# Positioning Error Impact Compensation through Data-Driven Optimization in
User-Centric Networks

Waseem Raza1, Fahd Ahmed Khan1, Muhammad Umar Bin Farooq1, Sabit Ekin2 and Ali
Imran13  1AI4Networks Research Center, School of Electrical & Computer
Engineering, University of Oklahoma, OK, USA
3James Watt School of Engineering, University of Glasgow, United Kingdom.
2School of Electrical and Computer Engineering, Texas A&M University, TX, USA.
Email: {waseem, fahd.khan, umar.farooq<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The performance of user-centric ultra-dense networks (UCUDNs) hinges on the
Service zone (Szone) radius, which is an elastic parameter that balances the
area spectral efficiency (ASE) and energy efficiency (EE) of the network.
Accurately determining the Szone radius requires the precise location of the
user equipment (UE) and data base stations (DBSs). Even a slight error in
reported positions of DBSs or UE will lead to an incorrect determination of
Szone radius and UE-DBS pairing, leading to degradation of the UE-DBS
communication link. To compensate for the positioning error impact and improve
the ASE and EE of the UCUDN, this work proposes a data-driven optimization and
error compensation (DD-OEC) framework. The framework comprises an additional
machine learning model that assesses the impact of residual errors and
regulates the erroneous data-driven optimization to output Szone radius,
transmit power, and DBS density values which improve network ASE and EE. The
performance of the framework is compared to a baseline scheme, which does not
employ the residual, and results demonstrate that the DD-OEC framework
outperforms the baseline, achieving up to a $23\%$ improvement in performance.

###### Index Terms:

User-centric ultra-dense networks (UCUDNs), Positioning errors, Machine
Learning, Residual learning, and multi-objective optimization.

##  I Introduction

Emerging wireless networks are adopting dense deployment to support an array
of applications with diverse requirements [1]. The cellular network is
required to simultaneously facilitate high data-rate extended reality
applications, ultra-reliable low-latency communication for intelligent
transportation systems, and energy-constrained devices with low data rates,
such as surveillance systems or body area networks [2]. The dense deployment
of base stations and heterogeneity of services make network management
increasingly complex as the number of configuration parameters grows
exponentially. In this context, Artificial Intelligence (AI) and Machine
Learning (ML) techniques demonstrate breakthrough performances, leading to an
era of AI-based wireless network configuration as an inevitable future [3, 4].
Although mathematical and analytical techniques are still useful, data-driven
models undoubtedly play a complementary role in improving future network
design and operational performance. Hence, data-driven modeling is employed
for mobility prediction, proactive handovers, network fault diagnosis, and
other tasks [5].

In addition to incorporating intelligence into emerging networks, new
architectures for 5G and beyond networks that can quickly adapt to changing
service requirements are being explored. One such elastic architecture is the
user-centric architecture with flexible service zones, which is gaining
recognition due to its promising benefits, such as lower interference,
improved area spectral efficiency, and better quality of experience (QoE) by
eliminating cell-edge users [6, 7]. These benefits are realized by serving
users within non-overlapping flexible service zones (Szones) based on their
service requirements. The Szone of a user-centric ultra-dense network (UCUDN)
and serving data base station (DBS) activation is controlled by a central
controller, which requires accurate user equipment (UE) and DBS location
information for efficient network operation and resource allocation.

While accurate location information is also necessary for base station (BS)
centric networks in tasks involving network automation and self-healing, its
importance increases in UCUDNs. Even a slight error in the position of the UE
or the DBS can result in scheduling UEs that are in close proximity to each
other or activating DBSs that are far from the UEs. Both scenarios can result
in high interference, which can have a detrimental effect on network
performance. Therefore, in real-world UCUDNs, where the central controller may
not have access to error-free locations of all UEs and DBS, the use of AI and
ML techniques to compensate for imperfect location information is essential
for its data-driven operations [8, 9].

Characterization of the impact of positioning errors in traditional BS-centric
networks has been carried out in several studies [10, 11, 12]. The reliability
of the data-based autonomous coverage estimation in the presence of UE and BS
positioning error was investigated and quantified in [10]. Onireti et. al.,
[11] demonstrated that inaccuracies in positioning techniques could result in
faulty associations, leading to degraded area spectral efficiency (ASE). This
issue is further exacerbated by the increased BS density in ultra-dense
networks. In [12], the authors demonstrated that an optimal bin width could
mitigate the effects of positioning and quantization errors, resulting in the
most accurate coverage estimate.

Most existing works on UCUDNs assume accurate location information for UEs and
DBS [9, 14, 8], and there has been very little work on investigating and
understanding the impact of positioning error. Erroneous position estimates
were considered in [13], where time-series forecasting was utilized to predict
the trend of received signal power, ASE, and energy efficiency (EE) values,
with varying three configuration and optimization parameters (COPs), namely,
transmit power, Szone radius, and DBS density. The prediction was restricted
within a prediction window, and only a limited number of COP combinations were
explored. Moreover, the error prediction was not utilized to determine the
optimal COP combination values, which jointly optimize the network
performance. In this work, we adopt a different approach compared to [13], and
an AutoML model is trained to learn the impact of position error on network
key performance indicators (KPIs). This error characterization is then
utilized to compensate for the joint KPI optimization and yield improved
network configuration values of transmit power, Szone radius, and DBS density.
The contribution of this paper is summarized below.

  * •

We propose a data-driven optimization and error compensation (DD-OEC)
framework to jointly maximize two KPIs, namely, ASE and EE, where UE service
is impacted by the positioning error. The framework adopts a data-driven
approach and utilizes an AutoML module for learning the relationship between
the COPs and the KPIs. The trained ML model is then employed for heuristic
multi-objective optimization.

  * •

We demonstrate that the candidate COP solution derived from data with
positioning errors is suboptimal. To mitigate this adverse impact of
positioning errors, we employ a novel AutoML model to learn the _residual_
error obtained from the difference between the current erroneous database and
a historical error-free database. This residual error is then utilized during
the optimization process to compute COP values that improve the corresponding
KPI values.

  * •

The performance of the proposed DD-OEC is compared with a baseline scheme that
does not take into account the _residual_. Simulation results reveal that the
proposed DD-OEC scheme is able to learn the residual and incorporate it to
perform better than the baseline scheme. Therefore, the DD-OEC optimization is
necessary to compensate for the error impact on the optimization solution.

(a) Positioning error modeling.

(b) The data-driven optimization and error compensation modeling (DD-OEC)
framework.

Fig. 1: Positioning error modeling, and the DD-OEC framework for the data-
driven optimization and error compensation COP-KPI optimization. This
framework includes synthetic data generation, AutoML based model fitting, and
multi objective optimization.

##  II System Model

We consider a UCUDN, where the DBSs and the UEs are assumed to be distributed
randomly over an area using a Poisson point process. The density of the DBSs
and the UEs is represented by $\lambda_{dbs}$ and $\lambda_{ue}$,
respectively. All the DBS are equipped with a single omnidirectional antenna
and transmit with the same power. A macro BS acts as the central controller,
collecting data from the UEs and scheduling them during each transmission time
interval. The controller creates virtual non-overlapping Szones, of radius
$R_{sz}$, around the scheduled UEs and activates a DBS within each Szone to
serve the respective UEs as shown in Fig. 1b. This ensures that there is at
least a distance greater than $2R_{sz}$ between the scheduled UEs. Moreover,
the UE is served by the DBS, having the best channel power, located within a
distance of $R_{sz}$ from the UE. Consequently, if the distance between UEs is
less than $2R_{sz}$, then they are scheduled in different TTIs based on
scheduling priority. This scheduling and activation criterion limits the
interference to the serviced UEs. Moreover, if a UE has no DBS within its
Szone, the network can adjust and increase $R_{sz}$ to serve the disadvantaged
UEs.

###  II-A Channel Model

We adopt the close-in two-slope path-loss model [15],

$PL(d)=-F(f,1m)_{dB}-10l_{1}\log
d-10l_{2}\log\left(\frac{d}{d_{t}}\right)u(d-d_{t}),$ (1)

where $d$ is the 3D distance between the UE and BS,
$F(f,1m)_{[dB]}=20\log_{10}\big{(}\frac{4\pi f}{c}\big{)}$ is the free space
pathloss in dB at a transmitter-receiver distance of 1 meter at carrier
frequency $f$ with $c$ representing the speed of light, $u(\cdot)$ denotes the
unit step function, $d_{t}$ is the breakpoint or threshold distance, $l_{1}$
and $l_{2}$ are the path loss exponents for a distance less than $d_{t}$ and
greater than $d_{t}$, respectively. The signal-to-interference plus noise
ratio (SINR) for UE $x$ is given as,

$\gamma_{x}=\frac{P_{x}G_{x}\chi_{x}10^{\frac{{PL}(d_{x})}{10}}}{N_{0}+\sum\limits_{\forall
i\in\mathbb{I}}P_{i}G_{i}\chi_{i}10^{\frac{{PL}(d_{i})}{10}}},$ (2)

where $P_{x}$ is the transmit power of the serving DBS, $G_{x}$ is the
downlink antenna gain of the serving DBS, $\chi_{x}$ denotes the channel
shadowing modeled as a log-normal distribution with $0$dB mean and $4$dB
variance, $d_{x}$ is the distance between the serving DBS and UE $x$. $N_{0}$
is the thermal noise power, $P_{i}$ and $G_{i}$ are the transmit power and
antenna gain of the $i$-th interfering DBS, respectively. $\chi_{i}$ shows the
channel shadowing for interfering BS $i$ modeled as a log-normal distribution
with $0$dB mean and $4$dB variance, and $d_{i}$ is the distance between UE $x$
and the $i$-th interfering DBS. $\mathbb{I}$ is the set of all interfering BS.

