aainstitutetext: Namtech, IIT Gandhinagar campus, Gandhinagar 382421,
Indiabbinstitutetext: CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon,
UMR 5822,
F-69622, Villeurbanne, France

# A Fuzzy Model of Quarks & Leptons

Pratik Chattopadhyay b and Florian Nortier<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

UV-completions of quantum field theories (QFT’s) based on string-inspired
nonlocality have been proposed to improve the high-energy behavior of local
QFT, with the hope of including gravity. One problematic issue is how to
realize spontaneous symmetry breaking without introducing an infinite tower of
ghosts in the perturbative spectrum. In this letter, a weakly nonlocal
extension of the Standard Model (SM) is proposed: the Fuzzy Standard Model
(FSM). It is a smooth deformation of the SM based on covariant star-products
of fields. This new formalism realizes electroweak symmetry breaking without
ghosts at tree-level, and it does not introduce any new elementary particles.
We argue that the FSM has several appealing theoretical and phenomenological
features that deserve to be investigated in future works.

###### Keywords:

Gauge Field Theories, Spontaneous Symmetry Breaking, Weak Nonlocality

##  1 Introduction

Embedding the electromagnetic and subatomic interactions – described by the
Standard Model (SM) of particle physics Halzen:1984mc  – with gravity into a
UV-complete quantum theory seems to go beyond the traditional framework of
local quantum field theory (QFT) Marshakov:2002ff ; Giddings:2011xs . Lorentz-
invariant string-inspired QFT’s have been proposed with ghost-free infinite-
derivative exponential form factors Buoninfante:2018mre  that exhibits weakly
nonlocal (WNL) features, with the hope to soften the UV-behavior in gravity
Modesto:2011kw ; Biswas:2011ar ; Biswas:2013cha ; Calcagni:2014vxa ;
Talaganis:2014ida ; Talaganis:2016ovm ; Abel:2019zou  and particle physics
Biswas:2014yia ; Ghoshal:2017egr ; Buoninfante:2018gce ; Ghoshal:2020lfd ;
Frasca:2020jbe ; Frasca:2020ojd ; Frasca:2021iip ; Frasca:2022vvp ;
Chatterjee:2023ehr ; Abu-Ajamieh:2023syy , while preserving analyticity and
unitarity at the perturbative level Efimov:1966ylf ; Pius:2016jsl ;
Carone:2016eyp ; Pius:2018crk ; Briscese:2018oyx ; Chin:2018puw ;
DeLacroix:2018arq ; Briscese:2021mob ; Koshelev:2021orf ; Buoninfante:2022krn
, with applications in: stability of the Higgs potential and vacuum decay
Krasnikov:1987yj ; Biswas:2014yia ; Ghoshal:2017egr ; Ghoshal:2022mnj ; Abu-
Ajamieh:2023syy ; dark matter physics Ghoshal:2018gpq ; grand unification and
anomalies Krasnikov:2020kgh ; Abu-Ajamieh:2023roj ; braneworlds
Nortier:2021six ; cosmology Koshelev:2023elc  and black hole physics
Buoninfante:2022ild . The formalism is manifestly gauge-invariant via the
Krasnikov-Terning scheme Krasnikov:1987yj ; Terning:1991yt , but not UV-finite
Shapiro:2015uxa . In order to deal with the UV-divergences, the traditional
power-counting theorems of perturbative renormalization Collins:1984xc  select
the more involved class of Kuz’min-Tomboulis form factors Kuzmin:1989sp ;
Tomboulis:1997gg . They interpolate between a (local) 2-derivative QFT in the
IR and a (local) higher-derivative QFT in the UV via a (nonlocal) infinite-
derivative window BasiBeneito:2022wux . In this case, one can build
gravitational or Yang-Mills theories that are superrenormalizable (or even UV-
finite with higher-derivative “killer operators” that do not add ghosts in
tree-level propagators) Kuzmin:1989sp ; Tomboulis:1997gg ; Modesto:2011kw ;
Modesto:2014lga ; Modesto:2015lna ; Modesto:2015foa ; Tomboulis:2015esa ;
Modesto:2016max ; Modesto:2017hzl ; Modesto:2017sdr ; Koshelev:2017ebj .

However, a well-known difficulty of this program arises when a scalar field
acquires a vacuum expectation value (vev): an infinite tower of ghosts pops up
above the nonlocal scale in the physical vacuum Barnaby:2007ve ; Galli:2010qx
; Gama:2018cda ; Hashi:2018kag ; Koshelev:2020fok ; Nortier:2023dkq . This is
a serious issue since, in the Glashow-Weinberg-Salam (GWS) model of
electroweak (EW) interactions Glashow:1961tr ; Weinberg:1967tq ; Salam:1968rm
, spontaneous electroweak symmetry breaking (EWSB) occurs via tachyon
condensation, aka the Higgs mechanism Englert:1964et ; Higgs:1964pj ;
Guralnik:1964eu . Fortunately, several models to realize a WNL Higgs mechanism
without ghosts have been proposed in the literature Hashi:2018kag ;
Modesto:2021okr ; Nortier:2023dkq . In this letter, we will use the formalism
with covariant star-products between fields, which has been recently proposed
by one of the authors in Ref. Nortier:2023dkq . Our goal is to build the Fuzzy
Standard Model (FSM): a WNL deformation of the SM that is ghost-free in the
physical EW vacuum. Since our analysis is at tree-level, we stay agnostic on
the precise choice of the infinite-derivative form factors involved in the
star-products.

Name | Elementary Field | Generation(s) | Spin | $\left(R_{C},R_{W}\right)_{Y}$
---|---|---|---|---
Gluons | $G_{\mu}=G_{\mu}^{a}\,\frac{\lambda_{a}}{2}$ | $1$ | $1$ | $\left(\mathbf{8},\mathbf{1}\right)_{0}$
Weak isospin bosons | $W_{\mu}=W_{\mu}^{a}\,\frac{\sigma_{a}}{2}$ | $1$ | $1$ | $\left(\mathbf{1},\mathbf{3}\right)_{0}$
Weak hypercharge boson | $B_{\mu}$ | $1$ | $1$ | $\left(\mathbf{1},\mathbf{1}\right)_{0}$
Left-handed quarks | $Q_{L}^{i}=\left(u_{L}^{i},\ d_{L}^{i}\right)^{T}$ | $3$ | $1/2$ | $\left(\mathbf{3},\mathbf{2}\right)_{+1/6}$
Right-handed $u$-type quarks | $u_{R}^{i}$ | $3$ | $1/2$ | $\left(\mathbf{3},\mathbf{1}\right)_{+2/3}$
Right-handed $d$-type quarks | $d_{R}^{i}$ | $3$ | $1/2$ | $\left(\mathbf{3},\mathbf{1}\right)_{-1/3}$
Left-handed leptons | $L_{L}^{i}=\left(\nu_{L}^{i},\ e_{L}^{i}\right)^{T}$ | $3$ | $1/2$ | $\left(\mathbf{1},\mathbf{2}\right)_{-1/2}$
Right-handed $e$-type leptons | $e_{R}^{i}$ | $3$ | $1/2$ | $\left(\mathbf{1},\mathbf{1}\right)_{-1}$
Higgs tachyon | $H=\left(H^{-},\ H^{0}\right)^{T}$ | $1$ | $0$ | $\left(\mathbf{1},\mathbf{2}\right)_{+1/2}$

Composite Bosonic Field | $\left(R_{C},R_{W}\right)_{Y}$
---|---
$\nabla_{\mu}G_{\nu}=\left(\nabla_{\mu}G_{\nu}\right)^{a}\frac{\lambda_{a}}{2}\,,\ \ \left(\nabla_{\mu}G_{\nu}\right)^{a}=\nabla_{\ c\mu}^{a}G_{\ \nu}^{c}=\left(\delta_{c}^{a}\partial_{\mu}+g_{3}\,f_{\ bc}^{a}G_{\ \mu}^{b}\right)G_{\nu}^{c}$ | $\left(\mathbf{8},\mathbf{1}\right)_{0}$
$\mathcal{G}_{\mu\nu}=\mathcal{G}_{\ \mu\nu}^{a}\frac{\lambda_{a}}{2}\,,\ \ \mathcal{G}_{\ \mu\nu}^{a}=\partial_{\mu}G_{\ \nu}^{a}-\partial_{\nu}G_{\ \mu}^{a}+g_{3}\,f_{\ bc}^{a}G_{\ \mu}^{b}G_{\ \nu}^{c}$ | $\left(\mathbf{8},\mathbf{1}\right)_{0}$
$\nabla_{\mu}W_{\nu}=\left(\nabla_{\mu}W_{\nu}\right)^{a}\frac{\sigma_{a}}{2}\,,\ \ \left(\nabla_{\mu}W_{\nu}\right)^{a}=\nabla_{\ c\mu}^{a}W_{\ \nu}^{c}=\left(\delta_{c}^{a}\partial_{\mu}+g_{2}\,\epsilon_{\ bc}^{a}W_{\ \mu}^{b}\right)W_{\ \nu}^{c}$ | $\left(\mathbf{1},\mathbf{3}\right)_{0}$
$\mathcal{W}_{\mu\nu}=\mathcal{W}_{\ \mu\nu}^{a}\frac{\sigma_{a}}{2}\,,\ \ \mathcal{W}_{\ \mu\nu}^{a}=\partial_{\mu}W_{\ \nu}^{a}-\partial_{\nu}W_{\ \mu}^{a}+g_{2}\,\epsilon_{\ bc}^{a}W_{\ \mu}^{b}W_{\ \nu}^{c}$ | $\left(\mathbf{1},\mathbf{3}\right)_{0}$
$\mathcal{B}_{\mu\nu}=\partial_{\mu}B_{\mu}-\partial_{\nu}B_{\nu}$ | $\left(\mathbf{1},\mathbf{1}\right)_{0}$
$\mathcal{D}_{\mu}H=\left(\partial_{\mu}-ig_{2}\,W_{\mu}-ig_{1}\,Y_{H}\,B_{\mu}\right)\cdot H$ | $\left(\mathbf{1},\mathbf{2}\right)_{+1/2}$
$H^{\dagger}\cdot H$ | $\left(\mathbf{1},\mathbf{1}\right)_{0}$

