Are Low-energy Data already Hinting at Five Dimensions?

Naoyuki Habaa, Nobuchika Okadab and Toshifumi Yamadaa

a Institute of Science and Engineering, Shimane University, Matsue 690-8504,
Japan

b Department of Physics and Astronomy, University of Alabama, Tuscaloosa,
Alabama 35487, USA

Abstract

Low-energy data, combined with renormalization group (RG) equations, can
predict new physics at far higher energy scales. In this paper, we consider
the possibility that the measured Higgs boson mass and top quark mass hint at
a five-dimensional gauge-Higgs unification (5D GHU) model at a scale above
TeV. We note that the vanishing of the Higgs quartic coupling and the
proximity of the top quark Yukawa coupling and weak gauge coupling at high
scales, inferred from the experimental data, are in harmony with 5D GHU,
because in 5D GHU models the Higgs quartic coupling is forbidden by the 5D
gauge symmetry and the Yukawa couplings and the weak gauge coupling originate
from a common 5D gauge coupling. Based on the above insight, we propose a 5D
GHU model where the Standard Model fermions are embedded in 5D fermions in a
way to tightly relate the top Yukawa coupling with the weak gauge coupling.
Also, the model predicts the presence of vector-like fermions (other than the
Kaluza-Klein modes), which can affect the RG evolutions of the 4D theory and
reconcile the scale of vanishing Higgs quartic coupling and that of equality
of the top Yukawa and weak gauge couplings, thereby achieving a successful
matching of the 4D theory with 5D GHU. We predict the vector-like fermion mass
and the compactification scale of 5D GHU from the conditions for the
successful matching.

##  1 Introduction

In the Standard Model (SM), the Higgs quartic coupling exhibits an interesting
property that it vanishes at an energy scale below the Planck scale along
renormalization group (RG) evolutions. This property is usually interpreted as
implying metastability of our vacuum [1] as the Higgs quartic coupling turns
negative above that scale. Another interpretation [2, 3, 4, 5] is that the SM
is matched, around the scale of vanishing Higgs quartic coupling, with a model
of gauge-Higgs unification [6, 7, 8, 9, 10, 11] 111 For early work on
extradimensional models, see, e.g., Refs. [12, 13].  in five dimensions (5D
GHU), where the SM Higgs field is identified with the fifth dimensional
component of a gauge field whose quartic coupling is forbidden by the 5D gauge
symmetry.

Although less noted, the top quark Yukawa coupling and the weak gauge coupling
also show an interesting feature that they become equal at the energy scale of
$O(10^{8})$ GeV. This suggests that the two couplings may have a common origin
in some new physics above that scale. Indeed, in 5D GHU, the SM Yukawa
interactions are part of an extension of the weak interaction and the Yukawa
couplings are tied with the weak gauge coupling.

Motivated by the above insights, we propose a model in which the scale where
the Higgs quartic coupling vanishes is reconciled with the scale where the top
quark Yukawa coupling equals the weak gauge coupling, and this common scale is
interpreted as the scale where 5D GHU emerges.

Our model is characterized by the direct embedding of the SM fermions into 5D
bulk fermions. Here, we introduce 5D fermions in 3 representation of
$SU(3)_{W}$ gauge group that extends the $SU(2)_{W}$ weak gauge group. After
orbifold compactification, one 5D fermion yields one isospin-doublet and one
isospin-singlet 4D fermions of opposite chiralities as the massless modes,
which are directly identified with an isospin-doublet and an isospin-singlet
SM quarks or leptons. Consequently, all the SM Yukawa couplings are identical
with the weak gauge coupling at this stage. Small Yukawa couplings other than
the top quark Yukawa coupling are reproduced by introducing 4D vector-like
fermions at an orbifold fixed point that mix with the massless modes of 5D
fermions 222 In the next section, we will show that it is the square sum of
the up-type and down-type quark Yukawa matrices that is related to the weak
gauge coupling. Hence, the bottom quark Yukawa coupling can be small without
help of the additional 4D vector-like fermions. . The above structure
materializes the idea that the Yukawa couplings and the weak gauge coupling
are basically the same entity and this fact is encoded in the value of the top
quark Yukawa coupling. Thus, this structure is aligned with our motivation to
interpret the scale of equality of the top Yukawa coupling and weak gauge
coupling, as the scale of 5D GHU.

Another characteristic of our model is the presence of vector-like fermions
(other than Kaluza-Klein modes) that is not arbitrary but is required by the
model structure. They are important not only theoretically, but also
phenomenologically; since their mass can be smaller than the compactification
scale, they can change the RG evolutions of the 4D theory and achieve its
matching with the 5D GHU theory. In fact, they can amend a discrepancy in the
SM between the scale of equality of the top Yukawa and weak gauge couplings
and the scale of vanishing Higgs quartic coupling, thereby making the matching
of the 4D and 5D theories successful.

It should be noted that our model is not intended to solve the hierarchy
problem of the Higgs mass, and the compactification scale can be much larger
than TeV scale.

Previously, the unification of the weak gauge coupling and top quark Yukawa
coupling in a high-scale 5D GHU model has been discussed in Ref. [14], but
only in a toy model. We in this paper investigate the unification of the two
couplings in a complete, realistic model, and further relate the unification
with the high-scale vanishing of the Higgs quartic coupling.

This paper is organized as follows. In Section 2, we present our model. In
Section 3, we determine the mass of the vector-like fermions and the
compactification scale of 5D GHU, by solving the RG equations of the 4D theory
made of the SM content and the vector-like fermions, and matching it with the
5D GHU theory. Section 4 summarizes the paper.

##  2 Model

We consider $SU(3)_{C}\times SU(3)_{W}\times U(1)_{V}$ gauge theory in a flat
5D spacetime whose fifth dimension is compactified on $S^{1}/Z_{2}$ orbifold.
We write the fifth coordinate as $y$, and the $S^{1}$ is obtained by the
identification of $y$ with $y+2\pi R$. The $S^{1}/Z_{2}$ orbifold is then
obtained by identifying $y$ with $-y$. $SU(3)_{C}$ is the SM QCD gauge group,
and $SU(3)_{W}\times U(1)_{V}$ incorporates the electroweak gauge groups. We
denote the $SU(3)_{W}$ gauge field by $(W_{\mu},W_{5})$ and the $U(1)_{V}$
gauge field by $(V_{\mu},V_{5})$, where $\mu=0,1,2,3$. The fields transform
under the $Z_{2}$ as

$\displaystyle W_{\mu}(y)$ $\displaystyle=PW_{\mu}(-y)P^{\dagger},\ \ \ \ \
W_{5}(y)=-PW_{5}(-y)P^{\dagger},$ $\displaystyle V_{\mu}(y)$
$\displaystyle=V_{\mu}(-y),\ \ \ \ \ V_{5}(y)=-V_{5}(-y),$ $\displaystyle\ \ \
\ \ \ \ P=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\\\ \end{pmatrix},$ (1)

and accordingly $SU(3)_{W}\times U(1)_{V}$ is broken into $SU(2)_{W}\times
U(1)_{W}\times U(1)_{V}$ at $y=0,\pi R$. Gauge group $U(1)_{W}\times U(1)_{V}$
is further broken into SM hypercharge $U(1)_{Y}$ by a brane-localized scalar
field in $({\bf 1},{\bf 1},-\frac{1}{2\sqrt{3}},\frac{1}{2\sqrt{3}})$
representation of $SU(3)_{C}\times SU(2)_{W}\times U(1)_{W}\times U(1)_{V}$,
denoted by $\phi$, that develops a vacuum expectation value (VEV).
Accordingly, the hypercharge $Q_{Y}$ is given in terms of the $U(1)_{W}$
charge $Q_{W}$ and the $U(1)_{V}$ charge $Q_{V}$ by

$\displaystyle Q_{Y}=\frac{1}{\sqrt{3}}(Q_{W}+Q_{V}).$ (2)

The four-dimensional components of the $SU(2)_{W}\times U(1)_{Y}$ gauge fields
have massless modes, which are identified with the SM electroweak gauge
fields. The fifth dimensional component of the $SU(3)_{W}/(SU(2)_{W}\times
U(1)_{W})$ gauge field has massless KK mode, which is identified with the SM
Higgs field, $H$, as

$\displaystyle W_{5}|_{\rm massless\ KK}\ =\
\frac{1}{\sqrt{2}}\begin{pmatrix}O&H\\\ H^{\dagger}&0\\\ \end{pmatrix}.$ (3)

We introduce three copies of four bulk 5D fermions in $({\bf\bar{3}},{\bf
3},-\frac{1}{\sqrt{3}})$, $({\bf 3},{\bf 3},0)$, $({\bf 1},{\bf
3},-\frac{2}{\sqrt{3}})$, $({\bf 1},{\bf 3},\frac{1}{\sqrt{3}})$
representations of $SU(3)_{C}\times SU(3)_{W}\times U(1)_{V}$, denoted by
$\Psi_{u^{c}}^{i}$, $\Psi_{d}^{i}$, $\Psi_{e}^{i}$, $\Psi_{\nu^{c}}^{i}$ with
$i=1,2,3$. The three copies correspond to the three generations of the SM. The
5D fermions transform under the $Z_{2}$ as 333 Since $\Psi_{u^{c}}^{i}$,
$\Psi_{d}^{i}$, $\Psi_{e}^{i}$, $\Psi_{\nu^{c}}^{i}$ are in 3 representation
of $SU(3)_{W}$, the same $P$ as Eq. (1) enters Eqs. (4)-(7).

$\displaystyle\Psi_{u^{c}}^{i}(-y)$ $\displaystyle=$
$\displaystyle-\gamma_{5}P\,\Psi_{u^{c}}^{i}(y),$ (4)
$\displaystyle\Psi_{d}^{i}(-y)$ $\displaystyle=$
$\displaystyle\gamma_{5}P\,\Psi_{d}^{i}(y),$ (5)
$\displaystyle\Psi_{e}^{i}(-y)$ $\displaystyle=$
$\displaystyle\gamma_{5}P\,\Psi_{e}^{i}(y),$ (6)
$\displaystyle\Psi_{\nu^{c}}^{i}(-y)$ $\displaystyle=$
$\displaystyle-\gamma_{5}P\,\Psi_{\nu^{c}}^{i}(y).$ (7)

The 5D fermions are summarized in Table 1.

