# Kaluza-Klein Spectroscopy from Neutron Oscillations
into Hidden Dimensions

Gia Dvali, Manuel Ettengruber, Anja Stuhlfauth  0009-0005-0920-379X Arnold
Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333
München, Germany  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany

###### Abstract

Neutrons and neutrinos are natural probes for new physics. Since they carry no
conserved gauge quantum numbers, both can easily mix with the fermions from
hidden sectors. A particularly interesting effect is the oscillation of a
neutron or a neutrino into a fermion propagating in large extra dimensions. In
fact, such a mixing has been identified as the possible origin of small
neutrino mass. In this paper, we study neutron oscillations into an extra-
dimensional fermion and show that this effect provides a resonance imaging of
the Kaluza-Klein tower. The remarkable feature of this phenomenon is its
generic nature: because of a fine spacing of the Kaluza-Klein tower, neutrons
at a variety of energy levels, both free or within nuclei, find a bulk
oscillation partner. In particular, the partner can be a Kaluza-Klein mode of
the same species that gives mass to the neutrino. The existence of bulk states
matching the neutron energy levels of nuclear spectra gives rise to tight
constraints as well as to potentially observable effects. For a free neutron,
we predict recurrent resonant oscillations occurring with the values of the
magnetic field correlated with the KK levels. We derive bounds on extra
dimensions from ultra-cold neutron experiments and suggest signatures for
refined measurements, which, in particular, can probe the parameter space
motivated by the Hierarchy Problem. Ultra-cold neutron experiments offer a
unique way of Kaluza-Klein spectroscopy.

##  I Introduction

The framework of large extra dimensions Arkani-Hamed:1998jmv ; Arkani-
Hamed:1998sfv  (hereafter referred to as ADD model) is motivated by the
solution to the Hierarchy Problem, the inexplicable smallness of the weak
interaction scale relative to the Planck mass, $M_{P}$. In this theory, the
fundamental cutoff of gravity, $M_{f}$, defined as the Planck mass of
$4+N$-dimensional theory, is lowered relative to the four-dimensional Planck
scale $M_{P}$. Correspondingly, all UV-sensitivities, including the one of the
Higgs mass, are cut off at the scale $M_{f}\ll M_{P}$. The lowering of the
cutoff can be understood in different languages.

First, this phenomenon has a clear geometric meaning. At separation $r$
smaller than the compactification radius $R$, the gravitational flux of point-
like sources spreads according to high-dimensional Gauss’s law.
Correspondingly, it is diluted in the large volume of extra-dimensional space,
usually referred to as “bulk”. Because of this dilution, the strength of four-
dimensional gravity that operates at large distances, $r\gg R$, is effectively
weakened: four-dimensional Newton’s constant is suppressed relative to the
fundamental one by the $N$-dimensional volume of the extra space, $V_{N}$. The
resulting relation between the high-dimensional and four-dimensional Planck
scales is

$M_{f}=\frac{M_{P}}{\sqrt{M_{f}^{N}V_{N}}}.$ (1)

For definiteness, we shall assume compactification on an $N$-torus of radii
$R_{1},R_{2}\dots R_{N}$, with corresponding volume
$V_{N}=(2\pi)^{N}R_{1}\dots R_{N}$.

An alternative interpretation of (1) was offered in Dvali:2007hz ;
Dvali:2007wp ; Dvali:2009ne , where it was pointed out that (1) represents a
manifestation of the following general relation,

$M_{f}=\frac{M_{P}}{\sqrt{N_{sp}}},$ (2)

where $N_{sp}$ is the number of particle species. This expression tells us
that in a $3+1$-dimensional theory with $N_{sp}$ particle species, the
fundamental scale of gravity is lowered to the species scale given by (2).
Indeed, noticing that $N_{sp}=M_{f}^{N}V_{N}$ is the number of Kaluza-Klein
(KK) species, it is clear that (2) reproduces relation (1). Thus, the ADD
model can be viewed as a geometric realization of the species effect (2),
describing it as the suppression by the volume.

In the ADD framework, all the Standard Model (SM) species are localized on a
brane with a $4$-dimensional world volume. The localization of the gauge
fields, such as the photon Dvali:1996xe , imposes the condition that no
charged particles (including the hypothetical ones beyond SM) can freely
propagate in the bulk.

The volume suppression feature is shared by the interactions between the
Standard Model particles and any other hypothetical bulk species, i.e., the
species that propagate in the extra space. This feature has several
interesting implications for the SM particles that carry no conserved gauge
charges. The two prominent low-energy candidates are the neutron Dvali:1999gf
and the neutrino Arkani-Hamed:1998wuz ; Dvali:1999cn ; Dienes:1998sb . The
feature of gauge neutrality allows both of these particles to serve as sort of
“portals” to hidden dimensions through their mixing with bulk degrees of
freedom.

Such mixing can shed new light on some of the long-standing puzzles of the
Standard Model, e.g., such as the origin of the neutrino mass. A solution to
the latter puzzle was proposed in Arkani-Hamed:1998wuz ; Dvali:1999cn ;
Dienes:1998sb  where it was suggested that a mixing of the SM (active)
neutrino with a bulk sterile fermion can be the origin of the neutrino’s small
mass. Just as in the case of gravity, the smallness of the neutrino mass is
the result of the volume suppression. This mechanism of neutrino mass
generation is accompanied by new phenomenological signatures, such as the
possibility of oscillations of SM active neutrinos into the KK tower of the
sterile bulk fermion Dvali:1999cn .

The possibility of a neutron portal into hidden dimensions was first pointed
out in Dvali:1999gf . As discussed in this work, neutron transitions into a
hidden state are a generic feature of the brane-world scenario. This is due to
the transportation of neutrons across the bulk by virtual brane bubbles, so-
called “baby branes”. In such a process, a neutron can tunnel across the bulk
to a nearby parallel brane.

For an observer inhabiting the world-volume of our brane, the process will
effectively be described as a transition of a neutron into a hidden particle.
This process is non-perturbative in its origin. Its probability is
exponentially suppressed but never zero, as it is permitted by all
conservation laws. The suppressed rate makes it naturally compatible with the
phenomenological bounds on neutron disappearance.

In the present paper, we shall investigate a different process of neutron
transitions into extra dimensions. Namely, via its mixing with a bulk fermion.
This process is interesting because of the following specifics. First, again,
the mixing between neutrons and bulk sterile fermions is expected to be rather
generic, since none of them carry any conserved SM gauge quantum numbers.

Secondly, there is a qualitative novelty with respect to theories in which the
neutron mixes with a partner of a fixed mass, such as the neutron from a
mirror SM Berezhiani:2005hv  or many hidden copies of the SM Dvali:2009ne . In
such theories, due to a single available partner, the resonance transition
requires coincidences that are difficult to control. This difficulty is most
prominent in the case of a single mirror copy. The mirror symmetry is
inevitably broken by environmental factors such as the nuclear binding energy
of the neutron, or an ambient magnetic field. As a result, the transition
becomes unobservably suppressed, unless one assumes certain coincidences
and/or cancellations of external factors, such as, for example, the magnetic
field in the hidden sector Berezhiani:2009ldq .

The transition of a neutron into extra dimensions offers a qualitatively
different picture. The reason is a finely spaced tower of KK states in the ADD
model. Because of the high density of KK states, neutrons in a wide spectrum
of energy levels find a nearly degenerate bulk partner into which they can
oscillate. A variety of bound or free neutrons can become portals into extra
space. This puts severe constraints on the parameters of the theory and at the
same time provides new signatures for experimental searches.

A particularly interesting signature comes from the recurrent resonant
oscillations of a free neutron into the KK modes, taking place for a sequence
of special values of the magnetic field. Unlike the scenarios with a single
oscillation partner with the fixed mass, in the case of extra dimensions, the
resonant transitions take place for the quantized values of the magnetic field
matching the KK spectrum. In this way, the neutron provides a magnetic imaging
of the KK tower.

Because of such effects, the scenario is subjected to non-trivial bounds from
current experiments with ultra-cold neutrons nEDM:2020ekj ; Ban:2023cja .
Fitting the data of these experiments, we derive bounds on the parameters of
extra dimensions. These bounds are already probing the parameter regimes
motivated by the Hierarchy Problem. We also discuss new signatures for future
refined measurements.

##  II General Framework

In this section, we review the basics of the ADD model Arkani-Hamed:1998jmv ;
Arkani-Hamed:1998sfv , preparing the basis for further analysis. An intrinsic
property of extra dimensions is the existence of KK modes. Each bulk particle
gives rise to a tower of four-dimensional massive modes. That means that each
extra momentum eigenstate, from a $3+1$-dimensional perspective, corresponds
to an independent massive particle. Because of the periodicity of extra
coordinates, the momenta (and correspondingly the $4$-dimensional masses) are
quantized. For $N$ extra dimensions, the masses of the KK modes $m_{\bm{k}}$
are labeled by a set of integers that can be represented as $N$-component
vector $\bm{k}=(k_{1},\dots,k_{N})$ and are given by

$m_{\bm{k}}\,=\,\sqrt{\frac{k_{1}^{2}}{R_{1}^{2}}+\dots+\frac{k_{N}^{2}}{R_{N}^{2}}}\,.$
(3)

The counting holds for each bulk field independently. It is therefore clear
that a single bulk field (e.g., the graviton) gives rise to $N_{\rm
KK}=M_{f}^{N}V_{N}$ four-dimensional species.

Let us review some constraints on ADD parameters. These come from various
phenomenological, astrophysical, and cosmological considerations, and have
been discussed already in Arkani-Hamed:1998sfv . More updated versions of
these constraints represent the refinements, in particular, in the light of
the improved experimental data. Among the phenomenological constraints, the
most important ones are the following two model-independent bounds.