Composite Fermionic Field | $\left(R_{c},R_{w}\right)_{Y}$
---|---
$\mathcal{D}_{\mu}Q_{L}^{i}=\left(\partial_{\mu}-ig_{3}\,G_{\mu}-ig_{2}\,W_{\mu}-ig_{1}\,Y_{Q}\,B_{\mu}\right)\cdot Q_{L}^{i}$ | $\left(\mathbf{3},\mathbf{2}\right)_{+1/6}$
$\mathcal{D}_{\mu}u_{R}^{i}=\left(\partial_{\mu}-ig_{3}\,G_{\mu}-ig_{1}\,Y_{u}\,B_{\mu}\right)\cdot u_{R}^{i}$ | $\left(\mathbf{3},\mathbf{1}\right)_{+2/3}$
$\overline{Q}_{L}\cdot\overline{H}\ \ \text{with}\ \ \overline{H}\equiv i\sigma_{2}\cdot H^{*}$ | $\left(\mathbf{\overline{3}},\mathbf{1}\right)_{-2/3}$
$\mathcal{D}_{\mu}d_{R}^{i}=\left(\partial_{\mu}-ig_{3}\,G_{\mu}-ig_{1}\,Y_{d}\,B_{\mu}\right)\cdot d_{R}^{i}$ | $\left(\mathbf{3},\mathbf{1}\right)_{-1/3}$
$\overline{Q}_{L}^{i}\cdot H$ | $\left(\mathbf{\overline{3}},\mathbf{1}\right)_{+1/3}$
$\mathcal{D}_{\mu}L_{L}^{i}=\left(\partial_{\mu}-ig_{2}\,W_{\mu}-ig_{1}\,Y_{L}\,B_{\mu}\right)\cdot L_{L}^{i}$ | $\left(\mathbf{1},\mathbf{2}\right)_{-1/2}$
$\overline{L}_{L}^{i}\cdot H$ | $\left(\mathbf{1},\mathbf{1}\right)_{+1}$
$\mathcal{D}_{\mu}e_{R}^{i}=\left(\partial_{\mu}-ig_{1}\,Y_{e}\,B_{\mu}\right)\cdot e_{R}^{i}$ | $\left(\mathbf{1},\mathbf{1}\right)_{-1}$

Table 1: Gauge quantum numbers of the elementary and composite fields in the
SM, based on the gauge group $SU(3)_{C}\times SU(2)_{W}\times U(1)_{Y}$, where
the subscripts $C$, $W$ and $Y$ refer to color, weak isospin and weak
hypercharge, respectively. $R_{C}$ and $R_{W}$ stands for the representations
of $SU(3)_{C}$ and $SU(2)_{W}$, respectively. $\sigma^{a}$ and $\lambda^{a}$
are the Pauli and Gell-Mann matrices, respectively. $\nabla_{\mu}$ and
$\mathcal{D}_{\mu}$ are the covariant derivatives acting on the adjoint and
fundamental representations, respectively.

This letter is organized as follows. In Sections 2, we discuss the
generalization of the covariant star-product of fields to include non-Abelian
gauge bosons, and then we use it to build a ghost-free EWSB model. In Section
3, we define the star-product between fermion fields to propose a WNL
deformation of the Yukawa sector. We briefly comment on the Cabibbo-Kobayashi-
Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv  and the status of
flavor-changing neutral currents (FCNC’s), related to the
Glashow–Iliopoulos–Maiani (GIM) mechanism Glashow:1970gm . In Section 4, we
summarize our results and give several points that deserve to be studied
elsewhere. When not specified, our QFT conventions are the same as in the
textbook Peskin:1995ev , and we remind the reader of the elementary and
composite fields appearing in the SM Lagrangian in Tab. 1.

##  2 Bosonic Fields

###  2.1 Pure Gauge Sector

Following the formalism introduced in Ref. Nortier:2023dkq , one wants to
define covariant star-products $\star_{\bullet}$ between gauge fields, as well
as their noncovariant avatars $\ast_{\bullet}$, with their associated real
entire function $\vartheta_{\bullet}(z)$ and fuzzy-plaquettes
$\eta_{\bullet}=\left(1/\Lambda_{\bullet}\right)^{2}$ that are defined in
terms of the nonlocal scales $\Lambda_{\bullet}$. All exponent functions
$\vartheta_{\bullet}(z)$ appearing in this letter satisfy
$\vartheta_{\bullet}(-z)=\vartheta_{\bullet}(z)$. The form factors are thus
entire functions without zeroes in $\mathbb{C}$, and can be developed as

$e^{\vartheta_{\bullet}(z)}=\sum_{n=0}^{+\infty}c_{2n}z^{2n}\,,$ (1)

with the same behavior111This condition is required to keep perturbative
unitarity for scattering amplitudes above the nonlocal scales
$\Lambda_{\bullet}$ Koshelev:2021orf . for both timelike and spacelike momenta
$z=p^{2}$. These form factors are WNL, i.e. in the local limit
$\eta_{\bullet}\rightarrow 0$, one retrieves Dirac distributions $\delta(z)$.
With the Minkowski metric $g_{\mu\nu}=\text{diag}(+1,-1,-1,-1)$, one
introduces the tensors $\eta_{\bullet}^{\mu\nu}=\eta_{\bullet}\,g^{\mu\nu}$ to
define the following star-products:

  * •

For the field of weak hypercharge bosons $B_{\mu}$, associated with the
Abelian gauge symmetry $U(1)_{Y}$ of coupling $g_{1}$, one has222We use
similar notations as in the definition of the Groenewold-Moyal product in
noncommutative QFT Hinchliffe:2002km .

$\displaystyle B_{\rho}(x)\star_{0}B_{\sigma}(x)$ $\displaystyle\equiv
B_{\rho}(x)\ast_{0}B_{\sigma}(x)$ (2) $\displaystyle\equiv
B_{\rho}(x)\,e^{\vartheta_{0}\left(\overleftarrow{\partial_{\mu}}\,\eta_{0}^{\mu\nu}\,\overrightarrow{\partial_{\nu}}\right)}\,B_{\sigma}(x)$
(3)
$\displaystyle=\left.e^{\vartheta_{0}\left(\partial_{\mu}^{y}\,\eta_{0}^{\mu\nu}\,\partial_{\nu}^{z}\right)}\,B_{\rho}(y)\cdot
B_{\sigma}(z)\right|_{y=z=x}$ (4)
$\displaystyle=\int\dfrac{d^{4}p_{i}\,d^{4}p_{j}}{(2\pi)^{8}}\
e^{-i\left(p_{i}+p_{j}\right)\cdot
x\,+\,\vartheta_{0}\left(\eta_{0}\,p_{i}\cdot
p_{j}\right)}\,\widetilde{B}_{\rho}(p_{i})\cdot\widetilde{B}_{\sigma}(p_{j})\,,$
(5)

where $\widetilde{f}(p)$ is the Fourier transform of the function $f(x)$.

  * •

For the field of weak isospin bosons $W_{\mu}$, associated with the non-
Abelian gauge symmetry $SU(2)_{W}$ of coupling $g_{2}$, one has

$\displaystyle W_{\rho}(x)\ast_{w}W_{\sigma}(x)$ $\displaystyle\equiv
W_{\rho}(x)\,e^{\vartheta_{w}\left(\overleftarrow{\partial_{\mu}}\,\eta_{w}^{\mu\nu}\,\overrightarrow{\partial_{\nu}}\right)}\,W_{\sigma}(x)\,,$
(6) $\displaystyle W_{\rho}(x)\star_{w}W_{\sigma}(x)$ $\displaystyle\equiv
W_{\rho}(x)\,e^{\vartheta_{w}\left(\overleftarrow{\nabla_{\mu}}\,\eta_{w}^{\mu\nu}\,\overrightarrow{\nabla_{\nu}}\right)}\,W_{\sigma}(x)$
(7)
$\displaystyle=\left.e^{\vartheta_{w}\left(\nabla_{\mu}^{y}\,\eta_{w}^{\mu\nu}\,\nabla_{\nu}^{z}\right)}\,W_{\rho}(y)\cdot
W_{\sigma}(z)\right|_{y=z=x}$ (8)
$\displaystyle=W_{\rho}\ast_{w}W_{\sigma}+\mathcal{O}\left(\dfrac{g_{2}}{\Lambda_{w}^{4}}\,W^{3}\right)\,,$
(9)

with the commutator $\left[\nabla_{\mu}^{y},\nabla_{\nu}^{z}\right]=0$ for
$y\neq z$.