###  II-B Key Performance Indicators:

The performance of the UCUDN is determined based on area spectral efficiency
and network energy efficiency. ASE is defined as the number of bits that are
transmitted to a UE from a DBS per unit bandwidth per unit area and is
mathematically expressed as $\theta_{se}=\frac{1}{A}\sum\limits_{\forall
x\in\mathbb{U}}\log_{2}(1+\gamma_{x})$, where $\mathbb{U}$ is the set of all
served UEs in the network and $A$ is the simulation area. EE is defined as
$\eta_{ee}=\frac{A}{P_{T}}\theta_{se}$, where $P_{T}$ is the total power
consumption of all CBS and DBS in the network following the network energy
consumption model in [16].

###  II-C Positioning Error Impact and its Modeling

In UCUDN, the central controller requires exact location information of the
UEs and DBSs for scheduling, DBS activation, and determining $R_{sz}$.
However, in practical systems, only estimates of the positions are
available111Although there can be various other sources of imperfections such
as channel modeling errors, quantization errors, etc. however, the scope of
this paper is limited to the impact of error in UE and DBS positions.. These
positioning errors affect UE scheduling, DBS activation, and Szone radius. For
instance, it can lead to the scheduling of UEs that are spaced less than
$2R_{sz}$ apart, thus, increasing the interference and lowering the QoE.
Similarly, a DBS can be activated which is outside the Szone of the UE, again
resulting in lower reference signal received power (RSRP) and increased
interference to other UEs.

For error modeling, we assume that the ideal position of a DBS or UE will be
uniformly distributed within the circular ring of error-radius, $R_{er}$,
centered around the position estimates. A lower value of $R_{er}$ will
indicate a more accurate estimate and vice versa. In the subsequent
discussion, we define _perceived distance_ as the distance between the UE and
DBS based on the position estimates and _actual distance_ as the true physical
distance. The former is shown by the subscript ‘e’, and the latter by ‘a’ in
the distance d in the following discussion.

To elaborate the positioning error impact, we assume a UE $x$ and two DBSs $A$
and $B$ as shown in Fig. 1(a). In the ideal case, referred as Case J, the
actual and reported positions of UE $x$ and DBS $A$ are the same; hence the
Szone is shown by a green solid circle in Fig. 1(a), and the SINR depends on
the actual distance $d_{a}^{A}$. However, with positioning errors, two
scenarios occurs based on the error magnitude, as shown in Fig. 1(a).

  * •

In low error regimes, shown as _Case L_ , despite the position errors, the UE-
DBS association is not changed and remains the same as the ideal case.
However, despite this ideal-like association, the central controller allocates
resources based on the perceived distance, $d_{e}^{A}$, which is different
from the actual distance $d_{a}^{A}$.

  * •

In the high error regimes, the positioning errors cause a change in UE-DBS
association, i.e., UE $x$ is now associated with the distant DBS $B$ instead
of DBS $A$ due to lower perceived distance $d_{e}^{B}$, as shown in _Case H_
in the figure. It is possible that in this case, the DBS $B$ might actually
lie outside of the S-zone. As a result, in addition to the sub-optimal
resource allocation (based on the perceived distance), the larger value of
$d_{a}^{B}$, leads to a significant deterioration of SINR at the UE.

Thus, from this discussion, it can be summarized that positioning errors can
result in significant variations in the perceived distances between UEs and
DBSs. This, in turn, can lead to suboptimal resource allocation and UE-DBS
association, negatively affecting the network’s KPIs of interest.

##  III Optimization Problem Formulation and DD-OEC Framework

In this Section, we present the optimization objective function and introduce
the proposed framework for data-driven optimization and error compensation
(DD-OEC).

###  III-A Optimization Problem Formulation and Algorithm

As mentioned earlier, our focus in this study is to enhance both the ASE and
EE. Since these two KPIs exhibit a certain degree of trade-off, we develop a
multi-objective optimization function for the data-driven optimization
process. The objective function is formulated as the weighted sum of the
normalized KPIs, which is specifically defined as,

$f_{obj}(\theta_{se},\leavevmode\nobreak\
\eta_{ee})=\alpha_{se}\frac{\theta_{se}}{\theta_{se}^{max}}+(1-\alpha_{se})\frac{\eta_{ee}}{\eta_{ee}^{max}},$
(3)

where $\alpha_{se}$ and $1-\alpha_{se}$ are the weight factors for ASE and EE,
respectively. These KPIs are functions of the COPs; DBS density, Szone radius,
and transmit power. Since maximizing both ASE and EE is our goal, the joint
optimization problem can be expressed as follows,

$\displaystyle\max_{\lambda_{dbs},\leavevmode\nobreak\
R_{sz},\leavevmode\nobreak\ P_{tx}}$ $\displaystyle
f_{obj}(\theta_{se},\leavevmode\nobreak\ \eta_{ee}),$ (4) s.t. $\displaystyle
0\leq\lambda_{dbs}^{min}\leq\lambda_{dbs}\leq\lambda_{dbs}^{max},$
$\displaystyle 0\leq R_{sz}^{min}\leq R_{sz}\leq R_{sz}^{max},$ $\displaystyle
0\leq P_{tx}^{min}\leq P_{tx}\leq P_{tx}^{max}.$

where, $\lambda_{dbs}^{min}$ and $\lambda_{dbs}^{max}$ denote the minimum and
maximum value of DBS density, $R_{sz}^{min}$ and $R_{sz}^{max}$ denote the
minimum and maximum value of Szone radius, and $R_{sz}^{min}$ and
$R_{sz}^{max}$ denote the minimum and maximum value of Szone radius,
respectively. These minimum and maximum values are selected based on domain
knowledge along with trial and testing in the simulator and are mentioned in
Table II. The optimization problem in (4) is non-convex, and therefore, we
resort to meta-heuristic algorithms, Simulated Annealing, and Genetic
Algorithm, for finding the optimal solution [17, 4].

Simulated annealing (SA) involves several parameters that specify its
annealing schedules, such as the initial temperature, cooling schedule, number
of iterations, and stopping criteria. The values of these parameters should be
chosen such that the objective function is sampled across the entire solution
space. Following the discussion in [17], we select an adaptive temperature
schedule that adjusts the rate of cooling based on the previous runs i.e.
$T_{k+1}=\frac{T_{k}}{1+\left(\ln\left(1+\delta\right)3\sigma T_{k}\right)}$
where, $\sigma$ is the standard deviation of the objective function and
$\delta$ is a small real number. Genetic algorithm (GA), on the contrary,
relies on bio-evolutionary operations like mutation, crossover, and natural
selection to direct the random search into a better solution space and
eventually achieve the best solution [4].

###  III-B Synthetic Data Generation

To achieve data-driven optimization, the initial step is to train an ML model
to learn the relationship between the COPs and network KPIs using data
gathered from the network. Subsequently, this ML model is utilized in
conjunction with the OE to determine the optimal COP values. To obtain the
necessary data, we generate synthetic COP-KPI data using UC-SyntheticNET, a
module of the 3GPP compliant system level simulator to model user-centric
networks [18]. The simulator considers spatial correlations, as well as
mobility, to generate realistic COP-KPI data. We enhance the simulator module
by incorporating errors in the positions of UEs and DBSs and simulating their
resulting effects on relevant KPIs. This enables the simulator to generate
both ideal and erroneous databases.

Since ideal data is unavailable at runtime in real networks, it cannot be used
for model training in the proposed DD-OEC framework. Instead, the ideal data
is considered as historical data, which was obtained in the past through
measurement data collection campaigns, such as drive tests. The DD-OEC
framework incorporates this data for residual computation to enhance
optimization performance. Furthermore, it is also used as a benchmark for
performance comparison.

Table I: AutoML-based Model Fitting Module Performance: Regression models; GB: Gradient Boosting, LG: Light GBM, RF: Random Forest, CB: Catboost, AB: Adaboost. | AutoML Model Name | RMSE Performance
---|---|---
| Case
---
Flag
| 1st
---
Model
| 2nd
---
Model
| 3rd
---
Model
| 1st
---
Model
| 2nd
---
Model
| 3rd
---
Model
ASE Model-E | GBR | LGR | RFR | 3.71E-5 | 3.78E-5 | 3.99E-5
EE Model-E | GBR | LGR | CBR | 2.02E-5 | 2.06E-5 | 2.14E-5
ASE Model-R | ABR | LGR | CBR | 1.02E-5 | 2.17E-5 | 2.34E-5
EE Model-R | LGR | ABR | CBR | 2.17E-5 | 2.46E-5 | 2.12E-5

###  III-C DD-OEC Framework

The block diagram of the proposed ML-aided DD-OEC optimization framework for
data-driven optimization is shown in Fig. 1(b). The selected optimization
algorithm and data generation methodology in the DD-OEC framework have been
previously discussed. Next, we discuss the remaining components, their
interconnections, and their reasoning. The DD-OEC framework trains two ML
models. The first ML model called _Model-E_ , is trained on the erroneous data
available at runtime in networks to learn the COP-KPI relation. The COP-KPI
relation learned by this model is inaccurate and sub-optimal due to
positioning errors. To address this issue, a second ML model, named _Model-R_
, is trained to learn the residual error for each input COP. The residual
signal is the difference in KPI values for the erroneous and ideal historical
data. The output of both models is combined during optimization to compensate
for the positioning error and output the COPs, which maximize the desired
network KPIs. The residual learning approach is particularly advantageous when
dealing with complex relationships between input and output data. Rather than
attempting to directly predict the complex, non-linear output, residual
learning focuses on learning the residual, which has a limited variation.
Training on data with low variance makes the training process more efficient
and effective. Furthermore, the impact of noise or errors in the data can be
mitigated, which is needed in our scenario [19].