Table 1: The gauge and $Z_{2}$ charges of the 5D fermions. $i$ is the flavor index with $i=1,2,3$. | $SU(3)_{C}$ | $SU(3)_{W}$ | $U(1)_{V}$ |  $Z_{2}$ transformation
---|---|---|---|---
$\Psi_{u^{c}}^{i}$ | ${\bf\bar{3}}$ | ${\bf 3}$ | $-\frac{1}{\sqrt{3}}$ | $\Psi_{u^{c}}^{i}(-y)=-\gamma_{5}P\,\Psi_{u^{c}}^{i}(y)$
$\Psi_{d}^{i}$ | ${\bf 3}$ | ${\bf 3}$ | $0$ | $\Psi_{d}^{i}(-y)=\gamma_{5}P\,\Psi_{d}^{i}(y)$
$\Psi_{e}^{i}$ | ${\bf 1}$ | ${\bf 3}$ | $-\frac{2}{\sqrt{3}}$ | $\Psi_{e}^{i}(-y)=\gamma_{5}P\,\Psi_{e}^{i}(y)$
$\Psi_{\nu^{c}}^{i}$ | ${\bf 1}$ | ${\bf 3}$ | $\frac{1}{\sqrt{3}}$ | $\Psi_{\nu^{c}}^{i}(-y)=-\gamma_{5}P\,\Psi_{\nu^{c}}^{i}(y)$

From the $Z_{2}$ charge assignments, we find that the following chiral fields
with $SU(3)_{C}\times SU(2)_{W}\times U(1)_{W}\times U(1)_{V}$ charges possess
massless KK modes:

$\displaystyle{\rm Right\mathchar 45\relax handed\ \ }\left({\bf\bar{3}},{\bf
2},\frac{1}{2\sqrt{3}},-\frac{1}{\sqrt{3}}\right){\rm\ \ component\ of\
}\Psi_{u^{c}}^{i}$ (8) $\displaystyle{\rm Left\mathchar 45\relax handed\ \
}\left({\bf\bar{3}},{\bf
1},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right){\rm\ \ component\ of\
}\Psi_{u^{c}}^{i}$ (9) $\displaystyle{\rm Left\mathchar 45\relax handed\ \
}\left({\bf 3},{\bf 2},\frac{1}{2\sqrt{3}},0\right){\rm\ \ component\ of\
}\Psi_{d}^{i}$ (10) $\displaystyle{\rm Right\mathchar 45\relax handed\ \
}\left({\bf 3},{\bf 1},-\frac{1}{\sqrt{3}},0\right){\rm\ \ component\ of\
}\Psi_{d}^{i}$ (11) $\displaystyle{\rm Left\mathchar 45\relax handed\ \
}\left({\bf 1},{\bf 2},\frac{1}{2\sqrt{3}},-\frac{2}{\sqrt{3}}\right){\rm\ \
component\ of\ }\Psi_{e}^{i}$ (12) $\displaystyle{\rm Right\mathchar 45\relax
handed\ \ }\left({\bf 1},{\bf
1},-\frac{1}{\sqrt{3}},-\frac{2}{\sqrt{3}}\right){\rm\ \ component\ of\
}\Psi_{e}^{i}$ (13) $\displaystyle{\rm Right\mathchar 45\relax handed\ \
}\left({\bf 1},{\bf 2},\frac{1}{2\sqrt{3}},\frac{1}{\sqrt{3}}\right){\rm\ \
component\ of\ }\Psi_{\nu^{c}}^{i}$ (14) $\displaystyle{\rm Left\mathchar
45\relax handed\ \ }\left({\bf 1},{\bf
1},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right){\rm\ \ component\ of\
}\Psi_{\nu^{c}}^{i}$ (15)

At this stage, there are chiral anomalies with respect to the $U(1)_{V}$ gauge
group. These anomalies should be cancelled, and to achieve the cancellation,
we introduce 4D chiral fermions of Table 2 at an orbifold fixed point $y=0$:

Table 2: The gauge charges and chirality of the 4D fermions localized at $y=0$. $i$ is the flavor index with $i=1,2,3$. | $SU(3)_{C}$ | $SU(2)_{W}$ | $U(1)_{W}$ | $U(1)_{V}$ | chirality
---|---|---|---|---|---
$\psi_{qR}^{i}$ | ${\bf 3}$ | ${\bf 2}$ | 0 | $\frac{1}{2\sqrt{3}}$ | Right-handed
$\psi_{\ell R}^{i}$ | ${\bf 1}$ | ${\bf 2}$ | 0 | $-\frac{3}{2\sqrt{3}}$ | Right-handed

The presence of 4D fermions $\psi_{qR}^{i}$, $\psi_{\ell R}^{i}$ is probably a
consequence of more fundamental physics that underlies the orbifolding. In the
present paper, however, we adhere to the orbifold picture and introduce the 4D
fermions by hand.

The most generic 5D action is

$\displaystyle S=\int{\rm d}^{4}x\int^{\pi R}_{-\pi R}{\rm d}y\ $
$\displaystyle-\frac{1}{2}{\rm
tr}\left[W_{MN}W^{MN}\right]-\frac{1}{4}V_{MN}V^{MN}$
$\displaystyle+i\overline{\Psi}_{u^{c}}^{i}\,\Gamma^{M}D_{M}\Psi_{u^{c}}^{i}+i\overline{\Psi}_{d}^{i}\,\Gamma^{M}D_{M}\Psi_{d}^{i}+i\overline{\Psi}_{e}^{i}\,\Gamma^{M}D_{M}\Psi_{e}^{i}+i\overline{\Psi}_{\nu^{c}}^{i}\,\Gamma^{M}D_{M}\Psi_{\nu^{c}}^{i}$
$\displaystyle+\delta(y)\left[i\psi_{qR}^{i\,\dagger}\,\bar{\sigma}^{\mu}D_{\mu}\,\psi_{qR}^{i}+i\psi_{\ell
R}^{i\,\dagger}\,\bar{\sigma}^{\mu}D_{\mu}\,\psi_{\ell R}^{i}\right.$
$\displaystyle\ \ \ -\psi_{qR}^{i\,T}\left\\{A_{ij}\ \phi\
i\sigma_{2}\,\Psi_{u^{c}}^{j}|_{R}+B_{ij}\ \phi^{\dagger}\
\Psi_{d}^{j}|_{L}^{*}\right\\}-\psi_{\ell R}^{i\,T}\left\\{C_{ij}\
\phi^{\dagger}\ \Psi_{e}^{j}|_{L}^{*}+E_{ij}\ \phi\
i\sigma_{2}\,\Psi_{\nu^{c}}^{j}|_{R}\right\\}$ $\displaystyle\left.\ \ \ +{\rm
H.c.}\ \right]$ (16)

where spacetime indices $M,N$ run as $M,N=0,1,2,3,5$, and $D_{M},D_{\mu}$
denote gauge-covariant derivatives. $\Psi_{u^{c}}^{j}|_{R}$,
$\Psi_{d}^{j}|_{L}$, $\Psi_{e}^{j}|_{L}$, $\Psi_{\nu^{c}}^{j}|_{R}$ denote the
4D-decomposed components of the 5D bulk fermions with the indicated chirality
that are in $SU(2)_{W}$ doublet representation, listed in Eqs.
(8),(10),(12),(14). $A_{ij},B_{ij},C_{ij},E_{ij}$ denote their coupling
constants with 4D localized fermions $\psi_{qR}^{i},\psi_{\ell R}^{i}$ and the
$U(1)_{W}\times U(1)_{V}$-breaking scalar $\phi$. After $\phi$ develops a VEV,
the fourth line of Eq. (16) yields a quark mass matrix, which can be recast by
a flavor rotation of $\psi_{qR}^{i}$ into

$\displaystyle\psi_{qR}^{T}\begin{pmatrix}M_{3\times 3}&O_{3\times
3}\end{pmatrix}\begin{pmatrix}U_{1}&U_{2}\\\ U_{3}&U_{4}\\\
\end{pmatrix}\begin{pmatrix}i\sigma_{2}\,\Psi_{u^{c}}|_{R}\\\
\Psi_{d}|_{L}^{*}\\\ \end{pmatrix}$ (17)

where $M_{3\times 3}$ is a $3\times 3$ diagonal matrix, $O_{3\times 3}$
denotes the $3\times 3$ null matrix, $\begin{pmatrix}U_{1}&U_{2}\\\
U_{3}&U_{4}\\\ \end{pmatrix}$ is a unitary matrix, and
$U_{1},U_{2},U_{3},U_{4}$ are its $3\times 3$ submatrices. We see that the
fields below are massless chiral modes,

$\displaystyle(U_{3})_{kj}\,i\sigma_{2}\,\Psi_{u^{c}}^{j}|_{R}+(U_{4})_{kj}\,\Psi_{d}^{j}|_{L}^{*}\
\equiv\ Q_{L}^{k\,*}.$ (18)

$Q_{L}^{k}$ are identified with the SM isospin-doublet quarks. (Remember that
$\Psi_{u^{c}}^{j}|_{R}$, $\Psi_{d}^{j}|_{L}^{*}$ have hypercharge $Q_{Y}=1/6$
after the breaking of $U(1)_{W}\times U(1)_{V}$ into $U(1)_{Y}$.) There also
are vector-like modes with mass $M_{3\times 3}$ that comprise

$\displaystyle\psi_{qR}^{k}\ \ \ {\rm and}\ \ \ \
(U_{1})_{kj}\,i\sigma_{2}\,\Psi_{u^{c}}^{j}|_{R}+(U_{2})_{kj}\,\Psi_{d}^{j}|_{L}^{*}\equiv
X_{qL}^{k\,*}.$ (19)

They represent new, vector-like quarks, which have the same charges as the SM
isospin-doublet quarks. Other fields $\Psi_{u^{c}}^{j}|_{L}$,
$\Psi_{d}^{j}|_{R}$ are identified with the SM up-type and down-type isospin-
singlet quarks.

The presence of the vector-like quarks Eq. (19) is an important prediction of
the model. As for theoretical aspects, their presence is not arbitrary but is
required by the model structure. Accordingly, their number and gauge charges
are uniquely fixed. As for phenomenological aspects, since their mass
$M_{3\times 3}$ can be smaller than the compactification scale, they can
modify the RG evolutions of SM parameters. This modification can fix the
discrepancy between the scale of vanishing Higgs quartic coupling and that of
equality of the top Yukawa and weak gauge couplings, achieving the matching of
the 4D theory with the 5D GHU theory.

The quark Yukawa interactions are derived from the 5D $SU(3)_{W}$ gauge
interaction as

$\displaystyle S\ \supset\ 2\pi R\int{\rm d}^{4}x\
i\frac{g_{W}}{\sqrt{2}}\,H\left\\{\Psi_{u^{c}}^{i}|_{L}^{\dagger}\,i\sigma_{2}\
(U_{3}^{\dagger})_{ik}\ Q_{L}^{k\,*}+\Psi_{d}^{i}|_{R}^{T}\
(U_{4}^{\dagger})_{ik}\ Q_{L}^{k\,*}\right\\}+{\rm H.c.},$ (20)

where $g_{W}$ denotes the $SU(3)_{W}$ gauge coupling. The up-type and down-
type quark Yukawa matrices, $Y_{u},Y_{d}$, are extracted as
$Y_{u}=i\frac{g_{W}}{\sqrt{2}}U_{3}^{*}$,
$Y_{d}=-i\frac{g_{W}}{\sqrt{2}}U_{4}^{*}$. Unfortunately, these Yukawa
matrices are unrealistic because the unitarity relation
$U_{3}^{\dagger}U_{3}+U_{4}^{\dagger}U_{4}=$diag$(1,1,1)$ gives
$Y_{u}^{\dagger}Y_{u}+Y_{d}^{\dagger}Y_{d}=\frac{g_{W}^{2}}{2}$diag$(1,1,1)$,
which contradicts the smallness of the 1st and 2nd generation Yukawa
couplings. To reproduce their smallness, we introduce two generations of 4D
vector-like fermions of Table 3 at $y=0$.