  * •

For the field of gluons $G_{\mu}$, associated with the non-Abelian gauge
symmetry $SU(3)_{C}$ of coupling $g_{3}$, one has

$\displaystyle G_{\rho}(x)\ast_{c}G_{\sigma}(x)$ $\displaystyle\equiv
G_{\rho}(x)\,e^{\vartheta_{c}\left(\overleftarrow{\partial_{\mu}}\,\eta_{c}^{\mu\nu}\,\overrightarrow{\partial_{\nu}}\right)}\,G_{\sigma}(x)\,,$
(10) $\displaystyle G_{\rho}(x)\star_{c}G_{\sigma}(x)$ $\displaystyle\equiv
G_{\rho}(x)\,e^{\vartheta_{c}\left(\overleftarrow{\nabla_{\mu}}\,\eta_{c}^{\mu\nu}\,\overrightarrow{\nabla_{\nu}}\right)}\,G_{\sigma}(x)$
(11)
$\displaystyle=G_{\rho}\ast_{c}G_{\sigma}+\mathcal{O}\left(\dfrac{g_{3}}{\Lambda_{c}^{4}}\,G^{3}\right)\,.$
(12)

The noncovariant star-product $\ast_{\bullet}$ is thus used as a notation to
ignore the perturbative expansion of the could of gauge bosons (_gauge cloud_
for short) that dresses the covariant star-product $\star_{\bullet}$ of fields
(cf. Fig. 1). Note that these star-products are nonassociative in general, and
one needs to keep this property in mind when dealing with operators involving
more than 1 star-product. One can generalize these definitions to any bosonic
field with the same gauge quantum numbers as the gauge fields, i.e. in the
adjoint representation. Therefore, in this letter, one attaches the star-
product definitions to the gauge quantum numbers of the elementary / composite
fields. Note that the spin of the bosons does not enter the definition of
these star-products.

Figure 1: _Feynman diagrams of the emission vertex of 4 gauge bosons from the
gauge clouds (double wiggly lines) by a gauge boson (left, single wiggly
line), a scalar (middle, dashed line) and a fermion (right, plain line)._

The pure gauge sector of the FSM is described by the Lagrangian

$\displaystyle\mathcal{L}_{\text{gauge}}$
$\displaystyle=-\dfrac{1}{2}\,\text{tr}\left[\mathcal{G}_{\mu\nu}\star_{c}\mathcal{G}^{\mu\nu}\right]-\dfrac{1}{2}\,\text{tr}\left[\mathcal{W}_{\mu\nu}\star_{w}\mathcal{W}^{\mu\nu}\right]-\dfrac{1}{4}\,\mathcal{B}_{\mu\nu}\star_{0}\mathcal{B}^{\mu\nu}$
(13) $\displaystyle\supseteq-\dfrac{1}{4}\,\mathcal{G}_{\
\mu\nu}^{a}\ast_{c}\mathcal{G}^{\ \mu\nu}_{a}-\dfrac{1}{4}\,\mathcal{W}_{\
\mu\nu}^{a}\ast_{w}\mathcal{W}^{\
\mu\nu}_{a}-\dfrac{1}{4}\,\mathcal{B}_{\mu\nu}\ast_{0}\mathcal{B}^{\mu\nu}\,,$
(14)

where the terms that are not displayed in the second line correspond to the
gauge cloud that does not contribute to the tree-level propagators. Since
$SU(3)_{C}$ is not affected by EWSB, one can already extract the gluon
propagator in the Feynman-’t Hooft gauge:

$\Pi_{G}^{ab\mu\nu}(p^{2})=\dfrac{-ie^{-\vartheta_{c}\left(\eta_{c}\,p^{2}\right)}}{p^{2}+i\epsilon}\,\delta^{ab}g^{\mu\nu}\,,$
(15)

which has only the canonical pole at $p^{2}=0$, aka the gluon, like in local
quantum chromodynamics.

###  2.2 Electroweak Symmetry Breaking

In the GWS model, EWSB occurs by tachyon condensation of the Higgs field,
where $SU(2)_{W}\times U(1)_{Y}\rightarrow U(1)_{em}$. In the physical EW
vacuum, the gauge symmetry $SU(2)_{W}\times U(1)_{Y}$ is nonlinearly realized.
As for the gauge fields, one introduces the covariant $\star_{h}$ and
noncovariant $\ast_{h}$ star-products between the fields in the
representations of the Higgs tachyon that are the complex conjugates of each
other:

$\displaystyle H(x)^{\dagger}\ast_{h}H(x)$ $\displaystyle\equiv
H^{\dagger}(x)\,e^{\vartheta\left(\overleftarrow{\partial_{\mu}}\,\eta^{\mu\nu}_{h}\,\overrightarrow{\partial_{\nu}}\right)}\,H(x)\,,$
(16) $\displaystyle H^{\dagger}(x)\star_{h}H(x)$ $\displaystyle\equiv
H^{\dagger}(x)\,e^{\vartheta\left(\overleftarrow{\mathcal{D}_{\mu}}\,\eta^{\mu\nu}_{h}\,\overrightarrow{\mathcal{D}_{\nu}}\right)}\,H(x)$
(17)
$\displaystyle=\left.e^{\vartheta\left(\mathcal{D}_{\mu}^{y}\,\eta^{\mu\nu}_{h}\,\mathcal{D}_{\nu}^{z}\right)}\,H^{\dagger}(y)\cdot
H(z)\right|_{y=z=x}$ (18)
$\displaystyle=H^{\dagger}(x)\ast_{h}H(x)+\mathcal{O}\left(\dfrac{g_{1}}{\Lambda_{h}^{4}}\,B|H|^{2}\right)+\mathcal{O}\left(\dfrac{g_{2}}{\Lambda_{h}^{4}}\,W|H|^{2}\right)\,,$
(19)

with the tensor $\eta_{h}^{\mu\nu}=\eta_{h}\,g^{\mu\nu}$, the covariant
derivative
$\mathcal{D}_{\mu}H^{\dagger}=\left(\mathcal{D}_{\mu}H\right)^{\dagger}$, the
commutator $\left[\mathcal{D}_{\mu}^{y},\mathcal{D}_{\nu}^{z}\right]=0$ if
$y\neq z$, and the gauge clouds (cf. Fig. 1) in the ellipses
$\mathcal{O}(\cdots)$. Note that the star-products are Hermitian symmetric.
One can then build the following Lagrangian for the kinetic and potential
terms of the Higgs tachyon:

$\displaystyle\mathcal{L}_{\text{Higgs}}$
$\displaystyle=\mathcal{D}_{\mu}H^{\dagger}\star_{h}\mathcal{D}^{\mu}H+\mu^{2}\,H^{\dagger}\star_{h}H-\lambda\,\left(H^{\dagger}\star_{h}H\right)\star_{0}\left(H^{\dagger}\star_{h}H\right)\,.$
(20)

With the quartic Higgs self-coupling, one has an example where the
nonassociativity of the star-products enters the game: one needs to explicitly
write the parentheses. Then, one performs the replacement
$\left(\star_{h}\mapsto\ast_{h}\right)$ in Eq. (20), i.e. one drops the gauge
cloud from the covariant star-products of Higgs tachyons. Indeed, one can see
that this gauge cloud does not contribute to the tree-level propagators after
EWSB by using the properties

$v\ast_{h}H(x)=v\cdot H(x)\ \ \ \text{and}\ \ \
v\star_{h}v=v^{2}+\mathcal{O}(W^{4})+\mathcal{O}(B^{4})\,,$ (21)

where $v$ is the vev of the Higgs field. As discussed in Ref. Nortier:2023dkq
and Fig. 2, without the condition $\vartheta_{h}(-z)=\vartheta_{h}(z)$, the
gauge cloud that dresses the product of Higgs vev’s starts in general at
$\mathcal{O}(W^{2})$ and $\mathcal{O}(B^{2})$. One can check that such
quadratic pointwise products of gauge bosons would contribute to the mass of
the weak bosons in the physical EW vacuum: it would spoil the ghost-free
property of the model.

Figure 2: _Feynman diagrams in unitary gauge, where blobs are a nonlocal
vertices, crosses are insertions of Higgs vev’s, single wiggly lines are gauge
bosons from the Lagrangian with the replacement $\star\mapsto\ast$, whereas
double wiggly lines are gauge bosons coming from the local expansion of the
gauge clouds. Left: the covariant derivatives in the quadratic term in Higgs
tachyons of Eq. (23) implies an interaction between a gauge boson with 2
insertions of Higgs vev’s, giving a nonlocal contribution to the mass term of
the gauge boson. Middle: when $\vartheta_{h}(-z)\neq\vartheta_{h}(z)$, the
gauge cloud in the Higgs potential of Eq. (20) implies in general a local
interaction between a gauge boson with 2 insertions of Higgs vev’s, giving a
local contribution to the mass term of the gauge boson. Right: when
$\vartheta_{h}(-z)=\vartheta_{h}(z)$, the insertion of Higgs vev’s from the
gauge cloud in the Higgs potential starts to interact with the gauge bosons
with a quartic self-interaction but does not contribute to the mass term of
the gauge boson._