Since, the main focus of this work is on optimization and error compensation,
and not on the ML model fitting, we resort to the off-the-shelf AutoML
library, PyCaret, which provides robust regression models for the given data.
The ML model fitting problem quite similar to a typical regression problem
with three features (DBS density, Szone radius, and transmit power) and two
target variables (ASE and EE), and we dedicate a separate model for each
target variable. All the standard pre and post-processing steps, like the
feature scaling, K-fold cross-validation, and hyper-parameters optimization,
are carried out within PyCaret. In addition, PyCaret can identify the best-
performing models among the diverse models that are simultaneously trained.
The best-performing models, both Model-E and Model-R, in terms of root mean
square error (RMSE) are given in Table I. The results in Table I show that the
family of boosting models with their typical hyperparameter values predefined
in PyCaret is best suited for the problem under consideration. Specifically,
the gradient boosting regressor (GBR) remains the best model with the minimum
RMSE values. Moreover, compared to the neural networks these models are faster
and requires relatively less data for training. The final key component of the
proposed framework is the OE, which employs two widely utilized heuristic
optimization algorithms, GA and SA. In the proposed framework, Model-R output
is added with Model-E output within the fitness evaluation functions, of both
SA and GA algorithms, for obtaining the COPs which maximize (3).

Table II: Parameters for database generations and simulations. Parameter Name | Value
---|---
DBS Density ($\lambda_{dbs}$) | 0.0005–0.0125
Transmit Power ($P_{tx}$) | 15–30 dBm
Szone Radius ($R_{sz}$) | 10–50 m
UEs and DBS deployment | Poisson Point Process
Positioning Error Distribution | Uniform
Shadowing Standard Deviation | 4
User Density($\lambda_{ue}$) | 0.0005
Network Area | 1km sq
Bin Size | 10
Simulation Size | 100 cycles
Error Radii | 15 m
Population Size | 24
Initial Temperature, $\delta$, $\sigma$ | 250, 0.0001, 0.01

(a) Convergence performance comparison for SA.

(b) Convergence performance comparison for GA.

(c) Required iterations for convergence.

Fig. 2: Performance of the DD-OEC framework with the optimization techniques
of SA and GA for $\alpha_{se}=0.5$.

##  IV Numerical Simulations & Performance Evaluation

The performance of the proposed DD-OEC framework is analyzed through numerical
simulations. To illustrate the benefit of learning the residual and performing
error compensation, the performance of the DD-OEC framework is compared with a
_baseline_ optimization framework, which does not process the residual and
optimizes based on the model solely trained on erroneous data. The performance
is evaluated in terms of the achieved objective function values, KPI values,
and the corresponding COP combinations. In addition, the number of iterations
for convergence is also evaluated. We simulate the user-centric network over a
geographic area of 1 km by 1 km. The values of the relevant parameters for
network simulation and data generation are given in Table II.  For data
generation, $10$ equally spaced values are taken within the ranges specified
in Table II, for each COP parameter. Moreover, as there are three COP
parameters, the length of both erroneous and ideal databases becomes $10^{3}$.
Finally, the parameters, $\delta$ and $\sigma$, describe the initial
temperature and rate of cooling for simulated annealing.

The convergence performance of SA and GA for both DD-OEC and baseline schemes
is shown in Fig. 2(a) and Fig. 2(b), respectively. These performance curves
are obtained after averaging the converged objective function values from
$1000$ Monte-Carlo trials. The COPs obtained after algorithm convergence are
validated on the simulator by giving the converged COPs as input to the
simulator and using the resulting KPIs to evaluate the objective function in
(4). In addition to this, the converged COPs are validated using an ML-based
model of the simulator trained using the ideal (without error) COP-KPI
combinations. These validated values are plotted as horizontal lines and
represent the maximum values after convergence. Moreover, the COPs obtained
through the baseline scheme are also validated in each iteration using the
ideal model.

Table III: Performance comparison of the DD-OEC framework with the baseline
scheme for different weights combinations.

Opt. Algo. Name |  Weight Case ($\alpha_{se}$, $1-\alpha_{se}$) | Baseline Optimization |  COPs Validation on Simulator | DD-OEC Optimization |  COPs Validation on Simulator |  Exec. Time (s)
---|---|---|---|---|---|---
| Objective
---
Function
Opt. COP Comb.
$[\lambda_{dbs},R_{sz},P_{tx}]$ |  | Objective
---
Function
Opt. COP Comb.
$[\lambda_{dbs},R_{sz},P_{tx}]$
SA | (0.25, 0.75) | 0.82=(0.21+0.61) | [5.17E-4, 13.33, 24.41] | 0.77=(0.22+0.55) | 0.85=(0.23+0.62) | [5.79E-4, 10.95, 18.34] | 0.84=(0.23+0.61) | 433
(0.50, 0.50) | 0.86=(0.45+0.41) | [5.74E-4, 11.62, 17.64] | 0.78=(0.44+0.34) | 0.87=(0.45+0.42) | [6.24E-4, 10.05, 20.64] | 0.87=(0.45+0.42) | 406
(0.75, 0.25) | 0.84=(0.61+0.23) | [5.16E-4, 12.94, 23.31] | 0.79=(0.61+0.18) | 0.92=(0.72+0.20) | [5.68E-4, 10.70, 16.35] | 0.91=(0.71+0.20) | 443
GA | (0.25, 0.75) | 0.88=(0.19+0.69) | [5.05E-4, 12.26, 17.23] | 0.72=(0.16+0.56) | 0.89=(0.18+0.71) | [5.89E-4, 11.23, 20.36] | 0.89=(0.18+0.71) | 313
(0.50, 0.50) | 0.94=(0.46+0.48) | [5.50E-4, 11.63, 15.25] | 0.92=(0.48+0.44) | 0.98=(0.49+0.49) | [5.35E-4, 10.16, 15.00] | 0.96=(0.49+0.47) | 306
(0.75, 0.25) | 0.88=(0.69+0.19) | [5.20E-4, 13.24, 23.31] | 0.81=(0.62+0.19) | 0.87=(0.72+0.15) | [5.75E-4, 11.07, 18.85] | 0.86=(0.72+0.14) | 316

It can be observed from Fig. 2(a) and Fig. 2(b) that even though the baseline
scheme improves the objective function value with erroneous data (see blue
curve) when these COPs are validated using the ideal model (red curve) or the
simulator, the objective function value is much lower. This indicates that
with erroneous data, the COPs obtained after optimization are highly sub-
optimal for the actual network. On the contrary, the COPs obtained via the DD-
OEC scheme achieve the maximum value of the objective function when validated
using the simulator. These results reveal that the mechanism of DD-OEC is
necessary for the attainment of a close-to-optimal COP combination, resulting
in a better network operating point.

Moreover, it can be observed that the convergence of GA differs from SA
convergence and is able to converge earlier. SA shows a rising trend because
of a high degree of variability of its convergence path in each run. Contrary
to this, the convergence path of GA is relatively less varying over multiple
runs. Furthermore, as GA is a population-based algorithm, which remains fixed
in each run, its initialization is not as random as that of SA, and it reaches
the optimum solution in relatively less time. In addition, the number of
generations/iterations required to reach the optimum solution varies in
different runs of the same algorithm. Hence, to provide more insights, we plot
these results using box plots in Fig. 2(c). An important observation to note
here is that the number of required iterations in the DD-OEC scheme is
slightly higher than the baseline scheme because the DD-OEC scheme needs to
combine the effect of both the Model-E and Model-R; hence it takes more time
to converge.

The simulation results with different values of $\alpha_{se}$ and
$1-\alpha_{se}$ are summarized and compared in Table III. For both the
baseline and proposed DD-OEC schemes, the achieved objective function value
along with the corresponding EE and ASE values and the respective COP values,
indicated by the triplet $[\lambda_{dbs}$, $R_{sz}$, $P_{tx}]$, are also shown
in Table III. Again, it can be noted that for both SA and GA, the objective
function value achieved for the baseline scheme is lower compared to the
objective function value for the DD-OEC scheme. When the baseline COPs are
validated using the simulator, the objective function value is lowered
further, highlighting the sub-optimality of the solution. On the contrary,
when the optimized COPs from the DD-OEC framework are validated on the
simulator, the utility degradation is minimal. Compared to the baseline
scheme, it can be noted that with the SA scheme, a gain up to $15\%$ is
observed, and with the GA scheme a gain of up to $23\%$ is achieved. This
highlights the need for the DD-OEC scheme and validates its effectiveness to
counter the impact of errors.

Furthermore, it can be noted that the optimized COPs for different cases
converge to lower values of $\lambda_{dbs}$ and $R_{sz}$ and medium values for
the transmit power. This is because, for the lower Szone sizes, relatively
more UEs are scheduled, which increases the ASE. A lower Szone radius results
in an increase in interference, however, this consequence is avoided by
reducing the transmit power. The lower value of the DBS density is due to the
limited UE density and this lower DBS density is sufficient to serve the
limited UE density. Comparing the different values of $\alpha_{se}$, it can be
noted that the maximum objective function value is obtained when both ASE and
EE are given equal weight as both components are normalized. Finally,
comparing GA and SA reveals that the former, with its population-based
solution-finding approach, can efficiently reach convergence, and the
converged objective function values are also higher. Hence, for this problem,
GA is the better choice.

##  V Conclusion

The errors in UE and DBS positions impact the performance of user-centric
networks. In particular, the performance of the data-driven optimization
solutions becomes sub-optimal when utilizing the data generated from the UEs
with erroneous positions. This paper proposes a data-driven framework to
minimize the adverse impact of UE and DBS positioning errors on the
performance of the user-centric network. The framework relies on an error
residual model trained on historical data, which can be collected from a drive
test campaign. Data-driven models from AutoML are trained on the current
erroneous data to learn the impact of parameters on spectral and energy
efficiency. A multi-objective optimization problem of joint maximization of
spectral and energy efficiency was formulated. We then utilize the current
erroneous model and the historical residual model in the OE to predict the
optimum COPs. Results indicate that the proposed DD-OEC method produces better
convergence compared to the baseline optimization method.

##  VI Acknowledgement

This work is supported by the National Science Foundation under Grant Numbers
1923669, 2323300, 1923295.

