Intro
Kaluza-Klein theory is a kind of theory with some extra dimensions, and they are often assumed to be compactified in a tiny scale. In this note, we will give a brief review of some famous models based on this theory, and their experimental constraints.
Kaluza-Klein Theory
The basic set-up of Kaluza-Klein theory is a higher dimensional action with a given geometric structure of those extra dimensions.
Considering an Einstein-Hilbert action in $d+n$ dimension with $d$ macroscopic and $n$ mesoscopic or microscopic, we have:
\[\hat{S}_{EH}=\frac{1}{2}\hat{M}_P^{d+n-2}\int\mathrm{d}^{d+n}x\sqrt{-\hat{g}}\hat{R},\]where hat means they are defined in $d+n$ dimension, and the power of $\hat{M}_P$ is determined by simply dimensional analysis. We will take $\eta=\mathrm{diag}(-,+,+,\cdots)$ and denote coordinates in $d+n$ dimension as $x^{\hat{\mu}}=(x^\mu,z^i)$. To see the relation between $\hat{M}_P$ and $M_P$ defined in $d$ macroscopic dimension, constraining metric $\hat{g}$ in $d$ dimension:
and integrating out $n$ extra dimension:
\[\begin{aligned} \hat{S}_{EH}&\simeq \frac{1}{2}\hat{M}_P^{d+n-2}\int\mathrm{d}^{d}x\mathrm{d}^{n}z\sqrt{-g}\sqrt{g_n}R+\cdots \\&=\frac{1}{2}V_n\hat{M}_P^{d+n-2}\int\mathrm{d}^{d}x\sqrt{-g}R+\cdots \\&=\frac{1}{2}M_P^{d-2}\int\mathrm{d}^{d}x\sqrt{-g}R+\cdots \\&\simeq S_{EH}, \end{aligned}\]where $\cdots$ denotes some other term beyond $d$ dimension E-H action. Thus, we can see the relationship between $M_P$ and $\hat{M}_P$:
We will denote $R\sim V_n^{1/n}$ as the characteristic scale of the extra dimension.
$T^n$ Compactification
The simplest Kaluza-Klein model is constructed by taking $n$ extra dimension as $T^n$. We will show the complete construction of $T^n$ K-K model in this subsection.
Scalar Field
We can begin with the easiest situation. The Lagrangian of a $d+n$ dimension massless scalar field is:
\[\hat{\mathscr{L}}_{scalar}=-\frac{1}{2}\partial_{\hat{\mu}}\phi\partial^{\hat{\mu}}\phi=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}\partial_i\phi\partial^i\phi\]notice that we have taken $\eta_{\hat{\mu}\hat{\nu}}$ as the background metric of $T^n$. We will assume that this field can propagate in the bulk, otherwise we need to add $\delta^n(z-z_0)$ in the Lagrangian (which we do need to in some models).
By compactification, $\phi$ should have the mode expansion:
\[\phi(x,z)=\sum_{\vec{n}}\phi^{\vec{n}}(x)\exp\left(\mathrm{i}\frac{2\pi\vec{n}\cdot\vec{z}}{R}\right)\]Substituting this into the Lagrangian, and integrating out the $n$ extra dimension, we can get $d$ dimensional effective Lagrangian:
\[\begin{aligned} \mathscr{L}_{scalar}&=\int\mathrm{d}^n{z}\:\hat{\mathscr{L}}_{scalar} \\&=\sum_{\vec{n}}\int\mathrm{d}^n{z}\:\exp\left[\mathrm{i}\frac{2\pi(\vec{n}+\vec{m})\cdot\vec{z}}{R}\right]\left\{-\frac{1}{2}\partial_\mu\phi^{\vec{n}}\partial^{\mu}\phi^{\vec{m}}+\frac{1}{2}\frac{4\pi^2\vec{n}\cdot\vec{m}}{R^2}\phi^{\vec{n}}\phi^{\vec{m}}\right\} \\&=V_n\sum_{\vec{n}}\left\{-\frac{1}{2}\partial_\mu\phi^{\vec{n}}\partial^{\mu}\phi^{-\vec{n}}-\frac{1}{2}m_{\vec{n}}^2\phi^{\vec{n}}\phi^{-\vec{n}}\right\}, \end{aligned}\]where we have $m_{\vec{n}}^2=4\pi^2n^2/R^2$.