The first is the bound on the fundamental scale,

$M_{f}\,\gtrsim 10{\rm TeV}\,,$ (4)

imposed by the LHC experiments ATLAS ATLAS:2021kxv  and CMS CMS:2021far , due
to non-observation of new particle resonances at the energy scale probed by
this collider. ADD predicts the existence of such resonances since the
scattering at momentum transfer $M_{f}$ saturates perturbative unitarity. The
unitarization requires the presence of new resonances. They shall come in the
form of the tower of quantum microscopic black holes and/or string resonances
Antoniadis:1998ig .

The second model-independent constraint comes from the modification of
Newton’s inverse square law. An immediate consequence of ADD theory is that
Newton’s inverse square law is modified at distances shorter than the radii of
the extra dimensions. For example, taking all radii equal to $R$, at a
separation $r\ll R$, Newton’s law changes into

$\frac{1}{M_{P}^{2}}\frac{1}{r}\,~{}\,\rightarrow\,~{}\,\frac{1}{M_{f}^{2+N}}\frac{1}{r^{1+N}}\,.$
(5)

This modification can be understood as the result of the exchange of the KK
tower of gravitons,

$\lim_{R\rightarrow\infty}\frac{1}{M_{P}^{2}}\sum_{k}\frac{e^{-m_{k}r}}{r}\,=\,\frac{1}{M_{f}^{2+N}}\frac{1}{r^{1+N}}\,,$
(6)

where masses of KK gravitons $m_{k}$ are given by (3). Each graviton mediates
a Yukawa interaction and the summation over them gives a power-law high-
dimensional Newtonian potential.

The physical meaning of the above equation is easy to understand. First notice
that, because of the exponential suppression of modes with $m_{k}>1/r$, at
each separation only the modes with $m_{k}<1/r$ contribute. For them, the
exponential e${}^{-m_{k}r}/r$ can be replaced by $1/r$. Thus, at any scale,
each light mode increases Newton’s force by one unit. At the same time, the
multiplicity of such modes, for $r\ll R$ is $\simeq(2\pi R/r)^{N}$ and the
potential can be approximated as

$\displaystyle\frac{1}{M_{P}^{2}}\sum_{k}$
$\displaystyle\frac{e^{-m_{k}r}}{r}\,\simeq\,\frac{(2\pi
R)^{N}}{M_{P}^{2}}\frac{1}{r^{1+N}}\,=\,$ (7) $\displaystyle=$
$\displaystyle\frac{1}{M_{f}^{2+N}}\frac{1}{r^{1+N}}\,.$

which is simply the $4+N$-dimensional Newtonian potential with the fundamental
scale $M_{f}$ that is related to $M_{P}$ via (1).

In the continuum limit $R\rightarrow\infty$, the equality becomes exact.
Notice that in the continuum limit the summation over a discrete KK tower
becomes an integration over the continuum of extra-dimensional momenta
$\bm{p}=\frac{\bm{k}}{R}$,

$\lim_{R\rightarrow\infty}\sum_{k}=V_{N}\int\frac{d^{N}\bm{p}}{(2\pi)^{N}}\,.$
(8)

From here it also follows (see the appendix) that the multiplicity of the KK
modes close to a given mass $m\gg 1/R$ is,

$Z\,\sim\,(2\pi Rm)^{N-1}\,.$ (9)

Non-observation of any modification of Newton’s law in the tabletop
experiments puts the following upper bound on the radius of the largest extra
dimension, ParticleDataGroup:2022pth ; Lee:2020zjt ; Adelberger:2009zz ;
Tan:2016vwu ,

$R_{max}\,\lesssim\,30\,\mu{\rm m}\,.$ (10)

On top of these constraints there exist others that emerge based on further
assumptions about the parameters. For instance, as shown in Arkani-
Hamed:1998sfv , the star-cooling bound can alter the value of $M_{f}$ or of
the radii, depending on $N$ and assuming that all radii are equal. However, as
shown in Dvali:1999cn  this alteration is avoided if some of the radii are
smaller. In this case, the supernova cooling constraints are weakened and the
bound on the largest extra dimension (10) and on $M_{f}$(4) are unaffected.

Similarly, the decay of relic KK gravitons produced either cosmologically or
via supernovae cooling, can alter the diffuse gamma ray spectrum. However, as
explained in Arkani-Hamed:1998sfv , the diffuse cosmic gamma-ray bound is
rather sensitive to the bulk physics and can easily be avoided, in particular,
in cases in which KK gravitons have other decay channels into the bulk species
(e.g., the ones inhabiting “fat” branes).

The diffuse gamma-ray bound was later reconsidered in Hannestad:2001jv  by
taking into account an additional contribution coming from KK gravitons
produced in supernova stars throughout the history of the Universe and was
concluded to be more severe. However, these authors put aside the loopholes
pointed out in the previous analyses of Arkani-Hamed:1998sfv  which
significantly weaken the bounds. Thereby, their refinements fall in the
category that is strongly sensitive to the details of the model, such as the
shape of the compact manifold and other unknowns of bulk physics.

There exists an extensive literature with several other important model-
dependent constraints, which we shall not discuss here due to lack of space.
These must be taken into account within specific realizations of the ADD
framework. However, the bottom line is that, currently, with no further
assumptions, the universal constraints are (4) and (10).

##  III Neutrino oscillations into large dimensions

Before we move to the discussion of the neutron, we wish to review the
oscillations of neutrinos into hidden dimensions studied in Dvali:1999cn .
This analysis was done within the theory of neutrino mass introduced in
Arkani-Hamed:1998wuz . In this paper, it was proposed that the ADD setup
provides a natural framework for generating small neutrino masses via their
mixing with bulk sterile partners. For definiteness, we shall consider a
single active SM neutrino species and correspondingly a single species of a
bulk sterile neutrino. We denote this bulk field by $\Psi$.

The field $\Psi(x,y)$ is a function of four space-time coordinates $x$ and $N$
extra space coordinates $y$. It transforms in the spinor representation of the
$4+N$-dimensional Poincare symmetry. For simplicity, we shall choose it as the
smallest representation of this sort. However, at a more fundamental level,
other considerations, e.g., the cancellation of gravitational anomalies, may
apply.

From the point of view of the four-dimensional Lorentz group acting on space-
time $x$, the bulk fermion $\Psi$ contains both chiralities. Moreover,
$4+N$-dimensional chirality is only defined in even space-time dimensions,
which in the present case implies $N=2n$ with $n$ an integer. There exists no
chirality for $N=2n-1$. The dimensionalities of the irreducible massless
spinors in $N=2n$ and $N=2n-1$ are the same and equal to $2^{2+n-1}$.

The reduction of a generic $4+N$-dimensional spinor into the irreducible
representations of ($4D$-Lorentz)$\times SO(N)$ symmetry has the following
schematic form,

$2^{2+n-1}\,\rightarrow\,2_{L}\times 2^{n-1}+2_{R}\times{2^{n-1}}\,,$ (11)

where numbers indicate the dimensionalities of the representations and only
the $4$-dimensional chiralities are labeled explicitly by $L,R$. The
chiralities with respect to the internal $SO(N)$-symmetry depend on the type
of the initial spinor, as discussed above.

Thus, a massless bulk fermion $\Psi$ under the $4$-dimensional Lorentz
symmetry decomposes into $2^{n-1}$ left-handed and $2^{n-1}$ right-handed
fermions:

$\Psi\,\rightarrow\,\sum_{A}\Psi_{L}^{(A)}\>\,+\,\sum_{\bar{A}}\Psi_{R}^{(\bar{A})}\,,$
(12)

where $A,\bar{A}$ label the basic $SO(N)$-spinors 111That is, indices
$A,\bar{A}$ label the types of spinors forming a proper complete basis in the
$SO(N)$ spinor space, not to be confused with a spinor index..

The above decomposition takes place for each KK level separately,

$\Psi(x,y)\,\rightarrow\,\frac{1}{\sqrt{V_{N}}}\sum_{\bm{k}}e^{\frac{i\bm{k}\bm{y}}{R}}\left(\sum_{A}\Psi_{\bm{k},L}^{(A)}(x)\>\,+\,\sum_{\bar{A}}\Psi_{\bm{k},R}^{(\bar{A})}(x)\right)\,.$
(13)

Notice that a canonically normalized bulk fermion $\Psi$ has mass dimension
$\frac{3+N}{2}$, whereas the KK modes $\Psi_{\bm{k}}$ have mass dimension
$\frac{3}{2}$.

Upon dimensional reduction from $4+N$ to $4$-dimensional space-time, the
extra-dimensional part of the Dirac operator,
$\bar{\Psi}\bm{\gamma}\bm{\partial}_{y}\psi$, provides the Dirac mass terms
for the KK modes. In this process, at each KK level, the left and right
chiralities pair up and produce the tower of massive Dirac fermions:

$\sum_{\bm{k}}m_{\bm{k}}\sum_{A=\bar{A}=1}^{2^{n-1}}\bar{\Psi}_{\bm{k},L}^{(A)}(x)\Psi_{\bm{k},R}^{(\bar{A})}(x)\,.$
(14)

The labeling $A=\bar{A}$ is assumed to be such that the spinors get filtered
through the high-dimensional $\bm{\gamma}$-matrix properly. That is, each KK
level $\bm{k}\neq 0$ gives rise to $2^{n-1}$ massive Dirac fermions.

The fermions corresponding to the KK level $\bm{k}\,=\,0$ remain massless. A
certain superposition of their right-handed components plays the role of the
right-handed partner of the SM neutrino, which endows the latter with a Dirac
mass. Let us discuss this effect in more detail.

The mass term for the neutrino arises from a Yukawa-type interaction involving
the left-handed lepton doublet $L=(\nu_{L},e_{L})$, the bulk fermion $\Psi$
and the Higgs doublet field $H=(H^{0},H^{-})$,

$\mathcal{L}_{\rm
int}\,=\,\frac{1}{M_{*}^{N/2}}H(x)\,\bar{L}(x)\,\Psi(x,y=0)\,.$ (15)

The brane is localized at $y=0$. The above expression is rather schematic and
requires further specification. First, the requirement imposed by the
consistency of the four-dimensional effective field theory is that it must be
invariant under the four-dimensional Poincare symmetry as well as the gauge
symmetry of the SM.

