# A Scale at 10 MeV, Gravitational Topological Vacuum, and Large Extra Dimensions

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School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden

Received: 21 June 2018
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Revised: 15 July 2018
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Accepted: 16 July 2018
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Published: 18 July 2018

(This article belongs to the Special Issue Gravity, Black Holes and Cosmology XXI)

We discuss a possible scale of gravitational origin at around 10 MeV, or ${10}^{-12}$ cm, which arises in the MacDowell–Mansouri formalism of gravity due to the topological Gauss–Bonnet term in the action, as pointed out by Bjorken several years ago. A length scale of the same size emerges also in the Kodama solution in gravity, which is known to be closely related to the MacDowell–Mansouri formulation. We particularly draw attention to the intriguing incident that the existence of six compact extra dimensions originated from TeV-scale quantum gravity as well points to a length scale of ${10}^{-12}$ cm, as the compactification scale. The presence of six such extra dimensions is also in remarkable consistency with the MacDowell–Mansouri formalism; it provides a possible explanation for the factor of ∼${10}^{120}$ multiplying the Gauss–Bonnet term in the action. We also comment on the relevant implications of such a scale regarding the thermal history of the universe motivated by the fact that it is considerably close to 1–2 MeV below which the weak interactions freeze out, leading to Big Bang Nucleosynthesis.

Bjorken points out in Ref. [1] that the MacDowell–Mansouri (MM) formulation of gravity [2] naturally reveals an induced scale of ∼10 MeV, or ∼${10}^{-12}$ cm, which he names after Zeldovich, inspired by Zeldovich’s seminal papers [3,4]. The MM formulation unifies the tetrad and spin connection of the first order Einstein–Cartan formalism, which take values in $SO(3,1)$, into a grand connection that lives in $SO(4,1)$ (or $SO(3,2)$ for a negative cosmological constant). The resulting action, through breaking the $SO(4,1)$ symmetry down to the $SO(3,1)$, yields the usual Einstein–Hilbert term, a cosmological constant, and the Gauss–Bonnet (GB) term, which is topological in four dimensions [5,6,7].

Intriguingly, a length scale of ${10}^{-12}$ cm, as noted in Ref. [8], is also encountered in the context of so-called the Kodama wavefunction in gravity [9,10,11,12,13], analogous to the Chern–Simons solution in Yang–Mills theory in four dimensions, which is also an important element in Loop Quantum Gravity [14,15]. Actually, there is known to be a connection between the inner product of Kodama states and the MM formalism; Ref. [13] points out that the topological terms arising in the (extended) MM action and the inner product are the same.

Bjorken, additionally, suggests six extra spatial dimensions, assumed to be compactified on this induced scale of ${10}^{-12}$ cm, simply to account for the large factor multiplying the MM action [1]; ∼${10}^{120}$, which, quite remarkably, also happens to be the infamous number often encountered in the cosmological constant problem [16,17,18,19].

In this paper, we emphasize that the TeV-scale quantum gravity picture with large extra dimensions (LED) [20,21,22,23,24,25,26] (known as the ADD model) naturally reveals a scale of ${10}^{-12}$ cm as the compactification scale, provided that the number of extra spatial dimensions is set to six, with no need for an ad hoc assumption of the corresponding length scale. In order to be consistent with the known physics up to the TeV-scale, we adopt the well-known approach that only the graviton is allowed to propagate throughout the bulk experiencing the extra dimensions, while the Standard Model (SM) fields are localized to the usual four dimensions. This, in this scenario, would introduce a deviation in the gravitational interactions on scales smaller than ${10}^{-12}$ cm; the gravitational interaction has so far been tested down to the scale of $0.01$ cm [27].

Moreover, we notice a combination of the “Bjorken–Zeldovich (BZ) scale” and the TeV scale, ${M}_{BZ}^{3}/{M}_{EW}^{2}\sim {10}^{-3}$ eV, which is in the order of the observed vacuum energy density in the present universe and in the ballpark of the anticipated neutrino masses [28,29]. Although it is most likely a coincidence, we present several toy models that illustrate its possible role as some sort of a see-saw-type suppression in obtaining the neutrino mass and the cosmological constant.

We also comment on possible other implications in cosmology. This scale is considerably close to 1–2 MeV below which the weak interactions freeze out, leading to Big Bang Nucleosynthesis (BBN). Premised on our current understanding of BBN, it is in general supposed that any deviation from the known radiation density around the decoupling temperature would change the time scale associated with BBN, and it is thus tightly constrained from the observations on the primordial abundances of light elements [30,31,32].1

Bjorken, in Ref. [1], discusses how a scale of ∼10 MeV is induced in the MM formalism through the GB topological term arising naturally in the formalism in addition to the usual Einstein–Hilbert action and a cosmological constant term.

The $SO(3,1)$ MM action, obtained through breaking the $SO(4,1)$ symmetry, is given as [1,2,5,6,7]
where the $\u03f5$ symbols denote Levi–Civita tensors, ${F}_{\mu \nu}^{ab}={R}_{\mu \nu}^{ab}-{H}_{0}^{2}\left({e}_{\mu}^{a}{e}_{\nu}^{b}-{e}_{\nu}^{a}{e}_{\mu}^{b}\right)$, ${R}_{\mu \nu}^{ab}={R}_{\mu \nu}^{\rho \sigma}{e}_{\rho}^{a}{e}_{\sigma}^{b}$ is the Riemann tensor, ${e}_{\mu}^{a}$ is the tetrad (vielbein), and $a,\mu =0,1,2,3$ are the indices of the internal $SO(3,1)$ space and the four dimensional space-time, respectively. ${H}_{0}$ is the Hubble constant.

