The Standard Model with one universal extra dimension

Abstract

Effects of universal extra dimensions on Standard Model observables first arise at the one-loop level. The quantization of this class of theories is therefore essential in order to perform predictions. A comprehensive study of the SU _C(3) × SU _L(2) × U _Y(1) Standard Model defined in a space-time manifold with one universal extra dimension, compactified on the oribifold $S^1/Z_2$, is presented. The fact that the four-dimensional Kaluza–Klein theory is subjected to two types of gauge transformations is stressed and its quantization under the basis of the BRST symmetry discussed. A SU _C(3) × SU _L(2) × U _Y(1)-covariant gauge-fixing procedure for the Kaluza–Klein excitations is introduced. The connection between gauge and mass eigenstate fields is established in an exact way. An exhaustive list of the explicit expressions for all physical couplings induced by the Yang–Mills, Currents, Higgs, and Yukawa sectors is presented. The one-loop renormalizability of the standard Green’s functions, which implies that the Standard Model observables do not depend on a cut-off scale, is stressed.

Appendices

Appendix A. The boson masses

The mass term for the gauge fields is given by

$$ \begin{array}{rll} {\cal L}^{\rm gauge}_{\rm mass}&=&\sum\limits_{n=0}\left\{\left[\frac{1}{2}\left(\frac{n}{R} \right)^2+ \frac{g^2v^2}{8}\right]\left(W^{(n)1}_\mu W^{(n)1\mu}+W^{(n)2}_\mu W^{(n)2\mu} \right)\vphantom{\left(\begin{array}{ccc} W^{(n)3\mu} \\ \, \\ B^{(n)\mu} \end{array}\right)}\right.\\ &&\left.\;\;\;\quad +\left(W^{(n)3}_\mu,B^{(n)}_\mu\right)M^{(n)} \left(\begin{array}{ccc} W^{(n)3\mu} \\ \, \\ B^{(n)\mu} \end{array}\right) \right\}, \end{array} $$

(A.1)

where

$$ M^{(n)}=\frac{1}{2} \left( \frac{n}{R} \right)^2I+M^{(0)} , $$

(A.2)

with I representing the 2 × 2 identity matrix and

$$ M^{(0)}=\frac{v^2}{8}\left(\begin{array}{ccc} \, \, g^2 & -gg' \\ \, \\ -gg' & \, \, g'^2 \end{array}\right). $$

(A.3)

The M ⁽ⁿ⁾ matrix is diagonalized by the well-known orthogonal matrix

$$ R=\left(\begin{array}{ccc} \, \, \, c_W & s_W \\ \, \\ -s_W & c_W \end{array}\right), $$

(A.4)

where s _W and c _W stand for the sine and cosine of the weak angle, given by tanθ _W  = g′/g. The mass eigenstate fields and their corresponding masses are given by

$$ W^{(n)+}_\mu =\frac{1}{\sqrt{2}}\left(W^{(n)1}_\mu-iW^{(n)2}_\mu \right), \quad n=0,1, ... , $$

(A.5)

$$ W^{(n)-}_\mu =\frac{1}{\sqrt{2}}\left(W^{(n)1}_\mu+iW^{(n)2}_\mu \right), \quad n=0,1, ... , $$

(A.6)

$$ m^2_{W^{(n)}}=\left(\frac{n}{R}\right)^2+m^2_{W^{(0)}}, \quad n=0,1, ... , $$

(A.7)

$$ Z^{(n)}_\mu =c_W W^{(n)3}_\mu -s_WB^{(n)}_\mu , \quad n=0,1, ... , $$

(A.8)

$$ A^{(n)}_\mu =s_W W^{(n)3}_\mu +c_WB^{(n)}_\mu , \quad n=0,1, ... , $$

(A.9)

$$ m^2_{Z^{(n)}}= \left(\frac{n}{R} \right)^2+m^2_{Z^{(0)}}, \quad n=0,1, ... , $$

(A.10)

$$ m^2_{\gamma^{(n)}}=\left(\frac{n}{R} \right)^2, \quad n=0,1, ... , $$

(A.11)

where $m_{W^{(0)}}$ and $m_{Z^{(0)}}$ are the SM masses for the W and Z gauge bosons, respectively.

