COSMOLOGICAL BARYON/LEPTON ASYMMETRY
IN TERMS OF KALUZA-KLEIN EXTRA DIMENSIONS

The Lagrangian of the scalar field and its interaction with the fermion field is chosen as simple as possible

In this work, we do not specify the form of the potential $V(\Phi)$ — it can be ignored and it is used only in subsection 4.3.

Fermions acting in the curved 6-dim space-time are described by action [7]

where $\Gamma^{\tilde{A}}$ is the flat gamma matrices, $G_{AB}$ is the metric and $h_{\tilde{A}}^{B}$ is the frame field. The latter relates to the metric as follows:

We choose the gamma matrices in 6 dimensions (two extra coordinates $y_{a}$ are denoted as $\theta,\varphi$) as follows

The covariant derivative of the fermion field is expressed in terms of the spin connection:

where the spin connection is expressed in terms of the Christoffel symbols (there is no torsion in our model) and the tetrads:

As is well known, the charge conservation is related to a Lagrangian invariance under a group transformation. In our case, we interpret such invariance as an extra space geometric isometry. Therefore, the extra space metric should be invariant under group transformations. Our analysis is based on metrics that possess U(1) isometry. The simplest example of such sort is the 2-dim apple-like metric

Here $g_{\mu\nu}\left(x^{\alpha}\right)$ is 4-dim metric, $r_{0}$ is the radius of compact 2-dim extra space, coefficient $\beta(\theta)$ defines geometry of extra space, parameterized by azimuthal angle $\theta$, and $b$ is the parameter, related to the deficit or excess of angle $\varphi$ in the extra space. This metric has been discussed in Ref. \refciteGogberashvili_2007 where the variety of analytical results were obtained. Also, such kind of manifolds have been revealed Refs. \refcite2016arXiv160907361R,Gani:2014lka,Rubin:2018ybs,Bronnikov:2020tdo as the static numerical solutions to the generalized Einstein equations. Some of such solutions can be approximated by metric (8) which strongly facilitate the analysis below.

The Dirac equation for a multidimensional fermion field has the form

where the tetrad

is associated with the metric (4),(8).

The solution to this equation can be decomposed as:

where $\psi_{jl}$ and $\xi_{jl}$ is 4-dim Dirac spinors, and $Y^{+}_{jl}$, $Y^{-}_{jl}$ is complete set of functions (that is the eigenfunctions of the corresponding operator, see A for definition). Number $j$ corresponds to the energy (mass) level of KK-tower and varies from 0 to infinity. We are interested only in massless modes $j=0$.

The integer number $l$ varies in the range $-b<2l<b$ (see A) and can be associated with the internal angular momentum. This number could describe the triplet of fermions in the ground state if the extra space angle excess varies in the interval $2<b\leq 4$.

Let us obtain the effective 4-dim action starting from its initial form (3). Using Kaluza-Klein decomposition (11) (see A, (37),(38)) and extra space tetrad (10) we get 4-dim action for the 4-dim fermions (KK-tower):

It can be seen that the masses of nonzero modes $j\neq 0$ are proportional to the inverse radius of the extra space $M_{j},M^{\prime}_{j}\sim h^{n}_{\tilde{m}}\sim 1/r_{0}$. We are not interested in massive modes $j\neq 0$, since they are extremely heavy due to small extra space size $r_{0}\ll 10^{-19}$ cm [18, 13].

Further, we consider only massless modes $\psi_{0l}=\psi_{l},\,\xi_{0l}=\xi_{l}$. As we have indicated above (and in A) there are three independent modes for the ground state $j=0$: $l=-1,0,1$. Thus, the low-energy effective action contains two 4-dimensional fermion triplets:

The fermions $\psi_{l}$ and $\xi_{l}$ are indistinguishable from the 4-dimensional point of view, so further we will focus on one of them: $\psi_{l}=\{\psi_{-1},\psi_{0},\psi_{+1}\}$.

The geometric symmetries give rise the conserved currents $\nabla_{A}J^{A}=0$ and the conserved charges associated with them [7]. Here we consider the compact symmetric extra dimensions. Let the extra–spatial metric $k_{mn}$ remains invariant under the infinitesimal isometries $y^{m}\rightarrow y^{m}+\xi^{m}(y)$, where $\xi^{m}(y)$ is the Killing vector field. According to the Noether’s theorem there is a conserved current associated with such invariance. This current is given by

for any matter field $\chi$. Here $L_{\text{m}}$ is a Lagrangian of matter. The corresponding conserved number

In the lower (vacuum) state, the metric of the extra space (8) comes to the unperturbed axially symmetrical configuration: $\beta(y)=\beta(\theta)$ [19]. The corresponding symmetry is just rotational symmetry along coordinate $\varphi$. It is characterized by the Killing vector field, which in spherical coordinates has the form [14, 20, 21]:

The conserved number $Q$, corresponding to the Killing vector (16), has the meaning of the total internal angular momentum of the fields along the coordinate $\varphi$:

where $J^{0}$ is time component of corresponding conserved current for action $S_{\Phi}+S_{\Psi}$. It has the following form:

In the rest of the section 3, we will consider only the fermion part of (18). In section 4 we explain why the fermion field alone is not enough to generate the lepton/baryon number, and describe the reason for introducing the scalar field.