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}=\frac{{M}_{Pl}^{2}}{64\pi {H}_{0}^{2}}\int {d}^{4}x\sqrt{-g}\phantom{\rule{4pt}{0ex}}\frac{1}{4}\phantom{\rule{0.277778em}{0ex}}{F}_{\mu \nu}^{ab}{F}_{\lambda \sigma}^{cd}\phantom{\rule{0.277778em}{0ex}}{\u03f5}_{abcd}\phantom{\rule{0.277778em}{0ex}}{\u03f5}^{\mu \nu \lambda \sigma}\phantom{\rule{0.277778em}{0ex}},\end{array}$$

Note that ${F}_{\mu \nu}^{ab}$ is the $SO(3,1)$ projection of the curvature ${F}_{\mu \nu}^{AB}$, constructed from the generalized connection ${A}_{\mu}^{AB}$ ($A=0,1,2,3,4$) that lives in a local $SO(4,1)$. ${A}_{\mu}^{AB}$ takes the following form. $\phantom{\rule{0.277778em}{0ex}}{A}_{\mu}^{4a}\equiv {H}_{0}{e}_{\mu}^{a}$ and ${A}_{\mu}^{ab}\equiv {w}_{\mu}^{ab}$, where w is the spin connection that lives in the $SO(3,1)$ group.

The action in Equation (1) yields
where the cosmological constant $\Lambda =3{H}_{0}^{2}$ as it ought to be, and the first two terms are the GB and the Einstein–Hilbert terms, respectively. Notice the factor ∼${10}^{120}$ in front of the GB term in Equation (2), also multiplying the total MM action in Equation (1), which happens to be the infamous number in the cosmological constant problem ($\frac{{M}_{Pl}^{4}}{64{\pi}^{2}{\rho}_{\Lambda}}\sim {10}^{120}$). The possible role of this factor of the MM action in the resolution of the cosmological constant problem has not been demonstrated yet, to the best of our knowledge.

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}=\frac{{M}_{Pl}^{2}}{8\pi}\int {d}^{4}x\sqrt{-g}\left(\frac{1}{32{H}_{0}^{2}}{R}_{\mu \nu}^{\alpha \beta}{R}_{\lambda \sigma}^{\gamma \delta}{\u03f5}_{\alpha \beta \gamma \delta}\phantom{\rule{0.277778em}{0ex}}{\u03f5}^{\mu \nu \lambda \sigma}+\frac{1}{2}R-\Lambda \right)\phantom{\rule{0.277778em}{0ex}},\end{array}$$

In the Friedmann–Robertson–Walker (FRW) background, where $d{s}^{2}=-d{t}^{2}+{a}^{2}\left(t\right)d{x}_{i}d{x}_{i}\phantom{\rule{0.277778em}{0ex}},$ the GB term in Equation (2) becomes
where $V\left(0\right)$ is given through time dependent volume of region of interest dominated by dark energy, $V\left(t\right)=V\left(0\right){a}^{3}=V\left(0\right){e}^{3{H}_{0}t}$. Since in the semiclassical approximation the action is just the phase of the wavefunction, and for a topological term like the GB term the phase takes values in units of $2\pi $, we can write the total amount of the action contributed by the GB term at time t, from Equation (3), as

$${\mathcal{S}}_{GB}=-\frac{{M}_{Pl}^{2}V\left(0\right)}{8\pi {H}_{0}^{2}}{\int}_{0}^{t}dt\frac{d}{dt}{\dot{a}}^{3}\phantom{\rule{0.277778em}{0ex}},$$

$$\begin{array}{c}\hfill |{\mathcal{S}}_{GB}|=\frac{{M}_{Pl}^{2}V\left(0\right){\dot{a}}^{3}}{8\pi {H}_{0}^{2}}\phantom{\rule{0.277778em}{0ex}}{|}_{0}^{t}\equiv 2\pi (N\left(t\right)-N\left(0\right))\phantom{\rule{0.277778em}{0ex}}.\end{array}$$

Then, some sort of number density can be defined as
which is time independent for the cosmological constant dominated space. Bjorken uses the term “darkness” for the quantity $N\left(t\right)$; we prefer to use the “Gauss–Bonnet number”.

$$\begin{array}{c}\hfill n\equiv \frac{N\left(t\right)}{V\left(t\right)}=\frac{{M}_{Pl}^{2}}{16{\pi}^{2}{H}_{0}^{2}}{\left(\frac{\dot{a}}{a}\right)}^{3}=\frac{{H}_{0}{M}_{Pl}^{2}}{16{\pi}^{2}}\equiv {\Lambda}_{BZ}^{3}\phantom{\rule{0.277778em}{0ex}},\end{array}$$

Once we put in the numerical factors, the Bjorken–Zeldovich scale yields

$${\Lambda}_{BZ}\sim 10\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}{l}_{BZ}=\frac{1}{{\Lambda}_{BZ}}\sim 2\times {10}^{-12}\phantom{\rule{0.277778em}{0ex}}\mathrm{cm}.\phantom{\rule{0.277778em}{0ex}}$$

${\Lambda}_{BZ}$ appears to be the scale up to which the MM formalism is valid. Next, we will see how a length scale of the same size comes about as the compactification scale of six extra dimensions originated from TeV-scale gauge-gravity unification. Considering this as the picture above ${\Lambda}_{BZ}$, $N\left(t\right)$ can be interpreted as an effective quantity, revealed below ${\Lambda}_{BZ}$ upon integrated-over extra dimensions. This scenario, as we will see, accurately explains the factor ${10}^{120}$ in the MM action as well.

In this section, we draw attention to an interesting incident regarding the onset of the scale of ${10}^{-12}$ cm from six compact extra (spatial) dimensions originated from TeV-scale gauge-gravity unification. If one imposes gauge-gravity unification at the TeV scale, the weakness of gravitational interactions can be explained via the existence of compact extra dimensions, large compared to the (inverse) TeV-scale [20,21,22,23,24,25,26].