Concerning the mass spectrum of the scalar fields, the up components of the Higgs doublets Φ⁽ⁿ⁾, denoted by φ ^(n)±, mix with the charged fields $W^{(n)\pm}_5\equiv \frac{1}{\sqrt{2}}\big(W^{(n)1}_5\mp iW^{(n)2}_5\big)$ as follows:

$$ \begin{array}{rll} {\cal L}^{\rm cs}_{\rm mass}&=&-(\phi^{(n)-},W^{(n)-}_5)\left(\begin{array}{ccc} \, \, \, \left({n}/{R}\right)^2 & im_{W^{(0)}}\!\left({n}/{R}\right)\\ \, \, \\ -im_{W^{(0)}}\!\left({n}/{R}\right)& m^2_{W^{(0)}} \end{array}\right) \left(\begin{array}{ccc} \phi^{(n)+} \\ \, \, \\ W^{(n)+}_5 \end{array}\right) \nonumber \\[6pt] \, \nonumber \\ &=&-m^2_{W^{(n)}}H^{(n)-}H^{(n)+}, \quad n=1,2,..., \end{array} $$

(A.12)

where the physical fields H ^(n)± and the pseudo-Goldstone bosons $G^\pm_{W^{(n)}}$ are related to the original gauge eigenstates through the following unitary transformation:

$$ \left(\begin{array}{ccc} H^{(n)+} \\ \, \\ G^+_{W^{(n)}} \end{array}\right)=\left(\begin{array}{ccc} c_\alpha & \, \, \, is_\alpha\\ \, \, \\ s_\alpha & -ic_\alpha \end{array}\right)\left(\begin{array}{ccc} \phi^{(n)+} \\ \, \\ W^{(n)+}_5 \end{array}\right), $$

(A.13)

with the angle α given by

$$ \tan\alpha=\frac{m_{W^{(0)}}}{\left({n}/{R}\right)}. $$

(A.14)

The pseudo-Goldstone bosons associated with the standard gauge fields $W^{(0)\pm}_\mu$ and $Z^{(0)}_\mu$ are, respectively, the up component of Φ⁽⁰⁾ and the imaginary part of the down component of Φ⁽⁰⁾:

$$ \Phi^{(0)}=\left(\begin{array}{ccc} G^+_{W^{(0)}} \\ \, \, \\ \frac{v+H^{(0)}+iG_{Z^{(0)}}}{\sqrt{2}} \end{array}\right), $$

(A.15)

where H ⁽⁰⁾ is the SM Higgs boson. On the other hand, the mass terms for the neutral fields $W^{(n)3}_5$, $B^{(n)}_5$, H ⁽ⁿ⁾, and $\phi^{(n)}_I$, the latter two fields being the real and imaginary parts of the down component of the Φ⁽ⁿ⁾ doublet, can be written as

$$ \begin{array}{rll} {\cal L}^{\rm ns}_{\rm mass}&=&-\frac{1}{2}m^2_{H^{(n)}}H^{(n)}H^{(n)} \nonumber \\&& -\frac{1}{2}\left(\phi^{(n)}_{\rm I},\hat{W}^{(n)3}_5\right)\nonumber \\ &&\times \left(\begin{array}{ccc} \left({n}/{R}\right)^2 & -m_{Z^{(0)}}\!\left({n}/{R}\right)\\ \, \, \\ -m_{Z^{(0)}}\!\left({n}/{R}\right) & m^2_{Z^{(0)}} \end{array}\right)\left(\begin{array}{ccc} \phi^{(n)}_{\rm I} \\ \, \\ \hat{W}^{(n)3}_5 \end{array}\right),\end{array} $$

(A.16)

where we have carried out the following rotation:

$$ \left(\begin{array}{ccc} W^{(n)3}_5 \\ \, \\ B^{(n)}_5 \end{array}\right)=\left(\begin{array}{ccc} \, \, \, c_W & s_W\\ \, \, \\ -s_W & c_W \end{array}\right)\left(\begin{array}{ccc} \hat{W}^{(n)3}_5 \\ \, \\ G_{A^{(n)}} \end{array}\right). $$

(A.17)