The invariance of (15) under the SM gauge symmetry uniquely fixes $\Psi$ to be
a gauge-neutral degree of freedom.

At the same time, the four-dimensional Poincare invariance implies that only
the right-handed components $\Psi_{\bm{k},R}^{(\bar{A})}(x)$ (or the charge
conjugates of the left-handed ones) of the bulk fermion $\Psi$ participate in
the above Yukawa coupling. However, this requirement does not fully specify
the form of the coupling because of the following reason.

Since the brane breaks the higher-dimensional Poincare symmetry, it has no
“obligation” to respect its $SO(N)$ subgroup. That is, in general, the brane
acts as a spurion that absorbs the $SO(N)$-spinor index. The pattern of this
breaking defines to which superposition of the bulk spinors
$\Psi_{R}^{(\bar{A})}$ the SM neutrino couples.

Within the effective field theory, this information is not available. It
depends on the origin of the brane and on the dynamics that localize the
$4$-dimensional chiral fermions on it. In our phenomenological study, we shall
treat this as an input. Note that the smallness of the mass of the SM neutrino
is largely insensitive to the precise form of the
$\Psi_{R}^{(\bar{A})}$-superposition to which it mixes since the mass comes
predominantly from the $\bm{k}=0$ mode. However, the mixing with higher energy
states in the KK tower will depend on it.

Upon taking into account the Higgs vacuum expectation value $\langle
H\rangle=(v,0)$, the above coupling reduces to a mass term that mixes the
active left-handed neutrino, with the tower of KK modes,

$\mathcal{L}_{\rm
int}\,=\,\alpha_{\nu}\,\bar{\nu}_{L}(x)\,\sum_{\bm{k}}\nu_{k,R}\,,$ (16)

where

$\alpha_{\nu}\,\equiv\,\frac{v}{\sqrt{M_{*}^{N}V_{N}}}\,,$ (17)

and

$\nu_{k,R}\,\equiv\,\sum_{\bar{A}}c_{\bar{A}}\Psi_{\bm{k},R}^{(\bar{A})}(x)$
(18)

denotes the superposition of spinors from each KK level to which $\nu_{L}$
mixes. In general, the coefficients $c_{A}$ can also depend on the level $k$.
We shall make a simplified assumption that the $\nu_{k,R}$ are the eigenstates
of the KK masses. In this case, the orthogonal superpositions will effectively
decouple from our problem and can be ignored.

Mixing of $\nu_{L}$ with the tower of $\nu_{k,R}$ generates the mass of
neutrino,
$m_{\nu}\simeq\alpha_{\nu}\sqrt{\sum_{\bar{A}}|c_{\bar{A}}|^{2}}\sim\alpha_{\nu}$,
which for $M_{*}\sim M_{f}$ is

$m_{\nu}\sim\frac{vM_{f}}{M_{P}}\,.$ (19)

For $M_{f}\sim 10$TeV, this value is in the right phenomenological ballpark.

The mass of the neutrino is generated predominantly through the mixing with
the right-handed components of the $\bm{k}=0$ level, which have no masses of
their own. However, mixing with higher members of the KK tower results in
oscillations of the neutrino into these states. This can have potentially
observable effects Arkani-Hamed:1998wuz ; Dvali:1999cn .

For simplicity, let us reproduce this effect for one relevant extra dimension
with radius $R$. In this analysis, we shall closely follow Dvali:1999cn . In
this case, the indices $A,\bar{A}$ are not required and we can write,

$\Psi(x,y)=\frac{1}{\sqrt{2\pi
R}}\sum_{k=-\infty}^{k=\infty}e^{\frac{iky}{R}}\left(\Psi_{k,L}(x)\>\,+\,\Psi_{k,R}(x)\right)\,.$
(20)

Notice that the hierarchy $M_{f}/M_{P}\sim 10^{15}$ is still maintained by
assuming the existence of additional $N-1$ dimensions of much smaller radii.
The corresponding KK excitations are heavy and effectively decouple from the
neutrino mass problem.

Because of the degeneracy under the reflection $k\rightarrow-k$ for each
$k\neq 0$ level, the active neutrino mixes with the modes
$\nu_{kR}\equiv\frac{1}{\sqrt{2}}(\psi_{k,R}+\psi_{-k,R})$, whereas the
orthogonal superpositions, $\frac{1}{\sqrt{2}}(\psi_{k,R}-\psi_{-k,R})$,
decouple and can be neglected.

Then, the part of the Lagrangian describing the relevant mass terms is

$\mathcal{M}_{\nu}=\alpha_{\nu}\nu_{L}\left(\nu_{0R}\,+\,\sqrt{2}\sum_{k=1}^{\infty}\nu_{kR}\right)+\sum_{k=1}^{\infty}\frac{k}{R}\bar{\nu}_{kL}\nu_{kR}\,.$
(21)

Thus, we have the mass matrix of the following form,

$\begin{pmatrix}0&\alpha_{\nu}&\sqrt{2}\alpha_{\nu}&\sqrt{2}\alpha_{\nu}&...\\\
0&0&0&0&...\\\ 0&0&1/R&0&...\\\ 0&0&0&2/R&...\\\ ...&...&...&...&...\\\
\end{pmatrix}.$ (22)

Each $k\neq 0$ mixes with SM neutrino with an angle given by
$\tan\varphi_{k}=\frac{\alpha_{\nu}R}{k}$. Thus, the modes with higher values
of $k$ quickly decouple and the main effect is concentrated in the part of the
KK tower with $k\sim 1$.

The active (left-handed) SM neutrino represents a superposition of the mass
eigenstates of the form

$\nu_{L}\,=\,\frac{1}{\mathcal{N}}\Big{(}\nu_{L}^{\prime}+\sum_{k=1}^{\infty}\frac{\alpha_{\nu}R}{k}\nu_{k,L}^{\prime}\Big{)},$
(23)

where $\mathcal{N}^{2}=1+\sum_{k}\frac{(\alpha_{\nu}R)^{2}}{k^{2}}\simeq 1$ is
a normalization factor. The mixing with the KK tower results in the
oscillations of the active flavor into the KK modes. The survival probability
is

$P_{\rm
surv}(t)=|\langle\nu_{L}|\nu_{L}(t)\rangle|^{2}=\frac{1}{\mathcal{N}^{4}}\Big{|}1+\sum_{k=1}^{\infty}\frac{(\alpha_{\nu}R)^{2}}{k^{2}}\exp{(\mathrm{i}\phi_{k})}\Big{|}^{2}.$
(24)

Thus, we obtain an interference of the infinite number of oscillating modes
with growing frequencies $\propto k^{2}$ and decreasing amplitudes $\propto
1/k^{2}$. Because of this, in practice, the higher-frequency modes can be
averaged out and only a few low-frequency modes are observable. Some
implications of this phenomenon, in particular for solar neutrinos, were
discussed in Dvali:1999cn . More updated experimental constraints can be found
in Machado:2011jt  and in subsequent papers, with the latest bounds appearing
in Forero:2022skg .

Notice that with the assumption of the alignment of the interaction (18) and
mass (14) eigenstates, the above-reproduced analysis of Dvali:1999cn  can be
straightforwardly generalized to an arbitrary number of relevant dimensions.

We shall now move to a discussion of an analogous story of neutron
oscillations into extra dimensions.

##  IV Neutron Oscillations into hidden dimensions

The main novelty of the present paper is the possibility of neutron
oscillations into extra dimensions. Similarly to the neutrino, the neutron
carries no conserved gauge charge and can freely mix with the bulk species. We
consider the oscillation of the neutron into KK modes of a bulk fermion. As in
the neutrino case, we shall denote this bulk fermion by $\Psi$. To start with,
we assume that $\Psi$ is massless.

We shall keep its origin generic. However, a particularly motivated
possibility would be the case in which $\Psi$ is the bulk partner of one of
the SM neutrinos that generates its mass via the above-discussed mechanism.
Consequently, the neutron would mix with the same KK-tower as the active
neutrino. Notice that such a mixing is totally safe from the neutrino mass
perspective. Because of a very large splitting between the masses of neutrino
and neutron, they predominantly mix with the highly separated sectors of the
KK spectrum. Correspondingly, they have essentially zero influence on one
another. This is a rather economic scenario. Nevertheless, we shall keep the
discussion maximally general.

It is reasonable to assume that the mixing between the neutron and the bulk
fermion originates from a more fundamental four-fermi interaction of the type,

$\mathcal{S}_{\rm int}=\int
dx^{4}\frac{1}{M_{*}^{2+N/2}}\overline{udd}\,\Psi\,+\,{\rm h.c.}\,,$ (25)

where $M_{*}$ is a scale. We shall not specify its origin and shall treat it
as a phenomenological parameter. In general, it is reasonable to expect that
$M_{*}$ is of order or higher than $M_{f}$.

In the effective low-energy theory below the scale of the QCD confinement, the
coupling (25) translates into an effective mass term that mixes the neutron
with $\Psi$. The effective Lagrangian is derived by the following replacement,

$udd\rightarrow\Lambda_{\rm QCD}^{3}\,n\,,$ (26)

where $n$ is the neutron field and $\Lambda_{\rm QCD}$ is of the order of the
QCD scale.