For two test objects placed within a distance $r\gg R$, the gravitational potential becomes $V\left(r\right)\sim \frac{{m}_{1}{m}_{2}}{{M}_{U}^{n+2}{R}^{n}}\frac{1}{r}\phantom{\rule{0.277778em}{0ex}},$ where ${M}_{U}$ is the unification scale of gauge and gravitational interactions, and R is the compactification scale of the extra dimensions. Imposing the requirement to get the right (reduced) Planck mass through the identification ${M}_{U}^{n+2}{R}^{n}={\overline{M}}_{Pl}^{2}$, and assuming ${M}_{U}\sim 1$ TeV, we obtain
where ${l}_{U}$ and ${l}_{P}$ are corresponding length scales for the TeV-scale and (reduced) Planck masses, respectively. As can be seen in Equation (7), for n = 6, we have
a remarkable agreement with the Bjorken–Zeldovich length scale, given in Equation (6), revealed in the MM formalism, discussed previously.

$$R=\frac{{l}_{U}^{1+2/n}}{{l}_{Pl}^{2/n}}\sim 2.0\times {\left(2.4\right)}^{2/n}\times {10}^{30/n-17}\mathrm{cm}\phantom{\rule{0.277778em}{0ex}},$$

$$R\sim 2.7\times {10}^{-12}\mathrm{cm}\sim {l}_{BZ}\phantom{\rule{0.277778em}{0ex}},$$

Remarkably, existence of six extra spatial dimensions compactified on a scale of ${10}^{-12}$ cm in the MM framework, as also noted in Ref. [1], could also explain the factor ${M}_{Pl}^{2}/64\pi {H}_{0}^{2}\phantom{\rule{0.277778em}{0ex}}\sim \phantom{\rule{0.277778em}{0ex}}{10}^{120}$, multiplying both the MM action given in Equation (1) and the GB term in the action given in Equation (2). Extending the internal symmetry of the general MM action from $SO(4,1)$ to $SO(10,1)$, and breaking the symmetry down to $SO(9,1)$, in analogy with Equation (1), the action symbolically becomes
where we suppress the complete version of the tensors, and the indices run over ten values instead of original four. We expect $\langle F\rangle $ to take a value around the order of the (square of the) energy scale that sets the strength for the effective four-dimensional gravity, i.e., the (reduced) Planck mass square, $\langle F\rangle \sim {\overline{M}}_{Pl}^{2}$ (or $\langle F\rangle \sim {\overline{M}}_{Pl}^{2}\sim {M}_{U}^{8}{R}^{6}$ in the context of large extra dimensions picture discussed above). On the other hand, each integrated-over extra dimension contributes a factor in the order of the corresponding length scale, i.e., $\int dy\sim {l}_{BZ}$. Therefore,
which accurately accounts for the factor ${10}^{120}$ in the MM action given in Equation (1).

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}\to \int {d}^{4}x\int \sqrt{-\tilde{g}}\phantom{\rule{0.277778em}{0ex}}d{y}_{1}\dots d{y}_{6}\phantom{\rule{0.277778em}{0ex}}{\left(F\right)}_{\left(i\phantom{\rule{0.277778em}{0ex}}\overline{\mu}\right)}^{5}\xb7{\u03f5}^{\left(i\right)}\xb7{\u03f5}^{\left(\overline{\mu}\right)}\phantom{\rule{0.277778em}{0ex}},\end{array}$$

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{MM}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\to & \underbrace{{({\overline{M}}_{Pl}^{2}{l}_{BZ}^{2})}^{3}}& \int {d}^{4}x\phantom{\rule{0.277778em}{0ex}}\sqrt{-g}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\left(F\right)}_{\left(a\phantom{\rule{0.277778em}{0ex}}\mu \right)}^{2}\xb7{\u03f5}^{\left(a\right)}\xb7{\u03f5}^{\left(\mu \right)}\phantom{\rule{0.277778em}{0ex}},\hfill \\ & \sim {10}^{120}& \end{array}$$

There are two main ways that the extra dimensions, in the context of the ADD model, would appear at the Large Hadron Collider (LHC). The first one is through the direct production of the graviton Kaluza–Klein (KK)-tower states, where the signal would appear as missing energy. The other signature would manifest itself via the exchange of virtual KK gravitons between the SM particles, which would give rise to enhancement in certain cross sections above the SM values [62,63]. Currently at the LHC, the lower limit for the gauge-gravity scale in the ADD model with six extra dimensions is set as ${M}_{U}>2.6$ TeV with 95% confidence level (CL) by the CMS experiment [64,65,66], which translates into an upper bound on the corresponding length scale as $R<0.8\times {10}^{-12}$ cm. This is still in the vicinity of the Bjorken–Zeldovich scale within an order of magnitude. Then, one wonders how robust the numerical agreement, given in Equation (8), is against the value of ${M}_{U}$. As can be seen in Equation (7), the outcome has some sensitivity against the value of ${M}_{U}$. Nevertheless, with a value of ${M}_{U}$ up to around 10 TeV, we still get a required length scale up to an order of magnitude, which is generally acceptable when a scale is under discussion, as displayed in Figure 1. If, for instance, we take ${M}_{U}=5$ TeV, the corresponding length scale becomes $R=0.3\times {10}^{-12}\mathrm{cm}$; or, for ${M}_{U}=10$ TeV, we have $R=0.1\times {10}^{-12}\mathrm{cm}\cong {d}_{\mathrm{proton}}$.

In the effort to understand the smallness of the cosmological constant, several numerical relations among the energy scales have been noticed (or proposed) in the literature that mimic a see-saw-type suppression mechanism [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82].

In the case of the existence of an energy scale of ∼10 MeV, the relevant combination we notice is ${\rho}_{\Lambda}^{1/4}\stackrel{?}{=}{M}_{BZ}^{3}/{M}_{EW}^{2}\sim {10}^{-3}\mathrm{eV}\phantom{\rule{0.277778em}{0ex}},$ where ${M}_{EW}\sim 1$ TeV. It is not straightforward to devise a realistic model yielding such a relation, since this requires a contribution in the amount of ${M}_{BZ}^{12}/{M}_{EW}^{8}$ in Lagrangian. Nevertheless, this type of term in the context of vacuum energy density contributions may be obtained in models where the cosmological constant problem is addressed by entertaining the possibility that the universe may be stuck in a false vacuum, split from the vanishing global vacuum in the amount of the cosmological constant [75,83].

When a new scale is under discussion, another question that comes to mind is the possibility of a (some sort of) see-saw mechanism that utilizes a relevant combination of the scales in the theory to explain the smallness of the neutrino mass. The relation ${M}_{BZ}^{3}/{M}_{EW}^{2}\sim {10}^{-3}$ eV is intriguing from this point of view as well, since it is in the vicinity of (at least one of the) the neutrino masses. Next, we will work on a hypothetical scenario just as an illustration of obtaining this combination in a model.