In the above expression, $G_{A^{(n)}}$ is the pseudo-Goldstone boson associated with the gauge boson $A^{(n)}_\mu$. In addition, the masses of the KK excitations of H ⁽⁰⁾ are given by

$$ m^2_{H^{(n)}}=\left(\frac{n}{R} \right)^2+m^2_{H^{(0)}}, $$

(A.18)

where $m_{H^{(0)}}$ is the SM Higgs mass. The above mass matrix can be diagonalized through the following orthogonal rotation:

$$ \left(\begin{array}{ccc} \phi^{(n)}_{\rm I} \\ \, \\ \hat{W}^{(n)3}_5 \end{array}\right)=\left(\begin{array}{ccc} \, \, \, c_\beta & s_\beta \\ \, \, \\ -s_\beta & c_\beta \end{array}\right)\left(\begin{array}{ccc} A^{(n)} \\ \, \\ G_{Z^{(n)}} \end{array}\right), $$

(A.19)

where $G_{Z^{(n)}}$ is the pseudo-Goldstone boson associated with the gauge KK mode $Z^{(n)}_\mu$ and A ⁽ⁿ⁾ represents a physical pseudoscalar field with mass given by $m_{A^{(n)}}=m_{Z^{(n)}}$. In addition,

$$ \tan \beta=\frac{m_{Z^{(0)}}}{\left({n}/{R}\right)}. $$

(A.20)

Notice that

$$ \frac{\tan\alpha}{\tan\beta}=c_W. $$

(A.21)

Appendix B. Definitions in the Currents sector

The covariant objects appearing in eq. (76) are given by

$$ \begin{array}{rll} (D_\mu F)^{(0)}_{\rm L}&=&D^{(0)}_\mu F^{(0)}_{\rm L}\\ &&-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_\mu +ig\frac{\sigma^i}{2}W^{(n)i}_\mu +ig'\frac{Y}{2}B^{(n)}_\mu \right)F^{(n)}_{\rm L}, \end{array} $$

(B.1)

$$ \begin{array}{rll} (D_\mu F)^{(n)}_{\rm L}&=&D^{(0)}_\mu F^{(n)}_{\rm L}-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_\mu+ig\frac{\sigma^i}{2}W^{(n)i}_\mu +ig'\frac{Y}{2}B^{(n)}_\mu \right)F^{(0)}_{\rm L} \\ && -\Delta^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_\mu+ig\frac{\sigma^i}{2}W^{(r)i}_\mu +ig'\frac{Y}{2}B^{(r)}_\mu \right)F^{(s)}_{\rm L}, \end{array} $$

(B.2)

$$ (D_\mu F)^{(n)}_{\rm R}=D^{(0)}_\mu F^{(n)}_{\rm R} -\Delta'^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_\mu+ig\frac{\sigma^i}{2}W^{(r)i}_\mu +ig'\frac{Y}{2}B^{(r)}_\mu \right)F^{(s)}_{\rm R}, $$

(B.3)

$$ (D_5F)^{(0)}_{\rm L}=\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_5+ig\frac{\sigma^i}{2}W^{(n)i}_5 +ig'\frac{Y}{2}B^{(n)}_5 \right)F^{(n)}_{\rm R}, $$

(B.4)

$$ (D_5F)^{(n)}_{\rm L}=\frac{n}{R}F^{(n)}_{\rm R} -\Delta'^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_5+ig\frac{\sigma^i}{2}W^{(r)i}_5 +ig'\frac{Y}{2}B^{(r)}_5 \right)F^{(s)}_{\rm R}, $$

(B.5)

$$ \begin{array}{rll} (D_5F)^{(n)}_{\rm R}&=&-\frac{n}{R}F^{(n)}_{\rm L}-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_5+ig\frac{\sigma^i}{2}W^{(n)i}_5 +ig'\frac{Y}{2}B^{(n)}_5 \right)F^{(0)}_{\rm L}\\ &&-\,\Delta'^{nrs}\left(ig_s\frac{\lambda^a}{2}G^{(r)a}_5+ig\frac{\sigma^i}{2}W^{(r)i}_5 +ig'\frac{Y}{2}B^{(r)}_5\right)F^{(s)}_{\rm L}, \end{array} $$

(B.6)

and

$$ (D_\mu \hat{f})^{(0)}_{\rm R}=D^{(0)}_\mu f^{(0)}_{\rm R}-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_\mu+ig'\frac{Y}{2}B^{(n)}_\mu \right)\hat{f}^{(n)}_{\rm R}, $$

(B.7)

$$ \begin{array}{rll} (D_\mu \hat{f})^{(n)}_{\rm R}&=&D^{(0)}_\mu \hat{f}^{(n)}_{\rm R}-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_\mu+ig'\frac{Y}{2}B^{(n)}_\mu \right)f^{(0)}_{\rm R} \\ &&-\,\Delta^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_\mu+ig'\frac{Y}{2}B^{(r)}_\mu \right)\hat{f}^{(s)}_{\rm R} , \end{array} $$