As previously, we expand $\Psi$ into the KK modes (13). The difference from
the neutrino case is that the neutron is a Dirac fermion. Therefore, both
chiralities, $n_{L}$, and $n_{R}$, are available for mixing with the
respective components of the KK tower:

$\displaystyle\Psi_{\bm{k},R}(x)\equiv$
$\displaystyle\frac{1}{\sqrt{\sum_{\bar{A}}|c_{\bar{A}}|^{2}}}\sum_{\bar{A}}c_{\bar{A}}\Psi_{\bm{k},R}^{(\bar{A})}(x)\,,$
$\displaystyle\Psi_{\bm{k},L}(x)\equiv$
$\displaystyle\frac{1}{\sqrt{\sum_{A}|c_{A}|^{2}}}\sum_{A}c_{A}\Psi_{\bm{k},L}^{(A)}(x)\,.$
(27)

We again make a simplified assumption that the combinations (IV) with
$A=\bar{A}$ are KK mass eigenstates. That is, modulo mixing with the neutron,
at each KK level $\bm{k}$ they form a four-dimensional Dirac fermion
$\Psi_{\bm{k}}\,\equiv\,\Psi_{\bm{k},L}(x)\,+\,\Psi_{\bm{k},R}(x)$ with mass
$m_{\bm{k}}$. In such a case, the orthogonal combinations decouple and we get
the following effective $4$-dimensional Lagrangian describing the Dirac
neutron, the KK tower of Dirac fermions $\Psi_{\bm{k}}$ and their mixing,

$\begin{split}\mathcal{L}=&\;\bar{n}\mathrm{i}\not{\partial}n-m_{n}\bar{n}n\\\
&+\sum_{k}\Big{(}\bar{\Psi}_{k}\mathrm{i}\not{\partial}\Psi_{k}-m_{k}\bar{\Psi}_{k}\Psi_{k}\Big{)}\\\
&+\alpha\sum_{k}\bar{n}\Psi_{k}+h.c.,\end{split}$ (28)

where $m_{n}$ is the SM mass of neutron and $\alpha$ is the mixing mass term,

$\alpha\equiv\frac{\Lambda_{\rm QCD}^{3}}{M_{*}^{2+N/2}\sqrt{V_{N}}}\,.$ (29)

We have absorbed the normalization factor $\sqrt{\sum_{A}|c_{A}|^{2}}$ into
the rescaling of parameters.

Notice that $\alpha$ is an extremely small mass scale. In particular, taking
into account that $M_{*}\geqslant M_{f}$ and the current experimental bound
(4), we get the following upper bound on $\alpha$,

$\alpha\lesssim 10^{-24}{\rm GeV}\,.$ (30)

This quantity is much below any other scale in the problem. This allows us to
study the oscillation picture perturbatively in $\alpha$.

The masses and mixings between the neutron and the bulk states are described
by the following mass matrix,

$\begin{pmatrix}m_{n}&\alpha&\alpha&\alpha\\\ \alpha&0&0&0\\\
\alpha&0&m_{\bm{k}}&0\\\ \alpha&0&0&m_{\bm{k^{\prime}}}\end{pmatrix}.$ (31)

Notice that because of the bound (30), we have $\alpha/m_{n}\lesssim
10^{-24}$. In the leading approximation, each KK mixes with the neutron with
an angle given by $\tan\varphi_{k}=\frac{\alpha}{\Delta m_{k}}$, where $\Delta
m_{\bm{k}}=|m_{n}-m_{\bm{k}}|$. Putting aside miraculous coincidences, the
smallest value of $\Delta m_{\bm{k}}$ is set by the level splitting between
the KK modes.

The picture is very similar to what we encountered in the neutrino case but
with two differences. First, the mass matrix is symmetric. Secondly, $\Delta
m_{\bm{k}}$ reaches the minimum not at the bottom of the KK spectrum but in
the region closest to the energy of the neutron.

Because of the mixing with the KK tower, the ordinary neutron, which is a SM
interaction eigenstate, is not an exact mass eigenstate. Labeling the mass
eigenstates by primes, the SM neutron represents a superposition of the mass
eigenstates of the following form,

$n\,=\,\frac{1}{\mathcal{N}}\Big{(}n^{\prime}+\sum_{\bm{k}}\frac{\alpha}{\Delta
m_{\bm{k}}}\Psi_{\bm{k}}^{\prime}\Big{)},$ (32)

where $\mathcal{N}^{2}=1+\sum_{k}\frac{\alpha^{2}}{\Delta
m_{k}^{2}}=1+\mathcal{O}(\alpha^{2})$ is a normalization factor. This state
evolves in time as,

$n(t)=\frac{1}{\mathcal{N}}\Big{(}n^{\prime}+\sum_{\bm{k}}\frac{\alpha}{\Delta
m_{\bm{k}}}e^{\mathrm{i}\phi_{\bm{k}}}\Psi_{\bm{k}}^{\prime}\Big{)},$ (33)

where $\phi_{\bm{k}}=|m_{n}-m_{\bm{k}}|t$. The corresponding survival
probability of the neutron is,

$P_{\rm surv}(t)=|\langle
n|n(t)\rangle|^{2}=\frac{1}{\mathcal{N}^{4}}\Big{|}1+\sum_{\bm{k}}\frac{\alpha^{2}}{\Delta
m_{\bm{k}}^{2}}\exp{(\mathrm{i}\phi_{\bm{k}})}\Big{|}^{2}.$ (34)

Similarly to the neutrino case, the oscillation takes place in a collection of
modes. The “resonant” modes correspond to KK levels that are closest to the SM
energy of the neutron, $m_{n}$. The more distant levels oscillate in
increasing frequencies and suppressed amplitudes. Averaging over all modes
except the level which is closest to $m_{n}$, we get the following expression
for the probability of the neutron oscillating into $\Psi$,

$P_{n\to\Psi}(t)=\frac{Z}{\mathcal{N}^{4}}\frac{4\alpha^{2}}{\Delta
m^{2}}\sin^{2}\Big{(}\frac{\Delta m}{2}t\Big{)}.$ (35)

where $\Delta m\equiv|m_{n}-m_{\bm{k}}|$ denotes the smallest mass splitting
and the factor $Z$ accounts for the corresponding degeneracy. For
visualization, see Fig. 1.

For a single extra dimension of radius $R$, we have $Z=2$ because the mass is
degenerate for $k$ and $-k$. Equivalently, the neutron mixes with the states
$\Psi_{k}\equiv\frac{1}{\sqrt{2}}(\psi_{k}+\psi_{-k})$, whereas the orthogonal
superpositions decouple. The corresponding survival probability is

$P_{n\to\Psi}(t)=\frac{1}{\mathcal{N}^{4}}\frac{8\alpha^{2}}{\Delta
m^{2}}\sin^{2}\Big{(}\frac{\Delta m}{2}t\Big{)}\,,$ (36)

where $\Delta m\leqslant 1/R$. Although “accidentally” $\Delta m$ may appear
arbitrarily below this upper bound, a natural value would be $\Delta m\sim
1/R$.

For a higher number of relevant extra dimensions, the level-splitting is as
small as $\Delta m\sim 1/(m_{n}R)^{2}$ (see the Appendix). The degeneracy
count is the following. For $N_{R}$ relevant dimensions of equal radii $R$
(again, $N_{R}$ must not be confused with the total number $N$ of large extra
dimensions, $N_{R}\leqslant N$), in general, the number $Z$ of KK states
within a gap $\Delta m$ satisfies,

$\frac{Z}{\Delta m}\sim\frac{1}{m_{n}}(m_{n}R)^{N_{R}}\,.$ (37)

Taking this into account, the survival probability for $N_{R}$ relevant
dimensions of radius $R$ can be presented in the following form

$P_{n\to\Psi}(t)\,\sim\,\frac{m_{n}}{\Delta
m}(m_{n}R)^{N_{R}}\left(\frac{\alpha}{m_{n}}\right)^{2}\sin^{2}\Big{(}\frac{\Delta
m}{2}t\Big{)}\,.$ (38)

The oscillation amplitude is the largest for the states with the smallest
$\Delta m$. Equipped with these equations, we shall next discuss the effect of
neutron disappearance.

Figure 1: A schematic description of the matching of the energy levels of free
and nuclear-bound neutrons with the KK spectrum of $\Psi$.

##  V Phenomenological bounds from neutron disappearance in nuclei

In the previous section, we discussed how in the ADD framework a neutron can
oscillate into particles propagating in extra dimensions. We shall now find
phenomenological constraints on such a process. We first study the case of a
massless bulk partner. In this case, the most severe constraints come from the
non-observation of the disappearance of neutrons from the nuclei.

Let us consider an atomic nucleus with several neutrons in it. The above-
derived formulas can be directly applied to such neutrons. The mass $m_{n}$
must be understood as the energy level of a neutron in the given bound state.
The neutron has a finite probability of transitioning into the bulk KK modes.
From the point of view of the SM observer, this process is viewed as the
disappearance of the neutron. The vacated nucleus will get de-excited. Namely,
a neutron from a higher level will occupy the state freed by the disappearing
neutron. The process of de-excitation is accompanied by the emission of a hard
photon, i.e., a photon of nuclear energy.

For large extra dimensions, the typical oscillation time, set by the KK level
splitting $\sim 1/\Delta m$, is much longer than the nuclear transition time.
In such a case, after the de-excitation, the process of coming back is no
longer possible, since the place of the original neutron is occupied.
Therefore, such a process would lead to a decay of atoms into isotopes with
fewer neutrons. This is not observed in nature and the experimental bound on
the lifetime of neutrons within nuclei is KamLAND:2005pen

$\tau_{n}\,>\,10^{30}\,{\rm y}\,\sim\,10^{62}\,{\rm GeV}^{-1}\,.$ (39)

This bound translates to a bound on the neutron disappearance rate in our
scenario. The rate can be derived from (36) as the average transition
probability per unit time,

$\lambda_{n}\,=\,\frac{2Z\alpha^{2}}{\Delta
m}=\frac{2Z\Lambda_{QCD}^{6}}{\Delta mM_{*}^{4+N}V_{N}},$ (40)

where we plugged-in the parameter $\alpha$ from Eq. (29). Using the expression
(38), it is also useful to present the rate as

$\lambda_{n}\,\sim\,m_{n}\left(\frac{\alpha^{2}}{m_{n}^{2}}\right)(m_{n}R)^{N_{R}}\,.$
(41)

Now, we demand that $\tau_{n}=1/\lambda_{n}>10^{30}$ y. This leads to the
following constraint,

$M_{*}^{4+N}>10^{21}({\rm GeV}\>R)^{N_{R}}M_{f}^{2+N}{\rm GeV}^{2}\,,$ (42)

where we have expressed $V_{N}$ through $M_{P}\sim 10^{19}$GeV and $M_{f}$ and
used the fact that $m_{n}\simeq$ GeV and $\Lambda_{QCD}\simeq 0.3$ GeV. We
also used the neutron lifetime bound (39) in GeV units.

