 Tilings with square tiles

Our tiles are shown in Fig. 1.

Figure 1: The tiles.

The sides drawn in black have one of two orientations and one of two colors,
green or red. Thus, each of the four pictures depicts $4^{4}=256$ different
tiles. In addition, these tiles can be rotated by angles that are multiples of
$90^{\circ}$.111It may seem that the second tile is obtained from the first by
$180^{\circ}$ rotation, and the fourth from the third. However, this is not
the case — with such a rotation, the slanted arrow coming from the upper right
corner would change its color. Tiles cannot be flipped. Thus, the total number
of tiles is $4^{6}=4096$. A specific tile is shown in Fig. 2.

Figure 2: A tile.

The local rules:

  1. (1)

Tiles are attached only side-to-side, and the shared sides must have matching
colors and orientations.

  2. (2)

Adjacent triangles must have different colors.

  3. (3)

We will call the _crossing_ centered at a given vertex the set of arrows
entering or leaving this vertex. The rule restricts possible crossings: only
the crossings shown in Fig. 3 are legal, as well as crossings obtained from
them by rotations by angles that are multiples of $45^{\circ}$.

Figure 3: Legal crossings.

Legal crossings can be described as follows: a pair of arrows passes through
the center of the crossing without changing color or direction. This pair of
arrows is called the _axis of the crossing_. There are also two arrows of
different colors perpendicular to the axis, entering the center. In addition
to these four arrows, there are four more, coming out of the center at an
angle of $45^{\circ}$ to the axis. If you go to the center of the crossing
perpendicular to the axis, then the green arrow goes to the right, and the red
arrow to the left.

######  Definition 1.

A _tiling_ is a set of tiles whose interiors are pair wise disjoint. A
_correct tiling by square tiles_ is a tiling of the whole plane with the above
tiles that satisfies these rules.

######  Theorem 1.

There is a correct tiling with square tiles, and any such tiling is non-
periodic.

The proof of the second part of this theorem uses the _unique composition
property_. In any tiling of the plane, tiles can be grouped in a unique way
into the so-called macro-tiles. Each macro-tile corresponds to some tile of
the original set, and when macro-tiles are replaced by the corresponding
tiles, the local rules are preserved.

This method will not be applied to the original tilings, but to tilings with
isosceles right triangle obtained from the original tilings as follows. Let us
cut each tile into 4 right-angled triangles along the diagonal arrows. We
obtain a tiling with triangles. The original local rules become local rules
for triangular tiles. To these rules, we add yet another rule that guarantees
that triangles join into square tiles. In any tiling, according to the rules
obtained, the tiles are divided into pairs of tiles with a common leg. Each
such pair will correspond to one triangular tile, and when replacing it with
the corresponding tile, the local rules will be preserved. This operation is
called _the composition_. Since the local rules are preserved, the triangles
in the resulting tiling should again join into square tiles. However, new
square tiles can be composed of triangles from different square tiles in the
original tiling. Therefore, composition can be defined only for tilings with
triangles, but not for tilings with square tiles.

A similar technique is also used in the proof of the non-periodicity of
Penrose tilings P3 by rhombuses of two shapes. The rhombuses are also
partitioned into triangles, which is called the Robinson decomposition.
Robinson rectangles can be grouped into larger rectangles, which again can be
grouped into rhombuses.

So, we move on to tilings with triangular tiles.

##  3 Tilings with triangles

###  3.1 Tiles and local rules

The tiles are isosceles right triangles colored red or green. Each side of a
tile has an orientation and a color, either red or green. Thus each tile is
specified by seven bits. In addition, tiles can be rotated by 90-degree
angles, so the number of tiles is $2^{9}=512$ up to shift. Two examples of
tiles are shown in Fig. 4.

Figure 4: Examples of triangular tiles.

Each correct tiling with square tiles corresponds to a tiling with triangles,
which is obtained as follows. We cut each square tile diagonally into four
triangles. This will result in a tiling with triangular tiles of the type
described above. Obviously, it satisfies the following local rules:

  1. (1)

The tiles are attached side by side, and the shared sides of adjacent tiles
have matching colors and orientations.

  2. (2)

Any two triangles with a common side have different colors.

  3. (3)

Only two types of crossings are allowed, $C_{4}$ and $C_{8}$ (Fig. 5).