Consider the polar representation of the Higgs tachyon around its vev as

$H=e^{\frac{i\pi^{a}(x)\sigma_{a}}{v}}\,H_{U}\,,\ \ \
H_{U}(x)=\dfrac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v+h(x)\end{pmatrix}\,,\ \ \
v=\sqrt{\dfrac{\mu^{2}}{\lambda}}\,,$ (22)

where $h(x)$ is the radial mode, aka the Higgs boson. In order to study the
physical spectrum, one chooses the unitary gauge, where the pion fields
$\pi^{a}(x)$ are eaten by the $W_{\mu}$ and $B_{\mu}$ gauge bosons, such that
$H(x)\equiv H_{U}(x)$. One can then extract the vector boson masses from the
quadratic term in Higgs tachyons with covariant derivatives:

$\displaystyle\mathcal{D}_{\mu}H_{U}^{\dagger}\ast_{h}\mathcal{D}^{\mu}H_{U}$
$\displaystyle=\dfrac{1}{2}\left[\begin{pmatrix}0&v\end{pmatrix}\cdot\left(\dfrac{g_{2}}{2}\,W_{\mu}^{a}\sigma_{a}+\dfrac{g_{1}}{2}B_{\mu}\right)\right]\ast_{h}\left[\left(\dfrac{g_{2}}{2}\,W_{\mu}^{b}\sigma_{b}+\dfrac{g_{1}}{2}B^{\mu}\right)\cdot\begin{pmatrix}0\\\
v\end{pmatrix}\right]$
$\displaystyle=\dfrac{v^{2}}{8}\left[g_{2}^{2}\left(W_{\mu}^{1}\ast_{h}W^{\mu}_{1}+W_{\mu}^{2}\ast_{h}W^{\mu}_{2}\right)\right.$
$\displaystyle+\left.\left(-g_{2}\,W_{\mu}^{3}+g_{1}\,B_{\mu}\right)\ast_{h}\left(-g_{2}\,W^{\mu}_{3}+g_{1}\,B^{\mu}\right)\right]\,.$
(23)

These mass terms mix the $W_{\mu}^{3}$ and $B_{\mu}$ fields, and one must
ensure that they combine with the kinetic terms in Eq. (14) to give ghost-free
propagators for the massive vector bosons of the form

$\dfrac{-ie^{-\vartheta_{\bullet}\left(\eta_{\bullet}\,p^{2}\right)}}{p^{2}-m_{\bullet}^{2}+i\epsilon}\left(g^{\mu\nu}-\dfrac{p^{\mu}p^{\nu}}{m_{\bullet}^{2}}\right)\,.$
(24)

For this to happen, the involved noncovariant star-products must be the same:

$\ast_{w}\equiv\ast_{0}\equiv\ast_{h}\ \ \Rightarrow\ \
\eta_{w}=\eta_{0}=\eta_{h}\,,\ \
\vartheta_{w}(z)=\vartheta_{0}(z)=\vartheta_{h}(z)\,.$ (25)

Otherwise, the WNL form factors do not factorize between the kinetic and mass
terms, and the ghost-free condition is spoiled Nortier:2023dkq .

Now, one can introduce the physical vector bosons:

$\displaystyle W_{\mu}^{\pm}=\dfrac{1}{\sqrt{2}}\left(W_{\mu}^{1}\mp
W_{\mu}^{2}\right)\,,\ \ \ \begin{pmatrix}Z_{\mu}\\\
A_{\mu}\end{pmatrix}=\begin{pmatrix}\cos\theta_{w}&-\sin\theta_{w}\\\
\sin\theta_{w}&\cos\theta_{w}\end{pmatrix}\begin{pmatrix}W^{3}_{\mu}\\\
B_{\mu}\end{pmatrix}\,.$ (26)

The weak mixing angle $\theta_{w}$ and the $U(1)_{em}$ gauge coupling $g_{e}$
are defined as usual:

$\displaystyle\tan\theta_{w}=\dfrac{g_{1}}{g_{2}}\,,\ \ \
g_{e}=g_{2}\,\sin\theta_{w}\,.$ (27)

The quadratic Lagrangian of the physical vector bosons reads

$\displaystyle-\dfrac{1}{4}\,\mathcal{A}_{\mu\nu}\ast_{w}\mathcal{A}^{\mu\nu}-\dfrac{1}{2}\,\mathcal{W}_{\mu\nu}^{+}\ast_{w}\mathcal{W}^{-\mu\nu}-\dfrac{1}{4}\,\mathcal{Z}_{\mu\nu}\ast_{w}\mathcal{Z}^{\mu\nu}$
$\displaystyle+\dfrac{M_{A}^{2}}{2}\,A_{\mu}\ast_{w}A^{\mu}+M_{W}^{2}\,W^{+}_{\mu}\ast_{w}W^{-\mu}+\dfrac{M_{Z}^{2}}{2}\,Z_{\mu}\ast_{w}Z^{\mu}\,,$
(28) $\displaystyle\text{with}\ M_{W}=g_{2}\,\dfrac{v}{2}\,,\
M_{Z}=\dfrac{M_{W}}{\cos\theta_{w}}\ \ \text{and}\ \ M_{A}=0\,,$ (29)

and the field strength tensors

$\mathcal{W}_{\mu\nu}^{\pm}=\partial_{\mu}W^{\pm}_{\nu}-\partial_{\nu}W^{\pm}_{\mu}\,,\
\ \ \mathcal{Z}_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}\,,\ \ \
\mathcal{A}_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,.$ (30)

From this quadratic Lagrangian, one gets the propagators for the photon
$A_{\mu}$ and the weak bosons $V_{\mu}=Z_{\mu}$ or $W_{\mu}^{\pm}$ in Feynman
and unitary gauges, respectively:

$\Pi_{A}^{\mu\nu}(p^{2})=\dfrac{-ie^{-\vartheta_{w}\left(\eta_{w}\,p^{2}\right)}}{p^{2}+i\epsilon}\,g^{\mu\nu}\,,\
\ \
\Pi_{V}^{\mu\nu}(p^{2})=\dfrac{-ie^{-\vartheta_{w}\left(\eta_{w}\,p^{2}\right)}}{p^{2}-M_{V}^{2}+i\epsilon}\left(g^{\mu\nu}-\dfrac{p^{\mu}p^{\nu}}{m_{V}^{2}}\right)\,.$
(31)

Concerning the Higgs boson $h(x)$, its quadratic Lagrangian is

$\displaystyle\mathcal{L}_{\text{Higgs}}\supseteq\dfrac{1}{2}\,\partial_{\mu}h\ast_{w}\partial^{\mu}h-\dfrac{M_{H}^{2}}{2}\,h\ast_{w}h\,,$
(32) $\displaystyle\text{with}\ M_{H}=\sqrt{2}\mu=\sqrt{2\lambda}v\,,$ (33)

and one gets the ghost-free propagator

$\Pi_{h}(p^{2})=\dfrac{ie^{-\vartheta_{w}\left(\eta_{w}\,p^{2}\right)}}{p^{2}-M_{H}^{2}+i\epsilon}\,.$
(34)

This analysis shows that there is no ghost at tree-level in the bosonic sector
of the proposed FSM. As discussed in detail in Ref. Nortier:2023dkq , the main
difference with the usual string-inspired models of the literature
Gama:2018cda ; Hashi:2018kag ; Koshelev:2020fok ; Nortier:2023dkq  is that
_both_ the kinetic term and the potential of the Higgs field are smeared via
the star-products. Instead, in string-inspired models, only the fields in
_either_ the quadratic _or_ the self-interaction terms are smeared, but not in
both. In addition, the other crucial properties for a ghost-free FSM are the
linearity and the Hermitian symmetry of the star-products, combined with the
properties of the gauge cloud in Eq. (21).

A very important observation for the phenomenology of the FSM is that the
tree-level expressions of the particle masses after EWSB are the same as in
the SM in terms of the Lagrangian parameters. This means that the difference
between the perturbative spectra of the 2 models arises only at loop-level.
This is in contrast with what happens when EWSB occurs in the string-inspired
formalism that is not ghost-free Gama:2018cda ; Hashi:2018kag ;
Krasnikov:2022xsi .

##  3 Fermionic Fields

###  3.1 Dirac & Weyl Fermions

Since the fundamental building blocks of the SM fermions are Weyl spinors,
where different chiralities belong to different representations of the EW
gauge group, one considers the Dirac $\gamma$-matrices in the chiral
representation, and the chirality projectors are given by

$P_{R}=\dfrac{1+\gamma^{5}}{2}\,,\ \ \ P_{L}=\dfrac{1-\gamma^{5}}{2}\,,\ \ \
\text{with}\ \ \ \gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\,.$ (35)

We work in the 4-component formalism where, for a spinor $\psi$, the chiral
decomposition $\psi=\psi_{L}+\psi_{R}$ is given by $\psi_{L}=P_{L}\,\psi$ and
$\psi_{R}=P_{R}\,\psi$.