We now wish to apply this bound to different cases distinguished by the number
and size of relevant dimensions. For this, let us first make the concept of
relevance more quantitative. We have $N_{R}$ relevant dimensions of size $R$
and $N-N_{R}$ subdominant ones of radii $\tilde{R}$. Both of these categories
contribute to (1). The distinction is that the contribution of the subdominant
dimensions to the rate (41) is less. That is,

$(m_{n}\tilde{R})^{N-N_{R}}\,\ll\,(m_{n}R)^{N_{R}}\,.$ (43)

Then taking into account the relation (1) and the fact that
$V_{N}=(R^{N_{R}}\tilde{R}^{N-N_{R}})$, we get

$(m_{n}\tilde{R})^{N-N_{R}}\ll\frac{M_{P}}{M_{f}}\left(\frac{m_{n}}{M_{f}}\right)^{N/2}.$
(44)

We can now apply the above to the following cases. The first is the case
$N_{R}=1$ with just one relevant extra dimension of radius $R$. The radius $R$
and $M_{f}$ are free parameters. Therefore, we set both to their current
experimental bounds, given by equations (10) and (4). The results for
different total numbers $N$ of extra dimensions are shown in Table 1.

$N$ |  $M_{*}[{\rm GeV}]$
---|---
3 |  $>3\cdot 10^{7}$
4 |  $>1\cdot 10^{7}$
5 |  $>5\cdot 10^{6}$
6 |  $>3\cdot 10^{6}$
Table 1: Bound on $M_{*}$ for one dominant $R$ with $M_{f}=10$ TeV and
$R=30\mu$m

Notice that condition (44) implies that for $N_{R}=1$ and the above choice of
$R$ and $M_{f}$, we must have $N>2$. In other words, with only two extra
dimensions and $M_{f}\sim 10$TeV, it is not possible to have one much shorter
than the other without conflicting with observations.

The second example we study is the case with $N$ equal size extra dimensions.
That is, in this case, $N_{R}\,=\,N$. Correspondingly, using (1), the r.h.s.
of (42) can be expressed via $M_{P}\sim 10^{19}$GeV, and the bound acquires a
simple numerical form,

$M_{*}^{4+N}\,>\,10^{59}{\rm GeV}^{4+N}\,.$ (45)

The resulting bounds on $M_{*}$ and corresponding values of $R$ are listed in
Table 2.

$N$ |  $R[\mu{\rm m}]$ |  $M_{*}[{\rm GeV}]$
---|---|---
2 |  $1.1$ |  $>7\cdot 10^{9}$
3 |  $1.6\cdot 10^{-5}$ |  $>3\cdot 10^{8}$
4 |  $5.5\cdot 10^{-8}$ |  $>2\cdot 10^{7}$
5 |  $2\cdot 10^{-9}$ |  $>4\cdot 10^{6}$
6 |  $2.2\cdot 10^{-10}$ |  $>8\cdot 10^{5}$
Table 2: Bound on $M_{*}$ for equal size extra dimensions.

Another instructive choice is $M_{*}\sim M_{f}$. In this case, (42) becomes

$M_{f}^{2}\,>\,10^{21}({\rm GeV}R)^{N_{R}}{\rm GeV}^{2}\,.$ (46)

Since $N_{R}\leqslant N$, from (1) we have $M_{f}\leqslant(M_{P}^{2}/(2\pi
R)^{N_{R}})^{1/(N_{R}+2)}$. Together with this inequality, equation (46)
becomes a bound on $R$. For example, for $N_{R}=1$ we get

$R\,\lesssim 10^{2}{\rm GeV}^{-1}\sim 10^{-8}\mu{\rm m}\,.$ (47)

Equivalently, the bound on $M_{f}$ is

$M_{f}\,\gtrsim\,10^{12}{\rm GeV}\,.$ (48)

##  VI Proton Decay

The possibility of neutron oscillations into a bulk fermion opens the door to
proton decay. If the high-dimensional mass of $\Psi$ is less than the mass
difference between the proton and the sum of electron and neutrino masses, the
tree-level decay process is possible. Via a virtual neutron exchange, the
proton can decay into a positron, the SM neutrino, and $\Psi$,

$p\rightarrow e^{+}+\nu+\Psi.$ (49)

Figure 2: Proton Decay via virtual neutron exchange. The SM particles are
confined to the brane, whereas the extra-dimensional particle, $\Psi$,
propagates in the bulk.

The process is depicted in Fig. 2. The rate of the process is given by

$\lambda_{p}\,\sim\,m_{p}\,\left(\frac{m_{n}}{v}\right)^{4}\,\left(\frac{\alpha}{m_{n}}\right)^{2}\,(m_{p}R)^{N_{R}}.$
(50)

It is useful to compare this rate to the neutron disappearance rate (41).
Taking into account that $m_{n}\simeq m_{p}$, we obtain that the proton decay
rate is suppressed relative to the neutron rate by an additional factor
$\left(\frac{m_{n}}{v}\right)^{4}\sim 10^{-9}$ coming from the $W$-boson
exchange:

$\lambda_{p}\,\sim\,\lambda_{n}\left(\frac{m_{n}}{v}\right)^{4}\,\sim\lambda_{n}10^{-9}\,.$
(51)

At the same time, the current bound on the proton lifetime for this channel,
$\tau_{p}>10^{30}$y Learned:1979gp , is the same as for the bound on neutron
disappearance (39). Correspondingly, the bound on the proton lifetime is
automatically satisfied, as long as the neutron bound is fulfilled.

Because of this, in terms of $M_{*}$ the proton gives a milder bound. E.g.,
assuming $M_{f}\sim 10$TeV, the bound is

$M_{*}^{4+N}>10^{12}(m_{p}R)^{N_{R}}\,M_{f}^{2+N}{\rm GeV}^{2}\,.$ (52)

Therefore, the improved accuracy of neutron disappearance experiments would
appear more promising for testing the discussed scenario.

##  VII Kaluza-Klein Spectroscopy from Free Neutron Oscillations

We now wish to discuss another experimentally-motivated domain of our
framework. In this domain, the dominant effect is the oscillation of a free
neutron into hidden dimensions. In this regime, the bulk partner of the
neutron, $\Psi$, has a high-dimensional mass, $\mu$. If the mass is above the
energy levels of the nuclear neutron, the stability of nuclei is unaffected.
Similarly, if $\mu$ is above the proton mass, there is no bound from proton
decay. Thus, if $\mu$ is within a window, $\sim$ MeV below the mass of a free
neutron, only the free neutron is experiencing oscillations into extra
dimensions. The transition amplitude can correspondingly be much larger.

Notice that for dimensions that satisfy $1/R\,\ll\,|m_{n}-\mu|$, the number
and the level-splitting of the available KK modes to which the neutron can
oscillate is essentially the same as for $\mu\,=\,0$.

For $N_{R}=2$ this level-splitting is sufficiently small to be scanned by an
external magnetic field available in current laboratory setups. This allows
for potentially-observable resonant $n-\Psi$ oscillations in correlation with
the features of the KK spectrum. Let us discuss this effect.

The amplitude of the oscillation of the neutron into the bulk particle is
suppressed by the minimal mass splitting between the neutron and the nearest
KK mode $\Delta m$. At the same time, the amplitude is enhanced by the KK
multiplicity factor $Z$.

The oscillation of a free neutron is effectively mapped onto a $2\times 2$
problem, in which the neutron mixes with a single state with the nearest mass,

$\begin{pmatrix}m_{n}&\sqrt{Z}\alpha\\\ \sqrt{Z}\alpha&m_{n}+\Delta
m\end{pmatrix}.$ (53)

In the absence of a parameter that could scan the neutron energy with a
precision of $\Delta m$, the oscillation amplitude is suppressed as given by
(38). However, an external magnetic field can serve as a scanner. A neutron
placed in a magnetic field $\bm{B}$, because of its magnetic moment
$\bm{\mu}_{n}$, experiences an additional shift in the energy level equal to
$\epsilon\,=\,\bm{\mu}_{n}\bm{B}$. Correspondingly, the mixing matrix becomes

$\begin{pmatrix}m_{n}+\epsilon&\sqrt{Z}\alpha\\\ \sqrt{Z}\alpha&m_{n}+\Delta
m\end{pmatrix}.$ (54)

Notice that, regardless of the value of the magnetic field, in the above
mixing matrix we must always assume $\epsilon\lesssim\Delta m$. If we increase
the magnetic field in the way that $\epsilon>\Delta m$, the whole problem will
be shifted to the KK level which is closest to the shifted energy of the
neutron $m_{n}+\epsilon$. Thus, an increase in the magnetic field above the
resonance value,

$B_{r}\equiv\Delta m/\mu_{n},$ (55)

results in the effective shift $m_{n}\rightarrow m_{n}\,+\,\epsilon$.

Because of this, for the array of magnetic field values, the neutron energy
comes in resonance with some of the KK levels, and an enhanced oscillation
amplitude is achieved.

For $N_{R}\,=\,2$ the levels are not exactly equally spaced. Because of this,
the effect is not strictly periodic but has a clear repetitive pattern fully
reflecting the KK spectrum.

This feature is very different from the theories in which the neutron has a
single oscillation partner, e.g., such as the dark neutrons from the hidden
standard model copies. We shall make a more explicit comparison with such
scenarios later.