Figure 5: Allowed crossings. The two sides painted purple must have the same
color (red or green). Crossings of the form $C_{4}$ can be rotated by
multiples of $90^{\circ}$, and crossings of the form $C_{8}$ can be rotated by
multiples of $45^{\circ}$.

Crossings of the form $C_{8}$ are exactly legal crossings for tilings with
square tiles. Crossings of the form $C_{4}$ appear at the centers of square
tiles when they are cut into triangles.

  4. (4)

If we go to the center of the $C_{4}$ crossing perpendicular to the axis, then
there should be a red triangle on the left, and a green triangle on the right.
The axis of the $C_{4}$ crossing is defined similarly to the axis of $C_{8}$,
as the pair of uni-color unidirectional arrows passing through the center.

We will call tilings that satisfy these local rules _correct triangle
tilings_. For tilings of _parts_ of the plane, we require that the third and
fourth conditions hold only for internal vertices.

######  Lemma 1.

(a) If a correct tiling $T$ with square tiles is cut into triangles, then a
correct tiling $T^{\prime}$ with triangles is obtained. (b) Conversely, any
correct triangular tiling $T^{\prime}$of the plane can be obtained from some
correct tiling $T$ with square tiles by cutting it into triangles. This tiling
$T$ is unique.

###### Proof.

(a) Conditions (1) and (2) for $T^{\prime}$ follow from conditions 1 and 2 for
$T$.

Condition (3): every crossing in $T^{\prime}$ either coincides with the
corresponding crossing in $T$ and hence is of the form $C_{8}$, or is the
crossing in the center of a tile in $T^{\prime}$ and hence is of the form
$C_{4}$.

Condition (4) is verified by hand.

(b) Since in a correct tiling of the plane by triangles there can only be a
crossing $C_{4}$ at the vertex of any right angle, its tiles can be grouped
into quadruples, each of which forms a square tile from our set. Obviously,
this partition is unique.

Since the local rules for tilings by squares essentially coincide with the
local rules for tilings by triangles, the tiling $T$ is correct. ∎

Thus, to prove Theorem 1 we have to prove the existence and non-periodicity of
correct tilings by triangles.

###  3.2 Substitution

To prove existence and non-periodicity of correct tilings by triangles,
consider the following substitution: each triangular tile is cut by its height
into two tiles similar to the original one with the ratio $\sqrt{2}$. The
height is oriented from the vertex of the right angle to the hypotenuse and
has the same color as the original tile. The triangle left from the height,
when viewed in the direction of the arrow, is called _left_ , it is colored
red, and the right triangle is called _right_ , it is colored green (Fig. 6).
All sides (including the two halves of the hypotenuse) of the original
triangle retain their color and orientation.

Figure 6: Substitution. The sides drawn in black can have any color and
orientation, which are carried over to the sides of the resulting tiles.

For example, applying the substitution to the left tile in Fig. 4, we will get
a tiling in Fig. 7.

Figure 7: An example of applying the substitution

Using this substitution, we define the operations of decomposition and
composition of the given tiling. To decompose a tiling, we apply the
substitution to all its tiles. The resulting tiling tiles the same part of the
plane with smaller tiles. This tiling is then stretched by a factor of
$\sqrt{2}$, and the resulting tiling is called the _decomposition_ the
original tiling. It is easy to see that the decomposition operation is
injective. Indeed, we call two tiles _siblings_ if they are obtained by
substitution from the same tile. Then for each tile $F$ there exists a unique
sibling $G$, provided we ignore the color and orientation of edges. Indeed, if
$F$ is a red tile, then the only its sibling is the green tile obtained from
$F$ by $90^{\circ}$ rotation _counterclockwise_ , with respect to the vertex
of the right angle of $F$. If $F$ is a green tile, then the rotation is
_clockwise_.

Thus, for each tiling, there is at most one tiling of which it is the
decomposition. If such a tiling exists, then it is called _the composition_ of
the original tiling. The decomposition of the tiling $T$ will be denoted by
$\sigma T$, and the composition by $\sigma^{-1}T$.

###  3.3 Supertiles

_A level $n$_ supertile is the $n$-fold decomposition of a single tile. Fig. 8
shows a level 6 supertile obtained by $6$-fold decomposition from the left
tile in Fig. 7.

Figure 8: Supertile $S_{6}$. Long arrows represent sequences of equally
oriented and equally colored short arrows

Supertiles of level $n$ will be denoted by $S_{n}$.

