Models where the space-time is not 3+1 dimensional but higher dimensional (generally 
dimensional, where n is the number of spacelike extra dimensions) are popular since the beginnings of the 20th century.
The fundamental scale of gravity need not to be the 4D “effective” Planck scale 
but a new scale 
(sometimes called 
), and it could be as low as 
. The observed Planck scale (related to the Newton constant 
) is then related to in 
dimensions by a relationship like the next equation:
![\[ \eta^2=\left(\dfrac{M_f}{M_D}\right)^2\sim\dfrac{1}{M_f^2R^n}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9d37f12d2cdcca53da310c163a8bb00b_l3.png "Rendered by QuickLaTeX.com")
Here, 
is the radius of the typical length of the extra dimensions. We can consider an hypertorus 
for simplicity (but other topologies are also studied in the literature). In fact, the coupling is 
if we choose 
. When we take more than one extra dimension, e.g., taking 
, the radius R of the extra dimension(s) can be as “large” as 1 millimeter! This fact can be understood as the “proof” that there could be hidden from us “large” extra dimensions. They could be only detected by many, extremely precise, measurements that exist at present or future experiments. However, it also provides a new test of new physics (perhaps fiction science for many physicists) and specially, we could explore the idea of hidden space dimensions and how or why is so feeble with respect to any other fundamental interaction.
According to the SM and the standard gravity framework (General Relativity), every group charged particle is localized on a 3-dimensional hypersurface that we could call “brane” (or SM brane). This brane is embedded in “the bulk” of the higher dimensional Universe (with 
extra space-like dimensions). All the particles can be separated into two categories: 1) those who live on the (SM) 3-brane, and 2) those who live “everywhere”, i.e., in “all the bulk” (including both the extra dimensions and our 3-brane where the SM fields only can propagate). The “bulk modes” are (generally speaking) quite “model dependent”, but any coupling between the brane where the SM lives and the bulk modes should be “suppressed” somehow. One alternative is provided by the geometrical factors of “extra dimensions” (like the one written above). Another option is to modify the metric where the fields propagate. This last recipe is the essence of non-factorizable models built by Randall, Sundrum, Shaposhnikov, Rubakov, Pavŝiĉ and many others as early as in the 80’s of the past century. Graviton and its “propagating degrees of freedom” or possible additional neutral states belongs to the second category. Indeed, the observed weakness of gravity in the 3-brane can be understood as a result of the “new space dimensions” in which gravity can live. However, there is no clear signal of extra dimensions until now (circa 2013, July).
The small coupling constant derived from the Planck mass above can also be used in order to explain the smallness of the neutrino masses! The left-handed neutrino 
having weak isospin and hypercharge is thought to reside in the SM brane in this picture. It can get a “naturally small” Dirac mass through the mixing with some “bulk fermion” (e.g., the right-handed neutrino or any other neutral fermion under the SM gauge group) which can be interpreted as a right-handed neutrino 
:
![\[ \mathcal{L}(m,Dirac)\sim h\eta H\bar{\nu}_L\nu_R\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-0972d367ede56d8041949dee15d79581_l3.png "Rendered by QuickLaTeX.com")
Here, 
are the two Higgs doublet fields and the Yukawa coupling, respectively. After spontaneous symmetry breaking, this interaction will generate the Dirac mass term
The right-handed neutrino has a hole tower of Kaluza-Klein relatives 
. The masses of these states are given by
and the couples with all KK state having the same “mixing” mass. Thus, we can write the mass lagrangian as
with
Are you afraid of “infinite” neutrino flavors? The resulting neutrino mass matrix M is “an infinite array” with structure:
![\[ \mathbb{M}=\begin{pmatrix}m_D &\sqrt{2}m_D &\sqrt{2}m_D &\ldots &\sqrt{2}m_D &\ldots \\ 0 &1/R &0 &\ldots &0 & \ldots\\ 0 & 0 &2/R & \ldots & 0 &\ldots \\ \ldots & \ldots & \ldots & \ldots & k/R & \ldots\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\end{pmatrix}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-df31a629c3b52ee509c13444cda6dac8_l3.png "Rendered by QuickLaTeX.com")
The eigenvalues of the matrix 
are given by a trascendental equation. In the limit where 
, or 
, the eigenvalues are 
, where 
and 
is a double eigenvalue (i.e., it is doubly degenerated). There are other examples with LR symmetry. For instance, 
right-handed neutrinos that, living on the SM brane, were additional neutrino species. In these models, it has been showed that the left-handed neutrino is exactly massless whereas the assumed bulk and “sterile” neutrino have a mass related to the size of the extra dimensions. These models produce masses that can be fitted to the expected values 
coming from estimations at hand with the neutrino oscillation data, but generally, this implies that there should be at least one extra dimension with size in the micrometer range or less!
The main issues that extra dimension models of neutrino masses do have is related to the question of the renormalizability of their interactions. With an infinite number of KK states and/or large extra dimensions, extreme care have to be taken in order to not spoil the SM renormalizability and, at some point, it implies that the KK tower must be truncated at some level. There is no general principle or symmetry that explain this cut-off to my knowledge.
May the neutrinos and the extra dimensions be with you!
See you in my next neutrinological post!
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