It is tempting to define different star-products for different chiralities,
where the kinetic terms would be

$\overline{\psi}_{L}\star_{L}\left(\dfrac{i}{2}\,\gamma^{\mu}\partial_{\mu}\psi_{L}\right)+\overline{\psi}_{R}\star_{R}\left(\dfrac{i}{2}\,\gamma^{\mu}\partial_{\mu}\psi_{R}\right)+\text{H.c.}\,.$
(36)

Nevertheless, mass terms mix chiralities, which imposes the same expression
for the noncovariant star-product:

$-M_{\psi}\left(\overline{\psi}_{L}\ast_{?}\psi_{R}+\overline{\psi}_{R}\ast_{?}\psi_{L}\right)\
\ \ \Rightarrow\ \ \ \ast_{?}\equiv\ast_{L}\equiv\ast_{R}\,,$ (37)

in order to get a ghost-free propagator for a Dirac fermion of the form

$\dfrac{ie^{-\vartheta_{\psi}\left(\eta_{\psi}\,p^{2}\right)}}{\gamma\cdot
p-M_{\psi}+i\epsilon}\,.$ (38)

As for the bosons, one defines the covariant $\star_{\psi}$ and noncovariant
$\ast_{\psi}$ star-products between 2 elementary/composite fermions in the
complex conjugate gauge representation of each other:

$\displaystyle\overline{\Psi}(x)\ast_{\psi}\Psi(x)$
$\displaystyle=\overline{\Psi}(x)\,e^{\vartheta\left(\overleftarrow{\partial_{\mu}}\,\eta_{\psi}^{\mu\nu}\,\overrightarrow{\partial_{\nu}}\right)}\,\Psi(x)\,,$
(39) $\displaystyle\overline{\Psi}(x)\star_{\psi}\Psi(x)$
$\displaystyle=\overline{\Psi}(x)\,e^{\vartheta\left(\overleftarrow{\mathcal{D}_{\mu}}\,\eta_{\psi}^{\mu\nu}\,\overrightarrow{\mathcal{D}_{\nu}}\right)}\,\Psi(x)$
(40) $\displaystyle=\overline{\Psi}(x)\ast_{\psi}\Psi(x)+\text{gauge
cloud}\,,$ (41)

with $\overline{\Psi}=\Psi^{\dagger}\gamma^{0}$, the covariant derivative
$\mathcal{D}_{\mu}\Psi^{\dagger}=\left(\mathcal{D}_{\mu}\Psi\right)^{\dagger}$,
and the tensor

$\eta_{\psi}^{\mu\nu}=\eta_{\psi}^{+}\,g^{\mu\nu}+i\eta_{\psi}^{-}\,\gamma^{\mu\nu}\,,$
(42)

that is defined with the symmetric and antisymmetric rank 2 tensors made with
the $\gamma$-matrices:

$g^{\mu\nu}=\dfrac{1}{2}\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}\,,\ \ \
\gamma^{\mu\nu}=\dfrac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]\,,$ (43)

respectively. Note that one can (a priori) associate 2 different fuzzy-
plaquettes $\eta_{\psi}^{\pm}=\left(1/\Lambda_{\psi}^{\pm}\right)^{2}$ to 1
fermion field. Again, fermions are dressed by gauge clouds (cf. Fig. 1). The
gauge-invariant kinetic terms for the fermions are given by

$\displaystyle\mathcal{L}_{\text{kinetic}}$
$\displaystyle=\overline{Q}_{L}^{i}\star_{q}\left(\dfrac{i}{2}\,\delta_{ij}\gamma^{\mu}\mathcal{D}_{\mu}Q_{L}^{j}\right)+\overline{d}_{R}^{i}\star_{q}\left(\dfrac{i}{2}\,\delta_{ij}\gamma^{\mu}\mathcal{D}_{\mu}d_{R}^{j}\right)+\overline{u}_{R}^{i}\star_{q}\left(\dfrac{i}{2}\,\delta_{ij}\gamma^{\mu}\mathcal{D}_{\mu}u_{R}^{j}\right)$
$\displaystyle+\overline{L}_{L}^{i}\star_{\ell}\left(\dfrac{i}{2}\,\delta_{ij}\gamma^{\mu}\mathcal{D}_{\mu}L_{L}^{j}\right)+\overline{e}_{R}^{i}\star_{\ell}\left(\dfrac{i}{2}\,\delta_{ij}\gamma^{\mu}\mathcal{D}_{\mu}e_{R}^{j}\right)+\text{H.c.}\,,$
(44)

where one has a star-product $\star_{q}$ for quarks and another one
$\star_{\ell}$ for leptons. We comment on the choice of the same star-product
for different generations in the following section.

###  3.2 Yukawa Sector

In order to write gauge-invariant Yukawa terms, with the covariant star-
products between chiral fermions in different representations of the EW gauge
group, one can use the composite fields made of 1 chiral fermion and 1 Higgs
tachyon (built with the usual pointwise product). For 1 generation, one can
propose the following Lagrangian with positive Yukawa couplings:

$\displaystyle\mathcal{L}_{\text{Yukawa}}^{(1)}$
$\displaystyle=-\lambda_{d}\left(\overline{Q}_{L}\cdot
H\right)\star_{q}d_{R}-\lambda_{u}\left(\overline{Q}_{L}\cdot\overline{H}\right)\star_{q}u_{R}-\lambda_{e}\left(\overline{L}_{L}\cdot
H\right)\star_{\ell}e_{R}+\text{H.c.}$ (45) $\displaystyle\supseteq-
m_{d}\,\overline{d}_{L}\ast_{q}d_{R}-m_{u}\,\overline{u}_{L}\ast_{q}u_{R}-m_{e}\,\overline{e}_{L}\ast_{\ell}e_{R}+\text{H.c.}\,,$
(46) $\displaystyle\text{with}\ \ \ m_{d}=\dfrac{\lambda_{d}v}{\sqrt{2}}\,,\ \
\ m_{u}=\dfrac{\lambda_{u}v}{\sqrt{2}}\,,\ \ \
m_{e}=\dfrac{\lambda_{e}v}{\sqrt{2}}\,,\ \ \ $ (47)

where the mass terms of the second line are extracted from the quadratic terms
involving the noncovariant star-products (like for the boson fields). Again,
the tree-level spectrum is ghost-free and is the same as in the SM with 1
generation.

The generalization to several generations with complex Yukawa couplings is
straightforward: one considers the Lagrangian

$\mathcal{L}_{\text{Yukawa}}=-\lambda_{d}^{ij}\left(\overline{Q}_{L}^{i}\cdot
H\right)\star_{q}d_{R}^{j}-\lambda_{u}^{ij}\left(\overline{Q}_{L}^{i}\cdot\overline{H}\right)\star_{q}u_{R}^{j}-\lambda_{e}^{ij}\left(\overline{L}_{L}^{i}\cdot
H\right)\star_{\ell}e_{R}^{j}+\text{H.c.}\,.$ (48)

We stress that it is necessary to use the same covariant star-products between
different generations that belong to the same gauge representation. Indeed,
the WNL form factors in the mass and kinetic terms need to factorize to keep
ghost-free propagators, as in Eq. (38). It follows that both the covariant and
noncovariant star-products are linear in flavor space, so the flavor
rotations:

$\displaystyle\text{Quarks:}\ \ \ d_{L}^{i}\mapsto U_{d}^{ij}d_{L}^{j}\,,\ \ \
d_{R}^{i}\mapsto W_{d}^{ij}d_{R}^{j}\,,\ \ \ u_{L}^{i}\mapsto
U_{u}^{ij}u_{L}^{j}\,,\ \ \ u_{R}^{i}\mapsto W_{u}^{ij}u_{R}^{j}\,,$ (49)
$\displaystyle\text{Leptons:}\ \ \ e_{L}^{i}\mapsto U_{e}^{ij}e_{L}^{j}\,,\ \
\ e_{R}^{i}\mapsto W_{e}^{ij}e_{R}^{j}\,,\ \ \ \nu_{L}^{i}\mapsto
U_{\nu}^{ij}\nu_{L}^{j}\,,$ (50)

can go through the star-products without complications. By using the usual
field redefinition procedure Peskin:1995ev , it is thus straightforward to
show that the FSM has the same qualitative features as the SM concerning
flavor mixing, which are encoded in the CKM matrix: there is only 1 physical
$CP$-violating phase coming from the Yukawa couplings (with 3 fermion
generations) and no FCNC’s at tree-level via the GIM mechanism.

##  4 Conclusion & Outlook

In this letter, we have proposed the FSM: a WNL extension of the SM based on
the covariant star-product formalism introduced in Ref. Nortier:2023dkq . We
have generalized the previous article by including non-Abelian gauge
symmetries and fermions. Our main achievement is to realize EWSB without
introducing ghosts in the physical EW vacuum, which is the main drawback of
the previous string-inspired attempts Gama:2018cda ; Hashi:2018kag ;
Koshelev:2020fok ; Nortier:2023dkq . As mentioned in the introduction, 2
alternative proposals have already been published that differ from our
approach in their construction:

  * •

_Gauge-Higgs Unification_ Hashi:2018kag : The Higgs sector arises from the
dimensional reduction of a 6D gauge theory. Our proposal is more minimal,
since we do not need to add extra compact dimensions, and there are no extra
elementary particles other than those already present in the SM.

  * •

_Tree-Duality_ Modesto:2021okr : This formalism shares several features with
our proposal: there are no new fields beyond the SM, and it is realized as a
WNL deformation of a local “mother theory”. However, it is built from the
constraint to have the same tree-level scattering amplitudes as the mother
theory Modesto:2021ief ; Modesto:2021soh ; Calcagni:2023goc .

Our construction looks promising, but several issues remain to be
investigated, both from the theoretical and phenomenological points of view.
From the theoretical side, the following points would be interesting to
address:

  * •

The inclusion of gravity (the modern motivation for WNL) with the covariant
star-product formalism deserves to be discussed elsewhere.

  * •

Our analysis remains at tree-level, and it is thus crucial to study the UV-
behavior of the quantum corrections in the FSM, as well as the issue of ghosts
at loop-level Shapiro:2015uxa . For these purposes, one needs to specify the
exponent functions $\vartheta_{\bullet}(z)$.