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Schwarzschild deformed supergravity background: possible geometry origin of
fermion generations and mass hierarchy

Boris L. Altshuler111E-mail addresses<EMAIL_ADDRESS>&<EMAIL_ADDRESS>
Theoretical Physics Department, Lebedev Physical Institute of the Russian
Academy of Sciences,
53 Leninsky Prospect, Moscow, 119991, Russia

Abstract: The problem of fermion masses hierarchy in the Standard Model is
considered on a toy model of a 10-dimensional space-time with a IIA
supergravity type background. Dirac equation on this background, after
compactification of extra 4- and 1-dimensional subspaces, gives the spectrum
of Fermi fields which profiles in 5 dimensions and corresponding Higgs
generated masses in 4 dimensions depend on the eigenvalues of Dirac operator
on the named compact subspaces. Schwarzschild Euclidean deformation of the
supergravity throat with the ”apple-shaped” conical singularity permits to
leave only three non-divergent angular modes interpreted as three generations
of the down-type quarks. Calculated ratios of their masses are close to the
experimentally observed values for a natural choice of parameters of
10-dimensional geometry. Equations for non-chiral modes coincide with the non-
relativistic Schrödinger equation for an electron moving in a Coulomb field;
the corresponding small fermion masses generated by the twisted boundary
conditions are expressed through the degenerate hypergeometric functions.

PACS numbers: 12.15.Ff

Keywords: Standard Model, quark sector, generations

##  1 Introduction

Models in warped extra dimensions are among the popular trends aimed at
resolving the Standar Model (SM) fermions mass hierarchy enigma. In the works
of this approach, based upon $5D$ Randall-Sundrum (RS) model [1], chiral SM
fermions ”live” in the bulk and their small masses $m_{f}$ in 4 dimensions
result from the overlaps of the fermionic bulk wave functions and the Higgs
field profile confined to the IR end of the slice of the RS $AdS_{5}$
background, see reviews [2] \- [4] and references therein. Then the calculated
values of masses of quarks or leptons strongly depend on the ratios
$c_{f}=M_{f}/k$ of bulk masses $M_{f}$ of corresponding Fermi fields to the
curvature $k$ of $AdS_{5}$: $m_{f}\sim\epsilon^{c_{f}-1/2}$, where
$\epsilon=10^{-16}$ is the Planck-TeV hierarchy parameter. Thus to get the
observed masses $m_{f}$ the special choice of values of parameters $c_{f}$ in
vicinity of 1/2 for every SM fermion is demanded. This fine-tuning of fermion
bulk masses is an essential drawback of models [2] \- [4]. We also pay
attention to papers [5], [6] where each Standard Model family ”lives” in its
own slice of the $5D$ bulk separated by additional 3-branes and fermion masses
in 4 dimensions essentially depend on the choice of parameters of the model,
and to paper [7], where fermions bulk masses are dynamically generated through
fermions interaction with the bulk scalar field, but again bulk Yukawa
coupling constants of this interaction are fine-tuned for every SM fermion.

In the present paper the Dirac equation with zero bulk mass on a ten-
dimensional background is considered where profiles of Fermi fields in the
$5D$ space-time remaining after compactification of (4 + 1) extra dimensions
are determined by eigenvalues $K^{(4)}_{l}$ of Dirac operator on compact
subspace $K^{(4)}$ and by the angular momentum $q$ with respect to the warped
$S^{1}$ dimension. In this way the present paper generalizes the approach of
[8] \- [10] where the named angular momentum enumerates fermion generations
and three fermionic families arise from a single Fermi field in six
dimensions. In the $6D$ model of [9] three fermion generations originate
thanks to the introduced codimension two brane and corresponding ”apple-
shaped” conical singularity of the extra two-dimensional sphere. We use the
same trick at the point of the Schwarzschild Euclidean ”horizon” of the
supergravity throat.

Beginning from Witten’s seminal work [11] it is known that the chiral fermions
of SM cannot be obtained from Dirac equation on $D$-dimensional space
$M_{4}\times\Sigma_{D-4}$ purely from the metric and spin connection on
compact extra space $\Sigma_{D-4}$. In order for the Dirac operator on extra
space to have physically necessary zero modes topologically non-trivial
background gauge fields must be introduced, as it is done, for example, in
[12] \- [14]. However Witten’s no-go theorem is not applicable in the case of
intermediate compactification of type $M^{(5)}\times S^{1}\times K^{(4)}\to
M^{(5)}$ when non-zero eigenvalues of Dirac operator on $K^{(4)}$ play the
role similar to the bulk fermion masses in lower dimensions, that is they, as
it was noted above, determine the bulk profiles of Fermi fields but do not
prevent the appearance of chiral fermions in 4 dimensions. We’ll demonstrate
how it works on the simplest example $K^{(4)}=S^{4}$.

The picture considered in this paper is a toy model that ignores the
$SU(3)\times SU(2)\times U(1)$ group nature of the Standard Model. We also
consider only the down-type quarks ($d,s,b$) that differ in $U(1)$ charge
associated with the rotational symmetry around the brane. Thus the results of
this paper are in a sense complementary to approach of [12] where only single
generation of the standard model spectrum is considered in a high-dimensional
model incorporating the full group nature of Standard Model.

##  2 Ten dimensional background

Let us begin with the action in $D$ dimensions for gravity described by metric
$g_{AB}$ ($A,B=0,1,2...(D-1)$), for scalar field $\phi$, $n$-form
$F_{(n)i_{1}i_{2}...i{n}}$ and with $\Lambda$-term, $F_{(n)}$ and $\Lambda$
interacting with $\phi$:

$S^{(D)}=M_{(D)}^{D-2}\int\left\\{R^{(D)}-\frac{1}{2}(\nabla\phi)^{2}-\frac{e^{\alpha\phi}}{2\,n!}F_{(n)}^{2}-2\Lambda
e^{-\frac{\alpha}{n-1}\phi}\right\\}\,\sqrt{-g^{(D)}}d^{D}x,$ (1)

where $M_{(D)}$ and $R^{(D)}$ are Planck mass and scalar curvature in $D$
dimensions. Following [15] we postulate the invariance of the action (1) under
the simultaneous scale transformation $g_{AB}\to e^{2\lambda}g_{AB}$,
$\phi\to\phi+(2(n-1)/\alpha)\lambda$, $M_{(D)}\to e^{-\lambda}M_{(D)}$
($\lambda=const$); this gives the coupling constant of $\phi$ in
$\Lambda$-term in (1). This Action is a modified and reduced version of the
written in Einstein frame bosonic part of the Action of IIA supergravity
(where $D=10$, $n=4$, $\alpha=1/2$, $\Lambda=0$, but plus to 4-form there are
non-zero 2- and 3-forms [16]).

The long time known [15], [17] \- [22] $p$-brane (fluxbrane) throat-like
solution of the dynamical equations following from action (1) looks in the
deep of the throat as:

$\begin{array}[]{c}ds_{(D)}^{2}=\left(\frac{r}{L}\right)^{2\beta(n-1)}\,\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+U(r)\,\left(\frac{T_{\theta}}{2\pi}\right)^{2}d\theta^{2}\right]+\\\
\\\
+\left(\frac{L}{r}\right)^{2\xi(n-1)}\,\left[\frac{dr^{2}}{U(r)}+\delta^{2}r^{2}d\Omega_{(n)}^{2}\right],\end{array}$
(2)

$e^{\phi}=e^{\phi_{0}}\left(\frac{r}{L}\right)^{\frac{2\alpha(n-1)}{\Delta}},\,F_{(n)}=Q_{(n)}dy_{1}\wedge...\wedge
dy_{n},\,U(r)=1-\left(\frac{r_{Sch}}{r}\right)^{n-1},$ (3)

where $\eta_{\mu\nu}$ is the most plus metric of the $p$-dimensional Minkowski
space-time $M_{(p)}$; $\mu,\nu=0,1,...(p-1)$; $T_{\theta}$ is period of
compact coordinate $S^{1}$ ($0<\theta<2\pi$); $r$ is isotropic coordinate
along the throat; $d\Omega_{(n)}^{2}$ is volume element on sphere $S^{n}$ of
unit radius, $y_{i}$ are angles of this sphere ($i=1,2,...n$); length $L$ and
dimensionless constant $\delta$ are expressed through $Q_{(n)}$,
$e^{\phi_{0}}$, $\Lambda$; $D=p+n+2$,

$\begin{array}[]{c}\beta=\frac{2(n-1)}{(p+n)\Delta},\,\,\,\,\,\,\xi=\frac{2(p+1)}{(p+n)\Delta},\,\,\,\,\,\,\Delta=\alpha^{2}+\frac{2(p+1)(n-1)}{p+n}.\end{array}$
(4)

In metric (2) $\delta=1$ if $\Lambda=0$ in action (1), non-zero $\Lambda$-term
in (1) results in $\delta\neq 1$, thus $\delta$ is a free parameter of the
considered model, its physical meaning is clarified below.

The Schwarzschild type modification of solution when $U\neq 1$ in (2) was
considered in [17] \- [20] in its Lorentzian version; here we use the
Euclidean ”time” version of this deformation.

In order to avoid unnecessary cluttering of formulas, we will consider in the
future only physically interesting values of dimensions $D=10$, $p=4$, $n=4$
when, according to (3), (4):

$\beta=\frac{3}{4\alpha^{2}+15},\qquad\xi=\frac{5}{4\alpha^{2}+15},\qquad
U=1-\left(\frac{r_{Sch}}{r}\right)^{3}.$ (5)

Let us now perform the coordinate transformation

$r\sim z^{-(15+4\alpha^{2})/(9-4\alpha^{2})}$

which takes metric (2) (with parameters (5)) to the more familiar Poincare-
like form, convenient for writing down the Dirac equation on this background:

$ds_{(10)}^{2}=\frac{1}{(kz)^{2s}}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+U(z)\,\left(\frac{T_{\theta}}{2\pi}\right)^{2}d\theta^{2}+\frac{dz^{2}}{U(z)}+\kappa^{2}z^{2}d\Omega_{(4)}^{2}\right],$
(6)

where

$U(z)=1-\left(\frac{z}{z_{IR}}\right)^{\gamma},\quad\gamma=\frac{3(15+4\alpha^{2})}{9-4\alpha^{2}},\quad
s=\frac{\gamma+3}{8},\quad\kappa=\delta\cdot\frac{3}{\gamma}.$ (7)

$k$ and $z_{IR}$ in (6), (7) are simply expressed through $L$ and $r_{Sch}$ in
(2), (3), there is no need to put down these expressions since in what follows
only background (6), (7) with its free dimensionless parameters $\gamma$ and
$\kappa$ will be used. Also, for completeness, we present the expression of
the scalar field in terms of $z$ and $\gamma$: $e^{\alpha\phi}\sim
z^{3(5-\gamma)/4}$, although this will not be needed in what follows.