######  Remark 1.

It can be proven by induction that only three (not 8) different side
orientations of inner red tiles and three different side orientations of inner
green tiles occur in supertiles (Fig. 9). Therefore, we can assume that the
total number of triangular tiles is $(3+3)\cdot 8\cdot 4=192$ (and not 512).

Figure 9: Orientations of the sides of tiles occurring in supertiles.

###  3.4 Existence of tilings satisfying local rules

First we prove the existence of correct tilings by triangles of the entire
plane. To do this, it suffices to prove that all supertiles satisfy the local
rules. Why is this enough? For example, we can argue as follows: we can verify
that the $S_{6}$ supertile in Fig. 8 contains strictly inside the tile in Fig.
7 on the left, from which this supertile was obtained by sixfold
decomposition. Therefore, if we apply the sixfold decomposition to the
supertile in Fig. 8, we will get a supertile $S_{12}$ including the tiling
$S_{6}$. Iterating this operation, we get a tower of supertiles

$S_{0}\subset S_{6}\subset S_{12}\subset S_{18}\subset\dots,$

whose union is a correct tiling of the entire plane. Thus, it suffices to
prove the following

######  Lemma 2.

Any supertile is a correct tiling.

###### Proof.

This is proved by induction. The induction base is trivial, since any tile
constitutes a correct tiling.

For the inductive step, we have to show that the decomposition of a correct
tiling $T$ is again correct. But there is a problem here. This is true for
tilings of the plane, but not for tilings of its parts. For example, a tiling
consisting of two triangles with a common leg in Fig. 10,

Figure 10: The decomposition a correct tiling of a part of the plane may not
be correct.

is correct (the third condition is satisfied because there are no internal
vertices), but its decomposition is not. To make the inductive step, it is
necessary to impose a restriction on the crossings at the boundary points. To
prove the lemma, it will suffice to require that the crossings with the center
at any vertex of the boundary, except for the extreme ones, are halves of the
legal crossings, that is, they have one of the two forms in Fig. 11.

Figure 11: Allowed crossings on the boundary of a supertile. The purple sides
must have the same color, red or green.

Now we can make the inductive step. Assume that $T$ is a supertile and is
correct. We have to show that so is its decomposition $\sigma T$. It is
obvious that the first condition (tiles border side by side and colors and
orientations of shared sides match) is inherited during decomposition.

Let us verify the second condition (adjacent triangles have different colors)
for $\sigma T$. Let $F$ be a tile in $\sigma T$ and let $G$ be its sibling.
W.l.o.g. assume that $F$ is a right, and hence green, tile (Fig. 12).

Figure 12: All neighbors of a green tile $F$ in the decomposition are red.

We need to prove that all neighbors of $F$ are red.

The tile $G$ is red by definition of the substitution. Besides $G$, the tile
$F$ can have other neighbors. Namely, if the hypotenuse of $I$ does not lie on
the edge of $T$, then the tiling $T$ contains the tile $E$ bordering $I$ along
the hypotenuse. As a result of cutting $E$, two tiles will appear, of which it
is the left, and therefore red, tile that borders $F$ along the leg.

Moreover, if the leg of the tile $I$ does not lie on the boundary of the
supertile, then the tiling $T$ contains the tile $H$, which is the reflection
of the tile $I$ with respect to this leg. Indeed, otherwise the crossing
centered at the vertex $A$ would be illegal (right and acute angles cannot
converge at a legal crossing). When the tile $H$ is decomposed, it is the
left, and therefore red, tile that is adjacent to $F$.

Let us check the third condition for the tiling $\sigma T$ (that all crossings
are legal). To do this, we need to check that during decomposition, a legal
crossing is converted into a legal one, and the same holds for half crossings
in Fig. 11. This is done by a direct verification. In addition, we need to
check that all the crossings at the new vertices are legal as well.

New vertices are the centers of hypotenuses of tiles from $T$. The hypotenuse
itself becomes the axis of the new crossing, and the new crossing is $C_{4}$
by the definition of the substitution.

Finally, we check the fourth condition. The crossings of the form $C_{4}$ are
only the crossings at the centers of the hypotenuses of the tiles in $T$, and
by the definition of substitution, all the left triangles in them are red and
the right ones are green.