Consider a hidden sector with a quantum-chromodynamics (QCD)-like gauge interaction, where the symmetry group is ${\mathcal{G}}_{h}\equiv SU\left(N\right)$. Besides the corresponding gauge bosons, consider a real scalar field $\varphi $ and a Dirac fermion F (for each family), both of which transform in some representation of ${\mathcal{G}}_{h}$, where the fermion does so vectorially (non-chirally). We assume that F has a confining scale of order 10 MeV and the SM fields are not charged under ${\mathcal{G}}_{h}$. The SM connects to the hidden sector through a portal coupling between the Higgs and the scalar $\varphi $. We also assume a discrete ${\tilde{Z}}_{2}$ symmetry that transforms F, $\varphi $, and the neutrinos in the following way. ${F}_{L\left(R\right)}\to \pm {F}_{L\left(R\right)}$, ${\nu}_{L\left(R\right)}\to \pm {\nu}_{L\left(R\right)}$, and $\varphi \to -\varphi $. Therefore, there are no mass or Yukawa-type terms (via the Standard Model Higgs) allowed at the tree level. The Yukawa-type terms involving the scalar $\varphi $ do not contribute as mass terms at tree-level either, since we assume that $\langle \varphi \rangle =0$ so that the $SU\left(N\right)$ symmetry remains unbroken.

However, a mass term can be induced through the effective operator ${\mathcal{L}}^{eff}\supset \frac{{c}_{\nu}}{{\Lambda}^{2}}\overline{\nu}\nu \overline{F}F\phantom{\rule{0.277778em}{0ex}},$ which is induced by integrating out the scalar field. We assume that a condensate forms, due to the possible nonperturbative characteristic of the $SU\left(N\right)$ vacuum, at ∼${M}_{BZ}$, breaking the ${\tilde{Z}}_{2}$ symmetry; $\langle \overline{F}F\rangle \sim {f}^{3}\sim {(a\xb7{M}_{BZ})}^{3}$, $\Lambda \sim {m}_{\varphi}\sim {\Lambda}_{EW}\sim 1$ TeV. The scalar $\varphi $ gets its mass via the portal coupling to the SM Higgs, i.e., $\lambda {\varphi}^{2}{H}^{\u2020}H$, which justifies its electroweak-scale mass. The extra fermion F acquires its mass via coupling to the condensate through the corresponding dimension-6 operator that yields a mass value on the order of the neutrino mass. The effective neutrino mass in this scenario becomes ${m}_{\nu}=\left({c}_{\nu}{a}^{3}\right)\phantom{\rule{0.277778em}{0ex}}{M}_{BZ}^{3}/{M}_{EW}^{2}\lesssim {10}^{-2}\mathrm{eV}\phantom{\rule{0.277778em}{0ex}},$ provided that ${c}_{\nu}{a}^{3}\lesssim 10$.

A scale around 10 MeV might be relevant also in terms of the thermal history of the universe. It is an energy scale considerably close to $T\sim 1-2$ MeV below which the weak interactions freeze out; the reaction rate $\Gamma \sim {G}_{F}^{2}{T}^{5}$ drops below the expansion rate $H\sim \sqrt{{g}^{*}}{T}^{2}/{M}_{Pl}$, where ${g}^{*}$ is given as ${g}^{*}={g}_{b}+(7/8){g}_{f}$ and ${g}_{b}\phantom{\rule{0.277778em}{0ex}}\left({g}_{f}\right)$ denotes the total number of the effective bosonic (fermionic) degrees of freedom at around the background temperature T. Consequently, primordial neutrinos and possibly cold dark matter—if it exists—decouple from the rest of the matter, and the ratio of neutrons to protons freezes out. Any increase from the known radiation density would bring forward the Big Bang Nucleosynthesis (BBN) and hence would cause a larger Helium abundance in the present universe [30,31,32]. Therefore, if there is some unrevealed physics associated with such a scale of 10 MeV, they may have direct implications on our understanding of BBN, which is consistent with the current observations on the primordial abundances of light elements.

Since the effective MM action in 4D reveals the Einstein gravity with a cosmological constant and the GB term that does not have any effects in the equations of motion in 4D, the formalism at first sight only defines the graviton. This seemingly does not cause any problem in terms BBN since the gravitational interaction rate, as well known, is significantly suppressed compared to the expansion rate, i.e., $\Gamma \sim {G}_{P}^{2}{T}^{5}\ll H$, at $T\sim 1$ MeV. However, this is the case only if there is no any other relevant degrees of freedom obtained from the original action based on $SO(4,1)$, in addition to the terms given in Equation (2). Recall that the generalized connection ${A}_{\mu}^{AB}$ living in $SO(4,1)$ has 40 components. As also mentioned in Ref. [1], one may wonder whether some of these degrees of freedom can be identified with the (bosonic) degrees of freedom of the SM2. Then, several leftover terms may possibly define additional light degrees of freedom. One may expect at first that the relevant interactions are supposed to be suppressed, similar to the case with gravitons. However, one should not forget the enormous factor of ${10}^{120}$ multiplying the action in 4D, possibly arising due to being integrating over extra dimensions. If such identifications related to the SM are possible, then it is probably because of this large factor, and the same factor may amplify some interactions regarding these new light degrees of freedom, making them interact frequently enough to be in equilibrium at $T\sim 1$ MeV. Then, the model would be in tension with the constraints coming from BBN.

In this paper, we aim to bring attention to the possibility of a gravitational scale at around ∼10 MeV, or ${10}^{-12}$ cm, induced in the (MM) extension to the Einstein–Cartan formalism in 4D, due to the topological GB term in the action, as suggested by Bjorken [1].