For currently available experimental setups, $\epsilon$ is a very small
quantity. For example, for earth’s magnetic field, $|\bm{B}_{e}|\simeq 0.5$G,
we have $\epsilon_{e}\sim 10^{-12}$eV. With the presently available most
powerful artificial magnets, it can be increased by a few orders of magnitude.

This is sufficient for scanning the KK spectrum for $N_{R}=2$ in an
interesting range of radii. For example, for the largest possible value of
extra dimensions $R=R_{max}$, given by experimental bound (10), the splitting
of closely spaced KK modes is $\Delta m\equiv 1/(2m_{n}R_{max}^{2})\simeq
2\times 10^{-14}$ eV. This value falls within the territory testable by
currently available experimental setups.

For $\alpha<|\epsilon-\Delta m|<\Delta m$, the oscillation amplitude is

$A\,\simeq\,\frac{\alpha^{2}}{|\epsilon-\Delta m|^{2}}\,,$ (56)

and the oscillation frequency is given by $\omega=|\epsilon-\Delta m|$. For
the resonance regime, $\alpha\leq|\epsilon-\Delta m|$, the amplitude becomes
order one, and the frequency is set by $\omega\sim\alpha$.

Notice again that for the regime $\epsilon\gg\Delta m$, we are shifting to a
new KK level which is closest to $m_{n}+\epsilon$, and the excess of
$\epsilon$ gets effectively reabsorbed in the shift of $m_{n}$.

We shall now confront the above dynamics with some experimental data. In order
to establish a dictionary, we shall recast the expression for the amplitude
(56) into a more user-friendly form. For this, we take into account that
$\epsilon\simeq 6\times 10^{-14}B\frac{{\rm eV}}{\mu{\rm T}}$ Tiesinga:2021myr
and that for $R=R_{max}$, we have $\Delta m\simeq 2\times 10^{-14}$ eV. We can
then write

$A\sim\frac{10^{27}\,\alpha^{2}\,{\rm eV}^{-2}}{|3\frac{B}{\mu{\rm
T}}-\frac{R_{max}^{2}}{R^{2}}|^{2}}\,.$ (57)

Since for resonant values of $B$, the defining factor is the difference of the
two terms in the denominator, we pay more precise attention to them, while
examining the order-one overall numerical coefficients less closely. We shall
use this formula for interpreting the two experimental results.

The first one nEDM:2020ekj , the ultra-cold neutron storage experiment,
constrains the disappearance of a neutron for two separate values of the
magnetic field: $B\simeq 10.20\pm 0.02\mu$T and $B\simeq 20.39\pm 0.04\mu$T.

If we assume $R=R_{\rm max}$, then for the above values of the magnetic field,
the first entry in the denominator (57) is larger than the second one by
factors of $30$ and $60$, respectively. This implies that the system will
shift to a higher KK level, absorbing the extra contribution from the magnetic
energy into the mass of a new KK partner. Excluding miraculous coincidences of
finer cancellation, the optimal KK level will satisfy

$|\epsilon-\Delta m|\sim\Delta m\,.$ (58)

Notice that the error in $\delta\epsilon=\epsilon\delta B/B$, due to an
inaccuracy of the magnetic field, which in both cases is $|\delta B/B|\simeq
2\times 10^{-3}$, is not sufficient for further reducing the difference
$|\epsilon-\Delta m|$ significantly.

Now, it is clear that the relation (58) will persist for $R\,<\,R_{\rm max}$.
Indeed, for $\frac{R_{max}^{2}}{R^{2}}\ll 3\frac{B}{\mu{\rm T}}$ the level
will always get shifted to the one that satisfies (58). At the same time, for
$\frac{R_{max}^{2}}{R^{2}}\,>\,3\frac{B}{\mu{\rm T}}$, the same condition is
satisfied without any shift. Correspondingly, the relation (58) is satisfied
for the entire parameter space probed by the experiment of nEDM:2020ekj .
Thus, for fitting the data, the equation (57) can be approximated as,

$A\sim 10^{27}\,\alpha^{2}\,\frac{R^{4}}{R_{max}^{4}}\,{\rm eV}^{-2}\,.$ (59)

Putting everything together, we get the following constraint on our
parameters,

$\,\alpha\,\frac{R^{2}}{R_{max}^{2}}\lesssim\,10^{-16}{\rm eV}\,.$ (60)

Expressing $\alpha$ in terms of $M_{*}$ and $M_{f}$, we can write,

$\,\frac{10{\rm
TeV}}{M_{f}}\left(\frac{M_{f}}{M_{*}}\right)^{2+N/2}\,\frac{R^{2}}{R_{max}^{2}}\lesssim\,\frac{1}{3}\,.$
(61)

Interestingly, by taking $M_{f}$ at its current experimental bound (4) of
$M_{f}\sim 10$TeV, while at the same time taking $M_{*}$ at its theoretical
one, $M_{*}\sim M_{f}$, equation (61) translates as the current experimental
bound on $R$ (10) imposed by the checks of Newtonian gravity. Having a
rigorous bound on the extra-dimensional setup from the cold neutron
experiments is rather remarkable.

We shall now move to the second experiment Ban:2023cja , with an ultra-cold
neutron beam. The important insight from this measurement is that the authors
scanned a wide range of the magnetic field within the interval in between
$B_{min}\simeq 50\,\mu$T and $B_{max}\simeq 1100\,\mu$T, with a step of
$\Delta B=3\mu$T. In order to translate the results of this experiment into
bounds on our parameters, we must distinguish the following cases.

In the regime

$\frac{R_{max}^{2}}{R^{2}}\,\gg\,3\frac{|B_{max}-B_{min}|}{\mu{\rm T}}\simeq
3150\,,$ (62)

or equivalently $\Delta m\gg\epsilon_{max}-\epsilon_{min}$, our transition
amplitude is essentially the same as without the magnetic field, (59), and the
bound is

$\,\alpha\,\frac{R^{2}}{R_{max}^{2}}\lesssim\,10^{-15}{\rm eV}\,.$ (63)

It has the same form as (60) but subject to (62).

In the opposite case,

$\frac{R_{max}^{2}}{R^{2}}\,<\,3\frac{|B_{max}-B_{min}|}{\mu{\rm T}}\,,$ (64)

the bound depends on the ratio between $\Delta m$ and the scanning step
$\Delta\epsilon$. In the denominator of expression (56) the smallest of the
two will enter. Because of this, for $R_{max}^{2}/R^{2}<9$, the bound is again
given by the expression (63).

For $R_{max}^{2}/R^{2}>9$, the situation is different since the $\Delta m$
term in (56) can be compensated by the accuracy of $\Delta\epsilon$.

The magnetic term $\epsilon$ can be gradually cranked up in small increments
$\Delta\epsilon$ all the way to $\Delta m$, sooner or later the resonant
transition will take place with $|\epsilon-\Delta m|\simeq\Delta\epsilon$. In
this case, the absolute value of $\Delta m$ (and thus dependence on $R$) drops
out from the transition amplitude and we have

$A\,\sim\,\frac{\alpha^{2}}{|\Delta\epsilon|^{2}}\sim\alpha^{2}10^{25}\,{\rm
eV}^{-2}\,.$ (65)

This imposes the following $R$-independent constraint on $\alpha$

$\,\alpha\,\lesssim\,10^{-14}{\rm eV}.$ (66)

In summary, the characteristic signature of neutron oscillations into extra
dimensions, which makes it very different from other proposals, is the
recurrence of the resonance amplitude for multiple values of the magnetic
field with steps $\Delta B\,=\,\frac{\Delta m}{\mu_{n}}$, where $\Delta m$ is
the mass splitting between the nearest KK levels. Such recurrent resonance
transitions are the prediction of the extra dimensional scenario, see Fig. 3
and Fig. 4.

The experiment Ban:2023cja  is effectively probing the interval

$0.8\mu\textrm{m}<R<10\mu\textrm{m}\,$ (67)

and correspondingly impose the bound (66) on $\alpha$.

Extensions of the upper and lower bounds of this interval can be achieved by
an increase of the magnetic field-range and the decrease of the step size,
respectively. For example, a decrease of the step size to $1/3\mu$T would
bring the upper bound to the level of (10) imposed by the measurements of
Newton’s law. An increase of the range of the scanned magnetic field would
correspondingly allow to probe smaller sizes of extra dimensions.

The above analysis makes the general tendency clear. The experiments with
finer scanning of wider ranges of the magnetic field can provide deeper probes
of physics of extra dimensions. Strikingly, the bounds (60), (63) and (66)
indicate that these experiments already probe the domain motivated by the
Hierarchy Problem.

Figure 3: A qualitative sketch of the transition amplitude as a function of
the magnetic field. The resonance transitions take place with steps $\Delta
B=\Delta m/\mu_{n}$. The differences in the heights are due to different
degeneracy factors of different KK levels. The repetitive but not strictly
periodic behavior is due to the nonuniform population of KK levels.  Figure 4:
Scanning with the magnetic field between two KK-levels. A resonance occurs
when the shifted energy level of the neutron comes close to a KK state with a
precision of $\mu_{n}\Delta B$.

##  VIII Comparing with oscillations into hidden copies of the neutron

In this section, we wish to confront the presented scenario with the previous
proposal Dvali:2009ne  in which the neutron also mixes with multiple partners
$n_{i},~{}i=1,2,...,N$ from $N$ hidden sectors ($N$ not to be confused with
the same notation of number of dimensions in ADD). These partners represent
neutron-like particles belonging to $N$ copies of the SM. The copies are
related by an exact permutation symmetry. Similarly to the ADD model, such a
scenario was originally motivated by the Hierarchy Problem, since the
existence of many SM copies lowers the cutoff of the theory Dvali:2007hz ;
Dvali:2007wp .