We note that for $\alpha=0$ in action (1) metric (6) describes the
Schwarzschild deformed $AdS_{6}\times S^{4}$ space-time where $\kappa$ is the
ratio of radius of $S^{4}$ to radius $k^{-1}$ of $AdS_{6}$. For $\alpha=1/2$
in (1) background (6) is the Schwarzschild deformed IIA supergravity metric.

To determine finally the background it is necessary to match $z_{IR}$ and
$T_{\theta}$. Following [17] \- [22] we change coordinate $z$ in (6), (7) to
$\tau$:

$z=z_{IR}\left[1-\frac{\gamma}{4}\,\left(\frac{\tau}{z_{IR}}\right)^{2}\right],$
(8)

which gives in vicinity of $z_{IR}$, $(z_{IR}-z)\ll z_{IR}$, for metric (6)
and for $U$ (7):

$\begin{array}[]{c}ds_{(10)}^{2}=\frac{1}{(kz_{IR})^{2s}}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\eta^{2}\tau^{2}\,d\theta^{2}+d\tau^{2}+\kappa^{2}z_{IR}^{2}d\Omega_{(4)}^{2}\right],\\\
\\\ \eta=\gamma\cdot\frac{T_{\theta}}{4\pi z_{IR}},\qquad
U=\frac{\gamma^{2}}{4}\,\left(\frac{\tau}{z_{IR}}\right)^{2}.\end{array}$ (9)

For $\eta=1$ we have the smooth IR-end of the throat, for $\eta\neq 1$ there
is conical singularity with the codimension two IR-brane at this point. As it
was shown in [9] in $6D$ model, in case $2<\eta<4$ there are three fermion
generations enumerated by the quantum number along angular $\theta$
coordinate. Thus $\eta$ (9) is one more, plus to $\gamma$ and $\kappa$ in (7),
free dimensionless parameter of the proposed model. We’ll see that physically
interesting results are received for $\gamma=9$ (that is $\alpha=\sqrt{3}/2$
in (1)), $\eta=3$ and $\kappa$ being adjusted to have the observed value of
the current mass of $b$-quark: $m_{b}=4.18(2)$ GeV.

Throat (6) is terminated at its IR end $z=z_{IR}$, or smoothly (for $\eta=1$),
or with the codimenrion two brane (for $\eta\neq 1$). According to the
conventional Randall-Sundrum approach we limit the throat (6) also from below
with the codimension one Planck UV-brane located at $z=z_{UV}=1/k$, so that:

$z_{UV}=\frac{1}{k}\leq z\leq z_{IR}.$ (10)

Again, according to the familiar Randall-Sundrum approach, the Planck-TeV
hierarchy ($\epsilon=10^{-16}$) is equal to the ratio of warp factors at the
UV and IR ends of the throat; for metric (6) this means:

$\epsilon=10^{-16}=\left(\frac{z_{UV}}{z_{IR}}\right)^{s}=(kz_{IR})^{-s},$
(11)

the value of $kz_{IR}$ given by this equation will be used below.

To conclude this section, a few remarks about UV and IR branes named above. At
the UV end the extrinsic curvature, that is logarithmic derivative over $z$ of
scale factors of subspaces $M_{(4)}$, $S^{1}$ and $S^{4}$ of metric (6), and
also derivative of scalar field $\phi(z)$ are non-zero, hence $Z_{2}$ symmetry
and corresponding Israel junction conditions on the codimension one Planck
brane $M_{(4)}\otimes S^{1}\otimes S^{4}$ demand anisotropy of the UV-brane
energy-momentum tensor. It is always possible to introduce the brane action
which dependence on space-time metric and on scalar field ensures the
fulfillment of the UV jump conditions. A more detailed discussion of this
topic is beyond the scope of the present paper.

The case of the codimension two brane $M_{(4)}\otimes S^{4}$ at the IR end of
the throat (6) is theoretically less trivial [23] \- [27], while in our case
it is technically more simple since there are no discontinuities on the brane
in the extrinsic curvatures of brane’s subspaces and in the derivative of
scalar field. Using near $z=z_{IR}$ the coordinate transformation $z\to\tau$
(8), it can be seen that dependence of scaling factors and of scalar field on
the radial variable $\tau$ have the form $a+b\tau^{2}$ ($a\neq 0$), that is
they smoothly change over the brane at $\tau=0$. This in turn means that
action of the codimension two IR-brane cannot depend on the field $\phi$ and
is of the isotropic Nambu-Goto type
$S^{br}_{IR}=-M^{8}_{(10)}\oint\sigma\sqrt{-h}d^{8}x$, where $h$ is
determinant of the induced metric, $\sigma$ is dimensionless brane’s tension.
Then non-zero components of the brane’s energy-momentum tensor
$T^{br\nu}_{\mu}\sim\sigma\delta^{\nu}_{\mu}$,
$T^{bri}_{k}\sim\sigma\delta^{i}_{k}$, and $T^{br}\sim 8\sigma$ ($T^{br}$ is
trace of the brane’s energy momentum tensor).

Although components $T^{br}_{AB}$ are proportional to delta function fixing
the position of IR-brane in 10 dimensions
($\delta(\tau)/[2\pi\sqrt{g_{\tau\tau}g_{\theta\theta}}]$ for metric (9)), the
components $(\mu\nu)$ and $(ik)$ of the RHS of the Einstein equations written
in the form $M^{8}_{(10)}R^{(10)}_{AB}=T^{br}_{AB}-\frac{1}{8}g_{AB}T^{br}$
identically vanish. Hence there is no delta function in the components
$R^{\mu}_{\nu}$, $R^{i}_{k}$ of the Ricci tensor and, as expected, no
discontinuity in corresponding extrinsic curvatures.

On the other hand the scaling factor before the $S^{1}$ subspace of metric (9)
is singular at $\tau=0$, its logarithmic derivative is equal to $1/\tau$. The
terms of the $\theta\theta$ component of the Einstein equations that include
the delta function $\delta(\tau)$ give the known relation between brane
tension and angle deficit factor: $\sigma=2\pi(1-\eta)$ [23] \- [27]. Thus the
choice of arbitrary parameter $\eta$ (9) of the model is equivalent to the
choice of the IR-brane dimensionless tension $\sigma$.

##  3 Dirac equation

Dirac equation with zero bulk mass in ten dimensions follows from the action

$S_{\Psi}=\int{\bar{\Psi}}_{(32)}{\tilde{\Gamma}}^{A}D_{A}\Psi_{(32)}\sqrt{-g_{(10)}}d^{10}x,$
(12)

where ${\tilde{\Gamma}}^{A}$ are $32\times 32$ gamma matrices in curved
10-dimensional space-time, $D_{A}$ are covariant derivatives with account of
spin connection. We first write down Dirac equation for the general metric of
type (6)

$ds_{(10)}^{2}=b^{2}(z)\eta_{\mu\nu}dx^{\mu}dx^{\nu}+P^{2}(z)d\theta^{2}+N^{2}(z)dz^{2}+a^{2}(z)d\Omega_{(4)}^{2},$
(13)

making the following choice of flat anti-commuting $\Gamma^{A}$ (cf. [28],
[29]):

$\begin{array}[]{c}\Gamma^{\mu}=\gamma^{\mu}\otimes\sigma^{1}\otimes
I_{4};\qquad\Gamma^{z}=\gamma_{5}\otimes\sigma^{1}\otimes I_{4};\\\
\Gamma^{\theta}=I_{4}\otimes\sigma^{2}\otimes\tau_{5};\qquad\Gamma^{i}=I_{4}\otimes\sigma^{2}\otimes\tau^{i},\end{array}$
(14)

where $\gamma^{\mu}$ are ordinary gamma matrices in Minkowski space-time,
$\tau^{i}$ are the same on $S^{4}$,
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$,
$\tau_{5}=\tau^{1}\tau^{2}\tau^{3}\tau^{4}$, $\sigma^{1,2}$ are Pauli
matrices, $I_{4}$ is $4\times 4$ unit matrix. Corresponding Dirac equation on
space-time (13) (prime means derivative over $z$):

$\begin{array}[]{c}\left[\frac{1}{b}\Gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}+\frac{1}{N}\Gamma^{z}\left(\frac{\partial}{\partial
z}+2\frac{b^{\prime}}{b}+\frac{1}{2}\frac{P^{\prime}}{P}+2\frac{a^{\prime}}{a}\right)+\frac{1}{P}\Gamma^{\theta}\frac{\partial}{\partial\theta}+\frac{1}{a}\Gamma^{i}\nabla_{i}\right]\Psi_{(32)}=0.\end{array}$
(15)

32-component spinor may be presented as a product of 8-component spinor
$\Psi_{(8)}$ in 6-dimensional space-time $\\{x^{\mu},\theta,z\\}$ and some
eigenfunction $\chi_{l}(y^{i})$ of Dirac operator $\Gamma^{i}\nabla_{i}$ on
sphere $S^{4}$ of unit radius:

$\Psi^{(l)}_{(32)}=\Psi^{(l)}_{(8)}(x^{\mu},\theta,z)\cdot\chi_{l}(y^{i});\quad\Gamma^{i}\nabla_{i}\chi_{l}=iK^{(4)}_{l}\chi_{l};\quad
K^{(4)}_{l}=\pm(l+2),$ (16)

$l=0,1,2...$ (eigenvalues of Dirac operator on sphere $S^{n}$ are:
$K^{(n)}_{l}=\pm(l+n/2)$) [30].