First, we point out that a scale of the same size, ${10}^{-12}$ cm naturally comes about in the context of large extra dimensions, originated from TeV-scale quantum gravity, as the compactification scale, if the number of extra spatial dimensions is set to six.3 Apparently, these two approaches can be combined, where the four-dimensional MM formalism is the effective theory after the six extra dimensions are integrated over, which also explains the factor ${10}^{120}$ in the MM action. Second, we discuss that existence of such a scale may play a role in the smallness of the cosmological constant and the neutrino mass; to this end, we refer to some toy models as illustrations that generate a seesaw-type suppression mechanism. Note that we do not claim in this paper that this mechanism can directly be accommodated into the MM formalism. Instead, the main intention of this paper should be taken as an attempt to point out several coincidences regarding a scale at around 10 MeV, or ${10}^{-12}$ cm; the possibility that they are non-accidental deserves attention. Finally, we comment on possible implications in cosmology in the context of Big Bang Nucleosynthesis.

We note that if Nature contains six extra dimensions with a compactification scale of ${10}^{-12}$ cm, it raises the question why no Kaluza–Klein excitations with masses with a starting value of ∼10 MeV have been observed so far. However, their elusiveness would not be unanticipated since these modes are expected to be relatively suppressed [100].

Currently at the LHC, the ADD model with six extra dimensions is excluded with 95% CL for values ${M}_{U}\u2a7d2.6$ TeV by the CMS experiment [64,65,66], equivalent to an upper bound on the corresponding compactification length scale, $R<0.8\times {10}^{-12}$ cm, which is in the ballpark of the Bjorken–Zeldovich scale within an order of magnitude. By the time the LHC searches are completed, we will have a compelling answer on the TeV-scale ADD model and hence on the MM-LED picture discussed in this paper. Note that a negative result along these lines does not necessarily invalidate Bjorken’s original proposal in Ref. [1], which is not obliged to connect to the TeV-scale ADD model, and yet which can still include six compact extra dimensions. The most definitive answer regarding Bjorken’s proposal will come from the gravitational inverse-square-law experiments. The current upper bound for the size of such extra dimensions, through the deviation in the effective gravitational interaction, is $0.01$ cm [27].

This work is supported in parts by the National Natural Science Foundation of China (NSFC) under Grant No. 11505067 and the Swedish Research Council under contract 621-2011-5107.

We would like to thank James D. Bjorken and Djordje Minic for their comments and suggestions regarding the manuscript.