Table 3: 4D vector-like fermions localized at $y=0$. Index $a$ runs as $a=1,2$. | $SU(3)_{C}$ | $SU(2)_{W}$ | $U(1)_{W}$ | $U(1)_{V}$
---|---|---|---|---
$\chi_{u}^{a}$ | ${\bf 3}$ | ${\bf 1}$ | $\frac{1}{\sqrt{3}}$ | $\frac{1}{\sqrt{3}}$
$\chi_{d}^{a}$ | ${\bf 3}$ | ${\bf 1}$ | $-\frac{1}{\sqrt{3}}$ | 0

4D vector-like fermions $\chi_{u}^{a}$, $\chi_{d}^{a}$ mix with the massless
KK modes of 5D fermions through the term

$\displaystyle-\Delta S=\int{\rm d}^{4}x\int^{\pi R}_{-\pi R}{\rm d}y\
\delta(y)\left[M_{u}^{aj}\
\chi_{u}^{a}|^{T}_{L}\,i\sigma_{2}\,\Psi_{u^{c}}^{j}|_{L}+M_{d}^{aj}\
\chi_{d}^{a}|_{L}^{\dagger}\,\Psi_{d}^{j}|_{R}+{\rm
H.c.}+\mu_{u}^{a}\,\bar{\chi}_{u}^{a}\chi_{u}^{a}+\mu_{d}^{a}\,\bar{\chi}_{d}^{a}\chi_{d}^{a}\right],$
(21)

which reduces the Yukawa couplings of two generations by the ratio
$\mu_{u}/M_{u}$ or $\mu_{d}/M_{d}$.

Even with the above mechanism to reduce the 1st and 2nd generation Yukawa
couplings, there is a tight connection between the 3rd generation Yukawa
couplings and the $SU(3)_{W}$ gauge coupling. Since the relation
$Y_{u}^{\dagger}Y_{u}+Y_{d}^{\dagger}Y_{d}=\frac{g_{W}^{2}}{2}$diag$(1,1,1)$
is still valid for the 3rd generation quarks, by neglecting the mixing between
the 3rd generation and other generation quarks, we get

$\displaystyle y_{t}^{2}+y_{b}^{2}\ \simeq\ \frac{1}{2}g_{W}^{2},$ (22)

where $y_{t},y_{b}$ denote the top and bottom quark Yukawa couplings. As
$y_{t}$ is dominantly large, and $g_{W}$ matches with the weak gauge coupling,
Eq. (22) materializes the idea that equality of the top quark Yukawa coupling
and the weak gauge coupling hints at 5D GHU.

As for the lepton sector, the charged lepton Yukawa matrix and the neutrino
Dirac Yukawa matrix are derived in the same fashion. The fourth line of Eq.
(16) gives a lepton mass matrix, which can be recast into the form

$\displaystyle\psi_{\ell R}^{T}\begin{pmatrix}M^{\prime}_{3\times
3}&O_{3\times 3}\end{pmatrix}\begin{pmatrix}V_{1}&V_{2}\\\ V_{3}&V_{4}\\\
\end{pmatrix}\begin{pmatrix}\Psi_{e}|_{L}^{*}\\\
i\sigma_{2}\,\Psi_{\nu^{c}}|_{R}\\\ \end{pmatrix}$ (23)

where $M^{\prime}_{3\times 3}$ is a $3\times 3$ diagonal matrix,
$\begin{pmatrix}V_{1}&V_{2}\\\ V_{3}&V_{4}\\\ \end{pmatrix}$ is a unitary
matrix, and $V_{1},V_{2},V_{3},V_{4}$ are its $3\times 3$ submatrices. The
massless chiral modes below,

$\displaystyle(V_{3})_{kj}\,\Psi_{e}^{j}|_{L}^{*}+(V_{4})_{kj}\,i\sigma_{2}\,\Psi_{\nu^{c}}^{j}|_{R}\
\equiv\ L_{L}^{k\,*},$ (24)

are identified with the SM isospin-doublet leptons. The vector-like modes with
mass $M^{\prime}_{3\times 3}$ that comprise

$\displaystyle\psi_{\ell R}^{k},$
$\displaystyle(V_{1})_{kj}\,(\Psi_{e}^{j}|_{L})^{*}+(V_{2})_{kj}\,i\sigma_{2}\,\Psi_{\nu^{c}}^{j}|_{R}\equiv
X_{\ell L}^{k\,*}$ (25)

represent new, vector-like leptons, which have the same charges as the SM
isospin-doublet leptons. Other fields $\Psi_{e}^{j}|_{R}$,
$\Psi_{\nu^{c}}^{j}|_{L}$ are identified with the SM charged leptons and
isospin-singlet neutrinos, respectively. The realistic charged lepton Yukawa
couplings and tiny active neutrino masses are obtained by introducing vector-
like fermions analogous to those of Table 3.

As with the vector-like quarks, the presence of the vector-like leptons Eq.
(25) is an important prediction of the model.

##  3 Estimation of Vector-like Fermion Mass and Compactification Scale

The presence of vector-like quarks Eq. (19) and vector-like leptons Eq. (25)
can modify the RG evolutions of SM parameters and amend the discrepancy
between the scale of vanishing Higgs quartic coupling and that of equality of
the top Yukawa and weak gauge couplings, enabling us to match the 4D theory
with the 5D GHU theory. Noting this fact, we estimate the mass of the vector-
like fermions with which the matching conditions of the 4D theory with the 5D
GHU theory are satisfied. Additionally, we obtain the compactification scale
from the matching scale.

The matching relates the radiatively-generated potential for the fifth
dimensional component of $SU(3)_{W}$ gauge boson in 5D GHU with the Higgs
potential in the 4D theory. It also relates the $SU(3)_{W}$ gauge coupling in
the former with the weak gauge coupling and quark Yukawa couplings in the
latter. For the Higgs potential, we adopt the general result [2] that the
scale of vanishing Higgs quartic coupling $\lambda$ at one loop equals the
compactification scale of 5D GHU models as

$\displaystyle\lambda\left(\frac{1}{2\pi R}\right)\ =\ 0.$ (26)

Note that Eq. (26) has been derived by considering one-loop threshold
corrections from all the Kaluza-Klein modes of the $SU(3)_{W}$ gauge field and
bulk fermions in a general 5D GHU model. Hence, we can use Eq. (26) without
further including threshold corrections from Kaluza-Klein modes. Note also
that the 4D fermions localized at $y=0$ do not alter Eq. (26) at one-loop
level because these fermions do not couple to the Higgs field at tree level.
For the weak gauge coupling and quark Yukawa couplings, we perform a tree-
level matching around the compactification scale as

$\displaystyle g_{W}$ $\displaystyle=g_{SU(2)_{W}}$
$\displaystyle\frac{1}{2}g_{W}^{2}$ $\displaystyle=y_{t}^{2}+y_{b}^{2}\ \ \ \
\ \ \ \ \ \ \ \ \ {\rm at}\ \mu=\frac{1}{2\pi R},$ (27)

where $g_{SU(2)_{W}}$ denotes the weak gauge coupling, and the second
condition is based on Eq. (22).

We numerically estimate the value of the vector-like fermion mass with which
Eqs. (26),(27) are simultaneously satisfied at one unique scale and the
matching is successful. This scale is then identified with the
compactification scale $1/(2\pi R)$. To this end, we solve the two-loop RG
equations [15, 16, 17] of the 4D theory (comprising the SM content and the
vector-like fermions) by varying the vector-like fermion mass. The masses of
the vector-like quarks and leptons are assumed degenerate, for simplicity. As
for the SM parameters used as inputs of the RG equations, since the top quark
pole mass significantly affects the RG evolution of the Higgs quartic
coupling, we vary it in the 3$\sigma$ range of the latest measurement of the
CMS Collaboration, $170.5\pm 0.8$ GeV [18] 444 The latest measurement of the
top quark pole mass of the ATLAS Collaboration is $171.1^{+1.2}_{-1.0}$ GeV
[19], which favors large values compared to the CMS result. . The strong gauge
coupling is fixed at the central value of the same measurement, and the other
parameters are fixed at their central values in accordance with Ref. [20]. The
pole masses of the top quark, Higgs particle, and $W,Z$ particles are
translated into the input values of the top Yukawa coupling, the Higgs quartic
coupling and the weak mixing angle for the RG equations by using the code
[21], based on the results of [22, 23, 24, 25, 26, 27, 28]. In Fig. 1, we give
a contour plot of

$\displaystyle\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)$ (28)

where

$\displaystyle\lambda\left(\mu_{\lambda}\right)\ =\ 0,$ (29)
$\displaystyle\frac{1}{2}g_{SU(2)_{W}}^{2}|_{\mu=\mu_{gy}}\ =\
y_{t}^{2}+y_{b}^{2}|_{\mu=\mu_{gy}},$ (30)

on the plane of the vector-like fermion mass $M_{\rm vec}$ versus the top
quark pole mass $M_{t}$.

Figure 1:  Contour plot of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)$ Eq.
(28), which is the logarithm of the ratio of the scale where the Higgs quartic
coupling vanishes and the scale where the quark Yukawa couplings and weak
gauge coupling satisfy Eq. (27). The vertical axis is the logarithm of the
vector-like fermion mass $\log_{10}(M_{\rm vec}/{\rm GeV})$, and the
horizontal axis is the top quark pole mass $M_{t}$. The range of the
horizontal axis corresponds to the 3$\sigma$ range of the top quark pole mass
measured by the CMS Collaboration, and the range between the two vertical
dashed lines corresponds to the 2$\sigma$ range. On the contour of
$\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)=0$, Eqs. (26) and (27) are
simultaneously satisfied at one scale and the matching is successful.

The contour of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)=0$ is the region
where Eqs. (26) and (27) are simultaneously satisfied at one scale and the
matching is successful. Thus, this contour is the model’s prediction on the
vector-like fermion mass $M_{\rm vec}$ and the precise top quark pole mass.
Since the contour exists only for $M_{t}\gtrsim 171.4$ GeV, if future
measurements of the top quark pole mass exclude this range of $M_{t}$, the
present model is falsified. To confirm the model, one should test the relation
between the vector-like fermion mass and the top quark pole mass given by the
contour. This is not possible in near-future collider experiments, since the
vector-like fermion mass is predicted to be above $10^{3}$ TeV. On the contour
of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)=0$, the value of
$\mu_{\lambda}=\mu_{gy}$, which corresponds to the compactification scale
$1/(2\pi R)$, varies from $10^{11.6}$ GeV to $10^{10.2}$ GeV from the upper-
left to the lower-right. The above range is the model’s prediction on the
compactification scale.

The contour of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)=0$ is interrupted
at $M_{t}\simeq 171.4$ GeV because for lower values of the top quark pole
mass, the Higgs quartic coupling remains positive along RG evolutions and
hence $\mu_{\lambda}$ is not defined. There are no contours in the area under
the contour of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)=0$ because the
Higgs quartic coupling remains positive there. The reason that the Higgs
quartic coupling tends to remain positive for smaller values of the vector-
like fermion mass is that the presence of the vector-like fermions increases
the weak gauge coupling along RG evolutions, which changes the beta function
of the Higgs quartic coupling positive. To visualize the situation, we show in
Fig. 2 the plot of $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)$ for a fixed
value of the top quark pole mass $M_{t}=172.1$ GeV. One sees that as the
vector-like fermion mass decreases,
$\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)$ increases and becomes about 0,
and then the plot line vanishes because $\mu_{\lambda}$ is not defined for
lower values of the vector-like fermion mass.