(B.8)

$$ (D_\mu \hat{f})^{(n)}_{\rm L}=D^{(0)}_\mu \hat{f}^{(n)}_{\rm L} -\Delta'^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_\mu+ig'\frac{Y}{2}B^{(r)}_\mu \right)\hat{f}^{(s)}_{\rm L} , $$

(B.9)

$$ (D_5\hat{f})^{(0)}_{\rm R}=\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_5+ig'\frac{Y}{2}B^{(n)}_5 \right){\kern-3pt} \hat{f}^{(n)}_{\rm L}, $$

(B.10)

$$ (D_5f)^{(n)}_{\rm R}=\frac{n}{R}\hat{f}^{(n)}_{\rm L}-\Delta'^{nrs}\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(r)a}_5+ig'\frac{Y}{2}B^{(r)}_5 \right){\kern-3pt} \hat{f}^{(s)}_{\rm L} , $$

(B.11)

$$ \begin{array}{rll} (D_5\hat{f})^{(n)}_{\rm L}&=&-\frac{n}{R}\hat{f}^{(n)}_{\rm R}-\left(ig_{\rm s}\frac{\lambda^a}{2}G^{(n)a}_5+ig'\frac{Y}{2}B^{(n)}_5 \right){\kern-3pt} f^{(0)}_\mathrm{R} \\ && -\,\Delta'^{nrs}\left(ig_s\frac{\lambda^a}{2}G^{(r)a}_5+ig'\frac{Y}{2}B^{(r)}_5\right){\kern-3pt} \hat{f}^{(s)}_\mathrm{R} . \end{array} $$

(B.12)

In the above expressions, an appropriate application of the covariant derivative is assumed. For example, $D^{(0)}_\mu e^{(0)}_{\rm R}=(\partial_\mu -ig'B^{(0)}_\mu Y/2 )e^{(0)}_{\rm R}$, but $D^{(0)}_\mu d^{(0)}_{\rm R}=(\partial_\mu-ig_sG^{(0)a}_\mu \lambda^a/2 -ig'B^{(0)}_\mu Y/2 )d^{(0)}_{\rm R}$.

Appendix C. The fermion masses

As commented in §2, the masses of the fermions emerge from both the Yukawa and the Currents sectors. The corresponding Lagrangian is given by

$$ \begin{array}{rll} -{\cal L}^{f}_{\rm mass}&=&\sum\limits_{a,b=1}^3\Big\{\left(\lambda_{eab}\bar{L}^{(0)}_{{\rm L}a}e^{(0)}_{{\rm R}b}+\lambda_{dab}\bar{Q}^{(0)}_{\mathrm{L}a}d^{(0)}_{\mathrm{R}b}\right){\kern-3pt} \Phi^{(0)}_0 \\ &&\;\qquad+\,\lambda_{uab}\bar{Q}^{(0)}_{{\rm L}a}u^{(0)}_{{\rm R}b}\tilde{\Phi}^{(0)}_0+\lambda_{eab}\left(\bar{L}^{(n)}_{{\rm L}a}\hat{e}^{(n)}_{{\rm R}b}+\bar{L}^{(n)}_{{\rm R}a}\hat{e}^{(n)}_{{\rm L}b} \right){\kern-3pt} \Phi^{(0)}_0 \\ &&\qquad\;+\,\lambda_{dab}\left(\bar{Q}^{(n)}_{{\rm L}a}\hat{d}^{(n)}_{{\rm R}b}+\bar{Q}^{(n)}_{{\rm R}a}\hat{d}^{(n)}_{{\rm L}b} \right){\kern-3pt} \Phi^{(0)}_0 \\ &&\qquad\;+\,\lambda_{uab}\left(\bar{Q}^{(n)}_{{\rm L}a}\hat{u}^{(n)}_{{\rm R}b}+\bar{Q}^{(n)}_{{\rm R}a}\hat{u}^{(n)}_{{\rm L}b} \right){\kern-3pt} \tilde{\Phi}^{(0)}_0\Big\}\\ &&+\sum_{a=1}^3\left\{\left(\frac{n}{R}\right)\bigg(\bar{L}^{(n)}_{a{\rm L}} L^{(n)}_{a{\rm R}}+\bar{\hat{e}}^{(n)}_{a{\rm R}}\hat{e}^{(n)}_{a{\rm L}}+\bar{Q}^{(n)}_{a{\rm L}}Q^{(n)}_{a{\rm R}}\right.\\ &&\qquad\quad\qquad\;\;\;+\left.\bar{\hat{d}}^{(n)}_{a{\rm R}}\hat{d}^{(n)}_{a{\rm L}} +\bar{\hat{u}}^{(n)}_{a{\rm R}}\hat{u}^{(n)}_{a{\rm L}}\bigg)\right\}\, +\, {\rm h.c.} \end{array} $$