From the phenomenological side, several directions are possible:

  * •

It has been argued in some toy models that string-inspired WNL provides a
mechanism to stabilize the weak scale (and more generally the Higgs potential)
under radiative corrections Krasnikov:1987yj ; Biswas:2014yia ;
Ghoshal:2017egr ; Abu-Ajamieh:2023syy , implying interests for looking at WNL
signatures at current particle physics experiments Biswas:2014yia ; Su:2021qvm
; Capolupo:2022awe ; Capolupo:2023kuu ; Abu-Ajamieh:2023txh . If also true in
the FSM, it might provide a concrete framework to discuss the implications on
the infamous gauge hierarchy problem.

  * •

The FSM offers a new scenario where deviations from the SM are not associated
with new particles, and it is worth understanding its underlying
phenomenology. Moreover, once Taylor expanded, the WNL form factors imply
corrections of $\mathcal{O}\left(1/\Lambda_{\bullet}^{4}\right)$ to the SM at
tree-level, i.e. dimension 8 operators, whereas one typically expects
$\mathcal{O}(1)$ contributions to the Wilson coefficients of dimension 6
operators by integrating out new heavy fields at tree-level Falkowski:2023hsg
.

  * •

The FSM does not introduce new operators violating the baryon and lepton
numbers at the perturbative level. Furthermore, in the limit of vanishing
Yukawa couplings, the FSM exhibits the same flavor symmetry group as the SM:

$U(3)_{Q}\times U(3)_{u}\times U(3)_{d}\times U(3)_{L}\times U(3)_{e}\,.$ (51)

Since the star-products introduce a WNL deformation of the local Yukawa
operators, the breaking effects are also controlled by the standard Yukawa
couplings, with subleading contributions coming from higher-dimensional
operators suppressed by the nonlocal scales. Therefore, the FSM offers a
framework where the contributions to the breaking of flavor symmetries are
naturally suppressed, for the same reason they are suppressed in the SM.

  * •

Some speculations in WNL theories about the existence of bound states made of
SM particles that are confined via the higher-derivative form factors can be
found in Refs. Tomboulis:1997gg ; Modesto:2015lna ; Mo:2022szw . If true in
the FSM, it would be interesting to study their phenomenology and experimental
signatures.

###### Acknowledgements.

Authors thank Anish Ghoshal (Warsaw U.) for useful discussions all along the
collaboration.

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# Maximizing Performance with Minimal Resources
for Real-Time Transition Detection

Zeynep Özge Orhan1,2, Andrea Dal Prete1,3, Anastasia Bolotnikova2,4,
Marta Gandolla3, Auke Ijspeert2, and Mohamed Bouri1 1 Zeynep Özge Orhan,
Andrea Dal Prete, Mohamed Bouri are with the REHAssist group, Ecole
Polytechnique Federale de Lausanne (EPFL). (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>mohamed.bouri@epfl.ch)2 Zeynep Özge Orhan,
Anastasia Bolotnikova and Auke Ijspeert are with the BioRobotics Laboratory,
Ecole Polytechnique Federale de Lausanne (EPFL). (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>auke.ijspeert@epfl.ch)3
Andrea Dal Prete and Marta Gandolla are with Politecnico di Milano, Department
of Mechanical Eng. (e-mail<EMAIL_ADDRESS>marta.gandolla@polimi.it)4 Anastasia Bolotnikova is with the Reconfigurable
Robotics Lab, Ecole Polytechnique Federale de Lausanne (EPFL). (e-mail:
<EMAIL_ADDRESS>
###### Abstract

Assistive devices, such as exoskeletons and prostheses, have revolutionized
the field of rehabilitation and mobility assistance. Efficiently detecting
transitions between different activities, such as walking, stair ascending and
descending, and sitting, is crucial for ensuring adaptive control and
enhancing user experience. We here present an approach for real-time
transition detection, aimed at optimizing the processing-time performance. By
establishing activity-specific threshold values through trained machine
learning models, we effectively distinguish motion patterns and we identify
transition moments between locomotion modes. This threshold-based method
improves real-time embedded processing time performance by up to 11 times
compared to machine learning approaches. The efficacy of the developed finite-
state machine is validated using data collected from three different
measurement systems. Moreover, experiments with healthy participants were
conducted on an active pelvis orthosis to validate the robustness and
reliability of our approach. The proposed algorithm achieved high accuracy in
detecting transitions between activities. These promising results show the
robustness and reliability of the method, reinforcing its potential for
integration into practical applications.

##  I Introduction

Locomotion mode identification involves recognizing various human activities,
such as walking, running, stair ascent, stair descent, and sitting [1]. This
field has significantly evolved over the years, with a focus on achieving
accurate human intention recognition.

Wearable sensors like inertial measurement units (IMUs), force sensors, and
electromyography (EMG) sensors have played a pivotal role in locomotion-mode
identification [2]. Researchers have commonly used feature extraction methods
to capture activity-specific characteristics, followed by classification
algorithms to identify these activities [3, 4]. Classification methods vary in
complexity, ranging from simple threshold-based (TH) approaches to advanced
deep learning techniques [5, 6].

Figure 1: The outdoor environment where the onboard transition detection tests
are performed for walk-stair ascend, stair ascend-walk, walk-sit, sit-walk,
walk-stair-descend, and stair descend-walk transitions

Reported accuracy varies across different studies, and depends on factors such
as choice of sensors, feature extraction methods, and classification
algorithms [7, 8]. Most of the time, the performance of the selected
techniques is analyzed based on accuracy, sensitivity, and specificity.
However, for onboard real-time implementation of suggested strategies, the low
computational cost is also important [9]. Since the performance of an
assistive device is strongly affected by the response time or delay, an
accurate but also fast transition detection strategy is essential [10, 4, 11].

Existing classification methods excel in identifying steady-state activities
for various locomotion modes [12, 2]. However, detecting the transitions
between different activities is crucial from the perspective of control of a
lower-limb exoskeleton to be able to assist promptly with changes in
locomotion, ensuring smooth human-robot interaction and enhancing the safety
and efficiency of lower-limb exoskeletons. Hence, a need to develop
specialized algorithms that focus on transition detection rather than
continuous activity recognition.

Most of the studies primarily rely on the sensors placed on the human body and
often lack real-time experimental results integrated into lower-limb assistive
devices [13, 14]. There are few studies with exoskeletons that conducted
experiments followed by offline recognition and analysis and even less with
online recognition [15].

In the field of lower limb prosthesis, multiple methodologies are developed to
achieve high real-time performance in recognizing different locomotion types
for intuitive prosthetic control [16, 17]. However, the identification of
transitions between locomotion modes with lower-limb exoskeletons has not been
studied as comprehensively in prosthetic systems. In the review article [15],
it is shown that only three studies investigated the transitions between
locomotion modes among the eight studies focused on orthoses or exoskeletons.
In a more recent review [18], the number of studies is increasing to detect
transitions between locomotion modes.

In Long et al., [19], the Support Vector Machine (SVM) algorithm is optimized
by particle swarm optimization to identify five locomotion modes with a knee
exoskeleton: level-ground walking, stair ascent/stair ascent, ramp
ascent/descent, and transitions between them based on ground reaction forces
and attitude and heading reference system. Human subject experiments achieved
low error rates, but the suggested system may not translate well to seamless
online control since the identification of transitions between certain modes
relies on the entire upcoming step.

In Wang et al., an LSTM-based model is implemented on a soft knee exoskeleton
to detect the level walking, stair ascent/descent, and the transitions between
them based on IMUs [20]. They report 98.2$\%$ for steady-state accuracy while
the recognition delay of all transitions is slightly less than one step. Zhou
et al. also studied the same three locomotion modes and transitions between
them [13] by using SVM with the data of IMUs for a unilateral knee
exoskeleton. However, these approaches [20, 13] rely on fixed-length sliding
time windows that shift over the gait time series, potentially limiting its
adaptability to variable walking speeds.

Liu and Wang, [21] implemented SVM to recognize the locomotion mode for a
unilateral knee exoskeleton together with recognition of sitting. Their
experimental results demonstrated that the real-time recognition with two IMUs
obtained satisfactory recognition accuracy and low delay time. A fuzzy logic-
based algorithm for locomotion and transition mode recognition based on the
uses of inertial sensors mounted on a hip joint exoskeleton is proposed in
[22]. The proposed approach is able to overcome the subject-dependent
parameters in data training, avoiding a training procedure per subject. In
contrast, subject-independent locomotion mode classification for a hip
exoskeleton is suggested by [23] based on a deep CNN. Although this suggested
strategy seems to be promising without fine-tuning subject-specific
parameters, it has only been validated in offline scenarios.

TH methods offer transparency and interpretability, allowing clear rules and
decision-making criteria. Since they are computationally efficient, they are
suitable for real-time applications. However, they may struggle with complex
patterns, have limited generalization, and require manual threshold creation.
Most of the traditional machine learning approaches can handle complicated
patterns, generalize well, and provide versatility with various algorithms.
Yet, they often require feature engineering and are sensitive to feature
quality, limiting scalability. On the other hand, deep learning approaches
automatically learn features from raw data, excel at capturing complex
patterns, and scale well. However, they need large amounts of labeled data,
are computationally demanding, and lack interpretability.

By considering the data availability and limited computational resources on
assistive devices, a TH method is proposed in this work. Instead of setting
thresholds via manual inspection, traditional machine learning approaches are
utilized considering the generalizability of the approach.