8-component spinor is in turn a couple of 4-component spinors $(\pm)$, each of
them consisting of right and left Weyl 2-spinors. Thus for definite eigenvalue
of Dirac operator on $S^{4}$ and definite angular mode $q$ ($q=0,1,2...$):

$\Psi^{(l,q)}_{(8)}=\left(\begin{array}[]{ccc}\psi_{R}^{+}(x)F_{R}^{+}(z)\\\
\psi_{L}^{+}(x)F_{L}^{+}(z)\\\ \psi_{R}^{-}(x)F_{R}^{-}(z)\\\
\psi_{L}^{-}(x)F_{L}^{-}(z)\end{array}\right)\cdot\frac{e^{iq\theta}}{b^{2}P^{\frac{1}{2}}a^{2}},$
(17)

here and below we omit the indexes $(l,q)$ of $\psi^{\pm}_{R,L}$ and profiles
$F^{\pm}_{R,L}$.

From the Dirac equations $(\gamma^{\mu}\partial_{\mu}-m^{\pm})\psi^{\pm}(x)=0$
for 4-spinors $\psi^{\pm}=\psi^{\pm}_{R}+\psi^{\pm}_{L}$ ($m^{\pm}$ are the
fermion masses in four dimensions), and with account of (16), (17) Dirac
equation (15) comes to four equations for profiles $F^{\pm}_{R,L}$:

$\left\\{\begin{array}[]{c}\left(\frac{1}{N}\frac{d}{dz}+\frac{q}{P}+\frac{K^{(4)}_{l}}{a}\right)F^{-}_{R}-\frac{m^{-}}{b}F^{-}_{L}=0,\\\
\\\
\left(\frac{1}{N}\frac{d}{dz}-\frac{q}{P}-\frac{K^{(4)}_{l}}{a}\right)F^{-}_{L}+\frac{m^{-}}{b}F^{-}_{R}=0,\end{array}\right.$
(18)

$\left\\{\begin{array}[]{c}\left(\frac{1}{N}\frac{d}{dz}-\frac{q}{P}-\frac{K^{(4)}_{l}}{a}\right)F^{+}_{R}-\frac{m^{+}}{b}F^{+}_{L}=0,\\\
\\\
\left(\frac{1}{N}\frac{d}{dz}+\frac{q}{P}+\frac{K^{(4)}_{l}}{a}\right)F^{+}_{L}+\frac{m^{+}}{b}F^{+}_{R}=0.\end{array}\right.$
(19)

We note that equations for ($+$) and ($-$) components of $6D$ 8-spinor
separate thanks to the choice (14) of higher-dimensional gamma-matrices.

Finally, taking the scale factors $b(z),P(z),N(z),a(z)$ from comparison of
metrics (13) and (6), multiplying equations (18), (19) by $N$, defining
constants $c_{l}$ which are analogous to $c_{f}=M_{f}^{bulk}/k$ in $5D$ models
of [2] \- [4], and passing to dimensionless quantities $t$, $\mu^{\pm}$:

$t=\frac{2\pi
z}{T_{\theta}};\quad\frac{\gamma}{2\eta}\,\epsilon^{\frac{1}{s}}=t_{UV}<t<t_{IR}=\frac{\gamma}{2\eta};\quad\mu^{\pm}=\frac{T_{\theta}}{2\pi}\cdot
m^{\pm};\quad c_{l}=\frac{K^{(4)}_{l}}{\kappa},$ (20)

(the lower and upper limits for the variable $t$ are determined from (9) -
(11), $\epsilon=10^{-16}$) we obtain from (18) the following system of
equations for the profiles $F^{-}_{R,L}(t)$:

$\left\\{\begin{array}[]{c}\left(\frac{d}{dt}+\frac{q}{U}+\frac{c_{l}}{t\,\sqrt{U}}\right)F^{-}_{R}-\frac{\mu^{-}}{\sqrt{U}}F^{-}_{L}=0,\\\
\\\
\left(\frac{d}{dt}-\frac{q}{U}-\frac{c_{l}}{t\,\sqrt{U}}\right)F^{-}_{L}+\frac{\mu^{-}}{\sqrt{U}}F^{-}_{R}=0.\end{array}\right.\quad
U=1-\left(\frac{2\eta\,t}{\gamma}\right)^{\gamma}$ (21)

Equations (19) for $F^{+}_{R,L}$ take a similar form with corresponding
changes in the signs of various terms. It is important to note that profiles
of zero modes of $F^{-}_{R}$, $F^{+}_{L}$ and of $F^{-}_{L}$, $F^{+}_{R}$
coincide which is easily seen from (18), (19) in case $m^{\pm}=0$; this will
be used in Sec. 6.

##  4 Three generations from one Fermi field

The coefficient in action (12) at the kinetic term in 4 dimensions
${\bar{\psi}}\gamma^{\mu}\partial_{\mu}\psi$ must be equal to one for each
mode $q$, $l$, $\pm$ and separately for $R$ and $L$ Weyl components. Thus,
with account of $\sqrt{g_{(10)}}=b^{4}PNa^{4}$ (we again use generic metric
(13)), $\Psi^{2}_{(8)}\sim(b^{4}Pa^{4})^{-1}$ (see (17)),
$\int{\bar{\chi}}_{l}\chi_{l^{\prime}}d\Omega_{(4)}=\delta_{l\,l^{\prime}}$,
$\int e^{i(q-q^{\prime})\theta}d\theta=2\pi\delta_{q\,q^{\prime}}$ the
following normalization condition of functions $F^{\pm}_{R,L}(z)$ must be
satisfied:

$2\pi\int_{z_{UV}}^{z_{IR}}\frac{N}{b}(F^{\pm}_{R,L})^{2}dz=2\pi\int_{z_{UV}}^{z_{IR}}\frac{1}{\sqrt{U}}(F^{\pm}_{R,L})^{2}dz=1.$
(22)

Transforming in this integral and in equations (18), (19) coordinate
$z\to\tau$ like in (8), and taking into account that near the $IR$ end of the
throat $F_{R,L}\sim\tau^{\pm q/\eta}$ (see (18), (19)) and $dz/\sqrt{U}\sim
d\tau$ (see expression for $U$ in (9)) we see that finitness of integral (22)
demands:

$\int_{0}\tau^{\pm\frac{2q}{\eta}}d\tau<\infty.$ (23)

For smooth $IR$ end of metric (6) or (9) ($\eta=1$) this integral is non-
divergent only for one mode $q=0$. In case

$2<\eta<4$

integral (23) is finite for three modes (three fermion generations) $q=0,\pm
1$. Thus, we reproduce here result of [9] but not for the extra 2-sphere as a
background but for the Schwarzschild deformed supergravity background (6),
(7).

##  5 Twisted BC: 3 small masses

In what follows we set $U=1$ in system (21) for $F^{-}_{R,L}$ and in similar
equations for $F^{+}_{L,R}$. As it is discussed in the end of Sec. 6, taking
into account $U\neq 1$, as in (7), (21), only slightly changes the calculated
mass ratios of quarks of different generations. However the numerical results
and their comparison with observations are not of particular interest here,
since the present paper considers just a toy model that neglects the group
nature of the Standard Model and includes only one type of quarks. To be
realistic, the discussed approach needs to be significantly improved, see
comments in the Conclusion.

For $U=1$ and $q=0$ system (21) coincides with equations in $5D$ models [2] \-
[4] for one Fermi field on a slice (10) of $AdS_{5}$ with an essential
difference, already noted in the Introduction, that in $5D$ models the values
of parameters $c_{f}=M^{bulk}_{f}/k$ are arbitrary for every fermion whereas
in (21) $c_{l}$ are geometrically motivated (see (20)). For $q\neq 0$ the
solutions of equations (21) differ in an interesting way from those in the
models [2] \- [4], this is what we will discuss now.

For $q\neq 0$, $m^{-}\neq 0$ in (18) (or equivalently $\mu^{-}\neq 0$ in
(21)), the following second-order equations for $F^{-}_{R,L}$ are obtained
from (21):

$\begin{array}[]{c}\frac{d^{2}F^{-}_{R}}{dt^{2}}-\left[\omega^{2}+\frac{2qc_{l}}{t}+\frac{c_{l}(c_{l}+1)}{t^{2}}\right]F^{-}_{R}=0,\\\
\frac{d^{2}F^{-}_{L}}{dt^{2}}-\left[\omega^{2}+\frac{2qc_{l}}{t}+\frac{c_{l}(c_{l}-1)}{t^{2}}\right]F^{-}_{L}=0,\end{array}\qquad\omega=\sqrt{q^{2}-(\mu^{-})^{2}}.$
(24)

It is curious to note that these equations coincide with the non-relativistic
Schrödinger equation for an electron moving in a Coulomb field, where $c_{l}$
in (24) play the role of the electron’s orbital momentum. The solutions of
these equations, as it is well known [31], are expressed through the
degenerate hypergeometric functions $\Phi(\alpha,\gamma;\rho)$.
Correspondingly for solutions of Eq-s (24), with account of the first order
connections (21) of $F^{-}_{R}$ and $F^{-}_{L}$, it is obtained ($A,B$ are two
arbitrary constants):

$\begin{array}[]{c}F^{-}_{R}=A\,\frac{\mu^{-}}{2\omega}\,\frac{1}{2c_{l}+1}\,\rho^{c_{l}+1}\,e^{-\frac{\rho}{2}}\,\Phi\left(c_{l}+\frac{qc_{l}}{\omega}+1,2c_{l}+2;\rho\right)+\\\
\\\
+B\,\,\rho^{-c_{l}}\,e^{-\frac{\rho}{2}}\,\Phi\left(-c_{l}+\frac{qc_{l}}{\omega},-2c_{l};\rho\right),\end{array}\qquad\rho=2\,\omega\,t$
(25)

$\begin{array}[]{c}F^{-}_{L}=A\,\rho^{c_{l}}\,e^{-\frac{\rho}{2}}\,\Phi\left(c_{l}+\frac{qc_{l}}{\omega},2c_{l};\rho\right)+\\\
\\\
+B\,\frac{\mu^{-}}{2\omega}\,\frac{1}{2c_{l}-1}\,\rho^{-c_{l}+1}\,e^{-\frac{\rho}{2}}\,\Phi\left(-c_{l}+\frac{qc_{l}}{\omega}+1,-2c_{l}+2;\rho\right).\end{array}\qquad\rho=2\,\omega\,t$
(26)

Solutions for $F^{+}_{R,L}$ differ from the presented solutions for
$F^{-}_{R,L}$ by the changing of signs of $q$ and $c_{l}$ in (25), (26).