Figure 2:  $\log_{10}\left(\mu_{\lambda}/\mu_{gy}\right)$ versus the logarithm
of the vector-like fermion mass, for a fixed value of the top quark pole mass
$M_{t}=172.1$ GeV. The plot line vanishes for lower values of the vector-like
fermion mass because $\mu_{\lambda}$ is not defined.

Interestingly, those parameter sets which yield $\mu_{\lambda}/\mu_{gy}=1$ are
critical in the sense that the Higgs quartic coupling nearly remains positive
for such sets. To illustrate this, we present in Fig. 3 the RG evolutions of
the Higgs quartic coupling, the square sum of Yukawa couplings and the weak
gauge coupling for a parameter set $(M_{t},\,\log_{10}(M_{\rm vec}/{\rm
GeV}))=(172.1$ GeV$,\,6.6)$. The solid purple, blue and green lines are the RG
evolutions of the Higgs quartic coupling, the Yukawa coupling square sum, and
half the weak gauge coupling squared, respectively. The dashed lines are their
RG evolutions if the 4D theory were valid at high scales. One confirms
$\mu_{\lambda}/\mu_{gy}=1$ and that the Higgs quartic coupling nearly remains
positive. For such parameter sets, even a slight decrease in the vector-like
fermion mass enhances the weak gauge coupling and the Higgs quartic coupling
along RG evolutions, rendering the Higgs quartic coupling non-vanishing.

Figure 3:  RG evolutions of the Higgs quartic coupling $\lambda$, the square
sum of Yukawa couplings $y_{t}^{2}+y_{b}^{2}$, and half the weak gauge
coupling squared $(1/2)g_{SU(2)_{W}}^{2}$ for $M_{t}=172.1$ GeV and
$\log_{10}(M_{\rm vec}/{\rm GeV})=6.6$, colored in purple, blue and green,
respectively. $\mu$ denotes the renormalization scale. The solid lines are the
RG evolutions in the current model, and the dashed lines are those if the 4D
theory were valid at high scales.

##  4 Summary

The vanishing of the Higgs quartic coupling and the proximity of the top
Yukawa and weak gauge couplings at high energy scales hint at 5D GHU. Based on
the above idea, we have proposed a model where the scale of vanishing Higgs
quartic coupling and equality of the square sum of the top and bottom Yukawa
couplings $y_{t}^{2}+y_{b}^{2}$ and half the weak gauge coupling squared
$(1/2)g_{SU(2)_{W}}^{2}$, is interpreted as the compactification scale of 5D
GHU. The model is characterized by the embedding of the SM fermions in 3
representation of the $SU(3)_{W}$ gauge group. This embedding leads to the
equality of $y_{t}^{2}+y_{b}^{2}$ and $(1/2)g_{SU(2)_{W}}^{2}$ that must be
satisfied at the compactification scale. The model structure necessitates the
presence of isospin-doublet vector-like quarks and leptons. These vector-like
fermions modify the RG evolutions of the 4D theory and reconcile the scale of
vanishing Higgs quartic coupling and that where
$y_{t}^{2}+y_{b}^{2}=(1/2)g_{SU(2)_{W}}^{2}$ holds, thereby achieving the
successful matching of the 4D theory with 5D GHU. Based on this property, we
have calculated the RG equations of the 4D theory and obtained a prediction
for the vector-like fermion mass, the precise top quark pole mass, and the
compactification scale. The prediction for the former two is given by the 0
contour in Fig. 1, and that for the compactification scale varies from
$10^{11.6}$ GeV to $10^{10.2}$ GeV as one moves from the upper-left to the
lower-right on the contour. About experimental testability of the model, if
future measurements of the top quark pole mass exclude the region above about
$171.4$ GeV, the model is ruled out. The model can be confirmed by testing the
relation between the vector-like fermion mass and the top quark pole mass
given by the 0 contour in Fig. 1, but this is not possible in near-future
collider experiments, since the vector-like fermion mass is predicted to be
above $10^{3}$ TeV.

## Acknowledgement

This work is partially supported by Scientific Grants by the Ministry of
Education, Culture, Sports, Science and Technology of Japan Nos. 17K05415 and
21H000761 (N.H.) and No. 19K147101 (T.Y.), and by the United States Department
of Energy Grant No. DE-SC0012447 (N.O.).

## Appendix: Matching Condition for the Higgs Quartic Coupling

We re-derive the one-loop matching condition for the Higgs quartic coupling in
Ref. [2] utilized in the main text,

$\displaystyle\lambda\left(\frac{1}{2\pi R}\right)\ =\ 0.$

First, we study a general $SU(3)_{W}$ gauge theory in a five-dimensional
spacetime compactified on $S^{1}/Z_{2}$ orbifold, with 5D bulk fermions in the
fundamental representation, and without 4D chiral fermions localized at $y=0$
or $\pi R$. The action with the gauge-fixing term is given by

$\displaystyle S=\int{\rm d}^{4}x\int^{\pi R}_{-\pi R}{\rm d}y\ $
$\displaystyle-\frac{1}{2}{\rm tr}\left[W_{MN}W^{MN}\right]-\frac{1}{\xi}{\rm
tr}\left[(\partial_{\mu}W^{\mu}-\xi\overline{D}_{5}W_{5})^{2}\right]$
$\displaystyle+2\,{\rm
tr}\left[b(\partial_{\mu}D^{\mu}-\xi\overline{D}_{5}D_{5})c\right]$
$\displaystyle+\frac{i}{2}\overline{\Psi}\,\Gamma^{M}D_{M}\Psi-\frac{i}{2}\left(D_{M}\overline{\Psi}\right)\Gamma^{M}\Psi,$
(31)

where $\xi$ is the gauge-fixing parameter, $b,c$ are ghost fields, and $\Psi$
represents multiple bulk fermions in the 3 representation of $SU(3)_{W}$ gauge
group. Also, in the first line, $\overline{D}_{5}$ denotes a covariant
derivative whose gauge field part is replaced by the VEV of $W_{5}$ field. We
restrict ourselves to the case when $\Psi$ is $Z_{2}$-even, as $Z_{2}$-odd
bulk fermions give the same contribution as $Z_{2}$-even ones to the one-loop
effective potential. The gauge field, ghost fields and bulk fermions transform
under the $Z_{2}$ as

$\displaystyle W_{\mu}(y)$ $\displaystyle=PW_{\mu}(-y)P^{\dagger},\ \ \ \ \
W_{5}(y)=-PW_{5}(-y)P^{\dagger},$ $\displaystyle b(y)$
$\displaystyle=Pb(-y)P^{\dagger},\ \ \ \ \ c(y)=Pc(-y)P^{\dagger},$
$\displaystyle\Psi(y)$ $\displaystyle=\gamma_{5}P\,\Psi(-y),$ $\displaystyle\
\ \ \ \ \ \ P=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\\\ \end{pmatrix}.$ (32)

The $Z_{2}$ transformation property above breaks $SU(3)_{W}$ into
$SU(2)_{W}\times U(1)_{W}$. The massless mode of the
$SU(3)_{W}/(SU(2)_{W}\times U(1)_{W})$ component of $W_{5}$ is identified with
a Higgs field. Unlike Ref. [2], we do not impose any non-trivial condition for
the periodic change $y\to y+2\pi R$.

We compute the one-loop effective potential for the $W_{5}$ VEV. To this end,
we extract the quadratic part of the action Eq. (31) in the presence of the
$W_{5}$ VEV parametrized as

$\displaystyle\langle W_{5}\rangle\ =\ \frac{1}{2gR}\begin{pmatrix}0&0&0\\\
0&0&a\\\ 0&a&0\\\ \end{pmatrix}.$ (33)

The quadratic part is found to be

$\displaystyle S|_{\rm quad}=$ $\displaystyle\int{\rm d}^{4}x\int^{\pi
R}_{-\pi R}{\rm d}y\ {\rm
tr}\left[W_{\mu}\left(\eta^{\mu\nu}\square-\left(1-\frac{1}{\xi}\right)\partial^{\mu}\partial^{\nu}\right)W_{\nu}-W_{\mu}\,\overline{D}_{5}^{2}\,W^{\mu}\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-W_{5}\square W_{5}+\xi\,W_{5}\,\overline{D}_{5}^{2}\,W_{5}+b\square
c-\xi\,b\,\overline{D}_{5}^{2}\,c\right]$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \
+i\overline{\Psi}\left(\gamma^{\mu}\partial_{\mu}+i\gamma_{5}\overline{D}_{5}\right)\Psi$
(34) $\displaystyle+$ $\displaystyle 2\left[\int{\rm d}^{4}x\ {\rm
tr}\left[2(\partial^{\mu}W_{\mu})W_{5}+W_{\mu}\overline{D}_{5}W^{\mu}-\xi\,W_{5}\overline{D}_{5}W_{5}\right]-\overline{\Psi}\gamma_{5}\Psi\right]^{y=\pi
R}_{y=0}$ (35)

where $\overline{D}_{5}$ is a covariant derivative whose gauge field part is
given by $\langle W_{5}\rangle$ of Eq. (33). From the $Z_{2}$ transformation
property and the periodicity, the fields obey the following boundary
conditions at $y=0,\pi R$:

$\displaystyle W_{\mu}(y)$ $\displaystyle=PW_{\mu}(y)P^{\dagger},\ \ \ \ \
W_{5}(y)=-PW_{5}(y)P^{\dagger},$ $\displaystyle\overline{D}_{5}W_{\mu}(y)$
$\displaystyle=-P\overline{D}_{5}W_{\mu}(y)P^{\dagger},\ \ \ \ \
W_{5}(y)=P\overline{D}_{5}W_{5}(y)P^{\dagger},$ $\displaystyle b(y)$
$\displaystyle=Pb(y)P^{\dagger},\ \ \ \ \ c(y)=Pc(y)P^{\dagger},$
$\displaystyle\overline{D}_{5}b(y)$
$\displaystyle=-P\overline{D}_{5}b(y)P^{\dagger},\ \ \ \ \
\overline{D}_{5}c(y)=-P\overline{D}_{5}c(y)P^{\dagger},$
$\displaystyle\Psi(y)$ $\displaystyle=\gamma_{5}P\,\Psi(y),$
$\displaystyle\overline{D}_{5}\Psi(y)$
$\displaystyle=-\gamma_{5}\overline{D}_{5}P\,\Psi(y)\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm at}\ y=0,\,\pi
R.$ (36)

Note that given the above boundary conditions, the boundary action Eq. (35)
vanishes, and the ghost term is Hermite. We calculate the eigenvalues of the
quadratic operators in the bulk Eq. (34) under the condition that the
eigenfunctions fulfill the boundary conditions Eq. (36). The eigenvalue
equations are given by

$\displaystyle\left(\eta^{\mu\nu}\square-\left(1-\frac{1}{\xi}\right)\partial^{\mu}\partial^{\nu}\right)W_{\nu}-\overline{D}_{5}^{2}\,W^{\mu}$
$\displaystyle=\lambda_{W_{\mu}}\,W^{\mu},$ $\displaystyle-\square
W_{5}+\xi\,\overline{D}_{5}^{2}\,W_{5}$
$\displaystyle=\lambda_{W_{5}}\,W_{5},$ $\displaystyle\square
c-\xi\,\overline{D}_{5}^{2}\,c$ $\displaystyle=\lambda_{c}\,c,$
$\displaystyle\square b-\xi\,\overline{D}_{5}^{2}\,b$
$\displaystyle=\lambda_{b}\,b,$
$\displaystyle\left(i\gamma^{\mu}\partial_{\mu}-\gamma_{5}\overline{D}_{5}\right)^{2}\Psi$
$\displaystyle=\lambda_{\Psi}\,\Psi.$ (37)