The author declares no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

- Bjorken, J. Darkness: What comprises empty space? Annalen der Physik
**2013**, 525, A67–A79. [Google Scholar] [CrossRef] - MacDowell, S.W.; Mansouri, F. Unified Geometric Theory of Gravity and Supergravity. Phys. Rev. Lett.
**1977**, 38, 739. [Google Scholar] [CrossRef] - Zeldovich, Y.B. Cosmological Constant and Elementary Particles. JETP Lett.
**1967**, 6, 316–317. [Google Scholar] [CrossRef] - Zel’dovich, Y.B.; Krasinski, A. The Cosmological constant and the theory of elementary particles. Sov. Phys. Uspekhi
**1968**, 11, 381–393. [Google Scholar] [CrossRef] - Freidel, L.; Starodubtsev, A. Quantum gravity in terms of topological observables. arXiv, 2005; arXiv:hep-th/0501191. [Google Scholar]
- Wise, D.K. MacDowell–Mansouri gravity and Cartan geometry. Class. Quantum Gravity
**2010**, 27, 155010. [Google Scholar] [CrossRef][Green Version] - Wise, D.K. Symmetric space Cartan connections and gravity in three and four dimensions. Symmetry Integr. Geom.
**2009**, 5, 080. [Google Scholar] [CrossRef] - Randono, A. A Mesoscopic Quantum Gravity Effect. Gen. Relativ. Gravit.
**2010**, 42, 1909–1917. [Google Scholar] [CrossRef] - Kodama, H. Specialization of Ashtekar’s Formalism to Bianchi Cosmology. Prog. Theor. Phys.
**1988**, 80, 1024. [Google Scholar] [CrossRef] - Smolin, L. Quantum gravity with a positive cosmological constant. arXiv, 2002; arXiv:hep-th/0209079. [Google Scholar]
- Witten, E. A Note on the Chern–Simons and Kodama wave functions. arXiv, 2003; arXiv:gr-qc/0306083. [Google Scholar]
- Randono, A. Generalizing the Kodama state. I. Construction. arXiv, 2006; arXiv:gr-qc/0611073. [Google Scholar]
- Randono, A. Generalizing the Kodama state. II. Properties and physical interpretation. arXiv, 2006; arXiv:gr-qc/0611074. [Google Scholar]
- Rovelli, C. Loop quantum gravity: The first twenty five years. Class. Quantum Gravity
**2011**, 28, 153002. [Google Scholar] [CrossRef] - Ashtekar, A.; Pullin, J. The Overview Chapter in Loop Quantum Gravity: The First 30 Years. arXiv, 2017; arXiv:gr-qc/1703.07396. [Google Scholar]
- Weinberg, S. The Cosmological Constant Problem. Rev. Mod. Phys.
**1989**, 61, 1–23. [Google Scholar] [CrossRef] - Carroll, S.M. The Cosmological constant. Living Rev. Relat.
**2001**, 4, 1. [Google Scholar] [CrossRef] [PubMed] - Polchinski, J. The Cosmological Constant and the String Landscape. In The Quantum Structure of Space and Time: Proceedings of the 23rd Solvay Conference on Physics, Brussels, Belgium, 1–3 December 2005; Cornell University Library: Ithaca, NY, USA, 2006; pp. 216–236. [Google Scholar]
- Padilla, A. Lectures on the Cosmological Constant Problem. arXiv, 2015; arXiv:hep-th/1502.05296. [Google Scholar]
- Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. The Hierarchy problem and new dimensions at a millimeter. Phys. Lett. B
**1998**, 429, 263–272. [Google Scholar] [CrossRef] - Antoniadis, I.; Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. New dimensions at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett. B
**1998**, 436, 257–263. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity. Phys. Rev. D
**1999**, 59, 086004. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Dimopoulos, S.; Kaloper, N.; Sundrum, R. A Small cosmological constant from a large extra dimension. Phys. Lett. B
**2000**, 480, 193–199. [Google Scholar] [CrossRef] - Antoniadis, I.; Dimopoulos, S.; Dvali, G.R. Millimeter range forces in superstring theories with weak scale compactification. Nucl. Phys. B
**1998**, 516, 70–82. [Google Scholar] [CrossRef] - Accomando, E.; Antoniadis, I.; Benakli, K. Looking for TeV scale strings and extra dimensions. Nucl. Phys. B
**2000**, 579, 3–16. [Google Scholar] [CrossRef] - Antoniadis, I.; Benakli, K.; Quiros, M. Direct collider signatures of large extra dimensions. Phys. Lett. B
**1999**, 460, 176–183. [Google Scholar] [CrossRef][Green Version] - Kapner, D.J.; Cook, T.S.; Adelberger, E.G.; Gundlach, J.H.; Heckel, B.R.; Hoyle, C.D.; Swanson, H.E. Tests of the gravitational inverse-square law below the dark-energy length scale. Phys. Rev. Lett.
**2007**, 98, 021101. [Google Scholar] [CrossRef] [PubMed] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; Bartolo, N.; et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**2016**, 594, A13. [Google Scholar] - Patrignani, C.; et al. [Particle Data Group]. Review of Particle Physics. Chin. Phys. C
**2016**, 40, 100001. [Google Scholar] - Carroll, S.M. Spacetime and Geometry: An Introduction to General Relativity; Pearson Education Inc.: London, UK, 2004. [Google Scholar]
- Weinberg, S. Cosmology; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Langacker, P. The Standard Model and Beyond; CRC Press: Boca Raton, FL, USA, 2010. [Google Scholar]
- Krasznahorkay, A.J.; Csatlós, M.; Csige, L.; Gácsi, Z.; Gulyás, J.; Hunyadi, M.; Kuti, I.; Nyakó, B.M.; Stuhl, L.; Timár, J.; et al. Observation of Anomalous Internal Pair Creation in
^{8}Be: A Possible Indication of a Light, Neutral Boson. Phys. Rev. Lett.**2016**, 116, 042501. [Google Scholar] [CrossRef] [PubMed] - De Boer, F.W.N.; Fröhlich, O.; Stiebing, K.E.; Bethge, K.; Bokemeyer, H.; Balanda, A.; Buda, A.; Van Dantzig, R.; Elze, T.W.; Folger, H.; et al. A deviation in internal pair conversion. Phys. Lett. B
**1996**, 388, 235–240. [Google Scholar] [CrossRef][Green Version] - De Boer, F.W.N.; van Dantzig, R.; van Klinken, J.; Bethge, K.; Bokemeyer, H.; Buda, A.; Muller, K.A.; Stiebing, K.E. Excess in e
^{+}e^{−}pairs near 9 MeV invariant mass. J. Phys.**1997**, G23, L85–L96. [Google Scholar] [CrossRef] - De Boer, F.W.N.; Bethge, K.; Bokemeyer, H.; van Dantzig, R.; van Klinken, J.; Mironov, V.; Muller, K.A.; Stiebing, K.E. Further search for a neutral boson with a mass around 9-MeV/c2. J. Phys.