For $q=0$ ($\omega=i\mu^{-}$ in (24) - (26)) solutions of Eq-s (21) are the
familiar expressions in Bessel functions [2] \- [4] ($A$, $B$ are another
arbitrary constants):

$\begin{array}[]{c}F^{-}_{R}=A\,\sqrt{t}\,J_{c_{l}+\frac{1}{2}}(\mu^{-}t)+B\,\sqrt{t}\,J_{-(c_{l}+\frac{1}{2})}(\mu^{-}t),\\\
\\\
F^{-}_{L}=A\,\sqrt{t}\,J_{c_{l}-\frac{1}{2}}(\mu^{-}t)-B\,\sqrt{t}\,J_{-(c_{l}-\frac{1}{2})}(\mu^{-}t).\end{array}$
(27)

It is known that Hermiticity of Dirac operator in (18) (or (21)) requires the
fulfillment of the boundary conditions (BC):

$F^{-}_{L}(z_{UV})\,F^{-}_{R}(z_{UV})=F^{-}_{L}(z_{IR})\,F^{-}_{R}(z_{IR})=0.$
(28)

For zero modes, $m^{-}=0$ in (18), this means: or $F^{-}_{L}\equiv 0$, or
$F^{-}_{R}\equiv 0$; we’ll discuss the zero modes option in the next Section.
In turn, BC $F^{-}_{L}(z_{UV})=F^{-}_{L}(z_{IR})=0$ (or
$F^{-}_{R}(z_{UV})=F^{-}_{R}(z_{IR})=0$) give a tower of Kaluza-Klein (KK)
heavy modes with masses $m^{-}_{n}$, their lower value is of order
$z^{-1}_{IR}$. To escape the observationally forbidden phenomena, like flavor
changing neutral currents generated by the exchange of these KK heavy modes,
$z_{IR}^{-1}$ must be above several TeV. Thus heavy modes can not be the low
mass quarks or leptons.

In the next section, the commonly used and corresponding to the Standard Model
ideology method of obtaining small quarks’ current masses in higher
dimensional theories by introducing a Higgs field located on or near the IR
brane will be applied. But now we’ll consider the appearance of small fermion
masses by imposing on solutions (25) - (27) the so called ”twisted” BC,
asymmetric on the two end points, first proposed in [32] (see also [29],
[33]):

$F^{-}_{L}(z_{UV})=F^{-}_{R}(z_{IR})=0.$ (29)

This BC excludes zero mode $m=0$, and it gives of course its tower of heavy
modes $m_{n}>z^{-1}_{IR}$. But twisted BC (29) also gives the single (for
every $q$, $l$ in (18), (19) or (21)) mode with small mass $m_{q,l}^{tw}$ for
which $m_{q,l}^{tw}z\ll 1$ everywhere on the interval (10).

For $q=0$ twisted BC gives (received from (27) with account that $\mu^{tw}t\ll
1$):

$m^{tw}_{q=0,l}=\frac{1}{z_{IR}}\,\sqrt{4c^{2}_{l}-1}\,\left(\frac{z_{UV}}{z_{IR}}\right)^{c_{l}-\frac{1}{2}},\qquad
c_{l}>\frac{1}{2},$ (30)

$z_{UV}/z_{IR}\ll 1$ (see (11)), from now on we’ll consider physically
interesting positive values of $c_{l}$ (20) which are slightly above $1/2$,
like $c_{f}$ in [2] \- [4]. Thus fermion mass (30) is really small, $m^{tw}\ll
z^{-1}_{IR}$.

For $q\neq 0$, expanding $\omega$ (24) in $\mu^{-}$,
$\omega=|q|[1-(\mu^{-})^{2}/2q^{2}]$, and applying twisted BC (29) to
solutions (25), (26) in the lowest order in $\mu^{-}$, we obtain following
expressions for $m^{tw}$, which differ for $q>0$ and $q<0$:

$m^{tw}_{q=1,l}=m^{tw}_{q=0,l}\cdot\frac{1}{\sqrt{\Phi\left(2c_{l}+1,2c_{l}+2;\frac{\gamma}{\eta}\right)}},$
(31)

$m^{tw}_{q=-1,l}=m^{tw}_{q=0,l}\cdot\frac{1}{\sqrt{\Phi\left(2c_{l}+1,2c_{l}+2;-\frac{\gamma}{\eta}\right)}},$
(32)

here the upper value of variable $t=\gamma/2\eta$, see (20), was used with
account of $\rho=2\omega t=2|q|t=2t$ for $|q|=1$.

In case the 4-dimensional compact subspace (the base) of 10-dimensional
supergravity background is sphere $S^{4}$, like it is supposed in (6), we have
$c_{l}=(l+2)/\kappa$ (see (16), (20)), the lowest and the next values of
$c_{l}$ are:

$c_{l=0}=\frac{2}{\kappa},\qquad c_{l=1}=\frac{3}{\kappa}.$ (33)

The possible physical meaning of $c_{l}$ we’ll discuss in the Conclusion.
Having in mind that in (30) the value of ratio $z_{UV}/z_{IR}$ is extremely
small (see (11)) and that KK scale $1/z_{IR}$ is somewhat above 1 TeV, it is
always possible to match the value of $\kappa<4$ and close to 4 in (33) so
that $c_{l=0}$ in the exponent in (30) was above and sufficiently close to
1/2. This permits to equalize the biggest of three masses $m^{tw}_{q=0,\pm 1}$
to, say, the mass of $b$-quark. In fact, the mass ratios of quarks of one and
the same generation given by formulas (31), (32) are in focus of our interest.
Substituting there in the RHS $c_{l}=c_{l=0}\approx 1/2$ and taking into
account the known identities for degenerate hypergeometric functions, we get:

$m^{tw}_{q=1,l=0}=m^{tw}_{q=0,l=0}\cdot\frac{\gamma}{\eta\sqrt{2}}\,\frac{1}{\sqrt{\left(\frac{\gamma}{\eta}-1\right)\,e^{\frac{\gamma}{\eta}}+1}},$
(34)

$m^{tw}_{q=-1,l=0}=m^{tw}_{q=0,l=0}\cdot\frac{\gamma}{\eta\sqrt{2}}\,\frac{1}{\sqrt{1-\left(\frac{\gamma}{\eta}+1\right)\,e^{-\frac{\gamma}{\eta}}}}.$
(35)

For the choice of constants $\gamma=9$ ($\alpha=\sqrt{3}/2$ in (1), see (7))
and $\eta=3$ (see Sec. 4), that is for $\gamma/\eta=3$, expressions (34), (35)
come to: $m^{tw}_{q=1,l=0}=m^{tw}_{q=0,l=0}\cdot 0.33$,
$m^{tw}_{q=-1,l=0}=m^{tw}_{q=0,l=0}\cdot 2.37$. Identifying
$m^{tw}_{q=-1,l=0}=m_{b}>m^{tw}_{q=0,l=0}=m_{s}>m^{tw}_{q=1,l=0}=m_{d}$ we see
that received ratios of masses of down-type quarks are rather far from the
observed values $m_{d}/m_{s}=\lambda^{2}$, $m_{s}/m_{b}=0.44\lambda^{2}$ where
$\lambda=0.2253(7)$ is the Wolfenstein parameter. And it is impossible to find
the value of $\gamma/\eta$ that allows to get from (34), (35) the observed
values of the quark mass ratios.

The conventional Higgs mechanism for obtaining small fermionic masses
discussed in the next Section proves to be more successful, although not
perfect as well, in frames of the considered toy model. .

##  6 Higgs mechanism and zero modes profiles

For $m^{\pm}=0$ in Eq-s (18), (19), or equivalently $\mu^{-}=0$ in Eq-s (21)
for $F^{-}_{R.L}$ and $\mu^{+}=0$ in similar equations for $F^{+}_{R,L}$, and
for $U=1$ in these equations we have four zero modes profiles

$\begin{array}[]{c}F^{-}_{R}=C^{-}_{R}\,e^{-qt}\,t^{-c_{l}},\qquad
F^{-}_{L}=C^{-}_{L}\,e^{qt}\,t^{c_{l}},\\\ \\\ \qquad
F^{+}_{R}=C^{+}_{R}\,e^{qt}\,t^{c_{l}},\quad\qquad
F^{+}_{L}=C^{+}_{L}\,e^{-qt}\,t^{-c_{l}}.\end{array}$ (36)

Since Higgs field is located on or near the $IR$ brane small fermion masses
are acquired by the fields which profiles have maximum at the other - $UV$ end
of the throat where $t\ll 1$ (see definition of $t$ in (20)). Thus, out of
four solutions (36) two proportional to $t^{-c_{l}}$ must be left (we remind
that $c_{l}>1/2$), while $C^{-}_{L}$ and $C^{+}_{R}$ are set equal to zero.
Which in turn ensures the fulfillment of the boundary conditions (28).