The solutions to the above equations, before the boundary conditions are
imposed, are in the form,

$\displaystyle W_{\mu}$ $\displaystyle=e^{\pm i\,p\,x}\
\Omega(y)\left\\{C^{TL}_{\mu}\cos(m\,y)+S^{TL}_{\mu}\sin(m\,y)\right\\}\Omega^{\dagger}(y)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ +e^{\pm i\,p\,x}\
\Omega(y)\left\\{C^{S}_{\mu}\cos(m\,y)+S^{S}_{\mu}\sin(m\,y)\right\\}\Omega^{\dagger}(y)$
$\displaystyle W_{5}$ $\displaystyle=e^{\pm i\,p\,x}\
\Omega(y)\left\\{C_{5}\cos(m\,y)+S_{5}\sin(m\,y)\right\\}\Omega^{\dagger}(y)$
$\displaystyle c$ $\displaystyle=e^{\pm i\,p\,x}\
\Omega(y)\left\\{C_{c}\cos(m\,y)+S_{c}\sin(m\,y)\right\\}\Omega^{\dagger}(y)$
$\displaystyle b$ $\displaystyle=e^{\pm i\,p\,x}\
\Omega(y)\left\\{C_{b}\cos(m\,y)+S_{b}\sin(m\,y)\right\\}\Omega^{\dagger}(y)$
$\displaystyle\Psi$ $\displaystyle=e^{\pm i\,p\,x}\
\Omega(y)\left\\{C^{L}\cos(m\,y)+S^{L}\sin(m\,y)\right\\}+e^{\pm i\,p\,x}\
\Omega(y)\left\\{C^{R}\cos(m\,y)+S^{R}\sin(m\,y)\right\\},$ (38)

where

$\displaystyle\Omega(y)=\exp\left[i\,g\int_{0}^{y}\langle W_{5}\rangle{\rm
d}y^{\prime}\right]=\exp\left[\frac{i}{2}\begin{pmatrix}0&0&0\\\ 0&0&a\\\
0&a&0\\\ \end{pmatrix}\frac{y}{R}\right].$ (39)

Here $p$ denotes a four-momentum, and $m$ denotes the mass of Kaluza-Klein
modes determined from the boundary conditions. The zero-modes correspond to
the case where $m=0$. The $C$’s and $S$’s are constants satisfying
$p^{\mu}C^{TL}_{\mu}=p^{\mu}S^{TL}_{\mu}=0$, $C^{S}_{\mu}\propto
S^{TL}_{\mu}\propto p_{\mu}$, $\gamma_{5}C^{L}=-C^{L}$,
$\gamma_{5}S^{L}=-S^{L}$, $\gamma_{5}C^{R}=C^{R}$, $\gamma_{5}S^{R}=S^{R}$.
The corresponding eigenvalues are given by

$\displaystyle\lambda_{W_{\mu}^{TL}}=-p^{2}+m^{2},\ \ \ \ \
\lambda_{W_{\mu}^{S}}=-p^{2}/\xi+m^{2},$
$\displaystyle\lambda_{W_{5}}=p^{2}-\xi m^{2},$
$\displaystyle\lambda_{c}=-p^{2}+\xi m^{2},$
$\displaystyle\lambda_{b}=-p^{2}+\xi m^{2},$
$\displaystyle\lambda_{\Psi}=-p^{2}+m^{2}.$ (40)

Now we impose the boundary conditions Eq. (36) on the solutions Eq. (38) and
determine the value of $m$. The boundary conditions are equivalent to the
following constraints on the constants:

$\displaystyle PC^{TL}_{\mu}P^{\dagger}=C^{TL}_{\mu},\ \ \ \ \ \ \ \ \ \
PS^{TL}_{\mu}P^{\dagger}=-S^{TL}_{\mu},$ $\displaystyle P\Omega(\pi
R)\left\\{C^{TL}_{\mu}\cos(m\,\pi R)+S^{TL}_{\mu}\sin(m\,\pi
R)\right\\}\Omega^{\dagger}(\pi R)P^{\dagger}$ $\displaystyle\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ =\Omega(\pi R)\left\\{C^{TL}_{\mu}\cos(m\,\pi
R)+S^{TL}_{\mu}\sin(m\,\pi R)\right\\}\Omega^{\dagger}(\pi R),$ $\displaystyle
P\Omega(\pi R)\left\\{-C^{TL}_{\mu}\sin(m\,\pi R)+S^{TL}_{\mu}\cos(m\,\pi
R)\right\\}\Omega^{\dagger}(\pi R)P^{\dagger}$ $\displaystyle\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ =-\Omega(\pi R)\left\\{-C^{TL}_{\mu}\sin(m\,\pi
R)+S^{TL}_{\mu}\cos(m\,\pi R)\right\\}\Omega^{\dagger}(\pi R)$ (41)
$\displaystyle\ \ \ \ \ \ \ \ \ {\rm(the\ same\ for\
}C^{S}_{\mu},\,S^{S}_{\mu},\,C_{c},\,S_{c},\,C_{b},\,S_{b},\ {\rm and\ the\
sign\ is\ flipped\ for}\ C_{5},\,S_{5}),$

and

$\displaystyle PC^{L}=-C^{L},\ \ \ \ \ \ \ \ \ \ PS^{L}=S^{L},$ $\displaystyle
P\Omega(\pi R)\left\\{C^{L}\cos(m\,\pi R)+S^{L}\sin(m\,\pi
R)\right\\}=-\Omega(\pi R)\left\\{C^{L}\cos(m\,\pi R)+S^{L}\sin(m\,\pi
R)\right\\},$ $\displaystyle P\Omega(\pi R)\left\\{-C^{L}\sin(m\,\pi
R)+S^{L}\cos(m\,\pi R)\right\\}=\Omega(\pi R)\left\\{-C^{L}\sin(m\,\pi
R)+S^{L}\cos(m\,\pi R)\right\\},$ $\displaystyle PC^{R}=C^{R},\ \ \ \ \ \ \ \
\ \ PS^{R}=-S^{R},$ $\displaystyle P\Omega(\pi R)\left\\{C^{R}\cos(m\,\pi
R)+S^{R}\sin(m\,\pi R)\right\\}=\Omega(\pi R)\left\\{C^{R}\cos(m\,\pi
R)+S^{R}\sin(m\,\pi R)\right\\},$ $\displaystyle P\Omega(\pi
R)\left\\{-C^{R}\sin(m\,\pi R)+S^{R}\cos(m\,\pi R)\right\\}=-\Omega(\pi
R)\left\\{-C^{R}\sin(m\,\pi R)+S^{R}\cos(m\,\pi R)\right\\}.$ (42)

In Eq. (41), non-zero solutions for the set
$\left(C^{TL}_{\mu}\cos(m\,y),\,S^{TL}_{\mu}\sin(m\,y)\right)$ exist when

$\displaystyle m=\frac{n}{R},\ \ \frac{n\pm a/2}{R},\ \ \frac{n\pm a}{R}\ \ \
\ \ \ \ \ (n=0,\pm 1,\pm 2,...),$ (43)

and when $n>0$ there are one solution for $m=n/R$, two solutions for each of
$m=(n\pm\frac{a}{2})/R$, and one solution for each of $m=(n\pm a)/R$. The
solutions when $n<0$ are degenerate with those when $n>0$. When $n=0$, there
are one solution for $m=0$, two common solutions for $m=\pm\frac{a}{2}/R$, and
one common solution for $m=\pm a/R$, which respectively correspond to the
photon, $W^{\pm}$ bosons, and $Z$ boson with wrong Weinberg angle. The values
of $m$ and the number of solutions for each $m$ are the same for
$(C^{S}_{\mu},\,S^{S}_{\mu})$, $(C_{c},\,S_{c})$, $(C_{b},\,S_{b})$, and
$(C_{5},\,S_{5})$.

In Eq. (42), non-zero solutions for the set
$\left(C^{L}\cos(m\,y),\,S^{L}\sin(m\,y)\right)$ exist when

$\displaystyle m=\frac{n}{R},\ \ \frac{n\pm a/2}{R}\ \ \ \ \ \ \ \ (n=0,\pm
1,\pm 2,...),$ (44)

and when $n>0$ there are one solution for $m=n/R$ and one solution for each of
$m=(n\pm\frac{a}{2})/R$. The solutions when $n<0$ are degenerate with those
when $n>0$. When $n=0$, there are one solution for $m=0$ and one common
solution for $m=\pm\frac{a}{2}/R$, which respectively correspond to one Weyl
fermion that does not have a Yukawa coupling with the Higgs field 555 In the
model of the main text, this fermion gains a vector-like mass with a 4D
fermion localized at an orbifold fixed point.  and one of a pair of Weyl
fermions that have a Yukawa coupling.

Non-zero solutions for the set
$\left(C^{R}\cos(m\,y),\,S_{R}\sin(m\,y)\right)$ exist when

$\displaystyle m=\frac{n}{R},\ \ \frac{n\pm a/2}{R}\ \ \ \ \ \ \ \ (n=0,\pm
1,\pm 2,...),$ (45)

and when $n>0$ there are one solution for $m=n/R$ and one solution for each of
$m=(n\pm\frac{a}{2})/R$. The solutions when $n<0$ are degenerate with those
when $n>0$. When $n=0$, there only is one common solution for
$m=\pm\frac{a}{2}/R$, which corresponds to one of a pair of Weyl fermions that
have a Yukawa coupling.

Substituting Eqs. (43)-(45) into Eqs. (40), we obtain the true eigenvalues.

From the eigenvalues and their duplications found above, the one-loop
effective potential for the $W_{5}$ VEV $a$ is computed as

$\displaystyle V_{\rm eff}(a)-V_{\rm eff}(0)$
$\displaystyle=-\frac{i}{2}\int\frac{{\rm
d}^{4}p}{(2\pi)^{4}}\sum_{n=1}^{\infty}\left[3\left\\{2\log\frac{-p^{2}+(n+\frac{a}{2})^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}+2\log\frac{-p^{2}+(n-\frac{a}{2})^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}\right.\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\log\frac{-p^{2}+(n+a)^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}+\log\frac{-p^{2}+(n-a)^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}\right\\}$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-2N_{\Psi}\left\\{2\log\frac{-p^{2}+(n+\frac{a}{2})^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}+2\log\frac{-p^{2}+(n-\frac{a}{2})^{2}/R^{2}}{-p^{2}+n^{2}/R^{2}}\right\\}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+3\left\\{2\log\frac{-p^{2}+(\frac{a}{2})^{2}/R^{2}}{-p^{2}}+\log\frac{-p^{2}+a^{2}/R^{2}}{-p^{2}}\right\\}-4N_{\Psi}\log\frac{-p^{2}+(\frac{a}{2})^{2}/R^{2}}{-p^{2}}.$
(46)

After summation over $n$ and a Wick rotation with $p^{0}=ip_{E}^{0}$, one gets

$\displaystyle V_{\rm eff}(a)-V_{\rm eff}(0)$
$\displaystyle=\frac{1}{2}\int\frac{{\rm
d}^{4}p_{E}}{(2\pi)^{4}}3\left\\{2\log\left(1+\frac{\sin^{2}(\pi\,\frac{a}{2})}{\sinh^{2}(\pi
R\,p_{E})}\right)+\log\left(1+\frac{\sin^{2}(\pi\,a)}{\sinh^{2}(\pi
R\,p_{E})}\right)\right\\}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \
-4N_{\Psi}\log\left(1+\frac{\sin^{2}(\pi\,\frac{a}{2})}{\sinh^{2}(\pi
R\,p_{E})}\right),$ (47)

where $p_{E}^{2}$ denotes the Wick-rotated momentum squared, and $p_{E}$
denotes its square root. Note that the above effective potential is finite.