In a sense, this scenario is a “Fourier transform” of the ADD solution meaning
that the dilution of gravity takes place in the space of species.
Correspondingly, solutions to certain puzzles offered by ADD, also find
counterparts in this framework. For example, a “dilution” of the neutrino mass
in the space of species can be achieved Dvali:2009ne ; Ettengruber:2022pxf .
Also, the particles in the hidden copies of SM can be dark matter Dvali:2009fw
; Dvali:2009ne  222In fact, this idea was originally proposed within the ADD
framework, with the role of exact copies of SM played by the parallel folds of
our brane Arkani-Hamed:1999rvc .

As discussed in Dvali:2009ne , one of the phenomenological consequences of the
scenario with $N$ SM copies can be the oscillation of a neutron into the
hidden partners, since the mixing $\alpha\sum_{i\neq j}\bar{n}_{i}n_{j}$ is
permitted by the gauge symmetries of all sectors. The only constraint,
$\alpha<m_{n}/N$, comes from unitarity.

This scenario exhibits some crucial differences from the present case. Because
of the exact permutation symmetry, the mass matrix has the form,

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}(m_{n}-\alpha)\,+\begin{pmatrix}1&1&1&1\\\ 1&1&1&1\\\
1&1&1&1\\\ 1&1&1&1\end{pmatrix}\,\alpha\,.$ (68)

Due to this structure, the oscillation dynamics can be reduced to a $2\times
2$ problem in which the neutron from our copy, $n_{1}$, mixes with the state
$n_{h}\equiv\frac{1}{\sqrt{N-1}}\sum_{j\neq 1}n_{j}$, via the following mass
matrix,

$\begin{pmatrix}m_{n}&\alpha\sqrt{N-1}\\\
\alpha\sqrt{N-1}&m_{n}+\alpha\sqrt{N-2}\\\ \end{pmatrix}\,.$ (69)

The rest of the orthogonal $N-2$ states decouple.

The resulting disappearance probability is

$P(t)\simeq\frac{4}{N}\sin^{2}\left(\frac{N\alpha t}{2}\right)\,.$ (70)

This picture is very different from the case of a neutron mixing with the KK
tower. The main difference is that the neutron possesses a single oscillation
partner in the form of a state $n_{h}$. For neutrons in all possible energy
states, this partner is unique and is fixed by the theory.

The phenomenology of free neutron oscillations is correspondingly very
different from the KK case, in particular, in the magnetic field dependence of
the resonant amplitude. In case of an oscillation into a single hidden
partner, the amplitude peaks around the resonant value of the magnetic field
$B_{r}\,=\,\alpha\sqrt{N-2}/\mu_{n}$, and diminishes in both directions as a
function of $|B-B_{r}|$.

This behavior goes in sharp contrast with the present scenario in which a
neutron oscillates into the KK tower. As already discussed, in this case, a
neutron of arbitrary energy $m_{n}+\epsilon$ finds its oscillation partner
among the KK states that are closest to it. Because of this effect, the
amplitude is recurrent in $B$ with the steps $\Delta m/\mu_{n}$.

This difference applies to a picture with an arbitrary number of SM copies
including the special case of $N=2$. This case is usually referred to as the
mirror SM. The oscillation between the neutrons of the two copies was studied
earlier in Berezhiani:2005hv . The additional difficulty that arises in this
case is the exact degeneracy of the diagonal masses between $n$ and its mirror
partner $n^{\prime}$. Because of this, the oscillations are suppressed by
potentials arising in our sector due to the environment, such as the earth’s
magnetic field which gives a large level splitting. Therefore, the transition
can take place only if one assumes the presence of an analogous magnetic field
in the hidden sector, which is outside of any experimental or theoretical
control, as it fully depends on the nature of the hidden sector. Regardless of
this issue, assuming the proper conditions, the resonance can only occur for
one specific value of the magnetic field which happens to match the hidden
one. This is different from the extra-dimensional oscillations of neutrons for
which the resonances happen for discrete values of the magnetic field that are
synchronized with the KK spectrum.

##  IX Neutron lifetime measurements

The possibility of free neutron oscillations into extra dimensions also
creates an urgency for more precise measurements of its lifetime. In fact,
some authors have argued that already existing measurements may indicate a
discrepancy that supports the existence of new channels of neutron
disappearance.

On the one hand, we possess the data coming from so-called ultra-cold neutron
lifetime measurements. They account for the missing neutrons from a given
number of free neutrons. The reported lifetime of the neutron is
$\tau_{1}=878s$ UCNt:2021pcg ; Ezhov:2014tna ; ParticleDataGroup:2022pth . On
the other hand, we have the data from beam experiments that look for the
protons resulting from ordinary decay of neutrons caused by the weak
interaction. These experiments report a lifetime of $\tau_{2}=888s$
Yue:2013qrc ; Nico:2004ie ; ParticleDataGroup:2022pth .

This difference between the reported neutron lifetimes prompted some attempts
to address it in the context of oscillations to mirror neutrons
Berezhiani:2005hv ; Berezhiani:2009ldq ; Berezhiani:2018eds ; Tan:2019mrj , or
decays of neutrons into other hypothetical particles Fornal:2018eol  (for some
cosmological implications, see, Karananas:2018goc ). However, doubts about the
rigor of the discrepancy have also been raised Czarnecki:2018okw .

Scrutinizing the validity of the puzzle is beyond the goal of the present
paper. Regardless, the neutron oscillations into extra dimensions provide an
additional motivation for improved precision measurements of the neutron
lifetime.

##  X Baryon and Lepton Numbers

Notice that the mixing of the SM neutrino with a bulk fermion (15) preserves
lepton number symmetry since $\Psi$ can be assigned one unit of lepton number.
Likewise, the mixing of the neutron with the bulk fermion $\Psi$ (25), fixes
the baryonic charge of $\Psi$ as equal to one. In the case where neutrino and
neutron mix with the same $\Psi$, one combination of the baryon and lepton
number is preserved depending on whether both mix with $\Psi$ or with relative
conjugates. Mixing with the same $\Psi$ leaves $B+L$ symmetry unbroken and
breaks $B-L$. Mixing with the conjugates $\Psi$ and $\Psi^{*}$ has the
opposite effect. Notice, as it was discussed in Arkani-Hamed:1998sfv , the
$B-L$ symmetry can be gauged in the bulk, resulting in gravity competing
forces from the exchange of a $B-L$ gauge boson. If $B-L$ is gauged, one of
the mixing operators must be suppressed by the vacuum expectation value of the
field that Higgses it.

We would also like to comment on the case when the bulk particle $\Psi$ that
mixes with neutron possesses a Majorana mass of the form $\mu_{M}\Psi C\Psi$.
This term breaks the baryon number by two units. Due to this, the exchange of
$\Psi$ can result in neutron-anti-neutron oscillations. In this case, the
neutron disappearance rate becomes correlated with the $n-\bar{n}$
oscillation. This case will not be discussed here further.

##  XI Cosmology

We would like to briefly comment on some straightforward cosmological
implications and constraints on the present scenario.

In the large extra-dimensional theory, there exists a standard list of
constraints shared by the bulk species Arkani-Hamed:1998sfv . In particular,
an important bound is coming from Big Bang Nucleosynthesis (BBN). In this
consideration, the key control parameter is the “normalcy temperature”. The
point is that at BBN temperatures the production of bulk species must be
sufficiently suppressed in order not to interfere with the standard evolution
of the Universe. The bulk species must neither enter in thermal equilibrium
with the SM particles nor be overproduced.

In the case of a massless bulk fermion $\Psi$ mixing with the neutrino, such
constraints were discussed in Dvali:1999gf . The bound in the present case is
milder since the neutron mixes with the sector of the KK tower which is much
heavier than the BBN temperature.

In particular, in the parameter regime relevant for free neutron oscillations
into the bulk fermion, the thermal production of $\Psi$ due to the
rescattering of quarks via the four-fermi operator (25) is effectively shut
off below the temperature of its mass $\mu\sim$ GeV.

Besides the constraints, cosmology can provide some strong motivations for the
existence of a neutron portal into extra dimensions in the form of its bulk
partner $\Psi$. The first one is baryogenesis via the mechanism introduced in
Dvali:1999gf . The idea is that the excess of baryonic change in our sector
(i.e., particles inhabiting the SM brane) is generated by its loss into extra
dimensions.

Even though a transition of a neutron into an extra dimension does not violate
baryon number, baryogenesis can still take place, provided the other two
Sakharov’s conditions ($CP$-violation and an out-of-equilibrium state) are
satisfied. This is because the bulk species can transport baryon number away
from our SM sector and “hide” it in the KK states.

In our case, such a process can be realized as an out-of-equilibrium
conversion of our baryons into the fermions $\Psi$. Since $\Psi$ is extremely
weakly interacting, the inverse decays are highly suppressed. The overall
effect can be a generation of net baryon number in our sector. Of course, the
exact amount of the missing baryonic charge is carried by the bulk species.
However, this charge is effectively inaccessible for our measurements. Hence,
the Universe appears to be asymmetric in baryon number.

The second natural implication of the bulk particle $\Psi$ is that its KK
states can play the role of dark matter. The generic feature of bulk modes is
that they interact with SM species with gravitational strength and,
correspondingly, are extremely long-lived. This makes them viable candidates
for dark matter. The possibility that KK gravitons of large extra dimensions
can be dark matter, was originally proposed in Arkani-Hamed:1998sfv  (a more
recent discussion of this idea can be found in Gonzalo:2022jac ).

Similarly, in our scenario, $\Psi$ can be a dark matter candidate. Of course,
the construction of a fully viable and predictive scenario requires a more
detailed cosmological investigation of the parameter space of our setup. This
is beyond the goals of the present paper.

##  XII Conclusions and outlook

In this paper, we pointed out that neutron experiments can provide a unique
spectroscopy of extra dimensions.