Constants $C^{-}_{R}$ and $C^{+}_{L}$ are equal to each other and are
determined from the normalization integral (22) which being rewritten through
variable $t$ (20), for $U=1$, and after substitution non-zero profiles (36)
takes a form ($C$ is any of two named constants):
$C^{2}T_{\theta}\int_{t_{UV}}^{t_{IR}}\,e^{-2qt}t^{-2c_{l}}dt=1$. Since the
main contribution to this integral comes from small $t$ where $qt\ll 1$
exponent in the integrand can be ignored and neglecting the small contribution
from the upper limit of integration we come to the normalization condition

$(C^{-}_{R})^{2}=(C^{+}_{L})^{2}=\frac{2c_{l}-1}{T_{\theta}}\,t^{2c_{l}-1}_{UV},$
(37)

which dependence on $c_{l}$ is familiar from the similar conditions in $5D$
models [2] \- [4].

The conventional way to introduce the SM Yukawa interactions is to add to
spinor action (12) the Higgs field mass term:

$S_{H}=\int{\bar{\Psi}}_{(32)}H\Psi_{(32)}\sqrt{-g_{(10)}}d^{10}x,$ (38)

where vacuum average of the Higgs field $\langle H\rangle$ ”lives” at the $IR$
end of the throat. After substitution of expressions for $\Psi_{(32)}$ from
(16), (17) and integration over five extra coordinates $\theta,y^{i}$, the
following formula for fermion masses is obtained from (38) for every ($l,q$)
component of $\Psi_{(32)}$:

$m_{q,l}=2\pi\int F^{-(l,q)}_{R}\,F^{+(l,q)}_{L}\,\langle H\rangle Ndz$ (39)

(we have restored here the indexes ($l,q$) of the profiles $F$; $N$ is lapse
function in metrics (13) or (6)). Finally, under the assumption $\langle
H\rangle=v\,\delta(z-z_{IR})/N$, writing down from (36), (37) expressions for
$F^{-}_{R}(t_{IR})$, $F^{+}_{L}(t_{IR})$, taking into account the
relationships (9), (20) between $T_{\theta}$, $z_{IR}$, $t_{IR}=\gamma/2\eta$
we get:

$m_{q,l}=v\,Y_{q,l}=v\,\frac{2c_{l}-1}{z_{IR}}\,\left(\frac{z_{UV}}{z_{IR}}\right)^{2c_{l}-1}\,e^{-q\cdot\frac{\gamma}{\eta}}.$
(40)

$Y_{q,l}$ are the diagonal elements of Yukawa matrix; the possibility to
generate the non-diagonal elements of $Y_{q,l}$ is discussed in the
Conclusion.

Masses of three fermion generations, enumerated by $q=0,\pm 1$, differ only in
the exponent in (40). Like in previous Section it is possible to match the
value of $c_{l=0}=2/\kappa>1/2$ (see (16), (20), (33)) so that biggest (for
$m_{q=-1,l=0}$) of these three masses be equal to mass of $b$-quark $m_{b}$.
Identifying $m_{q=0,l=0}=m_{s}$, $m_{q=1,l=0}=m_{d}$ the following ratios of
quark masses are obtained from (40):

$\frac{m_{d}}{m_{s}}=\frac{m_{s}}{m_{b}}=e^{-\frac{\gamma}{\eta}}.$ (41)

Let us again put $\gamma=9$ ($\alpha=\sqrt{3}/2$ in (1), see (7)) and $\eta=3$
(see Sec. 4); then we get $m_{d}/m_{s}=e^{-3}=0.050$ which exactly coincides
with the experimental value $(m_{d}/m_{s})_{exp}=0.050(7)$ at scale 2 GeV
[34]. Whereas experimental value of $(m_{s}/m_{b})_{exp}=2.22(25)10^{-2}$ is
approximately two times less then one predicted by (41). Another option is to
take in Action (1) the type IIA supergravity value $\alpha=1/2$ [16]. Then
$\gamma=6$, see (7), and taking $\eta$ slightly above 2 (which preserves 3
permitted generations, see Sec. 4) we again get the value of $m_{d}/m_{s}$
(41) equal to its observed value. However, these numbers should not be taken
seriously. As it was said, the correct numerical predictions cannot be
expected within the extremely simplified model of this paper.

It makes sense to note that, in contrast to the Dirac equation (15), where due
to the choice of gamma matrices (14), the equations for two 4-components of
the 8-component spinor (17) are separated (see (18), (19)), the mass Higgs
term of the action (38) entangles these 4-components, providing the necessary
product of right and left Weyl spinors - like in (39). We believe that this is
an important advantage of the model under consideration, since, for example,
in the $5D$ models [2] \- [4], right and left spinors are introduced
independently and are quite arbitrarily combined into the Higgs terms of the
Lagrangian, giving Yukawa interactions.

Now a few final remarks about the approximation $U=1$ in Eq-s (21) and,
accordingly, in all further expressions. It is interesting that coordinate
transformation (8) in the exact Eq-s (21), when $U\neq 1$, gives in the
vicinity of $z=z_{IR}$, that is when Dirac equation is written down on rather
simple metric (9), in case $\mu^{-}\neq 0$ (Sec. 5) the approximate equations
similar to (21) with $U=1$ where the roles of two parameters $q$, $c_{l}$ are
reversed. Correspondingly, these parameters must be swaped in Eq-s (24) and in
solutions (25), (26) of new equations. The conjugation of two approximate
solutions, one being valid near the $IR$ end and other near the $UV$ end of
the throat, in some intermediate point may allow one to perform more accurate
calculations of small masses (30) - (32). But this work hardly makes sense,
given the unrealistic values of these small masses.

As for the correction of zero-mode profiles (36) of Sec. 6, then again it is
not difficult to write down these profiles near $z=z_{IR}$, that is near
$\tau=0$, using the coordinate transformation (8) and metric (9). But now, in
contrast to the situation in Sec. 6, the zero-modes profiles are singular at
$\tau=0$ (although their norms are integrable for $q=0,\pm 1$, see Sec. 4).
Assumption $\langle H\rangle=v\,\delta(z-z_{IR})/N$ does not work now, to get
the Higgs generated masses of type (39), (40) it is necessary to put $\langle
H\rangle=const$ in some interval in vicinity of $z=z_{IR}$, like it was done,
for example, in [8]. Estimates show that this way of taking into account
$U\neq 1$ slightly changes the result (41) for the quark mass ratios. Of
course, it would be more correct to obtain the form of the bulk wave function
of the Higgs field from the solution of the corresponding dynamic equation,
like it is done in [7]. But it makes sense to bring the calculations to the
numerical results comparable with the experiment within the framework of a
realistic higher dimensional approach to the Standard Model. Perhaps the
present paper will contribute to the search for such a realistic approach.

##  7 Conclusion

The simplified model of this paper demonstrates the possibility of obtaining
physically reasonable results using the high dimensional supergravity
background. There are plenty of different backgrounds in the theory of
supergravity, thus there are vast possibilities to improve the model
considered above. In particular, it is possible to introduce the group nature
of Standard Model and thus to incorporate four types of fermions (up and down
quarks, charged leptons and neutrinos).

It can be hoped that in the improved model it will be possible to overcome the
significant difficulty of the considered approach, which consists in the fact
that the obtained ratios of the masses of fermions of different generations
(like $m_{d}/m_{s}$ and $m_{s}/m_{b}$ in (41)) are determined by the one and
the same factor, while experimentally these ratios differ significantly.
Perhaps the introduction of Maxwell field monopole configuration $F_{\theta
z}\neq 0$ considered, for example, in [8] may help to overcome this
difficulty?

Parameters $c_{l}$ (20) in Eq-s (21) may determine the mass scales of
different types of SM fermions fixed by different values of $c_{l}$ at
$l=0,1,2...$ due to the factor $(z_{UV}/z_{IR})^{2c_{l}-1}$, like in (40), and
like similar factor with replacement $c_{l}\to c_{f}=M^{bulk}_{f}/k$ in $5D$
models [2] \- [4]; the value of $z_{UV}/z_{IR}$ is determined here from (11),
(7). Values of $c_{l}$ are proportional to the eigenvalues $K^{(4)}_{l}$ of
Dirac operator on the compact 4D base $K^{(4)}$ of space-time (6). But it can
be hardly expected to obtain the realistic scales of masses of different types
of fermions for the most simple choice $K^{(4)}=S^{4}$ (when
$c_{l}=(l+2)/\kappa$, see (16), (20), (33)) as a 4-dimensional subspace of
supergravity metric (6).

A realistic model should include a more sophisticated subspace $K^{(4)}$ with,
generally speaking, topologically non-trivial background gauge fields (like,
for example, in [12] \- [14]), and with eigenvalues of Dirac operator on this
subspace giving physically more interesting set of non-zero parameters
$c_{l}$. In contrast to the works [12] \- [14] where the key role is played by
the zero-modes of Dirac operator on the subspace $K^{(4)}=CP^{2}$, in the
present paper it is shown that in order to obtain chiral fermions in four
dimensions, it is not necessary to have chiral fermionic modes on $K^{(4)}$.
This opens up wide mathematical possibilities for the search for physically
realistic models. In particular, it is possible that the proper choice of
subspace $K^{(4)}$ will permit to abandon the artificial parameter $\kappa\neq
1$ in (6) introduced with use of the $\Lambda$-term in action (1). To study
these possibilities is a task for future research.

We did not discuss the possible origin of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix, this would make sense in the model incorporating up and down quarks
which Yukawa matrices have non-diagonal elements. If Higgs field in (38)
depends on $\theta$ ($H\sim e^{ip\theta}$ [8] \- [10]) then (38) will generate
the non-diagonal elements ($q\neq q^{\prime}$) of Yukawa matrix. However,
consideration of this possibility, including the calculation of the CKM
matrix, is beyond the scope of the present paper. A surprising correlation
between the experimentally observed values of quark masses and the CKM matrix
mixing angles (the so-called ”flavor puzzle”) was considered in a recent
author’s paper [35], which is not related to the high dimensional models.

## Acknowledgments

Author is grateful to Mikhail Vysotsky for constant friendly consultations and
to Valery Rubakov, who tragically died in October 2022, for the inspiring
discussions on the ABC of Standard Model and its problems.

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