This shows that through compactification, we get a series of field with mass $m_{\vec{n}}$, which is known as K-K tower. The propagator is easy to get:
Vector Field
The treatment of fermions in Kaluza-Klein theory is rather subtle. We will deal with it later, and consider vector fields for now. It is easy to write down the Yang-Mills Lagrangian in $d+n$ dimension:
\[\begin{aligned} \hat{\mathscr{L}}_{vec}&=\mathrm{Tr}\:\left\{-\frac{1}{4}F_{\hat{\mu}\hat{\nu}}F^{\hat{\mu}\hat{\nu}}\right\} \\&=\mathrm{Tr}\:\left\{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}F_{\mu i}F^{\mu i}-\frac{1}{4}F_{ij}F^{ij}\right\} \end{aligned}\]where we have $F_{\hat{\mu}\hat{\nu}}=\partial_{\hat{\mu}}A_{\hat{\nu}}-\partial_{\hat{\nu}}A_{\hat{\mu}}-\mathrm{i}g[A_{\hat{\mu}},A_{\hat{\nu}}],\:A_{\hat{\mu}}=A_{\hat{\mu}}^aT_a$.
The gauge transformation is:
\[A'_{\hat{\mu}}=\frac{\mathrm{i}}{g}U^{-1}D_{\hat{\mu}}U,\:U(x^\mu,z^i)\in G,\]where $D_{\hat{\mu}}$ is the covariant derivative. We can get its infinitesimal formal:
Writing down the model expansion:
\[\begin{aligned} &A_{\hat{\mu}}(x,z)=\sum_{\vec{n}}A_{\hat{\mu}}^{\vec{n}}(x)\exp\left(\mathrm{i}\frac{2\pi\vec{n}\cdot\vec{z}}{R}\right) \\&\Omega(x,z)=\sum_{\vec{n}}\Omega^{\vec{n}}(x)\exp\left(\mathrm{i}\frac{2\pi\vec{n}\cdot\vec{z}}{R}\right) \end{aligned}\]and we can calculate the $d$ dimensional effective Lagrangian. The $F_{\mu i}$ can be expanded as:
where we have:
\[F^{\vec{n}}_{\mu i}(x)=\partial_{\mu}A_{i}^{\vec{n}}(x)-\mathrm{i}\frac{2\pi}{R}n_i A_{\mu}^{\vec{n}}(x)+g{f^{a}}_{bc}\sum_{\vec{k}}A^{b,\vec{k}}_{\mu}(x)A^{c,\vec{n}-\vec{k}}_{i}(x)T_{a}.\]Similarly, expanding $F_{ij}$ to the K-K mode:
where $F^{\vec{n}}_{ij}$ is:
and $F_{\mu\nu}$ is:
which is obvious. Substituting them into Lagrangian, we can get:
\[\begin{aligned} \mathscr{L}_{vec}&=\int\mathrm{d}^n z\:\hat{\mathscr{L}}_{vec} \\&=V_n\mathrm{Tr}\sum_{\vec{n}}\left\{-\frac{1}{4}F^{\vec{n}}_{\mu\nu}F^{\mu\nu,-\vec{n}}-\frac{1}{2}F^{\vec{n}}_{\mu i}F^{\mu i,-\vec{n}}-\frac{1}{4}F^{\vec{n}}_{ij}F^{ij,-\vec{n}}\right\} \end{aligned}\]after some calculation, we can get:
\[\begin{aligned} \mathscr{L}_{vec}^{\vec{n}}=\mathrm{Tr}&\left\{-\frac{1}{4}(\partial_{\mu}A_{\nu}^{\vec{n}}-\partial_{\nu}A_{\mu}^{\vec{n}})(\partial^{\mu}A^{\nu,-\vec{n}}-\partial^{\nu}A^{\mu,-\vec{n}})-\frac{1}{2}(\partial_{\mu}A_{i}^{\vec{n}}-\mathrm{i}\frac{2\pi}{R}n_i A_{\mu}^{\vec{n}})(\partial^{\mu}A^{i,-\vec{n}}+\mathrm{i}\frac{2\pi}{R}n^i A^{\mu,-\vec{n}})\right. \\&\left.-\frac{1}{4}\cdot\frac{4\pi^2}{R^2}(n_iA_j^{\vec{n}}-n_jA_{i}^{\vec{n}})(n^iA^{j,-\vec{n}}-n^jA^{i,-\vec{n}})+\mathscr{L}_{int}^{\vec{n}}\right\}. \end{aligned}\]Reorganizing the expression, we have:
