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

Abstract

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.

1. Introduction

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

2. The MacDowell–Mansouri Formalism and the Bjorken–Zeldovich Scale

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]

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}=\frac{M_{Pl}^2}{64\pi H_0^2}\int d^{4}x\sqrt{-g}\frac{1}{4}{F}_{\mu \nu }^{ab}{F}_{\lambda \sigma }^{cd}{ϵ}_{abcd}{ϵ}^{\mu \nu \lambda \sigma },\end{array}$$

(1)

where the $ϵ$ 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.

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. $A_{\mu }^{4a}\equiv H_0{e}_{\mu }^a$ 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

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}=\frac{M_{Pl}^2}{8\pi }\int d^{4}x\sqrt{-g}\left(\frac{1}{32H_0^2}{R}_{\mu \nu }^{\alpha \beta }{R}_{\lambda \sigma }^{\gamma \delta }{ϵ}_{\alpha \beta \gamma \delta }{ϵ}^{\mu \nu \lambda \sigma }+\frac{1}{2}R-\Lambda \right),\end{array}$$

(2)

where the cosmological constant $\Lambda =3H_0^2$ 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.

In the Friedmann–Robertson–Walker (FRW) background, where $d{s}^2=-d{t}^2+a^2\left(t\right)d{x}_{i}d{x}_i,$ the GB term in Equation (2) becomes

$${\mathcal{S}}_{GB}=-\frac{M_{Pl}^{2}V\left(0\right)}{8\pi H_0^2}{\int }_0^{t}dt\frac{d}{dt}{\dot{a}}^3,$$

(3)

where $V\left(0\right)$ is given through time dependent volume of region of interest dominated by dark energy, $V\left(t\right)=V\left(0\right)a^3=V\left(0\right)e^{3H_{0}t}$. 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

$$\begin{array}{c}\hfill |{\mathcal{S}}_{GB}|=\frac{M_{Pl}^{2}V\left(0\right){\dot{a}}^3}{8\pi H_0^2}{|}_0^t\equiv 2\pi (N\left(t\right)-N\left(0\right)).\end{array}$$

(4)

Then, some sort of number density can be defined as

$$\begin{array}{c}\hfill n\equiv \frac{N\left(t\right)}{V\left(t\right)}=\frac{M_{Pl}^2}{16{\pi }^2{H}_0^2}{\left(\frac{\dot{a}}{a}\right)}^3=\frac{H_0{M}_{Pl}^2}{16{\pi }^2}\equiv {\Lambda }_{BZ}^3,\end{array}$$

(5)

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”.

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

$${\Lambda }_{BZ}\sim 10\mathrm{MeV}\mathrm{or}{l}_{BZ}=\frac{1}{{\Lambda }_{BZ}}\sim 2\times {10}^{-12}\mathrm{cm}.$$

(6)

${\Lambda }_{BZ}$ 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.

3. Bjorken–Zeldovich Scale from Large Extra Dimensions

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 $V\left(r\right)\sim \frac{m_1{m}_2}{M_U^{n+2}{R}^n}\frac{1}{r},$ 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

$$R=\frac{l_U^{1+2/n}}{l_{Pl}^{2/n}}\sim 2.0\times {\left(2.4\right)}^{2/n}\times {10}^{30/n-17}\mathrm{cm},$$

(7)

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

$$R\sim 2.7\times {10}^{-12}\mathrm{cm}\sim l_{BZ},$$

(8)

a remarkable agreement with the Bjorken–Zeldovich length scale, given in Equation (6), revealed in the MM formalism, discussed previously.

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 $M_{Pl}^2/64\pi H_0^2\sim {10}^{120}$, 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

$$\begin{array}{c}\hfill {\mathcal{S}}_{MM}\to \int d^{4}x\int \sqrt{-\tilde{g}}d{y}_1\dots d{y}_6{\left(F\right)}_{\left(i\overline{\mu }\right)}^5·{ϵ}^{\left(i\right)}·{ϵ}^{\left(\overline{\mu }\right)},\end{array}$$

(9)

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,

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

(10)

which accurately accounts for the factor ${10}^{120}$ in the MM action given in Equation (1).

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 $R=0.1\times {10}^{-12}\mathrm{cm}\cong d_{\mathrm{proton}}$.

4. A “See-Saw” Relation for the Small Cosmological Constant and the Neutrino Mass?

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 ${\rho }_{\Lambda }^{1/4}\stackrel{?}{=}{M}_{BZ}^3/M_{EW}^2\sim {10}^{-3}\mathrm{eV},$ 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 ${\mathcal{L}}^{eff}\supset \frac{c_{\nu }}{{\Lambda }^2}\overline{\nu }\nu \overline{F}F,$ 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; $\langle \overline{F}F\rangle \sim f^3\sim {(a·M_{BZ})}^3$, $\Lambda \sim m_{\varphi }\sim {\Lambda }_{EW}\sim 1$ TeV. The scalar $\varphi$ gets its mass via the portal coupling to the SM Higgs, i.e., $\lambda {\varphi }^2{H}^{†}H$, 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 $m_{\nu }=\left(c_{\nu }{a}^3\right)M_{BZ}^3/M_{EW}^2\lesssim {10}^{-2}\mathrm{eV},$ provided that $c_{\nu }{a}^3\lesssim 10$.

5. More on the Relevance in Cosmology

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 $g_b\left(g_f\right)$ 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.

6. Discussion

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 $M_U⩽2.6$ 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].