From the 4-dim observer point of view, the fermion part of conserved current takes the form:

where we consider only $\psi_{l}$ fields in effective Lagrangian $\mathcal{L}_{4}$ (13). Right equation in (20) represents the effective 4-dim generator matrix of primordial 6-dim geometric extra space symmetry (for general explanation, see the B).

Associated conserved number $Q_{\Psi}$ takes the form:

where the capital letter $N_{\psi}$ denotes the 4-dim fermion number of the corresponding particle (the number of particles minus the number of antiparticles), which is positive for an excess of matter and negative for an excess of antimatter.

We see, that internal (extra space) momentum of the multidimensional fermion field is equal to the difference of fermion numbers of its effective 4-dim components with opposite index $l$ and it is conserved. We will interpret the internal momentum $Q_{\Psi}$ as lepton/baryon number.

The fermion action (3) is invariant under the global phase transformation $\Psi\rightarrow e^{i\theta}\Psi$. This symmetry, in contrast to the one discussed above, is the symmetry of the Lagrangian itself and does not relate to the symmetry of extra space metric. The global invariance is responsible for the conservation of the total multidimensional fermion number

The particles $\psi_{+1}$ and antiparticles $\psi_{-1}$ have positive lepton (baryon) number $Q_{\Psi}$, see (21). Therefore we identify these particles with leptons (baryons). A nonzero value $Q_{\Psi}$ points to a nonzero difference between the leptons (baryons) and antileptons (antibaryons) — the lepton (baryon) asymmetry.

The scalar field charge relates to its internal momentum in the extra space and has the form

following from the expression (18). Total charge

includes both the fermion and boson part. This charge is conserved after the inflation is finished and the extra space metric is stabilized. Under these conditions the total fermion number (22) is also conserved:

Thus, if we consider only massless fermions $\psi_{l}$, the field $\Phi$ can decay into 3 different modes: $l=-1,0,+1$. Because of the conservation laws (24), (25), the non-zero momentum $Q_{\Phi}$ (for example, with a positive sign) can transferred into the $\psi_{+1}$ mode with the simultaneous generation of a similar number of antifermions $\psi_{0}$. If we identify the lepton (baryon) number with the $+1$ mode, we obtain a mechanism for generating lepton (baryon) asymmetry.

Consider the Yukawa term in action (2) and expand it over the vacuum state of the scalar field $\Phi(y)=\Phi_{\text{vac}}(y)+\delta\Phi(y)$:

The second term describes the interaction of scalar field perturbations and the fermions. It is the scalar field perturbations $\delta\Phi$ that accumulate the total nonzero internal moment during inflation.

After decomposition and integration over the extra coordinates, the effective 4-dimensional interaction acquires the form

where functions $A^{lm}(x),B^{lm}(x)$ can be considered as classical 4-dimensional sources proportional to the scalar field fluctuations $\delta\Phi$:

The vertices corresponding to the interaction (27) describe the decay of the scalar field condensate $\delta\Phi$ into effective 4-dimensional fermions $\psi_{l},\xi_{l}$:

d1 {fmfgraph*} (60,37) \fmflefti1 \fmfrighto1,o2 \fmfvlabel=$A^{lm}$,l.a=180,l.d=.1wi1 \fmfvlabel=$\psi_{l}$,l.a=0,l.d=.05wo1 \fmfvlabel=$\overline{\psi}_{m}$,l.a=0,l.d=.05wo2 \fmfvlabel=$f$,l.a=110,l.d=.1ww1 \fmffermiono2,w1,o1 \fmfdashesi1,w1 \fmfdotw1 \fmfvdecor.shape=crossi1 {fmfgraph*} (60,37) \fmflefti1 \fmfrighto1,o2 \fmfvlabel=$B^{lm}$,l.a=180,l.d=.1wi1 \fmfvlabel=$\xi_{l}$,l.a=0,l.d=.05wo1 \fmfvlabel=$\overline{\xi}_{m}$,l.a=0,l.d=.05wo2 \fmfvlabel=$f$,l.a=110,l.d=.1ww1 \fmffermiono2,w1,o1 \fmfdashesi1,w1 \fmfdotw1 \fmfvdecor.shape=crossi1

We assumed zero masses of the fermions from the beginning. In theories with extra dimensions, there are a variety of ways to give mass to the fermions. For example, in Ref. \refciteneronov2002fermion the emergence of masses and mixing is associated with the presence of angular momentum in the extra space.

For our purposes, there is already a scalar field that naturally induces the Higgs mechanism. Consider the first term in (26):

In (29), we used the expansion of fermions in terms of eigenfunctions (11). The vacuum state $\Phi_{\text{vac}}$ is a homogeneous in 4 dimensions and stationary solution of the Klein-Gordon equation:

where $\Box_{M},\Box_{K}$ is the multidimensional and extradimensional d’Alembert operators, respectively. Due to the rotational symmetry of the vacuum state, $\Phi_{\text{vac}}(y)=\Phi_{\text{vac}}(\theta)$, which leads to the diagonalization of the matrices $M^{\pm}_{lm}\sim\int e^{i(l-m)\varphi}d\varphi\sim\delta_{lm}$.

The structure of eigenfunctions according to the B is as follows

Thus, 3 different masses arise, determined by the dependence of the eigenfunction on the azimuthal angle $\theta$. In addition, due to the azimuthal asymmetry of the apple-like extra space, a hierarchy arises $m_{+}<m_{0}<m_{-}$.