Neutrons can play a crucial role in exploring new particle species that
interact with the SM ultra-feebly: with a strength comparable to that of
gravity. Especially motivated are the frameworks in which the number of ultra-
feebly interacting species is large. This is because such frameworks address
the Hierarchy Problem by lowering the cutoff of the theory according to (2).
The two extreme representatives of such theories are the ADD model of large
extra dimensions Arkani-Hamed:1998jmv ; Arkani-Hamed:1998sfv  and the
framework with multiple exact copies of the SM Dvali:2007hz ; Dvali:2007wp ;
Dvali:2009ne . In the latter framework, oscillations of a neutron into hidden
copies of the SM have been studied in Dvali:2009ne . However, oscillations of
a neutron into large extra dimensions have not been studied previously. In the
present paper, we attempted to fill this gap.

Within the ADD framework, we studied the oscillations of a neutron into a
fermion $\Psi$ propagating in large extra dimensions. In particular, $\Psi$
can be the same bulk fermion that endows the SM neutrino with a naturally
small mass via the mechanism of Arkani-Hamed:1998wuz ; Dvali:1999cn . The
consequence of this mechanism is an oscillation of the neutrino into the KK
tower of the bulk sterile fermion Dvali:1999gf .

In the present setup, a similar oscillation into the KK species of bulk
fermion is experienced by the neutron. However, neutron oscillations exhibit
certain features that make them subject to special interest for a wide range
of experiments and phenomenological constraints. A unique feature of a neutron
mixing with a bulk species is that because of the high density and degeneracy
of the KK spectrum, neutrons in various energy states, both free or within
nuclei, find their oscillation partners. The threshold is set by the mass of
the bulk fermion $\Psi$ which is a theory parameter.

For the mass below the energy of a nuclear neutron, the main effect is neutron
disappearance from nuclei resulting in its de-excitation and the emission of a
hard photon. This imposes a variety of bounds on the parameters of the theory.
The associated proton decay rate is suppressed by another nine orders of
magnitude and is a subdominant effect. Therefore, the avenue for tightening
the constraints is via improving the precision of neutron disappearance
experiments as well as in scanning a larger diversity of samples since the
resonant energy levels depend on the features of the Kaluza-Klein spectrum.

For the mass of the bulk partner in a window between the energies of a bound
and a free neutron, the nuclei are stable. However, the resonant transition
can be observed for a free neutron. This gives an exciting possibility of
mapping the Kaluza-Klein spectrum by using a magnetic field as a scanner.
Unlike the scenarios in which the neutron has a single oscillation partner, in
the extra-dimensional case, the resonance takes place recurrently as a
function of a magnetic field. That is, the resonant values of the magnetic
field are quantized in one-to-one correspondence with the KK spectrum.

We analyzed our findings in the light of two recent experiments: the ultra-
cold neutron storage experiment nEDM:2020ekj  and the experiment with an
ultra-cold neutron beam Ban:2023cja . We found that both experiments impose
bounds on the parameters of extra dimensions. In fact, the bounds, such as
(60), (63) and (66), are already within the domain motivated by the Hierarchy
Problem.

Future experiments with improved accuracy and a wider range of the scanned
magnetic field will give a unique possibility of performing precision Kaluza-
Klein spectroscopy in neutron labs.

Acknowledgments — This work was supported in part by the Humboldt Foundation
under the Humboldt Professorship Award, by the European Research Council
Gravities Horizon Grant AO number: 850 173-6, by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2111 - 390814868, Germany’s Excellence Strategy
under Excellence Cluster Origins and the Sonderforschungsbereich SFB1258.

Disclaimer: Funded by the European Union. Views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union or European Research Council. Neither the European Union nor
the granting authority can be held responsible for them.

##  XIII Appendix

###  XIII.1 Mass Splitting in KK Tower for Equal Size Extra Dimensions

We show that for $N\geq 2$ with all radii equal to $R$, the mass splitting of
the KK tower (around the neutron mass) is proportional to

$\delta m\sim\frac{1}{R^{2}m_{n}}.$ (71)

This is correct for both the KK tower originating from a massless or a massive
extra-dimensional fermion.

We estimate $\delta m$ by calculating the mass difference between two special
states $(k+1,\dots,k+1)$ and $(k,\dots,k)$ with masses
$m_{k+1}\equiv\sqrt{N}(k+1)/R$ and $m_{k}\equiv\sqrt{N}k/R$, respectively and
then dividing their mass difference by the number of states in between the
two.

The mass difference between the two special states is

$m_{k+1}-m_{k}=\frac{\sqrt{N}}{R}.$ (72)

Now, we estimate the number of different levels in between them (i.e.,
ignoring the level degeneracies). That is, we want to find the number of
levels that fulfill

$m_{k}<m_{k_{1},...,k_{N}}<m_{k+1}.$ (73)

Using the expression (3) for the mass levels in the KK tower, this reduces to

$Nk^{2}<k_{1}^{2}+\dots+k_{N}^{2}<N(k+1)^{2}.$ (74)

For $N>3$, we can express every integer as a sum of integer squares (This is
actually a theorem: Lagrange theorem, see, e.g., hardy2008introduction .) This
is also very accurate for $N=3$, where we can express almost every integer as
a sum of squares, as well as for $N=2$ up to a log-factor landau1909handbuch .
Therefore, in all cases of our interest, the number of levels is essentially
given by the number of integers in between the mass states

$\\#{\rm mass-levels}=N(k+1)^{2}-Nk^{2}\approx 2Nk.$ (75)

Thus, the mass splitting of the KK tower is

$\delta m=\frac{m_{k+1}-m_{k}}{\\#{\rm mass-
levels}}\approx\frac{1}{2\sqrt{N}Rk}=\frac{1}{2R^{2}m_{k}}.$ (76)

Now set $m_{k}\approx m_{n}$ and we get

$\delta m\approx\frac{1}{2R^{2}m_{n}}.$ (77)

For $N\leq 3$, the above is essentially exact. For $N=2$ the level spacing is
less uniform but is an excellent approximation for averaged splitting.

Indeed, for $N=2$, the number of states in between $m_{k+1}$ and $m_{k}$ is
lower, because we cannot express every integer as a sum of two integer
squares. The number of integers between $0$ and $x$ that can be expressed as a
sum of two integer squares goes as $x/\sqrt{\log x}$ landau1909handbuch .
Hence, the number of integers we are looking for between $x_{1}$ and $x_{2}$
is proportional to

$\frac{x_{1}}{\sqrt{\log x_{1}}}-\frac{x_{2}}{\sqrt{\log x_{2}}}.$ (78)

For $x_{1}=Nk^{2}$ and $x_{2}=N(k+1)^{2}$ and $k\sim 10^{11}$ (remember $k\sim
m_{n}R/\sqrt{2}$), this number is $0.42k\approx k/2$.

Thus, for $N=2$, the mass splitting of the tower around $m_{k}\approx m_{n}$
is

$\delta m\approx\frac{4}{R^{2}m_{n}}.$ (79)

For a massive bulk fermion with mass $\mu$, we repeat the calculation with,

$m_{k}^{2}=\mu^{2}+\frac{k_{1}^{2}}{R_{1}^{2}}+\dots+\frac{k_{N}^{2}}{R_{N}^{2}}.$
(80)

The mass-splitting between the two special states is

$m_{k+1}-m_{k}\approx\frac{Nk}{R^{2}m_{k}}\,.$ (81)

For $N\geq 3$, the number of states in between the two special states is the
same as in the massless case, $\\#{\rm states}\approx 2Nk$. Correspondingly,
so is the level-splitting,

$\delta m\approx\frac{1}{2R^{2}m_{n}}.$ (82)

For $N=2$, the number of states is slightly different due to a smaller gap
$k^{2}\sim(m_{n}^{2}-\mu^{2})R^{2}/2\approx{\rm MeVGeV}R^{2}/2$, which limits
the maximal $k$, approximately by $k\sim 10^{9}$ instead of $k\sim 10^{11}$ of
the massless case. This however is a small difference. Using again Eq. (78),
$\\#{\rm mass-levels}\approx 0.46k\approx k/2$, we also get approximately the
same result as in the massless case, (79).

###  XIII.2 Degeneracy of States

Using the expression of KK masses Eq. (3),

$(mR)^{2}=k_{1}^{2}+\dots+k_{N}^{2},$ (83)

we map the degeneracy count onto a problem in number theory of counting the
number of different possibilities, $r_{N}(n)$, of integers $k_{1},\dots,k_{N}$
that satisfy $n=k_{1}^{2}+\dots+k_{N}^{2}$, with $n=(mR)^{2}$. The averaged
number of possibilities $r_{N}(n)$, will give us the degeneracy of states $Z$.

There is a “cheap” way of getting the answer by making a continuum
approximation. That is, for $R\rightarrow\infty$, the number of states $n(m)$
with mass $\leq m$ is

$n(m)=v_{N}(mR)^{N}\,,$ (84)

where $v_{N}$ is a volume of an unit $N$-ball. The number of states within the
interval of masses between $m+\Delta m$ and $m$ then is

$Z(m)=Nv_{N}\Delta m\,m^{N-1}R^{N}\,,$ (85)

which for $\Delta m=1/(2mR^{2})$ taken from (77) gives,

$Z(m)\approx\frac{N}{2}\,v_{N}(mR)^{N-2}\,.$ (86)

For an alternative count, we start by estimating all possibilities to solve
the equation $k_{1}^{2}+\dots+k_{N}^{2}\leq x$. This can be solved
geometrically, as it is just the volume of a ball,

$\sum_{n=0}^{x}r_{N}(n)=v_{N}x^{\frac{N}{2}}.$ (87)

To estimate the average of $r_{N}$ at some number $x$, we average the $a$
nearest values of $r_{N}$

$\displaystyle Z=$ $\displaystyle\frac{1}{a}(r_{N}(x-a+1)+\dots+r_{N}(x))$
(88) $\displaystyle=$
$\displaystyle\frac{v_{N}}{a}(x^{\frac{N}{2}}-(x-a)^{\frac{N}{2}})$ (89)
$\displaystyle\approx$ $\displaystyle\frac{N}{2}v_{N}x^{\frac{N}{2}-1}.$ (90)

In our case, we replace $x=(mR)^{2}$ and find for the degeneracy of states
given by (86).

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