\[\begin{aligned} \mathscr{L}_{vec,kin}^{\vec{n}}=\mathrm{Tr}&\left\{-\frac{1}{4}(\partial_{\mu}A_{\nu}^{\vec{n}}-\partial_{\nu}A_{\mu}^{\vec{n}})(\partial^{\mu}A^{\nu,-\vec{n}}-\partial^{\nu}A^{\mu,-\vec{n}})-\frac{1}{2}\cdot\frac{4\pi^2}{R^2}n^2A_{\mu}^{\vec{n}}A^{\mu,-\vec{n}}-\frac{1}{2}\partial_{\mu}A_{i}^{\vec{n}}\partial^{\mu}A^{i,-\vec{n}}\right. \\&\left.-\frac{1}{2}\cdot\frac{4\pi^2}{R^2}n^2A_{i}^{\vec{n}}A^{i,-\vec{n}}+\mathrm{i}\frac{\pi}{R}\left(n_i A^{\vec{n}}_{\mu}\partial^{\mu}A^{i,-\vec{n}}-n_i A^{-\vec{n}}_{\mu}\partial^{\mu}A^{i,\vec{n}}\right)+\frac{2\pi^2}{R^2}n_i A^{i,\vec{n}}n_j A^{j,\vec{-n}}\right\}. \end{aligned}\]Notice that for modes who have $\vec{n}\neq 0$, fields $A^{\vec{n}}_{\mu}$ are massive, which indicates some kinds of “Higgs” mechanism. Recalling the gauge transformation of $A_{\hat{\mu}}$, we have:
In order to do the quantization, we add the gauge fixing term to the Lagrangian:
\[\mathscr{L}_{GF}=\mathrm{Tr}\left\{-\frac{1}{2\xi}(\partial_{\mu}A^{\mu}-\xi\partial_{i}A^{i})^2\right\},\]and get the F-P ghost term:
\[\mathscr{L}_{gh}=-\int\mathrm{d}^{d+n}y\:\bar{c}_a(x)\mathcal{M}^{ab}(x,y)c_b(y)\]where we have:
\[\mathcal{M}_{ab}=\frac{\delta \mathcal{F}_{a}}{\delta \Omega^{b}}\simeq\frac{1}{g}\delta_{ab}(\partial_{\mu}\partial^{\mu}-\xi\partial_i\partial^i)-f_{abc}(\partial_{\mu}A^{c \mu}-\xi\partial_i A^{c i})\]Expanding in K-K modes:
\[\begin{aligned} \mathscr{L}^{\vec{n}}_{GF}&=-\frac{1}{2\xi}(\partial_{\mu}A^{\mu,\vec{n}}-\mathrm{i}\xi\frac{2\pi}{R}n_{i}A^{i,\vec{n}})(\partial_{\mu}A^{\mu,-\vec{n}}+\mathrm{i}\xi\frac{2\pi}{R}n_{i}A^{i,-\vec{n}}) \\&=-\frac{1}{2\xi}\partial_{\mu}A^{\mu,\vec{n}}\partial_{\mu}A^{\mu,-\vec{n}}-\frac{\xi}{2}\cdot\frac{4\pi^2}{R^2}(n_i A^{i,\vec{n}})(n_i A^{i,-\vec{n}})+\mathrm{i}\frac{\pi}{R}(n_i A^{i,\vec{n}}\partial_{\mu}A^{\mu,-\vec{n}}-n_i A^{i,-\vec{n}}\partial_{\mu}A^{\mu,\vec{n}}) \end{aligned}\]notice that when we sum overall $\vec{n}$, the mixing term in $\mathscr{L}^{\vec{n}}_{vec}$ will be eliminated by the last term in $\mathscr{L}^{\vec{n}}_{GF}$. Thus we can get the whole Lagrangian:
So we can read out the propagators. The zero mode of $A_{\mu}$ is just normal $d$ dimension Yang-Mills fields, and $\vec{n}\neq 0$ modes are the transverse components of massive vector fields; $A^{i}$ perpendicular to $n^i$ becomes massive scalars and proportional to $n^i$ is recognized as Goldstone boson and becomes the longitudinal component of $A^{\vec{n}}_{\mu}$.