The third and fourth conditions for crossings centered on the boundary are
verified in a similar way. ∎

###  3.5 Non-periodicity of tilings satisfying local rules

The non-periodicity of correct tilings follows from the fact that any tiling
of the plane has a structure in the following sense: for any $n$ it can be
partitioned uniquely into level $n$ supertiles. This is a simple consequence
of the following

######  Theorem 2.

Any correct tiling is composable and its composition is correct.

###### Proof.

Consider any tile of a given correct tiling $T$ and the vertex of its right
angle. The crossing at this vertex is $C_{4}$. At this crossing, each tile has
a sibling. Let us combine them into one, erasing the common leg, and leave all
the colors and orientations as they were. The hypotenuse of the resulting
triangle is formed by connecting two arrows on the axis of the crossing. Since
they are equally colored and oriented, the hypotenuse will be correctly
oriented and correctly colored. The resulting tiling is the composition of the
original one. Let us prove that it is correct.

We are given that $T$ is correct and need to prove that $\sigma^{-1}T$ is
correct. First we verify condition (1), that is, prove that $\sigma^{-1}T$ is
side-to-side and shared sides have matching colors and orientations. This is
obvious for the legs of the tiles from $\sigma^{-1}T$, since they are the
hypotenuses of the tiles from $T$, and by the condition (1) for $T$ the
hypotenuses are adjacent to the hypotenuses of the same color and orientation.

Now let us prove the same for the hypotenuses of tiles from $\sigma^{-1}T$.
Consider a tile $F$ from $\sigma^{-1}T$ (Fig. 13(a)). W.l.o.g. assume that it
is red. We need to show that the tile $G$ shown in Fig. 13(a) belongs to
$\sigma^{-1}T$. To this end, consider the midpoint of the hypotenuse, denoted
by the letter $A$. The crossing centered on $A$ in the tiling $T$ can only be
$C_{4}$, so there are two tiles in $T$ under the hypotenuse $A$, as shown in
Fig. 13(b). These two tiles, when composed, give the sought tile $G$.

Figure 13: Two types of neighborhoods of tiles in the composed tiling.

As a bi-product of this argument, we see that the colors of the tiles $F$ and
$G$ are different because the colors of the vertical arrows at the crossing
$C_{4}$ are different. Therefore, condition (2) is satisfied for triangles
$\sigma^{-1}T$ with a common hypotenuse. Let us verify condition (2) for tiles
that share a leg. Let the tiles $F$ and $G$ have a common leg (Fig. 13(c)).
Then their colors are equal to the colors of two arrows in the tiling $T$
(Fig. 13(d)), connecting the point $A$, the common vertex of right angles of
$F$ and $G$, with the centers of their hypotenuses. These two arrows form a
right angle and at their common origin (point $A$) there can only be the
crossing $C_{8}$ in the tiling $T$. By a routine check, we can see that any
two orthogonal arrows with a common origin at the center of the crossing
$C_{8}$ have different colors.

Let us verify condition (3). In the course of composition, the crossings
$C_{4}$ disappear. We need to show that crossings of the form $C_{8}$ are
transformed into themselves or into $C_{4}$. In all crossings $C_{8}$ the
sides of the tiles adjacent to the center of the crossing alternate —
hypotenuse, leg, hypotenuse, and so on. And the colors of the tiles also
alternate. Therefore, 4 options arise, depending on whether the alternation of
colors begins with red or green, and the alternation of sides — from the leg
or hypotenuse. These four options are shown in Fig. 14.

Figure 14: Four types of filling $C_{8}$ crossing.

Let us first prove that the first option is impossible. Indeed, the crossings
at the lower and upper blue points cannot be legal. Indeed, the axes of these
crossings must be directed vertically, since otherwise an arrow would go from
its center in the direction orthogonal to the axis. For the upper crossing,
the axis should be directed downwards, and for the lower one, upwards. But
then the colors of both triangles with a vertex at the blue points should be
opposite to the existing colors.

In the next two crossings, the siblings to all tiles are not adjacent to the
crossing. For this reason, when composed, the crossings do not change.
Finally, in the fourth case, the sibling for each tile is also adjacent to the
center. When decomposed, the eight tiles produce four tiles, and we get the
crossing $C_{4}$. In this case, condition (4) is satisfied for the resulting
crossing $C_{4}$. Indeed, in any $C_{8}$ crossing, the red arrow goes to the
left of the arrow entering its center, and the green arrow goes to the right.
Hence a red triangle is to the left of the arrow entering the center of the
obtained $C_{4}$ crossing, and a green one to the right. ∎

######  Theorem 3.

Any correct tiling of the plane by triangles is non-periodic. Consequently,
any correct tiling by square tiles is also non-periodic.