We are interested in the Higgs quartic coupling derived from the effective
potential Eq. (47). We have a formula,

$\displaystyle\frac{{\rm d}^{4}}{{\rm d}a^{4}}\int\frac{{\rm
d}^{4}p_{E}}{(2\pi)^{4}}\log\left(1+\frac{\sin^{2}(\pi\,a)}{\sinh^{2}(\pi
R\,p_{E})}\right)\ =\
\frac{3}{4\pi^{2}}\frac{1}{R^{4}}\log\left(4\sin^{2}(\pi\,a)\right).$ (48)

Applying Eq. (48) to Eq. (47), we find that the quartic coupling for the
physical Higgs boson $h$ is derived as

$\displaystyle\frac{{\rm d}^{4}}{{\rm d}h^{4}}V_{\rm
eff}(a)=(gR)^{4}\frac{{\rm d}^{4}}{{\rm d}a^{4}}V_{\rm eff}(a)$
$\displaystyle\ \ \ \ =\frac{3}{8\pi^{2}}g^{4}\left\\{3\cdot
2\cdot\frac{1}{2^{4}}\log\left(4\sin^{2}(\pi\,\frac{a}{2})\right)+3\log\left(4\sin^{2}(\pi\,a)\right)-4N_{\Psi}\cdot\frac{1}{2^{4}}\log\left(4\sin^{2}(\pi\,\frac{a}{2})\right)\right\\}.$
(49)

This should be compared with the Higgs quartic coupling derived from the one-
loop effective potential of a 4D model that contains $W^{\pm}$ bosons with
mass $m=\frac{a}{2}/R$, $Z$ boson with mass $m=a/R$, $N_{\Psi}$ Dirac fermions
with mass $m=\frac{a}{2}/R$, and a Higgs boson whose quartic coupling vanishes
at tree level. In the 4D model, the Higgs quartic coupling derived from the
one-loop effective potential reads

$\displaystyle\frac{{\rm d}^{4}}{{\rm d}h^{4}}V_{\rm pSM\,eff}$
$\displaystyle=\lambda(\mu)+\frac{3}{8\pi^{2}}\left\\{3\cdot
2\left(\frac{g}{2}\right)^{4}\log\frac{\frac{1}{4}g^{2}h^{2}}{\mu^{2}}+3g^{4}\log\frac{g^{2}h^{2}}{\mu^{2}}-4N_{\Psi}\left(\frac{g}{2}\right)^{4}\log\frac{\frac{1}{4}g^{2}h^{2}}{\mu^{2}}\right\\},$
(50)

where $\mu$ denotes the renormalization scale and $\lambda(\mu)$ is the Higgs
quartic coupling generated from RG evolutions. In Eq. (49), when $a\ll 1$ but
$\log a$ is still $O(1)$ (note that $a$ need not correspond to the true Higgs
VEV 246 GeV when deriving the matching condition for the Higgs quartic
coupling), one can make approximations of
$\log(4\sin^{2}(\pi\,a))\simeq\log(4\pi^{2}a^{2})$ and
$\log(4\sin^{2}(\pi\,\frac{a}{2}))\simeq\log(\pi^{2}a^{2})$, while
perturbation theory remains valid. Given these approximations, and noting that
$h$ in Eq. (50) is related to $a$ as $h=a/(gR)$, we obtain from the comparison
of Eqs. (49),(50) the following one-loop matching condition for the Higgs
quartic coupling:

$\displaystyle\lambda\left(\frac{1}{2\pi R}\right)\ =\ 0.$

In addition to the field content of the above general model, the model of the
main text contains extra $U(1)_{V}$ gauge group that is involved in the
breaking of $U(1)_{W}\times U(1)_{V}\to U(1)_{Y}$, and 4D chiral fermions
localized at $y=0$ that mix with 4D-decomposed components of the 5D bulk
fermions along the breaking. However, the presence of the $U(1)_{V}$ gauge
group and the 4D chiral fermions has negligible impact on the matching
condition $\lambda(\frac{1}{2\pi R})=0$, as shown below. The scale of the
$U(1)_{W}\times U(1)_{V}\to U(1)_{Y}$ breaking is given by the VEV of a scalar
$\phi$ localized at $y=0$. When coupling constants
$A_{ij},B_{ij},C_{ij},E_{ij}$ in Eq. (16) are $O(1)$, we have

$\displaystyle M_{\rm vec}\sim\langle\phi\rangle.$ (51)

As seen in Section 3, the model predicts $M_{\rm vec}<10^{7.5}$ GeV and
$1/(2\pi R)>10^{10.2}$ GeV, and hence there is a large hierarchy between
$\langle\phi\rangle$ and the compactification scale given by
$\langle\phi\rangle/(2\pi R)\lesssim 10^{-2.7}$. Consequently, the mixing of
the $U(1)_{V}$ and $U(1)_{W}$ gauge bosons and that of the 4D chiral fermions
and 4D-decomposed components of the 5D bulk fermions, induced by
$\langle\phi\rangle$, change the scale of $\lambda=0$ only by a negligible
amount of about $10^{-2.7}$ or below.

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# Distributed Agent-Based Collaborative Learning in Cross-Individual Wearable
Sensor-Based Human Activity Recognition

Ahmad Esmaeili  Computer and Information Technology
Purdue University
West Lafayette, IN, USA
<EMAIL_ADDRESS>Zahra Ghorrati  Computer and Information Technology
Purdue University
West Lafayette, IN, USA
<EMAIL_ADDRESS>Eric T. Matson  Computer and Information Technology
Purdue University
West Lafayette, IN, USA
<EMAIL_ADDRESS>
###### Abstract

The rapid growth of wearable sensor technologies holds substantial promise for
the field of personalized and context-aware Human Activity Recognition. Given
the inherently decentralized nature of data sources within this domain, the
utilization of multi-agent systems with their inherent decentralization
capabilities presents an opportunity to facilitate the development of
scalable, adaptable, and privacy-conscious methodologies. This paper
introduces a collaborative distributed learning approach rooted in multi-agent
principles, wherein individual users of sensor-equipped devices function as
agents within a distributed network, collectively contributing to the
comprehensive process of learning and classifying human activities. In this
proposed methodology, not only is the privacy of activity monitoring data
upheld for each individual, eliminating the need for an external server to
oversee the learning process, but the system also exhibits the potential to
surmount the limitations of conventional centralized models and adapt to the
unique attributes of each user. The proposed approach has been empirically
tested on two publicly accessible human activity recognition datasets,
specifically PAMAP2 and HARTH, across varying settings. The provided empirical
results conclusively highlight the efficacy of inter-individual collaborative
learning when contrasted with centralized configurations, both in terms of
local and global generalization.

###### Index Terms:

Human Activity Recognition, Collaborative Learning, Multi-agent Systems,
Wearable Sensors

##  I Introduction

Human Activity Recognition (HAR) is a field focused to inferring human
activities from raw time-series signals, involving the identification and
interpretation of human actions and behaviors through the analysis of data
obtained from body-worn and/or ambient sensors, specifically designed to
monitor the physical movements and gestures of individuals. The applications
of HAR span a broad spectrum, with notable utilization in Ambient Assisted
Living (AAL), well-being management, medical diagnosis, and particularly in
elderly care [1]. In the context of elderly care, for instance, HAR primarily
serves the purpose of identifying falls and monitoring Activities of Daily
Living (ADLs) among the elderly population, with a focus on delivering long-
term care, promoting well-being, and preserving their quality of life,
especially for those living independently.

The efficacy of everyday HAR applications relies on seamless integration and
utilization of wearable sensors, spanning from simple devices, such as
pedometers and accelerometers, to more computationally capable ones like
smartwatches. Traditionally, wearable sensors have served the purpose of
collecting monitoring data and gaining insights about individuals. However,
due to significant advancements in their computational capabilities in recent
years, coupled with the integration of machine learning techniques as in
practices like TinyML [2, 3], they have evolved into potent instruments for
real-time activity recognition and health assessment. Capitalizing on the
capabilities of these advanced devices, this paper introduces a collaborative
agent-based mechanism that allows distributed, on-device learning while
prioritizing privacy.

The complexity of HAR tasks is closely tied to various notable challenges,
including activity variability stemming from diverse human movements, data
quality and noise from sensor accuracy, scalability involving device and user
numbers, privacy and security for personal data, generalization to new
individuals, multimodal data fusion, and adaptability to evolving user
preferences and activities [4].

Distributed and decentralized methods in HAR offer the potential for
mitigating several of these challenges. For instance, through the distribution
of the learning process across networked devices and enabling on-device
processing, the need for centralized data storage is reduced and most privacy
concerns are alleviated. Additionally, decentralization offers the opportunity
to enhance scalability through sharing the computational burden among
individual devices and adaptability by allowing models to evolve in real-time
and better generalize across varying individuals and settings.

Multi-Agent Systems (MAS) are a class of distributed computational systems
where multiple autonomous agents independently perceive their environment and
engage in interactions towards individual or collective goals [5]. Unlike
traditional parallel methods, which often require a central coordinator, MAS
rely on distributed and collaborative approaches, enabling agents to work
together while maintaining individual decision-making capabilities. MASs’
inherent decentralization offers significant advantages for the design of
distributed and collaborative systems, where data privacy, adaptability, and
scalability are crucial considerations. Our proposed method seeks to achieve
decentralization by treating each individual as an autonomous agent, each with
its dedicated machine learning model and a private, locally stored dataset. In
this approach, every individual agent independently trains its personal model,
learning from its unique experiences. These agents then exchange insights with
their peers through a presumed communication channel connecting them,
contributing to the collective intelligence of the network.

The structure of the paper is outlined as follows: In Section II, we provide
an overview of noteworthy human activity recognition approaches that are
relevant to the objectives of this paper. Section III delves into the details
of the proposed agent-based learning method. Next, in Section IV, we explore
the characteristics and performance of the model we’ve put forward. To bring
our discussion to a close, Section V summarizes the key findings of this paper
and offers insights into potential directions for future research.

##  II Related Work

Traditionally, HAR has largely adhered to centralized paradigms, characterized
by the centralized collection, pre-processing, and storage of data for
training machine learning models. There are multiple recent surveys, such as
[6, 4, 7] to name a few, that discuss the notable deep learning-based
advancements in this domain and underscore the evolution of the field.
Centralized approaches, while effective in many respects, face limitations in
addressing privacy concerns and adapting to the dynamism of real-world
scenarios. Such limitations accompanied by the emergence of distributed
machine learning frameworks and the ubiquity of computationally capable IoT
and wearable devices have resulted in a transformative shift towards
distributed methods in recent years.