**2001**, G27, L29. [Google Scholar] [CrossRef] - De Boer, F. Anomalous internal pair conversion signaling elusive light neutral particles. AIP Conf. Proc.
**2006**, 802, 146–152. [Google Scholar] - Krasznahorkay, A.; de Boer, F.W.N.; Csatlós, M.; Csige, L.; Gácsi, Z.; Gulyás, J.; Hunyadi, M.; Ketel, T.J.; van Klinken, J.; Krasznahorkay, A., Jr.; et al. e
^{+}e^{−}pairs from a nuclear transition signaling an elusive light neutral boson. AIP Conf. Proc.**2005**, 802, 236. [Google Scholar] - Krasznahorkay, A.; Gacsi, Z.; Ketel, T.J.; van Klinken, J.; de Boer, F.W.N.; Gulyás, J.; Csige, L.; Krasznahorkay, A. Jr.; Hunyadi, M.; Vitez, A.; et al. Lepton pairs from a forbidden M0 transition: Signaling an elusive light neutral boson? Acta Phys. Polon. B
**2006**, 37, 239–244. [Google Scholar] - De Boer, F.W.N.; Fields, C.A. A Re-evaluation of Evidence for Light Neutral Bosons in Nuclear Emulsions. Int. J. Mod. Phys. E
**2011**, 20, 1787–1803. [Google Scholar] [CrossRef] - Wojtsekhowski, B.; Nikolenko, D.; Rachek, I. Searching for a new force at VEPP-3. arXiv, 2012; arXiv:hep-ex/1207.5089. [Google Scholar]
- Gulyas, J.; Ketel, T.J.; Krasznahorkay, A.J.; Csatlos, M.; Csige, L.; Gacsi, Z.; Hunyadi, M.; Krasznahorkay, A.; Vitez, A.; Tornyi, T.G. A pair spectrometer for measuring multipolarities of energetic nuclear transitions. Nucl. Instrum. Method.
**2016**, A808, 21–28. [Google Scholar] [CrossRef] - Feng, J.L.; Fornal, B.; Galon, I.; Gardner, S.; Smolinsky, J.; Tait, T.M.P.; Tanedo, P. Protophobic Fifth-Force Interpretation of the Observed Anomaly in
^{8}Be Nuclear Transitions. Phys. Rev. Lett.**2016**, 117, 071803. [Google Scholar] [CrossRef] [PubMed] - Feng, J.L.; Fornal, B.; Galon, I.; Gardner, S.; Smolinsky, J.; Tait, T.M.P.; Tanedo, P. Particle Physics Models for the 17 MeV Anomaly in Beryllium Nuclear Decays. arXiv, 2016; arXiv:hep-ph/1608.03591. [Google Scholar] [CrossRef]
- Gninenko, S.N.; Krasnikov, N.V.; Kirsanov, M.M.; Kirpichnikov, D.V. Missing energy signature from invisible decays of dark photons at the CERN SPS. Phys. Rev. D
**2016**, 94, 095025. [Google Scholar] [CrossRef][Green Version] - Gu, P.H.; He, X.G. Realistic model for a fifth force explaining anomaly in
_{8}Be_{∗}→_{8}Be e^{+}e^{−}Decay. Nucl. Phys. B**2017**, 919, 209–217. [Google Scholar] [CrossRef] - Jia, L.B.; Li, X.Q. The new interaction suggested by the anomalous
^{8}Be transition sets a rigorous constraint on the mass range of dark matter. Eur. Phys. J. C**2016**, 76, 706. [Google Scholar] [CrossRef] - Kitahara, T.; Yamamoto, Y. Protophobic Light Vector Boson as a Mediator to the Dark Sector. Phys. Rev. D
**2017**, 95, 015008. [Google Scholar] [CrossRef] - Ellwanger, U.; Moretti, S. Possible Explanation of the Electron Positron Anomaly at 17 MeV in
^{8}Be Transitions Through a Light Pseudoscalar. J. High Energy Phys.**2016**, 2016, 039. [Google Scholar] [CrossRef] - Chen, C.S.; Lin, G.L.; Lin, Y.H.; Xu, F. The 17 MeV Anomaly in Beryllium Decays and U(1) Portal to Dark Matter. arXiv, 2016; arXiv:hep-ph/1609.07198. [Google Scholar]
- Neves, M.J.; Helayel-Neto, J.A. TeV- and MeV-physics out of an SU(2) × U(1) × U(1) model. arXiv, 2016; arXiv:hep-ph/1609.08471. [Google Scholar]
- Kahn, Y.; Krnjaic, G.; Mishra-Sharma, S.; Tait, T.M.P. Light Weakly Coupled Axial Forces: Models, Constraints, and Projections. arXiv, 2016; arXiv:hep-ph/1609.09072. [Google Scholar] [CrossRef]
- Fayet, P. The light U boson as the mediator of a new force, coupled to a combination of Q, B, L and dark matter. Eur. Phys. J. C
**2017**, 77, 53. [Google Scholar] [CrossRef] - Neves, M.J.; Helayel-Neto, J.A. A Unified Hidden-Sector-Electroweak Model, Paraphotons and the X-Boson. arXiv, 2016; arXiv:hep-ph/1611.07974. [Google Scholar]
- Kozaczuk, J.; Morrissey, D.E.; Stroberg, S.R. Light Axial Vectors, Nuclear Transitions, and the
^{8}Be Anomaly. arXiv, 2016; arXiv:hep-ph/1612.01525. [Google Scholar] - Chiang, C.W.; Tseng, P.Y. Probing a dark photon using rare leptonic kaon and pion decays. Phys. Lett. B
**2017**, 767, 289–294. [Google Scholar] [CrossRef] - Krasnikov, N.V. The muon (g − 2) anomaly and a new light vector boson. arXiv, 2017; arXiv:hep-ph/1702.04596. [Google Scholar]
- Araki, T.; Hoshino, S.; Ota, T.; Sato, J.; Shimomura, T. Detecting the L
_{μ}− L_{τ}gauge boson at Belle II. Phys. Rev. D**2017**, 95, 055006. [Google Scholar] [CrossRef] - Benavides, R.; Munoz, L.A.; Ponce, W.A.; Rodriguez, O.; Rojas, E. Minimal non-universal EW extensions of the Standard Model: A chiral multi-parameter solution. arXiv, 2016; arXiv:hep-ph/1612.07660. [Google Scholar]
- Neves, M.J. The protophobic X-boson unified to the quantum electrodynamics. arXiv, 2017; arXiv:hep-ph/1704.02491. [Google Scholar]
- Delle Rose, L.; Khalil, S.; Moretti, S. Explanation of the 17 MeV Atomki Anomaly in a U(1)
^{′}-Extended 2-Higgs Doublet Model. arXiv, 2017; arXiv:hep-ph/1704.03436. [Google Scholar] [CrossRef] - Csaki, C. TASI lectures on extra dimensions and branes. From fields to strings: Circumnavigating theoretical physics. Ian Kogan memorial collection (3 volume set). arXiv, 2004; arXiv:hep-ph/0404096. [Google Scholar]
- Rizzo, T.G. Introduction to Extra Dimensions. AIP Conf. Proc.
**2010**, 1256, 27–50. [Google Scholar][Green Version] - Roy, A. Search for Dark Matter and Large Extra Dimensions in the Photon + MET Final State in pp Collisions at $\sqrt{s}$ = 13 TeV. Springer Proc. Phys.
**2018**, 203, 205–208. [Google Scholar] - Ghosh, S. Large Extra Dimensions Search in the Photon + MET Final State in pp Collisions at $\sqrt{s}$ = 13 TeV at CMS in LHC. Springer Proc. Phys.
**2018**, 203, 745–747. [Google Scholar] - Sirunyan, A.M.; et al. [The CMS Collaboration]. Search for new physics in the monophoton final state in proton-proton collisions at $\sqrt{s}$ = 13 TeV. J. High Energy Phys.
**2017**, 2017, 73. [Google Scholar] [CrossRef] - Banks, T. SUSY breaking, cosmology, vacuum selection and the cosmological constant in string theory. In Proceedings of the ITP Workshop on SUSY Phenomena and SUSY GUTS Santa Barbara, California, CA, USA, 7–9 December 1995. [Google Scholar]
- Cohen, A.G.; Kaplan, D.B.; Nelson, A.E. Effective field theory, black holes, and the cosmological constant. Phys. Rev. Lett.