(C.1)

In the flavour space, this Lagrangian can be written as follows:

$$ \begin{array}{rll} -{\cal L}^{f}_{\rm mass}&=&\bar{E}^{(0)}_{\rm L}\Lambda_e E^{(0)}_{\rm R}+\bar{E}^{(n)}_{\rm L}\Lambda_e \hat{E}^{(n)}_{\rm R}+\bar{E}^{(n)}_{\rm R}\Lambda_e \hat{E}^{(n)}_{\rm L}\\ &&+\left(\frac{n}{R}\right)\left(\bar{E}^{(n)}_\mathrm{L}E^{(n)}_{\rm R} +\bar{\hat{E}}^{(n)}_{\rm R}\hat{E}^{(n)}_{\rm L}+\bar{N}^{(n)}_{\rm L}N^{(n)}_{\rm R} \right) \\ &&+\,\bar{D}^{(0)}_{\rm L}\Lambda_\mathrm{d} D^{(0)}_{\rm R}+\bar{D}^{(n)}_{\rm L}\Lambda_\mathrm{d} \hat{D}^{(n)}_{\rm R}+\bar{D}^{(n)}_{\rm R}\Lambda_\mathrm{d} \hat{D}^{(n)}_{\rm L} \\ &&+\left(\frac{n}{R}\right)\left(\bar{D}^{(n)}_{\rm L}D^{(n)}_{\rm R} +\bar{\hat{D}}^{(n)}_{\rm R}\hat{D}^{(n)}_{\rm L} \right) \\ &&+\,\bar{U}^{(0)}_{\rm L}\Lambda_{\rm u} U^{(0)}_{\rm R}+\bar{U}^{(n)}_{\rm L}\Lambda_\mathrm{u} \hat{U}^{(n)}_{\rm R}+\bar{U}^{(n)}_{\rm R}\Lambda_\mathrm{u} \hat{U}^{(n)}_{\rm L} \\ &&+\left(\frac{n}{R}\right)\left(\bar{U}^{(n)}_{\rm L}U^{(n)}_{\rm R} +\bar{\hat{U}}^{(n)}_{\rm R}\hat{U}^{(n)}_{\rm L} \right)+{\rm h.c.}, \end{array} $$

(C.2)

where $\Lambda_{e,d,u}=({v}/{\sqrt{2}})\lambda_{e,d,u}$ are matrices in the flavour space. Additionally,

$$ \begin{array}{rll} E^{(n)}_{\rm L,R}&=&\left(\begin{array}{ccc} e^{(n)} \\ \, \, \\ \mu ^{(n)} \\ \, \\ \tau^{(n)} \end{array}\right)_{\rm L,R} , \quad n=0,1,... , \\ N^{(n)}_{\rm L,R}&=&\left(\begin{array}{ccc} \nu_e^{(n)} \\ \, \, \\ \nu_\mu ^{(n)} \\ \, \\ \nu_\tau^{(n)} \end{array}\right)_{\rm L,R} , \quad n=1,2,... , \end{array} $$

(C.3)

$$ D^{(n)}_{\rm L,R}=\left(\begin{array}{ccc} d^{(n)} \\ \, \, \\ s ^{(n)} \\ \, \\ b^{(n)} \end{array}\right)_{\rm L,R} , \qquad U^{(n)}_{\rm L,R}=\left(\begin{array}{ccc} u^{(n)} \\ \, \, \\ c ^{(n)} \\ \, \\ t^{(n)} \end{array}\right)_{\rm L,R} \, , \qquad n=0, 1,... , $$

(C.4)

$$ \hat{E}^{(n)}_{\rm L,R}=\left(\begin{array}{ccc} \hat{e}^{(n)} \\ \, \, \\ \hat{\mu}^{(n)} \\ \, \\ \hat{\tau}^{(n)} \end{array}\right)_{\rm L,R} , \qquad n= 1,2,... , $$