The Federated Learning (FL) framework, originally introduced in [8], has
served as the driving force behind a number of distributed contributions to
HAR in the literature. Within the FL paradigm, a shared model is trained
across a network of decentralized edge devices,such as smartphones or IoT
devices, without the necessity of exchanging raw data. Instead, these devices
compute model updates locally and send only these updates to a central server
for aggregation and model improvement while preserving data privacy. The
research conducted in [9, 10, 11, 12, 13, 14, 15] are among the noteworthy
research endeavors that have investigated the use of FL in HAR. While all
these FL-based contributions share the common principle of utilizing local
data on devices, they exhibit diversity in their specific focus, ranging from
intra-device model design to comprehensive system-scale analyses.

There are some other research contributions that exploit collaborative
approaches not for building and training models but for in the enhancement of
data. For instance, in [16], the authors have proposed a scheme called RG-RP
that uses a nonlinear function and a random projection matrix to perturb the
data of participants before sending it to a cloud service, where their
collaborative LSTM-CNN model is trained. In the Collaborative Self-Supervised
Learning (ColloSSL) technique presented in [17] leverages unlabeled data from
multiple devices worn by a user to learn high-quality features for HAR.

Multi-agent systems have also been previously explored as a viable approach to
address key HAR, with a particular focus on collaborative recognition and data
generation. The works reported in [18, 19, 20, 21, 22] are examples of the
agent-based approaches.

In [18], a high-level prototype multi-agent-based architecture is proposed for
human activity recognition, incorporating domain ontology and a set of six
agents responsible for tasks like individual localization and data collection
from diverse sources.

The work in [19] presents an agent-based signal processing approach for human
activity monitoring, centered on wireless body sensor networks. In this model,
agents represent sensor nodes, engaging in data collection, preprocessing, and
transmission to a central coordinator for subsequent analysis.

In [20], a platform called Cloud-assisted Agent-based Smart Home environment
(CASE) is introduced. Their three-layer client server architecture utilizes a
three-layer client-server architecture in which agents are responsible for
feature extraction, while the cloud takes charge of discovery tasks and the
creation of classifier agents.

The DCR method, as proposed in [21], usesa triad of agents with distinct roles
including data collection, segmentation and feature extraction, and activity
identification. Within the DCR framework, the collaboration is established
throughthe agents’ ability to request one another’s assistance in recognizing
observed activities, with results reported based on the trust levels they
mutually hold.

The work conducted in [22], the authors have introduced a multi-agent
attention-based deep learning algorithm named the Multiagent Transformer
Network (MATN) to address multimodal human activity recognition. In this
framework, agents represent distinct information sources, and activity classes
are predicted using a decentralized actor and centralized critic approach.

Conversely, the use of MAS in human activity data generation has been a
subject of research, exemplified by the work presented in [23, 24, 25]. The
primary objective of these studies is to address privacy concerns and data
insufficiency in HAR by providing realistic synthetic data through agent-based
simulation techniques.

The methodology presented in this paper distinguished itself from the
aforementioned approaches through its complete decentralization and relying
exclusively on wearable devices and their communication capabilities. In
essence, model updates and collaborations in this approach eschew the need for
a central or cloud-based server, hinging instead on the interactions between
an agent representing an individual and its immediate neighbors. Additionally,
diverging from other agent-based paradigms that assign distinct roles, the
proposed model harnesses collective intelligence among agents to enhance the
generalization capability of the entire network. The solution presented in
this paper is not bounded to any specific machine learning model to be used by
the agents, and it not only aligns intuitively with real-world environments
but also exhibits more flexibility, scaling adeptly to address the intricacies
of complex scenarios. The empirical validation in two multi-subject HAR
datasets of varying sizes demonstrates that the agents need not possess
similar datasets, whether in terms of size and distribution as it properly
accommodates the non-identically distributed (None-IID) and unbalanced nature
of the time series data collected by wearable sensors through inter-agent
collaboration and model updates.

##  III Proposed Model

This paper leverages a group of agents, each equipped with a deep learning
model, to classify activities based on the multivariate sensory data collected
from its corresponding individual. Figure 1 provides a high-level view of a
four-agent MAS utilized for collaborative HAR. In this model, each agent
functions as a processing unit representing an individual engaged in a range
of physical activities. These processing units can take the form of either
wearable devices with enhanced computational capabilities or dedicated
Inertial Measurement Units (IMUs) connected to various sensors. As depicted,
the activities of each individual may vary in both quantity and variability,
resulting in distinct local data. Having that said, the deep learning model
employed by each agent for recognizing the activities of its corresponding
individual maintains a uniform architecture across all peers and operates
within the same feature space.

Figure 1: High-level view of a four-agent MAS for collaborative HAR in the
proposed model.

Formally, let $\mathcal{A}=\\{a_{1},a_{2},\dots,a_{n}\\}$ represent the set of
agents, each corresponding to one of the $n$ individuals. We denote the MAS
network for activity recognition as
$\mathcal{N}\left<\mathcal{A},\mathcal{G}\right>$, where $\mathcal{G}$
signifies the interaction graph that delineates the connections between these
agents. In our approach, each agent $a_{i}$ is characterized by deep learning
model
$\mathcal{M}_{i}\left<\bm{\theta}_{i},\bm{\lambda}_{i}\right>:\mathbb{R}^{|\mathcal{S}_{i}|}\rightarrow\mathbb{Z}_{+}$
and dataset $\mathcal{D}_{i}=\\{(\bm{x}_{i}^{(t)},{y}_{i}^{(t)})\\}_{t}$,
where $\bm{\theta}_{i}$ and $\bm{\lambda}_{i}$ represent the agent’s learning
model parameters and hyperparameters, and $\bm{x}_{i}^{(t)}$ denotes the value
vector collected from $|\mathcal{S}_{i}|$ sensors worn by the agent while
performing activity $y_{i}^{(t)}$at time step $t$. Note that, for the sake of
presentation simplicity, we have assumed that activities are indexed using
positive integer values. The underlying assumption of uniformity across models
and feature spaces implies that:

$\forall a_{i},a_{j}\in\mathcal{A}:\mathcal{M}_{i}\simeq\mathcal{M}_{j}\text{
and }\mathcal{S}_{i}=\mathcal{S}_{j}$ (1)

where $\mathcal{M}_{i}\simeq\mathcal{M}_{j}$ signifies the structural
similarity between two models $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$.

Each agent $a_{i}$’s personal learning process involves the minimization of
the empirical risk function $\mathsf{R}_{i}$ defined based on the loss
function $\mathcal{L}_{i}$ as follows:

$\min_{\bm{\theta}_{i}}\left[\mathsf{R}_{i}(\mathcal{M}_{i}):=\frac{1}{|\mathcal{D}_{i}|}\sum_{(\bm{x}_{i}^{(t)},{y}_{i}^{(t)})\in\mathcal{D}_{i}}\mathcal{L}_{i}(\mathcal{M}_{i}(\bm{x}_{i}^{(t)});{y}_{i}^{(t)})\right]$
(2)

Assuming that each agent independently trains its model on its locally
collected sensory data, the resulting model may exhibit limited generality and
will struggle to recognize activities the individual has not encountered or
for which there is minimal data available. To address this challenge, we
harness collaboration by sharing model parameters among neighboring agents. In
essence, the proposed model seeks to approximate a near-optimal solution for
the aforementioned optimization problem by employing a sequence of local model
optimizations and scheduled aggregation of the models from connected peers.

Formally, we assume weighted directional connections between the agents and
define the interaction weights $\mathcal{W}$ as a square matrix as follows:

$\mathcal{W}:=\begin{bmatrix}w_{1,1}&w_{1,2}&\ldots&w_{1,n}\\\
w_{2,1}&w_{2,2}&\ldots&w_{2,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
w_{n,1}&w_{n,2}&\ldots&w_{n,n}\\\ \end{bmatrix}_{n\times n}$ (3)

where $w_{i,j}$ represents the weight that agent $a_{j}$ assigns to the
contribution from agent $a_{i}$ and is defines as follows:

$w_{i,j}=\begin{cases}0,&\text{If }(a_{j},a_{i})\notin\mathcal{G}\\\
w\in\mathbb{R}_{+},&\text{otherwise}\end{cases}$ (4)

Note that we have assumed the notation $(a_{j},a_{i})$ to specify a
directional communication link from agent $a_{j}$ to agent $a_{i}$ within the
graph $\mathcal{G}$. Practically, the agents only keep track of their
immediate neighbors. With that said, using a similar notation, we employ the
column vector $\bm{w_{i}}\in\mathbb{R}^{z}_{+}$ to represent the weights that
agent $a_{i}$ assigns to its $z$ immediate contributing neighbors.

In the proposed model, each agent uses weighted averaging as the aggregation
method. Formally, the aggregation function of the agent $a_{i}$ is denoted by
$\mathsf{F}_{i}$ and is defines as follows:

$\mathsf{F}_{i}(\bm{\Theta},\bm{w_{i}}):=\frac{1}{\bm{w_{i}}^{\top}\bm{1}_{z}}(\bm{\Theta}\bm{w_{i}})$
(5)

where $\bm{1}_{z}$ represents an $z$-dimensional column vector of ones, and
$\bm{\Theta}\in\mathbb{R}^{p\times z}$ is a matrix composed of model
parameters of length $p$ from agents $z$ neighbors. That is:

$\bm{\Theta}:=\begin{bmatrix}\bm{\theta}_{1}&\bm{\theta}_{2}&\ldots&\bm{\theta}_{z}\end{bmatrix}:=\begin{bmatrix}\theta_{1,1}&\theta_{1,2}&\ldots&\theta_{1,z}\\\
\theta_{2,1}&\theta_{2,2}&\ldots&\theta_{2,z}\\\
\vdots&\vdots&\ddots&\vdots\\\
\theta_{p,1}&\theta_{p,2}&\ldots&\theta_{p,z}\\\ \end{bmatrix}_{p\times z}$
(6)

The weights that agents assign to their peers can be adjusted dynamically, and
they may even be learned based on the interaction experiences of the agents.
In this paper, we consider bidirectional communication between the agents and
establish the weights based on the size of the datasets as follows:

$\mathcal{W}:=\begin{bmatrix}|\mathcal{D}_{1}|&|\mathcal{D}_{1}|&\ldots&|\mathcal{D}_{1}|\\\
|\mathcal{D}_{2}|&|\mathcal{D}_{2}|&\ldots&|\mathcal{D}_{2}|\\\
\vdots&\vdots&\ddots&\vdots\\\
|\mathcal{D}_{n}|&|\mathcal{D}_{n}|&\ldots&|\mathcal{D}_{n}|\\\
\end{bmatrix}_{n\times n}$ (7)

It’s worth emphasizing that these weights are communicated during
interactions, and the agents are not required to have omniscient access to all
the network weights in $\mathcal{N}$ beforehand.