**1999**, 82, 4971–4974. [Google Scholar] [CrossRef] - Kim, J.E. Model dependent axion as quintessence with almost massless hidden sector quarks. J. High Energy Phys.
**2000**, 2000, 016. [Google Scholar] [CrossRef] - Kiritsis, E. Supergravity, D-brane probes and thermal superYang–Mills: A Comparison. J. High Energy Phys.
**1999**, 1999, 010. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Hall, L.J.; Kolda, C.F.; Murayama, H. A New perspective on cosmic coincidence problems. Phys. Rev. Lett.
**2000**, 85, 4434–4437. [Google Scholar] [CrossRef] [PubMed] - Banks, T. Cosmological breaking of supersymmetry? Int. J. Mod. Phys. A
**2001**, 16, 910–921. [Google Scholar] [CrossRef] - Chang, L.N.; Minic, D.; Okamura, N.; Takeuchi, T. The Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem. Phys. Rev. D
**2002**, 65, 125028. [Google Scholar] [CrossRef] - Chang, L.N.; Minic, D.; Okamura, N.; Takeuchi, T. Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations. Phys. Rev. D
**2002**, 65, 125027. [Google Scholar] [CrossRef] - Barr, S.M.; Seckel, D. The Cosmological constant, false vacua, and axions. Phys. Rev. D
**2001**, 64, 123513. [Google Scholar] [CrossRef] - Berglund, P.; Hubsch, T.; Minic, D. Relating the cosmological constant and supersymmetry breaking in warped compactifications of IIB string theory. Phys. Rev. D
**2003**, 67, 041901. [Google Scholar] [CrossRef] - Hsu, S.D.H.; Zee, A. A Speculative relation between the cosmological constant and the Planck mass. Mod. Phys. Lett. A
**2005**, 20, 2699–2704. [Google Scholar] [CrossRef] - Urban, F.R.; Zhitnitsky, A.R. The Cosmological constant from the ghost: A Toy model. Phys. Rev. D
**2009**, 80, 063001. [Google Scholar] [CrossRef] - Urban, F.R.; Zhitnitsky, A.R. The cosmological constant from the QCD Veneziano ghost. Phys. Lett. B
**2010**, 688, 9–12. [Google Scholar] [CrossRef] - Urban, F.R.; Zhitnitsky, A.R. The QCD nature of Dark Energy. Nucl. Phys. B
**2010**, 835, 135–173. [Google Scholar] [CrossRef] - Chang, L.N.; Minic, D.; Takeuchi, T. Quantum Gravity, Dynamical Energy-Momentum Space and Vacuum Energy. Mod. Phys. Lett. A
**2010**, 25, 2947–2954. [Google Scholar] [CrossRef] - Chang, L.N.; Lewis, Z.; Minic, D.; Takeuchi, T. On the Minimal Length Uncertainty Relation and the Foundations of String Theory. Adv. High Energy Phys.
**2011**, 2011, 493514. [Google Scholar] [CrossRef] - Garretson, W.D.; Carlson, E.D. Could there be something rather than nothing? Phys. Lett. B
**1993**, 315, 232–238. [Google Scholar] [CrossRef][Green Version] - Lisi, A.G. An Exceptionally Simple Theory of Everything. arXiv, 2007; arXiv:hep-th/0711.0770. [Google Scholar]
- Nesti, F.; Percacci, R. Chirality in unified theories of gravity. Phys. Rev. D
**2010**, 81, 025010. [Google Scholar] [CrossRef] - Lisi, A.G.; Smolin, L.; Speziale, S. Unification of gravity, gauge fields, and Higgs bosons. J. Phys. A
**2010**, 43, 445401–445408. [Google Scholar] [CrossRef] - Ne’eman, Y. Irreducible Gauge Theory of a Consolidated Weinberg-Salam Model. Phys. Lett. B
**1979**, 81, 190–194. [Google Scholar] [CrossRef] - Fairlie, D.B. Higgs’ Fields and the Determination of the Weinberg Angle. Phys. Lett. B
**1979**, 82, 97–100. [Google Scholar] [CrossRef] - Fairlie, D.B. Two Consistent Calculations of the Weinberg Angle. J. Phys. G
**1979**, 5, L55. [Google Scholar] [CrossRef] - Ne’eman, Y.; Sternberg, S. Superconnections and internal supersymmetry dynamics. Proc. Natl. Acad. Sci. USA
**1990**, 87, 7875–7877. [Google Scholar] [CrossRef] [PubMed] - Ne’eman, Y.; Sternberg, S.; Fairlie, D. Superconnections for electroweak su(2/1) and extensions, and the mass of the Higgs. Phys. Rep.
**2005**, 406, 303–377. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Connes, A. Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I. Fortsch. Phys.
**2010**, 58, 553–600. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Connes, A. Resilience of the Spectral Standard Model. J. High Energy Phys.
**2012**, 2012, 104. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Connes, A.; van Suijlekom, W.D. Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification. J. High Energy Phys.
**2013**, 2013, 132. [Google Scholar] [CrossRef] - Aydemir, U.; Minic, D.; Takeuchi, T. The Higgs Mass and the Emergence of New Physics. Phys. Lett. B
**2013**, 724, 301–305. [Google Scholar] [CrossRef] - Aydemir, U.; Minic, D.; Sun, C.; Takeuchi, T. Higgs mass, superconnections, and the TeV-scale left-right symmetric model. Phys. Rev. D
**2015**, 91, 045020. [Google Scholar] [CrossRef] - Schwarz, J.H. Physical States and Pomeron Poles in the Dual Pion Model. Nucl. Phys. B
**1972**, 46, 61–74. [Google Scholar] [CrossRef] - Scherk, J.; Schwarz, J.H. Dual Field Theory of Quarks and Gluons. Phys. Lett. B
**1975**, 57, 463–466. [Google Scholar] [CrossRef] - Candelas, P.; Horowitz, G.T.; Strominger, A.; Witten, E. Vacuum Configurations for Superstrings. Nucl. Phys. B
**1985**, 258, 46–74. [Google Scholar] [CrossRef] - Han, T.; Lykken, J.D.; Zhang, R.J. On Kaluza–Klein states from large extra dimensions. Phys. Rev. D
**1999**, 59, 105006. [Google Scholar] [CrossRef]

1 | Recently, the Atomki group in Hungary reported an anomaly in the ${}^{8}$Be nuclear decay by internal ${e}^{+}{e}^{-}$ formation at an invariant mass ${m}_{\overline{e}e}\cong 17$ MeV, with a statistical significance of 6.8$\sigma $ [33]. See also Refs. [34,35,36,37,38,39,40,41,42] for the previous studies relevant to this observation. The observation has ignited interest in the high energy physics community to suggest explanations, some of which consider a hidden sector at around this energy scale whose effects have so far remained unnoticed [43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]. |

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