(C.5)

$$ \hat{D}^{(n)}_{\rm L,R}=\left(\begin{array}{ccc} \hat{d}^{(n)} \\ \, \, \\ \hat{s} ^{(n)} \\ \, \\ \hat{b}^{(n)} \end{array}\right)_{\rm L,R} , \qquad \hat{U}^{(n)}_{\rm L,R}=\left(\begin{array}{ccc} \hat{u}^{(n)} \\ \, \, \\ \hat{c} ^{(n)} \\ \, \\ \hat{t}^{(n)} \end{array}\right)_{\rm L,R} \, , \, \, \, \, n= 1,2, ... . $$

(C.6)

The mass eigenstates of the zero modes are determined by means of the standard unitary transformations

$$ N'^{(0)}_{\rm L}=V^e_{\rm L}N^{(0)}_{\rm L} , $$

(C.7)

$$ E'^{(0)}_{\rm L,R}=V^e_{\rm L,R}E^{(0)}_{\rm L,R} , $$

(C.8)

$$ D'^{(0)}_{\rm L,R}=V^d_{\rm L,R}D^{(0)}_{\rm L,R} , $$

(C.9)

$$ U'^{(0)}_{\rm L,R}=V^u_{\rm L,R}U^{(0)}_{\rm L,R} . $$

(C.10)

Notice that, as it is usual, the left-handed neutrinos are rotated in the same way as the left-handed charged leptons. Regarding the excited KK modes, we impose the following transformations:

$$ E'^{(n)}_{\rm L,R}=V^e_{\rm L}E^{(n)}_{\rm L,R} , \, \, \, \, \hat{E}'^{(n)}_{\rm L,R}=V^e_{\rm R}\hat{E}^{(n)}_{\rm L,R} , $$

(C.11)

$$ D'^{(n)}_{\rm L,R}=V^d_{\rm L}D^{(n)}_{\rm L,R} , \, \, \, \, \hat{D}'^{(n)}_{\rm L,R}=V^d_{\rm R}\hat{D}^{(n)}_{\rm L,R} , $$

(C.12)

$$ U'^{(n)}_{\rm L,R}=V^u_{\rm L}U^{(n)}_{\rm L,R} , \, \, \, \, \hat{U}'^{(n)}_{\rm L,R}=V^u_{\rm R}\hat{U}^{(n)}_{\rm L,R} . $$

(C.13)

We also demand that the neutrino excitations transform as the corresponding charged lepton excitations:

$$ \label{eq1} N'^{(n)}_{\rm L,R}=V^e_{\rm L}N^{(n)}_{\rm L,R} \,. $$

(C.14)

Once carried out these transformations, one obtains

$$ -{\cal L}^{f}_{\rm mass}={\cal L}^e_{\rm mass}+{\cal L}^d_{\rm mass}+{\cal L}^u_{\rm mass} , $$

(C.15)

where

$$ \begin{array}{rll} {\cal L}^e_{\rm mass}&=&\bar{E}'^{(0)}_{\rm L}M^{(0)}_e E'^{(0)}_{\rm R}+\left(\frac{n}{R}\right)\bar{\nu}^{(n)}_{\rm L}\nu^{(n)}_{\rm R} +\sum\limits_{a=1}^3\,m_{e^{(n)}_a}\, \left(\bar{e}'^{(n)}_{a{\rm L}}\, \, \, \bar{\hat{e}}'^{(n)}_{a{\rm L}} \right)\\ &&\times\left(\begin{array}{ccc} \cos\alpha^{(n)}_{e_a} & \sin\alpha^{(n)}_{e_a} \\ \, \, \\ \sin\alpha^{(n)}_{e_a} & \cos\alpha^{(n)}_{e_a} \end{array}\right) \left(\begin{array}{ccc} e'^{(n)}_{a{\rm R}} \\ \, \, \\ \hat{e}'^{(n)}_{a{\rm R}} \end{array}\right)+ {\rm h.c.} \end{array} $$