The specific steps that each agent follows during collaborative learning are
outlined in Algorithm 1. In line 1, the function ShouldContinue determines
when to conclude the training process. This function can be implemented using
criteria such as a fixed number of training epochs or performance on the
validation set as examples. The function IsFirstIteration in line 1 ensures
that the agent conducts an initial training on its private dataset before
initiating the collaboration process. In case that the agent is not in its
first training iteration, the receiving process and the collection of the
model parameters from its immediate neighbors are managed employing the
function CollectContributions in line 1, and the aggregation result is applied
to its current model. Finally, using the function SendModel in line 1, the
agent sends its model parameter values to its neighbors as part of the
collaboration. This algorithm is presented in a general manner, without
imposing any specific implementations. The next section of the paper provides
detailed insights into the implementations utilized for conducting the
experiments.

1 Procedure  _Learn(__)_

2 while  _ShouldContinue $()$_ do

3 if  _IsFirstIteration $()=False$_ then

4 $\bm{\Theta},\bm{w_{i}}\leftarrow\textsc{CollectContributions}()$

5 $\bm{\theta}_{i}\leftarrow\textsf{F}_{i}(\bm{\Theta},\bm{w_{i}})$

6

7 end if

8 $\bm{\theta}_{i}\leftarrow\textsc{Train}(\mathcal{M}_{i})$

9 for  _$z\in\textsc{Neighbors}()$ in parallel_ do

10 $\textsc{SendModel}(z,\theta_{i})$

11

12 end for

13

14 end while

15

16 end

Algorithm 1 The collaborative learning procedure run by each agent $a_{i}$ in
parallel.

##  IV Results and Discussion

As mentioned earlier, our collaborative learning model for HAR is not
constrained to a specific deep learning model. With that in mind, this section
establishes a simplified scenario to examine the system’s effectiveness and
behavior under various conditions. We intentionally selected a basic deep
learning model to underscore the collaboration aspects of the model. It is
important to note that employing more complex learning structures within an
agent’s design would likely yield higher performance.

###  IV-A Agent Design

As illustrated in Figure 2, the fundamental components of an agent consist of
three major elements: the _preparation/preprocessing_ unit, which performs
cleansing and windowing operations on the raw multi-variate time-series data
collected from the sensors, the _learning_ unit that holds the specification
of agents deep learning model $\mathcal{M}$ and handles the learning process,
and _collaboration & Aggregation_ component that controls the communications
with the neighbors and aggregates the model parameters as explained before.

In all our experiments, the deep learning model we utilized incorporates a
convolutional feature network, consisting of a 1-D convolutional layer and a
max-pooling layer, followed by a fully connected layer. The input channel of
the convolutional layer corresponds to the number of features in the input
data space, denoted as $|\mathcal{S}_{i}|$, with 64 output channels and a
filter size of 3. The kernel size used for the max-pooling layer is set to 2.

Figure 2: The building blocks of an agent’s architecture.

###  IV-B Datasets

We have studies the behavior of the proposal model on two publicly avaiable
HAR datasets explained below.

####  IV-B1 PAMAP2

Provided by [26], this dataset features data collected from 9 subjects (1
female and 8 males). These subjects wore 3 Inertial Measurement Units (IMUs)
on their wrists, chests, and ankles, in addition to a heart rate monitor,
while performing 18 different daily physical activities, including 6 optional
activities. The dataset encompasses all recorded values sampled at 100 Hz from
the sensors, resulting in a raw dataset with 52 columns for each of 3,850,505
instances. In our experiments, we focused on using 36 columns, which comprise
the sensory data from gyroscopes, magnetometers, and accelerometers.
Additionally, the experiments excluded more complex activities such as
“playing soccer,” “car driving,” “house cleaning,” and others, concentrating
instead on the classification of remaining 12 distinct activities.

####  IV-B2 HARTH

The Human Activity Recognition Trondheim (HARTH) dataset [27] is gathered from
22 subjects in a free-living setting wearing two 3-axial accelerometers on
their right thighs and lower backs. The dataset features 6,461,328 instances
sampled at a rate of 50Hz, and its 12 activities were annotated through
analyzing video recordings from a chest-mounted camera frame-by-frame. In our
experiments, we opted to use all 6 features from the original data and focused
on 5 locomotion activities: “walking,” “running,” “standing,” “sitting,” and
“lying.”

###  IV-C Experimental Settings

The collaborative HAR system proposed in this study was developed using the
pytorch framework, and all experiments were conducted on a computing cluster
node that featured an A10 GPU, 256 GB of memory, and a 32-core CPU. All agents
employ a sequence length of 100 time steps for segmenting the samples into
windows within the processing unit. They utilize mini-batch training with
epoch-based termination criteria in their learning units, employing a batch
size of 64 windows during the training process. For the sake of optimizing the
parameters of the neural network, we have used Adam method [28] with suggested
default settings $\alpha=0.001$, $\beta_{1}=0.9$, $\beta_{2}=0.999$, and
$\epsilon=10^{-8}$. Collaboration between agents and their neighbors takes
place at the end of each batch of training, and we have assumed a fully
connected bidirectional network topology among the agents, allowing each agent
to communicate with all others.

###  IV-D Results

We have assessed the effectiveness of collaborative learning in terms of both
global and local generalization capabilities. Our global generalization
results showcase how the trained models of the agents perform when applied to
data from individuals other than those they were initially trained on. In
contrast, local generalization examines how collaboration with peers impacts
the generalization of the trained model when applied to the agent’s private
data. We have used F1-score with macro averaging as the metric to measure
generalization.

To evaluate the global generalization of the collaborative learning approach
on the PAMAP2 dataset, data from subject 1 were designated as the test
dataset, while data from the remaining 8 subjects was allocated for training,
with each subject assigned to a separate agent. Similarly, for evaluating the
global generalization of the HARTH dataset, subjects 1, 2, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 17, and 20 were selected as training agents, and data
from subjects 16 and 18 were designated for testing. This partitioning was
carried out to ensure that the test subjects had samples for all performed
activities. Figures 3 and 4 provide an overview of the activity data
distributions for each used subject in the PAMAP2 and HARTH datasets,
respectively.

Figure 3: The data distribution for each activity within each subject in the
PAMAP2 dataset used in our experiments. Figure 4: The data distribution for
each activity within each subject in the HARTH dataset used in our
experiments.

To investigate the local generalization capability of the proposed learning
model on both PAMAP2 and HARTH datasets, we randomly divided the data for each
subject into 80% training and 20% testing sets. Agents representing the
subjects then collaboratively trained their models using the training set and
assessed local generalization on their respective test sets.

In both of the previously mentioned generalization assessments, we also
conducted a performance comparison between our proposed decentralized method
and a centralized approach. In the centralized approach, we followed the same
training and test dataset splits as described earlier, with the only
difference being that a single agent was given all the samples from the
training set to construct its model in isolation.

####  IV-D1 Result on PAMAP2

The results, which evaluate the effects of collaboration on both global and
local generalization measures of the agents at each training epoch, are
presented in Figures 5 and 6, respectively. Furthermore, Figure 7 offers an
average performance overview of the agents in the system under various
settings, facilitating a macro-level comparison of the results.

In addition to the evident positive impact of inter-agent collaboration on
both global and local generalization, two significant observations deserve
attention. The first observation pertains to the substantial effect of
collaboration on the agent representing subject 9. As illustrated in Figure 3,
this subject had no training samples for 11 out of the 12 activities.
Consequently, as observed in Figures 5(b) and 6, the agent exhibited very weak
performance, with F1-scores falling below 5% for both local and global
generalizations. However, thanks to the collaboration with other agents in the
network, the agent’s performance dramatically improved, even surpassing that
of most other agents despite its limited training data. This improvement can
be attributed to the agent having a comparable number of samples for the
relatively challenging-to-classify “rope_jumping” activity and the higher
accumulated weight of its peers during aggregation.

The second important finding is the overall higher performance of the proposed
decentralized method compared to the centralized approach in activity
recognition. As shown in Figure 7, collaboration not only led to a nearly 100%
increase in the average F1-scores of agents for both local and global
generalization but also resulted in performance surpassing that of a
centralized system in the final training epochs, despite having the same
training data and number of training batches.

(a) With Collaboration

(b) Without Collaboration

Figure 5: The global generalization of the proposed decentralized model on the
PAMAP2 dataset.

(a) With Collaboration

(b) Without Collaboration

Figure 6: The local generalization of the proposed decentralized model on the
PAMAP2 dataset.

(a) Global With Collaboration

(b) Global Without Collaboration

(c) Local With Collaboration

(d) Local Without Collaboration

Figure 7: The average performance comparison between the centralized and
decentralized approaches in terms of both local and global generalization on
the PAMAP2 dataset.

####  IV-D2 Result on HARTH

We conducted a similar series of experiments on the HARTH dataset and
illustrated the outcomes in Figures 8, 9, and 10. Our observations align with
those made in the PAMAP2 experiments, underscoring the substantial positive
impact of collaboration, especially for agents dealing with challenging
training datasets. As depicted in the figures, except for agent 15, which
represents subject 18 in the dataset, all agents exhibited notably improved
performance compared to the centralized approach, achieving an impressive
average F1-score increase of approximately 200% when leveraging collaboration.

One intriguing aspect revealed in the results is how collaboration affected
the highest-performing agent, agent 15. A comparison between results with and
without collaboration shows a decline in the agent’s performance, which we
believe can potentially be corrected as training progresses into higher epochs
and by incorporating more adaptive techniques for managing the weight
assignments during the aggregation process.

It is worth highlighting that these results were achieved using a simple deep
learning model within the agents, without extensive fine-tuning for the
experiments. Furthermore, since the network size in the HARTH experiment is
nearly twice that of the PAMAP2 experiment, it is expected that convergence
and improvements in the collaborative system would proceed at a slower pace,
contributing to the relatively pronounced performance oscillations observed as
the training epochs advanced.

(a) With Collaboration

(b) Without Collaboration

Figure 8: The global generalization of the proposed decentralized model on the
HARTH dataset.

(a) With Collaboration

(b) Without Collaboration

Figure 9: The local generalization of the proposed decentralized model on the
HARTH dataset.

(a) Global With Collaboration

(b) Global Without Collaboration

(c) Local With Collaboration

(d) Local Without Collaboration

Figure 10: The average performance comparison between the centralized and
decentralized approaches in terms of both local and global generalization on
the HARTH dataset.

##  V conclusion

This paper introduced a decentralized approach to Human Activity Recognition.
The proposed agent-based method leverages the collaborative capabilities of
individual agents, each equipped with its deep learning model, to collectively
recognize a diverse range of activities from multi-variate time series data.
We conducted comprehensive experiments on the PAMAP2 and HARTH datasets,
comparing the performance of our approach to a centralized model. The results
unequivocally demonstrate the positive impact of inter-agent collaboration,
significantly improving both local and global generalization capabilities.
Remarkably, the decentralized method’s performance surpassed that of the
centralized system, even in the presence of the same training data,
highlighting its potential for real-world applications.

For future work, there are several promising directions to explore. First, we
aim to investigate the scalability of our approach to larger networks with
more diverse agents, potentially including non-human sensors. Additionally, we
intend to delve into adaptive learning strategies, where agents dynamically
adjust their inter-agent connections and weights based on their experiences.
Privacy preservation is another critical avenue, as we plan to extend our work
to ensure robust data security within the collaborative framework.
Furthermore, we aim to test our approach on more diverse and complex datasets,
enabling us to validate its efficacy in a broader range of real-world
scenarios.

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