###### Proof.

Let $T$ be a correct tiling of the plane by triangles and $\sigma^{-1}T$ be
its composition. Assume that $a\neq 0$ is its period, that is $T=T+a$. Then
the vector $a/\sqrt{2}$ is the period of $\sigma^{-1}T$. Indeed,

$\sigma^{-1}T=\sigma^{-1}(T+a)=(\sigma^{-1}T)+a/\sqrt{2}.$

It is proved similarly that $a/2$ is the period of the double composition of
$T$. Repeating this argument many times, we get a tiling whose period is much
less than the size of the tiles, which is impossible. ∎

###  3.6 Substitution tilings

######  Definition 2.

A tiling is called a _substitution tiling_ (for the given substitution) if
every its finite fragment occurs in a supertile.

Since supertiles are correct tilings, any substitution tiling is correct. It
turns out that the opposite is also true.

######  Theorem 4.

Any correct tiling of the plane by triangles is a substitution tiling.

###### Proof.

Consider any fragment $P$ of a correct tiling $T$ of the plane. We have to
show that $P$ occurs in a supertile. To this end add in $P$ a finite number of
tiles from $T$ so that $P$ becomes an inner part of the resulting fragment $Q$
of $T$.

By Theorem 2 for any $k$ we can compose the given tiling $k$ times and the
resulting tiling $\sigma^{-k}T$ is correct. Call a _crown in $\sigma^{-k}T$
centered at a vertex $A$_ of a tile in $\sigma^{-k}T$ the set of all tiles
from $\sigma^{-k}T$ that include $A$. Since $\sigma^{-k}T$ is a correct
tiling, all its crowns have a form shown on Fig. 15 (see the proof of Th. 2).

Figure 15: Crowns in a correct tiling. Black and and purple sides can have any
colors and orientations. The purple sides must have matching colors and
orientations.

Consider the sets of the form $\sigma^{k}C$ where $C$ is a crown within the
tiling $\sigma^{-k}T$. As $k$ increases, these sets increase as well. If $k$
is large enough, then the set $Q$ is covered by a single such set, say by
$\sigma^{k}C$, that is, $Q\subset\sigma^{k}C$. As $\sigma^{-k}T$ is a correct
tiling, it can have only crowns shown on Fig. 15.

Let us show that all crowns from Fig. 15 appear in supertiles provided we
ignore colors and orientations of its outer sides (shown in black color on
Fig. 15). For crowns of the form (a) it can be verified by hand: all the four
such crowns occur within supertiles $S_{5}$ shown on Fig. 16.

Figure 16: On diagonals of supertiles $S_{5}$, there are all the four crowns
of the form (a).

Again it is easy to verify by hand that the substitution transforms the crowns
as follows: (a)$\to$(b)$\to$(c)$\to$(d)$\to$(c). Hence all the four crowns (b)
occur within supertiles $S_{6}$, the crowns (c) within supertiles $S_{7}$, and
the crowns (d) within supertile $S_{8}$.

It follows that the crown $C$, whose $k$-fold decomposition covers $Q$,
appears in a supertile, say in $S_{n}$. Therefore the tiling $\sigma^{k}C$
appears in the supertile $S_{n+k}$. Hence the patch $Q$ appears in that
supertile provided we ignore colors and orientations of its outer sides. Since
$P\subset Q$ and no side of the patch $P$ is an outer side of $Q$, we are
done. ∎

##  4 Acknowledgments.

The author is sincerely grateful to the participants of the Kolmogorov seminar
and of the International academic conference “Graphs, Games and Models”
(October 12-15, 2022, Maikop, Adygea) for helpful discussions.

## References

  * [1] 9
  * [2] Robert Berger, The undecidability of the domino problem, Memoirs of the American Mathematical Society, 66 (1966) 1–2.
  * [3] Bruno Durand, Leonid Levin, Alexander Shen. Local rules and global order, or aperiodic tilings. The Mathematical Intelligencer, 27:1 (2005) 64–68
  * [4] Roger Penrose, Pentaplexity, Eureka, 39 (1978) 16–22.
  * [5] Raphael Robinson, Undecidability and nonperiodicity of tilings in the plane, Invent. Math., 12 (1971) 177-209