Therefore, this study will focus on recognizing transitions between locomotion
types such as overground walking, sitting, slope ascend/descend, and stair
ascend/descend, while minimizing the computational resources on the required
processing power and time on lower-limb assistive devices for real-time
onboard implementation.

The major contributions of this study are the following.

  * •

The proposed threshold-based implementation offers a significant advancement
in real-time embedded performance, reducing computational costs while
maintaining high accuracy in real-time classification. By utilizing
conventional machine learning techniques, the threshold-setting process is
simplified, resulting in a more efficient and streamlined approach.

  * •

The efficacy of the developed finite-state machine is validated using data
collected from three different measurement systems: i) a Vicon camera motion
capture system, ii) goniometers, and force plates, and iii) an IMUs-based
motion capture system. This comprehensive validation ensures the reliability
and versatility of the proposed approach across different measurement
modalities, enhancing its applicability in practical scenarios.

  * •

Robustness and reliability of the suggested strategy to use in lower-limb
wearable robots are verified through the human subject experiments with an
active hip orthosis with a total of 10 healthy participants.

##  II Methods

###  II-A Training Dataset

Human gait kinematics and kinetics are studied for various distinct locomotion
scenarios in the literature together with several public datasets [24, 25, 26,
27, 28, 29, 30]. However, in most of the datasets, transitions between
activities are not considered. To be able to detect the transitions, the
public dataset provided by [31] is used to train the classifiers. This dataset
involves several activities including walking, stair ascending/descending,
sitting, and standing with continuous variations and transitions between
activities. The transitions between different locomotion modes are notated as
Walk to Sit (W $\rightarrow$ S), Sit to Walk (S $\rightarrow$ W), Walk to
Stair Ascend (W $\rightarrow$ SA), Stair Ascend to Walk (SA $\rightarrow$ W),
Walk to Stair Descend (W $\rightarrow$ SD), Stair Descend to Walk (SD
$\rightarrow$ W) and finally all the steady state cases are notated with SS-
mode.

Gait data were acquired with a 10-cameras Vicon T40 motion capture system on
ten healthy subjects. The training dataset is constructed as described in
[32].Of the available data, 72% of samples are taken for the training while
18% of samples are used for tests of each individual classifier. The remaining
10% of data is later used for validation of the method together with FSM.

###  II-B Features and State-Machine

In [32], they derive kinematics-based classification features called
instantaneous characteristic features (ICFs) based on thigh angle ($\theta$)
and thigh velocity ($\dot{\theta}$). Three ICFs are derived for transitions
between Walk and Sit, Walk and Stair Ascent, and Walk and Stair Descent at
three specific moments. The specific moments are defined as maximum hip
flexion (MHF), heel strike (HS), and $\theta_{10}$ (TH10). The MHF and HS are
used for locomotion transition classifications while TH10 is used for walk and
sit transition. ICF-1 is $\theta_{th}$ at MHF $\theta_{th,MHF}$, ICF-2 is the
difference $\theta_{th,MHF}$ \- $\theta_{th,HS}$ and ICF-3 is the sign of
$\dot{\theta}_{th}$, when $\theta_{th}=10^{\circ}$ as shown in Tab. I.

TABLE I: The derived features for transitions between distinct locomotion modes | Notation | Definition
---|---|---
W $\rightarrow$ S | ICF-3 | The sign of, $\dot{\theta}_{th}$, when $\theta_{th}=10^{\circ}$
S $\rightarrow$ W | ICF-3 | The sign of, $\dot{\theta}_{th}$, when $\theta_{th}=10^{\circ}$
SA $\rightarrow$ W | ICF-1 |  $\theta_{th}$ at MHF $\theta_{th,MHF}$
W $\rightarrow$ SA | ICF-1 |  $\theta_{th}$ at MHF $\theta_{th,MHF}$
W $\rightarrow$ SD | ICF-2 | The difference $\theta_{th,MHF}$ \- $\theta_{th,HS}$
SD $\rightarrow$ W | ICF-2 | The difference $\theta_{th,MHF}$ \- $\theta_{th,HS}$

In [32], the researchers employ a finite state machine (FSM) comprising four
distinct states — Sit, Walk, Stair Ascent, and Stair Descent — to decompose
the classification challenge into six transition detection problems, each
corresponding to the current activity. We are utilizing the same FSM
introduced by [32], as depicted in Fig. 2. Within the FSM, level walking,
standing, and ramp walking are grouped into a unified state. In the flowchart,
the ICFs detector will check for identifying distinct moments (e.g., MHF, HS)
containing various ICFs from the thigh angle and ground reaction forces to
predict the transitioning state.

Figure 2: The finite state machine suggested by Cheng et al. in with the
updated value of thigh boundary for ICF3 [32].

###  II-C Classification Algorithms

Several ML algorithms such as Linear Support Vector Machines (SVM), Logistic
Regression (LR), k-Nearest Neighbors, Naïve Bayes, Random Forest, and Gradient
Boosting are used to train the classifiers. After manually tuning the
parameters based on the accuracy results of each algorithm, the best-
performing one in terms of training accuracy is selected. Training and test
accuracy for the best performing approach for each transition are reported in
Table II.

cross validation??]

TABLE II: Training and test accuracies with selected algorithms for each classifier | Algorithm | Training Acc. | Test Acc.
---|---|---|---
W$\rightarrow$S | Linear SVM | 86.94% | 90.91%
S$\rightarrow$W | LR | 95.07% | 100.0%
W$\rightarrow$SA | LR | 91.44% | 89.36%
SA$\rightarrow$W | LR | 81.67% | 86.89%
W$\rightarrow$SD | LR | 93.20% | 94.38%
SD$\rightarrow$W | LR | 97.25% | 96.55%

###  II-D Threshold-based Implementation

Many of the assistive devices are resource-constrained embedded systems to
perform real-time operations. For assistive devices, it is critical to be
computationally efficient in the context of real-time embedded systems since
it enables quick and responsive control actions. To improve computational
efficiency and optimize resource usage, we are introducing TH implementation
for a practical and efficient solution. The computational efficiency, ease of
implementation, interpretability, and inherent robustness of a TH method, make
it an appealing choice for transition detection in the control of assistive
devices.

In the method proposed in [32], machine learning models were trained using 1-D
feature space and resulted in thresholds on ICFs for various transition
activities. However, deploying these classifiers on devices can introduce
additional computational costs, potentially causing delays in the system,
especially in resource-constrained environments. By performing offline
training, a threshold value is defined for each classifier. Therefore, the ML
models can be substituted by a straightforward $if/else$ condition based on
the trained thresholds while maintaining accuracy and ensuring efficient
performance in real-time embedded operation. The check for the classifiers
shown in Fig. 2, is therefore implemented via thresholds.

##  III Experiments

###  III-A Offline Experiments

####  III-A1 Testing on two different data sets

To evaluate the performance of the trained models, the classifiers are tested
with FSM by using two different human gait datasets. To simulate the
performance in real-time implementation, gait data is provided to the FSM
consecutively ”sample-by-sample” with the goal of mimicking upcoming sensor
data. Therefore, the detection of ICF, extraction of the corresponding
feature, and performing the binary classification are tested simultaneously in
this evaluation.

The first data is the 10% of the dataset provided by [31] that has been
described in Section II-A. The second gait dataset [33] contains gait data
from 22 healthy subjects for several locomotion modalities such as level-
ground and treadmill walking, stair ascent, stair descent, slope ascent, and
slope descent. Although multiple locomotion modes at different conditions are
available in this dataset, walk-to-sit and sit-to-walk transitions were not
included. Therefore the performance of sit-to-stand and stand-to-sit
transitions aren’t taken into account.

The accuracy results for the two datasets [31, 33] are presented in Table III.
Data from [31] is also used for testing and validation of the trained model.
However, as stated in Section II-A, $18\%$ of the data is separated for this
experiment.

TABLE III: Transition detection accuracy with two different datasets | Data from [31] | Data from [33]
---|---|---
W$\rightarrow$S | 90.00% | \- %
S$\rightarrow$W | 99.90% | \- %
W$\rightarrow$SA | 90.00% | 99.90%
SA$\rightarrow$W | 80.33% | 93.33%
W$\rightarrow$SD | 90.00% | 80.00%
SD$\rightarrow$W | 85.00% | 80.00%

####  III-A2 Results with IMU-based motion capture system

Figure 3: The indoor environment where the test is performed with an IMU-based
motion capture system

An offline test was conducted in an indoor environment with a single subject
to assess the method’s suitability for real-time implementation. The subject
was a 25-year-old female with a height of 167 cm and a weight of 65 kg.
Kinematic data during human gait were captured using the MVN Awinda motion
capture system by Xsens (Xsens Technologies, B.V., The Netherlands) [34],
which utilizes IMUs to accurately track motion.

To evaluate the classifiers, a predefined protocol was followed in a
controlled environment, incorporating sitting, overground walking, and stair
ascend/descend activities as shown in Fig. 3. The data collection procedure is
followed the following transitions: S$\rightarrow$W, W$\rightarrow$SD,
SD$\rightarrow$W, W$\rightarrow$SA, SA$\rightarrow$W, and finally
W$\rightarrow$S.

The collected data was input into the FSM in an offline way to analyze the
performance of ICF detection and classification for each transition. The Fig.
4 shows that for sit-to-walk, walk-to-stair descend, stair-descend-to-walk,
walk-to-stair ascend, stair ascend-to-walk, and walk-to-sit transitions ICFs
are detected, and all of the transitions are correctly classified.