(C.16)

$$ \begin{array}{rll} {\cal L}^d_{\rm mass}&=&\bar{D}'^{(0)}_{\rm L}M^{(0)}_{\rm d} D'^{(0)}_{\rm R}+\sum_{a=1}^3\,m_{d^{(n)}_a}\,\left(\bar{d}'^{(n)}_{a{\rm L}}\, \, \, \bar{\hat{d}}'^{(n)}_{a{\rm L}} \right) \\ &&\times\left(\begin{array}{ccc} \cos\alpha^{(n)}_{d_a} & \sin\alpha^{(n)}_{d_a} \\ \, \, \\ \sin\alpha^{(n)}_{d_a} & \cos\alpha^{(n)}_{d_a} \end{array}\right) \left(\begin{array}{ccc} d'^{(n)}_{a{\rm R}} \\ \, \, \\ \hat{d}'^{(n)}_{a{\rm R}} \end{array}\right) + {\rm h.c.} \end{array} $$

(C.17)

$$ \begin{array}{rll} {\cal L}^u_{\rm mass}&=&\bar{U}'^{(0)}_{\rm L}M^{(0)}_{\rm u} U'^{(0)}_{\rm R}+\sum\limits_{a=1}^3\,m_{u^{(n)}_a}\,\left(\bar{u}'^{(n)}_{a{\rm L}}\, \, \, \bar{\hat{u}}'^{(n)}_{a{\rm L}} \right) \\ &&\times\left(\begin{array}{ccc} \cos\alpha^{(n)}_{u_a} & \sin\alpha^{(n)}_{u_a} \\ \, \, \\ \sin\alpha^{(n)}_{u_a} & \cos\alpha^{(n)}_{u_a} \end{array}\right) \left(\begin{array}{ccc} u'^{(n)}_{a{\rm R}} \\ \, \, \\ \hat{u}'^{(n)}_{a{\rm R}} \end{array}\right)+ \,{\rm h.c.} \end{array} $$

(C.18)

In the above expressions, $M^{(0)}_{e}={\rm diag}(m_{e^{(0)}},m_{\mu^{(0)}}, m_{\tau^{(0)}})$, etc. In addition,

$$ \tan\alpha^{(n)}_{f_a}=\frac{m_{f^{(0)}_a}}{\left({n}/{R}\right)} , $$

(C.19)

where f _a stands for a charged lepton or quark. The mass eigenstates associated with the KK modes, which we shall denote by $\tilde{f}_a$ and $\tilde{\hat{f}}_a$, are given by the following unitary transformations:

$$ \left(\begin{array}{ccc} f'^{(n)}_{a{\rm L}} \\ \, \, \\ \hat{f}'^{(n)}_{a{\rm L}}\\ \end{array}\right)=V_{\rm L}\left(\begin{array}{ccc} \tilde{f}^{(n)}_{a{\rm L}} \\ \, \, \\ \tilde{\hat{f}}^{(n)}_{a{\rm L}}\\ \end{array}\right), \, \, \, \, \, \left(\begin{array}{ccc} f'^{(n)}_{a{\rm R}} \\ \, \, \\ \hat{f}'^{(n)}_{a{\rm R}}\\ \end{array}\right)=V_\mathrm{R}\left(\begin{array}{ccc} \tilde{f}^{(n)}_{a{\rm R}} \\ \, \, \\ \tilde{\hat{f}}^{(n)}_{a{\rm R}}\\ \end{array}\right), $$

(C.20)

where

$$ \begin{array}{rll} V_{\rm L}&=&\left(\begin{array}{ccc} \cos\left({\alpha^{(n)}_{f_a}}/{2}\right) & \, \, \sin\left({\alpha^{(n)}_{f_a}}/{2}\right) \\ \, \, \\ \sin\left({\alpha^{(n)}_{f_a}}/{2}\right) & -\cos\left({\alpha^{(n)}_{f_a}}/{2}\right) \\ \end{array}\right), \\ V_{\rm R}&=&\left(\begin{array}{ccc} \cos\left({\alpha^{(n)}_{f_a}}/{2}\right) & -\sin\left({\alpha^{(n)}_{f_a}}/{2}\right) \\ \, \, \\ \sin\left({\alpha^{(n)}_{f_a}}/{2}\right) & \, \, \cos\left({\alpha^{(n)}_{f_a}}/{2}\right) \\ \end{array}\right). \end{array} $$

(C.21)

$\tilde{f}_a$ and $\tilde{\hat{f}}_a$ states are degenerate, with mass given by

$$ m_{f^{(n)}_a}=\sqrt{\left(\frac{n}{R}\right)^2+m^2_{f^{(0)}_a}} . $$

(C.22)