Spinor Field
We now try to deal with fermions. The beginning is the $d+n$ dimensional massless Dirac Lagrangian:
where $\Gamma^{\hat{\mu}}$ satisfying $d+n$ dimensional Clifford algebra:
In order to do the Kaluza-Klein reduction, we first need to get the reduced form of Clifford algebra. Following the definitions in Ref.\cite{Freedman:2012zz}, we can decompose gamma matrix as:
\[\Gamma^{\mu}=\gamma^{\mu}\otimes\mathbf{1},\:\Gamma^{i}=\gamma_{*}\otimes\gamma^i\]where $\gamma^\mu$ and $\gamma^i$ are normal $d$ and $n$ dimensional gamma matrix, and we have assumed $d$ is even to define chirality matrix $\gamma_{*}=(-\mathrm{i})^{d/2+1}\gamma_{0}\gamma_1\cdots\gamma_{d-1}$. Since the chirality matrix is anti-commuting with $\gamma^{\mu}$, we can verify that:
and thus we can write down the decomposition of Dirac spinor\footnote{Notice that even in a trivial manifold $T^n$, which we consider here, the eigenfunctions of the Dirac operator have more energy degenerate than the eigenfunctions of the Laplacian operator used for the scalar and Yang-Mills fields above, especially for the zero modes. If we try to deal with non-trivial manifold in extra dimension, then the eigenfunctions of the Laplacian operator may also have extra degeneracy too.}:
\[\Psi(x,z)=\sum_k\psi_{k,0}(x)\otimes \zeta_{k,0}(z)+\sum_{\vec{n},s}\left[\psi_{+,\vec{n},s}(x)\otimes\zeta_{+,\vec{n},s}(z)+\psi_{-,\vec{n},s}(x)\otimes\zeta_{-,\vec{n},s}(z)\right]\]where $s$ takes from $1$ to $2^{\lfloor n/2\rfloor-1}$ denotes the spin degenerate, and $\zeta_{\pm,n,s}(z)$ satisfies that:
and $\zeta_{k,0}(z)$ satisfies $\mathrm{i}\gamma^{i}\partial_i\zeta_{k,0}(z)=0$, thus $\zeta_{k,0}$ are constant spinors and $k$ takes from $1$ to $2^{\lfloor n/2\rfloor}$.
For $T^n$ situation, $\lambda_n={2\pi|\vec{n}|}/{R}$ and $\zeta(z)$ is just the free particle solution:
where $u_{\pm,\vec{n},s}$ satisfies that:
and orthogonal relation:
\[\int\mathrm{d}^n z\:\left[u_{\alpha,\vec{m},s}\exp\left(\mathrm{i}\frac{2\pi\vec{m}\cdot\vec{z}}{R}\right)\right]^{\dagger}u_{\beta,\vec{n},s'}\exp\left(\mathrm{i}\frac{2\pi\vec{n}\cdot\vec{z}}{R}\right)=V_{n}\delta_{\alpha\beta}\delta_{\vec{m},\vec{n}}\delta_{ss'}.\]Substituting these to the Lagrangian, we get:
\[\begin{aligned} \hat{\mathscr{L}}_{dirac}&=\mathrm{i}\Psi^{\dagger}\Gamma^0\Gamma^{\hat{\mu}}\partial_{\hat{\mu}}\Psi \\&=\mathrm{i}\Psi^{\dagger}\Gamma^0\Gamma^{\mu}\partial_{\mu}\Psi+\mathrm{i}\Psi^{\dagger}\Gamma^0\Gamma^{i}\partial_{i}\Psi \end{aligned}\]Integrating out extra dimensions, we get the compactified Lagrangian:
\[\begin{aligned} \mathscr{L}_{dirac}&=\int\mathrm{d}^{n}z\hat{\mathscr{L}}_{dirac} \\&=V_n\left\{\sum_k\mathrm{i}\bar{\psi}_{k,0}\gamma^{\mu}\partial_{\mu}\psi_{k,0}+\sum_{\vec{n},s}\bar{\psi}_{+,\vec{n},s}(\mathrm{i}\gamma^{\mu}\partial_{\mu}-\lambda_{n}\gamma_{*})\psi_{+,\vec{n},s}+\sum_{\vec{n},s}\bar{\psi}_{-,\vec{n},s}(\mathrm{i}\gamma^{\mu}\partial_{\mu}+\lambda_{n}\gamma_{*})\psi_{-,\vec{n},s}\right\} \end{aligned}\]Notice that zero modes are massless, and others get a pseudo-scalar mass term. In a free theory, we can rewrite this pseudo-scalar mass to a normal mass term through a field redefinition, but if we couple these fermions to a gauge field, this pseudo-scalar will give us an anomaly term.