This evaluation also indicated that to detect the transitions W$\rightarrow$S
and S$\rightarrow$W the changing ICF-3 from when $\theta=10$ to while $\theta$
is in between 70-75 would increase the detection accuracy performance.

Figure 4: Collected data with IMU-based motion capture system and detected
transitions

####  III-A3 Onboard Bench-marking

Figure 5: Benchmarking results on two devices with the methods as machine
learning (LR and SVM) and TH implementation for each classifier separately

The computational costs of the TH method and the ML-based approach are
evaluated and compared on the BeagleBone Black. This embedded single-board
computer runs on a Debian-based Linux distribution that is specifically
optimized for embedded systems, ensuring efficient performance.

The collected data with the IMU-based motion capture system is sequentially
inputted to the FSM. When the binary classification is performed, the time
spent is recorded for each individual classifier and also for the FSM which
includes the detection of ICF, and extraction of the corresponding feature as
well. This process is performed for both of the methods with 100 transition
cycles for each classifier.

To show the importance of efficient onboard computing, the test is repeated on
both a notebook computer (Apple M1 processor, 8-core CPU, 7-core integrated
GPU with 8 GB RAM) and the embedded computer (BeagleBone Black, 1GHz ARM®
Cortex-A8, 512MB RAM). The results of the benchmarking tests for the two
methods are illustrated in Fig. 5.

###  III-B Experimental Protocol with Human Subjects

Figure 6: The active hip orthosis eWalk.

The hip orthosis eWalk is designed for partial assistance for the hip
flexion/extension movement in collaboration between the EPFL research group
Biorob-REHAssist and the company Sonceboz. eWalk has 2 active degrees of
freedom which are actuated with DC servo motors (Gyems RMD X8 Pro), as shown
in Fig.6. To allow natural walking, the abduction/adduction and the internal
rotation of the hip are not constrained to allow natural walking due to the
compliant nature of thigh segments.

The position is measured by an incremental encoder on the motor side for the
assistive device. The zero position of the joints is calibrated by the homing
technique. Ground reaction forces are measured by the flexible insole based on
force-sensing resistors. eWalk has an embedded computer (BeagleBone Black,
Texas Instruments, USA) that collects and keeps the log of all the sensory
data. Wireless communication with both devices is established through a Wi-Fi
module.

During the experiments, eWalk is utilized in passive mode due to the
transparent nature of the device with low mass and low friction.

During the tests, ten healthy subjects with the age of $25.6\pm 2.76$ years, a
height of $178.4\pm 8.1$ cm, and a body mass of $71.9\pm 9.68$ kg were asked
to perform the trial.

The experimental scenario included four locomotion modes (level-ground
walking, stair ascent, stair descent, and sit), and six locomotion transitions
were tested in the environment depicted in Fig 1. Locomotion transitions as W
$\rightarrow$ SA, SA $\rightarrow$ W, W $\rightarrow$ S, S $\rightarrow$ W, W
$\rightarrow$ SD and SD $\rightarrow$ W are performed in consecutive order
during the experiment. There were 15-step stairs. For all locomotion modes,
subjects were instructed to walk at their self-selected walking speed. The
sitting position on the bench is self-selected. Each trial is repeated 5 times
by each subject. Therefore each locomotion transition is tested 50 times in
total.

####  III-B1 Data Labeling

The ground truth data is labeled during the experimental procedure. The person
who is observing the subject during the trials puts the ground truth label for
each transition and steady-state locomotion type. This label is logged in
real-time together with the rest of the data of the exoskeletons.

####  III-B2 Performance of classifiers for transitions

The recognition accuracy results are reported in Fig. 7 for the experiments
with the eWalk exoskeleton. The recognition accuracy is defined as shown in
Eqn. 1:

$A=\frac{N_{\text{correctly detected transitions}}}{N_{\text{total
transitions}}}.100$ (1)

where $N_{\text{correctly detected transitions}}$ is the detected transitions
at the correct moment and $N_{\text{total transitions}}$ was the number of
total trials. If the method does not recognize the transition at the time of
transition, it is considered not detected, even if the steady state is
correctly classified.

Figure 7: The results real-time recognition accuracy (%) of locomotion
transitions from experiments with eWalk exoskeletons

##  IV Discussion

###  IV-A The importance of real-time performance

Classification delays, while possibly acceptable for health monitoring
applications, can become problematic in real-time control scenarios [32]. In
these scenarios, where locomotion mode classification is performed onboard in
real-time alongside a high-level controller, the computational load might not
only affect transition detection latency but may also influence controller
performance.

Haptic systems typically utilize a local 1 kHz control loop to overcome device
dynamics and display high-frequency haptic feedback to the user [35]. In
physical human-robot interaction (pHRI), for the robot to react rapidly in the
case of an impact or to be as transparent as possible when physically
collaborating with a human, its control loop should run at a minimum of 1 kHz
[36].

Given these considerations and the inherent limitations of computational
resources in embedded systems, it becomes imperative to seek methods that can
minimize computational costs while upholding classification accuracy. Note
that the accuracy outcomes of ML-based and TH methods are identical as long as
the selected ML model’s output values are utilized as the threshold values.
Consequently, TH methods demonstrate a notable reduction in computational
costs, as presented in Fig. 5. For example, for the W$\rightarrow$S
classifier, the median value of computational time changes from 0.2273 ms to
2.568 ms which is about making the threshold TH method 11 times faster.

The computational costs in terms of time for both of the methods on a notebook
computer are quite low. However, when assessing computational performance on
an embedded board, the TH method outperforms the ML-based approach in terms of
computational time, showcasing its efficiency in resource-constrained
settings.

###  IV-B Verification of the system

As presented in Table III, when all the data points are provided to FSM one by
one the overall accuracy is between $80\%-99.90\%$ in all of the cases. Since
the dataset [33] does not include W$\rightarrow$S, and S$\rightarrow$W
transitions, they couldn’t be reported.

Our experiments with the Xsens measurement system yielded highly promising
results as presented in SectionIII. It’s noteworthy that all transitions were
accurately classified, showcasing the robustness of our approach. However, a
subtle point deserves attention, as depicted in Fig. 4: the transition from W
$\rightarrow$ SA exhibited a one-step delay.

It’s essential to clarify that this delay isn’t indicative of a
misclassification; rather, it stems from the methodology employed. In this
particular test, we didn’t measure ground reaction forces. Instead, we relied
on foot position in the vertical axis to detect load and unload conditions.
Consequently, the observed one-step delay is a reflection of the limitations
in acquiring foot position data, as illustrated in Fig. 4.

These results collectively underscore the versatility and robustness of the
proposed method, further validated by its successful implementation across
three distinct measurement systems.

###  IV-C Real-time results with assistive devices

During the real-time locomotion recognition experiments, high accuracy is
reported as illustrated in Fig. 7. This result shows that the proposed
strategy could be utilized in real-time recognition with high accuracy and low
computational costs.

Figure 8: Thigh angle trajectories of measured with during W$\rightarrow$SD
when the exoskeleton joints are misaligned and aligned properly.

Although high accuracy is reported, the recognition accuracy in other
scenarios outperforms the results in SA$\rightarrow$W and W$\rightarrow$SD
during the experiments with eWalk. The main reason behind this is the proper
alignment of the hip flexion/extension joint to the user’s joint alignment.
Fig. 8 provide an example for a W$\rightarrow$SD case. Since the joint
positions of the exoskeleton were placed above the body ones due to the width
of the trunk, the thigh angle measurement was smaller than the actual one.
Therefore the range of motion of the joint is decreased with a smaller peak.
Therefore, the feature ($ICF-1=\theta_{th,MHF}-\theta_{th,HS}$) never
overtakes the threshold for the transition detection.

As stated in [37, 32], activity mode misclassifications most frequently occur
between walking and walk-to-stair transitions. To be able to reduce these
misclassifications, a conditional offset is suggested for steady-state modes
of walking and steady-state stairs in [37].

###  IV-D Limitations

Firstly, the open-source dataset used for training the classifiers was
collected from subjects who were not wearing a wearable assistive device. This
limits our understanding of how well the model can adapt to different levels
of assistance, as the kinematics of users can vary depending on the level of
assistance received. Additionally, since the measurements were obtained from
the sensors of the exoskeleton, any misalignment between human joints and
exoskeleton joints can impact the reliability of the data. Moreover, while the
proposed approach demonstrates satisfactory results with a subject-independent
model, it is important to consider that the accuracy of the implementation may
deteriorate when applied to target populations using exoskeletons, as their
gait dynamics may differ from those of healthy users.

##  V Conclusion

In conclusion, our study highlights the significance of enhancing real-time
embedded performance through a threshold-based approach, achieving an
efficient balance between computational costs and high classification
accuracy. The validation of utilized finite-state machine using data from
diverse measurement systems shows its versatility and reliability.

Furthermore, our approach’s practicality is demonstrated in the context of
lower-limb wearable robots, as evidenced by human subject experiments
involving an assistive device and a total of 10 healthy subjects. These
results not only highlight the method’s effectiveness but also its potential
to significantly impact the field of assistive technology. Future work should
focus on the verification of the suggested method with a target population of
the exoskeletons and also should incorporate a control strategy that
facilitates seamless human-robot interaction with the assistance of the
exoskeleton.

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